Simulation Modelling Practice and Theory 18 (2010) 1185–1198

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Simulation Modelling Practice and Theory

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

Bond graph based control of a three-phase inverterwith LC filter – Connection to passive and active loads

Roberto Sanchez a,b,c,d,*, Geneviève Dauphin-Tanguy a,c,d, Xavier Guillaud a,b,d, Frédéric Colas a,e

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 8146, F-59650 Villeneuve d’Ascq, Francee Arts et Métiers ParisTech, L2EP, F-59000 Lille, France

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

Article history:Received 1 December 2009Received in revised form 11 May 2010Accepted 30 May 2010Available online 2 June 2010

Keywords:Bond graphInverterPassive and active load

1569-190X/$ - see front matter � 2010 Elsevier B.Vdoi:10.1016/j.simpat.2010.05.016

* Corresponding author at: ECLille, L2EP, F-59650E-mail addresses: roberto.sanchez@ec-lille.fr (R

Guillaud), frederic.colas@ensam.eu (F. Colas).

This paper proposes a model of a three phase electrical inverter with a LC output filter indelta connection used in a renewable energy supply system. The concept of inverse bondgraph via bicausality is used for the control law design. The control law robustness is testedby connecting passive and active (induction machine) loads.

� 2010 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 offossil and nuclear fuels. Nowadays the renewable energy sources start to play an important role in power systems, and thegreat majority of them are integrated into the electrical network using power electronic converters.

Most of the time, distributed generation is connected to the grid as a simple current injector which means that the sourceneeds to be synchronized on an existing grid. In the present paper, the source gives a voltage source behavior. Indeed, thisdevice can create its proper network and deliver its power to any sort of load which can be connected to this grid. In theseconditions, we must add a storage element to a renewable source to guarantee the energy to be available when needed. Thissolution is very well known and has been used for many years in case of DC local network.

Recent studies are involved in developing important AC microgrid which may be connected to a powerful network or not.CERTS in United States [20] was one of the first laboratory to develop such a microgrid. Several European projects have beeninterested at this aim [16]. An interesting review of all these developments is proposed in [13]. Other papers have been pub-lished in the same area [6,3]. Even if lot of work has been done on these topics, the development of electrical network whosepower is delivered only by power electronic converter and energized by non conventional source is a real challenge. First, thesmaller and weaker the network is, the more accurate physical models of the different elements are needed. Furthermore,the generation of voltage source is much more complicated than with a synchronous machine.

In classical electrical machines, the voltage source is generated by the interaction of a rotating field with the statoricwindings. The sinusoidal form is guaranteed and the action on field current is controlling the root mean square value of

. All rights reserved.

Villeneuve d’Ascq, France. Tel.: +33 3 20 33 54 15.. Sanchez), genevieve.dauphin-tanguy@ec-lille.fr (G. Dauphin-Tanguy), xavier.guillaud@ec-lille.fr (X.

1186 R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198

the voltage. In case of power electronic devices, the voltage waveform delivered to the grid results from a closed loop oper-ation. The stability must be assessed for a very large range of loads. As said before, good behavior of this type of networkrequires different types of study: accurate modeling of new type of source, stability assessment . . . Bond graph methodologyis a generic tool which may answer to these different requirements. This paper may be considered as a first step in thisdirection.

The bond graph model of a three phase inverter has been addressed in several research papers, as for example, for themost recent publications: [15,1,7]. Each presented model has different characteristics and is used for different applications.They share the particularity of being established with an electronic switch (MTF) for modeling the three commutation cells ofthe converter, taking into account phase-neutral voltages. Here, the three phase electrical inverter is modeled with only twoelectronic switches, taking the phase–phase voltages [9]. The proposed model is more general since it is valid in case ofunbalanced or balanced mode.

To design the control law, the inverse bond graph model is derived using bicausality concept [4]; the notion of bicausalityintroduced by Gawthrop can be used to point out properties about inverse dynamics [18] and parameter estimation [19]. Aspecific controller is proposed combining the inverse model and a resonant controller.

The load in an electrical system can be of passive (RL load) or active (induction machine) type.The induction machine bond graph model has been addressed in several papers. The model can be represented in two

general ways: one using the arbitrary reference frame (based on Park’s transformation) [21,10,11], and the other, the naturalreference frame (three sinusoidal signals) [17,24]. Here we will use the second approach [2].

