Optimal Reactive Power Dispatch Using Ant Colony Optimization Algorithm

Transcription

1 Proceedngs of the 14 th Internatonal Mddle East Power Systems Conference (MEPCO 10), Caro Unversty, Egypt, December 19-21, 2010, Paper ID 315. Optmal Reactve Power Dspatch Usng Ant Colony Optmzaton Algorthm A. A. Abou El-Ela, A. M. Knawy and M. T. Mouwaf R. A. El Sehemy Department of electrcal Engneerng Department of electrcal engneerng Faculty of engneerng Faculty of engneerng Mnoufya Unversty Kafrelshekh Unversty Egypt Egypt Abstract Ths paper proposes a procedure for solvng the optmal reactve power dspatch (ORPD) problem usng ant colony optmzaton (ACO) algorthm. The objectve of the ORPD s to mnmze the transmsson lne losses under control and dependent varable constrants usng proposed senstvty parameters of reactve power that dependent on a modfcaton of Fast Decoupled Power Flow (FDPF) model. The ACO algorthm s appled to the IEEE standard 14-bus and a real power system at west delta network as a part of the Unfed Egyptan etwork (UE). Smulaton results show the capablty of the proposed ACO algorthm for the ORPD problem compared wth those obtaned usng the conventonal optmzaton technques as lnear programmng (P), genetc algorthm (A) and partcle swarm optmzaton (PSO) technque. Index Terms Ant colony optmzaton, reactve power dspatch. I. ITRODUCTIO ORPD s one of the optmzaton problems n power systems that optmzed the control varables by conventonal, ntellgence and modern optmzaton technques. The objectve functon of ORPD s to mnmze the transmsson lne losses by optmzng the control varables such as generaton voltages, swtchable reactve power and on-load tap changers (OTC) under control and dependent varable constrants. Subbaraj and Rajnarayanan [1] proposed self-adapted real coded genetc algorthm (SARA) to solve optmal reactve power dspatch (ORPD) problem. The selfadaptaton n real coded genetc algorthm (RA) ntroduced by applyng the smulated bnary crossover (SBX) operator, bnary tournament selecton and polynomal mutaton. The problem formulaton nvolves contnuous (generator voltages), dscrete (transformer tap ratos) and bnary (VAR sources) decson varables. Aruna and Devaraj [2] presented real coded genetc algorthm (RA) for solvng the mult-objectve ORPD problem n a power system. Modal analyss of the system used for statc voltage stablty assessment. oss mnmzaton and mzaton of voltage stablty margn taken as the objectves. enerator termnal voltages, reactve power generaton of the capactor banks and tap changng transformer settng taken as the optmzaton varables. Mahadevan and Kannan [3] solved the ORPD problem by mnmzng the actve power losses for a fxed economc power dspatch by adjustng the control varables usng partcle swarm optmzaton (PSO) technque. To overcome the drawback of premature convergence n PSO, a learnng strategy ntroduced n PSO, and ths approach called, comprehensve learnng partcle swarm optmzaton (CPSO) also appled to the proposed problem. Subbaraj and Rajnarayanan [4] proposed a two-phase hybrd PSO approach to solve ORPD problem. In ths hybrd approach, the phase-1 used PSO to explore the optmal regon and the phase-2 appled drect search as local optmzaton technque for fner convergence. Through few recent years, ACO algorthms are employed to solve optmzaton problems n dfferent felds wth more accurate and effcently soluton compared wth conventonal and other modern optmzaton algorthms. akawro and Erlch [5] presented a hybrd method for solvng voltage stablty constraned optmal reactve power dspatch (VSCORPD) problem, wth mnmzng the energy loss at the current tme nterval and the cost of adjustng dscrete control devces whle mantanng system securty and voltage stablty constrants wthn the permssble lmts. An artfcal neural network (A) s traned to approxmate the voltage stablty margn durng the optmzaton process, whle modfed ACO used to solve the VSCORPD. Abbasy and Hossen [6] proposed a novel ACO-based ORPD approach and four dfferent ACO algorthms, ncludng the smple ant system and three of ts drect successors, eltst ant system (AS), rank-based AS and mn AS used to solve the ORPD problem, wth