Constant-Parameter Voltage-Behind-Reactance Model of Six...

548 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 32, NO. 2, JUNE 2017

Constant-Parameter Voltage-Behind-ReactanceModel of Six-Phase Synchronous Machines

Navid Amiri, Student Member, IEEE, Seyyedmilad Ebrahimi, Student Member, IEEE,Mehrdad Chapariha, Member, IEEE, Juri Jatskevich, Fellow, IEEE, and Hermann W. Dommel, Life Fellow, IEEE

Abstract—Six-phase electrical machines have received sig-nificant attention in the literature due to their use in specialpurpose applications (e.g., aircraft, naval, and vehicular systems).Recently, such machines have also been considered for renewableenergy systems including wind generators. Modeling of suchmachines in commonly available transient simulation programsis not straightforward, especially when the machine modelis interfaced with external inductive network and/or powerelectronic converters. The available modeling approaches includethe classical qd0 model, the coupled-circuit-phase-domain andthe voltage-behind-reactance (VBR) models (each having itsinterfacing challenges). This paper extends the prior research inthis area and proposes a constant-parameter VBR model that hasa very convenient constant RL−branch interfacing circuit (evenfor salient pole machines), which makes it simple to implement inmost state-variable-based simulations programs. The presentedcomputer studies demonstrate significant numerical advantagesof the new model over the existing alternative models.

Index Terms—Constant-parameter voltage-behind-reactance(CPVBR) model, modeling and simulation, numerical efficiency,power system transients, six-phase synchronous machine.

I. INTRODUCTION

ROTATING electrical machines are the main devices usedfor electromechanical energy conversion in existing and

emerging power systems. Although the conventional three-phase machines are the dominant case, there has been an increas-ing interest in using machines with higher number of phases(e.g., five, six, nine, twelve, etc.) in special purpose applicationssuch as aircraft, naval, and vehicular power systems. Such ma-chines are also finding their role as wind generators in renewableenergy applications. Machines with higher number of phasescan offer reduced power per phase, reduce rating of power elec-tronic converters per phase, increased reliability, fault tolerantoperation, etc., [1]–[6].

This paper considers the six-phase synchronous machines,for which the coupled-circuit phase-domain (CCPD) [7] and

Manuscript received June 14, 2016; revised October 20, 2016; accepted De-cember 6, 2016. Date of publication December 14, 2016; date of current versionMay 18, 2017. This work was supported by the Natural Science and EngineeringResearch Council of Canada under the Discovery Grant. Paper no. TEC-00509-2016.

The authors are with the Electrical and Computer Engineering Depart-ment, The University of British Columbia, Vancouver, BC V6T 1Z4, Canada(e-mail: [email protected]; [email protected]; [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/TEC.2016.2639505

conventional qd0 [8], [9] models have been used for steady-state and transient analysis. The CCPD model is generallyderived based on the magnetically-coupled circuits of the statorwindings and rotor equivalent model [10]. This base model canbe directly interfaced with any external network, provided thatthe simulation program where the model is developed allowsimplementation of the rotor-position-depended (variable)inductances. The rotor is typically represented using severalequivalent damper windings and a field winding, all of whichresult in increased size of the machine inductance matrix.Moreover, since the machine inductance matrix is variable, theresulting CCPD model becomes computationally expensive asits parameters have to be recalculated at each time step in thetransient simulation program.

The traditional qd0 model for six-phase machines with twosets of abc stator windings can be obtained using the classi-cal rotor reference frame transformation [11], [12], which re-sults in constant parameter equivalent circuits in two qd planes.The mutual leakage inductance between the two sets of abcstator windings also creates a cross-coupling between the twoqd planes, respectively [11]. Similar to the conventional three-phase qd0 models [13], the six-phase qd0 model is difficult to in-terface with the external inductive networks, and such interfacein state-variable-based simulation programs is often resolvedusing artificial snubber circuits [14].

The modeling approach presented in this paper considersthe state-variable-based transient simulation programs such asPLECS [15] and SimPowerSystems [16] block set in MAT-LAB/Simulink [17]. The voltage-behind-reactance (VBR) for-mulation derived for three-phase machines [18]–[19] has beenextended to the six-phase synchronous machines in [20]–[21].The resulting VBR model has six-phase interfacing stator circuitin abc1abc2 coordinates, and the rotor subsystem is in imple-mented in dual qd012 planes. However, similar to the CCPD,such variable parameter VBR model will also have the rotor-position-depended inductances, and will be computationally ex-pensive.

To achieve a very efficient interfacing, this paper presentsa new constant-parameter voltage-behind-reactance (CPVBR)model for six-phase synchronous machines using the approachset forth in [22]–[23]. The proposed six-phase CPVBRmodel is shown to produce very accurate simulation resultsand offer advantages over the existing CCPD, qd0, and therecently presented variable parameter VBR models in terms ofsimulation efficiency and numerical accuracy. Moreover, due to

0885-8969 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications standards/publications/rights/index.html for more information.

AMIRI et al.: CONSTANT-PARAMETER VOLTAGE-BEHIND-REACTANCE MODEL OF SIX-PHASE SYNCHRONOUS MACHINES 549

Fig. 1. A simplified cross-section of a salient pole six-phase synchronousmachine, assuming a shift angle ζ between the two three-phase stator sets.

simple constant RL−branch interfacing circuit, the new modelcan be easily implemented in most commonly available state-variable-based simulation programs and toolboxes. To the bestof the author’s knowledge, the six-phase machine models withsuch properties have not been proposed in the prior literature.

II. MODEL FORMULATIONS AND EQUIVALENT CIRCUIT

To set the stage for the new model, a six-phase synchronousmachine with two sets of sinusoidally distributed symmetricthree-phase windings on the stator is assumed. For the pur-pose of this paper, the magnetic saturation is not considered. Asimplified cross sectional view is shown in Fig. 1. The statorwindings are denoted by the two sets abc1 and abc2, respec-tively, and the sets are assumed to be displaced by angle ζ.A salient rotor is assumed to have one damper winding along theq−axis, and one damper and a field winding along the d−axis[11]. All parameters are assumed to be referred to the statorside, and the motor sign convention is used. To be consistent

with typical nomenclature [13] and prior literature (e.g., [22]),throughout this paper, bold upper-case letters are used to denotematrices; bold lower-case letters denote vectors; and lower-caseitalic letters denote instantaneous quantities/variables, and otherparameters.

