CHAPTER 2 MULTI-OBJECTIVE GENETIC ALGORITHM (MOGA) FOR OPTIMAL POWER FLOW PROBLEM INCLUDING VOLTAGE STABILITY

Transcription

1 26 CHAPTER 2 MULTI-OBJECTIVE GENETIC ALGORITHM (MOGA) FOR OPTIMAL POWER FLOW PROBLEM INCLUDING VOLTAGE STABILITY 2.1 INTRODUCTION Voltage stablty enhancement s an mportant tas n power system operaton. Optmal Power Flow (OPF) can be used perodcally to determne the optmal settngs of the control varables to enhance the stablty level of the system. In ths wor, the optmal power flow problem s formulated as a mult-objectve optmzaton problem wth cost mnmzaton and mzaton of statc voltage stablty margn as the objectves. L-ndex of the load buses s used as the ndcator of voltage stablty. When an optmzaton problem nvolves more than one objectve functon, the tas of fndng one or more optmum solutons s nown as mult-objectve optmzaton (Deb 2001). Snce classcal search and optmzaton algorthms use a pont by pont approach, the outcome of usng a classcal approach s a sngle optmzed soluton. However, most real world problems nvolve smultaneous optmzaton of several often mutually concurrent objectves. For mult-objectve optmzaton, the preference based approach requres multple runs as many tmes as the number of desred optmal solutons. Mult- objectve Genetc Algorthm (MOGA) s proposed n ths chapter to solve the mult-objectve OPF problem.

2 VOLTAGE STABILITY INDEX There are a number of methods proposed n the lterature for voltage stablty evaluaton. One such method s L-ndex proposed by essel and Glavtch (1986). It s based on load flow analyss. Its value ranges from 0 (no load condton) to 1 (voltage collapse). The bus wth the hghest L-ndex value wll be the most vulnerable bus n the system. The L-ndex calculaton for a power system s brefly dscussed below: Consder a N-bus system n whch there are N g generators. The relatonshp between the voltage and current can be expressed by the followng expresson: I I G L Y Y GG LG Y Y GL LL V V G L (2.1) where I G, I L and V G, V L represent currents and voltages at the generator buses and load buses. Rearrangng the above equaton we get, V I L G Z LL GL F Y LG GG I V L G (2.2) where F LG = - [Y LL ] -1 [Y LG ] and matrx. LG = - [Y LL ] -1 [Y GL ] are the sub matrxes of the above hybrd The L-ndex of the j-th node s gven by the expresson,

3 28 L j N g V 1 F j j j V 1 j (2.3) where V Voltage magntude of - th generatng unt. V j Voltage magntude of j- th generatng unt. j Phase angle of the term F j. Voltage phase angle of - th generatng unt. j Voltage phase angle of j- th generatng unt. N g Number of generatng unts. The values of F j are obtaned from the matrx F LG. The L ndces for a gven load condton are computed for all the load buses and the mum of the L- ndces (L ) gves the proxmty of the system to voltage collapse. It was demonstrated that when a load bus approaches a voltage collapse stuaton, the L-ndex approaches one. Hence, for a systemwde voltage stablty assessment, the L-ndex s evaluated at all the load buses and the mum value of the L ndces gves an ndcaton of how far the system s from voltage collapse. Contngences such as transmsson lnes or generator outages often result n voltage nstablty n the power system. The system s sad to be secured f none of the contngences causes voltage nstablty n the system. The mum L-ndex of the system under a contngency gves a measure of the severty of that contngency. 2.3 PROBLEM FORMULATION The OPF problem s to optmze the steady state performance of a power system n terms of one or more objectve functons whle satsfyng several equalty and nequalty constrants. It s formulated as non lnear mult-objectve optmzaton problem. The objectves consdered here are

