Real-time power system simulation

M. Rafian, BSEE, PhDProf. M.J.H. Sterling, BEng, PhD, CEng, FIEEM.R. Irving, BEng, PhD

Indexing terms: Simulation, Power systems and plant

Abstract: This paper describes the theoreticalbasis and application of a simulation package forelectrical generation, transmission and distribu-tion systems. The software provides anenvironment for the development and testing ofonline analysis and control algorithms, and alsohas potential for application in operator training.Dynamic and algebraic models of power systemelements are described which are appropriate forstudy time scales of up to one day. Numericaltechniques are presented which obtain solutionswith acceptable accuracy in real time for mediumsized networks using minicomputer hardware.Typical results are given for a test network of 30substations which illustrate the dynamic behav-iour of the network under emergency conditions.

1 Introduction

The development of an accurate and reliable powersystem simulator offers a number of important advan-tages for application in electrical generation, transmissionand distribution systems control. A real-time simulationcan be used to provide a testbed for energy managementsoftware, and to provide a realistic operator trainingenvironment. If sufficient computational resources areavailable a simulation may also be initiated in real timeas part of an online security assessment facility.

Interest in online simulation has increased in recentyears and has attempted to consider the complete systemrather than individual elements. Two fundamentalapproaches to the problem have been adopted:

(a) using a low-flow technique to solve the networkequations and to evaluate the nodal voltages, which inturn are used in solving the network dynamic equations[1-3]

(b) solving the network dynamic equations simulta-neously and in conjunction with the network algebraicequations [4].

The former method can be faster, but it is less accurateespecially for the allocation of power demands to gener-ators. It also introduces problems if network islanding isto be considered because the loadfiow algorithm is notindependent of the network reference node. The latter

Paper 5277C (P9), first received 27th June and in revised form 17thNovember 1986Prof. Sterling and Dr. Irving are, and Dr. Rafian was formerly, with theSchool of Engineering and Applied Science, University of Durham,South Road, Durham DH1 3LE, United KingdomDr. Rafian is now with Everest Engineering Company, 9080 TelstarAvenue, El Monte, CA 91731, USA

method is numerically more stable, robust and inherentlycapable of handling islanding and fault analysis. It can beslower than the first method especially when the system isin the steady state and the simulator is designed for rela-tively short time scales, of the order of one second. Thesimulation technique described in this paper has beenbased on the second approach for a number of reasons:

(i) the simulator is more stable numerically and theresults are more accurate

(ii) the modelling of the network components and thesolution techniques can be chosen such that the simula-tor performs faster than real time for average sizedsystems using a minicomputer

(iii) there is a tight time coupling between the networkalgebraic and dynamic equations. In the alternative tech-nique the solution for the network equations and thedynamic equations are out of step at least by an integra-tion time step, which produces a weak coupling betweenthe algebraic and dynamic equations of the system

(iv) it is important that the simulator should becapable of simulating a split network efficiently, which isa characteristic of the second method.

This paper presents the detailed mathematical modellingof the system components together with a suitablenumerical technique for the solution of the derived equa-tions. The unique and important features of the simulatorare demonstrated by means of graphical results. It is alsoshown that the simulator can provide a suitableenvironment for operator training, including the possibil-ity of operating under various emergency conditions.

2 Generator and station dynamics

This Section describes the models for generators, excita-tion systems, governors, turbines and boilers which havebeen adopted. In selecting models the prime objective hasbeen to limit the burden of computation time for simula-tion, without sacrificing the accuracy of the model formedium- to long-term analysis of a system. The modelsare not detailed enough to allow the simulation of sub-transient and very fast transient effects, but are adequateto represent the system during a slow transient or dis-turbance, as well as under steady-state operation. It isworth noting, however, that the models are robust withrespect to any abrupt or transient changes of the system,but cannot provide the detailed behaviour of the systemduring the very short period following the disturbance.

2.1 Generator modelIn a balanced 3-phase system, the general phase equa-tions of a generator can be transformed into d-q axisequations, using the Park transformation technique. Themajor assumptions are (a) neglecting the harmonics

206 1EE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

above second order, (b) linearity in the magnetic circuit,and (c) independence of generator parameters from fre-quency. As the model is used to simulate slow transients,the main magnetic paths will not be highly saturated,also the percentage of harmonics in most modern gener-ators is negligible. The effect of frequency variation onthe generator parameters could be included if required.

The vector phase voltage and current based on directand quadrature parameters are:

~ jxq Iq - ra Ic

= coMafIf

(1)

(2)

(3)

whereVa

Ea

afM

coIf

xd, x

Id, I

= terminal voltage of phase a= excitation voltage or voltage behind gener-

ator phase impedance for phase a= mutual inductance between phase a and field

winding= rotor angular velocity, electric rad/s= field current= direct and quadrature axes reactances,

respectivelyq = direct and quadrature axes current com-

ponents, respectively= phase a current

Fig. 1 shows the vector diagram for eqns. 1 and 2. If lin-earity between the excitation current and the airgap flux

0 A

Fig. 1 Approximate vector diagram for a generator

produced is assumed, Maf is constant and Ea is a func-tion of rotor speed and field current. This assumptionmay cause considerable error in the fast transientanalysis. However, as the present generator model is partof a simulator which is not designed to simulate this typeof transient, the results are less affected by the saturationof the main magnetic path.

When the excitation voltage magnitude E and termin-al voltage magnitude V are known, direct and quadra-ture axis currents may be calculated from the followingequations (see Appendix 9.1):

(4)

(5)I VI (xd sin 3 — ra cos 3)

The active and reactive powers are of the form

|K|2(xa-xa)s in2<5 + 2

PG= {\E\\V\ir.™j+^anS)-\V\rm) ^

| V11E | (xa cos 3 - ra sin 3)

- | K | 2 ( x a c o s 2 3 + Xdsin2 3)

In modern turbogenerators the airgap is uniform and thevariation of flux from direct to quadrature axis does nothave an appreciable effect on generator performance, par-ticularly during small disturbances. In the model adoptedfor the simulation it is assumed that xq = xd. With thisassumption eqns. 4-7 reduce to

IE | xa - | V | (ra sin <5 + xd cos 3)h =

QG =

|K | (xas in

\E\\V\{ra

\E\\V\{xd

5-

cos

COS

H

3Y"

3

, cos

-rl+ xa

, r2

D-

sin

sin

H

5)

E\ra

-\v

-\V

2ra

2xd

(9)

(10)

(11)

Further simplification can be achieved if ra is assumed tohave a negligible value.

E — V cos 3

xd

V sin 3

EV . ePG = sin 3

EV cos 3 - V2

(12)

(13)

(14)

(15)

In the model adopted for the generators, the armatureresistances are not assumed to be zero and hence eqns.8-11 are used.

2.1.1 Electromechanical equations: A dynamic modelfor a generator must include the following electrome-chanical equations:

^ + D(co- coa) = Pm-Pe

or

(16)

(17)

d3— = co-co0 (18)

whereH = inertia constantPm = mechanical input power into a generator, p.u.Pe = electrical output power from a generator, p.u.D = damping coefficient, s/radcoa = 2n x average system (island) frequency, rad/s/ 0 = synchronous frequency, Hz3 = load angle, rad

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987 207

To integrate eqns. 17 and 18, Pe is substituted from eqn10 or 14 into eqn. 16. This will be discussed in furtherdetail in Section 3.

