Optimal Load Shedding for Voltage Stability Enhancement by Ant Colony Optimization

Transcription

1 Optma Load Shedng for Votage Stabty Enhancement by Ant Coony Optmzaton Worawat Naawro Insttute of Eectrc Power Systems (EAN) Unversty of Dusburg Essen Dusburg, Germany Istvan Erch Insttute of Eectrc Power Systems (EAN) Unversty of Dusburg Essen Dusburg, Germany Abstract Votage stabty has become a serous treat of modern power system operaton nowadays. To tace ths probem propery, oad shedng s one of the effectve countermeasures. However, ts consequences mght resut n huge technca and economc osses. Therefore, ths contro measure shoud be optmay and carefuy carred out. Ths paper proposes an ant coony optmzaton (ACO) based agorthm for sovng the optma oad shedng probem. Two prncpa concerns of the probem are addressed. The approprate oad buses for the shedng are dentfed by senstvtes of votage stabty margn wth respect to the oad change at fferent buses. Then, the amount of oad shedng at each bus s determned by appyng ACO to sove a nonnear optmzaton probem formuated n the optma power fow framewor. The performance of the proposed ACO based method s ustrated wth a crtca operatng conton of the IEEE 3-bus test system. Keywords- Load shedng; Ant coony optmzaton; Votage stabty;optma power fow I. INTRODUCTION In recent years, votage stabty has been reported as one of the reasons of the bacouts a over the word n the ast three decades []. Ths phenomenon occurs as a resut of a contngency, such as oss of an mportant transmsson ne or a major generator, nadequate reactve power support at crtca buses due to a hgh oang conton or a combnaton of the two aspects. Votage coapse s the fna consequence of votage nstabty. Ths phenomenon causes huge osses n many countres throughout the word. Therefore, t has receved sgnfcant attentons from many researchers both n academcs and ndustry. Generay votage stabty anayss nvoves two prncpa steps. Frst, the stance to the votage coapse pont, defned as votage stabty margn (VSM) must be assessed. In the second step, f the power system s vunerabe to the stabty probem, preventve and/or correctve contro actons may be requred []. Preventve contro actons can be acheved by readjustng the most effectve contros n order to provde the suffcent VSM. Correctve contro actons are on the other hand amed at restorng the stabe system operaton when subjected to severe sturbances. Correctve oad shedng may be an effectve means to acheve ths tas but t shoud be depoyed as the ast resort when other contro actons become exhausted or they are not fast enough to counteract the sturbance. In [3], the authors proposed an ntegrated method of MW/MVar management and mnmum oad shedng to satsfy votage stabty crteron. Unfortunatey, the oad shedng probem formuaton was not ceary scussed n the paper. In [4], the so-caed outage contnuaton power fow (OCPF) was proposed to consder the effects of branch and generator outages on votage stabty. The oad shedng strategy was aso deveoped based on sequenta updatng rues unt the power fow sovabty conton s met. In [5], the optma oad shedng probem s formuated n a more genera form to mnmze the tota power nterrupton cost whe mantanng system contons wth n ther respectve mts. Because the suffcent VSM to the coapse s of the most prmary concern of the oad shedng probem, however the VSM requrement was not consdered n [5]. The neggence of ths mportant constrant woud fnay resut n an neffectve operaton where the votage stabty may be st a probem even after the oad curtament. Therefore, we have formuated the oad shedng n ths paper cosey smar to [5] but an adtona constrant on VSM s adtonay ncuded. Gven the mpementaton fexbty and powerfu capabty of heurstc optmzaton methods apped n power system probems, ths paper appes ant coony optmzaton (ACO) ACO s the agorthm nspred by the foragng behavor of rea ants and was ntay proposed to sove combnatora optmzaton probems [6]. It s ony recenty that there have been attempts to extend ACO to optmze a functon n contnuous doman. Among these methods, the method proposed by Socha and Dorgo n [7] caed ACO R, whch reserves ntrnsc concepts of ant agorthm, s shown to be more effectve than other ant-reated agorthms n deang wth goba optmzaton probems. We have mofed the orgna ACO R to dea wth constraned optmzaton probems and apped t n ths paper. The rest of ths paper s organzed as foows. The frame wor of the proposed oad shedng s scussed n secton II. The mofed ACO R and constrant handng technque are expaned n secton III. The oad shedng probem s mathematcay formuated n secton IV. Smuaton resuts are

