Yutaka Matsuo and Akihiko Yokoyama. Department of Electrical Engineering, University oftokyo , Hongo, Bunkyo-ku, Tokyo, Japan

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1 Optmzaton of Installaton of FACTS Devce n Power System Plannng by both Tabu Search and Nonlnear Programmng Methods Yutaka Matsuo and Akhko Yokoyama Department of Electrcal Engneerng, Unversty oftokyo 7-3-, Hongo, Bunkyo-ku, Tokyo, Japan matsuo@syl.t.u-tokyo.ac.jp, yokoyama@syl.t.u-tokyo.ac.jp Abstract: FACTS devces such as phase shfters and seres capactors enable us to control actve power ow and to avod thermal constrants on transmsson lnes, resultng n an ncrease of the network loadablty and a reducton of producton cost. However where to place these devces s an essental matter because ther eects consderably depend on the locaton. From an economc pont of vew, the nvestment cost must be taken nto account as well as the reducton of the producton cost. Therefore we maxmze return on nvestment(roi) by searchng not only the best locaton of FACTS devces but also the ratng of each devce. Ths problem s expressed as combnatoral optmzaton problem nested by nonlnear optmzaton problem. To solve ths problem, tabu search ncorporated wth nonlnear programmng method s used. Numercal results are shown for a 4-lne test system. Keywords: FACTS devce, OPF, phase shfter, seres capactor, tabu search, nonlnear programmng I. INTRODUCTION Nowadays, t s becomng more and more dcult to obtan the rght of way for buldng new transmsson lnes. Therefore FACTS devces, whch can control the power ows and mprove the transmsson capacty of the current transmsson network, has been consdered to be placed nto the network. Especally n a meshed network, congeston problem of actve power ow becomes so crtcal that the actve power ow control s requred n order to mprove the transmsson capacty. In ths paper, we focus on two knds of FACTS devces, such as phase shfter(ps) and seres capactor(sc) for the above purpose. Installng a phase shfter can make the ow on the bottleneck lne pass through other paths, resultng n the ncrease of transmsson capacty. Whle nstallng a seres capactor can change reactance of the lne to reduce the overloaded ow. Usng these devces we can ncrease power producton wth cheaper cost and decrease the total producton cost[][2]. On the other hand, we must take nto account nvestment cost to acheve the above-mentoned reducton of the producton cost. From an economc pont of vew, return on nvestment(roi) s an mportant gure. For power system planners, t can be of great use to seek the nvestment plan wth large ROI[3]. Here the am of our research s to nd how to place FACTS devces, that s to say the number, the locaton and the ratng of multple phase shfters and/or seres capactors that gve the maxmal ROI. It s mportant to nd a sub-optmal soluton n a practcal case snce ths problem conssts of combnatoral optmzaton problem. II. OPTIMIZATION PROBLEM In ths optmzaton problem, DC model s used for power ow, and (N-) rule s consdered for securty. In DC model a phase shfter can be treated as a voltage source nserted to the branch. We also regard a seres capactor as a voltage source nserted to the branch. The voltage dvded by the current (.e. the actve power n AC model ) on the branch ndcates the equvalent resster value of the seres capactor(.e. the reactance of the seres capactor). Snce the current s derent n each (N-) case, the varable for the voltage n each (N-) case must be prepared sub-

2 ject to equal constrant of the reactances of the seres capactors n all (N-) cases. [objectve functon] where Max: ROI = Investment = X 2B + X 2B Return Investment = C 0 0 C Investment u 0I+ 0 I2+I3 max v I4+I5 max f max 0f max 2 X X penalty C= cost P + pcost L 2ND G 2ND L () [suject to] (Krchho's current law) P f = P 0 L ( 2 N) j (Krchho's voltage law) j 0 = + 0 x f ( 2 B) (producton constrant) P mn P P max ( 2 ND G) (PS angle constrant) (SC capacty constrant) 0 max 0 max max ( 2 B) f max ( 2 B) Seres capactors can be operated both n capactor mode and n reactor mode. (thermal constrant) 0f max f f max ( 2 B) (penalty cost of outage loads) L penalty L demand 0 L ; L penalty 0 ( 2 ND L ) (constrant related to PS and SC settng) Pu =0or ; P v =0or 2B u + 2B v 0 max u max 0 max v max ( 2 B) ((N-)rule) In case,...,case N where one transmsson lne s trpped o, (Krchho's current law), (Krchho's voltage law) and (thermal constrant) are mposed. And the capacty of a seres capactor must be equal n all cases. (SC capacty equal constrant) where C 0,C: FACTS devces f case = =...