Voltage Security Enhancement with Corrective Control Including Generator Ramp Rate Constraint

Transcription

1 Voltage Securty Enhancement wth Correctve Control Includng Generator Ramp Rate Constrant 1 P.ARUNA JEYANTHY, 2 D.DEVERAJ, 3 J.D.DARWIN 1,2 Department of Electrcal and Electroncs Engneerng 3 Department of Mechancal Engneerng 1,2 Kalasalngam Unversty, Tamlnadu,3 Loyala Insttute of Technology & Scence,Thovala, Kanyakumar, Anna Unversty INDIA 1 arunadarwn@yahoo.com, 2 deva230@yahoo.com, 3 jdarwa@yahoo.com Abstract: - Correctve acton for voltage stablty s one of the ssues whch the electrcal utltes care most about. Ths paper deals wth the development of optmzaton model that s capable of performng correctve control acton. Though the preventve control approach s preferred for the secure operaton of the system, correctve control can also be carred out as t s consdered economcal. Correctve control actons would mantan voltage stablty of the system n case of severe and unforeseen contngences. Under the correctve control strategy, control actons are not taken untl the contngency actually occurs. But a contngency plan s prepared n advance for the dentfed severe cases. Correctve control s actvated when a contngency has occurred endangerng voltage stablty. The objectve of ths paper s to acheve maxmum voltage stablty margn n the contngency state whle satsfyng system and equpment constrants. Also the generator ramp rate constrants are taken nto account for the system s correctve control capabltes after the outage has occurred. Partcle Swarm Optmzaton (PSO) algorthm s appled to solve ths optmzaton problem. The effectveness of ths algorthm s demonstrated through the Voltage Securty Enhancement n the IEEE 30-bus and IEEE 57-bus test systems. Key-Words: - L-ndex, ramp rate, Partcle swarm optmzaton (PSO), Optmal power flow (OPF) 1 Introducton Correctve acton s an ndspensable part of the onlne voltage stablty montorng system. It s to stablze an unstable power system, drectng the system trajectory onto a new stable equlbrum pont shortly after a severe contngency, such as the trppng of a heavly loaded transmsson lne or the outage of a large generatng unt. It can be formulated as a statc nonlnear optmzaton problem whch can be solved by the OPF. Generator ramp rates can sgnfcantly restrct the speed wth whch actve power s rerouted n the network. Hence they are taken as the addtonal control varable constrant of the optmzaton problem for the correctve acton. The use of ramp rate constrant to smulate the unt state and generaton changes s an effectve and acceptable approach n theoretcal developments of ndustral processes. Ths constrant ensures that output from each unt s wthn rampng range. The correctve control s actvated when a contngency has occurred endangerng voltage stablty. Most of the securty constraned OPF problems has assessed the voltage securty based on the ndces whch depends on load bus voltage magntudes. However voltage nstablty problems have been shown to occur n systems where voltage magntudes never declne below acceptable lmts. To measure the severty level of voltage stablty problems, a lot of performance ndces have been proposed [1]. They could be used on-lne or off-lne to help the operators to determne how close the system s to collapse. In general, these ndces amed at defnng a scalar magntude that can be montored as system parameters change wth fast computaton speed. They nclude senstvty factors [2,3], second order performance ndex [4,5], voltage nstablty proxmty ndex[6], sngular values and egen values [7,8,9] and so on. A methodology of control aganst voltage nstabltes based on sngular value decomposton s developed and presented n [10]. One of the dsadvantages of ths methodology s that large amount of CPU tme s requred n sngular value decomposton. Song et al presented a new concept of reactve reserve based contngency constraned OPF for enhancement of voltage stablty margn by ncreasng the mnmum egen value of load flow E-ISSN: X 269 Volume 12, 2017