2 scussed n secton V. Fnay, concung remars and outoos on potenta future wors are gven n secton VI. II. FRAMEWORK OF THE PROPOSED SCHEME The proposed agorthm of ths paper starts from the VSM assessment of the base case by the contnuaton power fow (CPF) method [8]. In the CPF smuaton of ths paper, oad demands at a buses are ncreased n a proportona way. System generaton s aso scaed up to match the ncreasng oad demand aong the CPF process. These ncreases can be wrtten as; PG = λp G () PD = λ PD ; QD = λqd where P G, P D and Q D are the base case generator and oad powers, respectvey; λ s the oang parameter. For ong-term stabty framewor, λ can be apped as a good ncator of votage stabty []. Aong the PV curve constructon process, λ s ncreased from zero to the mum oang conton pont λ. Let suppose f λ =.6, that means the oad powers can be ncreased 6% further before reachng the coapse pont. It s aso possbe to say that the system has 6% stabty (oad) margn. If λ <, t ncates an unstabe case due to the negatve stabty margn. For an unstabe case, oad shedng mght be an effcent souton. The oad shedng probem prncpay answers two questons, namey ocatons and amounts. The approprate oad shedng ocaton can be dentfed by senstvtes of λ wth respect to parameter change at the oad bus cacuated from [9]: [ ] G P G P λ P = v [ ] λ G v G Q v G P vg Q () P P GQ G Q P λ where G P and G Q are the matrces of agebrac equatons representng net rea and reactve power njecton at buses, respectvey. v and v GP G Q are the zero eft egenvector assocated wth G P and G Q, respectvey. () P s the parta dervatve of a matrx wth respect to P,where P s a scaar representng the parameter change at oad bus. () λ s the parta dervatve of a matrx wth respect to λ. ( G P ) λ and ( GQ ) λ correspond to the rght most coumn of the CPF augmented Jacoban matrx. Once the ocatons are dentfed, the amount of oad to be curtaed at each effectve each bus s determned by appyng ACO R to sove the probem formuated n the optma power fow (OPF) framewor. Statstca stues are undertaen to examne the effectveness of the proposed strategy. III. ANT COLONY OPTIMIZATION A. Agorthm As mentoned earer, one of the most recent ant-based goba optmzaton methods n the contnuous doman s the ACO R agorthm. In ths secton, the orgna ACO R s mofed to hande constraned optmzaton probems as demonstrated n []. From the goba optmzaton vewpont, conventona ant agorthms ntay deveoped to dea wth combnatora optmzaton probems, map the entre search space of every menson nto a scretzed and defnte graph, namey the pheromone tra. To generate a new ant souton at each constructon step, a screte probabty strbuton s used to seect the pont on the graph representng the candate vaue. On the other hand, ACO R defnes the entre search doman of each menson by a contnuous PDF. The orgna ACO R agorthm uses Gaussan erne PDF to mode mutpe promsng search regons. A Gaussan erne s defned as a weghted sum of snge PDFs defned by: x ( μ ) ( σ ) G ( x) = ω = g ( x) ω e (3) = = σ π where s the number of snge Gaussan PDF at constructon step; ω, μ, and σ are vectors of sze defnng the weghts, means and standard devatons assocated wth every nvdua Gaussan PDF at the constructon step. For each ant, a new varabe vaue can be deveoped at each constructon step by a random sampng technque of a gven PDF based on mean μ and standard devaton σ. Snce the goba optmzaton n a contnuous doman nvoves an ndefnte number of candate soutons, the pheromone tra concept of conventona ACO s no onger appcabe. Therefore, ACO R stores the nowedge ganed from prevous searches n a tabe