IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101, No. 9 September 1982 3303

IMPROVED DYNAMIC LOAD MODEL FORPOWER SYSTEM STABILITY STUDIES

F. John Meyer, Member IEEE Kwang Y. Lee, Member IEEEHiouston Lighting & Power Co. University of HoustonHIouston, Texas Houston, Texas

Abstract - The behavior of the load model in At present the load representation in typicalpower' system stability studies has been shown by stability programs is limited to a static load modelvarious investigators to be a very important factor where the load is represented as constant MVA,in determining the stability of power system constant current, constant impedance, or some com-elements. Load models which are presently available bination of the three. As an alternative, the loadin typical stability programs make it difficult if may also be modeled in detail as an induction ornot impossible to modei the actual dynamics of power synchronous motor. A-lthough the alternative existssystem load. Present load models allow easy static it is not feasible or possible to model the dynamicrepresentations, but require detailed individual behavior of each load.dynamic models such as induction or synchronous motormodels. Thus a simple load model that can represent Ihe attention in stability models has moved topower system composite load dynamics will be a defining dynamic load behavior as functions ofvaluable tool in the stability study. Presented voltage and frequency. The reasons for more accurateherein is a second order state space load model which load modeling include (1) a tendency to improvecan represent power system composite load dynamics. quantitative accuracy of the system simulation; (2)In addition, the parameters of this state space load the use of the digital. computer has made it easier tomodel can be identified, from actual system load mea- refine representation of all elements; (3) thesurements by a weighted least squares parameter iden- representation of extensive automatic controls intification process. power systems has made it essential to evaluate

contribution of all elements; (4) the extension ofINTRODUCTION the studies in simulated time such that it is nec-

essary to include the exact load effects. [1].Several authors [1,2,3,4] have recognized the

significance of load behavior to changes in voltage Nunerous stability investigations such as thoseas long as 25 years ago. However, most of the conducted by Detroit Edison [7] and Southernattention in stability studies have been directed California Edison [3] which found stability transfertoward improved generator and control system models. limits on interconnections, have demonstrated theUntil recently little modeling has been considered load model can change the stability limit on the tienecessary for power system loads since the effects of line by more than 50%. In general, the specificload models in stability studies were considered only cases where modeling load dynamnics are necessary aresecondary. (1) stu4y of an industrial plant, (2) study of an

area which has a large concentration of motors, (3)The stability study has becane a very important studies made to explain actual system breakdowns, (4)

analytical tool in planning the power system.- Today study of self excitation and sub-synchronousmore than ever environmental pressures, distant power resonance, and (5) study of applying shunt-reactiveplant siting, and economic constraints have forced power control devices as an aid to stability [1].the power system engineer to minimize the number of Ilecito [2] found from actual measurements that fortranmnission lines between the generators and the large voltage drops (25-75%), there is a noticeableload. Thus the trend has become one of increasing transient component of load that is as long or longertransfer impedances and reducing stability margins, than the voltage depression. The IEEE ComnitteeTo insure adequate stability margins exist in the report [7] on load modeling acknowledged that staticsystem, planners must resort to detailed modeling of load representations are only valid if the voltage ispower system behavior. greater than 0.5 per unit. The report concluded that

the only practical means of obtaining realistic loadIn this regard pow7er system engineers have voltage data for stability studies is by system

developed six (6) standard generator models, four (4) measurements.standard excitation system models, and two (2)standard turbine-governor models [5,5]. The PROPOSED DYNAMIC LOAD MODELsimulation process of calculating the statevariables, through the study period has also improved Existing static load models in stabilityvastly over the years. The load representation in programs ca-nnot accurately model the dynamics of moststability studies however has not changed appreciably power system load. These static models are usuallysince the first digital stability study. expressed by the equations

P(v,f) = Po (av2 + bv + c) (1 + g Af)

Q(v,f) = Qo (dv2 + ev + q) (1 + Af)

where constants "a" and "d" specify the per unit ofreal and reactive load that behaves as constantimpedance, constants "b"' and "e" specify the per unitof real and reactive load that behaves a.s constant

-- ~~~~current, and constants "c" and "q" specify the per82 WM 126-1 A paper recomended and approved by the unit of real and reactive load that behaves asIEEE Power System Engineering Committee of the IEEEPower Engineering Society for presentation at the IEEEPES 1982 Winter Meeting, New York, New Yorkc, January 31-February 5, 1982. Manuscript submitted August 31, 1981;made available for printing November 30, 1981.

