An Interactive Fuzzy Satisfying Method Based on Particle Swarm Optimization for Multi-Objective Function in Reactive Power Market

Transcription

1 An Interactve Fuzzy Satsyng Method Based on Partcle Swarm Optmzaton or Mult-Objectve Functon n Reactve Power Maret N. Tabrz*, E. Babae* (C.A.) and M. Mehdnejad* Abstract: Reactve power plays an mportant role n supportng real power transmsson, mantanng system voltages wthn proper lmts and overall system relablty. In ths paper, the producton cost o reactve power, cost o the system transmsson loss, nvestment cost o capactor bans and absolute value o total voltage devaton (TVD) are ncluded nto the objectve uncton o the power low problem. Then, by usng partcle swarm optmzaton algorthm (PSO), the problem s solved. The proposed PSO algorthm s mplemented on standard IEEE 14-bus and IEEE 57-bus test systems and wth usng uzzy satsyng method the optmal solutons are detered. The uzzy goals are quanted by denng ther correspondng membershp unctons and the decson maer s then ased to specy the desrable membershp values. The obtaned results show that solvng ths problem by usng the proposed method gves much better results than all the other algorthms. Keywords: Reactve power maret; Partcle swarm optmzaton; Total voltage devaton; Fuzzy satsyng method. 1. Introducton 1 Reactve power dspatch problem s mpressve on sae and economcal operaton o power systems. In act, t plays an mportant role or secure operaton o power systems. For ths reason, the reactve power dspatch has been o great nterest to researchers as well as system operators, especally ater the restructurng o the power ndustry [1]. Ths nterest s manly because o the sgncant eect that reactve power has on system securty gven ts close relatonshp wth the bus voltages throughout the power networ []. Tradtonally, reactve power dspatch has always been vewed by researchers as a power loss mzaton problem, subject to derent system constrants such as nodal real and reactve power balance, power generaton lmts and bus voltage lmts [, 3]. Mult-objectve optmzaton models have also been presented or the reactve power dspatch problem. In these models, the reactve power dspatch ncludes smultaneous mzaton o transmsson loss, voltage stablty ndex and voltage devaton [4-6]. In deregulated Iranan Journal o Electrcal & Electronc Engneerng, 016. Paper receved 4 May 015 and accepted 5 January 016. * The Authors are wth the Faculty o Electrcal and Computer Engneerng, Unversty o Tabrz, Tabrz, Iran. E-mal: n.tabrz9@ms.tabrzu.ac.r, e-babae@tabrzu.ac.r and m.mehdnejad1369@gmal.com. electrcty marets, the ndependent system operator (ISO) s responsble or the provson o ancllary servces that are necessary to support the transmsson o electrcal energy whle mantanng secure and relable operaton o the power system [4]. In deregulated electrcty marets, the reactve power ancllary servces can be provded based on a two-stage approach, namely, reactve power dspatch and reactve power procurement [7]. In [8], the problem o reactve power procurement by an ISO n deregulated electrcty marets has been presented. In [9], the techncal and economc ssues o deterng reactve power prcng structures n an open-access envronment have been exaed. In the compettve electrcty marets, the reactve power dspatch reers to short-term allocaton o reactve power needed rom generators, based on current system operatng condtons. The ndependent system operator s problem s to specy the optmal reactve power schedule or all provders based on a gven objectve that depends on system operatng condton. The ISO can use derent objectve unctons besdes the tradtonal transmsson loss mzaton such as mzaton o devatons rom contracted transactons [10] or mzaton o reactve power cost [11, 1]. In [13], an nteractve uzzy satsyng method based on evolutonary programg technque s proposed or economc emsson load dspatch o Iranan Journal o Electrcal & Electronc Engneerng, Vol. 1, No. 1, March