A. Coupled-Circuit Phase-Domain Model

Based on Fig. 1, the CCPD model can be can be expressedusing the following voltage and flux equations:

⎡⎢⎣

vsabc1

vsabc2

vrqd

⎤⎥⎦=

⎡⎢⎣

rs1s1 0 0

0 rs2s2 0

0 0 rrr

⎤⎥⎦+ p

⎡⎢⎣

λsabc1

λsabc2

λrqd

⎤⎥⎦ , (1)

⎡⎢⎣

λsabc1

λsabc2

λrqd

⎤⎥⎦ =

⎡⎢⎣Ls1s1(θr ) Ls1s2(θr ) Ls1r (θr )

Ls2s1(θr ) Ls2s2(θr ) Ls2r (θr )

Lrs1(θr ) Lrs2(θr ) Lrr

⎤⎥⎦

9 × 9

×

⎡⎢⎣isabc1

isabc2

irqd

⎤⎥⎦ . (2)

Here, the Heaviside’s operator is used to denote the timederivative, p = d/dt. In (2), the stator self and mutual inductancesare defined in (3)–(5) [11], shown at the bottom of the page. In(1), the rotor self-inductance matrix is:

Lrr =

⎡⎣

Llkq + Lmq 0 00 Llkd + Lmd Lmd

0 Lmd Llf d + Lmd

⎤⎦ . (6)

The remaining sub-matrices are

Ls2s2(θr ) = Ls1s1(θr − ζ), (7)

Ls2r (θr ) = Ls1r (θr − ζ), (8)

Ls1s1(θr )

=

⎡⎢⎣Lls1 + LA + LB cos(2θr ) −LA/2 + LB cos(2θr − 2π/3) −LA/2 + LB cos(2θr + 2π/3)−LA/2 + LB cos(2θr − 2π/3) Lls1 + LA + LB cos(2θr + 2π/3) −LA/2 + LB cos(2θr )−LA/2 + LB cos(2θr + 2π/3) −LA/2 + LB cos(2θr ) Lls1 + LA + LB cos(2θr − 2π/3)

⎤⎥⎦

3 × 3

(3)

Ls1s2(θr )

=

⎡⎣Lla1a2 + LA cos(ζ) + LB cos(2θr − ζ) Lla1b2 + LA cos(ζ + 2π/3) + LB cos(2θr − ζ − 2π/3)Lla1c2 + LA cos(ζ − 2π/3) + LB cos(2θr − ζ − 2π/3) Lla1a2 + LA cos(ζ) + LB cos(2θr − ζ + 2π/3)Lla1b2 + LA cos(ζ + 2π/3) + LB cos(2θr − ζ + 2π/3) Lla1c2 + LA cos(ζ − 2π/3) + LB cos(2θr − ζ)

Lla1c2 + LA cos(ζ − 2π/3) + LB cos(2θr − ζ + 2π/3)Lla1b2 + LA cos(ζ + 2π/3) + LB cos(2θr − ζ)Lla1a2 + LA cos(ζ) + LB cos(2θr − ζ − 2π/3)

⎤⎦

3 × 3

(4)

Ls1r (θr )

=

⎡⎣

Lmq sin(θr ) Lmd cos(θr ) Lmd cos(θr )Lmq sin(θr − 2π/3) Lmd cos(θr − 2π/3) Lmd cos(θr − 2π/3)Lmq sin(θr + 2π/3) Lmd cos(θr + 2π/3) Lmd cos(θr + 2π/3)

⎤⎦

3 × 3

(5)


Ls2s1(θr ) = LTs1s2(θr ), (9)

Lrs1(θr ) =23LT

s1r (θr ), (10)

Lrs2(θr ) =23LT

s2r (θr ), (11)

where LA and LB are defined as [13]:

LA =Lmd + Lmq

3, LB =

Lmd − Lmq

3(12)

The mutual leakage inductances in (4) are denoted by Lla1a2 ,Lla1b2 , . . . , Llc1c2 . These inductances are the result of magneticflux that does not cross the air gap but couples the three-phasestator sets. Assuming that every stator phase has the same wind-ing function, the following relation between the mutual leakageinductances can be established [11], [12]:

Lla1a2 = Llb1b2 = Llc1c2 , Lla1b2 = Llb1c2 = Llc1a2

Lla1c2 = Llb1a2 = Llc1b2 (13)

Finally, the electromagnetic torque is expressed in (14), shownat the bottom of the page.

The CCPD model derived in (1)–(14) has 9 × 9 inductancematrix. This model can be interfaced with arbitrary networksconnected to both stator and rotor circuits. However, accordingto (3)–(4), the inductance matrices are variable depending onthe rotor position.

B. qd0 Model

The qd0 model is derived by transforming the correspondingCCPD model to the rotor reference frame [11], wherein onetransformation per three-phase set is required. The transforma-tion can be expressed in matrix form as

fqd012 = K(θr )fabc1abc2 , (15)

where

K(θr ) = diag[K(θr ),K(θr − ζ)], (16)

fabc1abc2 = [fa1 , fb1 , fc1 , fa2 , fb2 , fc2 ]T , (17)

fqd012 = [fd1 , fq1 , f01 , fd2 , fq2 , f02 ]T . (18)

Here, f may represent voltage, current, and flux variables, andK is the Park’s 3 × 3 transformation matrix [13]:

K(θr ) =23

⎡⎢⎣

cos(θr ) cos(θr − 2π3 ) cos(θr + 2π

3 )

− sin(θr ) − sin(θr − 2π3 ) − sin(θr + 2π

3 )12

12

12

⎤⎥⎦ .

(19)The machine voltage and flux equations (1)−(2) can be trans-

formed to the rotor reference frame using (15), which results

Fig. 2. Transformed qd0 equivalent circuit of six-phase synchronous machinein rotor reference frame [11].

(20)–(37). Based on the transformed equations (20)−(37), anequivalent circuit for the machine qd0 model can be derived asdepicted in Fig. 2.