0018-9510/82/0900-3303$00.75 (C 1982 IEEE

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constant MVA. The constants "g" and "h" indicate the of current, the proposed load model can be expressedper mit change in real and reactive load for corres- asponding changes in frequency, Af, from nominal FFFfrequency f. The static load model has the x O 1 1 0 R1shortcaning of not accurately representing the power 1 = x + Rsystem load response when temtinal voltages are fromn x -a -a x O 1 V0.0 to 0.5 per unit. 2 1 2_--. 2_-- __-.-_ I__

m le detailed model of induction motors in L _ [ L_L._stability studies consists of three state equations I R c] c x d d

V

where both mechanical and electrical torque are + 1 i Lfunctions of the motor slip. Although the detailecl c c x d d Vmnodels are available for representing motor load __ 3 4 ._j2 ._Idynamics, the practicality of accurate representationis not possible because (1) data is not always (3)available, (2) very large number of motors are con- A further reduction of the [D] matrix can benected to the power system, (3) large differences accomplished by applying Ohms law for constantexist in the electrical and mechanical paraneters, impedance [ll The state space load model can beand (4) the induction motors are scattered diffusely linearized about the operating point Vo, I T.Thisin the distribution network [2]I final simplification allows the state vector x to be

initialized at zero and the weighted least squaresA new power system load model is proposed which identification process to be applied as the parameter

will be an improvement in the accuracy of the static estimator.and detailed dynamic load models. Thi s new loadmodel should meet the following re,quirements: The state space load model is expressed in final

form as(1) Model must be as simple as possible.(2) Model must accurately represent the dynam- - _ r F

ics of canposite power system load. 1 1I x1 | 1 0 AVR(3) Model must adapt easily to existing = ]

(4) Model must have paraneters which are 2 1 0 X AVidentified from actual power system loadmeasurements. LA ra3 a Fal A]rvR 3 4 li 7 Ct8 VRI

A load model is proposed in state space form to l 1 = + | Ilprovide parameters which can easily be determined by I I a a x lAVIan identi fication process and to facilitate L X2 L LX 2 8 7 L Uincorporation of the equations into the stability (4)progran solution. The proposed equations in statespace form are

PAREAMER IDENTIFICATIONx = Ax + Bu (1)

The eight parameters of the proposed load modely = Cx + Du, (2) can be identified by numerous identification pro-

cesses using actual load measurement responses towhere x is the state vector, u the input vector, and input disturbances. For simplicity a weighted leasty the output vector. squares parameter estimator was chosen to determine

the alphas ( al, .a8). Figure 1 illustrates a typicalThe state vector x is order two because this is identification process.

the minimum order to account for the dynamicsproduced by the induction motor swing equation.

The inputs to the proposed load model can be chosenas IVLI, 6L, and fL or load voltage magnitude, loadphase angle, and load frequency respectively. IAlternatively the rectangular components of voltageVR and VI may be chosen instead of voltage magnitude u(t) Real System + Yr(t)and phase angle. The state vector x has no phiysical Srneaning. The output vector y may be chosen as thereal and reactive power PL and QLr or as the real andreactive component of load current IR and IIrespectively. Sampling

Since the last two inputs 6L and fL arerelated, it is felt only one is needed to model theload dynamics. Alternately VR and VT, since they are YrWequivalent to IVL | and dL should be sufficient input Syte Moe Paaee Paaeeietmtquantities. A further reduction of the load model SsmMdl y aret Prebtiatcan be achieved by expressing it in a cannonical form I s pir_such as that expressed by Goodwin and Payne [10].*s koiChoosing the output vector as rectangular components

