Transent Stablty Constraned Optmal Power Flow Usng Improved Partcle Swarm Optmzaton Tung The Tran and Deu Ngoc Vo Abstract Ths paper proposes an mproved partcle swarm optmzaton method for transent stablty constraned optmal power flow (TSCOPF) problem. The transent stablty constrant should be taken nto consderaton for the soluton of the optmzaton problems n power systems. The formulas of TSCOPF are derved through the addton of rotor angle nequalty constrants nto optmal power flow relatonshps. The proposed IPSO s the partcle swarm optmzaton wth constrcton factor and the partcle s velocty guded by a pseudo-gradent. The pseudo-gradent s to determne the drecton of the partcles so that they can quckly move to optmal soluton. The proposed method has been appled on WSCC 3-generator, 9-bus system, IEEE 30-bus systems. The obtaned results usng the IPSO are compared wth those obtaned usng other modern technques for performance examnaton. Keywords Improved partcle swarm optmzaton, optmal power flow, transent stablty. 1. INTRODUCTION Optmal power flow (OPF) s an mportant tool for power system operaton, control and plannng. It was frst ntroduced by Dommel and Tnney (1968) [1]. OPF has become an mportant ssue to the researchers over past two decades and has establshed ts poston as one of the man tools for optmal operaton and plannng of modern power systems. The objectve of an OPF problem s to fnd the steady state operaton pont of generators n the system so as ther total generaton cost s mnmzed whle satsfyng varous generator and system constrants such as generator s real and reactve power, bus voltage, transformer tap, swtchable capactor bank, and transmsson lne capacty lmts. 1 In the OPF problem, the controllable varables usually determned are real power output of generators, voltage magntude at generaton buses, njected reactve power at compensaton buses, and transformer tap settngs. OPF wth transent stablty constrants s an extenson of the tradtonal OPF problems. In addton to the common constrants of OPF, the TSCOPF problems consder the dynamc stablty constrants of power system. When any of a specfed set of dsturbances occurs, a feasble operaton pont should wthstand the fault and ensure that the power system moves to a new stable equlbrum after the clear-ance of the fault wthout volatng equalty and nequalty constrants even durng transent perod. These condtons for all of the specfed credble contngences are called as transent stablty constrants. Transent stablty constraned optmal power flow s an effectve measure to coordnate the securty and economc of power system. TSCOPF s a large-scale nonlnear optmzaton problem wth both Tung Tran The and Deu Ngoc Vo (correspondng author) are wth Department of Power Systems, Faculty of Electrcal and Electronc Engneerng, Ho Ch Mnh Cty Unversty of Technology, Ho Ch Mnh Cty, Vetnam. Emal: vndeu@gmal.com. algebrac and dfferental equatons developed by Sauer and Pa (1998) [2], Kundur (1994) [3]. The TSCOPF problem has been solved by several conventonal methods such as: prmal-dual nteror-pont method [4], lnear programmng (LP) [5], etc the conventonal methods can fnd the optmal soluton for an optmzaton problem wth a very short tme. However, the man drawback of these methods s that they are dffcult to deal wth non-convex optmzaton problems wth non-dfferentable objectve. Moreover, these methods are also very dffcult for dealng wth large-scale problems due to large search space. Metaheurstc search methods recently developed have shown that they have capablty to deal wth ths complcated problem. Several meta-heurstc search methods have been also wdely appled for solvng the TSCOPF problem such as Evolutonary Programmng (EP) [6], Genetc Algorthm (GA) [7], Artfcal Bee Colony [8]... These meta-heurstc search methods can overcome the man drawback from the conventonal methods wth the problem not requred to be dfferentable. However, the optmal solutons obtaned by these methods for optmzaton problems are near optmum and qualty of the solutons s not hgh when they deal wth large -scale problems; that s the obtaned solutons may be local optmums wth long computatonal tme. In 1995, Eberhart and Kennedy suggested a partcle swarm optmzaton (PSO) method based on the analogy of swarm of brd flockng and fsh schoolng [9]. 