format caed the archve (T). Fg. Data structure of the archve souton Fg. shows the data structure of the souton archve T desgned to hande constraned optmzaton probems. The archve stores the set of good ant soutons that have been scovered from the prevous generatons. The frst part of T contans the set of candate soutons s, =,,, where s R n ; n s the probem menson. The next coumn of T stores the corresponng ftness vaue of th candate souton f(s ). The probabty of seectng the th souton as a mean p s recorded n the next coumn. The ast coumn hods the bnary ncatng feasbty status of the the th souton ( f feasbe and f nfeasbe). If the Gaussan erne s recty apped for random sampng, t s necessary to determne the nverse of

3 cumuatve strbuton functon (CDF), D - (x). However, ths s not aways straghtforward for an arbtrary PDF. Therefore, an aternatve sampng technque s used n ths paper n order to ncrease the mpementaton fexbty. Ths can be done n two steps. In the frst step, a snge component of the erne s probabstcay seected for each ant. The weght ω of the souton s the Gaussan PDF vaue wth mean of and standard devaton of q. It s computed accorng to: ( ) q ω = e (4) q π where q s a parameter of the agorthm and s the sze of souton archve. When q s sma, the soutons wth ower rans n the archve have very strong nfuences n gung new search rectons whereby a arger q aows the wder search versfcaton over the entre space. For each archve souton of ran n T, the corresponng probabty s cacuated by: ω p = ; r= ω =,,..., ; p p (5) r where p s the vector of probabty of seecton. Then, to generate an ant of the descent ant popuaton, the Rouette whee seecton method [] s apped to randomy seect whch candate souton of T shoud be set as the vector of mean vaues expressed by: μ = s j ; =,,...,nant ; j = Rouette(p) (6) where n s the probem menson; n ant s the sze of ant popuaton; Rouette(p) s the Rouette seecton functon wth p as the nput and returnng the seected ran. Standard devaton σ for every constructon step s cacuated from the average stance from the chosen souton s to the other soutons n T accorng to: se s σ = ξ ; =,,..., nant ; =,,..., n (7) e= where ξ s the pheromone evaporaton coeffcent; and n s the probem menson. Based on determned mean and standard devaton, a corresponng random varabe can be generated by a technque, such as Box and Mueer []. After a compete generaton, a soutons n T are raned accorng to ther feasbty status (the fourth coumn of Fg.), and ftness vaue (the second coumn of Fg.). Because the archve soutons wth ower rans have the greater nfuence n gung the search rectons, feasbe soutons are frst raned based on ther ftness vaues found by the sef-adaptve penaty scheme n [3]. The same process s repeated for nfeasbe soutons whch are paced n the ower porton of T. The same ranng procedure s repeated on the rest member of the ant popuaton. B. Constrant handng technque For constraned optmzaton probems, the attracton of a souton s s measured by the vaue of ftness functon nstead of the vaue of orgna objectve functon. Therefore, the objectve functon of ACO R s mofed to the sum of the stance vaue, d(x) and the penaty vaue, p(x). Mnmze f ( x) = d( x) + p( x) (8) The stance functon s defned as foows: v ( x) f rf = d( x) = (9) ( ) + f x v ( x) otherwse where r f s the rato of the number of feasbe soutons n the archve or ant popuaton. f (x) s the normazed vaue of f (x) and v (x) s the sum of normazed voaton of each constrant vded by the number of constrants. When there s no feasbe souton n the popuaton, the objectve functon s now to mnmze the constrant voaton. If there are feasbe soutons, then the stance vaue becomes the root mean square of the sum of the objectve vaue and constrant voatons. Ths process can hep mprove the search performance of ACO R because t gudes the ants to concentrate ony on the stance to feasbe space when there s a drought of feasbe soutons. When a number of feasbe soutons have been expored, the ants st contnues to search for further feasbe regon and smutaneousy trace the optma area. The second term of () s caed the penaty