Air-fuel Mixing and Fuel Flow Velocity Modeling of Multi ...
Modeling and Control of Z-Source Converter for Fuel Cell ...
Transcript of Modeling and Control of Z-Source Converter for Fuel Cell ...
Modeling and Control of Z-Source Converter
for Fuel Cell Systems (Technical Report)
Jin-Woo Jung, Ph. D Student Ali Keyhani, Professor
Mechatronic Systems Laboratory Department of Electrical and Computer Engineering
The Ohio State University
Columbus Ohio 43210
Tel: 614-292-4430
Fax: 614-292-7596
Date: May 03, 2004
TABLE OF CONTENTS
ABSTRACT
I. INTRODUCTION
II. MODELING OF FUEL CELL SYSTEMS
III. CONFIGURATION OF FUEL CELL SYSTEM
IV. CIRCUIT ANALYSIS
V. SYSTEM MODELING
VI. SPACE VECTOR PWM IMPLEMENTATION
VII. SIMULATION RESULTS
VIII. CONCLUSIONS
ACKNOWLEDGEMENT
REFERENCES
APPENDIX
A. MATLAB CODE FOR THE INITIALIZATION BEFORE THE SIMULINK SIMULATION
B. THE SIMULINK MODEL USED IN THE SIMULATION
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Abstract—This report presents a detailed circuit analysis and PWM implementation of a
fuel cell based Z-source converter using L and C components. The fuel cell system is
modeled by an electrical R-C circuit in order to include a slow dynamics of the fuel cells
and a voltage-current characteristic of a cell is also considered. A discrete-time state space
model of the Z-source converter is given for DSP implementation. A modified space vector
pulse width modulation is described in detail. To verify the effectiveness of the analyzed
circuit model and modified space vector PWM technique, various simulation results using
Matlab/Simulink are presented under a closed-loop control and Matlab/Simulink models
are attached in Appendix. Also, all Matlab/Simulink codes are given.
Index Terms—Z-source converter, boost converter, fuel cells, distributed generation systems.
I. INTRODUCTION
Interest in emerging power generation technologies such as wind turbines, photovoltaic arrays
and fuel cells is rapidly increasing due to global pollution problems [1-7]. The fuel cell systems
can always produce electric power regardless of climate conditions as long as hydrogen and
oxygen are supplied.
A fuel cell is an electrochemical device which converts chemical energy directly to electric
energy by an electrochemical reaction of hydrogen and oxygen. Furthermore, the fuel cell system
produces only electricity, water, and heat, and has 50 % efficiency for only electricity and could
reach 85 % in case of co-generation. Thus, fuel cells are a DC power source of safe, clean, and
efficient electric power generation. Types of the fuel cells are categorized according to the
electrolyte used: Proton-Exchange-Membrane (PEM), Phosphoric Acid (PA), Molten Carbonate
(MC), Solid Oxide (SO), Alkaline, and Zinc-Air (ZA) fuel cells, etc. Also, anticipated
applications of the fuel cells include stationary (buildings, hospitals, etc), residential (domestic
utility), transportation (fuel cell vehicles), portable power (laptop, cell phone) and distributed
power generation.
A detailed dynamic modeling of the fuel cell systems has been presented based on chemical
and physical processes [8-15]. The mathematical model of the reformer and stack is represented
by a simple R-C circuit for a conventional DC-DC boost converter in [16]. Lately, attention is
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put on design of the Power Electronic interface system that is inexpensive, reliable, small-sized
and light-weighted for residential applications [18-21]. In order to produce higher AC voltage
than the DC output voltage of the fuel cells, the conventional fuel cell systems in these papers
have used a DC/DC boost converter and a DC/AC inverter, and the fuel cells are modeled by a
DC voltage source. In reference [22], a new Z-source converter is proposed that does not need
any boost converter in order to step up a low DC output voltage of fuel cells to a higher DC
voltage and uses the fuel cell modeled by a constant DC voltage source and then analyzed its
operation under an open-loop control.
