Research Article Mathematical Modeling and Analysis of...
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Research ArticleMathematical Modeling and Analysis of Different VectorControlled CSI Fed 3-Phase Induction Motor Drive
Arul Prasanna Mark Rajasekaran Vairamani and Gerald Christopher Raj Irudayaraj
Department of EEE PSNA College Engineering amp Technology Dindigul Tamilnadu 624622 India
Correspondence should be addressed to Arul Prasanna Mark arulprasannapsnaceteduin
Received 2 April 2014 Accepted 26 May 2014 Published 9 July 2014
Academic Editor Kerim Guney
Copyright copy 2014 Arul Prasanna Mark et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Themain objective of this paper is to build a simplemathematical competentmodel that describes the circuits and interconnectionsof a 3-phase squirrel cage induction motor used for industrial applications This paper presents the detailed analysis of theoreticalconcepts used in mathematical modeling simulationrsquo and hardware implementation The objective of this work is to compare thedynamic performances of the vector control methods for CSI fed IM drives Based on the results dynamic performances of theproposed drives are individually analysed using the sensitivity tests The tests that are chosen for the comparison are step changesin the reference speed and torque of the motor drive Here the IM is mathematically modeled in different reference frames for inputoutput linearization (IOL) control field oriented control (FOC) and direct torque control method (DTC) which are designedusing hardware equivalent mathematical equations The most important contributions in this paper are mathematical simulationstructure of IM model in rotor flux frame using current and speed that were developed and implemented in MATLAB-SimulinkThe operation and performance of the different vector control methods are verified by simulation usingMATLABSIMULINK andexperimental results
1 Introduction
The alternating current (AC) motor especially asynchronousthree-phase induction motor (IM) has been the motor ofchoice in industrial settings for about the past half century aspower electronics can be used to control its output behaviorBefore that the direct current (DC) motor especially theseparately excited one was widely used because of its easyspeed and torque controllabilityThe IM is a rugged structuremotor because it is brushless and has very fewer internal partsthat needmaintenance or replacementThis makes it cheaperin comparison to other motors such as the DC motor ThusIM and its drive system have been gaining market share inindustry and even in alternative applications such as hybridelectric vehicles For variable speed electric motor applica-tions in low to moderate power pulse width modulation(PWM) with voltage source inverter (VSI) is usually usedHowever the switched voltages produce high voltage slopesover the stator windings which stress the insulations andcause bearing current problems A possible solution is to use
PWM with current source inverter (CSI) Phillips [1] Lipoand Cornell [2] Palaniappan et al [3] Kaimoto et al [4]Krishnan et al [5] Kazmierkowski and Koepcke [6] andHombu et al [7] tried to develop the CSI and make itsuitable for ac motor (especially IM) variable speed drivessince both stator current and voltage waveforms are close tothe sinusoidal waveform the above problems are reducedFurther CSI can be used for high power applications Inthe recent years the high performance CSI fed variablespeed drives especially IM drives have been dominated byvarious control methods The important methods are vectorbased 119881119891 control input output linearization (IOL) controlindirect field oriented control (IFOC) and direct torquecontrol (DTC) as each method has its own unique features119881119891 is by far the simplest drive since it requires no parameterknowledge and is essentially an open loop drive IOL is onetype of nonlinear state feedback control which is completelyinput-output decoupled at all times even in transients IFOCoffers similar dynamic torque control performance to that ofDC motors giving fast near step changes in machine torque
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 623982 13 pageshttpdxdoiorg1011552014623982
2 Journal of Applied Mathematics
and DTC provides simpler control architecture with a similardynamic performance as that of IFOC
Similar studies of vector controlled inverter fed drives arediscussed mostly with the voltage source inverters and veryfew papers are available with the study of vector controlledcurrent source inverters Veerachary [8] Salo and Tuusa [9]and Babaei and Heydari [10] This paper presents a compar-ative analysis of different mathematical modeling of vectorcontrolled CSI fed inductionmotor drives and in particularlythe dynamic responses of the drive motor speed againstchange in speed as well as the torque references in a singlesimulation cycle and verified experimentally and to ascertaindifferent mathematical dynamic IM modeling for CSI feddrives as Holmes et al [11] and to apply control methodson them The MATLAB s-functions are described to designIM modeling and drives Finally the above mentioned drivestrategies are compared as fairly as possible by comparingtheir dynamic responses when there is step change in thereferences (usually stator currents motor speed and loadtorque)
2 Mathematical Modeling of3-Phase Induction Motor Drive
In this section Mathematical model of IM is shown inFigure 1 and described In general IM is used to transformelectrical energy to mechanical energy It consists of elec-tric circuitry electromagnetic circuitry and electromechaniccircuitry Krishnan [12] The main objective of the motormodeling is to build a simple but competent model thatdescribes these circuits and their interconnectionsThe mainpath in themodel is in relationwith the voltage or current andthe stator phases in input and magnetic flux inside the motorand electromagnetic torque in the output
As mentioned earlier the squirrel cage IM is preparedfor industry applications The IM can either be supplied by avoltage source or a current sourceThese sources are assumedto be ideal thismeans that they can supply any desired voltageor current without losses According to Krause and Thomas[13] the basic equations of the IM are the stator voltageequation (1) the rotor voltage equation (2) the equationsof the flux linkages of the stator and the rotor windingsystems (3) and (4) and the torque equation (5) Considerthe following
V119904119904= 119877119904119894119904
119904+ 119904
119904 (1)
V119903119903= 0 = 119877
119903119894119903
119903+ 119903
119903 (2)
120582119886
119904= 120582119886
119898+ 119871119897119904119894119886
119904 (3)
120582119886
119903= 120582119886
119898+ 119871119897119903119894119886
119903 (4)
119879119890= [119862(
120587
2) 120582119886
119898]
119879
119894119886
119904 (5)
The equations are given in vector format Bose [14] Everyvector variable has two components the first component isparallel to the reference frame axis or ldquo119889rdquo axis and the secondcomponent is perpendicular to the reference axis or ldquo119902rdquo axis
The symbols are Vmdashvoltage 119894mdashcurrent 120582119898mdashflux linkage
119877mdashresistance and 119871119897mdashleakage inductance subscript ldquo119904rdquo
indicates the stator winding system ldquo119903rdquo the rotor windingsystem superscript ldquo119904rdquo indicates the stator reference frame(stator coordinates) ldquo119903rdquo the rotor reference frame andldquo119886rdquo any arbitrary reference frame 119879
119890is the electromagnetic
torque 119862(1205872) is a rotation matrix over 1205872 radians the dot( ) indicates a time derivative The stator voltage equationis restricted to the stator reference frame because of thedifferential of the stator flux vector whereas the rotor voltageequation is restricted to the rotor reference frame becauseof the differentiation The flux equations and the torqueequation can be defined in any reference frame denoted bysuperscript ldquo119886rdquo
It is evident from (1) to (5) that themotor voltage flux andtorque can be controlled easily by controlling the current ofthe motor So the CSI is the right choice to control the motordirectly because the inputs of the current fed IM model arethe stator currents
Before these equations are completed to obtain the IMmodel the reference frame of the model in relation to theproblem being investigated and the type of computer (analogor digital) are chosen In order to evaluate the usefulness ofeach reference frame the above equations are simulated usingMatlab software to predict the transient performance of anIM
The mathematical modeling of 3-phase induction motordrive which can be run at any of the three rotating referenceframe models namely stator rotating reference frame rotorrotating reference frame and synchronous rotating referenceframe in MatlabSimulink is shown below in Figure 2
3 Transient Performance of DifferentMathematical Reference Frame Models
Figure 3 shows the transient performance of ldquo119886rdquo phase and119889-axis stator currents in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds From Figure 3(a) it is evident that in statoror stationary reference frame stator 119889-axis current behavesexactly the same way as do the stator ldquo119886rdquo phase current ofthe motor If the stationary reference frame is used then thestator 119889-axis current is identical to the stator phase currentThis would be useful when interest is specifically confinedto stator variables for example variable speed stator-fedIM drives Similarly if the rotor reference frame is usedthen the rotor 119889-axis current will behave in exactly thesame manner as the rotor ldquo119886rdquo phase current This wouldbe useful when interest is confined to rotor variables onlyas for example variable speed rotor-fed IM drives FromFigure 3(b) it is evident that in rotor rotating reference framestator 119889-axis current does not behave as the stator phasecurrent FromFigure 3(c) it is apparent that in synchronouslyrotating reference frame stator 119889-axis current behaves likeDC quantity If the synchronously rotating reference frameis used then the stator and rotor current behave like DCquantity Thus the controller design is simple and is widelyused in AC drives
Journal of Applied Mathematics 3
Rotor
Stator
4
3
2
1
[0 0]
rot1
rot
1s
1s
-K-
-K-3
2
1
fr
fs
is
irfr = rotor flux
fs = stator flux
is = stator current
Te = electromagnetic torque
Te
ir = rotor current
Flux-current relationship+minus
minus
+minus
+minus
minus
+minus
+minus
times
times
r
Rr
Rs
wm
Gm1
Gm
Gs
Gr
wk
s
K lowast u
K lowast u
∙
(Lrl + Lm ) (Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
Lm (Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
Lm (Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
(Lsl + Lm)(Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
Figure 1 Induction motor modeling in arbitrary reference frame
dqreference
frame
Stator
Synchronous
Rotor
Speed
Angle
-K-
Torque
Speed
Frequency
dq
ab
dq2ab
Currents
Amplitude
ab
etakdq
ab2dq
ab2ABC
Step load pu
Mechanicalsystem
1s
1s
Inductionmotor
pu
-K--
f(u)
f(u)
f(u)
0
0
ABC2ab dqABC ab
etak
Variable-amplitude variable-frequency3-phase sinusoidal voltages
K lowast u
wm
wk
wmwm
wmwkws
ws
wo
sTe Te
Te
Te
Tl
Tl
i dq
i dq
i ab
i abi ABC
is
120579r
120579k
120579s
120579s
K lowast u
Figure 2 Mathematical modeling of 3-phase induction motor drive
Figure 4 shows the transient performance of 119902-axis statorcurrent and 119889-rotor flux in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds From Figures 4(a) and 4(b) it is apparentthat there is a coupling effect between stator 119902-axis currentand rotor 119889-axis flux So independent control is difficultFigure 4(c) shows that the rotor 119889-axis flux and stator 119902-axis current (control variables) have linear relationship with
torque (speed) Thus it is useful in the rotor flux oriented(RFO) control methods
Figure 5 shows the transient performance of 119902-axis statorcurrent and 119889-stator flux in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds Figures 5(a) and 5(b) state that there is acoupling effect between stator 119902-axis current and stator119889-axis flux and so independent control is difficult From
4 Journal of Applied Mathematics
minus10
0
10
0 05 1 15 2Time (s)
I ac
andI d
c(P
U)
(a)
minus10
0
10
0 05 1 15 2Time (s)
I ac
andI d
c(P
U)
(b)
0 05 1 15 2minus10
0
10
Stator a phase currentStator b-axis current
Step change in load torque
Time (s)
I ac
andI d
c(P
U)
(c)
Figure 3 Simulated results of ldquo119886rdquo phase and 119889-axis stator currents in different rotating reference frames
minus10
0
10
0 05 1 15 2Time in seconds
I qs
and
FLU
Xdr
(a)
minus10
0
10
0 05 1 15 2Time in seconds
I qs
and
FLU
Xdr
(b)
0 05 1 15 2minus10
minus5
0
5
Step change in load torque
Stator q-axis currentRotor d-axis flux
Time (s)
I qs
and
FLU
Xdr
(PU
)
(c)
Figure 4 Simulated results of 119902-axis stator current and 119889-axis rotorflux in different rotating reference frames
minus10
0
10
0 05 1 15 2Time (s)
I qs
and
FLU
Xds
(a)
minus10
0
10
0 05 1 15 2Time (s)
I qs
and
FLU
Xds
(b)
0 05 1 15 2minus10
minus5
0
5
Step change in load torque
Time (s)
Stator q-axis currentStator d-axis flux
I qs
and
FLU
Xds
(c)
Figure 5 Simulated results of 119902-axis stator current and 119889-axis statorflux in different rotating reference frames
Journal of Applied Mathematics 5
Figure 5(c) it is evident that the stator119889-axis flux and stator 119902-axis current (control variables) have linear relationship withtorque (speed) It is used in the stator flux oriented (SFO)control methods
The basic IM equations from Krause et al [15] aresimulated for three different reference frames usingMATLAB76 software and the results obtained are compared with eachreference frame From the comparison it is observed that ifthe stator rotating reference frame is used then stator 119889-axiscurrent is identical to that of the stator ldquo119886rdquo phase current asshown in Figure 3(a) which is used in stator fed IM drivesSimilarly in rotor reference frame rotor 119889-axis current isidentical to the rotor ldquo119886rdquo phase current and useful in rotor fedIM drives If the synchronous reference frame is used thenthe stator 119889-axis current behaves like DC quantity as shownin Figure 3(c) From Figures 4(c) and 5(c) it is evident thatsynchronous reference frame is useful in RFO and SFO drivecontrol methods In this paper synchronous reference frameis preferred to model the IM in 119881119891 IOL control methodsRFO control is used in IFOCmethod and SFO is used inDTCmethod
4 IOL Control for CSI Fed IM Drives
In this section a type of IOL control method for the CSIfed IM drive is described The controller is designed based
on pole placement technique of Wonham [16] to a 119889-119902 axisstate space linearized model of the drive which includes areduced order rotor current observer A control law has beendeveloped to improve the dynamic response of the drive byusing the concepts from Veerachary [8] which includes feedforward as well as feedback controls
41 Mathematical Modeling of IOL Control Method Theequivalent circuit for the CSI capacitor and IM in a 119889-119902 axis rotating reference frame is shown in Figure 6 Thecorresponding state equation (differential equations) withmotor flux linkages as variables is derived fromWu et al [17]
Veerachary [8] presented the linearized CSI-IM drivestate space model in synchronously rotating reference frameas follows
= 119860119909 + 119861119906 (6)
where
=119889119909
119889119905 (7)
119910 = 119862119909 (8)
where
119909 =
[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]
]
=119889
119889119905
[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]
]
119906 =
[[[[
[
Δ1198811015840
119889
Δ120596119890
120596119887
Δ119879119871
]]]]
]
119910 =[[
[
Δ119894119902119904
Δ120596119903
120596119887
]]
]
119860 =
[[[[[[[[[[[[[[
[
minus1199091015840
1199031199030120596119887
11990901199091015840119903minus 1199092119898
1199091198981198771015840
119903120596119887
11990901199091015840119903minus 1199092119898
minus12059611989001199091015840
119903119909119898(1 minus 119904)
11990901199091015840119903minus 1199092119898
minus1199091198981199091015840
1199031198941015840
1198891199030120596119887
11990901199091015840119903minus 1199092119898
1205961198871199091198981199030
11990901199091015840119903minus 1199092119898
minus11990901198771015840
119903120596119887
11990901199091015840119903minus 1199092119898
1205961198900(1199092
119898minus 11990411990901199091015840
119903)
11990901199091015840119903minus 1199092119898
11990901199091015840
1199031198941015840
1198891199030120596119887
11990901199091015840119903minus 1199092119898
1199041205961198900119909119898
1199091015840119903
1199041205961198900
minus1205961198871198771015840
119903
1199091015840119903
minus
120596119887(1199091198981198941199021199040+ 1199091015840
1199031198941015840
1199021199030)
1199091015840119903
1199091198981198941015840
1198891199030
21198670
1199091198981198941199021199040
21198670
]]]]]]]]]]]]]]
]
119861 =
[[[[[[[[[[[[[[[
[
1199091015840
119903120596119887
11990901199091015840119903minus 1199092119898
0 0
minus119909119898120596119887
11990901199091015840119903minus 1199092119898
minus1205961198871198941015840
11988911990300
0 minus
120596119887(1199091198981198941199021199040+ 1199091015840
1199031198941015840
1199021199030)
1199091015840119903
0
0 0 minus1
2119867
]]]]]]]]]]]]]]]
]
119862 = [1 0 0 0
0 0 0 1]
1198811015840
119889=120587119881dc
3radic3 119903
0= 119877119904+1205872
119877dc18
1199090= 119909119904+1205872
119909119889
18
(9)
6 Journal of Applied Mathematics
Rs
Rs
Rr
Vin Idi = 0
VqsLm
Lm
Ll
Ll
p 120582ds
p 120582qs p 120582qr
p 120582dr
+ minus
+minus
c
c
120596r 120582qr
120596r 120582qr
IdsIdi
IqsIqi
Vds
CSI
Iqi = MdIdc Vqs
Figure 6 Equivalent circuit of CSI-IM section
With usual state feedback control there can be the trackingerror in steady state To eliminate this error Smith andDavidson [18] introduced an integral control which includesfeedback as well as feed forward control In this work anoptimal control law is derived by using the concepts of Smithand Davidson [18] and Veerachary [8]
42 Design of Optimal Control Law A new multivariableoptimal controller incorporating state feedback as well asfeed forward control for fast regulation and stability of a CSIfed IM drive is presented in this section The design of thestate feedback controller is based on the industrial regulatortheory together with pole placement technique applied ona linearized 119889-119902 axes state space model of the drive andincludes a reduced order observer to estimate the inaccessiblestates like 119889-119902 axes rotor currents A feed forward control interms of reference and disturbance inputs has been addedto the feedback controller-observer to obtain faster dynamicresponse either from (10) or (11) as follows (Figure 7)
119906 = 1198701119909 + 119870
2int
119905
0
(119910 minus 119910119903) 119889119905 + 119870
119865119865[119889
119910119903
] (10)
or
119906 = 1198701(2times4)
[[[[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]]]]
]
+ 1198702(2times2)
[[[[[
[
int
119905
0
(119894119902119904minus 119894lowast
119902119904) 119889119905
int
119905
0
(120596119903minus 120596lowast
119903) 119889119905
]]]]]
]
+ 119870119865119865(2times3)
[[[[
[
119879119871
119894lowast
119902119904
120596lowast
119903
]]]]
]
(11)
5 IFOC for CSI Fed IM Drives
In this section a high performance current fed indirect RFOcontrol method is described for IMs The IM is mathe-matically modeled in the rotor flux reference frame that is
CSIfed IM
Reduced order
observer
yy
x
r = [ iqslowast120596rlowast] y= [ iqs
120596r
]u
u
y
y
= [Vd
120596e
]
K1(2times4)
K2(2times2)
KFF(2times3)
+ +++
minus
d = TL
int
u998400
Figure 7 Combined feed forward and integral feedback controller
presented The rotor flux orientation is also obtained In thelow speed region the rotor flux components can be synthe-sized more easily with the help of speed and current signalsDesign of indirect vector controller Blaschke [19 20] androtor flux estimator is discussed Implementation of IFOCto CSI fed IM drive and transient response analysis of othersimulation results are also discussed
51 MathematicalModeling of IFOControlMethod The rotorequations of the IM containing flux linkages as variables aregiven by Krause et al [15] as follows
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus (120596119890minus 120596119903) 120582119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ (120596119890minus 120596119903) 120582119890
119889119903= 0
(12)
where
120596sl = 120596119890 minus 120596119903 (13)
then (12) becomes
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus 120596sl120582
119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ 120596sl120582
119890
119889119903= 0
(14)
The resultant rotor flux linkage 120582119903 also known as the rotor
flux linkages phasor is assumed to be on the direct axisto reduce the number of variables in the equations by oneMoreover it relates to reality that rotor flux linkages are asingle variable Hence aligning d axis with rotor flux phasoryields
120582119903= 120582119890
119889119903 (15)
120582119890
119902119903= 0 (16)
119901120582119890
119902119903= 0 (17)
Substituting (15) into (14) results in the new rotor equationsas follows
119877119903119894119890
119889119903+ 119901120582119903= 0
119877119903119894119890
119902119903+ 120596sl120582119903 = 0
(18)
Journal of Applied Mathematics 7
The rotor currents in terms of the stator currents are derivedfrom (18) as
119894119890
119902119903= minus
119871119898
119871119903
119894119890
119902119904
119894119890
119889119903=120582119903
119871119903
minus119871119898
119871119903
119894119890
119889119904
(19)
119894119890
119889119904= 119894119891= [1 +
119871119903
119877119903
119901]120582119903
119871119898
119894119890
119889119904= 119894119891= [1 + 120591
119903119901]
120582119903
119871119898
(20)
120596sl =119877119903119871119898
119871119903
119894119890
119902119904
120582119903
120596sl =119877119903119871119898
119871119903
119894119879
120582119903
120596sl =119871119898
120591119903
119894119879
120582119903
(21)
The 119902- and 119889-axes currents are relabeled as torque (119894119879) and
flux producing (119894119891) components of the stator-current phasor
respectively 120591119903denotes rotor time constant Similarly by the
same substitution of the rotor currents from (19) into torqueexpression the electromagnetic torque is derived as
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904minus 120582119890
119902119903119894119890
119889119904)
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904)
119879119890= 119870119905119890120582119903119894119890
119902119904= 119870119905119890120582119903119894119879
(22)
where the torque constant119870119905119890is defined as
119870119905119890=3
2
119875
2
119871119898
119871119903
(23)
Note that the torque is proportional to the product of rotorflux linkage and the stator 119902-axis current This resemblesthe air gap torque expression of the DC motor which isproportional to the product of the field flux linkages and thearmature current If the rotor flux linkage is maintained asconstant the torque is proportional to the torque producingcomponent of the stator current This is in relation to theseparately excited DC motor with armature current controlwhere the torque is proportional to the armature currentwhen the field current is constant Further here too the timeconstant is measured in the order of a few milliseconds Therotor flux linkages and air gap torque given in (21) and (22)respectively complete the transformation of the IM into anequivalent separately-excited dc motor from a control pointof view
52 Design of IFO Controller The IFO controller was de-signed using the concepts of Vas [21] and Salo and Tuusa [9]
From that the stator-current phasor is the phasor sum of the119889- and 119902-axis stator currents in any frames and it is given as
119894119904= radic(119894119890
119902119904)2
+ (119894119890
119889119904)2
(24)
and the 119889119902 axes (2120601) to (3120601) abc phase current relationship isobtained from
[
[
119894119890
119902119904
119894119890
119889119904
]
]
=2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+4120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+4120587
3)
]]]
]
[
[
119894119886119904
119894119887119904
119894119888119904
]
]
(25)
And it can be written as
119894119902119889= [119879] [119894
119886119887119888] (26)
where
119894119902119889= [119894119890
119902119904119894119890
119889119904]119905
(27)
119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905
(28)
[119879] =2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+2120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+2120587
3)
]]]
]
(29)
where 119894119886119904 119894119887119904 and 119894
119888119904are the three phase stator currents Note
that the elements in the 119879 matrix are cosinusoidal functionsof electrical angle 120579
119891 The electrical field angle in this case is
that of the rotor flux-linkages phasor and is obtained as thesum of the rotor and slip angles as follows
120579119891= 120579119903+ 120579sl (30)
and the slip angle is obtained by integrating the slip speed andis given as
120579sl = int120596sl (31)
IFO controller developed from these derivations accepts thetorque and flux requests and generates the torque and fluxproducing components of the stator-current phasor and theslip-angle commandsThe command values are denoted withasterisks throughout this thesis From (20) (21) and (23) thecommand values of 119894
119879 119894119891 and 120596sl are obtained as follows
119894lowast
119879=
119879lowast
119890
119870119905119890120582lowast119903
= (2
3) (
2
119875)119879lowast
119890
120582lowast119903
119871119903
119871119898
(32)
119894lowast
119891= (1 + 119901
119871119903
119877119903
)120582lowast
119903
119871119898
(33)
120596lowast
sl =119877119903119871119898
119871119903
119894lowast
119879
120582lowast119903
(34)
8 Journal of Applied Mathematics
The command slip angle 120579lowastsl is generated by integrating 120596lowastsl The torque angle command is obtained as the arctangentof 119894lowast119879and 119894lowast119891 The field angle is obtained by summing the
command slip angle and rotor angleWith the torque and fluxproducing components of the stator current commands androtor field angle the 119902119889 axes current commands (119886119887119888 phasecurrent commands) are obtained as follows The relevantsteps involved in the realization of the IFO controller are asfollows
[
[
119894lowast
119902119904
119894lowast
119889119904
]
]
= [
[
cos 120579119891
sin 120579119891
minus sin 120579119891
cos 120579119891
]
]
[
[
119894lowast
119879
119894lowast
119891
]
]
(35)
[[[[
[
119894lowast
119886119904
119894lowast
119887119904
119894lowast
119888119904
]]]]
]
= [119879minus1
]
[[[[
[
119894lowast
119902119904
119894lowast
119889119904
0
]]]]
]
(36)
where
119879minus1
=
[[[[[[[
[
1 0 1
minus1
2minusradic3
21
minus1
2
radic3
21
]]]]]]]
]
(37)
By using (35) to (37) the stator 119902- and 119889-axes and abccurrent commands are derived as
119894lowast
119902119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119889119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 cos 120579lowast
119904
119894lowast
119886119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119887119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904minus2120587
3)
119894lowast
119888119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904+2120587
3)
(38)
where
120579lowast
119904= 120579119891+ 120579lowast
119879= 120579119903+ 120579lowast
sl + 120579lowast
119879 (39)
The implementation of IFOC on a CSI fed IM is shownin Figures 8 and 9 The torque command 119894lowast
119879is generated
as a function of the speed error signal generally processedthrough a PI controller The flux command 119894lowast
119891can be given
directly or as a function of speed The rotor position 120579119903can
be measured with an encoder and converted into necessarydigital information for feedbackThere has been a substantialamount of research in the development of rotor flux observersfor field orientation that are compensated for variationsin parameters by their feedback corrections Digital imple-mentation of integrators for the estimation of rotor flux ofan IM from the stator voltages and stator currents posesproblems associated with the offset in the sensor amplifiers
CSI
IM
PWMswitching
logic
Rotor speed
regulator
Stator current
generator
DC link currentias
lowast
ibslowast
i cslowast
120579riTlowast is
iflowast
iflowast and iT
lowast120582rlowast
120596rlowast
120596sllowast
120596r
Telowast
(1 + pLr
Rr
) 120582rlowast
Lm
RrLm
Lr
iTlowast
120582rlowast
4
3P
Telowast
120582rlowast
Lr
Lm
+minus
Figure 8 Simplified diagram for IFOC method
islowast = radic(iTlowast)2 +(iflowast)2 ias
lowast = |islowast|sin120579s
lowast|islowast|
ibslowast = |is
lowast|sin120579slowast minus
2120587
3
icslowast = |is
lowast|sin120579slowast +
2120587
3
120579r 120579r
iTlowast
iflowast
120579Tlowast
120579slowast
120596sllowast 120579sl
lowastint
iaslowast
ibslowast
i cslowast
tanminus1 iTlowast
iflowast
+++
Figure 9 Stator current generator
Traditional low-pass filters can replace the integrator Soin this work rotor flux estimated using low pass filter isdescribed The instantaneous flux linkage can be computedusing the measured d-axis stator current using (40) which isreferred to as the current model as follows
120582119903=119871119898119894119889119904
1 + 120591119903119904 (40)
The theoretical concepts of IFOC method for CSI fed IMdrive discussed in Sections 51 and 52 are verified throughsimulation using the Simulink toolbox of MATLAB Thedetailed simulations results are presented in Section 71
6 DTC for CSI Fed IM Drives
Variable speed drives are used in all industries to controlprecisely the speed of electric motors driving loads rangingfrom pumps and fans to complex drives on paper machinesrolling mill cranes and similar drives The most modernmethod for these drives is direct torque and stator flux vectorcontrol method usually called DTC The VSI fed version hasbeen realized in an industrial way by ABB by using thetheoretical background proposed by Takahashi and Ohmori[22] and Depenbrock [23] This solution is based both onfield oriented control (FOC) as well as on the direct self-control theory The idea is that motor flux and torque areused as primary control variables which is contrary to thewayin which traditional AC drives control input frequency andvoltage but is in principle similar to what is done with a DCdrive where it is much more straightforward to achieve
Journal of Applied Mathematics 9
Casadei et al [24] has described that by controllingmotor torque directly DTCprovides dynamic speed accuracyequivalent to closed loop AC and DC systems and torqueresponse times that are 10 times faster It is also claimedthat the DTC does not generate noise like that produced byconventional PWM AC drives This work describes a newmodular approach to implement DTC method on a CSIfed IM drive the simulation implementation procedure Thedynamic response of the drive is analyzed and presented
61 Mathematical Modeling of DTC Control Method TheDTC method is developed from Babaei and Heydari [10]From Krause et al [15] in the stationary reference frame thestator voltage and flux equation can be written as
V119904= 119877119904119894119904+119889120582119904
119889119905 (41)
120582119904= int (V
119904minus 119877119904119894119904) 119889119905 = 120582
120572119904+ 119895120582120573119904 (42)
10038161003816100381610038161205821199041003816100381610038161003816 = radic120582
2
120572119904+ 1205822
120573119904 (43)
ang120593119904= tanminus1 (
120582120573119904
120582120572119904
) (44)
where 119877119904is the stator resistance and 120582
119904is the stator flux
vector in the stationary reference frame |120582119904| and ang120593
119904are the
amplitude and position of stator flux vector respectivelyIn DTC of CSI fed IM it is desirable to determine the
stator current reference so that the torque and stator fluxfollow their reference values Babaei and Heydari [10] Theproposed DTC system is based on the stator flux oriented(SFO) reference frame In this rotating reference frame it iswritten as
120582119902119904= 0 120582
119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)
where 119889 and 119902 are the real and imaginary axes in the SFOreference frame Therefore the torque equation is rewrittenas
119879119890=3
2(119875
2)1
120596119887
(120582119889119904119894119902119904) (46)
where 119894119902119904is the imaginary part of the stator current vector
in the SFO reference frame Therefore if |120582lowast119904| and 119879lowast
119890are
stator flux magnitude and torque references respectively theimaginary part of the stator current reference vector whichleads to torque and stator flux magnitude references is asfollows
119894lowast
119902119904=2
3
1
119875
119879lowast
119890
1003816100381610038161003816120582lowast
119904
1003816100381610038161003816
(47)
Because of the existence of a capacitor in the controlsystem and its corresponding losses when (47) is used tocontrol the torque for all rotor speeds the control system willnot be accurate enough
Currentlimiter
Telowast
Te
|120582slowast|
2
3
1
P
PI
Telowast
|120582slowast |
+
+
+minus iqs
lowast
Figure 10 119902-axis reference current calculator
