Dynamic load model synthesisK.P. Wong, M.Sc, Ph.D., C.Eng., M.I.E.E., M.I.E.(Aust-), Prof. W. DerekHumpage, B.Sc, Ph.D., C.Eng., F.I.E.E., F.I.E.(Aust-), T.T. Nguyen, B.E.,

Ph.D., and K.K.K. Ho, B.E.

Indexing terms: Power transmission and distribution, Load and voltage regulation

Abstract: The paper develops methods by which groups of individual loads supplied from power networks maybe represented equivalently in composite form in evaluations of dynamic modes of network operation and thelimiting conditions of stable system operation related to them. Central to the methods developed is a formal,parameter-identification procedure by which values are found for the parameters which define an initially pos-tulated equivalent-load representation. All of the steps of analysis and modelling underlying this procedure aredeveloped in detail in the paper, leading to a systematic sequence of dynamic load model synthesis. In vali-dating the methods developed, they are applied in the paper to earlier system tests in which a total of 23auxiliary motors were isolated from the turbogenerator unit with which they are normally associated andsupplied through a transmission circuit from a neighbouring power station. Controlled short-circuits at one endof the transmission line precipitate dynamic operating conditions in which the principal transients of the aux-iliary drives are recorded. Test conditions were formulated which project the auxiliaries towards the limits oftheir capabilities in recovering to steady operating conditions on fault clearance. While these represent espe-cially demanding conditions for which to synthesise equivalent load models, it is shown in the paper that themethods it develops lead to a single composite model for all 23 auxiliary motors, and which, when included indynamic simulation of the system tests, closely reproduces the measured transients.

!* =

ln. =

List of principal symbols

Induction motor modelvds> vqs d- and

2 System tests and measurementsComprehensive data sets and site-test recordings are avail-able from earlier work [8] for the test system of Fig. 1, inwhich auxiliary motors were isolated from the turbo-generator unit with which they are normally associated,and supplied from a neighbouring power station. Of the

faultbusbar

-33kV

TCBI -66kV

66 kV

Fig. 1 Test system configurationO induction motorD static load

total of 23 motors included in the tests, 17 are supplied at3.3 kV and six at 415 V. In the test configuration of Fig. 1,transformers T2 and T3 in back-to-back connection assistin achieving a high-impedance infeed to the motor loadcomplex, together with the generator transformer 7\, the66 kV transmission line and transformer T4. Faults of con-trolled duration are applied through circuit breaker CBl,at the 66 kV busbar of the generating station providing thesupply to the motors under test.

Among the 3.3 kV motors, there are four primary-airfans, and, as these are identical machines, they can be rep-resented at the outset by a single equivalent motor. Incomposite load representation, this is an obvious first step,and one of which advantage is normally taken as a matterof course. Similarly, there are two cooling-water pumps,two induced-draft fans and two forced-draft fans. In eachcase, the motors are identical, and each pair of machinescan be represented by an equivalent single one. A prelimi-nary grouping of machines along these lines leads to atotal of eight subgroups of 3.3 kV motors and to four sub-groups of 415 V motors, as in Fig. 2. Parameter sets forthem are collected together in Table 3 of Appendix 10.1.From this Table, it will be seen that parameter values forthe individual subgroups vary very widely. In recoveringfrom the severe voltage depressions of short-circuit faultoperation, the dynamic characteristics of the separate sub-groups are, therefore, likely to vary correspondinglywidely. Moreover, the test conditions were chosen deliber-ately to provoke substantial motor transients, and toproject motors towards the limits of their capabilities inrecovering to steady operation on fault clearance. Adiverse range of dynamic characteristics within the totalload complex of Fig. 1, together with severe disturbanceconditions for which some of the individual loads are onthe verge of instability, combine to represent especiallydemanding conditions in load-model synthesis. As was thespecific intention in planning the site tests [8], theyprovide a representative, but exacting case, by which thevalidity of model synthesis procedures can closely betested. The principal system variables recorded in the testsare summarised in Table 1.

