Tractable Reserve Scheduling Formulations for Alternating Current Power Grids with Uncertain Generation

Abstract

The increasing penetration of wind power generation introduces uncertainty in the behaviour of electric power grids. This work is concerned with the problem of day-ahead reserve scheduling (RS) for power systems with high levels of wind power penetration, and proposes a novel set-up that incorporates an alternating current (AC) Optimal Power Flow (OPF) formulation. The OPF-RS problem is non-convex and in general hard to solve. Using a convex relaxation technique, we focus on systems with uncertain generation and formulate a chance-constrained optimization problem to determine the minimum cost of production and reserves. Following a randomization technique, we approximate the chance constraints and provide a-priori feasibility guarantees in a probabilistic sense. However, the resulting problem is computationally intractable, due to the fact that the computation time complexity grows polynomially with respect to the size of the power network and scheduling horizon. In this thesis, we first use the so-called scenario approach to approximate a convex set which contains almost surely the probability mass distribution of underlying random events. We rely on the special property of reserve scheduling problems which leads to linear constraint functions with respect to the uncertain parameters. We can therefore formulate a robust problem for only the vertices of the approximated set. Using the proposed approach, the number of scenarios is reduced significantly which is beneficial for the tractability. Such a formulation requires the power network state to only be feasible for all vertices of the convex approximated set. To even further relax such a requirement, we develop a novel RS formulation by considering the network state as a non-linear parametrization function of the uncertainty. By using a conic combination of matrices, only three positive semidefinite constraints per time step are considered. Unlike existing works in RS, our proposed parametrization has a practical meaning and is directly related to the distribution of reserve power. Such a reformulation yields a reduction in computational complexity of OPF-RS problems. Finally, we extend our results to a more realistic size of power grids, using sparsity pattern and spatiality (multi-area) decomposition of the power networks, leading to a decomposed semidefinite programming (SDP) problem. To solve the SDP in a distributed setting, we formulate a distributed consensus optimization problem, and then the alternating direction method of multipliers (ADMM) algorithm is employed to coordinate local OPF-RS problems between neighbouring areas. The theoretical developments in aforementioned cases were validated on a realistic benchmark system and a discussion on the tractability of the resulting optimization problems by means of computational time analysis is presented.