In order to have amore accurate torque control the blockdiagram shown in Figure 10 is used to calculate 119894lowast
119902119904This block
diagram is a combination of (47) and a simple PI controllerThis controller removes the steady-state error and leads to anaccurate control The coefficients of this controller must besmall enough so that it remains stable and the ripples in 119894lowast
119902119904
are limited because a ripple of 119894lowast119902119904results in a ripple of the
generated torque If only a PI is used 119894lowast119902119904
will have a largevalue of ripple and the torque response time will be verylargeTherefore in order to have a faster transient response oftorque it is necessary to use (47) In DTC method with VSIthe real part of voltage vector (V
119889) in the stator flux reference
frame leads to direct control of stator flux magnitudeTherefore using the real part of the stator current vector
in the stator flux reference frame it is possible to control theflux magnitude indirectly In order to obtain the real part ofthe stator current reference a simple PI controller is usedTheinput of this PI controller is the error that resulted from thecomparison between the reference stator flux magnitude andthe estimated flux value In order to control themotor currentin all cases especially in the starting mode the output ofthis PI controller must be limited between proper valuesTheoutput signal of the PI controller is the real part of referencestator current 119894lowast
119889119904
Therefore the stator current reference vector that leads tocontrol the IM stator flux and torque is as follows
119894lowast
119904(119889119902)= 119894lowast
119889119904+ 119895119894lowast
119902119904 (48)
1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 =radic119894lowast2
119889119904+ 119894lowast2119902119904 (49)
The obtained reference current vector is in the statorflux reference frame In order to generate it by SVM of theCSI it must be transferred to the stationary reference frameTo do that the stator flux position is used to transfer thisvector from the stator flux reference frame to the stationaryreference frame as follows
119894lowast
119904(120572120573)= 119894lowast
120572119904+ 119895119894lowast
120573119904= (119894lowast
119889119904+ 119895119894lowast
119902119904) sdot 119890119895120593119904
=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 ang (120593119904 + tanminus1 (119894lowast
119902119904
119894lowast
119889119904
))
(50)
This reference current vectormust be generated in theCSIfirst and then applied to the motor after removing its har-monics with filtering capacitorThis current has a nearly sinu-soidal waveform that leads to a nearly sinusoidal voltage inmotor terminals after feeding the motor and passing through
10 Journal of Applied Mathematics
Stator flux torque
estimator
PI
CSI
SVM IM
Gate Signals
PI
120582slowast
120591elowast
120582s
120582s
120591e
ang120593s
120591e
iqlowast
idlowast
dq
to120572120573
iaib
vb
va
|islowast| and idc
lowast
idclowast
+minus
+minus i120572
lowast
i120573lowast
Figure 11 Block diagram schematic of the proposed DTC
motor impedances The speed and stator flux control of anIM with DTC method is shown in Figure 11 Separate PIcontrollers are used to control the speed and flux The inputto the speed controller is the error caused by comparingthe reference speed and the rotor speed The output of thiscontroller is the imaginary part of stator current vector 119894lowast
119902119904
which is limited to a proper value
7 Results and Discussion
71 Simulation Results The simulationmodels of IOL IFOCand DTC control methods for CSI fed IM drive systemsare developed and simulated using MATLAB simulationsoftware From the simulation results dynamic performancesof the developed drives have been individually analyzedmainly using the sensitivity tests Here the results of all thedrives are compared with one another The tests that arechosen for the comparison are step changes in the referencespeed and torque of the motor drive
The first test is step change in reference speed which isuseful when trying to change from one speed to anotherThe next test is step change in reference torque that wouldbe useful in identifying the controller with a high torqueresponse Change in speed reference is the first test that isdone to compare the motor drives
The results of this test can be seen in the Figures 12 13 and14The speed command is varied from0 to its rated valueThespeed response of IOL drive is faster and has the settling timeof 05 second range while DTC and IFOC drives performsimilarly but slower than IOL (settling time of less than 1second)
The second test as mentioned earlier is the torque steptestThe results of this test can be seen in Figures 12 to 14Thetorque reference is varied from 0 to its rated value The speedresponse of IFOC drive is faster and has the settling timeof 02 seconds while IOL and DTC drives perform similarly(settling time of less than 05 second)
During this test the motor torque response of DTCdrive is almost instantaneous within the 02 second rangewhile IFOC drive closely follows behind it IOL and vectorbased 119881119891 have slower torque response than all other VectorControl techniques
Table 1 Prototype motor parameters
1 Hp 3 phase star connected 4 pole 415V 18 A 50Hz
Stator resistance 0087Ω Stator amp rotorleakage reactance 08119890minus3H
Rotor resistance 0228Ω Magnetizingreactance 347119890minus3H
72 Experimental Results Brief experimental results arepresented for IOL method and DTC method usingTMS320F2812 DSP based hardware setup Figure 15 showsthe schematic diagram for the hardware setup of theproposed laboratory prototype CSI fed induction motordrive The induction motor ratings are given in Table 1 Theinverter consists of CM75TL-12NF The gate driving signalis developed by using Texas instruments DSP processorTMS320F2812 DSP The software used to develop programsfor TMS320F2812 DSP is by code composer studio andMatlabSimulinkThe processor contains the program that isdownloaded to the computer
The clock speed of the DSP is 150MHz and it is capableof 32-bit operations The on board available flash memoryis 2048Mb It was created specifically for motor controloperation and therefore Parkrsquos and Clarkrsquos transformationsare conveniently built in Another convenient feature is thatit has sixteen 12-bit ADC pins that allow for a high degree ofprecision while taking many possible measurements
The above experimental result of IOL method of vectorcontrol technique in Figure 16 shows the fast settling timewith feed forward control compared to feedback techniqueSimilarly the experimental result shown in Figure 17 depictsa greater torque response within 02 seconds comparedto other vector control techniques With this experimentalverification it is clear that IOL is adaptive for speed responseand DTC is more suitable for torque response controlleddrives
From Tables 2 and 3 it seems that the dynamic speed andtorque performance of all four drives using change in speedreference It is evident from the table that IOL drive hasthe best speed response with DTC and IFOC occupyingpositions next to it It is also evident that DTC drive has thebest instantaneous torque response with IFOC and IOL nextto it
8 Conclusion
This paper presents a new approach in mathematical mod-eling of a 3-phase induction motor drive Mathematics con-cepts are used in developing the simulation and implementa-tion of IOL IFOC and DTC methods for CSI fed IM drivesIn this paper a rotor flux based reference frame controlmethod for PWM CSI fed IM with simplified simulationcircuits has been developed in which the motor speed andtorque follow the references closely in all the three vectorcontrol methods However a CSI fed induction motor drivehas some drawbacks such as slower dynamic response andbulky overall size due to the requirement of large smoothing
Journal of Applied Mathematics 11
15
minus05
0
05
1
Spee
d (P
U)
0 1 2 3 4 5Time (s)
Without FF controlWith FF control
(a)
0 1 2 3 4 5
minus2
0
2
Time (s)
Torq
ue (P
U)
Without FF controlWith FF control
(b)
Figure 12 Speed torque response of IOL control method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus4
minus2
0
24
Torq
ue (P
U)
Time (s)
Reference speedStep increase in speed command
Step decrease in speed command
Step increase in torque command
Step decrease in torque command Reference torque
Motor torque
Motor speed
Response due to change intorque reference
Response due to change in
speed reference
Figure 13 Speed torque response of IFOC method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus2minus1
012
Time (s)
Torq
ue (P
U)
Step decrease in speed command
Step increase in speed commandReference speed Motor speed
Response due to change intorque reference
Response due to change inspeed reference
Motor torque
Step decrease in torque commandReference torque
Step increase in torque command
Figure 14 Speed torque response of DTC method
inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying
TMS320F2812DSP
ADC
PWM
-EV
B
PWM
-EVA
Supplyfilter
3120601controlled
rectifier3120601 CSI inverter
3120601IM
3120601supply
Texasinstruments
DSP processor
Speed sensor output
(Nr) Rpm
CM75TL-12NFseries
IGBT (6-PAC)
Ldc
Is (Amps)
Figure 15 Schematic diagram of the hardware setup of CSI fed IM
Torque output with feedforward control
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Torque output withoutfeed forward control
Speed output with feedforward control
Speed output withoutfeed forward control
Figure 16 Experimental result torque-speed response of IOLcontrol method
applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 Journal of Applied Mathematics
and DTC provides simpler control architecture with a similardynamic performance as that of IFOC
Similar studies of vector controlled inverter fed drives arediscussed mostly with the voltage source inverters and veryfew papers are available with the study of vector controlledcurrent source inverters Veerachary [8] Salo and Tuusa [9]and Babaei and Heydari [10] This paper presents a compar-ative analysis of different mathematical modeling of vectorcontrolled CSI fed inductionmotor drives and in particularlythe dynamic responses of the drive motor speed againstchange in speed as well as the torque references in a singlesimulation cycle and verified experimentally and to ascertaindifferent mathematical dynamic IM modeling for CSI feddrives as Holmes et al [11] and to apply control methodson them The MATLAB s-functions are described to designIM modeling and drives Finally the above mentioned drivestrategies are compared as fairly as possible by comparingtheir dynamic responses when there is step change in thereferences (usually stator currents motor speed and loadtorque)
2 Mathematical Modeling of3-Phase Induction Motor Drive
In this section Mathematical model of IM is shown inFigure 1 and described In general IM is used to transformelectrical energy to mechanical energy It consists of elec-tric circuitry electromagnetic circuitry and electromechaniccircuitry Krishnan [12] The main objective of the motormodeling is to build a simple but competent model thatdescribes these circuits and their interconnectionsThe mainpath in themodel is in relationwith the voltage or current andthe stator phases in input and magnetic flux inside the motorand electromagnetic torque in the output
As mentioned earlier the squirrel cage IM is preparedfor industry applications The IM can either be supplied by avoltage source or a current sourceThese sources are assumedto be ideal thismeans that they can supply any desired voltageor current without losses According to Krause and Thomas[13] the basic equations of the IM are the stator voltageequation (1) the rotor voltage equation (2) the equationsof the flux linkages of the stator and the rotor windingsystems (3) and (4) and the torque equation (5) Considerthe following
V119904119904= 119877119904119894119904
119904+ 119904
119904 (1)
V119903119903= 0 = 119877
119903119894119903
119903+ 119903
119903 (2)
120582119886
119904= 120582119886
119898+ 119871119897119904119894119886
119904 (3)
120582119886
119903= 120582119886
119898+ 119871119897119903119894119886
119903 (4)
119879119890= [119862(
120587
2) 120582119886
119898]
119879
119894119886
119904 (5)
The equations are given in vector format Bose [14] Everyvector variable has two components the first component isparallel to the reference frame axis or ldquo119889rdquo axis and the secondcomponent is perpendicular to the reference axis or ldquo119902rdquo axis
The symbols are Vmdashvoltage 119894mdashcurrent 120582119898mdashflux linkage
119877mdashresistance and 119871119897mdashleakage inductance subscript ldquo119904rdquo
indicates the stator winding system ldquo119903rdquo the rotor windingsystem superscript ldquo119904rdquo indicates the stator reference frame(stator coordinates) ldquo119903rdquo the rotor reference frame andldquo119886rdquo any arbitrary reference frame 119879
119890is the electromagnetic
torque 119862(1205872) is a rotation matrix over 1205872 radians the dot( ) indicates a time derivative The stator voltage equationis restricted to the stator reference frame because of thedifferential of the stator flux vector whereas the rotor voltageequation is restricted to the rotor reference frame becauseof the differentiation The flux equations and the torqueequation can be defined in any reference frame denoted bysuperscript ldquo119886rdquo
It is evident from (1) to (5) that themotor voltage flux andtorque can be controlled easily by controlling the current ofthe motor So the CSI is the right choice to control the motordirectly because the inputs of the current fed IM model arethe stator currents
Before these equations are completed to obtain the IMmodel the reference frame of the model in relation to theproblem being investigated and the type of computer (analogor digital) are chosen In order to evaluate the usefulness ofeach reference frame the above equations are simulated usingMatlab software to predict the transient performance of anIM
The mathematical modeling of 3-phase induction motordrive which can be run at any of the three rotating referenceframe models namely stator rotating reference frame rotorrotating reference frame and synchronous rotating referenceframe in MatlabSimulink is shown below in Figure 2
3 Transient Performance of DifferentMathematical Reference Frame Models
Figure 3 shows the transient performance of ldquo119886rdquo phase and119889-axis stator currents in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds From Figure 3(a) it is evident that in statoror stationary reference frame stator 119889-axis current behavesexactly the same way as do the stator ldquo119886rdquo phase current ofthe motor If the stationary reference frame is used then thestator 119889-axis current is identical to the stator phase currentThis would be useful when interest is specifically confinedto stator variables for example variable speed stator-fedIM drives Similarly if the rotor reference frame is usedthen the rotor 119889-axis current will behave in exactly thesame manner as the rotor ldquo119886rdquo phase current This wouldbe useful when interest is confined to rotor variables onlyas for example variable speed rotor-fed IM drives FromFigure 3(b) it is evident that in rotor rotating reference framestator 119889-axis current does not behave as the stator phasecurrent FromFigure 3(c) it is apparent that in synchronouslyrotating reference frame stator 119889-axis current behaves likeDC quantity If the synchronously rotating reference frameis used then the stator and rotor current behave like DCquantity Thus the controller design is simple and is widelyused in AC drives
Journal of Applied Mathematics 3
Rotor
Stator
4
3
2
1
[0 0]
rot1
rot
1s
1s
-K-
-K-3
2
1
fr
fs
is
irfr = rotor flux
fs = stator flux
is = stator current
Te = electromagnetic torque
Te
ir = rotor current
Flux-current relationship+minus
minus
+minus
+minus
minus
+minus
+minus
times
times
r
Rr
Rs
wm
Gm1
Gm
Gs
Gr
wk
s
K lowast u
K lowast u
∙
(Lrl + Lm ) (Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
Lm (Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
Lm (Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
(Lsl + Lm)(Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
Figure 1 Induction motor modeling in arbitrary reference frame
dqreference
frame
Stator
Synchronous
Rotor
Speed
Angle
-K-
Torque
Speed
Frequency
dq
ab
dq2ab
Currents
Amplitude
ab
etakdq
ab2dq
ab2ABC
Step load pu
Mechanicalsystem
1s
1s
Inductionmotor
pu
-K--
f(u)
f(u)
f(u)
0
0
ABC2ab dqABC ab
etak
Variable-amplitude variable-frequency3-phase sinusoidal voltages
K lowast u
wm
wk
wmwm
wmwkws
ws
wo
sTe Te
Te
Te
Tl
Tl
i dq
i dq
i ab
i abi ABC
is
120579r
120579k
120579s
120579s
K lowast u
Figure 2 Mathematical modeling of 3-phase induction motor drive
Figure 4 shows the transient performance of 119902-axis statorcurrent and 119889-rotor flux in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds From Figures 4(a) and 4(b) it is apparentthat there is a coupling effect between stator 119902-axis currentand rotor 119889-axis flux So independent control is difficultFigure 4(c) shows that the rotor 119889-axis flux and stator 119902-axis current (control variables) have linear relationship with
torque (speed) Thus it is useful in the rotor flux oriented(RFO) control methods
Figure 5 shows the transient performance of 119902-axis statorcurrent and 119889-stator flux in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds Figures 5(a) and 5(b) state that there is acoupling effect between stator 119902-axis current and stator119889-axis flux and so independent control is difficult From
4 Journal of Applied Mathematics
minus10
0
10
0 05 1 15 2Time (s)
I ac
andI d
c(P
U)
(a)
minus10
0
10
0 05 1 15 2Time (s)
I ac
andI d
c(P
U)
(b)
0 05 1 15 2minus10
0
10
Stator a phase currentStator b-axis current
Step change in load torque
Time (s)
I ac
andI d
c(P
U)
(c)
Figure 3 Simulated results of ldquo119886rdquo phase and 119889-axis stator currents in different rotating reference frames
minus10
0
10
0 05 1 15 2Time in seconds
I qs
and
FLU
Xdr
(a)
minus10
0
10
0 05 1 15 2Time in seconds
I qs
and
FLU
Xdr
(b)
0 05 1 15 2minus10
minus5
0
5
Step change in load torque
Stator q-axis currentRotor d-axis flux
Time (s)
I qs
and
FLU
Xdr
(PU
)
(c)
Figure 4 Simulated results of 119902-axis stator current and 119889-axis rotorflux in different rotating reference frames
minus10
0
10
0 05 1 15 2Time (s)
I qs
and
FLU
Xds
(a)
minus10
0
10
0 05 1 15 2Time (s)
I qs
and
FLU
Xds
(b)
0 05 1 15 2minus10
minus5
0
5
Step change in load torque
Time (s)
Stator q-axis currentStator d-axis flux
I qs
and
FLU
Xds
(c)
Figure 5 Simulated results of 119902-axis stator current and 119889-axis statorflux in different rotating reference frames
Journal of Applied Mathematics 5
Figure 5(c) it is evident that the stator119889-axis flux and stator 119902-axis current (control variables) have linear relationship withtorque (speed) It is used in the stator flux oriented (SFO)control methods
The basic IM equations from Krause et al [15] aresimulated for three different reference frames usingMATLAB76 software and the results obtained are compared with eachreference frame From the comparison it is observed that ifthe stator rotating reference frame is used then stator 119889-axiscurrent is identical to that of the stator ldquo119886rdquo phase current asshown in Figure 3(a) which is used in stator fed IM drivesSimilarly in rotor reference frame rotor 119889-axis current isidentical to the rotor ldquo119886rdquo phase current and useful in rotor fedIM drives If the synchronous reference frame is used thenthe stator 119889-axis current behaves like DC quantity as shownin Figure 3(c) From Figures 4(c) and 5(c) it is evident thatsynchronous reference frame is useful in RFO and SFO drivecontrol methods In this paper synchronous reference frameis preferred to model the IM in 119881119891 IOL control methodsRFO control is used in IFOCmethod and SFO is used inDTCmethod
4 IOL Control for CSI Fed IM Drives
In this section a type of IOL control method for the CSIfed IM drive is described The controller is designed based
on pole placement technique of Wonham [16] to a 119889-119902 axisstate space linearized model of the drive which includes areduced order rotor current observer A control law has beendeveloped to improve the dynamic response of the drive byusing the concepts from Veerachary [8] which includes feedforward as well as feedback controls
41 Mathematical Modeling of IOL Control Method Theequivalent circuit for the CSI capacitor and IM in a 119889-119902 axis rotating reference frame is shown in Figure 6 Thecorresponding state equation (differential equations) withmotor flux linkages as variables is derived fromWu et al [17]
Veerachary [8] presented the linearized CSI-IM drivestate space model in synchronously rotating reference frameas follows
= 119860119909 + 119861119906 (6)
where
=119889119909
119889119905 (7)
119910 = 119862119909 (8)
where
119909 =
[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]
]
=119889
119889119905
[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]
]
119906 =
[[[[
[
Δ1198811015840
119889
Δ120596119890
120596119887
Δ119879119871
]]]]
]
119910 =[[
[
Δ119894119902119904
Δ120596119903
120596119887
]]
]
119860 =
[[[[[[[[[[[[[[
[
minus1199091015840
1199031199030120596119887
11990901199091015840119903minus 1199092119898
1199091198981198771015840
119903120596119887
11990901199091015840119903minus 1199092119898
minus12059611989001199091015840
119903119909119898(1 minus 119904)
11990901199091015840119903minus 1199092119898
minus1199091198981199091015840
1199031198941015840
1198891199030120596119887
11990901199091015840119903minus 1199092119898
1205961198871199091198981199030
11990901199091015840119903minus 1199092119898
minus11990901198771015840
119903120596119887
11990901199091015840119903minus 1199092119898
1205961198900(1199092
119898minus 11990411990901199091015840
119903)
11990901199091015840119903minus 1199092119898
11990901199091015840
1199031198941015840
1198891199030120596119887
11990901199091015840119903minus 1199092119898
1199041205961198900119909119898
1199091015840119903
1199041205961198900
minus1205961198871198771015840
119903
1199091015840119903
minus
120596119887(1199091198981198941199021199040+ 1199091015840
1199031198941015840
1199021199030)
1199091015840119903
1199091198981198941015840
1198891199030
21198670
1199091198981198941199021199040
21198670
]]]]]]]]]]]]]]
]
119861 =
[[[[[[[[[[[[[[[
[
1199091015840
119903120596119887
11990901199091015840119903minus 1199092119898
0 0
minus119909119898120596119887
11990901199091015840119903minus 1199092119898
minus1205961198871198941015840
11988911990300
0 minus
120596119887(1199091198981198941199021199040+ 1199091015840
1199031198941015840
1199021199030)
1199091015840119903
0
0 0 minus1
2119867
]]]]]]]]]]]]]]]
]
119862 = [1 0 0 0
0 0 0 1]
1198811015840
119889=120587119881dc
3radic3 119903
0= 119877119904+1205872
119877dc18
1199090= 119909119904+1205872
119909119889
18
(9)
6 Journal of Applied Mathematics
Rs
Rs
Rr
Vin Idi = 0
VqsLm
Lm
Ll
Ll
p 120582ds
p 120582qs p 120582qr
p 120582dr
+ minus
+minus
c
c
120596r 120582qr
120596r 120582qr
IdsIdi
IqsIqi
Vds
CSI
Iqi = MdIdc Vqs
Figure 6 Equivalent circuit of CSI-IM section
With usual state feedback control there can be the trackingerror in steady state To eliminate this error Smith andDavidson [18] introduced an integral control which includesfeedback as well as feed forward control In this work anoptimal control law is derived by using the concepts of Smithand Davidson [18] and Veerachary [8]
42 Design of Optimal Control Law A new multivariableoptimal controller incorporating state feedback as well asfeed forward control for fast regulation and stability of a CSIfed IM drive is presented in this section The design of thestate feedback controller is based on the industrial regulatortheory together with pole placement technique applied ona linearized 119889-119902 axes state space model of the drive andincludes a reduced order observer to estimate the inaccessiblestates like 119889-119902 axes rotor currents A feed forward control interms of reference and disturbance inputs has been addedto the feedback controller-observer to obtain faster dynamicresponse either from (10) or (11) as follows (Figure 7)
119906 = 1198701119909 + 119870
2int
119905
0
(119910 minus 119910119903) 119889119905 + 119870
119865119865[119889
119910119903
] (10)
or
119906 = 1198701(2times4)
[[[[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]]]]
]
+ 1198702(2times2)
[[[[[
[
int
119905
0
(119894119902119904minus 119894lowast
119902119904) 119889119905
int
119905
0
(120596119903minus 120596lowast
119903) 119889119905
]]]]]
]
+ 119870119865119865(2times3)
[[[[
[
119879119871
119894lowast
119902119904
120596lowast
119903
]]]]
]
(11)
5 IFOC for CSI Fed IM Drives
In this section a high performance current fed indirect RFOcontrol method is described for IMs The IM is mathe-matically modeled in the rotor flux reference frame that is
CSIfed IM
Reduced order
observer
yy
x
r = [ iqslowast120596rlowast] y= [ iqs
120596r
]u
u
y
y
= [Vd
120596e
]
K1(2times4)
K2(2times2)
KFF(2times3)
+ +++
minus
d = TL
int
u998400
Figure 7 Combined feed forward and integral feedback controller
presented The rotor flux orientation is also obtained In thelow speed region the rotor flux components can be synthe-sized more easily with the help of speed and current signalsDesign of indirect vector controller Blaschke [19 20] androtor flux estimator is discussed Implementation of IFOCto CSI fed IM drive and transient response analysis of othersimulation results are also discussed
51 MathematicalModeling of IFOControlMethod The rotorequations of the IM containing flux linkages as variables aregiven by Krause et al [15] as follows
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus (120596119890minus 120596119903) 120582119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ (120596119890minus 120596119903) 120582119890
119889119903= 0
(12)
where
120596sl = 120596119890 minus 120596119903 (13)
then (12) becomes
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus 120596sl120582
119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ 120596sl120582
119890
119889119903= 0
(14)
The resultant rotor flux linkage 120582119903 also known as the rotor
flux linkages phasor is assumed to be on the direct axisto reduce the number of variables in the equations by oneMoreover it relates to reality that rotor flux linkages are asingle variable Hence aligning d axis with rotor flux phasoryields
120582119903= 120582119890
119889119903 (15)
120582119890
119902119903= 0 (16)
119901120582119890
119902119903= 0 (17)
Substituting (15) into (14) results in the new rotor equationsas follows
119877119903119894119890
119889119903+ 119901120582119903= 0
119877119903119894119890
119902119903+ 120596sl120582119903 = 0
(18)
Journal of Applied Mathematics 7
The rotor currents in terms of the stator currents are derivedfrom (18) as
119894119890
119902119903= minus
119871119898
119871119903
119894119890
119902119904
119894119890
119889119903=120582119903
119871119903
minus119871119898
119871119903
119894119890
119889119904
(19)
119894119890
119889119904= 119894119891= [1 +
119871119903
119877119903
119901]120582119903
119871119898
119894119890
119889119904= 119894119891= [1 + 120591
119903119901]
120582119903
119871119898
(20)
120596sl =119877119903119871119898
119871119903
119894119890
119902119904
120582119903
120596sl =119877119903119871119898
119871119903
119894119879
120582119903
120596sl =119871119898
120591119903
119894119879
120582119903
(21)
The 119902- and 119889-axes currents are relabeled as torque (119894119879) and
flux producing (119894119891) components of the stator-current phasor
respectively 120591119903denotes rotor time constant Similarly by the
same substitution of the rotor currents from (19) into torqueexpression the electromagnetic torque is derived as
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904minus 120582119890
119902119903119894119890
119889119904)
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904)
119879119890= 119870119905119890120582119903119894119890
119902119904= 119870119905119890120582119903119894119879
(22)
where the torque constant119870119905119890is defined as
119870119905119890=3
2
119875
2
119871119898
119871119903
(23)
Note that the torque is proportional to the product of rotorflux linkage and the stator 119902-axis current This resemblesthe air gap torque expression of the DC motor which isproportional to the product of the field flux linkages and thearmature current If the rotor flux linkage is maintained asconstant the torque is proportional to the torque producingcomponent of the stator current This is in relation to theseparately excited DC motor with armature current controlwhere the torque is proportional to the armature currentwhen the field current is constant Further here too the timeconstant is measured in the order of a few milliseconds Therotor flux linkages and air gap torque given in (21) and (22)respectively complete the transformation of the IM into anequivalent separately-excited dc motor from a control pointof view
52 Design of IFO Controller The IFO controller was de-signed using the concepts of Vas [21] and Salo and Tuusa [9]
From that the stator-current phasor is the phasor sum of the119889- and 119902-axis stator currents in any frames and it is given as
119894119904= radic(119894119890
119902119904)2
+ (119894119890
119889119904)2
(24)
and the 119889119902 axes (2120601) to (3120601) abc phase current relationship isobtained from
[
[
119894119890
119902119904
119894119890
119889119904
]
]
=2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+4120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+4120587
3)
]]]
]
[
[
119894119886119904
119894119887119904
119894119888119904
]
]
(25)
And it can be written as
119894119902119889= [119879] [119894
119886119887119888] (26)
where
119894119902119889= [119894119890
119902119904119894119890
119889119904]119905
(27)
119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905
(28)
[119879] =2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+2120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+2120587
3)
]]]
]
(29)
where 119894119886119904 119894119887119904 and 119894
119888119904are the three phase stator currents Note
that the elements in the 119879 matrix are cosinusoidal functionsof electrical angle 120579
119891 The electrical field angle in this case is
that of the rotor flux-linkages phasor and is obtained as thesum of the rotor and slip angles as follows
120579119891= 120579119903+ 120579sl (30)
and the slip angle is obtained by integrating the slip speed andis given as
120579sl = int120596sl (31)
IFO controller developed from these derivations accepts thetorque and flux requests and generates the torque and fluxproducing components of the stator-current phasor and theslip-angle commandsThe command values are denoted withasterisks throughout this thesis From (20) (21) and (23) thecommand values of 119894
119879 119894119891 and 120596sl are obtained as follows
119894lowast
119879=
119879lowast
119890
119870119905119890120582lowast119903
= (2
3) (
2
119875)119879lowast
119890
120582lowast119903
119871119903
119871119898
(32)
119894lowast
119891= (1 + 119901
119871119903
119877119903
)120582lowast
119903
119871119898
(33)
120596lowast
sl =119877119903119871119898
119871119903
119894lowast
119879
120582lowast119903
(34)
8 Journal of Applied Mathematics
The command slip angle 120579lowastsl is generated by integrating 120596lowastsl The torque angle command is obtained as the arctangentof 119894lowast119879and 119894lowast119891 The field angle is obtained by summing the
command slip angle and rotor angleWith the torque and fluxproducing components of the stator current commands androtor field angle the 119902119889 axes current commands (119886119887119888 phasecurrent commands) are obtained as follows The relevantsteps involved in the realization of the IFO controller are asfollows
[
[
119894lowast
119902119904
119894lowast
119889119904
]
]
= [
[
cos 120579119891
sin 120579119891
minus sin 120579119891
cos 120579119891
]
]
[
[
119894lowast
119879
119894lowast
119891
]
]
(35)
[[[[
[
119894lowast
119886119904
119894lowast
119887119904
119894lowast
119888119904
]]]]
]
= [119879minus1
]
[[[[
[
119894lowast
119902119904
119894lowast
119889119904
0
]]]]
]
(36)
where
119879minus1
=
[[[[[[[
[
1 0 1
minus1
2minusradic3
21
minus1
2
radic3
21
]]]]]]]
]
(37)
By using (35) to (37) the stator 119902- and 119889-axes and abccurrent commands are derived as
119894lowast
119902119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119889119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 cos 120579lowast
119904
119894lowast
119886119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119887119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904minus2120587
3)
119894lowast
119888119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904+2120587
3)
(38)
where
120579lowast
119904= 120579119891+ 120579lowast
119879= 120579119903+ 120579lowast
sl + 120579lowast
119879 (39)
The implementation of IFOC on a CSI fed IM is shownin Figures 8 and 9 The torque command 119894lowast
119879is generated
as a function of the speed error signal generally processedthrough a PI controller The flux command 119894lowast
119891can be given
directly or as a function of speed The rotor position 120579119903can
