Planning Reconfigurable Reactive Control for Voltage Stability Limited Power Systems

Transcription

1 Plannng Reconfgurable Reactve Control for Stablty Lmted Power Systems Hafeng Lu, Member, IEEE, Lcheng Jn, Member, IEEE, James D. McCalley, Fellow, IEEE, Ratnesh Kumar, Fellow, IEEE, Venkataramana Ajjarapu, Fellow, IEEE, Ncola Ela, Member, IEEE Abstract--Ths paper proposes an optmzaton based method of plannng reactve power control for electrc transmsson systems to endow them wth the capablty of beng reconfgured to a secure confguraton under a lst of contngences. The overall objectve functon s to mnmze the nstallaton cost of new controls such as mechancally swtched capactors, whle satsfyng the requrements of stablty margn and magntude under a contngency lst. The backward/forward search algorthm wth lnear complexty s used to select canddate locatons for swtched capactors. Optmal locatons and amounts of new swtch controls are obtaned by solvng a sequence of mxed nteger programmng problems. The modfed New England 39-bus system and a North Amercan power system wth 6358 buses are adopted to llustrate the effectveness of the proposed method. Index Terms--Mxed nteger programmng, power system plannng, reactve power control, reconfguraton, stablty. I. INTRODUCTION PPROPRIATE long-term plannng to strengthen Atransmsson capablty s necessary to ncrease future relablty levels of the electrc transmsson system. There are three basc optons for strengthenng transmsson systems: () buld new transmsson lnes, (2) buld new generaton at strategc locatons, and (3) ntroduce addtonal control capablty. However, optons () and (2) have become less and less vable because of the dffculty n obtanng rght-of-way and expensve nvestment of the transmsson or generaton facltes. As a result, there s a sgnfcantly ncreased potental for applcaton of addtonal power system control to expand transmsson n the face of growng transmsson usage. In ths paper, we focus on plannng reconfgurable reactve power control to ncrease the stablty lmt and thus enhance transmsson capablty n stablty lmted systems. There are three basc problems to be addressed for plannng Ths work was supported by fundng from the Natonal Scence Foundaton and from the Offce of Naval Research under the Electrc Power Networks Effcency and Securty (EPNES) program, award ECS H. Lu s wth the Dvson of Market & Infrastructure Development, Calforna ISO, Folsom, CA USA (e-mal: hlu@caso.com). L. Jn, J. D. McCalley, R. Kumar, V. Ajjarapu, and N. Ela are wth the Department of Electrcal and Computer Engneerng, Iowa State Unversty, Ames, IA 500 USA (e-mal: lcjn@astate.edu; jdm@astate.edu; rkumar@astate.edu; vajjarap@astate.edu; nela@astate.edu). reconfgurable reactve power control: () when s system enhancement needed, (2) where to mplement the enhancement, and (3) how much s the reactve power control needed. We address the frst queston usng the technques of contnuaton power flow (CPF) [], [2], [3] and fast contngency screenng [4], [5]. The last two questons could be answered under an optmzaton framework. The problem addressed n ths paper s smlar to the reactve power plannng problem [6]-[]. enerally, the reactve power plannng problem can be formulated as a mxed nteger nonlnear programmng problem to mnmze the nstallaton cost of reactve power devces subject to a set of equalty and nequalty constrants. In the work descrbed n ths paper, however, we explctly target the plannng of reactve power controls,.e., reactve power devces ntended to serve as control response for contngency condtons. Instead of consderng only the most severe contngency or consderng several contngences sequentally [2], the proposed plannng method consders multple contngences smultaneously. Vaahed et al. n [3] presented a dynamc securty constraned OPF tool whch smultaneously consdered magntude constrants and stablty margn constrants under normal and contngency condtons. The tool was based on a three level herarchcal decomposton scheme where each sub-problem was solved by the nteror pont method. The CPF program was used to valdate the stablty margn. The objectve functon dd not nclude fxed cost. Yorno et al. n [4] proposed a mxed nteger nonlnear programmng formulaton for reactve power control plannng whch takes nto account the expected cost for collapse and correctve controls. The Benders decomposton technque was appled to obtan the soluton. Z. Feng et al. n [5] dentfed reactve power controls usng lnear optmzaton wth the objectve of mnmzng the control cost. The stablty margn senstvty [6], [7], [8] was used n the formulaton. Ths formulaton s sutable to the operatonal decson makng problem. However, fxed nstallaton costs of new controls should be ncluded n the long-term plannng problem. In ths paper, a comprehensve methodology s proposed for the long-term reactve power control plannng to mprove stablty and aganst contngences. Mechancally swtched capactors are used as the reactve power control means. Such devces have been used for post-contngency

