Research Article Optimal Power Flow Using Adaptive Fuzzy ...downloads.hindawi.com/journals/mpe/2013/975170.pdf · Optimal Power Flow Using Adaptive Fuzzy Logic Controllers AbdullahM.Abusorrah

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 975170 7 pageshttpdxdoiorg1011552013975170

Research ArticleOptimal Power Flow Using Adaptive Fuzzy Logic Controllers

Abdullah M Abusorrah

Electrical and Computer Engineering Department King Abdulaziz University Jeddah PO Box 80230 Jeddah 21589 Saudi Arabia

Correspondence should be addressed to Abdullah M Abusorrah aabusorrahhotmailcom

Received 27 December 2012 Accepted 13 January 2013

Academic Editor Jian Li

Copyright copy 2013 Abdullah M Abusorrah This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents an approach for optimum reactive power dispatch through the power network with flexible AC transmissionsystems (FACTSs) devices using adaptive fuzzy logic controller (AFLC) driven by adaptive fuzzy sets (AFSs) The membershipfunctions of AFLC are optimized based on 2nd-order fuzzy set specifications The operation of FACTS devices (particularly staticVAR compensator (SVC)) and the setting of their control parameters (119876SVC) are optimized dynamically based on the proposedAFLC to enhance the power system stability in addition to their main function of power flow control The proposed AFLC iscompared with a static fuzzy logic controller (SFLC) driven by a fixed fuzzy set (FFS) Simulation studies were carried out andvalidated on the standard IEEE 30-bus test system

1 Introduction

Optimal power flow (OPF) control in power systems hasa direct impact on system security and economic dispatchEarly OPF approaches were based on classical mathematicalprogramming methods and successfully showed their capa-bility in this field [1 2] A nonlinear programming (NLP)[1ndash3] linear programming (LP) [4 5] and Newton-basedmethod [6] have presented their competences in this area aswell Shahidehpour and Deeb [7] discussed the advantagesand drawbacks of most of the existing techniques On theother hand FACTS devices have become very popular inimproving the overall performance of power system underboth steady state and dynamic conditions In the literaturemany research works were carried out with FACTS devicesbeing included in the power system analysis and optimization[8ndash10] They were introduced to discuss the control strategyof FACTS devices using multiobjective OPF [10] Nowadaysthis target is applicable and achievable with the help of fastcontrollers FACTS to provide a greater operational flexibilityand more coordinated control of power flow [5 10] Theconcept of FACTS is made possible by the application ofhigh power electronic devices It is well recognized thatstability limits can be increased in real time by employingFACTS controllers [11 12] In addition to the steady statecontrol of power flow and voltage FACTS controllers can

also contribute to both large and small signal dynamicperformance of the power system [13 14]

FACTS controllers can be generally classified as seriescontrollers shunt controllers and combined series-shuntcontrollers [15 16] This paper addresses the problem ofoptimal sizing and control of static VAR compensator (SVC)as a FACTS device to attain certain objectives using FACTScontrollers for dynamic performance improvement More-over the optimal location of FACTS devices is an importantissue and depends on the objectives of the problem in handMahdad et al [10] introduced an 8 SVCs devices connectedto the IEEE 30-bus system in specific locations (at buses 1012 15 17 21 23 24 and 29) to enhance OPF at different severloading conditionsThis paperwill take the advantage of theselocations to examine the economic dispatch and the securityof the system as theminimization of voltage deviations at loadbuses

2 Problem Formulation

The aim of this paper is to investigate the power networkperformance under the following terms

(1) to develop an adaptive-fuzzy-logic-controller-(AFLC-) based method for optimal reactive power

2 Mathematical Problems in Engineering

(119876) of SVCs which can simultaneously optimizespecific objective function

(2) to investigate the effect of SVC controller on theperformance of the power system using adaptivefuzzy logic controller approach (AFLC)

21 Objective Function Consider119870119891 is the total fuel cost of agenerator operation and is given as

119870119891 =

NGsum

119894=1

119891119888119894 ($ℎ) (1)

where 119891119888119894 is normally given by a quadratic equation form interms of generator active power as follows