FIGURE 1 SYSTEM IDENTIFICATION USING PARAMETER ESTIMATION

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In the weighted least squares estimation (WISE),a penalty function on the output error squared ischosen as the loss function. The loss function is Turbine-Governorexpressed as Model

1T T= TLYr(t) - y(t)] W EYr(t) - y(t)] dt

where yr (t) is the output of the real systan to aknown input of the model to the same input function Pof time. The output vector y is expanded about oninitial operating point by using the Taylor seriesexpansion. Neglecting higher order terms and Generatordifferentiating with respect to a yields the equation E 6g' Model and Load

1- r~~~~~~~~~~~~~~~~~~~~~~~~~

yields the equation ModelStatic Load

State Space MIc Modelsddt SaX] = Load Model+(a

D _O o

as(X)I U p

L_ act ct0 (6) Fit R IOrITRCINOF STATE SPACE LOAD MODEL

Substitutin g into the loss fnction and optimizingsuch that this loss function is minimized yields analgebraic set of equations

-| i-l IC

d ax z- 2

~~L~"] ~~ L 1] ~C0 IDETUIEWD THE PARAMETE1 OF THE

1 act a aat

UN T IRACTI SPACE LOAD MODELtin itL l ss It is well koawn that the load dynamics of the

L~~ ~ ~ ~ ~ ~ c . (7 ) power system are primarily influenced by the numbrof induction motors connected in the network. There-

where i-l,. ....N are the discrete sampling points and W fore, it is technically acceptable to use anis a weighting matrix. Finally an iterative solution induction motor model's response to changes incan be formulated using the equation terminal voltage as the input data for the

s o sidentification process. The parameters of the stateck =ck-l +G Ank, space load model identified from this type of data

(8 ) are typical of that used to represent all powersystem load dynamics. Step changes in voltage were

where K is the iteration count and G is a gain investigated since they more closely resemble a faultmatrix. Equations (1), (2), (5), (6), (7), and (8) condition on the power system.are used to formulate the WISE identification pro-cess.* A computer program was written to incorporate

the weighted least squares estimator algorithmIM~PLEM4ENTATIONI OF THE LOAD t)DEL developed previously. This prograM identified the

state space load model parameters franthe inductionOne reason for choosing the state space load motor response. The required data input for the pro-

form and the change in current as output was to gram includes the number of samples, convergencefacilitate implementation into the typical transient tolerance, the input voltages in rectangular form atstability progrNa. Figure 2 illustrates in block each sampling instant, the output current indiawrm fomn the implementation of the state space rectangular form at each samploing instant, sapledload model into the stability program. Te time interval, aed gain matrix. Optionally theinteraction of the state space model with the network weighting matrix and initial estimates of a may beperfornance equations and the static load models is input if desired. Table I summarizes the optimumclearly illustrated. A change in the voltage at the alrea parameters identified fro the induction' motorload bus is input into the state space load model . model'ss response to step changes in terminalThe output of the moelaisn he change in the static voltages.constant current cc by the appropriate change AIc.

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TABLE I current responses of both the state space linear loadTABLE OF OPTIMUM ALPHAS IDENTIFIED model and the induction motor model t4 a +.2/50 step

TIME STEP. .002th lo UC Of j. pchange in voltage.

AS \E --B/- 0 AV.8-.L-00 AV -.2L AV .2/5"XSTEP-82 ISTE S 80 * STEP 85 f STE£S a85

1-394.68 -314.127 -513.977 -233.911.6

2 -35.54 -31.701 -31.068 -22.91

3-40.65 -72.17 -136.143 -138.75IA

4 -3.B0 5.899 4.089 -12.40

5109.94 106.586 90.285 96.32

6 4.813 4.548 -0.348 5.58 1.2\ 2

7 0.507 0.528 0.584 0.811

a4.931 4.960 4.990 5.015 1.0

.8~~~~~NIt should be noted that the identification of -

the model parameters is quite sensitive to the gain ,parameters, initial estimates of alphas, and con- kiitorstraint of a1 and a2. Reasonably good estimates P.

of P1 and a2 are required to assure convergence. U.

