Hybrid electrical power system modeling and management

Simulation Modelling Practice and Theory 25 (2012) 190–205

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Simulation Modelling Practice and Theory

journal homepage: www.elsevier .com/locate /s impat

Hybrid electrical power system modeling and management

R. Sanchez a,b,c,d, X. Guillaud b,⇑, G. Dauphin-Tanguy a,c,d

a Univ. Lille Nord de France, F-59000 Lille, Franceb ECLille, L2EP, F-59650 Villeneuve d’Ascq, Francec ECLille, LAGIS, F-59650 Villeneuve d’Ascq, Franced CNRS, UMR 8219, F-59650 Villeneuve d’Ascq, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 June 2010Received in revised form 30 June 2011Accepted 16 August 2011Available online 10 April 2012

Keywords:Bond graphCommutation systemSwitching power junction

1569-190X/$ - see front matter � 2011 Elsevier B.Vdoi:10.1016/j.simpat.2011.08.006

⇑ Corresponding author.E-mail addresses: [email protected] (R.

Tanguy).

A distributed electrical system composed of a primary source (alternator) and a renewablesource associated to a converter, both of them are coupled to an induction machine load ispresented. A causality conflict arises at the coupling node, due to the inductive nature ofeach device. To deal with the variable structure of the model (same devices have to be cou-pled or decoupled depending on specific conditions), switching power junction concept isused at the coupling node. Causality changes involve changes in alternator and inductionmachine models allowing the management of the structure commutations. Power flowmanagement is presented. Simulation results show the advantages of using bond graphmodels and the validity of the proposed models.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

Power systems have been developed over the years to supply a varying demand from centralized generation sources(hydraulic, fossil fuel, nuclear fuel). Nowadays the renewable energy sources start to play an important role in power sys-tems, and the great majority of them are integrated into the electrical network using power electronic converters.

Most of the time, distributed generation is connected into the grid as a simple current injector which means that thesource needs to be synchronized on an existing grid. In the presented paper, the source is given a voltage source behavior.Indeed, this device can create its proper network and deliver its power to any sort of load which can be connected to thisgrid. In these conditions, we must add a storage element to a renewable source to guarantee the availability of the energywhen needed.

Recent studies are involved in developing important AC microgrid with these kinds of power electronic converters with avoltage source behavior. CERTS in United States [13] was one of the first laboratory to develop such a microgrid. SeveralEuropean projects have been interested at this aim [10]. An interesting review of all these developments is proposed in[8]. Other papers have been published in the same area [3,2]. Even if a lot of work has been done on these topics, the devel-opment of electrical network whose power is delivered only by power electronic converter and energized by non conven-tional source is a real challenge.

One of the questions is the association of this power electronic source with a classical synchronous machine source asshown in Fig. 1. The different natures of the power sources may lead to the qualification of ‘‘hybrid’’ for the studied electricalpower system.

. All rights reserved.

Sanchez), [email protected] (X. Guillaud), [email protected] (G. Dauphin-

http://dx.doi.org/10.1016/j.simpat.2011.08.006

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http://dx.doi.org/10.1016/j.simpat.2011.08.006

http://www.sciencedirect.com/science/journal/1569190X

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Fig. 1. Association of classical source with a power electronic converter connected source.

R. Sanchez et al. / Simulation Modelling Practice and Theory 25 (2012) 190–205 191

As we can see the load may be energized either by both sources: the alternator or the power converter. Since the config-uration may change, we have to associate with this system a variable structure model, which will be studied using bondgraph modeling and Switching Power Junction (SPJ) concept at the coupling node. The use of this type of junctions isdiscussed extensively in [11,18].

As shown in Fig. 1, the power electronic converter is connected to the grid via a ‘‘connection inductance’’ and, most of thetime, the load is inductive as, for example, an induction machine. In term of modeling, this system presents some causalityproblem since the point of common coupling is placed between these three inductive elements. One of these devices has toimpose the voltage to the two others which means that this element has to be modeled with a derivative causality whereasthe two others may be modeled with an integral causality. As shown in this paper, this specific element may change withrespect to the breaker configuration. A simplified model is proposed in order to avoid this derivative causality which maycause simulation problems when the used solver does not manage it. A comparison between full model and reduced modelsimulation results will show the validity of this approach. It means that such reduced models could be used as models in theloop in real time simulation platforms without any simulation problems but with a sufficient confidence level.

