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Mathematics and Computers in Simulation 81 (2010) 225–238

Modeling and control of a cascaded doubly-fed induction generatorbased on dynamical equivalent circuits

N. Patin a,∗, E. Monmasson b, J.-P. Louis c

a University of Technology of Compiegne, Laboratoire d’Electromecanique (EA 1006), 60205 Compiegne, Franceb University of Cergy-Pontoise, SATIE (UMR CNRS 8029), 95031 Cergy-Pontoise, France

c ENS Cachan, UniverSud-Paris, SATIE (UMR CNRS 8029), 94235 Cachan, France

Received 2 October 2008; received in revised form 29 March 2010; accepted 10 April 2010Available online 2 May 2010

Abstract

This paper deals with the control of an autonomous cascaded doubly-fed induction generator operating in a variable speedconstant frequency mode. The proposed structure is a full stand-alone generating system dedicated to isolated grids in embeddedsystems or in small-scale renewable energy systems such as windmill and hydropower generators. The study is focused on theCDFIG. Its behavior against several design parameters (numbers of pole pairs and rotor interconnection) is recalled. A model, basedon dynamical equivalent circuits, is also given for the design of the controller. Finally, the synthesized controller is validated bysimulations and experimental results.© 2010 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Cascaded doubly-fed induction generator; Variable speed constant frequency; Isolated grid; Aircraft applications; Topology analysis;Modeling, Control

1. Introduction

Stationary and mobile applications require more and more electrical power. In these two cases, constant frequencyAC grids are usually used whereas, in many cases, primary mechanical power sources operate at variable speed. Thus,an adapted architecture of generating systems is required. The only kind of electromechanical generator, operatingat variable speed, which can be directly connected to a constant frequency grid is the doubly-fed induction machine(DFIG). However, some applications require high level reliability. As a consequence, cascaded doubly-fed inductiongenerator (CDFIG) is a good candidate for DFIG replacement because of its similar behavior without brushes and slingrings. Indeed, CDFIG has been already studied in various applications such as windmills [10], small-scale hydropower[6] and aircraft power supply [7]). This association of two doubly-fed induction machines (DFIMs), mechanically andelectrically coupled, can be justified on the basis of a functional inversion principle and the constraint forbidding slidingcontacts in targeted applications. This synthesis methodology will be developed in the following section (Section 2).Then, three parameters can be identified in the CDFIG:

∗ Corresponding author.E-mail address: [email protected] (N. Patin).

0378-4754/$36.00 © 2010 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2010.04.016

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226 N. Patin et al. / Mathematics and Computers in Simulation 81 (2010) 225–238

• Number of pole pairs p1 of the first DFIM (DFIM1).• Number of pole pairs p2 of the first DFIM (DFIM2).• Electrical coupling sequence between rotor windings of the two machines (two cases: direct and inverse coupling

sequences characterized using a coefficient λc = +1 and λc = −1, respectively).

In Section 2, the power flow through the two machines is studied in order to evaluate qualitatively the generatorefficiency (Section 4) for all cases of parameters values. Then, even if CDFIG modeling (Section 4) is as generativeas possible, it is oriented by these preliminary results, more particularly concerning coupling sequence between rotorwindings. So, a model of a realistic generator is established using a graphical representation of the system, adaptedto the end-user point of view: an electrical scheme. The representation proposed in this paper (dynamical equivalentcircuit representation) is not limited to the classical case of sinusoidal steady-state operation but it can be applied to atransient description of the system. The innovating aspect of this work concerns the design methodology allowing toestablish a hierarchical control structure on the basis of model inversion principles applied to the dynamical equivalentcircuit representation of the system (see Section 5). Finally, all this study is validated by simulations and experimentalresults (Section 6). All the parameters of this experimental set-up have been used in simulation using Matlab/Simulinkand the proposed controller has been validated and implemented in a real-time controller board.

