OPTIMAL POWER FLOW (OPF) problems or variants

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1 SUBMITTED 1 Distributed Optimal Power Flow using ALADIN Alexander Engelmann, Yuning Jiang, Tillmann Mühlpfordt, Boris Houska and Timm Faulwasser arxiv:18.863v1 [cs.dc] 3 Feb 18 Abstract The present paper applies the recently proposed Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method to solve non-convex AC Optimal Power Flow Problems (OPF) in a distributed fashion. In contrast to the often used Alternaring Direction of Multipliers Method (ADMM), ALADIN guarantees locally quadratic convergence for AC-OPF. Numerical results for IEEE 5 3 bus test cases indicate that ALADIN is able to outperform ADMM and to reduce the number of iterations by at least one order of magnitude. We compare ALADIN to numerical results for ADMM documented in the literature. The improved convergence speed comes at cost of increasing the communication effort per iteration. Therefore, we also propose a variant of ALADIN that uses inexact Hessians to reduce communication. Finally, we provide a detailed comparison of both variants of ALADIN to ADMM from an algorithmic and communication perspective. Index Terms Distributed Optimization, Optimal Power Flow, OPF, ALADIN, Alternating Direction of Multipliers Method, ADMM. I. INTRODUCTION OPTIMAL POWER FLOW (OPF) problems or variants thereof are employed in many power system contexts to ensure stable and economic system operation. In presence of line congestions for example, OPF problems are used to determine/re-dispatch generator set points. In the German power grid, the number of these re-dispatch events increased drastically in recent years due to the increasing penetration of renewables, phase-out of nuclear plants, and liberalized energy markets [7]. This trend illustrates the increasing importance of efficient and reliable OPF computation in daily grid operation. Whereas in the past, the distribution the grid level was not considered in OPF computations, nowadays this may lead to problems as renewable generation might cause violation of voltage limits and line congestions at the distribution grid level. Including the distribution grid to OPF problems under AC conditions can help to resolve this, yet doing so leads to increased problem size. These observations have triggered significant recent research activity on hierarchical, respectively, distributed algorithms for OPF; i.e. algorithms that split the overall problem into a number of smaller subproblems whose AE, TM and TF are with the Institute for Automation and Applied Computer Science, Karlsruhe Institute of Technology, Germany {alexander.engelmann, tillmann.muehlpfordt, timm.faulwasser}@kit.edu. YJ and BH are with the School of Information Science and Technology, ShanghaiTech University, Shanghai, China {borish, jiangyn}@shanghaitech.edu.cn. BH and TF acknowledge support of this joint research by the Deutsche Forschungsgemeinschaft Grant WO 56/4-1. TF acknowledges funding from the Baden-Württemberg Stiftung under the Elite Program for Postdocs. This work was also supported by the Helmholtz Association under the Joint Initiative Energy System 5 - A Contribution of the Research Field Energy. Manuscript received TBD. parallel solution may or may not be coordinated by a central entity [8]. 1 Given the relevance of solving OPF problems, it is not surprising that there exists a multitude of results on distributed algorithms for OPF under AC conditions; we refer to [8] for a recent overview. In principle, one can distinguish three main lines of research: (i) (ad-hoc) application of algorithms tailored to convex Nonlinear Programs (NLPs) thus in general losing convergence properties [14, 15]; (ii) convex relaxation of OPF by either inner or outer approximation of the feasible set [11, 5]; and (iii) application of distributed algorithms tailored to non-convex NLPs [13, 18]. The present paper will advocate a novel method along line (iii). But before doing so, we concisely review existing results for items (i) (iii). With respect to (i), the set of convex algorithms directly applied to OPF ranges from the Auxiliary Problem Principle [1, 3], the Predictor Corrector Proximal Multiplier Method [] to the popular Alternating Direction of Multipliers Method (ADMM) [14, ]. A number of recent works discusses ADMM in more detail and with different foci: exhaustive simulationbased convergence analysis [14], parameter update rules [15], and applicability to large-scale grids [16]. An advantage of all these methods is, that they usually only exchange primal variables corresponding to linear consensus constraints between the subproblems. However, the convergence rate is at most linear [4, 6]. Recently ADMM convergence results for problems with non-convex objective function of a special form (consensus and sharing problems) have been presented [17]. However, so far it is unclear whether AC OPF fits into that form and, to the best of the authors knowledge, there are no general convergence guarantees for AC OPF using ADMM Another subbranch of (i) proposes Optimality Condition Decomposition (OCD), see [9, 1] and [,, 33] to solve AC-OPF problems. This method aims to solve the first-order necessary conditions including nonlinear coupling constraints without any problem modification in a distributed fashion. In OCD, all subproblems receive primal and dual variables from neighboring regions and consider them as fixed parameters in each local optimization. In [1], a necessary condition for convergence to a first-order stationary point is discussed. However, to the best of our knowledge, it remains unclear whether this condition holds for arbitrary OPF problems [14]. Moreover, in [1], the convergence rate of this method is shown to be linear. With respect to (ii), convex outer approximations of the feasible set via Semi-Definite Programming (SDP) are considered in [3, 11, 9, 34]; i.e. the OPF problem is mapped 1 We remark that in context of numerical optimization for OPF problems the notions of distributed algorithms are not unified: While in the optimization literature distributed algorithms entail a central coordinating entity [4], in context of OPF such schemes are referred to as being hierarchical [8].

