An exact relaxation method for complementarity constraints of energy storages in power grid optimization problems

The increasing deployment of energy storages in power grid necessitates the consideration of their operational costs and constraints. However, energy storages introduce complementary constraints or binary variables, make the optimization problems non-convex and challenging to solve. To tackle this issue, we propose a generalized relaxation condition (GRC), which possesses a priori property. As long as the GRC is met, it is guaranteed that there is no simultaneous charging and discharging (SCD) in the relaxed optimization problems excluding the complementarity constraints. Moreover, we prove that the exact relaxation region on the locational marginal prices (LMPs) formed by the proposed GRC contains those of the other existing relaxation conditions. The proposed GRC can be satisfied for the majority of electricity price scenarios, including positive prices and mild negative prices. Beyond that, through introducing penalties for charging and discharging losses of storage into the objective function, the proposed GRC can be further ensured in all price scenarios. We also develop a criterion to set the penalty parameter properly, making the GRC satisfied. Afterward, a practical relaxation method is presented based on the proposed GRC. Finally, we propose a practical two-stage penalty parameter adjustment scheme to weaken the dependence of the proposed relaxation method on the prediction accuracy of the LMPs. The proposed exact relaxation condition is applicable to various scenarios, such as storage-participating economic dispatch, joint energy-reserve optimization and Volt/VAR optimization. Finally, comprehensive test cases verify the exactness and advantages of the proposed relaxation method.