Stochastic Weight Trade-Off Particle Swarm Optimization for Optimal Power Flow

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1 Stochastc Weght Trade-Off Partcle Swarm Optmzaton for Optmal Power Flow Luong Dnh Le and Loc Dac Ho Faculty of Mechancal-Electrcal-Electronc, Ho Ch Mnh Cty Unversty of Technology, HCMC, Vetnam Emal: Jrawadee Polprasert and Weeraorn Ongsaul Energy Feld of Study, School of Envronment, Resources and Development, Asan Insttute of Technology, Pathumtnan 1212, Thaland Emal: Deu Ngoc Vo and Dung Anh Le Department of Power Systems, Ho Ch Mnh Cty Unversty of Technology, HCMC, Vetnam Emal: Abstract Ths paper proposes a stochastc weght trade-off partcle swarm optmzaton () method solvng optmal power flow (OPF) problem. The proposed SWTPSO s a new mprovement of PSO method usng a stochastc weght trade-off for enhancng search ts search ablty. The proposed method has been tested on the IEEE 3 bus and 57 bus systems and the obtaned results are compared to those from other methods such as conventonal PSO, genetc algorthm (GA), ant colony optmzaton (ACO), evolutonary programmng (EP), and dfferental evoluton (DE) methods. The numercal results have ndcated that the proposed method s better than the others n terms of total fuel costs, total loss and computatonal tmes. Therefore, the proposed method can be a favorable method for solvng OPF problem. optmzaton (ACO), genetc algorthm (GA), mproved evolutonary programmng (IEP), tabu search (TS), smulated annealng (SA), etc. These methods have been effectvely for solvng the problem. In 1995, Eberhart and Kennedy suggested a partcle swarm optmzaton (PSO) method based on the analogy of swarm of brd flocng and fsh schoolng [1]. Due to ts smple concept, easy mplementaton, and computatonal effcency when compared wth mathematcal algorthm and other heurstc optmzaton technques, PSO has attracted many attentons and been appled n varous power system optmzaton problems such as economc dspatch [2]-[5], reactve power and voltage control [6]-[8], transent stablty constraned optmal power flow [9], and many others [1], [11]-[13]. In ths paper, a stochastc weght trade-off partcle swarm optmzaton () algorthm s proposed by mprovement of conventonal PSO method wth new parameter for better optmal soluton and faster computaton. The proposed method has been tested on the IEEE 3-bus system wth quadratc fuel cost functon and fuel cost functon wth valve pont effects fuel functon and the IEEE 57-bus system. The obtaned results are compared to those from many other methods n the lterature such as genetc algorthm GA [2], ant colony optmzaton (ACO) [21], mproved evolutonary programmng (IEP) [22], evolutonary programmng (EP) [23], gravtatonal search algorthm (GSA) [24], dfferental evoluton (DE OPF) [25], modfed dfferental evoluton (MDE OPF) [25], base-case [28], and Matpower [28]. Index Terms optmal power flow, partcle swarm optmzaton, stochastc weght trade-off, quadratc fuel functon, valve pont effects I. INTRODUCTION Optmal power flow (OPF) problem s the mportant fundamental ssues n power system operaton. In essence, t s the optmzaton problem and ts man objectve s to reduce the total generaton cost of unts whle satsfyng unt and system constrants. Although the OPF problem developed long tme ago but so far t has been extensvely studed due to ts mportance n power system operaton. There have been many methods developed to solve OPF problem from classcal methods such as Newton s method, gradent search, lnear programmng (LP), nonlnear programmng, quadratc programmng (QP), etc to methods based on artfcal ntellgence and evolutonary based methods such as ant colony II. The OPF problem can be descrbed as an optmzaton (mnmzaton) problem wth nonlnear objectve functon Manuscrpt receved July 1, 214; revsed November 21, 214. do: /joace OPTIMAL POWER FLOW PROBLEM 31