Table 1: Principal system variables recordedPoint of measurement in Fig. 1 Quantities recordedterminals of generator Gfield circuit of generator G66 kV terminals of transformer 7",

3.3 kV side of transformer 7"4

415 V side of transformer T5induction motors

voltagevoltage, currentvoltage, currentactive- and reactive-powervoltage, currentactive- and reactive-powervoltage, currentcurrent, speed

33kV

faultbusbar ] CBl

TT7

66kV

66kV

-3.3kV

415 V

Fig. 2 Test system with grouping of similar motorsO induction motor or motor groupD static load

3 Basis of model synthesis

In the detailed studies [8] undertaken prior to the site testsbeing carried out, each of the motor subgroups was rep-resented independently. Reference 8 indicates the close cor-relation between the results of the pre-test analyses andthose of direct system measurements. The present workhas drawn on the analysis and simulation facilities of theseearlier studies. The accuracy of representation in dynamicanalysis in the present work is, therefore, that which theearlier paper [8] has sought to confirm. From this basis,the objective of load-model synthesis in its application tothe system of Fig. 1 is that of reducing the number ofseparately represented machines. The requirements of loadgroupings in equivalent representations may differ con-siderably in different applications, but the case consideredhere is that of representing the total load complex in Fig. 1in equivalent form, as in Fig. 3. In the equivalent, there is asingle induction motor representation, together with asingle static load. For this postulated equivalent, it istherefore required in load-model synthesis to find par-ameter values for it, such that the equivalent closely repro-duces the combined response of the individual loads whichit represents. The validity of the equivalent can be assessed

180 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985

very directly by comparing the transients in voltage,current and active and reactive power for it with those ofeither multiple-machine simulation or site-test results.

33kV

66kVCB1

66kV

3.3kV

6Fig. 3 Test system with single equivalent load representationO equivalent single induction motor equivalent static load

In Fig. 4 is shown the site-test recording for the 3.3 kVbusbar voltage subsequent to a 3-phase-to-earth shortcircuit at the fault busbar in Fig. 1. Also shown in Fig. 4 isthe solution for the 3.3 kV busbar voltage from dynamicsimulation when all 12 motor subgroups are representedindependently, as in Fig. 2. In Fig. 5, a comparison is madebetween the measured slip transient for the boiler feedpump and the solution from computer analysis for thisvariable. In this case, the motor is represented directly, asthere is only a single boiler feed pump in the motor groupof Fig. 1.

The starting point for finding the load equivalent of Fig.3 is that of subdividing into equal-duration steps the totaltime period for which equivalent-model parameter identifi-cation is to be carried out.

In each step, the transient voltage is taken from Fig. 4,and used as a known or specified quantity in the equationsof the equivalent model of Fig. 3. From these equations,the current drawn from the supply busbar into the equiva-lent load model is found. This total busbar current, havingits induction motor and static load components, is now

3.0

2.5

a," 2.0?| 1.5| . o:> Q5

0.00.0 0.2 0.4 0.6 0.8 10 12 U 1.6 IB 20

time.s

Fig. 4 Voltage transient at 3.3 kV busbarfrom site-test measurementdynamic representation based on 12-motor subgroup of Fig. 2

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985

identified by ib(n). Here, n identifies the time step at whichthe busbar current is calculated. If im{ri) denotes the corre-sponding busbar current from either measurement in the

8.0

- 6 . 0EQ.T3 4.0

I 2.0.o

0.00.0 0.2 0.4 0.6 08 10 12 U 1.6 1.8 2JO

time.s

Fig. 5 Slip transient of boiler feed pumpfrom site-test measurementsdynamic simulation based on 12-motor subgroup of Fig. 2

system tests, or explicit system simulation, then an error Eis formed from

E= (1)

Simplified motor representations have been used suc-cessfully in certain applications [9], but, where the motorequivalent to be found is to be of d q axis form, the im(n)and ib(ri) are quantities in the d q axes. As dynamic simu-lation reproduces closely the measured responses of thetest system of Fig. 1, im(n) is derived in the present workfrom simulation in which it is available directly in d qcomponent form. Using recorded values of phase currentsrequires transformation to their d q components.

Typically, the period over which parameter identifica-tion is required is in the range 2-5 s, and a typical stepinterval duration is 10 ms. The upper limit denoted by Nin eqn. 1 specifies the number of step intervals of givenduration to span a specified total period for which synthe-sis is to be carried out. From the equations of the pro-posed equivalent model in Fig. 3 and the known busbarvoltage from Fig. 4, the total busbar current ib(n) is calcu-lated for each value of n. As im{n) is specified, the errorfunction E is found as a numerical value at each step inter-val. Of the variables recorded in the tests and summarisedin Table 1, it is those corresponding to the voltage andcurrent at the 3.3 kV supply busbar of Fig. 1, and asformed in dynamic simulation, which are used incomposite-load-model synthesis.

