EE292K Project Report: Stochastc Optmal Power Flow n Presence of Renewables Anubhav Sngla (05737385) Advsor: Prof. Danel O Nell June, 202 Abstract In ths work we propose optmal power flow problem maxmzng socal welfare n the presence of renewables. We use onlne learnng approach to do the stochastc optmzaton, whch also means we do not assume avalablty of pror knowledge of the dstrbuton functons nvolved. We restrc our work to power systems wth tree topology whch allowed us to solve the exact problem (no DC approxmaton requred). We also study optmal locatonal margnal prces and congeston cost of transmsson lne n presence of renewables. In addton we formulate OPF to take nto account durnal patterns n renewables. Introducton In deregulated electrcty market, electrcal energy s bought and sold lke any other comodty. In one of the model we have three key players n the market, generators, load and ndependent grd operator (IGO). Generators are sellers of electrcty, they submt ther cost functon to IGO, smlarly load are buyers and they submt ther utlty functon (or economc beneft functon) to IGO. Gven cost and utlty functons IGO decdes quantty of energy bought and sold from/to buyers and sellers. Whle makng these decsons, IGO takes nto account the physcal constrants assocated wth power flow and maxmum capacty of transmsson lnes between varous generators and loads. More specfcally IGO solves Socal Welfare maxmzaton problem, the outome also ncludes locatonal margnal prces (assocated wth each bus) and congeston cost for each transmsson lne [3]. There have been growng support for the clean generaton of energy, and addng renewable generators to the grd. Two man characterstcs of the renewable energy are: (a) It s qute cheaply produced, (b) t s non-determnstc, unrelable & varable. The renewable energy generaton mght also be supported by the government ncentves, whch means cost functon could be negatve valued (we would not consder ths case though). The unrelable and varable nature s more promnent n case of wnd and solar energy because weather condton changes and cannot be relably predcted. It s challengng to develop a good scheme for socal welfare. In general, the varablty n renewable generaton also leads to varablty n power njecton and LMP. When renewables are present t s more meanngful to maxmze the average socal welfare under average constrants on transmsson lne capacty. In general, we need to know dstrbuton functon of the renewable generator s capacty to solve a stochastc optmzaton
problem. But dstrbuton functon may not be avalable, n ths work we are proposng onlne learnng (or samplng) based method to solve such problem. More specfcally we are usng stochastc subgradent method to solve the optmzaton problem ([4], [5]). As we wll see later, such an approach s not adversly affected by number of renewables and we need not dscretse the renewable generaton capacty. Fnally we wll explore the durnal pattern [2] of renewables (e.g. wnd). It refers to the phenomenon that for specfc season wnd would start to blow and become calm at specfc tme of day. Snce such a pattern captures most of the varaton n the wnd, we can use rollng horzon method to maxmze socal welfare over just one day. The physcal constrants of the power system nvolves non-lnear functons [] whch lead to non-convex optmzaton problem. Tradtonally ths has been resolved by takng DC approxmaton [3]. DC approxmaton ncludes three key assumptons, (a) transmsson lnes are lossless (no resstve component), (b) bus voltage magntudes are close to unty (n per unt), (c) bus voltage phases are qute close to each other. These assumptons mght not hold, for nstance, renewables mght not have close phase angles due to absence of actual rotatng parts. Recent results [6] have shown that f power system network follow specfc topology (e.g. tree) then t mght be possble to transform the optmal power flow (OPF) problem to a convex problem and solve t exaclty. 