2 2 control [9]-[26]. Fast contngency screenng and contnuaton power flow technques are utlzed to determne nvestment years for the necessary network enhancements. A backward/forward search algorthm wth lnear complexty s used to select canddate control locatons. A sequence of mxed nteger programmng (MIP) usng stablty margn senstvtes and magntude senstvtes s proposed to obtan optmal reactve control locatons and amounts. The objectve functon of the MIP s to mnmze the total nstallaton cost ncludng fxed cost and varable cost of new controls whle satsfyng the requrements of stablty margn and magntude under normal and contngency condtons. The CPF program s utlzed to check the true stablty margn and magntude after each MIP. Ths teratve process s requred to account for system nonlneartes and to mprove the optmalty of the soluton. The branch-and-bound method s used to solve the MIP problem [27]. Because the optmzaton formulaton s lnear, t s fast, yet t provdes good solutons for large-scale power systems compared wth nonlnear optmzaton formulatons. The followng assumptons are made n ths paper: No new lnes or transformers are nstalled or consdered n the plannng expanson, and generaton expanson occurs only at exstng generaton facltes. These assumptons represent the extreme form of a relance on control to strengthen/expand transmsson capablty wthout buldng new transmsson lnes or strategcally stng new generaton. There exsts an equlbrum pont after a contngency. If ths assumpton s not satsfed, we frst restore the equlbrum pont by usng mnmum amount of reactve power controls [28], [29], [30]. The paper s organzed as follows. Some fundamental concepts of stablty margn and margn senstvty are ntroduced n Secton II. Secton III descrbes the proposed method of the long-term reactve power control plannng. Numercal results are dscussed n Secton IV. Secton V concludes. II. VOLTAE STABILITY MARIN AND MARIN SENSITIVITY The goal of the paper s to determne locatons and amounts for swtched capactors so as to enable mprove stablty margn. In ths secton, we formally defne the noton of stablty margn and ts senstvty to parameters, for we use such senstvtes n determnng the desred locatons and amounts. stablty margn s defned as the dstance between the collapse pont of the system power- (PV) curve and the forecasted total system real power load as shown n Fg.. The potental for contngences such as unexpected component (generator, transformer, transmsson lne) outages n an electrc power system often reduces the stablty margn [3], [32], [33]. We are nterested n fndng effectve and economcally justfed reactve power controls to steer operatng ponts far away from collapse ponts by havng a pre-specfed margn and a satsfactory level under a set of prescrbed contngences. Swtched capactors are adopted to mprove the stablty margn. We thnk of each such capactor as a controllable dscrete swtch that may be swtched n or out as needed. Fg. shows the stablty margn under dfferent operatng condtons and swtches (controls). V max V mn normal control contngency M0: normal stablty margn M: reduced stablty margn M2: ncreased stablty margn forecasted load Real Power Fg.. stablty margn for dfferent operatng condtons. One queston of the reactve power control plannng s how much of the control s needed for the requrement of a gven amount of margn ncrease. One way to answer the queston s to use the dea of the ( stablty) margn senstvty, whch s the change n stablty margn wth respect to a (small) change n power system parameter or control varable. In the followng, the analytcal expresson of the margn senstvty s gven. Detaled development of margn senstvty s found n [6], [7], [8], [34], [35]. Suppose that the steady state of the power system satsfes a set of equlbrum equatons expressed n the vector form F ( x, p, λ) = 0 () where x s the vector of state varables, p s any parameter n the power system steady state equatons such as the susceptance of shunt capactors or the reactance of seres capactors, and λ s the bfurcaton parameter whch s a scalar. At the collapse pont of the system PV curve, the value of the bfurcaton parameter s equal to λ. A specfed system scenaro can be parameterzed by λ as Pl = ( + Klpλ) P (2) l0 Ql = ( + Klqλ) Q (3) l0 Pgj = ( + Kgjλ) P (4) gj0 where P l0 and Q l0 are the ntal loadng condtons at bus at the base case, K lp and K lq are factors characterzng the load ncrease pattern, P gj0 s the real power generaton at bus j at the base case, K gj represents the generator load pck-up factor. The stablty margn can be expressed as M M2 M0

