Re Examination of Synchronous Machine Modeling Techniques for Electromagnetic Transients

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 3, AUGUST 2007 1221

Re-examination of Synchronous MachineModeling Techniques for Electromagnetic

Transient SimulationsLiwei Wang, Student Member, IEEE, Juri Jatskevich, Member, IEEE, and Hermann W. Dommel, Life Fellow, IEEE

Abstract—This paper re-examines the three synchronous ma-chine modeling techniques used for electromagnetic transient sim-ulations, namely, the model, phase-domain model, and voltage-behind-reactance model. Contrary to the claims made in severalrecent publications, these models are all equivalent in the contin-uous-time domain, as their corresponding differential equationscan be algebraically derived from each other. Computer studies ofa single-machine infinite-bus system demonstrate that all of thesemodels can be used for unsymmetrical operation of power sys-tems. The conversion of machine parameters is also discussed andis shown to have some impact on simulation accuracy, which is ac-ceptable for most cases. When the models are discretized and inter-faced with an EMTP-type network solution, the voltage-behind-re-actance model is shown to be the most accurate due to its advancedstructure.

Index Terms—Electromagnetic Transient Programs (EMTP),phase-domain model, synchronous machine, unbalanced fault,voltage-behind-reactance model.

I. INTRODUCTION

MODELING of synchronous machines has been an ac-tive research subject for quite a long time. Depending

on the objective of studies and the required level of fidelity, themodeling approaches may be roughly divided into three cat-egories: finite element (or difference) method [1]; equivalentmagnetic circuit approach [2], [3]; and coupled electric circuitapproach. This paper mainly considers the last approach, whichleads to a relatively small number of equations and has beenvery often utilized for predicting the dynamic responses of elec-trical machines in power system operations. To further simplifythe coupled electric circuit approach, the machine physical vari-ables are often transformed into quadrature and direct magneticrotor axes [4]–[7]. This approach is also known as the mod-eling of rotating machines, which is widely used in Electromag-netic Transient Programs (EMTP) [8] for power system tran-sient simulations and analysis. Currently, there are many EMTP-type programs that use general purpose synchronous machine

Manuscript received August 29, 2006; revised December 8, 2006. This workwas supported in part by the Natural Science and Engineering Research Council(NSERC) of Canada under the Discovery Grant and in part by the Power En-gineering Grant-In-Aid from BC Hydro and Powertech Labs, Inc. Paper no.TPWRS-00563-2006.

The authors are with the Department of Electrical and Computer Engi-neering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada(e-mail: [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPWRS.2007.901308

models based on the transformation. These models includethe Type-50 machine model in MicroTran (MT) [9], the Type-59machine model in EPRI/DCG EMTP [10] and ATP [11], and thesynchronous machine model in PSCAD/EMTDC [12]. The va-lidity of these models for both balanced and unbalanced powersystem operations has been confirmed by academic and indus-trial applications [13], [14].

The so-called phase-domain (PD) models, as the originalform of the coupled electric circuit machine models, wereproposed in [15] and [16] for the nodal analysis approach.These models achieved simultaneous solution of the machineelectrical variables and the network equations, they and im-prove numerical accuracy and stability [17], [18]. However,the time variant self and mutual inductances of the PD modelstator and rotor circuits complicate the modeling and increasethe computational cost.

To integrate the advantages of the PD model with the benefitof the transformation, the voltage-behind-reactance (VBR)machine model was introduced in [19] for the state variable ap-proach and extended for the EMTP-type solution in [20]. Inthe VBR model structure, the stator circuit is represented in

phase coordinates while the rotor equations are expressedin rotor reference frame. Similar to the PD model, simulta-neous solution of the network and machine electrical variablesis achieved, and numerical accuracy and simulation efficiencyare further improved upon.

Although all of the above-mentioned models rely on the sameset of assumptions [21]–[23], some researchers have questionedthe applicability of the model to unbalanced conditions. In[24]–[26], the authors claim that the machine model “as-sumes perfectly balanced operation conditions in machine” and“cannot accurately describe the transient and steady-state unbal-anced operation of the synchronous machine.” The authors of[27] and [28] made similar conclusions by comparing the sim-ulation results of the and PD models under unbalanced op-eration. These results seem to contradict the existing machinemodeling theories and undermine the long-established validityof the machine model for power system transient simulations.