The method used to identify the alpha parameters when l .o .09 .09 .o .10 .12 .14 .16 .1;no knowledge of initial estimates are available wasto use the default estimates of alpha (ai=l) and set FIGURE 3 MEALCURE EPMT.2L9OTEc MTAGEthe g ii of the gain matrix G equal to .01 forseveral iterations. This technique producedreasonable starting.values of a1 and a2 such thatthe gain can be increased to about .25. Finally,when the, maximum change in alpha is less than 2.5 itis time to set the gain equal to one. The gainparameter adjusted in this manner achieved .2convergence of the WISE identification routine eachtime.

0

As stated previously, it is also important to .1cornstrain a and (2 to prevent divergence of the _ _ .process. Divergence of the process is particularly h3 ,2 -I .2susceptable in the first few iterations if 0C and _-

a2 drift far away fran their optimum values. The P Do ,o /sensitivity toal andct2 is explained by the fact that U. -/these paraneters influence the eigenvalues of the X /_ motorload model which, therefore, characterize the load -.7 /dynamics. In order for the proposed load model to /be dynamically stable, a1 and a2 must be less than ,/ PIGURI 4zero. An additional constraint might be applied if - / REACTIVECURRENT RESPOMEoscillatory behavior is desired such as that X2/ ZVOLTAGECHAEexhibitied by the induction motor's response to step-,1 /changes in voltages. That constraint is to havecamplex roots of the characteristic equation in theleft half of the complex plane. This constraint is,therefore a:- 4 c,.

Application of the constraints in the WISEidentification program results in a much betterchance for convergence to optimum alphas.Experiepce demonstrated that the constraintsol < 0and a2 < 0 were sufficient for most paraeter The comparison in these figures as well as allidentifications studied. The additional constraint other cases shows that the state space linear model

ai < -4aC is a special condition not applicable in a is in good agreement with the induction motor model.general sense. The results of these comparisons clearly illustrate

the "WLSE" identification routine used converges toOMPARISON OF STATE SPACE A MIODEL optimun alpha parameters and the state space linear

RESPONSE TO INDCTIMN lOR iEPCSE load model expressed in a cannonical form canadequately represent the induction motor dynamics.

Figures 3 and 4 illustrate the real and reactive The typical power system problem investigatedwith a transient stability stidy usually involves athree phase fault applied to a netwrk bus at timezero, and clearing of the fault occurring from 4 to12 cycles. The load model in such studies isimportant as recognized in earlier sections.

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A voltage pulse described by 1.0 per mit APPLICATION OF THE STATE SPACE LINEARvoltage prior to time equal zero, .5 per unit voltage MDEL TO STABILITY STUDIESfor .06 seconds, and 1.04 per unit voltage thereafterwas applied to the induction motor model. Again, aninput-output file of voltage and current was used as The purpose of this paper is to develop ainput to the WESE program and the optimun alphas of dynamic load model which could be utilized in powerthe state space load model were identified. The system stability studies to more adequately reflectoptimun alphas identified after 32 iterations are as the dynamic characteristics of power system loadsfollows: than the conventional static load models.

= -15.361 = 89.588 Since actual load response data was not avail-able, data was generated to estimate the alpha

a2 = -19.713 a6 = 94.731 parameters. The data was generated using a transientstability program with induction motor and static

C3 = -38.688 cr7 = 0.457 models representing a known load composition. Thisexample will demonstrate that the proposed load model

o4 = 60.538 o8 = 4.943 can replace all conventional load models for powersystem studies.

Figures 5 and 6 illustrate the real and reactivecurrent response of both the induction motor and the A simple four bus, five transnission line, twostate space linear load model to a pulse change in generator test system was used to demonstrate theload terminal voltage. It is obvious fran both application of the state space load model infigures that the optimum alphas identified provide a transient stability studies. This test system isresponse for the proposed state space model that is illustrated in Figure 7.almost identical to the induction motor response.