The main advantages of bond graph tool for modeling can be summarized through few keywords, which make thisapproach quite specific and justify its use in the paper:

– It provides the modeler with a unified graphical language for representing with physical insight power exchanges, energydissipation and storage phenomena in dynamic systems of any physical domain.

– It allows the visualization of causality properties before writing equations according to chosen modeling hypotheses.– Some software exist with bond graph graphical editor exempting the modeler from writing global equations.

In Appendix A, some background recalls on bond graph methodology are proposed for non familiar readers.In a first part, the power converter model with its simplification is presented, and then the classical and the simplified

models of the synchronous machine are described. The induction machine model is also recalled and then, the whole andthe simplified models in their different configurations are simulated.

2. Presentation of the system

Fig. 2 shows under a word bond graph form connection of a renewable electrical source via a power electronic converterto a classical source (alternator). The two sources feed a load chosen here as an induction machine. The structure of the pre-sented system can be analyzed at different levels of complexity.

Large scale power is generated by three-phase alternators (synchronous generators), driven either by steam turbines,hydro turbines or gas turbines. In this paper the primary source is taken as a constant effort source.

Fig. 2. Electrical network presented in terms of word bond graph.

192 R. Sanchez et al. / Simulation Modelling Practice and Theory 25 (2012) 190–205

The power electronic converter interfaces the renewable energy source to the grid. The fluctuated nature of renewablesources makes necessary to introduce a storage device (i.e. batteries, super capacitors), which is not taken in considerationexplicitly in this paper. Therefore, we will consider the renewable energy source also as a constant source without takingcare of its fluctuation and regulation. As said in the introduction, we want to give to the power electronic source a voltagesource behavior. For this purpose, we use a LC output filter with a closed loop voltage control. Nevertheless, the introductionof inductors is necessary in order to control the power transferred from the renewable source to the electrical network. Thesethree phase inductors are called connection inductors. The mechanical load applied to the induction machine is consideredas a torque source.

The connection node (0-junctions) is the central part of the system. It can be analyzed in term of causality and switchingconditions. From a causality point of view, each single device has an inductive nature, which means that each element has aflow (current) output causality assignment that introduces an indeterminate causality (Fig. 3a) at the 0-junction. Differentsolutions may be proposed to solve this causality problem:

(a) The voltage is imposed by the power electronic converter and the connection inductors (Fig. 3b).(b) The voltage is imposed by the alternator (Fig. 3c).(c) The voltage is imposed by the induction machine (Fig. 3d). This configuration would not be valid here, because the

induction machine in the structure is considered as a load.

A 0s-junction (Switching Power Junction, SPJ [18]) is introduced to point out these different modes. It allows introducing acommutation logic when using associated mathematical equations.

In order to choose the device which imposes the voltage, we have to take into consideration the different possible con-figurations of the system:

1. In normal operation both sources, classical and renewable, are connected to the load. The classical source is chosen as thevoltage reference (Fig. 3c).

2. If the power electronic converter is tripped, the classical source is also chosen as the reference voltage (Fig. 3c).3. If the classical source is tripped, the power electronic converter may impose the reference (Fig. 3b).

It is important to say that the structure presented in Fig. 3 shows the interaction between only two sources (renewableand classical), but in reality more sources may be connected to the same node in order to supply the load power demand. Itwould involve new possibilities in the causality assignment but the principle would be the same.

Fig. 3. Different word bond graphs for causality assignments at the coupling junction.


3. Power converter model

3.1. Power converter with a LC filter

Bond graph models of a three phase converter have been addressed in many papers, such as: [9,4]. Each presented modelhas its own characteristics and is used for different applications. In this paper, we use the model presented in [16], but mod-ified and adapted for this application.

The word bond graph model of the converter connected to a RL load composed of two parts (load 1 and load 2) connectedvia a contactor (which is used for connecting the load 2 at the instant t) is depicted in Fig. 4. The converter + LC filter block isderived from [16].

Characteristics of the power converter model are:

– The renewable energy source is considered as a constant source (Se:us).– The LC filter is composed of three inductances with their associated resistances and three Delta connected capacitors.

