Reactive power compensation by a linear programming technique

Comput. & Elect. Engng Vol. 8, No. 3, pp. 175-186, 1981 Printed in Great Britain.

0045-7906/811030175--12502.00/0 © 1981 Pergamon Press Ltd.

R E A C T I V E P O W E R C O M P E N S A T I O N BY A LINEAR PROGRAMMING TECHNIQUE S. S. CHOi,J. B. X. DEVOTTA,K. C. KoH and E. LtM Department of Electrical Engineering,National University of Singapore, Singapore 0511

(Received 29 July 1980; receivedfor publication 22 May 1981) Abstract--This article presents a method of optimizingthe reactive compensation used in power systems to establish acceptable voltage profiles during period of abnormal loads and during foreseeable contingencies. The system equations, which are nonlinear, are first approximated to a linear form, and then linear programmingtechnique is applied to obtain the quasi-optimalsolution. An iterative procedure is then used to obtain results of acceptable accuracy. The main features of the proposed method are that both inductive and capacitative compensation is optimized and that the busbars where compensation is applied, can be selected to suit the users' operating constraints. 1. INTRODUCTION

An essential feature in the planning of a power system is the provision of reactive compensation for control of voltage profile. During periods of heavy load and at very light loads, some of the voltages in the system could fall outside acceptable limits. Also, during contingencies, such as the loss of a major transmission line or a generating unit, the busbar voltages could attain unacceptable values. Voltage control is usually achieved by means of tapchangingunder-load (TCUL) transformers, and switching in/out of shunt reactors and capacitors. The need for proper planning of both inductive and capacitative reactive compensation is highlighted when considering supply networks like that of Singapore. The local primary transmission network is entirely by h.v. cables. The reactive generation exceeds that of the load, especially during light loads, and shunt reactors are switched in to prevent excessive voltage rise. In the past, trial-and-error methods for solving the reactive compensation problem have been used. First, the actual voltage profile during any contingency is evaluated by an a.c. load flow program. Then a reactive compensation pattern is selected based on engineering judgement and past experience. Unlimited reactive generation/absorption is specified at some selected buses to yield the acceptable voltage profile pattern. The pattern of reactive compensation is then refined and minimized by an iterative method. This method is quite tedious and its effectiveness depends on the engineering judgement and skill of the user. Recently, there has been several papers in which a more analytic approach has been utilized to solve the problem [1-4]. Maliszewki[1] has proposed a method using linear programming and a.c. load flow. The limitations of Maliszewki's method are that only capacitative compensation has been considered and that provision for TCUL transformers in the network has not been included. A method based on linear models, discrete variables and selective iteration was proposed by Pretelt[2]. He uses a modified bus impedance approach and the best busbars are selected by sensitivity tests. Happ et aL [3] have presented a reactive optimization technique similar to that of Maliszewki[1], together with an assessment of the dynamic system performance. Although in[3], inductive compensation has been included, this is primarily for obtaining feasible solutions and hence the solution obtained is not optimal. In this paper, the method of[l] is reexamined with a view to including both capacitative and inductive compensation and simplifying the solution technique. Also, the possibility of using TCUL transformers to improve the system voltage profile is first explored before any nodal reactive compensation is attempted. 2. BRIEF DESCRIPTION-OF THE PROPOSED SOLUTION TECHNIQUE

The various steps involved in the proposed method are shown in Fig. 1. A brief description of the function of each block of Fig. I is as follows: CAEE Vol. 8, No. 3---B

175

176

S. S. CHO} et al, STApT

I

)

i

E£TABL/SH NO,OMAL VOLThGE LEVEL BY A/C LOAD FLOW

I CONSTRUCTCONTINGE,CYM~TR, ~USX ADM,T;ANCC 1. l

!B I

CONTINGENCY VOLTAGE LEVELS BY A C LOAD FLOW

YES

......