119891119888119894 = 119886119894 + 119887119894119901119866119894 + 1198881198941199012119866119894 (

$ℎ) (2)

where 119886119894 119887119894 and 119888119894 are the cost coefficients of the 119894th generatorMaintaining the voltage within tight control helps the

system to be stable and more secure The objective functionexpresses this idea by minimizing the sum of the voltagedeviations of all load buses This can be mathematicallyexpressed as follows

119870119881 =

NLsum

119894=1

(10038161003816100381610038161003816119881119871 119894 minus 119881ref

10038161003816100381610038161003816) (3)

where 119881ref is a specified reference value of the voltage magni-tude at load buses and usually set at 10 pu

22 Constraints The minimization of (3) is subject to anumber of equality and inequality constraints which can beanalyzed as follows

221 Equality Constraints (Power Flow Constraints) From(2) 119892 is the set of equality constraints representing the powerflow equations for 119894th bus and given by

119875119866119894 minus 119875119871119894 minus 119875Loss = 0

119876119866119894 minus 119876119871119894 minus 119876Loss = 0(4)

where (4) represents the active and reactive power injectionsfor the power balance equations

119875119866119894 and 119876119866119894 active and reactive powers of the 119894thgenerator respectively

119875119871119894 and119876119871119894 active and reactive powers of the 119894th loadbus respectively

119875Loss and 119876Loss total active and reactive networklosses

222 Inequality Constraints The inequality constraints onsecurity limits are presented as follows

Fuzzificationinterface

Fuzzy Fuzzy

Process outputand state

Knowledge base

Defuzzificationinterface

Inference systems

Actual controlnon fuzzy

Controlled systems

Figure 1 Generic structure of fuzzy logic controller

(1) The generator voltage (119881119866) constrainsgenerator voltage of each 119875119881 bus lays between theirlower and upper limits

119881min119866119894 le 119881119866119894 le 119881

max119866119894 119894 = 1 NG (5)

119876min119866119894 le 119876119866119894 le 119876

max119866119894 119894 = 1 NG (6)

where 119881min119866119894 and 119881

max119866119894 are minimum and maximum

voltages of 119894th generating unit respectively119876

min119866119894 and 119876

max119866119894 are minimum and maximum reactive

powers of 119894th generating unit respectively(2) Transformer tap (119879) constraint

Transformer taps are varied between lower and upperlimits as shown below

119879min

le 119879119894 le 119879max

119894 = 1 NT (7)

where 119879min and 119879max are minimum and maximum

tap settings of 119894th transformer respectively(3) Switchable VAR compensations (119876119862) constraints

shunt compensation units are subjected to their lowerand upper limits as follows

119876119862min le 119876119862119894 le 119876119862max 119894 = 1 NC (8)

where119876119862min and 119876119862max areminimumandmaximumVAR injection limits of 119894th shunt capacitor respec-tively

(4) Load bus voltage (119881119871119894) constraints

119881min119871 le 119881119871119894 le 119881

max119871 119894 = 1 NL (9)

where 119881min119871 and 119881


load voltages of 119894th bus respectively

3 Fuzzy Logic Controller

Fuzzy control systems are rule-based systems The outcomeof certain system can be corrected by a set of fuzzy rulesthat represent the fuzzy logic controller (FLC) techniqueTheFLC also provides a strategy which can change the linguisticcontrol approach based on expert knowledge to automaticcontrol ones Figure 1 shows the basic pattern of the FLCIt consists of a fuzzification interface a knowledge base adecision-making logic and defuzzification interface [17]

Mathematical Problems in Engineering 3

0 005 010

05

1

Voltage deviation error

0 2 4 60

05

1

Change error of voltage deviation

0 500

05

1

ZNSNL PS PL

PLPS

Z

Z

NS

NS

NL

NL

PS PL

minus01 minus005

minus2minus4minus6

minus50

Mem

bers

hip

grad

eM

embe

rshi

pgr

ade

Mem

bers

hip

grad

e

Reactive power (119876) of SVC 10

Figure 2 Membership Functions of inputs and output variables forSFLC

31 Global Input Variables The fuzzy input vector of eachstatic fuzzy logic controller (SFLC) has two variables whichcan briefly summarized by the voltage deviation error (119890V) andvoltage deviation change of error (Δ119890V)