The causality assignment (which cannot change due to the capacitors) in bonds vc1, vc2, vc3 gives to the power electronicconverter associated with the LC filter a voltage source behavior (actually this is its objective). Since a closed loop voltagecontrol is achieved, the source may be considered as an ideal voltage source. The consideration of an ideal source can be doneassuming a good robustness of the control and neglecting the harmonics effects [17]. Thus, the bond graph of Fig. 4 can behighly simplified and represented as shown in natural frame in Fig. 5a. This system can also be representing in Park referenceframe (Fig. 5b).

As presented in the introduction, we would like to impose the voltage to the point of common connection with thissource. In term of causality, it would induce a derivative causality on the connection inductance (as it is shown in Fig. 4).

As usually used in power system study, we can simplify the dynamic model of Fig. 5b to propose an algebraic one. Then,the dynamics are eliminated, the transient is canceled and the order of the model is reduced.

In Fig. 5b, bonds 5 and 6 are eliminated. We choose to change the causality assignment in 9 and 10. This model is called‘‘simplified’’.

A simulation is undertaken in order to show the differences between the complete model (taken as in Fig. 4, with a deriv-ative causality in the connection impedance) and the simplified model (replacing the connection impedance for the simpli-fied one shown in Fig. 5b). As shown in Fig. 4, a RL load of 10 kW and 1 kVAR (load 1) is connected to the source. At t = 0.5 s,another load (load 2) with the same characteristic is connected to the source to enhance the transient response.

Fig. 6a and b shows respectively the currents load for complete and simplified models. The transient effect in the com-plete model is eliminated in the simplified one.

The simplification operated here is obvious but it must be considered as an introduction to the simplification which willbe operated further on the alternator.

3.2. Power control principle

In case of a connection with an infinite voltage source, the power control loop is inspired from the power flow conceptbetween two sources (vc and v), connected by an impedance line Z = R + jX [14]. Assuming that the transmission line hassmall resistance compared to the reactance (R = 0), the active power is given by:

Fig. 4. Word bond graph of the power converter connected to a RL load.

Fig. 5. Ideal voltage source with connection impedance. Natural frame (three phases) (a). Park frame (b).

Fig. 6. Complete (a) and simplified (b) current responses, considering the ideal voltage source.


P1!2 ¼jVcjjV j

XsinðdcÞ ð1Þ

where Vc and V are respectively the RMS values for vc and v, and dc is the phase of Vc versus V. Expression (1) shows that asmall difference on the angle has a direct impact on the active power flow. The active power control is defining the value fordc angle as shown on the schematic diagram of Fig. 7.

Fig. 7. Power control loop block diagram.


In Fig. 7, kP is the gain in the controller, which has to be calculated taking into account the desired time response, and xest

is the grid pulsation.Taking the current, voltage and power control laws, the power electronic source shown in Fig. 4 is connected to an infinite

bus as shown Fig. 8.In order to show the power control loop behavior, the bond graph model of Fig. 8 is simulated considering a time response

of 1 ms for the current, 10 ms for the voltage and 1 s for the power control loops.As we are analyzing a three phase electrical system (it is well known that instantaneous power is in each bond), ac-

tive power is calculated by adding the three phase instantaneous powers. Fig. 9 shows the active power for differentpower references and verifies the behavior of the power control with the different references given for the completemodel.

4. Alternator model

4.1. Alternator model

The alternator model is defined in a 2-axis d–q reference frame [15]. The main assumptions considered in the model are:

– Magnetic hysteresis and magnetic saturation effect are neglected.– The stator winding are sinusoidally distributed along the air–gap.– The stator slots cause no appreciable variation of the rotor inductances with rotor position.

Damper windings are represented by two short circuit windings for each axis (sources vkd and vkq = 0). The alternatormodel is presented in Fig. 10 [15].

The I-multiport elements Md and Mq correspond to the magnetic couplings between self and mutual inductances of d-axisand q-axis winding, respectively. They are characterized by inductance matrices:

Md ¼Laad Lmd Lmd

Lmd Lfd Lmd

Lmd Lmd Lkd

0B@

1CA; Mq ¼

Laaq Lmq

Lmq Lkq

� �ð2Þ

where Laad, Lmd, Lfd and Lkd are respectively self, mutual, exciting and damper windings inductances for d-axis; and Laaq, Lmq

and Lkq self, mutual and damper windings inductances for q-axis. In Fig. 10, I-element J represents shaft and rotor moment ofinertia; the effort source Talt is the applied torque; Ra, Rf, Rkd and Rkq are electrical resistances in each axis; and finally sourcesvd and vq represent input voltages applied to the alternator. The bond graph model in Fig. 10 is called here ‘‘normal alternatormodel’’.