= f f case 2 case N the producton cost before and after placng ND: the set of all nodesb: the set of all branches ND G ;ND L : the subset of all generator nodes, load nodes u : 0- varable that represents whether the phase shfter on branch s set or unset v : 0- varable that represents whether the seres capactor on branch s set or unset cost : the cost per unt producton of generator pcost : load the penalty cost per unt nterrupted actve power of P : the actve power producton of generator (P =0,when =2 ND G ) P mn ;P max : mnmal/maxmal producton of generator L : the actve power suppled to load ( L = 0 when =2 ND L ) L demand,l penalty : the actve power demanded by load the nterrupted actve power ; max : the phase shfter angle/maxmal angle on branch : the equvalent voltage of the seres capactor on branch max : the maxmal compensaton reactance of the seres capactor on branch f : the actve power on branch from to j f max : the maxmal admssble actve power ow on branch : the voltage phase of node x : the reactance of branch I;I2;I3;I4;I5: constants It should be noted that maxmzaton of ROI leads to mnmzaton of C because the varables such as productons of generators, the phase shfter angle and the seres capactor capacty are ndependent of the faclty varables such as FACTS devce set/unset, the maxmal phase shfter angle and the maxmal capacty of seres capactor. Therefore C represents the producton cost for the economc load dspatch. III. OPTIMIZATION OF RATING OF FACTS DEVICE Ths problem conssts of the combnatoral optmzaton problem that seeks where to place FACTS devces nested by nonlnear optmzaton problem that seeks the ratng of each FACTS devce. To begn wth, we refer to optmzaton of the ratng of the FACTS devces when the locaton of each devce s settled. If the locaton s gven, the problem turns to be a constraned nonlnear optmzaton. To solve ths problem, we use SQP (Successve Quadratc Programmng) method, that s sad to be an ecent method to solve a general constraned nonlnear optmzaton problem. As related chap.iv, to determne the locaton of each devce we employ tabu search, by whch a sub-optmal soluton can be reached n a gven teraton number. Therefore to reduce computng tme of SQP method s essental for practcal use. A. Lmtaton of (N-) Rule In a transmsson network t s requred to satsfy thermal constrants of all branches n all (N-) cases. Though the total number of thermal constrants would 2

3 be very large, the constrants necessary to get consstent solutons s a small part of all thermal constrants, that concerns the weakest part of the network n some specc (N-) cases. Even f the other constrants are gnored, the soluton to be obtaned doesn't change. Therefore the proper lmtaton of the thermal constrants seems to play an mportant role to reduce the computng tme. Here a dynamc lmtaton of (N-) cases and branches under consderaton s needed. The followng algorthm s proposed: Denote t ( =; 2; :::; N) as each (N-) case and T as the set of t. Denote s j as the thermal constrants of branch j n (N-) case t and S as the set of s j.. Gve T nt and S nt, that s the ntal set of T and S.Set T = T nt and S = S nt. 2. Solve the constraned nonlnear optmzaton problem consderng only (N-) cases of t 2 T and thermal constrants of branch s j 2 S ( = ;...;N) by SQP method. 