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54Mathematics and Computers in Simulation XXV1 (1984)54-62

North-Holland

MODELLING AND SIMULATION OF MULTIAREA POWER SYSTEM LOAD-

FREQUEKY CONTROL

L. BASAfiEZ, J. RIERA and J. AYZA

Institut de CibernGtica, Universitut PoMcnica, Barcelona - 28, Spuin

A comparative analysis of stiff and elastic tie-line models used to simulate load-frequency control

of two-area interconnected power systems is performed. Then, after the extension of the models

to multi-area case, the lack of asymptotic stability of elastic tie-line models for certain

interconnected structures is shown. To overcome this situation, asymptotically stable compound

models with both stiff and elastic tie-lines are proposed.

1. INTROOUCT ION

load-frequency control of interconnected powersystems has been the object of a large number of

studies. Several dynamic models have been suggested

to analyse the behaviour of that kind of systems. In

general the models deal with phenomena that extend

their action from some seconds to few minutes and

assume that the load-frequency and reactive-voltage

control problems are decoupled in order to conjoin

reasonable precision and low complexity [l], 121.

Some models are based on the assumption that all

system areas have the same frequency, the inter-

connection between areas being stiff (31. These

models stand for the power interchange of each area

as the balance of all its tie-lines.

CXher models [4], [5], widely accepted, show

explicit!y the power interchange through each tie-line.

In these models, each area has its own frequency

(elastic interconnections).

This paper presents, firstly, a comparative analysis of

the behaviour of these two kinds of model for a two-

area sygtem. The analysis, via hybrid computer simul-

ation, leads to a clear interpretation of the relations

between the responses of the models.

Next, after the extension of the models to multiarea

case, it is shown that the model with elastic tie-lines

is not adequate to represent multiarea systems with

specific interconnected structures due to the inherentlack of asymptotic stability of the model in those

cases.

This consideration leads us to propose asymptotically

stable compound models with both stiff and elastic

tie-lines. Several possible configurations and a de-

tailed block diagram of the compound model of a

three-area system are also presented.

2. TWO-AREA MODELS

2.1. Model with stiff tie line.

This model is based on the assumption that trans-

mission lines within each individual control area, and

tie lines between areas, are completly stiff. Then, the

whole system can be characterized by a single

frequency. That is, all the generators of the system

swing in unison.

The real power deviation, A PIi, of the interchange

between area i and the rest of areas in the system, canbe obtained from the dynamic equation of the

generators and the fact that the overall exchange

balance between areas must be zero.

The model proposed in ref [3] can be improved by

introducing the load-frequency characteristic of the

areas in the dynamic equation of the power system and

representing in more detail the speed governor and the

turbine-generator of the power plants. Then, assuming

neglected line losses, the deviation of interchanged

real power can be written as

A P&-h P12=(A PC+A PDlxl- c+(J PG2-A PD2b - BAF

where d PGi is the incremental power generation of

area i, A PDi is the increment of load consumption in

the area, and ~1 6are coefficients given by

Hl.Prl

a=

Hl.Pr l+H2.Pr2

8 = 01a(D1+02)

where Hi is the equivalent inertia constant of

machines in area i, Di is the area load-frequency

characteristic and Pri is the rated area power.

The system frequency will given by

A F(s) KP=------

nn=2

!;( APGi- A PQ)

i=il+sTp

(1)

where Tp k f2Hl.Prl+2H2.Pr2 1

F+ D1+D2

and Kp 2 .-

01+D2

F* being the rated frequency. l

A block diagram of the two-area system model is

shown in figure 1, where blocks and nomenclature have

been taken from reference [C] to facilitate com-parison of models.

033%4754/84/$3.00 10 1984, IMACS/Els&er Science Publishers B.V. (North-Holland)



L Basaikzet al. / Power system control

Cocl~rdl~r Speed governor Turbm-Generatu I

Power System 1.2

Speed gove#no r tr he -Gene roIor

Figure 1: Block diagram of two-area system model with stiff tie-line.

2.2. Model with elastic tie-line.

The! two-area system mod81 with elastic tie-lines (fig.2) is based on the assumption that transmission lineswithin each individual control area are strong in

relation to ties between areas. So, a whole area can be

characterized by a single frequency. This implies that

where T12* k24Vll iV2l

. cas(6 1”“62”)

X12

is the synchronizing coefficient or electrical stiffness

of the tie-line; X12, its reactance and Vi=1 Vi1 eJ&l the

bus voltage of the line terminal i.

generators belonging to an area swing in unison but notnecessarily with generators of the other area. The elastic tie-line model improves the stiff tie-line

model because it supplies the individual frequency ofNeglecting l ine 10!3swes, the inc rementa l tie-line power, each area. Nevertheless transmission lines within

API, can be written as areas remain stiff.