In this paper, computer studies are conducted using the samesynchronous machine as in [24]–[26] to demonstrate that the

machine model, PD model, and VBR model all give iden-tical simulation results for unbalanced operation, provided thatthe simulation time-step is sufficiently small. This result isexpected as all of these models can be derived from each otherand are therefore equivalent in the continuous time domain. Theclaims made in [24]–[28] that the machine model cannot rep-resent unbalanced operations are unfounded. The difference in

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1222 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 3, AUGUST 2007

Fig. 1 Coupled-circuits model of a synchronous machine.

simulation results, however, may possibly arise from inconsis-tent machine parameters, problems with user interfaces, and/ormodel implementation. We also show that the use of equivalentcircuit parameters versus the manufacturer’s data (transient/sub-transient inductances and time constants) should not in generallead to significant differences in simulation results. However,one should be aware of existing procedures for parameter con-version and apply them with care.

This paper also shows that numerical properties of the con-sidered machine models differ when their respective differentialequations are discretized and interfaced with the network solu-tion. The VBR model is shown to be more accurate, even forsufficiently large integration time steps.

II. MACHINE MODELS

Prior to comparing the results, it is instructive to briefly re-view the various models discussed in this paper. According tothe commonly used assumptions [21]–[23], a general purposethree-phase synchronous machine may be represented as a cou-pled circuit, depicted in Fig. 1. For the purpose of discussion andwithout loss of generality, the stator circuit consists of three ar-mature windings, and the rotor circuit includes one field windingand one damper winding in axis and two damper windings inaxis, respectively. The axis is assumed to be leading the axisby 90 [23]. Motor convention is used in the machine’s voltageequations so that the stator currents flowing into the machinehave a positive sign in the voltage equations. The flux linkageof each winding is assumed to have the same sign as the currentflowing in that winding.

The mechanical subsystem is assumed to be represented as

(1)

(2)

Here, the operator , is the number of poles, is themoment of inertia, and are the developed electromagnetictorque and the mechanical torque, and and are the rotorposition and speed, respectively. Although the models consid-ered herein all are assumed to have the same mechanical sub-system represented by (1) and (2), the electromagnetic torque isexpressed differently in each case.

A. The Model

The voltage equations of the machine model can be repre-sented in terms of transformed variables as

(3)

where

(4)

(5)

(6)

(7)

Here, and are the winding voltages and currents, re-spectively, in rotor reference frame; is the field windingvoltage; is the speed voltage vector; and is the stator androtor resistance matrix.

The flux linkage equations are given as

(8)

where the machine self and mutual inductance matrix isconstant, due to Park’s transformation.

The electromagnetic torque is expressed as

(9)

B. Phase-Domain Model

The PD model is commonly expressed in terms of the ma-chine’s physical variables and phase coordinates. In particular,the voltage equation is expressed as

(10)

and the flux linkages are represented as

(11)

where the stator and rotor self and mutual inductance matrixdepends on the rotor position . The electromagnetic

torque is expressed in the machine variables as

(12)

C. Voltage-Behind-Reactance Model

The VBR machine model represents the stator variables inphase coordinates and the rotor variables in transformed

coordinates [19]. Here, the stator voltage equation is expressedas

(13)

WANG et al.: RE-EXAMINATION OF SYNCHRONOUS MACHINE MODELING TECHNIQUES 1223

where

(14)

(15)

(16)

(17)

(18)

The inductances and are calculated as

(19)

(20)

The voltage-behind-reactance term in (13) is defined as

(21)

where the rotor reference frame transformation matrix is

(22)

(23)

(24)

with

(25)

(26)

The rotor state equations are represented as

(27)

(28)

where

(29)

(30)

Here, denotes the rotor winding flux linkages; andare the magnetizing flux linkages in the and axes,

respectively.The equations for rotor speed and position in the VBR ma-

chine model are identical to (1) and (2). However, the electro-magnetic torque is calculated as [23]

(31)

D. Relationship Among Models

Although the PD model is based on straightforward stateequations describing the coupled circuit depicted in Fig. 1,its implementation in digital programs is complicated by thepresence of the rotor-position-dependent self and mutual in-ductances represented in (11). The model is algebraicallyderived from the PD model using Park’s transformation withno approximations [22], [23]. Because the model resultsin equations with constant coefficients, its implementation isrelatively simple, which explains the wide use of this model.However, the disadvantage of the model is its interface withEMTP network solutions, which was shown to result in a lossof numerical accuracy and stability for large time steps [20].