3. If the obtaned soluton satses all the constrants ncludng constrants whch are not consdered at the prevous step 2, accept t. 4. Otherwse remove thes j 2 S whch s not actve from S.If S turns to be, remove t from T. 5. Among the set of the volated constrants not ncluded n S, choose the most greatly volated constrants s j and add t to S. If S turns not to be, add t to T. 6. If volated constrants don't exst anymore or the element number of S s more than a gven number, go to 2. Otherwse go to 5. Though ths algorthm seems not to be ecent due to necessty to apply SQP method repeatedly, the number of (N-) cases or branches under consderaton decrease so drastcally that the computng tme would be short enough to apply ths algorthm. In ths algorthm the optonal procedure to cope wth the cyclng must be mplemented. B. Local Optmum Ths nonlnear functon has no local optmum n case of consderng only phase shfters but has local optmum theoretcally n some cases consderng seres capactors. However because an ntal pont rarely aects the obtaned soluton as shown n chap.v, we regard the rst vsted local optmum as a global optmum. In a larger network, t s necessary to avod beng struck nto the local optmum. IV. SEARCH FOR THE LOCATION OF FACTS DEVICE The problem to seek the locaton of FACTS devces s a combnatoral optmzaton problem. Obvously FACTS devces must be placed to reduce the ow on the ntal bottleneck, that may make a new bottleneck. Another FACTS devce to cope wth the new bottleneck may make another bottleneck. The locatons of the FACTS devces by whch the bottlenecks can be elmnated are to some degree bounded near the bottlenecks. The good locaton of FACTS devces would be alke to the other good solutons and the neghbor of the good locaton (e.g. addng, elmnatng or movng the FACTS devce ) can be also a good one. For ths reason the neghborhood search whch proceeds from current pont to the neghbor s ecent. But a smple way such as greedy neghborhood search would cause the problem of beng struck n a local optmum. Another problem s that the only one soluton can be obtaned by the method. Our model employs some approxmatons and the optmal soluton n our model s not always optmal n practce. It s better to show some sub-optmal solutons. Here we employ tabu search, that s based on neghborhood search and has a mechansm to escape from the local optmum. A. Tabu Search Tabu search s one of modern heurstc search methods for combnatoral optmzaton problems, based on neghborhood search wth local optma avodance, whch models human memory processes. Usng a tabu lst, that retans attrbutes of moves of the solutons, specc moves are prohbted for a certan number of teratons to avod trappng nto a local mnmum. Ths helps to avod cyclng, and serves also to promote a dversed search of the solutons[4]. In our problem, the locaton of FACTS devce s expressed as a bt strng that has the length equal to the number of all lnes and mples that a FACTS devce on each branch s set or unset. Two bt strngs are needed for phase shfters and seres capactors. The neghbor of the current pont s dened as addton or elmnaton of one phase shfter or seres capactor, that s the reverse of one bt. And the reversal of the current move s dened as tabu. In each teraton, after the locaton s determned by the bt strngs, ROI s obtaned by optmzaton of ratng of each devce usng SQP method. 3

4 Fg. : Flowchart B. Lower Boundary of Return In our problem, ROI s n general larger when we set fewer FACTS devces than when we set a large number of FACTS devces. In practce, however, solutons whch produce greater returns and not the largest ROI are often desrable. Therefore we settle the lower lmt of return and maxmze the ROI as long as the return s greater than a gven value, Return mn. The followng constrant s added to the problem n chap.ii. C 0 0 C Return mn (2) However t sometmes happens that the locaton of FACTS devce determned by the bt strngs can't meet the constrant even f the ratng of the FACTS devce s properly settled. In ths case the search pont should move toward where the above constrant wll be satsed. Therefore after the locaton s determned by the bt strngs, we rst maxmze the return by SQP method and f the above constrants are satsed, we maxmze ROI, that becomes an objectve functon value. Otherwse return dvded by a large constant turns to be an objectve functon value. Ths makes the search pont shft to make the return greater untl the above constrant s satsed and after that s satsed, ROI s maxmzed. The whole algorthm s shown n Fg.. V. NUMERICAL RESULTS The model power system s shown n Fg. 2. Tabu length s set to 3, upper lmt of the maxmal angle of a phase shfter s set