4 29 mnmzaton of fuel cost and mzaton of voltage stablty margn. Ths s acheved by proper adjustment of real power generaton, generator voltage magntude, reactve power generaton of capactor ban and transformer tap settng. Power flow equatons are the equalty constrants of the problems, whle the nequalty constrants nclude the lmts on real and reactve power generaton, bus voltage magntudes, transformer tap postons and lne flows. The expresson representng the objectve functons and the constrants are gven below: Objectve functons Here the goal s to determne the optmal values of generator actve power, generator bus bar voltages, transformer tap settngs and reactve power sources to enhance the system stablty level whle mnmzng the fuel cost and mzng the voltage stablty margn Mnmzaton of fuel cost The most commonly used objectve n the OPF problem formulaton s the mnmzaton of the total operaton cost of the fuel consumed for producng electrc power wthn a scheduled tme nterval. The ndvdual costs of each generatng unt are assumed to be the functon only of real power generaton and are represented by quadratc curves of the second order. The objectve functon for the whole power system networ can be expressed as the sum of the quadratc cost model at each generator as, Mnmze F N g 2 ( a P b P c ) $ hr (2.4) C g g / 1 where F C s the operatng fuel cost of power system a,b,c are the cost coeffcents of generator at bus

5 30 N g s the number of generatng unts Voltage stablty margn The other objectve of the OPF problem s the mzaton of the voltage stablty margn. Ths s acheved by mnmzng the L value. Ths s stated as, Mnmze L (2.5) The above objectves are subjected to the followng equalty and nequalty constrants Equalty constrants In a power system wth N B buses, at each bus, the sum of the total njected real and reactve power must be equal to zero. The specfed power s equal to the dfference between the power generaton and the load. These constrants are mathematcally represented as: P V N B j1 V ( G Cos B Sn ) 0 ; N 1 (2.6) j j j j j B Q V N B j1 V ( G Sn B Cos ) 0 ; N (2.7) j j j j j PQ The set of equatons formed by equaton (2.6) for all system buses except the slac bus and equaton (2.7) for all load buses consttute the load flow equatons. The load flow equaton determne the steady state condton of the power system networ for specfed generatons and load patterns, calculate voltages, phase angles and flows across the entre system. When solvng the power flow equatons teratvely, successve solutons wll have a

6 31 msmatch between the specfed and the njected power. So equaton (2.6) and (2.7) wll not be satsfed. Hence, a tolerance s specfed for the power flow solutons Inequalty constrants The nequalty constrants reflect the lmts on physcal devce n the power system as well as the lmts created to ensure securty.the nequalty constrants whch are generally consdered are upper and lower bus voltage lmts, mum lne loadng lmts and lmts on tap settngs. () Voltage lmts Too hgh or too low voltage magntudes could cause problems to the end user power apparatus or nstablty n the power system. The voltage magntude constrant s expressed as: mn V V V ; N (2.8) B () Generator reactve power capablty lmt The reactve power of a generator s an mportant measure of voltage magntude qualty, e.g. a low voltage ndcates a local shortage of reactve power. The upper and lower reactve power lmts are specfed as: mn Q Q Q ; N (2.9) g g g g () Capactor reactve power generaton lmt The reactve power generaton of capactor ban has a mum generatng capacty, above whch t s not feasble to generate due to techncal

7 32 or economcal reasons. Capactve reactve power generaton lmts are usually expressed as mum and mnmum reactve power outputs as: Q mn c c c Q Q ; N c (2.10) (v) Transformer tap settng lmt The flow of real power along the transmsson lne s determned by the angle of dfference of the termnal voltages and the flow of reactve power s determned manly by the magntude dfference of termnal voltages. The value s modfed n the search procedure among the exstng tap postons and expressed as: mn t t t ; N (2.11) T (v) Transmsson lne flow lmt The mum MVA values for transmsson lnes and transformers are gven due to lmtatons of the branch materal. Excessve power would damage the transmsson elements. Ths s stated as an nequalty constrant as: S S ; l N l l (2.12) l The equalty constrants are satsfed by runnng the power flow program. The actve power generaton (P g ) (except the generator at the slac bus), generator termnal bus voltages (V g ), transformer tap settngs (t ) and the reactve power generaton of the capactor ban (Q c ) are the control varables and they are self restrcted by the optmzaton algorthm. The actve power generaton at slac bus (P gs ), load bus voltage (V load ) and reactve power generaton (Q g ) are the state varables, and they are restrcted by addng a quadratc penalty term to the objectve functon. Aggregatng the objectves

