Internatonal Journal of Electroncs and Electrcal Engneerng Vol. 4, No., February 206 Applcaton of Modfed Subgradent Algorm Based on Feasble Values to Securty Constraned Non-Convex Economc Dspatch Problems w Prohbted Zones and Ramp Rates Fadıl Salh, Urazel Burak, and Tamyürek Bünyamn Department of Electrcal and Electroncs Engneerng, Esksehr Osmangaz Unversty, 26480 Esksehr, Turkey Emal: {sfadl, burazel, btamyurek}@ogu.edu.tr Abstract A securty constraned non-convex power dspatch problem w prohbted operaton zones and ramp rates s formulated and solved usng an teratve soluton meod based on e modfed subgradent algorm operatng on feasble values (F-MSG). Snce e cost functon, all equalty and nequalty constrants n e nonlnear optmzaton model are wrtten n terms of e bus voltage magntudes, e phase angles, e off-nomnal tap settngs, and e susceptance values of statc-var (SVAR) systems, ey can be taken as ndependent varables. The actual power system loss s ncluded n e soluton snce e load flow equatons are nserted nto e model as e equalty constrants. The proposed technque s tested on e IEEE 30-bus, 40 generator and 40 generator test systems and compared aganst e oer meods based on heurstc and determnstc algorms. The sgnfcant savng n e soluton tme s due to e elmnaton of e power flow calculatons from e meod except at e ntal step. because a power flow calculaton s requred for each possble soluton n e soluton populaton. Many meods have been developed and appled to solve e economc power dspatch problems and reported n e lterature so far. Some of ese meods are e shuffle frog leapng algorm [], e mxed nteger genetc algorm [2], e partcle swarm optmzaton based technques [3]-[7], e dfferental harmony search meod [8], e evolutonary and e dfferental evolutonary based meods [9], [0], e artfcal bee colony search meod [], e cuckoo search meod [2], e gravtatonal search and pattern search meod [3], e bogeography based optmzaton meods [4], mxed nteger programmng [5], [6], λlogc based algorm [7] and fnally nteror pont meods [8]. In all e references gven n above, except n references [], [5]-[8], e fuel cost rate functons are taken as non-convex polynomals whch nclude e valve-pont loadng effects of e generators. In many applcatons reported n lterature, transmsson lne losses are eer gnored or added nto e dspatch problem n two ways; by usng eer B-matrx loss formula or performng AC load flow. In references [4], [0], [5][7], e transmsson lne losses are not consdered n order to reduce e complexty of e problem. In references [5], [6], [8], []-[4] and [8] for example, e total system loss s calculated by usng B-matrx loss formula. Nevereless, e loss values n e optmal soluton ponts are e approxmate ones. Snce power flow soluton s not performed n ose studes, e constrants assocated w e bus voltage magntude, e transmsson lne capacty, e off-nomnal tap settngs of e tap changng transformers and e susceptance values of e SVAR systems are not properly ncluded n er optmzaton models. In references []-[3], [7] and [9], e AC power flow calculatons are performed to obtan e bus voltage magntudes and e phase angles. Alough e constrants for e off-nomnal tap settngs of e tap changng transformers and e susceptance values of SVAR systems are added nto e models gven n references [2], [3] and [8] e prohbted operaton zone constrants for e generators are not. Index Terms economc power dspatch, F-MSG algorm, non-convex fuel cost rate curves, prohbted operaton zones, ramp rates, securty constrants I. INTRODUCTION Economc dspatch problem n electrc power systems can be consdered as a constraned non-lnear optmzaton problem. The soluton of t gves e mnmum total actve power generaton cost rate where all equalty and nequalty constrants assocated w e problem are satsfed. The non-convex power dspatch problems consdered n recent lterature are mostly solved va dspatchng technques at employ evolutonary meods and use smplfed models of power systems. Alough e constrants assocated w e actve generatons of e unts are modeled n a detaled manner, e oer constrants of e exact models of e power systems such as e voltage magntude, transmsson lne loadablty, and so on are not employed most of e tme n e optmzaton models used by ose meods. If e exact models of e power systems were used n ose dspatch technques, e soluton tmes would ncrease. Ths s Manuscrpt receved January 6, 205; revsed Aprl 5, 205. 206 Int. J. Electron. Electr. Eng. do: 0.878/jeee.4..-8