As previously, we simplify the alternator model by canceling the stator dynamic. Thus, by setting _/Laad and _/Laaq equal tozero, the model order is reduced from 6 to 4. This is shown up on the bond graph in Fig. 11 with a causality change in bonds 7and 15.

As we can see in Fig. 11, the input voltage sources vq and vd of Fig. 10 have been changed by current sources id and iq, inorder to respect the causality change made. With a derivative causality in bonds 7 and 15, the relations for magnetic cou-plings between self and mutual inductances for d and q-axis change in the I-multiport as:

Fig. 8. Power converter into the infinite bus connection.

Fig. 9. Controlled active power converter.

Fig. 10. 2-Axis d–q reference frame alternator model.

Fig. 11. d–q-Axis simplified alternator model.


/Laad

ifd

ikd

0B@

1CA ¼

Laad � aLkdLfd�L2

md

ðLmdLkdÞ�L2md

LkdLfd�L2md

ðLmdLfdÞ�L2md

LkdLfd�L2md

�ðLmdLkdÞþL2md

LkdLfd�L2md

Lkd

LkdLfd�L2md

�Lmd

LkdLfd�L2md

�ðLmdLfdÞþL2md

LkdLfd�L2md

�Lmd

LkdLfd�L2md

Lfd

LkdLfd�L2md

0BBBBB@

1CCCCCA

id

/Lfd

/Lkd

0B@

1CA ð3Þ

/Laaq

ikq

� �¼

Laaq �L2

mq

Lkq

Lmq

Lkq

� Lmq

Lkq

1Lkq

0@

1A iq

/Lkq

!


where a ¼ ðL2mdLkdÞ þ ðL2

mdLfdÞ � 2Lmd.In both expressions, the first equation is a static relation, because the effort is set to zero (Fig. 11). These static relations

provide stator fluxes (/Laad, /Laaq) needed in MGY’s elements. The two neglected dynamics correspond to the stator winding,which means reflecting electric transient in stator voltage equations.

Mathematical validity of the simplification could be done via the calculation of the modes in the alternator model [7,1].Alternator equations being nonlinear, linearization of machine equations is thus required.

4.2. Alternator control laws

The classical control structure is shown in Fig. 12. It is composed of two parts:

– A PI controller is used to control the voltage at the point of common connection v. The controller acts on vfd in the exci-tation winding (voltage vd is chosen equal to zero).

– Another PI controller is used in the mechanical part. It corresponds to the speed control, which acts on the energy deliv-ered by the primary source. The dynamic of the primary source has not been taken into account in this model.

As for the power electronic converter, comparison between models is carried out connecting a RL load to the alternator.However, the inverse Parks transformation needs to be added in order to pass the 2-axis reference frame to a natural refer-ence (three phases).

Fig. 13 shows the complete model connected to a RL load (159 MW, 98 MVAR). This model is simulated with a large alter-nator of 187 MVA 13.3 kV at 50 Hz which parameters (see appendix) have been taken from [7]. The use of this alternator isjustified because comparing to a smaller one, the transitory oscillation are enhanced [7].

For the simulation a full load is used, and at time = 1 s a default is applied during 0.2 s.As shown in Fig. 14, simplified and modified models responses are nearly the same. Compared to the complete model, the

speed oscillations have been eliminated (Fig. 14a) and the current offset (Fig. 14b) also. It is notice that the general electro-mechanical behavior is the same for all these models.

5. Induction machine model

5.1. Induction machine model

Induction machines have been addressed in many papers. Models could be represented in two general ways: one using aPark reference frame [15,5,6], and the other one using the natural reference frame (three sinusoidal signals) [12].

An induction machine bond graph model in the Park 2-axis d–q reference frame related to the rotor is given in Fig. 15 [5].The assumptions considered for induction machine model are:

Fig. 12. Alternator with its control laws.

Fig. 13. Alternator with a load connection.

