Exploiting the Use of DC SCOPF Approximation to Improve Iterative AC SCOPF Algorithms

1 Exploi ing he use o DC SCOPF app oxima ion o imp o e i e a i e AC SCOPF algo i hms A. Ma ano Ma colini, F. Capi anescu, J.L. Ma inez Ramos Senio Membe , IEEE, and L. Wehenkel Abs ac —This pape ocuses on imp o ing he solu ion ech- niques o he AC SCOPF p oblem o ac i e powe dispa ch by using he DC SCOPF app oxima ion wi hin he SCOPF algo- i hm. Ou app oach b ings wo bene i s compa ed o benchma k SCOPF algo i hms: i speeds-up he solu ion o an i e a i e AC SCOPF algo i hm hanks o a mo e e icien iden i ica ion o binding con ingencies, and allows imp o ing he objec i e by an app op ia e choice o a limi ed numbe o co ec i e ac ions o each con ingency. The p oposed app oach is illus a ed on 5 es sys ems o 60, 118, 300, 1203, and 2746 buses. Index Te ms—secu i y-cons ained op imal powe low, DC app oxima ion, con ingency il e ing, mixed-in ege linea p o- g amming, nonlinea p og amming I. INTRODUCTION The Secu i y-Cons ained Op imal Powe Flow (SCOPF) is a nonlinea , non-con ex, la ge-scale, s a ic op imiza ion p oblem, wi h bo h con inuous and disc e e a iables [1], [2]. I compu es op imal p e en i e/co ec i e ac ions ha sa is y cons ain s o bo h he p e-con ingency con igu a ion as well as unde a se o pos ula ed con ingencies. Powe sys em enginee s gene ally sol e sepa a ely and sequen ially wo SCOPF p oblems, namely he ac i e powe dispa ch and he eac i e powe dispa ch [3]. The main ea- sons o his sepa a e compu a ion a e: hese p oblems ha e di e en objec i e unc ions (e.g. minimum gene a ion cos s. maximum eac i e powe ese es o minimum powe losses), he wo p oblems a e be e posed sepa a ely as he con ol a iables o one p oblem ha e gene ally li le impac on he cons ain s and he objec i e unc ion o he o he p oblem, and hey ha e a smalle size and a e hence mo e ac able. This pape ocuses on he imp o emen o he solu ion ech- niques o he AC SCOPF p oblem o ac i e powe dispa ch by explo ing he use o a DC SCOPF. We use as a benchma k he i e a i e AC SCOPF algo i hm p oposed in [4], [5]. Among he challenges o he SCOPF compu a ions [3], [6], [8] we deal wi h he iden i ica ion o he binding con ingencies and wi h he selec ion o a limi ed numbe o co ec i e ac ions. Re . [11] epo s a case whe e he e is a sa is ac o y ag ee- men be ween he binding con ingencies a he AC and DC SCOPF solu ion o a e y la ge eal-li e sys em in he con ex o ene gy p icing by loca ional ma ginal p ices. Mo i a ed by his encou aging esul and by he compu a ional speed o he A Ma ano Ma colini and J.L. Ma inez Ramos a e wi h he Depa - men o Elec ical Enginee ing, Uni e si y o Se ille, Spain (e-mail: ale- jand [email p o ec ed]; [email p o ec ed]). F. Capi anescu and L. Wehenkel a e wi h he Depa men o Elec ical Enginee ing and Compu e Science, Uni e si y o Li`ege, B4000 Li`ege, Belgium (e-mail: capi ane@mon e io e.ulg.ac.be; l.wehenke[email p o ec ed]). linea p og amming sol e s, a i s con ibu ion o his pape is o u he ca e ully assess he bene i o using he DC SCOPF inside he i e a i e AC SCOPF algo i hms. Inc easing le els o unce ain y in he con ex o day- ahead ope a ional planning and in aday ope a ion oge he wi h ongoing e o s o enhancing he ansmission sys em lexibili y (e.g. WAMS and FACTS) lead o an inc eased use o co ec i e con ol in e e y-day p ac ice, and he e o e yield a g owing need o he e ec i e coo dina ion o co ec i e and p e en i e con ols [8], [12], [13]. On he o he hand, elying on co ec i e con ol inc eases signi ican ly he com- plexi y o powe sys em ope a ion and also in oduces new eliabili y issues, ela ed o he complexi y o implemen a ion o co ec i e con ol and i s induced ailu e modes. Mos mode n powe sys ems ha e deployed dedica ed communica- ion channels be ween con ol cen es and con ol means (e.g. powe plan s) ha may allow he au oma ic implemen a ion o op imal solu ions in ol ing a la ge numbe o emedial ac ions. Howe e , an op imal solu ion in ol ing many emedial ac ions is di icul o unde s and (e.g. his is pa icula ly ue o ac ions wi h small magni ude), in e p e , alida e by he TSO expe ience, and hence us . Fu he mo e, du ing he au