2 SUBMITTED to a higher dimensional space wherein it becomes convex whenever a specific rank constraint is dropped. This relaxed and inflated problem can be solved using the above mentioned convex algorithms obtaining convergence guarantees. The crux of SDP relaxations of OPF problems is that the exactness of solutions (in terms of the original non-relaxed OPF problem) can so far only be guaranteed via structural assumptions: either on technical equipment like small transformer resistances or on the grid topology, e.g. radial grids [8, 4, 5, 6]. Finally, research line (iii) considers algorithms with certain convergence guarantees for non-convex problems. This includes approaches based on trust region and alternating projection methods with convergence guarantees at linear rate [18]. A distributed method based on interior point is proposed in [7], where the authors (similar to works on optimal control [31, 35]) decompose certain steps in of a centralized optimization method. Hence, the authors obtain due to equivalence to the corresponding centralized method promising numerical results even for very large cases. The present paper considers the recently proposed Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method [19] which fully exploits the distributed nature of OPF problems. Similar to ADMM, ALADIN performs sequence of local optimizations combine with a coordination step. This coordination step solves an equalityconstrained Quadratic Program (QP) composed of local QP approximations in each iteration. Thereby, all computationally expensive operations (i.e. non-convex minimizations as well as function and derivative evaluations) are performed locally. Moreover, the coordination step is computationally cheap as an equality-constrained QP results in a linear system of equations. Motivated by the fact that ALADIN has locally quadratic convergence properties [19], we proposed its application to OPF in a preceding conference paper [13] presenting results for the IEEE 5 bus problem without line limits. The contribution of the present paper is a more detailed investigation of the prospect of ALADIN for OPF problems. We present results for ALADIN for a series of IEEE test systems ranging from 5 to 3 buses. For the sake of comparison, we explicitly compare to ADMM results presented in [15]. Moreover, we show how inexact Hessians can be used to reduce the communication effort of ALADIN. The remainder of the paper is structured as follows: Section II states the AC-OPF problem. Section III, recalls AC-OPF problem in affinely coupled separable form and revisits the ALADIN algorithm. Extensive numerical case studies for ALADIN and ADMM are discussed Section IV. Finally, Section V compares ALADIN and ADMM in terms of their convergence properties and in terms of their communication effort. Notation Subscripts ( ) k,l describe nodal variables, subscripts ( ) i,j denote local variables, and superscripts ( ) k indicate ALADIN iterates. II. PROBLEM STATEMENT A. AC Optimal Power Flow Introduction Consider an electrical grid at steady state described by the triple (N, G, Y ), where N = {1,..., N} is the bus set, G N is the generator set and Y = G + jb C N N is the bus admittance matrix. Neglecting shunts for simplicity, the entries Y kl = G kl + jb kl of the bus admittance matrix are given by y km, if k = l, Y kl = m N \{k} y kl, if k l, where y kl C is the admittance of the transmission line connecting buses k and l. One bus r N is specified as reference bus for the voltage angles. The AC-OPF problem can be written as the following NLP c 1,k p k + c,k p k + c 3,k, (1a) min θ,v,p,q k G subject to v k v l (G kl cos(θ kl ) + B kl sin(θ kl )) = p k p d k, l N v k v l (G kl sin(θ kl ) B kl cos(θ kl )) = q k qk, d l N p k p k p k, k G, q k q k q k, k G, v k v k v k, k N, v r = 1, θ r =, (1b) (1c) (1d) with c 1,k > and θ kl = θ k θ l. In Problem (1) v k denotes the voltage magnitude, θ k denotes the voltage angle, p k and q k denote the active and reactive power injections, p d k and qk d denote the active and reactive power demands at bus k. Problem (1) aims to minimize the total generation cost subject to the power flow equations (1b), generation and voltage bounds (1c), and the reference constraint (1d). B. AC