Security Constrained Optimal Power Flow

Transcription

1 Security Constrained Optimal Power Flow 1. Introduction and notation Fiure 1 below compares te optimal power flow (OPF wit te security-constrained optimal power flow (SCOPF. Fi. 1 Some comments about tese different formulations: SCOPF solution will always ave a cost > OPF solution. If we inore losses, ten we can say tat an OPF solution differs from an EDC solution (economic dispatc calculation, i.e., no transmission only wen a normal transmission constraint becomes bindin. o Occurs wen normal flow moves from just < 1% to >1% of continuous ratin. SCOPF differs from an OPF solution only wen a continency transmission constraint becomes bindin. o Occurs wen post-continency flow moves from just < 1% to >1% of emerency ratin. 1

2 We will cane notation now. Instead of usin te notation (prime to indicate te constraints under continencies, we will subscript te constraints, were te subscript indicates te continency state. For example, te optimal power flow (OPF problem can be written as below. We will call tis problem P. P Min f Here, (x,u = represents te power flow equations; (x,u represents te branc-flow constraints; represent branc continuous ratin Te state variables x denote te bus voltae manitudes and anles under pre-continency condition Te index = indicates tis problem is posed for only te pre-continency condition, i.e., te condition wit no continencie Tus, tis problem is just te OPF. Now let s consider te security-constrained OPF (SCOPF. Its problem statement is iven as problem P p : P p Min f,1,2,..., c Notice tat tere are c continencies to be addressed in te SCOPF, and tat tere are a complete new set of constraints for eac of tese c continencie Observe: Eac set of continency-related equality constraints is exactly lie te oriinal set of equality constraints (tose for problem P, except it corresponds to te system wit an element removed.,1,2,..., c 2

3 Eac set of continency-related inequality constraints is exactly lie te oriinal set of inequality constraints (tose for problem P, except it corresponds to te system wit an element removed and, for branc flow constraints and for voltae manitudes, te limits will be differen Also notice tat te constraints are a function of x, te voltae manitudes and anles under te pre-continency (= and continency conditions (>1,2,,c, and u, te controls wic were set under te pre-continency conditions (=. 2. Reducin computation time for SCOPF Denote te number of constraints for te OPF, Problem P, as N. Assumption: Let s assume tat runnin time T of te aloritm we use to solve te above problem is proportional to te square of te number of constraints, i.e., N 2. For simplicity, we assume te constant of proportionality is 1, so tat T=N 2. So te SCOPF must deal wit te oriinal N constraints, and also anoter set of N constraints for every continency. Terefore, te total number of constraints for Problem P p is N+cN=(c+1N. Under our assumption tat runnin time is proportional to te square of te number of constraints, ten te runnin time will be proportional to [(c+1n] 2 =(c+1 2 N 2 =(c+1 2 T. Wat does tis mean? It means tat te runnin time of te SCOPF is (c+1 2 times te runnin time of te OPF. So if it taes OPF 1 minute to run, and you want to run SCOPF wit 1 continencies, it will tae you 11 2 minutes, or 1,21 minutes to run te SCOPF. Tis is 17 ours, about 1 wee!!!! 3

4 Many systems need to address 1 continencie Tis would tae about 2 years! So tis is wat you do.. Solve OPF = (normal condition Solve OPF =1 (continency 1 Solve OPF = (continency 2 Solve OPF = (continency 3 Solve OPF =c (continency c Fi. 1: Decomposition solution stratey Te solution stratey first solves te OPF (master problem and ten taes continency 1 and re-solves te OPF, ten continency 2 and resolves te OPF, and so on (tese are te subproblem For any continency-opfs wic require a redispatc, relative to te = OPF, an appropriate constraint is enerated, at te end of te cycle, tese constraints are atered and applied to te = OPF. Ten te = OPF is resolved, and te cycle starts aain. Experience as it tat suc an approac usually requires only 2-3 cycle Denote te number of cycles as m. Eac of te individual problems as only N constraints and terefore requires only T minute Tere are (c+1 individual problems for every cycle. Tere are m cycle So te amount of runnin time is m(c+1t. 4

5 If c=1 and m=3, T=1 minute, tis approac requires 33 minute Tat would be about 5 ours (instead of 1 wee. If c=1 and m=3, T=1 minute, tis approac requires about 5 ours (instead of 2 year In addition, tis approac is easily parallelizable, i.e., eac individual OPF problem can be sent to its own CPU. Tis will save even more time. Fiure 2 [1] compares computin time for a 6-bus system (Fi. 2a and a 24 bus test system (Fi. 2b. Te comparison is between a full SCOPF, a decomposed SCOPF (DSCOPF, and a decomposed SCOPF were individual OPF problems are sent to separate CPU Tis ind of aloritm is formalized as Benders decomposition. Fi. 2 Tere is a ric literature on usin decomposition metods for solvin SCOPF and SCUC. Searcin on Benders and (optimal power flow or unit commitment returns 54 its in IEEE Xplore. 5

6 3. Preventive v Corrective In tis section, we desire to distinuis between two inds of security-related actions, i.e., two inds of control. Consider te security-state diaram of Fi. 3. Fi. 3 Preventive control is an action taen to move from te alert state to te normal state. Preventive control is taen to prevent an undesirable operatin condition from occurrin if a continency occur Since tere is no immediate consequence of suc a state, preventive control is not typically concerned wit ow muc time a particular action require Corrective control is an action taen to move from te emerency state to te alert state or from te emerency state to te normal state. Since an emerency state is experiencin an existin undesirable operatin condition, it is important to move out of te emerency state quicly. As a result, corrective control is eavily concerned wit ow muc time a particular action will tae. 6