Fig. 14. Alternator speed (a) and current (b) responses for a connected load.


– Magnetic hysteresis and magnetic saturation effect are neglected.– The stator winding are sinusoidally distributed along the air–gap.– The stator slots cause no appreciable variation of the rotor inductances with rotor position.

The inductance matrices for I-elements Mimd and Mimq are given by:

Mimd ¼Ls Lm

Lm Lr

� �¼ Mimq ð4Þ

where Ls, Lm and Lr are respectively stator self inductance, mutual inductance between the stator and rotor, and rotor selfinductance. I-element Jim represents the shaft and rotor moment of inertia; Rs and Rr are stator and rotor resistances, p isthe number of pair of poles and x corresponds to 2pf, with f the network frequency.

The simplified induction machine model (stator dynamics neglected) is formulated in the same way as for the alternatorsimplified model. Causality change is required in bonds 4 and 19. Fig. 16 shows the stator d–q-axis simplified model of theinduction machine.

Matrix expressions in the I-elements are changed for:

/sd;q

ird;q

� �¼

Ls � L2m

Lr

LmLr

� LmLr

1Lr

0@

1A isd;q

/rd;q

!ð5Þ

Fig. 15. 2-Axis reference frame induction machine.

Fig. 16. Stator d, q-axis simplified induction machine.


Causality changes reduce from 5 to 3 the number of independent state variables in the model. The dynamic equation in thestator part is changed for a static one, and the effort is set equal to zero in these bonds.

6. Distributed electrical system

Alternator, power electronic converter and induction machine are associated as depicted in Fig. 17, to compose the dis-tributed electrical system.

Three SPJ (junctions 0vi, with i = 1, 2, 3) represent the three phase electrical network connection nodes. Park transforma-tion is necessary to convert the electrical d, q quantities to a three phase system (and vice versa). Fig. 17 shows up the sim-plified alternator model which imposes the effort in the SPJ because it is considered here as the main source in thedistributed system.

As mentioned previously, several operating modes may occur. We associate different SPJ’s conditions to these modes:

� Switching condition k1 becomes false when a fault occurs in the alternator or per maintenance purposes. This condition in0s-junctions involves:

– Bonds 1, 2, 3 from alternator and 7, 8, 9 from connection inductors are switched off when condition (k1) becomes false inthe 0vi SPJ.

– Simultaneously bonds 4, 5, 6 from the simplified connection inductors are switched on.– In the same instant SPJ, bonds 15, 16 are switched off in 0c1, 0c2 and 17, 18 are activated.

Since the alternator is tripped, the voltage is imposed with the renewable energy source. The change from connectioninductors to simplified connection inductors is necessary during this mode. The power control is also disconnected to moveto a voltage control mode. This allows feeding the necessary power in the load only with the converter.

Fig. 17. Bond graph model of the distributed electrical system.


� Switching condition k2 is true when the converter is connected to the network. This last condition allows connecting anddisconnecting the converter. In the distributed electrical system, special care should be taken at the moment of connec-tion. An estimation of the phase via a PLL (phase locked loop) is necessary. In [19] a complete review of different PLL isgiven. As said previously the choice of the model depends on the connection of the alternator on the microgrid or not.� Switch condition k3 allows connecting the induction machine. This means that bonds 10, 11 and 12 are validated when

condition becomes true; otherwise the induction machine is disconnected. In each case, it is possible to use a complete ora simplified model.

The sequence switching conditions are described hereafter.

1. The alternator is started. In order to proceed with the alternator phase estimation, the converter is kept disconnected (allconditions are false).

2. The induction machine is connected (t = 3 s).3. Nominal torque is applied on the induction machine (t = 6 s).4. Connection of the converter (t = 8.5 s).5. The three elements are connected and a default arise (voltage sag) and the alternator tripped (t = 12 s).6. Alternator reconnection (t = 16 s).

Fig. 17 regroups also the different control laws.The use of simplified and normal connection inductors in the bond graph model shows the different steps of the proce-

dure explicitly.

Fig. 18. Alternator and IM speed. Complete (a) and simplified (b) models.

Fig. 19. Active power flows in the system. Complete and simplified models.


Simulation results were obtained with software 20Sim�. We use an alternator of 31 kVAR, 400 V at 50 Hz, and an induc-tion machine of 3 Hp, 220 V at 50 Hz. Machine parameters have been taken from [7], and are given in the appendix.