mnmzng the real power loss under decson and dependent varables constrants. In ths paper, an approach s proposed to solve the ORPD problem usng the ACO algorthm, whch based on the behavor of real ants for searchng the shortest route between the colony and the source of food based on the ndrect communcaton meda, called pheromone. The mnmzaton of the transmsson lne losses are consdered as an objectve functon wth control and dependent varable constrants. II. PROBEM FORMUATIO The ORPD problem can be expressed as an optmzaton problem wth the transmsson lne losses that are a functon n the generator voltages and swtchable reactve power, whch are defned as: 960

2 MnΔP ( P / V ) + ( P / Q ) 1 1 P s the objectve functon of the change n transmsson lne losses. V and Q are the changes n the control varables of generaton voltages and swtchable reactve power, respectvely. s the number of generaton buses. s the number of buses whch the swtchable reactve sources are located. ow, the changes n transmsson lne losses wth respect to changes n generator voltage can be derved based on the load flow equatons as; also; S I (2) * j 1 {[ cosδ + B snδ ] + j[ snδ B cosδ ]} V j S P + jq (3) From equatons (2) and (3), the actve power at bus can be formulated as: P [ cos δ + B sn δ ] V j j 1 At, j, equaton (4) can be rewrtten as; P 2 + V [ cos δ + B sn δ ] V j j 1, j ow, the values of P / V and P j / V can be calculated by dfferentatng equaton (5) as: [ cos δ B sn ] V j P / V δ j 2V + + j 1, j j j [ cos δ B sn δ ] P / V j Y -1 / Z +j B ; s the lne admttance between buses and j. δ and δ j are the angles of the voltages at buses and j, respectvely. Also, the change n transmsson lne losses due to changes n swtchable reactve power can be wrtten as: ( P / V )( V Q ) j (7) P / Q / (8) V / Q : s the change of swtchable voltage buses wth respect to change n swtchable reactve power at that bus. The objectve functon (1) s subjected to the followng constrants: a) Control varable constrants eneraton voltage constrants The change n generator voltage must be wthn ther permssble lmts as: mn Δ V (9) (1) (4) (5) (6) V nt mn mn nt nt s the ntal value of the generator voltage at each generator bus. mn V and V are the mnmum and mum of generaton voltages, respectvely. V man and V are the mnmum and mum changes n generaton voltages, respectvely. Swtchable reactve power constrants The change n swtchable reactve power must be wthn ther permssble lmts as: mn Δ Q (10) Δ Q mn mn nt nt Δ Q Q nt s the ntal value of the swtchable reactve power. Q mn and Q are the mnmum and mum of swtchable reactve power, respectvely. Q man and Q are the mnmum and mum changes n swtchable reactve power, respectvely. b) Dependent varable constrants oad voltage constrants The voltage at each load bus must be wthn ther permssble lmts as: Δ V (11) V nt mn mn mn nt nt s the ntal value of the voltage at each load bus. V mn and V are the mnmum and mum of load bus voltages, respectvely. V man and V are the mnmum and mum changes n load bus voltages, respectvely. eneraton reactve power constrants The reactve power of each generator must be wthn ther permssble lmts as: Δ Q Δ Q Δ Q (12) mn mn mn nt nt nt Q s the ntal value of reactve power at generaton buses. 961

3 Q mn and Q are the mnmum and mum values of reactve power at generaton buses, respectvely. man Q and Q are the mnmum and mum changes n reactve power at generaton buses, respectvely. III. PROPOSED SESITIVITY PARAMETERS The FDPF method s one of the load flow methods that assumed the changes n actve power flows due to changes n voltage phase angles are hgher than the changes due to voltage magntudes. On the other hand, the changes n reactve power flows due to changes n voltage magntudes are hgher than the changes due to voltage phase angles. So, t can be wrtten as: [ P/V] [B'] [ δ] and [ Q/V] [B"] [ V] [ P] and [ Q] are the vectors of actve and reactve power msmatches, respectvely. [B'] and [B"] are the susceptance matrces. [ δ] and [ V] are the vectors of changes n voltage angles and magntudes, respectvely. In ths paper, the ORPD problem s based on the second equaton of the FDPF as: [ Q/V] [B"] [ V] (13) B" s