IEEE Transactions on Power Apparatus and Systems, Vol. PAS-103, No. 4, April 1984

AN AGGREGATE INDUCTION MOTOR MODEL FOR INDUSTRIAL PLANTS

Graham J. RogersSystem Planning DivisionOntario Hydro, Toronto, Canada

John Di MannoSystem Planning DivisionOntario Hydro, Toronto, Canada

Robert T.H. Alden, S.M. IEEEDept. of Elect. and Comp. EngineeringMcMaster University, Hamilton, Canada

Abstract

A technique is developed for deriving an equi-valent load model to represent the dynamic and steadystate effects of a system consisting of inductionmotors and constant impedance loads interconnected by anetwork. Motor parameters are calculated from standardspecifications, and together with network and load dataare incorporated into an admittance matrix which isused to define most of the parameters of the load equi-valent. The inertia and running slip are chosen tominimize the error between the transient response ofthe system and its equivalent. Step responses arecalculated using a state approach. Adequacy of theequivalent is validated by comparing the response ofthe original system and the equivalent to simulated busvoltage change, transformer outage, and remote andlocal balanced faults using a transient stabilityprogram.

1. INTRODUCTION

It has long been recognised that the power systemload variation with system voltage must be accountedfor in accurate load flow and transient stabilitystudies. Over the years [.1], system tests have beenperformed which show that, in the steady-state,following a small change in voltage magnitude; the realpower and reactive power loads may be modelled by:

P= P 1V 'n

Q Q ,vmMany current programs allow for load representation inthis form or extensions of it to include the effects ofsystem frequency.

In highly stressed areas of a power system thistype of model may be inadequate because of its neglectof the dynamic nature of the power system load. Thisis reinforced by Concordia and Ihara in their recentexcellent overview paper [2]. Even in residentialareas a significant proportion of the total load iscomprised of the small motors required in refrigera-tors, space-heaters and air-conditioners. Generally,two types of power system loads can be identified:

- large industrial plants-;- aggregates of small industrial, residential and

commercial loads.

The second type of load is not amenable to direct

83 SM 377-9 A paper recommended and approvedby the IEEE Power System Engineering Committee ofthe IEEE Power Engineering Society for presentationat the IEEE/PES 1983 Summer Meeting, Los Angeles,California, July' 17-22, 1983. Manuscript submittedFebruary 2, 1983; made available for printing May2, 1983.

mathematical modelling. Models have been synthesisedfrom dynamic tests results. The procedure used byShackshaft et al. [3] is particularly good since bothmajor modes (mechanical and electrical) are excited bythe disturbance used in their test (a small voltagemagnitude and voltage phase change).

Iliceto and Capasso [4] examined in 1974 therepresentation of both' types of loads for voltagemagnitude variation and proposed an heuristic approach.Abdel Hakim and Berg [5] demonstrated in 1976 a singleunit steady state equivalent for parallelled inductionmotors. Richards and Tan [6] in 1979 applied parameterestimation for a single motor equivalent of a radiallyconnected group. In all of these studies, voltagephase changes were not considered. Sabir and Lee [7]have recently applied parameter estimation to obtain acomposite industrial load model from system transientresponses.

In this paper we address the general problem ofdeveloping a single machine equivalent of a fullyconnected subsystem containing induction motors andstatic loads. Load models of large industrial plantscan generally be synthesised from data obtainable fromplant engineers. A single line diagram of the plantand details of major dynamic and static loads are anecessary starting point. Since it. may be difficult toobtain explicit equivalent circuit paramet-ers, wepresent a method for estimating these from. commonlyavailable nane plate data. It is assuned that the loadmay be represented by a single induction motor inparallel with a static load, and the parameters areselected so that both steady state and dynamicresponses of. the model, to a small step change involtage magnitude and phase, are as close as possibleto that of the original.

The paper thus discusses the development of adetailed plant representation and subsequentaggregation to a simple equivalent model.