vaue. Ths term s very hepfu at a gven generaton of ACO R to dentfy whch ant can hep the exporaton. The dea of ths term s to ensure that the most usefu ant both n terms of nfeasbe soutons (at the eary stage) and feasbe soutons (toward the end) are assgned ower penates reatve to other ant soutons. Therefore p(x) can be defned as shown beow: p( x) = ( r f ) X ( x) + rf Y ( x) () where f r = X ( x) = f v ( x) otherwse () Y ( x) = f x s feasbe () f ( x) f x s nfeasbe IV. PROBLEM FORMULATION The souton of optma oad shedng nvoves the determnaton of the effectve ocatons and optma oad reductons subject to varous system constrants. Ths optmzaton tas can be carred out n two stages: pannng and operaton. In the pannng stage, system behavors of fferent scenaros are anayzed and f necessary fferent contro strateges may be determned. Durng the operaton, an optmzaton agorthm s used to suggest the effcent operaton scheme as per grd requrements. A. Probem Formuaton In the OPF framewor, the man objectve of optmzaton s to mnmze the cost of power nterrupton at buses: Mnmze subject to Δp f ( Δp = d ) C n s λ p (3)

4 a) Load bus votage mts Base conton u mn L, b ul, b u L, b n pq (4) Max. oang conton mn ul, m ul, m u L, m b) Lne power fow mts Base conton s mn L, b sl, b s L, b n (5) Max. oang conton mn sl, m sl, m s L, m c) Fxed power factor Δp Δq = p n s (6) q d) Aowabe oad curtament mn Δ p Δp Δp n s (7) e) Votage stabty margn mt N λ N λ λ + Δp + Δq.6 (8) = p = q where C s the power nterrupton cost at bus ($/W) ; n pq s the set of oad (PQ) buses; n s the set of transmsson nes; n s s the set of effectve oad buses seected for oad shedng. In ths paper, the contro varabes are the actve power oad curtament at effectve buses represented by Δp. and the dependent varabes sted n (4)-(5). To smpfy the probem, power factor at the oad shedng buses are mantaned by proportonatey curtang the reactve power oad Δq accorng to (6), where p and q are nta actve and reactve power demand of bus, respectvey. The vaue of λ s cacuated based on the near estmaton technque n []. Because the power system may become unstabe (λ<) after a severe sturbance, therefore the oad shedng agorthm must be abe to brng the system bac to the boundary of stabe operaton (λ=). However, t may not be necessary n the practca vewpont to guarantee a great stance to the coapse. Therefore, the mum stabty margn of 6% s set (λ=.6). B. Impementaton The mpementaton steps of the proposed ACO R based agorthm can be wrtten as foows; Step : At the generaton Gen =; store ACO R parameters and randomy ntaze nvduas wthn respectve mts and save them n the archve. Step : For each nvdua n the archve, evauate the orgna objectve as shown n (3) and determne the corresponng λ from the mdde term of (8). Step 3: To mantan constant power factor at oad buses, reactve power demand s adtonay curtaed (assumed at no cost) accorng to (6). Step 4: Run power fow to determne oad bus votages and cacuate ne power fows n (4) and (5) at base- and mum oang conton (at the λ found from step ). Step 5: Evauate the ftness of each nvdua based on the strategy secton III.B. Step 6: Sort nvduas of the archve based on feasbty and ftness vaues. Step 7: To generate ant popuaton, perform random sampng based on n the method scussed n secton III.A and evauate the corresponng ftness accorng to steps -5. Step 8: Sort nvduas of the ant popuaton based on feasbty and ftness vaues. Step 9: Fnd the generaton (oca) best x oca and goba best x goba ant based on the foowng crtera; a) Any feasbe souton s preferred to any nfeasbe souton; b) Between two feasbe soutons, the one havng better objectve vaue s preferred; c) Between two nfeasbe soutons, the one havng smaer ftness vaue (smaer constrant voaton) s preferred. Step : Store x oca and x goba Step : In the archve, update the nvduas by repacng a pre-specfed number of worse soutons (n rp ) by n rp better ant soutons, reevauate the ftness, and resort the archve. Step : Increase the generaton counter Gen = Gen+. Step 3: If one of stoppng crteron have not been met, repeat steps 7-. In ths paper, two stoppng