Optimal Charging of Electric Vehicles With Uncertain Departure Times: A Closed-Form Solution

Abstract

Abstract-In this paper, we show that an uncertain departure time significantly changes the analysis in optimizing the charging schedule of electric vehicles. We also obtain a closed-form solution for the stochastic optimization problem that is formulated to schedule charging of electric vehicles with uncertain departure times in presence of hourly time-of-use pricing tariffs.Keywords: Electric vehicles, uncertain departure time, timeshiftable loads, stochastic optimization, closed-form solution. I. BACKGROUNDAssume that time is divided into T time slots. Consider an electric vehicle (EV) charging station with multiple charging outlets, c.f. [1], [2]. At each charging outlet, let α ≥ 1 denote the time slot at which an electric vehicle plugs in to the charging outlet. Also let e denote the total energy needed to charge the electric vehicle. In this section, we assume that the driver indicates her departure time β ≤ T . As an example, if a Chevrolet Volt with a depleted battery plugs in to the charging outlet at 10:00 AM and the driver indicates that she will depart at 2:00 PM and she expects a fully charged battery, then we have α = 10, β = 14, and e = 16 kWh. Let x[t] denote the scheduled energy usage for charging at time slot t. To finish charging before the departure time β, it is required that(1)Let µ[t] denote the price of electricity at time slot t. The cost to finish charging is. If all parameters are known, then the cost is minimized if we choose x[τ ] = e, where τ = arg min α≤t≤β µ[t]; and we set x[t] = 0 for t = τ . Here, τ is the time slot between α and β with the lowest price. Note that, the above analysis works at each charging outlet, regardless of the number of charging outlets at the charging station. An exception is when the charging station is very large and thus price-maker, as we point out in Section IV. II. ANALYSIS UNDER UNCERTAIN DEADLINENext, assume that the deadline β is not known. Once an EV plugs in, its target charge level e and start time α are identified. However, the charging station may not know when the EV will depart. Nevertheless, the charging station needs to operate in a way that it minimizes the cost of electricity.The system setup is as follows. At the beginning of each time slot t ≥ α, if the user indicates that the EV will departThe authors are with the Department of Electrical Engineering, University of California, Riverside, CA, USA, e-mail: {hamed, ghamkhari}@ee.ucr.edu. This work was supported by NSF grants ECCS 1253516, ECCS 1307756, and CNS 1319798. The corresponding author is H. Mohsenian-Rad.at the end of the current time slot, then we must setbecause we must assure finishing charging before the user's departure as required by (1). Otherwise, i.e., if the user does not indicate that it will depart at the end of the current time slot t, then the charging station faces the uncertainty that the EV may depart at any of the future time slots t + 1, . . . , T . At each time slot t ≥ α, let π k|t P r{β = k | β > t} denote the conditional probability that the user will depa...

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