Optimal Power Flow Separable Reformulation For the sake of completeness, we recall the reformulation of the AC-OPF Problem (1) in affinely coupled separable form amenable to distributed optimization [13]. The IEEE 5-bus test case depicted in Fig. 1 serves as a tutorial example detailing our notation. Due to space limitations, we assume that auxiliary buses connecting two neighboring regions are already introduced and added to the bus set N, cf. [13] for details. By doing so, the bus set is enlarged and different from the bus set used in the previous section. For example, the bus set of the IEEE 5 bus case shown in Fig. 1 enlarges from 5 to 13 buses by including auxiliary buses. The enlarged bus set N is partitioned into R = {1,..., R} local bus sets N i = {n i 1,..., n i N i } N such that we have N i = N, and for all i, j R with i j holds N i N j =. Additionally, we introduce the set of corresponding auxiliary buses A N N that contains all auxiliary bus However, including them poses no significant challenge to the method proposed in the present paper since shunts only change the numerical value of Y kk, cf. [3].

3 SUBMITTED 3 6 p.u p.u. N 1 (1, 11).4 p.u. 1 3 (6, 7) (1, 13) generator consumer N (8, 9) 4 p.u. 1.7 p.u. 3 p.u. 5. p.u. 3 p.u. 4 p.u. Fig. 1. Decomposed IEEE 5-bus test case with three local bus sets N 1 = {1, 5, 6, 1, 1}, N = {, 3, 7, 8}, N 3 = {4, 9, 11, 13} (black), auxiliary bus pairs A = {(6, 7), (8, 9), (1, 11), (1, 13)} (green) and line limits depicted in red. pairs located at borders between two neighboring regions. We assign nodal variables χ k = [ θ k v k p k q k ] R 4 to all buses k N. All nodal variables χ k from region i R are concatenated in local variables x i = [χ... χ ] n i 1 n i N i R ni with n i = 4N i. Doing so, we formulate local objective functions f i : R ni R for all regions i R by f i (x i ) := k G i c 1,k p k + c,k p k + c 3,k, where G i = N i G are local generator sets. The power flow equations (1b) and slack constraints (1d) are formulated as local nonlinear equality constraints h i : R ni R n hi. Finally, we arrive at the affinely coupled separable reformulation of (1): min x s.t. f i (x i ) A i x i = λ, (a) (b) h i (x i ) = κ i i R, (c) x i x i x i η i i R. (d) Here, x = [x 1,..., x R ] R nx stacks the local decision vectors x i and λ, κ i, η i denote the dual variables (multipliers) of the affine coupling constraints, local equality constraints and local bounds respectively. At all auxiliary bus pairs (k, l) A, we enforce consensus of the corresponding auxiliary buses by θ k = θ l, v k = v l, p k = p l, q k = q l. (3) These equalities lead to the affine consensus constraint (b). The box constraints (d) collect local bounds on active/reactive power injections and voltage magnitudes for all regions. Algorithm 1 ALADIN based Distributed OPF Initialization: Initial guess (z, λ ), choose Σ i, ρ, µ, ɛ. Repeat: 1) Parallelizable Step: Solve the local NLPs x k i = arg min x i [x i,x i] f i (x i ) + (λ k ) A i x i + ρk x i zi k Σ i s.t. h i (x i ) = κ k i. (4) ) Termination Criterion: If A i x k i ɛ and x k z k ɛ, (5) return x = x k. 3) Sensitivity Evaluations: Compute local gradients gi k = f i (x k i ), Hessian approximations Bk i {f i (x k i ) + κ i h i(x k i )} and constraint Jacobians Ck i = h i(x k i ). 4) Consensus Step: Solve the coordination QP min x,s s.t. { 1 x i Bi k x i + gi k A i (x k i + x i ) = s λ QP, C k i x i = i R, ( x i ) j = j A k i i R, xi } + (λ k ) s + µk s obtaining x k and λ QP. 5) Line Search: Update primal and dual variables by z k+1 z k + α k 1(x k z k ) + α k x k, λ k+1 λ k + α k 3(λ QP λ k ), (6) with α1, k α, k α3 k from [19]. If full step is accepted, i.e. α1 k = α k = α3 k = 1, update ρ k and µ k by { ρ k+1 (µ k+1 r ρ ρ k (r µ µ k ) if ρ k < ρ (µ k < µ) ) = ρ k (µ k. ) otherwise III. ALADIN BASED DISTRIBUTED OPF Now, we describe a variant of ALADIN for solving Problem () in a distributed fashion. As summarized in Algorithm 1, ALADIN consists of five main steps: 1) Solve the decoupled NLPs (4) in parallel. ) If the solution satisfies the termination criterion (5) (ɛ is choosen by the user), the iteration is terminated with the solution from the local NLPs, x = x k. 3) If not, we compute gradients gi k, Hessian approximations Bi k and constraint Jacobians Ci k. Note that these computations are fully parallelizable. 3 4) We construct the consensus QP (6) based on local sensitivities and the active sets A k i = { j (x k i ) j = x i or x i } 3 In case of derivative based solvers, the sensitivities can be obtained from the local solvers for (4) avoiding explicit evaluation.