oma ic implemen a ion, he TSO may lack ocusing on wha happens on he g id (and is hence mo e p one o e o ) when dealing wi h a la ge numbe o quickly mo ing powe injec ions. Finally, i he numbe o ac ions o implemen is oo la ge and some communica ion channels ail, he TSO may no be able o use he ypical back-up solu ion based on phone calls. Fo hese easons, o educe he complexi y o implemen a ion and educe he p obabili y o ailu e o co ec i e con ol, one possible app oach is o impose o each con ingency a bound on he numbe o co ec i e ac ions used in he e en ha i would happen. Indeed, he need o limi ing he numbe o co ec i e con ol ac ions has al eady been pu o wa d by many au ho s [6]–[10]. In cu en p ac ice, he choice o an app op ia e subse o co ec i e ac ions is ypically de ined in a heu is ic way based on enginee ing judgemen and o -line s udies, yielding a ixed lis o co ec i e ac ions ha a e hen plugged in o he sub- sequen SCOPF calcula ions [6], [8]–[10]. Because he mos e icien co ec i e ac ions o each con ingency may change in unp edic able ways (e.g. due o he inc easing a iabili y o load pa e ns, ne wo k opology, gene a o s dispa ch) and because he se o possible co ec i e ac ions g ows (e.g. due o he pene a ion o dispe sed gene a ion and demand side managemen possibili ies) his app oach may lead o sub- op imal esul s in e ms o ma ke e iciency and eliabili y. The e o e, he limi ed lis o co ec i e ac ions used by he 2 SCOPF should ideally be compu ed in an au oma ic ashion o each con ingency and o each ope a ing scena io. Wi hin his con ex , a ew pape s ha e p oposed echniques o limi ing he numbe o con ol ac ions in an OPF [8], i.e. o one sys em s a e, whe eas only one app oach has been epo ed o he SCOPF p oblem [15]. This app oach adop s a DC g id model [16], [17] and looks o a limi ed numbe o opological maneu e s as p e en i e ac ions only bu does no ex end he analysis o he AC SCOPF. A second con ibu ion o his pape is he ex ension o he concep s p oposed o he OPF p oblem in [14] o he AC SCOPF p oblem wi h limi ed numbe o co ec i e ac ions. We also explo e a DC mixed-in ege linea p og amming (MILP) app oxima ion o his p oblem in o de o iden i y app op ia e co ec i e ac ions o each con ingency. We assess he in e es o ou app oach o compu ing combina ions o p e en i e and co ec i e gen- e a o e-dispa ches bu he app oach may be ex ended o o he con ol ac ions (e.g. phase shi ing ans o me , ne wo k swi ching, e c.). The ield o applica ion o ou app oach is mainly day-ahead and in aday planning o ope a ion. I may be used o ensu e he he mal secu i y o a uni commi men solu ion o nex 12-24 hou s pe iod and also o eac o non- an icipa ed changes in he in aday ope a ing scena io. The es o he pape is o ganized as ollows. Sec ion II p esen s he p oblem o mula ion and he solu ion echnique. Sec ion III p o ides nume ical expe imen s wi h he p oposed app oach. Sec ion IV concludes. II. SECURITY-CONSTRAINED OPTIMAL POWER FLOW WITH LIMITED NUMBER OF CORRECTIVE ACTIONS A. S a emen o he p oblem We conside he SCOPF p oblem o ac i e powe dispa ch wi h limi ed numbe o co ec i e ac ions o ace line ou age con ingencies. The p oblem is called he ea e SCOPF-LNCA, and is o mula ed as ollows: min P0 gi ,P k gi ,sk i X i∈G ciP0 gi (1) subjec o: Pk gi −Pli −X j∈Bk i Pk ij(Vk i, V k j, θk i, θk j) = 0, ∀i∈ N,∀k∈ {0} ∪ C (2) Qk gi −Qli −X j∈Bk i Qk ij(Vk i, V k j, θk i, θk j) = 0, ∀i∈ N,∀k∈ {0} ∪ C (3) Ik ij (Vk i, V k j, θk i, θk j)≤Imax k ij ,∀i, j ∈ N,∀k∈ {0} ∪ C (4) Pmin gi ≤Pk gi ≤Pmax gi ,∀i∈ G,∀k∈ {0} ∪ C (5) Qmin gi ≤Qk gi ≤Qmax gi ,∀i∈ G,∀k∈ {0} ∪ C (6) −sk i∆Pi≤Pk gi −P0 gi ≤sk i∆Pi,∀i∈ G,∀k∈ C (7) X i∈G sk i≤Nk,∀k∈ C (8) sk i∈ {0,1},∀i∈ G,∀k∈ C,(9) whe e, supe sc ip 0 ( esp. k) e e s o he base case ( esp. con ingency ks a e), Cis he se o pos ula ed con ingencies, Gis he se o gene a o s, Nis he se o buses, Bk iis he se o b anches connec ed o bus iin s a e kand accoun s o he line ou ages in he pos -con ingency s a es, ciis he ac i e powe cos o gene a o i,Nkis he maximum numbe o co ec i e ac ions ha he sys em ope a o wishes o implemen , sk iis a bina y a iable desc ibing he s a us o gene a o i o he pos - con ingency s