2 System Model We model power system as network wth node representng bus and edge representng transmsson lne. We assume that each bus s ether a generator bus or load bus. A generator could be ether renewable or non-renewable Fgure. Note that a bus, wth multple generators or loads connected to t, can be modeled by splttng the bus nto multple busses and connectng them through low mpedence transmsson lne. Let us assume that there are N busses, ndexed from to N and J transmsson lnes, ndexed from to J (transmsson lne can also be ndexed by two bus ndexes at ether end of t). Also, set of all load busses as B L, set of all renewable generator busses as B R and set of all non-renewable generator busses as B NR. Let P R be the power njected nto the system at bus. Ths would typcally be postve for generator bus and negatve for load bus. Snce we want to maxmze socal welfare, let us defne cost and utlty functons. If bus s the load bus let at P = x, U (x) be ts utlty functon. Smlarly, f bus s the generator bus, let C (x) be the cost of P = x power generaton. For smplcty of notaton, for any bus, we defne utlty and cost functon as: C (x) = U (x) Then the socal welfare s gven as B L U (P ) B R,B NR C (P ) = N C (P ). Let V = V e jθ be the voltage at bus, and V be correspondng vector. Let Y = G+jB (Y C N N ) be the admttance matrx of the network of transmsson lnes. Element of Y, Y mn = G mn + jb mn, are lnearly related to admttance of transmsson lne (connectng bus-m and bus-n) y mn = g mn + jb mn. Let I be the current njected nto the network at bus, and I s correspondng vector. Bus current and bus voltages are related as, I = YV. The bus power njecton P s constrant by bus voltages as, P = N k= V V k (G k cosθ k + B k snθ k ). The dfference between power generated and power consumed s the heat dss- 2
pated n the transmsson lnes. The heat loss n transmsson lne (m, n) as a functon of bus voltage s gven as L mn (V) = V m V n 2 g mn 3 Optmal Power Flow (OPF) In ths secton we wll go through some schemes of optmal power flow based on socal welfare maxmzaton. 3. Determnstc OPF We formulate socal welfare maxmzaton optmal power flow problem as follows: mnmze V,P C (P ) () subject to: P = V V k (G k cosθ k + B k snθ k ), =,..., N (2) k= L (V) l max, =,..., J (3) V mn V V max, =,..., N (4) P P max, B R (5) P 0, B L (6) Note that objectve () corresponds to maxmzng socal utlty. Equaton (2) represent physcal constrant on P (θ k = θ θ k ), t also mples that suppled power equals power consumed and heat loss. There s physcal lmt to how much heat can be dsspated n transmsson lne wthout any damage, l max n (3) s that lmt. Ths constrant s sometmes [3] approxmated as maxmum power transmsson capacty of lne. Equaton (4) s bus voltage magntude lmt constrant, whch s assumed to be unty n some lnear models. P max n (5) s maxmum power generaton capacty of renewable generators, for nstance n case of wnd mll P max depends on wnd speed. Fnally, (6) means that load cannot nject power nto the system. The optmzaton problem n ths form s hard to solve. Interestngly, a subset of ths general problem can be tranformed nto a convex problem. Let W S N + be a hermtan symmetrc postve semdefnte matrx of rank (W mn beng ts element), then we do followng change of varables (to make the functons lnear): W = VV H (7) V V k (G k cosθ k + B k snθ k ) = R( W k Yk (8) k= k= L mn (W) = g mn (W + W kk W k W k ) (9) 3