4 SUBMITTED Fig.. Convergence of ALADIN (solid) and ADMM (dashed) for the 5-bus system with and without considering line limits. detected by the local NLPs. Note that there are no inequality constraints in the QP (6). Thus, solving this problem is equivalent to solving a linear system of equations yielding a computationally cheap numerical operation [3]. 4 5) Apply the globalization strategy proposed in [19] to update x and λ. We remark that in practice full steps are often accepted and the line search can be omitted. Finally, the parameters ρ and µ are updated. Remark 1 (Inexact Hessians): Instead of computing exact Hessians in Step 3), one can also use approximation techniques for Bi k based on previous gradient evaluations. Here we use the blockwise and damped Broyden-Fletcher-Goldfarb-Shanno (BFGS) update. In contrast to standard BFGS, the damped version ensures positive definiteness of the Bi k s to preserve the convergence properties of ALADIN, cf. [19, 3]. The BFGS update formula is defined by B k+1 i Bi k s k i Bi ksk i = Bi k Bk i sk i sk i + rk i rk i s k i ri k with s k i = xk+1 i x k i and rk i = θk (g k+1 i (λ k+1 ) gi k(λk+1 ))+ (1 θ k )Bi ksk i, where gk i (λ) = ρk (zi k xk i ) A i λ are the gradients of the Lagrangians [3]. The damping parameter θ k is computed by the update rule given in [3, p. 537]. We note that employing BFGS reduces the need for communication within ALADIN. Instead of the Hessian matrix, it suffices to communicate the gradient of the Lagrangian and to perform the BFGS step at the coordination entity that handles the QP (6). IV. NUMERICAL RESULTS The presentation of our results is divided into three parts: First, we show considerable performance differences of AL- ADIN and ADMM for a motivating 5-bus example depicted in Fig. 1. Second, we show that ALADIN performs well for larger grids (3-bus, 57-bus), also when inexact Hessians are used. 4 ALADIN requires the Hessian approximations B k i to be positive definite to ensure convergence [19]. To ensure positive definiteness, we toggle the sign of all negative eigenvalues of all B k i s (regularization). (7) Third, we apply ALADIN to the 118 and 3-bus test cases, and compare our results to variants of ADMM published in the literature. Subsequently, all units are given in p.u. with respect to the base power of 1 MVA. The following quantities are computed a posteriori in order to visualize the convergence behavior of Algorithm 1 (where the superscript k denotes the iteration index): The consensus violation Ax k with A = [A 1,..., A R ] indicating to which extend voltages and powers at auxiliary buses match. The distance to the minimizer x k x indicating how far the current power/voltage iterates are from their optimal value. Thereby, x is the true minimizer obtained by solving () centrally. The inf-norm r k of the dual residual r k = { fi (x k i ) + A i λ k + h i (x k i )κ k i + η k i }, measuring the first-order optimality condition violation. The cost suboptimality f k f illustrating performance losses. We initialize the primal guess for voltage magnitudes to 1 p.u.; all other values are set to zero initially (flat start). All dual variables needed in ALADIN or ADMM are initialized with zero. We compare ALADIN and ADMM in terms of number of iterations and communication effort. 5 Our implementation relies on the CasADi toolbox [1] running with MATLAB R16a and IPOPT [36] as solver for the local NLPs. The true minimizers x are obtained by solving problem () centrally with IPOPT. A. 5-bus with line congestions First, we consider the IEEE 5-bus case including line limits as shown in Fig. 1. We partition the grid into three regions in a way such that it is expected to be hard for distributed 5 We omit investigating computation times since they are highly dependent non-algorithm specific factors like the programming language, code structure, compiler, and executing machine.