The corresponding stator voltage equations in rotor referenceframe are:

vsq1 = rsisq1 + ωrλsd1 + pλsq1 , (20)

vsd1 = rsisd1 − ωrλsq1 + pλsd1 , (21)

vs01 = rsis01 + pλs01 , (22)

vsq2 = rsisq2 + ωrλsd2 + pλsq2 , (23)

vsd2 = rsisd2 − ωrλsq2 + pλsd2 , (24)

vs02 = rsis02 + pλs02 . (25)

The rotor voltage equations are:

0 = rkq ikq + pλkq , (26)

0 = rkdikd + pλkd , (27)

vf d = rf dif d + pλf d . (28)

The stator flux linkages are:

λsq1 = λmq + Llm (isq1 + isq2) + Llisq1 − Lldq isd2 , (29)

λsd1 = λmd + Llm (isd1 + isd2) + Llisd1 + Lldq isq2 , (30)

λs01 = L01is01 + (Lla1a2 + Lla1b2 + Lla1c2)

× (is01 + is02), (31)

λsq2 = λmq + Llm (isq1 + isq2) + Llisq2 + Lldq isd1 , (32)

Te(θr ) =P

2

{−1

2

([isabc1isabc2

])T ([Ls1s1(θr ) Ls1s2(θr )Ls2s1(θr ) Ls2s2(θr )

]− LlsI6 × 6

)[isabc1isabc2

]+([

isabc1isabc2

])T [Ls1r (θr )Ls2r (θr )

]irqd

}.

(14)


λsd2 = λmd + Llm (isd1 + isd2) + Llisd2 − Lldq isq1 , (33)

λs02 = L02is02 + (Lla1a2 + Lla1b2 + Lla1c2)

× (is01 + is02). (34)

The rotor flux linkages can be expressed as

λkq = λmq + Llkq ikq , (35)

λkd = λmd + Llkdikd , (36)

λf d = λmd + Llf dif d . (37)

In (29)–(37), the magnetizing fluxes are defined as

λmq = Lmq (isq1 + isq2 + ikq ), (38)

λmd = Lmd(isd1 + isd2 + ikd + if d). (39)

Note that due to the mutual leakage flux and the angle dis-placement ζ between the stator three-phase sets, the stator mag-netic axes d1 and q2 become coupled. Similarly, the axes q1 andd2 are also coupled. This coupling is also depicted in Fig. 2 andit can be calculated as [11]:

Llm = La1a2 cos ζ + La1b2 cos(

ζ +2π

3

)

+ La1c2 cos(

ζ − 2π

3

), (40)

Lldq = La1a2 sin ζ + La1b2 sin(

ζ +2π

3

)

+ La1c2 sin(

ζ − 2π

3

). (41)

The electromagnetic torque of the machine can also be cal-culated using the following equation:

Te =32

P

2[(isq1 + isq2)Lmd(isd1 + isd2 + ikd + if d)

−(isd1 + isd2)Lmq (isq1 + isq2 + ikq )] . (42)

Equations (20)–(42) and equivalent circuit in Fig. 2 definethe qd0 model for six-phase synchronous machine [11]. Notethat this qd0 state model has the rotor and stator fluxes as thestate variables and currents as its outputs. Because of that, it cannot be directly interfaced to an inductive or power electronicnetwork without snubbers [14].

C. VBR Model [19]–[20]

Derivation of VBR formulation starts from the qd0 model.For completeness, this model is also briefly presented here. Therotor fluxes and stator currents are selected as the state variables[17]. Substituting the rotor currents in (38)–(39) with (35)–(37),

the magnetizing fluxes can be obtained as:

λmq = L′′mq

(λkq

Llkq+ isq1 + isq2

), (43)

λmd = L′′md

(λkd

Llkd+

λf d

Llf d+ isd1 + isd2

). (44)

Here, the sub-transient inductances are:

L′′mq =

(1

Lmq+

1Llkq

)−1

, (45)

L′′md =

(1

Lmd+

1Llkd

+1

Llf d

)−1

. (46)

Using (26)–(28), the rotor currents (35)–(37), and magnetiz-ing fluxes (43)–(44), the following state equations for machinerotor fluxes can be obtained:

pλkq =rkq

Llkq

[(L′′

mq

Llkq− 1)

λkq + L′′mq (isq1 + isq2)

], (47)

pλkd =rkd

Llkd

[(L′′

md

Llkd− 1)

λkd + L′′md

(λf d

Llf d+ (isd1 + isd2)

)], (48)

pλf d =rf d

Llf d

[(L′′

md

Llf d− 1)

λf d + L′′md

(λkd

Llkd+ (isd1 + isd2)

)]+ vf d . (49)

Substituting the stator fluxes in (20)–(25) with (29)–(34), andusing (43)–(44) for the magnetizing fluxes and (47)–(49) for therotor state equation, the stator voltage equation can be re-writtenin the following matrix form:

⎡⎢⎢⎢⎢⎣

vsq1

vsd1

vsq2

vsd2

⎤⎥⎥⎥⎥⎦

= rs

⎡⎢⎢⎢⎢⎣

isq1

isd1

isq2

isd2

⎤⎥⎥⎥⎥⎦

+ pL′′qd12

⎡⎢⎢⎢⎢⎣

isq1

isd1

isq2

isd2

⎤⎥⎥⎥⎥⎦

+ωr

⎡⎢⎢⎢⎢⎣

λsd1 − λ′′sd1

λ′′sq1 − λsq1



⎤⎥⎥⎥⎥⎦

+

⎡⎢⎢⎢⎢⎣

e′′sq1

e′′sd1

e′′sq2

e′′sd2

⎤⎥⎥⎥⎥⎦

(50)

where the sub-transient inductance matrix is defined in (51),shown at the bottom of the page, and the sub-transient

L′′qd12 =

⎡⎢⎢⎣

Ll + L′′mq + Llm 0 L′′

mq +Llm −Lldq

0 Ll + L′′md + Llm Lldq L′′

md + Llm

L′′mq + Llm Lldq Ll + L′′

mq + Llm 0−Lldq L′′

md + Llm 0 Ll + L′′md + Llm

⎤⎥⎥⎦

4 × 4

(51)


back-emf voltages are:

e′′sq1 = ωrλ′′d1 +

rkqL′′mq

L2lkq

(L′′

mq

Llkq− 1)