The time response for voltage, speed alternator and also the power control with the power electronic converter is 1 s. Ta-ble 2, in appendix gives the numerical values for the controllers.

In Fig. 18, speeds in both machines are presented for the complete (Fig. 18a) and simplified (Fig. 18b) models. It is shownthat the small oscillations have been eliminated when using the simplified model. Also, the time taken for simulating thecomplete model is three times the time taken for the simplified ones.

Fig. 19 shows up the active power in each device.In the model, converter active power is set to Pref = 1 kW. For the interval time from 0 to 3 s, there is not power in

any device. Induction machine demands a very big power peak when connected during the starting transient in the ma-chine. As the induction machine demands principally reactive power, before and when the nominal torque is applied (3–8.5 s), the active power is given only by the alternator. At time = 8.5 s (when the converter is connected), the power ab-sorbed by the induction machine is distributed between the converter (1 kW) and the alternator (0.87 kW). In theinterval time when the alternator is switched off, all the power in the induction machine is fed by the converter,and the alternator power is zero. Finally, the same conditions as before are recovered, when the alternator is connectedagain.

Table 1Generalized variables for several physical domains.

Power variables Energy variables

Effort e Flow f Momentum p Displacement q

Mechanical (Transl) Force, F Velocity v Momentum p Displacement xMechanical (Rot.) Torque s Angular velocity x Angular momentum h Angle hHydraulic Pressure P Volume flow rate Q Pressure momentum pp Volume VElectrical Voltage u Current i Flux linkage k Charge q

Table 2Passive and active elements.

Elements BG symbol General relation Linearrelation

Physical elements

Resistor R R e = /R(f) e = R.f Electrical resistor, mechanical damper, dashpot, friction, hydraulic restriction

R f ¼ /�1R ðeÞ f = G.e = e/R

CapacitorC

C e ¼ /�1C ðR

fdtÞ integral e = 1/CR

fdt Electrical capacitor, mechanical spring,torsion bar, tank, accumulator

C f = d/C(f)/dt derivat. f = C.de/dt

Inertia I I f ¼ /�1I ðR

edtÞ integral f = 1/IR

edt Electrical inductance, mass, inertia

I e = d/I(f)/dt derivat. e = I.df/dt

Sources Se e(t) given, Voltage supply, pressure source, gravity, current supply, pump

Sf f(t) given,

Sensors De e(t) measured, f(t) = 0 Voltmeter, force sensor, pressure sensor, ammeter, flow rate sensor,tachometerDf f(t) measured, e(t) = 0


7. Conclusions and future work

A distributed electrical system has been presented. The modeling of such a system in terms of bond graphs enlightenedcausality assignment conflicts at the coupling node between devices. Causality purpose justified some modifications in alter-nator and connection inductor models. Complete or simplified model may be used for induction machine in each situation.

Power flows transferred and absorbed by each device were analyzed and presented in the simulation results. It wasshown up that the converter supplies power when the alternator is switched off. This was enlightened in the bond graphwith the necessity to cancel the power control loop in the converter.

The switch power junction concept has been used for studying the variable structure model, leading to a commutationvector condition which could be used for coordination proposes. In practice a distributed electrical system implies differentpossibilities from the point of view of connection and disconnection elements, which compose the complete system. In thepaper has been treated only one case, but the dynamic behavior and variable structure of a distributed system makes nec-essary local and global supervision of the system. In this context, as a next step the supervision using bond graphs needs tobe treated, introducing more connected renewable sources.

Also, stability analysis could be performed taking in consideration different power references in the converter.

Acknowledgment

The authors thank the support given by the Conacyt, Mexico.

Appendix A. Some recalls on bond graph methodology

A bond graph consists of subsystems linked together by half arrows, representing power bonds. They exchange instanta-neous power at places called ports. The variables that are forced to be identical when two ports are connected are the powervariables, considered as functions of time. The various power variables are classified in a universal scheme, and called eithereffort e(t) or flow f(t). Their product P(t)=e(t) � f(t) is the instantaneous power flowing between the ports. Two other types ofvariables, called energy variables, turn out to be important in describing dynamic systems: the momentum p(t) =

Re(t)dt and

the displacement q(t) =R

f(t)dt in generalized notation. Table 1 shows power and energy variables for several physicaldomains.