2.2 Automatic voltage regulator (AVR) modelThe AVR model adopted is the simplified version of thetype I model proposed in Reference 5. This model hasbeen used very often for transient studies (e.g. Reference6). The basic difference between this model and an alter-native proposed in Reference 7 is that the effect of shorttime constants on the simulated results is ignored. Thereare regulator, exciter and feedback time constants in thetype 1 model of Reference 7. The regulator time constantis around 0.1 s; hence its function can be reduced to animmediate amplification subject to limit constraints. InReferences 8 and 9 the time constant for the exciter,based on the manufacturers' design data, is claimed tohave a maximum of 0.15 s. The effect of time constants inthat range on the midterm transient analysis is negligible.The time constant for the feedback loop in Reference 8 isless than 0.1 s although in other References it is found tobe longer; nonetheless the amplification coefficient is sosmall that its effect is minimal.

The excitation voltage is applied to the field of a gen-erator, whereas in the generator model the voltagebehind the generator impedance is required, that is theno-load terminal voltage. If the effect of saturation isneglected, the excitation voltage is related to the fieldvoltage through a transfer function with open-circuit fieldtransient time constant T'd0. This time constant isbetween 5 and 10 s for the majority of generators and isincluded in the AVR model which is shown in Fig. 2.

KA/

JV,,,

^ Efd1

1

*Tdo Sb generator

Fig. 2 Block diagram for the adopted AVR model

The mathematical model of Fig. 2 is

AV = Vref - VT

Efd = KAAV VR{min)^Efd^VR{max)

dE KA(Vref -VT)-E

dt

where

T'd

(19)

(20)

(21)d0

E = open-circuit terminal voltage magnitude or exci-tation voltage

Efd = applied field voltageT'do = open-circuit field transient time constantVref = prespecified reference voltage (magnitude)KA = excitation gain

Initially the value of Vref should be set to

_ E° 0

ref = JT+ VT(22)

where E° and V% are the initial excitation and terminalvoltages for a generator.

2.3 Governor modelFig. 3 shows the block diagram for the governor modelused in the simulator. This is a similar model to the

general model proposed for the speed governing systemsin Reference 10 for steam turbines, except that thederivative of the input power to the turbine PGV isnot constrained. In Reference 11 in place of a simplegain K to amplify the speed error, a transfer function

GV(max)

turbine andgenerator

'GVCmin)

Acu

Fig. 3 Block diagram of the governor for steam turbines

K{\ + ST2)/(l + STJ is introduced, but Tx and T2 arezero for the governors without acceleration feedback. Themathematical equations corresponding to Fig. 3 are

Aco = co — co0

AP = Pset - KAco

dPGV AP-P GV

dt

(23)

(24)

(25)

whereK = governor gain = 100/(percent of steady-state

speed regulation), for co in p.u.= 100/(percent of steady-state speed regulation)

x co0, for co in rad/s7̂ = governor time constantPset = power set point

The model shown in Fig. 4 was suggested in Reference 10for hydroelectric systems. As T2 is of the order of 0.1 s,

A oo

Fig. 4 Block diagram of the governor for hydroturbines

for slow transients the model may be simplified to thatshown in Fig. 5, where

Aco = co0 — co

dAP KAco - AP

dt

PGV = -AP

(26)

(27)

(28)

Fig. 5 Modified block diagram of the governor for hydroturbines

2.4 Turbine models

2.4.1 Steam turbines: For steam turbines, the modelwhich has been selected represents a 3 stage single reheat

208 IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

turbine which is similar to the general model suggested inReference 10. The model with the equivalent block

shaft

Fig. 6 Block diagram for steam turbine model

diagram is shown in Fig. 6, wherePGV = output power from control valveTCH = steam chest time constant (including the high

pressure (HP) stage of the turbine timeconstant)

TRH = reheater time constant (including the inter-mediate-pressure (IP) stage of turbine timeconstant)

= steam storage or cross-over time constant(including the low-pressure (LP) stage ofturbine time constant)

FHP = HP turbine power fractionFIP = IP turbine power fractionFLP = LP turbine power fractionPm = equivalent generator input mechanical power

Tco

m q g p pThe equations that represent the block diagram of Fig.6b may be written as follows:

dPHP

dt

dPLP _dt

PGv-iTCH

PHP-fTRH

PIP-P

>HP

>

LP

Tco

+ PIP + LP

(29)

(30)

(31)

(32)

2.4.2 Hydroturbines: A simple model for an ideal hydro-turbine is discussed in Reference 10 and is adopted here.

governorPGV

valve

1-5TW

U0.5STw

Fig. 7 Block diagram for hydroturbine model

Fig. 7 shows the block diagram of this model, whereTw = water time constantPGV = equivalent input power to turbinePm = equivalent input power to generator

PGV ~ GV

dt O.5TW

PGy is calculated from eqn. 28.

1EE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

(33)

2.5 Boiler modelVarious boiler models for mid-term to long-term stabilityanalysis are discussed in References 12 and 13. Reference14 gives a survey of some of these models.

The boiler model presented here is a low-order modelfor a drum type or once-through boiler with an integralboiler-turbine control system. The block diagram of themodel is shown in Fig. 8, where

X, = pressure controller multiplying factorTA, TB = time constants for combustion controllerTD = time delay for fuel to reach the boilerTv = fuel time constantCB = boiler storage time constantK2 = factor denoting the pressure dropK3 = gain factor to regulate the effect of frequency

deviation on boiler response

The typical parameters can be found in Appendix 9.2.To write the lst-order differential equations of the

model shown in Fig. 8, the combustion controller and thefuel system is expanded as shown in Fig. 9.

T'A = TJ10

APR = K;ATP

= APR + TRAPRdt

ABC,dt T'A

AFD{t) = ABCl(t_tD)

dASF AFP - ASF

dt ~ Tv

TA ABCX - ABC2

(34)

(35)

(36)

(37)

(38)

(39)

whereSFFDPR

= steam production= virtual steam production= firing rate setting

BCX = fire intensityBC2 = an auxiliary variable

The equations that formulate the boiler storage, throttlepressure, and change in the flow of steam into steamchest according to the boiler response are as follows:

dADP AFS - APHP

dt CB

PHP = PHP-(Pset-K2A(o)

dPf PGV + K.ADP - PHP

(40)

(41)

(42)

(43)

dt TCH

ATP = ADP - K2 APHP

whereDP = throttle pressureTP = drum pressure

Eqn. 42 replaces eqn. 29 to allow for the effect of pressurechange in the boiler on the steam flow into the turbine.

It should be noted that the above equations specify thechange in the various quantities and therefore to find theabsolute value of any of these variables (except for eqn.42), the initial values must be added to the calculatedquantities.

In a steam power station the appropriate governor,steam turbine and boiler models must be chosen depend-ing on the type of the plant (i.e. coal fired, oil fired, gasfired etc.). For fast response plant the boiler model couldbe simplified (see References 11, 15 and 16) and for some

209

plants a simpler version of the turbine model can bechosen. For hydroelectric stations the overall model is

The effect of the voltage tap changer on the self andmutual admittances of busbars i and j are as follows:

A (HP steam flow)

A (throttle pressure)

M1*STA)(USTB)5(USTA/10)

e"STD

WSTV

AR

combustion controller fuelsystem

(pressureloss)

governor

valve

steamstorage

Fig. 8 Block diagram for boiler model

turbine andgenerator

loadfrequencycontrol

electricsystemnetworkand load

much simpler, because it contains only the turbine andthe governor model.

