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Distributed AC Optimal Power Flow using ALADIN *
Proceedings of the 20th World Congress Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th World Congress Proceedings of the 20th World The International of Congress Automatic Control Toulouse, France,Federation July 9-14, 2017 The International Federation of Automatic Control Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 5536–5541
Distributed AC Optimal Power Distributed AC Optimal Power Distributed AC Optimal Power Distributedusing AC Optimal Power ALADIN using ALADIN using using ALADIN ALADIN ∗
Flow Flow Flow Flow
Alexander Engelmann ∗ Tillmann M¨ uhlpfordt ∗∗ Alexander Engelmann M¨ u hlpfordt ∗ Tillmann ∗ ∗ ∗∗ ∗∗ Alexander Engelmann Tillmann M¨ u hlpfordt Yuning Jiang Boris Houska Timm ∗ ∗∗ ∗∗ Alexander Engelmann Tillmann M¨ uFaulwasser hlpfordt ∗ ∗∗ Yuning Jiang Faulwasser ∗∗ Boris Houska ∗∗ Timm Yuning Yuning ∗Jiang Jiang ∗∗ Boris Boris Houska Houska ∗∗ Timm Timm Faulwasser Faulwasser ∗ ∗ Institute for Applied Computer Science, ∗ Institute for Applied Computer Science, for Applied Computer Science, Karlsruhe Institute Technology, Karlsruhe, Germany ∗ Institute Institute forof Applied Computer Science, Karlsruhe Institute of Technology, Karlsruhe, Germany Karlsruhe Institute of Technology, Karlsruhe, Germany {alexander.engelmann, Karlsruhe Institute of Technology, Karlsruhe, Germany {alexander.engelmann, {alexander.engelmann, tillmann.muehlpfordt, timm.faulwasser}@kit.edu {alexander.engelmann, tillmann.muehlpfordt, timm.faulwasser}@kit.edu ∗∗ timm.faulwasser}@kit.edu Information Science and Technology, ShanghaiTech, China ∗∗ School of tillmann.muehlpfordt, timm.faulwasser}@kit.edu Information Science and Technology, ShanghaiTech, China ∗∗ School of tillmann.muehlpfordt, School of Information Science and {borish, jiangyn}@shanghaitech.edu.cn ∗∗ School of Information Science and Technology, Technology, ShanghaiTech, ShanghaiTech, China China {borish, jiangyn}@shanghaitech.edu.cn {borish, {borish, jiangyn}@shanghaitech.edu.cn jiangyn}@shanghaitech.edu.cn Abstract: This paper investigates the distributed solution of non-convex ac power flow Abstract: This paper investigates the distributed solution of non-convex ac power flow Abstract: This paper investigates the solution of non-convex ac optimization problems in electrical grids. Specifically, the application of a Abstract: This paperarising investigates the distributed distributed solutionwe of consider non-convex ac power power flow flow optimization problems arising in electrical grids. Specifically, we consider the application of a optimization problems arising in electrical grids. Specifically, we consider the application of recently proposed Augmented Lagrangian Based Alternating Direct Inexact Newton (aladin) optimization problems arising Lagrangian in electrical Based grids. Alternating Specifically, Direct we consider theNewton application of aa recently proposed Augmented Inexact (aladin) recently proposed Augmented Lagrangian Based Alternating Direct Inexact Newton (aladin) scheme toproposed ac optimal power flow problems. Based Using standard reformulations, we Newton show how aladin recently Augmented Lagrangian Alternating Direct Inexact (aladin) scheme to ac optimal power flow problems. Using standard reformulations, we show how aladin scheme to ac optimal power flow standard we show how aladin can be applied to electrical gridsproblems. of genericUsing topology. We reformulations, draw upon an ieee 5-bus system to scheme to ac optimal power flow problems. Using standard reformulations, we show how aladin can be applied to electrical grids of generic topology. We draw upon an ieee 5-bus system to can be applied to electrical grids of generic topology. We draw upon an ieee 5-bus system demonstrate that aladin offers the potential to outperform common admm schemes. can be applied to aladin electricaloffers gridsthe of potential generic topology. We draw upon admm an ieeeschemes. 5-bus system to to demonstrate that to outperform common demonstrate that aladin offers the potential to outperform common admm schemes. demonstrate that aladin offers the potential to outperform common admm schemes. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Distributed Optimal Power Flow, aladin, opf, admm, Distributed Optimization Keywords: Keywords: Distributed Distributed Optimal Optimal Power Power Flow, Flow, aladin, aladin, opf, opf, admm, admm, Distributed Distributed Optimization Optimization Keywords: Distributed Optimal Power Flow, aladin, opf, admm, Distributed Optimization 1. INTRODUCTION are directly handled by receiving optimal decision values 1. are directly handled by receiving optimal decision values 1. INTRODUCTION INTRODUCTION are directly handled by optimal decision values and Lagrange multipliers from all neighboring 1. INTRODUCTION are handled by receiving receiving optimal decisionregions values and directly Lagrange multipliers from all neighboring regions Lagrange multipliers from all neighboring regions and considering these values as fixed numbers in each The electrical energy production is currently undergoing a and considering Lagrange multipliers from all neighboring regions these values as fixed numbers in each The electrical energy production is currently undergoing aa local and considering these values as fixed numbers in optimization. Conejo et al. (2002) state a necessary The electrical energy production is currently undergoing paradigm shift: few large-scale centralized energy producconsidering these values as (2002) fixed numbers in each each The electrical energy productioncentralized is currently undergoing a and local optimization. Conejo et al. state aa necessary paradigm shift: few large-scale energy produclocal optimization. Conejo et al. (2002) state necessary for convergence to aal.kkt point. However, it is paradigm shift: few large-scale large-scale centralized energy produc- condition ers are being replaced by small-scale distributed generation local optimization. Conejo et (2002) state a necessary paradigm shift: few centralized energy produccondition for convergence to a kkt point. However, it is ers are being replaced by small-scale distributed generation condition for convergence to aa holds kkt point. However, it is unclear whether this condition for opf problems ers are being replaced by small-scale distributed generation units. In view of this transition, various challenging control condition for convergence to kkt point. However, it in is ers areIn being replaced by small-scale distributed generation unclear whether this condition holds for opf problems in units. view of this transition, various challenging control unclear whether this condition holds for opf problems in any case (Erseghe, 2015). units. In view of this transition, various challenging control problems arise that call for the development of distributed unclear whether this condition holds for opf problems in units. In view ofthat this transition, various challenging control any case (Erseghe, 2015). problems for of distributed (Erseghe, 2015). problems arise arisemethods that call callsuitable for the the development development of non-convex distributed any optimization for large-scale any case case (Erseghe, 2015). schemes to ac opf problems has problems arise that call for the development of distributed The application of admm optimization methods suitable for large-scale non-convex methods suitable for large-scale non-convex The application of admm schemes to ac opf problems has optimization problems. In the case of electrical grids, ac The application of admm schemes to opf has optimization problems. methods suitable for large-scale non-convex been investigated in different papers 2014a,b; In the case of electrical grids, ac The application of in admm schemes to ac ac(Erseghe, opf problems problems has optimization problems. In the case of electrical grids, ac been investigated different papers (Erseghe, 2014a,b; optimal power flow (opf) problems are of particular inbeen investigated in different papers (Erseghe, 2014a,b; optimization problems. In the case ofare electrical grids, ac Liu et al., 2015). However, admm is a distributed scheme optimal power flow (opf) problems of particular inbeen investigated in different papers (Erseghe, 2014a,b; optimal power flow (opf) problems are of particular inLiu et al., 2015). However, admm is a distributed scheme terest. Inpower these flow problems, one determines optimal power et However, admm aa distributed scheme optimal (opf) one problems are ofoptimal particular in- Liu for convex problems (Bertsekas andis 1989), while terest. problems, power et al., al., 2015). 2015). However, admm isTsitsiklis, distributed scheme terest. In In these these problems, one determines determines optimal power Liu for convex problems (Bertsekas and Tsitsiklis, 1989), while injections for allproblems, power generation units such that operafor convex problems (Bertsekas and Tsitsiklis, 1989), while terest. In these one determines optimal power the power flow equations in ac opf are non-convex. Nevinjections for all power generation units such that operafor convex problems (Bertsekas and Tsitsiklis, 1989), while injections for all power generation units such that operathe power flow equations in ac opf are non-convex. Nevtion cost is minimized. the power flow equations in ac opf are non-convex. Nevinjections for all power generation units such that opera- ertheless, admm works well in several non-convex cases. tion cost is minimized. the power admm flow equations in ac opf are non-convex non-convex.cases. Nevtion cost is minimized. ertheless, works well in several ertheless, admm works well in several non-convex cases. tion is minimized. (2015) proposed a parameter update rule to enThe cost present paper investigates the distributed solution of Erseghe ertheless, admm works well in several non-convex cases. Erseghe (2015) proposed a parameter update rule to enThe present paper investigates the distributed solution of Erseghe (2015) proposed a parameter update rule to force convergence for non-convex ac opf. Another extenThe present paper investigates the distributed solution of ac opf problems. Early works by Kim and Baldick (1997, (2015) proposed a parameter update rule extento enenThe present paperEarly investigates theKim distributed solution of Erseghe force convergence for non-convex ac opf. Another ac opf problems. works by and Baldick (1997, force convergence for ac Another extenrelying on sequential convexification admm, was ac opf opfare problems. Early works by Kim Kim and and Baldick (1997, 2000) motivated by works computational benefits of (1997, paral- sion, force convergence for non-convex non-convex ac opf. opf. and Another extenac problems. Early by Baldick sion, relying on sequential convexification and admm, was 2000) are motivated by computational benefits of paralsion, sequential and admm, by on Magn´ usson etconvexification al. (2015). While 2000) are are motivated by solution computational benefits of paralparal- proposed lelized andmotivated distributed schemes. The numerical sion, relying relying on sequential convexification andconvergence admm, was was 2000) by computational benefits of proposed by Magn´ u sson et al. (2015). While convergence lelized and distributed solution schemes. The numerical proposed by Magn´ u sson et al. (2015). While convergence conditions are investigated therein, the convergence rate lelized and distributed solution schemes. The numerical results are promising, yet no statements on convergence proposed by Magn´ u sson et al. (2015). While convergence