2. Architecture synthesis

An architecture of electrical generator can be synthesized using a functional inversion methodology: the onlyvariables which are initially defined are the outputs of the system. In the case studied here, a constant frequency outputvoltage is wanted. Thus, due to the inductive behavior of induction (and synchronous) machines, parallel capacitorsconnected to the grid are required. Then, a variable speed constant frequency (VSCF) generator must be introducedbecause of speed variations of the prime mover in such applications. A good candidate for VSCF operation is thewell-known doubly-fed induction machine (DFIM) because stator frequency can be maintained constant in spite ofspeed variations thanks to opposite variations of the rotor frequency, controlled by a Voltage Switching Inverter (VSI).This classical structure, which is currently used in windmills is not completely adapted in the aircraft environment dueto slip rings in the DFIM, prohibited in this case. So, another solution must be investigated.

If the DFIM is conserved, rotor windings must be supplied by another three-phase AC machine: another DFIM isintroduced as it is shown in Fig. 1. It can be seen that these two doubly-fed induction machines can be directly connectedand finally integrated, giving us a complete brushless solution called cascaded doubly-fed induction machine—CDFIM(more precisely here Generator—CDFIG). Notice that the global structure of the generator based on a single doubly-fedinduction generator can be conserved with the CDFIG: the stator windings of the DFIM no. 1 are connected to a VSIand this inverter is supplied by a PWM rectifier connected to a Permanent Magnet Synchronous Machine (PMSM)allowing full stand-alone capabilities to the complete VSCF generator.

Fig. 1. Full stand-alone generator based on two cascaded doubly-fed induction machines (DFIMs).

N. Patin et al. / Mathematics and Computers in Simulation 81 (2010) 225–238 227

The study presented in the following section is focused on the CDFIG. Indeed, it is necessary to analyze the behaviorof this structure with respect to its main parameters (p1, p2 and λc).

3. CDFIG analysis

This study is based on a simplified model of the doubly-fed induction machine where copper/iron losses and magneticleakages are neglected. Thus, a DFIM is characterized by the following equations⎧⎪⎨

⎪⎩ωs = ωr + p�

Pr = −s · Ps

Ps + Pr = Pm

(1)

corresponding to a super-synchronous motor convention, illustrated by Fig. 2 where

• ωs is the stator frequency (rad/s),• ωr is the rotor frequency (rad/s),• � is the mechanical speed (rad/s),• p is the number of pole pairs of the DFIM,• Ps is the (entering) stator power,• Pr is the (entering) rotor power,• Pm is the (outgoing) mechanical power,• s is the slip ratio defined as follows

s� ωr

ωs

(2)

In order to avoid all ambiguity between the two DFIM of the cascade, 1 and 2 subscripts will be employed for allquantities in the system. Then, several relations must be introduced to describe the rotor interconnections:

ωr1 = λc · ωr2 (3)

Fig. 2. Doubly-fed induction machine in super-synchronous motor convention.


Fig. 3. Cascaded doubly-fed induction generator (CDFIG) power flow convention.

Fig. 4. Dynamical equivalent circuit representation of the doubly-fed induction machine (DFIM) in its stator dq reference frame.

where λc is the interconnection coefficient between the rotor windings, defined as follows

λc ={

+1 for a direct coupling sequence

−1 for an inverse coupling sequence(4)

and

Pr1 = −Pr2 (5)

due to the same convention used for the two machines (super-synchronous motor convention) as it can be seen in Fig. 3.All configurations have been studied in [8] and it has been shown that the inverse coupling sequence is the most

interesting solution with p2 ≥ p1. Notice that the test bench used in this paper is based on two DFIMs with the samenumber of pole pairs (p1 = p2 = 2).

4. Dynamical modeling using dynamical equivalent circuits

4.1. Single DFIM dynamical model

A dynamical equivalent circuit of the doubly-fed induction machine can be easily deduced from dq equations of themachine in an arbitrary reference frame (stator reference frame here) as it is shown in Fig. 4. Notice that it describethe machine not only for steady-state operation but it is also valuable in transient mode operations. However, severalcomponents of the well-known steady-state equivalent circuit can be identified:

• magnetizing inductance, noted Lcs,• leakage inductance σLcr,• ideal transformer (transformer turns ration m),• and obviously, stator and rotor resistances, respectively Rs and Rr.


Fig. 5. Complex conjugate transformer introduced by the inverse coupling sequence between rotor windings of the CDFIG.