NO

NO

{

TAP SETTING ADJUSTMENT

1

1

A/C

1 J

LOAD FLOW

1

~S NO NO

L

D

I SELECt'ION

I

_

OF PREFISRR,ED

DETERMINE THE OPTIMAL L/NEAp

BUSES

A Q USING

PROGRAMMING

COMPENSATED VOLTAGE LEVELS BY A/C LOAD FLOW

lF

NO

YES

PRINT PESULT

5 TOP

)

Fig. 1. Flow chart schematic of proposed technique,

In Block A, the a.c. load flows program using the Gauss-Siedel iterative technique is used to determine the normal voltages of the system. In Block B, for a specified contingency (e.g. loss of a transmission line), the new bus admittance matrix is formed. Load pattern is fed in as data into the program. The voltage levels

Reactive power compensation by a linear programmingtechnique

177

during the contingency or abnormal load conductions are calculated by a rerun of the a.c. load flow program. The program then automatically uses the facility offered by the TCUL transformers existing in the network to meet the specified voltage constraints, as shown in Block C. It enters the successive stages only when all the constraints are not satisfied. In Block D, buses at which reactive compensation cannot be applied due to economic or physical reasons, are singled out. These are penalized by using a large weighting factor in the cost function to be minimized (see next section). This allows the user the flexibility of varying the location of the reactive compensation and also to obrain a realistic solution for systems having location constraints, The simplex method[5] is used, as shown in Block E, to obtain the minimum additional reactive power necessary to restore the voltages to lie within the preset bounds. Since the procedure is only approximate, the a.c. load flow is rerun to check whether the solution is acceptable (Block F). The mathematical formulation of the problem is described in the next section. 3. PROBLEM FORMULATION

The purpose of the study is to determine the minimum reactive compensation, both inductive and capacitative, required in a power system to obtain acceptable voltage profiles during abnormal conditions. For small changes of voltage magnitude, the corresponding changes in the injected nodal reactive powers can be approximated by the following expressions: n

aE,

=

xoAO

(1)

where AEi denotes the change (in p.u.) in voltage magnitude at bus i, AQi is the change in reactive power flow (in p.u.) into bus j. xii is the (i, j)th element of the network bus reactance matrix. Different interpretations can be attached to the index "j" in (1). In almost all the presently available methods on reactive compensation[l, 3], "j" has been taken to include all the n buses. In this study, however, "j" refers only to those buses where reactive compensation is to be considered. This permits the user the flexibility of not considering those buses where reactive compensation is not feasible for some specified reasons. The problem of reactive compensation is not completely described by (1). In a practical situation, the reactive compensation provided at the selected buses should be such that the resulting voltage profile lies within specified bounds. This last point is illustrated by Fig. 2. 3.1 Preliminary [ormulation Consider the case in which, during a contingency, none of the system busbar voltages exceed their specified upper bounds, see Fig. 2(a), i.e. Econ.i ~ Eu. ~for all i. In such a case it is evident that capacitative compensation has to be provided at appropriate buses to raise those busbar voltages which are less than the specified lower bound. The objective function to be

EH,

Econ , ~ •

i

} -

.4 Emin i

EH, i Enor,

~Emax, t EL, i Ecol~ , i

l

Eno6 ~1 E m a x , i

~_Ernin, i EL/i -

(a)

(b) Fig. 2(a) Case of undervoltage, (b) Case of overvoltage.

s. s.

178

CHOI

el a/.

minimized is the amount of capacitative reactive power added into the system, i.e. H

=

2

CjAQ;iwtched

j=l

where Cj is the weighing factor at bus j and AQjiispatched is the dispatched corrective reactive power needed at bus j. H therefore gives a measure of the cost involved in the compensation. For capacitative compensation,