lelized and distributed solution schemes. The numerical conditions are investigated therein, the convergence rate results promising, yet on investigated appears to are be slow. results are are are promising, yet no no statements statements on convergence convergence guarantees made. Furthermore, three main directions conditions conditions are investigated therein, therein, the the convergence convergence rate rate results are promising, yet no statements on convergence appears to be slow. guarantees are made. Furthermore, three main directions appears to be slow. guarantees are made. Furthermore, three main directions of research on distributed ac opf can be distinguished: appears to be slow. guarantees are made. Furthermore, threebemain directions A third line of research applies admm to a convex relaxof on ac of research research Condition on distributed distributed ac opf opf can can(ocd), be distinguished: distinguished: third line of research applies admm to a convex relaxOptimality Decomposition Alternating A A third research applies to convex relaxof research on distributed ac opf can be distinguished: ation of line the of original non-convex ac opf sdp Optimality Condition Decomposition (ocd), Alternating A third line of research applies admm admm to aasolved convexby relaxOptimality Condition Decomposition (ocd), Alternating ation of the original non-convex ac opf solved by sdp Direction of Multipliers Method (admm), and convex reation of the original non-convex ac opf solved by sdp Optimality Condition Decomposition (ocd), Alternating (Low, 2014a; Zhang et al., 2015; Dall’Anese et al., 2013; Direction of Multipliers Method (admm), and convex reation of the original non-convex ac opf solved by2013; sdp Direction of Multipliers Method (admm), and convex re(Low, 2014a; Zhang et al., 2015; Dall’Anese et al., laxations relying on Semidefinite Programming (sdp). (Low, 2014a; Zhang et al., 2015; Dall’Anese et al., 2013; Direction relying of Multipliers Method (admm), and convex re- Peng and Low, 2014; Bai et al., 2008). For the relaxed laxations on Semidefinite Programming (sdp). (Low, 2014a; Zhang etBai al., et 2015; Dall’Anese et al.,relaxed 2013; laxations relying on Semidefinite Programming (sdp). Peng and Low, 2014; al., 2008). For the and Low, Bai 2008). relaxed laxations Semidefinite problem, has guaranteed properties. The ocd relying methodon was introducedProgramming by Conejo et (sdp). al. (2002, Peng Peng and admm Low, 2014; 2014; Bai et et al., al., convergence 2008). For For the the relaxed problem, admm has guaranteed convergence properties. The ocd method was introduced by Conejo et al. (2002, problem, admm has guaranteed convergence properties. However, the difficulty lies in recovering the solution of the The ocd ocd method was introduced by(2003); ConejoArnold et al. al. (2002, (2002, 2006) andmethod is used was in Nogales et al. et al. problem, admm has guaranteed convergence properties. The introduced by Conejo et However, the difficulty lies in recovering the solution of the 2006) and is used in Nogales et al. (2003); Arnold et al. However, the difficulty lies in recovering the solution of original problem from the solution of the relaxed problem. 2006) and is used in Nogales et al. (2003); Arnold et al. (2007).and In isthese works, nonlinear couplingArnold constraints the difficulty lies solution in recovering the solution of the the 2006) used works, in Nogales et al. (2003); et al. However, original problem from the of the relaxed problem. (2007). In these nonlinear coupling constraints original problem from the of (2014b) showed the recoverability canproblem. only be (2007). In In these these works, works, nonlinear nonlinear coupling coupling constraints constraints Low original problem from that the solution solution of the the relaxed relaxed problem. (2007). Low (2014b) showed that the recoverability can only be TF is indebted to the Baden-W¨ Low (2014b) recoverability only be guaranteed forshowed radial that grids,the which is a strongcan restriction urttemberg Stiftung for the finanLow (2014b) showed that the recoverability can only be TF is indebted to the Baden-W¨ guaranteed for radial grids, which is a strong restriction urttemberg Stiftung for the finan guaranteed for radial grids, which is a strong restriction with respect to the grid topology. Recently, Madani et al. cial support of this by the Elite Programme Postdocs. TF is indebted to research the Baden-W¨ urttemberg Stiftung for the finan guaranteed for radial grids, whichRecently, is a strong restriction TF is indebted to research the Baden-W¨ urttemberg Stiftung for the finanwith respect to the grid topology. Madani et al. cial support of this by the Elite Programme Postdocs. with respect to grid topology. Recently, Madani et TF BH are supported by by thethe Deutsche Forschungsgemeinschaft, cial and support of this research Elite Programme for Postdocs. (2015) have shown certain technical conditions with respect to the the that grid under topology. Recently, Madani et al. al. cial support of this research Elite Programme for Postdocs. TF and BH are supported by by thethe Deutsche Forschungsgemeinschaft, (2015) have shown that under certain technical conditions Grants 2056/1 and WO 2056/4-1. YJForschungsgemeinschaft, and BH are supported TF and WO BH are supported by the Deutsche (2015) have shown that under certain technical conditions the sdp approach may yield near optimal solutions for TF and WO BH are supported by the Deutsche (2015) have shown that under certain technical conditions Grants 2056/1 and WO 2056/4-1. YJForschungsgemeinschaft, and BH are supported the sdp approach may yield near optimal solutions for by the WO National Natural Science Foundation (NSFC), Nr. Grants 2056/1 and WO 2056/4-1. YJ andChina BH are supported the sdp approach meshed Grants 2056/1 and WO 2056/4-1. YJ andChina BH are supported by the WO National Natural Science Foundation (NSFC), Nr. the sdp grids. approach may may yield yield near near optimal optimal solutions solutions for for meshed grids. 61473185, as wellNatural as ShanghaiTech University,China Grant-Nr. F-0203by the National Science Foundation (NSFC), Nr. meshed grids. by the National Science Foundation (NSFC), Nr. 61473185, as wellNatural as ShanghaiTech University,China Grant-Nr. F-0203meshed Finally, grids. one could also attempt to parallelize or even dis14-012. This also supportedUniversity, by the Helmholtz Association 61473185, as work well was as ShanghaiTech Grant-Nr. F-020361473185, as work well was as ShanghaiTech Grant-Nr. F-0203Finally, one could also attempt to parallelize or even dis14-012. This also supportedUniversity, by the Helmholtz Association Finally, one could also attempt parallelize or even disunder Joint Initiative System 2050 – A Contribution of 14-012.the This work was also“Energy supported by the Helmholtz Association tribute many operations of an to existing large-scale optiFinally, one could also attempt to parallelize or even dis14-012. This work was also“Energy supported by the Helmholtz Association under the Joint Initiative System 2050 – A Contribution of tribute many operations of an existing large-scale optithe Research Field Energy”. under the Joint Initiative “Energy System 2050 – A Contribution of tribute many operations of an existing large-scale optiunder the Joint Initiative “Energy System 2050 – A Contribution of the Research Field Energy”. tribute many operations of an existing large-scale optithe Research Field Energy”.
the Research Field Energy”. Copyright © 2017, 2017 IFAC 5716Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 5716 Copyright ©under 2017 responsibility IFAC 5716Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 5716 10.1016/j.ifacol.2017.08.1095
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Alexander Engelmann et al. / IFAC PapersOnLine 50-1 (2017) 5536–5541
mization method. There exists a large number of articles proposing such methods, which could also be applied to ac opf—at least in principle. For example, most of the operations of standard sequential quadratic programming methods can be distributed (Necoara et al., 2009; TranDinh et al., 2013), although in this case a coupled largescale QP (or another type of convex optimization problem) needs to be solved in every iteration. Similarly, standard augmented Lagrangian methods have been used for largescale optimization for a long time (Powell, 1969) and there exist mature optimization software packages implementing these methods, also for multi-core architectures (Conn et al., 1992; Gould et al., 2004). A more complete review of decomposition methods for convex and non-convex optimization algorithms can be found in Hamdi and Mishra (2011). Hours and Jones (2015) have proposed yet another way to distribute certain operations of a trust-region algorithm in context of ac opf with promising results. Recently, Houska et al. (2016) developed the Augmented Lagrangian Based Alternating Direction Inexact Newton (aladin) method for solving distributed non-convex optimization problems. Instead of attempting to parallelize as many operations as possible of an existing nlp solver, aladin solves—similar to admm—decoupled nlp problems as part of its iteration. In contrast to admm, aladin has favorable local and global convergence properties for nonconvex problems (Houska et al., 2016). The contribution of this paper is two-fold: first, in Sections 2 and 3 we reformulate the ac opf problem in a way such that aladin can be applied. Second, in Section 4, we present simulation results, which show that for meshed grids aladin converges faster and more reliably than admm—at least for our case study. From the viewpoint of distributed optimization algorithm development, this paper contributes a relevant case study that is inspired by power grid applications of current interest. 2. PROBLEM STATEMENT This section introduces the ac opf problem and a reformulation thereof that is amenable to distributed optimization algorithms. 2.1 Optimal Power Flow Consider an electrical grid described by a node index set N = {1, . . . , N }, a line index set L ⊆ N × N , a generator index set G ⊆ N , line conductances gkl and line susceptances bkl , for all (k, l) ∈ L. For non-connected nodes {(k, l) ∈ N × N | (k, l) ∈ / L} the parameters gkl , bkl are zero. Shunt elements of lines are neglected—the error is expected to be small since shunt currents are much smaller than currents flowing across the line. The nodal steady state χnk for all nodes k ∈ N is described by four physical values χnk = [θk vk pk qk ] ∈ R4 , (1) namely the voltage angle θk , voltage magnitude vk , injected active and reactive power pk , qk at node k. We collect all nodal (steady) states in the vector n χn = χn ∈ R4N . (2) 1 , . . . , χN
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The elements of the nodal steady states χnk are coupled by the power flow equations in polar form vk vl (gkl cos(θkl ) + bkl sin(θkl )) + pdk = pk , l∈N
vk
l∈N
vl (gkl sin(θkl ) − bkl cos(θkl )) + qdk = qk ,
(3)
where θkl = θk − θl . The power demands for active and reactive power, i.e. pdk and qdk , are assumed to be constant and given. In opf problems one tries to minimize the cost of energy production and grid operation while satisfying the power flow equations (3) and ensuring certain technical operation constraints. Hence, we deal with an optimization problem min f (χn ) n χ
(4) h(χn ) ≤ 0, n g(χ ) = 0. We assign nodal cost functions fkn (χnk ) = ak p2k + bk pk + ck , ak > 0 (5) to all generators k ∈ G describing the cost of power generation. The total cost function f (χn ) is defined as (6) fkn (χnk ). f (χn ) = s.t.