8 33 and constrants, the problem can be mathematcally formulated as a multobjectve non- lnear constraned problem as follows: Mnmze F T [ F, L ] (2.13) C subject to the constrants (2.6) (2.12). 2.4 MULTI-OBJECTIVE OPTIMIZATION Many real world problems nvolve the smultaneous optmzaton of several objectve functons. Generally, these functons are noncommensurable and often conflctng objectves. follows: In general, a mult-objectve optmzaton problem s formulated as Mn Max f ( x) 1... / (2.14) N obj Subject to: where g h j ( x) 0, j 1,... M ( x) 0, 1,... f s the th objectve functon. (2.15) x s the decson vector that represents a soluton N obj s the number of objectves M and are the numbers of equalty and nequalty constrants. Mult-objectve optmzaton wth such conflctng objectve functons gve rse to a set of optmal solutons, nstead of one optmal soluton. The reason for the optmalty of many solutons s that no one can be consdered to be better than any other wth respect to all objectve functons. These optmal solutons are nown as Pareto-Optmal solutons.

9 34 Mult-objectve optmzaton uses a concept of domnaton by comparng between two solutons. If a feasble soluton s not domnated by any other feasble solutons of the mult-objectve optmzaton problem, a soluton s sad to be a non-domnated soluton. The followng procedure can be adopted to fnd a set of non domnated solutons. x(1) domnates x(2) f x(1) s no worse than x(2) n all objectves and x(1) s strctly better than x(2) n at least one objectve. Soluton x(1) s sad to domnate x(2) or x(1) s sad to be non-domnated by x(2) f both condtons are true (Deb 2001). Domnance s an mportant concept n mult-objectve optmzaton. In case of a two objectve mnmzaton problem (f 1 and f 2 ) f 1 s sad to domnate f 2 f no component of f 1 s greater than the correspondng component of f 2 and at least one component s smaller. Accordngly, we can say that a soluton x 1 domnates x 2, f f(x 1 ) domnates f(x 2 ). A set of optmal solutons n the decson space whch are not domnated by other solutons, s called Pareto set and ts mage on the objectve space s called Pareto front. Fgure 2.1 llustrates the man concept of Pareto optmalty for sample two objectve mn-mn problems. The data for ths fgure s taen from the followng reference Deb (2001). The set of non- domnated soluton s referred as Pareto optmal front. Pareto optmal ponts are also nown as non- domnated or non- nferor ponts that are n the relatonshp of trade- off solutons.

10 35 Domnated solutons f 2 (mnmze) Pareto front Pareto Solutons f 1 (mnmze) Fgure 2.1: Man concept of Pareto optmalty In general, the goal of a mult-objectve optmzaton algorthm s not only gude the search towards the Pareto-optmal front but also mantan populaton dversty n the set of non-domnated solutons. The commonly appled methods for solvng the mult-objectve optmzaton algorthm are weghted sum method, -constrant method and goal programmng methods. Some of the dffcultes assocated wth the classcal optmzaton methods are as follows: An algorthm has to be appled many tmes to fnd multple Pareto-optmal solutons. Most algorthms demand some nowledge about the problem beng solved. Some algorthms are senstve to the shape of the Paretooptmal front. The spread of Pareto-optmal solutons depends on effcency of the sngle objectve optmzer.

11 36 Recently, the studes on evolutonary algorthms have shown that these algorthms can be effcently used to elmnate most of the dffcultes of classcal methods Mult-objectve Genetc Algorthm (MOGA) Genetc algorthms (Goldberg 1989) are search algorthms based on the prncple of natural genetcs and evoluton. They combne soluton evaluaton wth randomzed, structured exchanges of nformaton between solutons to obtan optmalty. Startng wth an ntal populaton, the genetc algorthm explots the nformaton contaned n the present populaton and explores new ndvduals by generatng offsprng usng the three genetc operators namely, reproducton, crossover and mutaton whch can then replace members of the old generaton. Genetc algorthms mantan a populaton of soluton structures throughout the process; therefore they are not lmted by the soluton of ntal guesses. In ths way the entre soluton space may be explored and multple solutons detected. Evaluaton of the ndvduals n the populaton s accomplshed by calculatng the objectve functon value of the problem usng the parameter set. The result of the objectve functon calculaton s used to calculate the ftness value of the ndvdual. Ftter chromosomes have hgher probabltes of beng selected for the next generaton. After several generatons, the algorthm converges to the best chromosome, whch hopefully represents the optmum or near optmal soluton. Genetc Algorthms have proven to be a useful approach to address a wde a varety of optmzaton problems. Beng a populaton-based approach, GA s well suted to solve the mult-objectve optmzaton problems. In ths wor, mult-objectve genetc algorthm s appled to solve the mult-objectve OPF problem. Fgure 2.2 shows the flow chart of the GAbased algorthm for solvng the optmzaton problem.