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, reactve power and voltage control, transent stablty constraned optmal power flow and many others. In ths paper, a newly mproved partcle swarm optmzaton (IPSO) method s proposed for solvng transent stablty constraned optmal power flow problem. The proposed IPSO s the partcle swarm 87
optmzaton wth constrcton factor and the partcle s velocty guded by a pseudo-gradent. The pseudogradent s to determne the drecton for the partcles so that they can quckly move to optmal soluton. The proposed method has been tested on WSCC 3-generator, 9-bus system, IEEE 30-bus systems and the obtaned results are compared to those from Dfferental Evoluton (DE), Trajectory Senstvtes (TS), Tme Doman Smulaton (TDS), Genetc algorthm (GA), Evolutonary programmng (EP). 2. PROBLEM FORMULATION 2.1. OPF formulaton The OPF problem was defned n the early 1960 s as an extenson of conventonal economc dspatch to determne the optmal settngs for control varables n a power network wth respect to varous constrants. The OPF s a statc constraned nonlnear optmzaton problem, whose development has closely followed advances n numercal optmzaton technques and computer technology. The OPF s a nonlnear optmzaton problem wth nonlnear objectve functon and nonlnear constrants. The objectve of the OPF problem s to mnmze s to optmze the objectve functons whle satsfyng several equalty and nequalty constrants [10]. Mathematcally, the problem s formulated as follows: subject to Mn f(x,u) (1) g(x,u) = 0 (2) h(x,u) 0 (3) where f s the objectve functon to be mnmzed, g s the set of equalty constrants, and h s the set of nequalty constrants. Vectors x and u, the parameters of these functons, are called the state varable vector and control varable vector, respectvely. 2.2. Objectve functon The objectve functon s defned as the total fuel cost of the system wth fuel cost curve approxmated as a quadratc functon of generator real power output: N g Mn F P a b P c P 1 2 ( g ) g g (4) where P g s the real power output of generatng unts ; N g s the number of generatng unts; a, b and c are fuel cost coeffcents of generatng unt. 2.3. Equalty constrants The equalty n the OPF are defned as equalty constrants: N b P P V V [ G cos( ) B sn( )] g d j j j j j j 1 1,..., N (5) b N b Q Q Q V V [ G cos( ) B sn( )] g c d j j j j j j 1 1,..., N (6) where N b s the number of buses; P d, Q d are the real and reactve power demands at bus, respectvely; V s the voltage magntude of the th bus, G j s the transfer conductance between bus and j, B j s the transfer susceptance between bus and j, and j s the voltage angle dfference between bus and j. 2.4. Inequalty constrants a. Lmts at generaton buses b P P P ; 1,..., N (7) g,mn g g, max g Q Q Q ; 1,..., N (8) g,mn g g, max g V V V ; 1,..., N (9) g,mn g g, max g b. Capacty lmts for swtchable shunt capactor banks: Q Q Q ; 1,..., N (10) c,mn c c, max c c. Transformer tap settngs constrants T T T ; k 1,..., N (11) k,mn k k, max t 2.5. Transent stablty constrants The transent stablty problem s explaned through a range of algebrac equatons. The oscllaton equatons of th generator: 0 (12) M ( ) (13) 0 Pm Pe D where s the rotor angle of the th generator, s the rotor speed of the th generator, M s the moment of nerta of the th generator, D s the dampng constant of the th generator, P m s the mechancal nput power of the th generator, P e s the electrcal output power of the th generator, and 0 s the synchronous speed. The poston of the center of nerta (COI) s defned as follows: COI N g 1 N g 1 M The nequalty constrants of the transent stablty are formulated as follows: COI max max 1,..., N g (15) 3. 3. PSO ALGORITHMS FOR TSCOPF M 3.1. Conventonal Partcle Swarm Optmzaton (14) Partcle swarm optmzaton (PSO) provdes a populaton-based search procedure n whch ndvduals 88