the susceptance matrx as: B B " " 1 1 / j 1, j X ; 1,2,..., 1 " B 1/ X ;, j 1,2,..., 1 j X : s the lne reactance between buses and j. ow, equaton (13) can be rewrtten n terms of control and dependent varables as: / V B B (14) / V B B The senstvty parameters between control and dependent varables can be derved based on equaton (14) as fellows: a) The load bus voltages related to control varables o changes n the reactve power at load-bus ( Q 0) The senstvty parameters relatng the changes n loadbus voltages due to the changes n generaton-bus voltages are gven as: Δ V S 1 [ S ] [ B ] [ B ] o changes n generaton voltages ( V 0) (15) The senstvty parameters relatng to the changes n load-bus voltages due to the changes n swtchable reactve power sources can be gven as: Δ V S (16) 1 1 [ S ] [ B ] V and, V s the vector of bus voltages that connected to VAR sources. b) eneraton reactve power related to control varables The senstvty parameters relatng to the changes n generated reactve power due to the changes n generatonbus voltages can be wrtten as: [ B ] + [ B ] Δ Q / V (17) By substtutng from equaton (15) nto (17), we obtan: Δ Q S (18) { [ ] } S B + B S The senstvty parameters relatng to the changes n generated reactve power due to the changes n swtchable reactve power sources can be wrtten as: 1 [ B][ S ] + [ B] / V (19) By substtutng from equaton (16) nto (19), we get: Δ Q S 1 [ S ] {[ B ][ S ] + [ B ]}[ S ] (20) ow, the proposed senstvty between the control and the dependent varables can be formulated based on equatons (15), (16), (18) and (20) n a compact matrx form as: S S Q S Q Δ S Δ IV. ACO AORITHM (21) A random amount of pheromone s deposted n each rout after each ant completes t s tour, anther antes attract to the shortest route accordng to the probablstc transton rule that depends on the amount of pheromone deposted and a heurstc gude functon as equal to the nverse of the dstance between begnnng and endng of each route. The probablstc transton rule of ant k to go from cty to cty j can be expressed as: P α β [ τ ( t) ] [ η ( t) ] α [ τ ( t) ] [ η ( t) ] k ( t) β q q q ; j, q k (22) τ s the pheromone tral deposted between cty and j by ant k, η s the vsblty or sght and equal to the nverse of the dstance or the transton cost between cty and j ( η 1/d ). α and β are two parameters that nfluence the relatve weght of pheromone tral and heurstc gude functon, respectvely. If α0, the closest ctes are more lkely to be selected that correspondng to a classcal greedy algorthm. On the contrary, f β0, the probablty wll be 962

4 depend on the pheromone tral only. These two parameters should be tuned wth each other, q s the ctes that wll be vsted after cty. r k s a tabu lst n memory of ant that recodes the ctes whch wll be vsted to avod stagnatons. After each tour s completed, a local pheromone update s determned by each ant dependng on the route of each ant as n equaton (23), after all ants attractve to the shortest route, a global pheromone update s consdered to show the nfluence of the new addton deposts by the other ants that attractve to the best tour as shown n equaton (24): ( 1 ρ ) τ ( ρτ o τ ( t + 1) t) + (23) ( 1 ρ ) τ ( t) + εδτ ( ) τ ( t + 1) t (24) τ (t+1) s the pheromone after one tour or teraton, ρ s the pheromone evaporaton constant, ε s the elte path weghtng constant, τ o 1 / d s the ncremental value of pheromone of each ant. Whle, τ s the amount of pheromone for elte path as: Δ τ ( t ) 1 / (25) d best d best s the shortest tour dstance. V. ACO AORITHM FOR ORPD PROBEM ACO algorthm s appled to solve the ORPD problem as an optmzaton technque wth control and dependent varable constrants where artfcal ants travels n search space to fnd the shortest route that havng the strongest pheromone tral and a mnmum objectve functon. Our objectve n ths paper s to mnmze the change n power losses as descrbed n (1) wth control varable constrants n (9) and (10), and dependent varable constrants n (11) and (12). So that, the heurstc gude functon s the nverse of the ndvdual change n losses of each ant that postoned n the reasonable lmt of the control varable to the vsblty of each