2. DETAILED REPRESENTATION OF INDUSTRIAL PLANT

The plant is considered to be radially connectedto the rest of the system, and is assuned to take thegeneral form of an interconnected system with inductionmotor and static loads distributed within it as shownin Fig. 1. Large synchronous motors should be modelledin detail in any transient stability study in whichplant dynamics are important and are thus notconsidered to be part of the aggregate load.

Fig. 1: System Representation

0018-9510/84/0400-0683$01.00 1984 IEEE

683

684Induction motors in the plant are represented

using the concepts developed by Brereton., Lewis andYoung [8 ]. For this, an equivalent circuit and aknowledge of the inertia of the motor and its load arerequired. Not all motor manufacturers provide equi-valent circuits for their machines but the necessaryparameters may be estimated from a knowledge of thestandard specifications of such machines. This isshown in Appendix A together with the calculation ofthe operating value of the slip.

Static loads are taken to be constant impedancesbut there is no difficulty in extending this repre-sentation to the normal exponent form discussed in theintroduction.

2.1 Linearised Plant Equations

Because the disturbance (a change in bus voltage)to be applied to the plant is assumed to be small, thedetailed plant equations can be linearised about thenormal working point and the response (correspondingchanges in power flow to the plant) obtained by lineardynamic analysis rather than from a step-by-stepnumerical integration.

The transformers, static loads and interconnectinglines of the plant are represented by an admittancematrix in which only the connecting and motor terminalnodes remain explicitly. Thus:

Ai] Y Y 1 AVi0 0 = oo on1 01 (2)AI Y Y AV-sj L no nnj L-MJ

where I and V are the injected current and voltage atthe corfhectinP bus, I and V are the vectors of nmotor currents and teranal voi%ages, etc.

It is shown in Appendix B that each motor isrepresented by a series impedance R + jX', an internalnode whose voltage is V!; ands forsn machines,differential equations of the form:

FAV E E 1] AV' FF r 1-i ni n2 -m + n(3AS En3 0 J s[AJ

Equation (2) is rewritten with the internal motor.voltages V? replacing the terminal voltages V , and theadmittanceW augmented by the series impedances Thus:

(4)A 0l | Y00 YOm [a op-s Lmo imm --in

Using the lower part of (4), we eliminate AI in (3)-sand obtain the following equation in state space form:

AV l, A A vVI [B v1 1 1 21 -m + 1 [LAs LA3 A4J ASj B2 (5)

The injected. power at the connection node is writtenas:

AP IId q AVd Vd Vq AdAQJ -I Id AVq V I AI

= G AV + G A II O V O

(6)(7)

Using the upper part of (4), we can eliminate AI in(7) . Thus: 0

FAP]0

= G Y AV' + [G + G Y ]AVAQ v Om

-m LI v oo oLo AV'1

= [C1,0] _m + [D][AV0]As

(8)

(9)where C =G Y1 v om

D =G + G YI v 00

Thus (5) and (9) constitute the state equation set forthe plant consistent with the initial objective:

Ax = AAx + BAu(10)

Ay = CAx + DAu

VI vqdo

2.2 Response of the Plant

The response to any input vector u(t) is given by:

ty(t) = C{4(t) x(0) + f 0(t-T) B u(T)dT} + D U(T) (11)

0

where (t) = U exp(At) U is the plant transitionmatrix.

U is the matrix of eigenvectors of A.

A is the matrix of eigenvalues of A.

For the plant step response, x(0) is zero and u(T) isconstant leading to

y(t) = f-CA (I - 4(t))B + D} u (12)

INDUCTION

EXTERNAL MOTORSYSTEM STATIC

LOAD

Fig. 2: Plant Aggregate Model

where

A = E + F Y and B = F Y1 ni n1 mm 1 nl moA = E2 n2A = E + F Y3 n3 n2 mmA = 04

2=

n2 mo

JR5 XXtG B Xm

I~~~~so

Fig. 3: Aggregate Model Equivalent Circuit

3. THE AGGREGATE LOAD MODEL

We assume that the plant may be represented by asingle induction motor in parallel with a static load,as shown in Fig. 2. Eight parameters are required tocompletely define this aggregate model as illustratedin Fig. 3. Two of these, G and B, describe the staticload and six, R , X, X , R , s and H describe themotor. All paPameters %xceit f6r s and H can beobtained so that in the steady state tRe load drawn bythe aggregate model is identical to that using thedetailed plant representation. The remaining two areselected to optimize the step response match.