3 3 M = n = P l n P l0 = n = λ K P (5) = lp l0 The senstvty of the stablty margn wth respect to the control varable at locaton, S, s n M λ S = = KlpP (6) l0 p p = If the collapse s due to a saddle-node bfurcaton, the bfurcaton pont senstvty wth respect to the control varable p evaluated at the saddle-node bfurcaton pont s λ wfp = (7) p wf λ where w s the left egenvector correspondng to the zero egenvalue of the system Jacoban F x, F λ s the dervatve of F wth respect to the bfurcaton parameter λ, and F p s the dervatve of F wth respect to the control varable p. If the collapse s due to a lmt-nduced bfurcaton [36], the bfurcaton pont senstvty wth respect to the control varable p evaluated at the crtcal lmt pont (as opposed to at the saddle-node bfurcaton pont) s F p w λ = E p (7 ) p F λ w Eλ where E(x, λ, p) = 0 s the lmt equaton representng the bndng control lmt (.e. Q - Q max = 0 representng generator reaches ts reactve power lmt), E λ s the dervatve of E wth respect to the bfurcaton parameter λ, and E p s the dervatve of E wth respect to the control varable p, w s the nonzero row vector orthogonal to the range of the Jacoban J of the equlbrum and lmt equatons where F x J =. Ex The stablty margn senstvtes are updated at each backward/forward search step n Secton III.D, and at each optmzaton teraton n Sectons III.E and III.F to account for system nonlneartes. III. ALORITHM OF REACTIVE POWER CONTROL PLANNIN The proposed reactve power control plannng approach requres 5 basc steps: () development of generaton/load growth future, (2) stablty assessment by fast screenng and CPF, (3) tme determnaton to enhance the system, (4) selecton of canddate control locatons, and (5) development of reactve power control plan by solvng the orgnal and successve MIP problems. The frst three steps are descrbed brefly, wth more emphass placed on the fourth and ffth steps. A. Development of eneraton and Load rowth Future In ths step, the generaton/load growth future s dentfed, where the future s characterzed by a load growth percentage for each load bus and a generaton allocaton for each generaton bus. The dentfed generaton/load growth future can be easly mplemented n the CPF program [] by parameterzaton as shown n (2), (3) and (4). B. Stablty Assessment by Fast Contngency Screenng and CPF We can use the CPF program to calculate the stablty margn of the system under each contngency. However, the CPF algorthm s tme-consumng. If many contngences must be assessed, the calculaton tme s large. The margn senstvty can be used to speed up the contngency analyss procedure. Frst, the CPF program s used to calculate the stablty margn at the base case, the margn senstvty wth respect to lne admttances S l, and the margn senstvty wth respect to bus power njectons S pq. For crcut outages, the resultng stablty margn s estmated as (0) M = M + Sl l (8) where M (k) s the stablty margn under contngency k, M (0) s the stablty margn at the base case, and l s the negatve of the admttance vector for the outaged crcuts. For generator outages, the resultng stablty margn s estmated as (0) M = M + S pq pq (9) where Δpq s the negatve of the output power of the outaged generators. Then the contngences are ranked from most severe to least severe accordng to the value of the estmated stablty margn. After the ordered contngency lst s obtaned, each contngency s evaluated startng from the most severe one usng the CPF program. Evaluaton termnates after encounterng a certan number of sequental contngences havng stablty margn greater than or equal to the requred value, where the number depends on the sze of the contngency lst. A smlar dea has been used n on-lne rskbased securty assessment [5]. Note that the stablty margn senstvtes are used for contngency rankng purpose n ths step. The true stablty margn under each contngency s calculated usng the CPF program. C. Determnaton of Expanson Year Assumng postve load growth but wthout transmsson system enhancement, the stablty margn under normal and contngency condtons decreases wth tme. The year when the stablty margn becomes less than the requred value s the tme to enhance the transmsson system by addng reactve power controls. D. Stage : Selecton of Canddate Control Locatons An mportant step n the reactve power control plannng problem s the selecton of the canddate locatons for reactve power control devces. Canddate control locatons may be chosen based on experence and/or the relatve value of lnear senstvtes of new control devces. In ths case, however, there s no guarantee that the selected canddate control locatons are suffcent to provde needed stablty margn for all pre-defned contngences. On the other hand, the computatonal burden to solve the MIP problem presented n Sectons III.E and III.F may be hgh f all buses n a large power system are selected as canddates. To dentfy a suffcent but not excessve number of locatons, the

4 backward/forward search algorthm wth lnear complexty proposed n [37] can be used to fnd canddate locatons for swtched capactors. It s assumed that the capacty of the swtched capactor at each locaton s fxed at the maxmum allowable value n ths stage. The man steps to select approprate canddate reactve power control locatons are as follows: ) Choose an ntal set of control locatons usng the bsecton approach for each dentfed contngency possessng unsatsfactory stablty margn accordng to the followng 3 steps: a) Rank the feasble control locatons accordng to the numercal value of margn senstvty n descendng order wth locaton havng the largest margn senstvty and locaton n havng the smallest margn senstvty. b) Estmate the stablty margn wth top half of the controls swtched n as n / 2 M = X S + M (0) est = max where (k ) M s the estmated stablty margn and est n /2 s the largest nteger less than or equal to n/2. If the estmated stablty margn s greater than the requred value, then reduce the number of control locatons by one half, otherwse ncrease the number of control locatons by addng one half of the remanng. c) Contnue n ths manner untl we dentfy the mnmum set of control locatons that satsfes the stablty margn requrement. 