Given values for this error function at each step inter-val, and also its gradient (Appendix 10.6), a standard mini-misation procedure [10] can be used to find values of theparameters of the model in Fig. 3 which minimise the errorfunction. In a single pass of the complete procedure, thevalue of the error function of eqn. 1 is found from its com-ponents for each step interval. From this value, and that ofits gradient, the minimisation procedure finds a set of par-ameter values for the equivalent model of Fig. 3 based onerror-function minimisation. In successive passes, the pro-cedure is then repeated until checks in the minimisationalgorithm confirm that no further improvement in par-ameter identification can be made. The closeness withwhich the equivalent can reproduce the combined responseof the individual loads it represents can then be tested byusing in dynamic simulation the parameter values whichthe minimisation procedure gives. A very direct assessmentof the validity of equivalents can be achieved on this basis.

181

For the system of Fig. 2, detailed dynamic analysis, inwhich each load subgroup is represented separately, repro-duces responses which closely correlate [8] with measuredresponses. On replacing the individual representations withthe single one of Fig. 3, and with parameters for it foundby a minimisation sequence, the closeness with which theresponse of the equivalent reproduces the combinedresponse of the constituent loads can be assessed andquantified directly.

In addition to the error function given in eqn. 1, severalother forms are possible. Experimental work has beencarried out in which each of the main alternatives has beenconsidered and used in parameter identification, includingthat in which an error function is formed from a com-bination of active and reactive power. However, the formin eqn. 1 is that which appears to be the most satisfactoryone for general use.

4 Equivalent model4.1 GeneralFor the model proposed in Fig. 3 as one which is to rep-resent equivalently the multiple-machine load of Fig. 1, itis now required to develop a model formulation fromwhich numerical values may be found for the busbarcurrent ib(n) and used in the error function of eqn. 1. Theequivalent proposed in Fig. 3 has in it a single inductionmotor and a single static load. Equation formulations arenow developed for each of these components.

4.2 Induction motor modelFrom the derivation of Appendix 10.2, an induction-motormodel suitable for the purposes of load-model synthesismay most directly be considered as having three mainparts. First, there are the stator circuits for which the equa-tions are of nondifferential form. Next, there are the rotorcircuits and their equations in vector differential form;and, finally, in the mechanical axes, there is the scalar dif-ferential equation of motion. These separate equationsystems can be summarised as in the following:

4.2.1 Nondifferential stator equations: If vs is the vectorof d- and ^-components of motor terminal voltage, is is thecorresponding vector of stator current components, and \jiris the vector of rotor flux linkages, then

vs = A}\fr + Zis (2)In Appendix 10.2 is given the derivation leading to the Aand Z matrices of eqn. 2 together with their explicit forms.

4.2.2 Motor differential equations: These reduce to thefirst-order form:

in which the Q and F matrices are given in eqns. 36 and 37of Appendix 10.2.

4.2.3 Equation of motion: Denoting the instantaneousslip by s, it is shown in Appendix 10.3 that the equation ofmotion can be expressed in the form:

1 (4)

In eqn. 4, M is the inertia constant, cof is the nominalangular frequency. The first term in the brackets of eqn. 4represents the mechanical torque, where K is a constantand y is the load-torque index. The second term represents

the electromagnetic torque, in which the kf coefficient isdefined in eqn. 41 of Appendix 10.3 and the ^* vector isdefined in eqn. 42.

4.3 Parameters to be foundEqn. 4 indicates three of the parameters of the model ofSection 4.2 for which parameter values are to be found.These are the inertia constant M, the load-torque constantK and the load-torque index y. Contained in the kf coeffi-cient of eqn. 4 is the magnetising inductance Lm and therotor inductance Lrr. The remaining inductance coefficientin Appendix 10.2 is that of the stator inductance Lss.Finally, in the model equations of Appendix 10.2, there arethe stator and rotor resistances Rss and Rrr, respectively.Collecting together in vector JC parameters for whichvalues are to be found, gives

x = M, K, y, Lm, Lrr, Lss, Rss, Rrr

In the minimisation sequence by which values for theunknown parameters are found, vector JC is the vector ofvariables.