API12 = Tl2* VA fldt - J A f2dt)

APD,(4

Controlter Speed Governor Turbine-Generator

Codroller

. 82 -

‘*%2

t Speed GOV~~I-KWR2 APD+sI

I - __. --F---- -

Figure 2: Block diagram of two-area system model with elastic tie-line.



56 L. &satlez et al. / Power systeM ontd

2.3. Comparative analysis of the models.

The extended use of both models makes interesting thecomparison of their dynamic responses. In this way thecomparison of the frequency of both models will bespe&ally relevant. _

From (l), the stiff tie-line model frequency can beexpressed by

AFKP

= - 1 (APGl+APG2)-(APDl+A PD2))l+sTp

The frequency of thie elastic tie-line model is [4] 2.4.1. Two identical area system response.

AFi zKPi

(A PGi-A F’Di-A PIi) i = 1,2l+sTpi

Defining a weighted average, AFa, of both indiviaualarea frequencies of the elastic model:

$‘F, 2 ?.IAFl+x 2 AF2 x1+x2 = 1 (2)

and assuming the same instantaneous pc>wergenerationin both stiff and elastic models, it follows that

i?F = AF, (3)

Q,vhen h 1 = _-

Kp2(1+Tpld

Kpl(l+Tp2&Kp2(1+Tpl~)

a 2=

Kp+l+Tp2”)

Kpl(ltTp2s)+Kp2(1+TpIs) (4)

Therefore E_F can be expressed in terms of hF1 end

4F2 by means of a dynamic weighted average, theweighting factors being a function of the Laplacetransform variable s.

In tne case that Tp >jlfactors will be reduce a to

and T,$>l, the weighting

(5)

Analoglously, for Tpl<<l and Tp2g< 1, we would have

Kp2Al= _-

Kpl+Kp2

KPlx2= --

KPltKP2

(6)

Expre&ons (2) to (6) show that tha fraquancy of thastiff model can be interpreted as standing for theaverage behaviour of the different freqU?nCieS Of the

system.

2.4. Simulation results.

With the aim of comparing their behaviour and inorder to verify the above relation betweenfrequencies, simulation of the two modal8 of two-areasystem has been carried out, Rerrultr of two ca8e8 arc)presented: identical areas and different araaa.

For this case, data have been taken from (41, and thefollowing parameter values have been choosan

$=Dl+l/Rl pu MW/HZ Kll=K12=0,8

Q=D2+l/R2 pu MW/HZ 6” = SO Hz

APROG@ PROG2 = 0,Ol pu MW

Figure 3 shows the frequency and tie line powerresponse of the two models for a step Load of 0,Ol puMW applied to area 1. The same figure shows AFacomputed from (2) with X 1=X2=0,5. It is verified thatAF and 4Fa match quite well

Figure 4 shows the response of the model to a 0,Ol puMW step load in scheduled net interchange. It can beagain verified that AF and AFa have the same timeevolution.

2.4.2. Two different area system response.

In this case the system data values are the same as

before excepting the rated power and the area load-frequency characteristic of area 2, which are now

Pr2=1000 MW, D2=16,66.lO-’ pu MW/HZ.

Simulation results are shown in figures 5 and 6 for

respectively 0,01 p.u. MW step load and scheduledinterchange increases. Again A Fa, computed now with“1 = l/3, shows the same bahaviour asA F.

213 and x2 =

Summarizing, though the behaviour of both models

seems quite different, the frequency of the stiff modelis the average of individual frequencies of the area8 ofthe elastic one. The oscillations in the area frequency

response are compensated in the average which givesthe same result as the stiff model,

These conclusions, a8 it will be seen later, areparticulary interesting in tha interpretation of theresponse of compound models of multi-area systema,



L Bade2 et al. / Power system control 57

a)AWbltl I

API (hr(Wpu)I

Figure 3: Two identical area system. Frequency and tie-lines powor responses tc; step loaddisturbance in area 1.

a) Elastic model

al

b) Stiff model

h)

5 10 15

-t-

t ISI

AF IHz)

Figure 4: Two identical area system. Frequency and tie-lines Fewer responses to a stbp in theachedulad power interchange.

a) Elastic model b) Stiff model



L Basdiez et 01. / Power system amtroi

[email protected]

- 0.016-.

u

Figure 5: Two different area system. Frequency and tie-lines power responses to a loaddisturbance in area 1.

a) Elastic model b: Stiff model

Af’1 (MWpu!