λkq

− rkqL′′2mq

L2lkq

(isq1 + isq2), (52)

e′′sd1= −ωrλ′′q1 +

(rkdL

′′2md

L3lkd

− rkdL′′2md

Llkd+

rf dL′′2md

LlkdL2lf d

)λkd

+

(rf dL

′′2md

L3lf d

− rf dL′′md

L2lf d

+rkdL

′′2md

Llf dL2lkd

)λf d

−(

rkdL′′2md

L2lkd

+rf dL

′′2md

L2lf d

)(isd1 + isd2) + vf d (53)

e′′sq2 = ωrλ′′d2 +

rkqL′′mq

L2lkq

(L′′

mq

Llkq− 1)

λkq

− rkqL′′2mq

L2lkq

(isq1 + isq2), (54)

e′′sd2 = − ωrλ′′q2 +

(rkdL

′′2md

L3lkd

− rkdL′′md

L2lkd

+rf dL

′′2md

LlkdL2lf d

)λkd

+

(rf dL

′′2md

L3lf d

− rf dL′′md

L2lf d

+rkdL

′′2md

Llf dL2lkd

)λf d

−(

rkdL′′2md

L2lkd

+rf dL

′′2md

L2lf d

)(isd1 + isd2)

+L′′

md

Llf dvf d . (55)

In (52)–(55), the sub-transient fluxes are defined as:

λ′′q1 = λ′′

q2 =L′′

mq

Llkqλkq , (56)

λ′′d1 = λ′′

d2 =L′′

md

Llkdλkd +

L′′md

Llf dλf d . (57)

Applying the inverse transformation to (50) yields the volt-age equation in six-phase abc1abc2 coordinates, which can bewritten as

vsabc1abc2 = rs isabc1abc2 + (pL′′abc1abc2(θr )) isabc1abc2

+ L′′abc1abc2(θr ) (pisabc1abc2) + e′′sabc1abc2 .

(58)

Here, the variable inductance matrix is:

L′′abc1abc2(θr ) = K(θr )−1L′′

dq12K(θr ). (59)

which is also given in (B1) in Appendix B. Equation (59) rep-resents the desired interfacing circuit model that can be rep-resented using six RL-branches behind the current-controlledvoltage sources as depicted in Fig. 3. Unlike the qd0 machinemodel, the VBR model can be easily interfaced with any ex-ternal network without the need for snubber circuits. Comparedto the CCPD model, the size of the variable inductance matrixdecreases from 9 × 9 to 6 × 6, which reduces computationalburden.

Fig. 3. Implementation of the proposed CPVBR model with constant-parameter RL branch interfacing circuit.

The electromagnetic torque (42) is also valid for the VBRmodel. The VBR model defined by (47)–(55) is an exact equiv-alent to the CCPD defined in Section II–A and the qd0 modelpresented in Section II–B, as it was derived using only al-gebraic manipulations and no approximations. Moreover, therotor-position-dependent inductance matrix in (58) implies thatthe interfacing branches will have variable and coupled induc-tances, which complicates the implementation of this model aswell as making it computationally expensive [18].

III. CONSTANT-PARAMETER VBR MODEL

The variable inductances in VBR machine model are theresult of different sub-transient inductances along rotor q– andd–axes in (50), making the inductance matrix (51) variable whenit is transformed back to abc1abc2 coordinates (59). In orderto make the interfacing equation (58) constant parameter (andalso the corresponding interfacing circuit), one can modify (50)so that stator voltages would have the same sub-transient in-ductances in both q– and d–axes. The extra terms resulted frommaking the q– and d–axes sub-transient inductances equal arethen added to the sub-transients back emf voltages, as shownin (60):

⎡⎢⎢⎣

vsq1vsd1vsq2vsd2

⎤⎥⎥⎦ = rs

⎡⎢⎢⎣

isq1isd1isq2isd2

⎤⎥⎥⎦+ pL′′′

qd12

⎡⎢⎢⎣

isq1isd1isq2isd2

⎤⎥⎥⎦

+ ωr

⎡⎢⎢⎣





⎤⎥⎥⎦ ...

... − (L′′md − L′′

mq )

⎡⎢⎢⎣

p(isq1 + isq2)−ωr (isq1 + isq2)p(isq1 + isq2)−ωr (isq1 + isq2)

⎤⎥⎥⎦

+

⎡⎢⎢⎣

e′′sq1e′′sd1e′′sq2e′′sd2

⎤⎥⎥⎦ , (60)


where the modified matrix L′′′qd12 in (60) is given in (61),

shown at the bottom of the page, and has equal values in q– andd−axes. Also, based on (60), the new sub-transient back-emfvoltage vector can be defined containing the leftover saliencyterms from the q−axis sub-transient inductance:⎡⎢⎢⎣

e′′′sq1e′′′sd1e′′′sq2e′′′sd2

⎤⎥⎥⎦ =

⎡⎢⎢⎣

e′′sq1e′′sd1e′′sq2e′′sd2

⎤⎥⎥⎦− (L′′

md − L′′mq )

⎡⎢⎢⎣

p(isq1 + isq2)−ωr (isq1 + isq2)p(isq1 + isq2)−ωr (isq1 + isq2)

⎤⎥⎥⎦ .

(62)Based on (61)–(62), the interfacing circuit equation becomes:

vsabc1abc2 = rs isabc1abc2 + p (L′′′abc1abc2 isabc1abc2)

+ e′′′sabc1abc2 , (63)

where the matrix L′′′abc1abc2 , as shown in (B5) in Appendix B, is

constant and obtained from L′′′qd12 as

L′′′abc1abc2 = K(θr )−1L′′′

dq12K(θr ). (64)

It should be noted that the leftover term at the end of (62)compensates for the sub-transient saliency and contains thederivatives of the stator currents in q−axis (isq1 and isq2)and their resulting rotational-induced voltages on the machined−axis. The rotational-induced voltage terms in (62) are easilycalculated using the rotor speed and the stator currents. How-ever, obtaining the stator current derivatives along q−axis ismore challenging. Based on (29) and (32), it is possible tocalculate the sum of q−axis stator currents derivatives as:

p (isq1 + isq2) =1

2Llm + Ll· p {(λsq1 + λsq2)

−2λmq − Llqd (isd2 − isd1)} . (65)

However, (65) also contains the q−axis stator flux linkagesand the d−axis stator currents. Substituting (30) and (33), andusing (20)–(25), after some algebraic manipulations, (65) can bere-written in terms of stator currents, rotor fluxes, and voltages,as presented in (66), shown at the bottom of the page. Eq. (66) isthen used to calculate the new sub-transient back-emfs in (62).