be measured with an encoder and converted into necessarydigital information for feedbackThere has been a substantialamount of research in the development of rotor flux observersfor field orientation that are compensated for variationsin parameters by their feedback corrections Digital imple-mentation of integrators for the estimation of rotor flux ofan IM from the stator voltages and stator currents posesproblems associated with the offset in the sensor amplifiers
CSI
IM
PWMswitching
logic
Rotor speed
regulator
Stator current
generator
DC link currentias
lowast
ibslowast
i cslowast
120579riTlowast is
iflowast
iflowast and iT
lowast120582rlowast
120596rlowast
120596sllowast
120596r
Telowast
(1 + pLr
Rr
) 120582rlowast
Lm
RrLm
Lr
iTlowast
120582rlowast
4
3P
Telowast
120582rlowast
Lr
Lm
+minus
Figure 8 Simplified diagram for IFOC method
islowast = radic(iTlowast)2 +(iflowast)2 ias
lowast = |islowast|sin120579s
lowast|islowast|
ibslowast = |is
lowast|sin120579slowast minus
2120587
3
icslowast = |is
lowast|sin120579slowast +
2120587
3
120579r 120579r
iTlowast
iflowast
120579Tlowast
120579slowast
120596sllowast 120579sl
lowastint
iaslowast
ibslowast
i cslowast
tanminus1 iTlowast
iflowast
+++
Figure 9 Stator current generator
Traditional low-pass filters can replace the integrator Soin this work rotor flux estimated using low pass filter isdescribed The instantaneous flux linkage can be computedusing the measured d-axis stator current using (40) which isreferred to as the current model as follows
120582119903=119871119898119894119889119904
1 + 120591119903119904 (40)
The theoretical concepts of IFOC method for CSI fed IMdrive discussed in Sections 51 and 52 are verified throughsimulation using the Simulink toolbox of MATLAB Thedetailed simulations results are presented in Section 71
6 DTC for CSI Fed IM Drives
Variable speed drives are used in all industries to controlprecisely the speed of electric motors driving loads rangingfrom pumps and fans to complex drives on paper machinesrolling mill cranes and similar drives The most modernmethod for these drives is direct torque and stator flux vectorcontrol method usually called DTC The VSI fed version hasbeen realized in an industrial way by ABB by using thetheoretical background proposed by Takahashi and Ohmori[22] and Depenbrock [23] This solution is based both onfield oriented control (FOC) as well as on the direct self-control theory The idea is that motor flux and torque areused as primary control variables which is contrary to thewayin which traditional AC drives control input frequency andvoltage but is in principle similar to what is done with a DCdrive where it is much more straightforward to achieve
Journal of Applied Mathematics 9
Casadei et al [24] has described that by controllingmotor torque directly DTCprovides dynamic speed accuracyequivalent to closed loop AC and DC systems and torqueresponse times that are 10 times faster It is also claimedthat the DTC does not generate noise like that produced byconventional PWM AC drives This work describes a newmodular approach to implement DTC method on a CSIfed IM drive the simulation implementation procedure Thedynamic response of the drive is analyzed and presented
61 Mathematical Modeling of DTC Control Method TheDTC method is developed from Babaei and Heydari [10]From Krause et al [15] in the stationary reference frame thestator voltage and flux equation can be written as
V119904= 119877119904119894119904+119889120582119904
119889119905 (41)
120582119904= int (V
119904minus 119877119904119894119904) 119889119905 = 120582
120572119904+ 119895120582120573119904 (42)
10038161003816100381610038161205821199041003816100381610038161003816 = radic120582
2
120572119904+ 1205822
120573119904 (43)
ang120593119904= tanminus1 (
120582120573119904
120582120572119904
) (44)
where 119877119904is the stator resistance and 120582
119904is the stator flux
vector in the stationary reference frame |120582119904| and ang120593
119904are the
amplitude and position of stator flux vector respectivelyIn DTC of CSI fed IM it is desirable to determine the
stator current reference so that the torque and stator fluxfollow their reference values Babaei and Heydari [10] Theproposed DTC system is based on the stator flux oriented(SFO) reference frame In this rotating reference frame it iswritten as
120582119902119904= 0 120582
119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)
where 119889 and 119902 are the real and imaginary axes in the SFOreference frame Therefore the torque equation is rewrittenas
119879119890=3
2(119875
2)1
120596119887
(120582119889119904119894119902119904) (46)
where 119894119902119904is the imaginary part of the stator current vector
in the SFO reference frame Therefore if |120582lowast119904| and 119879lowast
119890are
stator flux magnitude and torque references respectively theimaginary part of the stator current reference vector whichleads to torque and stator flux magnitude references is asfollows
119894lowast
119902119904=2
3
1
119875
119879lowast
119890
1003816100381610038161003816120582lowast
119904
1003816100381610038161003816
(47)
Because of the existence of a capacitor in the controlsystem and its corresponding losses when (47) is used tocontrol the torque for all rotor speeds the control system willnot be accurate enough
Currentlimiter
Telowast
Te
|120582slowast|
2
3
1
P
PI
Telowast
|120582slowast |
+
+
+minus iqs
lowast
Figure 10 119902-axis reference current calculator
In order to have amore accurate torque control the blockdiagram shown in Figure 10 is used to calculate 119894lowast
119902119904This block
diagram is a combination of (47) and a simple PI controllerThis controller removes the steady-state error and leads to anaccurate control The coefficients of this controller must besmall enough so that it remains stable and the ripples in 119894lowast
119902119904
are limited because a ripple of 119894lowast119902119904results in a ripple of the
generated torque If only a PI is used 119894lowast119902119904
will have a largevalue of ripple and the torque response time will be verylargeTherefore in order to have a faster transient response oftorque it is necessary to use (47) In DTC method with VSIthe real part of voltage vector (V
119889) in the stator flux reference
frame leads to direct control of stator flux magnitudeTherefore using the real part of the stator current vector
in the stator flux reference frame it is possible to control theflux magnitude indirectly In order to obtain the real part ofthe stator current reference a simple PI controller is usedTheinput of this PI controller is the error that resulted from thecomparison between the reference stator flux magnitude andthe estimated flux value In order to control themotor currentin all cases especially in the starting mode the output ofthis PI controller must be limited between proper valuesTheoutput signal of the PI controller is the real part of referencestator current 119894lowast
119889119904
Therefore the stator current reference vector that leads tocontrol the IM stator flux and torque is as follows
119894lowast
119904(119889119902)= 119894lowast
119889119904+ 119895119894lowast
119902119904 (48)
1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 =radic119894lowast2
119889119904+ 119894lowast2119902119904 (49)
The obtained reference current vector is in the statorflux reference frame In order to generate it by SVM of theCSI it must be transferred to the stationary reference frameTo do that the stator flux position is used to transfer thisvector from the stator flux reference frame to the stationaryreference frame as follows
119894lowast
119904(120572120573)= 119894lowast
120572119904+ 119895119894lowast
120573119904= (119894lowast
119889119904+ 119895119894lowast
119902119904) sdot 119890119895120593119904
=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 ang (120593119904 + tanminus1 (119894lowast
119902119904
119894lowast
119889119904
))
(50)
This reference current vectormust be generated in theCSIfirst and then applied to the motor after removing its har-monics with filtering capacitorThis current has a nearly sinu-soidal waveform that leads to a nearly sinusoidal voltage inmotor terminals after feeding the motor and passing through
10 Journal of Applied Mathematics
Stator flux torque
estimator
PI
CSI
SVM IM
Gate Signals
PI
120582slowast
120591elowast
120582s
120582s
120591e
ang120593s
120591e
iqlowast
idlowast
dq
to120572120573
iaib
vb
va
|islowast| and idc
lowast
idclowast
+minus
+minus i120572
lowast
i120573lowast
Figure 11 Block diagram schematic of the proposed DTC
motor impedances The speed and stator flux control of anIM with DTC method is shown in Figure 11 Separate PIcontrollers are used to control the speed and flux The inputto the speed controller is the error caused by comparingthe reference speed and the rotor speed The output of thiscontroller is the imaginary part of stator current vector 119894lowast
119902119904
which is limited to a proper value
7 Results and Discussion
71 Simulation Results The simulationmodels of IOL IFOCand DTC control methods for CSI fed IM drive systemsare developed and simulated using MATLAB simulationsoftware From the simulation results dynamic performancesof the developed drives have been individually analyzedmainly using the sensitivity tests Here the results of all thedrives are compared with one another The tests that arechosen for the comparison are step changes in the referencespeed and torque of the motor drive
The first test is step change in reference speed which isuseful when trying to change from one speed to anotherThe next test is step change in reference torque that wouldbe useful in identifying the controller with a high torqueresponse Change in speed reference is the first test that isdone to compare the motor drives
The results of this test can be seen in the Figures 12 13 and14The speed command is varied from0 to its rated valueThespeed response of IOL drive is faster and has the settling timeof 05 second range while DTC and IFOC drives performsimilarly but slower than IOL (settling time of less than 1second)
The second test as mentioned earlier is the torque steptestThe results of this test can be seen in Figures 12 to 14Thetorque reference is varied from 0 to its rated value The speedresponse of IFOC drive is faster and has the settling timeof 02 seconds while IOL and DTC drives perform similarly(settling time of less than 05 second)
During this test the motor torque response of DTCdrive is almost instantaneous within the 02 second rangewhile IFOC drive closely follows behind it IOL and vectorbased 119881119891 have slower torque response than all other VectorControl techniques
Table 1 Prototype motor parameters
1 Hp 3 phase star connected 4 pole 415V 18 A 50Hz
Stator resistance 0087Ω Stator amp rotorleakage reactance 08119890minus3H
Rotor resistance 0228Ω Magnetizingreactance 347119890minus3H
72 Experimental Results Brief experimental results arepresented for IOL method and DTC method usingTMS320F2812 DSP based hardware setup Figure 15 showsthe schematic diagram for the hardware setup of theproposed laboratory prototype CSI fed induction motordrive The induction motor ratings are given in Table 1 Theinverter consists of CM75TL-12NF The gate driving signalis developed by using Texas instruments DSP processorTMS320F2812 DSP The software used to develop programsfor TMS320F2812 DSP is by code composer studio andMatlabSimulinkThe processor contains the program that isdownloaded to the computer
The clock speed of the DSP is 150MHz and it is capableof 32-bit operations The on board available flash memoryis 2048Mb It was created specifically for motor controloperation and therefore Parkrsquos and Clarkrsquos transformationsare conveniently built in Another convenient feature is thatit has sixteen 12-bit ADC pins that allow for a high degree ofprecision while taking many possible measurements
The above experimental result of IOL method of vectorcontrol technique in Figure 16 shows the fast settling timewith feed forward control compared to feedback techniqueSimilarly the experimental result shown in Figure 17 depictsa greater torque response within 02 seconds comparedto other vector control techniques With this experimentalverification it is clear that IOL is adaptive for speed responseand DTC is more suitable for torque response controlleddrives
From Tables 2 and 3 it seems that the dynamic speed andtorque performance of all four drives using change in speedreference It is evident from the table that IOL drive hasthe best speed response with DTC and IFOC occupyingpositions next to it It is also evident that DTC drive has thebest instantaneous torque response with IFOC and IOL nextto it
8 Conclusion
This paper presents a new approach in mathematical mod-eling of a 3-phase induction motor drive Mathematics con-cepts are used in developing the simulation and implementa-tion of IOL IFOC and DTC methods for CSI fed IM drivesIn this paper a rotor flux based reference frame controlmethod for PWM CSI fed IM with simplified simulationcircuits has been developed in which the motor speed andtorque follow the references closely in all the three vectorcontrol methods However a CSI fed induction motor drivehas some drawbacks such as slower dynamic response andbulky overall size due to the requirement of large smoothing
Journal of Applied Mathematics 11
15
minus05
0
05
1
Spee
d (P
U)
0 1 2 3 4 5Time (s)
Without FF controlWith FF control
(a)
0 1 2 3 4 5
minus2
0
2
Time (s)
Torq
ue (P
U)
Without FF controlWith FF control
(b)
Figure 12 Speed torque response of IOL control method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus4
minus2
0
24
Torq
ue (P
U)
Time (s)
Reference speedStep increase in speed command
Step decrease in speed command
Step increase in torque command
Step decrease in torque command Reference torque
Motor torque
Motor speed
Response due to change intorque reference
Response due to change in
speed reference
Figure 13 Speed torque response of IFOC method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus2minus1
012
Time (s)
Torq
ue (P
U)
Step decrease in speed command
Step increase in speed commandReference speed Motor speed
Response due to change intorque reference
Response due to change inspeed reference
Motor torque
Step decrease in torque commandReference torque
Step increase in torque command
Figure 14 Speed torque response of DTC method
inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying
TMS320F2812DSP
ADC
PWM
-EV
B
PWM
-EVA
Supplyfilter
3120601controlled
rectifier3120601 CSI inverter
3120601IM
3120601supply
Texasinstruments
DSP processor
Speed sensor output
(Nr) Rpm
CM75TL-12NFseries
IGBT (6-PAC)
Ldc
Is (Amps)
Figure 15 Schematic diagram of the hardware setup of CSI fed IM
Torque output with feedforward control
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Torque output withoutfeed forward control
Speed output with feedforward control
Speed output withoutfeed forward control
Figure 16 Experimental result torque-speed response of IOLcontrol method
applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
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Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 3
Rotor
Stator
4
3
2
1
[0 0]
rot1
rot
1s
1s
-K-
-K-3
2
1
fr
fs
is
irfr = rotor flux
fs = stator flux
is = stator current
Te = electromagnetic torque
Te
ir = rotor current
Flux-current relationship+minus
minus
+minus
+minus
minus
+minus
+minus
times
times
r
Rr
Rs
wm
Gm1
Gm
Gs
Gr
wk
s
K lowast u
K lowast u
∙
(Lrl + Lm ) (Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
Lm (Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
Lm (Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
(Lsl + Lm)(Lsl lowast Lrl + Lsl lowast Lm + Lrl lowast Lm)
Figure 1 Induction motor modeling in arbitrary reference frame
dqreference
frame
Stator
Synchronous
Rotor
Speed
Angle
-K-
Torque
Speed
Frequency
dq
ab
dq2ab
Currents
Amplitude
ab
etakdq
ab2dq
ab2ABC
Step load pu
Mechanicalsystem
1s
1s
Inductionmotor
pu
-K--
f(u)
f(u)
f(u)
0
0
ABC2ab dqABC ab
etak
Variable-amplitude variable-frequency3-phase sinusoidal voltages
K lowast u
wm
wk
wmwm
wmwkws
ws
wo
sTe Te
Te
Te
Tl
Tl
i dq
i dq
i ab
i abi ABC
is
120579r
120579k
120579s
120579s
K lowast u
Figure 2 Mathematical modeling of 3-phase induction motor drive
Figure 4 shows the transient performance of 119902-axis statorcurrent and 119889-rotor flux in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds From Figures 4(a) and 4(b) it is apparentthat there is a coupling effect between stator 119902-axis currentand rotor 119889-axis flux So independent control is difficultFigure 4(c) shows that the rotor 119889-axis flux and stator 119902-axis current (control variables) have linear relationship with
torque (speed) Thus it is useful in the rotor flux oriented(RFO) control methods
Figure 5 shows the transient performance of 119902-axis statorcurrent and 119889-stator flux in each reference frame when thereis a step change in the reference torque from 0 to full loadat 14 seconds Figures 5(a) and 5(b) state that there is acoupling effect between stator 119902-axis current and stator119889-axis flux and so independent control is difficult From
4 Journal of Applied Mathematics
minus10
0
10
0 05 1 15 2Time (s)
I ac
andI d
c(P
U)
(a)
minus10
0
10
0 05 1 15 2Time (s)
I ac
andI d
c(P
U)
(b)
0 05 1 15 2minus10
0
10
Stator a phase currentStator b-axis current
Step change in load torque
Time (s)
I ac
andI d
c(P
U)
(c)
Figure 3 Simulated results of ldquo119886rdquo phase and 119889-axis stator currents in different rotating reference frames
minus10
0
10
0 05 1 15 2Time in seconds
I qs
and
FLU
Xdr
(a)
minus10
0
10
0 05 1 15 2Time in seconds
I qs
and
FLU
Xdr
(b)
0 05 1 15 2minus10
minus5
0
5
Step change in load torque
Stator q-axis currentRotor d-axis flux
Time (s)
I qs
and
FLU
Xdr
(PU
)
(c)
Figure 4 Simulated results of 119902-axis stator current and 119889-axis rotorflux in different rotating reference frames
minus10
0
10
0 05 1 15 2Time (s)
I qs
and
FLU
Xds
(a)
minus10
0
10
0 05 1 15 2Time (s)
I qs
and
FLU
Xds
(b)
0 05 1 15 2minus10
minus5
0
5
Step change in load torque
Time (s)
Stator q-axis currentStator d-axis flux
I qs
and
FLU
Xds
(c)
Figure 5 Simulated results of 119902-axis stator current and 119889-axis statorflux in different rotating reference frames
Journal of Applied Mathematics 5
Figure 5(c) it is evident that the stator119889-axis flux and stator 119902-axis current (control variables) have linear relationship withtorque (speed) It is used in the stator flux oriented (SFO)control methods
The basic IM equations from Krause et al [15] aresimulated for three different reference frames usingMATLAB76 software and the results obtained are compared with eachreference frame From the comparison it is observed that ifthe stator rotating reference frame is used then stator 119889-axiscurrent is identical to that of the stator ldquo119886rdquo phase current asshown in Figure 3(a) which is used in stator fed IM drivesSimilarly in rotor reference frame rotor 119889-axis current isidentical to the rotor ldquo119886rdquo phase current and useful in rotor fedIM drives If the synchronous reference frame is used thenthe stator 119889-axis current behaves like DC quantity as shownin Figure 3(c) From Figures 4(c) and 5(c) it is evident thatsynchronous reference frame is useful in RFO and SFO drivecontrol methods In this paper synchronous reference frameis preferred to model the IM in 119881119891 IOL control methodsRFO control is used in IFOCmethod and SFO is used inDTCmethod
4 IOL Control for CSI Fed IM Drives
In this section a type of IOL control method for the CSIfed IM drive is described The controller is designed based
on pole placement technique of Wonham [16] to a 119889-119902 axisstate space linearized model of the drive which includes areduced order rotor current observer A control law has beendeveloped to improve the dynamic response of the drive byusing the concepts from Veerachary [8] which includes feedforward as well as feedback controls
41 Mathematical Modeling of IOL Control Method Theequivalent circuit for the CSI capacitor and IM in a 119889-119902 axis rotating reference frame is shown in Figure 6 Thecorresponding state equation (differential equations) withmotor flux linkages as variables is derived fromWu et al [17]
Veerachary [8] presented the linearized CSI-IM drivestate space model in synchronously rotating reference frameas follows
= 119860119909 + 119861119906 (6)
where
=119889119909
119889119905 (7)
119910 = 119862119909 (8)
where
119909 =
[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]
]
=119889
119889119905
[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]
]
119906 =
[[[[
[
Δ1198811015840
119889
Δ120596119890
120596119887
Δ119879119871
]]]]
]
119910 =[[
[
Δ119894119902119904
Δ120596119903
120596119887
]]
]
119860 =
[[[[[[[[[[[[[[
[
minus1199091015840
1199031199030120596119887
11990901199091015840119903minus 1199092119898
1199091198981198771015840
119903120596119887
11990901199091015840119903minus 1199092119898
minus12059611989001199091015840
119903119909119898(1 minus 119904)
11990901199091015840119903minus 1199092119898
minus1199091198981199091015840
1199031198941015840
1198891199030120596119887
11990901199091015840119903minus 1199092119898
1205961198871199091198981199030
11990901199091015840119903minus 1199092119898
minus11990901198771015840
119903120596119887
11990901199091015840119903minus 1199092119898
1205961198900(1199092
119898minus 11990411990901199091015840
119903)
11990901199091015840119903minus 1199092119898
11990901199091015840
1199031198941015840
1198891199030120596119887
11990901199091015840119903minus 1199092119898
1199041205961198900119909119898
1199091015840119903
1199041205961198900
minus1205961198871198771015840
119903
1199091015840119903
minus
120596119887(1199091198981198941199021199040+ 1199091015840
1199031198941015840
1199021199030)
1199091015840119903
1199091198981198941015840
1198891199030
21198670
1199091198981198941199021199040
21198670
]]]]]]]]]]]]]]
]
119861 =
[[[[[[[[[[[[[[[
[
1199091015840
119903120596119887
11990901199091015840119903minus 1199092119898
0 0
minus119909119898120596119887
11990901199091015840119903minus 1199092119898
minus1205961198871198941015840
11988911990300
0 minus
120596119887(1199091198981198941199021199040+ 1199091015840
1199031198941015840
1199021199030)
1199091015840119903
0
0 0 minus1
2119867
]]]]]]]]]]]]]]]
]
119862 = [1 0 0 0
0 0 0 1]
1198811015840
119889=120587119881dc
3radic3 119903
0= 119877119904+1205872
119877dc18
1199090= 119909119904+1205872
119909119889
18
(9)
6 Journal of Applied Mathematics
Rs
Rs
Rr
Vin Idi = 0
VqsLm
Lm
Ll
Ll
p 120582ds
p 120582qs p 120582qr
p 120582dr
+ minus
+minus
c
c
120596r 120582qr
120596r 120582qr
IdsIdi
IqsIqi
Vds
CSI
Iqi = MdIdc Vqs
Figure 6 Equivalent circuit of CSI-IM section
With usual state feedback control there can be the trackingerror in steady state To eliminate this error Smith andDavidson [18] introduced an integral control which includesfeedback as well as feed forward control In this work anoptimal control law is derived by using the concepts of Smithand Davidson [18] and Veerachary [8]
42 Design of Optimal Control Law A new multivariableoptimal controller incorporating state feedback as well asfeed forward control for fast regulation and stability of a CSIfed IM drive is presented in this section The design of thestate feedback controller is based on the industrial regulatortheory together with pole placement technique applied ona linearized 119889-119902 axes state space model of the drive andincludes a reduced order observer to estimate the inaccessiblestates like 119889-119902 axes rotor currents A feed forward control interms of reference and disturbance inputs has been addedto the feedback controller-observer to obtain faster dynamicresponse either from (10) or (11) as follows (Figure 7)
119906 = 1198701119909 + 119870
2int
119905
0
(119910 minus 119910119903) 119889119905 + 119870
119865119865[119889
119910119903
] (10)
or
119906 = 1198701(2times4)
[[[[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]]]]
]
+ 1198702(2times2)
[[[[[
[
int
119905
0
(119894119902119904minus 119894lowast
119902119904) 119889119905
int
119905
0
(120596119903minus 120596lowast
119903) 119889119905
]]]]]
]
+ 119870119865119865(2times3)
[[[[
[
119879119871
119894lowast
119902119904
120596lowast
119903
]]]]
]
(11)
5 IFOC for CSI Fed IM Drives
In this section a high performance current fed indirect RFOcontrol method is described for IMs The IM is mathe-matically modeled in the rotor flux reference frame that is
CSIfed IM
Reduced order
observer
yy
x
r = [ iqslowast120596rlowast] y= [ iqs
120596r
]u
u
y
y
= [Vd
120596e
]
K1(2times4)
K2(2times2)
KFF(2times3)
+ +++
minus
d = TL
int
u998400
Figure 7 Combined feed forward and integral feedback controller
presented The rotor flux orientation is also obtained In thelow speed region the rotor flux components can be synthe-sized more easily with the help of speed and current signalsDesign of indirect vector controller Blaschke [19 20] androtor flux estimator is discussed Implementation of IFOCto CSI fed IM drive and transient response analysis of othersimulation results are also discussed
51 MathematicalModeling of IFOControlMethod The rotorequations of the IM containing flux linkages as variables aregiven by Krause et al [15] as follows
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus (120596119890minus 120596119903) 120582119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ (120596119890minus 120596119903) 120582119890
119889119903= 0
(12)
where
120596sl = 120596119890 minus 120596119903 (13)
then (12) becomes
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus 120596sl120582
119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ 120596sl120582
119890
119889119903= 0
(14)
The resultant rotor flux linkage 120582119903 also known as the rotor
flux linkages phasor is assumed to be on the direct axisto reduce the number of variables in the equations by oneMoreover it relates to reality that rotor flux linkages are asingle variable Hence aligning d axis with rotor flux phasoryields
120582119903= 120582119890
119889119903 (15)
120582119890
119902119903= 0 (16)
119901120582119890
119902119903= 0 (17)
Substituting (15) into (14) results in the new rotor equationsas follows
119877119903119894119890
119889119903+ 119901120582119903= 0
119877119903119894119890
119902119903+ 120596sl120582119903 = 0
(18)
Journal of Applied Mathematics 7
The rotor currents in terms of the stator currents are derivedfrom (18) as
119894119890
119902119903= minus
119871119898
119871119903
119894119890
119902119904
119894119890
119889119903=120582119903
119871119903
minus119871119898
119871119903
119894119890
119889119904
(19)
119894119890
119889119904= 119894119891= [1 +
119871119903
119877119903
119901]120582119903
119871119898
119894119890
119889119904= 119894119891= [1 + 120591
119903119901]
120582119903
119871119898
(20)
120596sl =119877119903119871119898
119871119903
119894119890
119902119904
120582119903
120596sl =119877119903119871119898
119871119903
119894119879
120582119903
120596sl =119871119898
120591119903
119894119879
120582119903
(21)
The 119902- and 119889-axes currents are relabeled as torque (119894119879) and
flux producing (119894119891) components of the stator-current phasor
respectively 120591119903denotes rotor time constant Similarly by the
same substitution of the rotor currents from (19) into torqueexpression the electromagnetic torque is derived as
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904minus 120582119890
119902119903119894119890
119889119904)
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904)
119879119890= 119870119905119890120582119903119894119890
119902119904= 119870119905119890120582119903119894119879
(22)
where the torque constant119870119905119890is defined as
119870119905119890=3
2
119875
2
119871119898
119871119903
(23)
Note that the torque is proportional to the product of rotorflux linkage and the stator 119902-axis current This resemblesthe air gap torque expression of the DC motor which isproportional to the product of the field flux linkages and thearmature current If the rotor flux linkage is maintained asconstant the torque is proportional to the torque producingcomponent of the stator current This is in relation to theseparately excited DC motor with armature current controlwhere the torque is proportional to the armature currentwhen the field current is constant Further here too the timeconstant is measured in the order of a few milliseconds Therotor flux linkages and air gap torque given in (21) and (22)respectively complete the transformation of the IM into anequivalent separately-excited dc motor from a control pointof view
52 Design of IFO Controller The IFO controller was de-signed using the concepts of Vas [21] and Salo and Tuusa [9]
From that the stator-current phasor is the phasor sum of the119889- and 119902-axis stator currents in any frames and it is given as
119894119904= radic(119894119890
119902119904)2
+ (119894119890
119889119904)2
(24)
and the 119889119902 axes (2120601) to (3120601) abc phase current relationship isobtained from
[
[
119894119890
119902119904
119894119890
119889119904
]
]
=2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+4120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+4120587
3)
]]]
]
[
[
119894119886119904
119894119887119904
119894119888119904
]
]
(25)
And it can be written as
119894119902119889= [119879] [119894
119886119887119888] (26)
where
119894119902119889= [119894119890
119902119904119894119890
119889119904]119905
(27)
119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905
(28)
[119879] =2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+2120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+2120587
3)
]]]
]
(29)
where 119894119886119904 119894119887119904 and 119894
119888119904are the three phase stator currents Note
that the elements in the 119879 matrix are cosinusoidal functionsof electrical angle 120579
119891 The electrical field angle in this case is
that of the rotor flux-linkages phasor and is obtained as thesum of the rotor and slip angles as follows
120579119891= 120579119903+ 120579sl (30)
and the slip angle is obtained by integrating the slip speed andis given as
120579sl = int120596sl (31)
IFO controller developed from these derivations accepts thetorque and flux requests and generates the torque and fluxproducing components of the stator-current phasor and theslip-angle commandsThe command values are denoted withasterisks throughout this thesis From (20) (21) and (23) thecommand values of 119894
119879 119894119891 and 120596sl are obtained as follows
119894lowast
119879=
119879lowast
119890
119870119905119890120582lowast119903
= (2
3) (
2
119875)119879lowast
119890
120582lowast119903
119871119903
119871119898
(32)
119894lowast
119891= (1 + 119901
119871119903
119877119903
)120582lowast
119903
119871119898
(33)
120596lowast
sl =119877119903119871119898
119871119903
119894lowast
119879
120582lowast119903
(34)
8 Journal of Applied Mathematics
The command slip angle 120579lowastsl is generated by integrating 120596lowastsl The torque angle command is obtained as the arctangentof 119894lowast119879and 119894lowast119891 The field angle is obtained by summing the
command slip angle and rotor angleWith the torque and fluxproducing components of the stator current commands androtor field angle the 119902119889 axes current commands (119886119887119888 phasecurrent commands) are obtained as follows The relevantsteps involved in the realization of the IFO controller are asfollows
[
[
119894lowast
119902119904
119894lowast
119889119904
]
]
= [
[
cos 120579119891
sin 120579119891
minus sin 120579119891
cos 120579119891
]
]
[
[
119894lowast
119879
119894lowast
119891
]
]
(35)
[[[[
[
119894lowast
119886119904
119894lowast
119887119904
119894lowast
119888119904
]]]]
]
= [119879minus1
]
[[[[
[
119894lowast
119902119904
119894lowast
119889119904
0
]]]]
]
(36)
where
119879minus1
=
[[[[[[[
[
1 0 1
minus1
2minusradic3
21
minus1
2
radic3
21
]]]]]]]
]
(37)
By using (35) to (37) the stator 119902- and 119889-axes and abccurrent commands are derived as
119894lowast
119902119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119889119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 cos 120579lowast
119904
119894lowast
119886119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119887119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904minus2120587
3)
119894lowast
119888119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904+2120587
3)
(38)
where
120579lowast
119904= 120579119891+ 120579lowast
119879= 120579119903+ 120579lowast
sl + 120579lowast
119879 (39)
The implementation of IFOC on a CSI fed IM is shownin Figures 8 and 9 The torque command 119894lowast
119879is generated
as a function of the speed error signal generally processedthrough a PI controller The flux command 119894lowast
119891can be given
directly or as a function of speed The rotor position 120579119903can
be measured with an encoder and converted into necessarydigital information for feedbackThere has been a substantialamount of research in the development of rotor flux observersfor field orientation that are compensated for variationsin parameters by their feedback corrections Digital imple-mentation of integrators for the estimation of rotor flux ofan IM from the stator voltages and stator currents posesproblems associated with the offset in the sensor amplifiers
CSI
IM
PWMswitching
logic
Rotor speed
regulator
Stator current
generator
DC link currentias
lowast
ibslowast
i cslowast
120579riTlowast is
iflowast
iflowast and iT
lowast120582rlowast
120596rlowast
120596sllowast
120596r
Telowast
(1 + pLr
Rr
) 120582rlowast
Lm
RrLm
