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A HYBRID MODEL FOR OPTIMAL POWER FLOWINCORPORATING FACTS DEVICES

Narayana Prasad Padby M. A. R. Abdel-Moamen P. K. Trivedi B. DasDept. of Electrical Engineering

Univers ity of RoorkeeRoorkee -247667IndiaEmail: ~

Abstract: Optimal power flow (OPF) is one among the mostimportant algorithms available to utility for generating least costgeneration patterns in a power system satisfying transmission andoperational constraints. In day to day life, the forecasted loads usedin classical OPF algorithms are increasing with time and are also notcompletely free from errors. Increase in load demands leads tooverloading of the transmission lines and forecasted errors causeloss of optimality. So classical optimal power flow algorithms maynot be able to provide optimal solutions and they are limited tofixed type of cost characteristic curves and can not handle frequentincrease and variation in load demands. in this poper a ncw hybridmodel for OPF incorporating FACTS devices has been proposed toovercome the above said diftlculties. In the proposed model loaddemands, generation outputs and cost of generation are treated asfuzzy variables. la the first stage fuzzy dynamic programming hasbeen used to determine a set of feasible generation patterns. In thesecond stage dc power flow algorithm has been used to determine aset of valid sub-optimal generation patterns satisfying transmissionnnd opcmtionnl constraints. Finally in the third stage over loadingof transmission lines can be eliminated by installing FACTSdevices in the system. To verify the validity of the proposed modela 5 generator and 10-bus test system with and without modificationhas been used. It was found that the solutions obtained are quiteencouraging and suitable for modem deregulated environment.Key Words: Optimal Power Flow, Fuzzy Logic atui Facts Devices

I. INTRODUCTION

Optimal power flow is one of the most importantoperational functions of the modern clay energy managementsystem. The purpose of the optimal power flow is to find theoptimum generation among the existing units, such that thetotal generation cost is minimized while simultaneouslysatisfying the power baluncc equations and various otlwrconstraints in the system. The most common constraints inthe operation of a power transmission system are constraintson the voltage magnitudes of the buses and constraints on thereactive power genemted by the generators. For reliableoperation of a power system, the voltage magnitudes of thebuses and the reactive power[ 1] generated by the generatorsare constrained to stay within specified limits. However, dueto the increasing load demand, more and more power isrequired to be pushed over the existing transmission lines butincrease in the power flow above a certain operating level

decreases the overall stability of the system[2]. To achievebetter power flow control over the transmission lines withoutviolating the stability margin of the system, application oflexible ac transmission system (FACTS) technology iscurrently being pursued very intensively [3]. FACTStechnology is essentially the art and science of achievingbetter controllability over various electrical quantities in apower transmission system. Different FACTS device, such astatic var compensator (SVC), solid state synchronouscompensator (STATCOM), thyristor controlled seriescapacitor (TCSC), unified power flow controller (UPFC) etcare among the most potential controllers for application topower system to achieve better controllability. SVC andSTATCOM essentially control the voltage of a bus in asystem. TCSC essentially controls power flow over a line andUPFC controls both the bus voltage and power flow over aline.

If the transmission capacity constraints areincluded[4], there is a possibility that the optimum generationpoint may shifi to satisfy the transmission line capacityconstraints. At any operating point of a transmission systemthere must be some transmission lines through which theactual power flow is quite small compared to theiroperational limits. If the power flow through the underutilized lines can be increased by putting a TCSC in the lines,then power flow in the over-loaded lines may be diverted tothose under utilized lines and consequently, the power flowin the overloaded lines may reduce and ultimately comedown to a level wkch is below their operating limits.

II. PROBLEM FORMULATION

Mathematically, the optimal power flow problem[5] can beformulated as follows :The objective is to minimize the total cost of generation. i.e.

minimize

FT= ~Fi(PGi )iafwhere,

(1.1)

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FT = Total cost of generation($/hr)Fi(fGi) = The cost of PGigeneration by ih generator($/hr)NG = Number of generatorsFi(pd = aip~ibi PGi+Ci($~) (1.2)Where, ai, bi, Ciare the cost coefficients for i generator.