2 Jacoban so as to mantan desred voltage profle[11]. A nonlnear least square optmzaton algorthm for voltage stablty margn mprovement usng L-ndex method s presented n [12]. A voltage stablty ndex called L-ndex based on the power flow soluton s dscussed n [13]. Ths ndex ranges from 0 to 1. The bus wth the hghest L-ndex wll be the most vulnerable bus n the system. The modal analyss technque to compute the voltage stablty level of the system s developed n [14]. The correctve control for the voltage stablty of complex power systems based on Prmal- Dual Interor pont method s dscussed n [15]. The control actons to counter the ll effects of sudden dsturbances are dealt n [16]. Wang et al solved both preventve and correctve control problems for satsfyng a certan level of the voltage stablty margn, but the condton at the base soluton after applyng the controls s not taken nto account. A senstvty based heurstc tool to determne correctve acton, so as to help the system operator n the reactve power flow control problem s stated n [17]. The correctve acton for couplng optmzaton and dynamc smulaton of voltage nstabltes wth an teratve approach s presented n [18]. Several evolutonary algorthm based technques have been proposed to solve OPF and voltage securty enhancement problems [19-22]. In ths paper, PSO algorthm s used to solve the correctve control of voltage securty enhancement problem formulated as an optmzaton system, wth the mnmzaton of the maxmum L-ndex as the objectve functon. The PSO algorthms are nsenstve to scalng of the desgn varables. They are easly parallelzed for concurrent processng and are dervatve free. It has very few algorthms parameters and s very effcent n global search algorthms [23-27]. 2 Voltage Securty Enhancement There are varous methods of determnng the voltage collapse proxmty ndcator. One such method s the L-ndex of the load buses n the system proposed n [13]. It s based on load flow analyss and ts value ranges from 0 (no load condton) to 1 (voltage collapse). The bus wth the hghest L ndex value wll be the most vulnerable bus n the system. The L- ndex calculaton for a power system s brefly dscussed below: Consder a N-bus system n whch there are N g generators. The relatonshp between voltage and current can be expressed by the followng expresson: I YGG YGL G V = G I L Y V L LG YLL (1) where I G, I L and V G, V L represents currents and voltages at the generator buses and load buses. Rearrangng the above equaton we get, V L Z LL FLG I L = V G K GL YGG V G (2) F = Y 1 Y (3) where [ ] [ ] K LG LG LL LG 1 [ Y ] [ Y ] = (4) LL The L-ndex of the j-th node s gven by the expresson, L j N g = 1 GL ( + δ ) V = 1 F j θ δ j (5) V j where V voltage magntude of - th generator V j Voltage magntude of j- th generator. θ Phase angle of the term F j. δ Voltage phase angle of - th generator unt. δj Voltage phase angle of j- th generator unt. N g Number of generatng unts The values of F j are obtaned from the matrx F LG. The L ndces for a gven load condton are computed for all the load buses and the maxmum of the L ndces gves the proxmty of the system to voltage collapse. It was demonstrated that when a load bus approaches a voltage collapse stuaton, the L-ndex approaches one. Hence for a systemwde voltage stablty assessment, the L-ndex s evaluated at all load buses and the maxmum value of the L ndces gves an ndcaton of how far the system s from voltage collapse. Contngences such as transmsson lne or generator outages often result n voltage nstablty n power system. The system s sad to be secured f none of the contngences causes voltage nstablty n the system. The maxmum L-ndex of the system under a contngency gves a measure of severty of that contngency. 3 Ramp rate Constrant Whle consderng the correctve acton formulaton, ramp constrants or couplng constrants are of the general form: h u, (6) ( ) 0 u w Correctve control acton nvolves changng the control varables of the system n response to contngency occurrence wthn pre-specfed lmts. Ths process s also known as post contngency correctve reschedulng. The use of ramp rate constrants to smulate the unt state and generaton E-ISSN: X 270 Volume 12, 2017