a e k( he gene a o can be used o co ec i e con ol i sk i= 1 and is o he wise ozen o i s p e-con ingency alue), ∆Piis he maximum amoun o powe ha a gene a o can edispa ch ollowing a con ingency, Vk i( esp. θk i) is he ol age magni ude ( esp. angle) a bus iin s a e k, he o he no a ions being sel -explana o y. The objec i e unc ion (1) is minimum gene a ion cos in he p e-con ingency s a e bu o he objec i es could be used (e.g. minimum de ia ion om he ma ke solu ion). Equali y cons ain s (2,3) a e powe low equa ions in p e- con ingencyand pos -con ingencys a es. Inequali y cons ain s (4) a e b anch cu en limi s in p e-con ingency and pos - con ingency s a es. Inequali y cons ain s (5,6) a e physical limi s o gene a o ’ ac i e and eac i e powe s. The limi a ion o he numbe o co ec i e ac ions in pos - con ingency s a es is modeled by cons ain s (7)-(9). No e ha co ec i e ac ions compu ed wi h he SCOPF- LNCA app oach o each con ingency me ely ensu e he easibili y o all pos -con ingency con ol p oblems gi en he esul ing p e en i e mode gene a ion eschedulings. Feasibil- i y being g an ed by eezing hese p e en i e con ols, co - ec i e con ols o each con ingency may hen be compu ed sepa a ely by sol ing an app op ia e se o OPF p oblems ac- co ding o any objec i e unc ion o in e es , while espec ing he cons ain s on he numbe o co ec i e con ols. We sol e he SCOPF-LNCA p oblem in wo s eps. We i s sol e he MILP app oxima ion o he p oblem so as o de e mine he op imal se {i∈ G :sk i= 1}o co ec i e ac ions o each con ingency k∈ C. Then we sol e a classical SCOPF by allowing only hese la e con ol ac ions. B. Fo mula ion o he DC MILP SCOPF app oxima ion o he SCOPF-LNCA p oblem We compu e he app op ia e co ec i e ac ions o be used in he AC SCOPF by sol ing he ollowing DC MILP SCOPF app oxima ion o he SCOPF-LNCA (1)-(9): min P0 gi ,P k gi ,sk i X i∈G ciP0 gi (10) 3 Cpb AC SCOPF Ccyes no secu i y analysis con ingency il e ing DC SCOPF C;{˜sk i} secu i y analysis AC OPF C;{˜sk i} Cc;{˜sk i} Cpb;{˜sk i}Cpb;{sk i}o {˜sk i} con ingency il e ing STOP Cc=∅? Cs Cpb ← Cpb ∪ Cs C Cpb Fig. 1. Flowcha o he i e a i e SCOPF algo i hm a ian s subjec o: Pk gi −Pli −X j∈Bk i θk i−θk j Xij = 0,∀i∈ N,∀k∈ {0} ∪ C (11) −Imax k ij ≤θk i−θk j Xij ≤Imax k ij ,∀i, j ∈ N ,∀k∈ {0} ∪ C (12) Pmin gi ≤Pk gi ≤Pmax gi ,∀i∈ G,∀k∈ {0} ∪ C (13) −sk i∆Pi≤Pk gi −P0 gi ≤sk i∆Pi,∀i∈ G,∀k∈ C (14) X i∈G sk i≤Nk,∀k∈ C (15) sk i∈ {0,1},∀i∈ G,∀k∈ C,(16) whe e (11) a e he DC powe low equa ions [16], [17], Xij is he b anch eac ance, he o he no a ions ha ing he same meaning as in he SCOPF-LNCA p oblem. Because he DC app oach assumes ha all he ol ages a e 1 p.u., hen he alues in p.u. o b anch cu en s and ac i e powe lows coincide. C. I e a i e SCOPF algo i hms Figu e 1 p o ides he lowcha o he i e a i e SCOPF algo i hm a ian s compa ed in his pape . In his igu e he p oposed app oach, called he ea e P-SCOPF, is depic ed wi h con inuous lines, while he benchma k SCOPF algo i hm [4], [5], called he ea e B-SCOPF, is depic ed by showing in dashed lines i s i s s eps (uppe le pa o he lowcha ). Figu e 1 also shows he a ious se s o con ingencies in inpu and ou pu o each module. The no a ions o hese se s o con ingencies a e u he explained in Table I. TABLE I DEFINITION OF VARIOUS CONTINGENCY SETS Cbse o binding con ingencies Cpb se o po en ially binding con ingencies Ccse o c i ical con ingencies (i.e. ha iola e cons ain s) Csse o con ingencies selec ed by he il e C a se o alse ala ms in il e ing Cni se o binding con ingencies no iden i ied by he il e We dis inguish be ween wo uses o he DC SCOPF wi hin he P-SCOPF app oach: 1) In p e en i e-only mode SCOPF 1o when he co ec- i e ac ion se s a e chosen be o ehand (i.e. o ixed alues o s a uses sk i= ˜sk iin (1)-(9)), he DC SCOPF eplaces he h ee i s s eps o he B-SCOPF, namely he OPF, he Secu i y Analysis (SA), and he Con- ingency Fil e ing (CF). Thus a he i s i e a ion o he algo i hm he AC SCOPF is ed wi h he binding con ingencies om he DC SCOPF solu ion (se Cpb). A he subsequen i e a ions bo h B-SCOPF and P- SCOPF app oaches coincide. In pa icula hey use he non-domina ed con ingency (NDC) echnique o con- ingency selec ion [4], [5]. In his case he DC SCOPF (10)-(16) is a linea p og amming p oblem. 