Internatonal Journal of Electroncs and Electrcal Engneerng Vol. 4, No., February 206 In e lterature, classcal determnstc meods are appled to soluton of varous power dspatch problems [5]-[8]. In ose solutons, actve generatons of e unts are taken as ndependent varables. Because of at, e total reactve power generaton load balance constrant and e reactve power generaton lmts for e generators are not handled. Besdes e power system loss s eer gnored [5]-[7] or s ncorporated nto e soluton process va reference bus penalty factors at are obtaned from Jacoban matrx of load flow soluton [8]. Snce determnstc meods especally based on classcal gradent meod can have dffculty n fndng e absolute mnmum soluton n e non-convex cost curve case, valve-pont loadng effects are not consdered n references [5]-[8]. Also e prohbted operaton zones and e ramp rates of e generators are gnored n [8] n order reduce e non-convex character of e problem. Furermore, e securty constrants assocated w e bus voltage magntude, e transmsson lne capacty, e off-nomnal tap settngs of e tap changng transformers and e susceptance values of e SVAR systems are not properly ncluded n er optmzaton models. The non-convex power dspatch problems are mostly solved va e evolutonary meods []-[4]. Alough e constrants assocated w e actve generatons of e unts are modeled n detaled manner, e oer constrants such as e reactve power generatons of e reactve power sources, e transmsson lne capactes, e bus voltage magntudes, and e off-nomnal tap ratos are not generally modeled n e optmzaton models at are used by em. The actve power system loss s modeled eer va approxmate B-matrx loss formula or not modeled at all. The actve power generatons are taken as e decson (ndependent) varables. If e exact actve and reactve power balance constrants, e transmsson lne capacty constrants, and e bus voltage magntude constrants are desred to be ncluded n ose models, e reactve power generatons of e unts, susceptance values of SVAR systems, and off-nomnal tap rato values (f ere are SVAR systems and off-nomnal tap rato transformers n e system) should be added nto e decson varable set and a power flow soluton must be performed for each possble soluton n e populaton set. Snce e transmsson lne capacty constrants, e bus voltage magntude constrants and actve and reactve power generaton constrants of e slack generator cannot be handled durng e producton of decson varables, ey are possbly consdered as penalty terms n e formulaton of ftness functon. Consequently, f all e constrants of economc dspatch problem are consdered n e soluton technques based on evolutonary meods, e number of decson varables wll ncrease, e expresson of ftness functon for each soluton wll become more complex, and a load flow soluton must be performed to calculate e ftness value of each possble soluton. It s because e populaton sze and e number teratons (generaton) wll ncrease compared to ose models at use a smple model of e consdered power system for e same level of soluton accuracy. As a result, e soluton tme wll be hgher an what s generally gven for e optmzaton models where e smple mode of e power system s used. Applcaton of e F-MSG meod n a non-convex securty constraned dspatch problem s gven n reference [9]. Anoer applcaton of e F-MSG meod where t s used n e soluton of securty constraned non-convex economc dspatch problem of an electrc power area at ncludes e lmted energy supply ermal unts s gven n reference [20]. In reference [2], a securty-constraned