In this paper the operation of SVCdevices and the settingof their control parameters (119876SVC) are optimized dynamicallybased on the proposed AFLC to enhance power systemstability in addition to their main function of power flowcontrol The eight fuzzy controllers are then designed to findthe optimal values of119876SVC10119876SVC12119876SVC15119876SVC17119876SVC21119876SVC23 119876SVC24 and 119876SVC29 Two static fuzzy controllers arethen designed one with five linguistic variables using fixedfuzzy sets for each input variable see Figure 2 the secondSFLC has three linguistic variables with fixed fuzzy sets foreach input variable and output variable based on five fuzzysets indicated by the solid lines as shown in Figure 2 Thefigure uses several linguistic variables PL (positive large) PS(positive small) Z (zero) NS (negative small) NL (negativelarge) as indicated in Tables 1 and 2

The fuzzy input vector of each AFLC consists of theprevious variables used in SFLCwith three linguistic variablesusing adaptive fuzzy sets Each input and output variablefuzzy set uses only three linguistic variables (LVs) as shownin Figure 3 In this figure the fuzzy set of the related variablesusedwith SFLC is indicated by the solid lines while the dottedlines represent the simulation results of the fuzzy set whenusing the AFLC LVs used are PL (positive large) Z (zero)and NL (negative large) as indicated in Table 2

4 Fuzzy Controllers Optimization

In this study the adaptive fuzzy set derived from GA isemployed for the adaptive fuzzy logic controller (AFLC)AFLC uses three fuzzy sets for input and output variableIt means that the full rule base is 9 rules Reference [18]identified the SFLC as an FLC with a fixed structure of fuzzyset Figure 2 shows a case of five fuzzy sets indicated by the

0 005 010

05

1


0 2 4 60

05

1


0 500

05

1

minus01 minus005

minus2minus4minus6

minus50

Mem

bers

hip

grad

eM

embe

rshi

pgr

ade

Mem

bers

hip

grad

e


Figure 3Membership Functions of inputs andoutput variablesTheSFLC indicated by the solid lines while the dotted lines for AFLC

Table 1 The lookup table which defines the relation between inputand output variables in a fuzzy set form for five fuzzy sets for SFLC

Voltage deviationerror (119890V)

Change error of voltage deviation (Δ119890V)NL NS Z PS PL

NL NL NL NL NS ZNS NL NL NS Z PSZ NL NS Z PS PLPS NS Z PS PL PLPL Z PS PL PL PL

Table 2 The lookup table which defines the relation between inputand output variables in a fuzzy set form for three fuzzy sets for SFLCand AFLC


Change error of voltage deviation (Δ119890V)NL Z PL

NL NL NL ZZ NL Z PLPL Z PL PL

solid lines The rules have the general form given by thefollowing statement

If vector 119890V is NS and Δ119890V is the change in 119890V which is Zthen 119876SVC10 is NS

In addition the membership function (mf119894) is expressedas follows mf119895 isin NB NS Z PS and PB as in the staticfuzzy case On the other hand the output space has fivedifferent fuzzy sets The adaptation technique is employed tomanage the parameters of the input and output fuzzy sets inorder to control the variation in operating conditions

Moreover the adapted genetic algorithm with adjustingpopulation size (AGAPOP) is used to optimize the member-ship function parameters of the FLCs The simulation resultsusing the AFLC controller are denoted in dotted lines asshown in Figure 3 and will be described later The AGAPOP


FLC for SVC 29

FLC for SVC 10

Start

Initialization

Randomly generatechromosomes of MFs for FLCs

Forgeneration = 1 max gen

For119896 = 1 pop size

Reproduction based onAGAPOP approach Mutation

Crossover

Yes No119869 gt Tolerance Record required

information

Stop

MFs

MFs

FLC for SVC 29

FLC for SVC 10119876SVC 10

119876SVC 29

Δ11989011990710

Δ11989011990729

11989011990710

11989011990710

middot middot middot

Calculate (119869max )