However, a detailed dynamic model of the fuel cells in conjunction with a detailed circuit of Z-
source converter with PWM implementation has not been studied [22-23]. In this report, we
focus on an electrical analysis of the fuel cell based three-phase PWM inverter with an electrical
R-C circuit model to present the slow dynamic response of a fuel cell. In our analysis, a discrete-
time state space model of the Z-source converter is given for design of a closed-loop control
system using digital signal processor (DSP). To validate the proposed method, simulation studies
are performed for a test bed of an AC 208 V/60 Hz/10 kVA system.
II. MODELING OF FUEL CELL SYSTEMS
Hydrogen gas required for the fuel cell systems to produce DC power is indirectly derived
from a reformer using fuels such as natural gas, propane, methanol, gasoline or from the
electrolysis of water, or can be directly obtained from stored hydrogen or hydrogen pipeline.
Fig. 1. Configuration of the fuel cell systems.
In case of indirectly producing the hydrogen by the reformer, the fuel cell generation systems
consist of three parts shown in Fig. 1: a reformer, stacks, and power converters. The reformer
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produces hydrogen gas from fuels and then provides it for the stacks. The stacks have many unit
cells which are stacked in series to generate a higher voltage needed for their applications
because one single cell that consists of electrolyte, separators, and plates, produces
approximately 0.7 V DC. Also it generates DC electric power by an electrochemical reaction of
hydrogen and oxygen. The power converters convert a low voltage DC from the fuel cell to a
high voltage DC or a sinusoidal AC.
Fig. 2. Block diagram of reformer and stacks.
For dynamic modeling of the fuel cells, the reformer and stack are further described because a
dynamic response of the fuel cell systems is determined by them. Fig. 2 shows a block diagram
of reformer and stacks to illustrate their operation. First of all, the reformer that produces the
hydrogen for power generation requested from the load significantly affects dynamic response of
the fuel cell system because it takes several to tens of seconds to convert the fuel into the
hydrogen depending on the demand of the load current. Thus, in order to investigate an overall
operation of fuel cell powered systems, the dynamics of the reformer may be represented as a
simple model like a first order time delay circuit [16] or a second order model [17].
On the other hand, the response of the stacks is considered to much faster respond to
hydrogen from the reformer and oxygen from the air rather than that of the reformer. In Fig. 2,
the output of the stacks shows different voltage-current polarization curves for varying hydrogen
mass flow rate: i.e., the maximum cell current and stack voltage increase as the hydrogen mass
flow rate increases. As a consequence, a dynamic response of the reformer and stacks and a
characteristic of cell current-stack voltage need to be modeled for more realistic analysis of the
fuel cell powered systems.
In this report, a simple R-C circuit model is used to realize slow dynamics caused by a
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chemical/electrical response of the reformer and stacks [16]. Fig. 3 shows an electrical
equivalent circuit model used for the fuel cells. As shown in Fig. 3, the reformer and stack are
modeled by Rr and Cr, and Rs and Cs, respectively.
Fig. 3. Electrical equivalent circuit model of the fuel cell.
Further, voltage-current characteristic of the stack is considered. Even if the maximum cell
current or stack voltage for each hydrogen flow rate should be represented as a sharp drop of cell
voltage due to primarily starvation of the hydrogen as shown in Fig. 2, the V-I characteristic in
Fig. 4 is used for a simplified circuit model in this study.
Fig. 4. Voltage-current characteristic of a cell.