3 changes s an effectve and acceptable approach n the vew of theoretcal developments. In practcal systems, the operatng range of all on-lne unt s restrcted by ther ramp rate lmts due to physcal operatng lmtatons. These constrants recognze that the range of adjustment of certan control s determned by ther settng at the tme of contngency. They act as a brdge between the base and the post contngency case. In the algorthm they are modelled as u u w w =1,..., k (7) where and are the lower and upper ramp rate lmts. The ramp rate of the generator s usually defned as the percentage of the generator capacty. where k-1...c represents the post contngency state. u 0 s the preventve control varable. T k s the assumed tme for correctve control. du/dt max represents the ramp rate of correctve control. 4 Mathematcal Problem Formulaton Enhancng voltage stablty under contngency can be acheved through mnmzng the voltage stablty ndcator L-ndex values at every bus of the system and consequently the global power system L-ndex. Ths s acheved through reschedulng of control varables. L-ndex gves a scalar number to each load bus. Ths ndex uses nformaton on a normal power flow and s n the range of zero (no load case) to 1 (voltage collapse). Ths s mathematcally stated as Mnmze L max (8) Subject to 4.1 Equalty Constrants P V N B j= 1 Q V V ( G Cosθ + B Snθ ) = 0, N 1 (9) N B j= 1 j j V ( G Sn B Cosθ ) = 0, N θ (10) The equalty constrants are satsfed by runnng the power flow program. 4.2 Inequalty Constrants The nequalty constrants are the physcal and operatng lmts whch must be satsfed by correctve control soluton. These constrants are Voltage lmt mn max V V V ; N B (11) Generator reactve power lmt mn max Q Q Q ; N (12) g g g B B PQ SVC reactve power generaton lmt mn max Qc Qc Qc ; N c Transformer tap settng lmt mn max tk tk tk ; k NT Transmsson lne flow lmt max Sl Sl ; l N l Ramp rate constrant (13) (14) (15) du uk uo Tk* max (16) dt From the above formulaton t s found that the voltage securty enhancement problem s a combnatoral non-lnear optmzaton problem. The actve power generaton (P g ) and generator termnal bus voltages (V g ) are the control varables and they are self restrcted by the optmzaton algorthm. The actve power generaton at the slack bus (P sl ), load bus voltage (V load ) and reactve power generaton (Q g ) are the state varables and are restrcted by addng a quadratc penalty term of the objectve functon. 5 Partcle Swarm Optmzaton (PSO) The PSO s a populaton based optmzaton algorthm. Its populaton s called a swarm and each ndvdual s called a partcle. The PSO algorthm works on the socal behavor of partcles n the swarm. It fnds the global best soluton by smply adjustng the trajectory of each ndvdual toward ts own best locaton and toward the best partcle of the entre swarm at each tme step [23,24]. The partcle updates ts velocty and poston wth the followng equatons k + 1 k k k V = W * V + C1 * rand() 1 *( pbest S ) + C2 * rand() 2 * gbest S ) Wmax Wmn W = Wmax * ter (18) termax k +1 k k +1 S = S + V (19) Usually the constant weghtng factor or the acceleraton coeffcentsc 1, C2 = 2, control how far a partcle moves n a sngle teraton. The nerta weght W s used to control the convergence behavor of the PSO. The sutable selecton of the nerta weght provdes a balance between global and local exploraton, and the explotaton of results n a lesser number of teratons on an average to fnd a suffcent optmal soluton. As orgnally developed, W max and W mn are often set to 0.9 and 0.4. rand( ) 1 and rand( ) 2 are two separately generated unformly dstrbuted numbers n the range [0,1]. pbest s the best prevous poston of the E-ISSN: X 271 Volume 12, 2017