2) Refne canddate control locatons for each dentfed contngency possessng unsatsfactory stablty margn usng a backward/forward search algorthm wth complexty lnear n the number of feasble reactve power controls. The backward/forward search algorthm begns at the ntal control confguraton and searches from that confguraton n a prescrbed drecton, ether backward or forward. We gve only the backward search algorthm here snce the forward search algorthm s smlar. Consder the graph where each node represents a confguraton of dscrete controls, and two nodes are connected f and only f they are dfferent n one control. The graph has 2 n nodes where n s the number of feasble controls. We pctorally conceve of ths graph as consstng of layered groups of nodes, where each successve layer (movng from left to rght) has one more control on (or closed ) than the layer before t, and the t th layer (where t=0,,n) conssts of a number of nodes equal to n!/t!(n-t)!. Fg. 2 llustrates the graph for the case of 4 controls. The backward search algorthm has 4 steps. 2.) Select the node correspondng to all controls n the ntal set that are closed. 2.2) For the selected node, use the CPF program to check f the stablty margn requrement s satsfed for the concerned contngency on the lst. If not, then stop, the soluton corresponds to the prevous node (f there s a prevous node, otherwse no soluton exsts). 2.3) For the selected node, update margn senstvtes and elmnate (open) the control that has the smallest margn senstvty. We denote ths as control : { S Ω } = arg mn () where Ω ={set of closed controls for the selected node}, c S s the margn senstvty wth respect to the susceptance of shunt capactors or the reactance of seres capactors under contngency k, at locaton. 2.4) Choose the neghborng node correspondng to the control beng off. If there s more than one control dentfed n step 3,.e., >, then choose any one of the controls n to elmnate (open). Return to step 2. If step 2 of the above procedure results n no soluton n the frst teraton, then no prevous node exsts. In ths case, we extend the graph n the forward drecton. 3) Obtan the fnal canddate control locatons as the unon of nodes for whch stablty margn s satsfed, as found n step 2) for every dentfed contngency. Pre-contngency state (0000) Post-contngency state, no swtches on (000) (000) (000) (000) Fg. 2. raph for 4-swtch problem. c (00) (00) (00) (00) (00) (00) (0) (0) (0) (0) () All swtches on E. Stage 2: Formulaton of Intal MIP In stage, we fnd the canddate locatons for swtched capactors. We use the ntal MIP to estmate the locatons and amounts of swtched capactors from stage canddates. The MIP mnmzes control nstallaton cost whle satsfyng the requrements of stablty margn and magntude. The multple contngences dentfed n Secton III.B are treated smultaneously n the optmzaton formulaton. mnmze F = C B + C q (2) Ω v f subject to S B + M M, k (3) r Ω mn max V j SV, j, B + V j V j, j, Ω k ) 0 B B, k k 4 (4) ( (5)

5 5 B mn q B B q (6) max q = 0, (7) The decson varables are B (k), B, and q. Varable defnton follows, C f s fxed nstallaton cost and C v s varable cost of swtched capactors, B : sze of the swtched capactor at locaton, q = f locaton s selected for reactve power control expanson, otherwse, q =0, Ω : set of canddate locatons to nstall swtched capactors, the superscrpt k represents the contngency causng nsuffcent stablty margn and/or volaton problems, B : sze of the swtched capactor to be swtched n at locaton under contngency k, (k S : senstvty of the stablty margn wth respect ) to the sze of the swtched capactor at locaton under contngency k, M (k) : stablty margn under contngency k and wthout controls, M r : requred stablty margn, the subscrpt j represents the bus havng volaton problems, S : senstvty of the magntude at bus j wth V, j, respect to the sze of the swtched capactor at locaton under contngency k [38], V j (k) : magntude at bus j under contngency k and wthout controls, V j mn : mnmum allowable at bus j, V j max : maxmum allowable at bus j, B mn : mnmum sze of the swtched capactor at locaton, and B max : maxmum sze of the swtched capactor at locaton. The nequaltes n (3)-(4) are the constrants of stablty margn and magntude respectvely. For the problem havng k stablty margn volaton and m magntude volaton and n selected canddate control locatons, there are n(k+2) decson varables and k+2m+3n+2kn constrants. The backward/forward search algorthm provdes effectve ways to lmt the number of canddate control locatons and thus to reduce the computatonal burden for solvng the above MIP formulaton. The branch-and-bound method s used to solve ths MIP problem. The output of the MIP problem s the control locatons and amounts