5 SUBMITTED Fig. 3. Convergence behavior of ALADIN for the IEEE 3-bus and 57-bus test cases using exact Hessians (solid) and inexact Hessians (dashed, here BFGS). algorithms like ALADIN or ADMM. Specifically, there is a generation center in the west containing cheap generators and no consumers. Hence, all power has to be transferred to consumption centers in the east, which results in a high power exchange between regions. Moreover, line constraints are active at the transmission lines (1, ) and (4, 5) connecting west an east. Several works using ADMM for OPF do not consider line congestions [14, 15]. To the best of the authors knowledge the work [16] is one of the few that explicitly considers line constraints. In the following, line constraints are considered as limits on the magnitude of the apparent power where p kl + q kl s kl, (8) p kl = v kg kl + v k v l (G kl cos(θ k θ l ) + B kl sin(θ k θ l )), q kl = v kb kl v k v l (B kl cos(θ k θ l ) G kl sin(θ k θ l )). Due to non-convexity of these constraints they are difficult to handle; especially when they are located at lines connecting regions. Note that bounds on the apparent power (8) can also be expressed in the framework of (d) by introducing additional decision variables. The values of the tuning parameters for ALADIN are summarized in Table III in the Appendix. We choose the weighting matrices Σ i in Algorithm 1 as diagonal matrices to obtain similar scaling in the consensus variables. The entries for the voltage magnitudes and for voltage angles are chosen as 1, the entries for the power injections are set to 1. Fig. shows the convergence behavior of ALADIN and ADMM considering the IEEE 5-bus case for two settings: In the first setting line limits are neglected while in the second there are line congestions at the lines (1, ) and (4, 5) with apparent power limitations of 4 MVA and 18 MVA respectively, cf. Fig. 1. To foster a fair comparison, the penalty parameters ρ for ADMM are chosen based on parameter sweeps aiming for fast convergence. Without line congestions, ADMM converges slower than ALADIN. However, with slight abuse of optimality and consensus, applicable solutions can be obtained within around 5 iterations assuming that underlying frequency controllers account for the remaining power mismatch. In case of active line limits, ADMM needs at least 15 iterations even for approximate solutions whereas ALADIN converges within 5 iterations to very high accuracy. This indicates the potential of ALADIN to outperform ADMM especially in case of active line limits. While ALADIN converges to high accuracies with a seemingly quadratic rate, see Fig., similar statements cannot be made for ADMM. B. 3-bus and 57-bus with inexact Hessians Next, we compare the performance of ALADIN with exact Hessians to ALADIN with inexact Hessian for larger grids. Specifically, we use approximations based on the BFGS formula (7). Inexact Hessians reduce the per-step communication effort, which is advantageous. The employed grid partitioning for the considered IEEE 3- and 57-bus test cases are taken from [14], and listed in Table IV in the Appendix for selfcontainment. To improve numerical convergence we add a quadratic regularization for the reactive power injection to the local objective functions f i (x i ) = f i (x i ) + γ k N i q k with γ nonnegative in the rest of the paper. This regularization follows the technical motivation to keep reactive power $ injections small. We choose γ = 1 hr (p.u.) which is around 1 % of the quadratic coefficient of the active power injections c 1,k. Fig. 3 depicts the convergence behavior of ALADIN with exact and inexact Hessians. 6 In either case, ALADIN converges 6 The centralized minimizer x is computed here including the regularization into the objective of ().