4.4 Calculating is(n)Subject to values of the d- and ^-components of statorvoltage in vs being available from measurement or simula-tion, it is required to solve eqns. 2-4 to give the d- and^-components of stator current in the single equivalentinduction-motor representation as one of the componentsof the total busbar current ib{n). The second component isthat for the static load i,, which is considered in Section4.5. In calculating is(n), the differential equations for therotor circuit, together with the differential equation ofmotion, are first combined with a numerical integrationsequence based on the trapezoidal rule. As in Appendix10.4, this gives, for the rotor circuits,

ij,r(n)=J{n-l) + as(n) (5)For the equation of motion, the procedure gives

s(n) = g(n - 1) +2Mco, - s(n)Y -

(6)In eqn. 6, h is the duration of the step interval which nidentifies.

To eqns. 5 and 6 can now be added the nondifferentialequations for the stator. As these are valid at each stepinterval, they can now be expressed in the form

vJLn) = A+JLn) + Zis{n) (7)Beginning with known values for vs(n) for all n, eqns. 5-7solve iteratively without complication. The sequence con-verges rapidly with no more than 4 or 6 iterations usuallybeing required. A Newton-Raphson form of solution hasbeen developed, but, in practice, repeated passes througheqns. 5-7 in the form given appears most directly to fulfilsolution requirements. At convergence at each step inter-val, numerical values are available for is(n).

4.5 Static loadGiven the busbar voltage at the load supply point frommeasurement or simulation, vs(n), the static load currentit{n) is available directly from

il{n)=Ylvs{n) (8)where Yt is the load admittance. The total busbar currentib(n), as used in the error function of eqn. 1, is then avail-

182 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985

able fromit{n) (9)

dent parameters, and, if they are segregated to vector p,then

Also, the static load representation introduces two furtherparameters for which values are to be found. These are theload conductance G and susceptance B. For the equivalentinduction motor and static load together, the vector ofunknown parameters now extends to

x< = M, K, y, Lm, Lrr, Lss, Rss, Rrr, G, B (10)

4.6 Predisturbance conditionsFor the static load component of the proposed model, theequations in predisturbance operating conditions are thesame as those in the period of dynamic operation forwhich model synthesis is required. Corresponding tosteady operating conditions in the induction motor equa-tions, however, p = 0. The one set of parameters mustsatisfy the equations of the single induction-motor modelin eqns. 2-4, and also the particular form which the equa-tions take when, in predisturbance operation, p = 0, andfor which it is convenient to identify variables by subscript0. On this basis, the stator equations in predisturbanceoperation become

v,o = A&ro + Ziao (11)From eqn. 3, the rotor equations when pij/r = 0 take theform

0 = (12)When ps = 0, the equation of motion, that of eqn. 4,becomes

As each of the vector eqns. 11 and 12 expands to twoscalar equations, there is a total of five equations insteady-state operation. These are arranged in a form suit-able for solution in Appendix 10.5.

4.7 Initial-condition values in calculating is(n)Also related to predisturbance operating conditions are thestarting values from which the solution of the differentialequations in forming is(n), as in Section 4.4, can begin.Measured values of stator current are available from whichto form initial-condition values for is(ri) in eqns. 5 and 6 inSection 4.4. The remaining initial-condition valuesrequired are those for the rotor flux linkages in vector ^*and for the rotor slip. It is shown in Appendix 10.5 howthese initial-condition values can be found from a solutionof the predisturbance form of the induction-motor equa-tions.

4.8 Dependen t and independen t parametersSolving the particular form of the induction-motor equa-tions when p = 0, as in Appendix 10.5, allows initial-condition values ipdr0,

during the course of parameter identification of 100%-150% of initially estimated values, have arisen frequentlyin the present work.

6 Validation and applicationIn providing a close measure of validation of the synthesisprocedure developed, parameter values from the identifica-tion sequence of Section 5 have been included in dynamicsimulation of the system of Fig. 3, which includes thesingle equivalent model for the total load complex of Fig.1. Solution values from this simulation are compared inFigs. 6-9 with those where all motor subgroups in Table 3of Appendix 10.1 are represented explicitly. In turn, themultimachine simulation has been extensively validated bydrawing on the earlier site-test results, as in Figs. 4 and 5,and, in much greater detail, in the correlation studies ofReference 8. In Figs. 6-9, solutions based on the equivalentand those from the multimachine simulation agree closely

3.0

2.5

Si 2 02| 1 5

| 10

0.0 02 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 20time.s

Fig. 6 Voltage transient at 3.3 kV busbardynamic simulation based on the 12-motor subgroup of Fig. 2dynamic simulation based on equivalent