Refer to [6] for more detals. Now consder followng modfed optmzaton problem: mnmze W,P C (P ) (0) subject to: P = R(dag(WY H )) µ () L (W) l max, =,..., J λ (2) V 2 mn W V 2 max, =,..., N (3) P P max, B R (4) P 0, B L (5) W S N + (6) [Rank(W) = ] (7) Here S N + s set of all hermtan symmetrc postve semdefnte matrces and dag(a) s vector formed from dagonal entres of A. Ths problems s equvalent to the one dscussed before (). If we relax Rank(W) = constrant then ths problem becomes convex convex optmzaton problem. [6] showed that under certan condtons the optmal pont satsfes the rank constrant. The set of suffcent condtons are: (a) C (x) should be convex and ncreasng; (b) The network topology should be tree; (c) If two buses are connected by transmsson lne, both should not have lower bound on P ; We would restrct our problem/system model to satsfy these condtons. Note that to ensure that (c) s satsfed we dd not add constrant P 0 for generator bus. Thus t s possble to have the optmal pont as P < 0 for generator, whch has nterpretaton that f t s economcally vable generator bus can consume power (ts cost/utlty would be zero). Here λ and µ are the dual varables of the correspondng constrants. The optmal value of daul varable provdes useful economc nformaton. For our case, µ gves margnal prce at bus (Locatonal Margnal Prce - LMP). λ s the congeston cost for the transmsson lne, n other words, t s the margnal ncrease n optmal socal utlty per unt ncrease n transmsson lne capacty (or more precsely, per unt ncrease n l max ). Wthout any constrants on transmsson lne loss (2), the optmal LMP at each bus tend to be close to each other. Wth the lne loss constrant (beng actve), LMP tend to separate from each bus and wth the dfference of the order of congeston cost λ. Wthout constrant (4) the LMP s close to margnal cost,.e. µ C (x)/ x x=p. Constrant (4) can be thought of as value of cost functon becomng extremely hgh around P max. Thus f constrant (4) s actve the LMP could be hgher then margnal cost. 3.2 Stochastc OPF The maxmum power generaton capacty of renewables vary wth tme.e. (t) s functon of tme (n case of wnd mll, t depends on varyng wnd condtons). One way to formulate optmal power flow problem s to solve above problem (0) ndependently for every tme nstant, snce we know the present renewable capacty we can solve the problem exactly. In ths case constrants would be satsfed for each tme nstants, whch may not be requred for all the constrants. For nstance, as long as the average heat dsspaton s below certan level nstantaeneous heat loss could go very hgh. 4 P max
Thus we want to formulate OPF to reflect average objectve and constrants. Let the random varable (vector) X takes the value of P max, B R. We formulate the followng OPF problem: mnmze W(X),P(X) ( N ) E X C (P (X)) subject to: P(X) = R(dag(W(X)Y H )) E X (L (W(X))) l max, =,..., J λ V 2 mn W (X) V 2 max, =,..., N P (X) P max (X), B R P (X) 0, B L W(X) S N + Note that soluton of ths problem, W(X), P(X), s functon of X. In theory, t can be solved f we assume X to be dscrete random varable wth known dstrbuton functon, snce the problem remans convex. And snce we get to know value of the renewable generaton for current tme nstant (X 0 ), the optmal power flow values could be found, W(X 0 ), P(X 0 ). But the problem scales badly wth some system parameter. Let number of dscrete levels of P max be n s and number of renewable generator be n r. Then X can take (n s ) nr values and t becomes computatonally nfeasble to solve the OPF. Also, n many cases we do not have knowledge of dstrbuton functon of X. We thus propose an onlne learnng approach to solve ths problem. The teratve algorthm s gven as: mnmze W t,p t C (P t ) + J λ t ( L (W t ) l max) (8) subject to: P t = R(dag(W t Y H )) (9) V 2 mn W t V 2 max, =,..., N (20) P t P max, t, B R (2) P t 0, B L (22) W t S N + (23) λ t+ = λ t + α t ( L (W t ) l max) (24) For every tme slot t optmzaton problem s solved (8) and λ s updated for next tme slot t +. Over the tme the algorthm learns the dstrbuton of X, and λ converges to optmal value. Here we are essentally usng stochastc subgradent method ([4], [5]) to solve the problem, detals are furnshed n Appendx (Secton 5.). 5