Figure 4 Flowchart of AGAPOP algorithm for optimizing MFs

approach uses the function (119865 = 1(1 + 119869)) where (119869) is theobjective function to be minimized The proposed techniqueapplies its operators and functions to identify the valuesof fuzzy set parameters to attain a better system dynamicperformance These parameters values initiate the optimumvalue of control actions with a significant improvementin the rising time oscillations and overshoot percentage(POS) Figure 4 represents the flowchart of the AGAPOP thatoptimizes the fuzzy set parameters of FL controllers [19]

41 Fuzzy Set Parameters Representation in GA The fuzzy setparameters of FL are formulated by the AGAPOP techniqueThe values of static fuzzy set parameters is primarily activatethe fuzzy set parameters of FL controllers where (Δc =

[119888min 119888max] and Δ120590 = [120590min 120590max] for Gaussian) represent theranges of the suitable values for each fuzzy set shape formingparametermore detail can be found in [19] Gaussian shape ischosen in order to show how the parameters of fuzzy sets areformulated and coded in the chromosomes The minimumperformance criteria 119869 are [18]

119869 = int

119879

0(1205721 |119890 (119905)| + 1205731

100381610038161003816100381610038161198901015840(119905)

10038161003816100381610038161003816+ 1205741

1003816100381610038161003816100381611989010158401015840(119905)

10038161003816100381610038161003816) 119889119905 (10)

where 119890(119905) is the voltage deviation error (119890V) and (Δ119890V) is thechange of error of voltage deviation The parameters (1205721 1205731and 1205741) are weighting coefficients

42 The Coded Parameters Rules The coded parameters areadjusted based on their constraints to shape a chromosomeof the population [17]

5 Selection Strategy

The selection policy usually manages how to pick individualsto be parents for new ldquochildrenrdquo and applies some selection

1198752

1198751

119875119871

Figure 5 Roulette wheel Selection Scheme [18]

weight [18] Assuming an appropriate population consists of119871chromosomes after procreation which are all originally ran-domized [19] Figure 5 demonstrates how each chromosomeis calculated and linked with fitness and the present popula-tion experiences the reproduction procedure to generate thefollowing population The possibility on the roulette wheel isadaptive and specified by 119875119897sum119875119897 as in the following [18]

119875119897 =1

119869119897 119897 isin 1 119871 (11)

where 119869119897 is the model routine encoded in the chromosome

51 Crossover and Mutation Operators The crossover isapplied and the mating pool is created Then the mutationprocedure is utilized followed by AGAPOP algorithm givenin [18] Finally the overall fitness of population is enhancedThe process is repeated for maximum allowable number ofgenerations when the termination condition is achievedThisprocess is described in the flowchart illustrated in Figure 4

52The Application of AFLC for Optimal Reactive Power Con-trol GA deals with the problem of optimizing a nonlinearobjective function in the presence of nonlinear equality andinequality constraints The main goal here is to construct theoptimum GA algorithms (AFLC) for SVCs control

521 Objective Function One objective function was createdas follows

1198651 = minimization of voltage deviations (12)

522 Network Constraints 119876119894 represents the reactive powervalue at any bus 119894 whereas Δ119876119894 is the change in thereactive power which will be obtained from the SVC at bus119894 Moreover the reactive power value at bus 119894 should bewithin the permissible limits (ie between119876119894min and119876119894max)Additionally the voltage limits stated in (5) and (9) should beconsidered as constraints

6 Results and Discussions

To study and design the system controller with 1198651 as a mainobjective function the following emergencies were applied toit

(1) overload test up to 132(2) load rejection test up to 80

In each test the optimum values of the control variables(119880) for all SVCrsquos are determined using the AFLC and SFLC


5 10 15 20 25 30

Typical voltage levels with SVCs

5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

111

09

01001

1

0

minus1

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)

