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Publication# AC OPF in radial distribution networks – Part I: On the limits of the branch flow convexification and the alternating direction method of multipliers

Konstantina Christakou, Jean-Yves Le Boudec, Mario Paolone, Dan-Cristian Tomozei

*Elsevier, *2017

Article

Article

Résumé

The optimal power-flow problem (OPF) has always played a key role in the planning and operation of power systems. Due to the non-linear nature of the AC power-flow equations, the OPF problem is known to be non-convex, therefore hard to solve. During the last few years several methods for solving the OPF have been proposed. The majority of them rely on approximations, often applied to the network model, aiming at making OPF convex and yielding inexact solutions. Others, kept the non-convex nature of the OPF with consequent increase of the computational complexity, inadequateness for real time control applications and sub-optimality of the identified solution. Recently, Farivar and Low proposed a method that is claimed to be exact for the case of radial distribution systems under specific assumptions, despite no apparent approximations. In our work, we show that it is, in fact, not exact. On one hand, there is a misinterpretation of the physical network model related to the ampacity constraint of the lines’ current flows. On the other hand, the proof of the exactness of the proposed relaxation requires unrealistic assumptions and, in particular, (i) full controllability of loads and generation in the network and (ii) no upper-bound on the controllable loads. We also show that the extension of this approach to account for exact line models might provide physically infeasible solutions. In addition to the aforementioned convexification method, recently several contributions have proposed OPF algorithms that rely on the use of the alternating direction method of multipliers (ADMM). However, as we show in this work, there are cases for which the ADMM-based solution of the non-relaxed OPF problem fails to converge. To overcome the aforementioned limitations, we propose a specific algorithm for the solution of a non-approximated, non-convex OPF problem in radial distribution systems. In view of the complexity of the contribution, this work is divided in two parts. In this first part, we specifically discuss the limitations of both BFM and ADMM to solve the OPF problem.

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Typical optimal controls of power systems, such as scheduling of generators, voltage control, losses reduction, have been so far commonly investigated in the domain of high-voltage transmission networks. However, during the past years, the increased connection of distributed energy resources (DERs) in power distribution systems results in frequent violations of operational constraints in these networks and has raised the importance of developing optimal control strategies specifically applied to these systems. In particular, two of the most important control functionalities that have not yet been deployed in active distribution networks (ADNs) are voltage control and lines congestion management. Usually, this category of problems has been treated in the literature by means of linear approaches applied to the dependency between voltages and power flows as a function of the power injections. On the one hand, recent progress in information and communication technologies, the introduction of new advanced metering devices such as phasor measurement units and the development of real-time state estimation algorithms present new opportunities and will, eventually, enable the deployment of processes for optimal voltage control and lines congestion management in distribution networks. On the other hand, ADNs exhibit specific peculiarities that render the design of such controls compelling. In particular, it is worth noting that the solution of optimal problems becomes of interest only if it meets the stringent time constraints required by real-time controls and imposed by the stochasticity of DERs, in particular photovoltaic units (PVs), largely present in these networks. Moreover, control schemes are meaningful for implementation in real-time controllers only when convergence to an optimal solution is guaranteed. Finally, control processes for ADNs need to take into account the inherent multi-phase and unbalanced nature of these networks, as well as the non-negligible R/X ratio of longitudinal parameters of the medium and low-voltage lines, together with the influence of transverse capacitances. Taking into consideration the aforementioned requirements, the distribution management systems (DMSs) need to be updated accordingly in order to incorporate optimization processes for the scheduling of the DERs. This chapter starts with a general description of a centralized DMS architecture that includes voltage control and lines congestion management functionalities. Then, the formulation of the corresponding optimal control problems is described, based on a linearized approach linking control variables, e.g., power injections, transformers tap positions, and controlled quantities, e.g., voltages, current flows, by means of sensitivity coefficients. Computation processes for these sensitivity coefficients are presented in Sections 8.2 and 8.3. Finally, in Section 8.4, we provide case studies of optimal voltage control and lines congestion management targeting IEEE distribution reference networks suitably modified to integrate distributed generation.

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2004Konstantina Christakou, Jean-Yves Le Boudec, Mario Paolone, Dan-Cristian Tomozei

The optimal power-flow problem (OPF) has always played a key role in the planning and operation of power systems. Due to the non-linear nature of the AC power-flow equations, the OPF problem is known to be non-convex, therefore hard to solve. Most proposed methods for solving the OPF rely on approximations (e.g., of the network model) that render the problem convex, but that consequently yield inexact solutions. Recently, Farivar and Low proposed a method that is claimed to be exact for the case of radial distribution systems under specific assumptions, despite no apparent approximations. In our work, we show that it is, in fact, not exact. On one hand, there is a misinterpretation of the physical network model related to the ampacity constraint of the lines’ current flows and, on the other hand, the proof of the exactness of the proposed relaxation requires unrealistic assumptions related to the unboundedness of specific control variables. Therefore, there is a need to develop algorithms for the solution of the non-appproximated OPF problem that remains inherently non- convex. Recently, several contributions have proposed OPF algorithms that rely on the use of the alternating-direction method of multipliers (ADMM). However, as we show in this work, there are cases for which the ADMM-based solution of the non-relaxed OPF problem fails to converge. To overcome the aforementioned limitations, we propose a specific algorithm for the solution of a non-approximated, non-convex OPF problem in radial distribution systems.

2015