3 Network model

A power system consists of the following main com-ponents: generating stations, transformers, transmissionlines, reactive compensators and loads.

Models for the first item were presented in the preced-ing Section. In this Section the models for the remainingitems will be discussed.

3.1 TransformersIn considering the generally accepted equivalent circuit ofa transformer, it is assumed that the magnetising reac-tance of the transformer may be neglected. The resistance

ATPrT-lAPR- ^ K i p — •

A B C I USTA

USTyASF

Fig. 9 Block diagram for combustion controller

of the transformer windings is often ignored, but may beincluded in the present model. The leakage reactancewhich represents a transformer with sufficient accuracyduring slow transients is included together with acomplex tap changing facility. The resulting model isshown in Fig. 10.

node i

Fig. 10

"i< " k node j

Single-line model for a tapped transformer

It is assumed that i is the sending and j is the receivingbusbar for transformer k. The equivalent resistance andleakage reactance of the transformer windings are rk andxk, respectively. The tap changing turns ratio and thephase shifting are assumed to be referred to the sendingend busbar.

The equivalent admittance of the transformer is there-fore:

l

rk+jxk

= Gk-jBk (44)

(45)

ya = t2kyk (46)

Vjj = Vk (47)

For a phase shift <j)k, the self and mutual admittances ofbusbars i andy will be affected in the following way:

ya = yk (48)

yjj = yk (49)

y u = Gk cos 4>k + Bk sin d>k —j(Bk cos <bk — Gk sin d>k)

(50)

j/j-,- = Gk cos 0k — Bk sin 0k — 7(J5fc cos <f)k + Gk sin <j>k)

(51)

The above equations are derived in Appendix 9.3. For atransformer with tap changing and phase shifter thechanges in the admittances are

yjj = yk

yu = Gkn + Bk n -j(Bk n - ck n )yyi = GkT

rk- Bk Tk -j(Bk Tk + Gk T

lk)

where

Trk = tk cos 4>k a n d Tl

k = tk sin 4>k

(52)

(53)

(54)

(55)

(56)

210

Eqns. 54 and 55 can be generalised to the following form:

yij=Gij-jBij (57)

y..= Gji-jBji (58)

where

.. = ljk 1 k — (j£ i k \vv)

j i k k k k V /

fl,, = B k n + Gftn (62)

Eqns. 57 and 58 have the same form as transformeradmittance eqn. 44 or a branch equation, except that Gand B are modified to allow for tap changing and phaseshifting effects and that ytj ± yj{.

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

A transformer can either have an automatic mechan-ism to control the tap and phase shifter settings or thechanges could be implemented by the operator in thecontrol centre. In the former case the voltage tap positionis changed automatically to maintain the voltage of abusbar as close as possible to a prespecified value. Thisbusbar can be i or j in Fig. 10 or any busbar in thesystem. The change in tap position may be formulated asfollows:

tnew _ foldlk — rfc

- I yspec _ y \Ck\V m Vm) (63)

wherem = number of the controlled busbarck = gain factor

For the phase shift the control is on the flow of power inthe transformer branch or any other branch in thesystem. The change in the phase shifter step is of the form

= 4>lld (64)

where

m = number of the controlled branchdk = gain factor

3.2 Transmission linesTransmission lines are represented using a n model asshown in Fig. 11, whererij + Jxij = impedance of line y

s, = Sj = — = shunt susceptance of line ij/2

(65)

x{j, s,, and Sj are functions of frequency, and so if thefrequency of the system deviates from the normal valuethe series and parallel impedance of the line will change.The changes are cumulative in the sense that frequencybelow normal decreases the series impedance whileincreasing the parallel impedance. The combined effectcauses an increase in the receiving end voltage.

node i r00O0(T node j

Fig. 11 Single-line model for a transmission line

To adjust the line impedances with frequency, the fre-quencies for the sending and the receiving busbars areaveraged

(66)

(67)Si =

wheref{,fj = frequency at busbars i and j , respectivelyfset = set frequency for the system or subsystemxfj = impedance of line ij at frequency/m

sf = susceptance of line ij at frequency fset

3.3 Static compensatorsStatic compensators are commonly used to regulate thereactive power and improve the power factor in a powersystem. Shunt reactors or capacitors may be installed inload centres or on transmission lines. The reactance ofthe compensators is, of course, frequency dependent andtherefore they can be modelled as admittances varyingwith the frequency of the connected busbar.

If a compensator is connected to busbar i, its imped-ance and reactive power are:

Qc=Yc\Vi\2

QL=YL\V,\2

where

Bo = 2nCfset

Xo = 2nLfset

(68)

(69)

(70)

(71)

(72)

(73)

3.4 LoadsIn power system modelling, much attention has beenpaid to generator modelling, although the impact of theload models on the system is of equal importance. Forexample, the behaviour of motor loads under transientconditions is comparable to that of generators. Althoughit is simple to model certain types of load, usually it isvery difficult to find the exact composition of the aggre-gate load. It has been common practice to model a loadas (a) constant power, (b) constant current and (c) con-stant impedance or a combination of (a), (b) and (c). Thisform of load modelling may be satisfactory for certaintypes of load under specific conditions, but, in general,the load cannot be satisfactorily modelled as either (a), (b)or (c).

The load power varies generally with respect tovoltage and frequency. The effect of voltage angle onreactive load power is zero while its effect on the activeload power is of a transient nature which persists for lessthan a second [17]. This is also verified by the field testsand reported in Reference 18. The effect of voltage angleon the load model can therefore be reasonably ignored. Itremains, therefore, to include the effect of the voltagemagnitude and frequency on the load model. The generalform of load variation with voltage magnitude may beapproximated by

7 ^ 1 +JQoo Vo

(74)

whereP o 9 Qo = initial active and reactive power at initial

frequency and voltage Vo

Vo = initial voltageP, Q = active and reactive power at set frequency

and voltage magnitude Vn, m = rational coefficients representing the degree

of dependency of active and reactive powerwith voltage magnitude

In eqn. 74, for example, n = m = 0 represents constantpower, n = m = 1 constant current and n = m = 2 con-stant impedance. As in each substation the load aggre-gate usually is a combination of various types of load (i.e.domestic, industrial, agricultural, etc.) eqn. 74 in generalform is a summation of fractions of p0 and q0 with their

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987 211

(75)

respective coefficients m, and n,.

V0\J + Jqot\Wo\

wherek = number of types of loads.

It is shown in Reference 19 that, for small changes involtage, from the measurement of AI/AV for every typeof load and knowing the percentage of each load, theoverall value of n and m may be calculated, but for largechanges in voltage, eqn. 75 must be used. There is no testevidence that n and m for any type of load will remainconstant for the whole range of voltage variations. Forexample, motor loads that can be modelled as constantpower (n = 0), for small changes in voltage, exhibit con-stant impedance (n = 2) behaviour at very low voltages.Therefore, it is possible to conclude that the values of nand m vary with voltage in general, but the functionswhich represent this dependency cannot be establishedowing to the lack of test data. The simplest practice is tochange all or some loads to constant impedance typewhenever the related voltage is below a threshold level.

The variation of load with frequency is even more dif-ficult to determine. Tests are difficult to perform on a realsystem and results are difficult to interpret. The motorload, for instance, not only changes with frequency, butalso with the rate of change of frequency. A simple butreasonably accurate assumption is to allow the power tovary linearly with frequency. The frequency sensitivity ofloads also varies; active load power generally decreaseswith decreasing frequency whereas the reactive loadpower increases with decreasing frequency.