3.1 The Aggregate Model

For the equivalent induction motor, the model ofBrereton, Lewis and Young is again used. See AppendixB for details. The static load is assumed to beconstant impedance. Because the two parts of theaggregate load are connected directly to the interfacebus, they can be considered separately and any changein real and reactive power, due to a change in busvoltage, is directly additive.

Thus for the motor:

FAV'l E E2 FAVJ [F] AISIm mj + 1 (13)[As [E0J [As] LFJ

and the following two phasor equations:

V = (R + j Xt)I + V' (14)s s s s m

P + j Q =V I* (15)m m s s

and for the static load

p + j Q (G - jB) IV2 (16)5 5Linearising and eliminating AI gives the model instate space form. Following the same procedure as forthe plant, we rewrite (14) and substitute into (13) toeliminate AI . Thus:

s

I =Y [V - VI] where Y1 s s-s s s m s XI R

FAV, [(E-FFY) EJ [AV'l [FYs] [AV(and As =1(E3-F2Y) J LAS + (17)

Similarly, we make the same substitution into (7) toobtain:

FAPi= GI AV + G aILAQI= [-Gv Y5][AVt] + [G0 + G Y ][A V] (18)

Equations (17) and (18) are of the form:Ax = A Ax + B Au

m m

Ay = C Ax + D Aum m

which is the same as (10) for the full representationfor the plant.

6853.2 Response of the Model

Similarly to that of the plant, the response ofthe model to a step change in voltage magnitude andphase is

y (t) = f-C A 1 (I - 4 (t)) B + D I umm m m m mz

where o (t) is the model transition matrix.m

( 1 9)

3.3 Static Load Parameters

The static load in the aggregate model is chosenas the sum of the static loads in the plant.

M 2Thus G = Z P ./:V

i= S1 S

M 2and B = E Q */V,

i=l Si S

(20)

(21 )

3.4 Induction Motor Electrical Parameters

The stator impedance of the single equivalentinduction motor must reflect the impedance of thenetwork of the plant as well as the motor impedances.It is calculated as the impedance to the network at theinterface bus with the machine voltages behindtransient reactance (V') set to zero and with staticloads neglected. This ensures identical initialresponse to a voltage change and using equation (4) wecan write:

R + j XI = 1/(Y - Y Y Y )s s .00 om mm mo (22)

The real and reactive power supplied to the equivalentmotor must be the difference between the total suppliedto the plant and that supplied to the static load.Thus:

M M

Pm + j Qm = T - 1 s i T 1 Si1=1~ ~ 1=giving tan 6 = Qm/Pm

(23)

(24)

where 0 is the motor power factor angle. If we assumethat ,V = 1 pu and that the equivalent motor isoperatirSg at full load with II = 1 pu, then thevoltage behind transient reactanc is given by:

(V' + jV') = (1 - R cos 8 - XI sin i)d q s s+ j(-X' cos 0 + R sin 6)

s s

It is shown in Appendix B that:

(25)

RX = {I cos 6 - R } / I sin 6 - X' }

0

Rand r = Xs5 5

0

(26)

(27)VI V'I

R5 q + XI } /I R - X qt }d dv~At full load s is approximately equal to R and takingR /s as unity for a starting value, (26) End (27) maybg Pterated to give consistent values for X and(R /s).

r o

The magnetizing reactance X is given by:m

686x = (x (X - X' )) 0 5m s s s

(28)

and the leakage reactance (assumed equally distributedbetween stator and rotor) is:

x = X - XQ s m (29)

3.5 Induction Motor Mechanical Parameters

Two induction motor parameters, the per unitinertia constant (H) and the initial slip (s ), remainundefined. They are chosen to optimize tI?e dynamicperformance of the model. The change in real andreactive power, owing to a change in voltage magnitudeand phase, for the aggregate is compared to that fromthe detailed plant representation. A generalisedoptimization routine, using the finite differenceLevenberg-Marquardt algorithm [9] readily available asan IMSL library subroutine, is used to determine H ands to minimize the least square error between the tworesponses defined in (12) and (19).