Figure 6 Load bus voltage magnitudes the optimum VAR of eachSVC and voltage bus error as a function of load bus number usingAFLC technique

techniques that made the load bus voltages at 1 pu with avariation of plusmn 5

In order to investigate the controller design performancethe proposed techniques have been applied to the IEEE 30-bus system

61 Overload Test (Loading Factor = 132) The increasein loads up to 132 overload (Loading factor = 132) wascarried out at the same time for all network loads With theapplication of the objective functions under investigationFigure 6 shows the load bus voltagemagnitudes the optimumVAR of each SVC and voltage bus error as a function of loadbus number using AFLC technique

The SFLC technique is used for the same system con-dition Figure 7 describes the system performance Thepresented figures show the variation of each SVC VARs withoverloadThe results shows some remarkable comments suchas

(1) the network bus voltages are within the acceptable(1 pu plusmn 5)

(2) SVC 1 2 worked as capacitors SVC 29 worked asan inductor

62 Load Rejection Test (Loading Factor = 08) In this caseall network loads were partially rejected and the techniquewas applied to determine the optimum VAR and systemlossesThe sameprevious techniques are applied to show theireffectiveness AFLC technique helped the system to remainwithin its limits as shown in Figure 8 compared with SFLC asshown in Figure 9

It is noted from the presented figure that SVCs 12 15and 23 worked as capacitor while the remain SVCs workedas inductors Moreover the network bus voltages are withinthe suitable (1 pu plusmn 5)

5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

111

09

010

minus01

1

0

minus1

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)

Figure 7 Load bus voltage magnitudes the optimum VAR of eachSVC and voltage bus error as a function of load bus number usingSFLC technique

5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

11

109

02

0

minus02

02

0

minus02

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)


The SFLC achieved a remarkable conclusion shown inFigure 9 Furthermore the system active losses in this casewere less than the initial state

7 Conclusion

The SVC is considered as an effective tool for shunt reactivecompensation devices for the use on high voltage powersystems The main purpose of the current study is to presenta new design of controllers with the advantage of AFLC andSFLC techniques that control the VAR flow in the powersystem network The designed control parameters affect thequantity of reactive power injected to or taken from the


5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

11

109

02

0

minus02

02

01

0

minus01

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)


network during normal and emergency conditions In thispaper an adaptive fuzzy set is introduced and tested through asimulation program The proposed adaptive fuzzy controllerdriven by genetic algorithm improves the simulation resultsit shows the superiority of the adaptive fuzzy controllerFurthermore the effectiveness of the suggested AFLC withadaptive fuzzy set design as a remarkable approach canbe observed Specifications of parameter constraints relatedto inputoutput reference fuzzy sets are based on 2nd-order fuzzy sets The GA with changeable crossover andmutation probabilities rates analyzes and solves the problemof constrained nonlinear optimization The computationaltime of the FLC is decreased by the proposed AFLC usingAFS where the number of rules is reduced from 25 to 9rules In addition the proposed technique adaptive FLCdriven by genetic algorithm has reduced the fuzzy modelcomplexity The proposed techniques would be particularlya useful tool to assist the system operator in making controldecisions to improve the voltage profiles (with overvoltagesand undervoltages) in the system The method has beensuccessfully tested on standard IEEE 30-bus system

Acknowledgments

The author would like to thank Professor Yusuf Al-Turki andDr Abdel-Fattah Attia for their help and guidance duringthis work Also this paper was funded by the Deanshipof Scientific Research (DSR) King Abdulaziz UniversityJeddah The author therefore acknowledges with thanks theDSR technical and financial support

References

[1] M Huneault and F D Galiana ldquoA survey of the optimal powerflow literaturerdquo IEEE Transactions on Power Systems vol 6 no2 pp 762ndash770 1991

[2] W Dommel and F Tinney ldquoOptimal power flow solutionsrdquoIEEE Transactions on Power Apparatus and Systems vol PAS-87 pp 1866ndash11876 1968

[3] M O Mansour and T M Abdel-Rahman ldquoNon-linear varoptimization using decomposition and coordinationrdquo IEEETransactions on Power Apparatus and Systems vol 103 no 2pp 246ndash255 1984