called partcles change ther poston (state) wth tme. In a PSO system, partcles fly around n a multdmensonal search space. Durng the flght, each partcle adjusts ts poston accordng to ts own experence (personal best: pbest), and accordng to the experence of a neghborng partcle (global best: gbest), leadng to the best poston encountered by tself and ts neghbor [11]. The modfed velocty and poston of each partcle are calculated: v v c rand (pbes t x ) ( k 1) ( k ) ( k ) ( k ) d d 1 1 d d c rand (gbes t x ) (16) ( k ) ( k ) 2 2 d d x x v ( k 1) ( k ) ( k 1) d d d (17) where the constants c 1 and c 2 are cogntve and socal parameters, respectvely and rand 1 and rand 2 are the random values n [0, 1]. 3.2. Concept of Pseudo-Gradent The man dea of the pseudo-gradent s determnng the drecton of each ndvdual n populaton based methods to solve non-convex optmzaton problems wth nondfferentable functons [12]. The advantage of the pseudo-gradent s that t can provde a good drecton n the search space of a problem wthout requrng the objectve functon to be dfferentable. For n-dmenson optmal problem wth nondfferentable functon f(x), the pseudo-gradent g p (x) s defned as follow [13]: Supposed that x k = [x k1, x k2,, x kn ] s a pont n the search space of the problem and t moves to another pont x l. There are two abltes for ths movement by consderng the value of the objectve functon at these two ponts. If f(x l ) < f(x k ), the drecton from x k to x l s defned as the postve drecton. The pseudo-gradent at pont xl s determned by: T g ( x ) [ ( x ), ( x ),..., ( x )] (18) p 1 l1 l 2 ln where (x l ) s the drecton ndcator of element x movng from pont k to pont l defned by: x 1,f x l 0,f x x (19) 1,f x k l l k l x If f(x l ) f(x k ), the drecton from x k to x l s defned as the negatve drecton. The pseudo-gradent at pont x l s determned by: 1 x k g ( x ) 0 (20) p From the defnton, f the value of the pseudo-gradent g p (x l ) 0, t mples that a better soluton for the objectve functon could be found n the next step based on the drecton ndcated by the pseudo-gradent g p (x l ) at pont l. Otherwse, the search drecton at ths pont should be changed due to no mprovement of the objectve functon n ths drecton. 3.3. Improved Partcle Swarm Optmzaton The IPSO here s the PSO wth constrcton factor enhanced by the pseudo-gradent for speedng up ts convergence process. The purpose of the pseudo-gradent s to gude the movement of partcles n postve drecton so that they can quckly move to the optmzaton. In the PSO wth constrcton factor (Clerc & Kennedy, 2002) [14], the velocty of partcles s determned as follows: v c rand ( pbes t x ) ( k ) ( k) ( k) ( k 1) d 1 1 d d d C ( k) ( k) c2 rand 2 ( pbes td x d ) v 2 2 4 1 2 (21) 2 C ; where c c, 4 (22) The factor has an effect on the convergence characterstc of the system. But f the factor ncreases and makes the constrcton C decrease producng dversfcaton, t leads to slower convergence. Thus, the typcal of factor s 4.1. PSO wth constrcton factor mproved performance for a wde range of problems and was appled n varous techncal feld. For mplementaton of the pseudo-gradent n PSO, the two consdered ponts correspondng to x k and x l n search space of the pseudo-gradent are the partcle s poston at teratons k and k+1 those are x (k) and x (k+1), respectvely. Therefore, the updated poston for partcles n (17) s rewrtten by: 3.4. Implementaton of IPSO for TSCOPF problem The overall procedure of the proposed IPSO for solvng the TSCOPF