Demand Response in Radial Distribution Networks: Distributed Algorithm

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1 Demand Response n Radal Dstrbuton Networks: Dstrbuted Algorthm (Invted Paper) Na L, Ljun Chen and Steven H. Low Abstract Demand response has recently become a topc of actve research. Most of work however consders only the balance between aggregate load and supply, and abstracts away the underlyng power network. In ths paper, we study demand response n a radal dstrbuton network, by formulatng t as an optmal power flow problem that maxmzes the aggregate user utltes and mnmzes the supply cost and the power lne losses, subject to the power flow constrants and operatng constrants. We propose a fully dstrbuted algorthm for the users to coordnate ther demand response decsons through local communcaton wth ther neghbors so as to acheve the optmum. Numercal examples wth the real-world dstrbuton crcuts are provded to complement our theoretcal analyss. Index Terms Demand response, Dstrbuted algorthm, Branch flow model, Optmal power flow problem, Dstrbuton networks I. INTRODUCTION Load management s ncreasngly needed to mprove power system effcency and ntegrate ever-ncreasng renewable generaton [1], [2]. A large lterature have developed dfferent schemes for load management; see, e.g., [3] [7]. Most of work however consders only the balance between aggregate load and supply, and abstracts away the underlyng power network and the assocated power flow constrants and operatng constrants. As a result, the schemes proposed may end up wth an electrcty consumpton/sheddng decson that would volate those network and operatng constrants. There are some recent work on load management that take nto consderaton the physcal network constrants; see, e.g., [8] [11]. But they usually use the bus njecton model for the electrcty network and propose locaton-based margnal prcng schemes for load management, whch s more sutable for the transmsson system. In ths paper, we study optmal load management n the presence of the network and operatng constrants for the radal dstrbuton networks, usng the branch flow model [12], [13] for the electrcty network rather than the bus njecton model. The branch flow model focuses on currents and powers on the branches. It has been used manly for modelng dstrbuton crcuts that are usually radal. Specfcally, we formulate load management problem as an optmal power flow (OPF) problem whose objectve s to maxmze the aggregate user utlzes N. L and S. H. Low are wth the Dvson of Engneerng and Appled Scences, Calforna Insttute of Technology, Pasadena, CA 91125, USA (Emal: {nal, slow}@caltech.edu). L. Chen s wth the College of Engneerng and Appled Scences, Unversty of Colorado, Boulder, CO 80309, USA (Emal: ljun.chen@colorado.edu). and mnmze the supply cost and the power lne losses, subject to the power flow constrants and operatng constrants such as the voltage regulaton constrant and power njecton constrants. Snce the resultng OPF problem s non-convex and thus dffcult to solve, we propose a convex relaxaton of the optmzaton problem, and dscuss whether the relaxaton can be exact and under what condtons. Convexty of the optmzaton problem s usually requred for the development of computatonally-effcent and dstrbuted algorthms for system operatons. In a prevous paper [14], we consder a scenaro where the radal dstrbuton network s served by a sngle loadservng entty (LSE), whch coordnates the end users demand response decsons by settng the rght prces, and propose a dstrbuted algorthm for the LSE to fnd such prce sgnals. Ths algorthm requres two-way communcaton between the LSE and each user, and at each teraton, the LSE s requred to solve a large OPF problem. In ths paper, we nstead develop a fully dstrbuted OPF algorthm for demand response, where the end users make and coordnate ther local demand response decsons through local communcaton wth ther neghbors. Ths demand response scheme requres two-way communcaton only between the end users that are drectly connected n the dstrbuton network, and each user only needs to solve a small optmzaton problem. We develop ths demand response algorthm based on a well-known dstrbuted algorthm, Predctor Corrector Proxmal Multpler (PCPM) [15]. Provded that the convex relaxaton of the OPF problem for demand response s exact, the algorthm s guaranteed to converge to the global optmum of the problem. The rest of the paper s organzed as follows. In Secton II we formulate the optmal demand response problem, ntroduce the PCPM algorthm, and dscuss convex relaxaton of the optmzaton problem. In Secton III, we develop a fully decentralzed algorthm for demand response. In Secton IV, we provde numercal examples to complement the theoretcal analyss, usng a real-word dstrbuton crcut. Notatons: V, v : complex voltage on bus wth v = V 2 ; s = p + q : complex net load on bus ; I j, l j : complex current from buses to j wth l j = I j 2 ; S j = P j +Q j : complex power flowng out from buses to bus j; z j = r j + x j : mpedance on lne (, j);