00 0.2 0.4 0.6 0.8 10 1.2 1.4 1.6 1.8 2.0time.s

Fig. 7 Current transient at 3.3 kV busbarfrom site-test measurementdynamic simulation based on 12-motor subgroup of Fig. 2

dynamic simulation based on equivalent

>COCO

"6

5.0

4.0

3.0

| | 2 O| o" 1.0oB oo

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 18 2.0time.s

Fig. 8 Active-power transient at 3.3 kV busbardynamic simulation based on 12-motor subgroup of Fig. 2dynamic simulation based on equivalent

for the principal variables of voltage, current, active powerand reactive power at the 3.3 kV busbar of Fig. 1.

a 6.0

^ 5.0

1 40in

J 3.02 2.0

1.0

00- oo~ -1.0uo-2.0

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 20time.s

Fig. 9 Reactive-power transient at 3.3 kV busbardynamic simulation based on 12-motor subgroup of Fig. 2dynamic simulation based on equivalent

14.0

12.0

10.0

8.0

5* 6 0a.

4.0

2.0

000.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

time.s

Fig. 10 Slip transientscooling-water pump

boiler feed pumpequivalent induction motor model

In Fig. 10 is shown the transient slip of the equivalentinduction motor, together with slip transients for two ofthe largest subgroups in the total load complex. In termsof the slip transient, the equivalent confirms that the totalload recovers to steady operation, following fault clearancefor the particular prefault and disturbance conditions towhich Fig. 10 relates.

7 Conclusions

The paper develops systematic methods by which the par-ameters of postulated load models may be found frommeasurements for any aggregate of individual loads sup-plied from a power network. To allow the validity of themethods to be investigated closely, they have been appliedin this paper to the particular load configuration of earliersite tests. The constituent loads forming the total onewhich is to be represented in equivalent form are here welldefined; and it is feasible to represent each directly, and indetail, for the purposes of dynamic analysis of the test con-ditions. Solutions from computer analysis, on this basis,correlate closely with the results from instrumented sitetests in substantiating the analysis method and in confirm-ing the closeness of system dynamic representation onwhich it is based. As in Figs. 6-9, the single equivalentmodel derived, when it is included in dynamic analysis torepresent all induction motors and the static load, as inFig. 3, closely reproduces the site-test conditions and the

184 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985

results of analysis when each individual load is separatelyrepresented. On this basis, a close and very direct vali-dation of the methods developed has been possible in thepresent work.

Although applied here specifically to the test system ofFig. 1, it will be clear that the methods are equally applic-able to any composite load group for which measurementsare available. Initial estimates for motor parameters canusefully be made on the basis of a simplified steady-statemodel. For example, eqn. 62 of Section 10.5 can be used tofind initial values for rotor resistance and initial slip, andeqn. 64 can be used to find the initially estimated value forthe magnetising reactance. Similar relationships are avail-able from which starting values for the remaining par-ameters of the model may be estimated. Measured valuesin phase co-ordinate form are re-expressed in terms oftheir d q components using a discrete-time sequencewhich implements the phase variable to d q componenttransformation. This is an aspect of continuing work in theapplication of formal load-model-synthesis methods to aseries of system tests which include load types which havepronounced fluctuations in their operation.

Constituent loads of particular types may suggest theform of the initially-postulated model for which parametervalues are to be found, but, for any chosen form, par-ameter values may be found for it using the methods whichthis paper seeks to report.

8 Acknowledgments

The authors are grateful to the West Australian RegionalComputing Centre for running their programs during thedevelopment of the model synthesis procedure of thepresent paper. Grateful acknowledgment is made to theState Energy Commission of Western Australia for

support, and, in particular, for the award of a ResearchFellowship to Dr. Nguyen and the award of a Postgradu-ate Research Studentship to Mr. Ho.