The electromechanical conversion is localized in the back e.m.f. er. More precisely, this back e.m.f. is composedof two terms

er = jp�(M · iμ + σLcr · ir

)= em + ear

(6)

where

• em = jp� · M · iμ is precisely the term which corresponds to the electromechanical conversion• ear = jp� · σLcr · ir, which can be seen as an armature reaction e.m.f.

4.2. Rotor interconnection modeling

In the cascaded doubly-fed induction generator, two doubly-fed induction machines are connected by their rotorwindings. As it has been presented in Section 3, the inverse coupling sequence has a real interest for generation systemsapplications.

Finally, a rotor interconnection model is proposed within this framework in Fig. 5. It is illustrated by an idealtransformer with a complex transformer turns ratio, noted mc and defined as follows

mc (θ) = e−j(p1+p2)θ = e−j�pθ (7)

This transformer is also characterized by conjugate operations introduced in Fig. 5. This is why this transformer iscalled below “complex conjugate transformer”.

4.3. CDFIG model

Notice that for the CDFIG modeling, variables associated to DFIM1 and DFIM2 are distinguished by 1 and 2 sub-scripts. An initial model based on a simple interconnection of all the building blocks models be be directly established.However, the two rotor circuits of the DFIMs include two leakage inductances which are associated to the same statevariable. Indeed, the rotor currents ir1 and ir2 are linked as it is shown in Fig. 5. Thus, it is mandatory to merge thesetwo inductances into a single one which represents the global rotor leakage inductance of the CDFIG. Moreover, itcan be noticed that it would be impossible to separate practically DFIM1 and DFIM2 leakages and only the globalleakage inductance can be identified in such an integrated machine. On the basis of operations described in [8], thisinitial dynamical equivalent circuit is transformed into the one proposed in Fig. 6.

Fig. 6. CDFIG dynamical equivalent circuit.


Fig. 7. Block diagram of the CDFIG (equivalent to 6).

In this model, four new elements are introduced, defined as follows

Rrt = Rr1 + Rr2 (8)

Lrt = σ1Lcr1 + σ2Lcr2 (9)

et = j�(p1M1 · iμ1 + p2M2 · mci

∗μ2

) − jp1Lrt� · ir (10)

where ir is the common rotor current in the merged rotor model of the CDFIG (notice that ir = ir1) and a globalcomplex conjugate transformer with a complex transformer turns ratio noted mt and defined as follows

mt = mc · m2 (11)

where m2 is the (real) transformer turns ratio of DFIM2. On the basis of this graphical representation, main propertiesof the system are highlighted:

• System order: n = 6 (3 complex state variables iμ1, iμ2 and ir1)• Since θ and � are not considered as state variables but only parameters of this system, the CDFIG is a Linear

Parameter Varying (LPV) system• Two complex inputs are identified: vs1 and vs2• Two complex outputs: is1 and is2

The dynamical equivalent circuit of the CDFIG is also translated into a block diagram as it is shown in Fig. 7 whichcan be implemented with Matlab/Simulink and used for simulations, allowing to validate theoretically controlledstrategies developed in Section 5.

4.4. Grid model and complete system

In the targeted application, the CDFIG is connected to an autonomous grid. In order to properly control grid voltagesvs2, parallel capacitors are connected to the stator 2 windings of the CDFIG. Then, loads connected to this grid can bemodeled using an equivalent current source as it is shown in Fig. 8.

Fig. 8. Interconnexion between the CDFIG and the autonomous grid.


Fig. 9. Global model including the CDFIG and its load.

Notice that model can take into account both linear and nonlinear loads and the grid voltage dynamics is characterizedby the following equation:

dvs2

dt= − 1

C

(is2 + iL

)(12)

where iL is the load(s) current.Since the inverter connected to stator 1 windings is supposed to be an ideal voltage source, grid and CDFIG models

are enough to describe the system behavior. A global high level representation of the interconnection between thesetwo elements is proposed in Fig. 9.

Notice that the mechanical part of the system is included in this model but the generator torque tgen can be neglected:

tgen � tres (13)

thus, the mechanical speed and angle, respectively � and θ, do not depend on the electrical power consumed by theloads connected to the CDFIG. Thus, these two variables are considered as varying parameters of the CDFIG.