1. Theoretical EMF or ideal voltage (1.16 V) 2. Region of Activation Polarization (Reaction Rate Loss)
3. Region of Ohmic Polarization 4. Region of Concentration Polarization (Gas Transport Loss)
III. CONFIGURATION OF FUEL CELL SYSTEM
Fig. 5 shows total system diagram of Z-source converter with a fuel cell that consists of a
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reformer, stacks, a fuel cell processor, a PWM inverter DSP controller, a Z-source converter, and
a load. As illustrated in Fig. 5, the fuel processor controls the reformer to produce hydrogen for
power requested from the PWM inverter DSP controller, and monitors the stack current and
voltage. The PWM inverter DSP controller communicates with the fuel cell processor to equalize
power available from the stack to power requested by the load, and controls the Z-source
converter, and senses output voltages/currents for a feedback control.
Fig. 5. Total system diagram of Z-source converter with a fuel cell.
Fig. 6 shows system configuration with Z-source converter used as power electronic interface
system for fuel cell systems. The filter capacitors (Cf) to eliminate harmonics of the inverter
output voltage due to the PWM technique are added to the conventional Z-source converter. In
Fig. 6, the system consists of a fuel cell, a diode, impedance components (L1, 2 and C1, 2), a three-
phase inverter, an output filter (Lf and Cf), and a 3-phase load. The diode between the fuel cell
and Z-source converter is required to prevent a reverse current that can damage the fuel cell.
The Z-source converter is based on a new concept different from a conventional DC/AC
power converter. In the conventional 3-phase voltage source inverter (VSI), the shoot-through
that both switches in a leg are simultaneously turned on must be avoided because it causes a
short circuit. So the traditional three-phase VSI has eight basic switching vectors that consist of
six active vectors V1-V6 which impress the DC-link voltage on the load and two zero vectors V0,
V7 which do not impress the DC-link voltage on the load. Meanwhile, the Z-source converter has
one more vector (i.e., the shoot-through zero vector) besides eight basic switching vectors. It
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utilizes the shoot-through in order to directly boost a DC source voltage without a boost DC/DC
power converter because topology of the Z-source converter makes the boost feature stated
above possible. Also, a boosted voltage rate absolutely depends on total duration (Ta) of the
shoot-through zero vectors over one switching period (Tz).
Fig. 6. System configuration with Z-source converter.
IV. CIRCUIT ANALYSIS
To explain the operating principle of the Z-source converter, the equivalent circuit model of
the Z-source converter is shown in Fig. 7 when Fig. 6 is viewed from the DC-link. Fig. 7 (a)
shows the equivalent circuit of the Z-source converter in the shoot-through zero vectors, while
Fig. 7 (b) shows that of the Z-source converter in the non-shoot-through switching vectors.
If we assume that inductor L1 and capacitor C1 are equal to inductor L2 and capacitor C2,
respectively, we can obtain the following equations from one of shoot-through zero vectors and
one of non-shoot-through switching vectors.
VC1 = VC2 and vL1 = vL2. (1)
Case 1: one of shoot-through zero vectors (Fig. 7 (a))
vL1 = VC1, vf = 2VC1, and vi = 0. (2)
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Case 2: one of non-shoot-through switching vectors (Fig. 7 (b))
loop : vL1 = vf – VC1 = Vin – VC1
loop : vi = VC1 – vL1 = 2VC1 – Vin, (3)
where, Vin is the output voltage of the fuel cell.
(a) (b)
Fig. 7. Equivalent circuit of Z-source converter.
(a) In the shoot-through zero vectors. (b) In the non-shoot-through switching vectors.