2 II. PROBLEM FORMULATION & PRELIMINARY A. Problem formulaton Consder a radal dstrbuton crcut that conssts of a set N of buses and a set E of dstrbuton lnes connectng these buses. We ndex the buses n N by = 0, 1,..., n, and denote a lne n E by the par (, j) of buses t connects and the ndex denotes the bus that s closer to the feeder. Bus 0 denotes the feeder, whch has fxed voltage but flexble power njecton to balance the loads; each of the other buses N\{0} represents an aggregator that can partcpate n demand response. For convenence we call aggregator as user, whch actually represents a or a group of customers that are connected to bus and jon the demand response system as a sngle entty. For each lnk (, j) E, let z j = r j + x j be the mpedance on lne (, j), and S,j = P,j + Q,j and I,j the complex power and current flowng from bus to bus j. At each bus N, let s = p +q be the complex load and V the complex voltage. As customary, we assume that the complex voltage V 0 on the feeder s gven and fxed. The branch flow model, frst proposed n [12], models power flows n a steady state n a radal dstrbuton network: for each (, j) E, P,j 2 + Q2,j = l,j, (1) v P,j = P j,h + r,j l,j + p j, (2) Q,j = h:(j,h) E h:(j,h) E Q j,h + x,j l,j + q j, (3) v v j = 2(r,j P,j + x,j Q,j ) (r 2,j + x 2,j)l,j, (4) where l,j := I,j 2, v := V 2. Each user N\{0} acheves certan utlty f (p ) when ts (real) power consumpton s p. The utlty functon f ( ) s usually assumed to be contnuous, nondecreasng, and concave. Furthermore, there are the followng operatng constrants for each N\{0}: v v v, = 1,, n, (5) q q q, = 1,, n, (6) p p p, = 1,, n. (7) The electrcty s delvered from the man grd to the radal dstrbuton network through the feeder (.e., the bus 0). The total (real) power supply P 0 s gven by P 0 := j:(0,j) E P 0,j. We consder a stuaton where the power supply P 0 s constraned by an upper bound P 0,.e., P 0 = P 0,j P 0. (8) j:(0,j) E Under such a stuaton, we would lke to desgn a dstrbuted mechansm to gude each user to choose a proper load p, so as to ) meet the supply constrant (8) as well as the power flow constrants and operatng constrants lsted n (1) - (7) and ) maxmze the aggregate user utltes and mnmze the power supply costs and power lne losses. Ths demand response problem s formulated as the followng optmal power flow problem (OPF): n OPF: max f (p ) C 0 (P 0 ) ρ r,j l,j =1 s.t. (1) (8), (,j) E where ρ s a trade off parameter. Throughout the paper, we assume that the feasble set of ths problem s nonempty. In the followng, we wll develop a fully dstrbuted OPF algorthm for demand response, where the end users make and coordnate ther local demand response decsons through local communcaton wth ther neghbors. B. A decentralzed optmzaton algorthm: predctor corrector proxmal multpler (PCPM) Ths paper we focus on usng the decentralzed algorthm, predctor corrector proxmal multpler (PCPM) [15] to develop a dstrbuted demand response scheme. Consder the followng convex problem: mn x X,y Y f(x) + g(y) (9a) s.t. Ax + By = C (9b) Introduce the Lagrangan varable z for constrant (9b). The algorthm PCPM s gven as follows: 1) Intally set k 0 and randomly choose ntal (x 0, y 0, z 0 ). 2) For each k 0, update a vrtual varable ẑ k := z k + γ(ax k By k C) Here γ > 0 s a constant parameter. 3) Based on the vrtual varable ẑ k, update x, y accordng to: x k+1 y k+1 = arg mn x X {f(x) + (ẑk ) T Ax + (1/(2γ)) x x k 2 }, = arg mn y Y {g(y) + (ẑk ) T By + (1/(2γ)) y y k 2 }. 4) z s updated accordng to z k+1 = z k + γ(ax k+1 + By k+1 C). 5) k k + 1, and go to step 2). From the algorthm, we see that PCPM s hghly decomposable. In terms of convergence, t has been shown n [15] that as long as strong dualty holds for the convex problem (9), the algorthm wll converge to a prmal-dual optmal soluton (x, y, z ) for suffcent small postve γ. C. Convexfcaton of Problem OPF OPF s non-convex due to the quadratc equalty constrants n (1) and thus dffcult to solve. Moreover, most decentralzed algorthms requre convexty to ensure convergence, e.g., PCPM as descrbed n II-B. We therefore consder the followng convex relaxaton of OPF: n ROPF: max f (p ) C 0 (P 0 ) ρ r,j l,j =1 (,j) E s.t. (2) (7) P 2,j + Q2,j v l,j, (, j) E, (10)