1

Figure 6: Two different area system. Frequency and tie-lines power responses to a step in the

scheduled power interchange

a) Elastic :n.odelb) Stiff model



L Basa~~ezt al. / Powersystem control 59

3. MODELS FOR MULTI-AREA SYSTEMS

3.1. Ext8neion of the mock18 to multi-area case.

The models described in the preceding paregraph caq

be euily extended to include multi-area cases. TheWt tie-b9 power incre4nent of area i of a muiti-arearyetommodelled by atiff inter connection will be

APQ = (Affij- APDi)-ai Ei(APGi-APDi) -SiAF

whereq =Hi.Pq

r Wi.Pri)$a=1 8 i=D~~iCDii Cfi i=O

i

The p8r8metOr8 p 8d Kp will now be eXpI%?fBed by

c (OHi.Pri)

A1 ’

“=;a CD; KpiA

i c Dii i

Figure 7 shows the part of the block diagram cor-

responding to tranamisaion and tie-lifU38 of 8 thrce-

area 8y8t8m with rtiff interconnection!&

APD3M

I

Figure 7: Block diagram of tha stiff tie-line model for

8 multi-area power syattsm.

The exteneion of the elastic model is also straight-

forward. The net incremental power interchange of

area i can be written, as in the two-are8 case, by the

sum of the incremental power through the tie-lines of

that area, that is API i - c {T***( /Afidt-/A f*dt) }

j‘I J

2ll I Vi I I v j Iwith Tij* = -_.-.- .

xij

COS(di*C-6 j”)

Figure 6 shows the part of a block diagram of the

elastic model corresponding to the tie-lines inter-

connecting area 1 with areas 2, 3 and 4 of a power

system.

Unlike the stiff model the elastic one supplies the

individuai power through each tie-line and not only the

net power interchange of each area.

AF, Is)

Figure 8: Block diagram of the elastic tie-line model

for 8 multi-area power system.

3.2. Stability problems

As has b88n pointed out, the elastic model works

properly in the two-area case giving more information

than the corresponding stiff model. Nevertheless, in

the m&i-area case, and with certain tie-line con-

figurations, a isck of asymptotic stability can appear.

For instance, the simulation of a three-area system

model with elastic interconnections in a triangular

configuration, Fig. 9, displays, even in the absence of

di8tUrbanC8S, 8 continuously incressing interchange of

power, but keeping up the area balance nearly equal to

the net. scheduled power interchange and making thefrequency errors practically null.



60 L Basafiez et al. / Power system control

Figure 9: Three area system in triangular con-figurhtian

The Zlastic model is then useless for simulation inthese cases, its undesirable behaviour being due to theexistence of at least a zero eiqenvalue in the system

matrix. This situation and the ineludible presence ofamplifier offsets in analog simulation or numerical

noise in digital simulation produces that increase onthe power tie-line. The zero eigenvalue is due to theinterconnection configuration and the way in whichthe tie-lines are modelled.

The stability analysis of this system can be performed

defining the state vector:

2 = [ .1Fl, ‘1F2,A F3, APIl2, API23, APII3, APCI, APC2,

L”, C3,A XE1,1IXE2, nXE3, APG1, (5PG2, APG3]T

Rows 4, 5 and 6 of A system matrix will be

T12* -T12* 0 0 . . . 0

0 T23+ -T23* 0 . . . 0

TI3* 0 -T13* 0 . . . 0

leading matrix A be singular, independently of Tij*

values. Then, this system has at least a zero eigen-value and is not asymptotically stable.

Multi-area elastic models can have, or have not, zero

eigenvalues due to their tie-lines configuration. Figure10 a) shows some examples of multi-area systems that

could lead to asymptotically stable models. In opposi-tion the mode!s of Figure 10 b) have at least a zero

eigenvalue because of the existence of closed paths

through the tie-lines and areas and, then, are not ableto be used as models for computer simulation.

3.3. Compound models

The la:!-. of asymptotic stability of certain elastic tie-

line models can be handled in different ways. Onepossible solution is to reject the simplificative hypo-thesis assumed in settling the above models, makinguse of more detailed and complex representat ion of

the power system [6]. This solution, feasible in somecases, will be in general expensive and time-consuming.