5. Controller synthesis

5.1. Principles of control using model inversion

The synthesis methodology proposed here is based on the direct model of the system presented above. Indeed, acontroller can be seen as the inverse model of the plant to control [1–5].

A state-space representation of a generic nonlinear system (covering a large class of actual systems) is introduced

dxdt

= f (x) + g(x) · u+ � (14)

where x, u and � are state, input and disturbance vectors, respectively.The aim of the controller is to drive the state vector with a given reference xref. If stability is achieved, error

cancellation between x and its reference is obtained with

dxdt

= Am ·(

xref − x)

(15)

Consequently, it leads to the following expression of the input vector

u = g(x)−1 ·(−f (x) + Am ·

(xref − x

)− �

)(16)

Three terms can be identified in this expression. It can be also noticed that u is calculable only if g(x) can be inverted.

u=ucomp+uCL+udcom (17)


with ⎧⎪⎪⎪⎨⎪⎪⎪⎩

ucomp = −f (x) · g(x)−1

uCL = Am · g(x)−1 · (xref − x

)udcom = −g(x)−1 · �

(18)

where

• ucomp is a compensation term of a vector state function,• uCL is a closed-loop term allowing to regulate the state vector to its reference,• udcom is a compensation term of disturbances, which must be either estimated/observed or neglected.

5.2. CDFIG control overview

In the previous paragraph, it has been assumed that the output vector was equal to the state vector. In fact, in theCDFIG model, outputs correspond to measurable currents:

• stator 1 current is1,• and stator 2 current is2,

whereas no state variables can be measured:

• magnetizing current iμ1,• magnetizing current iμ2,• rotor current ir1.

A physical analysis of the CDFIG, on the basis of its dynamical equivalent circuit representation, leads to thefollowing assessments:

• magnetizing current iμ1 and rotor current ir1 cannot be controlled separately because they are associated to parallelbranchs connected to a common voltage source (inverter) via a small resistance Rs1.

• as it is usually supposed, Rs2 can be neglected and then, magnetizing current iμ2 is strongly coupled to the statorvoltage vs2.

Thus, an overview of the CDFIG controller can be obtained with these two remarks:

• only one regulation loop has to be made for the rotor current ir1 (iμ1 can be then considered as a slave variable),• and this regulation loop is then controlled (with its reference) by an outer regulation loop dedicated to the output

voltage regulation, as it is already done for a single doubly-fed induction generator in [9].

5.3. Rotor current estimation

As it has been established in the previous paragraph, the rotor current has to be controlled. However, this currentis not measurable, this is why an estimator is required. A simple open-loop estimator is proposed here, based on thefollowing expressions:⎧⎪⎪⎪⎨

⎪⎪⎪⎩iμ1 = 1

Lcs1

∫ (vmeass1 − Rs1i

meass1

) · dt

ir1 = 1

m1

(iμ1 − imeas

s1

) (19)


Fig. 10. Inner rotor current loop.

Fig. 11. {cdfig + rotor current loop} simplified equivalent circuit.

where vmeass1 and imeas

s1 are two measurements. Notice that CDFIG parameters (Rs1, Lcs1 and m1) are supposed to beidentified.

5.4. Rotor current loop

On the basis of the inversion principle of dynamical models, rotor current state variable can be directly derived fromthe voltage applied to the rotor inductance Lrt . Thus, it can be noticed that

vLRt = mte∗μ2 − m1eμ1 − et = Lrt

d

dt

[ir1

] + Rrtir1

and vs1 is calculated in order to control vLRt as shown in Fig. 10.Notice that vLRt is the output of a vectorpi controller. Using this strategy allows the cancellation of the compensation

errors. Thereafter, the cdfig controlled by this rotor current loop can be described with the help of a simplified model(Fig. 11). In this figure, eμ2 and et are estimation terms of values given in Eq. (10).

Notice that vLRt is the output of a vectorpi controller. Using this strategy allows the cancellation of the compensationerrors, thus it yields

Thereafter, the cdfig controlled by this rotor current loop can be described with the help of a simplified model(Fig. 11). In this figure, eμ2 and et are estimation terms of values given in Eq. (10).


Fig. 12. Stator equivalent circuit in the field synchronous frame.