to 20 and the upper lmt Fg. 2: Intal load ow condton n model system of the compensaton reactance of a seres capactor s set to be 2.0 p.u. whch s equal to the reactance of branches except 3-C and A-C. The constant n the objectve functon s gven as I=I4=20500, I2=2.8, I3=4.7 and I5=0.4. The lower lmt of return s xed as 0, 20 or 40. Results are shown n Tables,2 and 3. Snce only one load pattern s used n ths study, the absolute value of the operatng angle of a phase shfter s equal to the maxmal angle of the phase shfter, so s a seres capactor. Therefore only the operatng angle/capacty s wrtten n the tables. Return and nvestment value s not the actual one, therefore the ROI doesn't havethe practcal meanng of how much year to take to collect the nvestment cost. In these tables, the branch wth multple lnes such as A-B means placng the FACTS devce on one of the multple lnes. For lack of space, only the case of Table s dscussed below: Computng tme was around 9 mnutes on Sun Ultra compatble wth Ultra SPARC 200MHz CPU. The number of teraton n tabu search was8 4

5 Table : The best 5 solutons (Return mn =0) (7.89) :236pu SC on A-C (2.5) :76 PS on A-C (2.5) :97 PS on (25.47) :26 PS on (37.69) 7873 :88 PS on 3-C 00:236pu SC on A-C Table 2: The best 5 solutons (Return mn =20) :97 PS on (25.47) :26 PS on (37.69) 7873 :88 PS on 3-C 00:236pu SC on A-C :26 PS on (37.85) :48 PS on 3-C :26 PS on (37.85) 7994 :99 PS on 3-C :76 PS on A-C :26 PS on (26.89) :00 PS on 3-C because the cyclng problem occurred. Totally 702 locatons of FACTS devces were tested, that s qute small because the total number of locatons of two or one FACTS devce s about 700 and that of three or less s about n ths 4-lne network. The obtaned solutons from st best to 4th best are valdated to be the same as the best 4 obtaned n the all search space to place two or one FACTS devces. Ths network has 4 (N-) cases and about 700 thermal constrants. By lmtaton of (N-) cases, the average number of (N-) cases and thermal constrants under consderaton was respectvely 6. and 6.4 for one teraton. In the ntal state of the network, the bottleneck lne s the energzed lnes of branch B-C wth one lne open of branch B-C. And FACTS devces make the ow on branch A-B,B-C go through branch A-C, resultng n the ncrease of transmsson capacty. Consequently branch A-C turns to be a bottleneck, and more ncrease of transmsson capacty through the left lower corrdor, that s branch 3-C, s requred. As a result the ow toward branch 3-C causes a new bottleneck on branch 9-0 wth branch 5-9 open or branch 0-3 wth branch 9-3 open. Therefore t s necessary to place FACTS devces around node 3 for the best use of branch 3-C. To nvestgate the local optmum n SQP method, Table 3: The best 5 solutons (Return mn =40) :79 PS on (40.47) : PS on 3-C :97pu SC on :59 PS on 3-C (42.37) 36 :34pu SC on 9-3 :34pu SC on :86 PS on (40.70) :38 PS on 9-3 6:5 PS on 3-C :85 PS on (42.89) :72 PS on 3-C 2:00pu SC on :24 PS on (40.70) :38 PS on 9-0 6:5 PS on 3-C we changed the ntal soluton of SQP method but the best 5 solutons n Table don't change. VI. CONCLUSION Ths paper has presented the optmzaton method based on tabu search and non-lnear programmng methods of the locaton and the ratng of the FACTS devces from an economc pont of vew. Where to set the FACTS devces and how much nvestment should be pad wll become an mportant matter because the eect of the FACTS devces would change drastcally due to the locaton. Our further work s to nvestgate for a larger-scale network and multple load condtons. VII. REFERENCES [] P.Patern, S.Vtet, A.Gard, M.Bena and A.Yokoyama, \Optmal Set of Phase Shfters for Avodng Thermal Constrants usng Genetc Algorthm", ICEE-97, A- 03, Matsue, Japan, July 28-August, 997 [2] P.Patern, S.Vtet, M.Bena and A.Yokoyama, \Optmal Locaton of Phase Shfters n the French Network by Genetc Algorthm", IEEE PES SM, PE- 078-PWRS , 998 [3] Y.Matsuo and A.Yokoyama, \Optmzaton Method of Phase Shfter Locaton n Power System Plannng ", Proc. of Annual Meetng n Power Energy Socety, IEE of Japan, No.88, 998 (n JAPANESE) [4] V.J.Rayward-Smth I.H.Osman, C.R.Reeves and G.D.Smth, MODERN HEURISTIC SEARCH METH- ODS: Wley, 996 5