9 References1 SHACKSHAFT, G., SYMONS, O.C., and HADWICK, J.G.:

'General-purpose model of power-system loads', Proc. IEE, 1977, 124,(8), pp. 715-723

2 'Determining load characteristics of transient performanceVol. 4'.EPRI Report RP849-1, 1981

3 SABIR, S.A.Y., and LEE, D.C.: 'Dynamic load models derived fromdata acquired during system transients', IEEE Trans., 1982, PAS-101,(9), pp. 3365-3372

4 MEYER, F.J., and LEE, K.Y.: 'Improved dynamic load model forpower system stability studies', ibid., 1982, PAS-101, (9), pp. 3303-3309

5 LEE, C.C., and TAN, O.T.: 'A weighted-least-squares parameter esti-mator for synchronous machines', ibid., 1977, PAS-96, (1), pp. 97-101

6 RICHARDS, G.G., and TAN, O.T.: 'Induction motor load aggre-gation for transient stability studies by constrained parameter estima-tion'. IEEE Summer Power Meeting, Vancouver, BC, 1979, PaperA79482-1

7 ROGERS, G.J., DI MANNO, J., and ALDEN, R.T.H.: 'An aggregateinduction motor model for industrial plants'. IEEE Trans., 1984,PAS-103, (4), pp. 683-690

8 HUMPAGE, W.D., DURRANI, K.E., and CARVALHO, V.F.:'Dynamic-response analysis of interconnected synchronous-asynchronous-machine groups', Proc. IEE, 1969, 116, (12), pp. 2015-2027

9 HOFFENBERG, M.S.: 'Simplified stability analysis of a distributionsystem in a chemical plant with induction motors', ibid., 1975, 122, (4),pp. 421-427

10 FLETCHER, R.: 'A new approach to the variable metric algorithm',Comput. J., 1970, 13, pp. 317-322

10 Appendixes

10.1 Parameters of motors in system testsIn Table 3 is collected together the parameters of themotor subgroups of Fig. 2.

Table 3: Motor parametersMotorsubgroup

1 Pulverised-fuel mill2 Primary-air fan3 Ash-sluice pump4 Extraction pump5 Induced-draft fan6 Forced-draft fan7 Boiler-feed pump8 Cooling-water pump9 Mixed-conveyor pump*

10 Tools compressor 111 Tools compressor 212 Feed-heater drains pump

Numberof motorsinsubgroup

441122123111

Total HPofsubgroup

HP

440504175

90828414

1275880

5290

17016

Rss

p.u.0.02920.02140.07620.3300.00950.02750.00550.01930.3380.1190.1162.209

x,.

p.u.

0.1580.2040.6691.4610.1430.2530.0770.1271.9950.9750.679

14.26

p.u.0.0290.0220.0680.2680.0120.0250.00490.0210.4890.1870.1132.58

Xlr

p.u.0.2850.2520.8250.8360.1590.2990.1490.2432.521.432.14

10.74

Xm

p.u.6.54

11.2145.4954.5

8.5214.7

7.734.89

121.963.4649.97

423.0

H{M/nf)

kWs/kVA1.2637.320.2310.068.502.891.660.4480.02070.05650.05720.0067

Loadtorqueindex

122222221002

Resistance values are expressed on a base of 1.00 MVA.Reactance values are calculated at 50 Hz on a 1.00 MVA base.Xls and Xlr are the stator and rotor leakage reactances, respectively.Inertia constant values in kWs/kVA are referred to a common base of 1.00 MVA.* Also sealing water pump and distilled-water pump.

10.2 Induction motor model: electrical axesExpressed in the synchronously-rotating frame of reference, the equation set for a single induction motor, as in theequivalent of Fig. 3, arranges to the form;

Vds (O' / Lmp-(o.

__LmP ! \

On partitioning eqn. 17, the stator equation set becomes

PKP~K

o (18)-((Of - (Or)L^ Rrr J liqr]

(33)

Here, p\l/ds and pi/^s are the rates of change of stator fluxlinkages in the d- and g-axes as given by:

Using eqns. 23 and 24 to eliminate idr and iqr from eqn. 33for vdr = 0 and vqr = 0, and defining the rotor slip s froms = ((of- cor)/cof, gives:

p\j/ds -p\jjqs

For theiAdr and

\j/dr =

0V =

= Lss Pi

rotor>/V are= *-*

Lmiqs

ds + Lm

qs ' Lm

circuits,formed

+ Lrr idr+ Lrr iqr

Pidr

Piqr

, flux linkagesfrom

(19)(20)

in the d- and g-axes,

(21)(22)

R" '-scof

SCOKrL

'rr

In terms of \j/dr and \pqr, the rotor currents idr and iqr arethen available from eqns. 21 and 22 using

IA. _ I'rr Lrr

dr T j ldsL>-- Li-

(23)

(24)