[4] M Lebow R Rouhani R Nadira et al ldquoA hierarchical approachto reactive volt ampere (VAR) optimization system planningrdquoIEEE Transactions on Power Apparatus and Systems vol PAS-104 no 8 pp 2051ndash2057 1985

[5] A Bhattacharya and P K Chattopadhyay ldquoBiogeography-basedoptimization for solution of optimal power flow problemrdquo inProceedings of the 7th Annual International Conference on Elec-trical EngineeringElectronics Computer Telecommunicationsand Information Technology (ECTI-CON rsquo10) pp 435ndash439 May2010

[6] D I Sun B Ashley B Brewer A Hughes and W F TinneyldquoOptimal power flow by Newton approachrdquo IEEE Transactionson Power Apparatus and Systems vol PER-4 no 10 pp 2864ndash2880 1984

[7] S M Shahidehpour and N I Deeb ldquoAn Overview of thereactive power allocation in electric power systemsrdquo ElectricMachines and Power Systems vol 18 no 6 pp 495ndash518 1990

[8] A E Hammad ldquoComparing the voltage control capabilities ofpresent and future VAr compensating techniques in transmis-sion systemsrdquo IEEE Transactions on Power Delivery vol 11 no1 pp 475ndash481 1996

[9] K R Padiyar FACTS Controllers in Power Transmission andDistribution New Age International Pvt Ltd Publishers NewDelhi India 1st edition 2007

[10] B Mahdad K Srairi and T Bouktir ldquoOptimal power flowfor large-scale power system with shunt FACTS using efficientparallel GArdquo International Journal of Electrical Power andEnergy Systems vol 32 no 5 pp 507ndash517 2010

[11] E W Kimbark ldquoImprovement of system stability by switchedseries capacitorsrdquo IEEE Transactions on Power Systems Appara-tus and Systems vol PAS-5 no 2 pp 180ndash188 1986

[12] L Gyugi C D Schauder S L Williams T R Rietman D RTorgerson and A Edris ldquoThe Unified power flow controllera new approach to power transmission controlrdquo IEEE Transac-tions on Power Delivery vol 10 no 2 pp 1085ndash1093 1995

[13] G Rogers Power System Oscillations Kluwer 2000[14] C D Schauder L Gyugyi M R Lund et al ldquoOperation of uni-

fied power flow controller (UPFC) under practical constraintsrdquoIEEE Transactions on Power Delivery vol 13 no 2 pp 630ndash6371998

[15] N G Hingorani and L Gyugyi Understanding FACTS Con-cepts and Technology of Flexible AC Transmission System IEEEPress 2000

[16] R M Mathur and R K Verma Thyristor-Based FACTS Con-trollers For Electrical Transmission Systems IEEE Press 2002

[17] A-F Attia A Yusuf Al-Turki and F Hussein Soliman ldquoGeneticalgorithm-based fuzzy controller for improving the dynamicperformance of self-excited induction generatorrdquo Arabian Jour-nal For Science and Engineering vol 37 no 3 pp 665ndash682 2012

[18] A Yusuf Al-Turki A-F Attia andH F Soliman ldquoOptimizationof fuzzy logic controller for supervisory power system stabiliz-ersrdquo Acta Polytechnica vol 52 no 2 pp 7ndash16 2012


[19] A F Attia E Mahmoud H I Shahin and A M OsmanldquoA modified genetic algorithm for precise determination thegeometrical orbital elements of binary starsrdquo New Astronomyvol 14 no 3 pp 285ndash293 2009

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Stochastic AnalysisInternational Journal of


(119876) of SVCs which can simultaneously optimizespecific objective function

(2) to investigate the effect of SVC controller on theperformance of the power system using adaptivefuzzy logic controller approach (AFLC)

21 Objective Function Consider119870119891 is the total fuel cost of agenerator operation and is given as

119870119891 =

NGsum

119894=1

119891119888119894 ($ℎ) (1)

where 119891119888119894 is normally given by a quadratic equation form interms of generator active power as follows

119891119888119894 = 119886119894 + 119887119894119901119866119894 + 1198881198941199012119866119894 (