non-convex pumped-storage hydraulc unt schedulng problem s solved va F-MSG algorm agan. In ese ree dfferent applcatons of e F-MSG meod; e actual transmsson lne losses are added nto e dspatch problem va formulatng e AC load flow equatons as equalty constrants. What s more, e valve-pont loadng effects on e generators cost rate curves are also consdered n e solutons, but e prohbted operaton zone and ramp rate constrants are not. The F-MSG s a determnstc soluton meod. It can solve securty constraned non-convex power dspatch problems w prohbted operaton zones and ramp rates. It s especally sutable to solve non-convex dspatch problems where exact model of e consdered power system (optmal power flow problem) s used. Snce power flow calculaton s not used n e calculaton process (except ntal step), e soluton tme becomes lower an ose of produced by oer algorms mentoned n recent lterature. Detaled explanaton about e F-MSG meod can be found n reference [9]. In s paper, applcaton of e F-MSG meod s extended to non-convex dspatch problems w prohbted operaton zones and ramp rates. Outperformance of e F-MSG algorm w respect to some oer economc dspatch algorms based on heurstc and determnstc meods mentoned n recent lterature s demonstrated on some well known test systems. In ose test systems, prohbted operaton zones and ramp rates of e generator are consdered and exact or approxmate model of power systems are used. II. PROBLEM FORMULATION A nonlnear optmzaton model for an economc power dspatch problem can be descrbed as follows: subject to Mn F ( ) T F P NG P P p 0 Load, j jn B Q Q q 0,,2,, N Load, j jn B mn 2 P P P pz pz P pz max pz P P, N npz G () (2) (3) 206 Int. J. Electron. Electr. Eng. 2
Internatonal Journal of Electroncs and Electrcal Engneerng Vol. 4, No., February 206 P P P DR mn mn 0 max(, ) P P P UR max max 0 mn(, ) (4) mn mn max max Q Q Q Q Q Q,, N N (5) Q Q man man p p p,, l L l L (6) l l l l mn max U U U,,2,, N, ref, vc (7) mn mn max max a a a a, a, N N (8) tap tap mn max b b b, N (9) svar svar svar svar Note at e actve power generaton of e unt P should satsfy one of e nequaltes shown n (3). In oer words, P should not be contaned by any of e closed prohbted zone sets P pzm, pz m, m, 2, n. pz The meanngs of e symbols used n s paper are gven n lst of e symbols secton. A. Determnaton of Lne Flows and Power Generatons To express e total cost rate functon n terms of ndependent varables of e proposed optmzaton model, e lne flows need to be wrtten n terms of e bus voltages, e off-nomnal tap settngs, and e susceptance values of SVAR systems (see () and (2)). The necessary equatons, gvng e actve and reactve power flows ( p, q ) over e lne at s connected j j between buses and j n terms of e ndependent varables, can be found n reference [9]. Usng ose equatons and (2), e actve and reactve power generatons of e unt connected to bus can be calculated as below: P PLoad p (0) j and jn B Q Q q Load j jn B () Also, e total loss of e network can be calculated as P ploss j pj p j (2) LOSS p (3) N jn, j The non-convex cost rate functon of e taken as 2 F ( P ) b c P d P where,,, mn e sn g P P, N G b c d e and j unt s (4) g are constant coeffcents. The sne term n (4) s added to e cost rate curve to reflect e valve pont loadng affect. The non-convex total cost rate s en determned as F ( ) ( / ) T F P R h NG (5) B. Convertng Inequalty Constrants nto Equalty Constrants. Snce e F-MSG algorm requres at all constrants should be expressed as n equalty constrant form, e nequalty constrants n e optmzaton model should be converted nto correspondng equalty constrants. The meod descrbed below s used for s purpose snce t does not add any extra ndependent varable (lke n e slack varable approach) nto e optmzaton model. It s erefore e soluton tme of e consdered dspatch problem s reduced furer. A double sded nequalty x x x can be wrtten as e followng two nequaltes: h ( x ) ( x x ) 0, h ( x ) ( x x ) 0 (6) Then we can rewrte e above nequaltes as a