12 37 Intalze the populaton Run the load flow program Evoluton (ftness assgnment) Set =1 Selecton, crossover and mutaton Form the new populaton Run the load flow program Evoluton (Ftness Assgnment) =+1 No Meetng the stoppng rule Output Yes Fgure 2.2 Flow chart of GA approach

13 Algorthm of MOGA MOGA s an extenson of the classcal GA (Fonseca and Flemng. 1995). The man dfference between a conventonal GA and a MOGA les n the assgnment of ftness to an ndvdual. The rest of the algorthm s the same as that n a classcal GA. The detals of the MOGA are descrbed below: In a MOGA, frst, each soluton s checed for ts domnaton n the populaton. To a soluton, a ran equal to one plus the number of solutons that domnate soluton s assgned: r = 1+ (2.16) In ths way, non-domnated solutons are assgned a ran equal to 1, snce no soluton would domnate a non-domnated soluton n a populaton. Once the ranng s performed, a raw ftness to a soluton s assgned based on ts ran. To perform ths, frst the rans are sorted n the ascendng order of magntude. Then a raw ftness s assgned to each soluton by usng a lnear (or any other) mappng functon. Usually, the mappng functon s chosen so as to assgn the ftness between N (for the best ran soluton) and 1 (for the worst ran soluton). Thereafter, the solutons of each ran are consdered at a tme and ther raw ftness s are averaged. Ths average ftness s now called the assgned ftness to each soluton of the ran. Ths emphaszes the nondomnated solutons n the populaton. In order to mantan the dversty among non-domnated solutons, nchng among solutons of each ran are ntroduced. The nche count s found wth the followng equaton nc ( r ) Sh( d j j1 ) (2.17) where ( r ) s the number of solutons n ran r and Sh ( d j ) s the sharng functon of the two solutons and j.

14 39 The sharng functon Sh d ) s calculated usng the objectve functon value as the dstance metrc, ( j Sh( d j d 1 ( ) 0 j share ) f d j otherwse share (2.18) share s the sharng parameter whch sgnfes the mum dstance between any two solutons before they can be consdered to be n the same nche. s a scalng factor less than or equal to 1. d j s the normalzed dstance between two solutons and j n a ran, and s calculated usng 1 M ( ) ( j ) 2 f 2 ( ) mn f d j (2.19) 1 f f where f and f mn are the mum and mnmum objectve functon value of the th objectve. The shared functon taes a value n [0, 1] dependng on the values of d j and share. The shared ftness value s calculated by dvdng the assgned ftness to a soluton by ts nche count. Although all solutons of any partcular ran have an dentcal ftness, the shared ftness value of a soluton resdng n a less crowded regon has a better shared ftness. Ths produces a large selecton pressure for poorly represented solutons n any ran. Hence wth scalng factor value equal to 1 compute the sharng functon value and nche count for all populaton members usng the equaton (2.17) and (2.18). Thus, t s clear that n a populaton, a soluton may not get any sharng effect

15 40 from some solutons and may get partal sharng effect from few solutons and wll get a full effect from tself. It s mportant to note that nche count s always greater than or equal to 1. Then calculate the shared ftness usng F ' j F j nc j.dvdng the assgned ftness value by the nche count reduces the ftness of each soluton. In order to eep the average ftness of the solutons n a ran the same as that before sharng, these ftness values are scaled usng equaton (2.20) so that ther average shared ftness value s the same as the average assgned ftness value. F sc j F ( r ) F j ( r ) 1 F ' ' j (2.20) where ' j F s the shared ftness and (r ) s the number of solutons n ran r. Ths procedure s contnued untl all rans are processed. Thereafter, selecton, crossover and mutaton operators are appled to create a new populaton. Wth each ndvdual represented as a strng of ntegers and floatng pont numbers, selecton process remans the same as the classcal GA, but the crossover and mutaton operators are appled varable by varable. Here, ftness proportonate selecton and non-unform mutaton are used for genetc operaton. 2.5 GA IMPLEMENTATION When applyng GA to solve a partcular optmzaton problem, three man ssues needs to be addressed: () Representaton of the decson varables () Ftness evaluaton () Applcaton of genetc operators These ssues are explaned n the subsequent secton.