In the present model for the load it is assumed thatthe dependency of the load on frequency is linear, butnonlinear dependency, if needed, can also be easilyimplemented.

Including the effect of frequency variation in the loadmodel, eqn. 75 may be rewritten in the following form:

r f i K ' Y + fPi(f-fse.)

+ jq0iVo\)

where

JPi Qf, Jqi

dq1

(76)

(77)

fi the frequency of busbar i connected to the load.

4 Numerical techniques

From the models developed in Sections 2 and 3, two setsof equations have emerged, namely differential and alge-braic equations. These two sets of equations are interde-pendent and therefore the solution technique mustprovide a simultaneous solution. There are two funda-mental methods for the numerical solution of coupleddifferential and algebraic equations. In the first methodthe differential equations are numerically integratedexplicitly with the algebraic equations solved as a sub-problem. A typical algorithm of this type is the Runge-Kutta integration technique. In the second method animplicit technique is adopted in which the differential and

algebraic equations are considered together rather thansuccessively. To facilitate this, the differential equationsmust be transformed into algebraic equations, andappended to the existing set of algebraic equations. Theimplicit trapezoidal method is a useful technique in thesecond category, and is known to be numerically stable,even when time constants in the model are much smallerthan the integration time step [20]. As a relatively largeintegration time step together with robustness and rea-sonable accuracy are the important factors in selecting anumerical technique for the present application, it wasfound that the implicit trapezoidal method is highly suit-able for the numerical solution of the differential equa-tions presented in Section 2.

4.1 Implicit trapezoidal methodThe implicit trapezoidal method is based on the dis-cretisation of a function with the assumption that thefunction is linear during an integration time step. For aset of differential equations of the following form:

Y=f(Y, U, Z) (78)

the application of the above concept means that the func-tion, at time t, has the value

Y dx (79)

As it is assumed that the function is linear between t andt — At it follows that

I (80)

Substitution of eqn. 79 in eqn. 80 results in algebraicequations for the interval At:

(Y (81)

or

t ^ t , Vt_At, Zt_Ar)} (82)+ f{f(Yt, Vt, Zt)

where/denotes the derivative of function F in the generalform.

This method is self starting, so provided that the valueof the function at t = t0 + is known, eqn. 82 can besolved.

4.2 Application of trapezoidal technique todifferential equations of Section 2

The first step in the application of the technique is towrite the differential equations in the form of eqn. 78.Therefore the independent variables must be defined.These variables for the generator i and the associatedboiler and steam turbine prime mover are

(83)T = [©„ 8t, £ , , PGVi, PHPi, PIPi, PL

For a hydrogenerator eqn. 83 changes to

= icDi,di,Ei,APi,Pm^ (84)

where T denotes the transpose.The independent variables of the network are the real

and imaginary components of node voltages. Forinstance, at node i, the variables are ebi and fbi.

Each of the equations developed in Section 2, must

212 IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

therefore be a function of the above variables or con-stants. Hence the dependent variables should be replacedby known functions of the independent variables, asfollows.

In eqn. 16 Pe is substituted for from eqn. 14 (in Appen-dix 9.4, the general form of eqn. 16 by replacing Pe fromeqn. 10 is derived).

l£l I V\ 1Pe = ' " ' sin (<5£ -Sy) = — Im (EV*) (85)

Xd Xd

whereIm = imaginary component of a complex variable* denotes the complex conjugate.

But

V = | K|(cos Sv +j sin by) = eb +jfb (86)

E = \E\(cos3E+j sin<5£) (87)

Therefore

Pe = - (- /„ cos SE + eb sin dE) (88)xd

Substituting eqns. 32 and 88 into eqn. 17 results in

Current Law to each node, that is:

dcozi

x ( - / „ cos SE + eb sin SE) - D(co - coa) > (89)

The remaining differential equations for that generatorare eqns. 18, 21, 25, 30, 31, 40 and 42 which are onlyfunctions of independent variables. The differential equa-tions 36, 37 and 39 are integrated separately using therectangular integration formula. This allows a reductionin the number of differential equations per boiler withoutaffecting the accuracy of the calculation because the timeconstants TA, T'A, TB, and Tv are in the order of seconds(See Appendix 9.2).

For a hydrogenerator, eqn. 89 is of the form

dt H

x (-fb cos 5E + eb sin SE) - D{co - coa) j> (90)

The remaining equations for a hydrogenerator are eqns.18, 21, 27 and 33.

Application of the trapezoidal technique to differentialequations 18, 25, 30, 31, 40 and 42 is straightforward andfollows the same principle described above. In eqn. 21, VT

is not a control variable, so it must be replaced by acontrol variable, namely, the generator busbar voltage,that is:

dE _KA[Vref-(e2bi+f2

bi)l<2l-E

dt T'd0(91)

If Efd is not within the limits of eqn. 20, eqn. 91 must bereplaced by the following equation:

dE VR-Edt ~ T'

(92)

4.3 Network equationsThe equations representing the network are algebraicequations derived from direct application of Kirchhoff s

(93)

where n is the total number of connections to node i, i.e.lines, generators, loads etc.

node i node /

si 4=

Fig. 12 Single-phase model for a line

For each type of connection the current equation mustbe derived. These are as follows.

4.3.1 Lines: Suppose node i is connected to node; via aline as shown in Fig. 12.

(94)Iu = Vt Yc{ij) + (Vt - V}YU

where

results in

or

(ebi +jfbi - ebj -jfbjUGij -jBu)

(95)

(96)

hj = (ebi - ebj)Gij +/bI(fl0- - S.) - / w f l y

+ j{eblSt - Bu) + (fbi - / w ) G y + ebj fly} (97)

Similar equations can also be derived for node;.

4.3.2 Generators: Fig. 13 shows the equivalent diagramwhere generator g is connected to node i, where i is the

node i

Fig. 13 Single-phase model for generator g

terminal node for the generator g with excitation voltageof Eg and impedance of rag + jxdg.

The equivalent voltage components for the generatorexcitation voltage are

(98)

(99)

eg = Eg cos Sg

Jg g g

Following the same procedure as in Section 4.3.1 resultsin

lig = (ebi - Eg cos dg)Gg + ( /„ - E , sin 6j)B9

+ j{(fu ~ Eg sin dg)Gg - (ebi - Eg cos 5g)Bg) (100)

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987 213

where equation of a generator is of the form

' ag and Bn =Idg <i

(101)

4.3.3 Loads: Fig. 14 shows the equivalent diagram whenload / is connected to node i, where s, represents the

I;;V = 0

n o d e i ' •-« ' I "I"

Fig. 14 Single-phase model for load I

complex load. The variation of a load with the voltagewas discussed in Section 3 and the relation is of the form

where

0/ Jo/O ,M

« = 5 =

(102)

(103)

(104)

multiplying the numerator and denominator by ebi +jfbi

results in

- p0lfbt\Vtr2} (105)

4.3.4 Static compensators: For the static compensatoror shunt reactor m, Fig. 15 shows the equivalent diagram.As the impedance of a shunt reactor is independent ofvoltage, the current Iim is of the form

Iim=Vl{jBm) = -fbiBm+jebiBn (106)

4.4 Formation of Jacobian MatrixThe Newton-Raphson algorithm is a powerful methodfor the solution of a set of nonlinear equations. It is

mode i

Fig. 15 Single-phase model for static compensator m

robust provided that an initial guess is available which isreasonably close to the solution and its rate of con-vergence is quadratic. For a set of equations of the form

[F(Y, V, Z)] = 0 (107)

The solution can be obtained by successive evaluations ofthe following equation:

[(7, K,Z)*+1] = [(y, V, Zf]

-[JkTl[F(Y, V, Zf] (108)where

k = interation index[ / ] = Jacobian matrix

The elements of the Jacobian matrix are the partial deriv-atives of eqn. 107 with respect to the independent vari-ables defined in Section 4.2.