Since (66) requires stator voltages vsq1 ,vsq2 ,vsd1 , andvsd2 , (62) and (66) contain algebraic feed-through fromstator voltages to the sub-transient back-emfs. At the same time,stator branch voltages (63) are also directly/algebraically depen-dent on the sub-transient back-emfs, which creates an algebraic

Fig. 4. Case study system with a six-phase synchronous generator feeding arectifier load and a three-phase ac grid.

loop and makes the formulation implicit. This algebraic loopmay be relaxed [22] by introducing an intermediate variable vs

that contains all stator voltages in (66) as

vs = vsq1 + vsq2 +Lldq

Ll(vsd2 − vsd1), (67)

and passing it through a low-pass filter

v′s =

11 + τf s

vs. (68)

Thereafter, v′s is used in (66) instead of vsq1 , vsq2 , vsd1 ,

and vsd2 to calculate the stator currents derivatives. Similar tothe three-phase VBR model, the filter pole (−1/τf ) should bechosen to give an accurate estimate of vs up to certain frequencywithout introducing large eigenvalue into the system. Interestedreader will find the detailed procedure with example in [23].

Finally, based on (63), the interfacing circuit of the six-phaseCPVBR is shown in Fig. 3. The interfacing inductances in eachphase are constant as shown in (B5) in Appendix B.

IV. CASE STUDY AND SIMULATION RESULTS

A. Test System and Simulated Study

To verify the proposed CPVBR model, the benchmark sys-tem [11] is used where a six-phase synchronous machinewith 30 degrees shift (ζ) between the two three-phase statorsets is considered. The machine is assumed to operate as agenerator driven by a turbine prime mover that supplies me-chanical torque. For the purpose of investigating the models’interfacing properties, the generator is connected to a networkconsisting of a six-pulse rectifier through set abc2 and a three-phase ac system through set abc1, as shown in Fig. 4. The

L′′′qd12 =

⎡⎢⎢⎣

Ll + L′′md + Llm 0 L′′

md + Llm −Lldq

0 Ll + L′′md + Llm Lldq L′′

md + Llm

L′′md + Llm Lldq Ll + L′′

md + Llm 0−Lldq L′′

md + Llm 0 Ll + L′′md + Llm

⎤⎥⎥⎦

4 × 4

(61)

p(isq1 + isq2) =

(Ll

L2l + 2Ll(Llm + L′′

mq) − L2ldq

).

{−rs

(isq1 + isq2 +

Lldq

Ll(isd2 − isd1)

)+ ωr

(λsd1 + λsd2 − Lldq

Ll(λsq2 − λsq1)

)

+ 2L′′

mqrkq

LlkqLlkq

[(L′′

mq

Llkq− 1)

λkq + L′′mq (isq1 + isq2)

]− vsq1 − vsq2 − Lldq

Ll(vsd2 − vsd1)

}. (66)


neutral points of the two wye-connected three-phase sets areisolated and floating. The three-phase ac grid is represented byits Thevenin equivalent impedance Zgrid and source vgabc . Theremaining system parameters are summarized in Table IV, inAppendix A.

The power system of Fig. 4 has been implemented in MAT-LAB/Simulink [17] software using PLECS [15] toolbox. Forthe purpose of comparison, four machine models have been im-plemented: qd0 model, CCPD model, basic VBR model (withvariable inductances), and the proposed CPVBR model. Forconsistency, all the subject models have been executed on aPC with Intel R© CoreTM i7-4700 @ 2.4 GHz processor usingvariable time step solver ode45 with relative and absolute errortolerances set to 10−4. To assess the numerical accuracy of thesubject models, the reference solution has been obtained usingthe CCPD model with relative and absolute error tolerances setto a very small value of 10−7 and a maximum step size limitedto 20 μs.

Since the qd0 model does not allow direct connection to ex-ternal inductive networks, according to methodology [14], sixun-grounded snubber resistors connected in parallel with themachine stator terminals have been used to interface this modelwith the network assumed in Fig. 4. It should be noted thatchoosing larger snubber resistors improves accuracy of such in-terfacing at the cost of having larger system eigenvalues andlower numerical efficiency. Here, the snubber resistors are se-lected to be 120 Ω (which corresponds to 100 pu, with themachine nominal power and voltage taken as the base values),to limit the snubber branch current to approximately 1% of thenominal stator current.

For the proposed CPVBR model, the filter pole (−1/τf ) in(68) should be sufficiently large in order to have good accu-racy of this approximation in steady-state and fast transientsup to certain frequency. Since the machine is connected to a60 Hz grid and a diode-rectifier, the stator voltage and cur-rent experience constantly excited transients with significant 5th

and 7th harmonics due to rectifier switching. Therefore, evenin steady-state, the stator voltages in qd0 frame [i.e. vsq1 , vsq2 ,vsd1 , vsd2 , and consequently vs in (68)] contain the frequency of360 Hz associated with the 7th harmonic. Hence, according tothe methodology set forth in [23], to achieve a solution accuracywithin 1% at 360 Hz, the pole (−1/τf ) is selected to be −2000.Choosing a larger pole for the low-pass filter improves accuracyat the cost of making the model numerically stiffer.

B. Case Study

The considered transient study assumes that the six-phasesynchronous generator is operating in steady-state delivering100 kW of total power to its loads. The prime mover turbine issupplying input mechanical torque of 500 Nm, which is assumedconstant during the study. The machine three-phase set abc1provides approximately 40 kW to the ac grid. The set abc2 isfeeding 60 kW into a rectifier supplying dc system representedby a 2.5 Ω resistive load. At t = 0.5 s, a single phase-to-groundfault occurs on phase a1 of the generator as depicted in Fig. 4.

Fig. 5. System transient response to a single phase-to-ground fault on phasea1 as predicted by the subject models: a) and b) phase currents for set 1 and set2, c) electromagnetic torque, d) rotor speed, and e) rectifier dc voltage.