Clearly the set of reactive compensation (AQ~isparched ) should be such that the resulting changes in the bus voltages should be greater than the minimum increase (Emi,,;) and at the same time, less than the maximum increase (E,,,,,.i ) , for all i. Translated into mathematical form,

and

see Fig. 2(a). The objective function (2) is used to provide a measure of the cost involved in the reactive compensation during a contingency. If reactive compensation is undesirable or difficult to implement physically at bus k, than C, may be assigned a relatively large value (say, 100) while each Cj(j# k) is assigned a much smaller value (say, unity). This is to ensure that AQiispatched will assume a negligible value in the final solution of the linear programming search. The above paragraphs describe the basic approach used in [ 1,3]. The dual problem of the case considered so far is the one in which, during the contingency, the busbar voltages rise, such that none of these voltages are less than the lower specified bounds, i.e. E,,, i 2 EL.i for all i. This can be solved in a similar manner to that shown in[l] or[3] except that AQ~rpnrched is now treated as inductive instead of capacitative compensation. See Fig. 2(b). It should be noted that modifications to the basic approach described above has been proposed in[3]. This is in order to overcome the convergence problem encountered in some numerical examples. Defining a new term AQP”’ which is the total additional amount of corrective reactive power required at bus j during a given contingency and ~Ql,otal I

=

AQtjispatched

+

AQP’

scheduled

J

(5)

The objective function (2) is modified to become

H

=

2 ,f,. ,:,,,I - -$

E

.

AQy"' sched”‘a’

subject to the constraints

(7) For capacitative compensation the voltage constraints are now given by

$

xii

(AQytai - AQT’ schedu’ed) 3 IE,,,,,,,j

Reactivepowercompensationby a linearprogrammingtechnique

179

and R

{AntOtal- AOnot schedu'ed)~
Xij ~--'.: j

(8)

For inductive compensation the voltage constraints are (A/.)total

Xo~-.,c j

F()not scheduled~. _< -- IErain, i I

- - .,c j

=

and n

(A {')total xij,-,~s - AQ7 °'

scheduled)

>" -IEma~.il

(9a)

=

or n

~= x0.(AQ~O,scheduled _ A()total ~1~- J//min,,I and n

xii(AQ~Ot

scheduled

AQitotal ) <~IEmax.il.

-

(9b)

=

In the objective function (6), ~ is assigned a small value (say 0.01). This is introduced solely to avoid convergence problems in the linear programming stage of the solution technique. AQ},Otscheduledshould theoretically have a weighting factor of zero[3]. Therefore the solution technique of[3] only optimizes the capacitative reactive power while inductive reactive power is introduced only to obtain a feasible solution. 3.2 Minimization of both capacitative and inductive reactive power In order to minimize both the capacitative and inductive reactive power at the same time, it is now proposed that the problem is reformulated in the following manner: H

=

~, CjAQs

(10)

j=l

where m is the total number of the bus where reactive compensation is to be considered. The voltage constraints are m

xij ' A Q i

~

IAEmin, il

and m

= x,j .AQj

laEmax.,I

(11)

with AQ~ I> 0 for all j.

(12)

In this formulation, AQj is neither capacitative nor inductive but it is treated purely as a decision variable in the linear programming stage of the solution procedure. To distinguish whether AQ~ should be added to (i.e. AQi is capacitative) or subtracted from (i.e. AQi is

180

S.S.

CHOl et al.

inductive), the net reactive power at bus j depends on the voltage level as bus j. As explained in Section 2 at the end of Block E, an a.c. load flow is performed. The voltage at bus j is then compared with its prescribed limits. Should it fall below its lower limit Er, j, then AQi is capacitative and its value is therefore added to the net reactive power injected at bus j. Conversely, if the voltage level exceeds its upper limit EH, i then AQj is inductive and is subtracted from the net reactive power at bus j, The last consideration also enables further simplification to be made in the constraint equations by assuming a common upper and lower bound for all busbars, i.e. (11) becomes

(13)

Xq Qj >1 IAEmir~, i

i

where for capacitance reactive compensation, AEmin is given by the difference between EL and Eco,,i; while for inductive reactive compensation, AEmi, is given by the difference between E , and Econ,j. See Fig. 3(a, b) respectively. In adopting (12), the dimensionality of the constraint equations is halved and the solution time on the computer reduced. Thus (10), (12) and (13) are the equations used in the final version of the reactive power compensation problem. 4. NUMERICAL EXAMPLES