Substituting for idr and iqr in eqn. 18 using eqns. 23-24,

3fLn K

-cof[Lss--L^

03

(25)

On discounting the pif/ds and p\pqs terms, eqn. 25 reduces tothe compact form

ZLwhere

ls lds> lqs

K = tdr, ^ar

A =0

LTr0

\_ Xeq ! Rss]and

(26)

(27)(28)(29)

(30)

(31)

(32)

Using for eqn. 34 the compact form

Ptr = Qtr + Ksthen

Q =

F =

SCOf

LmRrr

SCO

Rr

LmRrr

(35)

(36)

(37)

10.3 Induction motor model: equation of motionThe acceleration of the motor shaft pcor, due to an imbal-ance between the motor electromagnetic torque Te and themechanical torque of the shaft load Tm, is given by

, = IT. - (38)In eqn. 38, M = H/nf, where H is the inertia constant inkWs/kVA, as in Table 3 of Appendix 10.1.The electromagnetic torque Te is available in terms ofrotor and stator components of current from

Te = cof Lm[ids iqr - iqs idr] (39)On using eqns. 23 and 24 to eliminate idr and iqr fromeqn. 39,

Te = kfWis (40)

(41)where

kf = cofLJLrr

is=ids,iqs (43)The dependence on shaft speed of the mechanical loadtorque Tm is taken into account by using

Tm = K(l - s (44)

From the partitioning of eqn. 17, the voltage/current equa-tions for the rotor circuits are given by:

Here, K is the load torque constant, s is the rotor slip andy is the load torque index.

Using the expression for Te in eqn. 40, and that for Tm ineqn. 44, the acceleration pcor is given by

(45)

186 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985

Alternatively, as s = (cof cor)/cof, ps = pov/coy, and so

- s)y kf\]/?is~] (46)

70.4 Solving differential equationsThe differential equations of the rotor circuits and the dif-ferential equation of motion are solved in the present workby their substitution into a numerical integration sequencebased on the trapezoidal rule. If n denotes the time step innumerical integration and h the step duration, then thevector of rotor flux linkages ^r(n) is found from

1) + \ M + p+r(n - 1)]Using eqn. 35 in Appendix 10.2,

v, = Qtr + Fisthen gives

ilfr(n)=f(n-l) + Cis(n)where:

(47)

(48)

(49)

(50)

(51)

Similar steps of substitution for the equation of motion ineqn. 46 of Section 10.3 give

s(n) = g{n -

(52)where

g(n - 1) = s(n - 1) +2Ma>,

- s(n - Mn - l) /> - 1)] (53)Eqns. 49 and 52 can be solved as nondifferential equationsin which a numerical integration algorithm has now beenembedded.

10.5 Induction motor equations in the steady-stateUsing eqn. 25 in Appendix 10.2, eqn. 11 in Section 4.6expands to

s--rL)iqso (54)

(55)

(56)

(57)

2 I ; i_ P ;~~ r I lds0 "T Kss lqs0

Li,

Similarly, eqn. 12 in Section 4.6 has the explicit form

Rrr , , Rrr0 = VdrO ~ S0 (Of VqrO + ^m 'dsO

Lrr Lrr

0 = s0cof \pdrO - -f- iJ/qrO + SL-IL iqsQL'TT L'rr

IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985

For the equation of motion in the steady state, as in eqn.13 in Section 4.6,

0 = - sof - (58)While it is possible to solve these five equations as a non-linear system to give ipdr0, i/^r0, s0, K and Lss, the initialslip s0 can be found directly from the known operatingconditions prior to fault onset. For an initially estimatedor subsequently calculated value of the stator resistanceRss, the power in the rotor circuits Pr is available fromVsIf RSS\IS\2, where the Vs and /s phasors are knownfrom steady-state measurements. If Ir is the rotor current,then

Pr= \Ir\2RJs0 (59)Also from the induction motor equations in the steadystate,

Ir=jXMIJ(Rrr/s0+jXrr) (60)Substituting for Ir from eqn. 60 into eqn. 59 and usingy = RJs0 gives

P,=

On rearranging eqn. 61,

yx2m\is\2y2- rr = 0

(61)

(62)

For known 7S, eqn. 62 selves to give y, following whichs0 = Rrr/y. In the event of the solution of eqn. 62 notgiving a positive value for y corresponding to

2Pr Xj\h\2 Xrr

then Xm is re-estimated from

l2PrXrrm V | / s l 2

(63)