Because of this, we propose the adoption of “corn-pound’ models to simulate interconnected systems

when the corresponding elastic models are notasymptotically stable.

A l A l

\

4

88

2 4

W

Al A2

2%L 5

Al A2

x x% A3

4 " 2

583& A3

Figure 10: Some multi-area power system con-figurations.a) Elastic model allowedb) Elastic model not allowed

In a compound model several control areas are groupedinto %ectors”, the areas in a sector having the s a m e

frequency and tie-lines interconnecting those areasbeing considered stiff. On the contrary, tie-lines inter-connecting areas of different sectors are modelled aselastic. Note that each area remains a control area inspite of belonging to a sector.

To assume stiff tie-lines in lieu of elastic ones is

reasonable when frequency differences betweenseveral areas are of the some order of magnitude asthose existing within areas. In these cases it is

realistic to consider that a serious error is notcommited considering those areas forming a sectorand having the same frequency.

When previous assumption does not apply, compound

models can still be used, if a loss of information of thesystem behaviour is accepted. Then, as it has been

show 1 in paragraphs 2.3 aQd 2.4, a sector frequencycould be interpreted as the weighted average of the

frequency of the areas belonging to the sector.

In order to ensure compound asymptotic stability, tie-

lines interconnecting sectors must not form closed-paths. (Fig. 11).

A compound model has been successfully applied todescribe the load-frequency control behaviour of theCatalan system, composed gross0 modo by twocompanies (E.N.H.E.R. and F.E.C.S.A.) strongly inter-connected and geographically overlapped. This systemis described as two-area sector, and the remainderSpanish power companies (R.&C.) and the Frenchsystem (E.F.) are considered control areas withindividual frequencies (Fig, 12).

A detailed compound model block diagram of a three-

area system in a triangular configuration has been

shown in Figure.13. Two of the areas are stiff inter-connected while remainder tie-lines ar e elastic.



L Baudez et al. / Power system control 61

A\ *:

8%A'r' A'j

a)

b)

Figure 11: Some multi-area power system configura-tions.a) Compound model allowedb) Compound model not allowedSupraindex identifies sector, subdindexarea.

Figure 12: Compound model structure used toanalyze the Catalan Power System.

4. CONCLUSIONS

Load-frequency control models of interconnectedpower system with elastic tie-line representation are aclear improvement upon models with stiff tie-lines. In

fact elastic models provide more detailed informationabout the system behaviour because they supply thetime evolution of the frequency of each individualcontrol area and the power interchanged through eachtie-line.

Nevertheless elastic models can be unsuitable whenused in computer simulation of certain multi-areapower systems. Closed tie-line paths lead to theexistence of at least a zero eigenvalue making themodel not asymptotically stable.

Models with stiff tie-lines do not present that stabilityproblem and, although they do not allow to know someindividual system variables, they describe properly the

I!lobal system behaviour, since the frequency suppliedy the model can be interpreted as a dynamic

weighted average of the frequency of the differentareas.

In order to avoid stability problems, compound modelswith both elastic and stiff tie-lines can be used. Inthese models several areas having nearly the samefrequency are considered interconnected via stiff tie-lines and grouped into a sector. Thus the modelsupplies the frequency of each sector as the dynamicweighted average of the frequency of the areas

belonging to it.

Compound models have been successfully applied to

the analysis of the load-frequency control strategy tobe used in the Catalan power system.

REFERENCES

QUAZZA, G., (1977), Large scale controlproblems in electric power systems, Automati-ca, Vol. 13, pp. 579-593.

STERLING, M.J.H., (1974), Power SystemControl, TEE Cantrol Engineering, serie 6.

ERBACHER, W., (1965), Theoretic bases for thestudy of load-frequency control by analog

computers. Technical notes num. 296. Regul-

ation working group of UCPTE.

ELGERD, O., FOSHA, C.S., (1970), OptimumMegawatt-frequency control of multi-areaelectric energy systems. IEEE T. PAS-89, num.

4, pp. 556-563.

ELGERD, O., (1971), Electric energy systemstheory: an introduction, McGraw-Hill.

DAVISQN, E., TRIPATHI, N., (Feb. 1980), Des-centralized tuning regulators: an application tosolve the load end frequency control problemfor a large power system. Large Scale Systems,

Vol. 1, num.1. North-Holland Publishing

Company.



62 L. asafiezet al. / Power syste.mcontrol

4

1 1-- _l+sT~~ - l + s T T ,

1

“2

Figure 13: Block diagram of the compound model of a three-area system in triangular configuratron.

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