We assume that Tz = Ta + Tb, where Tz: switching period, Ta: total duration of shoot-through
zero vectors over Tz, and Tb: total duration of non-shoot-through switching vectors over Tz. If we
use the fact that the average voltage of the inductors over one switching period (Tz) is equal to
zero in steady state, we can obtain the following equation from (2) and (3)
( )
in
z
a
z
a
inab
bC
T
z
CinbCaLLL
V
TT
TT
VTT
TV
TVVTVT
dtvvV z
⋅−
−=
−=⇒
=−⋅+⋅
=== ∫
21
1
0
1
0
11111
. (4)
Also, the average DC-link voltage can be expressed below
8
( )1
10
20Cin
ab
b
z
inCbaT
iii VVTT
TT
VVTTdtvvV z =−
=−⋅+⋅
=== ∫ . (5)
From (5), the average DC-link voltage (Vi) is equal to capacitor voltage (VC1), so the
measured VC1 can be used to regulate the DC-link voltage. Next, we can calculate the peak DC-
link voltage using (3) and (4)
ininab
zinCDCp VKV
TTT
VVV ⋅=−
=−= 1_ 2 , (6)
where, 121
1≥
⋅−=
−=
z
aab
z
TTTT
TK , (7)
K is called a boost factor and is always more than one in order to obtain the boosted DC voltage
compared to the output DC voltage of the fuel cell. Using (6), a peak phase voltage of inverter
output can be written as
22_
_inDCp
paV
KMV
MV ⋅⋅=⋅= , (8)
where, M denotes the modulation index.
Finally, from (8) we can know that the peak phase voltage (Va_p) of inverter output definitely
depends on both the modulation index (M) and the boost factor (K), and from (7) the boost factor
(K) is determined by a ratio Ta/Tz.
( MTT
TTTTT
z
b
z
abaz =+=⇒+= 1 ) . (9)
Moreover, sum of the modulation index (M=Tb/Tz) and the ratio Ta/Tz is always equal to unity
because Tb/Tz is related to the non-shoot-through vectors while Ta/Tz is to the shoot-through zero
vectors as expressed in (9).
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V. SYSTEM MODELING
Fig. 6 can be simplified as shown in Fig. 8 for an analytic modeling of Z-source converter
with fuel cells. In Fig. 8, the system consists of a DC voltage source (Vdc), a three-phase inverter,
an output filter (Lf and Cf), and a 3-phase load. Note that a fuel cell, a diode (D), and impedance
components (L1, 2 and C1, 2) are replaced with a DC-link voltage source (Vdc) for circuit
modeling, and Vdc is also equal to the average DC-link voltage VC1 or VC2.
Fig. 8. Simplified circuit model of Z-source converter.
The simplified circuit model described in Fig. 8 uses the following quantities. The inverter
output line-to-line voltage is represented by the vector Vi = [ViAB ViBC ViCA]T. The inverter output
currents are iiA, iiB, and iiC. From these currents, a vector can be defined as Ii = [iiAB iiBC iiCA]T =
[iiA−iiB iiB−iiC iiC−iiA]T. Also, the load line to line voltage and phase current vectors can be
represented by VL = [VLAB VLBC VLCA]T and IL = [iLA iLB iLC]T, respectively.
On the L-C output filter, the following current and voltage equations are derived:
i). Current equations:
( )
( )
( )⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
−−=
−−=
−−=
LALCf
iCAf
LCA
LCLBf
iBCf
LBC
LBLAf
iABf
LAB
iiC
iCdt
dV
iiC
iCdt
dV
iiC
iCdt
dV
31
31
31
31
31
31
, (10)
10
ii). Voltage equations:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
−=
−=
−=
LCAf
iCAf
iCA
LBCf
iBCf
iBC
LABf
iABf
iAB
VL
VLdt
di
VL
VLdt
di
VL
VLdt
di
11
11
11
. (11)
Rewrite (10) and (11) into a vector form, respectively:
Lf
if
i
Lif
if
L
LLdtd
CCdtd
VVI
ITIV
113
13
1
−=
−=
. (12)
where, . ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−=
101110
011
iT
For implementation of SVPWM, the three-phase system represented by the above state
equations can be transformed to a stationary qd reference frame that consists of the horizontal (q)
and vertical (d) axes. The mathematical relationship between these two reference frames is
abcsqd fKf =0 , (13)
where, ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−−−
=2/12/12/1
232302/12/11
32
sK , fqd0=[fq fd f0]T, fabc=[fa fb fc]T, and f denotes either a voltage or a
current variable.