2) In SCOPF applica ions in co ec i e-also mode wi h an op imiza ion-based choice o a limi ed numbe o co ec i e ac ions (i.e. he ull SCOPF-LNCA) he DC SCOPF eeds he AC SCOPF also wi h he co ec i e ac ion se s {sk i}, chosen in a p ope way by sol ing he MILP (10)-(16), in addi ion o he binding con ingencies a he solu ion o his p oblem. No e ha in o de o speed-up he compu a ions he DC SCOPF in i s wo o ms, LP and MILP, i is implemen ed using he same algo i hm as he B-SCOPF (i.e. elying on: OPF, SA, CF, and DC SCOPF including only po en ially binding con ingencies). III. EXPERIMENTAL VALIDATION BY SIMULATION A. Desc ip ion o he es sys ems In his sec ion we p esen ep esen a i e nume ical esul s ob ained wi h he p oposed app oaches on i e es sys ems: a 60-bus sys em, which is a modi ied a ian o he No dic32 sys em [19], he IEEE118 and he IEEE 300 sys ems [20], a modi ied old planning model o he RTE ( he F ench TSO) sys- em o 1203 buses, and a win e peak load model o he Polish powe sys em [21]. A summa y o hei cha ac e is ics is gi en in Table II, whe e: |N|,|G|,|D|,|B|,|L|,|T |,|S|, and |C| deno e he numbe o : buses, gene a o s, loads, b anches, lines, all ans o me s, shun elemen s, and pos ula ed con ingencies, espec i ely. We conside o each sys em a con ingency se composed o all single line ou ages excluding hose ha would lead o a spli ing o he sys em in o sepa a e islands. 1The p e en i e-only mode is a pa icula case o he SCOPF-LNCA (1)-(9) ob ained o sk i= 0,∀i∈ G,∀k∈ C, excep o he gene a o s pa icipa ing in equency egula ion. 4 TABLE II TEST SYSTEMS SUMMARY sys em |N | |G| |D| |B| |L| |T | |S| |C| No dic32 60 23 22 81 57 31 12 33 IEEE118 118 54 91 186 175 11 14 166 IEEE300 300 69 198 411 282 129 14 174 1203-bus 1203 177 767 1797 1394 403 11 1029 2746-bus 2746 370 2024 3279 3107 172 0 2468 B. Sol e s used The AC SCOPF o mula ion is handled by using he in e io -poin based NLP sol e desc ibed in [22] and ollows he i e a i e app oach desc ibed in [4], [5]. The DC SCOPF in ei he linea p og amming o m o mixed-in ege linea p og amming o m is sol ed on he GAMS pla o m [23] using he CPLEX sol e . CPLEX im- plemen s a dual simplex algo i hm o sol ing he linea p og amming p oblem and a b anch and cu algo i hm o he mixed-in ege p oblem [24]. All es s ha e been pe o med on a PC Pen ium IV, 1.9- GHz, 2-GB RAM. C. Compa isons o he P-SCOPF and B-SCOPF app oaches We conside he SCOPF-LNCA p oblem (1)-(9) o mula ed in p e en i e mode o all sys ems bu he 2746-bus sys em, whe e we use he co ec i e-also mode. We conside a ew cases (deno ed o “case 1” o “case 4”) o each es sys em which di e by he o al sys em load (e.g. no mal load, peak load, e c.) and/o b anch he mal limi s. Table III p o ides a compa ison be ween he P-SCOPF and B-SCOPF app oaches in e ms o hei abili y o iden i y, a he i s i e a ion o he i e a i e SCOPF algo i hm, he binding con ingencies a he AC SCOPF solu ion. In he P-SCOPF ( esp. B-SCOPF) he po en ially binding con ingencies a e p o ided by he DC SCOPF ( esp. non-domina ed con ingency (NDC) il e ing echnique [4], [5]). The a ious se s o con- ingencies ha e been explained in Table I. To compensa e possible non-iden i ied con ingencies, due o he DC model app oxima ion, he DC SCOPF app oach also includes in he po en ially binding con ingencies se some nea -binding con ingencies (e.g. ha would lead o a line loading o mo e han 98%). We can obse e ha he DC SCOPF app oach p o ides excellen esul s, he binding con ingencies being co ec ly iden i ied in 10 ou o 11 cases (i.e. when |Cni|= 0) while in oducing a easonably small numbe o alse ala ms. On he o he hand, in he case 4 o he IEEE118 sys em, he DC SCOPF app oach does no iden i y 2 binding con ingencies. Acco ding o he DC SCOPF app oach he loading o he binding line he mal limi , a he AC SCOPF solu ion, o hese con ingencies is o 97.48 %and 94.49 %, espec i ely. This misma ch be ween he wo app oaches is due o h ee ac o s: hese con ingencies lead o a signi ican amoun o losses, he losses a e compensa ed by he slack gene a o only, and hese wo lines a e loca ed e y closely