Lr
iTlowast
120582rlowast
4
3P
Telowast
120582rlowast
Lr
Lm
+minus
Figure 8 Simplified diagram for IFOC method
islowast = radic(iTlowast)2 +(iflowast)2 ias
lowast = |islowast|sin120579s
lowast|islowast|
ibslowast = |is
lowast|sin120579slowast minus
2120587
3
icslowast = |is
lowast|sin120579slowast +
2120587
3
120579r 120579r
iTlowast
iflowast
120579Tlowast
120579slowast
120596sllowast 120579sl
lowastint
iaslowast
ibslowast
i cslowast
tanminus1 iTlowast
iflowast
+++
Figure 9 Stator current generator
Traditional low-pass filters can replace the integrator Soin this work rotor flux estimated using low pass filter isdescribed The instantaneous flux linkage can be computedusing the measured d-axis stator current using (40) which isreferred to as the current model as follows
120582119903=119871119898119894119889119904
1 + 120591119903119904 (40)
The theoretical concepts of IFOC method for CSI fed IMdrive discussed in Sections 51 and 52 are verified throughsimulation using the Simulink toolbox of MATLAB Thedetailed simulations results are presented in Section 71
6 DTC for CSI Fed IM Drives
Variable speed drives are used in all industries to controlprecisely the speed of electric motors driving loads rangingfrom pumps and fans to complex drives on paper machinesrolling mill cranes and similar drives The most modernmethod for these drives is direct torque and stator flux vectorcontrol method usually called DTC The VSI fed version hasbeen realized in an industrial way by ABB by using thetheoretical background proposed by Takahashi and Ohmori[22] and Depenbrock [23] This solution is based both onfield oriented control (FOC) as well as on the direct self-control theory The idea is that motor flux and torque areused as primary control variables which is contrary to thewayin which traditional AC drives control input frequency andvoltage but is in principle similar to what is done with a DCdrive where it is much more straightforward to achieve
Journal of Applied Mathematics 9
Casadei et al [24] has described that by controllingmotor torque directly DTCprovides dynamic speed accuracyequivalent to closed loop AC and DC systems and torqueresponse times that are 10 times faster It is also claimedthat the DTC does not generate noise like that produced byconventional PWM AC drives This work describes a newmodular approach to implement DTC method on a CSIfed IM drive the simulation implementation procedure Thedynamic response of the drive is analyzed and presented
61 Mathematical Modeling of DTC Control Method TheDTC method is developed from Babaei and Heydari [10]From Krause et al [15] in the stationary reference frame thestator voltage and flux equation can be written as
V119904= 119877119904119894119904+119889120582119904
119889119905 (41)
120582119904= int (V
119904minus 119877119904119894119904) 119889119905 = 120582
120572119904+ 119895120582120573119904 (42)
10038161003816100381610038161205821199041003816100381610038161003816 = radic120582
2
120572119904+ 1205822
120573119904 (43)
ang120593119904= tanminus1 (
120582120573119904
120582120572119904
) (44)
where 119877119904is the stator resistance and 120582
119904is the stator flux
vector in the stationary reference frame |120582119904| and ang120593
119904are the
amplitude and position of stator flux vector respectivelyIn DTC of CSI fed IM it is desirable to determine the
stator current reference so that the torque and stator fluxfollow their reference values Babaei and Heydari [10] Theproposed DTC system is based on the stator flux oriented(SFO) reference frame In this rotating reference frame it iswritten as
120582119902119904= 0 120582
119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)
where 119889 and 119902 are the real and imaginary axes in the SFOreference frame Therefore the torque equation is rewrittenas
119879119890=3
2(119875
2)1
120596119887
(120582119889119904119894119902119904) (46)
where 119894119902119904is the imaginary part of the stator current vector
in the SFO reference frame Therefore if |120582lowast119904| and 119879lowast
119890are
stator flux magnitude and torque references respectively theimaginary part of the stator current reference vector whichleads to torque and stator flux magnitude references is asfollows
119894lowast
119902119904=2
3
1
119875
119879lowast
119890
1003816100381610038161003816120582lowast
119904
1003816100381610038161003816
(47)
Because of the existence of a capacitor in the controlsystem and its corresponding losses when (47) is used tocontrol the torque for all rotor speeds the control system willnot be accurate enough
Currentlimiter
Telowast
Te
|120582slowast|
2
3
1
P
PI
Telowast
|120582slowast |
+
+
+minus iqs
lowast
Figure 10 119902-axis reference current calculator
In order to have amore accurate torque control the blockdiagram shown in Figure 10 is used to calculate 119894lowast
119902119904This block
diagram is a combination of (47) and a simple PI controllerThis controller removes the steady-state error and leads to anaccurate control The coefficients of this controller must besmall enough so that it remains stable and the ripples in 119894lowast
119902119904
are limited because a ripple of 119894lowast119902119904results in a ripple of the
generated torque If only a PI is used 119894lowast119902119904
will have a largevalue of ripple and the torque response time will be verylargeTherefore in order to have a faster transient response oftorque it is necessary to use (47) In DTC method with VSIthe real part of voltage vector (V
119889) in the stator flux reference
frame leads to direct control of stator flux magnitudeTherefore using the real part of the stator current vector
in the stator flux reference frame it is possible to control theflux magnitude indirectly In order to obtain the real part ofthe stator current reference a simple PI controller is usedTheinput of this PI controller is the error that resulted from thecomparison between the reference stator flux magnitude andthe estimated flux value In order to control themotor currentin all cases especially in the starting mode the output ofthis PI controller must be limited between proper valuesTheoutput signal of the PI controller is the real part of referencestator current 119894lowast
119889119904
Therefore the stator current reference vector that leads tocontrol the IM stator flux and torque is as follows
119894lowast
119904(119889119902)= 119894lowast
119889119904+ 119895119894lowast
119902119904 (48)
1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 =radic119894lowast2
119889119904+ 119894lowast2119902119904 (49)
The obtained reference current vector is in the statorflux reference frame In order to generate it by SVM of theCSI it must be transferred to the stationary reference frameTo do that the stator flux position is used to transfer thisvector from the stator flux reference frame to the stationaryreference frame as follows
119894lowast
119904(120572120573)= 119894lowast
120572119904+ 119895119894lowast
120573119904= (119894lowast
119889119904+ 119895119894lowast
119902119904) sdot 119890119895120593119904
=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 ang (120593119904 + tanminus1 (119894lowast
119902119904
119894lowast
119889119904
))
(50)
This reference current vectormust be generated in theCSIfirst and then applied to the motor after removing its har-monics with filtering capacitorThis current has a nearly sinu-soidal waveform that leads to a nearly sinusoidal voltage inmotor terminals after feeding the motor and passing through
10 Journal of Applied Mathematics
Stator flux torque
estimator
PI
CSI
SVM IM
Gate Signals
PI
120582slowast
120591elowast
120582s
120582s
120591e
ang120593s
120591e
iqlowast
idlowast
dq
to120572120573
iaib
vb
va
|islowast| and idc
lowast
idclowast
+minus
+minus i120572
lowast
i120573lowast
Figure 11 Block diagram schematic of the proposed DTC
motor impedances The speed and stator flux control of anIM with DTC method is shown in Figure 11 Separate PIcontrollers are used to control the speed and flux The inputto the speed controller is the error caused by comparingthe reference speed and the rotor speed The output of thiscontroller is the imaginary part of stator current vector 119894lowast
119902119904
which is limited to a proper value
7 Results and Discussion
71 Simulation Results The simulationmodels of IOL IFOCand DTC control methods for CSI fed IM drive systemsare developed and simulated using MATLAB simulationsoftware From the simulation results dynamic performancesof the developed drives have been individually analyzedmainly using the sensitivity tests Here the results of all thedrives are compared with one another The tests that arechosen for the comparison are step changes in the referencespeed and torque of the motor drive
The first test is step change in reference speed which isuseful when trying to change from one speed to anotherThe next test is step change in reference torque that wouldbe useful in identifying the controller with a high torqueresponse Change in speed reference is the first test that isdone to compare the motor drives
The results of this test can be seen in the Figures 12 13 and14The speed command is varied from0 to its rated valueThespeed response of IOL drive is faster and has the settling timeof 05 second range while DTC and IFOC drives performsimilarly but slower than IOL (settling time of less than 1second)
The second test as mentioned earlier is the torque steptestThe results of this test can be seen in Figures 12 to 14Thetorque reference is varied from 0 to its rated value The speedresponse of IFOC drive is faster and has the settling timeof 02 seconds while IOL and DTC drives perform similarly(settling time of less than 05 second)
During this test the motor torque response of DTCdrive is almost instantaneous within the 02 second rangewhile IFOC drive closely follows behind it IOL and vectorbased 119881119891 have slower torque response than all other VectorControl techniques
Table 1 Prototype motor parameters
1 Hp 3 phase star connected 4 pole 415V 18 A 50Hz
Stator resistance 0087Ω Stator amp rotorleakage reactance 08119890minus3H
Rotor resistance 0228Ω Magnetizingreactance 347119890minus3H
72 Experimental Results Brief experimental results arepresented for IOL method and DTC method usingTMS320F2812 DSP based hardware setup Figure 15 showsthe schematic diagram for the hardware setup of theproposed laboratory prototype CSI fed induction motordrive The induction motor ratings are given in Table 1 Theinverter consists of CM75TL-12NF The gate driving signalis developed by using Texas instruments DSP processorTMS320F2812 DSP The software used to develop programsfor TMS320F2812 DSP is by code composer studio andMatlabSimulinkThe processor contains the program that isdownloaded to the computer
The clock speed of the DSP is 150MHz and it is capableof 32-bit operations The on board available flash memoryis 2048Mb It was created specifically for motor controloperation and therefore Parkrsquos and Clarkrsquos transformationsare conveniently built in Another convenient feature is thatit has sixteen 12-bit ADC pins that allow for a high degree ofprecision while taking many possible measurements
The above experimental result of IOL method of vectorcontrol technique in Figure 16 shows the fast settling timewith feed forward control compared to feedback techniqueSimilarly the experimental result shown in Figure 17 depictsa greater torque response within 02 seconds comparedto other vector control techniques With this experimentalverification it is clear that IOL is adaptive for speed responseand DTC is more suitable for torque response controlleddrives
From Tables 2 and 3 it seems that the dynamic speed andtorque performance of all four drives using change in speedreference It is evident from the table that IOL drive hasthe best speed response with DTC and IFOC occupyingpositions next to it It is also evident that DTC drive has thebest instantaneous torque response with IFOC and IOL nextto it
8 Conclusion
This paper presents a new approach in mathematical mod-eling of a 3-phase induction motor drive Mathematics con-cepts are used in developing the simulation and implementa-tion of IOL IFOC and DTC methods for CSI fed IM drivesIn this paper a rotor flux based reference frame controlmethod for PWM CSI fed IM with simplified simulationcircuits has been developed in which the motor speed andtorque follow the references closely in all the three vectorcontrol methods However a CSI fed induction motor drivehas some drawbacks such as slower dynamic response andbulky overall size due to the requirement of large smoothing
Journal of Applied Mathematics 11
15
minus05
0
05
1
Spee
d (P
U)
0 1 2 3 4 5Time (s)
Without FF controlWith FF control
(a)
0 1 2 3 4 5
minus2
0
2
Time (s)
Torq
ue (P
U)
Without FF controlWith FF control
(b)
Figure 12 Speed torque response of IOL control method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus4
minus2
0
24
Torq
ue (P
U)
Time (s)
Reference speedStep increase in speed command
Step decrease in speed command
Step increase in torque command
Step decrease in torque command Reference torque
Motor torque
Motor speed
Response due to change intorque reference
Response due to change in
speed reference
Figure 13 Speed torque response of IFOC method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus2minus1
012
Time (s)
Torq
ue (P
U)
Step decrease in speed command
Step increase in speed commandReference speed Motor speed
Response due to change intorque reference
Response due to change inspeed reference
Motor torque
Step decrease in torque commandReference torque
Step increase in torque command
Figure 14 Speed torque response of DTC method
inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying
TMS320F2812DSP
ADC
PWM
-EV
B
PWM
-EVA
Supplyfilter
3120601controlled
rectifier3120601 CSI inverter
3120601IM
3120601supply
Texasinstruments
DSP processor
Speed sensor output
(Nr) Rpm
CM75TL-12NFseries
IGBT (6-PAC)
Ldc
Is (Amps)
Figure 15 Schematic diagram of the hardware setup of CSI fed IM
Torque output with feedforward control
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Torque output withoutfeed forward control
Speed output with feedforward control
Speed output withoutfeed forward control
Figure 16 Experimental result torque-speed response of IOLcontrol method
applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Applied Mathematics
minus10
0
10
0 05 1 15 2Time (s)
I ac
andI d
c(P
U)
(a)
minus10
0
10
0 05 1 15 2Time (s)
I ac
andI d
c(P
U)
(b)
0 05 1 15 2minus10
0
10
Stator a phase currentStator b-axis current
Step change in load torque
Time (s)
I ac
andI d
c(P
U)
(c)
Figure 3 Simulated results of ldquo119886rdquo phase and 119889-axis stator currents in different rotating reference frames
minus10
0
10
0 05 1 15 2Time in seconds
I qs
and
FLU
Xdr
(a)
minus10
0
10
0 05 1 15 2Time in seconds
I qs
and
FLU
Xdr
(b)
0 05 1 15 2minus10
minus5
0
5
Step change in load torque
Stator q-axis currentRotor d-axis flux
Time (s)
I qs
and
FLU
Xdr
(PU
)
(c)
Figure 4 Simulated results of 119902-axis stator current and 119889-axis rotorflux in different rotating reference frames
minus10
0
10
0 05 1 15 2Time (s)
I qs
and
FLU
Xds
(a)
minus10
0
10
0 05 1 15 2Time (s)
I qs
and
FLU
Xds
(b)
0 05 1 15 2minus10
minus5
0
5
Step change in load torque
Time (s)
Stator q-axis currentStator d-axis flux
I qs
and
FLU
Xds
(c)
Figure 5 Simulated results of 119902-axis stator current and 119889-axis statorflux in different rotating reference frames
Journal of Applied Mathematics 5
Figure 5(c) it is evident that the stator119889-axis flux and stator 119902-axis current (control variables) have linear relationship withtorque (speed) It is used in the stator flux oriented (SFO)control methods
The basic IM equations from Krause et al [15] aresimulated for three different reference frames usingMATLAB76 software and the results obtained are compared with eachreference frame From the comparison it is observed that ifthe stator rotating reference frame is used then stator 119889-axiscurrent is identical to that of the stator ldquo119886rdquo phase current asshown in Figure 3(a) which is used in stator fed IM drivesSimilarly in rotor reference frame rotor 119889-axis current isidentical to the rotor ldquo119886rdquo phase current and useful in rotor fedIM drives If the synchronous reference frame is used thenthe stator 119889-axis current behaves like DC quantity as shownin Figure 3(c) From Figures 4(c) and 5(c) it is evident thatsynchronous reference frame is useful in RFO and SFO drivecontrol methods In this paper synchronous reference frameis preferred to model the IM in 119881119891 IOL control methodsRFO control is used in IFOCmethod and SFO is used inDTCmethod
4 IOL Control for CSI Fed IM Drives
In this section a type of IOL control method for the CSIfed IM drive is described The controller is designed based
on pole placement technique of Wonham [16] to a 119889-119902 axisstate space linearized model of the drive which includes areduced order rotor current observer A control law has beendeveloped to improve the dynamic response of the drive byusing the concepts from Veerachary [8] which includes feedforward as well as feedback controls
41 Mathematical Modeling of IOL Control Method Theequivalent circuit for the CSI capacitor and IM in a 119889-119902 axis rotating reference frame is shown in Figure 6 Thecorresponding state equation (differential equations) withmotor flux linkages as variables is derived fromWu et al [17]
Veerachary [8] presented the linearized CSI-IM drivestate space model in synchronously rotating reference frameas follows
= 119860119909 + 119861119906 (6)
where
=119889119909
119889119905 (7)
119910 = 119862119909 (8)
where
119909 =
[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]
]
=119889
119889119905
[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]
]
119906 =
[[[[
[
Δ1198811015840
119889
Δ120596119890
120596119887
Δ119879119871
]]]]
]
119910 =[[
[
Δ119894119902119904
Δ120596119903
120596119887
]]
]
119860 =
[[[[[[[[[[[[[[
[
minus1199091015840
1199031199030120596119887
11990901199091015840119903minus 1199092119898
1199091198981198771015840
119903120596119887
11990901199091015840119903minus 1199092119898
minus12059611989001199091015840
119903119909119898(1 minus 119904)
11990901199091015840119903minus 1199092119898
minus1199091198981199091015840
1199031198941015840
1198891199030120596119887
11990901199091015840119903minus 1199092119898
1205961198871199091198981199030
11990901199091015840119903minus 1199092119898
minus11990901198771015840
119903120596119887
11990901199091015840119903minus 1199092119898
1205961198900(1199092
119898minus 11990411990901199091015840
119903)
11990901199091015840119903minus 1199092119898
11990901199091015840
1199031198941015840
1198891199030120596119887
11990901199091015840119903minus 1199092119898
1199041205961198900119909119898
1199091015840119903
1199041205961198900
minus1205961198871198771015840
119903
1199091015840119903
minus
120596119887(1199091198981198941199021199040+ 1199091015840
1199031198941015840
1199021199030)
1199091015840119903
1199091198981198941015840
1198891199030
21198670
1199091198981198941199021199040
21198670
]]]]]]]]]]]]]]
]
119861 =
[[[[[[[[[[[[[[[
[
1199091015840
119903120596119887
11990901199091015840119903minus 1199092119898
0 0
minus119909119898120596119887
11990901199091015840119903minus 1199092119898
minus1205961198871198941015840
11988911990300
0 minus
120596119887(1199091198981198941199021199040+ 1199091015840
1199031198941015840
1199021199030)
1199091015840119903
0
0 0 minus1
2119867
]]]]]]]]]]]]]]]
]
119862 = [1 0 0 0
0 0 0 1]
1198811015840
119889=120587119881dc
3radic3 119903
0= 119877119904+1205872
119877dc18
1199090= 119909119904+1205872
119909119889
18
(9)
6 Journal of Applied Mathematics
Rs
Rs
Rr
Vin Idi = 0
VqsLm
Lm
Ll
Ll
p 120582ds
p 120582qs p 120582qr
p 120582dr
+ minus
+minus
c
c
120596r 120582qr
120596r 120582qr
IdsIdi
IqsIqi
Vds
CSI
Iqi = MdIdc Vqs
Figure 6 Equivalent circuit of CSI-IM section
With usual state feedback control there can be the trackingerror in steady state To eliminate this error Smith andDavidson [18] introduced an integral control which includesfeedback as well as feed forward control In this work anoptimal control law is derived by using the concepts of Smithand Davidson [18] and Veerachary [8]
42 Design of Optimal Control Law A new multivariableoptimal controller incorporating state feedback as well asfeed forward control for fast regulation and stability of a CSIfed IM drive is presented in this section The design of thestate feedback controller is based on the industrial regulatortheory together with pole placement technique applied ona linearized 119889-119902 axes state space model of the drive andincludes a reduced order observer to estimate the inaccessiblestates like 119889-119902 axes rotor currents A feed forward control interms of reference and disturbance inputs has been addedto the feedback controller-observer to obtain faster dynamicresponse either from (10) or (11) as follows (Figure 7)
119906 = 1198701119909 + 119870
2int
119905
0
(119910 minus 119910119903) 119889119905 + 119870
119865119865[119889
119910119903
] (10)
or
119906 = 1198701(2times4)
[[[[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]]]]
]
+ 1198702(2times2)
[[[[[
[
int
119905
0
(119894119902119904minus 119894lowast
119902119904) 119889119905
int
119905
0
(120596119903minus 120596lowast
119903) 119889119905
]]]]]
]
+ 119870119865119865(2times3)
[[[[
[
119879119871
119894lowast
119902119904
120596lowast
119903
]]]]
]
(11)
5 IFOC for CSI Fed IM Drives
In this section a high performance current fed indirect RFOcontrol method is described for IMs The IM is mathe-matically modeled in the rotor flux reference frame that is
CSIfed IM
Reduced order
observer
yy
x
r = [ iqslowast120596rlowast] y= [ iqs
120596r
]u
u
y
y
= [Vd
120596e
]
K1(2times4)
K2(2times2)
KFF(2times3)
+ +++
minus
d = TL
int
u998400
Figure 7 Combined feed forward and integral feedback controller
presented The rotor flux orientation is also obtained In thelow speed region the rotor flux components can be synthe-sized more easily with the help of speed and current signalsDesign of indirect vector controller Blaschke [19 20] androtor flux estimator is discussed Implementation of IFOCto CSI fed IM drive and transient response analysis of othersimulation results are also discussed
51 MathematicalModeling of IFOControlMethod The rotorequations of the IM containing flux linkages as variables aregiven by Krause et al [15] as follows
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus (120596119890minus 120596119903) 120582119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ (120596119890minus 120596119903) 120582119890
119889119903= 0
(12)
where
120596sl = 120596119890 minus 120596119903 (13)
then (12) becomes
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus 120596sl120582
119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ 120596sl120582
119890
119889119903= 0
(14)
The resultant rotor flux linkage 120582119903 also known as the rotor
flux linkages phasor is assumed to be on the direct axisto reduce the number of variables in the equations by oneMoreover it relates to reality that rotor flux linkages are asingle variable Hence aligning d axis with rotor flux phasoryields
120582119903= 120582119890
119889119903 (15)
120582119890
119902119903= 0 (16)
119901120582119890
119902119903= 0 (17)
Substituting (15) into (14) results in the new rotor equationsas follows
119877119903119894119890
119889119903+ 119901120582119903= 0
119877119903119894119890
119902119903+ 120596sl120582119903 = 0
(18)
Journal of Applied Mathematics 7
The rotor currents in terms of the stator currents are derivedfrom (18) as
119894119890
119902119903= minus
119871119898
119871119903
119894119890
119902119904
119894119890
119889119903=120582119903
119871119903
minus119871119898
119871119903
119894119890
119889119904
(19)
119894119890
119889119904= 119894119891= [1 +
119871119903
119877119903
119901]120582119903
119871119898
119894119890
119889119904= 119894119891= [1 + 120591
119903119901]
120582119903
119871119898
(20)
120596sl =119877119903119871119898
119871119903
119894119890
119902119904
120582119903
120596sl =119877119903119871119898
119871119903
119894119879
120582119903
120596sl =119871119898
120591119903
119894119879
120582119903
(21)
The 119902- and 119889-axes currents are relabeled as torque (119894119879) and
flux producing (119894119891) components of the stator-current phasor
respectively 120591119903denotes rotor time constant Similarly by the
same substitution of the rotor currents from (19) into torqueexpression the electromagnetic torque is derived as
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904minus 120582119890
119902119903119894119890
119889119904)
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904)
119879119890= 119870119905119890120582119903119894119890
119902119904= 119870119905119890120582119903119894119879
(22)
where the torque constant119870119905119890is defined as
119870119905119890=3
2
119875
2
119871119898
119871119903
(23)
Note that the torque is proportional to the product of rotorflux linkage and the stator 119902-axis current This resemblesthe air gap torque expression of the DC motor which isproportional to the product of the field flux linkages and thearmature current If the rotor flux linkage is maintained asconstant the torque is proportional to the torque producingcomponent of the stator current This is in relation to theseparately excited DC motor with armature current controlwhere the torque is proportional to the armature currentwhen the field current is constant Further here too the timeconstant is measured in the order of a few milliseconds Therotor flux linkages and air gap torque given in (21) and (22)respectively complete the transformation of the IM into anequivalent separately-excited dc motor from a control pointof view
52 Design of IFO Controller The IFO controller was de-signed using the concepts of Vas [21] and Salo and Tuusa [9]
From that the stator-current phasor is the phasor sum of the119889- and 119902-axis stator currents in any frames and it is given as
119894119904= radic(119894119890
119902119904)2
+ (119894119890
119889119904)2
(24)
and the 119889119902 axes (2120601) to (3120601) abc phase current relationship isobtained from
[
[
119894119890
119902119904
119894119890
119889119904
]
]
=2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+4120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+4120587
3)
]]]
]
[
[
119894119886119904
119894119887119904
119894119888119904
]
]
(25)
And it can be written as
119894119902119889= [119879] [119894
119886119887119888] (26)
where
119894119902119889= [119894119890
119902119904119894119890
119889119904]119905
(27)
119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905
(28)
[119879] =2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+2120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+2120587
3)
]]]
]
(29)
where 119894119886119904 119894119887119904 and 119894
119888119904are the three phase stator currents Note
that the elements in the 119879 matrix are cosinusoidal functionsof electrical angle 120579
119891 The electrical field angle in this case is
that of the rotor flux-linkages phasor and is obtained as thesum of the rotor and slip angles as follows
120579119891= 120579119903+ 120579sl (30)
and the slip angle is obtained by integrating the slip speed andis given as
120579sl = int120596sl (31)
IFO controller developed from these derivations accepts thetorque and flux requests and generates the torque and fluxproducing components of the stator-current phasor and theslip-angle commandsThe command values are denoted withasterisks throughout this thesis From (20) (21) and (23) thecommand values of 119894
119879 119894119891 and 120596sl are obtained as follows
119894lowast
119879=
119879lowast
119890
119870119905119890120582lowast119903
= (2
3) (
2
119875)119879lowast
119890
120582lowast119903
119871119903
119871119898
(32)
119894lowast
119891= (1 + 119901
119871119903
119877119903
)120582lowast
119903
119871119898
(33)
120596lowast
sl =119877119903119871119898
119871119903
119894lowast
119879
120582lowast119903
(34)
8 Journal of Applied Mathematics
The command slip angle 120579lowastsl is generated by integrating 120596lowastsl The torque angle command is obtained as the arctangentof 119894lowast119879and 119894lowast119891 The field angle is obtained by summing the
command slip angle and rotor angleWith the torque and fluxproducing components of the stator current commands androtor field angle the 119902119889 axes current commands (119886119887119888 phasecurrent commands) are obtained as follows The relevantsteps involved in the realization of the IFO controller are asfollows
[
[
119894lowast
119902119904
119894lowast
119889119904
]
]
= [
[
cos 120579119891
sin 120579119891
minus sin 120579119891
cos 120579119891
]
]
[
[
119894lowast
119879
119894lowast
119891
]
]
(35)
[[[[
[
119894lowast
119886119904
119894lowast
119887119904
119894lowast
119888119904
]]]]
]
= [119879minus1
]
[[[[
[
119894lowast
119902119904
119894lowast
119889119904
0
]]]]
]
(36)
where
119879minus1
=
[[[[[[[
[
1 0 1
minus1
2minusradic3
21
minus1
2
radic3
21
]]]]]]]
]
(37)
By using (35) to (37) the stator 119902- and 119889-axes and abccurrent commands are derived as
119894lowast
119902119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119889119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 cos 120579lowast
119904
119894lowast
119886119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119887119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904minus2120587
3)
119894lowast
119888119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904+2120587
3)
(38)
where
120579lowast
119904= 120579119891+ 120579lowast
119879= 120579119903+ 120579lowast
sl + 120579lowast
119879 (39)
The implementation of IFOC on a CSI fed IM is shownin Figures 8 and 9 The torque command 119894lowast
119879is generated
as a function of the speed error signal generally processedthrough a PI controller The flux command 119894lowast
119891can be given
directly or as a function of speed The rotor position 120579119903can
be measured with an encoder and converted into necessarydigital information for feedbackThere has been a substantialamount of research in the development of rotor flux observersfor field orientation that are compensated for variationsin parameters by their feedback corrections Digital imple-mentation of integrators for the estimation of rotor flux ofan IM from the stator voltages and stator currents posesproblems associated with the offset in the sensor amplifiers
CSI
IM
PWMswitching
logic
Rotor speed
regulator
Stator current
generator
DC link currentias
lowast
ibslowast
i cslowast
120579riTlowast is
iflowast
iflowast and iT
lowast120582rlowast
120596rlowast
120596sllowast
120596r
Telowast
(1 + pLr
Rr
) 120582rlowast
Lm
RrLm
Lr
iTlowast
120582rlowast
4
3P
Telowast
120582rlowast
Lr
Lm
+minus
Figure 8 Simplified diagram for IFOC method
islowast = radic(iTlowast)2 +(iflowast)2 ias
lowast = |islowast|sin120579s
lowast|islowast|
ibslowast = |is
lowast|sin120579slowast minus
2120587
3
icslowast = |is
lowast|sin120579slowast +
2120587
3
120579r 120579r
iTlowast
iflowast
120579Tlowast
120579slowast
120596sllowast 120579sl
lowastint
iaslowast
ibslowast
i cslowast
tanminus1 iTlowast
iflowast
+++
Figure 9 Stator current generator
Traditional low-pass filters can replace the integrator Soin this work rotor flux estimated using low pass filter isdescribed The instantaneous flux linkage can be computedusing the measured d-axis stator current using (40) which isreferred to as the current model as follows
120582119903=119871119898119894119889119904
1 + 120591119903119904 (40)
The theoretical concepts of IFOC method for CSI fed IMdrive discussed in Sections 51 and 52 are verified throughsimulation using the Simulink toolbox of MATLAB Thedetailed simulations results are presented in Section 71
6 DTC for CSI Fed IM Drives
Variable speed drives are used in all industries to controlprecisely the speed of electric motors driving loads rangingfrom pumps and fans to complex drives on paper machinesrolling mill cranes and similar drives The most modernmethod for these drives is direct torque and stator flux vectorcontrol method usually called DTC The VSI fed version hasbeen realized in an industrial way by ABB by using thetheoretical background proposed by Takahashi and Ohmori[22] and Depenbrock [23] This solution is based both onfield oriented control (FOC) as well as on the direct self-control theory The idea is that motor flux and torque areused as primary control variables which is contrary to thewayin which traditional AC drives control input frequency andvoltage but is in principle similar to what is done with a DCdrive where it is much more straightforward to achieve
Journal of Applied Mathematics 9
Casadei et al [24] has described that by controllingmotor torque directly DTCprovides dynamic speed accuracyequivalent to closed loop AC and DC systems and torqueresponse times that are 10 times faster It is also claimedthat the DTC does not generate noise like that produced byconventional PWM AC drives This work describes a newmodular approach to implement DTC method on a CSIfed IM drive the simulation implementation procedure Thedynamic response of the drive is analyzed and presented
61 Mathematical Modeling of DTC Control Method TheDTC method is developed from Babaei and Heydari [10]From Krause et al [15] in the stationary reference frame thestator voltage and flux equation can be written as