The outline of this paper is as follows. In the second section, a presentation of the global system is given. In Section 3, thebond graph model of a three phase inverter with LC filter is proposed and some causality problems are discussed. A two stagecontrol law for the inverter is given in Section 4. In Section 5, experimental and simulation results for a passive load areshown and a robustness study is archived. The induction machine model is presented in Section 6. Section 7 shows exper-imental and simulation results for an active load.

2. Presentation of the global system

Fig. 1 shows the global scheme of a renewable energy source connected to a load. This structure is the basic one and canbe analyzed at different levels of complexity.

The renewable energy source can be a series of photovoltaic panels, a wind turbine associated with storage elements; itmay also be a cogeneration unit with a microturbine. It delivers the primary power in the DC bus which will be convertedinto AC power via the power converter. The introduction of a capacitor on the DC bus is necessary in order to allow decou-pling between the harmonics generated by both converters.

The electronic power converter can be considered as an amplifier. It is composed of three pairs of electronic switches(Mosfet, IGBT, etc.). Each pair of switches is called a commutation cell.

In the filter part, the basic one is known as a LR filter, which consists only of an inductance in series with a resistance; itcan be taken as a base for more complex filters, simply by adding components, for example LC one (Fig. 1) composed of oneinductance and resistance in series with a capacitor in parallel. Each filter configuration has its own characteristics and isused for specific requirements or analysis from the point of view of stability and harmonics analysis.

In Fig. 1, the currents in inductances Lf1, Lf2, Lf3 and the voltages of capacitors C1, C2, C12 are controlled. Thus, the behaviorof this system may be assimilated to a voltage source, contrary to the classical LR filter structure which is assimilated to acurrent source. This system can create a small microgrid since it controls the voltage magnitude and frequency.

The load will be chosen here firstly as passive (LR) and later as active (induction machine).

Fig. 1. Global electrical system.

R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198 1187

3. Models of the inverter and the filter

In this paper, the model of the inverter shows up explicitly only two of the three commutation cells, the third one fixingthe voltage reference. Only two phase to phase voltages um1 and um2 have to be considered for control [9]. The current if3 islinearly dependent on the two others (if1 and if2).

We are focusing on the average behavior of the converter. Furthermore, we are not interested in the switches themselves,but on the meantime impact of the converter on the LC filter.

Thus, we define two control signals m1 and m2 as:

m1 ¼um1

usand m2 ¼

um2

usð1Þ

Fig. 2. Bond graph model of the three phase inverter with a LC filter.

Fig. 3. Bond graph model with the connection of a passive load.

1188 R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198

The primary source of the converter in the DC bus is represented as a constant voltage source us, by assuming that it hasno variation or fluctuation.

The three phase LC filter is composed of three inductances with their associated resistance and three Delta connectedcapacitors. Fig. 2 shows the bond graph model of the inverter.

In Fig. 2, elements I: Lf3 and C: C12 have a derivative causality. These elements could be assigned to integral causality but itwould introduce a causality loop with unity gain between bonds 12, 15, 17, 18, 20 and 14; in order to avoid this problem it isnecessary to put these two elements in derivative causality.

The bond graph model explicitly points out the physical insight of the system with the two elements in derivative cau-sality (Lf3 and C12) and four elements in integral causality which means a fourth order model.

Finally, Fig. 3 shows the bond graph for the whole system in case of a connection of LR elements (passive load in star con-nection) on the LC filter.

The four detectors (Df: if1, Df: if2, De: uc1, De: uc2,) shown in Fig. 3 are associated with the sensors.

4. Control of the inverter and the filter

As shown previously, the three-phase LC filter is a highly coupled 4th order system with only two control signals m1, m2.In order to give a voltage source behavior to the LC filter, we have to control the three-phase voltage supplied by the asso-ciated capacitors. For this aim, a specific algorithm has to be designed. It is based on bicausal bond graph [5], and is designedon the inverse bond graph.