k∈G
Active and reactive power injections pk and qk for all generators k ∈ G are restricted to stay in certain bounds pk ≤ pk ≤ p¯k , (7) q k ≤ qk ≤ q¯k ,
due to the technical limits of the generators. Voltage magnitudes for all nodes k ∈ N have to remain bounded as well, i.e. v k ≤ vk ≤ v¯k . (8) Constraints (7) and (8) are collected in the inequality constraint h of optimization problem (4). As the power flow equations (3) depend on voltage angle differences θkl , one reference bus r ∈ N with (9) θr = 0, is needed to avoid multiple solutions of the power flow equations. Furthermore, one voltage magnitude reference bus r ∈ N with (10) vr = const, is considered. The injected powers pi and qi for all nongenerator nodes i ∈ N \ G are fixed to zero. These constraints together with the reference constraints (9) and (10), as well as the power flow equations (3) are stacked to the equality constraint function g : R4N → RL of optimization problem (4), where L is the number of equality constraint functions. For sake of compact notation, we consider all constraints as inequality constraints in the remainder. 2.2 Distributed opf Formulation First, we recall distributed consensus optimization problems. Second, we formulate ac opf in this form. For the scope of this work, a problem in consensus form is defined as, cf. (Boyd et al., 2011, Sec. 7),
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min
f (x) =
x
s.t.
fi (xi ) Ni
i∈R
hi (xi ) ≤ 0, Ai xi = b,
i ∈ R,
(11)
vk ejθk
k
Nic gkl
bkl 2
2
c pcim , qim c c vim ejθim
m
gkl 2
bkl 2
Njc
c pcjm , qjm c c vjm ejθjm
l
Nj vl ejθl
i∈R
where R = {1, . . . , R} is the index set of subproblems. 1 The global state vector x = [x ∈ RP is 1 , . . . , xR ] Ri composed of local state vectors xi ∈ R , i ∈ R. Here Ri is the dimension of xi and P = i∈R Ri is the dimension of x. The cost function f should be separable, i.e. a sum of (local) cost functions fi that depend only on xi with fi : RRi → R, i ∈ R. The global inequality constraint h from (4) should also be separable. Thus, in (11) it is composed of functions hi : RRi → RMi h(x) := h1 (x1 ) , . . . , hR (xR ) , where each hi depends only on xi , and Mi is the number of inequality constraints in region i. Coupling between the subsystems is due to the consensus constraint Ax = b, where A := [A1 , . . . , AR ] , with Ai ∈ RK×Ri and K is the number of consensus constraints. The right-hand side of the consensus constraint b ∈ RK is a constant. To obtain the desired consensus form (11) from the centralized opf problem (4), we partition the node index set of the grid, N , into geographically motivated regions R = {1, . . . , R}. 2 We define local node index sets Ni = {n1 , . . . , nNi } ⊆ N , i ∈ R, satisfying Ni = N .
auxiliary node m
Fig. 1. Splitting a line connecting two regions (k, l) ∈ Lc into segments and introducing an auxiliary node m ∈ C with auxiliary state ζm . function is doubled, this destroys the separability of the global cost function (6). 3 Auxiliary Nodes In view of the above, we introduce auxiliary nodes with the corresponding index set C = {1, . . . , C} at each border between two regions as shown in Fig. 1. We split lines connecting two regions Lc = {(k, l), (l, k) ∈ L | k = l, (k ∈ Ni ) ∧ (l ∈ / Nj ), i, j ∈ R)} in two parts of equal lengths, hence the parameters for each line segment bkl , gkl are half the value of the whole crossregion line (Fig. 1). The index set of the auxiliary nodes belonging to region i ∈ R is given by Ci = {c1 , . . . , cCi } ⊆ C. We denote the augmented nodal index set of region i by Nic . It is obtained from Nic = Ni ∪ Ci for all i ∈ R, i.e. the union of the local node index set and the index set of the auxiliary nodes belonging to that region (Fig. 1). Furthermore, we introduce consensus constraints for all auxiliary nodes m ∈ C that force the states of the auxiliary nodes to be consistent in both regions i and j c c = vj,m , vi,m c c θi,m = θj,m ,
i∈R
Furthermore, the local node index sets are disjoint, i.e. Ni ∩ Nj = ∅, i, j ∈ R, i = j. The local (physical) state χi is composed of the nodal steady states χnk of all nodes k in region i, i.e. n ∈ R4Ni , i ∈ R. (12) χi = χn 1 , . . . , χN i with χnk from (1) for all nodes k ∈ Ni .
The local objective function fi from (11) is composed of the nodal cost functions (5), namely fi (χi ) := (13) fkn (χnk ), i ∈ R, k∈Ni ∩G
which remains quadratic and positive semidefinite in χ. In contrast, the inequality constraint function h cannot be expressed in the desired separable form from (11) directly, as each power flow equation for node k depends on the state of its own node and on the states of all neighboring nodes. One common approach to overcome this difficulty is introducing auxiliary variables for states that appear as argument in more than one hi , and adding corresponding consensus constraints. Note that applying this directly to the unmodified problem can lead to difficulties: if a variable involved in a local cost 1 Note that there is no consensus on how to write an optimization problem in consensus form. For example, Boyd et al. (2011) refers to the setting x1 = x2 = . . . = xn , which is a special case of (11). 2 There exist also other motivations besides geographical location to partition the grid, e.g. computational aspects (Guo et al., 2015).
pci,m = −pcj,m , c c , qi,m = −qj,m
The states of the auxiliary nodes in region i, i.e. c c c vi,m pci,m qi,m ∈ R4 , m ∈ Ci , ζi,m = θi,m are stacked in one local auxiliary state ∈ R4Ci , i ∈ R. ζi = ζi,1 , . . . , ζi,C i
(14a)
The global augmented state is thus given by 4(Ni +Ci ) ∈ R i∈R . x = x 1 , . . . , xR
(14c)
The augmented state xi is obtained from stacking χi from (12) and ζi , i.e. xi = χ ∈ R4(Ni +Ci ) , i ∈ R. (14b) i , ζi
Based on the above reformulations we obtain the local cost functions fi and the local inequality constraints hi from (11), i ∈ R. Specifically, the nodal cost (5) becomes a local cost fi (χi ), cf. (13), and the local inequalities hi (xi ) follow from the power flow equations (3) and box constraints (7) and (8) for the augmented grid, i.e. the original grid together with the auxiliary nodes. Crucially, the introduction of the auxiliary nodes allows for separable local inequality constraint functions hi . The consensus constraint depends only on the local auxiliary state ζi and becomes Ax = 0. 3 There is an alternative way of decomposition without relying on auxiliary nodes (Erseghe, 2014b).
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3. ALADIN
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Algorithm 1: aladin Algorithm.
Next, we recall the Augmented Lagrangian Alternating Direction Inexact Newton method (aladin), introduced by Houska et al. (2016). The aladin method—shown in Algorithm 1—is a distributed optimization method for non-convex problems. It is initialized with a guess of the primal solution z 0 of (11), a Lagrange multiplier vector for the consensus constraint λ0 , a penalty parameter ρ > 0, positive definite weighting matrices Σi , and a penalty parameter µ > 0. In Step 1), all local subproblems i ∈ R are solved, i.e. the decoupled parts of the Powell-Hestenes augmented Lagrangian function of Problem (11) subject to local inequality constraints hi (xi ). This yields local solutions xki and Lagrange multipliers κi corresponding to the local inequality constraints hi (xi ). 4 In Step 2), a quadratic approximation of the local augmented Lagrangian composed of a gradient gik and a Hessian Hik is computed. Furthermore, linear approximations of the active inequality constraint vectors Ci∗k are calculated. In Step 3), a centralized quadratic program (qp) is composed as follows: Upon receiving all local approximations, the Hessian approximations Hik as well as the gradients gik and the approximation of the active inequality constraints Ci∗k are used to obtain a quadratic approximation of the global augmented Lagrangian around xk . The Hessian H k , the gradient g k , and the inequality constraint approximation C ∗k are given by k g k := g1k , . . . , gR , ∗k ∗k ∗k , C := C1 , . . . , CR k H k := diag(H1k , . . . , HR ). For numerical reasons, the consensus constraint Ax = 0 is not directly treated as an equality constraint (Houska et al. (2016)). Instead, a slack variable s containing the constraint residual is introduced and relaxed to the objective function. After a line search in Step 4), the global variable z and the Lagrange multiplier are updated using the solution of the qp in Step 5). A detailed description of the line search rule in Step 4) is left out due to space limitations.
Houska et al. (2016) provide a proof of global convergence to local minimizers for aladin applied to general nonconvex problems under certain non-restrictive conditions (Thm. 2). These convergence properties are a major advantage of aladin compared to most other distributed optimization algorithms like admm. For special choices of Hi = ρA i Ai , Σi = Ai Ai , Ci = 0 and µ → ∞, admm is almost recovered as a special case of aladin. The algorithms differ only slightly in the parameter update rule for λ and the fact that for admm several multiplier vectors λi are maintained instead of one for aladin (Houska et al., 2016). Note that the computational effort for each step and the communication need of the above variant of aladin is higher compared to admm. The centralized step in admm is just an averaging step compared to the need for solving a linear equation system of the qp in aladin. In the local 4 Note that the specific local subproblems from aladin are the local problems from (11) considering (14).
∗
Result: x Input: z 0 , λ0 , ρ, Σi , µ,
k = 0, x0 = ∞;
while Ax − b 1 > and ρ Σi (xi − zi )1 > do 1) Solve local NLPs ρ xki =argmin fi (xi ) + λk Ai xi + xi − zik 2Σi 2 xi s.t. hi (xi ) ≤ 0 | κki 2) Compute local gradients & Hessians for QP gik =∇fi (xki ) Hik
k =∇2 (fi (xki ) + κk i hi (xi )) if (hi (xki ))j = 0 ∇(hi (xki ))j ∗k Ci,j = 0 otherwise 3) Solve centralized QP µ 1 ∆xk = argmin ∆x H k ∆x + g k ∆x + λ s + s22 2 2 ∆x,s
s.t.