3.3 Optmal Dstrbuton Dfferent dstrbuton on P max would lead to dfferent optmal value of socal welfare. We would lke to know how the optmal value depends on the dstrbuton, keepng average (of renewable generator capacty) constant. Thus we solve the OPF where we modfy maxmum power constrant to be on an average. mnmze W(X),P(X) ( N ) E X C (P (X)) (25) subject to: P(X) = R(dag(W(X)Y H )) (26) E X (L (W(X))) l max, =,..., J λ (27) V 2 mn W (X) V 2 max, =,..., N (28) E X (P (X) P max (X)) 0, B R (29) P (X) 0, B L (30) W(X) S N + (3) It can be shown that optmal pont of ths problem are ndependent of X, that s, (W (X), P (X)) = (W, P ) (Secton 5.2). Thus an optmal dstrbuton would be one wth zero varance. It also suggests that optmal value may decrease on ncreasng varance of the dstrbuton. Another vewpont to look at ths s that f suffcent storage s avalable then there s no effect of ths wnd (say) varablty, we can acheve the same socal welfare as n the case of constant wnd. 3.4 Rollng Horzon The wnd speed s not dentcally dstrbuted for all tme throughout the day. For example, wnd may start blowng later n mornng and calm down around nght. Thus wnd may exhbt durnal pattern [2]. Hence t would be relevant to formulate the problem whch s coupled over just fnte number of tme slots spannng a day. Let fnte number of tme slots of a day be ndexed from to T, 6
mnmze W(t),P(t) T t= C (P (t)) (32) subject to: P(t) = R(dag(W(t)Y H )) t =,..., T (33) T L (W(t)) l max + δ L, =,..., J T (34) t= V 2 mn W (t) V 2 max, =,..., N (35) t P (t) P max (t) + τ= (P max (τ) P (τ)) + δ P, B R, t =,..., T (36) P (t) 0, B L, t =,..., T (37) W(t) S N +, t =,..., T (38) Here the objectve s to maxmze sum of socal welfare across all tme slots (32). (34) s constrant on heat dsspaton averaged across all tme slots (gnore δ L for now). In ths problem we also assumed avalablty of battery for storage of renewable energy, so that excess power can be used n future tme slots. Thus P (t) at present cannot exceed (36) capacty at present P max (t) plus excess energy stored n past t τ= (P max (τ) P (τ)). In the above problem the ndex t = represent present tme slot and t > represent future, we know the value of P max (t = ) and use estmates for P max (t), t = 2,..., T (rollng horzon). Smlarly, we nterpret optmal soluton, P (t = ) beng fxed power njecton for present, and P (t > ) just beng estmates. As we move ahead n tme the number of tme slots over whch we optmze decreases and eventually toward the end of the day t reduces to sngle tme slot (T = ). Also, δ P n (36) s excess power left from past tme slot (n a way t < s past) of that day and δ L s assocated wth past heat loss. Note that n ths case a more optmal approach would be a problem whch s coupled over multple days (rather than just multple tme slot n one day). But, most of the varaton n the wnd s captured by durnal pattern wthn a day so even ths approach should be close to optmal. 