Distributed optimization of the stochastic load of residential heat pumps for demand response

• A stochastic integrated system-level optimization problem is developed. • An ADMM-based solution strategy is proposed. • The demand for energy as well as for reserve capacity are simultaneously optimized. • The load uncertainty is revealed and controlled ahead of real time. • System-level cost reductions are achieved while guaranteeing thermal comfort. The main aim of this paper is to illustrate the added value at system level of explicitly accounting for the closed-loop feedback aspect in the open-loop optimal control problem of model predictive control strategies for thermostatically controlled loads participating in demand response programs, when subject to uncertainties. To this end, an integrated system-level optimization problem is set up, merging an economic dispatch problem to represent the supply side and an open-loop stochastic optimal control problem incorporating affine disturbance feedback to represent the demand side. The incorporation of affine disturbance feedback enables the simultaneous optimal scheduling of the demand for electrical energy, reserve capacity and real-time flexibility required to guarantee thermal comfort at the demand side, thereby disclosing very valuable information for an aggregator or system operator, since the load uncertainty can be revealed and controlled ahead of real time. To solve the mathematically complex integrated problem, a distributed solution strategy based on the alternating direction method of multipliers is developed. With the help of an illustrative case study, it is demonstrated that the day-ahead coordination of the demand for reserve capacity in addition to the energy demand is able to reduce the system operating cost while guaranteeing thermal comfort, and hence, enables a more cost-efficient electrification of the residential heating sector. Cost reductions up to 10.7 % are shown to be achievable for a demand side consisting of 900 000 flexible heat pumps combined with low-temperature radiators.