The above minimization problem is subjected tocertain system constraints. The most common constraints are

q Active Power Balance Ecmation for the SvstemThe total generation in the system must be equal tototal load plus the total loss in the system, i.e.

where, PIO,~= Total active load in the systemP,.,, = Total active power loss in the system

q Limits on the Outtmts of the Generation UnitsThe output of each generating unit must be withinsome specified minimum and maximum limits, i.e.P&i < pGi < P: ,fori= 1,2, . ..NG (1.4)where,PCli = The unit MW generated by i generntorP~ = The specified minimum MW generation by

ith generatorP ax = Thespecified maximum MW generation byGI

i* generator. O~eratin~ Line ConstraintsThe power flow over a transmission line should notexceed the specified maximum limit because ofstabili ty considerations, i.e.

PJj~ Pyax i=l,2.., . .. .. ...n (1.5)j=l,2 ...........nwhere,pij = active power flowing in line joining ih & jh busp ~ = maximum allowable active power flow inline joining i & j busn = number of kmscs in the system

III. PROPOSED MODELHowever, in practical scenario, the load demand

not always known precisely [(i], rather an approximate valuof the load demand is more likely to be known. In othewords, there is an uncertainty or vagueness associated withthe system load demand. This uncertainty can not badequately expressed by a crisp variable. On the other handthis vagueness can be adequately handled in fiuzzy set theoryThis theory provides a strict mathematical frame work iwhich vague conceptual phenomenon can be studiedrigorously. In this theory, the variables, functions, etcconnected with the imprecise phenomenon to be studied arexpressed as fuzzy variables and fuzzy functions.Consequently, to solve optimal power flow under impreciselydefined variables such as load demand etc., the objectivefunction and all or some of the constraints need to bexpressed as fuzzy objective function and fuzzy constraints.This uncertainty in load demand and generation output anproduction cost are treated as a fuzzy variables and differentsuitable membership functions are chosen.

In this paper, the system load demand has beenclassified into three categories ; low, medium and highConsequently, the total generation capacity of an individualgenerator is also classified into three categories. Themembership functions of generation capacity levels versusdifferent load levels have been shown in Fig. 1.1.The membership function chosen for the cost is written as.

pC(i,j) = exp(-Ac(i, j))

where,AC (i,,= C(i, j) - Cm,.(i, j )

Cmin(i,j)whereWC(i, j) = The cost membership value for ih bus

generator~witb jh discrete generation (P~ij)C(i, j) = cost per MW for i* bus generator with jh

discrete generation given by the expressionC(i, j) = ai P~ij + bi + ci/p~ij

C,,,i,,(i, j ) = minimum cost per MW for ih generatorbus with j ciiscrete generation,

(1.6)

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\

LoadLow Medium High

Generation(MW)

ow (XA ,G(xl~ ,G,x)l\

Pmm P P P

Medium

,G(x~ ;pm= ,G(;A

P I,,llm ]>,,,u

J

High

-h ;1 ,;;,> ~~ ,G::>p,,,w

P,,,i,, PIlmx Pmm ma% min

Figure 1.1: Membership functions for generation level (p~(.)) versus load demand levels(P)

Load demands and MW generation of the generatorshave been classified into three categories based on thefollowing formula.. Load in the range : (L,,,,,,) - (L,,,im+ 0,25LJ,)

+ low range,. Loads in the range :(L~in + 0.25L,,J - (L~u - 0.25 L,,,)+ medium range

q Loads in the range : (L.,n, - 0,25 L,,,) - (Ln,m)-+ high range

where,

L~i + minimum load demand of the systemL.m -+ maximum road demand of the systemL~i~ = (L~flX - L~in)

The generators have been classified into threedifferent categories, based on the following criteria.i) Low range : P~.X 150 MWWhere, Pc,mw -+ maximum generating capacity of agenerator.

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IV. PROPOSED HYBRID MODEL ALGORITHM Step 10: Check the line and other constraints.The step by step algorithm of the proposed hybrid model is Step 11 : If the line and other constraints are within theirpresented as follows: individual limits, then generation and cost of the set (Pol,Step 1: Read all inputs data. PG2- . . . .P~o) is belongs to the most economic solution.