$ℎ) (2)

where 119886119894 119887119894 and 119888119894 are the cost coefficients of the 119894th generatorMaintaining the voltage within tight control helps the

system to be stable and more secure The objective functionexpresses this idea by minimizing the sum of the voltagedeviations of all load buses This can be mathematicallyexpressed as follows

119870119881 =

NLsum

119894=1

(10038161003816100381610038161003816119881119871 119894 minus 119881ref

10038161003816100381610038161003816) (3)

where 119881ref is a specified reference value of the voltage magni-tude at load buses and usually set at 10 pu

22 Constraints The minimization of (3) is subject to anumber of equality and inequality constraints which can beanalyzed as follows

221 Equality Constraints (Power Flow Constraints) From(2) 119892 is the set of equality constraints representing the powerflow equations for 119894th bus and given by

119875119866119894 minus 119875119871119894 minus 119875Loss = 0

119876119866119894 minus 119876119871119894 minus 119876Loss = 0(4)

where (4) represents the active and reactive power injectionsfor the power balance equations

119875119866119894 and 119876119866119894 active and reactive powers of the 119894thgenerator respectively

119875119871119894 and119876119871119894 active and reactive powers of the 119894th loadbus respectively

119875Loss and 119876Loss total active and reactive networklosses

222 Inequality Constraints The inequality constraints onsecurity limits are presented as follows

Fuzzificationinterface

Fuzzy Fuzzy

Process outputand state

Knowledge base

Defuzzificationinterface

Inference systems

Actual controlnon fuzzy

Controlled systems

Figure 1 Generic structure of fuzzy logic controller

(1) The generator voltage (119881119866) constrainsgenerator voltage of each 119875119881 bus lays between theirlower and upper limits

119881min119866119894 le 119881119866119894 le 119881

max119866119894 119894 = 1 NG (5)

119876min119866119894 le 119876119866119894 le 119876

max119866119894 119894 = 1 NG (6)

where 119881min119866119894 and 119881


voltages of 119894th generating unit respectively119876

min119866119894 and 119876

max119866119894 are minimum and maximum reactive

powers of 119894th generating unit respectively(2) Transformer tap (119879) constraint

Transformer taps are varied between lower and upperlimits as shown below

119879min

le 119879119894 le 119879max

119894 = 1 NT (7)

where 119879min and 119879max are minimum and maximum

tap settings of 119894th transformer respectively(3) Switchable VAR compensations (119876119862) constraints

shunt compensation units are subjected to their lowerand upper limits as follows

119876119862min le 119876119862119894 le 119876119862max 119894 = 1 NC (8)

where119876119862min and 119876119862max areminimumandmaximumVAR injection limits of 119894th shunt capacitor respec-tively

(4) Load bus voltage (119881119871119894) constraints

119881min119871 le 119881119871119894 le 119881

max119871 119894 = 1 NL (9)

where 119881min119871 and 119881


load voltages of 119894th bus respectively

3 Fuzzy Logic Controller

Fuzzy control systems are rule-based systems The outcomeof certain system can be corrected by a set of fuzzy rulesthat represent the fuzzy logic controller (FLC) techniqueTheFLC also provides a strategy which can change the linguisticcontrol approach based on expert knowledge to automaticcontrol ones Figure 1 shows the basic pattern of the FLCIt consists of a fuzzification interface a knowledge base adecision-making logic and defuzzification interface [17]


0 005 010

05

1


0 2 4 60

05

1


0 500

05

1

ZNSNL PS PL

PLPS

Z

Z

NS

NS

NL

NL

PS PL

minus01 minus005

minus2minus4minus6

minus50

Mem

bers

hip

grad

eM

embe

rshi

pgr

ade

Mem

bers

hip

grad

e








0 005 010

05

1


0 2 4 60

05

1


0 500

05

1

minus01 minus005

minus2minus4minus6

minus50

Mem

bers

hip

grad

eM

embe

rshi

pgr

ade

Mem

bers

hip

grad

e
















FLC for SVC 29

FLC for SVC 10

Start

Initialization





Crossover


information

Stop

MFs

MFs

FLC for SVC 29


119876SVC 29

Δ11989011990710

Δ11989011990729

11989011990710

11989011990710







119869 = int

119879

0(1205721 |119890 (119905)| + 1205731

100381610038161003816100381610038161198901015840(119905)