Rewrite (12) in the stationary qd reference frame below:
11
Lqdf
iqdf
iqd
Lqdiqdf
iqdf
Lqd
LLdtd
CCdtd
VVI
ITIV
11
31
31
−=
−=
, (14)
where, Tiqd = [KsTiKs-1]row, column, 1,2 =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡ −
13
13
11
23 .
Thus, assuming the parameters of all passive components are constant, the given plant model
can be expressed as the following continuous-time state space equation for a linear time-
invariant (LTI) system
)()()()( tttt EdBuAXX ++=& , (15)
where,
14×
⎥⎦
⎤⎢⎣
⎡=
iqd
Lqd
IV
X ,
44
2222
2222
013
10
×
××
××
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−=
IL
IC
f
fA ,
24
22
2210
×
×
×
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡= I
L f
B , , [ ] ⎥⎦
⎤⎢⎣
⎡==
×id
iqiqd V
V12
Vu
242203
1
×× ⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−= iqd
fCT
E , . ⎥⎦
⎤⎢⎣
⎡== ×
Ld
LqLqd i
i12][Id
Notice that the line to line load voltage VLqd and inverter output current Iiqd are the state
variables of the system, the inverter output line-to-line voltage Viqd is the control input (u), and
the load current ILqd is defined as the disturbance (d).
For realization of digital control, the (15) can be converted into a discrete-time state space
equation:
)()()()1( *** kkkk dEuBXAX ++=+ , (16)
12
where, , , . zTe AA =* ∫ −= zz
T T de0
)(* ττ BB A ∫ −= zz
T T de0
)(* ττ EE A
The above discrete-time state space plant model (16) can be used for design of a feedback
control system using digital signal processor (DSP) or microcontroller.
VI. SPACE VECTOR PWM IMPLEMENTATION
Most of pulse-width modulation (PWM) schemes can be used for the Z-source converter. In
this report, a space vector PWM technique (SVPWM) is used to implement the Z-source
converter because of less harmonic distortion in the output voltages and more efficient use of
supply voltage.
Fig. 9. Basic space vectors and switching patterns.
For realization of SVPWM, a three-phase voltage or current vector in the abc reference frame
is transformed into a vector in the stationary qd coordinate frame. Fig. 9 shows eight possible
switching vectors of on and off patterns for the three upper power transistors that feed the three-
phase DC to AC inverter. Six non-zero vectors (V1 - V6) forms the axes of a hexagonal, and two
zero vectors (V0 and V7) are at the origin. Also, the vectors divide the plane into six sectors, and
the angle between any adjacent two non-zero vectors is 60 degrees.
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(a) Sector 1. (b) Sector 2.
Fig. 10. Modified SVPWM implementation.
To generate the same voltage as ῡref, we should determine three switching durations (T1, T2,
T0) using the most adjacent two voltage vectors within a constant period (Tz). Assuming the case
of the sector 1 in Fig. 9, the following equation is derived
)600,()3/sin()3/cos(
32
01
32
)sin()cos(
)(
21
2211
0
1
2
0 0
1
21
211
°≤≤
⎥⎦
⎤⎢⎣
⎡⋅⋅⋅+⎥
⎦
⎤⎢⎣
⎡⋅⋅⋅=⎥
⎦
⎤⎢⎣
⎡⋅⋅⇒
⋅+⋅=⋅∴
++= ∫∫∫ ∫+
+
αππ
αα
where
VTVTVT
VTVTVT
VdtVdtVV
dcdcrefz
refz
T
TT
TT
T
T T
ref
zz
. (17)
Therefore, each switching time interval can be calculated:
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
+−=
⋅⋅=
−⋅⋅=
dc
ref
z
z
z
V
Vawhere
TTTT
aTT
aTT
32,
)()3/sin(
)sin()3/sin(
)3/sin(
210
2
1
παπ
απ
. (18)
Based on the above conventional SVPWM technique to calculate T1, T2, and T0, a new
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duration (T = Ta/3) should be added to the traditional SVPWM in order to boost the DC-link
voltage of the Z-source converter and to generate the sinusoid AC output voltage. As mentioned
in the previous section, the rate of DC-link boosted voltage is determined by the total duration
(Ta) of shoot-through zero vectors that at once turn on both power switches in a leg. Fig. 10
shows both the conventional and modified switching patterns for the Z-source converter at sector
1, 2. In Fig. 10, each phase leg still switches on and off once per switching cycle (Tz), and each
phase has only one shoot-through zero state (T) during one period (Tz) in any sector without any
change of total zero vectors (V0, V7, and T) and total nonzero switching vectors (V1 – V6). Even
if the output voltage of inverter and DC-link voltage can be controlled by adjusting Ta, the
maximum available shoot-through interval (Ta) to boost the DC-link voltage (vi) is restricted by
the zero vector duration (T0/2) which is determined by the modulation index (M = a⋅(4/3)).