The corresponding transients as predicted by the subject modelsare shown in Fig. 5 for a 2 second simulated study. Fig. 5 showsthat the fault mostly affects the set abc1, due to containing thefaulty phase. Meanwhile, due to the mutual leakage inductancesbetween the two sets, the fault disturbance also propagates intothe set abc2 causing the reduction and ripple of the rectifier dcvoltage as shown in Fig. 5 (e).

Fig. 6 shows magnified view of the generator phase volt-ages, currents, and output torque during the fault transient. Ascan be observed in Fig. 6, all the four subject models pre-dict the transients that are visibly very close to the reference.To examine the performance of models even further, the mag-nified views from Fig. 6 at the fault moment are shown inFig. 7, which reveals a noticeable difference among the mod-els in terms of number of steps taken and accuracy. As it canbe seen in Fig. 7, the qd0 model visibly deviates from thereference solution and takes significantly larger number of timesteps as compared to other models. This deficiency of qd0 modelis attributed to the artificial interfacing snubbers that draw theadditional small currents introducing solution error, as well asmaking the model numerically stiff and cause the variable stepsolver to choose much smaller time steps. At the same time, the


Fig. 6. Magnified view of selected variables showing the details of the faulttransient obtained by the subject models for: a) phase currents of abc1 set; b)phase currents of abc2 set; c) stator phase voltages of abc1 set; d) stator voltagesof abc2 set; and e) electromagnetic torque.

CCPD, basic VBR and CPVBR models are directly interfacedand achieve highly accurate results with significantly fewer timesteps.

C. Computational Efficiency and Accuracy

To further assess the computational efficiency of the subjectmodels, the CPU time taken and the number of time steps forthe 2 second transient study are summarized in Table I. As canbe seen in Table I, the qd0 model takes the highest numberof steps (484, 570) and longest CPU time (51s). This perfor-mance is due to high stiffness created by the artificial snubbers.The directly interfaced CCPD, basic VBR and CPVBR mod-els are solved with considerably fewer number of time steps(10, 536), (8, 427) and (9, 464), respectively. The CCPD andbasic VBR models have rotor-position-dependent (variable) in-ductances of the interfacing circuit, which puts a computationalburden on the solver due to re-calculating the system impedancematrix at each time step. However, the inductance matrix of

Fig. 7. Magnified view of the transient response of the subject models com-pared to the reference solution at the fault moment for a) stator phase ‘a1’ andb) ‘a2’ currents and c) electromagnetic torque.

TABLE ICPU TIME AND NUMBER OF STEPS FOR THE 2 SECOND STUDY FOR THE

SUBJECT MODELS

Model Simulation Time Number of steps

qd0 model with snubbers 51 s 484, 570CCPD model(with variable inductance) 16.62 s 10, 536VBR model(with variable inductance) 14.51 s 8, 427CPVBR model 3.3 s 9, 464

the basic VBR model is 6 × 6, whereas this matrix is 9 ×9 for the CCPD model. This results in a CPU time of 14.51s for the basic VBR and 16.62 s for the CCPD model. Fi-nally, the CPVBR model achieves the fastest solution due to itsconstant parameter interfacing circuit, taking only 3.3 s. How-ever, the superior performance of CPVBR model is achievedat the cost of approximating the voltage in (68) to relax thealgebraic loop, which introduces a small numerical error in thesolution.

To compare the numerical accuracy of the subject models,the errors in the simulation results for stator phase a1 currentisa1 and electromagnetic torque Te are summarized inTable II. Without loss of generality, this paper considers the2-norm cumulative error evaluated over the 2 second solution


TABLE IICOMPARISON OF 2-NORM ERROR IN THE SIMULATION RESULTS

Model Stator Current a1 error Torque error

qd0 model with snubbers 1.4% 1.8%CCPD model(with variable inductance) 0.01% 0.006%VBR model(with variable inductance) 0.01% 0.009%CPVBR model 0.7% 0.73%

TABLE IIICOMPARISON OF THE LARGEST EIGENVALUE AND NUMBER OF STATE

VARIABLES

Model Largest eigenvalue Number of state variables

qd0 model with snubbers −9.07 × 105 44CCPD model(with variable inductance) −1,191 35VBR model(with variable inductance) −447 35CPVBR model − 1,783 36

interval. The 2-norm error of the solution trajectory x with re-spect to reference solution xref is defined as [24]:

‖e‖2 =‖xref (t) − x(t)‖2

‖xref (t)‖2. (69)

According to Table II, the qd0 model has the largest numer-ical error (1.4% for the stator phase a1 current and 1.8% forthe output torque) due to the artificial snubbers. The CCPD andbasic VBR produce most accurate results (0.01% error for sta-tor phase a1 current for both models, and 0.006% and 0.009%for the torque calculated by the CCPD and basic VBR mod-els, respectively). These small errors are mainly due to thesolver discretization. Table II also verifies that the CPVBRmodel achieves a very good accuracy (0.7% error in statorphase a1 current and 0.73% in torque). It is worth mention-ing that the performance of CPVBR model is a compromisebetween accuracy and stiffness (i.e. the model can be mademore accurate by choosing a faster low-pass filter pole, whichcomes at the cost of numerical stiffness and slows down thesimulation).

To see the impact on model stiffness, the largest eigenvalueand number of state variables of the overall power systemwith each of the subject machine models are summarized inTable III. Table III shows that the qd0 model has the largest num-ber of state variables (44) as well as largest eigenvalue (−9.07× 105) due to the snubbers. The effect of this eigenvalue can beclearly seen on the number of time steps and CPU time taken bythe qd0 model. The CCPD model has the minimum number ofstate variables (35) with the largest system eigenvalue (−1,191).The basic VBR model uses the same number of state variables(35) but improves the eigen-structure compared to the CCPDmodel [18], which results in reduction of the largest eigenvalue(−447) and making this model more efficient. The first orderlow-pass filter in the CPVBR model adds one state variable

compared to the CCPD and basic VBR models (36 states). Thelargest eigenvalue in the CPVBR model (−1,783) is associatedwith the low-pass filter pole selection in (68). The value of theselected pole is affected by the maximum transient frequencyin which the accuracy of the model should remain within ac-ceptable range (under 1% error in this case). Since the machineis connected to a diode rectifier, the large pole in the CPVBRmodel corresponds to the maximum desired frequency of 360 Hzaccording to [23]. However, in spite of having a larger eigen-value, the constant parameter interface of the CPVBR modelsignificantly reduces its computational cost by eliminating thenetwork impedance matrix re-calculations at every time step.The result is superior simulation performance of the CPVBRmodel compared to the CCPD and basic VBR models in terms ofCPU time.