4. VALIDATION BY TRANSIENT SIMULATION

In order to validate this technique, a number ofstudies were conducted on a computer model which hasbeen used previously in *NPCC (Northeast PowerCoordinating Council) investigations. This model [10]consists of 39 buses, 46 lines and 10 generatorsrepresenting the New England 345 kV system and is shownin Fig. 4. A portion of the load at bus 23 is assumedto be the subsystem, containing five interconnectedinduction motors, shown in Fig. 5. The transformerreactances are assumed to be 9% on the appropriatemachine rating. Lines 1-4 have resistances of .001 pu,reactances of .0055 pu, and charging admittances of.0008 pu and line 5 twice these values, all on 100 MVAbase. The standard specifications for these motors areshown in Table 1.

Table 1: Motor SpecificationsM1,M3,M4 M2,M5

Stator Voltage (V)Output Power (kW)EfficiencyPower FactorFull Load SlipSync Speed (r2d/sec)Inertia (kg.m )Starting Current (amps)Torque Ratio (Starting/Rated)

40004178.962 5.910.0179377. 01 014520.82

400011 94.9433.900.0083125.77813661.10

Fig. 5: 5 Motor Subsystem

4.1 Parameter Evaluation

The machines have been chosen, deliberately, to bedissimilar and hence difficult to model by a singlemachine. The plant motor parameters were computed andare given in Table 2.

The linearized response of both the plant and itsaggregate model to a step change in voltage magnitudeand phase at bus 41 (the response bus) is shown in Fig.6. The aggregate model parameters are also given inTable 2, the mechanical parameters having been chosento minimize the least square error between theseresponses.

Using both detailed representation and aggregatemodel the response of the system was computed, using atransient stability program, to a number of differentdisturbances.

4.2 Transformer Switching

This disturbance is a simulation of the test usedby Shackshaft et al. to determine load characteristics.Two paralleled transformers have staggered taps causinga circulating current. When one transformer is dis-connected, a step change in voltage magnitude and phase

Table 2: Motor Parameters (based on 20 MVA, 4kV)

Motor H s R R X Xo r s m

M1,M3,M4 .359 .0220 .0816 .0408 .402 10.0

M2,M5 0.0308 .0102 .1266 .450 1.40 31.5

Aggregate 1.141 .0216 .0226 .0160 .168 3.14

4-

3-

2-PU

0

IN

A\QLA

0

- PLAN T---MODELFIG'S 6-9

.1 SEC .2 .3Fig. 4: NPCC Study System

Fig. 6: Unit Step Response Comparison

54 -

3 -MW \2-- \

MVAR

0--1 7

Fig.- 7:

.1 SEC .2 -.

Transformer Switching Test Response

is produced at the low tension bus. In this simula-tion, Ti is opened. The response of the two models isshown in Fig. 7.

4.3 Remote Three Phase Fault

A three phase fault cleared in 120 ms is simulatedat bus 29 (Fig. 4). The voltage disturbance at theload bus is shown in Fig. 8a and the response of thetwo load models in Fig. 8b. The response of the restof the system was identical with both load models.

4.4 Local Three Phase Fault

A three phase fault cleared after 120 ms is alsosimulated at bus 21 (Fig. 4). The disturbance at theload bus and the response of the two load models isshown in Fig. 9. Again the response of the rest of thesystem was identical with both load models.

-r1.0DEG5040-30-20-1 0-0

-10

Pu_ _ _

MAGNITUDE-.9

-ANGLE

, ,~~ I

5. DISCUSSION 687

The ability of the aggregate model to adequatelyrepresent the five machine plant in the three transientsimulations can be seen in Figs. 7 to 9. The inherentlimitation of a reduced order model is suggested inFig. 6 where the oscillation characteristics areslightly different but difficult to observe graphi-cally. If we evaluate the plant state matrix given in(5), we find that the mode corresponding to the 4.2 MWmotors has w = 3.8 hz and i = .75, and for the 1.2 MWmotors, w =n5. 6 hz and = . 20. The response of theplant is strongly influenced by the smaller machineswhose higher frequency, less damped oscillations, siton top of those of the larger machines. The aggregatemodel, due to its lower order, must reflect theproperties of the larger machines which dominate theenergy flows between the plant and the rest of thesystem.