TABLE I. SWITCHING TIME DURATION AT EACH SECTOR Sector Upper (S1, S3, S5) Lower (S4, S6, S2)
1 S1 = T1 + T2 + T0 /2 + T S3 = T2 + T0 /2 S5 = T0 /2 − T
S4 = T0 /2 S6 = T1 + T0 /2 + T S2 = T1 + T2 + T0 /2 + 2T
2 S1 = T1 + T0 /2 S3 = T1 + T2 + T0 /2 + T S5 = T0 /2 − T
S4 = T2 + T0 /2 + T S6 = T0 /2 S2 = T1 + T2 + T0 /2 + 2T
3 S1 = T0 /2 − T S3 = T1 + T2 + T0 /2 + T S5 = T2 + T0 /2
S4 = T1 + T2 + T0 /2 + 2T S6 = T0 /2 S2 = T1 + T0 /2 + T
4 S1 = T0 /2 − T S3 = T1 + T0 /2 S5 = T1 + T2 + T0 /2 + T
S4 = T1 + T2 + T0 /2 + 2T S6 = T2 + T0 /2 + T S2 = T0 /2
5 S1 = T2 + T0 /2 S3 = T0 /2 − T S5 = T1 + T2 + T0 /2 + T
S4 = T1 + T0 /2 + T S6 = T1 + T2 + T0 /2 + 2T S2 = T0 /2
6 S1 = T1 + T2 + T0 /2 + T S3 = T0 /2 − T S5 = T1 + T0 /2
S4 = T0 /2 S6 = T1 + T2 + T0 /2 + 2T S2 = T2 + T0 /2 + T
To help understand implementation of the space vector PWM modified for the Z-source
converter, the switching time of the upper switches and the lower switches in a 3-phase inverter
is summarized in Table I. Note that when the shoot-through duration (T) is equal to zero, the
switching time of each power switch for the Z-source converter is exactly the same as that for the
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conventional one.
VII. SIMULATION RESULTS
In order to verify the effectiveness of the analyzed circuit model and modified SVPWM
implementation, a simulation test bed using Matlab/Simulink is constructed for an AC 208 V (L-
L)/10 kVA, and simulation studies are performed under a closed-loop control. The system
parameters are given in Table II.
TABLE II. SYSTEM PARAMETERS FOR SIMULATIONS Fuel Cell Output Voltage Vin = 150 – 250 V
Desired Average DC-link Voltage VC2 = 360 V Output Rated Power Pout = 10 kVA
Impedance Components L1 = L2 = 200 uH, C1 = C2 = 1000 uF
Inverter Output Filters Lf = 580 uH, Cf = 380 uF
AC Output Voltage VL, RMS = 208 V (L-L), f = 60 Hz
Switching/Sampling Period Tz = 1/(5.4 kHz)
From (7)-(9) and Table II, the parameters K, M, a, and T can be calculated theoretically below
by (19):
i). When Vin = 150 V and P = 10 kW:
Ta/Tz = 0.3684, K = 3.799, M = 0.6316, a = 0.4737,
T = 22.1 µsec.
ii). When Vin = 250 V and P = 5 kW:
Ta/Tz = 0.2340, K = 1.88, M = 0.766, a = 0.5745,
T = 14.44 µsec.
