Multi-parametric programming with constraint relaxation for the optimal operation of micro-grids integrating renewables

Motivated by the integration of distributed renewable energy sources in the electrical grid, we address the optimal operation of a micro-grid that has to adhere to a daily power exchange profile agreed in advance with the grid, by compensating local deviations that may occur along the day using dispatchable generators, loads, and storage units. We propose a solution that is based on explicit Model Predictive Control (eMPC) with constraint relaxation. More specifically, we formulate the problem as a multi-parametric quadratic program (mp-QP) whose solution is the optimal power to be dispatched expressed as a function of a parameter vector representing the actual power consumption/generation profile, which can then be assessed during the day via reliable predictions. Preferred operating regions for the involved dispatchable units are encoded in the mp-QP by introducing soft constraints, which are relaxed using an exact penalty approach when determining the eMPC solution so that they are violated but only when needed to recover feasibility. Performance of the proposed approach is shown on a self-consumption numerical example.