16 Representaton of the decson varables Whle solvng optmzaton problems usng the MOGA, each ndvdual n the populaton represents a canddate soluton. In the optmal power flow problem, the elements of the soluton consst of the control varables, namely; Generator actve power (P g ), Generator bus voltage (V g ), reactve power generated by the capactor (Q c ), and transformer tap settngs (t ). These varables are represented n ther natural form, that s generator actve power, generator bus voltage magntude, and reactve power generaton of capactor are represented as floatng pont numbers, whereas, the transformer tap settng, beng a dscrete quantty wth tappng ranges of ± 10% and a tappng step of p.u s represented by (0, 1,.8). Wth ths representaton, a typcal chromosome for the optmal power flow problem loos as: P g2 P g3 P gn V g1 Bnary coded GA dscretzes the search space by usng a codng of the problem varables n bnary strngs. However, the codng of real valued varables n fnte length strngs causes the nablty to acheve arbtrary precson n the obtaned soluton, fxed mappng of problem varables and nherent hammng clff problem assocated wth bnary codng. Whle presentng the dscrete varables usng bnary strngs, f the number of values s not a power of 2, then some of the bnary codes wll be redundant and so they should be taen care of through some remappng operaton. Ths may deterorate the performance of the algorthm. Hence to overcome these dffcultes the decson varables are represented n ther natural form. Thus the use of floatng pont numbers and ntegers to represent the soluton allevates the dffcultes assocated wth the bnary-coded GA for real varables (Herrera et.al.1998). Also, wth the drect representaton of the V g 2 V gn Q c1 Q c2 Q cn t 1 t 2 t

17 42 soluton varables, the computer space requred to store the populaton s reduced Ftness Evaluaton Genetc algorthm searches for the optmal soluton by mzng a gven ftness functon, and therefore an evaluaton functon whch provdes a measure of the qualty of the problem soluton must be provded. Each ndvdual n the populaton s evaluated accordng to ts ftness whch s defned as the non-negatve fgure of mert to be mzed. It s assocated manly wth the objectve functon. In the OPF problem here the objectve s to mnmze the operatng cost and L of the system smultaneously whle satsfyng the equalty and nequalty constrants. GAs are essentally unconstraned search procedures n the gven representaton space. For each ndvdual, the equalty constrants are satsfed by runnng the power flow program. The actve power generaton (P g ) (except the generator at the slac bus), generator termnal bus voltages (V g ) are the control varables and they are self restrcted by the optmzaton algorthm. The lmt on actve power generaton at the slac bus (P gs ), load bus voltages (V load ), reactve power generaton (Q g ) and lne flow (S l ) are satsfed through the penalty approach. The penalty terms for the slac bus generator actve power lmt volaton (PS ), load bus voltage-lmt volaton,(pv ), reactve power generaton lmt volaton,(pq g ), and lne flow lmt volaton (PL l ) are defned by the followng equatons: PS 0 S S PS PS PS PS 2 2,,, f PS f PS otherwse PS PS mn (2.21)

18 43 PV PQ PL g l l V V q q S V V Q Q l g g V V Q Q S mn g mn g 2 l , f V, f V V V, otherwse, f Q g, f Q g Q mn Q, otherwse, f S l S l, otherwse g mn g (2.22) (2.23) (2.24) where S, v, q, and l are the penalty factors for the slac bus power output, bus voltage lmt volaton, generator reactve power lmt volaton and lne flow volaton respectvely. Thereafter, all the penalty terms are added together to get the overall penalty functon. PF N C 0 P S N N C PQ 0 1 PV N N C g 0 1 PQ N N C l 0 l1 PL l (2.25) The penalty functon s added to each of the objectve functons to get the new objectve functons. The GA s usually desgned to mze the ftness functon, whch s a measure of the qualty of each canddate soluton. Hence, n ths wor, the ftness s taen as the nverse of the new objectve functon Genetc Operators The GA uses three man operatng schemes namely, selecton, crossover and mutaton The Selecton Scheme Selecton s a method whch stochastcally pcs ndvduals from the populaton accordng to ther ftness; the hgher the ftness, the more