A(xk + ∆x) − s = 0
| λkQP
C ∗k ∆x = 0 4) Line search Details in Houska et al. (2016) → α1k , α2k , α3k , otherwise α1 = α2 = α3 = 1 5) Update z k+1 ← z k + α1k (xk − z k ) + α2k ∆xk ,
λk+1 ← λk + α3k (λkQP − λk ), k ←k+1 end
steps the gradient and Hessian approximations have to be computed and sent to the central entity, which takes more communication effort than just sending the local solutions and the local multipliers for admm. Note, however, that there are variants of aladin with fixed Hessian matrix approximations, in which all linear algebra operations are prepared in an initialization or pre-conditioning step such that aladin has the same communication cost as admm, cf. Houska et al. (2016). The disadvantage of fixing the matrices Hi is that aladin in that case converges at most linearly. 4. SIMULATION We consider the modified ieee-5 bus systems shown in Fig. 2. The grid is clustered into three regions R = {1, 2, 3} connected by four auxiliary nodes C = {I, . . . , IV}. Despite its small size, this grid poses an interesting benchmark problem for distributed opf algorithms: First, the grid is strongly meshed, which can drive distributed algorithms to divergence. Hence, often radial grids are used for testing the algorithms (Peng and Low, 2014). Second, as there is no consumer in region 1, we have a generation center in the west and a consumption center in the east with borders between regions separating the centers. This unbalanced distribution of loads results in large amounts of exchanged power between regions, which is an additional difficulty for distributed algorithms as consensus on the amount of exchanged powers is required. We remark a difference to the original test case from Li and Bo (2010). Therein, two generators at node 1 are considered, while we consider
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generator consumer
N3
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with rµ , rρ > 1. The update factors pµ and pρ are chosen to pµ = pρ = 5, the maximum values to ρ¯ = 5 · 103 , µ ¯ = 150 · 103 and the initial values µ0 and ρ0 are assigned to 1 · 103 . It is advisable to penalize the components of the decision vector inversely proportional to their range of deviation from the nominal value. Here, the normalized deviations for the injected active and reactive powers pk and qk are in the range of [1, 10] p.u., the values for the voltage angles and magnitudes θk and vk in the range of [10−3 , 10−2 ] rad and [10−2 , 10−1 ] p.u., respectively. Hence, we choose the diagonal entries of the Σi ’s related to pi and qi to 1 and the entries related to θ and v to the value of 100. All other entries of the Σi ’s are set to zero. The penalty parameter for admm is chosen as ρ = 103 . Fig. 3 shows the injected active and reactive powers pi , qi in p.u. for aladin and admm over the iteration index k. 5 For aladin, all values converge to their optimal value in around 7 iterations. By contrast, it takes admm 20 iterations for the active power and at least 25 iterations for reactive power to converge to an acceptable level. Fig. 4 shows the progress of the 2-norm of the primal residual Axk 2 , the global objective residual (f (xk ) − f (x∗ )), the 2-norm of the deviation of the decision vector from the optimal value x − x∗ 2 , and the 2-norm of the violation of the first order necessary kkt-conditions for a minimum r = ∇f (xk ) + A λk + ∇h(xk )κk 2 . For admm, the convergence behavior is depicted for three different values of the penalty parameter ρ. For large ρ = 104 , admm reaches a fast consensus but the convergence to the optimal value x∗ is slow. Similarly, the kkt residual decreases slowly. On the other hand, for ρ = 5 · 102 admm
2 0
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k
Fig. 3. Convergence of injected active and reactive powers for aladin (left) and admm with ρ = 103 (right). ×104 2
100
Axk 2
We rely on the symbolic framework CasADi (Andersson (2013)) and Matlab R2016a. As mentioned in (Houska et al., 2016, Thm. 2), the penalty parameters ρ and µ have to be large enough to obtain a convergence guarantee for aladin. As in our case large penalization parameters cause numerical problems in the beginning of a simulation, we introduce parameter update rules k k ¯ ρ rρ if ρk < ρ¯ µ rµ if µk < µ k+1 , ρ = µk+1 = k ρk µ
30
k
-2
10-3
ALADIN ADMM
10-6 10
15
ρ = 104
0 -2
5
20
ρ = 103
5
k
xk − x∗ 2
only one generator here for symmetry reasons. Moreover, we consider quadratic cost functions instead of linear ones. Furthermore, shunt elements of the lines and line congestion constraints are neglected.
The nominal values are Vnom = 230 kV and Snom = 100 MVA.
2
k
Fig. 2. Modified ieee 5-node test case (Li and Bo, 2010).
5
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Fig. 4. Convergence of primal residual, objective function residual, suboptimality and kkt residual for aladin and admm for ρ ∈ {5 · 102 , 103 , 104 }. takes more iterations, yet the distance to the optimal value x∗ decreases faster. Thus, admm struggles to converge to the exact solution. This can be seen from the rather big gap between admm and aladin in the deviation of the actual solution xk from the optimal value x∗ and especially in the kkt residual (Fig. 4). This behavior is a typical drawback of admm and also observed in other applications of admm (Boyd et al., 2011, Sec. 3.2.2). Hence, the penalty parameter is chosen to ρ = 103 to obtain fast convergence. In comparison, the convergence rate of aladin seems relatively insensitive to the choice of ρ (not shown here due to space limitations). Apart from the existence of convergence guarantees for aladin the question arises, whether the faster convergence justifies the additional cost for communication and computation per step. Note that any attempt to answer this question based on a single case study would be ambiguous and could lead to wrong conclusions. Both aladin and admm are generic optimization algorithms offering the potential
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to scale-up well and work for a large class of power grids. The purpose of this case study is to show that aladin is a promising alternative to admm for solving non-convex ac opf problems. Thus, future work will focus on larger grids, refined models with additional non-convex line congestion constraints, and tailored variants of aladin. 5. CONCLUSIONS & FUTURE WORK This paper investigates the application of aladin to distributed opf problems involving grids of generic topology, i.e. we consider the general case of meshed grids. The advantages of aladin are its convergence guarantees and the expected fast convergence behavior for general non-convex problems. In a comparative case study on a modified ieee 5-bus test case aladin converges faster than commonly employed admm schemes. Future work will evaluate the performance of the algorithm for larger grids and it will involve highly non-convex line congestion constraints. REFERENCES Andersson, J. (2013). A General-Purpose Software Framework for Dynamic Optimization. PhD thesis, Arenberg Doctoral School, KU Leuven, Department of Electrical Engineering (ESAT/SCD) and Optimization in Engineering Center, Kasteelpark Arenberg 10, 3001-Heverlee, Belgium. Arnold, M., Knopfli, S., and Andersson, G. (2007). Improvement of OPF decomposition methods applied to multi-area power systems. In Power Tech, 2007 IEEE Lausanne, 1308–1313. Bai, X., Wei, H., Fujisawa, K., and Wang, Y. (2008). Semidefinite programming for Optimal Power Flow problems. International Journal of Electrical Power & Energy Systems, 30(67), 383 – 392. Bertsekas, D.P. and Tsitsiklis, J.N. (1989). Parallel and Distributed Computation: Numerical Methods. Prentice Hall, NJ. Boyd, S., Parikh, N., Chu, E., Peleato, B., and Eckstein, J. (2011). Distributed optimization and statistical learning via the Alternating Direction Method of Multipliers. Found. Trends Mach. Learn., 3(1), 1–122. Conejo, A.J., Castillo, E., Minguez, R., and Garcia-Bertrand, R. (2006). Decomposition Techniques in Mathematical Programming: Engineering and Science Applications. Springer Science & Business Media, Berlin. Conejo, J.A., Nogales, J.F., and Prieto, J.F. (2002). A decomposition procedure based on approximate Newton directions. Mathematical Programming, 93(3), 495–515. Conn, A., Gould, N., and Toint, P. (1992). LANCELOT: a FORTRAN Package for Large-scale Nonlinear Optimization (Release A). Springer, Berlin. Dall’Anese, E., Zhu, H., and Giannakis, G.B. (2013). Distributed Optimal Power Flow for smart microgrids. IEEE Transactions on Smart Grid, 4(3), 1464–1475. Erseghe, T. (2014a). A distributed algorithm for fast Optimal Power Flow regulation in smart grids. In IEEE International Conference on Smart Grid Communications (SmartGridComm), 2014, 31– 36. Erseghe, T. (2014b). Distributed Optimal Power Flow using ADMM. IEEE Transactions on Power Systems, 29(5), 2370–2380. Erseghe, T. (2015). A distributed approach to the OPF problem. EURASIP Journal on Advances in Signal Processing, 2015(1), 1–13. Gould, N., Orban, D., and Toint, P. (2004). Galahad, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization. ACM Trans. Math. Software, 29(4), 353–372. Guo, J., Hug, G., and Tonguz, O. (2015). Impact of partitioning on the performance of decomposition methods for AC Optimal Power Flow. In Innovative Smart Grid Technologies Conference (ISGT), 2015 IEEE Power Energy Society, 1–5.
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Hamdi, A. and Mishra, S.K. (2011). Decomposition methods based on augmented Lagrangians: A survey. In S.K. Mishra (ed.), Topics in Nonconvex Optimization: Theory and Applications, 175–203. Springer, New York, NY. Hours, J.H. and Jones, C.N. (2015). An Alternating trust region algorithm for distributed linearly constrained nonlinear programs, application to the AC Optimal Power Flow. arXiv:1502.03777. Houska, B., Frasch, J., and Diehl, M. (2016). An augmented Lagrangian based algorithm for distributed nonconvex optimization. SIAM Journal on Optimization, 26(2), 1101–1127. Kim, B.H. and Baldick, R. (1997). Coarse-grained distributed Optimal Power Flow. IEEE Transactions on Power Systems, 12(2), 932–939. Kim, B.H. and Baldick, R. (2000). A comparison of distributed Optimal Power Flow algorithms. IEEE Transactions on Power Systems, 15(2), 599–604. Li, F. and Bo, R. (2010). Small test systems for power system economic studies. In IEEE PES General Meeting, 1–4. Liu, J., Benosman, M., and Raghunathan, A.U. (2015). Consensusbased distributed Optimal Power Flow algorithm. In Innovative Smart Grid Technologies Conference (ISGT), 2015 IEEE Power Energy Society, 1–5. Low, S.H. (2014a). Convex relaxation of Optimal Power Flow part I: Formulations and equivalence. IEEE Transactions on Control of Network Systems, 1(1), 15–27. Low, S.H. (2014b). Convex relaxation of Optimal Power Flow part II: Exactness. IEEE Transactions on Control of Network Systems, 1(2), 177–189. Madani, R., Sojoudi, S., and Lavaei, J. (2015). Convex relaxation for Optimal Power Flow problem: Mesh networks. IEEE Transactions on Power Systems, 30(1), 199–211. Magn´ usson, S., Weeraddana, P.C., and Fischione, C. (2015). A distributed approach for the optimal power-flow problem based on ADMM and sequential convex approximations. IEEE Transactions on Control of Network Systems, 2(3), 238–253. Necoara, I., Savorgnan, C., Tran, D.Q., Suykens, J., and Diehl, M. (2009). Distributed nonlinear optimal control using sequential convex programming and smoothing techniques. In Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, 543–548. Nogales, F.J., Prieto, F.J., and Conejo, A.J. (2003). A decomposition methodology applied to the multi-area optimal power flow problem. Annals of Operations Research, 120(1), 99–116. Peng, Q. and Low, S.H. (2014). Distributed algorithm for Optimal Power Flow on a radial network. In 53rd IEEE Conference on Decision and Control, 167–172. Powell, M. (1969). A method for nonlinear constraints in minimization problems. Optimization, (R. Fletcher, ed.). Academic Press, London. Tran-Dinh, Q., Necoara, I., Savorgnan, C., and Diehl, M. (2013). An inexact perturbed path-following method for Lagrangian decomposition in large-scale separable convex optimization. SIAM Journal on Optimization, 23(1), 95–125. Zhang, B., Lam, A.Y.S., Domnguez-Garca, A.D., and Tse, D. (2015). An optimal and distributed method for voltage regulation in power distribution systems. IEEE Transactions on Power Systems, 30(4), 1714–1726.