Gi 140 MW 2 66 MW

BUS 1 BUS4

2.0-

1.3

23 160

[ P154. MW+2z -100%

PlQR IEALREEE2ND9T~SarEw LTM pZM5, P I.50% a z I00

+mM8.5 MVARPER UNIT -100 MVA

FROM BUS TO BUS R X CMVAl 2 ~~~~~.05.50 .201 3 ~~~~~.08.80 .30

S ~ ~ ~ ~ ~~~~~~~~23.03 .30 .12

1.6

.02 .04 .08, .12 14 12 4 .06 .5 21.8 3 4 .05 .50 .20

t. *6 \ ~~~~~~~ ~ ~~~~FIGURE7 TWOMACHINE,ONEu. *> MOTOR EXAMPLE COMPOSITE STATE SPACE LOAD MODEL

. . . . . n. . . . . . . * . * .3 .0 *.80_.3

O .X2 .04 2 .06.10 .12 .14 .16 .1612

-12 Bus 2 was chosen to demonstrate the ability of, ~~~~~~~thestate space model to replace ccinposite bus loads.-1.5/ The load model at bus 2 consisted of 45.8 + jl8.S in-

-12 duction motor load, 27E 1 + j21 .5 constant impedance-21 load, and 27.1 + jO constant current load. To effect

a significant change in voltage at bus 2, a fault wsplaced in the netwoerk. The current response at bus 2

FIOQU*R 6 REACTIVECURRENT RESOS TO.5PER U#ITPIJLE CHAO1N VOLTAG

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was observed and the resultant input-output charac- To demonstrate the usefulness of the proposedteristic was used to determine the alpha parameters state space load model, several stability studiesof the linear state space load model. were run using the alternate alpha values which were

identified frcn the sensitivity studies. The averageIt is intuitively obvious that the location and values of the alpha parameters fron all the

duration of the fault on the network have consider- sensitivity studies were used to represent the stateable effect on the input-output characteristics of space load model in one stability study. In anotherthe load response, and therefore, the alphas stability study, the alpha parameters wkich differedidentified. For this reason a sensitivity analysis the most fran the average values ware used toof fault location and duration was performed on the represent the state space load model. Figure 9four bus example system. The sensitivity analysis illustrates the angle 61 2 responses for the stateincluded placing three phase fault on each bus in the space load model using tie various alpha paranetersnetwork for a duration of .2 seconds (12 cycles) to fran the sensitivity studies. Note the stabilityshow the sensitivity on fault location, and also studies of varying alpha parameters demonstrate theplacing a .1 second fault (6 cycles) on bus l to proposed state space load model can adequately modelillustrate sensitivity on fault duration. Figure 7 the dynamics of the composite bus load. Thedemonstrates the voltage magnitude response at bus 2 excellent response of the state space load model indue to: faults at varying locations and duration. the sensitivity studies is due to the stabilizingoptimum alpha parameters were identified for each effect of the constant admittance terms 07 and a8.study. The alphas detemined are listed in Table II. The additional parameters cl , ... oas add theThe average value of optimum alphas was coinputed from dynamics to the constant admittance terms. Thisthe four studies. combination in the state space load model provides

good results duplicating the response of the actualThe sensitivity studies of varying the fault lo- load.

cation and the fault duration illustrated that thealphas identified were sensitive to both. Figure 7and Table II demonstrate that the alphas determinedare most sensitive to the shape of the voltage soA- \odmagnitude response as expected. Faults at bus 3 and g -C phbus 4 as well as the .1 second duration fault on bus X.KXXfAn.Aphu1 have similar voltage response as indicated in

0 -

h WostAphFigure 8; Table II also indicates the alphaparameters determined for these three cases are more 12

similar in value

E

50

' X~~~~~~~~~~~~~~~~~~~~~or

1.0 E

EII

30

0 .2w-fm a b

A 1,~~~~~~~~~~~~~~~~~~~~1

.2.

.2Se Ful . Schut 2 1echut .1 Se AFal

TIME-SECONJDSFIGURE 8 VOLTAGEMAGNITUDERESPONSEATSIJS2 FOR VARIOUSNETWORKFAUJLTS a f.8 10 .

