More on Serity Constrained Optimal Power Flow 1. Notation In te last lass we represented te OPF and te SCOPF as below. We will ange notation now. Instead of sing te notation prime to indiate te onstraints nder ontingenies we will sbsript te onstraints were te sbsript indiates te ontingeny state. For eample te optimal power flow OPF problem an be written as below. We will all tis problem P. Min f P s. t. g Here g represents te power flow eqations and g represents te line-flow onstraints. Te state variables denote te bs voltage magnitdes and angles nder pre-ontingeny onditions. Te inde indiates tis problem is posed for only te pre-ontingeny ondition i.e. te ondition wit no ontingenies. Ts tis problem is jst te OPF. 1
Now let s onsider te serity-onstrained OPF SCOPF. Its problem statement is given as problem P p : Min f P p s. t. g 12... 12... Notie tat tere are ontingenies to be addressed in te SCOPF and tat tere are a omplete new set of onstraints for ea of tese ontingenies. Ea set of ontingeny-related eqality onstraints is eatly lie te original set of eqality onstraints tose for problem P eept it orresponds to te system wit an element removed. Ea set of ontingeny-related ineqality onstraints is eatly lie te original set of ineqality onstraints tose for problem P eept it orresponds to te system wit an element removed and for line flow onstraints and for voltage magnitdes te limits will be different. Also notie tat te onstraints are a fntion of te voltage magnitdes and angles nder te pre-ontingeny and ontingeny onditions >12 and te ontrols wi were set nder te pre-ontingeny onditions. 2. Reding omptation time for SCOPF Denote te nmber of onstraints for te OPF Problem P as N. Assmption: Let s assme tat rnning time T of te algoritm we se to solve te above problem is proportional to te sqare of te nmber of onstraints i.e. N 2. For simpliity we assme te onstant of proportionality is 1 so tat TN 2. So te SCOPF mst deal wit te original N onstraints and also anoter set of N onstraints for every ontingeny. Terefore te total nmber of onstraints for Problem P p is N+N+1N. 2
Under or assmption tat rnning time is proportional to te sqare of te nmber of onstraints ten te rnning time will be proportional to [+1N] 2 +1 2 N 2 +1 2 T. Wat does tis mean? It means tat te rnning time of te SCOPF is +1 2 times te rnning time of te OPF. So if it taes OPF 1 minte to rn and yo want to rn SCOPF wit 1 ontingenies it will tae yo 11 2 mintes or 121 mintes to rn te SCOPF. Tis is 17 ors abot 1 wee!!!! Many systems need to address 1 ontingenies. Tis wold tae abot 2 years! So tis is wat yo do.. Solve OPF normal ondition Solve OPF 1 ontingeny 1 Solve OPF ontingeny 2 Solve OPF ontingeny 3 Solve OPF ontingeny Fig. 1: Deomposition soltion strategy Te soltion strategy first solves te OPF master problem and ten taes ontingeny 1 and re-solves te OPF ten ontingeny 2 and resolves te OPF and so on tese are te sbproblems. For any ontingeny-opfs wi reqire a redispat relative to te OPF an appropriate onstraint is generated at te end of te yle tese onstraints are gatered and applied to te OPF. Ten te OPF is resolved and te yle starts again. Eperiene as it tat s an approa sally reqires only 2-3 yles. 3
Denote te nmber of yles as m. Ea of te individal problems as only N onstraints and terefore reqires only T mintes. Tere are +1 individal problems for every yle. Tere are m yles. So te amont of rnning time is m+1t. If 1 and m3 T1 minte tis approa reqires 33 mintes. Tat wold be abot 5 ors instead of 1 wee. If 1 and m3 T1 minte tis approa reqires abot 5 ors instead of 2 years. In addition tis approa is easily parallelizable i.e. ea individal OPF problem an be sent to its own CPU. Tis will save even more time. Figre 2 [1] ompares ompting time for a 6-bs system Fig. 2a and a 24 bs test system Fig. 2b. Te omparison is between a fll SCOPF a deomposed SCOPF DSCOPF and a deomposed SCOPF were individal OPF problems are sent to separate CPUs. Tis ind of algoritm is formalized as Benders deomposition. 4
Fig. 2 Tere is a ri literatre on sing deomposition metods for solving SCOPF and SCUC. Searing on Benders and optimal power flow or nit ommitment retrns 54 its in IEEE Xplore. 3. Preventive vs. Corretive In tis setion we desire to distingis between two inds of serity-related ations i.e. two inds of ontrol. We reall te serity-state diagram of Fig. 3 onsidered before. Fig. 3 5
Preventive ontrol is an ation taen to move from te alert state to te normal state. Preventive ontrol is taen to prevent an ndesirable operating ondition from orring if a ontingeny ors. Sine tere is no immediate onseqene of s a state preventive ontrol is not typially onerned wit ow m time a partilar ation reqires. Corretive ontrol is an ation taen to move from te emergeny state to te alert state or from te emergeny state to te normal state. Sine an emergeny state is eperiening an eisting ndesirable operating ondition it is important to move ot of te emergeny state qily. As a reslt orretive ontrol is eavily onerned wit ow m time a partilar ation will tae. 4. Preventive SCOPF Te preventive SCOPF is te one we ave already posed as problem P p repeated below for onveniene. Min f P p s. t. g 12... 12... As already mentioned in Setion 1. te onstraints are a fntion of te voltage magnitdes and angles nder te pre-ontingeny and ontingeny >12 onditions and te ontrols wi were set nder te pre-ontingeny onditions. Te fat tat te ontrols were restrited to teir pre-ontingeny ondition settings ts denoted maes tis a preventive SCOPF. 6