sngle equalty constrant form as follows: h eq max0, ( x x ) ( x ) max 0, 0 max0, ( x x ) If x x x ( x x ) 0 (7), t s obvous at ( x x ) 0, and x x x x max 0, 0,, max 0, 0. So, e nequalty constrants n (6) can be represented by e correspondng sngle equalty constrant n (7). In s paper, e double sded nequalty constrants gven n (5)-(9) are converted nto e correspondng sngle equalty constrants n s manner. By e same reasonng, e unon of two sded nequaltes shown n (3) can be converted nto e correspondng sngle equalty constrant at s gven n (8). mn max0, ( P P ), max0, ( P pz ) max 0, ( ) eq pz P h ( P ) mn, 0 (8) max0, ( P pz ) 2 max0, ( pz P ) npz max max 0, ( P P ) N It should be noted at when P takes an nfeasble value, all quanttes nsde e square brackets n (8) become postve and erefore e equalty constrant s not satsfed. In e opposte case, once P takes a feasble value, one of e quanttes contaned by e square brackets becomes zero, so e equalty constrant s satsfed n s case. III. THE F-MSG ALGORITHM The ndependent (decson) varables of e meod are made up voltage magntudes and phase angles of e buses (except reference bus), e tap settngs of e off- G 206 Int. J. Electron. Electr. Eng. 3
Internatonal Journal of Electroncs and Electrcal Engneerng Vol. 4, No., February 206 nomnal tap rato transformers and e susceptance values of e SVAR systems n e network. The meod uses an augmented LaGrange functon at s called as sharp LaGrange functon. The F-MSG algorm proposed to solve e dspatch problem descrbed n Secton 2 and based on e modfed subgradent meod based on feasble values s gven n reference [9] n detaled manner. The reader should refer to reference [9] to examne e F-MSG algorm. IV. hybrd SFLA-SA. The optmal total cost rate and soluton tme (ST) values produced by e F-MSG and e oer meods are lsted n Table II for comparson. The best total cost rate produced by e F-MSG s 7.725, 5.8396, 5.45276 and 5.27 R/h less an ose produced by SA, PSO, SFLA and hybrd SFLA-SA, respectvely. Smlarly, e soluton tme of e best total cost rate soluton of e F-MSG s 9.33,.9375,.894 and.689 tmes smaller an ose gven by SA, PSO, SFLA and hybrd SFLASA, respectvely. We can conclude at e F-MSG meod outperforms e oers n terms of bo e total cost rate and e soluton tme. Some ntermedate results obtaned from applcaton of e F-MSG algorm to e dspatch problem usng e frst ntal data set s shown n Table III. The total cost rate s decreased from e ntal value of 005.934 R/h to 826.3639 R/h n 3 outer loop teratons where eght of em gve a feasble soluton. The algorm stops at e 3 outer loop snce 4 becomes less an (=ε2). Because of s, e last feasble soluton, whch s 829.3639 R/h found at e 3 outer loop teraton, s taken as e optmal total cost rate value [9]. The change of e total cost rate values (feasble/nfeasble) versus number of outer loop teratons durng each soluton procedure are shown n Fg.. Convergence of e F-MSG algorm to e same optmal total cost rate value for dfferent ntal data sets s clearly seen n Fg.. It s also seen from Table II at e hghest soluton tme produced by e F-MSG algorm s much lower an e best of soluton tmes produced by e oer meods. The optmal generatons, tap ratos and susceptances of SVAR systems are shown n Table IV. We see from e table at generaton, tap rato, and SVAR systems susceptance constrants are met at e soluton ponts. NUMERIC EXAMPLE In s secton, e proposed technque s gong to be tested on non-convex and convex dspatch problems of test systems whch were solved va heurstc and determnstc soluton meods prevously. The test systems nclude e IEEE 30-bus, e 40 generator and 40 generator test systems. The smulaton program s coded n Matlab 6.. The CSP problem appears n e rd step of e F-MSG algorm s solved by GAMS 2.5 w Conopt type solver [9]. A PC w Intel Core 2 Duo 2.20GHz CPU and 4GB RAM s used for e soluton of e dspatch problems. A. Solvng Economc Non-Convex Dspatch Problem of IEEE 30-Bus Test System w F-MSG. The detaled nformaton about e IEEE 