crteron are set up. The agorthm stops f the mum number of generatons s reached (Gen = Gen ) or there s no souton mprovement over a specfed number of generatons. V. SIMULATION RESULTS The IEEE 3-bus system s used to test the effectveness of the proposed agorthm. The test system used n ths study has sx generaton buses, oad buses, 4 transformers and 4 transmsson nes. The networ topoogy, generator, oad and transmsson ne data can found n [4]. The reactve power sources are connected to buses and 3. The system oang s ncreased to tmes of the base case to MW where a votage profes and ne fows are wthn the mts and the corresponng λ s The N- contngency anayss was conducted to dentfy the most crtca ne. It reveas that the outage of ne connected between buses 8 and 7 resuts n an unstabe case where λ=.7533 (see Fg.4).Ths means that the system s beng

5 drven to nstabty. If no contro actons are depoyed, coapse s nevtabe. The software pacage namey, PSAT [5], s used as the smuaton too. Fg. shows the VSM senstvty wth respect to actve and reactve power demand changes at each oad bus cacuated at an operatng pont cosed to the sadde node bfurcaton. Buses wth hgh senstvtes are very effectve for the VSM enhancement. Therefore, fve buses wth the hghest senstvtes (oad buses number 7- corresponng to buses 3, 4,6,9 and 3, respectvey) are seected to partcpate n the oad shedng program. By pre-screenng the effectve buses, the computaton efforts can be sgnfcanty reduced because of fewer decson varabes are requred for the optmzaton. Load bus senstvty Actve power Reactve power 5 5 Load bus number Fg. Senstvty of oad buses Costs of power nterrupton ncurred by power consumers n fferent sectors accorng to [6] are gven n Tabe A. of appenx. The permssbe range of oad shedng at buses and oad confguraton are sted n Tabe A.. From ths tabe, oad bus 3, as an exampe, has the confguraton of.6t+.+.r. Ths means 6% of tota demand of ths bus comes from the transportaton (t) sector, % from the ndustra () sector and % from the resdenta (r) sector. Costs per W power nterruptons at every bus showng fferent cost characterstcs are aso sted n Tabe A.. The ACO R agorthm deveoped n MATLAB s apped for sovng the NLP optmzaton probem defned n () to (6). The ACO R parameter settngs used n ths study are tuned based on expermenta nowedge and sted n Tabe I. TABLE I ACO R PARAMETER SETTINGS Parameter Vaue Archve sze () 4 Number of ants (n ant) Number of repancement (n rp) 8 Convergence rate factor (q). Pheromone evaporaton (ξ).99 The votage stabty constrant n (8) s determned from the near approxmaton of VSM subject to oad reductons. Fg.3 (a) shows the comparson between actua VSMs determned by CPF and estmated vaues of random oad shedng contons n order to examne the accuracy of (6). Corresponng absoute estmaton errors are cacuated as shown n Fg.3 (b) and good accuraces are demonstrated. Votage stabty margn Estmaton error (%) (a) Estmaton Actua Load shedng condton number mn=.6847 mean=.6659 = std=.88 (b) Load shedng condton number Fg. 3 Votage stabty margn vs. oad shedng (a) comparson (b) estmaton error Foowng the optmzaton process, the PV profe of the most crtca bus (bus 3) obtaned by the CPF s potted n Fg.4 aganst pre- and post-contgency (wth no contro actons) contons. It s demonstrated that the proposed ACO R technque s abe to restore votage stabty of the system whe mantanng a number of constrants wthn ther mts. Votage at bus 3 (p.u.) Pre-contngency Post-contngency After oad shedng Votage stabty margn Fg. 4 PV curves of fferent operatng contons Objectve vaue Percent of occurance.5 x Functon evauaton 3 (a) (b) Objectve vaue x 4 Fg. 5 ACO R performance (a) convergence property (average of 5 ndependent runs) (b) hstogram of optma objectve vaues