o he slack gene a o . Anyway, despi e hese excellen esul s, cases whe e no all binding con ingencies a e iden i ied a e o be expec ed as TABLE III COMPARISON OF THE ACCURACY OF DC SCOPF AND NDC APPROACHES TO IDENTIFY THE BINDING CONTINGENCIES AT THE SCOPF SOLUTION es AC SCOPF DC SCOPF NDC case |Cb| |Cpb| |Cni| |C a| |Cpb| |Cni| |C a| |Cc| No dic32 sys em case 1 5 12 0 7 7 1 3 16 case 2 3 9 0 6 7 0 4 9 case 3 5 11 0 6 7 2 4 18 case 4 4 7 0 3 9 0 5 18 IEEE118 sys em case 1 9 13 0 4 19 3 13 107 case 2 10 13 0 3 12 3 5 87 case 3 8 13 0 5 17 1 10 104 case 4 8 9 2 3 11 1 4 90 IEEE300 sys em case 1 4 7 0 3 9 0 5 11 case 2 3 6 0 3 5 0 2 6 1203-bus sys em case 1 4 6 0 2 4 1 1 8 case 2 5 10 0 5 6 1 2 29 2746-bus sys em case 1 4 5 0 1 6 0 2 8 ound ou in [11], e.g. due o he eac i e powe lows which also con ibu e o b anches cu en a e neglec ed, lossless g id assump ion o he DC model, e c. On he o he hand he NDC app oach ails in 6 ou o 11 cases o iden i y all he binding con ingencies a he i s i e a ion o he i e a i e SCOPF algo i hm. This is mos o he imes due o a binding con ingency does no iola e any cons ain a ha s age o he algo i hm and o a less ex en due o a binding con ingency is il e ed ou by mis ake. The la e si ua ion a ises especially when he numbe o c i ical con ingencies |Cc|is la ge (e.g. as is he case in he IEEE118 sys em). No e ha i e a i e NDC SCOPF app oach assumes ha ew loops may be needed o iden i y all binding con ingencies especially when hese a e ha mless a he i s i e a ion. We compa e bo h app oaches in e ms o o e all SCOPF CPU ime solu ion. By looking a Table IV we no ice ha he be e accu acy o iden i y binding con ingencies o he DC SCOPF app oach leads o a smalle numbe o loops on he i e a i e algo i hm, and hence a smalle numbe o calls o he secu i y analysis module, ansla es consequen ly in o a signi ican gain o compu a ional ime in almos all cases. E en i he SA is implemen ed using pa allel compu a ions, as is he case on con ol cen es, he compu a ional ad an age o he p oposed app oach will s ill pe sis , as demons a ed in Tables V and VI. This gain is less impo an only in cases whe e ei he bo h app oaches iden i y all binding con ingen- cies a he i s i e a ion (e.g. in cases 2 and 4 o he No dic32 sys em) o whe e he DC SCOPF ails iden i ying all binding con ingencies (e.g. his happens only in case 4 o he IEEE118 bus sys em). Fu he mo e bo h app oaches may bene i om a u he educ ion o he CPU imes o he AC SCOPF module i sel hanks o a mo e e icien implemen a ion. In o de o enable he compa ison be ween bo h app oaches conce ning he compu a ional e o o each ask o he SCOPF algo i hm p esen ed in Fig. 1, Table V ( esp. Table VI) p o ides he samples o CPU imes o each module o he 5 TABLE IV OVERALL CPU TIMES (S)OF BOTH APPROACHES AND TIME REDUCTION THANKS TO THE USE OF THE PROPOSED APPROACH es case B-SCOPF P-SCOPF ime educ ion (%) No dic32 sys em case 1 9.17 5.53 39.7 case 2 4.03 4.48 -11.2 case 3 9.36 5.27 43.7 case 4 4.57 4.03 11.8 IEEE118 sys em case 1 51.4 18.9 63.2 case 2 58.6 19.3 67.1 case 3 45.3 19.2 57.6 case 4 35.09 30.19 14.0 IEEE300 sys em case 1 57.2 37.6 34.3 case 2 48.9 33.16 32.2 1203-bus sys em case 1 1102.4 554.3 49.7 case 2 1667.4 824.8 50.5 2746-bus sys em case 1 2456.1 1488.1 39.4 TABLE V SAMPLE OF CPU TIMES (S)FOR THE P-SCOPF APPROACH es case i e DC SCOPF AC SCOPF SA CF ime (s) No dic32 sys em case 1 1 0.12 5.2 0.21 - 5.53 IEEE118 sys em case 4 1 0.35 10.2 3.06 0.0 30.19 2 - 13.6 2.98 - IEEE300 sys em case 2 1 0.46 32.6 10.10 - 33.16 1203-bus sys em case 1 1 8.4 238.7 307.2 - 554.3 2746-bus sys em case 1 1 12.4 611.9 863.8 - 1488.1 i e a i e P-SCOPF ( esp. B-SCOPF) algo i hm o one case o each es sys em. D. Choosing a limi ed numbe o co ec i e ac ions by he MILP DC SCOPF 1) Mo i a ion o he need o upda e au oma ically he se o co ec i e ac ions: Figu e 2 p o ides he alue o he AC SCOPF objec i e o a ious se s o co ec i e ac ions p oposed by he MILP DC SCOPF. We conside wo ope a ing poin s unde no mal load and peak load, espec i ely. The AC SCOPF objec i e e e s o no mal load condi ions. The SCOPF objec i e equal o 1 is ob ained using he whole se o 22 co ec i e ac ions allowed o each among he 33 con ingencies. This igu e shows ha i he se o co ec i e ac ions de i ed o he peak load condi ions is also used o eed he SCOPF p oblem o he no mal load condi ions he solu ions ob ained a e sys ema ically sub-op imal. In pa icula he solu ion sub-op imali y is unaccep able o N= 2. These expe imen s suppo he need o upda e au oma ically he se o co ec i e ac ions o each an icipa ed ope a ing poin . 