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Optimal price rebates for demand response under power flow constraints

Demand side participation is essential for a realtime energy balance in today's electricity grid, especially in the presence of highly uncertain renewable sources of energy such as wind and solar. In this paper, we consider a price rebate approach for demand response where an electric utility company can offer real-time price rebates to the consumers to reduce the load consumption. We study the problem of computing near-optimal prices (or rebates) to offer to the customers to reduce the consumption in the presence of power flow constraints and transmission losses on the distribution grid. To the best of our knowledge, this is the first work that optimizes demand response under an AC power flow model. The main challenge in this problem arises from the non-convexity of AC power flow constraints and also the uncertainty in price elasticity of the demand. We formulate an SDP based convex relaxation of the power flow constraints and give an iterative procedure to compute the offer prices to minimize the total expected cost. We conduct numerical experiments to compare the performance of our heuristic with other optimization approaches including using DC power flow model or no power flow model at all. Our computational study shows that the performance of our AC power flow based heuristic is significantly better than the other approaches. Unlike the DC power flow constraints, the AC power flow constraints model transmission losses. Therefore, we can optimize the offer prices based on the topology of the grid and leverage both the actual load reduction as well as the reduction in the transmission losses.