Computing global optimal solutions of the secondary voltage regulation problem

Optimization problems arising in management of transmission and distribution power grids constitute a broad class of problems with an important characteristic: the power flow constraints. These nonlinear, nonconvex constraints are very challenging for general-purpose optimization solvers, and modeling them with an eye for practical solvability is nontrivial. We consider the secondary Voltage Regulation (VR) problem for a transmission power grid, which consists in determining the set-point of the available voltage controllers, which are described by both continuous and discrete variables, subject to power flow constraints and power demand/supply constraints across the network, while minimizing a given cost function, i.e., the average squared deviation of all voltage magnitudes. Due to the nonconvex constraints, VR falls under the class of Mixed-Integer Nonlinear Optimization (MINLO) problems. We propose several optimization models for VR that can be tackled by exact, general-purpose MINLO solvers, which find a global optimum by resorting to spatial branch-and-bound algorithms. The modeling step is crucial to find global solutions in reasonable time: we study the impact of several modeling techniques on the ability of a MINLO solver to find tight lower bounds (good convex relaxations). We use a local solver for finding feasible solutions in relatively short time. We run our models on a real-world instance arising from an Italian regional transmission power grid, with 205 buses and 232 edges. We discuss the impact of our techniques in finding good solutions and tight lower bounds.