For the formulation of the inverse bond graph, it is necessary to change the flow detectors Df (supposed ideal) of the ori-ginal bond graph for sources (named Se0Sf, because they impose zero effort-no null flow to the inverse model), then prop-agate bicausality (in only one line of power transfer) from this source (SS) to the input source of the original bond graphwhich becomes a detector in the inverse bond graph [18].

With the inverse bond graph, we design the structure of the control in open loop. The decoupling actions are defined (in-verse matrix and disturbance compensation). The open loop structure is then extended to a closed loop control by fixing thedynamics of errors.

There are two different ways to compute the controller parameters:

� The first way assumes that the dynamics in the I: Lfi elements are faster, compared with the dynamics in the C-elements.Thus, only the voltages in C1 and C2 are controlled in closed loop; the voltage in the dependent C12 element is directlyrelated to the two others.� The other way considers the two loops in cascade, one for the current in the Lfi and another one for the voltage in Ci-ele-

ments. In this paper, the second approach has been chosen, for performance purpose.

The calculation of the control laws in the model is made in two steps, the first one for the currents and the next one for thevoltages.

Fig. 4 shows up the inverse bond graph, which allows deducing the open loop current control law. The two current sen-sors are inverted simultaneously via bicausal bonds and two disjoint bicausal paths are drawn to the two desired inversemodel outputs corresponding to the two control signals (m1, m2), which proves that the model is invertible [18].

The I-elements are in derivative causality. Eq. (2) is derived from the inverse bond graph of Fig. 4, by setting that thenumerical values of Rfi and Lfi are the same for the 3 phases. These values have been taken as estimated values bLf and bRf

of the actual system parameters. It gives:

m1

m2

� �¼ 1

us

� � bRf þ bLfddt

� �2 11 2

� �if 1

if 2

� �þ

uc1

uc2

� �� �ð2Þ

For establishing the closed loop control law, we set up the dynamics of the error (ei ¼ ifrefi� ifi

) in both expressions, as_ei þ kp1ei ¼ 0, where kp1 is a proportional gain to be fixed. Expression (2) becomes (3) as:

m1

m2

� �¼ 1

us

� � bRf þ bLfddt

� �2 11 2

� �if 1 � e1

if 2 � e2

� �þ

uc1

uc2

� �� �ð3Þ

Finally,

m1

m2

� �¼ 1

us

� �2 11 2

� � bRfif 1

if 2

� �þ bLf

ddt

if 1ref

if 2ref

� �þ bLf kp1

if 1ref � if 1

if 2ref � if 2

� �þ

uc1

uc2

� �� �� �ð4Þ

Expression (4) is represented in Laplace domain as:

m1

m2

� �ðsÞ¼ 1

us

� �ðsÞ

2 11 2

� � bRfif 1

if 2

� �ðsÞþ bLf s

if 1ref

if 2ref

� �ðsÞþ bLf kp1

if 1ref � if 1

if 2ref � if 2

� �ðsÞþ

uc1

uc2

� �ðsÞ

! !ð5Þ

Fig. 4. Inverse bond graph for calculation of the current control law.

Fig. 5. Inverse bond graph for the voltage control law.

R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198 1189

In expression (5), gain kp1 is calculated to get a 2500 rad/s dynamic for the current loop. It will be chosen the same for allphases. The desired performances for the closed loop model are given hereafter.

As for the determination of the current control, the voltage control law is calculated; it provides the current reference forEq. (5). Fig. 5 shows the inverse bond graph used for the determination of the current reference.

For deriving Eq. (6) from Fig. 5, we follow the same procedure as previously. Two bicausal disjoint paths exist from oneSeiSf0 to one De0Dfi. The model is thus invertible. It leads to

if 1ref ¼ C1ddtð2uc1 � uc2Þ þ il1

if 2ref ¼ C2ddtð2uc2 � uc1Þ þ il2

ð6Þ

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Eq. (6) shows up the voltage control law for the converter. Taking into account the fact that the three capacitors have thesame numerical capacitances (C1 = C2 = C12 = C) and using the estimated value bC , expression (6) becomes:

if 1ref

if 2ref

� �¼

2 11 2

� �C

ddt

uc1ref

uc2ref

� �þ Ckp2

uc1ref � uc1

uc2ref � uc2

� �� �þ

il1

il2

� �ð7Þ

As in Eq. (4), kp2 is a proportional gain to be fixed. It is changed for a resonant controller given Eq. (7) [23,8].