ant. Whle, heurstc gude functon of the problem s the nverse of the total change n losses at teraton t +1, as: η ( t + 1) 1/ ( P / V ) + ( P / Q ) (26) 1 1 In ACO algorthm, a search space creates wth dmensons of stages on number of control varables and states or the randomly dstrbuted values of control varables wth n a reasonable threshold. Artfcal ants leaves colony to search randomly n the search space based on the probablty n (22) to complete a tour matrx that conssts of the postons of ants wth the same dmenson of the search space. Then, tour matrx s appled on the objectve functon to fnd a heurstc gude functon to fnd the best soluton and update local and global pheromone to begn a next teraton. System parameters are adjusted by tral and error to fnd the best values of theses parameters. The ACO algorthm can be appled to solve the ORPD problem usng the followng seven steps: Step 1: Intalzaton Insert the lower and upper boundares of each control varable [( V mn, V ) and ( Q mn, Q )], system parameters, and create a search space wth a dmensons of number of control varables ( V, Q ) and the length of randomly dstrbuted values wth the same dmenson of the ntal pheromone that contans elements wth very small equal values to gve all ants wth the same chance of searchng. Step 2: Provde frst poston Each ant s postoned on the ntal state randomly wthn the reasonable range of each control varable n a search space wth one ant n each control varable n the length of randomly dstrbuted values. Step 3: Transton rule Each ant decde to vst a next poston n the range of other control varables accordng to the probablty transton rule n equaton (22) that depends on the amount of pheromone deposted and the vsblty that s the nverse of objectve functon (26). the effect of pheromone and vsblty on each other depends on the two parameters α and β. Step 4: ocal pheromone updatng ocal updatng pheromone s dfferent from ant to other because each ant takes a dfferent route. The ntal pheromone of each ant s locally updated as n (23). Step 5: Ftness functon After all ants attractve to the shortest path that havng a strongest pheromone, the best soluton of the objectve functon s obtaned Step 6: lobal pheromone updatng Amount of pheromone on the best tour becomes the strongest due to attractve of ants for ths path. Moreover, the pheromone on the other paths s evaporated n tme. Step 7: Program termnaton The program wll be termnated when the mum teraton s reached or the best soluton s obtaned wthout the ants stagnatons. The proposed procedure steps are shown n Fg. 1. A. Test Systems VI. APPICATIOS The standard IEEE 14-bus system [7] and a real power system at West Delta network as a part of the Unfed Egyptan etwork (UE) [8] are used to show the capablty of ACO algorthm for solvng solve the ORPD problem. The MVA base s taken 100 and the cost of power losses s assumed 0.07 E.P. /KWh whle, the cost of reactve power s assumed 15 E.P. /KVar. 963

5 Start Intalzaton; Insert parameters, control lmts and ntal pheromone enerate the ntal poston randomly of each ant Apply state transton rule Apply local pheromone updatng rule Intal value of load voltages V Vmn o Ftness functon evaluaton Apply global pheromone updatng rule Max ter. s reached Yes oad buses Fg. 3 Intal values of load voltages for west delta system B. Results and Comments The results of the ACO algorthm are obtaned usng the Matab code verson 7.1 that setup on a Pentum 4, 3.0 Hz PC, 0.99 B of RAM. Stop Fg. 1 Flow chart the proposed ACO algorthm The results obtaned are compared wth those obtaned usng a conventonal lnear programmng (P), A and PSO technque. The best values of ACO algorthm parameters are α 1, β5 ρ0.5 and ε5. The load flow s done usng the ewton-raphson load flow to get the values of power loss that are and MW for 14-bus and west delta system, respectvely. Fgures 2 and 3 show the mnmum, mum lmts and ntal values of control and dependent varables for 14-bus test system and West Delta system. IEEE 14-bus test system Tables I, II and Fg. 4 show the ORPD usng dfferent optmzaton technques. Table I shows the results of control varables (V and Q ) and dependent