V119904= 119877119904119894119904+119889120582119904
119889119905 (41)
120582119904= int (V
119904minus 119877119904119894119904) 119889119905 = 120582
120572119904+ 119895120582120573119904 (42)
10038161003816100381610038161205821199041003816100381610038161003816 = radic120582
2
120572119904+ 1205822
120573119904 (43)
ang120593119904= tanminus1 (
120582120573119904
120582120572119904
) (44)
where 119877119904is the stator resistance and 120582
119904is the stator flux
vector in the stationary reference frame |120582119904| and ang120593
119904are the
amplitude and position of stator flux vector respectivelyIn DTC of CSI fed IM it is desirable to determine the
stator current reference so that the torque and stator fluxfollow their reference values Babaei and Heydari [10] Theproposed DTC system is based on the stator flux oriented(SFO) reference frame In this rotating reference frame it iswritten as
120582119902119904= 0 120582
119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)
where 119889 and 119902 are the real and imaginary axes in the SFOreference frame Therefore the torque equation is rewrittenas
119879119890=3
2(119875
2)1
120596119887
(120582119889119904119894119902119904) (46)
where 119894119902119904is the imaginary part of the stator current vector
in the SFO reference frame Therefore if |120582lowast119904| and 119879lowast
119890are
stator flux magnitude and torque references respectively theimaginary part of the stator current reference vector whichleads to torque and stator flux magnitude references is asfollows
119894lowast
119902119904=2
3
1
119875
119879lowast
119890
1003816100381610038161003816120582lowast
119904
1003816100381610038161003816
(47)
Because of the existence of a capacitor in the controlsystem and its corresponding losses when (47) is used tocontrol the torque for all rotor speeds the control system willnot be accurate enough
Currentlimiter
Telowast
Te
|120582slowast|
2
3
1
P
PI
Telowast
|120582slowast |
+
+
+minus iqs
lowast
Figure 10 119902-axis reference current calculator
In order to have amore accurate torque control the blockdiagram shown in Figure 10 is used to calculate 119894lowast
119902119904This block
diagram is a combination of (47) and a simple PI controllerThis controller removes the steady-state error and leads to anaccurate control The coefficients of this controller must besmall enough so that it remains stable and the ripples in 119894lowast
119902119904
are limited because a ripple of 119894lowast119902119904results in a ripple of the
generated torque If only a PI is used 119894lowast119902119904
will have a largevalue of ripple and the torque response time will be verylargeTherefore in order to have a faster transient response oftorque it is necessary to use (47) In DTC method with VSIthe real part of voltage vector (V
119889) in the stator flux reference
frame leads to direct control of stator flux magnitudeTherefore using the real part of the stator current vector
in the stator flux reference frame it is possible to control theflux magnitude indirectly In order to obtain the real part ofthe stator current reference a simple PI controller is usedTheinput of this PI controller is the error that resulted from thecomparison between the reference stator flux magnitude andthe estimated flux value In order to control themotor currentin all cases especially in the starting mode the output ofthis PI controller must be limited between proper valuesTheoutput signal of the PI controller is the real part of referencestator current 119894lowast
119889119904
Therefore the stator current reference vector that leads tocontrol the IM stator flux and torque is as follows
119894lowast
119904(119889119902)= 119894lowast
119889119904+ 119895119894lowast
119902119904 (48)
1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 =radic119894lowast2
119889119904+ 119894lowast2119902119904 (49)
The obtained reference current vector is in the statorflux reference frame In order to generate it by SVM of theCSI it must be transferred to the stationary reference frameTo do that the stator flux position is used to transfer thisvector from the stator flux reference frame to the stationaryreference frame as follows
119894lowast
119904(120572120573)= 119894lowast
120572119904+ 119895119894lowast
120573119904= (119894lowast
119889119904+ 119895119894lowast
119902119904) sdot 119890119895120593119904
=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 ang (120593119904 + tanminus1 (119894lowast
119902119904
119894lowast
119889119904
))
(50)
This reference current vectormust be generated in theCSIfirst and then applied to the motor after removing its har-monics with filtering capacitorThis current has a nearly sinu-soidal waveform that leads to a nearly sinusoidal voltage inmotor terminals after feeding the motor and passing through
10 Journal of Applied Mathematics
Stator flux torque
estimator
PI
CSI
SVM IM
Gate Signals
PI
120582slowast
120591elowast
120582s
120582s
120591e
ang120593s
120591e
iqlowast
idlowast
dq
to120572120573
iaib
vb
va
|islowast| and idc
lowast
idclowast
+minus
+minus i120572
lowast
i120573lowast
Figure 11 Block diagram schematic of the proposed DTC
motor impedances The speed and stator flux control of anIM with DTC method is shown in Figure 11 Separate PIcontrollers are used to control the speed and flux The inputto the speed controller is the error caused by comparingthe reference speed and the rotor speed The output of thiscontroller is the imaginary part of stator current vector 119894lowast
119902119904
which is limited to a proper value
7 Results and Discussion
71 Simulation Results The simulationmodels of IOL IFOCand DTC control methods for CSI fed IM drive systemsare developed and simulated using MATLAB simulationsoftware From the simulation results dynamic performancesof the developed drives have been individually analyzedmainly using the sensitivity tests Here the results of all thedrives are compared with one another The tests that arechosen for the comparison are step changes in the referencespeed and torque of the motor drive
The first test is step change in reference speed which isuseful when trying to change from one speed to anotherThe next test is step change in reference torque that wouldbe useful in identifying the controller with a high torqueresponse Change in speed reference is the first test that isdone to compare the motor drives
The results of this test can be seen in the Figures 12 13 and14The speed command is varied from0 to its rated valueThespeed response of IOL drive is faster and has the settling timeof 05 second range while DTC and IFOC drives performsimilarly but slower than IOL (settling time of less than 1second)
The second test as mentioned earlier is the torque steptestThe results of this test can be seen in Figures 12 to 14Thetorque reference is varied from 0 to its rated value The speedresponse of IFOC drive is faster and has the settling timeof 02 seconds while IOL and DTC drives perform similarly(settling time of less than 05 second)
During this test the motor torque response of DTCdrive is almost instantaneous within the 02 second rangewhile IFOC drive closely follows behind it IOL and vectorbased 119881119891 have slower torque response than all other VectorControl techniques
Table 1 Prototype motor parameters
1 Hp 3 phase star connected 4 pole 415V 18 A 50Hz
Stator resistance 0087Ω Stator amp rotorleakage reactance 08119890minus3H
Rotor resistance 0228Ω Magnetizingreactance 347119890minus3H
72 Experimental Results Brief experimental results arepresented for IOL method and DTC method usingTMS320F2812 DSP based hardware setup Figure 15 showsthe schematic diagram for the hardware setup of theproposed laboratory prototype CSI fed induction motordrive The induction motor ratings are given in Table 1 Theinverter consists of CM75TL-12NF The gate driving signalis developed by using Texas instruments DSP processorTMS320F2812 DSP The software used to develop programsfor TMS320F2812 DSP is by code composer studio andMatlabSimulinkThe processor contains the program that isdownloaded to the computer
The clock speed of the DSP is 150MHz and it is capableof 32-bit operations The on board available flash memoryis 2048Mb It was created specifically for motor controloperation and therefore Parkrsquos and Clarkrsquos transformationsare conveniently built in Another convenient feature is thatit has sixteen 12-bit ADC pins that allow for a high degree ofprecision while taking many possible measurements
The above experimental result of IOL method of vectorcontrol technique in Figure 16 shows the fast settling timewith feed forward control compared to feedback techniqueSimilarly the experimental result shown in Figure 17 depictsa greater torque response within 02 seconds comparedto other vector control techniques With this experimentalverification it is clear that IOL is adaptive for speed responseand DTC is more suitable for torque response controlleddrives
From Tables 2 and 3 it seems that the dynamic speed andtorque performance of all four drives using change in speedreference It is evident from the table that IOL drive hasthe best speed response with DTC and IFOC occupyingpositions next to it It is also evident that DTC drive has thebest instantaneous torque response with IFOC and IOL nextto it
8 Conclusion
This paper presents a new approach in mathematical mod-eling of a 3-phase induction motor drive Mathematics con-cepts are used in developing the simulation and implementa-tion of IOL IFOC and DTC methods for CSI fed IM drivesIn this paper a rotor flux based reference frame controlmethod for PWM CSI fed IM with simplified simulationcircuits has been developed in which the motor speed andtorque follow the references closely in all the three vectorcontrol methods However a CSI fed induction motor drivehas some drawbacks such as slower dynamic response andbulky overall size due to the requirement of large smoothing
Journal of Applied Mathematics 11
15
minus05
0
05
1
Spee
d (P
U)
0 1 2 3 4 5Time (s)
Without FF controlWith FF control
(a)
0 1 2 3 4 5
minus2
0
2
Time (s)
Torq
ue (P
U)
Without FF controlWith FF control
(b)
Figure 12 Speed torque response of IOL control method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus4
minus2
0
24
Torq
ue (P
U)
Time (s)
Reference speedStep increase in speed command
Step decrease in speed command
Step increase in torque command
Step decrease in torque command Reference torque
Motor torque
Motor speed
Response due to change intorque reference
Response due to change in
speed reference
Figure 13 Speed torque response of IFOC method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus2minus1
012
Time (s)
Torq
ue (P
U)
Step decrease in speed command
Step increase in speed commandReference speed Motor speed
Response due to change intorque reference
Response due to change inspeed reference
Motor torque
Step decrease in torque commandReference torque
Step increase in torque command
Figure 14 Speed torque response of DTC method
inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying
TMS320F2812DSP
ADC
PWM
-EV
B
PWM
-EVA
Supplyfilter
3120601controlled
rectifier3120601 CSI inverter
3120601IM
3120601supply
Texasinstruments
DSP processor
Speed sensor output
(Nr) Rpm
CM75TL-12NFseries
IGBT (6-PAC)
Ldc
Is (Amps)
Figure 15 Schematic diagram of the hardware setup of CSI fed IM
Torque output with feedforward control
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Torque output withoutfeed forward control
Speed output with feedforward control
Speed output withoutfeed forward control
Figure 16 Experimental result torque-speed response of IOLcontrol method
applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 5
Figure 5(c) it is evident that the stator119889-axis flux and stator 119902-axis current (control variables) have linear relationship withtorque (speed) It is used in the stator flux oriented (SFO)control methods
The basic IM equations from Krause et al [15] aresimulated for three different reference frames usingMATLAB76 software and the results obtained are compared with eachreference frame From the comparison it is observed that ifthe stator rotating reference frame is used then stator 119889-axiscurrent is identical to that of the stator ldquo119886rdquo phase current asshown in Figure 3(a) which is used in stator fed IM drivesSimilarly in rotor reference frame rotor 119889-axis current isidentical to the rotor ldquo119886rdquo phase current and useful in rotor fedIM drives If the synchronous reference frame is used thenthe stator 119889-axis current behaves like DC quantity as shownin Figure 3(c) From Figures 4(c) and 5(c) it is evident thatsynchronous reference frame is useful in RFO and SFO drivecontrol methods In this paper synchronous reference frameis preferred to model the IM in 119881119891 IOL control methodsRFO control is used in IFOCmethod and SFO is used inDTCmethod
4 IOL Control for CSI Fed IM Drives
In this section a type of IOL control method for the CSIfed IM drive is described The controller is designed based
on pole placement technique of Wonham [16] to a 119889-119902 axisstate space linearized model of the drive which includes areduced order rotor current observer A control law has beendeveloped to improve the dynamic response of the drive byusing the concepts from Veerachary [8] which includes feedforward as well as feedback controls
41 Mathematical Modeling of IOL Control Method Theequivalent circuit for the CSI capacitor and IM in a 119889-119902 axis rotating reference frame is shown in Figure 6 Thecorresponding state equation (differential equations) withmotor flux linkages as variables is derived fromWu et al [17]
Veerachary [8] presented the linearized CSI-IM drivestate space model in synchronously rotating reference frameas follows
= 119860119909 + 119861119906 (6)
where
=119889119909
119889119905 (7)
119910 = 119862119909 (8)
where
119909 =
[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]
]
=119889
119889119905
[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]
]
119906 =
[[[[
[
Δ1198811015840
119889
Δ120596119890
120596119887
Δ119879119871
]]]]
]
119910 =[[
[
Δ119894119902119904
Δ120596119903
120596119887
]]
]
119860 =
[[[[[[[[[[[[[[
[
minus1199091015840
1199031199030120596119887
11990901199091015840119903minus 1199092119898
1199091198981198771015840
119903120596119887
11990901199091015840119903minus 1199092119898
minus12059611989001199091015840
119903119909119898(1 minus 119904)
11990901199091015840119903minus 1199092119898
minus1199091198981199091015840
1199031198941015840
1198891199030120596119887
11990901199091015840119903minus 1199092119898
1205961198871199091198981199030
11990901199091015840119903minus 1199092119898
minus11990901198771015840
119903120596119887
11990901199091015840119903minus 1199092119898
1205961198900(1199092
119898minus 11990411990901199091015840
119903)
11990901199091015840119903minus 1199092119898
11990901199091015840
1199031198941015840
1198891199030120596119887
11990901199091015840119903minus 1199092119898
1199041205961198900119909119898
1199091015840119903
1199041205961198900
minus1205961198871198771015840
119903
1199091015840119903
minus
120596119887(1199091198981198941199021199040+ 1199091015840
1199031198941015840
1199021199030)
1199091015840119903
1199091198981198941015840
1198891199030
21198670
1199091198981198941199021199040
21198670
]]]]]]]]]]]]]]
]
119861 =
[[[[[[[[[[[[[[[
[
1199091015840
119903120596119887
11990901199091015840119903minus 1199092119898
0 0
minus119909119898120596119887
11990901199091015840119903minus 1199092119898
minus1205961198871198941015840
11988911990300
0 minus
120596119887(1199091198981198941199021199040+ 1199091015840
1199031198941015840
1199021199030)
1199091015840119903
0
0 0 minus1
2119867
]]]]]]]]]]]]]]]
]
119862 = [1 0 0 0
0 0 0 1]
1198811015840
119889=120587119881dc
3radic3 119903
0= 119877119904+1205872
119877dc18
1199090= 119909119904+1205872
119909119889
18
(9)
6 Journal of Applied Mathematics
Rs
Rs
Rr
Vin Idi = 0
VqsLm
Lm
Ll
Ll
p 120582ds
p 120582qs p 120582qr
p 120582dr
+ minus
+minus
c
c
120596r 120582qr
120596r 120582qr
IdsIdi
IqsIqi
Vds
CSI
Iqi = MdIdc Vqs
Figure 6 Equivalent circuit of CSI-IM section
With usual state feedback control there can be the trackingerror in steady state To eliminate this error Smith andDavidson [18] introduced an integral control which includesfeedback as well as feed forward control In this work anoptimal control law is derived by using the concepts of Smithand Davidson [18] and Veerachary [8]
42 Design of Optimal Control Law A new multivariableoptimal controller incorporating state feedback as well asfeed forward control for fast regulation and stability of a CSIfed IM drive is presented in this section The design of thestate feedback controller is based on the industrial regulatortheory together with pole placement technique applied ona linearized 119889-119902 axes state space model of the drive andincludes a reduced order observer to estimate the inaccessiblestates like 119889-119902 axes rotor currents A feed forward control interms of reference and disturbance inputs has been addedto the feedback controller-observer to obtain faster dynamicresponse either from (10) or (11) as follows (Figure 7)
119906 = 1198701119909 + 119870
2int
119905
0
(119910 minus 119910119903) 119889119905 + 119870
119865119865[119889
119910119903
] (10)
or
119906 = 1198701(2times4)
[[[[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]]]]
]
+ 1198702(2times2)
[[[[[
[
int
119905
0
(119894119902119904minus 119894lowast
119902119904) 119889119905
int
119905
0
(120596119903minus 120596lowast
119903) 119889119905
]]]]]
]
+ 119870119865119865(2times3)
[[[[
[
119879119871
119894lowast
119902119904
120596lowast
119903
]]]]
]
(11)
5 IFOC for CSI Fed IM Drives
In this section a high performance current fed indirect RFOcontrol method is described for IMs The IM is mathe-matically modeled in the rotor flux reference frame that is
CSIfed IM
Reduced order
observer
yy
x
r = [ iqslowast120596rlowast] y= [ iqs
120596r
]u
u
y
y
= [Vd
120596e
]
K1(2times4)
K2(2times2)
KFF(2times3)
+ +++
minus
d = TL
int
u998400
Figure 7 Combined feed forward and integral feedback controller
presented The rotor flux orientation is also obtained In thelow speed region the rotor flux components can be synthe-sized more easily with the help of speed and current signalsDesign of indirect vector controller Blaschke [19 20] androtor flux estimator is discussed Implementation of IFOCto CSI fed IM drive and transient response analysis of othersimulation results are also discussed
51 MathematicalModeling of IFOControlMethod The rotorequations of the IM containing flux linkages as variables aregiven by Krause et al [15] as follows
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus (120596119890minus 120596119903) 120582119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ (120596119890minus 120596119903) 120582119890
119889119903= 0
(12)
where
120596sl = 120596119890 minus 120596119903 (13)
then (12) becomes
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus 120596sl120582
119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ 120596sl120582
119890
119889119903= 0
(14)
The resultant rotor flux linkage 120582119903 also known as the rotor
flux linkages phasor is assumed to be on the direct axisto reduce the number of variables in the equations by oneMoreover it relates to reality that rotor flux linkages are asingle variable Hence aligning d axis with rotor flux phasoryields
120582119903= 120582119890
119889119903 (15)
120582119890
119902119903= 0 (16)
119901120582119890
119902119903= 0 (17)
Substituting (15) into (14) results in the new rotor equationsas follows
119877119903119894119890
119889119903+ 119901120582119903= 0
119877119903119894119890
119902119903+ 120596sl120582119903 = 0
(18)
Journal of Applied Mathematics 7
The rotor currents in terms of the stator currents are derivedfrom (18) as
119894119890
119902119903= minus
119871119898
119871119903
119894119890
119902119904
119894119890
119889119903=120582119903
119871119903
minus119871119898
119871119903
119894119890
119889119904
(19)
119894119890
119889119904= 119894119891= [1 +
119871119903
119877119903
119901]120582119903
119871119898
119894119890
119889119904= 119894119891= [1 + 120591
119903119901]
120582119903
119871119898
(20)
120596sl =119877119903119871119898
119871119903
119894119890
119902119904
120582119903
120596sl =119877119903119871119898
119871119903
119894119879
120582119903
120596sl =119871119898
120591119903
119894119879
120582119903
(21)
The 119902- and 119889-axes currents are relabeled as torque (119894119879) and
flux producing (119894119891) components of the stator-current phasor
respectively 120591119903denotes rotor time constant Similarly by the
same substitution of the rotor currents from (19) into torqueexpression the electromagnetic torque is derived as
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904minus 120582119890
119902119903119894119890
119889119904)
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904)
119879119890= 119870119905119890120582119903119894119890
119902119904= 119870119905119890120582119903119894119879
(22)
where the torque constant119870119905119890is defined as
119870119905119890=3
2
119875
2
119871119898
119871119903
(23)
Note that the torque is proportional to the product of rotorflux linkage and the stator 119902-axis current This resemblesthe air gap torque expression of the DC motor which isproportional to the product of the field flux linkages and thearmature current If the rotor flux linkage is maintained asconstant the torque is proportional to the torque producingcomponent of the stator current This is in relation to theseparately excited DC motor with armature current controlwhere the torque is proportional to the armature currentwhen the field current is constant Further here too the timeconstant is measured in the order of a few milliseconds Therotor flux linkages and air gap torque given in (21) and (22)respectively complete the transformation of the IM into anequivalent separately-excited dc motor from a control pointof view
52 Design of IFO Controller The IFO controller was de-signed using the concepts of Vas [21] and Salo and Tuusa [9]
From that the stator-current phasor is the phasor sum of the119889- and 119902-axis stator currents in any frames and it is given as
119894119904= radic(119894119890
119902119904)2
+ (119894119890
119889119904)2
(24)
and the 119889119902 axes (2120601) to (3120601) abc phase current relationship isobtained from
[
[
119894119890
119902119904
119894119890
119889119904
]
]
=2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+4120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+4120587
3)
]]]
]
[
[
119894119886119904
119894119887119904
119894119888119904
]
]
(25)
And it can be written as
119894119902119889= [119879] [119894
119886119887119888] (26)
where
119894119902119889= [119894119890
119902119904119894119890
119889119904]119905
(27)
119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905
(28)
[119879] =2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+2120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+2120587
3)
]]]
]
(29)
where 119894119886119904 119894119887119904 and 119894
119888119904are the three phase stator currents Note
that the elements in the 119879 matrix are cosinusoidal functionsof electrical angle 120579
119891 The electrical field angle in this case is
that of the rotor flux-linkages phasor and is obtained as thesum of the rotor and slip angles as follows
120579119891= 120579119903+ 120579sl (30)
and the slip angle is obtained by integrating the slip speed andis given as
120579sl = int120596sl (31)
IFO controller developed from these derivations accepts thetorque and flux requests and generates the torque and fluxproducing components of the stator-current phasor and theslip-angle commandsThe command values are denoted withasterisks throughout this thesis From (20) (21) and (23) thecommand values of 119894
119879 119894119891 and 120596sl are obtained as follows
119894lowast
119879=
119879lowast
119890
119870119905119890120582lowast119903
= (2
3) (
2
119875)119879lowast
119890
120582lowast119903
119871119903
119871119898
(32)
119894lowast
119891= (1 + 119901
119871119903
119877119903
)120582lowast
119903
119871119898
(33)
120596lowast
sl =119877119903119871119898
119871119903
119894lowast
119879
120582lowast119903
(34)
8 Journal of Applied Mathematics
The command slip angle 120579lowastsl is generated by integrating 120596lowastsl The torque angle command is obtained as the arctangentof 119894lowast119879and 119894lowast119891 The field angle is obtained by summing the
command slip angle and rotor angleWith the torque and fluxproducing components of the stator current commands androtor field angle the 119902119889 axes current commands (119886119887119888 phasecurrent commands) are obtained as follows The relevantsteps involved in the realization of the IFO controller are asfollows
[
[
119894lowast
119902119904
119894lowast
119889119904
]
]
= [
[
cos 120579119891
sin 120579119891
minus sin 120579119891
cos 120579119891
]
]
[
[
119894lowast
119879
119894lowast
119891
]
]
(35)
[[[[
[
119894lowast
119886119904
119894lowast
119887119904
119894lowast
119888119904
]]]]
]
= [119879minus1
]
[[[[
[
119894lowast
119902119904
119894lowast
119889119904
0
]]]]
]
(36)
where
119879minus1
=
[[[[[[[
[
1 0 1
minus1
2minusradic3
21
minus1
2
radic3
21
]]]]]]]
]
(37)
By using (35) to (37) the stator 119902- and 119889-axes and abccurrent commands are derived as
119894lowast
119902119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119889119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 cos 120579lowast
119904
119894lowast
119886119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119887119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904minus2120587
3)
119894lowast
119888119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904+2120587
3)
(38)
where
120579lowast
119904= 120579119891+ 120579lowast
119879= 120579119903+ 120579lowast
sl + 120579lowast
119879 (39)
The implementation of IFOC on a CSI fed IM is shownin Figures 8 and 9 The torque command 119894lowast
119879is generated
as a function of the speed error signal generally processedthrough a PI controller The flux command 119894lowast
119891can be given
directly or as a function of speed The rotor position 120579119903can
be measured with an encoder and converted into necessarydigital information for feedbackThere has been a substantialamount of research in the development of rotor flux observersfor field orientation that are compensated for variationsin parameters by their feedback corrections Digital imple-mentation of integrators for the estimation of rotor flux ofan IM from the stator voltages and stator currents posesproblems associated with the offset in the sensor amplifiers
CSI
IM
PWMswitching
logic
Rotor speed
regulator
Stator current
generator
DC link currentias
lowast
ibslowast
i cslowast
120579riTlowast is
iflowast
iflowast and iT
lowast120582rlowast
120596rlowast
120596sllowast
120596r
Telowast
(1 + pLr
Rr
) 120582rlowast
Lm
RrLm
Lr
iTlowast
120582rlowast
4
3P
Telowast
120582rlowast
Lr
Lm
+minus
Figure 8 Simplified diagram for IFOC method
islowast = radic(iTlowast)2 +(iflowast)2 ias
lowast = |islowast|sin120579s
lowast|islowast|
ibslowast = |is
lowast|sin120579slowast minus
2120587
3
icslowast = |is
lowast|sin120579slowast +
2120587
3
120579r 120579r
iTlowast
iflowast
120579Tlowast
120579slowast
120596sllowast 120579sl
lowastint
iaslowast
ibslowast
i cslowast
tanminus1 iTlowast
iflowast
+++
Figure 9 Stator current generator
Traditional low-pass filters can replace the integrator Soin this work rotor flux estimated using low pass filter isdescribed The instantaneous flux linkage can be computedusing the measured d-axis stator current using (40) which isreferred to as the current model as follows
120582119903=119871119898119894119889119904
1 + 120591119903119904 (40)
The theoretical concepts of IFOC method for CSI fed IMdrive discussed in Sections 51 and 52 are verified throughsimulation using the Simulink toolbox of MATLAB Thedetailed simulations results are presented in Section 71
6 DTC for CSI Fed IM Drives
Variable speed drives are used in all industries to controlprecisely the speed of electric motors driving loads rangingfrom pumps and fans to complex drives on paper machinesrolling mill cranes and similar drives The most modernmethod for these drives is direct torque and stator flux vectorcontrol method usually called DTC The VSI fed version hasbeen realized in an industrial way by ABB by using thetheoretical background proposed by Takahashi and Ohmori[22] and Depenbrock [23] This solution is based both onfield oriented control (FOC) as well as on the direct self-control theory The idea is that motor flux and torque areused as primary control variables which is contrary to thewayin which traditional AC drives control input frequency andvoltage but is in principle similar to what is done with a DCdrive where it is much more straightforward to achieve
Journal of Applied Mathematics 9
Casadei et al [24] has described that by controllingmotor torque directly DTCprovides dynamic speed accuracyequivalent to closed loop AC and DC systems and torqueresponse times that are 10 times faster It is also claimedthat the DTC does not generate noise like that produced byconventional PWM AC drives This work describes a newmodular approach to implement DTC method on a CSIfed IM drive the simulation implementation procedure Thedynamic response of the drive is analyzed and presented
61 Mathematical Modeling of DTC Control Method TheDTC method is developed from Babaei and Heydari [10]From Krause et al [15] in the stationary reference frame thestator voltage and flux equation can be written as
V119904= 119877119904119894119904+119889120582119904
119889119905 (41)
120582119904= int (V
119904minus 119877119904119894119904) 119889119905 = 120582
120572119904+ 119895120582120573119904 (42)
10038161003816100381610038161205821199041003816100381610038161003816 = radic120582
2
120572119904+ 1205822
120573119904 (43)
ang120593119904= tanminus1 (
120582120573119904
120582120572119904
) (44)
where 119877119904is the stator resistance and 120582
119904is the stator flux
vector in the stationary reference frame |120582119904| and ang120593
119904are the
amplitude and position of stator flux vector respectivelyIn DTC of CSI fed IM it is desirable to determine the
stator current reference so that the torque and stator fluxfollow their reference values Babaei and Heydari [10] Theproposed DTC system is based on the stator flux oriented(SFO) reference frame In this rotating reference frame it iswritten as
120582119902119904= 0 120582
119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)
where 119889 and 119902 are the real and imaginary axes in the SFOreference frame Therefore the torque equation is rewrittenas
119879119890=3
2(119875
2)1
120596119887
(120582119889119904119894119902119904) (46)
where 119894119902119904is the imaginary part of the stator current vector
in the SFO reference frame Therefore if |120582lowast119904| and 119879lowast
119890are
stator flux magnitude and torque references respectively theimaginary part of the stator current reference vector whichleads to torque and stator flux magnitude references is asfollows
119894lowast
119902119904=2
3
1
119875
119879lowast
119890
1003816100381610038161003816120582lowast
119904
1003816100381610038161003816
(47)
Because of the existence of a capacitor in the controlsystem and its corresponding losses when (47) is used tocontrol the torque for all rotor speeds the control system willnot be accurate enough
Currentlimiter
Telowast
Te
|120582slowast|
2
3
1
P
PI
Telowast
|120582slowast |
+
+
+minus iqs
lowast
Figure 10 119902-axis reference current calculator
In order to have amore accurate torque control the blockdiagram shown in Figure 10 is used to calculate 119894lowast
119902119904This block
diagram is a combination of (47) and a simple PI controllerThis controller removes the steady-state error and leads to anaccurate control The coefficients of this controller must besmall enough so that it remains stable and the ripples in 119894lowast
119902119904
are limited because a ripple of 119894lowast119902119904results in a ripple of the
generated torque If only a PI is used 119894lowast119902119904
will have a largevalue of ripple and the torque response time will be verylargeTherefore in order to have a faster transient response oftorque it is necessary to use (47) In DTC method with VSIthe real part of voltage vector (V
119889) in the stator flux reference
frame leads to direct control of stator flux magnitudeTherefore using the real part of the stator current vector
in the stator flux reference frame it is possible to control theflux magnitude indirectly In order to obtain the real part ofthe stator current reference a simple PI controller is usedTheinput of this PI controller is the error that resulted from thecomparison between the reference stator flux magnitude andthe estimated flux value In order to control themotor currentin all cases especially in the starting mode the output ofthis PI controller must be limited between proper valuesTheoutput signal of the PI controller is the real part of referencestator current 119894lowast
119889119904
Therefore the stator current reference vector that leads tocontrol the IM stator flux and torque is as follows
119894lowast
119904(119889119902)= 119894lowast
119889119904+ 119895119894lowast
119902119904 (48)
1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 =radic119894lowast2
119889119904+ 119894lowast2119902119904 (49)