6 The average convergence property obtaned from 5 ndependent runs s shown n Fg. 5 (a). It s obvous that the proposed agorthm s capabe of scoverng the optma souton at a very fast speed. Statstca evauaton has been performed and the hstogram of the optma objectve vaue s depcted n Fg. 5(b). It s qute obvous that the ACO R neary converges to the same souton. The optma resuts of each ndpendent run, the fna λ after oad shedng and CPU tme are averaged and gven n Tabe II. Statstca vaues of the fna objectve vaues are shown n Tabe III. TABLE II OPTIMAL RESULTS AND SIMULATION TIME Optma contro varabes (W) λ Tme (s) Δp d3 Δp d4 Δp d6 Δp d9 Δp d TABLE III STATISTICAL DATA Mn Mean Max Std CONCLUSION Ths paper presents an ant coony optmzaton (ACO) based agorthm for optma oad shedng probem to enhance power system votage stabty. The proposed method s fexbe to study technca and economc aspects of the probem. The former goa s accompshed by anayzng senstvtes of the votage stabty margn wth respect to power demand changes at fferent buses. Ony few effectve oad buses are seected to partcpate n the oad shedng program. Cost of power nterrupton s mnmzed to acheve the second requrement. The recent ACO varant for goba search n contnuous doman namey ACO R s mofed to hande constraned optmzaton probems. The deveoped ACO R s apped to sove the optmzaton probem formuated n the optma power fow (OPF) framewor wth the fu consderaton of varous networ constrants. It s shown from the smuaton resuts that the proposed method can effectvey mprove votage stabty of the power system. The deveoped ACO R aso processes at a fast speed. Statstca stues based on mutpe ndependent runs aso revea that ACO R s a qute robust too because of ts abty to generate neary dentca resuts. Because the present ACO R agorthm was ntay deveoped to sove unconstraned optmzaton probems, therefore some conceptua mofcatons coud be very usefu when handng constraned optmzaton probems. If some of parameters, whch normay requre tunng, were emnated, the agorthm woud become more powerfu. These are our current doman of nvestgaton. APPENDIX TABLE A. INTERRUPTION COST IN DIFFERENT SECTORS Interrupton cost ($/W) Transportaton (t) Industra () Commerca (c) Resdenta (r) TABLE A. LOAD SHEDDING DATA Bus ΔP D,mn(pu) ΔP D,(pu) Confguraton Cost ($/ W) 3.3.5c+.5r t r c+.4r t+.+.r 66.8 REFERENCES [] Votage stabty assessment: concepts, practses and toos, Aug., [onne] [] T.Van Cutsem and C. Vournas, Votage stabty of eectrc power systems, Kuwer Academc Pubsher, 998 [3] C.M. Affonso, L.C.P. da Sva, F.G.M. Lma and S. Soares, MW and Mvar management on suppy and demand sde for meetng votage stabty margn crtera, IEEE Trans. on Power Systems, vo.9, no.3, pp , 4 [4] H.Song, S.D. Par and B. Lee, Determnaton of oad shedng for power fow sovabty and outage contnuaton power fow (OCPF), IEE Proc. Gener Transm Dstrb, vo. 53, no.3, pp. 3-35, 6 [5] T. Amraee, A.M Ranjbar, B. Mazafar and N. Sadat, An enhanced under votage oad shedng scheme to provde votage stabty, Eec. Power Syst. Res.,vo.77, 7, pp [6] M. Dorgo and T.Stütze, Ant coony optmzaton, MIT press, Cambrdge, MA, 4 [7] K. Socha and M. Dorgo, Ant coony optmzaton for contnuous domans, European Journa of Operatona Research, vo.85, 8, pp [8] V.Ajjarapu, C.Chrsty, The contnuaton power fow: a too to study steady state votage stabty, IEEE Trans. on Power Systems, vo.7, no., pp.46-43, 99 [9] V. Ajjarapu, Computatona technques for votage stabty assessment and contro, Sprnger-Verag, 6 [] W.Naawro and I. Erch, Votage securty assessment and contro usng a hybrd ntegent method, In.Proc. IEEE Power Tech 9, Bucharest, Romana, Juy 9 [] D.E. Godberg, Genetc Agorthms n Search, Optmzaton and Machne Learnng, Kuwer Academc Pubshers, Boston, MA.,989 [] G.E.P. Box and M.E. Müer, A note on the generaton of reandom norma devates, Annas of mathematca statstcs, vo. 9, no., pp.6-6, 958 [3] B.Tessema, G.G. Yen, A sef adaptve penaty functon based agorthm for constraned optmzaton, n proc. IEEE Congress on Evoutonary Computaton, 6, 6- Juy 6, pp [4] H.Saadat, Power system anayss, McGraw-H, 999 [5] F. Mano, L. Vanfrett, J. C. Morataya, An Open Source Power System Vrtua Laboratory: The PSAT Case and Experence, IEEE Trans on Educaton, vo. 5, no., pp. 7-3, Feb. 8. [6] P.J. Baducc, J.M. Roop, L.A. Schenben, J.G. Desteese and M.R. Wemar, Eectrca power nterrupton cost estmates for nvdua ndustres, sectors and U.S. economy, Pacfc Northwest Natona Lab, Feb.