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Distributed AC Optimal Power Distributed AC Optimal Power Distributed AC Optimal Power Distributedusing AC Optimal Power ALADIN using ALADIN using using ALADIN ALADIN ∗
Flow Flow Flow Flow
Alexander Engelmann ∗ Tillmann M¨ uhlpfordt ∗∗ Alexander Engelmann M¨ u hlpfordt ∗ Tillmann ∗ ∗ ∗∗ ∗∗ Alexander Engelmann Tillmann M¨ u hlpfordt Yuning Jiang Boris Houska Timm ∗ ∗∗ ∗∗ Alexander Engelmann Tillmann M¨ uFaulwasser hlpfordt ∗ ∗∗ Yuning Jiang Faulwasser ∗∗ Boris Houska ∗∗ Timm Yuning Yuning ∗Jiang Jiang ∗∗ Boris Boris Houska Houska ∗∗ Timm Timm Faulwasser Faulwasser ∗ ∗ Institute for Applied Computer Science, ∗ Institute for Applied Computer Science, for Applied Computer Science, Karlsruhe Institute Technology, Karlsruhe, Germany ∗ Institute Institute forof Applied Computer Science, Karlsruhe Institute of Technology, Karlsruhe, Germany Karlsruhe Institute of Technology, Karlsruhe, Germany {alexander.engelmann, Karlsruhe Institute of Technology, Karlsruhe, Germany {alexander.engelmann, {alexander.engelmann, tillmann.muehlpfordt, timm.faulwasser}@kit.edu {alexander.engelmann, tillmann.muehlpfordt, timm.faulwasser}@kit.edu ∗∗ timm.faulwasser}@kit.edu Information Science and Technology, ShanghaiTech, China ∗∗ School of tillmann.muehlpfordt, timm.faulwasser}@kit.edu Information Science and Technology, ShanghaiTech, China ∗∗ School of tillmann.muehlpfordt, School of Information Science and {borish, jiangyn}@shanghaitech.edu.cn ∗∗ School of Information Science and Technology, Technology, ShanghaiTech, ShanghaiTech, China China {borish, jiangyn}@shanghaitech.edu.cn {borish, {borish, jiangyn}@shanghaitech.edu.cn jiangyn}@shanghaitech.edu.cn Abstract: This paper investigates the distributed solution of non-convex ac power flow Abstract: This paper investigates the distributed solution of non-convex ac power flow Abstract: This paper investigates the solution of non-convex ac optimization problems in electrical grids. Specifically, the application of a Abstract: This paperarising investigates the distributed distributed solutionwe of consider non-convex ac power power flow flow optimization problems arising in electrical grids. Specifically, we consider the application of a optimization problems arising in electrical grids. Specifically, we consider the application of recently proposed Augmented Lagrangian Based Alternating Direct Inexact Newton (aladin) optimization problems arising Lagrangian in electrical Based grids. Alternating Specifically, Direct we consider theNewton application of aa recently proposed Augmented Inexact (aladin) recently proposed Augmented Lagrangian Based Alternating Direct Inexact Newton (aladin) scheme toproposed ac optimal power flow problems. Based Using standard reformulations, we Newton show how aladin recently Augmented Lagrangian Alternating Direct Inexact (aladin) scheme to ac optimal power flow problems. Using standard reformulations, we show how aladin scheme to ac optimal power flow standard we show how aladin can be applied to electrical gridsproblems. of genericUsing topology. We reformulations, draw upon an ieee 5-bus system to scheme to ac optimal power flow problems. Using standard reformulations, we show how aladin can be applied to electrical grids of generic topology. We draw upon an ieee 5-bus system to can be applied to electrical grids of generic topology. We draw upon an ieee 5-bus system demonstrate that aladin offers the potential to outperform common admm schemes. can be applied to aladin electricaloffers gridsthe of potential generic topology. We draw upon admm an ieeeschemes. 5-bus system to to demonstrate that to outperform common demonstrate that aladin offers the potential to outperform common admm schemes. demonstrate that aladin offers the potential to outperform common admm schemes. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Distributed Optimal Power Flow, aladin, opf, admm, Distributed Optimization Keywords: Keywords: Distributed Distributed Optimal Optimal Power Power Flow, Flow, aladin, aladin, opf, opf, admm, admm, Distributed Distributed Optimization Optimization Keywords: Distributed Optimal Power Flow, aladin, opf, admm, Distributed Optimization 1. INTRODUCTION are directly handled by receiving optimal decision values 1. are directly handled by receiving optimal decision values 1. INTRODUCTION INTRODUCTION are directly handled by optimal decision values and Lagrange multipliers from all neighboring 1. INTRODUCTION are handled by receiving receiving optimal decisionregions values and directly Lagrange multipliers from all neighboring regions Lagrange multipliers from all neighboring regions and considering these values as fixed numbers in each The electrical energy production is currently undergoing a and considering Lagrange multipliers from all neighboring regions these values as fixed numbers in each The electrical energy production is currently undergoing aa local and considering these values as fixed numbers in optimization. Conejo et al. (2002) state a necessary The electrical energy production is currently undergoing paradigm shift: few large-scale centralized energy producconsidering these values as (2002) fixed numbers in each each The electrical energy productioncentralized is currently undergoing a and local optimization. Conejo et al. state aa necessary paradigm shift: few large-scale energy produclocal optimization. Conejo et al. (2002) state necessary for convergence to aal.kkt point. However, it is paradigm shift: few large-scale large-scale centralized energy produc- condition ers are being replaced by small-scale distributed generation local optimization. Conejo et (2002) state a necessary paradigm shift: few centralized energy produccondition for convergence to a kkt point. However, it is ers are being replaced by small-scale distributed generation condition for convergence to aa holds kkt point. However, it is unclear whether this condition for opf problems ers are being replaced by small-scale distributed generation units. In view of this transition, various challenging control condition for convergence to kkt point. However, it in is ers areIn being replaced by small-scale distributed generation unclear whether this condition holds for opf problems in units. view of this transition, various challenging control unclear whether this condition holds for opf problems in any case (Erseghe, 2015). units. In view of this transition, various challenging control problems arise that call for the development of distributed unclear whether this condition holds for opf problems in units. In view ofthat this transition, various challenging control any case (Erseghe, 2015). problems for of distributed (Erseghe, 2015). problems arise arisemethods that call callsuitable for the the development development of non-convex distributed any optimization for large-scale any case case (Erseghe, 2015). schemes to ac opf problems has problems arise that call for the development of distributed The application of admm optimization methods suitable for large-scale non-convex methods suitable for large-scale non-convex The application of admm schemes to ac opf problems has optimization problems. In the case of electrical grids, ac The application of admm schemes to opf has optimization problems. methods suitable for large-scale non-convex been investigated in different papers 2014a,b; In the case of electrical grids, ac The application of in admm schemes to ac ac(Erseghe, opf problems problems has optimization problems. In the case of electrical grids, ac been investigated different papers (Erseghe, 2014a,b; optimal power flow (opf) problems are of particular inbeen investigated in different papers (Erseghe, 2014a,b; optimization problems. In the case ofare electrical grids, ac Liu et al., 2015). However, admm is a distributed scheme optimal power flow (opf) problems of particular inbeen investigated in different papers (Erseghe, 2014a,b; optimal power flow (opf) problems are of particular inLiu et al., 2015). However, admm is a distributed scheme terest. Inpower these flow problems, one determines optimal power et However, admm aa distributed scheme optimal (opf) one problems are ofoptimal particular in- Liu for convex problems (Bertsekas andis 1989), while terest. problems, power et al., al., 2015). 2015). However, admm isTsitsiklis, distributed scheme terest. In In these these problems, one determines determines optimal power Liu for convex problems (Bertsekas and Tsitsiklis, 1989), while injections for allproblems, power generation units such that operafor convex problems (Bertsekas and Tsitsiklis, 1989), while terest. In these one determines optimal power the power flow equations in ac opf are non-convex. Nevinjections for all power generation units such that operafor convex problems (Bertsekas and Tsitsiklis, 1989), while injections for all power generation units such that operathe power flow equations in ac opf are non-convex. Nevtion cost is minimized. the power flow equations in ac opf are non-convex. Nevinjections for all power generation units such that opera- ertheless, admm works well in several non-convex cases. tion cost is minimized. the power admm flow equations in ac opf are non-convex non-convex.cases. Nevtion cost is minimized. ertheless, works well in several ertheless, admm works well in several non-convex cases. tion is minimized. (2015) proposed a parameter update rule to enThe cost present paper investigates the distributed solution of Erseghe ertheless, admm works well in several non-convex cases. Erseghe (2015) proposed a parameter update rule to enThe present paper investigates the distributed solution of Erseghe (2015) proposed a parameter update rule to force convergence for non-convex ac opf. Another extenThe present paper investigates the distributed solution of ac opf problems. Early works by Kim and Baldick (1997, (2015) proposed a parameter update rule extento enenThe present paperEarly investigates theKim distributed solution of Erseghe force convergence for non-convex ac opf. Another ac opf problems. works by and Baldick (1997, force convergence for ac Another extenrelying on sequential convexification admm, was ac opf opfare problems. Early works by Kim Kim and and Baldick (1997, 2000) motivated by works computational benefits of (1997, paral- sion, force convergence for non-convex non-convex ac opf. opf. and Another extenac problems. Early by Baldick sion, relying on sequential convexification and admm, was 2000) are motivated by computational benefits of paralsion, sequential and admm, by on Magn´ usson etconvexification al. (2015). While 2000) are are motivated by solution computational benefits of paralparal- proposed lelized andmotivated distributed schemes. The numerical sion, relying relying on sequential convexification andconvergence admm, was was 2000) by computational benefits of proposed by Magn´ u sson et al. (2015). While convergence lelized and distributed solution schemes. The numerical proposed by Magn´ u sson et al. (2015). While convergence conditions are investigated therein, the convergence rate lelized and distributed solution schemes. The numerical results are promising, yet no statements on convergence proposed by Magn´ u sson et al. (2015). While convergence lelized and distributed solution schemes. The numerical conditions are investigated therein, the convergence rate results promising, yet on investigated appears to are be slow. results are are are promising, yet