V. CONCLUSIONS

Recently, a voltage-behind-reactance (VBR) formulation hasbeen presented for three-phase machines models and six-phasesynchronous machines as a methodology of improving thenumerical properties of machine models and their interfac-ing with external networks in various simulation packages.In this paper, a constant-parameter VBR (CPVBR) model forsix-phase synchronous machines has been proposed for thefirst time.

The desirable constant-parameter RL-branch interfacing cir-cuit is achieved by removing the dynamic saliency in thesub-transient inductances and appropriately compensating thesub-transient back emf sources. This formulation also involvesapproximation and relaxation of algebraic loop using a low-pass filter in order to yield an explicit formulation. However,the resulting CPVBR possesses significant advantages over al-ternative/existing models (i.e. CCPD, basic VBR, and qd0) interms of its interfacing and numerical efficiency, which comeswith only a slight reduction of accuracy as compared to the di-rectly interfaced models (CCPD and basic VBR). The simpleand constant interfacing circuit of CPVBR model also makesit more straightforward to implement the six-phase machinemodels in various simulation programs.

APPENDIX A

TABLE IVMACHINE PARAMETERS [11]

Parameters Value

Rated Phase Voltage / Power 240 V / 100 kVARated Rotor Speed / Torque 1800 rpm / 530 N.mLm d , Lm q , Ll s 3 mH, 1.4 mH, 150 μHLl f d , Ll k q , Ll k d 120 μH, 180 μH, 140 μHrs , rr k q , rr k d , rf d 16 mΩ , 2.5 mΩ , 2.3 mΩ , 1.6 mΩLa 1 a 2 , La 1 b 2 , La 1 c 2 43 μH, −43 μH, 0 HRc a b l e , Lc a b l e , Rg r i d , Lg r i d 0.05 Ω , 50 μH, 0.05 Ω , 50 μHRl o a d , Ld c , Cd c 2.5 Ω , 10 mH, 6 mF


APPENDIX B

where:

F1(x, y) = L ′′m d +L ′′

m q

3 cos(x) + L ′′m d −L ′′

m q

3 cos(2θ + y), (B2)

F2(x) = Llm cos(ζ + x) − Lldq sin(ζ + x), (B3)

By defining G1(x) as:

G1(x) = 2L ′′m d

3 cos(x), (B4)

the constant sub-transient interfacing inductance matrix in the CPVBR model can be expressed as:


REFERENCES

[1] E. Levi, “Multiphase electric machines for variable-speed applica-tions,” IEEE Trans. Ind. Electron., vol. 55, no. 5, pp. 1893–1909,May 2008.

[2] L. Parsa and H. A. Toliyat, “Fault-tolerant five-phase permanent magnetmotor drives,” in Proc. IEEE Ind. Appl. Soc. Annu. Meet., Seattle,WA,USA, 2004, pp. 1048–1054.

[3] G. J. Atkinson, B. C. Mecrow, A. G. Jack, D. J. Atkinson, P. Sangha,and M. Benarous, “The analysis of losses in high-power fault-tolerantmachines for aerospace applications,” IEEE Trans. Ind. Appl., vol. 42,no. 5, pp. 1162–1170, Sep./Oct. 2006.

[4] C. L. Ferreira and R. W. G. Bucknall, “Modelling and real-time sim-ulation of an advanced marine full-electrical propulsion system,” inProc. IEE Power Electron. Mach. Drives Conf., Edinburgh, U.K., 2004,pp. 574–579.

[5] E. Levi, M. Jones, and S. N. Vukosavic, “A series-connected two-motor six-phase drive with induction and permanent magnet ma-chines,” IEEE Trans. Energy Convers., vol. 21, no. 1, pp. 121–129,Mar. 2006.

[6] C. B. Jacobina, V. F. M. B. Melo, N. Rocha, and E. R. C. da Silva,“Six-phase machine conversion system with three- and single-phase se-ries converters,” IEEE Trans. Ind. Appl., vol. 50, no. 6, pp. 3846–3856,Nov./Dec. 2014.

[7] J. Figueroa, J. Cros, and P. Viarouge, “Generalized transformations forpolyphase phase-modulation motors,” IEEE Trans. Energy Convers.,vol. 21, no. 2, pp. 332–341, Jun. 2006.

[8] D. C. White and H. H. Woodson, Electromechanical Energy Conversion.New York, NY, USA: Wiley, 1959.

[9] R. H. Nelson and P. C. Krause, “Induction machine analysis for arbitrarydisplacement between multiple winding sets,” IEEE Trans. Power App.Syst., vol. PAS-93, no. 3, pp. 841–848, May 1974.

[10] J. R. Marti and K. W. Louie, “A phase-domain synchronous generatormodel including saturation effects,” IEEE Trans. Power Syst., vol. 12,no. 1, pp. 222–229, Feb. 1997.

[11] R. F. Schiferl and C. M. Ong, “Six phase synchronous machine withAC and DC stator connections, part I: Equivalent circuit representationand steady-state analysis,” IEEE Trans. Power App. Syst., vol. PAS-102,no. 8, pp. 2685–2693, Aug. 1983.

[12] R. F. Schiferl and C. M. Ong, “Six phase synchronous machine withAC and DC stator connections, part II: Harmonic studies and a proposeduninterruptible power supply scheme,” IEEE Trans. Power App. Syst.,vol. PAS-102, no. 8, pp. 2694–2701, Aug. 1983.

[13] P. C. Krause, O. Wasynczuk, S. D. Sudhoff, and S. Pekarek, Analysisof Electric Machinery and Drive Systems, 3rd ed. Piscataway, NJ, USA:IEEE Press, 2013.

[14] L. Wang et al., “Methods of interfacing rotating machine models intransient simulation programs,” IEEE Trans. Power Del., vol. 25, no. 2,pp. 891–903, Apr. 2010.