16
3/4/3/1)/(
/211
2/
/21/1
2
1
1
21
a
zazb
zainC
inCza
inza
za
az
az
ab
bCC
TTMaTTTTM
TTK
VVVVTT
VTT
TTTT
TTTT
TVV
=∴⋅=⇒−==∴⋅−
=⇒−⋅−
=∴
⋅⋅−
−=
−−
=−
==
. (19)
150 200 250 300 3500
0.2
0.4
Ta/Tz
Vin
150 200 250 300 3500
2
4
KTa/Tz
K
Fig. 11. Relationship between Vin, Ta/Tz, and K.
From Table II and (19), relationship between Vin, Ta/Tz, and K is shown in Fig. 11. As we
expect, Ta/Tz and K decrease as the output voltage (Vin) of the fuel cell increases.
Fig. 12. Simulink model for an overall system.
17
Fig. 13. Simulink model for a PWM inverter DSP controller.
Based on these calculated parameters, the simulations are implemented under a linear load for
two cases: a load increase and a load decrease. From Fig. 4, when the load increases from 5 kW
to 10 kW, we assume that the output voltage of the fuel cell is changed from 250 V to 150 V. On
the other hand, when the load decreases from 10 kW to 5 kW, the output voltage of the fuel cell
is changed from 150 V to 250 V. In addition, we assume the parameters of the reformer and
stack: Rr = 0.05 Ω and Cr = 42.8 mF, Rs = 0.02 Ω and Cs = 4.2 mF. Fig. 12 shows the Simulink
model for an overall system, while Fig. 13 shows the Simulink model for a PWM inverter DSP
controller which consists of discrete-time PI voltage/current controllers and a DC-link PI
controller.
18
0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170
1
2
S1
0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170
1
2
S4
0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170
1
2
S3
0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170
1
2
S6
0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170
1
2
S5
0.0167 0.0168 0.0168 0.0169 0.0169 0.017 0.017 0.0170
1
2
S2
Time [sec]
Fig. 14. Gating signals for six power switches.
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10
100
200
300
400
Vin
, VC2
[V]
VinVC2
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-400
-200
0
200
400
VLAB
, VLBC
, VLCA
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-50
0
50
Time [sec]
i LA, i
LB, i
LC [A]
Fig. 15. Simulation results when Vin = 250 V and P = 5kW.
19
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10
100
200
300
400
Vin
, VC2
[V]
VinVC2
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-400
-200
0
200
400
VLAB, VLBC, VLCA
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-50
0
50
Time [sec]
i LA, i
LB, i
LC [A]
Fig. 16. Simulation results when Vin = 150 V and P = 10 kW.
Fig. 14 shows PWM signals of six power switches and the shoot-through that both power
switches in a leg are simultaneously turned on is definitely shown. Fig. 15 and 16 show the
simulation waveforms in steady state without considering the slow dynamics of the fuel cell. Fig.
17 and 18 show the simulation results under the load increase and decrease, respectively. In Fig.
17 and 18, each figure indicates: (a) Power request at 70 msec., (b) Output voltage of the fuel
cell (Vin) and capacitor (VC2), (c) Load line to line voltages (VLAB, VLBC, VLCA), and (d) Load
phase currents (iLA, iLB, iLC). As shown in Fig. 17 and 18 (b), (d), a good voltage regulation of
capacitor voltage (VC2) is presented in spite of a slow change of the fuel cell output voltage and a
load change.
20
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10
5
10
15
P [k
W]
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10
200
400
Vin
, VC2
[V] Vin
VC2
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-400
-200
0
200
400
VLAB
, VLBC
, VLCA
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-50
0
50
Time [sec]
i LA, i
LB, i
LC [A]
Fig. 17. Simulation results when the load increases.