30-bus test system data can be found n web page of Unversty of Washngton. Please refer to reference [] for detaled generator data. The bus numbered as s chosen as e reference bus and ts voltage s taken as.05 0 pu. The lower and upper lmts of voltage magntudes for all busses, except e reference bus, are taken as 0.95pu and.05pu, respectvely. Also e lower and upper lmts of all off-nomnal transformer tap settngs are taken as 0.9 and., respectvely. Smlarly, e lower and upper lmts for susceptances values of all SVAR systems are taken as 0.0pu and 0.pu, respectvely. In addton, e parameters of e F-MSG algorm are chosen as α=250, λ=, ε= 0-5, ε2=, M=250, =2500, =[0,0, 0,0]( 07), =00R/h, c 2500 00 R / h,and u [0, 0,...0, 0]( 07) (k ) k [9]. The same dspatch problem s solved ree tmes va e F-MSG meod by usng ree dfferent ntal data sets. The same parameters are used n all solutons. The selected actual ntal actve and reactve generatons, tap ratos, per-unt susceptance values of SVAR systems for ree dfferent startng ponts are gven n Table I. To obtan e ntal cost rate and e bus voltage values for each ntal data set, a load flow soluton s carred out by usng each data set. The calculated ntal total cost rate values for each ntal data set are also shown n Table I. The non-convex dspatch problem of IEEE 30-bus test system, where e prohbted operaton zones of generatng unts are consdered, was prevously solved and e results were presented n reference []. The soluton was performed usng four dfferent meods: smulated annealng (SA), partcle swarm optmzaton (PSO), shuffled frog leapng algorm (SFLA), and TABLE I. SELECTED THREE DIFFERENT SET OF INITIAL ACTUAL GENERATIONS, TAP RATIOS, PER-UNIT SUSCEPTANCE VALUES OF SVAR SYSTEMS AND THE CORRESPONDING INITIAL TOTAL COST RATE VALUES FOR THE DISPATCH PROBLEM OF IEEE30-BUS TEST SYSTEM set-2 set-3 2.50 89. 30.52 64.03 PG 5 80.00 50.00 60.00 5.00 QG 5 5.00 PG 8 0.00 20.00 QG 8 QG3 30.00 30.00 0.00 25.00 0.00 35.00 5.00 a,(6 9) a2,(6 0) a5,(4 2) a36,(28 27) bsvar0 bsvar 24 005.934 969.2790 907.002 QG PG 2 QG 2 PG QG PG3 FT(R/h) http://www.ee.washngton.edu/research/pstca/pf30/pg_tca30bus.htm 206 Int. J. Electron. Electr. Eng. set- 77.87 0.45 PG 4
Internatonal Journal of Electroncs and Electrcal Engneerng Vol. 4, No., February 206 TABLE II. COMPARISON OF THE OPTIMAL TOTAL COST RATE AND SOLUTION TIME VALUES PRODUCED BY THE F-MSG METHOD WITH THOSE OF FOUND VIA THE OTHER METHODS. Meod Optmal total cost rate value, (R/h) F-MSG Set- Set-2 Set-3 SA PSO SFLA Hybrd SFLA-SA 829.3639 829.4442 829.4655 836.5364 835.4786 834.866 834.6339 ST (sec) 6.32 5.65 5. 52.32 3.62 30.72 27.57 TABLE III. SOME INTERMEDIATE RESULTS OBTAINED FROM APPLICATION OF THE F-MSG ALGORITHM TO THE DISPATCH PROBLEM OF IEEE 30-BUS TEST SYSTEM. THE PROHIBITED GENERATION ZONES ARE CONSIDERED. n H n(r/h) Feasble/ Infeasble n F (R/h) T n+(r/h) H n+ n+(r/h) k p q 0 005.934 - - - - - - - 000 Feasble 986.99-00 900 9 0 2 900 Feasble 887.0934-00 800 5 0 2 3 800 Infeasble - +50 850 3 2 4 850 Feasble 849.7078-25 825 2 3 5 825 Infeasble - +2.5 837.5 2 2 3 6 837.50 Feasble 834.7300-6.25 83.25 2 2 4 7 83.25 Feasble 83.0760-3.25 828.25 2 5 8 828.25 Infeasble - +.5625 829.6875 3 5 9 829.6875 Feasble 829.978-0.7825 828.90625 3 6 0 828.90625 Infeasble - +0.390625 829.296875 4 6 829.296875 Infeasble - +0.95325 829.492875 5 6 2 829.492875 Feasble 829.4503-0. 09765625 829.3945325 5 7 3 829.3945325 Feasble 829.3639-0. 