2 and nonlnear constrants. The general OPF problem can be expressed as follows: Mnmze F (u, x) (1) Subject to g (u, x) = (2) h (u, x) (3) where F (u, x) s the objectve functon, g (u, x) represents the equalty constrants, h (u, x) represents the nequalty constrants, and u s the vector of the control varables such as generated actve power, generaton bus voltage magntudes, transformers taps, etc), and x s state varables such as reactve power, load bus voltage magntude, bus voltage angle, etc). The essence of the optmal power flow problem resdes n reducng the objectve functon and smultaneously satsfyng the load flow equatons (equalty constrants) wthout volatng the nequalty constrants. The fuel cost of generators n form of quadratc functon s gven by: N G 2 ( ) ( G G ) 1 F x a b P c P (4) where, N G s the number of generators ncludng the slac bus, P G s the generated actve power at bus, a, b and c are the unt costs curve for th generator. The smooth quadratc fuel cost functon wthout valve pont loadngs of the generatng unts are gven by (4), where the valve-pont effects are gnored. The generatng unts wth mult-valve steam turbnes exhbt a greater varaton n the fuel-cost functons. Snce the valve pont results n the rpples, a cost functon contans hgher order nonlnearty. Therefore, the equaton (4) should be replaced by (5) for consderng the valve-pont effects. The snusodal functons are thus added to the quadratc cost functons as follows. F P a b P c P e f P P (5) 2 ( ) sn( (,mn )) where e and f are the fuel cost coeffcents of the th unt wth valve pont effects. The shape of fuel cost functon wth valve loadng effects s gven tn Fg. 1. Whle mnmzng the cost functon, t s necessary to mae sure that the generaton stll supples the load demands plus losses n transmsson lnes. Usually the power flow equatons are used as equalty constrants [14]. P (, ) ( ) P V PG P D Q Q ( V, ) ( QG Q ) D where actve and reactve power njecton at bus are defned n the followng equaton N B (6) (7) P ( V, ) VV G cos B sn j j j j j j 1 N B (8) Q ( V, ) VV G sn cos B j j j j j j 1 The nequalty constrants of the OPF reflect the lmts on physcal devces n power systems as well as the lmts created to ensure system securty. The most usual types of nequalty constrants are upper bus voltage lmts at generatons and load buses, lower bus voltage lmts at load buses, reactve power lmts at generaton buses, maxmum actve power lmts correspondng to lower lmts at some generators, maxmum lne loadng lmts, and lmts on tap settng. The nequalty constrants on the problem are as follows: Generaton constrant: Generator voltages, real power outputs, and reactve power outputs are restrcted by ther upper and lower bounds: P P P for = 1, 2,....., N G (9) G,mn G G,max Q Q Q for = 1, 2,....., N G (1) G,mn G G,max V V V for = 1, 2,....., N G (11) G,mn G G,max Shunt VAR constrant: Shunt VAR compensatons are restrcted by ther upper and lower bounds: Q Q Q for = 1, 2,....., N C (12) C,mn C C,max where N C s the number of shunt compensators. Tap changer constrant: Transformer tap settngs are restrcted by ther upper and lower bounds: T T T for = 1, 2,....., N T (13),mn,max where N T s the number of transformer taps. Securty constrant: Voltage magntudes at load buses are restrcted by ther upper and lower bounds as follows V V V for = 1, 2,....., N L (14) L,mn L L,max Fgure 1. Example cost functon wth 6 valves [14] where N L s the number of load buses. 32