2) Illus a ion o he app oach: Figu e 3 plo s he alue o he AC SCOPF objec i e o inc easing alues o he numbe o co ec i e ac ions and o wo anges o co ec i e ac ions. We do no ex end he analysis beyond N= 7 as TABLE VI SAMPLE OF CPU TIMES (S)FOR THE B-SCOPF APPROACH es case i e AC SCOPF SA CF ime (s) No dic32 sys em case 1 1 0.17 0.33 0.0 9.172 3.9 0.24 0.0 3 4.3 0.23 - IEEE118 sys em case 4 1 0.37 3.32 0.0 35.092 11.2 3.10 0.0 3 14.1 3.00 - IEEE300 sys em case 2 1 1.17 10.64 0.0 48.9 2 26.9 10.28 - 1203-bus sys em case 1 1 2.3 310.5 0.0 1102.42 51.3 306.1 0.0 3 126.9 305.3 - 2746-bus sys em case 1 1 4.1 863.8 0.0 2456.1 2 724.4 863.8 0.0 1 1.01 1.02 1.03 1.04 1.05 2 3 4 5 6 7 SCOPF objec i e o no mal load (pu) numbe o co ec i e ac ions allowed N co ec i e ac ions de i ed o no mal load co ec i e ac ions de i ed o peak load Fig. 2. No dic32 sys em: AC SCOPF objec i e o a ious se s o co ec i e ac ions de i ed unde wo ope a ing condi ions he objec i e o he p oposed app oach becomes p ac ically equal o he objec i e o SCOPF ha employs all possible co ec i e ac ions. The SCOPF objec i e wi h he whole se o 22 co ec i e ac ions allowed o each con ingency is 1.000 ( esp. 1.0128) o he la ge ( esp. smalle ) ange o co ec i e ac ions. We can obse e ha in bo h cases he objec i e unc ion dec eases as he numbe o co ec i e ac ions inc eases which demons a es he e ec i eness o he app oach. Fu he mo e, he choice o co ec i e ac ions by he MILP DC SCOPF is consis en as he la ge he amoun o co ec i e ac ions he be e he objec i e. Figu e 4 p o ides a compa ison be ween he p oposed app oach and an al e na i e app oach in e ms o quali y o he objec i e. The di e en ea u e o he la e app oach is ha he se o co ec i e ac ions o an i e a ion includes he se o co ec i e ac ions a he p e ious i e a ion. This means ha he se o co ec i e ac ions o N= 4 is de e mined by sol ing successi ely he MILP p oblems o N= 2, N= 3, and N= 4 while looking only o he nex con ol ac ion o be added o he exis ing se . The igu e shows ha 6 1 1.01 1.02 1.03 1.04 1.05 2 3 4 5 6 7 SCOPF objec i e (pu) numbe o co ec i e ac ions allowed N co ec i e ac ions dP=0.2*(Pmax-Pmin) co ec i e ac ions dP=0.1*(Pmax-Pmin) Fig. 3. No dic32 sys em: AC SCOPF objec i e e sus he numbe o co ec i e ac ions allowed 1.005 1.01 1.015 1.02 1.025 1.03 2 3 4 5 6 7 SCOPF objec i e (pu) numbe o co ec i e ac ions allowed N p oposed app oach al e na i e app oach Fig. 4. No dic32 sys em: AC SCOPF objec i e e sus he numbe o co ec i e ac ions allowed by wo app oaches he p oposed app oach sligh ly ou pe o ms he al e na i e app oach. On he o he hand, Figu e 5 shows ha he p oposed app oach becomes slowe han he al e na i e echnique when Ninc eases while, as expec ed, he ime equi ed o he al e na i e app oach is a he insensi i e o N. Figu es 4 and 5 aken oge he hus highligh he ade-o be ween he solu ion quali y and he compu a ional ime o hese wo app oaches. Figu e 4 allows he sys em ope a o o assess he bes ade- o be ween he objec i e and he numbe o co ec i e ac ions allowed, o in o he wo ds, he sub-op imali y implied by using smalle numbe s o con ol ac ions and whe he he e is enough oom o maneu e in he case whe e some con ol ac ions would ail. 3) Discussion abou he solu ion sub-op imali y: Since we use a linea app oxima ion o he o iginal MINLP AC SCOPF p oblem we could expec ha he p o ided se s o co ec i e ac ions lead o sub-op imal solu ions. Ve y ecen esea ch [18] epo s ha nowadays some signi ican MINLP sol e s a e unable o sol e o op imali y p oblems simila o ou o a la ge eal-li e sys em gi en ha he ime cons ain o p o iding he solu ion o an AC SCOPF in day-ahead planning is a mos a ew hou s. As a ma e o 1 1.5 2 2.5 3 3.5 4 4.5 5 2 3 4 5 6 7 CPU ime (s) numbe o co ec i e ac