The VBR model was originally derived from PD andmodels without any approximation by appropriately applyingthe rotor reference frame only to selected terms [19]. Therefore,the model, the PD model, and the VBR model can be derivedfrom each other and are all equivalent in the continuous timedomain. However, the numerical properties of these modelsdiffer when their equations are discretized using a particularintegration scheme. A detailed analysis of these models in [19]and [20] shows that the VBR model is more accurate than thePD model due to rescaled and improved eigenvalues. To makea fair comparison among the models here, (1) and (2) have beendiscretized in the same way using implicit trapezoidal rule, andthe mechanical variables and are predicted using linearextrapolation to avoid the solution of the machine nonlinearequations. This method is commonly used in EMTP-typeprograms and is justified on the facts that the time constant ofthe mechanical system is much larger than that of the electricalsystem.

III. INPUT DATA CONVERSION

For most EMTP-type programs the self and mutual in-ductances of stator and rotor circuits are internally utilizedto calculate the equivalent circuit parameters and solve themachine equations [22]. However, these internal circuit param-eters are not directly determined from the test measurements.Instead, the transient and subtransient inductances and timeconstants are first defined to fit the field test data (manu-facturer’s data) with acceptable accuracy, from which thecorresponding internal circuit parameters are calculated usinga parameter conversion procedure.

There are several commonly used parameter conversionmethods in the literature [29]–[32]. In [29], a classical (ap-proximate) parameter conversion procedure is presented thatassumes that the flux linkages do not change instantly following


a short-circuit test, and that and are much larger thanand , respectively, so that the subtransient period

elapses much faster than the transient period. Based on theseassumptions, the classical parameter conversion is summarizedin Appendix A. The conversion is quite straightforward and hasbeen widely used, although the possibility of a noticeable errorwas reported in [30].

A more accurate parameter conversion method proposed byCanay [30] and Dommel [31] (a refinement of Canay’s dataconversion) is given in Appendix B. This method is based ona set of more accurate definitions of transient and subtransientinductances and time constants in terms of machine circuit pa-rameters. For consistency, a brief review of these definitionsis included here. The detailed derivations can be found in [22,Appendix].

The transient and subtransient inductances and time constantsare derived using an eigenvalue and eigenvector approach tothe machine’s differential equations. Without loss of generality,the procedure of deriving short-circuit time constants is givenbelow. A similar procedure is used with the open-circuit timeconstants. If the armature resistance is assumed to be zero, thestator state equations for the short-circuit test can be expressedas [31]

(32)

and

(33)

The solution of (32) and (33) is given by

(34)

and

(35)

where is defined by the initial condition . Herethe solution is slightly different from [22, Appendix VI], as themotor convention is used in the machine voltage equations. Therotor state equations for the axis are formulated as

(36)

and

(37)

where

(38)

(39)

(40)

After and are expressed in terms of , , and ,the final equation has the following form [22, Appendix VI.11]

(41)

where is system matrix and is the forcing function vector.The short-circuit time constants , are obtained as the

negative reciprocals of the eigenvalues of the matrix [22, Ap-pendix VI.11]. The short-circuit currents and can be ob-tained by solving (41) analytically. The solutions of andinclude three parts: the steady state, the transient associated with

, and the subtransient associated with . Therefore, the so-lution for also contains three parts, since

(42)

After substituting the analytical solutions of and into(42), the transient part of associated with can be repre-sented as [22, Appendix VI.27a]

(43)Using (35), the transient part of can be further expressed as

(44)which can then be matched with the amplitude of the transientpart of as read from the short-circuit measurement. Therefore,the transient inductance can be obtained from

(45)

Similarly, the subtransient inductance is obtained from

(46)If , is much larger than the period of oscillation of

(i.e., , ), the transient andsubtransient inductances may be approximated as

(47)

(48)

When only the open-circuit or short-circuit time constants areknown, the other pair can be calculated using the solution of (47)and (48) or (45) and (46), which correspond to Canay’s data con-version [30] and its refinement [31], respectively. Both of theseapproaches are explicit and neglect the armature winding resis-tance to decouple the definitions of test parameters in the direct


Fig. 2. Simulation results with time-step of 50 �s.

and quadrature axes. In [32], the researchers removed this as-sumption and proposed an implicit procedure for calculating thecircuit parameters from the known axes’ transient and sub-transient time constants and eigenvalues. However, the calcula-tion results in [32] show that Canay’s data conversion is quiteaccurate for typical machine data.