for all k contngences and the combned control locaton and amount. Then the network confguraton s updated by swtchng n the controls under each contngency. After that, the stablty margn and the magntude are recalculated usng the CPF program to check f suffcent margn s acheved and magntude s wthn the lmts for each concerned contngency. Ths step s necessary because the stablty margn and the magntude nonlnearly depend on control varables, and our MIP algorthm uses lnear senstvtes to estmate the effect of varatons of control varables on the stablty margn and the magntude. As a result, there may be contngences that volate the requrements of stablty margn and/or magntude after updatng the network confguraton accordng to results of the ntal MIP problem. Also, the obtaned soluton may not be optmal after solvng the ntal MIP problem. The reactve control amount and locaton can be further refned by recomputng magntude, magntude senstvtes, stablty margn and margn senstvtes (wth updated network confguraton) under each concerned contngency, and solvng successve MIP problems wth updated nformaton, as descrbed n the next subsecton. F. Stage 3: Formulaton of Successve MIP The successve MIP s formulated to mnmze the total nstallaton cost of swtched capactors subject to the constrants of the requrements of stablty margn and magntude, as follows: mnmze F = C B + C q (9) Ω v subject to S ( B B ) + M M, k Ω f r ( B B ) V j max + V j, j k mn V j SV, j,, Ω (20) (2) 0 B B, k (22) mn max B q B B q (23) q = 0, (24) (k ) The decson varables are B, q and B. Varable defnton follows, B : new sze of the swtched capactor at locaton, q : new bnary locaton varable for swtched capactor at locaton, S : updated senstvty of the stablty margn wth respect to the sze of the swtched capactor at locaton under contngency k, (k ) B : new sze of the swtched capactor at locaton under contngency k, M : updated stablty margn under contngency k, S V, j, : updated senstvty of the magntude at bus j wth respect to the sze of the swtched capactor at locaton under contngency k, and (k ) V j : updated magntude at bus j under contngency k. The above MIP wll be re-executed untl all concerned contngences have satsfactory stablty margn and magntude and there s no sgnfcant movement of the decson varables. The senstvtes, stablty margn and magntude are updated at each MIP to account for

6 6 system nonlneartes. There may be stuatons where ths refnement process results n cyclng of capactors n and out wth each refnement. If ths s the case, the algorthm wll termnate after a pre-specfed number of teratons. Among the executed teratons, the mnmum cost soluton satsfyng the requrements of stablty margn and magntude mght be useful f such a soluton exsts. The overall procedure for the reactve power control plannng s shown n Fg. 3 whch ntegrates the above mentoned steps. Start Develop generaton/load growth future stablty assessment by fast screenng and CPF Stage : Fnd canddate locatons for swtched capactors usng the lnear complexty search algorthm Stage 2: Use ntal MIP to estmate control locatons and amounts Update control locatons and amounts Valdate stablty margn and magntude Performance crtera satsfed No No Update margn and magntude senstvtes Stage 3: Use successve MIP wth updated nformaton to refne control locatons and amounts Update control locatons and/or amounts Fg. 3. Flowchart for the reactve power control plannng. IV. NUMERICAL RESULTS Yes Converge A. Results of a Small System The proposed method has been appled to the modfed New England 39-bus system shown n Fg. 4. The orgnal real and reactve power loads n [39] are multpled.38 tmes wth assocated ncreases n real and reactve power capactes of generatng unts proportonal to ther orgnal value n order to stress the system. In the smulatons, the followng condtons are mplemented unless stated otherwse: Loads are modeled as constant power; Reactve power output lmts of generators are modeled; In computng stablty margn, the power factor of the load remans constant when the load ncreases, and load and generaton ncrease are proportonal to ther base case value; The system MVA base s 00; End Yes Requred stablty margn s assumed to be 0% of system load; V mn = 0.9 pu and V max =. pu; Swtched shunt capactors are adopted to mprove the stablty margn and profle; The parameter values adopted n the optmzaton problem are gven n Table I. The compensaton equpment costs are from [32] Fg. 4. Modfed New England 39-bus test system TABLE I PARAMETER VALUES ADOPTED IN OPTIMIZATION PROBLEM Shunt capactor (500 kv) Shunt capactor (230 kv) Varable cost ($ mllon/00 Mvar) Fxed cost ($ mllon) Maxmum sze.5.5 Mnmum sze B mn=0-3 B mn=0-3 Consderng all N- contngences (excludng radal transmsson lne outages), usng the algorthm presented n Secton III.B, 3 generator contngences (3, 32 and 35) cause the post-contngency stablty margn less than 0% as shown n Table II. The outages of the largest two generators 38 and 39 create unsolvable cases whch was addressed n [30]. The stablty margns under all nonradal transmsson lne outages are greater than 0%. Table II also shows the rank of the generator contngences. We wll use the proposed method to plan swtched shunt capactors to ncrease the stablty margn for the frst 3 contngences n Table II. Margn senstvtes to shunt capactors under the three contngences are shown n Table III. TABLE II 35