3 III. IMPROVEMENT OF PSO B. Durng the study PSO algorthm we found that the expresson affects the ablty of the algorthm convergence manly fallng nto the velocty updatng (16). In ths expresson, the two components ncludng the cogntve factor c1 and the socal factor c2 are ndependent on each other. If both coeffcents are too large or small, there wll be effects the convergence of the algorthm. In the case both coeffcents are too large, the search space s too far beyond the see regon, mang dffculty for the algorthm convergence whle both factors are too small, the search space s too narrow, leadng to nexact optmal results. To solve these problems, we propose several mprovements to the PSO algorthm. Among the mproved PSO methods, the stochastc weght trade-off PSO () [29] s proposed to solve the optmal power flow problem due to the goal of balancng between partcle experences and socal relatonshps as follows: 1) Improvement of r1 and r2 coeffcents r1 and r2 coeffcents are also the addtonal factors (1 r1) and (1-r2) as n expresson (17). The terms (1-r2) r1 and (1-r1) r2 wll mprove the algorthm effcency to converge faster to the optmal soluton. When both r1 and r2 are too large or too small, the term (1-r2) r1 wll lead to an mbalance for the algorthm. It s smlar to the case for the term (1-r1) r2. Ths method enables the algorthm to create a balance between the two components of personal experences and learnng from the communty. 2) Improvement c1 and c2 coeffcents The coeffcents c1 and c2 now are not constant as the orgnal PSO algorthm. We are recommend the tme varyng coeffcents c1 () and c2 () as n (18) and (19). The two expressons wll create the value of c1 and c2 factors large at the ntalzaton and decrease them untl the maxmum number of teratons reached. When startng, the algorthm searches for large space to put to the best possble area. At the algorthm termnaton, the c1 and c2 factors gude the algorthm converge to the optmal result. A. Overvew of the PSO The conventonal PSO was orgnally ntroduced by Kennedy and Eberhart as an optmzaton technque nspred by swarm ntellgence such as brd flocng, fsh schoolng, and even human socal behavor. Partcles representng canddate solutons change ther postons wth tme through search space. Durng the flght, each partcle adjusts ts poston accordng to ts own experence and the experence of neghborng partcles as a constructve cooperaton by mang use of the best postons encountered by tself and ts neghbors [1]. The poston mechansm of the partcles n the search space s updated by addng the velocty vector to ts poston vector as gven n equaton (2) and as llustrated n Fg. 2 [15]. Let X = (x1,, xn) and V = (v1,, vn) be partcle poston and ts correspondng velocty n a n-dmensonal search space, respectvely. The best poston acheved by a partcle s recorded and denoted by Pbest ( x1pbest,..., xnpbest ). The best partcle among all partcles n the populaton s represented as Gbest ( x1gbest,..., xngbest ). The updated velocty and poston of a partcle can be calculated by: X 1 X V 1 Stochastc Weght Trade-off (15) where V+1 s the velocty of ndvdual at teraton +1 gven by: V 1 V c1r1 ( Pbest X ) c2 r2 (Gbest X ) (16) X poston of ndvdual at teraton, X+1 poston of ndvdual at teraton +1, V velocty of ndvdual at teraton, c1 cogntve factors, c2 socal factors, Pbest the best poston of ndvdual untl teraton, Gbest the best poston of the group untl teraton, r1, r2 random numbers between and 1. V 1 rv 1 (1 r2 )c1 ( )r1 ( Pbest X ) + (1 r1 )c2 ( )r2 (Gbest X ) c1 ( ) c1mn c2 ( ) c2 mn (c2 max c2 mn ) max r1, r2 random numbers between and 1, c1mn and c1max ntal and fnal cogntve factors, c2mn and c2max ntal and fnal socal factors, max maxmum teraton number, current teraton number. Fgure 2. Concept of a searchng pont by PSO [15] (c1max c1mn ) max 33 (17) (18) (19)