IV. CASE STUDIES

In this paper, we used two EMTP software packages,MicroTran and ATP, both with the build-in synchronousmachine models based on the traditional model. ThePD and VBR models were also implemented using a nodalanalysis approach in order to compare their performances.A single-machine infinite-bus system is used in which thesynchronous machine parameters were obtained from [21] andare summarized in Appendix C. In this section, the machinereference circuit parameters were used for all machine modelsin Sections IV-A and IV-C. Different parameter conversionmethods are used in Section IV-B, and the results are comparedwith the reference circuit parameters.

In the following transient study, the synchronous machine ini-tially operates at no load. At , a single-phase-to-ground fault is applied at the machine terminals (phase ). Thetransient responses produced by the various models are plottedin Figs. 2–9. The transient studies of balanced operation have

Fig. 3. Damper winding current i using different data conversion methods.

Fig. 4. Magnified plot of i using different data conversion methods.

Fig. 5. Stator current i with time step of 1 ms.

been previously considered in [20] and are not included heredue to space limitations.

A. Small Time-Step Study

The considered study of an unbalanced fault is first sim-ulated using a relatively small time step, . Theresulting transients of the stator currents , , field current


Fig. 6. Stator current i with time step of 1 ms.

Fig. 7. Field current i with time step of 1 ms.

Fig. 8. Electromagnetic torque T with time step of 1 ms.

, and electromagnetic torque are depicted in Fig. 2.The reference solutions were obtained using the machinemodel implemented in MATLAB/Simulink and solved with theRunge–Kutta fourth-order method using a time step of 1 .The simulation results obtained by the MicroTran, ATP, PD,and VBR models are overlaid with the reference solutions. Ascan be observed from Fig. 2, the transient responses producedby all models coincide and converge to the reference solutions.

Fig. 9. Magnified plot of i with time step of 1 ms.

This clearly demonstrates that these models are all equivalentand predict the unbalanced operations with acceptable accuracyfor the given time step, which is contrary to the conclusionsmade in [24]–[28].

B. Impact of Parameter Conversion Methods

The authors of [24]–[26] suggested that the different simula-tion results obtained by the and PD models may be causedby “different data based on their frames of reference” [24] andlisted the machine’s internal circuit data and test data in the Ap-pendices of [24], [25]. However, these two sets of data are equiv-alent, as is shown in [21]. To investigate this point further, weimplemented the same system using MicroTran with the input ofthe test parameters and reference circuit parameters. Internally,MicroTran converts the test parameters into the circuit parame-ters using one of the three conversion methods as described inSection III. As expected, the deviation among the final circuitparameters was not very significant. The most noticeable differ-ence was observed in the axis parameters, which are summa-rized in Table I. The corresponding transient responses are vis-ibly the same as in Fig. 2 and are not shown here due to spacelimitations. As Table I reveals, the most noticeable differencesexists in the parameters of winding. Therefore, the tran-sient observed in current is shown in Fig. 3. A magnifiedfragment of Fig. 3 is also shown in Fig. 4 for better compar-ison. As can be seen in Figs. 3 and 4, the impact of various dataconversion methods is small, and all transients are reasonablyclose to the reference solution obtained using the original circuitparameters. The classical method (dashed line) gives the mostdeviation from the reference (solid line), with successive im-provements by Canay’s method (dash-dotted line) and its minorrefinement (long-dashed line), all consistent with the accuracyof the parameters summarized in Table I. Therefore, the impactof parameter conversion methods on the transient responses isinsignificant. The different simulation results in [24]–[26] ob-tained using the model and the phase-domain model may becaused by inconsistent data or problematic model implementa-tion.

C. Large Time-Step Study

To further compare the numerical properties and robustnessof the machine models, a larger time step is ap-


TABLE IDATA CONVERSION FOR Q AXIS IN OHM

Fig. 10. Propagation of numerical error in i for different time steps.

plied to the same test case as in Sections IV-A and IV-B. Thesame variables are plotted in Figs. 5–8. As can be observed inFigs. 5–8, the results produced by the various models visiblydiverge from the reference solution. The transient responses ob-tained by MicroTran’s machine model Type-50 (dotted line, MTlegend) diverge from the reference solutions with the largesterror among the considered models, whereas ATP’s machinemodel Type-59 was found not convergent at such a large timestep. The problem of convergence and accuracy of modelsis attributed to their interface with the external network and isdiscussed in detail in [17], [18], and [20]. At the same time,the VBR and PD models still produce reasonably accurate andconvergent simulation results. A more detailed fragment of thetransient observed in is shown in Fig. 9, which is a magni-fied plot of Fig. 6. As Fig. 9 (and Figs. 5–8) shows, the VBRmodel produces the most accurate results among the consideredmodels, due to its advanced model structure [19], [20].