ions allowed N p oposed app oach al e na i e app oach Fig. 5. No dic32 sys em: CPU ime (s) e sus he numbe o co ec i e ac ions allowed by wo app oaches ac , e en o he No dic32 sys em he combina o ial p oblem is e y la ge since we would ha e o ix |G|×|C| = 23×33 = 759 bina y a iables which leads o explo e a signi ican sub- space o he whole space o 2759 possible combina ions o co ec i e ac ions s a uses. In o de o assess he deg ee o sub-op imali y o he p ob- lem we ha e ied a ious heu is ics in he MILP DC SCOPF (e.g. sligh ly lowe ing he MVA limi s, sligh ly inc easing he load o compensa e o he lossless assump ion o he DC model, e c.) and sol ed he AC SCOPF wi h di e en se s o co ec i e ac ions. We ha e no iced ha o a gi en numbe o co ec i e ac ions he p oposed app oach may lead o sligh ly di e en solu ions bu no heu is ic leads o consis en ly be e solu ions. Ne e heless hese solu ions do no di e signi ican ly in e ms o he objec i e unc ion. Fu he mo e, as shown in Fig. 3, using me ely 5 co ec i e ac ions allows one o ob ain a alue o he objec i e unc ion which is al eady e y close o ha o he SCOPF ob ained when using he whole se o 22 co ec i e ac ions o each con ingency. Since adequa e MINLP sol e s canno comply wi h ou compu a ional ime cons ain s, especially on la ge eal-li e sys ems, we belie e ha ou app oach o selec au oma ically he se o co ec i e ac ions o each con ingency p o ides easonable esul s. 4) Assessmen o compu a ional ime o he DC MILP SCOPF app oach: In o de o assess in ealis ic condi ions he compu a ional ime equi ed by he DC MILP SCOPF solu ion we conside he 1203-bus sys em and he 2746-bus sys em. In hese simula ions we assume ha he se o he po en ially mos e icien possible co ec i e ac ions can be educed, e.g. hanks o he TSO expe ise, o 15 candida e gene a o shi s among he 143 ( espec i ely 71) dispa chable gene a o s in he 1203-bus sys em ( espec i ely he 2746-bus sys em). Table VII epo s he CPU imes o he DC MILP SCOPF o inc easing numbe s No allowed co ec i e ac ions among hose 15 candida es. The SCOPF includes only he six binding con ingencies a he DC SCOPF solu ion (see Table III) o he 1203-bus sys em, and he eigh c i ical con ingencies o he 2746-bus sys em. 7 TABLE VII SAMPLE OF CPU TIMES (S)FOR THE DC MILP SCOPF FOR VARIOUS NUMBERS NOF ALLOWED CORRECTIVE ACTIONS sys em N= 2 N= 3 N= 4 N= 5 1203-bus 17.4 19.0 17.2 14.6 2746-bus 23.3 27.4 25.8 28.5 TABLE VIII INFLUENCE OF THE NUMBER OF CANDIDATE CORRECTIVE CONTROLS (K) AND OF ALLOWED ONES (N)ON CPU TIME (S)OF THE DC MILP SCOPF N=K= 15 K= 30 K= 45 K= 60 K= 75 0 11.3 11.4 11.4 11.4 11.4 5 14.6 25.6 30.3 44.6 349.9 10 12.5 17.8 25.0 26.3 27.5 To gain u he insigh , Table VIII epo s he CPU imes on he 1203-bus sys em o g owing numbe s o candida e co ec i e con ol ac ions (K) and o allowed ones (N). In o de o assess he ex a compu a ional ime in ol ed by he MILP combina o ial sea ch we also p o ide he CPU imes o he SCOPF p e en i e mode ( he la e co esponds o he case N= 0, whe e he MILP educes o a LP p oblem). We obse e ha o a gi en alue o N, he compu a ional ime inc eases gene ally slowly as he numbe Ko candida e co ec i e con ols g ows. On he o he hand, o a ixed numbe Ko candida e con ols, he CPU ime dec eases when we inc ease he numbe No allowed con ols om N= 5 o N= 10, which sugges s ha he “p ac ical” complexi y o he MILP p oblem is no di ec ly linked o he numbe o can- dida e combina ions ha would be explo ed by an exhaus i e sea ch p ocedu e (in ou case, all possible subse s o size N chosen among Kcandida es). A possible explana ion is ha s a e-o - he-a MILP b anch-and-cu algo i hm has a much be e han wo s -case complexi y in ou p ac ical con ex . These expe imen s indica e ha he addi ional compu a- ional ime o sol ing he DC MILP SCOPF on op o he AC SCOPF is gene ally accep ably small. Ne e heless cases whe e he MILP solu ion ime becomes la ge due o he combina o ial explosion migh appea (e.g. as sugges ed by he las column o N= 5) bu o una ely hey can be kep ac able hanks o he p e-selec ion, on a long e m ho izon, o a se o candida e emedial ac ions o app op ia e size based on he TSO expe ience. IV. CONCLUSION This pape has p