The proposed technique described in Section 3 has been tested on several numerical examples from literature [6-8]. In this paper, only an example from [7] is included as most of the salient points which concern the proposed method can be adequately demonstrated. The single-line diagram of the system is as shown in Fig. 4. The power system has 14 buses with 16 transmission lines. Bus 1 is the swing-bus while buses 2-4 are voltage control buses. Nominal system loading, line and transformer data are given in Table 1. Figure 4 also shows the voltages at various buses under the nominal load condition. As the main concern of this work is on reactive compensation, only reactive power flow is indicated on Fig. 4. It will be subsequently assumed that the acceptable range of the bus voltage is between 0.95 and i.05 p.u. 4.1 Line 9 out

Inspection of Fig. 4 shows that line 9 is the most heavily loaded line in terms of reactive power flow. Hence, removal of this line from the network will most probably produce the greatest reorganisation in the reactive flow in the resulting network. As a result, the changes in bus voltages are likely to be most severe. Therefore, outage of line 9 is studied for each of the following cases: (i) Bus voltage correction without adjusting the tap-settings of the TCUL-transformers. Reactive compensation is introduced at all buses which have voltages outside the acceptable limits. Bus no. 9 is assumed to be a preferred bus where reactive compensation is not permitted. (ii) Bus voltage correction by first adjusting the tap settings of TCUL-transformers on lines 10 and 13 followed by reactive compensation at the remaining buses which have voltage values outside the acceptable limits. (iii) Similar to case (ii) except that reactive compensation is only introduced at two selected buses, namely buses 9 and 13. Econ i

EH, i

F AErnin

EH, l

T

Enor, Enor,~ ...................................................................................

EL, i

Eco.,

A Emin

EL , (a)

(b)

Fig. 3(a) Voltage constraint--~Qi capacitative, (b) Voltage constraint--~Qi inductive.

181

Reactive power compensation by a linear programming technique tO

9

2

(~)-1" "

18.6~6

®

5

135

9.8S 7

(I.02)

/:

I

l

99

(I 0"0?)

(?D t.'99

t /.O02)

®

®

•

t:

(/-009)

®

@

• 963 )

,~ I r" e<

®

@

/"/@f

6

=

_~I@

I.'I

(I)

!

-I

.IL. 19

13

®

@ •

-I

(

]

LINE

NO.

BASE

CASE

R EA C 71VE (~

fCUL

_ 1.35

®

28.79

26.82

f 09"/2)

(0.967)

a(~

(I.016]( l D

t~

_ II/4

/8!

1.73

{ ~.04)

4

--

(102)

o BUS

VOLfAOE

POWER

FLOW

IN IN

RU. M VA r

TlCtANSF'ORMER , TAP- RATIO

o : b

a:b

Fig. 4. Load-flow results--nominal load conditions.

Column 3 of Table 2 shows the bus voltages after line 9 is removed from the network. Buses 9, 10, 11, 13 and 14 have voltages below 0.95 p.u. Tables 2-4 show the optimal solutions to the three different cases considered above. The new TCUL tap settings, amount of reactive compensation required and where it is introduced for each case is included in each Table. Note that: (a) Without changing the tap setting of the TCUL transformers (case (i)), the minimum total reactive power required is 75.8 MVAr. However, if TCUL settings are adjusted prior to the introduction of reactive compensation, the total VAR requirement is reduced to 70.43 MVAr. Also, for this case the total number of buses where reactive power needs to be introduced has been reduced from 4 to 3. (b) For case (iii), Table 4, it is seen that the total reactive power required is increased to 76.56 MVAr. This is as expected since the number of buses where reactive compensation is permitted has been reduced.

182

S.S. CHol et aL Table l(a) Transmission line and transformer data

LINE

SERIES IMPEDANCE

TAP LIMITS

SHUNT ADMITTANCE

Tmin

Tma x I

1

0.067+j0.20

0.0+j0.042

2

0.067+J0.20

0.0+j0.042

3

0.000+j0.12

O.O+jO.O00

&

0.350+j0.42

O.O+jO.O07

5

0.067+j0.20

O.O+jO.Oa2

6

0.035+j0.10

0.0+j0.021

7

0.000+j0.12

O.O+jO.O00

8

0.350+jO.&2

0.0+,i0.007

9

0.035+j0.I0

0.0+j0.021

I0

0.000+j0.12

O.O+jO.O00

-

t

-

o.9

i i

i .!