[15] Piecewise Linear Electrical Circuit Simulation (PLECS) User Manual,Version 3.7, Plexim GmbH, Zurich, Switzerland, 2015. [Online]. Avail-able: www.plexim.com

[16] SimPowerSystems: Model and Simulate Electrical Power Systems User’sGuide, The MathWorks, Inc., Natick, MA, USA, 2015. [Online]. Avail-able: http://www.mathworks.com

[17] Simulink Dynamic System Simulation Software Users Manual,MathWorks, Inc., Natick, MA, USA, 2015. [Online]. Available:http://www.mathworks.com

[18] S. D. Pekarek, O. Wasynczuk, and H. J. Hegner, “An efficient and accuratemodel for the simulation and analysis of synchronous machine/convertersystems,” IEEE Trans. Energy Convers., vol. 13, no. 1, pp. 42–48,Mar. 1998.

[19] A. M. Cramer, B. P. Loop, and D. C. Aliprantis, “Synchronous ma-chine model with voltage-behind-reactance formulation of stator and fieldwindings,” IEEE Trans. Energy Convers., vol. 27, no. 2, pp. 391–402,Jun. 2012.

[20] N. Amiri, S.M. Ebrahimi, J. Jatskevich, and H.W. Dommel, “Voltage-behind-reactance model of six-phase synchronous machine consideringstator mutual leakage inductance,” in Proc. 2015 Int. Conf. Power Syst.Transients, Cavtat, Croatia, Jun. 2015, pp. 1–6.

[21] N. Amiri, S.M. Ebrahimi, M. Chapariha, J. Jatskevich, and H.W. Dom-mel, “Voltage-behind-reactance model of six-phase synchronous ma-chines considering stator mutual leakage inductance and main flux sat-uration,” Elsevier J. Elect. Power Syst. Res., vol. 138, pp. 155–164, Sep.2016.

[22] M. Chapariha, F. Therrien, J. Jatskevich, and H. W. Dommel, “Explicitformulations for constant-parameter voltage-behind-reactance interfacingof synchronous machine models,” IEEE Trans. Energy Convers., vol. 28,no. 4, pp. 1053–1063, Dec. 2013.

[23] F. Therrien, M. Chapariha, and J. Jatskevich, “Pole selection procedure forexplicit constant-parameter synchronous machine models,” IEEE Trans.Energy Convers., vol. 29, no. 3, pp. 790–792, Sep. 2014.

[24] W. Gautchi, Numerical Analysis: An Introduction. Boston, MA, USA:Birkhauser, 1997.

Navid Amiri (S’11) received the B.Sc. and M.Sc.degrees in electrical engineering in the field of powerand electrical machines from Isfahan University ofTechnology, Isfahan, Iran, in 2008 and 2011, re-spectively. He is currently working toward the Ph.D.degree at The University of British Columbia, Van-couver, BC, Canada.

His research interests include efficient modeling ofelectric machines, electric drives, electromechanicalenergy conversion systems, electric machine design,and power electronics.

Seyyedmilad Ebrahimi (S’13) received the B.Sc.and M.Sc. degrees in electrical engineering fromSharif University of Technology, Tehran, Iran, in2010 and 2012, respectively. He is currently workingtoward the Ph.D. degree at The University of BritishColumbia, Vancouver, BC, Canada. His research in-terests include modeling and analysis of power elec-tronic converters and electrical machines, applicationof power electronics to power systems, high-voltagedc systems, control of power systems, and simulationof electromagnetic transients.

Mehrdad Chapariha (S’08–M’15) received theB.Sc. and M.Sc. degrees in electrical engineeringfrom Isfahan University of Technology, Isfahan, Iran,and the Ph.D. degree in electrical and computer en-gineering from The University of British Columbia,Vancouver, BC, Canada, in 2006, 2009, and 2013, re-spectively. He is currently with the Protection and Au-tomation Group of Quanta Technology, Toronto, ON,Canada, as a Senior Consultant, where he is work-ing on the development and application of industry-leading techniques for wide-area protection studies.

His research interests include modeling and simulation of power systems, ap-plications of power electronics in power systems, and electrical machines.


Juri Jatskevich (M’99–F’17) received the M.S.E.E.and Ph.D. degrees in electrical engineering from Pur-due University, West Lafayette, IN, USA, in 1997 and1999, respectively.

Since 2002, he has been a faculty member atThe University of British Columbia, Vancouver, BC,Canada, where he is currently a Professor of electricaland computer engineering. His research interests in-clude power electronic systems, electrical machinesand drives, and modeling and simulation of electro-magnetic transients. From 2008 to 2013, he served

as an Associate Editor of the IEEE TRANSACTIONS ON POWER ELECTRONICS,and is currently the Editor-in-Chief of the IEEE TRANSACTIONS ON ENERGY

CONVERSION, and the Editor of the IEEE POWER ENGINEERING LETTERS. In2009–2010, he Chaired the IEEE CAS Power Systems and Power ElectronicCircuits Technical Committee. He is also Chairing the IEEE Task Force onDynamic Average Modeling, under Working Group on Modeling and Analysisof System Transients Using Digital Programs.

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

From 1959 to 1966, he was at the Technical Uni-versity of Munich, and from 1966 to 1973, he waswith the Bonneville Power Administration, Portland,OR, USA. Since 1973, he has been at The Universityof British Columbia, Vancouver, BC, Canada, wherehe is currently a Professor Emeritus.

本文献由“学霸图书馆-文献云下载”收集自网络，仅供学习交流使用。

学霸图书馆（www.xuebalib.com）是一个“整合众多图书馆数据库资源，

提供一站式文献检索和下载服务”的24 小时在线不限IP

图书馆。

图书馆致力于便利、促进学习与科研，提供最强文献下载服务。

图书馆导航：

图书馆首页文献云下载图书馆入口外文数据库大全疑难文献辅助工具

http://www.xuebalib.com/cloud/

http://www.xuebalib.com/

http://www.xuebalib.com/cloud/


http://www.xuebalib.com/vip.html

http://www.xuebalib.com/db.php

http://www.xuebalib.com/zixun/2014-08-15/44.html


Constant-Parameter Voltage-Behind-Reactance Model of Six...

Documents

Transcript of Constant-Parameter Voltage-Behind-Reactance Model of Six...