CðsÞ ¼ n2s2 þ n1sþ n0

s2 þx2n

ð8Þ

The natural frequency xn is adjusted to the input signal frequency in order to obtain an infinite open loop gain at thisfrequency [14].

In Laplace domain, expression (7) becomes Eq. (9).

if 1ref

if 2ref

� �ðsÞ¼

2 11 2

� � bCsuc1ref

uc2ref

� �ðsÞþ bC n2s2 þ n1sþ n0

s2 þx2n

� �uc1ref � uc1

uc2ref � uc2

� �ðsÞ

!þ

il1

il2

� �ðsÞ

ð9Þ

The parameter calculation in the resonant controller is achieved supposing the current loop dynamic is infinite. If weassimilate the current to its reference, the model becomes a sixth order one: 2 orders for the capacitors and 4 orders forthe 2 resonant controllers. The 6 poles are placed as following:

P1;2 : 500 ðrad=sÞ p3;4 : 500� i314 ðrad=sÞ p5;6 : 500� i314 ðrad=sÞ

This choice gives the model an approximate 10 ms response time.The controller block diagram of the complete model is shown in Fig. 6. It is important to notice that the structure of the

control law contains a feed-forward with a derivative action on the references. The reference signals are sinusoidal and theycan be derived analytically without any problem.

The filter parameters are not supposed to have a strong evolution because this filter is a part of the voltage source itself.On the contrary, the load may have a very large parameter evolution.

A robustness study is undertaken in respect with the evolution of the three-phase RL load. Two different power factors(PF = 0, 0.8) are considered with an evolution from no load case to nominal point (5 kW).

The robustness study is done with the pole plot, which can be taken directly from the simulator. The numerical values forcontrollers and model are given in the appendix. The pole plot is drawn assuming a slight error on the parameters of thefilter: 10% between the model and the controller parameters, in order to take into account the uncertainties due to parameterestimation (see appendix).

For the first case, when a power factor equal to zero is taken, the reactive power Q is null. Therefore, the load absorbs onlythe active power P. Fig. 7 shows the pole plot of the complete model.

We can notice a very slight deviation of the eight poles from 0 to 5 kW loading. The most important deviation is observedfor the double pole located near 2000 rad/s.

An inductive 0.8 power factor is now taken, the apparent power S gives the active and reactive power that is expected inthe load. For this case, we have considered a variation from no load case to 5 kVA of apparent power. The two added polescorrespond to the load. Since the L/R ratio does not change, the pole relative to the load is not moving (Fig. 8).

As in the first case, the variation of the load has no significant impact on the pole position; therefore the robustness of thecontrolled model is verified, in both cases.

Fig. 6. Inverse model controller block diagram in sinusoidal reference.

Fig. 7. Pole plot for a power factor PF = 0.

Fig. 8. Pole plot for a power factor PF = 0.8.

Fig. 9. Pole plot for a power factor PF = 0.8 – unbalanced load.

R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198 1191

As a final case in the robustness study, let us consider the worst case which is given when an unbalanced load is con-nected. In this case, a bi-phase load is considered, it means that one of the three phases is disconnected leaving only twoof them.

From Fig. 9, the pole placed next to 300 rad/s (load pole) is now a single pole. Also, it can be noticed that only one of thedouble poles (next to 2000 rad/s) is displaced, the other ones stay unchanged when the apparent power increases.

After verifying the robustness in the control law proposed via the pole plots, we present the experimental and simulationresults for the passive load in the next section.

1192 R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198

5. Simulation and experimental results for a passive load

The simulation results were obtained with software 20Sim�.Fig. 10 show respectively the load voltage (10a) and current (10b) responses obtained in simulation, when an unbalanced

series LR load (which absorbs 4 kVA) is applied at time = 0.717 s. A very good quality is obtained for the voltage even whenthe unbalanced load is applied on the voltage source.

The experimental implementation, were done on a real time simulator as shown in Fig. 11. A virtual inverter with LC filteris embedded in a real simulator (RT-LAB) and external dSPACE 1103 is used as controller to implement the current and volt-age closed loop controls.