04882825 5 8 TABLE IV. SOLUTION POINT ACTUAL GENERATIONS, THE TOTAL ( a ) lne number,( bus tobus ) ACTIVE LOSS, THE TAP RATIOS THE SUSCEPTANCES OF SVAR SYSTEMS AND THE TOTAL COST RATE VALUES FOR EACH INITIAL DATA SET FOR THE DISPATCH PROBLEM OF IEEE 30-BUS TEST SYSTEM set- set-2 set-3 P G 29.82 29.7796 29.582 Q -50.4049 5.5686-3.08 G P 29.543 30.0076 28.929 G2 Q 55.3226 24.4666 7.305 G2 P 5.4059 0 6.8843 G5 Q 40.8463 27.7390 2.0065 G5 P 0.0000 0.0054 0.0007 G8 Q 86.9862 72.0524 3.236 G8 P 0.0000 0.0000 0.0000 G Q 6.5357 2.7800 9.2252 G P 2.0004.9999 2.0000 G3 Q 3.6599 3.368 4.7889 G3 P LOSS 3.3657 3.4046 3.2606 a 90 8 0.996,(6 9) a2,(6 0) 0.9984 0 55 a5,(4 2) 0.9920 0.9886 0.9942 a36,(28 27) 0.99 0.997 b svar0 0.02 0.023 0.007 b svar 24 0.026 0.02 0.08 F (R/h) T 829.3639 829.4442 829.4655 Fgure. Change of e total cost rate values (feasble/nfeasble) versus number of outer loop teratons for e dspatch problem of IEEE 30-bus test system B. Solvng Non-Convex Economc Dspatch Problem of 40 Generator Test System w F-MSG Please refer to reference [0] for detaled generator data about 40 generators Korean power system. In e dspatch problem consdered n reference [0], a smple model of e power system s consdered. The total cost rate of e system s mnmzed under equalty constrant 40 F ( P ) P, where P LOAD stands for e total LOAD system actve load. The ramp rates of e generators are consdered n addton to prohbted operatng zones and e valve pont effects n e optmzaton model. 206 Int. J. Electron. Electr. Eng. 5
Internatonal Journal of Electroncs and Electrcal Engneerng Vol. 4, No., February 206 We solved e dspatch problem va our dspatch meod by usng e same ntal actve generatons and total system actve load, PLOAD=49342MW, whch are gven n reference [0]. The parameters of e F-MSG algorm [9] are chosen as α=.5, λ=.5, ε=0.005, ε2=000, M=250, u =[0,0, 0,0] 4), c =27500, 2500 [0, 0,...0, 0](( 07) =00000R/h, 00 R / h, and (k ) k. The dspatch problem contnuous quck group search optmzer (CQGSO) [22] and dfferental evoluton based on truncated Lévy-type flghts and populaton dversty measure (DEL) [0] prevously reported n lterature. The soluton pont total cost rate and soluton tme values produced by e FMSG, and e oer meods mentoned n e above are gven n Table V. It s seen from e table at all e soluton meods, except GSO, gve almost e same total cost rate value but e soluton tme produced by e FMSG s 5.345, 3.563,.97 and.28 tmes smaller an ose produced by CCPSO, CTPSO, GSO and CQGSO, respectvely. consdered n s secton was solved by means of partcle swarm optmzaton w bo chaotc sequences and crossover operaton algorm (CCPSO), partcle swarm optmzaton w e proposed constrant treatment strategy (CTPSO), group search optmzer (GSO), TABLE V. THE SOLUTION POINT TOTAL COST RATE AND SOLUTION TIME VALUES PRODUCED BY THE F-MSG, AND SOME OTHER METHODS FOUND IN THE RECENT LITERATURE. Meod F-MSG CCPSO CTPSO GSO CQGSO DEL Optmal total cost rate (R/h) 65796.345 657962.730 657962.730 7285.68 657962.727 657962.77 ST (sec) 28.06 50 00 53.80 3.67 - C. Solvng Non-Convex Economc Dspatch Problem of 40 Generator Test System w F-MSG. Please refer to reference [5] for detaled generator data about 40 generator power test system. In e dspatch problem consdered n reference [5], a smple lossless model of e power system s consdered. Convex cost rate functons are taken for each generator. The total cost rate of e system s mnmzed under e followng V. In s paper, a power dspatch technque based on e F-MSG algorm s proposed to solve e securty constraned non-convex dspatch problems w prohbted zones and ramp rates. The proposed dspatch technque s tested on IEEE 30-bus, 40 generator and 40 generator test systems. Outperformance of e proposed technque aganst ten dspatch technques at are based on evolutonary and determnstc meods and reported n e recent lterature s demonstrated on e selected test systems. The advantages of e proposed meod can be summarzed as follows: ) The exact optmzaton model of an electrc power system can be used n e proposed soluton technque. 2) Snce e bus voltage magntudes and angles, e off-nomnal tap ratos, and e susceptance values of SVAR systems are taken as ndependent (decson) varables, all constrants consdered n e dspatch problem are handled n e same model easly. 3) Due to e selecton of ndependent varables, bo actve and reactve power optmzaton processes are carred out smultaneously. 4) Alough e proposed meod s a determnstc one, t can solve non-convex securty constraned dspatch problems due to ts way of search and e formaton of ts sharp augmented LaGrange functon. 