An exact relaxation of AC-OPF problem for battery-integrated power grids

Renewable energy resources and power electronics-interfaced loads introduce fast dynamics in distribution networks. These dynamics cannot be regulated by slow conventional solutions and require fast controllable energy resources such as Battery Energy Storage Systems (BESSs). To compensate for the high costs associated to BESSs, their energy and power management should be optimized. In this paper, a convex iterative optimization approach is developed to find the optimal active and reactive power setpoints of BESSs in active distribution networks. The objective is to minimize the total cost of energy purchase from the grid. Round-trip and life-time characteristics of BESSs are modelled accurately and integrated into a relaxed and exact formulation of the AC power flow, resulting into a Modified Augmented Relaxed Optimal Power Flow (MAROP) problem. The feasibility and optimality of the solution under the grid security limits and technical constraints of BESSs is proven analytically. A 32-bus IEEE test benchmark is used to illustrate the performance of the developed approach in comparison to the alternative approaches existing in the literature

Global Optimisation for Energy System

The goal of global optimisation is to find globally optimal solutions, avoiding local optima and other stationary points. The aim of this thesis is to provide more efficient global optimisation tools for energy systems planning and operation. Due to the ongoing increasing of complexity and decentralisation of power systems, the use of advanced mathematical techniques that produce reliable solutions becomes necessary. The task of developing such methods is complicated by the fact that most energy-related problems are nonconvex due to the nonlinear Alternating Current Power Flow equations and the existence of discrete elements. In some cases, the computational challenges arising from the presence of non-convexities can be tackled by relaxing the definition of convexity and identifying classes of problems that can be solved to global optimality by polynomial time algorithms. One such property is known as invexity and is defined by every stationary point of a problem being a global optimum. This thesis investigates how the relation between the objective function and the structure of the feasible set is connected to invexity and presents necessary conditions for invexity in the general case and necessary and sufficient conditions for problems with two degrees of freedom. However, nonconvex problems often do not possess any provable convenient properties, and specialised methods are necessary for providing global optimality guarantees. A widely used technique is solving convex relaxations in order to find a bound on the optimal solution. Semidefinite Programming relaxations can provide good quality bounds, but they suffer from a lack of scalability. We tackle this issue by proposing an algorithm that combines decomposition and linearisation approaches. In addition to continuous non-convexities, many problems in Energy Systems model discrete decisions and are expressed as mixed-integer nonlinear programs (MINLPs). The formulation of a MINLP is of significant importance since it affects the quality of dual bounds. In this thesis we investigate algebraic characterisations of on/off constraints and develop a strengthened version of the Quadratic Convex relaxation of the Optimal Transmission Switching problem. All presented methods were implemented in mathematical modelling and optimisation frameworks PowerTools and Gravity