3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~. 1.0 -0.01008 002 02

TIME-SECONDSPIGURE 9 SENSITIVITY STUDOYANGLE RESPONSE (' l2

FOR EXAMPLE 1 AILPHAS OF TABLIE 6. 2.2 1

TABLE I I

OPTIRJR ALFHAS FOR EXAMPLEFAULTS OF EXAMPLE 1 COCI=LSIOIW

.2 Sec Fault .2 Sec Fault .2 Sec Fault .1Se

FaultAlpha/Case on Bus 1 on Bus 3 on Bus 4 an Bus 1 Avg

Due to the lack of sufficient loaod models this1 -68.37 -70.37 -57.27 -61.45 -43 paper addressed developing a model which could

2 -3.78-0.231.29 1.91 ~~adequately reflect the dynamics of power system loads2 -3.78 -0.23 -1.29 -1.91 and be identified fran actual system response data.3 1.02 -0.05 -0.068 -0.02 0.224 0.12 -0.002 0.025 -.9 0.013 A linear st;;fat spMeacIeloa moevil was4 proposed5 -1.82 -0.12 -0.12 -0.03 -0.52 which was secondl order cannonical form requiring

6 -0.33 -0.006 -.038 -0.13 -0.11 eight unknown parameters. A weighted least squares7 0.64 0.79 0.78 0.72 0.72 identification program was written to identify the8 0.32 0.23 0.32 0.3 0.31 unknrown parameters of the proposed state space modlel

fran known measurements of load current response to

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changes in terminal wltage. The induction motor 8) D. S. Brereton, D. G. Lewis, and C. C. Young,model was used to develop data from which the state "Representation of Induction - Motor Loadsspace load model parameters could be identified. The During Power-System Stability Studies."parameters identified produce state space load model AIEE Transaction, Volume 76, August, 1957,response almost identical to the induction motor re- PP 451-460.sponse for the same changes in voltage.

9) M. Katkin and G. J. Berg, "Dynamic Single-UnitThe state space load model was proven to ade- Representation of Induction Motor Groups.

quately model camposite bus load and a system dynamic "IEEE Transactions on Power Apparatus andequivalent [11], and to be conpatible with existing Systems, Volume PAS-95 No. 1, January/transient stability programs. These applications de- February, 1976, PP. 155-164.monstrate excellent potential for using this model infuture power system stability studies where data can 10) R. Payne and G. Goodwin, Dynamic System Identi-be measured with respect to the load or external cation. Academic Press, New York, 1977.system response due to large changes in voltage inthe system under study. 11) F. J. Meyer, "Dynamic State Space Power System

Load Model ," University of Houston, M. S.BIBLIOGRAPHY Thesis, 1980.

12) P. Eykhoff, Systen Identification, Paramter and1) C. Concordia, "Representation of Loads", IEEE State Estimation. John Wiley & Sons,

Power Engineering Society 1975 Winter London, England, 1975.Meeting Paper 75 CH0970-4-PWR, "Symposiumon Adequacy and Philosophy of Modeling: 13) C. C. Lee and 0. T. Tan, "A TWeighted LeastDynanic System Performance," PP. 41-45. Squares Parameter Estimator For Synchroous

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December,1972, PP. 2470 2479.

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4) W. Mauricio and A. Semlyen", Effect of Load Char-acteristic on the Dynamic Stability ofPower Systems, "IEEE Transactions on Powerbar~atus and Systems, Volume PAS-91, No.6,Noverber/December, 1972, PP. 2295-2304.

5) IEEE Committee Report, "Computer 'Representationof Excitation System." IEEE Transaction onPower AWaratus and Systems, Vol. PAS-87,No. 6, June, 1968, PP. 1460-1464.

6) IEEE Camnittee Report, "Dynamic ?I^odels for Steamand Hydro Turbines in Power System Studies,Power System Studies, "Paper T73-089-0,IEEE PES Meeting, New York, January, 1973.

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