**Charging scheduling of single Electric Vehicles**

In this subsection, we present a generic formulation of the scheduling problem of single EV charging. Here we assume the state of charging (SOC) is linear in the power consumptions and discharging is forbidden. Since the real-time electric price is released every 5 minutes or 15 minutes, we assume time is slotted2 and indexed by t. Assume the scheduler knows the charging cost and the EV is attached to the charger from t = 0 to t = T − 1. The scheduling problem can be formulated as a optimization problem as follows.

where u[t] is the charging rate of the EV within time slot t, (·) the charging cost as a function of the charging rate, j the required charging amount, T the plug-in duration (or the lead time on arrival), and the maximum charging rate. In general, the charging cost function (·) would include the energy cost of purchasing power, the cost offset by the renewable energy, and the regulation price. If the cost function is linear, i.e., (u[t]) = c[t]u[t], the problem (2) becomes a linear program, which can be easily solved by “Simplex” method or other commercial solver.

Now we present some characteristics of the optimal solution of (2.2) under the linear cost assumption. In this case, assume there exists an optimal charging profile {u[t]} = {u[0], · · · , u[T − 1]} and u[], u[] ∈ {u[t]} such that 0 < u[], u[] < . It follows that

If the sum of power, (u[]+u[]), is kept constant, then this becomes a linear equation with respect to u[]. The extreme occur only at both ends of the range of u[]. Thus there exists an optimal scheduling such that u[t] should be either 0 or . This result implicates that charging control should be an on/off control at the maximum charging rate to minimize the charging cost.In (2), the future charging cost function (·) is assumed known by the scheduler. However, the cost is in general stochastic. The charging cost is basically the electricity price offset by the local renewable energy. The electricity price is determined by the operation status of the power grid and the demand.

The renewable energy is significantly affected by the weather. The randomness is one of the major difficulties in the scheduling problem. Different approaches have been proposed to deal with the randomness in charging cost, EV arrival and charging capacity, including dynamic programming, Markov decision processes, model predictive control, and so on. Some of these approaches will be discussed in the following chapters.