4 th partcle. gbest s the global best poston among all the partcles n the swarm. ter s the current teraton number. ter max s the maxmum teraton number. The velocty of the partcle on each dmenson s clamped to the range [-V max, V max ] to reduce the possblty of the partcle leavng the feasble space. It determnes the resoluton or ftness, wth whch the regons between the present poston and the target poston are searched. If V max s too hgh, partcles may fly past good solutons. If V max s too small, partcles may be trapped n local optma, unable to move far enough s only one populaton n an teraton that moves towards the global optmal pont to reach the better poston n the problem space [25-27]. 5.1 PSO Algorthm Step 1: Step 2: Step 3: Step 4: Step 5: Intal search ponts and veloctes are randomly generated for each of the three varables between ther upper and lower bounds. The objectve for each set of partcles s evaluated based on the ftness functon. If the constrants are volated, penalty s added. Assgn the partcle s poston to the pbest poston, and ts ftness to the pbest ftness. Identfy the best among the pbests as the gbest. New veloctes and new search ponts (drectons) are formulated usng the equatons (17) to (19). Objectves correspondng to the new search ponts and veloctes are evaluated. Step 6: Compare the best current ftness evaluaton wth the populaton s gbest. If the current value s better than the gbest, reset the gbest to the current best poston and ftness value. Step 7: If the teraton reaches the maxmum number, then ext. Otherwse go to step PSO mplementaton Representaton Each ndvdual n the PSO populaton represents the canddate soluton. The elements of that soluton consst of all the optmzaton varables of the problem. Wth drect representaton of the soluton varables, the computer memory requred to store the populaton s reduced Evaluaton functon The functon of each ndvdual n the populaton evaluated accordng to ts ftness, whch s defned as the non-negatve fgure of mert to be maxmzed. It s assocated manly wth the objectve functon. In ths problem, the objectve s to maxmze voltage stablty margn;.e, mnmze the L max whle satsfyng the equalty and nequalty constrants equaton (9) to (16). For each ndvdual, the equalty constrants are satsfed by runnng the Newton- Raphson algorthm, and the constrants on the state varables are taken nto consderaton by addng the penalty functon to the objectve functon. Snce the PSO maxmzes the ftness functon, the mnmzaton objectve functon f s transformed nto a ftness functon to be maxmzed as Ftness = k/f where k s a large constant 6 Smulaton Results The proposed PSO-based approach was appled to the IEEE 30-bus and IEEE 57-bus test systems for voltage securty enhancement, under normal and contngency states. The real and reactve loads are scaled up accordng to predetermned weghtng factors to analyze the system under a stressed condton. The L-ndces for a gven load condton are computed for all the load buses and the maxmum of the L-ndces gves the proxmty of the system to a voltage collapse. Generaton exctaton, statc VAR compensators and transformer tap settngs are consdered as control varables for voltage stablty mprovement. The detals of the IEEE test data are taken from [28]. 6.1 Case 1 PSO-OPF for base case The IEEE 30- bus system has 6 generator buses, 24 load buses and 41 transmsson lnes of whch four branches are (6-9),(6-10),(4-12) and (28-27) wth tap settng transformers. The upper and lower voltage lmts at all buses except the slack bus are taken as 1.10 p.u and 0.95 p.u respectvely. The slack bus voltage s fxed at ts specfed value of 1.06 p.u. The PSO based algorthm was tested wth dfferent parameter settngs and the best results are obtaned wth the followng settng No: of generatons : 50 Populaton sze : 50 C 1 : 2 C 2 : 2 W max : 0.9 W mn : 0.4 E-ISSN: X 272 Volume 12, 2017