2 thermal plants wth non-smooth uel cost and emsson level unctons n coordnaton wth mult-reservor cascaded hydro plants. An nteractve uzzy satsyng method or solvng an economc emsson load dspatch problem s presented n [14]. In ths paper, a new ramewor that denes the reactve power dspatch problem to sut the ISO requrements n the compettve electrcty marets s proposed. The model sees to mze the ISO s total payments that nclude payments or mprovng TVD, reactve power dspatched rom servce provders and payments assocated wth the ncrease n total system losses. Interactve uzzy satscng method or multobjectve nonlnear programg s presented, by consderng that the decson maer has uzzy goals or each o the objectve unctons. Ater deterng the membershp unctons or each o the objectve unctons the completed () problem s solved, and the decson maer s suppled wth the correspondng Pareto optmal soluton and the trade-o rates between the membershp unctons. The rest o the paper s organzed as ollows: The mathematcal ormulaton o the reactve power prcng and TVD cost s presented. Then, a bre overvew o the PSO algorthm s descrbed. In nally, the smulaton results are presented and dscussed. Problem ormulaton In ths paper, the objectve uncton consst o two uncton. The rst objectve uncton s to mze the system actve power loss and overall producton cost o reactve power whch ncludes reactve power producton cost o generators and captal cost o capactors. The second objectve uncton s to mprove the voltage prole..1 Frst Objectve Functon.1.1 Cost o System Transmsson Loss The reactve support wll aect the transmsson loss. The cost uncton o transmsson loss and P Loss are consdered as ollows: C(P ) P (1) Loss Loss N TL Loss G V Vj VV j j 1 P ( cos ) () where s prce o electrcty; P Loss s the total actve power loss; G s the conductance o the th branch connected between the th and the jth bus; N TL s number o transmsson lnes; V, V j are the voltage magntude o the th and the jth bus, respectvely; j s the admttance angle o the transmsson lne connected between the th and the jth bus..1. Cost o Generator s Reactve Power The generators provde reactve support by consug or producng reactve power when operatng at leadng or laggng power actors, respectvely. The producton o reactve power may requre a decrease o real power output. Opportunty cost s the lost benet o ths decrease o real power output o the generator. Opportunty cost depends on supply and demand n maret, so t s hard to detere ts exact value. Hence, ths paper consders the opportunty cost o generator reactve power producton as modeled n [15]: C(Q G ) C( SG, ) C( SG, QG ) (3) where s the reactve power ecency rate (usually between 5% and 10 %), S G, s the mum apparent power n th bus and Q G the reactve power o generator n th bus. In (3), C (S G, ) and C( SG, QG ) are obtaned as ollows: C(S ) abs cs (4) G, G, G, C( S Q ) ab( S Q ) G, G G, G c( S Q ) G, G.1.3 Cost o Capactor Compensaton The charge or usng capactors s assumed proportonal to the amount o the reactve power output purchased and can be expressed as [16]: CCj ( QCj ) rj QCj (6) where r j and Q Cj are the reactve cost and amount purchased at locaton j, respectvely. The producton cost o the capactor s assumed as ts captal nvestment return, whch can be expressed as ts deprecaton rate. For example, the nvestment cost o a capactor s $/MVA, and ther le span and average worng rate are 15 years and /3, respectvely, the cost or deprecaton rate o the capactor can be calculated by: nvestment cost $11600 $0.134 r j (7) operatng hours MVAh 3 Thereore, the rst objectve uncton s proposed as mzng the summaton o reactve power producton costs, produced by generators and capactor bans and cost o power loss as ollows: 1 C( Ploss ) C(Q G ) CCj (Q Cj ) (8) N g jnc where Ng s the number o generators, N c the number o buses whch capactor bans are nstalled. (5). Second Objectve Functon..1 Improvement o Voltage Prole Treatng the bus voltage lmts as constrants n reactve power dspatch oten results n settng all the voltages toward ther mum lmts ater optmzaton, whch means the power system lacs the requred reserves to provde reactve power durng contngences. One o the eectve ways to avod ths stuaton s to choose the mzaton o the absolute 66 Iranan Journal o Electrcal & Electronc Engneerng, Vol. 1, No. 1, March 016