(64)

With s0 now available, either eqn. 56 or eqn. 57 isredundant. On using eqn. 56, together with eqns. 54 and55,

10.6 The gradientFrom among the induction motor equations, it is possibleto form the gradient analytically, and then to evaluate it asrequired. Alternatively, the gradient may be formednumerically. Both methods have been developed in thepresent work. The procedure for finding the gradientnumerically is given here, in which each of the independentparameters is adjusted one at a time. If the error functionchanges by AE^OL, /?) when the first of the independent

variables is adjusted by Aal5 then the corresponding com-ponent of the gradient is formed from Ax(a, /?)/Aax. Thisis then repeated for each of the remaining independentparameters, and the gradient vector C(a, /?) is formed fromthese separate evaluations to give

A2(a,0)Aa Aa

(67)

where m is the number of independent parameters.

Abstracts of papers published in other Parts of the IEE PROCEEDINGSThe following papers of interest to readers of IEE Proceedings Part C, Generation, Transmission & Distribution haveappeared in other Parts of the IEE Proceedings:The effect of atmospheric humidity on the corona extinctionvoltage of circular cylindersJ. RUNGISIEE Proc. A, 1985, 132, (3), pp. 136-138The corona extinction voltage for two cylindrical conduc-tors and a vibration damper as a function of atmospherichumidity using radio influence voltage measurements hasbeen determined. For the 5.0 cm conductor and the vibra-tion damper, the corona extinction voltage increases withincreasing humidity. It is postulated that this is due to thecapture of the free electrons by water-vapour molecules.For the smaller 2.8 cm conductor, the corona extinctionvoltage decreases with increasing relative humidity. Thisdifferent behaviour is ascribed to surface conditions, butmore data is required to verify this.

Electric field measurements in long gap discharge usingPockels deviceK. HIDAKA and PROF. Y. MUROOKAIEE Proc. A, 1985,132, (3), pp. 139-146The optical method using a Pockels device has been usedto investigate an electric field distorted by space chargesdue to discharge development in air. A Pockels deviceplaced in the discharge area has the advantage of directlymeasuring such a field. This method is successfully appliedto the measurement of the electric field in and around theleader column propagating in a rod-plane gap of 1 m sub-jected to positive switching impulse voltage of 0.4 MV, and1.03 MV in a rod-plane gap of 3 m. Moreover, in the 1 mgap system, the leader propagation is effectively controlledby using a conductive thread so that the leader columnpasses through the Pockels device and the exact electricfield in the leader column is measured. The electric field atthe device increases with the approach of the leadercolumn and reaches its peak value on the arrival of the tipof the leader column. Finally, it falls to a low constant

value during the leader propagation. From the results, it isfound that the peak field at the tip of the leader column is0.8 ~ 1.4 MV/m and that, subsequently, the low field inthe leader column is 0.1 ~ 0.5 MV/m, notwithstanding thegap spacing of 1 ~ 3 m. Furthermore, the electric field cal-culated theoretically by using a space-charge model whoseshape is a cylinder of 0.5 m in radius together with a hemi-sphere at the head of it agrees with the measured field.

Multiconductor transmission-line model for the line-end coilof large AC machinesP.G. McLAREN and H. ORAEEIEE Proc. B, Electr. Power Appl., 1985, 132, (3), pp.149-156The paper describes how the PE2D finite-elements fieldpackage is used to calculate the distributed L and C par-ameters of a multiconductor transmission-line model forthe line-end coil of a large AC motor. Under steady-stateAC conditions, the package produces field distributionsfrom which self and mutual inductances and capacitancescan be calculated. Distributed shunt and series losses areincluded in the model, and there is no need for anyassumptions about surge velocities in evaluating the coilresponse. Separate parameters are calculated for the slotand overhang regions and the various sections of uniformmulticonductor transmission lines are joined in series torepresent the complete coil. The coil response is then com-puted using a Fourier transform technique. The inputwaveform is broken down into its spectral componentsand the steady-state transmission-line equations are solvedat each component frequency. The response at each fre-quency is then used in the inverse Fourier transform toarrive at the solution in the time domain. Responses calcu-lated from the model are compared to test responses on alaboratory model of the line-end coil and are found to bein good agreement. Several parameter variations are theninvestigated on the theoretical model.

188 IEE PROCEEDINGS, Vol. 132, Pt. C, No. 4, JULY 1985