3 where the equalty constrants (1) are relaxed to the nequalty constrants (10). ROPF provdes a lower bound on OPF. For an optmal soluton X := (P, Q, l, v, p, q ) of ROPF, f the equalty n (10) s attaned at X, then X s also a soluton to OPF. We call ROPF an exact relaxaton of OPF f every soluton to ROPF s also a soluton to OPF, and vce versa. In prevous work [13], [16], we have studed whether and when ROPF s an exact relaxaton of OPF for the radal networks. It s shown n [13] that the relaxaton s exact when there are no upper bounds on the loads. However, removng upper bounds on the loads may be unrealstc, especally n the context of demand response. It s shown n [16] that the relaxaton s exact provded that nstead there are no upper bounds on the voltage magntudes and certan other condtons hold, whch are verfed to hold for many real-world dstrbuton systems. Moreover, the upper bounds on the voltage magntudes for the relaxaton soluton are characterzed [16]. The beneft of convexty s that convexty does not only facltates the desgn of effcent prcng schemes for power market and demand response, but t also facltates the development of tractable, scalable and dstrbuted algorthms for system operatons. Hence the condtons for exact relaxaton of OPF to ROPF specfed n [16] s mportant for our demand response desgn. In the rest of the paper, we wll assume that ROPF s an exact relaxaton of OPF and strong dualty holds for ROPF. When ROPF s an exact relaxaton of OPF, we can just focus on solvng the convex optmzaton problem ROPF. III. A FULLY DECENTRALIZED ALGORITHM In ths paper, we develop a fully dstrbuted OPF algorthm for demand response, where the end users make and coordnate ther local demand response decsons through local communcaton wth ther neghbors. Specfcally, we assume that each user has certan computaton ablty to decde a set of local varables of the OPF. The composton of those varables determnes the global status of the power flow over the dstrbuton network. We also assume that there s two way communcaton avalable between any two users that are drectly connected n the dstrbuton network. In the decentralzed OPF algorthm, at each teraton each user makes decsons about the local varables, communcate those decsons wth neghbors, and then update ther local varables and repeat the process. Before establshng the algorthm, let us defne the local decson varables for each user. Let π() be the parent of bus and δ() be the drect chldren of bus. The local decson varables for each buses are: For bus 0, P 0, v 0, where v 0 s fxed by conventon. For bus > 0, P π(),, Q π(),, l π(),, p,q,v,ˆv. Here ˆv s bus s estmaton about ts parent s voltage v π(). To smplfy the notatons, we denote P π(),, Q π(),, l π(), as P, Q, l ; and r π(),, x π(), as r, x. Wth the new notatons, OPF can be rewrtten as: max n f (p ) C 0 (P 0 ) =1 s.t. P 0 = j:(0,j) E P = j δ() Q = j δ() P j, ˆv = v π(), N\ {0} P 0 P, n r l =1 (11a) (11b) P j + r l + p, N\ {0} (11c) Q j + x l + q, N\ {0}(11d) v v v, N\ {0} p p p, N\ {0} q q q, N\ {0} (11e) (11f) (11g) (11h) (11) P 2 + Q2 l, N\ {0} (11j) ˆv ˆv v = 2(r P + x Q ) (r 2 + x 2 )l, N\ {0}, (11k) The new formulaton has the followng propertes that can be utlzed for the desgn of dstrbuted algorthms: The objectve functon (11a) s fully decomposable. Constrants (11b-11e) are lnear coupled constrants but each constrant only constrans local nformaton, namely that each constrant s defned over the local varables of one node and ts drect neghbors over the radal network. Constrants (11f-11k) are just local constrants that are defned over bus s local decson varables. Then we can apply algorthm PCPM to defne a decentralzed algorthm. We wll use PCPM to decouple those lnear coupled constrants (11b-11e). Let the Lagrangan dual varable correspondng to constrant (11b) be λ 0 and dual varables correspondng to constrants (11c-11e) be λ, θ, ω for each N\{0}. In the followng dstrbuted algorthm, node 0 takes charge of updatng λ 0 and node N\{0} takes charge of updatng λ, θ, ω. Now let us ntroduce the dstrbuted demand response algorthm whch converges to a global optmal soluton of the OPF. 1) Intally set k 0. Node 0 randomly chooses P0 k and λ k 0 and node N\{0} randomly chooses P k, Qk, l k, pk,qk,vk,ˆvk and the dual varables λ k, θk, ωk. Each node N\{0} send the prmal varables P k, Qk, lk to ts parent π(), and each node N except leaves n the network send v k to ts chldren. Note that v0 k s fxed for any k. 2) For each k 0, node 0 send a vrtual dual sgnal ˆλ k 0 := λ k 0 + γ(p0 k j:(0,j) E P j k ) to ts chldren; and each node N\{0} except the leaves send the followng

