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Accelerated Methods for the SOCP-Relaxed Component-Based Distributed Optimal Power Flow

The alternating current (AC) optimal power flow (ACOPF) problem consists of finding the least-cost dispatch of power from generators to satisfy the load at all buses in a way that is governed by physical laws, such as Ohm's Law and Kirchhoff's Law, and other technical restrictions, such as transmission line thermal limit constraints. The ACOPF is a core element of power system operation, which compels the electricity industry to seek efficient methods to solve it. Therefore, the research community has focused on improving interior-point methods (IPM) to compute feasible solutions efficiently. However, these methods are not privacy preserving and fail to scale on applications that require discrete variables. To this end, an increased attention is given to distributed methods as they can be scalable and privacy preserving.

There is a plethora of existing works on distributed ACOPF. These can be broadly classified into three categories, dual decomposition methods, optimality conditions decomposition (OCD) methods and sparse semidefinite programming (SDP) decomposition methods. The dual decomposition techniques underlying the dual-decomposition-based distributed ACOPF methods in the literature can in turn be classified into two categories: region-based decompositions and component-based decompositions. The focus of this study revolves around the latter decomposition techniques because they distribute the computation across every component in the network (generators, transformers, loads, buses, lines etc.) and are flexible enough to incorporate discrete decision variables to suit a wide variety of optimization applications in power systems and future grids. On the downside, dual-decomposition-based ACOPF methods have no theoretical guarantee of convergence because the (primal) ACOPF problem is nonconvex. Nonetheless, in our previous work in [1], we showed that fine-tuning different penalty parameters for each consensus constraint can result in a convergence to near-optimal (most likely globally optimal) solutions.

Against this background, and motivated by the electricity industry's real-time decision-making applications, this paper investigates three algorithms for accelerating the convergence of component-based distributed ACOPF. In more detail, the first algorithm is inspired by the fast (alternating direction method of multipliers) ADMM with restart method proposed in [2] for weakly convex problems, which is itself an adaptation of an accelerated gradient-descent method initially proposed by Nesterov. However, this method requires a central authority as the restart rule relies on a combined residual, which measures both the primal and dual errors simultaneously. The second algorithm is the over-relaxed ADMM analyzed in [3]. This method is fully decentralized. The third algorithm uses the bundle method [4] to update the Lagrange multipliers (dual variables). The bundle method, which is a variant of the proximal cutting plane method, discards old cutting planes (subgradients) to reduce the size of the (QP) quadratic program solved at each iteration. In this work, we show that only two cutting planes per iteration are enough to substantially speed-up the convergence of ADMM. However, in contrast to the second method, the bundle method is not fully decentralized as it requires collecting the values of each dual subproblem in order to compute the next dual point. More interestingly, since the convergence of ADMM is highly sensitive to the choice penalty parameters, the convergence of the three algorithms is further improved by incorporating adaptive consensus ADMM in which the penalty parameters are automatically tuned without a central oversight [5]. Specifically, adaptive consensus ADMM sets the penalty parameters at each node (bus) by estimating the local curvature of the dual subproblems, in the aim of automatically balancing the residual magnitudes. This method is inspired by the Barzilai-Borwein spectral step-size estimation, which yields a fast convergence and relative insensitivity to the initial stepsize and ill-conditioning.

Finally, we adopt an alternative ACOPF formulation in which the nonconvexties are contained only in the constraints which model the steady-state physics of power flows across transmission lines. As a result, this formulation relishes closed-form solutions for the buses and generators subproblems, when decomposed in terms of components. Problems with closed-form solutions are faster to compute compared to when they are solved using a numerical solver. Furthermore, we leverage the flexibility of the alternative ACOPF formulation to demonstrate the three methods on cases where the nonconvex line subproblems are relaxed into (i) second-order cone and (ii) semidefinite programs, respectively.

Author(s):

Sleiman Mhanna    
The University of Sydney
Australia

Gregor Verbic    
The University of Sydney
Australia

Archie Chapman    
The University of Sydney
Australia

 

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