In the developrnent of the aggregate model, theassumption of full load may seem limiting at first butis consistent with normal induction motor usage. Indeveloping the plant model, it is assumed that largesynchronous machines will be radially connected to the

DE4030201 0I

I I I

.2 .4 .6 SEC(a) LOAD B3US VOLTAGE RESPONSE

-1 0

I -.0-.8PU

MAGNITUDE

).6ANGLE

.2 .4 .6 SECI

T(a)LOAD BUS VOLTAGE RESPONSE

_ 815 -

-10

5 14

Fig. 9: Local Fault Simulation Responses

I

7

G

--T

688supp] y and can be modelled in detail separately. Themethod could be extended to incorporate small synch-ronous machines if necessary with the aggregate modelunchanged in form as noted by Sabir and Lee [7]. Theinertia of both motor and load is the one parameterlikely to be unavailable. For large units, thisinformation may be obtainable from the manufacturers,whereas for small or older units, it may be necessaryto estimate from similar known units or knowledge ofdimensions, etc. The method can be extended toidentify an aggregate model fromn a transformerswitching test.

6. CONCLUSION

This paper has presented a technique for repre-senting an industrial plant, consisting of inductionmotors and static loads interconnected by an arbitrarynetwork, by a single equivalent induction motor and asingle static load. The effectiveness of the equiva-lencing has been demonstrated by means of transientstability simulations.

7. ACKNOWLEDGEMENT

This work was performed as part of an M.Eng.program at McMaster University. The financial supportof the Natural Sciences and Engineering ResearchCouncil of Canada is gratefully acknowledged.

8. REFERENCES

[1] IEEE Working Group Report, "System Load Dynamics -Simulation Effects and Determination of LoadConstants". IEEE Trans. PAS-92 March/April 1973,pp. 600-609.

[2] C. Concordia and S. Ihara, "Load Representation inPower System Stability Studies", IEEE Trans.PAS-101, April 1982, pp. 969-977.

[3] G. Shackshaft, O.C. Symons, J.G. Hadwick,"General-Purpose Model Power-System Loads", IEEProc., Vol. 124, August, 1977, pp. 715-723.

[4] F. Iliceto, A. Capasso, "Dynamic Equivalents ofAsynchronous Motor Loads in System StabilityStudies", IEEE Trans. PAS-93, Sept./Oct. 1974, pp.1650-1659.

[5] M.M. Abdel Hakim, G.J. Berg, "Dynamic Single UnitRepresentation of Induction Motor Groups", IEEETrans. PAS-95, Jan./Feb., 1976, pp. 155-165.

[6] G.G. Richards, O.T. Tan, "Induction Motor LoadAggregation for Transient Stability Studies byConstrained Parameter Estimation", IEEE PES SummerMeeting, Paper No. A. 79-482-1, Vancouver, B.C.,July 1979.

[7] S.A.Y. Sabir, D.C. Lee, "Dynamic Load ModelsDerived from Data Acquired During SystemTransients", IEEE Trans. PAS-101, Sept. 1982, pp.3365-3372.

[8] D.S. Brereton, D.G. Lewis, C.C. Young, "Represen-tation of Induction Motor Loads During PowerSystem Stability Studies, AIEE Trans. PAS-76, PartIII, August 1957, pp. 451-461.

[9] K.M. Brown, J.E. Dennis, "Derivative Free Ana-logues of the Levenberg-Marquardt and GaussAlgorithms for Non-linear Least Squares Approxi-mations" , Numerische Mathematic, Vol. 18, 1972,

pp. 289-297.