Fig. 13. Outer grid voltage loop.

5.5. Grid voltage regulation

The cdfig is not a natural voltage source. So, parallel capacitors are required if a true voltage source is needed.Moreover, harmonic voltage disturbances are reduced because of their low impedance at high frequencies in comparisonwith the high impedance of the cdfig (and its rotor current loop). The simplified dynamical equivalent circuit of thecontrolled generator and its load is presented Fig. 12. It includes:

• the cdfig and its rotor current loop• filtering parallel capacitors• linear and/or nonlinear load described with an equivalent current source

Table 1Testbench parameters

dfims parameters DFIM 1 DFIM 2

Numbers of pole pairs 2 2Rated power 2.2 kW 4 kWStator resistances 0.81� 1.67�

Rotor resistances 0.83� 0.25�

Stator cyclic inductances 138 mH 296 mHRotor cyclic inductances 20.9 mH 21.1 mHStator/rotor mutual inductances 33.3 mH 50.7 mHCoefficient of dispersion 0.117 0.077Linear load parametersResistance R 40�

Inductance L 1 mHFiltering capacitor C 1 μ F


Fig. 14. Experimental testbench.

An external loop has to be added in order to control the grid voltages. Notice that a magnetizing current iμ2 regulationis not required because this state variable is strongly coupled to the grid voltage vs2. If Rs2 is neglected, it gives

vs2 ≈ eμ2 (20)

Keeping in mind the inversion principle, the designed outer loop is presented Fig. 13. It yields

As shown in the next section, this strategy allows taking into account nonlinear loads.

Fig. 15. Rotor current regulation at variable speed.


6. Simulations and experimental results

6.1. Testbench description

All the simulation parameters are given in Table 1. They are based on the machines used in the testbench presentedin Fig. 14.

The simulation on utility grid uses an ideal model of the grid which is considered as a three-phase source voltage(230 V, 50 Hz).

Fig. 16. vscf control of the cdfig.

Fig. 17. Rotor currents (i.e. stator 1 currents) regulation—experimental results.


6.2. Simulations

In a first simulation, the rotor current loop is tested at variable speed. Notice in Fig. 15 that during the acceleration(500 rpm/s), the rotor currents does not exactly follow its references.

In Fig. 16, the stator 1 currents and grid voltages are shown. We can see that the grid frequency is maintainedequal to 50 Hz in spite of the speed variations (±33% of the synchronous speed). Thus, the vscf operation is verified.Moreover, we can see that the stator 1 power is limited to a fraction of the grid power (−Ps2) for the whole speed rangeand is equal to zero at the synchronous speed.

6.3. Experiments

As it is shown in Fig. 17, sub- and super-synchronous operations correspond to a mechanical speed inferior andsuperior to 750 rpm, respectively (for a grid frequency equal to 50 Hz).

When the rotor currents control is achieved, grid voltage regulation can be performed, allowing to maintain voltageamplitude in spite of large variations of the load consumption.

7. Conclusions

The cascaded doubly-fed induction machine studied in this paper has been successfully controlled as a variablespeed constant frequency generator. The control strategy has been synthesized using a rigorous methodology basedon model inversion principles. For this purpose, a user-friendly modeling formalism (dynamical equivalent circuitrepresentations), adapted to electrical generators, has been used. Within this framework, a hierarchical control structurehas been established and validated not only in simulations but also in experiments.

References

[1] A. Bouscayrol, M. Pietrzak-David, P. Delarue, R. Pena-Eguiluz, P.-E. Vidal, X. Kestelyn, Weighted control of traction drives with parallel-connected AC machines, IEEE Transactions on Industrial Electronics 53 (6) (2006) 400–408, Dec.

[2] G. Georgiou, B. Le Pioufle, Combined magnetizing flux oriented control of the cascaded doubly-fed induction machine, in: Proceedings ofEPE’91, vol. 3, Firenze, Italy, 1991, p. 42.

[3] J.P. Hautier, P.J. Barre, The causal ordering graph—a tool for modelling and control law synthesis, Studies in Informatics and Control Journal13 (December (4)) (2004) 265–283.

[4] B. Hopfensperger, Doubly-fed AC machines: classification and comparison, in: Proceedings of EPE’01, CD-ROM, Graz, Austria, August,2001.