D. Error Behavior

In order to show the numerical accuracy of machine models atdifferent time steps, the same study was simulated several timesusing integration time steps from 10 to 1 ms. Herein, withoutloss of generality, only the phase current is considered dueto space limitation. The relative error between the reference so-lution (as defined in Section IV-A) and a given numericalsolution trajectory is calculated using the 2-norm [33] as

(49)

The errors are calculated for different time-step sizes, and theresults are shown in Fig. 10.

As can be seen in Fig. 10, the solutions of all models areconvergent to the reference solution when the time step issufficiently small. However, the simulation accuracy degradesrapidly when the time step is increased. A similar trend isobserved with other variables. For , MicroTran’s

model results in a cumulative error of about 35%, whereasATP’s Type-59 model was not convergent with a time steplarger than 300 (under default error tolerance). Based onthese studies, it appears that models may be used withacceptable accuracy with a time-step as large as 100 to 200 s.

Although the PD and VBR models considered here use pre-diction of mechanical variables in the same way as traditional

models (e.g., Type-50 model in MicroTran, Type-59, andUniversal Machine models in ATP), these models achieve si-multaneous solution of machine electrical variables and networkvariables and therefore produce stable and more accurate re-sults. The VBR model achieves higher numerical accuracy, asshown in Figs. 9 and 10. This improved numerical accuracy isattributed to better-scaled eigenvalues of this model formulation[19], [20]. In particular, the eigenvalues of the discretized modeldetermine the behavior and/or propagations of local errors. In-terested readers may find a more detailed analysis of numericalaccuracy and system eigenvalues in [34] and [35].

V. DISCUSSION

The models discussed in this paper belong to a group of gen-eral-purpose, lumped-parameter models that may be used withvarious EMTP-type packages to carry out simulations of typicalpower system transient studies with balanced and/or unbalancedconditions on the system side. However, sometimes there is aneed to model and implement more detailed and/or internal phe-nomena, such as stator inter-turn and/or inter-windings faults.

The general purpose models (e.g., , PD, and VBR) are allbased on the circuits that assume symmetrical stator phase wind-ings as depicted in Fig. 1. These models are therefore not di-rectly suitable for studying internal faults, since special con-sideration of the type of fault being examined is required. Tomodel the transient phenomena associated with internal faults ina single phase, at least one of the stator windings must be rep-resented in more detail; that is, this phase must be partitionedinto smaller sections such that the fault is then applied to an ap-propriate portion of the winding. If the internal fault involvesmore than one phase, then the affected windings must be parti-tioned accordingly to implement the required phenomena. Forexample, the authors of [36] use a partitioning of the stator wind-ings into interior and exterior sections in order to study the statorwinding internal faults. Since the PD model formulation doesnot require any symmetry among the phases, this model formu-lation can be readily extended to represent the synchronous ma-chine’s internal faults, as proposed in [37]–[39], at the expenseof including the required winding subsections and increasing thedimensions of the inductance matrix accordingly.

When the stator phase windings are not symmetric, the mainadvantage of applying the transformation, which previouslyresulted in constant matrices, is no longer present, and the trans-formed inductance and resistance matrices will depend on therotor position [23]. Thereafter, straightforward transformationof stator equations into the rotor reference frame may offer


little if any advantage over the direct coupled circuit formula-tion (i.e., the PD model). It might be possible to partition allthree stator phases into several equal subsections and still ob-tain some benefit by transforming the stator equations into the

rotor reference frame. The authors have not seen this reportedin the literature, however, and more research is required to con-firm this possibility.

Incorporating the effect of magnetic saturation is anotherimportant aspect of improving the accuracy and range ofapplication of synchronous machines models. Using themachine models, magnetic saturation may be representedwith accuracy acceptable for power-system transient analysis[21]–[23], [40]–[43]. For EMTP-type solutions, a piece-wiselinear approximation is often used to represent the magneticsaturation characteristics of the models [22], [40]. Thenecessary machine parameters may be obtained using thestandstill frequency response (SSFR) and/or online frequencyresponse (OLFR) measurements [41]. Higher-order machinemodels have been proposed to fairly accurately represent thedistributed-parameter nature of the rotor and saturation effects[43], with the model parameters derived from the measuredsaturation characteristics and the SSFR. The model pro-posed in [43] offers more accuracy than is typically requiredfor studying power system transients at low frequencies, whichcomes at the expense of additional complexity.