Step 12: If line constraints violated, select the next lower of(a) Generation units, P&n ,P ~, for i = 1,2 . .. . . .NG menlbersllip ~ ,,~vand go to step 8.(b) Transmission line limits p ~ for i = 1,2 . ... . ..n Step 15: Install the FACTS devices in the lines with overloading and repeat step 1 to 12.

j= 1,2 . . .....n(c) Cost coefficients ai, b,, c, for i = 1,2,.. ..N(3 V. RESULTS AND DISCUSSION(d) Line parameters such as resistance, reactance, line To illustrate the application of the proposed model acharging susceptance and FACTS device parameters for 5 generator, 10-bus system has been considered(referline between i{h and j{~ bus. appendix A). OPF problem has been solved using different

Step 2: Set PIW,= O. membership functions for generators and their correspondingcost timctions taking into account of uncertainties in the loadStep 3: Pflay = PIO.~+ Plw, demands.Step 4: Set the generation of all the generators (PGl,P~2,P~3,.. . . . .P~~) such that

Step 5 : ( ) and ~G(i>j) or all eetermine PC i, jgeneration patterns obtained from step 4 and calculate~R (i, j) for each p(j,, i ENG USing the following formula.

PRk (i,j)=minPcd, k(i,j))fork= 1,2,3 . . .. . ..NG

Step 6: Calculate the set membership value (SMV) for allthe feasible combinations and arrange them in the decendingorder.

It has been found that observed that for the mostoptimum generation pattern, the total generation cost is139.078583 units but unfortunately two lines, namely lines 1-9 and 3-7 are violating the maximum power flow constraints,However, it is observed that for the 36h sub-optimalgeneration pattern, all the system constraints are satisfied.The results under this condition are shown in Table 1.3 and1.4.

To explore the possibility of using TCSC to relievethe overloaded lines, OPF problem has been solved using1CSC at various lines with vurying compensation levels. Itwas found that for two TCSC installations, all the systemconstraints are satisfied. These two cases are :a) TCSC installed on the line 1-6 with 50%

compensation and().Psmvw =maximum ( ~1 1, J , p2(i,j), . . . .. . .. . p~(i,j)) b) TCSC installed on the line 2-6 with 30%compensation levels.forw=l,2,3 , . . . . . number of generation sets From these above case studies, the total generation

Step 7 : Select the generation pattern corresponc@g to the costs are less than that obtained at the operating conditionhighest membership value(obtained from step 6) and run the given in Table 1.1. Hence both these cases are economicallyDC power flow and calculate the corresponding p~~ viable as in both these cases, the cost of TCSC installationwould be offset by the savings accrued in the generation cost.Step 8: Compute error= lPl~, - P~~ 1. However, in this @ase(a) the generation cost is less than thecost in case (b). So solution set (a) is preferable as the savingsStep 9: If error >6 (to]crancc), set P,,),, = in this case is higher and consequently, the cost of the TCSCP~J~ and go to installation would be offset more quickly. I Ience, from thestep 3. Or otherwise go to step 10. above discussion, TCSC with 50% compensation on the line

1-6 is recommended.

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Table 1.1 : Optimum generations and their correspondingcosts.

Unit No. Generation Generating cost (units)(MW)

1, 85.00 29.8169992. 70.00 33.8390013. 55.00 26.1617514. 55.00 28.2525015. 59.110260 24.008343

Table 1.2: Line flows for the optimum generation patternLine joining Line flow (pu)

Bus Bus1 2 0.13469412 6 0.1463181 9 0.3088092 3 0.0938312 6 0,189410 ~3 7 0,3123214 7 0.1876014 8 0.1311575 6 0.0633405 10 0.2555406 9 0.1604118 10 0.0200839 10 0.127892

Table-1.3 : Generation patterns m~d their correspondingcosts without violating line constraints.

Unit Generation Generating cost (units)No. (MW)

I 115.00 29.6370012 70.00 33.8390013 55.00 26.1617514 40.00 26.760005 44.110260 22.684479

Table 1.4: Line flows corresponding tothe generation pattern of Table 1.3

Line joining Line flow (p.u)Bus Bus1 2 0.12859401 6 0.17687001 9 0.27136702 3 0.07099602 6 0.21802103 7 0.28781104 7 0.21248704 8 0.10617605 6 0.1450005 10 0.17386806 9 0.00998708 10 0.0467229 10 0.0658310

VI. CONCLUSIONSIn this paper, OPF has been solved considering th

uncertainty in the load demand. The uncertainty in the loademand has been incofiorated in the study by the usefuzzy logic. To gain insights regarding the advantagedemerits of using fuzzy dynamic programming an attemphas been made to solve the OPF problem. Also, possibilityusing TCSCS to satisfy the transmission capacity constraintsin the OPF has also been expIored.