problem s addressed as follows: Step 1: Input system data, contngency set; choose the controllng parameters for IPSO ncludng number of partcles NP, maxmum number of teratons ITmax, cogntve and socal acceleraton factors c 1 and c 2 Step 2: Create ntal partcles postons and veloctes Step 3: For each partcle, calculate value of the dependent varables based on power flow soluton and evaluate the ftness functon F pbestd. Determne the global best value of ftness functon F gbest Step 4: Set pbest to the ntal poston for each partcle and gbest for the best poston of all partcles. For transent stablty volaton evaluaton (15), transent-stablty smulaton s used to produce the generator rotor responses. The maxmum rotor angle devaton from the COI, among all generators and contngences, s then used to compute a transent-stablty penalty. 89
Step 5: Set the pseudo-gradent assocated wth partcles to zero. Set teraton counter k = 1. Step 6: Calculate new velocty v (k) d and update poston x (k) d for each partcle usng (21) and (23), respectvely. Step 7: Solve power flow based on the newly obtaned value of poston for each partcle. Step 8: Evaluate ftness functon for each partcle wth the newly obtaned poston. Compare the calculated FT to FT pbest to to update the best poston of each partcle. Step 9: Pck up the best poston of all partcles to update the global best ftness functon FTgbest and the global best poston gbest Step 10: Calculate the new pseudo-gradent for each partcle based on ts two latest postons correspondng to x (k) d and x (k-1) d. Step 11: If k < MAXITER, k = k + 1 and return to Step 6. Otherwse, stop. A flowchart for overall procedure of the IPSO for solvng the TSCOPF problem s also depcted n Fg 1. 4. NUMERICAL RESULTS The proposed IPSO s tested on WSCC 3-generator, 9- bus system, IEEE 30-bus systems. In all tested system, the upper and lower of voltage lmts are set to 1.1pu and 0.95pu, respectvely. The upper and lower of transformer tap changers are set to 1.1pu and 0.9pu, respectvely. The transformer taps and swtchable capactor banks are dscrete wth a changng step of 0.01pu and 0.1MVar, respectvely. The algorthm of ths method was programmed by MATLAB R2009b n 2.4 GHz, 3, personal computer. 4.1. WSCC 3-generator, 9-bus system The WSCC 3-generator, 9-bus system s shown n Fg 2 and the system data are gven n [2]. The upper and lower lmts of all of the generator voltage magntudes are set at 1.10 p.u. and 1.00 p.u., respectvely. The upper and lower lmts of the voltage magntudes of the other buses are also set at 1.10 p.u. and 0.90 p.u., respectvely. For ths test system, the OPF and TSCOPF problems are solved for 2 fault cases. The step tme of the ntegraton s 10 ms for the transent stablty smulaton and the smulaton perod s taken nto consderaton as 5.0s. Here, max s set to 200 0 for the WSCC 3-generator, 9-bus system. Case 1: There s no transent stablty constrant n ths optmzaton problem. The objectve n ths optmzaton problem s to mnmze the total fuel cost of the entre power system to subject the generator constrants. Fg. 1. Flowchart of IPSO for TSCOPF problem. Fg. 2. One-lne dagram of the WSCC 3-generator, 9-bus system. 90
The optmal power flow soluton wthout transent stablty lmts has been obtaned usng IPSO. The man objectve s to mnmze the total fuel cost of the entre system. The results of the proposed algorthm are compared wth the algorthms of: Dfferental Evoluton (DE) [15], Trajectory Senstvtes (TS) [16], Tme Doman Smulaton (TDS) [17]. The comparson tables of the smulaton results by dfferent optmzaton technques for Case 1 are gven n Table1. From ths table we can say that the value of the fuel cost obtaned by the proposed algorthm s less than the results obtaned from others n the above problem. Table 1: Comparson of smulaton results for Case 1 Table 2: Comparson of smulaton results for Case 2 Method DE [15] TS [16] TDS [17] IPSO P g1 (MW) 130.94 170.20 117.85 116.61 P g2 (MW) 94.46 48.94 103.50 105.89 P g3 (MW) 93.09 98.74 96.66 93.29 V g1 (p.u.) 0.9590 1.000 1.05 1.028 V g2 (p.u.) 1.0139 1.000 1.05 1.068 V g3 (p.u.) 1.0467 1.000 1.04 1.056 FC ($/hr) 1140.06 1179.95 1134.01 1132.37 Method DE [15] TS [16] TDS [17] IPSO P g1 (MW) 105.94 106.19 105.94 106.11 P g2 (MW) 113.04 112.96 113.04 114.26 P g3 (MW) 99.29 99.20 99.24 96.60 V g1 (p.u.) 1.05 1.00 1.05 1.035 V g2 (p.u.) 1.05 1.00 1.05 1.029 V g3 (p.u.) 1.04 1.00 1.04 1.033 FC ($/hr) 1132.30 1132.59 1132.18 1131.75 Fg. 4. Convergence characterstcs of IPSO for Case 2. Fg. 3. Convergence characterstcs of IPSO for Case 1. Case 2: A 3-phase to ground fault at bus 7 and n lne 7-5 n the system. The above fault was cleared by openng the contacts of the crcut breakers by 0.35 sec. The soluton obtaned from ths case satsfes transent stablty lmt. The mnmum fuel cost values and power generatons are compared wth other optmzaton methods Dfferental Evoluton (DE) [15], Trajectory Senstvtes (TS)[16], Tme Doman Smulaton (TDS) [17] are gven n Table 2; the convergence characterstcs n Fg.4. Case 3: A 3-phase to ground fault at bus 9 and n lne 9-6 n the system. The above fault was cleared at 0.3sec by openng the contacts of the nearby crcut breakers. Here also, the mnmum fuel cost values and power generatons are compared wth other optmzaton methods: Dfferental Evoluton (DE) [15], Trajectory Senstvtes (TS)[16], Tme Doman Smulaton (TDS) [17] gven n Table 3. From there, t s observed that the fuel cost values and power generatons obtaned by the proposed method are less than those others. The convergence characterstc s shown n Fg 5. Table 3: Comparson of smulaton results for Case 3 Method DE [15] TS [16] TDS [17] IPSO P g1 (MW) 130.01 164.38 120.01 120.23 P g2 (MW) 127.17 112.44 121.13 119.83 P g3 (MW) 60.72 41.00 76.84 76.95 V g1 (p.u.) 1.0495 1.000 1.05 1.019 V g2 (p.u.) 1.0481 1.000 1.05 1.039 V g3 (p.u.) 1.0327 1.000 1.04 1.024 FC ($/hr) 1148.58 1179.95 1137.82 1135.25 91
s. The step tme of the ntegraton s at 10 ms for the transent stablty smulaton and the smulaton perod s taken nto consderaton as 5.0s. The obtaned solutons are compared wth other optmzaton methods: Genetc algorthm (GA) [7], Evolutonary programmng (EP) [19]. From ths Table 4 we can say that the value of the fuel cost obtaned by the proposed algorthm s less than the results obtaned from other two. The convergence characterstc s shown n Fg 7. Table 4: Comparson of smulaton results for Case 4 Fg. 5. Convergence characterstcs of IPSO for Case 3. 4.2. IEEE 30-bus system The IEEE 30-bus test system contans 41 transmsson lnes, 6 generators, and 4 transformers, as shown n Fgure 6. The system data were taken from [18] and the data for the generators n the test system are gven n Table 3. The total actve load and reactve load of the system s 189.2 MW and 107.2 MVar, respectvely. Here, max s set to 50 0 for the IEEE 30-bus system. Method GA [7] EP [19] IPSO P g1 (MW) 41.88 50.25 41.10 P g2 (MW) 56.38 38.86 58.32 P g3 (MW) 22.94 17.96 25.39 P g4 (MW) 37.63 27.33 32.49 P g5 (MW) 16.7 20.29 18.24 P g6 (MW) 16.53 37.25 16.34 T1 (buses 6 9) 1.01 1.02 1.01 T2 (buses 6 10) 0.95 0.99 0.96 T3 (buses 4 12) 1.00 0.97 1.01 T4 (buses 27 and 28) 0.97 1.04 0.97 FC ($/hr) 585.62 585.83 585.10 Fg. 7. Convergence characterstcs of IPSO for Case 4. Fg. 6. IEEE 30-bus system one-lne dagram. The followng fault case s studed for the test system: Case 4: A 3-phase to ground fault at bus 2 and n lne 2-5 n the system. The fault clearng tme s taken as 0.18 5. CONCLUSION The proposed IPSO method has been effcently mplement for solvng the TSCOPF. The proposed IPSO method s a smple mprovement from the PSO method wth constrcton factor by ntegratng the pseudogradent n to partcle s velocty to enhance ts search capablty. The pseudo-gradent speeds up partcles n search space n case they are on a rght drecton. Smulatons are carred out wth WSCC 3-generator, 9- bus system, IEEE 30-bus systems then compared wth other methods. The results obtaned by the proposed method outperform the other methods n terms of soluton qualty and computaton effcency. Therefore, 92