no no statements statements on convergence convergence guarantees made. Furthermore, three main directions conditions conditions are investigated therein, therein, the the convergence convergence rate rate results are promising, yet no statements on convergence appears to be slow. guarantees are made. Furthermore, three main directions appears to be slow. guarantees are made. Furthermore, three main directions of research on distributed ac opf can be distinguished: appears to be slow. guarantees are made. Furthermore, threebemain directions A third line of research applies admm to a convex relaxof on ac of research research Condition on distributed distributed ac opf opf can can(ocd), be distinguished: distinguished: third line of research applies admm to a convex relaxOptimality Decomposition Alternating A A third research applies to convex relaxof research on distributed ac opf can be distinguished: ation of line the of original non-convex ac opf sdp Optimality Condition Decomposition (ocd), Alternating A third line of research applies admm admm to aasolved convexby relaxOptimality Condition Decomposition (ocd), Alternating ation of the original non-convex ac opf solved by sdp Direction of Multipliers Method (admm), and convex reation of the original non-convex ac opf solved by sdp Optimality Condition Decomposition (ocd), Alternating (Low, 2014a; Zhang et al., 2015; Dall’Anese et al., 2013; Direction of Multipliers Method (admm), and convex reation of the original non-convex ac opf solved by2013; sdp Direction of Multipliers Method (admm), and convex re(Low, 2014a; Zhang et al., 2015; Dall’Anese et al., laxations relying on Semidefinite Programming (sdp). (Low, 2014a; Zhang et al., 2015; Dall’Anese et al., 2013; Direction relying of Multipliers Method (admm), and convex re- Peng and Low, 2014; Bai et al., 2008). For the relaxed laxations on Semidefinite Programming (sdp). (Low, 2014a; Zhang etBai al., et 2015; Dall’Anese et al.,relaxed 2013; laxations relying on Semidefinite Programming (sdp). Peng and Low, 2014; al., 2008). For the and Low, Bai 2008). relaxed laxations Semidefinite problem, has guaranteed properties. The ocd relying methodon was introducedProgramming by Conejo et (sdp). al. (2002, Peng Peng and admm Low, 2014; 2014; Bai et et al., al., convergence 2008). For For the the relaxed problem, admm has guaranteed convergence properties. The ocd method was introduced by Conejo et al. (2002, problem, admm has guaranteed convergence properties. However, the difficulty lies in recovering the solution of the The ocd ocd method was introduced by(2003); ConejoArnold et al. al. (2002, (2002, 2006) andmethod is used was in Nogales et al. et al. problem, admm has guaranteed convergence properties. The introduced by Conejo et However, the difficulty lies in recovering the solution of the 2006) and is used in Nogales et al. (2003); Arnold et al. However, the difficulty lies in recovering the solution of original problem from the solution of the relaxed problem. 2006) and is used in Nogales et al. (2003); Arnold et al. (2007).and In isthese works, nonlinear couplingArnold constraints the difficulty lies solution in recovering the solution of the the 2006) used works, in Nogales et al. (2003); et al. However, original problem from the of the relaxed problem. (2007). In these nonlinear coupling constraints original problem from the of (2014b) showed the recoverability canproblem. only be (2007). In In these these works, works, nonlinear nonlinear coupling coupling constraints constraints Low original problem from that the solution solution of the the relaxed relaxed problem. (2007). Low (2014b) showed that the recoverability can only be TF is indebted to the Baden-W¨ Low (2014b) recoverability only be guaranteed forshowed radial that grids,the which is a strongcan restriction urttemberg Stiftung for the finanLow (2014b) showed that the recoverability can only be TF is indebted to the Baden-W¨ guaranteed for radial grids, which is a strong restriction urttemberg Stiftung for the finan guaranteed for radial grids, which is a strong restriction with respect to the grid topology. Recently, Madani et al. cial support of this by the Elite Programme Postdocs. TF is indebted to research the Baden-W¨ urttemberg Stiftung for the finan guaranteed for radial grids, whichRecently, is a strong restriction TF is indebted to research the Baden-W¨ urttemberg Stiftung for the finanwith respect to the grid topology. Madani et al. cial support of this by the Elite Programme Postdocs. with respect to grid topology. Recently, Madani et TF BH are supported by by thethe Deutsche Forschungsgemeinschaft, cial and support of this research Elite Programme for Postdocs. (2015) have shown certain technical conditions with respect to the the that grid under topology. Recently, Madani et al. al. cial support of this research Elite Programme for Postdocs. TF and BH are supported by by thethe Deutsche Forschungsgemeinschaft, (2015) have shown that under certain technical conditions Grants 2056/1 and WO 2056/4-1. YJForschungsgemeinschaft, and BH are supported TF and WO BH are supported by the Deutsche (2015) have shown that under certain technical conditions the sdp approach may yield near optimal solutions for TF and WO BH are supported by the Deutsche (2015) have shown that under certain technical conditions Grants 2056/1 and WO 2056/4-1. YJForschungsgemeinschaft, and BH are supported the sdp approach may yield near optimal solutions for by the WO National Natural Science Foundation (NSFC), Nr. Grants 2056/1 and WO 2056/4-1. YJ andChina BH are supported the sdp approach meshed Grants 2056/1 and WO 2056/4-1. YJ andChina BH are supported by the WO National Natural Science Foundation (NSFC), Nr. the sdp grids. approach may may yield yield near near optimal optimal solutions solutions for for meshed grids. 61473185, as wellNatural as ShanghaiTech University,China Grant-Nr. F-0203by the National Science Foundation (NSFC), Nr. meshed grids. by the National Science Foundation (NSFC), Nr. 61473185, as wellNatural as ShanghaiTech University,China Grant-Nr. F-0203meshed Finally, grids. one could also attempt to parallelize or even dis14-012. This also supportedUniversity, by the Helmholtz Association 61473185, as work well was as ShanghaiTech Grant-Nr. F-020361473185, as work well was as ShanghaiTech Grant-Nr. F-0203Finally, one could also attempt to parallelize or even dis14-012. This also supportedUniversity, by the Helmholtz Association Finally, one could also attempt parallelize or even disunder Joint Initiative System 2050 – A Contribution of 14-012.the This work was also“Energy supported by the Helmholtz Association tribute many operations of an to existing large-scale optiFinally, one could also attempt to parallelize or even dis14-012. This work was also“Energy supported by the Helmholtz Association under the Joint Initiative System 2050 – A Contribution of tribute many operations of an existing large-scale optithe Research Field Energy”. under the Joint Initiative “Energy System 2050 – A Contribution of tribute many operations of an existing large-scale optiunder the Joint Initiative “Energy System 2050 – A Contribution of the Research Field Energy”. tribute many operations of an existing large-scale optithe Research Field Energy”.
the Research Field Energy”. Copyright © 2017, 2017 IFAC 5716Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 5716 Copyright ©under 2017 responsibility IFAC 5716Control. Peer review of International Federation of Automatic Copyright © 2017 IFAC 5716 10.1016/j.ifacol.2017.08.1095
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Alexander Engelmann et al. / IFAC PapersOnLine 50-1 (2017) 5536–5541
mization method. There exists a large number of articles proposing such methods, which could also be applied to ac opf—at least in principle. For example, most of the operations of standard sequential quadratic programming methods can be distributed (Necoara et al., 2009; TranDinh et al., 2013), although in this case a coupled largescale QP (or another type of convex optimization problem) needs to be solved in every iteration. Similarly, standard augmented Lagrangian methods have been used for largescale optimization for a long time (Powell, 1969) and there exist mature optimization software packages implementing these methods, also for multi-core architectures (Conn et al., 1992; Gould et al., 2004). A more complete review of decomposition methods for convex and non-convex optimization algorithms can be found in Hamdi and Mishra (2011). Hours and Jones (2015) have proposed yet another way to distribute certain operations of a trust-region algorithm in context of ac opf with promising results. Recently, Houska et al. (2016) developed the Augmented Lagrangian Based Alternating Direction Inexact Newton (aladin) method for solving distributed non-convex optimization problems. Instead of attempting to parallelize as many operations as possible of an existing nlp solver, aladin solves—similar to admm—decoupled nlp problems as part of its iteration. In contrast to admm, aladin has favorable local and global convergence properties for nonconvex problems (Houska et al., 2016). The contribution of this paper is two-fold: first, in Sections 2 and 3 we reformulate the ac opf problem in a way such that aladin can be applied. Second, in Section 4, we present simulation results, which show that for meshed grids aladin converges faster and more reliably than admm—at least for our case study. From the viewpoint of distributed optimization algorithm development, this paper contributes a relevant case study that is inspired by power grid applications of current interest. 2. PROBLEM STATEMENT This section introduces the ac opf problem and a reformulation thereof that is amenable to distributed optimization algorithms. 2.1 Optimal Power Flow Consider an electrical grid described by a node index set N = {1, . . . , N }, a line index set L ⊆ N × N , a generator index set G ⊆ N , line conductances gkl and line susceptances bkl , for all (k, l) ∈ L. For non-connected nodes {(k, l) ∈ N × N | (k, l) ∈ / L} the parameters gkl , bkl are zero. Shunt elements of lines are neglected—the error is expected to be small since shunt currents are much smaller than currents flowing across the line. The nodal steady state χnk for all nodes k ∈ N is described by four physical values χnk = [θk vk pk qk ] ∈ R4 , (1) namely the voltage angle θk , voltage magnitude vk , injected active and reactive power pk , qk at node k. We collect all nodal (steady) states in the vector n χn = χn ∈ R4N . (2) 1 , . . . , χN
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The elements of the nodal steady states χnk are coupled by the power flow equations in polar form vk vl (gkl cos(θkl ) + bkl sin(θkl )) + pdk = pk , l∈N
vk
l∈N
vl (gkl sin(θkl ) − bkl cos(θkl )) + qdk = qk ,
(3)
where θkl = θk − θl . The power demands for active and reactive power, i.e. pdk and qdk , are assumed to be constant and given. In opf problems one tries to minimize the cost of energy production and grid operation while satisfying the power flow equations (3) and ensuring certain technical operation constraints. Hence, we deal with an optimization problem min f (χn ) n χ
(4) h(χn ) ≤ 0, n g(χ ) = 0. We assign nodal cost functions fkn (χnk ) = ak p2k + bk pk + ck , ak > 0 (5) to all generators k ∈ G describing the cost of power generation. The total cost function f (χn ) is defined as (6) fkn (χnk ). f (χn ) = s.t.