6 SUBMITTED Fig. 4. Convergence behavior of ALADIN for the IEEE 118 and 3-bus test cases. in less than 4 iterations to high accuracy (at least 1 4 for all convergence criteria). Furthermore, Fig. 3 shows that ALADIN with inexact Hessians needs just slightly more iterations than ALADIN with exact Hessians. One can observe that the convergence rate for ALADIN with inexact Hessians seems to be faster than linear. This observation is consistent with provably superlinear convergence for other methods using BFGS [3]. C. 118-bus and 3-bus ALADIN vs. ADMM For the IEEE 118-bus and 3-bus test cases, we compare our ALADIN results (with exact Hessians) to ADMM results documented in the literature [14, 15]. Supposing that the authors thereof chose the parameters and update rule carefully to ensure fast convergence, this provides a useful comparison. To facilitate comparability, we adopt the grid partitioning from [14] for the 118-bus case. Unfortunately the partitioning for the 3-bus case is not given in [14], such that we choose our own partitioning listed in Table IV. Using ALADIN we obtain the numerical results for the 118- bus and 3-bus case shown in Fig. 4. In either case, ALADIN shows fast convergence to a high level of accuracy for all convergence criteria. In [14, 15] the main convergence criterion is taken to be the infinity norm of the primal gap Ax k < ɛ. Adopting this criterion allows a direct comparison between ALADIN and ADMM results from [14, 15]; for ɛ = 1 4 the results are summarized in Table I. 7 ALADIN converges around one order of magnitude faster while much higher accuracies in optimality are obtained. 7 We remark that primal feasibility does not ensure convergence to a minimizer, cf. [6, Sec ] for an ADMM-specific discussion. This lack of optimality guarantees can be observed seen in the numerical results in [14, 15]. However, in practice small optimality gaps are often accepted. Nontheless one has to be aware of that using Ax k ɛ does not imply convergence of the reactive power injections to the optimal ones since the sensitivity of the objective function value with respect to the reactive power injections is much smaller than the sensitivity to active power injections. This can be verified by comparing the dual variables for active and reactive power injections, cf. [5, Chap. 3..3]. TABLE I COMPARISON OF TUNED ADMM FROM [15] AND ALADIN EMPLOYING Ax k 1 4 AS CONVERGENCE CRITERION ONLY. ADMM [14, 15] ALADIN N # iter f f # iter f f % % % % % % % % V. DISCUSSION ALADIN VS. ADMM Our numerical results from Section IV using ALADIN seem promising. However, compared to ADMM there is an increased per-step communication effort when employing AL- ADIN. Thus, we now turn towards discussing how to trade-off convergence behavior and guarantees versus per-step communication effort. 1) Convergence Properties: ADMM and ALADIN exhibit differences in known convergence guarantees. In case of ADMM, for strictly convex problems, a linear convergence rate can be achieved under rather mild assumptions like Lipschitz continuity of the gradient and regularity assumptions on the affine constraints [1]. In case of convex problems, sublinear convergence is achieved [1]. For the non-convex case, convergence can only be guaranteed for special problem classes, where to the best of the authors knowledge it is not clear whether AC OPF belongs to them [17]. However, this does not mean that ADMM does not work for non-convex OPF. Yet, one has to be aware that ADMM does not necessarily converge to a local minimizer, or converge at all. Nevertheless, in practice, ADMM works well in many cases but often gains slow practical convergence rates especially if high accuracies are needed [6]. This is in accordance with the simulation results from Section V.