4 Smulaton Results For smulaton we used the power system netwrok shown n Fgure, wth number of bus N = 2, number of transmsson lnes J =. Transmsson lne admttance y mn were unformly dstrbuted wthn range [2 : 8] j[8 : 30] per unt. The margnal utlty/cost functon ( U (x)/ x) for load/generators has been shown n Fgure 2. The utlty functon for the load were chosen to be log functon and cost functon of the generator were quadratc functon. Cost functon of renewable were chosen to be sgnfcant lower than the cost functon of non renewables. We used dfferent dstrbutons for P max but the average was kept same, E(P max ) = 5 per unt. The maxmum lmt on lne heat dsspaton s taken as, l max = 0.5 per unt. The maxmum bus voltage constrant s V max =.2 per unt. Fnally all the problems formulated are convex and cvx was used to solve them. 7
4. Determnstc OPF smulaton Determnstc OPF was solved wth fxed value of P max = 0.5 for every renewable. Fgure 3 shows the optmal power njecton thus obtaned, to get some perspectve margnal utlty/cost s also shown. Note that, renewables accounts for most of the power flowng nto the network because of low cost. Also loads wth hgher utlty functon tends to get more power from the system. Fgure 4 shows locatonal margnal prces at varous busses and congeston cost at transmsson lnes. Note bus- & bus-4 separated wth congested transmsson lne-3 have LMPs wth huge dfference. Intutvely, bus- s connected to renewable generator and bus-4 s connected to load wth hghest utlty functon value, causng huge power flow from bus- to bus-4 and hence congeston on lne-3. 4.2 Stochastc OPF smulaton At each teraton (tme) of stochastc OPF P max was drawn from exponental dstrbuton wth E(P max ) = 5. Wnd s known to follow exproxmately raylegh dstrbuton, hence exponental dstrbuton for the maxmum power capacty. Fgure 5 shows convergence of the lagrange multplers λ thss wth teraton counts. Snce we used stochastc subgradent method, t may not converge exactly, but stll gets very close to optmal value. Fgure 6 shows varaton of renewable generaton P max and LMP µ 4 at load bus. Here we note that at the tme of hgh renewable generaton the LMP at bus-4 falls and vce-versa. Snce cost of renewable s much less, an ncrease n generated power leads to more power to flow to the load whch leads to lower LMP. Fgure 7 shows varaton n lne loss wth tme (for couple of lnes). Note that nstantaeneous lne loss can go hgher than maxmum heat dsspaton lmt (l max = 0.5) but average stays below, ths leads to hgher average socal welfare. The effect of dfferent dstrbutons of P max or more specfcally effect of dfferent standard devaton n P max. Assumng t to be normally dstrbuted about same mean we plotted optmal value of socal welfare for dfferent standard devatons, Fgure 8. In general, socal welfare tends to decrease wth ncrease n standard devaton. We also observed that congeston cost tends to ncrease wth ncrese n standard devaton. 5 Appendx 5. Onlne Learnng Let us reformulate the dual problem as follows: 8
G(λ) = mnmze W(X),P(X) E X ( N ) C (P (X)) + J λ E X (L (W(X)) l max ) (39) subject to: P(X) = R(dag(W(X)Y H )) (40) maxmze λ V 2 mn W (X) V 2 max, =,..., N (4) P (X) P max (X), B R (42) P (X) 0, B L (43) W(X) S N + (44) G(λ) (45) We decouple the problem over values of X, so that for all X we solve followng : mnmze W(X),P(X) C (P (X)) + J λ (L (W(X)) l max ) (46) subject to: P(X) = R(dag(W(X)Y H )) (47) V 2 mn W (X) V 2 max, =,..., N (48) P (X) P max (X), B R (49) P (X) 0, B L (50) W(X) S N + (5) and use subgradent method (teratve nature denoted by ndex t) to maxmze G(λ). λ t+ = λ t + α t E X (L (W(X)) l max ) (52) we can use stochastc subgradent, that s, we need not take E X () at each teraton but t wll average out tself f X s drawn randomly: λ t+ = λ t + α t (L (W(X)) l max ) (53) Fnally we assume that the values of random varable over tme s ergodc process and hence can replace X by t. 5.2 Optmal Dstrbuton Let us assume that we can store energy from renewables, for ths case followng problem defnaton would be more approprate: 9