7 5. Flly orretive SCOPF Te flly orretive SCOPF is posed below as problem P 1. P 1 g g f Min s t 12... 12..... Tis problem is onsidered orretive bease post-ontingeny 12 ontrols are allowed to move in order to satisfy te post-ontingeny onstraints. Te problem is onsidered flly orretive bease we allow post-ontingeny onstraints to be satisfied independent of pre-ontingeny onditions i.e. g and > do not inlde as an argment. Te effet of tis is tat we are not allowing preventive ontrol in tis problem sine we do not allow movement of pre-ontingeny ontrols in order to satisfy post-ontingeny onstraints. 6. Preventive-orretive SCOPF Te preventive-orretive SCOPF is posed below as problem P 2. P 2 g g f Min s t 12... 12... 12..... Δ
Here te amont of orretive ontrol tat an be epended is limited by an amont and te pre-ontingeny ontrol setting trog te last onstraint. Te following observations sold be made: 1. Te rigt-and side of te last onstraint is te imm ange for te post-ontingeny ontrol variables. It is ompted as a prodt of te assmed time orizon allowed for orretive ations T and an assmed rate typially imm of ange of ontrol variables in response to ontingeny d /dt i.e. d Δ T 12... dt 2. Te post-ontingeny ontrol levels do not appear in te objetive fntion i.e. te only vales tat affet te objetive fntion are. 3. If tere are no violated post-ontingeny onstraints ten will be seleted based only on te objetive fntion and te pre-ontingeny onstraints. 4. If tere are violated post-ontingeny onstraints te algoritm will try to satisfy tem sing only post-ontingeny ontrol levels bease tis does not affet te objetive fntion. Tis is sing te orretive ontrol part of te algoritm. 5. If te violated post-ontingeny onstraints annot be satisfied sing only post-ontingeny ontrol levels ten te algoritm will se pre-ontingeny ontrol levels to satisfy tem. Tis is sing te preventive ontrol part of te algoritm. 6. It is important to realize tat te reason we se orretive ontrol first and preventive ontrol only if neessary is tat Te orretive ontrol is pereived not to ost very m if te ontingeny ors bease te ontingeny state is not epeted to last very long. In addition te 8
ontingeny liely will not or in wi ase te orretive ontrol will ost noting at all! In ontrast to te previos bllet any ange to preontingeny ontrol variables a preventive ontrol moves te system away from te optimal eonomi point independent of weter a ontingeny ors or not and terefore tis ange will always ost money! 7. Bease te post-ontingeny ontrol levels are not inlded in te objetive fntion it is possible to find different orretive ontrols tat will provide feasibility for te same objetive fntion vale. Ts we see tat te preventive-orretive SCOPF an ave mltiple soltions. To distingis between te varios soltions one old add post-ontingeny ontrol osts to te objetive fntion bt sine te ontingenies migt or migt not appen one wold ave to ondition tose post-ontingeny ontrol osts for ea ontingeny on te probability of tat ontingeny. 7. Ris-based preventive-orretive OPF We saw in te slides presented in te last lass tat te ris-based OPF relative to te SCOPF removed te post-ontingeny ineqality onstraints and replaed tem wit a ris onstraint. Model 1: SCOPF min Sbjet to: g P ' min min f P P ' P ' Model 2: RBOPF min g P min f P Sbjet to: P Ris P RMAX 9
We an apply te same idea to or preventive-orretive SCOPF as indiated below denoted as Problem P 2. Min f s. t. g g 12... P 2 Ris... Ris Ris 1 Δ 12... Problem P 2 wold indeed be an interesting problem to eplore frter. One sold start tis proess by reviewing te literatre at least te following ones: [1 2 3 4] togeter wit te itations in tese for pbliations. 8. Ris-based LMPs Wen eploring te LP-OPF we ave seen ow te OPF relates to LMPs as te Lagrange mltipliers on te power flow eqations. I derive te traditional form of te LMPs in a set of notes tat I ave loaded on te web. I will not go trog tat ere. Referene [5] derives tem nder te RB-OPF. Te soltions are given below. 1
Te ris-based LMP denoted RLMP above may be more attrative tan te deterministi LMPs denoted DLMP above in tree main ways: Control of ris level is niform Prie signal for ris-relief is more effetive LMPs are less volatile Tis is anoter interesting area to eplore. [1] Y. Li Deision maing nder nertainty in power system sing Benders deomposition. PD Dissertation Iowa State University 28. [2] F. Capitanes and L. Weenle Improving te Stetement of te Corretive Serity- Constrained Optimal Power Flow Problem IEEE Trans on Power Systems Vol. 22 No. 2 May 27. [3] F. Capitanes and L. Weenle a new iterative approa to te orretive serityonstrained optimal power flow problem IEEE Trans. On Power Systems Vol. 22 No. 4 Nov 28. [4] F. Capitanes T. Van Ctsem and L. Weenle Copling Optimization and Dynami Preventive-Corretive Control of Voltage Instability IEEE Trans. On Power Systems Vol. 24 No. 2 May 29. [5] F. Xiao Ris Based Mlti-objetive Serity Control and Congestion Management PD Dissertation Iowa State University 27. 11