The capacitor voltages (v1, v2, v3) are transmitted to the high bandwidth power amplifier which applies the voltages to theRL load.

The currents flowing into the load are measured and introduced into the real time simulation model via an ADC (analog-ical to digital converter), the load behaving as a three phased current source.

The same numerical values as for simulation are considered for the control law.The conditions are similar as for simulation. The unbalanced load is applied as a step at 0.717 s approximately. The three

single phase voltages are shown in Fig. 12a.

(a) (b)

Fig. 10. Simulation responses: – output voltages (a) – current load (b).

Fig. 11. Scheme of the experimental set up.

(a) (b)

Fig. 12. Experimental responses step change at 0.717 s: – output voltages (a) – current load (b).

R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198 1193

As expected, it can be noticed in Fig. 12a that the unbalanced load has no impact on the voltages.Experimental and simulation results are very similar. We want to improve the robustness study by applying a very strong

constraint on the system: starting an induction machine. This is the topic of the next section.

6. Induction machine model

The active load used to test the proposed control algorithm is an induction machine. The induction machine bond graphmodel is given in Fig. 13 [2].

The bond graph model shows up stator (Rs1, Rs2, Rs2) and rotor (Rr1, Rr2, Rr3) three-phase resistors, self and leakage induc-tances between stator and rotor (IC-multiport) and mechanical part (I-element and torque source T).

(1) Stator part: The assumptions considered are:� Resistive and inductive effects are only taken into account and capacitive effects are neglected. Thus, stator currents are

represented separately (is1, is2, is3).� The three-phase windings are sinusoidally distributed circuits with Ns equivalent turns and balanced Rs resistance.� Star or Delta connection stator windings can be feasible in the model.

(2) Rotor part: Likewise, the assumptions are:� Resistive and inductive effects are only taken into account.� Three-phase windings are sinusoidally distributed circuits with Nr equivalent turns and balanced Rr resistance.� The three windings are short circuited (source Se = 0).

Fig. 13. Induction machine bond graph model.

1194 R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198

(3) Self and leakage inductances part: Relations of self and leakage inductances between the stator and rotor are in the IC-multiport. For a magnetically linear system, the flux linkages can be expressed as:

w1;2;3s

w1;2;3s

!¼

Ls LM

LTM Lr

� �i1;2;3s

i1;2;3r

� �ð10Þ

The winding inductances are given by:

Ls ¼Lls þ Lms � 1

2 Lms � 12 Lms

� 12 Lms Lls þ Lms � 1

2 Lms

� 12 Lms � 1

2 Lms Lls þ Lms

0B@1CA ð11Þ

Lr ¼Llr þ Lmr � 1

2 Lmr � 12 Lmr

� 12 Lmr Llr þ Lmr � 1

2 Lmr

� 12 Lmr � 1

2 Lmr Llr þ Lmr

0B@1CA ð12Þ

Mutual inductances between stator and rotor windings are:

LM ¼ Lsr

cosðhÞ cos h� 2p3

� �cos hþ 2p

3

� �cos hþ 2p

3

� �cosðhÞ cos h� 2p

3

� �cos h� 2p

3

� �cos hþ 2p

3

� �cosðhÞ

0B@1CA ð13Þ

Fig. 14. Bond graph model with the connection of an active load.

Fig. 15. Actual induction machine.

R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198 1195

In the above equations, Lls and Lms are respectively leakage and magnetizing inductances of stator windings; Llr and Lmr arethe equivalents for the rotor windings; and Lsr is the mutual inductance between stator and rotor windings. h is the rotorposition.

(a)

(b)

(c)

Fig. 16. Simulation: – Induction machine speed (a) – current responses in the induction machine (b) – converter output voltages applied to the inductionmachine (c).

1196 R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198

(4) Mechanical part: It is composed of the rotor inertia J, the load torque T and the electromagnetic torque Te, as:

Fig. 17inducti

Jddt

X ¼ Te � T ð14Þ

where O is the rotor speed.The electromagnetic torque is expressed as a function of stator and rotor currents, and of stator–rotor mutual inductance

derivatives with respect to rotor position h, according to:

(a)

(b)

(c)

. Experimental: – induction machine speed (a) – current responses in the induction machine (b) – amplifier output voltages applied to the actualon machine (c).