[10] P.L. Dandeno, R.L. Hauth, R.P. Schulz, "Effects ofSynchronous Machine Modeling in Large Scale SystemStudies", IEEE Trans. PAS-92, Mar./Apr., 1973, pp.574-582.

[11] G.J. Rogers, D.S. Benaragama, "An Induction MotorModel with Deep Bar Effect and Leakage InductionSaturation", Archiv fur Elektrotechnik, Vol. 60,1978, pp. 193-201.

APPENDIX

A. Motor Parameters from Specifications

In this section, we compute estimates of the motorparameters for use in the equations describing theoriginal plant. These are determined from standardspecifications as illustrated in Table 1. Most ofthese are readily available or can be estimated fromsimilar units. The inertia constant must include theload and may have to be determined experimentally. Thebasic equations in per unit for output power, inputpower, and shaft torque are:

(Al)P = n P = T (1 -s)0 in

P = V I co sOin s sT = I' R /s

r r

(A2)

( A3)To compute values for the equivalent circuit para-meters, we regard the friction and windage losses aspart of the load. The rated efficiency is used tocompute input quantities and the complex power base(S ), The electrical losses (represented by R and R )ar4 assumed to be a fixed known proportion of Ehe totgllosses (we have assumed .75) and the effective effi-ciency (tn ) is increased accordingly. We also make theapprox imaEion that:

I = I cosG (A4)r s

To evaluate R and X at rated conditions, where V =1.0 pu and I r= 1.0 AI, we equate P. in (Al) and (12)and then subsstitute for T using (A+)n and for I using(AA4) to obtain: r

I2 cos O(R /s)(1-s)/ne = VsI0s0which upon simplification, yields an expression for Rras:.

R = zl s/[(1-s)coso] pu ( A5)Consistent with (A4), tan 0 R /(sX )

r m

which yields upon substitution into (A5), an expressionfor X as:

m

Xm ne/[(1-s) sin0] pu (A6)To evaluate R at rated conditions we write an expres-sion for the selectrical losses, and simplify by sub-stituting using (A1-A3). Thus:

I2 R + I2R = P (1 -n )s s r r in

R = cose[l - n /(l-s)] pu5 e (A7)The combined leakage reactance at starting (X s) iscomputed (assuming rated stator voltage and neglectingX ) as:m

XQ = [I-2 (R + Rrs) 2 .5 (A8)

where the value of the rotor resistance at starting(R ) may be computed from the ratio of starting toraEed torque (F ) and the starting current (I ) notingT ssagain that the rated current is 1.0 pu:

2R = F Rr/(s I2 )rs T r S5 (A9)The effect of leakage reactance saturation duringstarting is represented by the following describingfunction [11]:

D(F ) = [sin (F ) + F(Cl - F2) ]2/7,I I I I

where the current functional (F ) is defined orassumption that saturation staris at 3 timescurrent:

F =3I /I =3/II s Ss ss

Assuming that 50% of the leakage is assumed satuand also -to be equally divided between statorrotor:

x X /[1 + D(F )] puisIThe inertia constant H is calculated from themoment of inertia J and the synchronous speed to a

H = J 2 /2 S secs b

In the actual system, since the motor voltagesunlikely to be 1 pu, we must calculate the operslip (s ), consistent with the operating point dmined Prom the load flow computation. Thedelivered to the motor (P ) and its derivative wrt(P') can be written as:

s

P = V Y (A14) and P = V2Yts s r s s r

where Y is the real part of the input admittBoth P rand P' are thus evaluated as explicit funcof slip. F1llowing the load flow, both the svoltage (V ) and real power delivered to the busare known, therefore, the operating value of slibe determined using Newton's iterative method:

k+1 ks S - (P - P )/P' puo 0 s b s

(A1O)

689To develop the third order model for an inductionmotor, we assume that:

(a) Vds = and 'qVs = 0(b) vdr = 0 and v .= 0dr ~~qr

( B9)(BlO)

(c) v' = (-X /L )T and v? = (-X /L )T (Bl1)d m r qr q m r dr(d) X' = X - X2/X (B12)

s s m r

where (a) implies that stator -transients are negli-gible, (b) that the rotor is shorted, (c) defines a newconvenient variable set, and (d) is the normal"transient" definition.