3 devatons o all the actual bus voltages rom ther desred voltages as an objectve uncton. Mnmzaton o TVD o load buses can allow the mprovement o voltage prole [17]. Ths objectve uncton can be ormulated as ollows: re ( TVD) V V (9) NL re where, V s the desred voltage magntude value at bus whch s usually set to 1.0 p.u..3 Mult Objectve Functon The proposed model sees to mze the ollowng objectve uncton : 1 w (1 w) (10) 1 base base where w s weghted coecent, 1base and base are base value o rst and second objectve uncton, respectvely..4 System Constrants.4.1 Equalty Constrant The reactve and real power balance equatons are the equalty constrants o optmal reactve power dspatch problem and are expressed as ollows: N B P P V V [ G cos( ) G D j j j j 1 or N B B j sn( j )] 0; 1,...,N Q Q V V [ G sn( ) G D j j j j 1 B B B j cos( j )] 0 or 1,..., N (11) (1) where G j and B j are the real and magnary part o the jth entry o the admttance matrx, respectvely. P D and Q D are the actve and reactve load demand o the th bus, respectvely. P G and Q G represent the actve and reactve power generaton o the th bus, respectvely. V, V j are the voltage magntude o the th and the jth bus, respectvely. and j represent the phase angle o the th and the jth bus voltages, respectvely.4. Inequalty Constrants The reactve power source capacty restrctons, transormer tap settng lmts, reactve generaton restrcton, bus voltage restrcton and power low through the transmsson lnes restrcton are used as nequalty constrants. In reactve power dspatch problem, the tap poston o transormers, generator bus voltages and the amount o the reactve power source nstallatons are the ndependent varables and these nequalty constrants are mathematcally expressed as [4]: V G V ; 1,..., G V G N G (13) Q C Q ; 1,..., C Q C N C (14) T T T ; 1,..., NT (15) where V and V are the mum and G G mum generator voltage o the th bus, respectvely. Q C and Q C are the mum and mum reactve power njecton o the th shunt compensator, respectvely. T and T are the mum and mum tap settng o the th transmsson lne, respectvely. N T s the number o tap changng transormers and N C s the number o shunt compensators. The reactve power output o generators, load voltages and transmsson lne loadng are the dependent varables and they are restrcted by ther upper and lower lmts as ollows: V V V ; 1,..., N (16) L L L L Q Q Q ; 1,..., N (17) G G G G SL S ; 1,..., L N TL (18) where, V L and V L are the mum and mum voltage o the th load bus, respectvely. Q G and Q G are the mum and mum reactve power generaton o the th generator bus, respectvely. S s L the mum apparent power low n the th lne and N L s the number o load buses..5 Fuzzy Satsyng Method Fuzzy satsyng (or ()) method s a popular technque or selecton o the best soluton among the obtaned Np Pareto optmal solutons [18]. Suppose we have a problem wth N objectves to be mzed. The lnear membershp uncton or the n-th soluton o the -th objectve uncton s dened as: n 1 n n n (19) n 0 or 1,..., N ; n 1,..., N where and are mum and mum values o the objectve uncton n solutons o Pareto optmal set. represents the optmalty degree o the n n-th soluton o the -th objectve uncton. The membershp uncton o n-th soluton can be calculated usng the ollowng equaton: n n n ( 1,..., N ); (0) or n 1,..., N p The soluton wth the mum weaest membershp uncton s the best soluton. The p Tabrz et al: An Interactve Fuzzy Satsyng Method Based on Partcle Swarm Optmzaton or 67

