Abstract-In this paper, we show that an uncertain departure time significantly changes the analysis in optimizing the charging schedule of electric vehicles (EVs). We also obtain a closed-form solution for the stochastic optimization problem that is formulated to schedule charging of EVs with uncertain departure times in presence of hourly time-of-use pricing tariffs. Index Terms-Closed-form solution, electric vehicles, stochastic optimization, time-shiftable loads, uncertain departure time. A SSUME that time is divided into T time slots. Consider an electric vehicle (EV) charging station with multiple charging outlets (see [1] , [2] ). At each charging outlet, let α ≥ 1 denote the time slot at which an EV plugs in to the charging outlet. Also let e denote the total energy needed to charge the EV. 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 A.M. and the driver indicates that she will depart at 2:00 P.M. 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)  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.  Next, 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 depart at the end of the current time slot, then we must set because 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 Pr{β = k | β > t} denote the conditional probability that the user will depart in a future time slot k, knowing that it will not depart by the end of the current time slot t. We have T k=t+1 π k|t = 1. The expected cost of electricity to finish charging is calculated as where the equality is due to the departure-time requirement in (2). The question is: how can we minimize the above cost? Answering the above question is trivial at t = T, because from (2), we have no choice but setting x[T] = e − X [T]. However, to find the answer for t = α, . . . , T − 1, we need to first define a new notation φ [t] Theorem 1: At the beginning of time slot t = α, . . . , T − 1, if the user indicates that it will be departing at the end of
Meanwhile, Mohsenian-Rad and Ghamkhari REF considers the case where the departure times of EVs are uncertain.
Optimal Charging of Electric Vehicles With Uncertain Departure Times: A Closed-Form Solution
IEEE Transactions on Smart Grid
IEEE Transactions on Smart Grid
Engineering Computer Science