7 7 VOLTAE STABILITY MARIN UNDER CONTINENCIES stablty Contngency margn (%) (rank) (). Outage of the generator at bus (3) (2). Outage of the generator at bus (2) (3). Outage of the generator at bus () (4). Outage of the generator at bus (8) (5). Outage of the generator at bus (4) (6). Outage of the generator at bus (7) (7). Outage of the generator at bus (6) (8). Outage of the generator at bus (5) Canddate buses TABLE III MARIN SENSITIVITIES TO SHUNT CAPACITORS Margn sens. under cont. () Margn sens. under cont. (2) Margn sens. under cont. (3) Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus The canddate control locatons are determned based on the lnear search algorthm presented n Secton III.D. The best seven canddate buses to nstall swtched shunt capactors are buses 5, 6, 7, 0,, 2, and 3. Note that the canddate locatons dentfed by the backward/forward search algorthm to nstall swtched shunt capactors to ncrease the loadablty n the system level are buses 0,, 2 and 3 nstead of those near bus 35 under the outage of the generator at bus 35 because buses 0,, 2, and 3 have relatvely large margn senstvtes as shown n Table III. Ths result can be supported by the fact that the bottom left area n the New England 39-bus system s a weak area wth load more than generaton. On the other hand, the bottom rght area n the system s relatvely strong. It can mantan load and generaton balance even under the outage of the generator at bus 35. The superorty of the dentfed reactve control locatons over others s also verfed by comparng the ncrease n stablty margn for a same amount of reactve control at dfferent locatons usng the CPF program. For these canddate locatons, the optmzaton based reactve power control plannng algorthm presented n Sectons III.E and III.F was carred out. After three MIP teratons, the overall teratve procedure converges and the requrements of stablty margn and magntude are satsfed under the planned reactve controls. Table IV gves the verfed results of the reactve power control plannng wth the CPF program. Table V shows the results where the optmal allocatons for swtched shunt capactors are.500 p.u.,.500 p.u., p.u.,.86 p.u. and.500 p.u. at buses 5, 6, 7,, and 2 respectvely. The total nstallaton cost s $9.05 mllon for the reactve control allocatons. Note that the total cost s $9.42 mllon and the system has satsfactory stablty margn and magntude after solvng the ntal MIP. The proposed teratve reactve control plannng procedure reduces the total cost whle satsfyng the requrements of stablty margn and magntude. TABLE IV VERIFIED VOLTAE MANITUDE AND VOLTAE STABILITY MARIN Contngency Wthout shunt compensaton stablty margn (%) Max/Mn Wth shunt compensaton stablty margn (%) Max/Mn (). Outage of the generator at bus / /0.90 (2). Outage of the generator at bus / /0.90 (3). Outage of the generator at bus / /0.94 TABLE V REACTIVE CONTROL ALLOCATIONS WITH SHUNT CAPACITORS Iteraton number for MIP Locatons for Maxmum Overall optmal Soluton to Soluton to Soluton to shunt cap. sze lmt control allocaton cont. () cont. (2) cont. (3) Bus Bus Bus Bus Bus The proposed reactve control plannng method consders multple contngences smultaneously. Table VI lsts the result of ndependent control allocaton for each contngency and the unon of the control allocatons. Comparng Table V wth Table VI, t s clear that the overall cost of control allocaton when consderng multple contngences smultaneously s less than that when consderng each contngency ndependently. The computaton n the proposed reactve control plannng method s done only to (a) calculate magntude, magntude senstvtes, stablty margn and margn senstvtes and (b) solve the optmzaton. It s only n calculatng magntude, magntude senstvtes, stablty margn and margn senstvtes that we must deal wth the full sze of the power system. The CPU tme for the reactve control plannng of the modfed New England 39- bus system s 0.53 seconds on a standard 2.2 Hz machne. 3

8 8 Locatons for shunt cap. TABLE VI CONTROL ALLOCATION FOR EACH CONTINENCY Maxmum sze lmt Unon of control allocatons Control allocaton for cont. () Control allocaton for cont. (2) Control allocaton for cont. (3) Bus Bus Bus Bus Bus Bus Bus The computatonal performance of the proposed reactve control plannng method s examned aganst a hgher system loadng level and a system lower loadng level. For the hgher system loadng level, the system real and reactve power loads are multpled.0 tmes. The post-contngency stablty margn and magntude decrease when the load ncreases as shown n Table VII. The overall teratve procedure converges at the thrd teraton. The requrements of stablty margn and magntude are satsfed under the planned controls as shown n Table VII. Table VIII shows the optmal allocatons for swtched shunt capactors under the hgher loadng level. TABLE VII VERIFIED VOLTAE