7 4. Preventive SCOPF Te preventive SCOPF is te one we ave already posed as problem P p, repeated below for convenience. P p Min f,1,2,..., c As already mentioned in Section 1., te constraints are a function of x, te voltae manitudes and anles under te pre-continency (= and continency (>1,2,,c conditions, and u, te controls wic were set under te pre-continency conditions (=. But wat maes tis a preventive SCOPF?,1,2,..., c Te fact tat te controls are restricted to teir pre-continency condition settins, tus denoted u, maes tis a preventive SCOPF. Tat is, we must position te power system wile in te normal condition (i.e., no continency to prevent operatin conditions followin occurrence of a continency to exceed emerency ratin Note tat ere and elsewere in tese notes, continency refers to any continency on a specified continency list but not a continency tat is not on te specified continency lis Tat is, we do not account for all possible continencies but rater just a certain set, enerally all N-1 continencie 7

8 5. Fully corrective SCOPF Te fully corrective SCOPF is posed below as problem P c1. P c1 Min f u u K 1,2,..., c 1,2,..., c 1,2,..., c Here, K is a vector of very lare positive number Observations: Tis problem is considered corrective because postcontinency (=1,2, c controls u are allowed to move in order to satisfy te post-continency constraint Te problem is considered fully corrective because we allow post-continency constraints to be satisfied independent of precontinency conditions, i.e., and, >, include u as an arument instead of u (compare to prob P p. Two implications: o Post-continency controls may move as muc as needed, witin bounds of te control capabilities (usually te & min en values to satisfy post-continency constraint Tis is made possible by settin te elements of K to +, i.e., te difference between post-continency control levels u and pre-continency control levels u are unrestricted. o Post-continency controls must move as muc as needed, witin bounds of te control capabilities (usually te & min en values to satisfy post-continency constraint In oter words, we do not allow preventive control in tis problem, i.e., we do not allow settin pre-continency controls u to satisfy post-continency constraint So postcontinency constraints must be satisfied entirely by postcontinency control. 8

9 Because te post-continency conditions are independent of precontinency variables, Problem P c1 is really Problem P (te OPF, unless one of te continency problems >1 is infeasible. In tis case, te entire problem is considered infeasible because tere is a continency wic leads to an infeasible condition and cannot be made feasible no matter wat post-continency control we tae. Question: Wat is te order of tese problems, P, P p, P c1, in terms of decreasin production cost? Answer: Cost(P p >Cost(P c1 =Cost(P. Question: Wat is te order of tese problems, P, P p, P c1, in terms of increasin system ris? Here, we must imaine tat we ave an acceptable measure of system ri Answer: Ris(P p <Ris(P c1 <Ris(P. Observe: Production costs decrease as ris increase 6. Preventive-corrective SCOPF Te preventive-corrective SCOPF is posed below as problem P c2. Min f P c2 u u u 1,2,..., c 1,2,..., c 1,2,..., c Here, te amount of corrective control tat can be expended is limited by an amount u and te pre-continency control 9

10 settin u trou te last constrain Te followin observations are made: 1. Te rit-and side of te last constraint, u, is te imum cane for te post-continency control variable It is computed as a product of te assumed time orizon allowed for corrective actions T and an assumed rate (typically imum of cane of control variables in response to continency, du /dt, i.e., du u T 1,2,..., c dt 2. Te post-continency control levels u do not appear in te objective function, i.e., te only values tat affect te objective function are u. 3. If tere are no violated post-continency constraints, ten u will be selected based only on te objective function and te pre-continency constraint 4. If tere are violated post-continency constraints, te aloritm will try to satisfy tem usin only post-continency control levels u, because tis does not affect te objective function. Tis is usin te corrective control part of te aloritm. 5. If te violated post-continency constraints cannot be satisfied usin only post-continency control levels u, ten te aloritm uses pre-continency control levels u to satisfy tem. Tis is usin te preventive control part of te aloritm. 6. It is important to realize tat te reason we use corrective control first, and preventive control only if necessary, is tat Te corrective control is perceived not to cost very muc if te continency occurs, because te continency state is not expected to last very lon. In addition, te continency liely will not occur, in wic case te corrective control will cost notin at all! In contrast to te previous bullet, any cane to precontinency control variables u, a preventive control, 1

11 moves te system away from te optimal economic point independent of weter a continency occurs or not, and terefore, tis cane will always cost money! 7. Because post-continency control levels u are not included in te objective function, it is possible to find different corrective controls tat will provide feasibility for te same objective function value. Tus, we see tat te preventivecorrective SCOPF can ave multiple solution To distinuis between te various solutions, one can add postcontinency control costs to te objective function, but since te continencies mit or mit not appen, one must condition tose post-continency control costs for eac continency on te continency probability. Tis problem is provided below as Problem P c3. c Min p f p f 1 P c3 u u u 1,2,..., c 1,2,..., c 1,2,..., c [1] Y. Li, Decision main under uncertainty in power system usin Benders decomposition. PD Dissertation, Iowa State University,