5 The optmal values of the control varables from the algorthm are gven n the Table 1. The algorthm took 77 sec to reach the optmal soluton. Correspondng to these control varables, t was found that there was no lmt volaton. The convergence characterstcs are gven n Fgure 1. T 12 T 15 T 36 Q C10 Q C12 Q C15 Q C17 Q C20 Q C21 Q C Fg.1 Convergence dagram of IEEE 30-bus system Table 1 Results of PSO-OPF optmal control varables Q C24 Q C29 Cost($/hr) Control varables Varable settng P 1 P 2 P 5 P 8 P 11 P 13 V 1 V 2 V 5 V 8 V 11 V 13 T L max Case () Contngency state schedulng The PSO algorthm reaches a mnmum L-ndex value of for the base case. To analyze the system under dsturbance, contngency analyss was conducted for all the lnes. From the contngency analyss, the frst fve severe lne outages L-ndex values are determned. After dentfyng the severe contngency lnes the ramp constrant values are ncluded and the L-ndex values are determned. Wth the ncluson of generator ramp rate constrant the voltage securty enhancement values are tabulated n Table 2. From the table, t s found that the L-ndex value decreases rapdly n the correctve control approach wthout any volatons. Ths shows that the voltage stablty s mproved after the applcaton of ths algorthm. As an llustraton the optmal values wth correctve control for lne outages 1-2, 9-10, 4-12 and 6-7 are gven n Table 3. In order to analyze the system under stressed condtons, actve and reactve powers of each bus are multpled by Correspondng to ths settng, the L-ndces of all the load buses are computed. From the contngency analyss, lne outage 1-2 and 9-10 wth the L max values of and has been found to be severe. The PSO algorthm was appled to enhance E-ISSN: X 273 Volume 12, 2017

6 the voltage stablty under contngency state. The voltage stablty enhancement values before and after the contngences are stated n Table 4. From the table, t s found that the value of L max decreases and voltage stablty s mproved after the applcaton of the algorthm. Table 2 varables Results of PSO-based optmal control Control varables P 1 P 2 P 5 P 8 P 11 P 13 Contngency (lne 1-2 outage) Correctve control Table 3 Results of optmzaton under contngency state for IEEE 30-bus system for base case loaded condton Contngency lne L max value (Before optmzaton) L max value (After optmzaton) Table 4 Results of optmzaton under contngency state for IEEE 30-bus system for 125% loaded condton Contngency lne L max value (Before optmzaton) L max value (After optmzaton) V 1 V 2 V 5 V 8 V 11 V 13 SVC 1 SVC 2 SVC 3 SVC 4 SVC 5 L max Cost($/hr) IEEE 57-bus test system The IEEE 57-bus system has 7 generators, 50 load buses, 80 transmsson lnes, 5 synchronous condensers and 17 tap changng transformers. The base load of the system s 1272 MW and 298 MVAR. The PSO based algorthm was tested wth dfferent parameter settngs and the best results are obtaned wth the followng settng: No: of generatons : 70 Populaton sze :50 C 1 :2 C 2 :2 W max :0.9 W mn :0.4 The optmal settngs for the base case are lsted n Table 5. The sngle lne contngency analyss s performed n IEEE 57-bus system. Based on the contngency study lne outage (46-47) was dentfed as severe case wth L max value of respectvely. Buses 30, 31, 32, 33 and 34 were selected for reactve power njecton. The result of the PSO-based algorthm for voltage securty enhancement s summarzed n Table 6. From the table t s found that voltage stablty level of the system has mproved after the applcaton of the proposed algorthm. Ths E-ISSN: X 274 Volume 12, 2017

7 shows the effectveness of the proposed algorthm n solvng the contngency constraned voltage securty problems. Fgure 2 represents the convergence dagram. outage optmzaton Table 5 Control varable settngs for IEEE 57-bus system control varables P 1 P 2 P 3 P 6 P 8 P 9 P 12 V 1 V 2 V 3 V 6 V 8 V 9 V 12 T 19 T 20 T 31 T 35 T 36 T 37 T 41 T 46 T 54 T 58 T 59 T 65 T 66 T 71 T 73 T 76 T 80 Q 30 Q 32 Q 31 Q 33 Q 34 L max Varable settngs Table 6 System Performance for IEEE 57-bus test system Lne Before After optmzaton Fg.2 Convergence dagram of IEEE 57-bus system Table 7 Comparson of optmal values n prevous work n the lterature Method Optmal Value ($/hr) Gradent Approach[28] Hybrd Evolutonary Programmng[19] Refned GA[20] Improved Evolutonary Programmng [21] Proposed method Concluson The voltage securty enhancement problem s solved by PSO algorthm wth mnmzaton of L max value as the objectve functon. The algorthm was proposed to dentfy the optmal control varable settng under normal and contngency state. The proposed algorthm was demonstrated on IEEE -30 bus and IEEE 57-bus test system wth generator ramp rate lmts as an addtonal constrant. Results show that E-ISSN: X 275 Volume 12, 2017