!

11

0.350+j0.42

O.O+jO.O07

12

0.067+j0.20

0.0+j0.042

13

0.000+j0.12

O.O+jO.O00

l&

0.350+jO.a2

O.O+jO.OO7

15

0.034+j0.I0

0.0+j0.021

16

O.03&+jO.10

0.0+j0.021

i

0.9

I .!

I

1

Table l(b) Input data for nominal load conditions

BUS NO

VOLTAGE (p.u.)

GENERATION MW

MVAr

LOAD MW

MVAr

Qmin

Q max

MVAr

MVAr

I

1.04+j0.0

2

1.02+j0.0

200.0

150.0

100.0

3

1.00+jO.O

0.0

0.0

1O0.0

4

1.02+j0.O

200.0

150.0

100.0

I

5

200.0

56.2

100.0

6

0.0

0.0

0.0

0.0

7

O.O

O.O

0.0

0.0

8

0.O

O.0

0.0

0.0

9

O.O

0.O

50.0

20.0

10

O.O

0.O

5O.O

20.0

11

O.O

O.0

25.0

10.0

12

O.0

O.O

25.0

0.0

13

O.0

0.O

50.0

25.0

14

0.0

O.0

5O.O

25 .U

~0. L)

Reactivepowercompensationby a linear programmingtechnique

183

Table2. Summaryof resultsfor case(i) t

BUS

Enor(P.U. )

ECoN(P. U. )

ECON(P.U. )

I

1.0400

1.0400

1.0400

2

1.0200

1.0200

1.0200

3

1.0200

1.0200

1.0200

4

1.0200

1.0200

1.0200

5

1.0088

1.0021

1.0058

6

1.0161

1.0261

~.~290

7

1.0065

1.0102

1.0135

8

1.0311

1.0266

1.0299

9

.9585

.9464

.9514

lO

.9642

.9498

.9570

11

.9514

.9499

.9658

12

.9634

.9674

.9690

13

.9669

.8010

.9813

14

.9720

.7433

.9500

Bus

~ Qj(MVAr)

10

5.14 capacitative

11 13

5.44 capaciZative 52.79 capacitative

14

12.44 capacitative

!

E con

- final voltage profile

4.2 Line 2 out When Line 2 is taken out from the network, it can be seen from Table 5 that the voltage at bus 7 is raised to 1.065 p.u. while the voltages at buses 9 and 10 are lowered to below 0.95 p.u. As explained earlier, previous methods proposed by other researchers will be unsuitable to solve problems of this type, i.e. where there is a mixture of over- and under-voltages. Using the computer program based on the new formulation described in Section 3, it is found that 7.96 MVAr inductive compensation is required at bus 7 while 11.34 and 33.82 MVAr capacitative compensation are required at buses 9 and 10 respectively. Therefore, the total amount of reactive power required is 52.12 MVAr. 5. CON~2LUSIONS A technique has been proposed whereby the minimum amount of reactive compensation, both capacitative and inductive, can be determined to establish acceptable voltage profiles during period of abnormal load or foreseeable contingencies. The combined use of linear programming method and a.c. load flow computers have been shown to give satisfactory solution to the problem. Further refinement and extension of the proposed technique is possible in the following areas. With the present method the decision variable AQj has been assumed to be continuous,

184

S.S. C8ol et aL

Table 3. Summary of results for case (ii) THE TAP SEFTING OF TCUL TR&NSFORMER

**

THE OFF-NORMINAL TURNS RATIO OF f/P OF LINE (10) IS SET TO .98750 OFF-NORMINAL TI~NS R~TIO OF T/F OF LINE (13) IS SET TO 1.00625 THE

AFTER CHANGING THE TAPPING OF T/F BUS 5 6 7

Econ(P.U. )

9 10 11 12

.0027 .0166 .0108 .032a .9467 .9502 .9533 .9624

13 I~

.801 o .7433

w

BUS

Enor(P.U. )