A few basic types of elements are required in order to represent models in a variety of energy domains. Table 2 regroupsbasic 1-port elements which respectively dissipate power (R), store energy (I,C) and supply power (sources). The detectorsare used for sensors, supposed to be ideal (no power dissipated).

Table 3Junction elements.

Elements BG symbol General relation Physical interpretation

0-Junction

n

i

2

1 0

e1 = � � � = en = ei Common effort junctionaifi = �a1f1 � � � � � anfn

a1 = +1, a2 = ai = an = �1

1-Junctioni

n

2

1 1

f1 = � � � = fn = fi Common flow junctionaiei = �a1e1 � � � � �anen

TF

2 1

TF

TF

e2 = 1/me1, f1 = 1/mf2 Electrical transformer, lever, gear pair, hydraulic rame1 = me2, f2 = mf1

GY1 2

GY

GY

f2 = 1/re1, f1 = 1/re2 Hall effect sensor, gyroscope, voice coil, DC motore1 = rf2, e2 = rf1

Table 4Parameters for the system.

ConverterLf = 1 mH, Rf = 0.1 X, C = 20 lF, Ll = 5 mH, Rl = 1 X, us = 760 V

Alternator 31 kVARRa = 0.214 X, Rfd = 0.118 X, Rkd = 0.5715 X, Rkq = 0.4981 X, Laad = 1.05 mH, Lmad = 24.3 mH, Lmaq = 11.6 mH, Lfd = 2.24 mH, Lkd = 3.02 mH, Lkq = 2.04 mH,

Ja = 0.22, pa = 2

Induction machineRs = 0.435, Rr = 0.816 X, Ls = Lr = 71.3 mH, Lm = 69.3 mH, Jim = 89 m, pim = 2

Alternator 180 MVARRa = 29 mX, Rfd = 0.509 mX, Rkd = 11.9 mX, Rkq = 20 mX, Laad = 0.309 mH, Lmad = 32.16 mH, Lmaq = 0.971 mH, Lfd = 0.307 mH, Lkd = 0.4907 mH,

Lkq = 1.03 mH, Ja = 3.89e6, pa = 20

Table 5Parameters for the alternator controllers.

Alternatorkvolt = 0.09, Tivolt = 1, kspeed = 1.84, Tispeed = 0.233


A causal stroke, placed perpendicularly to the bond, shows up the way the constitutive relations in an element have to bewritten, as shown Table 2.

Table 3 regroups the junction structure elements with their causality restrictions. They are power conservative.The state variables obtained from a bond graph model are the energy variables associated with the dynamic I and C ele-

ments; the state vector is defined as x ¼ ½ ptI qt

C �t . Input u and output y variables are associated respectively with the sources

(control inputs MSe or MSf modulated by a state or a measurement feedback, or Se or Sf) and the detectors De and Df (vari-ables to be controlled).

The state equation is systematically deduced from the bond graph model in preferential integral causality (BGI).

NomenclaturePower converter
Rli connection resistance Lli connection inductance vCi voltage output converter m1, m2 control signals us constant voltage source vCref reference voltage tension vdc voltage axis d (Park frame) vqc voltage axis q (Park frame)


ifi
current filter dc phase of Vc versus V xest grid pulsation P active power Pref active power reference Alternator Ra stator resistance Rfd exciting resistance Rkd resistance axis d Rkq resistance axis q vfd exciting voltage Md/q magnetic couplings (axis d/q) vfd exciting voltage /Laad/Laaq stator fluxes (axis d/q) Laad self inductance (stator) Lfd exciting inductance Lkd damper winding inductance (axis d) Lkq damper winding inductance (axis q) Lmd mutual inductance (axis d) Lmq mutual inductance (axis q) Talt applied torque J inertia Induction machine x 2pf with f the network frequency Mimd/imq magnetic couplings (axis d/q) /s/r,d/q fluxes of stator/rotor in axis d/q Rs rotor resistance Rr stator resistance Ls self stator inductance Lr self rotor inductance Lm mutual inductance p number of pair of poles Tm torque Jim inertia
Table 4 gives the numerical values used in the complete model.Table 5 contains the numerical parameters for the alternator controllers.

References

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Hybrid electrical power system modeling and management

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