Battery storage, demand charges, multi-objective optimization, solar power
In this paper, we propose an optimization framework for optimal energy storage, in the form of batteries, by residential customers. Our goal is to determine the value of battery storage to those customers whose electricity bills consist of both Time-of-Use charges ($/kWh, with different rates for on-peak and off-peak hours) and demand charges ($/kW, proportional to the peak rate of consumption in a month). The customers may have access to a local power generating source in the form of solar PhotoVoltaic (PV). In order to quantify the benefits from the battery storage, we pose a battery optimization problem which minimizes the monthly electricity bill 30 poffÂ âˆ‘kâˆˆoff q(k)Î”t +Â 30 pon âˆ‘kâˆˆon q(k)Î”t + pd supkâˆˆon q(k), where poff, pon,Â pd are the off-peak, on-peak and demand prices, and q(k) is the power delivered by the utility company to the customer. We consider this power to be used according to q(k) = qb(k) + qa(k) - qsolar(k), where qa is the power consumed by the appliances, qsolar is the power provided by the solar PV, andÂ qb is the power given to or taken from the battery. We assume that the rate of the energy stored in the battery is proportional toÂ qb and the stored energy is bounded by the batteryâ€™s capacity (kWh). Furthermore, we account for the battery degradation by modeling the batteryâ€™s capacity as a function of the number of charging/discharging cycles and the depth of discharge. Because of the presence of demand charges (supk q(k)), the objective function of our battery optimization problem is not separable in time - a property (time separability) which is a sufficient for the dynamic programming algorithm to converge to an optimal solution. We establish a provably convergent algorithm for the non-separable optimization problem in the following two steps. First, we replaceÂ supk q(k) in the objective function using the following approximation Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â supkâˆˆon q(k) = q(k) lâˆž = (âˆ‘kâˆˆon q(k)p)1/p for some large p. Then, we construct a multi-objective problem (a class of optimization problems involving at least two objective functions to be minimized simultaneously) defined by a parameterized set of dynamic programs expressed in terms of the time-separable functions J1(q) = âˆ‘k q(k) J2(q) = âˆ‘kâˆˆon q(k)p Each of these parameterized dynamic programs can be solved using the standard dynamic programming algorithms. The set of solutions to these parameterized problems form a Pareto front - a set which is guaranteed to contain the solution to the original battery optimization problem as p â†’ âˆž. We apply our algorithm to multiple scenarios described by a range battery sizes, solar generation levels and appliances loads to quantify the savings from the batteries for a wide range of residential customers. The proposed approach can be potentially used to: 1) Model customers response to changes in electricity prices; 2) Quantify the benefits of energy storage to utility companies.