The obtained reference current vector is in the statorflux reference frame In order to generate it by SVM of theCSI it must be transferred to the stationary reference frameTo do that the stator flux position is used to transfer thisvector from the stator flux reference frame to the stationaryreference frame as follows
119894lowast
119904(120572120573)= 119894lowast
120572119904+ 119895119894lowast
120573119904= (119894lowast
119889119904+ 119895119894lowast
119902119904) sdot 119890119895120593119904
=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 ang (120593119904 + tanminus1 (119894lowast
119902119904
119894lowast
119889119904
))
(50)
This reference current vectormust be generated in theCSIfirst and then applied to the motor after removing its har-monics with filtering capacitorThis current has a nearly sinu-soidal waveform that leads to a nearly sinusoidal voltage inmotor terminals after feeding the motor and passing through
10 Journal of Applied Mathematics
Stator flux torque
estimator
PI
CSI
SVM IM
Gate Signals
PI
120582slowast
120591elowast
120582s
120582s
120591e
ang120593s
120591e
iqlowast
idlowast
dq
to120572120573
iaib
vb
va
|islowast| and idc
lowast
idclowast
+minus
+minus i120572
lowast
i120573lowast
Figure 11 Block diagram schematic of the proposed DTC
motor impedances The speed and stator flux control of anIM with DTC method is shown in Figure 11 Separate PIcontrollers are used to control the speed and flux The inputto the speed controller is the error caused by comparingthe reference speed and the rotor speed The output of thiscontroller is the imaginary part of stator current vector 119894lowast
119902119904
which is limited to a proper value
7 Results and Discussion
71 Simulation Results The simulationmodels of IOL IFOCand DTC control methods for CSI fed IM drive systemsare developed and simulated using MATLAB simulationsoftware From the simulation results dynamic performancesof the developed drives have been individually analyzedmainly using the sensitivity tests Here the results of all thedrives are compared with one another The tests that arechosen for the comparison are step changes in the referencespeed and torque of the motor drive
The first test is step change in reference speed which isuseful when trying to change from one speed to anotherThe next test is step change in reference torque that wouldbe useful in identifying the controller with a high torqueresponse Change in speed reference is the first test that isdone to compare the motor drives
The results of this test can be seen in the Figures 12 13 and14The speed command is varied from0 to its rated valueThespeed response of IOL drive is faster and has the settling timeof 05 second range while DTC and IFOC drives performsimilarly but slower than IOL (settling time of less than 1second)
The second test as mentioned earlier is the torque steptestThe results of this test can be seen in Figures 12 to 14Thetorque reference is varied from 0 to its rated value The speedresponse of IFOC drive is faster and has the settling timeof 02 seconds while IOL and DTC drives perform similarly(settling time of less than 05 second)
During this test the motor torque response of DTCdrive is almost instantaneous within the 02 second rangewhile IFOC drive closely follows behind it IOL and vectorbased 119881119891 have slower torque response than all other VectorControl techniques
Table 1 Prototype motor parameters
1 Hp 3 phase star connected 4 pole 415V 18 A 50Hz
Stator resistance 0087Ω Stator amp rotorleakage reactance 08119890minus3H
Rotor resistance 0228Ω Magnetizingreactance 347119890minus3H
72 Experimental Results Brief experimental results arepresented for IOL method and DTC method usingTMS320F2812 DSP based hardware setup Figure 15 showsthe schematic diagram for the hardware setup of theproposed laboratory prototype CSI fed induction motordrive The induction motor ratings are given in Table 1 Theinverter consists of CM75TL-12NF The gate driving signalis developed by using Texas instruments DSP processorTMS320F2812 DSP The software used to develop programsfor TMS320F2812 DSP is by code composer studio andMatlabSimulinkThe processor contains the program that isdownloaded to the computer
The clock speed of the DSP is 150MHz and it is capableof 32-bit operations The on board available flash memoryis 2048Mb It was created specifically for motor controloperation and therefore Parkrsquos and Clarkrsquos transformationsare conveniently built in Another convenient feature is thatit has sixteen 12-bit ADC pins that allow for a high degree ofprecision while taking many possible measurements
The above experimental result of IOL method of vectorcontrol technique in Figure 16 shows the fast settling timewith feed forward control compared to feedback techniqueSimilarly the experimental result shown in Figure 17 depictsa greater torque response within 02 seconds comparedto other vector control techniques With this experimentalverification it is clear that IOL is adaptive for speed responseand DTC is more suitable for torque response controlleddrives
From Tables 2 and 3 it seems that the dynamic speed andtorque performance of all four drives using change in speedreference It is evident from the table that IOL drive hasthe best speed response with DTC and IFOC occupyingpositions next to it It is also evident that DTC drive has thebest instantaneous torque response with IFOC and IOL nextto it
8 Conclusion
This paper presents a new approach in mathematical mod-eling of a 3-phase induction motor drive Mathematics con-cepts are used in developing the simulation and implementa-tion of IOL IFOC and DTC methods for CSI fed IM drivesIn this paper a rotor flux based reference frame controlmethod for PWM CSI fed IM with simplified simulationcircuits has been developed in which the motor speed andtorque follow the references closely in all the three vectorcontrol methods However a CSI fed induction motor drivehas some drawbacks such as slower dynamic response andbulky overall size due to the requirement of large smoothing
Journal of Applied Mathematics 11
15
minus05
0
05
1
Spee
d (P
U)
0 1 2 3 4 5Time (s)
Without FF controlWith FF control
(a)
0 1 2 3 4 5
minus2
0
2
Time (s)
Torq
ue (P
U)
Without FF controlWith FF control
(b)
Figure 12 Speed torque response of IOL control method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus4
minus2
0
24
Torq
ue (P
U)
Time (s)
Reference speedStep increase in speed command
Step decrease in speed command
Step increase in torque command
Step decrease in torque command Reference torque
Motor torque
Motor speed
Response due to change intorque reference
Response due to change in
speed reference
Figure 13 Speed torque response of IFOC method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus2minus1
012
Time (s)
Torq
ue (P
U)
Step decrease in speed command
Step increase in speed commandReference speed Motor speed
Response due to change intorque reference
Response due to change inspeed reference
Motor torque
Step decrease in torque commandReference torque
Step increase in torque command
Figure 14 Speed torque response of DTC method
inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying
TMS320F2812DSP
ADC
PWM
-EV
B
PWM
-EVA
Supplyfilter
3120601controlled
rectifier3120601 CSI inverter
3120601IM
3120601supply
Texasinstruments
DSP processor
Speed sensor output
(Nr) Rpm
CM75TL-12NFseries
IGBT (6-PAC)
Ldc
Is (Amps)
Figure 15 Schematic diagram of the hardware setup of CSI fed IM
Torque output with feedforward control
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Torque output withoutfeed forward control
Speed output with feedforward control
Speed output withoutfeed forward control
Figure 16 Experimental result torque-speed response of IOLcontrol method
applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Applied Mathematics
Rs
Rs
Rr
Vin Idi = 0
VqsLm
Lm
Ll
Ll
p 120582ds
p 120582qs p 120582qr
p 120582dr
+ minus
+minus
c
c
120596r 120582qr
120596r 120582qr
IdsIdi
IqsIqi
Vds
CSI
Iqi = MdIdc Vqs
Figure 6 Equivalent circuit of CSI-IM section
With usual state feedback control there can be the trackingerror in steady state To eliminate this error Smith andDavidson [18] introduced an integral control which includesfeedback as well as feed forward control In this work anoptimal control law is derived by using the concepts of Smithand Davidson [18] and Veerachary [8]
42 Design of Optimal Control Law A new multivariableoptimal controller incorporating state feedback as well asfeed forward control for fast regulation and stability of a CSIfed IM drive is presented in this section The design of thestate feedback controller is based on the industrial regulatortheory together with pole placement technique applied ona linearized 119889-119902 axes state space model of the drive andincludes a reduced order observer to estimate the inaccessiblestates like 119889-119902 axes rotor currents A feed forward control interms of reference and disturbance inputs has been addedto the feedback controller-observer to obtain faster dynamicresponse either from (10) or (11) as follows (Figure 7)
119906 = 1198701119909 + 119870
2int
119905
0
(119910 minus 119910119903) 119889119905 + 119870
119865119865[119889
119910119903
] (10)
or
119906 = 1198701(2times4)
[[[[[[[[[[
[
Δ119894119902119904
Δ1198941015840
119902119903
Δ1198941015840
119889119903
Δ120596119903
120596119887
]]]]]]]]]]
]
+ 1198702(2times2)
[[[[[
[
int
119905
0
(119894119902119904minus 119894lowast
119902119904) 119889119905
int
119905
0
(120596119903minus 120596lowast
119903) 119889119905
]]]]]
]
+ 119870119865119865(2times3)
[[[[
[
119879119871
119894lowast
119902119904
120596lowast
119903
]]]]
]
(11)
5 IFOC for CSI Fed IM Drives
In this section a high performance current fed indirect RFOcontrol method is described for IMs The IM is mathe-matically modeled in the rotor flux reference frame that is
CSIfed IM
Reduced order
observer
yy
x
r = [ iqslowast120596rlowast] y= [ iqs
120596r
]u
u
y
y
= [Vd
120596e
]
K1(2times4)
K2(2times2)
KFF(2times3)
+ +++
minus
d = TL
int
u998400
Figure 7 Combined feed forward and integral feedback controller
presented The rotor flux orientation is also obtained In thelow speed region the rotor flux components can be synthe-sized more easily with the help of speed and current signalsDesign of indirect vector controller Blaschke [19 20] androtor flux estimator is discussed Implementation of IFOCto CSI fed IM drive and transient response analysis of othersimulation results are also discussed
51 MathematicalModeling of IFOControlMethod The rotorequations of the IM containing flux linkages as variables aregiven by Krause et al [15] as follows
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus (120596119890minus 120596119903) 120582119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ (120596119890minus 120596119903) 120582119890
119889119903= 0
(12)
where
120596sl = 120596119890 minus 120596119903 (13)
then (12) becomes
119877119903119894119890
119889119903+ 119901120582119890
119889119903minus 120596sl120582
119890
119902119903= 0
119877119903119894119890
119902119903+ 119901120582119890
119902119903+ 120596sl120582
119890
119889119903= 0
(14)
The resultant rotor flux linkage 120582119903 also known as the rotor
flux linkages phasor is assumed to be on the direct axisto reduce the number of variables in the equations by oneMoreover it relates to reality that rotor flux linkages are asingle variable Hence aligning d axis with rotor flux phasoryields
120582119903= 120582119890
119889119903 (15)
120582119890
119902119903= 0 (16)
119901120582119890
119902119903= 0 (17)
Substituting (15) into (14) results in the new rotor equationsas follows
119877119903119894119890
119889119903+ 119901120582119903= 0
119877119903119894119890
119902119903+ 120596sl120582119903 = 0
(18)
Journal of Applied Mathematics 7
The rotor currents in terms of the stator currents are derivedfrom (18) as
119894119890
119902119903= minus
119871119898
119871119903
119894119890
119902119904
119894119890
119889119903=120582119903
119871119903
minus119871119898
119871119903
119894119890
119889119904
(19)
119894119890
119889119904= 119894119891= [1 +
119871119903
119877119903
119901]120582119903
119871119898
119894119890
119889119904= 119894119891= [1 + 120591
119903119901]
120582119903
119871119898
(20)
120596sl =119877119903119871119898
119871119903
119894119890
119902119904
120582119903
120596sl =119877119903119871119898
119871119903
119894119879
120582119903
120596sl =119871119898
120591119903
119894119879
120582119903
(21)
The 119902- and 119889-axes currents are relabeled as torque (119894119879) and
flux producing (119894119891) components of the stator-current phasor
respectively 120591119903denotes rotor time constant Similarly by the
same substitution of the rotor currents from (19) into torqueexpression the electromagnetic torque is derived as
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904minus 120582119890
119902119903119894119890
119889119904)
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904)
119879119890= 119870119905119890120582119903119894119890
119902119904= 119870119905119890120582119903119894119879
(22)
where the torque constant119870119905119890is defined as
119870119905119890=3
2
119875
2
119871119898
119871119903
(23)
Note that the torque is proportional to the product of rotorflux linkage and the stator 119902-axis current This resemblesthe air gap torque expression of the DC motor which isproportional to the product of the field flux linkages and thearmature current If the rotor flux linkage is maintained asconstant the torque is proportional to the torque producingcomponent of the stator current This is in relation to theseparately excited DC motor with armature current controlwhere the torque is proportional to the armature currentwhen the field current is constant Further here too the timeconstant is measured in the order of a few milliseconds Therotor flux linkages and air gap torque given in (21) and (22)respectively complete the transformation of the IM into anequivalent separately-excited dc motor from a control pointof view
52 Design of IFO Controller The IFO controller was de-signed using the concepts of Vas [21] and Salo and Tuusa [9]
From that the stator-current phasor is the phasor sum of the119889- and 119902-axis stator currents in any frames and it is given as
119894119904= radic(119894119890
119902119904)2
+ (119894119890
119889119904)2
(24)
and the 119889119902 axes (2120601) to (3120601) abc phase current relationship isobtained from
[
[
119894119890
119902119904
119894119890
119889119904
]
]
=2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+4120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+4120587
3)
]]]
]
[
[
119894119886119904
119894119887119904
119894119888119904
]
]
(25)
And it can be written as
119894119902119889= [119879] [119894
119886119887119888] (26)
where
119894119902119889= [119894119890
119902119904119894119890
119889119904]119905
(27)
119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905
(28)
[119879] =2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+2120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+2120587
3)
]]]
]
(29)
where 119894119886119904 119894119887119904 and 119894
119888119904are the three phase stator currents Note
that the elements in the 119879 matrix are cosinusoidal functionsof electrical angle 120579
119891 The electrical field angle in this case is
that of the rotor flux-linkages phasor and is obtained as thesum of the rotor and slip angles as follows
120579119891= 120579119903+ 120579sl (30)
and the slip angle is obtained by integrating the slip speed andis given as
120579sl = int120596sl (31)
IFO controller developed from these derivations accepts thetorque and flux requests and generates the torque and fluxproducing components of the stator-current phasor and theslip-angle commandsThe command values are denoted withasterisks throughout this thesis From (20) (21) and (23) thecommand values of 119894
119879 119894119891 and 120596sl are obtained as follows
119894lowast
119879=
119879lowast
119890
119870119905119890120582lowast119903
= (2
3) (
2
119875)119879lowast
119890
120582lowast119903
119871119903
119871119898
(32)
119894lowast
119891= (1 + 119901
119871119903
119877119903
)120582lowast
119903
119871119898
(33)
120596lowast
sl =119877119903119871119898
119871119903
119894lowast
119879
120582lowast119903
(34)
8 Journal of Applied Mathematics
The command slip angle 120579lowastsl is generated by integrating 120596lowastsl The torque angle command is obtained as the arctangentof 119894lowast119879and 119894lowast119891 The field angle is obtained by summing the
command slip angle and rotor angleWith the torque and fluxproducing components of the stator current commands androtor field angle the 119902119889 axes current commands (119886119887119888 phasecurrent commands) are obtained as follows The relevantsteps involved in the realization of the IFO controller are asfollows
[
[
119894lowast
119902119904
119894lowast
119889119904
]
]
= [
[
cos 120579119891
sin 120579119891
minus sin 120579119891
cos 120579119891
]
]
[
[
119894lowast
119879
119894lowast
119891
]
]
(35)
[[[[
[
119894lowast
119886119904
119894lowast
119887119904
119894lowast
119888119904
]]]]
]
= [119879minus1
]
[[[[
[
119894lowast
119902119904
119894lowast
119889119904
0
]]]]
]
(36)
where
119879minus1
=
[[[[[[[
[
1 0 1
minus1
2minusradic3
21
minus1
2
radic3
21
]]]]]]]
]
(37)
By using (35) to (37) the stator 119902- and 119889-axes and abccurrent commands are derived as
119894lowast
119902119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119889119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 cos 120579lowast
119904
119894lowast
119886119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119887119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904minus2120587
3)
119894lowast
119888119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904+2120587
3)
(38)
where
120579lowast
119904= 120579119891+ 120579lowast
119879= 120579119903+ 120579lowast
sl + 120579lowast
119879 (39)
The implementation of IFOC on a CSI fed IM is shownin Figures 8 and 9 The torque command 119894lowast
119879is generated
as a function of the speed error signal generally processedthrough a PI controller The flux command 119894lowast
119891can be given
directly or as a function of speed The rotor position 120579119903can
be measured with an encoder and converted into necessarydigital information for feedbackThere has been a substantialamount of research in the development of rotor flux observersfor field orientation that are compensated for variationsin parameters by their feedback corrections Digital imple-mentation of integrators for the estimation of rotor flux ofan IM from the stator voltages and stator currents posesproblems associated with the offset in the sensor amplifiers
CSI
IM
PWMswitching
logic
Rotor speed
regulator
Stator current
generator
DC link currentias
lowast
ibslowast
i cslowast
120579riTlowast is
iflowast
iflowast and iT
lowast120582rlowast
120596rlowast
120596sllowast
120596r
Telowast
(1 + pLr
Rr
) 120582rlowast
Lm
RrLm
Lr
iTlowast
120582rlowast
4
3P
Telowast
120582rlowast
Lr
Lm
+minus
Figure 8 Simplified diagram for IFOC method
islowast = radic(iTlowast)2 +(iflowast)2 ias
lowast = |islowast|sin120579s
lowast|islowast|
ibslowast = |is
lowast|sin120579slowast minus
2120587
3
icslowast = |is
lowast|sin120579slowast +
2120587
3
120579r 120579r
iTlowast
iflowast
120579Tlowast
120579slowast
120596sllowast 120579sl
lowastint
iaslowast
ibslowast
i cslowast
tanminus1 iTlowast
iflowast
+++
Figure 9 Stator current generator
Traditional low-pass filters can replace the integrator Soin this work rotor flux estimated using low pass filter isdescribed The instantaneous flux linkage can be computedusing the measured d-axis stator current using (40) which isreferred to as the current model as follows
120582119903=119871119898119894119889119904
1 + 120591119903119904 (40)
The theoretical concepts of IFOC method for CSI fed IMdrive discussed in Sections 51 and 52 are verified throughsimulation using the Simulink toolbox of MATLAB Thedetailed simulations results are presented in Section 71
6 DTC for CSI Fed IM Drives
Variable speed drives are used in all industries to controlprecisely the speed of electric motors driving loads rangingfrom pumps and fans to complex drives on paper machinesrolling mill cranes and similar drives The most modernmethod for these drives is direct torque and stator flux vectorcontrol method usually called DTC The VSI fed version hasbeen realized in an industrial way by ABB by using thetheoretical background proposed by Takahashi and Ohmori[22] and Depenbrock [23] This solution is based both onfield oriented control (FOC) as well as on the direct self-control theory The idea is that motor flux and torque areused as primary control variables which is contrary to thewayin which traditional AC drives control input frequency andvoltage but is in principle similar to what is done with a DCdrive where it is much more straightforward to achieve
Journal of Applied Mathematics 9
Casadei et al [24] has described that by controllingmotor torque directly DTCprovides dynamic speed accuracyequivalent to closed loop AC and DC systems and torqueresponse times that are 10 times faster It is also claimedthat the DTC does not generate noise like that produced byconventional PWM AC drives This work describes a newmodular approach to implement DTC method on a CSIfed IM drive the simulation implementation procedure Thedynamic response of the drive is analyzed and presented
61 Mathematical Modeling of DTC Control Method TheDTC method is developed from Babaei and Heydari [10]From Krause et al [15] in the stationary reference frame thestator voltage and flux equation can be written as
V119904= 119877119904119894119904+119889120582119904
119889119905 (41)
120582119904= int (V
119904minus 119877119904119894119904) 119889119905 = 120582
120572119904+ 119895120582120573119904 (42)
10038161003816100381610038161205821199041003816100381610038161003816 = radic120582
2
120572119904+ 1205822
120573119904 (43)
ang120593119904= tanminus1 (
120582120573119904
120582120572119904
) (44)
where 119877119904is the stator resistance and 120582
119904is the stator flux
vector in the stationary reference frame |120582119904| and ang120593
119904are the
amplitude and position of stator flux vector respectivelyIn DTC of CSI fed IM it is desirable to determine the
stator current reference so that the torque and stator fluxfollow their reference values Babaei and Heydari [10] Theproposed DTC system is based on the stator flux oriented(SFO) reference frame In this rotating reference frame it iswritten as
120582119902119904= 0 120582
119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)
where 119889 and 119902 are the real and imaginary axes in the SFOreference frame Therefore the torque equation is rewrittenas
119879119890=3
2(119875
2)1
120596119887
(120582119889119904119894119902119904) (46)
where 119894119902119904is the imaginary part of the stator current vector
in the SFO reference frame Therefore if |120582lowast119904| and 119879lowast
119890are
stator flux magnitude and torque references respectively theimaginary part of the stator current reference vector whichleads to torque and stator flux magnitude references is asfollows
119894lowast
119902119904=2
3
1
119875
119879lowast
119890
1003816100381610038161003816120582lowast
119904
1003816100381610038161003816
(47)
Because of the existence of a capacitor in the controlsystem and its corresponding losses when (47) is used tocontrol the torque for all rotor speeds the control system willnot be accurate enough
Currentlimiter
Telowast
Te
|120582slowast|
2
3
1
P
PI
Telowast
|120582slowast |
+
+
+minus iqs
lowast
Figure 10 119902-axis reference current calculator
In order to have amore accurate torque control the blockdiagram shown in Figure 10 is used to calculate 119894lowast
119902119904This block
diagram is a combination of (47) and a simple PI controllerThis controller removes the steady-state error and leads to anaccurate control The coefficients of this controller must besmall enough so that it remains stable and the ripples in 119894lowast
119902119904
are limited because a ripple of 119894lowast119902119904results in a ripple of the
generated torque If only a PI is used 119894lowast119902119904
will have a largevalue of ripple and the torque response time will be verylargeTherefore in order to have a faster transient response oftorque it is necessary to use (47) In DTC method with VSIthe real part of voltage vector (V
119889) in the stator flux reference
frame leads to direct control of stator flux magnitudeTherefore using the real part of the stator current vector
in the stator flux reference frame it is possible to control theflux magnitude indirectly In order to obtain the real part ofthe stator current reference a simple PI controller is usedTheinput of this PI controller is the error that resulted from thecomparison between the reference stator flux magnitude andthe estimated flux value In order to control themotor currentin all cases especially in the starting mode the output ofthis PI controller must be limited between proper valuesTheoutput signal of the PI controller is the real part of referencestator current 119894lowast
119889119904
Therefore the stator current reference vector that leads tocontrol the IM stator flux and torque is as follows
119894lowast
119904(119889119902)= 119894lowast
119889119904+ 119895119894lowast
119902119904 (48)
1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 =radic119894lowast2
119889119904+ 119894lowast2119902119904 (49)
The obtained reference current vector is in the statorflux reference frame In order to generate it by SVM of theCSI it must be transferred to the stationary reference frameTo do that the stator flux position is used to transfer thisvector from the stator flux reference frame to the stationaryreference frame as follows
119894lowast
119904(120572120573)= 119894lowast
120572119904+ 119895119894lowast
120573119904= (119894lowast
119889119904+ 119895119894lowast
119902119904) sdot 119890119895120593119904
=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 ang (120593119904 + tanminus1 (119894lowast
119902119904
119894lowast
119889119904
))
(50)
This reference current vectormust be generated in theCSIfirst and then applied to the motor after removing its har-monics with filtering capacitorThis current has a nearly sinu-soidal waveform that leads to a nearly sinusoidal voltage inmotor terminals after feeding the motor and passing through
10 Journal of Applied Mathematics
Stator flux torque
estimator
PI
CSI
SVM IM
Gate Signals
PI
120582slowast
120591elowast
120582s
120582s
120591e
ang120593s
120591e
iqlowast
idlowast
dq
to120572120573
iaib
vb
va
|islowast| and idc
lowast
idclowast
+minus
+minus i120572
lowast
i120573lowast
Figure 11 Block diagram schematic of the proposed DTC
motor impedances The speed and stator flux control of anIM with DTC method is shown in Figure 11 Separate PIcontrollers are used to control the speed and flux The inputto the speed controller is the error caused by comparingthe reference speed and the rotor speed The output of thiscontroller is the imaginary part of stator current vector 119894lowast
119902119904
which is limited to a proper value
7 Results and Discussion
71 Simulation Results The simulationmodels of IOL IFOCand DTC control methods for CSI fed IM drive systemsare developed and simulated using MATLAB simulationsoftware From the simulation results dynamic performancesof the developed drives have been individually analyzedmainly using the sensitivity tests Here the results of all thedrives are compared with one another The tests that arechosen for the comparison are step changes in the referencespeed and torque of the motor drive
The first test is step change in reference speed which isuseful when trying to change from one speed to anotherThe next test is step change in reference torque that wouldbe useful in identifying the controller with a high torqueresponse Change in speed reference is the first test that isdone to compare the motor drives
The results of this test can be seen in the Figures 12 13 and14The speed command is varied from0 to its rated valueThespeed response of IOL drive is faster and has the settling timeof 05 second range while DTC and IFOC drives performsimilarly but slower than IOL (settling time of less than 1second)
The second test as mentioned earlier is the torque steptestThe results of this test can be seen in Figures 12 to 14Thetorque reference is varied from 0 to its rated value The speedresponse of IFOC drive is faster and has the settling timeof 02 seconds while IOL and DTC drives perform similarly(settling time of less than 05 second)
During this test the motor torque response of DTCdrive is almost instantaneous within the 02 second rangewhile IFOC drive closely follows behind it IOL and vectorbased 119881119891 have slower torque response than all other VectorControl techniques
Table 1 Prototype motor parameters
1 Hp 3 phase star connected 4 pole 415V 18 A 50Hz
Stator resistance 0087Ω Stator amp rotorleakage reactance 08119890minus3H
Rotor resistance 0228Ω Magnetizingreactance 347119890minus3H
72 Experimental Results Brief experimental results arepresented for IOL method and DTC method usingTMS320F2812 DSP based hardware setup Figure 15 showsthe schematic diagram for the hardware setup of theproposed laboratory prototype CSI fed induction motordrive The induction motor ratings are given in Table 1 Theinverter consists of CM75TL-12NF The gate driving signalis developed by using Texas instruments DSP processorTMS320F2812 DSP The software used to develop programsfor TMS320F2812 DSP is by code composer studio andMatlabSimulinkThe processor contains the program that isdownloaded to the computer
The clock speed of the DSP is 150MHz and it is capableof 32-bit operations The on board available flash memoryis 2048Mb It was created specifically for motor controloperation and therefore Parkrsquos and Clarkrsquos transformationsare conveniently built in Another convenient feature is thatit has sixteen 12-bit ADC pins that allow for a high degree ofprecision while taking many possible measurements
The above experimental result of IOL method of vectorcontrol technique in Figure 16 shows the fast settling timewith feed forward control compared to feedback techniqueSimilarly the experimental result shown in Figure 17 depictsa greater torque response within 02 seconds comparedto other vector control techniques With this experimentalverification it is clear that IOL is adaptive for speed responseand DTC is more suitable for torque response controlleddrives
From Tables 2 and 3 it seems that the dynamic speed andtorque performance of all four drives using change in speedreference It is evident from the table that IOL drive hasthe best speed response with DTC and IFOC occupyingpositions next to it It is also evident that DTC drive has thebest instantaneous torque response with IFOC and IOL nextto it
8 Conclusion
This paper presents a new approach in mathematical mod-eling of a 3-phase induction motor drive Mathematics con-cepts are used in developing the simulation and implementa-tion of IOL IFOC and DTC methods for CSI fed IM drivesIn this paper a rotor flux based reference frame controlmethod for PWM CSI fed IM with simplified simulationcircuits has been developed in which the motor speed andtorque follow the references closely in all the three vectorcontrol methods However a CSI fed induction motor drivehas some drawbacks such as slower dynamic response andbulky overall size due to the requirement of large smoothing
Journal of Applied Mathematics 11
15
minus05
0
05
1
Spee
d (P
U)
0 1 2 3 4 5Time (s)
Without FF controlWith FF control
(a)
0 1 2 3 4 5
minus2
0
2
Time (s)
Torq
ue (P
U)
Without FF controlWith FF control
(b)
Figure 12 Speed torque response of IOL control method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus4
minus2
0
24
Torq
ue (P
U)
Time (s)
Reference speedStep increase in speed command
Step decrease in speed command
Step increase in torque command
Step decrease in torque command Reference torque
Motor torque
Motor speed
Response due to change intorque reference
Response due to change in
speed reference
Figure 13 Speed torque response of IFOC method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus2minus1
012
Time (s)
Torq
ue (P
U)
Step decrease in speed command
Step increase in speed commandReference speed Motor speed
Response due to change intorque reference
Response due to change inspeed reference
Motor torque
Step decrease in torque commandReference torque
Step increase in torque command
Figure 14 Speed torque response of DTC method
inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying
TMS320F2812DSP
ADC
PWM
-EV
B
PWM
-EVA
Supplyfilter
3120601controlled
rectifier3120601 CSI inverter
3120601IM
3120601supply
Texasinstruments
DSP processor
Speed sensor output
(Nr) Rpm
CM75TL-12NFseries
IGBT (6-PAC)
Ldc
Is (Amps)
Figure 15 Schematic diagram of the hardware setup of CSI fed IM
Torque output with feedforward control
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Torque output withoutfeed forward control
Speed output with feedforward control
Speed output withoutfeed forward control
Figure 16 Experimental result torque-speed response of IOLcontrol method
applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 7
The rotor currents in terms of the stator currents are derivedfrom (18) as
119894119890
119902119903= minus
119871119898
119871119903
119894119890