8 the PSO algorthm s well suted for obtanng the best soluton and s effectve for voltage securty enhancement n the normal and contngency states. References: [1] Mansour, Y. Voltage Stablty of Power Systems: Concepts, Analytcal Tools and Industry Experence, IEEE Press, [2] Carpnell, G., Laura, D. and Varlone, P. Voltage stablty analyss n unbalanced power systems by optmal power flow, IEE Proc. Generaton Transmsson Dstrbuton, Vol. 153, No. 3, 2006, pp [3] Aumuller, C. and Saha, T. K. Analyss and assessment of large scale power system voltage stablty by a novel senstvty based method, Proc. of IEEE/PES Summer Meetng, Vol. 3, 2002, pp [4] Berzz, A., Bresest, P., Marannno, P., Montagna, M., Cors, S. and Pccn, G. Securty enhancement aspects n the reactve-voltage control, IEEE Transactons of Power Systems, 1995, pp [5] Berzz, A., Fnazz, P. and Dos, D. Frst and second order methods for voltage collapse assessment and securty enhancement, IEEE Trans. Power System, Vol. 13, No. 2, 1998, pp [6] Nanba, M., Huang, Y., Ka, T. and Iwamoto, S. Studes on VIPI based control methods for mprovng voltage stablty, Proc. PSCC, Vol.2, 1996, pp [7] Ca L-Jun and Erlch Istvan, Power system statc voltage stablty analyss consderng all actve and reactve power controls - sngular value approach, Proc. of IEEE Power Tech, 2007, pp [8] Schlueter, R.A., Lu, S. Z. and Klan, K. B. Justfcaton of the voltage stablty securty assessment and dagnostc procedure usng a bfurcaton subsystem method, IEEE Trans. Power System, Vol. 15, No. 3, 2000, pp [9] Wang, Y., Da Slva, L. C. P., Xu, W. and Zhang, Y. Analyss of ll-condtoned power-flow problems usng voltage stablty methodology, IEE Proc. Gener. Transm. Dstrbuton, Vol.148, No. 5, 2001, pp [10] Tranucht, A. and Thomas, R.J. A posturng strategy aganst voltage nstablty n Electrc power systems, IEEE Transactons on Power Systems, Vol. 3, No. 1, 1998, pp [11] Song, H. Reactve Reserve-Based Contngency Constraned optmal Power Flow for Enhancement of Voltage Stablty Margns, IEEE Transacton on Power System, Vol. 18, No. 4, [12] Banslal, Thukaram, D. and Parthasarathy, K, Optmal reactve power dspatch algorthm for voltage stablty mprovement, Electrcal power and energy systems. Vol. 18, no.7, 1996, pp [13] Kessel, P. and Glavtch, H. Estmatng the voltage stablty of a power system, IEEE Trans. Power Delvery, Vol.1, No. 3, 1986 pp [14] Gao, B., Morsan G.K. and Kundur, P. Voltage Stablty Evaluaton usng Modal Analyss, IEEE Transactons on Power Systems, Vol.7, No.1992, pp [15] Yue Yuan, Kejun Qan and Xuehong Wen. Dscusson about the correctve control for voltage stablty of complex power systems based on Prmal Dual Interor pont method, Int. Conference on PST,, 2006, pp [16] Sarosh TaluMar and Ramesh, V. C. A mult-agent technque for contngency constraned optmal power flows, IEEE Transactons on Power Systems, Vol. 9, No. 2, 1994, pp [17] Angel, L.Trgo, Jose, L.Martnez, Jesus Rquelne and Esther Romero. A heurstc technque to determne correctve actons for reactve power flows, Electrc Power System Research, Vol. 81, ssue 1, January [18] Florn Captanescu, Threey Vancutsen, and Loys Wehenkel, Couplng optmzaton and dynamc smulaton for preventve, correctve control of voltage nstablty Transactons on power systems, Vol. 24, No.2, [19] Paranjoth S R, and Anburaja K, Optmal power flow usng refned genetc algorthm, Electrc power components and systems, Vol.30,2002, pp [20] Yuryvch J and Wong K P, Evolutonary programmng based optmal power flow algorthm, IEEE Transactons on Power System, Vol.14, No.4, 1999, pp [21] Somasundaram P, Kuppusamy K and Dev,R.P.K, Evolutonary programmng based securty constraned optmal power flow, Electrc power system research, Vol.72, 2004 pp E-ISSN: X 276 Volume 12, 2017

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