4 C. TABLE I. Procedure of OPF Problem The mplementaton of algorthm to solve OPF problem can be descrbed as follows: Varable Vg1 (pu) Vg2 (pu) Vg5 (pu) Vg8 (pu) Vg11 (pu) Vg13 (pu) T11 (pu) T12 (pu) T15 (pu) T36 (pu) Qc1 (MVAr) Qc24 (MVAr) CPU Tme (s) Total Cost ($/h) Step 1: Choose the populaton sze, the number of generatons and coeffcents c1mn, c1max, c2mn, c2max. Step 2: Intalze the velocty and poston of all partcles by randomly settng ther values wthn the pre-specfed boundares. Set the value of partcle postons to Pbest and the partcle correspondng to the best case to Gbest. Step 3: Set the teraton counter = 1 and partcle counter = 1. Step 4: For each partcle, solve AC power flow usng Newton Raphson s method. Step 5: Evaluate the ftness functon for each partcle accordng to the objectve functon. Step 6: Compare partcle s ftness evaluaton wth ts Pbest. If the current value s better than Pbest, set Pbest to the current value. Identfy the partcle wth the neghborhood wth the best success so far, and assgn ts ndex to Gbest. Step 7: Update the partcle velocty by usng the global best and ndvdual best of each partcle accordng to (17). Step 8: Update partcle poston by usng (15). Step 9: If < total number of partcles, = + 1 and return to Step 4. Step 1: If < Number of teratons, set = 1 and = + 1, return to Step 4. Step 11: Stop the algorthm. IV. Mn Max Optmal Soluton To verfy the feasblty of the proposed method, the standard IEEE 3-bus system [17] has been used to test the OPF problem. The system lne and bus data are gven n [18]. The system has sx generators located at buses 1, 2, 5, 8, 11, and 13 and four transformers wth off-nomnal tap rato n lnes 6-9, 6-1, 4-12, and The cost curve coeffcents are gven n [19]. Table I shows the optmal dspatches of the generators. Also note that all outputs of generator are wthn ts permssble lmts. The obtaned results of the are compared wth those of other methods n Table II ncludng GA [2], ACO [21], IEP [22], and EP [23]. In Table II, t s observed that algorthm gves better total cost than other methods n a fester manner. These results have shown that the proposed method s feasble and ndeed capable of acqurng better soluton. Fg. 3 shows the convergence characterstc of. NUMERICAL RESULTS The proposed method s tested on two systems ncludng the IEEE 3-bus system wth quadratc fuel functon and valve pont effects and the IEEE 57-bus system wth quadratc fuel functon. The algorthm of the method s coded n Matlab platform and run on a 2.5 GHz wth 4 GB of RAM PC. The control parameters of the method for all test systems are selected as follows: the cogntve and socal parameters are respectvely set to c1() and c2() wth c1max = c2max = 2.5, c1mn = c2mn =.5, the velocty lmt coeffcent s set to.15 (R =.15), the maxmum number of teratons ITmax s set to 2; the number of partcles Np s set to 15 for the IEEE 3-bus system and 25 for the IEEE 57-bus system. All penalty factors n the ftness functon are set to 16. For each system, the proposed method s run 1 ndependent trals and the obtaned optmal results are compared to those from other methods. The obtaned results for the systems nclude mnmum total cost, power losses, and computatonal tme. TABLE II. RESULT COMPARISON FOR THE IEEE 3 BUS SYSTEM WITH QUADRATIC FUEL COST FUNCTION Varable GA [2] ACO [21] IEP [22] CPU Tme (s) (mnutes) Total Cost ($/h) EP [23] Case 2: The IEEE 3-bus system wth valve pont effects fuel functon. In ths case, the generatng unts of buses 1 and 2 are consdered to have the valve-pont effects on ther fuel cost characterstcs. The fuel cost coeffcents of these generators are taen from [28]. The fuel cost coeffcents Case 1: The IEEE 3 bus system wth quadratc fuel cost functon OPTIMAL SOLUTION FOR THE IEEE 3-BUS SYSTEM WITH QUADRATIC FUEL COST FUNCTION 34