19 44 chance an ndvdual has to be selected for the next generaton. There are a number of methods proposed n the lterature for the selecton operaton. Ftness proportonate selecton s used n ths wor. Ftness proportonate selecton (Eshelman and Schaffer 1993), also nown as roulette-wheel selecton, s a genetc operator used n genetc algorthms for selectng potentally useful solutons for recombnaton. In ths approach, the ftness level s used to assocate a probablty of selecton wth each ndvdual chromosome. If f s the ftness of the ndvdual n the populaton, ts probablty of beng selected s f N P j1 f j (2.26) where N s the number of ndvduals n the populaton. Ths could be magned to be smlar to a roulette wheel n a casno. Usually a proporton of the wheel s assgned to each of the possble selectons based on ther ftness value. Ths could be acheved by dvdng the ftness of a selecton by the total ftness of all the selectons, thereby normalzng them to 1. Then, a random selecton s made smlar to how the roulette wheel s rotated. Whle canddate solutons wth a hgher ftness are less lely to be elmnated, there s stll a chance that they may be elmnated. Wth ftness proportonate selecton there s a chance that some weaer solutons may survve the selecton process; ths s an advantage, snce though a soluton may be wea, t may nclude some component whch could prove useful followng the recombnaton process. The analogy to a roulette wheel can be envsaged by magnng a roulette wheel n whch each canddate soluton represents a pocet on the wheel; the sze of the pocets are proportonate to the probablty of selecton of the soluton. Selectng N chromosomes from the populaton s equvalent to playng N games on the roulette wheel, as each canddate s drawn ndependently.

20 The Crossover Scheme The crossover operator s manly responsble for brngng dversty n the populaton. Crossovers for real parameter GAs have the nterestng feature of havng tunable parameters that can be used to modfy ther exploraton power. In the proposed approach each ndvdual n the populaton conssts of two types of varables: real and nteger. Hence a two-part crossover whch taes advantage of the specal structure of the problem representaton was developed. Frst, the two parents are represented on the floatng pont and nteger parts. The BLX (Blended crossover operator) (Eshelman and Schaffer 1993, Devaraj 2005, 2010) s used for real varables and the standard sngle pont crossover s appled for the nteger part. Fgure 2.3 represents the BLX- crossover operaton for the one dmensonal case. In the fgure, u 1 and u 2 are the selected ndvduals and u mn and u are the lower and upper lmts respectvely and I= (u 2 -u 1 ). In the BLX- crossover, the offsprng (y) s sampled from the space [e 1, e 2 ] as follows: mn e1 r ( e 2 e1 ) ; f u y u y repeat samplng ; otherwse where : where : e e 1 2 u u 1 2 ( u 2 ( u r unform random 2 u 1 u 1 ) ) number [0,1 ] (2.27) I u mn u e 1 e 2 u 1 u 2 Fgure 2.3 Schematc representaton of the BLX- Crossover

21 46 In a number of test problems, the nvestgators have observed that = 0.5 provdes good results. In ths crossover operator the locaton of the offsprng depends on the dfference n parent solutons. If both parents are close to each other, the new pont wll also be close to the parents. On the other hand, f the parents are far from each other, the search s more le a random search. Ths property of a search operator allows us to consttute an adaptve search The Mutaton Scheme The mutaton operator s used to nject new genetc materal nto the populaton. Mutaton randomly changes the new offsprng. In ths wor, the Non Unform Mutaton operator (Deb 2001) s appled to the mxed varables wth some modfcatons. Frst a varable s selected from an ndvdual randomly. If the selected varable s a real number, t s set to a unform random number between the varable s lower and upper lmts..e, f the selected varable s u wth the range u mn, u, two random numbers are generated and the result u 1 s calculated as u 1 u u u u u u mn.. 1 r 1 r 1 p 1 M 1 p 1 M q q f r f r (2.28) where p s the generaton number, q s a non unform mutaton parameter and M s the mum generaton number. On the other hand, f the selected varable s an nteger, the randomly generated floatng pont number s truncated to the nearest nteger.