119902119904
119894119890
119889119903=120582119903
119871119903
minus119871119898
119871119903
119894119890
119889119904
(19)
119894119890
119889119904= 119894119891= [1 +
119871119903
119877119903
119901]120582119903
119871119898
119894119890
119889119904= 119894119891= [1 + 120591
119903119901]
120582119903
119871119898
(20)
120596sl =119877119903119871119898
119871119903
119894119890
119902119904
120582119903
120596sl =119877119903119871119898
119871119903
119894119879
120582119903
120596sl =119871119898
120591119903
119894119879
120582119903
(21)
The 119902- and 119889-axes currents are relabeled as torque (119894119879) and
flux producing (119894119891) components of the stator-current phasor
respectively 120591119903denotes rotor time constant Similarly by the
same substitution of the rotor currents from (19) into torqueexpression the electromagnetic torque is derived as
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904minus 120582119890
119902119903119894119890
119889119904)
119879119890=3
2
119875
2
119871119898
119871119903
(120582119890
119889119903119894119890
119902119904)
119879119890= 119870119905119890120582119903119894119890
119902119904= 119870119905119890120582119903119894119879
(22)
where the torque constant119870119905119890is defined as
119870119905119890=3
2
119875
2
119871119898
119871119903
(23)
Note that the torque is proportional to the product of rotorflux linkage and the stator 119902-axis current This resemblesthe air gap torque expression of the DC motor which isproportional to the product of the field flux linkages and thearmature current If the rotor flux linkage is maintained asconstant the torque is proportional to the torque producingcomponent of the stator current This is in relation to theseparately excited DC motor with armature current controlwhere the torque is proportional to the armature currentwhen the field current is constant Further here too the timeconstant is measured in the order of a few milliseconds Therotor flux linkages and air gap torque given in (21) and (22)respectively complete the transformation of the IM into anequivalent separately-excited dc motor from a control pointof view
52 Design of IFO Controller The IFO controller was de-signed using the concepts of Vas [21] and Salo and Tuusa [9]
From that the stator-current phasor is the phasor sum of the119889- and 119902-axis stator currents in any frames and it is given as
119894119904= radic(119894119890
119902119904)2
+ (119894119890
119889119904)2
(24)
and the 119889119902 axes (2120601) to (3120601) abc phase current relationship isobtained from
[
[
119894119890
119902119904
119894119890
119889119904
]
]
=2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+4120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+4120587
3)
]]]
]
[
[
119894119886119904
119894119887119904
119894119888119904
]
]
(25)
And it can be written as
119894119902119889= [119879] [119894
119886119887119888] (26)
where
119894119902119889= [119894119890
119902119904119894119890
119889119904]119905
(27)
119894119886119887119888= [119894119886119904119894119887119904119894119888119904]119905
(28)
[119879] =2
3
[[[
[
cos 120579119891
cos(120579119891minus2120587
3) cos(120579
119891+2120587
3)
sin 120579119891
sin(120579119891minus2120587
3) sin(120579
119891+2120587
3)
]]]
]
(29)
where 119894119886119904 119894119887119904 and 119894
119888119904are the three phase stator currents Note
that the elements in the 119879 matrix are cosinusoidal functionsof electrical angle 120579
119891 The electrical field angle in this case is
that of the rotor flux-linkages phasor and is obtained as thesum of the rotor and slip angles as follows
120579119891= 120579119903+ 120579sl (30)
and the slip angle is obtained by integrating the slip speed andis given as
120579sl = int120596sl (31)
IFO controller developed from these derivations accepts thetorque and flux requests and generates the torque and fluxproducing components of the stator-current phasor and theslip-angle commandsThe command values are denoted withasterisks throughout this thesis From (20) (21) and (23) thecommand values of 119894
119879 119894119891 and 120596sl are obtained as follows
119894lowast
119879=
119879lowast
119890
119870119905119890120582lowast119903
= (2
3) (
2
119875)119879lowast
119890
120582lowast119903
119871119903
119871119898
(32)
119894lowast
119891= (1 + 119901
119871119903
119877119903
)120582lowast
119903
119871119898
(33)
120596lowast
sl =119877119903119871119898
119871119903
119894lowast
119879
120582lowast119903
(34)
8 Journal of Applied Mathematics
The command slip angle 120579lowastsl is generated by integrating 120596lowastsl The torque angle command is obtained as the arctangentof 119894lowast119879and 119894lowast119891 The field angle is obtained by summing the
command slip angle and rotor angleWith the torque and fluxproducing components of the stator current commands androtor field angle the 119902119889 axes current commands (119886119887119888 phasecurrent commands) are obtained as follows The relevantsteps involved in the realization of the IFO controller are asfollows
[
[
119894lowast
119902119904
119894lowast
119889119904
]
]
= [
[
cos 120579119891
sin 120579119891
minus sin 120579119891
cos 120579119891
]
]
[
[
119894lowast
119879
119894lowast
119891
]
]
(35)
[[[[
[
119894lowast
119886119904
119894lowast
119887119904
119894lowast
119888119904
]]]]
]
= [119879minus1
]
[[[[
[
119894lowast
119902119904
119894lowast
119889119904
0
]]]]
]
(36)
where
119879minus1
=
[[[[[[[
[
1 0 1
minus1
2minusradic3
21
minus1
2
radic3
21
]]]]]]]
]
(37)
By using (35) to (37) the stator 119902- and 119889-axes and abccurrent commands are derived as
119894lowast
119902119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119889119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 cos 120579lowast
119904
119894lowast
119886119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119887119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904minus2120587
3)
119894lowast
119888119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904+2120587
3)
(38)
where
120579lowast
119904= 120579119891+ 120579lowast
119879= 120579119903+ 120579lowast
sl + 120579lowast
119879 (39)
The implementation of IFOC on a CSI fed IM is shownin Figures 8 and 9 The torque command 119894lowast
119879is generated
as a function of the speed error signal generally processedthrough a PI controller The flux command 119894lowast
119891can be given
directly or as a function of speed The rotor position 120579119903can
be measured with an encoder and converted into necessarydigital information for feedbackThere has been a substantialamount of research in the development of rotor flux observersfor field orientation that are compensated for variationsin parameters by their feedback corrections Digital imple-mentation of integrators for the estimation of rotor flux ofan IM from the stator voltages and stator currents posesproblems associated with the offset in the sensor amplifiers
CSI
IM
PWMswitching
logic
Rotor speed
regulator
Stator current
generator
DC link currentias
lowast
ibslowast
i cslowast
120579riTlowast is
iflowast
iflowast and iT
lowast120582rlowast
120596rlowast
120596sllowast
120596r
Telowast
(1 + pLr
Rr
) 120582rlowast
Lm
RrLm
Lr
iTlowast
120582rlowast
4
3P
Telowast
120582rlowast
Lr
Lm
+minus
Figure 8 Simplified diagram for IFOC method
islowast = radic(iTlowast)2 +(iflowast)2 ias
lowast = |islowast|sin120579s
lowast|islowast|
ibslowast = |is
lowast|sin120579slowast minus
2120587
3
icslowast = |is
lowast|sin120579slowast +
2120587
3
120579r 120579r
iTlowast
iflowast
120579Tlowast
120579slowast
120596sllowast 120579sl
lowastint
iaslowast
ibslowast
i cslowast
tanminus1 iTlowast
iflowast
+++
Figure 9 Stator current generator
Traditional low-pass filters can replace the integrator Soin this work rotor flux estimated using low pass filter isdescribed The instantaneous flux linkage can be computedusing the measured d-axis stator current using (40) which isreferred to as the current model as follows
120582119903=119871119898119894119889119904
1 + 120591119903119904 (40)
The theoretical concepts of IFOC method for CSI fed IMdrive discussed in Sections 51 and 52 are verified throughsimulation using the Simulink toolbox of MATLAB Thedetailed simulations results are presented in Section 71
6 DTC for CSI Fed IM Drives
Variable speed drives are used in all industries to controlprecisely the speed of electric motors driving loads rangingfrom pumps and fans to complex drives on paper machinesrolling mill cranes and similar drives The most modernmethod for these drives is direct torque and stator flux vectorcontrol method usually called DTC The VSI fed version hasbeen realized in an industrial way by ABB by using thetheoretical background proposed by Takahashi and Ohmori[22] and Depenbrock [23] This solution is based both onfield oriented control (FOC) as well as on the direct self-control theory The idea is that motor flux and torque areused as primary control variables which is contrary to thewayin which traditional AC drives control input frequency andvoltage but is in principle similar to what is done with a DCdrive where it is much more straightforward to achieve
Journal of Applied Mathematics 9
Casadei et al [24] has described that by controllingmotor torque directly DTCprovides dynamic speed accuracyequivalent to closed loop AC and DC systems and torqueresponse times that are 10 times faster It is also claimedthat the DTC does not generate noise like that produced byconventional PWM AC drives This work describes a newmodular approach to implement DTC method on a CSIfed IM drive the simulation implementation procedure Thedynamic response of the drive is analyzed and presented
61 Mathematical Modeling of DTC Control Method TheDTC method is developed from Babaei and Heydari [10]From Krause et al [15] in the stationary reference frame thestator voltage and flux equation can be written as
V119904= 119877119904119894119904+119889120582119904
119889119905 (41)
120582119904= int (V
119904minus 119877119904119894119904) 119889119905 = 120582
120572119904+ 119895120582120573119904 (42)
10038161003816100381610038161205821199041003816100381610038161003816 = radic120582
2
120572119904+ 1205822
120573119904 (43)
ang120593119904= tanminus1 (
120582120573119904
120582120572119904
) (44)
where 119877119904is the stator resistance and 120582
119904is the stator flux
vector in the stationary reference frame |120582119904| and ang120593
119904are the
amplitude and position of stator flux vector respectivelyIn DTC of CSI fed IM it is desirable to determine the
stator current reference so that the torque and stator fluxfollow their reference values Babaei and Heydari [10] Theproposed DTC system is based on the stator flux oriented(SFO) reference frame In this rotating reference frame it iswritten as
120582119902119904= 0 120582
119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)
where 119889 and 119902 are the real and imaginary axes in the SFOreference frame Therefore the torque equation is rewrittenas
119879119890=3
2(119875
2)1
120596119887
(120582119889119904119894119902119904) (46)
where 119894119902119904is the imaginary part of the stator current vector
in the SFO reference frame Therefore if |120582lowast119904| and 119879lowast
119890are
stator flux magnitude and torque references respectively theimaginary part of the stator current reference vector whichleads to torque and stator flux magnitude references is asfollows
119894lowast
119902119904=2
3
1
119875
119879lowast
119890
1003816100381610038161003816120582lowast
119904
1003816100381610038161003816
(47)
Because of the existence of a capacitor in the controlsystem and its corresponding losses when (47) is used tocontrol the torque for all rotor speeds the control system willnot be accurate enough
Currentlimiter
Telowast
Te
|120582slowast|
2
3
1
P
PI
Telowast
|120582slowast |
+
+
+minus iqs
lowast
Figure 10 119902-axis reference current calculator
In order to have amore accurate torque control the blockdiagram shown in Figure 10 is used to calculate 119894lowast
119902119904This block
diagram is a combination of (47) and a simple PI controllerThis controller removes the steady-state error and leads to anaccurate control The coefficients of this controller must besmall enough so that it remains stable and the ripples in 119894lowast
119902119904
are limited because a ripple of 119894lowast119902119904results in a ripple of the
generated torque If only a PI is used 119894lowast119902119904
will have a largevalue of ripple and the torque response time will be verylargeTherefore in order to have a faster transient response oftorque it is necessary to use (47) In DTC method with VSIthe real part of voltage vector (V
119889) in the stator flux reference
frame leads to direct control of stator flux magnitudeTherefore using the real part of the stator current vector
in the stator flux reference frame it is possible to control theflux magnitude indirectly In order to obtain the real part ofthe stator current reference a simple PI controller is usedTheinput of this PI controller is the error that resulted from thecomparison between the reference stator flux magnitude andthe estimated flux value In order to control themotor currentin all cases especially in the starting mode the output ofthis PI controller must be limited between proper valuesTheoutput signal of the PI controller is the real part of referencestator current 119894lowast
119889119904
Therefore the stator current reference vector that leads tocontrol the IM stator flux and torque is as follows
119894lowast
119904(119889119902)= 119894lowast
119889119904+ 119895119894lowast
119902119904 (48)
1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 =radic119894lowast2
119889119904+ 119894lowast2119902119904 (49)
The obtained reference current vector is in the statorflux reference frame In order to generate it by SVM of theCSI it must be transferred to the stationary reference frameTo do that the stator flux position is used to transfer thisvector from the stator flux reference frame to the stationaryreference frame as follows
119894lowast
119904(120572120573)= 119894lowast
120572119904+ 119895119894lowast
120573119904= (119894lowast
119889119904+ 119895119894lowast
119902119904) sdot 119890119895120593119904
=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 ang (120593119904 + tanminus1 (119894lowast
119902119904
119894lowast
119889119904
))
(50)
This reference current vectormust be generated in theCSIfirst and then applied to the motor after removing its har-monics with filtering capacitorThis current has a nearly sinu-soidal waveform that leads to a nearly sinusoidal voltage inmotor terminals after feeding the motor and passing through
10 Journal of Applied Mathematics
Stator flux torque
estimator
PI
CSI
SVM IM
Gate Signals
PI
120582slowast
120591elowast
120582s
120582s
120591e
ang120593s
120591e
iqlowast
idlowast
dq
to120572120573
iaib
vb
va
|islowast| and idc
lowast
idclowast
+minus
+minus i120572
lowast
i120573lowast
Figure 11 Block diagram schematic of the proposed DTC
motor impedances The speed and stator flux control of anIM with DTC method is shown in Figure 11 Separate PIcontrollers are used to control the speed and flux The inputto the speed controller is the error caused by comparingthe reference speed and the rotor speed The output of thiscontroller is the imaginary part of stator current vector 119894lowast
119902119904
which is limited to a proper value
7 Results and Discussion
71 Simulation Results The simulationmodels of IOL IFOCand DTC control methods for CSI fed IM drive systemsare developed and simulated using MATLAB simulationsoftware From the simulation results dynamic performancesof the developed drives have been individually analyzedmainly using the sensitivity tests Here the results of all thedrives are compared with one another The tests that arechosen for the comparison are step changes in the referencespeed and torque of the motor drive
The first test is step change in reference speed which isuseful when trying to change from one speed to anotherThe next test is step change in reference torque that wouldbe useful in identifying the controller with a high torqueresponse Change in speed reference is the first test that isdone to compare the motor drives
The results of this test can be seen in the Figures 12 13 and14The speed command is varied from0 to its rated valueThespeed response of IOL drive is faster and has the settling timeof 05 second range while DTC and IFOC drives performsimilarly but slower than IOL (settling time of less than 1second)
The second test as mentioned earlier is the torque steptestThe results of this test can be seen in Figures 12 to 14Thetorque reference is varied from 0 to its rated value The speedresponse of IFOC drive is faster and has the settling timeof 02 seconds while IOL and DTC drives perform similarly(settling time of less than 05 second)
During this test the motor torque response of DTCdrive is almost instantaneous within the 02 second rangewhile IFOC drive closely follows behind it IOL and vectorbased 119881119891 have slower torque response than all other VectorControl techniques
Table 1 Prototype motor parameters
1 Hp 3 phase star connected 4 pole 415V 18 A 50Hz
Stator resistance 0087Ω Stator amp rotorleakage reactance 08119890minus3H
Rotor resistance 0228Ω Magnetizingreactance 347119890minus3H
72 Experimental Results Brief experimental results arepresented for IOL method and DTC method usingTMS320F2812 DSP based hardware setup Figure 15 showsthe schematic diagram for the hardware setup of theproposed laboratory prototype CSI fed induction motordrive The induction motor ratings are given in Table 1 Theinverter consists of CM75TL-12NF The gate driving signalis developed by using Texas instruments DSP processorTMS320F2812 DSP The software used to develop programsfor TMS320F2812 DSP is by code composer studio andMatlabSimulinkThe processor contains the program that isdownloaded to the computer
The clock speed of the DSP is 150MHz and it is capableof 32-bit operations The on board available flash memoryis 2048Mb It was created specifically for motor controloperation and therefore Parkrsquos and Clarkrsquos transformationsare conveniently built in Another convenient feature is thatit has sixteen 12-bit ADC pins that allow for a high degree ofprecision while taking many possible measurements
The above experimental result of IOL method of vectorcontrol technique in Figure 16 shows the fast settling timewith feed forward control compared to feedback techniqueSimilarly the experimental result shown in Figure 17 depictsa greater torque response within 02 seconds comparedto other vector control techniques With this experimentalverification it is clear that IOL is adaptive for speed responseand DTC is more suitable for torque response controlleddrives
From Tables 2 and 3 it seems that the dynamic speed andtorque performance of all four drives using change in speedreference It is evident from the table that IOL drive hasthe best speed response with DTC and IFOC occupyingpositions next to it It is also evident that DTC drive has thebest instantaneous torque response with IFOC and IOL nextto it
8 Conclusion
This paper presents a new approach in mathematical mod-eling of a 3-phase induction motor drive Mathematics con-cepts are used in developing the simulation and implementa-tion of IOL IFOC and DTC methods for CSI fed IM drivesIn this paper a rotor flux based reference frame controlmethod for PWM CSI fed IM with simplified simulationcircuits has been developed in which the motor speed andtorque follow the references closely in all the three vectorcontrol methods However a CSI fed induction motor drivehas some drawbacks such as slower dynamic response andbulky overall size due to the requirement of large smoothing
Journal of Applied Mathematics 11
15
minus05
0
05
1
Spee
d (P
U)
0 1 2 3 4 5Time (s)
Without FF controlWith FF control
(a)
0 1 2 3 4 5
minus2
0
2
Time (s)
Torq
ue (P
U)
Without FF controlWith FF control
(b)
Figure 12 Speed torque response of IOL control method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus4
minus2
0
24
Torq
ue (P
U)
Time (s)
Reference speedStep increase in speed command
Step decrease in speed command
Step increase in torque command
Step decrease in torque command Reference torque
Motor torque
Motor speed
Response due to change intorque reference
Response due to change in
speed reference
Figure 13 Speed torque response of IFOC method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus2minus1
012
Time (s)
Torq
ue (P
U)
Step decrease in speed command
Step increase in speed commandReference speed Motor speed
Response due to change intorque reference
Response due to change inspeed reference
Motor torque
Step decrease in torque commandReference torque
Step increase in torque command
Figure 14 Speed torque response of DTC method
inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying
TMS320F2812DSP
ADC
PWM
-EV
B
PWM
-EVA
Supplyfilter
3120601controlled
rectifier3120601 CSI inverter
3120601IM
3120601supply
Texasinstruments
DSP processor
Speed sensor output
(Nr) Rpm
CM75TL-12NFseries
IGBT (6-PAC)
Ldc
Is (Amps)
Figure 15 Schematic diagram of the hardware setup of CSI fed IM
Torque output with feedforward control
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Torque output withoutfeed forward control
Speed output with feedforward control
Speed output withoutfeed forward control
Figure 16 Experimental result torque-speed response of IOLcontrol method
applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Applied Mathematics
The command slip angle 120579lowastsl is generated by integrating 120596lowastsl The torque angle command is obtained as the arctangentof 119894lowast119879and 119894lowast119891 The field angle is obtained by summing the
command slip angle and rotor angleWith the torque and fluxproducing components of the stator current commands androtor field angle the 119902119889 axes current commands (119886119887119888 phasecurrent commands) are obtained as follows The relevantsteps involved in the realization of the IFO controller are asfollows
[
[
119894lowast
119902119904
119894lowast
119889119904
]
]
= [
[
cos 120579119891
sin 120579119891
minus sin 120579119891
cos 120579119891
]
]
[
[
119894lowast
119879
119894lowast
119891
]
]
(35)
[[[[
[
119894lowast
119886119904
119894lowast
119887119904
119894lowast
119888119904
]]]]
]
= [119879minus1
]
[[[[
[
119894lowast
119902119904
119894lowast
119889119904
0
]]]]
]
(36)
where
119879minus1
=
[[[[[[[
[
1 0 1
minus1
2minusradic3
21
minus1
2
radic3
21
]]]]]]]
]
(37)
By using (35) to (37) the stator 119902- and 119889-axes and abccurrent commands are derived as
119894lowast
119902119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119889119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 cos 120579lowast
119904
119894lowast
119886119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin 120579lowast
119904
119894lowast
119887119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904minus2120587
3)
119894lowast
119888119904=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 sin(120579lowast
119904+2120587
3)
(38)
where
120579lowast
119904= 120579119891+ 120579lowast
119879= 120579119903+ 120579lowast
sl + 120579lowast
119879 (39)
The implementation of IFOC on a CSI fed IM is shownin Figures 8 and 9 The torque command 119894lowast
119879is generated
as a function of the speed error signal generally processedthrough a PI controller The flux command 119894lowast
119891can be given
directly or as a function of speed The rotor position 120579119903can
be measured with an encoder and converted into necessarydigital information for feedbackThere has been a substantialamount of research in the development of rotor flux observersfor field orientation that are compensated for variationsin parameters by their feedback corrections Digital imple-mentation of integrators for the estimation of rotor flux ofan IM from the stator voltages and stator currents posesproblems associated with the offset in the sensor amplifiers
CSI
IM
PWMswitching
logic
Rotor speed
regulator
Stator current
generator
DC link currentias
lowast
ibslowast
i cslowast
120579riTlowast is
iflowast
iflowast and iT
lowast120582rlowast
120596rlowast
120596sllowast
120596r
Telowast
(1 + pLr
Rr
) 120582rlowast
Lm
RrLm
Lr
iTlowast
120582rlowast
4
3P
Telowast
120582rlowast
Lr
Lm
+minus
Figure 8 Simplified diagram for IFOC method
islowast = radic(iTlowast)2 +(iflowast)2 ias
lowast = |islowast|sin120579s
lowast|islowast|
ibslowast = |is
lowast|sin120579slowast minus
2120587
3
icslowast = |is
lowast|sin120579slowast +
2120587
3
120579r 120579r
iTlowast
iflowast
120579Tlowast
120579slowast
120596sllowast 120579sl
lowastint
iaslowast
ibslowast
i cslowast
tanminus1 iTlowast
iflowast
+++
Figure 9 Stator current generator
Traditional low-pass filters can replace the integrator Soin this work rotor flux estimated using low pass filter isdescribed The instantaneous flux linkage can be computedusing the measured d-axis stator current using (40) which isreferred to as the current model as follows
120582119903=119871119898119894119889119904
1 + 120591119903119904 (40)
The theoretical concepts of IFOC method for CSI fed IMdrive discussed in Sections 51 and 52 are verified throughsimulation using the Simulink toolbox of MATLAB Thedetailed simulations results are presented in Section 71
6 DTC for CSI Fed IM Drives
Variable speed drives are used in all industries to controlprecisely the speed of electric motors driving loads rangingfrom pumps and fans to complex drives on paper machinesrolling mill cranes and similar drives The most modernmethod for these drives is direct torque and stator flux vectorcontrol method usually called DTC The VSI fed version hasbeen realized in an industrial way by ABB by using thetheoretical background proposed by Takahashi and Ohmori[22] and Depenbrock [23] This solution is based both onfield oriented control (FOC) as well as on the direct self-control theory The idea is that motor flux and torque areused as primary control variables which is contrary to thewayin which traditional AC drives control input frequency andvoltage but is in principle similar to what is done with a DCdrive where it is much more straightforward to achieve
Journal of Applied Mathematics 9
Casadei et al [24] has described that by controllingmotor torque directly DTCprovides dynamic speed accuracyequivalent to closed loop AC and DC systems and torqueresponse times that are 10 times faster It is also claimedthat the DTC does not generate noise like that produced byconventional PWM AC drives This work describes a newmodular approach to implement DTC method on a CSIfed IM drive the simulation implementation procedure Thedynamic response of the drive is analyzed and presented
61 Mathematical Modeling of DTC Control Method TheDTC method is developed from Babaei and Heydari [10]From Krause et al [15] in the stationary reference frame thestator voltage and flux equation can be written as
V119904= 119877119904119894119904+119889120582119904
119889119905 (41)
120582119904= int (V
119904minus 119877119904119894119904) 119889119905 = 120582
120572119904+ 119895120582120573119904 (42)
10038161003816100381610038161205821199041003816100381610038161003816 = radic120582
2
120572119904+ 1205822
120573119904 (43)
ang120593119904= tanminus1 (
120582120573119904
120582120572119904
) (44)
where 119877119904is the stator resistance and 120582
119904is the stator flux
vector in the stationary reference frame |120582119904| and ang120593
119904are the
amplitude and position of stator flux vector respectivelyIn DTC of CSI fed IM it is desirable to determine the
stator current reference so that the torque and stator fluxfollow their reference values Babaei and Heydari [10] Theproposed DTC system is based on the stator flux oriented(SFO) reference frame In this rotating reference frame it iswritten as
120582119902119904= 0 120582
119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)
where 119889 and 119902 are the real and imaginary axes in the SFOreference frame Therefore the torque equation is rewrittenas
119879119890=3
2(119875
2)1
120596119887
(120582119889119904119894119902119904) (46)
where 119894119902119904is the imaginary part of the stator current vector
in the SFO reference frame Therefore if |120582lowast119904| and 119879lowast
119890are
stator flux magnitude and torque references respectively theimaginary part of the stator current reference vector whichleads to torque and stator flux magnitude references is asfollows
119894lowast
119902119904=2
3
1
119875
119879lowast
119890
1003816100381610038161003816120582lowast
119904
1003816100381610038161003816
(47)
Because of the existence of a capacitor in the controlsystem and its corresponding losses when (47) is used tocontrol the torque for all rotor speeds the control system willnot be accurate enough
Currentlimiter
Telowast
Te
|120582slowast|
2
3
1
P
PI
Telowast
|120582slowast |
+
+
+minus iqs
lowast
Figure 10 119902-axis reference current calculator
In order to have amore accurate torque control the blockdiagram shown in Figure 10 is used to calculate 119894lowast
119902119904This block
diagram is a combination of (47) and a simple PI controllerThis controller removes the steady-state error and leads to anaccurate control The coefficients of this controller must besmall enough so that it remains stable and the ripples in 119894lowast
119902119904
are limited because a ripple of 119894lowast119902119904results in a ripple of the
generated torque If only a PI is used 119894lowast119902119904
will have a largevalue of ripple and the torque response time will be verylargeTherefore in order to have a faster transient response oftorque it is necessary to use (47) In DTC method with VSIthe real part of voltage vector (V
119889) in the stator flux reference
frame leads to direct control of stator flux magnitudeTherefore using the real part of the stator current vector
in the stator flux reference frame it is possible to control theflux magnitude indirectly In order to obtain the real part ofthe stator current reference a simple PI controller is usedTheinput of this PI controller is the error that resulted from thecomparison between the reference stator flux magnitude andthe estimated flux value In order to control themotor currentin all cases especially in the starting mode the output ofthis PI controller must be limited between proper valuesTheoutput signal of the PI controller is the real part of referencestator current 119894lowast
119889119904
Therefore the stator current reference vector that leads tocontrol the IM stator flux and torque is as follows
119894lowast
119904(119889119902)= 119894lowast
119889119904+ 119895119894lowast
119902119904 (48)
1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 =radic119894lowast2
119889119904+ 119894lowast2119902119904 (49)
The obtained reference current vector is in the statorflux reference frame In order to generate it by SVM of theCSI it must be transferred to the stationary reference frameTo do that the stator flux position is used to transfer thisvector from the stator flux reference frame to the stationaryreference frame as follows
119894lowast
119904(120572120573)= 119894lowast
120572119904+ 119895119894lowast
120573119904= (119894lowast
119889119904+ 119895119894lowast
119902119904) sdot 119890119895120593119904
=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 ang (120593119904 + tanminus1 (119894lowast
119902119904
119894lowast
119889119904
))
(50)
This reference current vectormust be generated in theCSIfirst and then applied to the motor after removing its har-monics with filtering capacitorThis current has a nearly sinu-soidal waveform that leads to a nearly sinusoidal voltage inmotor terminals after feeding the motor and passing through
10 Journal of Applied Mathematics
Stator flux torque
estimator
PI
CSI
SVM IM
Gate Signals
PI
120582slowast
120591elowast
120582s
120582s
120591e
ang120593s
120591e
iqlowast
idlowast
dq
to120572120573
iaib
vb
va
|islowast| and idc
lowast
idclowast
+minus
+minus i120572
lowast
i120573lowast
Figure 11 Block diagram schematic of the proposed DTC
motor impedances The speed and stator flux control of anIM with DTC method is shown in Figure 11 Separate PIcontrollers are used to control the speed and flux The inputto the speed controller is the error caused by comparingthe reference speed and the rotor speed The output of thiscontroller is the imaginary part of stator current vector 119894lowast
119902119904
which is limited to a proper value
7 Results and Discussion
71 Simulation Results The simulationmodels of IOL IFOCand DTC control methods for CSI fed IM drive systemsare developed and simulated using MATLAB simulationsoftware From the simulation results dynamic performancesof the developed drives have been individually analyzedmainly using the sensitivity tests Here the results of all thedrives are compared with one another The tests that arechosen for the comparison are step changes in the referencespeed and torque of the motor drive
The first test is step change in reference speed which isuseful when trying to change from one speed to anotherThe next test is step change in reference torque that wouldbe useful in identifying the controller with a high torqueresponse Change in speed reference is the first test that isdone to compare the motor drives
The results of this test can be seen in the Figures 12 13 and14The speed command is varied from0 to its rated valueThespeed response of IOL drive is faster and has the settling timeof 05 second range while DTC and IFOC drives performsimilarly but slower than IOL (settling time of less than 1second)