4 correspondng membershp uncton o ths soluton ( ), s calculated as ollows: 1 (,..., N P ) (1) 3 Proposed Methodology The PSO s one o the algorthms based on swarm ntellgence and ntroduced by Kennedy and Eberhart n 1995 or the rst tme [19]. The PSO s based on swarm ntellgence (SI) and models the swarm behavors such as brds locng and shes schoolng [0]. In PSO, the populaton s conssted rom canddate solutons whch called partcles. In PSO, each partcle moves n the search space wth a velocty accordng to ts own prevous best soluton and ts group s prevous best soluton. Each partcle updates ts poston and velocty wth the ollowng equatons: X ( t 1) X ( t) C V( t 1) () where X (t) and V (t) are vectors representng the poston and velocty o the -th partcle, respectvely and C s the constrcton actor and can be calculated as ollows: C (3) 4 The velocty o every partcle wll be updated by usng the ollowng equaton. V j ( t 1) V j ( t) c1r1j ( pbj X j ( t)) (4) cr j ( gbj X j( t)) where j ϵ 1,, ; d represents the dmenson o the partcle; w s nerta weght; c 1 and c are cogntve and socal component acceleraton coecents, respectvely; r 1j, r j are two unorm random sequences sampled rom [0, 1]; pb j s the personal best poston ound by the th partcle; gb j s the best poston ound by the entre swarm so ar and equals to C 1 +C. The PSO has been proven to be very eectve or statc and dynamc optmzaton problems. 4 Smulaton Results and Dscussons In ths paper, the proposed method s appled to IEEE 14-and 57-bus standard test power systems or the soluton o reactve power maret problem. The proposed algorthm s mplemented by usng the MATLAB 7.0 sotware on a PC wth Intel(R) Core(TM) 3-330M CPU.0GHz GB RAM. Set populaton sze o PSO s 100 and the number o mum teratons s IEEE 14-Bus System The standard IEEE 14-bus system s conssts o two generators (at the buses 1, ), twenty transmsson lnes and three branches under load tap settng transormer branches. The possble reactve power compensaton bus s 9. In ths case, 1base and base are $, p.u, respectvely. The proposed mult-objectve uncton s solved by usng PSO algorthm. Thus the Eq. (10) wll be optmal or changes n w rom zero to 1 n steps o 0.1 and present the set o solutons or 1 and. These solutons are called the set o Pareto optmal solutons. Ths set s non-convex soluton n the all space search or 1 and that between two derent answers cannot preer one over the other. The pareto dagram s shown n Fg. 1. For choosng the best soluton rom the Pareto dagram, the uzzy satsyng method s used. The membershp uncton or 1 and are dened wth usng Eq. (19). Table 1 Degree o optmzaton satsacton or each soluton. W F1 F µ 1 µ Mn (µ1,µ) F Iranan Journal o Electrcal & Electronc Engneerng, Vol. 1, No. 1, March 016

5 Fg. 1 The Pareto optmal ront o case 1. Fg. The Pareto optmal ront o case. The set o solutons or 1, and ts membershp unctons (μ 1, μ ) are presented n Table 1. The mum value o the membershp uncton or each o the unctons 1 and, or changes n w are located n the last column ( (μ 1, μ )). Wth usng uzzy satsyng method, the reported mum value or (μ 1, μ ) n last column Table.1 s chosen as Pareto optmal solutons. In ths case, the optmal soluton s obtaned or w = 0.9. The optmal soluton or 1, and ts related control varables such as generator voltage, transormers tap, shunt capactor are presented n Table. I the goal s to optmze the uncton 1, the optmal value s $ or the control varables lsted n the rst column o Table. I the goal s to optmze the uncton, the optmal value s or the control varables lsted n the thrd column o Table. In latest column, the optmal solutons are presented or mult objectve uncton. Accordng ths table, the obtaned best solutons or 1 and are $ and , respectvely. 