4 vrtual sgnals to ts chldren: ˆλ k = λ k + γ P k Pj k + r l k + p k, j δ() ˆθ k = θ k + γ Q k Q k j + x l k + q k ; j δ() and each node N\{0} send the followng vrtual sgnals to ts parent: ˆω k = ω k + γ(ˆv k v k π() ). Here γ > 0 s a constant parameter. 3) Each node update ther local prmal varables accordng to the followng rules. Case 1:Node 0 solves the followng problem: mn P 0 C 0 (P 0 ) + ˆλ k 0P γ P 0 P0 k 2 s.t. P 0 P. The optmal P 0 s set as 0. Case 2: Each node such that (0, ) E, solves the followng problem: mn f (p ) + r l ˆλ k 0P + ˆλ (P r l p ) +ˆθ (Q x l q ) + ˆω ˆv ˆω j v j:(,j) E + 1 ( (P P k ) 2 + (Q Q k ) 2 2γ +(l l k ) 2 + (p p k ) 2 + (q q k ) 2 +(v v k ) 2 + (ˆv ˆv k ) 2) over P, Q, l, p, q, v, ˆv s.t. (11g 11k) The optmal P, Q, l, p, q, v, ˆv s set as, Q k+1, l k+1, p k+1, q k+1, v k+1, ˆv k+1. Case 3: Each node such that (0, ) E solves the followng problem: mn f (p ) + r l λ π() P θ π() Q +ˆλ (P r l p ) + ˆθ (Q x l q ) +ˆω ˆv ˆω j v j:(,j) E + 1 ( (P P k ) 2 + (Q Q k ) 2 2γ +(l l k ) 2 + (p p k ) 2 + (q q k ) 2 +(v v k ) 2 + (ˆv ˆv k ) 2) over P, Q, l, p, q, v, ˆv s.t. (11g 11k) The optmal P, Q, l, p, q, v, ˆv s set as, Q k+1, l k+1, p k+1, q k+1, v k+1, ˆv k+1. 4) Each node N\{0} send the prmal varables, Q k+1, l k+1 to ts parent π(), and each node N except leaves n the network send v k+1. Note that v0 k s fxed as v for any k. Then node 0 update the dual sgnal λ k+1 0 := λ k 0 + γ(p0 k+1 j:(0,j) E j ) to ts chldren; and each node N\{0} except the leaves update the followng varables: λ k+1 θ k+1 =λ k + γ =θ k + γ Q k+1 j δ() j δ() ω k+1 =ω k + γ(ˆv k+1 v k+1 π() ). j + r l k+1 + p k+1 Q k+1 j + x l k+1 + q k+1,, 5) k k + 1, and go to step 2). For suffcently small γ, the algorthm wll converge to the optmal solutons. Notce that n the dstrbuted algorthm, each node only needs determne a few varables by solvng a small optmzaton problem. IV. CASE STUDY Ths secton provdes numercal examples to complement the analyss n prevous sectons. We apply the algorthm developed n Secton III to a practcal dstrbuton crcut of the Southern Calforna Edson (SCE) wth 56 buses, as shown n Fg. 1. The correspondng network data ncludng the lne mpedances, the peak MVA demand of loads, and the nameplate capacty of the shunt capactors and the photovoltac generatons can be found n [17]. Note that there s a photovoltac (PV) generator located at bus 45. Snce the focus of ths paper s to study demand response n power networks, so n the smulaton we remove the PV generator. Prevous work [16] has shown that ths 56-bus crcut satsfes the suffcent condtons for the exact relaxaton of OPF to ROPF. Therefore, we can apply the proposed algorthm for the demand response n ths crcut. In the smulaton, the user utlty functon f (p ) s set to the quadratc form f (p ) = a (p p ) 2 + a ( p ) 2 where a s randomly drawn from [2, 5]. For each bus, set p and q to the peak demand and p to the half of the peak demand. If there s no shunt capactor attached to bus, we set q to the half of the peak demand as well, and f there s shunt capactor attached, we set q to the negatve of the nameplate capacty. We set γ = 0.01, and P 0 = 2.5MVA. Fg. 2 shows the dynamcs of the dstrbuted algorthm proposed n Secton III. We see that the algorthm converges fast for ths dstrbuton system. Notce that snce at each teraton step, each node only needs to solve a small optmzaton problem and the algorthm s hghly parallel, the total runnng tme s very fast. We also solve problem ROPF by usng CVX toolbox [18], whch mplements a centralzed algorthm, and verfy that t gves the same soluton as our dstrbuted algorthm. We further verfy that the optmal soluton of ROPF s a feasble pont of OPF,.e., ROPF s an exact relaxaton of OPF. V. CONCLUSION In ths paper, we have studed demand response n the radal dstrbuton network wth power flow constrants and

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