k∈G
Active and reactive power injections pk and qk for all generators k ∈ G are restricted to stay in certain bounds pk ≤ pk ≤ p¯k , (7) q k ≤ qk ≤ q¯k ,
due to the technical limits of the generators. Voltage magnitudes for all nodes k ∈ N have to remain bounded as well, i.e. v k ≤ vk ≤ v¯k . (8) Constraints (7) and (8) are collected in the inequality constraint h of optimization problem (4). As the power flow equations (3) depend on voltage angle differences θkl , one reference bus r ∈ N with (9) θr = 0, is needed to avoid multiple solutions of the power flow equations. Furthermore, one voltage magnitude reference bus r ∈ N with (10) vr = const, is considered. The injected powers pi and qi for all nongenerator nodes i ∈ N \ G are fixed to zero. These constraints together with the reference constraints (9) and (10), as well as the power flow equations (3) are stacked to the equality constraint function g : R4N → RL of optimization problem (4), where L is the number of equality constraint functions. For sake of compact notation, we consider all constraints as inequality constraints in the remainder. 2.2 Distributed opf Formulation First, we recall distributed consensus optimization problems. Second, we formulate ac opf in this form. For the scope of this work, a problem in consensus form is defined as, cf. (Boyd et al., 2011, Sec. 7),
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min
f (x) =
x
s.t.
fi (xi ) Ni
i∈R
hi (xi ) ≤ 0, Ai xi = b,
i ∈ R,
(11)
vk ejθk
k
Nic gkl
bkl 2
2
c pcim , qim c c vim ejθim
m
gkl 2
bkl 2
Njc
c pcjm , qjm c c vjm ejθjm
l
Nj vl ejθl
i∈R
where R = {1, . . . , R} is the index set of subproblems. 1 The global state vector x = [x ∈ RP is 1 , . . . , xR ] Ri composed of local state vectors xi ∈ R , i ∈ R. Here Ri is the dimension of xi and P = i∈R Ri is the dimension of x. The cost function f should be separable, i.e. a sum of (local) cost functions fi that depend only on xi with fi : RRi → R, i ∈ R. The global inequality constraint h from (4) should also be separable. Thus, in (11) it is composed of functions hi : RRi → RMi h(x) := h1 (x1 ) , . . . , hR (xR ) , where each hi depends only on xi , and Mi is the number of inequality constraints in region i. Coupling between the subsystems is due to the consensus constraint Ax = b, where A := [A1 , . . . , AR ] , with Ai ∈ RK×Ri and K is the number of consensus constraints. The right-hand side of the consensus constraint b ∈ RK is a constant. To obtain the desired consensus form (11) from the centralized opf problem (4), we partition the node index set of the grid, N , into geographically motivated regions R = {1, . . . , R}. 2 We define local node index sets Ni = {n1 , . . . , nNi } ⊆ N , i ∈ R, satisfying Ni = N .
auxiliary node m
Fig. 1. Splitting a line connecting two regions (k, l) ∈ Lc into segments and introducing an auxiliary node m ∈ C with auxiliary state ζm . function is doubled, this destroys the separability of the global cost function (6). 3 Auxiliary Nodes In view of the above, we introduce auxiliary nodes with the corresponding index set C = {1, . . . , C} at each border between two regions as shown in Fig. 1. We split lines connecting two regions Lc = {(k, l), (l, k) ∈ L | k = l, (k ∈ Ni ) ∧ (l ∈ / Nj ), i, j ∈ R)} in two parts of equal lengths, hence the parameters for each line segment bkl , gkl are half the value of the whole crossregion line (Fig. 1). The index set of the auxiliary nodes belonging to region i ∈ R is given by Ci = {c1 , . . . , cCi } ⊆ C. We denote the augmented nodal index set of region i by Nic . It is obtained from Nic = Ni ∪ Ci for all i ∈ R, i.e. the union of the local node index set and the index set of the auxiliary nodes belonging to that region (Fig. 1). Furthermore, we introduce consensus constraints for all auxiliary nodes m ∈ C that force the states of the auxiliary nodes to be consistent in both regions i and j c c = vj,m , vi,m c c θi,m = θj,m ,
i∈R
Furthermore, the local node index sets are disjoint, i.e. Ni ∩ Nj = ∅, i, j ∈ R, i = j. The local (physical) state χi is composed of the nodal steady states χnk of all nodes k in region i, i.e. n ∈ R4Ni , i ∈ R. (12) χi = χn 1 , . . . , χN i with χnk from (1) for all nodes k ∈ Ni .
The local objective function fi from (11) is composed of the nodal cost functions (5), namely fi (χi ) := (13) fkn (χnk ), i ∈ R, k∈Ni ∩G
which remains quadratic and positive semidefinite in χ. In contrast, the inequality constraint function h cannot be expressed in the desired separable form from (11) directly, as each power flow equation for node k depends on the state of its own node and on the states of all neighboring nodes. One common approach to overcome this difficulty is introducing auxiliary variables for states that appear as argument in more than one hi , and adding corresponding consensus constraints. Note that applying this directly to the unmodified problem can lead to difficulties: if a variable involved in a local cost 1 Note that there is no consensus on how to write an optimization problem in consensus form. For example, Boyd et al. (2011) refers to the setting x1 = x2 = . . . = xn , which is a special case of (11). 2 There exist also other motivations besides geographical location to partition the grid, e.g. computational aspects (Guo et al., 2015).
pci,m = −pcj,m , c c , qi,m = −qj,m
The states of the auxiliary nodes in region i, i.e. c c c vi,m pci,m qi,m ∈ R4 , m ∈ Ci , ζi,m = θi,m are stacked in one local auxiliary state ∈ R4Ci , i ∈ R. ζi = ζi,1 , . . . , ζi,C i
(14a)
The global augmented state is thus given by 4(Ni +Ci ) ∈ R i∈R . x = x 1 , . . . , xR
(14c)
The augmented state xi is obtained from stacking χi from (12) and ζi , i.e. xi = χ ∈ R4(Ni +Ci ) , i ∈ R. (14b) i , ζi
Based on the above reformulations we obtain the local cost functions fi and the local inequality constraints hi from (11), i ∈ R. Specifically, the nodal cost (5) becomes a local cost fi (χi ), cf. (13), and the local inequalities hi (xi ) follow from the power flow equations (3) and box constraints (7) and (8) for the augmented grid, i.e. the original grid together with the auxiliary nodes. Crucially, the introduction of the auxiliary nodes allows for separable local inequality constraint functions hi . The consensus constraint depends only on the local auxiliary state ζi and becomes Ax = 0. 3 There is an alternative way of decomposition without relying on auxiliary nodes (Erseghe, 2014b).
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3. ALADIN
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Algorithm 1: aladin Algorithm.
Next, we recall the Augmented Lagrangian Alternating Direction Inexact Newton method (aladin), introduced by Houska et al. (2016). The aladin method—shown in Algorithm 1—is a distributed optimization method for non-convex problems. It is initialized with a guess of the primal solution z 0 of (11), a Lagrange multiplier vector for the consensus constraint λ0 , a penalty parameter ρ > 0, positive definite weighting matrices Σi , and a penalty parameter µ > 0. In Step 1), all local subproblems i ∈ R are solved, i.e. the decoupled parts of the Powell-Hestenes augmented Lagrangian function of Problem (11) subject to local inequality constraints hi (xi ). This yields local solutions xki and Lagrange multipliers κi corresponding to the local inequality constraints hi (xi ). 4 In Step 2), a quadratic approximation of the local augmented Lagrangian composed of a gradient gik and a Hessian Hik is computed. Furthermore, linear approximations of the active inequality constraint vectors Ci∗k are calculated. In Step 3), a centralized quadratic program (qp) is composed as follows: Upon receiving all local approximations, the Hessian approximations Hik as well as the gradients gik and the approximation of the active inequality constraints Ci∗k are used to obtain a quadratic approximation of the global augmented Lagrangian around xk . The Hessian H k , the gradient g k , and the inequality constraint approximation C ∗k are given by k g k := g1k , . . . , gR , ∗k ∗k ∗k , C := C1 , . . . , CR k H k := diag(H1k , . . . , HR ). For numerical reasons, the consensus constraint Ax = 0 is not directly treated as an equality constraint (Houska et al. (2016)). Instead, a slack variable s containing the constraint residual is introduced and relaxed to the objective function. After a line search in Step 4), the global variable z and the Lagrange multiplier are updated using the solution of the qp in Step 5). A detailed description of the line search rule in Step 4) is left out due to space limitations.
Houska et al. (2016) provide a proof of global convergence to local minimizers for aladin applied to general nonconvex problems under certain non-restrictive conditions (Thm. 2). These convergence properties are a major advantage of aladin compared to most other distributed optimization algorithms like admm. For special choices of Hi = ρA i Ai , Σi = Ai Ai , Ci = 0 and µ → ∞, admm is almost recovered as a special case of aladin. The algorithms differ only slightly in the parameter update rule for λ and the fact that for admm several multiplier vectors λi are maintained instead of one for aladin (Houska et al., 2016). Note that the computational effort for each step and the communication need of the above variant of aladin is higher compared to admm. The centralized step in admm is just an averaging step compared to the need for solving a linear equation system of the qp in aladin. In the local 4 Note that the specific local subproblems from aladin are the local problems from (11) considering (14).