To demonstrate the method by which the Jacobianmatrix elements are derived the sub-Jacobian matrix fora generator is found, as follows.

The differential equations of a generator are rewrittenin the algebraic form of eqn. 81. For instance, the speed

substitution of eqn. 89 into eqn. 109 gives

F{(o) = Ft{(D) + Ft.Jco) + C = 0

where

(109)

(110)

cos <5£{f_Ar)

s i n <>E(t-At) ~~ FHPFHP^-AD

( nf0DAt\t-At)

C =nf0 Dcoa At

H

(111)

(112)

Ft-Ati*00) is calculated from the initial values of the vari-ables and will remain unchanged for k iterations. C is anindependent constant.

Ft(co) = cot + ^—- \ Dcot + — (fm cosZH I xd

b(t)

~ FIP *IP(,t) LP *LP(t) (113)

The Jacobian elements for the above function are

dF(co)

dco= 1+KD

dF(co) KE .—— = (-fb sin <5£ + eb cos

do xd

dF(co)

dP= -KF,

HP

dF(co)

dPlP

8P(114)

LP

dF{co) KE dF{co) KE . c— — = — — = sin dE

deb xd Sfb xd

-T7T- = — (fb cos 5E + eb sin 5E)ob xd

where

k =nf0At

2H(115)

214 IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

o <

<N ^

3:a.

<3:

h <1

Uj

0 ,

+

•a

<

1

o

<1 u

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987 215

The equations of a generator results in the sub-Jacobianmatrix of eqn. 116. It is further assumed that the gener-ator is connected to node i and for the sake of clarity, thesubscript for the generator is omitted.

The elements of the Jacobian matrix related to thenetwork equations can be derived from the algebraicequations of Section 4.3. Each node has two columns andtwo rows in the Jacobian matrix because eqn. 93 is acomplex variable, where each of its real and imaginaryparts have derivations with respect to real and imaginarycomponents of the nodal voltages. If the Jacobian matrixis first derived for a node connected to a line, a generator,a load and a shunt reactor, its application to specificnetwork configurations is then straightforward.

(a) If a line connects node i to j , the elements of theJacobian matrix related to node i are derived from eqn.97:

= tfij — a,dlR

debi

dIR

debj

dl\t

debi

' •

Si

D

1

-Bu

diR

dfbi

dIR

dfbj

dl\i

dfbi

(117)

= -Gu

dfbj

where IR = real (/,-_,-) and IM = imaginary (Jy).(b) For generator g connected to node i, the Jacobian

elements can be derived from eqn. 100:

d39= \Eg\(Gg sin dK - B. cos 5K)

(118)

= -Gg sin Sg + Bg cos 5

where the following relation holds between IRG, IMG andIig from eqn. 100.

(119)

(c) For load / connected to node i, the Jacobian ele-ments can be derived from eqn. 105:

' „ = IR, +jIM, (120)

(121)

(d) For shunt reactor m connected to node i, the Jaco-bian elements can be derived from eqn. 106:

(122)

= 0

dl(123)

Mm = 0

(e) If the transformer k is connected between nodes iand), the Jacobian elements are similar to case (a), exceptthat the effect of the tap changer and the phase shiftermust be included as discussed in Section 3.1.

= Gk 7T - Bb T[Rk(i)

(124)

' M k(i)

~ lk

= B, Ti - Gt Ti

= Gfc i t Bl. j l

It is worth noting that if the tap changer and phaseshifter are referred to node i (as is the case for eqn. 124),then for node j the Jacobian elements are not those givenby eqn. 124, but are of the form

dl

(125)

The mutual Jacobian elements between nodes i and j inthe case where they are connected via a phase shiftingtransformer are not symmetric, i.e.

debi

i) i dlRkjj)de

(126)bj

In the following general form of the Jacobian matrix,only the elements related to node i are shown. It isassumed that node i is connected to node j via a trans-mission line and transformer k, to generator g, to load /and to shunt reactor m. For any network the elements ofthe matrix may be found by following the same pro-cedure and taking into account the network topology.

216 IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

generators etc., whereupon the rows and columns of eband fb must be modified to include the appropriate ele-ments of the other components.

As may be seen from eqn. 127, the Jacobian matrixis highly sparse and therefore sparse factorisation tech-niques are used to solve eqn. 108.

5 Ancillary calculations

5.1 Measurement and bad data simulationTo simulate the telemetered measurements and to allowfor the error caused by the transducers and data commu-nication, two types of random error are added to the cal-culated quantities, namely static and dynamic errors. Toapply a random error, normally distributed randomnumbers with zero mean value and unit standard devi-ation are generated [21]. The equation used to producesuch a random number is:

RN =±± (128)

whereH = mean value of each probability density function

of each random number (fi = 0.5)a = standard deviation (a = 1/12)R = uniformly distributed random number

(0 < R < 1)N = number of uniformly distributed numbers

The static error varies with the full scale reading of thetransducers and the error percentage for each transducer.

>max(i) (129)

whereESt(i)

RN{i)

= static error for the measured quantity i= percentage error for the measured quantity i= full scale reading for the transducer of the

measured quantity i= random error calculated from eqn. 128

The static error is independent of time and remains con-stant during the course of calculation.

Dynamic errors are calculated for each measuredquantity and for every time interval. The dynamic error isadded to account for data transmission errors, its valuedepends on the maximum of the telemetered quantityand the percentage error.

RN(i)t Emax(i) (130)

whereEdya)t = dynamic error for the measured quantity i at

time tEd(v> = percentage error for the measured quantity iRN{i)t= random error calculated from eqn. 128 at time

tFor example if V{ is the calculated voltage magnitude atbusbar i, the telemetered magnitude of that voltage at thecontrol centre at time t has the value:

v. — v.r i(measured)r ri(calculated)t

Jst(Vi) (131)

Est(vi) is calculated once, whereas Edy(Vi)t is calculatedevery time the measured value is transmitted to thecontrol centre. To simulate bad data, certain measure-ments are corrupted by multiplication with an analogueerror parameter.

218

5.2 ProtectionThere are two types of protection in a power system.First, protection against likely damage caused by theoccurrence of sudden faults in the system. This type ofprotection is a fast transient phenomena and is thereforeoutside the scope of the present simulation. The secondform of protection acts during slow transients or understeady operation of the system. This includes protectionagainst (a) overspeed of generators, (b) under frequency,and (c) transmission line overload.

For the above protection, it is assumed that the relayshave a quadratic time inverse characteristic. Also eachrelay acts outside a deadband zone and there is a timedelay between successive operation of relays, for any typeof protection. For overspeed and overload protection therelays act so as to isolate the relevant generator or trans-mission line. For underfrequency protection dependingon the percentage of the frequency drop, to avoid theoverspeed, the loads are graded so that the most effectiveload is disconnected. Every protected load is assumed tobe equipped with an underfrequency relay that monitorsthe frequency of the busbar connected to that load.

