A two-level off-grid electric distribution problem on the continuous space

We introduce a two-level distribution network design problem to serve a set of demand points. At the higher level, primary facilities with source capabilities feed secondary facilities over tree networks. At the lower level, both the primary and secondary facilities serve customers within a coverage distance over star networks. The problem has wide applications especially in the spatial planning of energy distribution networks, where the coverage distance constraints may be associated with the loss of electric power over distance. The facilities can be located anywhere on the continuous space in our greenfield development problem. We formulate an optimization problem that determines the number, types, and locations of the facilities as well as the lower and higher-level networks to minimize a distribution cost. We also propose a heuristic solution method that solves a discrete counterpart and then improves its solution by moving facilities around on the continuous space. When the number of demand points is large, the discrete counterpart is also hard to solve. Therefore, we propose a decomposition method for its solution: We first solve a decentralized single-level problem with secondary facilities and then form the higher-level network. We perform numerical experiments to demonstrate the benefits of our discrete problem decomposition and facility location adjustments on the continuous space.