Consideration of uncertainty in the load demanresults into higher generation cost at the most feasibleoperating point compared to the cost when the uncertaintythe load demand is not considered. Whereas the generationpattern when the load uncertainty is considered is morreliable than the generation pattern obtained without thconsideration of the uncertainty in load demand. With the usof TCSC, the total generating cost comes out to be less thathat obtained without TCSC. Hence the savings obtainedbecause of the difference in generation cost would offset thcost of TCSC implementation.

tVII, REFERENCES

1. Precdavichit, P., and Srivastava, S. C., Optinxd ReactivePower Dispa[ch Considering FACTS Devices, ElectricalPower Systcms Research, Vol. 46, pp. 251-257, 1998.

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2. Gotham, fl. J., and 1Iaydt, G. T., { Pcrww Flow Control in Systemswith FACTS devices, Electrical Machines and Power Systems,vol. 26, pp. 951-962, 1998,3. Hingortsni, N., Flexible AC Transmission, IEEE Spcctlum, Vol.30, No. 4, pp. 40-45, April, 1993.

4. Ji. Yrrmr Fan, Lan Zhang, Real Time Economic Dispatch withLine Flow and Emission Constraints Using QuadraticProgramming, IEEE Transirctions on Power Systems, Vol. 13,I+o. 2, pp. 320-324, May 1998, ,

5. Wood, A.J. Wollenberg, B F., Power Generation Operation andControl, John Wiley and Sons, 1984.

6. Zimermann, H. J., FUZZV Sets, Decision Makirre and Exoert

Table-A3: Load DistributionBus # Load Bus # Load1 0.3 +jO.1 6 0.6 +jO.152 0.4 +jO.15 7 0.2 +jO.13 0.2 +jO. I 8 0.4 +jO.14 0.3 -tjO.15 9 0.2 +jO. I5 0.3 +jO.1 10 0.6 +jO.1Systems, Khrwer Academ;c Publishers, 1987. -

APPENDIX ATEST DAFA BIOGRAPHY

TTable-Al: Line Characteris tics Naravana Prasad Padhv obtained his degree of engineering andMaster of Engineering in1990 and 1993,respectively. in 1997, heobkrined his Pb. t). dcgrccfrom Anna University,Chennai, India. He joinedBirla Institute ofTechnology & Science ason Assistant Pmfcssor in1997. I I C is presently withthe faculty of Electrical

Line#12

Bus #

1-21-61-9

Impedance

0.02 + jO.080.06 +jO.250.04 +jO.16 z.01 0.60.02 0.30.02 0.30.02 0.30.02 0.334 2-32-63-7 0,06 +jO.250.06 +jO.250.06 +jO.25 ,. I ...ifI Engineerin~, University of~~:; P Roorkee, Roorkee.567 0.02 0.30.02 0.30.02 0.30.02 0.3

0.02 0.30.01 0.60.02 0.30.025 0.2

He taught course in Basic Electrical Engineering, Power Systems anArtificial Intelligence. His field of interest is Artificial IntelligenceAppl icat ions to Power System Optimization Problem.4-7 0.04 +jO.16

0.06 +jO.250.04 +jO.16

8 4-89 5-6 M. A. R. Abdc+-hloamcn, P. K. Trivcdi and Il. I)as arc working as

research scholar, post graduate student and faculty in the powesystems group, department of electrical engineering respectively.-1o

6-90.06 +jO.250.02 +jO.080.04 +jO.16

118-109-1o3 0.08 +jO.32

Table-A2: Unit CharacteristicsI Unit I Bus# 1 Cost function Pmax I Pmin 1I I I

1 6 20i-7p+ 1.2P~ 1.2 0.12 7 20+9p+l.l P2 1.2 0.1I 318 21+9p+0.7P2 i~I 4]9 23+9p+l.0P2 1.2 0.1, , II 5 10 19+8p+0.8P2 1.2 0.1

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