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10
5
10
15
P [k
W]
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.10
200
400
Vin
, VC2
[V] Vin
VC2
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-400
-200
0
200
400
VLAB, VLBC, VLCA
0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1-50
0
50
Time [sec]
i LA, i
LB, i
LC [A]
Fig. 18. Simulation results when the load decreases.
VIII. CONCLUSIONS
In this report, we have described in detail the circuit modeling and modified PWM
implementation of the Z-source boost converter for the fuel cell systems. A slow dynamics of the
21
fuel cells and a voltage-current characteristic of a cell are also considered for more realistic
applications. To verify the analyzed circuit model and modified SVPWM method, various
simulation results for two load conditions have been presented from Fig. 14 to Fig. 18 for a
system that needs a 3-phase AC 208 V (L-L)/60 Hz/10 kVA.
ACKNOWLEDGMENT
This work is supported in part by the National Science Foundation under the grant
ECS0105320.
22
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24
Appendix
A. Matlab Code for the initialization before the Simulink simulation
(Matlab version 6.1: filename: Z_source_PI.m)
% Written by Jin-Woo Jung, 04/03/2004
clear all;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Initialize the plant model parameters %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% fundamental output frequency
f=60; % Hz
w=2*pi*f; % rad/s
% Power rating and limit of fuel cell output voltage
P= 10e3; % Power rating
Vin_min= 150; % minimum voltage of fuel cell output
Vin_max= 360; % maximum voltage of fuel cell output
% Sampling time
Tz=1/60/90; % PWM frequency = 5.4 kHz
% Desired average DC-link voltage (VC2)
Vdc=360; % desired capacitor voltage
% Z-source impedance
L= 200e-6; % L-C impedance
C= 1000e-06; % L-C impedance
25
% Circuit parameters of L-C output filter
Lf= 580e-6; % Inductor
Cf= 380e-6; % Capacitor
Ilimit=3*10e3/120*sqrt(2); % 300% inverter current limit
%Shoot-through period
T0_min= 0; % Vin= 360 V
T0_max = 0.3684; % Vin= 150 V
% Voltage limit for SVPWM
Vmax_DSMC=Vdc*2/sqrt(3); % Line to line voltage limit
Vmax=Vdc*2/3; % Phase voltage limit
% Set desired output voltage (reference voltage)
VLLp= 120*sqrt(6); % line to line peak value (208 RMS)
% Convert abc to qd0 (stationary reference frame)
sr3_2=sqrt(3)/2;
Ks=[1 -0.5 -0.5; 0 sr3_2 -sr3_2; 0.5 0.5 0.5]*2/3;
Ksqd=Ks(1:2, :); % ignore 0 axis
Ksqd3=[Ksqd zeros(2,3) zeros(2,3)
zeros(2,3) Ksqd zeros(2,3)
zeros(2,3) zeros(2,3) Ksqd]; % Transformation matrix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Three PI controller gains %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% PI voltage controller
kp_v= 10; % P gain
ki_v= 0.1; % I gain
26
% PI current controller
kp_i= 1.5; % P gain
ki_i= 0.1; % I gain
% PI DC-link voltage controller
kp= 0.01; % P gain
ki= 0.1; % I gain
upper= 1.15; % Upper limit for saturation
lower_0= 0.8; % Lower limit for saturation
lower_1= 0; % Lower limit for saturation
% Program End
27
B. The Simulink model used in the simulation
(Matlab version 6.1: filename: Z_Source_Inv_PI.mdl)
B-1 Model of Total System Configuration
28
B-5-1 Model of PI Voltage Controller
B-5-2 Model of PI Current Controller
B-5-3 Model of PI DC-link Voltage Controller
31
B-5-4 Model of abc to qd Transformation
B-5-5 Model of SVPWM Implementation
B-5-5-1 Model of Vref and angle
32