4. IEEE 57-Bus System The standard IEEE 57-bus system conssts o seven generators (at the buses 1,, 3, 6, 8, 9, 1), eghty transmsson lnes and teen branches under load tap settng transormer branches. The possble reactve power compensaton buses are 18, 5 and 53. In ths case, 1base and base are 90.5 $ and p.u, respectvely. The PSO s used or solvng the proposed multobjectve uncton and the Pareto optmal ront o the solutons s obtaned as depcted n Fg.. The selecton o nal soluton usng uzzy satsyng approach s the next step ater ndng the Pareto optmal ront. The attrbutes o Pareto optmal ront solutons are descrbed n Table 3. The set o solutons or 1,, ts membershp unctons (μ 1, μ ) and (μ 1, μ ) or derent value o w are presented n Table 3. Wth usng uzzy satsyng method, the optmal soluton s obtaned n w = 0.8. The smulaton results or best soluton are shown n Table 4. In Table 4, the goal s to optmze the uncton 1, the optmal value s $ and the goal s to optmze the uncton, the optmal value s obtaned The optmum values or the unctons 1, or reported control varables are presented n columns and 3, respectvely. In latest column or w = 0.8, the optmal solutons are presented or mult objectve uncton. Table Smulaton results or best solutons. Varable Best Soluton 1 Best Soluton Best Soluton 1, Vg Vg Vg Vg Vg QC T T T ($) Conclusons In the study o the reactve power margnal prce n ths paper, the reactve power producton costs o generators and captal cost o capactors and mprovement o voltage prole are consdered n the objectve uncton o power low problem. In ths paper, nteractve uzzy satscng method usng the completed () problems has proposed n order to deal wth the uzzy goals o the decson maer n the nonlnear mult-objectve uncton. In ths nteractve scheme, ater deterng the membershp unctons, the satscng soluton o the decson maer can be derved by updatng desre membershp values based on the current values o the membershp unctons together wth the trade-o rates between the membershp unctons. In ths way the satscng soluton or the decson maer can be derved ecently rom among a Pareto optmal soluton set by updatng desre membershp values. The PSO algorthm s used to nd the optmal soluton o mult-objectve uncton. The valdty and eectveness o the proposed method s vered by usng standard IEEE 14-bus and IEEE 57- Tabrz et al: An Interactve Fuzzy Satsyng Method Based on Partcle Swarm Optmzaton or 69

6 bus test systems. In case 14 bus IEEE, the optmal soluton or 1 and are $ and , respectvely and n case 57 bus IEEE, the optmal soluton or 1 and are $ and , respectvely. Table 3 Degree o optmzaton satsacton or each soluton. W F1 F µ1 µ Mn (µ1,µ) F Table 4 Smulaton results or best solutons. Varable best soluton best soluton best soluton 1 1, Varable best soluton best soluton best soluton 1 Vg T Vg T Vg T Vg T Vg T Vg T Vg T QC T QC T QC T T T T T T ($) , Reerences [1] C. Canzares, K. Bhattacharya, I. El-Samahy, H. Haghghat, J. Pan and C. Tang, Re-denng the reactve power dspatch problem n the context o compettve electrcty marets, Generaton, Transmsson & Dstrbuton, IET, Vol. 4, No., pp , 010. [] M. El-Kady, B. Bell, V. Carvalho, R. Burchett, H. Happ and D. Verath, Assessment o real-tme 70 Iranan Journal o Electrcal & Electronc Engneerng, Vol. 1, No. 1, March 016

7 optmal voltage control, Power Systems, IEEE Transactons on, Vol. 1, No., pp , [3] J. Qu and S. Shahdehpour, new approach or mzng power losses and mprovng voltage prole, Power Systems, IEEE Transactons on, Vol., No., pp , [4] B. Mandal and P. K. Roy, Optmal reactve power dspatch usng quas-oppostonal teachng learnng based optmzaton, Internatonal Journal o Electrcal Power & Energy Systems, Vol. 53, pp , 013. [5] C. T. Su and C. T. Ln, Fuzzy-based voltage/reactve power schedulng or voltage securty mprovement and loss reducton, Power Delvery, IEEE Transactons on, Vol. 16, No., pp , 001. [6] N. Grudnn, Reactve power optmzaton usng successve quadratc programg method, Power Systems, IEEE Transactons on, Vol. 13, No. 4, pp , [7] I. El-Samahy, K. Bhattacharya and C. Cañzares, A uned ramewor or reactve power management n deregulated electrcty marets, Conerence A uned ramewor or reactve power management n deregulated electrcty marets. IEEE, pp , 006. [8] K. Bhattacharya and J. Zhong, Reactve power as an ancllary servce, Power Systems, IEEE Trans. on, Vol. 16, No., pp , 001. [9] S. Hao and A. Papalexopoulos, Reactve power prcng and management, Power Systems, IEEE Trans. on, Vol. 1, No. 1, pp , [10] J. Zhong and K. Bhattacharya, Toward a compettve maret or reactve power, Power