The transient impedance form is developed by. applying(B9) to (Bl) and (B2) to eliminate ' and T s sub-stituting (B5) and (B6) to eliminate Ps and Yq -sub-stituting for i and i using a reaPranged ?7) and(B8), eliminatidnrg 'V %nd T with the change ofvariable in (Bl1), agg finally simplifying using (B12)and dropping the s subscript since all quantities referto the stator. The resulting equation is:

are or in phasorform: V = R +jX') I +Vs s s s m

(B13)

(BB14)Thus we have separated the impedance drop which is analgebraic equation (since stator transients areneglected) from the remaining equations which involvetime derivatives. The companion state equations aredeveloped by applying (B10) to (B3) and (B4), substitu-ting for i and i using the rearranged version of(B7) and (C9), repfWcing 'V and ' by v' and v' byusing (Bll) and its derivalrive, anAr finally dropAingthe s subscript and rearranging. The resultingequations are:

(Al 6)

The reactive power drawn by the motor equivalentcircuit at slip s is computed using the imaginarycounterpart of (Al4). Since this in general may bedifferent from the actual reactive consumptionspecified in the loadflow, the static load is adjustedaccordingly.

B. Machine Equations

The (pu) voltage equations for a single rotorwinding induction motor in d,q coordinates are [6,8 ]:

v =R i -T + (B)ds s ds qs dsv = R i +w'V +' CB2)qs s qs ds qs

vd= Rr idr - wsqr + dr (B3)v = R i + Wsd + rqr r qr dr qr (B4)

The corresponding flux linkage equations are:

'V = L i + L i (B5)ds s ds m dr

' = L i + L i (B6)qs s qs m qr

V = L i + L i (B7)dr m ds r dr' = L i + L i (B8)qr m qs r qr

2v = (-R /L )v' + swv' - (L X R /L )i .d r r d q m m r r qv' = -scv' - (R /L )v' + (L X R /L2) iq d r r q m m r r d

(B15)(Bl6)

Expressing (B15) and (B16) in phasor form, we obtain:Ct= (-R /X - js)wV' + j(R /X )(X -X')wIm r r m r r s s s

and the linearized form of (15) and (16) is:Avl = (-wR /X )Av' + s wAv' + wv' Asd r r d o q qo

- (wR /X )(X -X') Airr s s q

Av' = - s wAv' - (wR /X )Av' - wv' Asq o d r r q do

(Bi 7)

(B18)

+ (R /X )CX -X') -i (B19)r r s s d

Equation (B17) is used to develop equations (26) and(27) in the main text for the evaluation of R and XUnder steady state conditions, V' = 0 and we assumethat X = Xs and 'I = 1 which ifiMplies that:r a s

I = cosO- j sinO5 and

m j R (X - XI) (cosO - jsinO)/(Rr + j s X )m r s s r 0(B20)( B21 )

Rationalizing and taking the ratio of imaginary to realparts yields:

V ' R cos0- s X sin9q r o s

VI R sin6 - s X cosOd r o s(B22)

690Similarly, the substitution of (B20) into (B14) with V= 1 yields: s

V R sine - X' cosOq s s

- 1 - R cosO - X' sinOd s s( B23)

Equating (B22) and (B23) yields equation (26). Re-grrangement of (B14) and substitution into (B19) withV' = 0 and V = 1 yields:m s

R RV' = j r(X - X')/((( r Rm s s s s s0 0

R- X XI' ) + jIX ( r + R ))

0 (B24)which implies that

V' R R /s -X X'q s r o s sV I X7(R + s/) (825 )d s s r o

Rearrangement of (B25) yields equation (27).