1 2 5 4 5 6 7 8 9 10 11 12 13 14

1.0400 1.0200 t.0000 1.0200 1.0088 1.0161 1.0065 1.0311 .9585 .9642 .951& .9634 .9669 .9720

Bus 9

E

con

Econ( p.u. ) 1.0400 1.0200 1.0000 1.0200 !.0021 1.0261 1.0102 1.0266 .9464 .9498 .9499 .Q67q .8010 .7433

5.19 capacitative

13

b2,81 capacitative

14

12.43 capacitative

-

con

-

1.0adO 1.0200 1.000u 1.0200 1.0052 1.0185 1.0108 1.0324 .9538 .9552 .9533 .9634 .9813 .9500

a ~j(MVAr)

voltage profile after tap-changing

h

E

Scon(P.U. )

final voltage profile

S. S. Cnol tt al.

185

Table4. Summaryofresultsforcase(iii) I

BUS

Enor(P'U')

Econ(P'U')

Econ(P'U')

1

1.0400

1.0400

1.0400

2

1.0200

1.0200

1.0200

3

1.0000

1.0000

1.0000

4

1.0200

1.0200

1.0200

5

1.0088

1.0001

1.0045

6

1.0161

1.0261

1.0180

7

1.0065

1.o102

1.0108

8

1.0311

1.0266

1.0324

9

.9585

.9464

.9518

10

.9642

.9498

.9538

11

.9514

.9499

.9533

12

.9634

.9674

.9631

13

.9669

.8010

.9956

14

.9720

.7433

.9514

Bus 9 13

Qj (MVAr) 3.75 capacitative 72.81 capacitative

l

Eco n - final voltage profile

186

S. S. CHOI et al. Table 5. Summary of results for the case of line 2 out B

BUS

Enor(P.U. )

Econ(P. u. )

Econ(P.U. )

1

t.0aO0

1.0400

1.0400

2

1.0200

1.0200

1.0200

5

1.0000

1.0000

1.0000

z,

1.0200

1.0dO0

~.0200

5

1.0101

.9721

1.0148

6

1.0172

1.0192

/.OzO0

7

1.0184

1.0645

1.0452

8

1.0266

1.0290

.0270

9

.9633

.8343

.~50u

10

.9068

.8793

.97/b

11

.9542

.9788

.=ibS1

12

.9643

.96ai

.9660

13

.9645

.9671

.96/0

14

.9684

.9717

.9!1o

Bus

~j(MVAr)

7

7.96 inductive

9

11.54 capacitatzve

10

52.82 capacitative

E' - final voltage profile con

whereas in some practical cases AQj can only assume discrete values. The method of mixed integer programming method could be used for such cases. In this study the selection of the nodes where compensation is to be applied has been chosen based on engineering judgement. However, in order to exploit further the potential of this method, it is best that a sensitivity test be incorporated into the program so as to identify those nodes at which reactive compensation would be most effective.

REFERENCES 1. R. M. Maliszewki, L. L. Garver and A. J. Wood, Linear programming as an aid in planning kilover requirements, IEEE Trans. Power Apparatus and Systems, PAS-87 (12) (!%8). 2. A. M. Pretelt, Automatic allocation of network capacitors, IEEE Trans. Power Apparatus and Systems, PAS-90, (I) (1971). 3. H. H. Happ and K. A. Wirgau, Static and dynamic VAR compensation in system planning, IEEE Trans. Power Apparatus and Systems PAS-97 (5) (1978). 4. K. Ramachandran and J. Sharma, A. Mixed integer programme for optimal static capacitor allocation in power systems, Comput. Elect. Engng 6 (4) (1978). 5. S. I. Gass, Linear Programming: Methods and applications, McGraw-Hill, New York f1%91. 6. G. W. Stagg and A. H. EI-Abiad, Computer Methods in Power System Analysis. McGraw-Hill, New York (1968). 7. P. M. Anderson, Analysis o[ Faulty Power System, 1st Edn. Iowa State University Press, Ames, Iowa (1973). 8. L. L. Freris et al., Investigation of the load flow problems. IEE Proc., p. 115 (1968).