The PD model proposed in [16] also provides a flexible andpotentially accurate way to represent variable reluctance andsaturation effects along different flux paths. However, the com-plexity of representing magnetic saturation in PD formulation isin general much higher than it is in models. Implementationof magnetic saturation with the VBR model has been previouslyconsidered in [44], where the authors have shown that incorpo-rating the saturation in the axis only may provide an adequatematch with the hardware measurements. The VBR model for-mulation may provide different options for including magneticsaturation, with accuracy similar to that of conventional andPD models. However, this subject requires more thorough con-sideration that is beyond the scope of this paper and deserves adedicated publication, which the authors will pursue in the nearfuture.

On the other hand, the finite-element-based and/or mag-netic-equivalent-circuit-based models are naturally much bettersuited for studying internal machine details from the initialdesign stage. Various internal faults and magnetic saturation,including very fine structural/material details, can be studiedusing these models. These approaches should be used insteadof low-order models (i.e., the , PD, and VBR) whenever veryhigh fidelity simulations of a single machine are required. Theresulting models, however, are computationally very expensive.

For typical power system transient simulations, however, theaccuracy achieved by low-order models has been generally con-sidered acceptable [21]. Even a more accurate model [43]may appear too complex and expensive when more than onemachine is being considered in the system.

VI. CONCLUSION

Several full-order synchronous machine models used forEMTP-type solutions are investigated in this paper. Presentedcomputer studies of a single-phase-to-ground fault show that

the model, PD model, and VBR model are all equivalent inthe continuous time domain. They can therefore all be used forstudying power system transients with unbalanced (as well asbalanced) operation, provided the time step is sufficientlysmall. For large time steps, the VBR model is shown to havegood stability property and to provide the most accurate results.The input data conversion procedures discussed are shown tohave a noticeable but minor impact on simulation accuracy,with results equivalent and/or acceptable for most applications.The conclusions stated in [24]–[28] with regard to the unsuit-ability of the machine model for unbalanced operations havenot been confirmed.

APPENDIX A

Classical data conversion [45, equations (15)–(28)]: The mu-tual inductance in the axis, is assumed to be known, or iscalculated by

(A1)

The unknown circuit parameters in the axis can be calculatedfrom the transient and subtransient inductances and time con-stants as

(A2)

(A3)

(A4)

(A5)

APPENDIX B

Canay’s data conversion [22]: Two time constants, and ,are defined as

(B1)

(B2)

Assuming is known, the time constants and can becalculated using the following equations:

(B3)

(B4)

where

(B5)

The parallel combination of and can then be calculatedas

(B6)


The circuit parameters can be determined as

(B7)

(B8)

(B9)

(B10)

Similar procedures apply to axis parameters.

APPENDIX C

Synchronous machine parameters [21]: 555 MVA, 24 kV, 0.9pf, 2 poles, 3600 r/min,

Reference circuit parameters (p.u.):

Test parameters (p.u.):

ACKNOWLEDGMENT

The authors would like to thank their friends and colleaguesin the UBC Power Group as well as the reviewers for providingvaluable comments.

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Liwei Wang (S’04) received the M.S. degree inelectrical engineering from Tianjin University,Tianjin, China, in 2004. He is currently pursuing thePh.D. degree in electrical and computer engineeringat the University of British Columbia, Vancouver,BC, Canada.

His research interests include electrical machines,power, and power electronic systems simulation.

Juri Jatskevich (M’99) received the M.S.E.E.and Ph.D. degrees from Purdue University, WestLafayette, IN, in 1997 and 1999, respectively.

He stayed at Purdue, as well as consulting for PCKrause and Associates, Inc., until 2002. Since 2002,he has been an Assistant Professor of electrical andcomputer engineering at the University of BritishColumbia, Vancouver, BC, Canada. His research in-terests include electrical machines, power electronicsystems, and simulation.

Hermann W. Dommel (LF’01) was born inGermany in 1933. He received the Dipl.-Ing. andDr.-Ing. degrees in electrical engineering from theTechnical University Munich, Munich, Germany, in1959 and 1962, respectively.

From 1959 to 1966, he was with the Technical Uni-versity Munich and, from 1966 to 1973, with Bon-neville Power Administration, Portland, OR. Since1973, he has been with the University of British Co-lumbia, Vancouver, BC, Canada.

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