oposed wo me hods o imp o e he solu ion echniques o he AC SCOPF p oblem o ac i e powe dispa ch. These me hods ely on he solu ion o he DC SCOPF app oxima ion o he o iginal p oblem. Using hese me hods in an i e a i e AC SCOPF algo i hm exhibi s wo ad an ages: •a signi ican speed-up o he solu ion hanks o a mo e e icien iden i ica ion o he binding con ingencies a he op imum; • he au oma ic and p ope choice o a limi ed numbe o co ec i e ac ions o each con ingency a a gene ally low compu a ional cos is a be e solu ion han he cu en sys em ope a o p ac ice which do no adap hese subse s o con ol a iables o he si ua ion a hand. The e ec i eness o he p oposed app oaches has been ex ensi ely alida ed on a ious es sys ems up o 2746 buses. Al hough we illus a ed he app oach only o line ou ages and o gene a o edispa ch as emedial ac ions, his app oach cons i u es a gene ic amewo k ha may include o he ypes o con ingencies as well as any o he ype o use ul eme- dial ac ions (e.g. opological swi ching, phase shi e angle changes, e c.). This app oach is lexible in he sense ha i o e s he use he possibili y o de ine he maximum numbe o sough emedial ac ions ha she/he conside s easible o i s p oblem o in e es . Fu u e wo k could add ess he inco po a ion o o he in ege a iables in o he op imiza ion p oblem in a simila ashion (e.g. ne wo k swi ching among he se o allowed co ec i e ac ions and gene a o s a -up decisions among he se o allowed p e en i e mode con ol ac ions), and o he ypes o secu i y conce ns (e.g. ol age and ansien s abili y). ACKNOWLEDGMENTS We hank RTE-F ance o allowing us o use and publish esul s wi h hei da a. The p ojec is a esul o a collabo a ion spawn by Eu opean FP7 p ojec PEGASE, whose unding is kindly acknowledged. A. Ma ano Ma colini and J.L. Ma - inez Ramos like o hank he inancial suppo p o ided by he Go e nmen o Andalusia, Spain, unde g an TEP-5170. This pape p esen s esea ch esul s o he Belgian Ne wo k DYSCO, unded by he In e uni e si y A ac ion Poles P o- g amme, ini ia ed by he Belgian S a e, Science Policy O ice. The scien i ic esponsibili y es s wi h he au ho s. REFERENCES [1] O. Alsac and B. S o , “Op imal load low wi h s eady-s a e secu i y”, IEEE T ans. PAS, ol. PAS-93, no. 3, 1974, pp. 745-751. [2] A.J. Mon icelli, M.V.P. Pe ei a, and S. G an ille, “Secu i y-cons ained op imal powe low wi h pos -con ingency co ec i e escheduling”, IEEE T ans. Powe Sys ., ol. PWRS-2, no. 1, Feb ua y 1987, pp. 175-182. [3] B. S o , O. Alsac, and A.J. 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Alejand o Ma ano Ma colini was bo n in A gen ina in 1977. He ecei ed he elec ical enginee ing deg ee om he Uni e si y o Malaga, and he Ph.D. deg ee om he Uni e si y o Se ille, Spain, in 2010. He is cu en ly an assis an p o esso a he Uni e si y o Se ille. His p ima y a eas o in e es a e ol age s abili y, powe sys em con ol and ope a ion, and op imiza ion applied o powe sys em enginee ing. Flo in Capi anescu was bo n in Romania in 1973. He g adua ed in Elec ical Powe Enginee ing om he Uni e si y “Poli ehnica” o Bucha es in 1997. He ob ained he Ph.D. deg ee om he Uni e si y o Li`ege in 2003. His main esea ch in e es s a e in powe sys ems ope a ion, planning, and con ol wi h pa icula emphasis on op imiza ion me hods and ol age s abili y. Jose Luis Ma inez Ramos (SM’04) was bo n in Dos He manas, Spain, in 1964. He ecei ed his Ph.D. deg ee in elec ical enginee ing in 1994. Since 1990 he has been wi h he Depa men o Elec ical Enginee ing, Uni e si y o Se ille, whe e he is cu en ly ull P o esso . His p ima y a eas o in e es a e ac i e and eac i e powe op imiza ion, powe sys em analysis and con ol, and elec ici y ma ke s. Louis Wehenkel g adua ed in Elec ical Enginee ing in 1986 and ecei ed he Ph.D. deg ee in 1990, bo h om he Uni e si y o Li`ege, whe e he is ull P o esso o Elec ical Enginee ing and Compu e Science. His esea ch in e es s lie in he ields o s ochas ic me hods o sys ems and modeling, op imiza ion, machine lea ning and da a mining, wi h applica ions in complex sys ems, in pa icula la ge scale powe sys ems planning, ope a ion and con ol, indus ial p ocess con ol, bioin o ma ics and compu e ision.