∗
Result: x Input: z 0 , λ0 , ρ, Σi , µ,
k = 0, x0 = ∞;
while Ax − b 1 > and ρ Σi (xi − zi )1 > do 1) Solve local NLPs ρ xki =argmin fi (xi ) + λk Ai xi + xi − zik 2Σi 2 xi s.t. hi (xi ) ≤ 0 | κki 2) Compute local gradients & Hessians for QP gik =∇fi (xki ) Hik
k =∇2 (fi (xki ) + κk i hi (xi )) if (hi (xki ))j = 0 ∇(hi (xki ))j ∗k Ci,j = 0 otherwise 3) Solve centralized QP µ 1 ∆xk = argmin ∆x H k ∆x + g k ∆x + λ s + s22 2 2 ∆x,s
s.t.
A(xk + ∆x) − s = 0
| λkQP
C ∗k ∆x = 0 4) Line search Details in Houska et al. (2016) → α1k , α2k , α3k , otherwise α1 = α2 = α3 = 1 5) Update z k+1 ← z k + α1k (xk − z k ) + α2k ∆xk ,
λk+1 ← λk + α3k (λkQP − λk ), k ←k+1 end
steps the gradient and Hessian approximations have to be computed and sent to the central entity, which takes more communication effort than just sending the local solutions and the local multipliers for admm. Note, however, that there are variants of aladin with fixed Hessian matrix approximations, in which all linear algebra operations are prepared in an initialization or pre-conditioning step such that aladin has the same communication cost as admm, cf. Houska et al. (2016). The disadvantage of fixing the matrices Hi is that aladin in that case converges at most linearly. 4. SIMULATION We consider the modified ieee-5 bus systems shown in Fig. 2. The grid is clustered into three regions R = {1, 2, 3} connected by four auxiliary nodes C = {I, . . . , IV}. Despite its small size, this grid poses an interesting benchmark problem for distributed opf algorithms: First, the grid is strongly meshed, which can drive distributed algorithms to divergence. Hence, often radial grids are used for testing the algorithms (Peng and Low, 2014). Second, as there is no consumer in region 1, we have a generation center in the west and a consumption center in the east with borders between regions separating the centers. This unbalanced distribution of loads results in large amounts of exchanged power between regions, which is an additional difficulty for distributed algorithms as consensus on the amount of exchanged powers is required. We remark a difference to the original test case from Li and Bo (2010). Therein, two generators at node 1 are considered, while we consider
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generator consumer
N3
6
IV
I 170MW
0
520MW
300MW
20
30
40
50
10
20
with rµ , rρ > 1. The update factors pµ and pρ are chosen to pµ = pρ = 5, the maximum values to ρ¯ = 5 · 103 , µ ¯ = 150 · 103 and the initial values µ0 and ρ0 are assigned to 1 · 103 . It is advisable to penalize the components of the decision vector inversely proportional to their range of deviation from the nominal value. Here, the normalized deviations for the injected active and reactive powers pk and qk are in the range of [1, 10] p.u., the values for the voltage angles and magnitudes θk and vk in the range of [10−3 , 10−2 ] rad and [10−2 , 10−1 ] p.u., respectively. Hence, we choose the diagonal entries of the Σi ’s related to pi and qi to 1 and the entries related to θ and v to the value of 100. All other entries of the Σi ’s are set to zero. The penalty parameter for admm is chosen as ρ = 103 . Fig. 3 shows the injected active and reactive powers pi , qi in p.u. for aladin and admm over the iteration index k. 5 For aladin, all values converge to their optimal value in around 7 iterations. By contrast, it takes admm 20 iterations for the active power and at least 25 iterations for reactive power to converge to an acceptable level. Fig. 4 shows the progress of the 2-norm of the primal residual Axk 2 , the global objective residual (f (xk ) − f (x∗ )), the 2-norm of the deviation of the decision vector from the optimal value x − x∗ 2 , and the 2-norm of the violation of the first order necessary kkt-conditions for a minimum r = ∇f (xk ) + A λk + ∇h(xk )κk 2 . For admm, the convergence behavior is depicted for three different values of the penalty parameter ρ. For large ρ = 104 , admm reaches a fast consensus but the convergence to the optimal value x∗ is slow. Similarly, the kkt residual decreases slowly. On the other hand, for ρ = 5 · 102 admm
2 0
40
50
30
40
50
2 0 -2
10
20
30
40
50
10
20
k
k
Fig. 3. Convergence of injected active and reactive powers for aladin (left) and admm with ρ = 103 (right). ×104 2
100
Axk 2
We rely on the symbolic framework CasADi (Andersson (2013)) and Matlab R2016a. As mentioned in (Houska et al., 2016, Thm. 2), the penalty parameters ρ and µ have to be large enough to obtain a convergence guarantee for aladin. As in our case large penalization parameters cause numerical problems in the beginning of a simulation, we introduce parameter update rules k k ¯ ρ rρ if ρk < ρ¯ µ rµ if µk < µ k+1 , ρ = µk+1 = k ρk µ
30
k
-2
10-3
ALADIN ADMM
10-6 10
15
ρ = 104
0 -2
5
20
ρ = 103
5
k
xk − x∗ 2
only one generator here for symmetry reasons. Moreover, we consider quadratic cost functions instead of linear ones. Furthermore, shunt elements of the lines and line congestion constraints are neglected.
The nominal values are Vnom = 230 kV and Snom = 100 MVA.
2
k
Fig. 2. Modified ieee 5-node test case (Li and Bo, 2010).
5
4
0 10
3 N2
300MW
2
q [p.u.]
2
200MW
fk − f∗
1
II
q [p.u.]
N1
N2c
4
ρ = 5 · 102
10
15
20
15
20
k
100
103
r
III
600MW
6
p [p.u.]
4
p [p.u.]
5
400MW
10-3
100
10-6 5
10
k
15
20
10-3
5
10
k
Fig. 4. Convergence of primal residual, objective function residual, suboptimality and kkt residual for aladin and admm for ρ ∈ {5 · 102 , 103 , 104 }. takes more iterations, yet the distance to the optimal value x∗ decreases faster. Thus, admm struggles to converge to the exact solution. This can be seen from the rather big gap between admm and aladin in the deviation of the actual solution xk from the optimal value x∗ and especially in the kkt residual (Fig. 4). This behavior is a typical drawback of admm and also observed in other applications of admm (Boyd et al., 2011, Sec. 3.2.2). Hence, the penalty parameter is chosen to ρ = 103 to obtain fast convergence. In comparison, the convergence rate of aladin seems relatively insensitive to the choice of ρ (not shown here due to space limitations). Apart from the existence of convergence guarantees for aladin the question arises, whether the faster convergence justifies the additional cost for communication and computation per step. Note that any attempt to answer this question based on a single case study would be ambiguous and could lead to wrong conclusions. Both aladin and admm are generic optimization algorithms offering the potential
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to scale-up well and work for a large class of power grids. The purpose of this case study is to show that aladin is a promising alternative to admm for solving non-convex ac opf problems. Thus, future work will focus on larger grids, refined models with additional non-convex line congestion constraints, and tailored variants of aladin. 5. CONCLUSIONS & FUTURE WORK This paper investigates the application of aladin to distributed opf problems involving grids of generic topology, i.e. we consider the general case of meshed grids. The advantages of aladin are its convergence guarantees and the expected fast convergence behavior for general non-convex problems. In a comparative case study on a modified ieee 5-bus test case aladin converges faster than commonly employed admm schemes. Future work will evaluate the performance of the algorithm for larger grids and it will involve highly non-convex line congestion constraints. REFERENCES Andersson, J. (2013). A General-Purpose Software Framework for Dynamic Optimization. PhD thesis, Arenberg Doctoral School, KU Leuven, Department of Electrical Engineering (ESAT/SCD) and Optimization in Engineering Center, Kasteelpark Arenberg 10, 3001-Heverlee, Belgium. Arnold, M., Knopfli, S., and Andersson, G. (2007). Improvement of OPF decomposition methods applied to multi-area power systems. In Power Tech, 2007 IEEE Lausanne, 1308–1313. Bai, X., Wei, H., Fujisawa, K., and Wang, Y. (2008). Semidefinite programming for Optimal Power Flow problems. International Journal of Electrical Power & Energy Systems, 30(67), 383 – 392. Bertsekas, D.P. and Tsitsiklis, J.N. (1989). Parallel and Distributed Computation: Numerical Methods. Prentice Hall, NJ. Boyd, S., Parikh, N., Chu, E., Peleato, B., and Eckstein, J. (2011). Distributed optimization and statistical learning via the Alternating Direction Method of Multipliers. Found. Trends Mach. Learn., 3(1), 1–122. Conejo, A.J., Castillo, E., Minguez, R., and Garcia-Bertrand, R. (2006). Decomposition Techniques in Mathematical Programming: Engineering and Science Applications. Springer Science & Business Media, Berlin. Conejo, J.A., Nogales, J.F., and Prieto, J.F. (2002). A decomposition procedure based on approximate Newton directions. Mathematical Programming, 93(3), 495–515. Conn, A., Gould, N., and Toint, P. (1992). LANCELOT: a FORTRAN Package for Large-scale Nonlinear Optimization (Release A). Springer, Berlin. Dall’Anese, E., Zhu, H., and Giannakis, G.B. (2013). Distributed Optimal Power Flow for smart microgrids. IEEE Transactions on Smart Grid, 4(3), 1464–1475. Erseghe, T. (2014a). A distributed algorithm for fast Optimal Power Flow regulation in smart grids. In IEEE International Conference on Smart Grid Communications (SmartGridComm), 2014, 31– 36. Erseghe, T. (2014b). Distributed Optimal Power Flow using ADMM. IEEE Transactions on Power Systems, 29(5), 2370–2380. Erseghe, T. (2015). A distributed approach to the OPF problem. EURASIP Journal on Advances in Signal Processing, 2015(1), 1–13. Gould, N., Orban, D., and Toint, P. (2004). Galahad, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization. ACM Trans. Math. Software, 29(4), 353–372. Guo, J., Hug, G., and Tonguz, O. (2015). Impact of partitioning on the performance of decomposition methods for AC Optimal Power Flow. In Innovative Smart Grid Technologies Conference (ISGT), 2015 IEEE Power Energy Society, 1–5.
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