[5] A. Isidori, Non-linear Control Systems, 2nd edition, Springer-Verlag, Berlin, 1989.[6] S. Kato, N. Hoshi, K. Ogushi, Small-scale hydropower, IEEE Industry Applications Magazine 9 (July–August (4)) (2003) 3–38.[7] T. Ortmeyer, W. Borger, Control of cascaded doubly-fed machines for generator applications, in: IEEE Transactions on Power Apparatus and

Systems, vol. PAS-103, No. 9, September, 1984, pp. 2564–2571.[8] N. Patin, J.-P. Louis, E. Monmasson, Cascaded doubly-fed induction generator—state-space modelling and performance analysis, in: Proceed-

ings of Electrimacs’05 Conference, CD-ROM, Hammamet, Tunisia, 17–20 April 2005, 2005.[9] N. Patin, E. Monmasson, J.-P. Louis, Control of a stand-alone variable speed constant frequency generator based on a doubly-fed induction

machine, EPE Journal 16 (December (6)) (2006) 37–43.[10] F. Runcos, R. Carlson, A.M. Oliveira, P. Kuo-Peng, N. Sadowski, Performance Analysis of a brushless doubly-fed cage induction generator,

in: Nordic Wind Power Conference, Chalmers University of Technology, March, 2004.

Nicolas Patin was born in Châteauroux, France, in 1979. He graduated from the Ecole Normale Supérieure (ENS) deCachan, Cachan, France, in 2004. Since September 2004, he is working toward the PhD degree in electrical engineering atthe ENS de Cachan in the Laboratoire Systèmes et Applications des Technologies de l’Information et de l’Energie (SATIE),ENS de Cachan/CNRS, Cachan, France. His main research topic consists in the study and the control of electrical generatorsdedicated to autonomous grids, especially the brushless structures obtained by association of doubly-fed induction machines.


Eric Monmasson was born in Limoges, France, in 1966. He received the Ing and PhD degrees from the Ecole NationaleSupérieure d’Ingénieurs d’Electrotechnique d’Electronique d’Informatique et d’Hydraulique de Toulouse, Toulouse, France,in 1989 and 1993, respectively. He is currently a full professor in the Institut Universitaire Professionnalisé de Génie Elec-trique et d’Informatique Industrielle, University of Cergy-Pontoise, Cergy-Pontoise, France. His current research interestsin the Laboratoire Systèmes et Applications des Technologies de l’Information et de l’Energie (SATIE, UMR CNRS 8029)are the advanced control of electrical motors and generators and FPGAs for industrial control. Prof. Monmasson is theGeneral Chairman of ELECTRIMACS 2011 Conference.

Jean-Paul Louis was born in Toulouse, France, in 1945. He received the Ing degree from the Ecole Nationale Supérieured’Electricité et de Mécanique de Nancy, Nancy, France, in 1968, the PhD degree from the University of Nancy I, Nancy,France, in 1972, and the Doctor d’Etatdegree from the Polytechnic National Institute of Lorraine (I.N.P.L.),Vandoeuvre,France, in 1981. He was a Maítre-assistant at the Technological Institute of the University of Nancy (I.U.T.) and a researcherin two laboratories of the I.N.P.L., the Groupe de Recherche en Electronique et Electrotechnique de Nancy (G.R.E.E.N.)and the Centre de Recherche en Automatique de Nancy (C.R.A.N.). In 1983, he became a professor at the Ecole NormaleSupérieure de Cachan (E.N.S. Cachan), Cachan, France. He was Chief of the Electrical Engineering Teaching Department(1983–1992), and researcher at the Laboratoire d’Electricité Signaux et Robotique (L.E.Si.R.-S.A.T.I.E.), which becamethe Laboratoire Systèmes et Applications des Technologies de l’Information et de l’Energie (UMR CNRS 8029). From1992 to 2001, he was Associate Director, then Director of this laboratory. His current research interests are in the fields ofmodeling of electrical machines, converter control, and control of electrical systems

science (1)

Education

following section section

constant frequency grid

cdfig modeling section

electrical power

experimentalresults

generatorefciency section

operatingat variable

induction machines dfims