Chance-Constrained Distributed Optimization with Shared States

Interconnected systems have attracted significant attention in numerous engineering applications such as energy, water, and oil, as well as gas distribution networks. However, due to their high complexity, it is enormously difficult to ensure a reliable and effective cooperation of such interconnected systems under limited communication and interaction capacities. In addition, uncertainty consideration poses further challenges in developing an efficient distributed optimization approach to such interconnected systems. Furthermore, satisfying the constraints of shared states between subsystems under uncertainty leads to conflict issues and has not been properly studied yet. This study proposes a chance-constrained distributed optimization approach to the optimal operation of interconnected systems by considering conflicting reliability levels of satisfying shared state constraints. A compromised reliability level for such constraints is determined by an averaged weighting. We establish that the optimal cost is Lipschitz-continuous with respect to the compromised reliability level, providing a theoretical basis for quantifying the marginal cost of reliability, and we show that the compromise is robust to perturbations in the subsystem weights. We develop a numerical solution framework based on the inner–outer approximation method. For the efficient computation of the high-dimensional integrals, we use Hoeffding’s inequality to determine a suitable sample size. The optimal operation of three interconnected energy distribution networks is used as a case study to demonstrate the proposed approach.