Systems, IEEE Transactons on,vol. 17, No. 4, pp , 00. [11] J. W. Lamont and J. Fu, Cost analyss o reactve power support, Power Systems, IEEE Trans. on, Vol. 14, No. 3, pp , [1] S. Hao, A reactve power management proposal or transmsson operators, Power Systems, IEEE Trans. on, Vol. 18, No. 4, pp , 003. [13] M. Basu, An nteractve uzzy satsyng method based on evolutonary programg technque or multobjectve short-term hydrothermal schedulng, Electrc Power Systems Research, Vol. 69, No., pp , 004. [14] P. Hota, R. Charabart and P. Chattopadhyay, Economc emsson load dspatch through an nteractve uzzy satsyng method, Electrc Power Systems Research, Vol. 54, No. 3, pp , 000. [15] Y. Zhao, M. R. Irvng and Y. Song, A cost allocaton and prcng method or reactve power servce n the new deregulated electrcty maret envronment, Conerence A cost allocaton and prcng method or reactve power servce n the new deregulated electrcty maret envronment, IEEE, pp. 1-6, 005. [16] A. Ketab, A. Albabaee and R. Feullet Applcaton o the ant colony search algorthm to reactve power prcng n an open electrcty maret, Internatonal Journal o Electrcal Power & Energy Systems, Vol. 3, No. 6, pp. 6-68, 010. [17] W. Zhang and Y. Lu, Mult-objectve reactve power and voltage control based on uzzy optmzaton strategy and uzzy adaptve partcle swarm, Internatonal Journal o Electrcal Power & Energy Systems, Vol. 30, No. 9, pp , 008. [18] A. Soroud, R. Care, N. Hadjsad and M. Ehsan, Probablstc dynamc mult-objectve model or renewable and non-renewable dstrbuted generaton plannng, IET generaton, transmsson & dstrbuton, Vol. 5, No. 11, pp , 011. [19] R. C. Eberhart and J. Kennedy, A new optmzer usng partcle swarm theory, Conerence A new optmzer usng partcle swarm theory, New Yor, Vol. 1, pp , [0] J. Kennedy, J. F. Kennedy and R. C. Eberhart, Swarm ntellgence, Morgan Kaumann, 001. Navd Tabrz was born n Tabrz, Iran. He receved the B.Sc. and the M.Sc. degree n electrcal engneerng rom the Tabrz Unversty, Tabrz, Iran, n 010 and 015, respectvely. Hs research nterests nclude power system protecton, lexble ac transmsson systems and power system operaton. Ebrahm Babae was born n Ahar, Iran, n He receved the B.Sc. degree n Electronc Engneerng and the M.Sc. degree n Electrcal Engneerng rom the Department o Engneerng, Unversty o Tabrz, Tabrz, Iran, n 199 and 001, respectvely, graduatng wth rst class honors. He receved the Ph.D. degree n Electrcal Engneerng rom the Department o Electrcal and Computer Engneerng, Unversty o Tabrz, n 007. In 004, he joned the Faculty o Electrcal and Computer Engneerng, Unversty o Tabrz. He was an Assstant Proessor rom 007 to 011, an Assocate Proessor rom 011 to 015 and has been Proessor snce 015. He s the author o more than 300 journal and conerence papers. He also holds 17 patents n the area o power electroncs. Hs current research nterests nclude the analyss and control o power electronc converters and ther applcatons. Dynamc power system, power system transents. Pro. Babae has been the Edtor-n-Che o the Journal o Electrcal Engneerng o the Unversty o Tabrz, snce 013. He s also currently an Assocate Edtor o the Tabrz et al: An Interactve Fuzzy Satsyng Method Based on Partcle Swarm Optmzaton or 71

8 IEEE Transactons on Industral Electroncs. He s a Guest Edtor or a specal ssue on Recent Advances n Multlevel Inverters and ther Applcatons n the IEEE Transactons on Industral Electroncs. In 013, he was the recpent o the Best Researcher Award rom o the Unversty o Tabrz. Pro. Babae has been ncluded n the Top One Percent o the World s Scentsts and Academcs accordng to Thomson Reuters' lst n 015. energy management. Mehd Mehdnejad was born n Mandoab, Iran. He receved the B.Sc. degrees n electrcal power engneerng rom Isalmc Azad Unversty, Bonab Branch, Bonab, Iran n 013 and the M.Sc. degree rom the Tabrz Unversty, Tabrz, Iran, n 015. Hs research areas nclude power system plannng, dstrbuton networs, and 7 Iranan Journal o Electrcal & Electronc Engneerng, Vol. 1, No. 1, March 016