R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198 1197

Te ¼X3

x¼1

isx

X3

y¼1

iry

dLsrx;y

dhð15Þ

By replacing h by phrm, where hrm is the mechanical position of the rotor and p is the number of the pole pairs, Te can thusbe expressed as:

Te ¼ ð is1 is2 is3 Þ �d

dhrmðLMðhrmÞÞ3�3 �

ir1

ir2

ir3

0B@1CA � p ð16Þ

Fig. 14 shows the connection of an induction machine as a load for the converter.As LC filter output voltages are phase-phase voltages (u1, u2, u3), bonds 4, 5 and 6 are necessary to introduce single-volt-

ages (v1, v2, v3) in the induction machine. MTFi elements represent ideal switches (Boolean modulated transformers) intro-duced in order to connect at time = tIM the induction machine.

7. Experimental and simulation results for an active load

An induction machine of 3Hp, 220 V phase-phase 50 Hz, is used in the simulation. The machine parameters given inappendix have been taken from Ref. [12]. They do not fit exactly with the used induction machine unknown ones, whichexplains the amplitude differences between experimentation and simulation results.

The induction machine used for experimentation (Fig. 15) is of 4HP, 380 V phase–phase 50 Hz, 1425 rpm.The same parameters are used for the converter in simulation and experimentation.It is well known that at starting phase the induction machine stator current is very important, and thus it may have a

large impact on voltages controlled by the power converter. Figs. 16 and 17 show the simulation and experimentationresults.

When the induction machine is connected at 1.7 s, we observe little oscillations in the speed response (Fig. 16a) due to theelectromechanical pole of the machine. In the simulation a step load is applied at time = 9.5 s.

Fig. 16b presents the very large current circulating in the phase of the machine during the starting operation. We can no-tice on Fig. 16c that this has no significant consequence on the voltage waveforms which proves a very good quality of thecontrol and a good robustness in case of non linear disturbance.

From experimentation, the induction machine currents are shown in Fig. 17b. The impact of current on the voltage maybe considered as negligible (Fig. 17c) when applying the load torque.

8. Conclusions

Lots of works have still been developed on microgrid applications. These kinds of networks, which will be probably devel-oped in the future for specific applications, are very weak and moreover may be fed either by classical electrical sources or bypower electronic devices. Indeed, it is most important to get a very accurate modeling of such networks and of the differentconnected devices. Furthermore, specific control algorithms have to be developed. Bond graph modeling may be a good toolfor a global approach to achieve such studies. This paper is a first step which proposes modeling and control law for a simplemicrogrid composed of an induction machine connected to a three-phase LC filter. A general model for three-phase powerconverter was proposed, available either in balanced situations or unbalanced ones. The LC filter was a 4th order model andthe control law derived from the inverse bond graph using bicausal bond graph methodology leads to a robust control law.Some simulation and experimental tests proved the robustness of the developed system in the case of an induction machinestarting on this network.

Some future work are yet under progress and use the bond graph properties for solving causal problem when asso-ciating different devices on the same network [22]. Moreover it will be possible to develop energetic modeling of dif-ferent types of sources (wind turbine, micro turbine . . .) and propose a unified modeling approach for all thesecomplex systems.

Acknowledgement

The authors would like to thank the support given by Conacyt, Mexico.

Appendix A

See Tables 1 and 2.

Table 1Numerical values for controllers and model.

ModelLf = 1.1 mH Rf = 0.11 X C = 22 lF Ll = 0.0704 H Rl = 22.77 X

Values for the control lawbLf ¼ 1 mH bRf ¼ 0:1X bC ¼ 20lF n2 ¼ 0:033 n1 ¼ 18:15 n0 ¼ 4413 kp1 ¼ 2500

Table 2Numerical parameters for an induction machine.

Number of poles pair 2 Stator inductance (Lls) 0.0024 HStator resistance (Rs) 0.435 X Rotor inductance (Llr) 0.0024 HRotor resistance (Rr) 0.816 X Leakage inductance (Lms, Lmr) 0.0832 HInertial mass (J) 0.089 kg m2 Mutual inductance (Lsr) 0.0832 H

1198 R. Sanchez et al. / Simulation Modelling Practice and Theory 18 (2010) 1185–1198

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