5.3 Generator run -up and run -downGenerators are controlled by load frequency control(LFC) between their maximum and minimum outputs.Therefore when a generator is requested to be discon-nected from the system by unit commitment, the dispatchprogram sets the output target to the minimum level ofgeneration for that generator, whereupon the LFC hasthe task of ramping down the generator to its minimumoutput in a specified time set by the dispatch program. Acommand is then issued to the simulator to disconnectthe generator. The simulator must first run the generatordown to nearly zero output and thereafter disconnectsthe generator either directly or through operator inter-vention. Similarly when the generator is going to partici-pate in the load, after being synchronised andreconnected, the simulator has the task of running it upto its minimum output and then hand over the control ofthe generator to the LFC.

The rate of run-down is usually faster than the rate ofrun-up by a factor of 5-10 times.

5.4 Line flow calculationsThe line flows are needed for overload protection andtelemetry measurements. The flow equations may bederived from Fig. 9. The power flowing from busbar i tobusbar j is

Pij +JQij =

where

and Vj = ebj + jfbi

(132)

(133)

(134)

When voltages are substituted by their equivalentcomplex components, eqn. 134 may be rewritten as

Pij = (ebi-ebj)(ebiGij-fbiBiJ)

+ (/w -Mfti Gtj + ebi By) (135)

Qij = (fbi-fbj)(ebiGij+fbiBiJ)

- (eu - ebj)(fbi Gbi - ebi BtJ) + \ V, \2St (136)

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

The power flowing from busbar j to busbar i can be cal-culated from the following equations:

ji = (ehJ ~ e^e^Gtj - / w B f / )

+ (fbj -fMjGij + ebjBi})

ji = ifbJ -ft&ebjGij +fbJBt} - (ebj - ebi)

(137)

(138)

5.5 Transformer power flow calculationsThe equations for power flow in a transformer withouttap changer and phase shifter are similar to eqns. 135 and136. When a transformer is equipped with a tap changerand a phase shifter or either of them, the flow is calcu-lated as follows [22].

From Fig. 10 if transformer k is connected betweennodes i and j , the power flow from node i to j is

(139)

(140)

(141)

(142)

(143)

(144)

The power flow from node./ to i is

where tk is the complex tap changing coefficient.From eqn. 56

tk = n+jTk

The power flow equations based on Section 3.1 variablesare derived in Appendix 9.5.

5.6 Topology determination and islandingThe topology or connectivity of an electric power systemchanges periodically as a result of deliberate switching orprotection operation. To simulate the system it is neces-sary to provide numerical solution algorithms with time-varying topological information. This includes lists ofnetwork elements which are energised, information onnodes (connected sections of busbar), and details of anynetwork islands (isolated groups of connected nodes).

Sullivan in Reference 23 has introduced a simple algo-rithm for the determination of nodal topology, which isalso applicable to the problem of detecting and identify-ing islands. The algorithm (in the case of islanddetermination) can be briefly described as follows:

(i) initially label node N as belonging to islandN, I(N) <- JV, for every N

(ii) for every energised branch, consider the nodes Nt

and N2 which it connects, and replace

O, 7(JV2))

7(N2)

(iii) repeat (ii) until no further changes occur(iv) renumber the islands to obtain a consecutive set(v) stop.

Various indexing arrays may then be constructed to forma mapping between the physical plant records and the

LCK i

substation 5Fig. 16 Test systemSystem consists of 30 nodes, 73 busbars, 6 generators, 4 transformers with tap changers, 41 lines, 68 links, 72 circuit breakers, 23 loads and 2 static compensators^^m i node i Gi generator /« - ^ i busbar i Ti transformer i

i line i Ci Static compensator i- • — / - - link i LDi Load i

1EE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987 219

present nodal and island structure. Although Sullivan'salgorithm is quite efficient, it is interesting to note thatthe number of iterations required is influenced by theorder in which the branches are considered.

When islanding occurs, the topology must be able toisolate the passive islands, i.e. islands without gener-ation. If a passive island has an isolated generator, andthe generator is reconnected, the topology should becapable of reincluding the island into the calculation.

6 Results

The test system is an extended version of the 30-nodeIEEE system [24]. Each node in the test system consistsof a number of busbars connected via links. Each linkcan have one or more circuit breakers, and represents thecoupling circuits between busbars or other connectionpoints. The IEEE system is shown in Fig. 16 togetherwith generators, transformers, loads, and static com-pensators. To give an example of a nodal configuration,the details of nodes 5 and 18 are also shown. In general asubstation may contain more than one node with thenumber of nodes for each substation depending on theoperating conditions.

To demonstrate the accuracy and robustness of thesimulator, it is subjected to the severe system transientslisted in Table 1. In this test the simulator is not con-trolled by the energy management software, hence theloading condition, the governors valve set points, and thegenerators excitation voltage set points are maintained attheir initial settings throughout the test. The change inloads due to voltage and frequency variation and theeffect of frequency on the line parameters are included.

Fig. 17 shows the power outputs and speeds of threegenerators. The generators are selected to demonstratethe effect of islanding on the system performance. Twotypes of islanding are imposed on the system: (a) apassive island with no generation which is discarded by

Table 1: Scenario of events

the topology and is equivalent to the simultaneous dis-connection of lines and loads, and (b) active islands which

Item Time Type of action

1 00:04:00 Load LD4 is disconnected2 00:08:00 Load LD4 is reconnected3 00:10:00 Line 1 is disconnected4 00:14:00 Line 1 is reconnected5 00:18:00 Generator G2 is disconnected6 00:22:00 Generator G2 is synchronised and reconnected7 00:26:00 Lines 32, 22, 19 and links 42 and 43 in

substation 12 are disconnected (a passiveisland is created)

8 00:30:00 Links 42 and 43 are reconnected9 00:33:00 Line 15 is disconnected (system is divided into

two islands)10 00:37:00 Lines 10 and 41 are disconnected11 00:38:20 Load LD10 (10 MW) is disconnected by under

frequency protection (Fig. 17 only)12 00:39:00 Line 33 is disconnected (island 1 is further

divided into two islands. The system is splitinto three islands)

13 00:44:00 Command to reconnect line 32 afterresynchronising islands 1 and 2 is issued

14 00:44:52 Islands 1 and 2 are resynchronised, line 32breaker is closed

15 00:49:00 Lines 19, 22 and 15 are reconnected16 00:52:00 Command to reconnect line 33 after

resynchronising the remaining island with therest of system is issued

17 00:54:12 Island 3 is resynchronised with the rest ofsystem

18 00:58:00 Lines 10 and 41 are reconnected

Items 14, 15, 17 and 18 are associated with Fig. 20 only.