varables (Q ). Whle, Fg. 4 shows the other dependent varables of load bus voltages (V ) that elmnate the voltage volatons usng A, PSO and ACO. Whle, buses 6 and 8 are stll volated ther lmts usng the conventonal P. Table II shows the results of power loss, cost of power loss (CP), root mean square of load-bus voltage devaton wth respect to the flat voltage (V.D.) rms, reactve power reserve (RPR) and cost of the connected reactve power (CRP). In Table II and Fg. 4, the ACO algorthm has mnmum values of power loss, CP, (V.D.) rms and CRP compared wth other technques. In addton, t has the mum value of RPR compared wth other technques. Intal value of load voltages V oad buses Fg. 2 Intal values of load voltages for 14-bus system TABE I COTRO AD DEPEDET VARIABES USI DIFFERET TECHIQUES FOR 14-BUS SYSTEM. Varables P A PSO ACO V V Q Q Q Q TABE II A COMPARISO BETWEE DIFFERET OPTIMIZATIO TECHIQUES FOR 14-BUS SYSTEM. Varables P A PSO ACO Power oss (pu.) CP ( EP) (V.D.)rms RPR CRP ( EPx103)

6 oad bus voltage (pu.) V Vmn. P A PSO ACO oad buses Fg. 4 oad bus voltage usng dfferent technques for 14-bus system West delta system Tables III, IV and Fg. 5 show the ORPD usng dfferent optmzaton technques. Table III shows the results of control varables (V ) and dependent varables (Q ), whle Fg. 5 shows the other dependent varables of load bus voltage (V ). In Table IV, the ACO algorthm has mnmum values of power loss wth mnmum cost and mnmum voltage devaton compared wth other technques. Also, the load voltages based on ACO algorthm are the nearest to the flat voltage compared to the other technques. TABE III COTRO AD DEPEDET VARIABES USI DIFFERET TECHIQUES FOR WEST DETA SYSTEM. Varables P A PSO ACO V V V V V V V V Q Q Q Q Q Q Q Q TABE IV A COMPARISO BETWEE DIFFERET OPTIMIZATIO TECHIQUES FOR WEST DETA SYSTEM Varables P A PSO ACO Power oss (pu.) CP ( EP) (V.D.)rms oad bus voltage (pu.) V Vmn P A PSO ACO oad buses Fg. 5 oad bus voltage usng dfferent technques for west delta system VII. COCUSIO Ths paper presents an ORPD based on ACO algorthm. The objectve of the ORPD s to mnmze the transmsson lne losses under control and dependent varable constrants whch are proposed senstvty parameters of reactve power that depend on a modfcaton of Fast Decoupled Power Flow (FDPF) model. Three test systems have been used to show the capablty of the proposed algorthm compared wth other conventonal P, A and PSO. The ACO algorthm leads to mnmum power losses wth mnmzng of voltage devaton, cost of swtchable reactve power and mzng reactve power reserve compared to the other proposed algorthms. So, the proposed ACO algorthm gves more accurate and effcently soluton for ORPD problem.. Moreover, the proposed algorthm represents a potental tool to ad the power system operators n the on-lne envronment. REFERECES [1] P. Subbaraj and P.. Rajnarayanan, "Optmal Reactve Power Dspatch Usng Self-Adapted Real Coded enetc Algorthm", Electrc Power Systems Research, Vol. 79, o. 2, pp , February [2] P. Aruna Jeyanthy and D. Devaraj, "Optmal Reactve Power Dspatch for Voltage Stablty Enhancement Usng Real Coded enetc Algorthm", Internatonal Journal of Computer and Electrcal Engneerng, Vol. 2, o. 4, pp , August [3] K. Mahadevan and P. S. kannan, "Comprehensve earnng Partcle Swarm Optmzaton for Reactve Power Dspatch Electrc Power Systems Research, Vol. 10, o. 2, pp , March [4] P. Subbaraj and P.. Rajnarayanan, "Hybrd Partcle Swarm Optmzaton Based Optmal Reactve Power Dspatch", Internatonal Journal of Computer Applcatons, Vol. 1, o. 5, pp , February [5] W. akawro and I. Erlch, "A Hybrd Method for Voltage Stablty Constraned Optmal Reactve Power Dspatch", IEEE Power and Energy Socety eneral Meetng, pp. 1-8, July [6] A. Abbasy and S. H. Hossen, "Ant Colony Optmzaton-Based Approach to Optmal Reactve Power Dspatch: A Comparson of Varous Ant Systems", IEEE Power Engneerng Socety (PES) Conference and Exposton, pp. 1-8, July, South Afrca, [7] Washngton Unversty Webste: [8] A. A. Abou El-Ela, S. M. Allam and M. M. Shatla, "Maxmal Optmal Benefts of Dstrbuted eneraton Usng enetc Algorthms ", Electrc Power Systems Research, Vol. 80, o. 7, pp , July