10038161003816100381610038161003816+ 1205741

1003816100381610038161003816100381611989010158401015840(119905)

10038161003816100381610038161003816) 119889119905 (10)





1198752

1198751

119875119871



119875119897 =1

119869119897 119897 isin 1 119871 (11)












5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

111

09

01001

1

0

minus1

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)










5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

111

09

010

minus01

1

0

minus1

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)


5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

11

109

02

0

minus02

02

0

minus02

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)



7 Conclusion



5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

11

109

02

0

minus02

02

01

0

minus01

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)



Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 005 010

05

1


0 2 4 60

05

1


0 500

05

1

ZNSNL PS PL

PLPS

Z

Z

NS

NS

NL

NL

PS PL

minus01 minus005

minus2minus4minus6

minus50

Mem

bers

hip

grad

eM

embe

rshi

pgr

ade

Mem

bers

hip

grad

e








0 005 010

05

1


0 2 4 60

05

1


0 500

05

1

minus01 minus005

minus2minus4minus6

minus50

Mem

bers

hip

grad

eM

embe

rshi

pgr

ade

Mem

bers

hip

grad

e
















FLC for SVC 29

FLC for SVC 10

Start

Initialization





Crossover


information

Stop

MFs

MFs

FLC for SVC 29


119876SVC 29

Δ11989011990710

Δ11989011990729

11989011990710

11989011990710







119869 = int

119879

0(1205721 |119890 (119905)| + 1205731

100381610038161003816100381610038161198901015840(119905)

10038161003816100381610038161003816+ 1205741

1003816100381610038161003816100381611989010158401015840(119905)

10038161003816100381610038161003816) 119889119905 (10)





1198752

1198751

119875119871



119875119897 =1

119869119897 119897 isin 1 119871 (11)












5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

111

09

01001

1

0

minus1

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)










5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

111

09

010

minus01

1

0

minus1

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)


5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

11

109

02

0

minus02

02

0

minus02

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)



7 Conclusion



5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

11

109

02

0

minus02

02

01

0

minus01

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)



Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










FLC for SVC 29

FLC for SVC 10

Start

Initialization





Crossover


information

Stop

MFs

MFs

FLC for SVC 29


119876SVC 29

Δ11989011990710

Δ11989011990729

11989011990710

11989011990710







119869 = int

119879

0(1205721 |119890 (119905)| + 1205731

100381610038161003816100381610038161198901015840(119905)

10038161003816100381610038161003816+ 1205741

1003816100381610038161003816100381611989010158401015840(119905)

10038161003816100381610038161003816) 119889119905 (10)





1198752

1198751

119875119871



119875119897 =1

119869119897 119897 isin 1 119871 (11)












5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

111

09

01001

1

0

minus1

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)










5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

111

09

010

minus01

1

0

minus1

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)


5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

11

109

02

0

minus02

02

0

minus02

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)



7 Conclusion



5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

11

109

02

0

minus02

02

01

0

minus01

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)



Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

111

09

01001

1

0

minus1

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)










5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

111

09

010

minus01

1

0

minus1

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)


5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

11

109

02

0

minus02

02

0

minus02

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)



7 Conclusion



5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

11

109

02

0

minus02

02

01

0

minus01

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)



Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 10 15 20 25 30


5 10 15 20 25 30

5 10 15 20 25 30

Bus number

Bus number

Bus number

11

109

02

0

minus02

02

01

0

minus01

Bus v

olta

ge(p

u)

Bus e

rror

volta

ge119876

SVCs

(pu

)



Acknowledgments


References




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Optimal Power Flow Using Adaptive Fuzzy ...downloads.hindawi.com/journals/mpe/2013/975170.pdf · Optimal Power Flow Using Adaptive Fuzzy Logic Controllers AbdullahM.Abusorrah