315.0315.0315.0

1.41.31.0

311.5« 311.5S 311.50 0.41 0.65* 0.5

308.0308.0308.0-0.60.00.0

(ii)

00:00:01 00:15:01 00:30:01 00:45:01lime (8th February 1985)

01:00:01

Fig. 17 Simulator response to system transients listed in Table 1 whenit is not controlled by EMS

(i) generator Gl speed, rad/s(ii) generator G4 speed, rad/s(iii) generator G6 speed, rad/s(iv) generator Gl electrical power output, p.u.(v) generator G4 electrical power output, p.u.(vi) generator G6 electrical power output, p.u.Base for power is 100 MVAScales are in order of curve number, i.e. the top scale is for curve (i) etc.

each contain at least one generator. A scenario gener-ation function executes the commands listed in Table 1and the successive commands make the system split intothree islands. The results of the simulation are shown inFig. 17. Each generator in this test is in one island, hencethe speed of each generator is an indication of the relatedisland frequency. Attempts to resynchronise the islandsfail because there is no load frequency controller to regu-

40.09.09.88.0

$35.75S 6.5c 7.4§ 6.0

31.54.05.04.0

V (ii)

(i)

00:01:30 00:02:00 00:02:30 00:03:00 00:03:30time(8th February 1985)

Fig. 18 Effect of voltage magnitude and frequency on loads active andreactive power(i) load LD4 active power, MW(ii) load LD4 reactive power, MVAr(iii) load LD10 active power, MW(iv) load LD10 reactive power, MVArAt time 00:02:00 the system is split into two islands. Load LO4 is in an islandwith frequency above 50 Hz and load LD10 is in another island with frequencybelow 50 HzScales are in order of curve number, i.e. the top scale is for curve 1 etc.

220 IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

late the input power to the prime movers as a result offrequency deviation in each island.

To demonstrate the effect of frequency and voltagemagnitude on the loads Fig. 18 is presented. It shows the

0.0050.0400.0600.0400.2000.240

-0.015« 0.010% 0.020o -0.040"5 0.030§ 0.050

-0.035-0.020-0.020-0.120-0.140-0.14(

( i )

(iv)

(v)(vi)

(ii)

(iii)

:01:00 00:07:15 00:13:30 00:19:45time(8th February 1985)

00:26:00

Fig. 19 Boiler response to system transients(i) change in throttle pressure for Gl , p.u.(ii) change in boiler steam production for Gl , p.u.(iii) change in boiler fuel consumption for Gl , p.u.(iv) change in throttle pressure for G6, p.u.(v) change in boiler steam production for G6, p.u.(vi) change in boiler fuel consumption for G6, p.u.At time 00:03:00 the system is split into two islandsScales are in order of curve number, i.e. the top scale is for curve 1 etc.

effect of a simultaneous change in voltage magnitude andfrequency caused by the system split on the active andreactive loads. Load 4 in Fig. 18 is in island 1 which hasa frequency above nominal and load 10 is in island 2with a frequency below nominal. The transients last forabout 10 s but the final load powers differ from initialvalues. The oscillations which persist after the fast tran-sient and are more noticeable in curves (iii) and (iv) aredue to frequency variations caused by boiler dynamics inboth islands.

To illustrate the effect of change in the system fre-quency and the governor valve set points on throttlepressure, boiler fuel consumption, and steam production,these quantities are plotted for two generators in twoislands. One of the generators is set to operate above syn-chronous speed and the other one below that speed. Fig.19 shows the result and the timing details.

The present simulator is used as a testbed to developenergy management software. The results in Fig. 20 areproduced under similar operational conditions to thoseof Fig. 17, except that the simulator is controlled by theenergy management software. The quantities whichresemble the measurements in an actual system are calcu-lated by the simulator and transmitted to the controlcomputer. These measurements are used by a state esti-mator to estimate the unmeasured quantities which arenecessary for the control software such as active powerdispatch, rescheduling, and load frequency control. Theoutputs of the control software are then transmitted tothe simulator computer.

The effect of the control software on the simulator per-formance is evident from the differences between Figs. 17and 20. First the active power dispatch has reduced theoverall generation cost by about 0.5% through eco-nomical redistribution of the load power among gener-ators. Secondly, as the governor set points are controlled

by the LFC, the system operates in the vicinity ofnominal frequency. This effect can be observed through-out the test but more clearly so when the islands are setto be resynchronised. In Fig. 17, lack of control over the

315.0315.0315.0

1.41.21.0

312.5« 312.5S 312.5.2 0.4§ 0.45* 0.5

310.0310.0310.0-0.6-0.30.0

( i )

(iii)

00:01:30 00:16:30 00:31:30 00:46:30 01:01:30time (8th February 1985)

Fig. 20 Simulator response to system transients listed in Table 1 whencontrolled by EMS(i) generator Gl speed, rad/s(ii) generator G4 speed in rad/s(iii) generator G6 speed, rad/s(iv) generator Gl electrical power output, p.u.(v) generator G4 electrical power output, p.u.(vi) generator G6 electrical power output, p.u.Base for power is 100 MVAScales are in order of curve number, i.e. the top scale is for curve 1 etc.

system, makes the resynchronisation impossible. Whenthe LFC controls the governor valve positions, however,the resynchronisation takes place according to the planand the system recovers from the imposed transients andsettles into a new steady state.

7 Conclusion

The paper has presented a real-time simulator which isspecifically designed to simulate a power system for time-scales in excess of 1 s. The simulator, however, remainsstable for timescales less than 1 s, but as the short timeconstants for some of the system elements are assumed tobe zero, the results produced for fast transients of lessthan a second are less accurately represented. Thedetailed modelling of power system elements togetherwith the algorithm to solve the mathematical models arepresented.

Other supporting software which is used by the simu-lator are referred to and discussed. Models for steady andslow transient protection relays, equivalent telemetrymeasurement, and synchronoscope device are also dis-cussed.

Using a test system with multibusbar nodes, it isshown that the simulator remains stable when the systemis subjected to any practical transients and step changesand the results obtained have an acceptable accuracy.The results show that the simulator can be used toprovide a testbed for energy management software. Thesimulator is also useful for operator training in the pre-sence of suitable man-machine interface facilities.

1EE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987 221

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25 PES Working Group on operator training (WG78-4): 'Bibliographyof power systems operator training resources', IEEE Trans., 1983,PAS-102, (1), pp. 233-237

26 BIGLARI, H., CASHAR, E.E., HEMMAPLARDH, K., LEE, D.K.,RAMCHANDANI, H., and SACKETT, S.A.: 'A dispatcher trainingsimulator design with multipurpose interfaces', ibid., 1985, PAS-104,(6), pp. 1276-1280

27 LOTFALIAN, M., SCHLUETER, R., IDIZIOR, D., and RUSCHE,P.: 'Inertial, governor, and AGC/economic dispatch load flow simu-

lations of loss of generation contingencies', ibid., 1985, PAS-104,(11), pp. 3020-3028

28 GARY, D.H., and SANGIOVANNI-VIGENTELLI, A.L.: 'A surveyof third-generation simulation techniques', Proc. IEEE, 69, (10), Oct.1981, pp. 1264-1280

29 HAPP, H.H., POTTLE, C, and WIRGAU, K.A.: 'Future computertechnology for large power system simulation'. IFAC, 15, Aug. 1979,pp. 621-629

30 MAGEE, D., FLYNN, F., WEHLAGE, P., WRIGHT, J., andLEHMAN, R.K.: 'A large two computer dispatcher training simula-tion', IEEE Trans., 1985, PAS-104, (6), pp. 1433-1438

31 HEMMAPLARDH, K., HAMMOND, R.A., CATE, E.G., andSACKETT, S.A.: 'Applications of dynamic models in dispatchertraining simulator and in other system dynamic performancestudies', ibid., 1985, PAS-104, (6), pp. 1349-1355

32 SAIKAWA, K., GOTO, M., IMAMURA, Y., TAKATO, M, andKANKE, T.: 'Real time simulation system of large scale powersystem dynamics for a dispatcher training simulator', ibid., 1984,PAS-103, (12), pp. 3496-3501