the proposed PG-PSO could be a useful and favorable method for solvng the non-dfferentable problem n power systems. ACKNOWLEDGEMENT Ths research s funded by Ho Ch Mnh Cty Unversty of Technology (HCMUT) under grant number T- T- 2015-15. REFERENCES [1] Dommel, H. W., & Tnney, W. F. (1968). Optmal power flow solutons. IEEE transactons on power apparatus and systems (10), 1866-1876. [2] Sauer, P. W., & Pa, M. A. (1998). Power system dynamcs and stablty.urbana. [3] Kundur, P. (1994). Power system stablty and control (Vol. 7). N. J. Balu, & M. G. Lauby (Eds.). New York: McGraw-hll. [4] Xa, Y., Chan, K. W., & Lu, M. (2005, January). Drect nonlnear prmal-dual nteror-pont method for transent stablty constraned optmal power flow. In Generaton, Transmsson and Dstrbuton, IEE Proceedngs- (Vol. 152, No. 1, pp. 11-16). IET. [5] Gan, D., Thomas, R. J., & Zmmerman, R. D. (2000). Stablty-constraned optmal power flow. IEEE Transactons on Power Systems, 15(2), 535-540. [6] Tangpatphan, K., & Yokoyama, A. (2008, October). Evolutonary programmng ncorporatng neural network for transent stablty constraned optmal power flow. In Power System Technology and IEEE Power Inda Conference, 2008. POWERCON 2008. Jont Internatonal Conference on (pp. 1-8). IEEE. [7] Mo, N., Zou, Z. Y., Chan, K. W., & Pong, T. Y. G. (2007). Transent stablty constraned optmal power flow usng partcle swarm optmsaton. Generaton, Transmsson & Dstrbuton, IET, 1(3), 476-483. [8] AYAN, K., & KILIÇ, U. (2013). Soluton of transent stablty-constraned optmal power flow usng artfcal bee colony algorthm. Turksh Journal of Electrcal Engneerng & Computer Scences, 21(2), 360-372. [9] Kennedy, J., & Eberhart, R. C. (1995). Partcle swarm optmzaton. In IEEE Internatonal Conference on Neural Networks (Vol. 11, p. 27). Perth, Australa. [10] Ongsakul, W., & Tantmaporn, T. (2006). Optmal power flow by mproved evolutonary programmng. Electrc Power Components and Systems, 34(1), 79-95.. [11] Le, D. A., & Vo, D. N. (2012, December). Optmal reactve power dspatch by pseudo-gradent guded partcle swarm optmzaton. In IPEC, 2012 Conference on Power & Energy (pp. 7-12). IEEE. [12] Pham, D. T., & Jn, G. (1995). Genetc algorthm usng gradent-lke reproducton operator. Electroncs Letters, 31(18), 1558-1559. [13] Wen, J. Y., Wu, Q. H., Jang, L., & Cheng, S. J. (2003). Pseudo-gradent based evolutonary programmng. Electroncs Letters, 39(7), 631-632. [14] Eberhart, R. C., & Sh, Y. (2000). Comparng nerta weghts and constrcton factors n partcle swarm optmzaton. In Evolutonary Computaton, 2000. Proceedngs of the 2000 Congress on (Vol. 1, pp. 84-88). IEEE. [15] Ca, H. R., Chung, C. Y., & Wong, K. P. (2008). Applcaton of dfferental evoluton algorthm for transent stablty constraned optmal power flow. IEEE Transactons on Power Systems, 23(2), 719-728. [16] Nguyen, T. B., & Pa, M. A. (2003). Dynamc securty-constraned reschedulng of power systems usng trajectory senstvtes. IEEE Transactons on Power Systems, 18(2), 848-854. [17] Zárate-Mñano, R., Van Cutsem, T., Mlano, F., & Conejo, A. J. (2010). Securng transent stablty usng tme-doman smulatons wthn an optmal power flow. IEEE Transactons on Power Systems, 25(1), 243-253. [18] Alsac, O., & Stott, B. (1974). Optmal load flow wth steady-state securty. IEEE Transactons on Power Apparatus and Systems, (3), 745-751. [19] K. Tangpatphan and A. Yokoyama (2008). Transent stablty constraned optmal power flow usng evolutonary programmng. 16th PSCC, Glasgow, Scotland, July 14-18. 93
94 T. T. Tran and D. N. Vo / GMSARN Internatonal Journal 10 (2016) 87-94