The second test as mentioned earlier is the torque steptestThe results of this test can be seen in Figures 12 to 14Thetorque reference is varied from 0 to its rated value The speedresponse of IFOC drive is faster and has the settling timeof 02 seconds while IOL and DTC drives perform similarly(settling time of less than 05 second)
During this test the motor torque response of DTCdrive is almost instantaneous within the 02 second rangewhile IFOC drive closely follows behind it IOL and vectorbased 119881119891 have slower torque response than all other VectorControl techniques
Table 1 Prototype motor parameters
1 Hp 3 phase star connected 4 pole 415V 18 A 50Hz
Stator resistance 0087Ω Stator amp rotorleakage reactance 08119890minus3H
Rotor resistance 0228Ω Magnetizingreactance 347119890minus3H
72 Experimental Results Brief experimental results arepresented for IOL method and DTC method usingTMS320F2812 DSP based hardware setup Figure 15 showsthe schematic diagram for the hardware setup of theproposed laboratory prototype CSI fed induction motordrive The induction motor ratings are given in Table 1 Theinverter consists of CM75TL-12NF The gate driving signalis developed by using Texas instruments DSP processorTMS320F2812 DSP The software used to develop programsfor TMS320F2812 DSP is by code composer studio andMatlabSimulinkThe processor contains the program that isdownloaded to the computer
The clock speed of the DSP is 150MHz and it is capableof 32-bit operations The on board available flash memoryis 2048Mb It was created specifically for motor controloperation and therefore Parkrsquos and Clarkrsquos transformationsare conveniently built in Another convenient feature is thatit has sixteen 12-bit ADC pins that allow for a high degree ofprecision while taking many possible measurements
The above experimental result of IOL method of vectorcontrol technique in Figure 16 shows the fast settling timewith feed forward control compared to feedback techniqueSimilarly the experimental result shown in Figure 17 depictsa greater torque response within 02 seconds comparedto other vector control techniques With this experimentalverification it is clear that IOL is adaptive for speed responseand DTC is more suitable for torque response controlleddrives
From Tables 2 and 3 it seems that the dynamic speed andtorque performance of all four drives using change in speedreference It is evident from the table that IOL drive hasthe best speed response with DTC and IFOC occupyingpositions next to it It is also evident that DTC drive has thebest instantaneous torque response with IFOC and IOL nextto it
8 Conclusion
This paper presents a new approach in mathematical mod-eling of a 3-phase induction motor drive Mathematics con-cepts are used in developing the simulation and implementa-tion of IOL IFOC and DTC methods for CSI fed IM drivesIn this paper a rotor flux based reference frame controlmethod for PWM CSI fed IM with simplified simulationcircuits has been developed in which the motor speed andtorque follow the references closely in all the three vectorcontrol methods However a CSI fed induction motor drivehas some drawbacks such as slower dynamic response andbulky overall size due to the requirement of large smoothing
Journal of Applied Mathematics 11
15
minus05
0
05
1
Spee
d (P
U)
0 1 2 3 4 5Time (s)
Without FF controlWith FF control
(a)
0 1 2 3 4 5
minus2
0
2
Time (s)
Torq
ue (P
U)
Without FF controlWith FF control
(b)
Figure 12 Speed torque response of IOL control method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus4
minus2
0
24
Torq
ue (P
U)
Time (s)
Reference speedStep increase in speed command
Step decrease in speed command
Step increase in torque command
Step decrease in torque command Reference torque
Motor torque
Motor speed
Response due to change intorque reference
Response due to change in
speed reference
Figure 13 Speed torque response of IFOC method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus2minus1
012
Time (s)
Torq
ue (P
U)
Step decrease in speed command
Step increase in speed commandReference speed Motor speed
Response due to change intorque reference
Response due to change inspeed reference
Motor torque
Step decrease in torque commandReference torque
Step increase in torque command
Figure 14 Speed torque response of DTC method
inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying
TMS320F2812DSP
ADC
PWM
-EV
B
PWM
-EVA
Supplyfilter
3120601controlled
rectifier3120601 CSI inverter
3120601IM
3120601supply
Texasinstruments
DSP processor
Speed sensor output
(Nr) Rpm
CM75TL-12NFseries
IGBT (6-PAC)
Ldc
Is (Amps)
Figure 15 Schematic diagram of the hardware setup of CSI fed IM
Torque output with feedforward control
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Torque output withoutfeed forward control
Speed output with feedforward control
Speed output withoutfeed forward control
Figure 16 Experimental result torque-speed response of IOLcontrol method
applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 9
Casadei et al [24] has described that by controllingmotor torque directly DTCprovides dynamic speed accuracyequivalent to closed loop AC and DC systems and torqueresponse times that are 10 times faster It is also claimedthat the DTC does not generate noise like that produced byconventional PWM AC drives This work describes a newmodular approach to implement DTC method on a CSIfed IM drive the simulation implementation procedure Thedynamic response of the drive is analyzed and presented
61 Mathematical Modeling of DTC Control Method TheDTC method is developed from Babaei and Heydari [10]From Krause et al [15] in the stationary reference frame thestator voltage and flux equation can be written as
V119904= 119877119904119894119904+119889120582119904
119889119905 (41)
120582119904= int (V
119904minus 119877119904119894119904) 119889119905 = 120582
120572119904+ 119895120582120573119904 (42)
10038161003816100381610038161205821199041003816100381610038161003816 = radic120582
2
120572119904+ 1205822
120573119904 (43)
ang120593119904= tanminus1 (
120582120573119904
120582120572119904
) (44)
where 119877119904is the stator resistance and 120582
119904is the stator flux
vector in the stationary reference frame |120582119904| and ang120593
119904are the
amplitude and position of stator flux vector respectivelyIn DTC of CSI fed IM it is desirable to determine the
stator current reference so that the torque and stator fluxfollow their reference values Babaei and Heydari [10] Theproposed DTC system is based on the stator flux oriented(SFO) reference frame In this rotating reference frame it iswritten as
120582119902119904= 0 120582
119889119904=10038161003816100381610038161205821199041003816100381610038161003816 ang120593119904 (45)
where 119889 and 119902 are the real and imaginary axes in the SFOreference frame Therefore the torque equation is rewrittenas
119879119890=3
2(119875
2)1
120596119887
(120582119889119904119894119902119904) (46)
where 119894119902119904is the imaginary part of the stator current vector
in the SFO reference frame Therefore if |120582lowast119904| and 119879lowast
119890are
stator flux magnitude and torque references respectively theimaginary part of the stator current reference vector whichleads to torque and stator flux magnitude references is asfollows
119894lowast
119902119904=2
3
1
119875
119879lowast
119890
1003816100381610038161003816120582lowast
119904
1003816100381610038161003816
(47)
Because of the existence of a capacitor in the controlsystem and its corresponding losses when (47) is used tocontrol the torque for all rotor speeds the control system willnot be accurate enough
Currentlimiter
Telowast
Te
|120582slowast|
2
3
1
P
PI
Telowast
|120582slowast |
+
+
+minus iqs
lowast
Figure 10 119902-axis reference current calculator
In order to have amore accurate torque control the blockdiagram shown in Figure 10 is used to calculate 119894lowast
119902119904This block
diagram is a combination of (47) and a simple PI controllerThis controller removes the steady-state error and leads to anaccurate control The coefficients of this controller must besmall enough so that it remains stable and the ripples in 119894lowast
119902119904
are limited because a ripple of 119894lowast119902119904results in a ripple of the
generated torque If only a PI is used 119894lowast119902119904
will have a largevalue of ripple and the torque response time will be verylargeTherefore in order to have a faster transient response oftorque it is necessary to use (47) In DTC method with VSIthe real part of voltage vector (V
119889) in the stator flux reference
frame leads to direct control of stator flux magnitudeTherefore using the real part of the stator current vector
in the stator flux reference frame it is possible to control theflux magnitude indirectly In order to obtain the real part ofthe stator current reference a simple PI controller is usedTheinput of this PI controller is the error that resulted from thecomparison between the reference stator flux magnitude andthe estimated flux value In order to control themotor currentin all cases especially in the starting mode the output ofthis PI controller must be limited between proper valuesTheoutput signal of the PI controller is the real part of referencestator current 119894lowast
119889119904
Therefore the stator current reference vector that leads tocontrol the IM stator flux and torque is as follows
119894lowast
119904(119889119902)= 119894lowast
119889119904+ 119895119894lowast
119902119904 (48)
1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 =radic119894lowast2
119889119904+ 119894lowast2119902119904 (49)
The obtained reference current vector is in the statorflux reference frame In order to generate it by SVM of theCSI it must be transferred to the stationary reference frameTo do that the stator flux position is used to transfer thisvector from the stator flux reference frame to the stationaryreference frame as follows
119894lowast
119904(120572120573)= 119894lowast
120572119904+ 119895119894lowast
120573119904= (119894lowast
119889119904+ 119895119894lowast
119902119904) sdot 119890119895120593119904
=1003816100381610038161003816119894lowast
119904
1003816100381610038161003816 ang (120593119904 + tanminus1 (119894lowast
119902119904
119894lowast
119889119904
))
(50)
This reference current vectormust be generated in theCSIfirst and then applied to the motor after removing its har-monics with filtering capacitorThis current has a nearly sinu-soidal waveform that leads to a nearly sinusoidal voltage inmotor terminals after feeding the motor and passing through
10 Journal of Applied Mathematics
Stator flux torque
estimator
PI
CSI
SVM IM
Gate Signals
PI
120582slowast
120591elowast
120582s
120582s
120591e
ang120593s
120591e
iqlowast
idlowast
dq
to120572120573
iaib
vb
va
|islowast| and idc
lowast
idclowast
+minus
+minus i120572
lowast
i120573lowast
Figure 11 Block diagram schematic of the proposed DTC
motor impedances The speed and stator flux control of anIM with DTC method is shown in Figure 11 Separate PIcontrollers are used to control the speed and flux The inputto the speed controller is the error caused by comparingthe reference speed and the rotor speed The output of thiscontroller is the imaginary part of stator current vector 119894lowast
119902119904
which is limited to a proper value
7 Results and Discussion
71 Simulation Results The simulationmodels of IOL IFOCand DTC control methods for CSI fed IM drive systemsare developed and simulated using MATLAB simulationsoftware From the simulation results dynamic performancesof the developed drives have been individually analyzedmainly using the sensitivity tests Here the results of all thedrives are compared with one another The tests that arechosen for the comparison are step changes in the referencespeed and torque of the motor drive
The first test is step change in reference speed which isuseful when trying to change from one speed to anotherThe next test is step change in reference torque that wouldbe useful in identifying the controller with a high torqueresponse Change in speed reference is the first test that isdone to compare the motor drives
The results of this test can be seen in the Figures 12 13 and14The speed command is varied from0 to its rated valueThespeed response of IOL drive is faster and has the settling timeof 05 second range while DTC and IFOC drives performsimilarly but slower than IOL (settling time of less than 1second)
The second test as mentioned earlier is the torque steptestThe results of this test can be seen in Figures 12 to 14Thetorque reference is varied from 0 to its rated value The speedresponse of IFOC drive is faster and has the settling timeof 02 seconds while IOL and DTC drives perform similarly(settling time of less than 05 second)
During this test the motor torque response of DTCdrive is almost instantaneous within the 02 second rangewhile IFOC drive closely follows behind it IOL and vectorbased 119881119891 have slower torque response than all other VectorControl techniques
Table 1 Prototype motor parameters
1 Hp 3 phase star connected 4 pole 415V 18 A 50Hz
Stator resistance 0087Ω Stator amp rotorleakage reactance 08119890minus3H
Rotor resistance 0228Ω Magnetizingreactance 347119890minus3H
72 Experimental Results Brief experimental results arepresented for IOL method and DTC method usingTMS320F2812 DSP based hardware setup Figure 15 showsthe schematic diagram for the hardware setup of theproposed laboratory prototype CSI fed induction motordrive The induction motor ratings are given in Table 1 Theinverter consists of CM75TL-12NF The gate driving signalis developed by using Texas instruments DSP processorTMS320F2812 DSP The software used to develop programsfor TMS320F2812 DSP is by code composer studio andMatlabSimulinkThe processor contains the program that isdownloaded to the computer
The clock speed of the DSP is 150MHz and it is capableof 32-bit operations The on board available flash memoryis 2048Mb It was created specifically for motor controloperation and therefore Parkrsquos and Clarkrsquos transformationsare conveniently built in Another convenient feature is thatit has sixteen 12-bit ADC pins that allow for a high degree ofprecision while taking many possible measurements
The above experimental result of IOL method of vectorcontrol technique in Figure 16 shows the fast settling timewith feed forward control compared to feedback techniqueSimilarly the experimental result shown in Figure 17 depictsa greater torque response within 02 seconds comparedto other vector control techniques With this experimentalverification it is clear that IOL is adaptive for speed responseand DTC is more suitable for torque response controlleddrives
From Tables 2 and 3 it seems that the dynamic speed andtorque performance of all four drives using change in speedreference It is evident from the table that IOL drive hasthe best speed response with DTC and IFOC occupyingpositions next to it It is also evident that DTC drive has thebest instantaneous torque response with IFOC and IOL nextto it
8 Conclusion
This paper presents a new approach in mathematical mod-eling of a 3-phase induction motor drive Mathematics con-cepts are used in developing the simulation and implementa-tion of IOL IFOC and DTC methods for CSI fed IM drivesIn this paper a rotor flux based reference frame controlmethod for PWM CSI fed IM with simplified simulationcircuits has been developed in which the motor speed andtorque follow the references closely in all the three vectorcontrol methods However a CSI fed induction motor drivehas some drawbacks such as slower dynamic response andbulky overall size due to the requirement of large smoothing
Journal of Applied Mathematics 11
15
minus05
0
05
1
Spee
d (P
U)
0 1 2 3 4 5Time (s)
Without FF controlWith FF control
(a)
0 1 2 3 4 5
minus2
0
2
Time (s)
Torq
ue (P
U)
Without FF controlWith FF control
(b)
Figure 12 Speed torque response of IOL control method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus4
minus2
0
24
Torq
ue (P
U)
Time (s)
Reference speedStep increase in speed command
Step decrease in speed command
Step increase in torque command
Step decrease in torque command Reference torque
Motor torque
Motor speed
Response due to change intorque reference
Response due to change in
speed reference
Figure 13 Speed torque response of IFOC method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus2minus1
012
Time (s)
Torq
ue (P
U)
Step decrease in speed command
Step increase in speed commandReference speed Motor speed
Response due to change intorque reference
Response due to change inspeed reference
Motor torque
Step decrease in torque commandReference torque
Step increase in torque command
Figure 14 Speed torque response of DTC method
inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying
TMS320F2812DSP
ADC
PWM
-EV
B
PWM
-EVA
Supplyfilter
3120601controlled
rectifier3120601 CSI inverter
3120601IM
3120601supply
Texasinstruments
DSP processor
Speed sensor output
(Nr) Rpm
CM75TL-12NFseries
IGBT (6-PAC)
Ldc
Is (Amps)
Figure 15 Schematic diagram of the hardware setup of CSI fed IM
Torque output with feedforward control
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Torque output withoutfeed forward control
Speed output with feedforward control
Speed output withoutfeed forward control
Figure 16 Experimental result torque-speed response of IOLcontrol method
applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Applied Mathematics
Stator flux torque
estimator
PI
CSI
SVM IM
Gate Signals
PI
120582slowast
120591elowast
120582s
120582s
120591e
ang120593s
120591e
iqlowast
idlowast
dq
to120572120573
iaib
vb
va
|islowast| and idc
lowast
idclowast
+minus
+minus i120572
lowast
i120573lowast
Figure 11 Block diagram schematic of the proposed DTC
motor impedances The speed and stator flux control of anIM with DTC method is shown in Figure 11 Separate PIcontrollers are used to control the speed and flux The inputto the speed controller is the error caused by comparingthe reference speed and the rotor speed The output of thiscontroller is the imaginary part of stator current vector 119894lowast
119902119904
which is limited to a proper value
7 Results and Discussion
71 Simulation Results The simulationmodels of IOL IFOCand DTC control methods for CSI fed IM drive systemsare developed and simulated using MATLAB simulationsoftware From the simulation results dynamic performancesof the developed drives have been individually analyzedmainly using the sensitivity tests Here the results of all thedrives are compared with one another The tests that arechosen for the comparison are step changes in the referencespeed and torque of the motor drive
The first test is step change in reference speed which isuseful when trying to change from one speed to anotherThe next test is step change in reference torque that wouldbe useful in identifying the controller with a high torqueresponse Change in speed reference is the first test that isdone to compare the motor drives
The results of this test can be seen in the Figures 12 13 and14The speed command is varied from0 to its rated valueThespeed response of IOL drive is faster and has the settling timeof 05 second range while DTC and IFOC drives performsimilarly but slower than IOL (settling time of less than 1second)
The second test as mentioned earlier is the torque steptestThe results of this test can be seen in Figures 12 to 14Thetorque reference is varied from 0 to its rated value The speedresponse of IFOC drive is faster and has the settling timeof 02 seconds while IOL and DTC drives perform similarly(settling time of less than 05 second)
During this test the motor torque response of DTCdrive is almost instantaneous within the 02 second rangewhile IFOC drive closely follows behind it IOL and vectorbased 119881119891 have slower torque response than all other VectorControl techniques
Table 1 Prototype motor parameters
1 Hp 3 phase star connected 4 pole 415V 18 A 50Hz
Stator resistance 0087Ω Stator amp rotorleakage reactance 08119890minus3H
Rotor resistance 0228Ω Magnetizingreactance 347119890minus3H
72 Experimental Results Brief experimental results arepresented for IOL method and DTC method usingTMS320F2812 DSP based hardware setup Figure 15 showsthe schematic diagram for the hardware setup of theproposed laboratory prototype CSI fed induction motordrive The induction motor ratings are given in Table 1 Theinverter consists of CM75TL-12NF The gate driving signalis developed by using Texas instruments DSP processorTMS320F2812 DSP The software used to develop programsfor TMS320F2812 DSP is by code composer studio andMatlabSimulinkThe processor contains the program that isdownloaded to the computer
The clock speed of the DSP is 150MHz and it is capableof 32-bit operations The on board available flash memoryis 2048Mb It was created specifically for motor controloperation and therefore Parkrsquos and Clarkrsquos transformationsare conveniently built in Another convenient feature is thatit has sixteen 12-bit ADC pins that allow for a high degree ofprecision while taking many possible measurements
The above experimental result of IOL method of vectorcontrol technique in Figure 16 shows the fast settling timewith feed forward control compared to feedback techniqueSimilarly the experimental result shown in Figure 17 depictsa greater torque response within 02 seconds comparedto other vector control techniques With this experimentalverification it is clear that IOL is adaptive for speed responseand DTC is more suitable for torque response controlleddrives
From Tables 2 and 3 it seems that the dynamic speed andtorque performance of all four drives using change in speedreference It is evident from the table that IOL drive hasthe best speed response with DTC and IFOC occupyingpositions next to it It is also evident that DTC drive has thebest instantaneous torque response with IFOC and IOL nextto it
8 Conclusion
This paper presents a new approach in mathematical mod-eling of a 3-phase induction motor drive Mathematics con-cepts are used in developing the simulation and implementa-tion of IOL IFOC and DTC methods for CSI fed IM drivesIn this paper a rotor flux based reference frame controlmethod for PWM CSI fed IM with simplified simulationcircuits has been developed in which the motor speed andtorque follow the references closely in all the three vectorcontrol methods However a CSI fed induction motor drivehas some drawbacks such as slower dynamic response andbulky overall size due to the requirement of large smoothing
Journal of Applied Mathematics 11
15
minus05
0
05
1
Spee
d (P
U)
0 1 2 3 4 5Time (s)
Without FF controlWith FF control
(a)
0 1 2 3 4 5
minus2
0
2
Time (s)
Torq
ue (P
U)
Without FF controlWith FF control
(b)
Figure 12 Speed torque response of IOL control method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus4
minus2
0
24
Torq
ue (P
U)
Time (s)
Reference speedStep increase in speed command
Step decrease in speed command
Step increase in torque command
Step decrease in torque command Reference torque
Motor torque
Motor speed
Response due to change intorque reference
Response due to change in
speed reference
Figure 13 Speed torque response of IFOC method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus2minus1
012
Time (s)
Torq
ue (P
U)
Step decrease in speed command
Step increase in speed commandReference speed Motor speed
Response due to change intorque reference
Response due to change inspeed reference
Motor torque
Step decrease in torque commandReference torque
Step increase in torque command
Figure 14 Speed torque response of DTC method
inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying
TMS320F2812DSP
ADC
PWM
-EV
B
PWM
-EVA
Supplyfilter
3120601controlled
rectifier3120601 CSI inverter
3120601IM
3120601supply
Texasinstruments
DSP processor
Speed sensor output
(Nr) Rpm
CM75TL-12NFseries
IGBT (6-PAC)
Ldc
Is (Amps)
Figure 15 Schematic diagram of the hardware setup of CSI fed IM
Torque output with feedforward control
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Torque output withoutfeed forward control
Speed output with feedforward control
Speed output withoutfeed forward control
Figure 16 Experimental result torque-speed response of IOLcontrol method
applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 11
15
minus05
0
05
1
Spee
d (P
U)
0 1 2 3 4 5Time (s)
Without FF controlWith FF control
(a)
0 1 2 3 4 5
minus2
0
2
Time (s)
Torq
ue (P
U)
Without FF controlWith FF control
(b)
Figure 12 Speed torque response of IOL control method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus4
minus2
0
24
Torq
ue (P
U)
Time (s)
Reference speedStep increase in speed command
Step decrease in speed command
Step increase in torque command
Step decrease in torque command Reference torque
Motor torque
Motor speed
Response due to change intorque reference
Response due to change in
speed reference
Figure 13 Speed torque response of IFOC method
0
05
1
Spee
d (P
U)
0 1 2 3 4 5minus2minus1
012
Time (s)
Torq
ue (P
U)
Step decrease in speed command
Step increase in speed commandReference speed Motor speed
Response due to change intorque reference
Response due to change inspeed reference
Motor torque
Step decrease in torque commandReference torque
Step increase in torque command
Figure 14 Speed torque response of DTC method
inductor at its dc source end and also a CSI drive systemwithon-line control strategy is more complex compared to aVSI drive due to the CSI gating requirements But depend-ing on some particular applications its advantages can attimes outweigh its disadvantages hence rendering it as asuitable choice for certain applications From the resultsit is concluded that DTC drive is the top most performerfor wide range of torque control applications (load varying
TMS320F2812DSP
ADC
PWM
-EV
B
PWM
-EVA
Supplyfilter
3120601controlled
rectifier3120601 CSI inverter
3120601IM
3120601supply
Texasinstruments
DSP processor
Speed sensor output
(Nr) Rpm
CM75TL-12NFseries
IGBT (6-PAC)
Ldc
Is (Amps)
Figure 15 Schematic diagram of the hardware setup of CSI fed IM
Torque output with feedforward control
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Torque output withoutfeed forward control
Speed output with feedforward control
Speed output withoutfeed forward control
Figure 16 Experimental result torque-speed response of IOLcontrol method
applications) with respect to change in speed as well as torquereferences IOL drive is the good performer for speed controlapplications with respect to change in speed reference andIFOC drive is the good performer for speed control applica-tions with respect to change in torque reference Hence forindustrial drive applications with the requirement of torque
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Journal of Applied Mathematics
Step increase in speed command
Reference torque
Step increase in torque command
Step decrease in speedcommand
Actual motor torque
Torque reference
Speed reference
Actual speed
radsec Actual motor speed 120596m = 140
1 Div = 50Nm
1 Div = 50Nm
1 Div = 50 rads
1 Div = 50 rads
Figure 17 Experimental result of torque-speed response of DTCcontrol method
Table 2 Transient response results for step change in speedreference command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1152 134 163IOL 566 79 26IFOC 958 454 142DTC 798 323 139
Motor torque responseVf 1121 632 214IOL 954 82 194IFOC 958 75 275DTC 520 26 149
Table 3 Transient Response results for Step change in Torquereference Command
Controlmethods
Settling time(119905119904) in ms
Rising time 119905119903
in msMaximum peakovershoot (119872
119901)
Motor speed responseVf 1133 44 132IOL 421 4 5IFOC 263 117 3DTC 496 272 7
Motor torque responseVf 1235 391 153IOL 892 75 81IFOC 215 54 151DTC 132 36 103
to be controlled direct torque control method is the mostefficient way to reach the desired operation of the drive tohandle the load If speed is to be controlled then IOLmethod
is efficient with change in speed reference and IFOC methodis efficient with change in torque reference
Symbols
120596119887 Base speed
119903
119903 Change in rotor flux wrt in rotor reference frame
119904
119904 Change in stator flux wrt in stator reference frame
119870119894 Current controller gain
120591119894 Current controller time constant120591119894119891 Current filter time constant
120582119889119903 119889-axis rotor flux
120582119890
119889119903 119889-axis rotor flux in synchronous reference frame
120582119889119904 119889-axis stator flux
119894dc 119868119889 DC link current119871dc DC link inductance119881dc DC link voltage120572 Delay angle or firing angle120582119902119903 119902-axis rotor flux
120582119890
119902119903 119902-axis rotor flux in synchronous reference frame
120582119902119904 119902-axis stator flux
120582119903 Rotor flux
119871119903 Rotor inductance
1199091015840
119903 Rotor reactance referred from stator
1198771015840
119903 Rotor resistance referred from stator
120596119903 Rotor speed
120579sl Slip angle120596sl Slip speed120591119904 Speed controller time constant
119870119905119890 Torque coefficient
119894120572119904 120572-axis stator current in stationary reference frame
119894120573119904 120573-axis stator current in stationary reference frame
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] K P Phillips ldquoCurrent-source converter for AC motor drivesrdquoIEEE Transactions on Industry Applications vol 8 no 6 pp679ndash683 1972
[2] T A Lipo and E P Cornell ldquoState-variable steady-state analysisof a controlled current induction motor driverdquo IEEE Transac-tions on Industry Applications vol IA-11 no 6 pp 704ndash7121975
[3] R Palaniappan J Vithayathil and S K Datta ldquoPrinciple of adual current converter for ACmotor drivesrdquo IEEE Transactionson Industry Applications vol IA-15 no 4 pp 445ndash452 1979
[4] M Kaimoto M Hashi T Yanase and T Nakano ldquoPerformanceimprovement of current source inverter-fed induction motordrivesrdquo IEEE Transactions on Industry Applications vol 18 no6 pp 703ndash711 1982
[5] R Krishnan J F Lindsay andV R Stefanovic ldquoDesign of angle-controlled current source inverter-fed induction motor driverdquoIEEE Transactions on Industry Applications vol 19 no 3 pp370ndash378 1983
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Applied Mathematics 13
[6] M P Kazmierkowski and H-J Koepcke ldquoA simple control sys-tem for current source inverter-fed induction motor drivesrdquoIEEE Transactions on Industry Applications vol IA-21 no 4 pp617ndash623 1985
[7] M Hombu S Ueda and A Ueda ldquoA current source GTOinverter with sinusoidal inputs and outputsrdquo IEEE Transactionson Industry Applications vol 23 no 2 pp 247ndash255 1987
[8] M Veerachary ldquoOptimal control strategy for a current sourceinverter fed induction motorrdquo Computers and Electrical Engi-neering vol 28 no 4 pp 255ndash267 2002
[9] M Salo and H Tuusa ldquoA vector-controlled PWM current-source-inverter-fed induction motor drive with a new statorcurrent control methodrdquo IEEE Transactions on Industrial Elec-tronics vol 52 no 2 pp 523ndash531 2005
[10] M Babaei and H Heydari ldquoDirect torque control of pulsewidth modulation current source inverter-fed induction motorby novel switching methodrdquo Electric Power Components andSystems vol 38 no 5 pp 514ndash532 2010
[11] D G Holmes B P McGrath and S G Parker ldquoCurrent regu-lation strategies for vector-controlled induction motor drivesrdquoIEEE Transactions on Industrial Electronics vol 59 no 10 pp3680ndash3689 2012
[12] R Krishnan Electric Motor Drives Modeling Analysis and Con-trol Prentice Hall of India 2002
[13] P C Krause and C H Thomas ldquoSimulation of symmetricalinduction machineryrdquo IEEE Transaction on Power Apparatusand Systems vol 84 no 11 pp 1038ndash1053 1965
[14] B K Bose Power Electronics and Motor Drives Advances andTrends Academic Press 2006
[15] P C Krause O Wasynczuk and S D Sudhoff Analysis of Elec-tric Machines and Drive Systems Wiley-IEEE Press New YorkNY USA 2002
[16] W H Wonham ldquoOn pole assignment in multi-input control-lable linear systemsrdquo IEEE Transactions on Automatic Controlvol 12 no 6 pp 660ndash665 1967
[17] B Wu G R Slemon and S B Dewan ldquoEigenvalue sensitivityanalysis of GTO-CSI induction machine drivesrdquo IEEE Transac-tions on Industry Applications vol 30 no 3 pp 767ndash775 1994
[18] H W Smith and E J Davison ldquoDesign of industrial regulatorsIntegral feedback and feed forward control PROCIEErdquo Pro-ceedings of the Institution of Electrical Engineers vol 119 no 8pp 1210ndash1216 1972
[19] F Blaschke ldquoA new method for the structural decoupling ofAC induction machinesrdquo in Proceedings of the IFAC pp 1ndash15Duesseldorf Germany October 1971
[20] F Blaschke ldquoThe principle of field-orientation as applied tothe transvector closed-loop control system for rotating-fieldmachinesrdquo Siemens Revenue vol 34 pp 217ndash220 1972
[21] P Vas Electrical Machines and Drives Oxford University PressLondon UK 1992
[22] I Takahashi and Y Ohmori ldquoHigh-performance direct torquecontrol of an induction motorrdquo IEEE Transactions on IndustryApplications vol 25 no 2 pp 257ndash264 1989
[23] M Depenbrock ldquoDirect self-control (DSC) of inverter-fedinduction machinerdquo IEEE Transactions on Power Electronicsvol 3 no 4 pp 420ndash429 1988
[24] D Casadei G Serra A Tani and L Zarri ldquoAssessment of directtorque control for inductionmotor drivesrdquo Bulletin of the PolishAcademy of Sciences Technical Sciences vol 54 no 3 pp 237ndash254 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of