33 LATIMER, J.R., and MASIELLO, R.D.: 'Design of a dispatchertraining system'. 1977 Power Industry Computer Conference,Toronto, Ontario, Canada, 24th-27th May 1977, pp. 87-93

34 JOETTEN, R., WESS, T., WALTERS, J., RING, H., andBJOERNSSON, B.: 'A new real-time simulator for power systemstudies', IEEE Trans., 1985, PAS-104, (9), pp. 3496-3501

35 JANOFSKY, E.B., and DURKIN, C.J.: 'Dispatcher simulator train-ing at consolidated Edison', ibid., 1981, PAS-100, (7), pp. 3213-3216

36 ARRILLAGA, J., GRAHAM, J.C., and HISHA, H.: 'Micro-processor-controlled HVDC simulator', IEE Proc. C, Gen., Trans. &Distrib., 1984,131, (5), pp. 197-203

37 DINELEY, J.L., and FENWICK, P.J.: 'The effects of prime-moverand excitation control of the stability of large steam turbine gener-ators', IEEE Trans., 1974, PAS-93, (5), pp. 1613-1623

38 PETERSON, N.M., and SCOTT MEYER, N.: 'Automatic adjust-ment of transformer and phase-shifter taps in the Newton powerflow', ibid., 1971, PAS-90, (1), pp. 103-108

39 ADIBI, M.M., HIRSCH, M., and JORDAN, J.A.: 'Solutionmethods for transient and dynamic stability', Proc. IEEE, 1974, 62,pp. 951-958

40 DOMMEL, H.W., and SATO, N.: 'Fast transient stability solu-tions', IEEE Trans., PAS-91, 1972, pp. 1643-1650

41 FONG, J., and POTTLE, C : 'Parallel processing of power systemanalysis problems via simple parallel microcomputer structures',ibid., 1978, PAS-97, (5). pp. 1834-1841

42 OHYMA, T., WATANABE, A., NISHIMINA, K., and TSURUTA,S.: 'Voltage dependence of composite loads in power systems', ibid.,1985, PAS-104, pp. 3064-3073

43 KENT, M., SCHMUS, W., MCCRACKIN, F.A., and WHEELER,L.M.: 'Dynamic modelling of loads in stability studies', ibid., 1969,PAS-88, pp. 756-763

44 BERG, G.J.: 'Power system load presentation', Proc. IEE, 1973,120, pp. 344-348

45 RIBERIO, J.R., and LANGE, F.J.: 'A new aggregation method fordetermining composite load characteristics', IEEE Trans., 1982,PAS-101, pp. 2869-2875

46 ILICETO, F., CEYHAN, A., and RUCKSTUHL, G.: 'Behaviour ofloads during voltage dips encountered in stability studies. Field andlaboratory tests', ibid., 1972, PAS-91, pp. 2470-2479

47 DURBECK, R.C.: 'Simulation of five load shedding schedules', ibid.,1970, PAS-89, pp. 959-966

48 ANDERSON, P.M., and NANAKORN, S.: 'An analysis and com-parison of certain low-order boiler models', ISA Trans., 1975, 14,(1), pp. 17-23

49 AHNER, D.J., BEMELLO, F.P., DYER, C.E., and SUMMER,V.C.: 'Analysis and design of controls for a once-thru boiler throughdigital simulation'. ISA Proceedings of 9th National Power instru-mentation Symposium, 1966, pp. 11-30

50 HERGET, C.J., and PARK, C.V.: 'Parameter identification andverification of low order boiler models', IEEE Trans., 1976, PAS-95,pp. 1153-1158

51 FARMER, E.D., DUCKWORTH, S., and LAING, W.D.: 'Powerplant response identification for on-line system control'. 7th PSCCConference, Lausanne, Switzerland, 1981, pp. 959-969

52 LAUBLI, F., and FENTON, F.H.: 'The flexibility of the super-critical boiler as a partner in power system design and operation.Parts 1 and 2', IEEE Trans., 1971, PAS-90, pp. 1719-1733

53 YOUNG, C.C.: 'Equipment and system modelling for large scalestability studies', ibid., 1972, PAS-91, pp. 99-109

54 TEWARSON, R.P., and CHEN, D.Q.: 'A method for solving alge-braic systems consisting of linear and nonlinear equations', Int. J.Numer. Methods Eng., 1985, 21, pp. 1577-1581

55 TINNEY, W.F., BRANDWAJN, V., and CHAN, S.M.: 'Sparsevector methods', IEEE Trans., 1985, PAS-104, pp. 295-301

222 IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987

9 Appendixes

9.1 Curren t in phase a of generator gFrom Fig. 13 the current in phase a of generator g can becalculated from

Ia = (Eg - (145)

substituting for Ia from eqn. 2, and for Eg and V{ fromeqns. 86 and 87, respectively, results in

q ,\(cos Sg + j sin S9) - ebi -jfbi}Yg (146)

Substituting for Yg from eqn. 101 and separating the realand imaginary parts results in eqns. 4 and 5.

92 Typical model parametersTypical parameters used for the models developed inSection 2 are:

(a) generatorsRa = 0.0025 p.u. Xd = 0.8 p.u.H = 4 s D = 0.2 s/rad

(b) governors and AVRsTc = 0.3 s KA = 10.0

T'd0 = 5 s K = 0.06 MW s/rad(c) steam turbines

TCH = 0.3 s TRH = 7.0 sTco = 0.2 s PHP = 30%FIP = 40% FLP = 30%

for hydro turbines see Reference 10(d) boiler

Tv = 25 sK2 = 0.095CB = 200 sT, = 70 s

TD = 60 siCx = 0.89Kc = 0.02TB = 90 s

K3 = 0.5

(e) For line and transformer parameters see Reference24.

9.3 Admittance of a transformerFrom Fig. 10, the admittance of transformer k without aphase shifter is

Yk = Y(cos 9k -j sin 9k) = Gk -jBk

where

0fc = tan"1 (xjrk)

Y=l/(rk+jxk)

For a transformer with a phase shift of <pk

Yu = Y{cos (0k - 4>k) -j sin (9k - cj>k)}

YJt = Y{cos (6k + 4>J -j sin (9k + <j>k)}

(147)

(148)

(149)

expanding the sine and cosine terms in the above equa-tions and combining the result with eqn. 147 results in

Yu = tk(Gk cos <l>k + Bk sin <£k)

-jtk{Bk cos 4>k - Gk sin (f>k) (150)

Substituting eqn. 56 in the above equation yields eqn. 54.

9.4 Generator outputThe armature resistance of generators are included in thedeveloped simulator. The generator power output istherefore of the form

1

+(151)

Substituting for V from eqn. 86 results in

Pe = — 2 ieb(ra cos SE + xd sin dE)

+ fb(ra sin dE - xd cos <5£) + (e2 +f2)ra} (152)

9.5 Power flow equationsSubstituting for Yk from eqn. 44, and for V{ and Vi fromeqns. 138 results in

Pn = Ikehi-rkfbi (153)

(154)

where

I^ErjG. + EijB,

l[ = E\jBk-E\jGk

E\j = tk ebj — T'kfbj

E\j = tkfbi — TjJbj + Tlkebj

The power flow from node; to node i is

Pij ~ Ik ebj ~ Ikfbj

Qji = Irkebj + Ikfbj

where

(155)

(156)

)i —fbj ~ Tlfhi — Tkebi

IEE PROCEEDINGS, Vol. 134, Pt. C, No. 3, MAY 1987 223