5 of the remanng generators have the same values as of the test Case 1. wth BASE-CASE [28], MATPOWER [28], and conventonal PSO as presented n Table VI. The result comparson has demonstrated that the proposed method can gve the lowest producton cost wthn reasonable tme. The convergence characterstc of the best fuel cost result obtaned from the approach s shown n Fg. 4. TABLE III. OPTIMAL SOLUTION FOR THE IEEE 3-BUS SYSTEM WITH VALVE POINT EFFECTS Varable Vg1 (pu) Vg2 (pu) Vg5 (pu) Vg8 (pu) Vg11 (pu) Vg13 (pu) T11 (pu) T12 (pu) T15 (pu) T36 (pu) Qc1 (MVAr) Qc24 (MVAr) CPU Tme (s) Total Cost ($/h) Fgure 3. Convergence characterstc wth quadratc fuel functon for the IEEE 3-bus system Table III shows the generaton outputs of the best soluton. We have also observed that the soluton obtaned by always satsfes the equalty and nequalty constrants. The result comparson n Table IV has ndcated that the algorthm gves better results than other methods wth the percentage as follow IEP [22] 3.3%, GSA [24].82%, DE OPF [25].96%, MDE OPF [25].93%. Therefore, the proposed SWTPSO s very effectve for solvng the OPF problem wth valve pont loadng effects. TABLE IV. Mn Varable CPU Tme (s) Total Cost ($/h) Optmal Soluton RESULTS FOR THE VALVE POINT EFFECTS FOR IEEE 3BUS SYSTEM WITH DIFFERENT METHOD Case 3: The IEEE 57 bus system To evaluate the effectveness and effcency of the proposed approach n solvng larger power system, a standard IEEE 57-bus test system s consdered. The IEEE 57-bus system conssts of 7 generaton buses, 5 load buses, and 8 branches. The generators are located at buses 1, 2, 3, 6, 8, 9, and 12 and 15 transformers are located at branches 19, 2, 31, 37, 41, 46, 54, 58, 59, 65, 66, 71, 73, 76, and 8. The system has also 3 swtchable capactor bans nstalled at buses 18, 25, and 53. For dealng wth ths system, there 31 control varables to be handled ncludng real power output of 6 generators except the generator at the slac bus, voltage at 7 generaton buses, tap changer of 15 transformers, and reactve power output of 3 swtchable capactor bans. The total load demand of system s MW and MVAR. The bus data, lne data, cost coeffcents, and mnmum and maxmum lmts of real power generatons are taen from [26], [27]. The maxmum and mnmum values for voltages of all generator buses and transformer tap settngs are consdered to be 1.1 and.9 n p.u. The maxmum and mnmum values for voltages of all load buses are 1.6 and.94 n p.u [24]. Table V shows the optmal soluton of the problem by the conventonal PSO and methods. The mnmum cost obtaned by ths algorthm s compared Max IEP [22] GSA [24] (mnutes) DE OPF MDE OPF [25] [25] (s) TABLE V. OPTIMAL SOLUTIONS FOR THE IEEE 57-BUS SYSTEM Varable Pg3 (MW) Pg6 (MW) Pg9 (MW) Pg12 (MW) Vg1 (pu) Vg2 (pu) Vg3 (pu) Vg6 (pu) Vg8 (pu) Vg9 (pu) Vg12 (pu) T19 (pu) T2 (pu) 35 PSO Varable T31 (pu) T37 (pu) T41 (pu) T46 (pu) T54 (pu) T58 (pu) T59 (pu) T65 (pu) T66 (pu) T71 (pu) T73 (pu) T76 (pu) T8 (pu) Qc18 (MVAr) Qc25 (MVAr) Qc53 (MVAr) PSO

6 TABLE VI. RESULT COMPARISON FOR THE IEEE 57-BUS SYSTEM Methods BASE-CASE [28] MATPOWER [28] PSO Total cost ($/h) CPU tme (s) [6] [7] [8] [9] [1] [11] [12] [13] Fgure 4. Convergence characterstc for the IEEE 57-bus system [14] V. CONCLUTION In ths paper, the stochastc weght trade-off partcle swarm optmzaton method has been presented to solve the OPF problem. The mproved has advantages such as smple algorthm and easy to use. Moreover, the algorthm can be mplemented n the whole problem search space rather than ndvdual ponts, leadng faster poston updatng functon of partcles. The proposed method has been tested on the IEEE 3 bus and 57 bus systems and the obtaned results are compared to those from many other methods n the lterature. The numercal results show the algorthm s flexblty and capablty n fndng the optmal soluton. Therefore, the proposed can be very favorable for solvng OPF problem, especally for large scale systems wth nonconvex objectve functon. 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