5) In e proposed soluton technque, a load flow calculaton s carred out w e selected ntal actve and reactve generatons, e ntal offnomnal tap ratos, and e susceptance values of SVAR systems just to obtan e ntal bus voltage magntudes and phase angles. No more load flow calculaton s needed n e subsequent stages of e proposed dspatch technque. The 40 equalty constrant F (P ) P LOAD, where PLOAD stands for e total system actve load, e ramp rates and prohbted operaton zones of e generators are consdered. We solved e dspatch problem va our dspatch meod by usng e same ntal actve generatons and total system actve load, 7000 MW, whch are gven n reference [5]. The parameters of e F-MSG algorm used n soluton of e problem are chosen as α=.5, λ=.5, ε=, ε2=, M=250, u =[0,0, 0,0] [0, 0,...0, 0] ( 4), ( 07) =5000, 2500 =500R/h, 00 R and / h, (k ) k [9]. The dspatch problem consdered n s secton was solved by means of mxed nteger quadratc programmng (MIQP), whch s a determnstc meod, prevously [5]. The soluton pont total cost rate and soluton tme values produced by e F-MSG, and MIQP are gven n Table VI. It s seen from e table at bo meods gve almost e same total cost rate value, but e soluton tme produced by e F-MSG s lower an e one produced by MIQP. c TABLE VI. OPTIMAL COST RATE AND SOLUTION TIME VALUES PRODUCED BY F-MSG AND MIQP Meod MIQP F-MSG Optmal total cost rate (R/h) 00767.6872 00767.644 ST (sec) 0.86 0.50 206 Int. J. Electron. Electr. Eng. DISCUSSION AND CONCLUSION 6
Internatonal Journal of Electroncs and Electrcal Engneerng Vol. 4, No., February 206 selected ntal generatons do not even need to satsfy all e constrants of e dspatch problem. Due to e reasons gven n e above and n tem 4, e soluton tme of e proposed dspatch technque s much lower an ose of e soluton technques based on evolutonary meods once especally e exact model of e power system s employed. 6) The proposed dspatch technque can be appled to hgh dmensonal dspatch problems snce e F- MSG meod can solve dspatch problems w hgh number of ndependent varables. APPENDIX LIST OF SYMBOLS R : A fcttous monetary unt. N : Set at contans all buses to whch a generator s G connected. N : Set at contans all buses to whch a reactve power Q source s connected. N : Set at contans all buses drectly connected to bus. B N svar : Set at contans all svar systems n e network. L : Set at contans all lnes n e network. U : Voltage magntude of bus. b : Susceptance of e svar system connected to bus. svar a : Off-Nomnal tap settng value of tap settng faclty at bus. p, q : Actve and reactve power flows from bus to j j bus j at bus border, respectvely. p : Actve power flow on lne l. l P, Q : Actve and reactve power generatons of e unt, respectvely. P, Q : Actve and reactve loads of e bus, Load, Load, respectvely. : Total actve power loss n e network. P LOSS F( P ): Actve power generaton cost rate functon of e generaton unt. F : Total actve power generaton cost rate of e system. T mn max P, P : Lower and upper actve generaton lmts of e generaton unt, respectvely. 0 P : Intal actve generaton of e generaton unt. UR, DR : Ramp-Up and ramp-down rate lmts of e unt, respectvely. pzm, pz : Lower and upper lmts of e m m prohbted zone for e unt s actve power generaton, respectvely. n : Number of prohbted zones for e generatng pz unt. Prohbted zones are numbered n such a way at pz ( m) pzm, m 2,3,..., n. pz REFERENCES [] T. Nknam, M. R. Narman, and R. Azzpanah-Abarghooee, A new hybrd algorm for optmal power flow consderng prohbted zones and valve pont effect, Energy Convers. Manage., vol. 58, pp. 97-206, 200. [2] Z.-L. Gang and R.-F. Chang, Securty-Constraned optmal power flow by mxed-nteger genetc algorm w armetc operators, n Proc. IEEE Power Engneerng Socety General Meetng, Montreal, Quebec, Canada, Jun. 2006, pp. -8. [3] Z.-L. Gang and X.