22 47 After mutaton, the new generaton s complete, and the procedure begns agan wth the ftness evaluaton of the populaton. 2.6 SIMULATION RESULTS The proposed GA approach for solvng the optmal power flow optmzaton problem was appled to the IEEE 30-bus test system. Fgure 3.4 represent the schematc dagram of the IEEE 30-bus system. The IEEE 30- bus system has 6 generator buses, 24 load buses and 41 transmsson lnes of whch four branches are (6-9),(6-10),(4-12) and (28-27) wth tap settng transformers. The upper and lower voltage lmts at all buses except the slac bus are taen as 1.10 p.u and 0.95 p.u respectvely. The slac bus voltage s fxed at ts specfed value of 1.06 p.u. The generator cost coeffcents and the lne parameters are taen from Alsac and Scot (1974). Fgure 2.4 IEEE 30-bus system

23 48 The possble locatons for reactve power sources are buses 10, 12, 15, 17, 20, 21, 23, 24 and 29. Two dfferent cases were consdered for smulaton and the results are presented. Case (): OPF usng GA Here the GA-based algorthm was appled to dentfy the optmal control varables of the system under base-load condton, wth cost mnmzaton objectve and wthout consderng the voltage stablty of the system. The upper and lower voltage lmts at all the busbars except the slac bus were taen as 1.10 and 0.95 respectvely. The slac busbar voltage was fxed to ts specfed value 1.06 p.u. Here the contngences are not consdered and the GA- based algorthm was appled to fnd the optmal schedulng of the power system for the base case loadng condton gven n (Alsac and Scot 1974). Generator actve power outputs, generator bus voltages, transformer tap settngs and reactve power generaton of capactor ban were taen as the optmzaton varables. The optmzaton varables are represented as floatng pont numbers and ntegers n the GA populaton. The ntal populaton was randomly generated between the varable s lower and upper lmts. Tournament selecton was appled to select the members of the new populaton. Blend crossover and mutaton were appled on the selected ndvduals. The performance of the GA generally depends on the GA parameters used, n partcular the crossover and mutaton probabltes n the ranges and respectvely was therefore evaluated. It was run wth dfferent control parameter settngs and the mnmzaton soluton was obtaned wth the followng parameter settng: Populaton sze : 50 Crossover rate : 0.9 Mutaton rate : 0.01 Maxmum generatons : 150

24 49 To llustrate the convergence of the algorthm, the relatonshp between the best ftness value of the results and the average ftness are plotted aganst the generaton number n Fgure 2.5. From the fgure t can be seen that the proposed method converges towards the optmal soluton very qucly. It can be seen that the fuel cost reduces rapdly n the frst 10 generatons of GA. Durng ths stage, the GA manly concentrates on fndng feasble solutons to the problem. Then the value decreases slowly and settles down near the optmum value wth the most of the ndvduals n the populaton reachng that pont. The proposed GA too 80 seconds to complete the 150 generatons. After 150 generatons t was found that all the ndvduals have reached almost the same ftness value. Ths shows that GA has reached the optmal soluton. The optmal values of the control varables along wth the mnmum cost obtaned are gven n Table 2.1. Correspondng to ths control varable settng, t was found that there are no lmt volatons n any of the state varables. The mnmum cost soluton obtaned by the proposed approach and the mnmum cost reported n the lterature are presented n Table 2.2. From the table t s found that the mnmum cost obtaned by the proposed method s less than the values reported n many papers. Ths shows the powerfulness of the proposed wor.