The equation of motion in pu form is:

2Hw(A) r =p - P (B26)Sbi n s(eL (826)

Substituting w wt (l-s) and P= (v'i + v'ii) gives:r s ~~e d d q q

2Hs = P - (v'i + v'i )L d d q q

The linearized form is:

2HAs = - (i Av' + i Av' + v' Ai + v' Ai )do d qo q do d qo q

(B27)

(B28)

The three linearized equations, using (B17, B18, B28)can be written as:

FvI -wR? /X S w WV' rv'1tvd =rr 0 qo d

A v'=I-s w -wR /X -wv' AV'1q 0 r r do qAs -i /2H -i q2H 0 Asdo qo _

0-(wR /X )(X -XI' ) i

r r s s d+ (wR /X )(X -X') 0 Ai

r r s s_qj

-vI /2H -v' /21[ do qo

( B29)

or, in partitioned form, we represent one motor as:

LV'JFE E 1AV'J F [A1FIlIm I 1 21 m li 1 A (B30)[AS LE3 0 As] F21

where, for example [AV'] = [Av', Av? Tm d q

and [E ] =- wR /X s X

r r o

- s X - wR /X_-

o r r

Similarly for n motors, (B24) is written as:

Lij nln[30 A [n2] S(B31 )

where, for example

[AV'] = [Avy', v', Avd' AV' ..Av'Iand -m d q' d2' q2'* qn

[E ]l = diagEE illE12, E1 ... El ], etc.

T. J. Hammons (Glasgow University, Glasgow, Scotland, U.K.): TheAuthors are to be complimented on presenting techniques for represen-ting industrial plant comprised of extensive induction motor and staticimpedance loads interconnected in an arbitrary network by a singleequivalent induction motor and an equivalent static impedance load.They have demonstrated the effectiveness of their algorithms by perfor-ming detailed transient simulations on an industrial subsystem situatedin a typical power system where good dynamic and steady-state cor-respondence of real system and reduced equivalent system response hasbeen achieved.

However, examination of the subsystem depicted in the paper showsthat both the reactance and resistance of the subsystem interconnec-tions (lines Ll-L5) are relatively small in comparison with the combinedreactance and resistance of each motor and its associated transformerrespectively. Reproducing results of a comparative study in which theimpedances of lines Ll-L5 (but not line admittances) are neglected, par-ticularly in respect of responses depicted in Fig. 6 and 7 would enhancethe presented results. It will be noted that with the above approxima-tion, the subsystem of Fig. 5 reduces effectively to just two motors atbusbar 40 if each identical induction motor in the subsystem is assumedto carry an identical load.

Could the Authors also state from the numerous computations theyhave made whether good correspondence between real system andreduced equivalent system responses would be expected if subsystem in-duction motor loads etc. of widely differing characteristics met with inpractice were located at adjacent main network load busbars (busbars21, 24, etc.)? Would the correlation be as good if the loads on inductionmotors Ml-M5 should differ significantly? In practice, some inductionmotors may run light while others will operate at high load. The paperwould be enhanced if more comparisons from the numerous studieswhich the Authors have performed could be made.

The Authors are to be congratulated on the excellent clarity ofpresentation of their work.Manuscript received August 9, 1983.

G. J. Rogers, J. Dimanno, and R. T. H. Alden: The line impedancesare indeed small in our example and may be even smaller in a trulyrepresentative industrial plant. The method is applicable, as we havedemonstrated, to an arbitrarily interconnected system which can in-clude: loads which are not separated by impedances, loads at differentvoltage levels, and radially connected loads. We have used computerprograms based on this method to determine single machine equivalentsfor station service loads at Ontario Hydro generating plants and alsofor industrial plants using information available on single linediagrams. In these cases, the derived models have provided agreementwith test data.With regard to widely differing load levels and characteristics of ad-jacent induction motors, we note that large drive units tend to be

operated close to full load in consideration of capital costs andoperating penalties on powerfactor and efficiency. In our experience,many plants have similar responses to system transients and we have nothad occasion to represent adjacent, very different units. While thealgorithm accommodates partial loading and variation in characteristicsby suitable selection of parameters, it is readily apparent that anymethod used to represent widely differing dynamic characteristics (highorder) using a single dynamic element (low order) cannot be expected toyield good correspondence.

In closing, we thank Dr. Hammons for his interest and commentsand look forward to continuing dialogue and progress in this area ofpower system load representation.Manuscript received September 19, 1983.