-H. Lu, New constrcton partcle swarm optmzaton for securty constraned optmal power flow soluton, n Proc. Internatonal Conference on Intellgent Systems Applcatons to Power Systems, Kaohsung, Tawan, Nov. 2007, pp. -6. [4] V. K. Jadoun, N. Gupta, K. R. Naz, and A. Swarnkar, Dynamcally controlled partcle swarm optmzaton for largescale nonconvex economc dspatch problems, Internatonal Transactons on Electrcal Energy Systems, 204. [5] N. Ghorban, S. Vakl, E. Babae, and A. Sakhavat, Partcle swarm optmzaton w smart nerta factor for solvng nonconvex economc load dspatch problems, Internatonal Transactons on Electrcal Energy Systems, vol. 24, pp. 20-33, 204. [6] T. Nknam, H. D. Mojarrad, and H. Z. Meymand, A new partcle swarm optmzaton for non-convex economc dspatch, European Transactons on Electrcal Power, vol. 2, pp. 656-679, 20. [7] T. Nknam, M. R. Narman, and M. Jabbar, Dynamc optmal power flow usng hybrd partcle swarm optmzaton and smulated annealng, Internatonal Transactons on Electrcal Energy Systems, vol. 23, pp. 975-00, 203. [8] A. Rajagopalan, V. Sengoden, and R. Govndasamy, Solvng economc load dspatch problems usng chaotc selfadaptve dfferental harmony search algorm, Internatonal Transactons on Electrcal Energy Systems, 204. [9] S. S. Reddy, P. R. Bjwe, and A. R. Abhyankar, Faster evolutonary algorm based optmal power flow usng ncremental varables, Int. J. Elec. Power Energy Syst., vol. 54, pp. 98-20, 204. [0] L. dos S. Coelho, T. C. 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Internatonal Journal of Electroncs and Electrcal Engneerng Vol. 4, No., February 206 [20] S. Fadıl, A. Yazıcı, and B. Urazel, A securty-constraned economc power dspatch technque usng modfed subgradent algorm based on feasble values and pseudo scalng factor for a power system area ncludng lmted energy supply ermal unts, Electr. Pow. Compo. Syst., vol. 39, no. 6, pp. 748-768, 20. [2] S. Fadıl and B. Urazel, Soluton to securty-constraned nonconvex pumped-storage hydraulc unt schedulng problem by modfed subgradent algorm based on feasble values and pseudo water prce, Electrc Power Components and Systems, vol. 4, no. 2, pp. -35, 203. [22] M. Morad-Dalvand, B. Mohammad-Ivatloo, A. Najaf, and A. Rabee, Contnuous quck group search optmzer for solvng non-convex economc dspatch problems, Electr. Pow. Syst. Res., vol. 93, pp. 93-05, 202. Burak Urazel was born n Eskşehr, Turkey, n 984. He receved hs B.Sc. degree from Anadolu Unversty, Eskşehr, Turkey, n 2008, hs M.Sc. degree from Eskşehr Osmangaz Unversty, Eskşehr, Turkey, n 20. He s currently workng as a research assstant and dong hs Ph.D. n Eskşehr Osmangaz Unversty. Hs man felds of research are optmzaton meods and er applcaton n power systems. Bunyamn Tamyurek was born n Artvn, Turkey, n 970. He receved e B.S. degree n Electrcal Engneerng from Yldz Techncal Unversty, Istanbul, Turkey, n 99, and e M.S. and Ph.D. degrees n Electrc Power Engneerng from Rensselaer Polytechnc Insttute, Troy, NY, n 996 and 200, respectvely. From 200 to 2003, he was w e Amercan Electrc Power, Corporate Technology Development Department, Groveport, OH, where he was nvolved n e evaluaton and testng of e sodum sulfur (NAS) battery systems. In 2004, he joned as a Faculty n e Department of Electrcal and Electroncs Engneerng, Esksehr Osmangaz Unversty, Esksehr, Turkey. Hs current research nterests nclude hgh-powerqualty swtch-mode rectfers, energy storage systems, UPS systems, frequency converters, and photovoltac nverters. Salh Fadıl was born n Istanbul, Turkey, n 959. He receved e B.Sc. and M.Sc. degrees n electrc power engneerng from Istanbul Techncal Unversty n 982 and 985, respectvely. He receved e Ph. D. degree from Washngton State Unversty, Wa, USA n 992. He s currently workng as a professor at Electrcal and Electroncs Engneerng Department of Eskşehr Osmangaz Unversty, Esksehr, Turkey. Hs research nterests nclude economc operaton of electrc power systems, electrc power system relablty modelng and small scale mcroprocessor based systems 206 Int. J. Electron. Electr. Eng. 8