7 REFERENCES 7 For ALADIN, convergence can be guaranteed without relying on a convexity assumption of the objective or the constraints [19, Thm. ]. Only mild assumptions w.r.t. the penalty parameter as well as Lipschitz continuity are required. In case of inexact Hessians, superlinear convergence, and in case of exact Hesssians even quadratic convergence can be achieved [19, Sec. 7]. This comes at the cost of an increased per-step communication, and the need for a central coordinating entity that has to solve the coupling QP. ) Communication Effort: The main conceptual difference between ALADIN and ADMM is that ALADIN uses secondorder information whereas ADMM only communicates local primal solutions. More specifically, ALADIN relies on communicating local sensitivities, i.e. g i, B i, C i, A i. for all regions i R, cf. Step 3) of Algorithm 1. The gradients g i are of dimension n i, the (symmetric) Hessians B i of dimension n i n i, and the Jacobians of the power flow equations collected in C i are of dimension (n i /) n i. Note that n i = 4N i, where N i is the number of buses in region i, cf. Section II-B. Additionally, the vector of binaries indicating the active bounds A i, which are of dimension 3ni 4, has to be communicated (bounds on power injections and voltages). Hence, the total forward communication need for ALADIN comprises n i + ni(ni+1) + n i floats and 3ni 4 binaries. The block BFGS updated described in Section III reduces the total communication need as follows. Instead of having to communicate the Hessians H i which lead to the quadratic term ni(ni+1), BFGS requires to communicate only the n i - dimensional gradients of the Lagrangian. Hence, the reduced forward communication need becomes n i + n i + n i floats and 3ni 4 binaries. After solving the QP (6), primal and dual steps for the consensus constraint are broadcast to the subproblems. The number of consensus constraints should typically be smaller than the decision variables since otherwise the original problem might be infeasible. Hence, the number of Lagrange multipliers is upper bounded by n i, and we obtain an upper bound for the backward communication effort of n i + n i floats. For ADMM, only the minimizers of the local problems have to be communicated in both directions. Hence, we obtain equal forward and backward communication need of n i floats. Remark (Floats vs. Binaries): Observe that communicating a binary value is much cheaper than communicating floats (1 bit vs. 3 or 64 bits). Hence, counting the floats is usually sufficient to approximately determine communication effort. Finally, Table II summarizes the results of this section concisely comparing convergence properties, convergence speed and communication effort for ADMM and both ALADIN variants. VI. CONCLUSION & OUTLOOK This paper has investigated the application of the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) to distributed AC OPF problems. The presented numerical results for grids of different sizes underpin the potential of ALADIN for AC-OPF and experimentally validate TABLE II COMPARISON OF ALGORITHMIC PROPERTIES ADMM ALADIN ALADIN BFGS Conv. guarantee no yes yes Exp. conv. rate (linear) quadratic superlinear Forw. comm. (float) n i n i (n i +3) n i (n i +4) Backw. comm. (float) n i n i n i its theoretical properties like a quadratic local convergence rate and convergence guarantees. Compared to ADMM, ALADIN is able to reduce the number of iterations by at least one order of magnitude. This comes at cost of an increased per step communication effort which can be reduced by virtue of using inexact Hessians obtained via BFGS updates. By doing so, we slightly increase the needed number of iterations but ALADIN remains faster and more accurate than ADMM. While the present paper focused primarily on comparing ALADIN with ADMM, a detailed comparison with other distributed schemes would be of interest. Moreover, future work will consider multi-stage OPF problems including storages and generator ramps. From an algorithmic and communication point of view, it would be interesting to reduce the communication effort even more, e.g. by formulating the coordination QP in the coupling variables only. APPENDIX TABLE III PARAMETERIZATION OF ALADIN FOR SHOWN IEEE TEST CASES. [ N ρ ρ r ρ µ µ r µ γ N N i ] $ hr (p.u.) TABLE IV GRID PARTITIONING (EXCLUDING AUXILIARY BUSES). 5 {1, 5}, {, 3}, {4} 3 {1 8, 8}, {9 11, 17, 1, } {4 7, 9, 3}, {1 16, 18-, 3} 57 {4 6, 3 33}, {1, 1, 16, 17, 51}, {8, 9, 11, 41 43, 55 57} {13, 14, 46 5}, {34 37, 39, 4}, {7, 7 9, 5 54}, {19 3, 38, 44}, {1 6, 15, 18, 45} 118 {1 3, , 117}, {33 67}, {68 81, 116, 118}, {8, 11} 3 {1 1}, {11 }, {1 3}

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