mnmze W,P ( N ) E C (P ) subject to: E(L k ) l max 0 E(P P max ) 0 P = R{dag(WY H )} µ W S N + The E() n above formulaton can be replaced by tme average: mnmze W,P subject to: T T T ( T N ) C (P t ) t= T L t k lmax 0 t= T t= (P t P max t ) 0 P = R{dag(WY H )} µ (54) (55) W S N + Snce above problem s convex, λ, δ, such that followng problem has same soluton: mnmze W,P subject to: T ( T N ) C (P t ) t= + P = R{dag(WY H )} µ λ ( T ) ( T M L t k lmax + δ j T t= j= T t= (P t P max t ) ) W S N + whch can be smplfed as: (56) T T mnmze C (P t ) + t= W,P t M λ j L t j + j= M δ P t λ j l max j= δ P max t (57) Ths has two mplcatons, (a) Optmzaton problem at each tme nstant can be solved ndependently from other tme nstants (b) Optmzaton problem at each tme nstants are dentcal..e. soluton P t = P t2 = P. Thus we can solve followng optmzaton problem wth ntex t dropped: 0
mnmze W,P subject to: C (P ) L k l max P E(P wnd ) (for renewable) P 0 (for load bus) P = R{dag(WY H )} µ W S N + (58) References [] J.J. Granger and W.D. Stevenson. Power System Analyss. McGraw-Hll seres n electrcal and computer engneerng: Power and energy. McGraw-Hll, 994. [2] H. Holttnen. Hourly wnd power varatons n the Nordc countres. Wnd Energy, 2005. [3] Mngha Lu. A framework for congeston management analyss. PhD thess, Champagn, IL, USA, 2005. AAI38239. [4] N.Z. Shor. Mnmzaton methods for non-dfferentable functons. Sprnger seres n computatonal mathematcs. Sprnger-Verlag, 985. [5] N.Z. Shor. Nondfferentable Optmzaton and Polynomal Problems. Nonconvex Optmzaton and Its Applcatons. Kluwer, 998. [6] Baosen Zhang and D. Tse. Geometry of feasble njecton regon of power networks. In Communcaton, Control, and Computng (Allerton), 20 49th Annual Allerton Conference on, pages 508 55, sept. 20.
5 4 3 2 tx lnes load generator wnd mll 0 2 3 4 5 0 0.2 0.4 0.6 0.8.2.4.6.8 2 Fgure : Power system network wth N = 2 busses (ndexed from top to bottom and left to rght) and J = transmsson lnes, B L = {4, 8, 0, 2}, B NR = {2, 3, 9, }, B R = {, 5, 6, 7} 2
00 90 4 Margnal Cost and Margnal Utlty functons Load: U( x)/ x Renewable: C(x)/ x Non-Renewable: C(x)/ x 80 70 8 2 60 0 50 3 40 30 9 2 20 0 5 67 0 0 2 3 4 5 6 7 8 9 0 x Fgure 2: Margnal Cost/Utlty functons for generators/loads, labels at start of the curve represent the bus number. 3
00 90 4 Optmal Power Flow from dfferent Busses Load: U( x)/ x Renewable: C(x)/ x Non-Renewable: C(x)/ x 80 70 8 2 60 0 50 3 40 30 9 2 20 0 5 67 0 0 2 3 4 5 6 7 8 9 0 x Fgure 3: The power njecton at varous bus has been shown usng dark lnes on margnal curves. 4
5 4 3 2 0 2 33 tx lnes load generator wnd mll 88.0 3 35 5.7 30 27 32 45 35 37 3 4 50 50 5 5 0 0.2 0.4 0.6 0.8.2.4.6.8 2 Fgure 4: LMP are shown near the bus and congeston cost are shown near center of lne 00 90 80 70 60 λ 50 40 Lne 3 Lne 4 Lne 6 Lne 7 30 20 0 0 0 000 2000 3000 4000 5000 6000 7000 8000 9000 0000 Iteratons (tme) Fgure 5: Convergence of lagrange multpler representng congeston cost of some lnes 5
55 Margnal Prce at Bus 4, µ 4 50 45 0 0 20 30 40 50 60 70 80 90 00 tme Power capacty at Bus, P max 30 25 20 5 0 5 0 0 0 20 30 40 50 60 70 80 90 00 tme Fgure 6: Fgure shows the varaton of LMP at bus-4 (load bus) wth varatons of wnd (and hence P max ) 6
0.9 0.8 0.7 Lne 3 Lne 9 Lne 0 l max 0.6 Lne Loss, L 0.5 0.4 0.3 0.2 0. 0 0 0 20 30 40 50 60 70 80 90 00 tme Fgure 7: Varaton n lne loss wth tme 5495 5490 5485 Optmal Socal Wellfare 5480 5475 5470 5465 5460 0 0.5.5 2 2.5 3 3.5 4 max Standard Devaton of P Fgure 8: Effect of standard devaton of renewables on socal welfare 7