MANITUDE AND VOLTAE STABILITY MARIN UNDER THE HIHER LOADIN LEVEL Contngency Wthout shunt compensaton stablty margn (%) Max/Mn Wth shunt compensaton stablty margn (%) Max/Mn (). Outage of the generator at bus / /0.90 (2). Outage of the generator at bus / /0.9 (3). Outage of the generator at bus / /0.94 TABLE VIII CONTROL ALLOCATIONS UNDER THE HIHER LOADIN LEVEL Iteraton number for MIP Maxmum Overall optmal Soluton to Soluton to Soluton to Locatons for sze lmt control allocaton cont. () cont. (2) cont. (3) shunt cap. Bus Bus Bus Bus Bus For the lower system loadng level, the system real and reactve power loads are multpled 0.96 tmes. The postcontngency stablty margn and magntude ncrease when the load decreases as shown n Table IX. After three MIP teratons, the overall teratve procedure converges and the requrements of stablty margn and magntude are satsfed under the planned controls as shown n Table IX. Table X shows the optmal allocatons for swtched 3 shunt capactors under the lower loadng level. TABLE IX VERIFIED VOLTAE MANITUDE AND VOLTAE STABILITY MARIN UNDER THE LOWER LOADIN LEVEL Contngency Wthout shunt compensaton stablty margn (%) Max/Mn Wth shunt compensaton stablty margn (%) Max/Mn (). Outage of the generator at bus / /0.90 (2). Outage of the generator at bus / /0.90 (3). Outage of the generator at bus / /0.92 Locatons for shunt cap. TABLE X CONTROL ALLOCATIONS UNDER THE LOWER LOADIN LEVEL Overall optmal Maxmum sze lmt Soluton to cont. () Soluton to cont. (2) Iteraton number for MIP 3 Soluton to cont. (3) control allocaton Bus Bus Bus Bus B. Results of a Large Sze System In ths secton, the applcaton of the proposed method to a North Amercan power system wth 6358 buses, 7724 branches and 82 generators s presented. The total generaton of the system s MW and the total load s MW. In the smulatons, the followng condtons are mplemented: In computng stablty margn, generaton s ncreased n area 40 and load s ncreased n areas 4, 22, 24, 26 and 30; V mn = 0.9 pu and V max =. pu; Requred stablty margn s assumed to be 200 MW. Among the 74 consdered contngences, there are 3 contngences causng the stablty margn less than 200 MW as shown n Table XI. Ten canddate locatons are selected to nstall swtched shunt capactors. Margn senstvtes to shunt capactors under the three contngences are shown n Table XII. TABLE XI VOLTAE STABILITY MARIN UNDER CONTINENCIES FOR A LARE SIZE POWER SYSTEM stablty Contngency number margn (MW) () (2) (3) (4) (5) (6) (7) (8) (9) (0) TABLE XII

9 9 MARIN SENSITIVITIES TO SHUNT CAPACITORS FOR A LARE SIZE POWER SYSTEM Canddate buses Margn sens. under cont. (2040) Margn sens. under cont. (2067) Margn sens. under cont. (2072) Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Table XIII shows the results of the optmal allocatons for swtched shunt capactors. Table XIV gves the verfed results of the reactve power control plannng wth the CPF program. After four MIP teratons, the overall teratve procedure converges and the requrements of stablty margn and magntude are satsfed under the planned controls. The CPU tme for the reactve control plannng of the large system s seconds on a standard 2.2 Hz machne. TABLE XIII REACTIVE CONTROL ALLOCATIONS FOR A LARE SIZE POWER SYSTEM Locatons for shunt cap. Maxmum sze lmt Overall optmal control allocaton Soluton to cont. (2040) Soluton to cont. (2067) Soluton to cont. (2072) Bus Bus Bus Bus TABLE XIV VERIFIED VOLTAE MANITUDE AND VOLTAE STABILITY MARIN FOR A LARE SIZE POWER SYSTEM Contngency Number Wthout shunt compensaton Max/Mn stablty margn (MW) Wth shunt compensaton stablty margn (MW) Max/Mn () / /0.99 (2) / /0.99 (3) / /0.99 V. CONCLUSION Iteraton number for MIP Ths paper presents an optmzaton based method of plannng reactve power controls n electrc power transmsson systems to satsfy the requrements of stablty margn and magntude under normal and contngency condtons. The planned reactve power controls are capable to serve as control response for contngences. The backward/forward search algorthm wth lnear complexty s used to select canddate locatons for reactve power controls. 4 Optmal locatons and amounts of new reactve power controls are obtaned by solvng a sequence of MIP problems. The proposed algorthm can handle a large-scale power system because t sgnfcantly reduces the computatonal burden by utlzng the senstvtes of stablty margn and magntude. The effectveness of the method s llustrated usng the modfed New England 39-bus system and a North Amercan power system wth 6358 buses. The results show that the method works satsfactorly to plan reactve power controls. We envson that the method to be used n generatng less costly transmsson renforcement solutons n comparson to solutons whch nclude new transmsson, helpng to dentfy more optons durng the plannng process. ACKNOWLEDMENT The authors would lke to thank the revewers for comments and questons that helped the authors mprove the qualty of ths paper. REFERENCES [] V. Ajjarapu and C. Chrsty, The contnuaton power flow: a tool for steady state stablty analyss, IEEE Trans. Power Syst., vol. 7, pp , Feb [2] C. A. Canzares