A Demand-Response Calculus with Perfect Batteries

We consider an electricity consumer equipped with a perfect battery, who needs to satisfy a non-elastic load, subject to external control signals. The control imposes a time-varying upper-bound on the instantaneous energy consumption (this is called ``Demand-Response via quantity"). The consumer defines a charging schedule for the battery. We say that a schedule is feasible if it successfully absorbs the effects of service reduction and achieves the satisfiability of the load (making use of the battery). Our contribution is twofold. (1) We provide explicit necessary and sufficient conditions for the load, the control, and the battery, which ensure the existence of a feasible battery charging schedule. Furthermore, we show that whenever a feasible schedule exists, we can explicitly define an online (causal) feasible schedule. (2) For a given arrival curve characterizing the load and a given service curve characterizing the control, we compute a sufficient battery size that ensures existence of an online feasible schedule. For an arrival curve determined from a real measured trace, we numerically characterize the sufficient battery size for various types of service curves.