25 50 Table 2.1 Results of GAOPF optmal control varables Control varables Varable settng P 1 (MW) P 2 (MW) P 5 (MW) P 8 (MW) P 11 (MW) P 13 (MW) V 1 (P.U) V 2 (P.U) V 5 (P.U) V 8 (P.U) V 11 (P.U) V 13 (P.U) T T T T Q C10 (MVAr) Q C12 (MVAr) Q C15 (MVAr) Q C17 (MVAr) 5.0 Q C20 (MVAr) 0.0 Q C21 (MVAr) 5.0 Q C23 (MVAr) Q C24 (MVAr) 5.0 Q C29 (MVAr) 5.0 Cost ($/hr)

26 51 Fgure 2.5 Convergence of the GAOPF algorthm for IEEE 30-bus test system Table 2.2 Comparson of the mnmum cost obtaned by dfferent methods for the IEEE30-bus system Method Gradent approach (Alsac, 1974) Hybrd evolutonary programmng (Yuryevch et.al. 1999) Refned Genetc Algorthm (Paranjoth et.al. 2002) Evolutonary Programmng (Somasundaram et.al. 2004) Improved Evolutonary programmng (Ongsaul et.al. 2006) Proposed method Mnmum cost ($/hr)

27 52 Case (): OPF usng MOGA Next, the proposed algorthm was run wth mnmzaton of fuel cost and mzaton of voltage stablty margn as the objectves. Ftness proportonate selecton was appled to select the members of new populaton. Blend crossover and non unform mutaton were appled on the selected ndvduals. The performance of GA generally depends on the GA control parameter used, n partcular, the crossover and mutaton probabltes respectvely. It was appled by consderng several sets of parameters n order to enhance ts capablty to provde acceptable trade-offs close to the Pareto optmal front. The optmal values of the MOGA were obtaned wth the followng parameter settngs: Generatons : 100 Populaton sze : 50 Crossover rate : 0.8 Mutaton rate : 0.01 Varable : 25 Fgure: 2.6 represent the Pareto-optmal front curve. It s worth mentonng that the proposed approach produces nearly 25 Pareto optmal solutons n a sngle run that have satsfactory dversty characterstcs and span over the entre Pareto optmal front. In the Pareto front, the extreme ponts represent the mnmum fuel cost and mum voltage stablty margn. The values of the control varable correspondng to these extreme solutons are gven n Table: 2.3.Correspondng to ths control varable, t s found that there s no lmt volaton n any of the state varables. It can be concluded that the proposed approach s capable of explorng more effcent and non-nferor solutons of mult-objectve optmzaton problem. It can be observed that the non domnated solutons are dverse and well dstrbuted over the Pareto front. Ths shows the effectveness of the proposed approach n solvng the optmal power flow problem.

28 Fgure 2.6 Pareto Optmal Front for fuel cost v s L Table 2.3 Control varables for IEEE 30-bus system Extreme Solutons Control varables Mnmum operatng cost soluton Maxmum voltage stablty margn soluton P 1 (MW) P 2 (MW) P 5 (MW) P 8 (MW) P 11 (MW) P 13 (MW) V 1 (P.U) V 2 (P.U) V 5 (P.U) V 8 (P.U) V 11 (P.U) V 13 (P.U) T T T T Q C10 (MVAr) 2 2 Q C12 (MVAr) 2 2 Q C15 (MVAr) 5 5 Q C17 (MVAr) 5 5 Q C20 (MVAr) 5 6 Q C21 (MVAr) 5 5 Q C23 (MVAr) 0 0 Q C24 (MVAr) 0 0 Q C29 (MVAr) 5 5 Cost ($/hr) L

29 CONCLUSIONS Ths chapter presents a MOGA algorthm approach to obtan the optmum values of the optmal power flow, ncludng voltage stablty enhancement. Voltage stablty enhancement s performed by the L-ndex method. It has consdered as an optmzaton crteron, the mnmzaton of fuel cost and the L-ndex value. The effectveness of the proposed method s demonstrated on the IEEE 30- bus system wth promsng results. The results show that EAs are effectve tools for handlng mult-objectve optmzaton where multple Pareto-optmal solutons can be found n one smulaton run. In addton, the dversty of the non-domnated solutons s preserved. The proposed algorthm performed well when t was used to characterze the Pareto optmal front of the mult-objectve power flow problem. From the smulaton wor, t s concluded that the MOGA performs better than the other methods. Ths approach s able to generate hgh qualty solutons, wth more stable convergence characterstcs than smple genetc algorthms. It can be concluded that the MOGA has the potental to solve dfferent multobjectve power systems optmzaton problems.