and F. L. Alvarado, Pont of collapse and contnuaton methods for large AC/DC systems, IEEE Trans. Power Syst., vol. 8, pp. -8, Feb [3] H. D. Chang, A. J. Flueck, K. S. Shah, and N. Balu, CPFLOW: a practcal tool for tracng power system steady-state statonary behavor due to load and generaton vacatons, IEEE Trans. Power Syst., vol. 0, pp , May 995. [4] S. reene, I. Dobson, F. L. Alvarado, Contngency rankng for collapse va senstvtes from a sngle nose curve, IEEE Trans. Power Syst., vol. 4, pp , Feb [5] M. N, J. D. McCalley, V. Vttal, S. reene, C. Ten, V. S. anugula, and T. Tayyb, Software mplementaton of onlne rsk-based securty assessment, IEEE Trans. Power Syst., vol. 8, pp , Aug [6] W. Xu, Y. Mansour, and P.. Harrngton, Plannng methodologes for stablty lmted power systems, Electrcal Power and Energy Systems, vol. 5, pp , Aug [7] K. H. Abdul-Rahman and S. M. Shahdehpour, Applcaton of fuzzy sets to optmal reactve power plannng wth securty constrants, IEEE Trans. Power Syst., vol. 9, pp , May 994. [8] V. Ajjarapu, P. L. Lau, and S. Battula, An optmal reactve power plannng strategy aganst collapse, IEEE Trans. Power Syst., vol. 9, pp , May 994. [9] J. A. Momoh, S. X. uo, E. C. Ogbuobr, and R. Adapa, The quadratc nteror pont method solvng power system optmzaton problems, IEEE Trans. Power Syst., vol. 9, pp , Aug [0] K. Y. Lee, X. Ba, and Y. M. Park, Optmzaton method for reactve power plannng by usng a modfed smple genetc algorthm, IEEE Trans. Power Syst., vol. 0, pp , Nov [] W. D. Rosehart, C. A. Canzares, and V. H. Quntana, Effect of detaled power system models n tradtonal and -stabltyconstraned optmal power flow problems, IEEE Trans. Power Syst., vol. 8, pp.27-35, Feb [2] O. O. Obadna and. J. Berg, Var plannng for power system securty, IEEE Trans. Power Syst., vol. 4, pp , May 989. [3] E. Vaahed, Y. Mansour, C. Fuchs, S. ranvlle, M. L. Latore and H. Hamadanzadeh, Dynamc securty constraned optmal power flow/var plannng, IEEE Trans. Power Syst., vol. 6, pp , Feb [4] N. Yorno, E. E. El-Araby, H. Sasak, and S. Harada, A new formulaton for FACTS allocaton for securty enhancement aganst collapse, IEEE Trans. Power Syst., vol. 8, pp. 3-0, Feb

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Sngh, Web-based securty costs analyss n electrcty markets, IEEE Trans. Power Syst., vol. 20, pp , May [36] I. Dobson and L. Lu, collapse precptated by the mmedate change n stablty when generator reactve power lmts are encountered, IEEE Trans. Crcuts Syst., vol. 39, pp , Sept [37] H. Lu, L. Jn, J. D. McCalley, R. Kumar, and V. Ajjarapu, Lnear complexty search algorthm to locate shunt and seres compensaton for enhancng stablty, n Proc North Amercan Power Symposum, pp [38] A. J. Wood and B. F. Wollenberg, Power eneraton Operaton and Control. New York: John Wley & Sons, Inc., 996. [39] M. A. Pa, Energy Functon Analyss for Power System Stablty. Boston: Kluwer Academc Publshers, 989. Hafeng Lu (M 08) receved the B.S. and M.S. degrees n electrcal engneerng from Zhejang Unversty, Hangzhou, Chna, n 2000 and 2003, respectvely, and the Ph.D. degree n electrcal engneerng from Iowa State Unversty, Ames, n Currently, he s wth the Dvson of Market & Infrastructure Development at Calforna ISO. Lcheng Jn (M 08) receved the B.S. and M.S. degrees n electrcal engneerng from Zhejang Unversty, Hangzhou, Chna, n 2000 and 2003, respectvely. She s currently pursng the Ph.D. degree n the Department of Electrcal and Computer Engneerng at Iowa State Unversty, Ames. Her research nterests are power system stablty analyss, control, and optmzaton. James D. McCalley (F 04) receved the B.S., M.S., and Ph.D. degrees from eorga Insttute of Technology, Atlanta, n 982, 986, and 992, respectvely. Currently he s a Professor n the Electrcal and Computer Engneerng Department at Iowa State Unversty, Ames, where he has been snce 992. He was wth Pacfc as and Electrc Company from 986 to 990. Ratnesh Kumar (F 07) receved the B.Tech. degree n electrcal engneerng from the Indan Insttute of Technology, Kanpur, Inda, n 987 and the M.S. and Ph.D. degrees n electrcal and computer engneerng from the Unversty of Texas at Austn, n 989 and 99, respectvely. He s currently a Professor n the Department of Electrcal and Computer Engneerng, Iowa State Unversty, Ames. Hs prmary research nterests are dscrete-event and hybrd systems and ther applcatons. Venkataramana Ajjarapu (F 08) receved hs Ph.D. degree n Electrcal Engneerng from the Unversty of Waterloo, Ontaro, Canada, 986. Currently, he s a Professor n the Department of Electrcal and Computer Engneerng at Iowa State Unversty, Ames. Hs present research s n the area of reactve power plannng, stablty analyss, and nonlnear phenomena. Ncola Ela (M 02) receved the Laurea degree n electrcal engneerng from Poltecnco d Torno, n 987 and the Ph.D. degree n electrcal engneerng and computer scence from the Massachusetts Insttute of Technology n 996. Presently he s an Assocate Professor n the Department of Electrcal and Computer Engneerng at Iowa State Unversty. Hs research nterests nclude computatonal methods for controller desgn and hybrd systems.