In this section, three strategies to attack the EV charging station problem are discussed:
In the eMPC and OCCF strategies, uncertain parameters such as EVs’ initial SoC at the arrival and their arrival times are considered. Both strategies are based on MPC formulations. The MPC strategy (also known as Receding Horizon Control) makes explicit use of a plant model to obtain the optimal control signal by minimising an objective function. MPC exploits forecast values together with new information to establish the future evolution of the system, handling the constraints in an efficient way. The main advantages of the MPC strategy are: (i) it introduces feed-forward control implicitly, to compensate disturbances and measurement noise rejection; (ii) it is not conceptually complex to treat the constraints over inputs, states, outputs and slew-rate variables; (iii) single-variable and multi-variable cases are easily treated; and (iv) it is appropriate to address single-objective and multi-objective control, and signal following.
3.2. Economic Model Predictive Control
A novel formulation for the cost minimisation strategy based on a MPC algorithm is presented in this section. The MPC path to the solution of the charging station problem looks for adjustments in the injected power
, at every time slot
k, in each charger
i, considering the time-variant energy prices
, the uncertainty in the EV arrival time and SoC at the arrival time, i.e.,
and
. Hence, the optimality here refers to a charging profile that minimises the EVCS operation costs, while guaranteeing for all EV
the minimum
at the departure time
. Thus, by recalling Equation (
3), Equation (
4) and Equation (
7), it holds:
Notice that the dynamic constraint considers the reported arrival time
in the request, and a random initial state
given with a uniform distribution between the minimum
and the SoC at the request time
. Therefore, the actual arrival SoC is lower than the one at the request, i.e.,
. In addition, the actual arrival time
is known within the time interval previous to the connection of the EV to the charger.
In Equation (11), the aggregator problem is an optimal control strategy in open loop. Specifically, for the kth time slot, is the state variable corresponding to the ith charger SoC, while is the commanded variable corresponding to the delivered power profile. It is worthwhile to notice that the problem parameters in Equation (11) are affected by uncertainties on arrival time and SoC of the EVs. An open loop strategy cannot consider the future unknown behaviors, leading to possible unfeasibilities in the optimal solution. Therefore, the solution strategy should take these uncertain behaviors into account, and possibly recompute the control signal, even at each time step if needed. For this reason, a model predictive control strategy is proposed, following the receding horizon principle. In a nutshell, the idea is to compute, at time k, an optimal control sequence, over the complete time-interval, e.g., , taking into account the current and future constraints. Nevertheless, only the first step in the resulting optimal control sequence is applied. Then, in the next time slot , once the chargers’ information is updated with the new measures, the aggregator recomputes the sequences, thus iterating the process.
Thereupon, in the eMPC problem framework, finding the problem solution requires to analyse, at each time slot, the system dynamics, and the future energy prices, while taking into account the current SoC and the arrival and departure times for each vehicle, as per Equation (11). Notice that both uncertain parameters (the arrival time and the initial SoC) are revealed when the EV arrives at EVCS and is plugged in. After this, there is no more uncertainty in the EV state, leading to an appropriate power profile schedule. As indicated in
Section 1.3, using time slots of the order of minutes, the possible variations with respect to the scheduled values are compensated in the first time slot of the MPC algorithm.
The parameters (price sequence
, reported arrival time
, and SoC at the request
) are assumed to be known, while
depends on the maximum power that either the charger can deliver, or the EV can accept. Moreover, the actual
is known when the EV connects to the charger; then, the prediction model is developed considering the expected value
for each EV (see
Figure 2a). Regarding
, it is determined by using the requested times
and
. Notice that both the arrival time and arrival SoC of each EV can be different at the actual connection time step.
The eMPC strategy looks for input sequences minimising the total cost of the EVCS, as per Equation (11), in a time window of
H hours, for each EV. To this aim, the dynamic constraint Equation (11b), i.e., when
while
, can be expressed in an extended form as:
where
Hence, the evolution of all
, throughout the prediction horizon
H, reads:
where
Notably, the system in Equation (
14) is time variant, since
in the prediction horizon has a switching behavior. In addition,
is the initial condition, it contains the current EV SoC when
and zero when
. Moreover, the cost function in Equation (
11a) is linear in
, and the dynamic Equation (
14) is a linear equality constraint in
. Furthermore, the other constraints in Equation (11) are linear inequalities that bind the feasible region described as a polytope. Then, the aggregator deals with a Linear Programming (LP) convex problem, which can be efficiently solved by Simplex or interior point methods.
Furthermore, it is noteworthy that the devised eMPC strategy might be affected by feasibility issues related to the charging time. As a matter of fact, it is assumed that the resulting charging time, when the eMPC formulation is employed, must be greater or equal than in the Minimum Time charging case. Thereby, the optimal control problem Equation (11) is said to be feasible if and only if:
In short, Equation (16) implies that the time an EV spends plugged-in (from
to
) is at least enough to charge it with maximum power
. From the feasibility condition Equation (16), how the economic MPC strategy will generally increase the time spent at the charging station premises appears, although a certain reduction of the recharge operating costs is guaranteed. Regarding the uncertain parameters, they must be inside the feasible region; otherwise, the problem is not feasible and the EV cannot be charged up to the minimum state
. However, an EV that arrives too late with respect to the request is still allowed to be charged, without guaranteeing that the minimum
will be reached.
In order to assess the complexity of the problem, it can be noticed that:
the size of the decision and the state variables, and respectively, is ;
the number of constraints in is , for each time slot, half for the lower bounds, and half for the upper bounds;
the number of constraints in is , for each time slot; equally allocated among the lower and the upper bounds, and the charger dynamics;
the number of constraints in
, related to the minimum SoC requirement
at the departure, is
I.
It is evident that the problem complexity grows linearly with H. This implies that, by scaling up the number of chargers, the number of constraints and decision variables would also increase accordingly, possibly impinging on the optimisation solution efficiency.
3.3. Optimal Control with Minimum Cost and Maximum Flexibility
In this subsection, a novel strategy for the charging station problem solution, based on flexibility maximisation, is proposed. The aim of this novel strategy is to offer a power flexibility capacity to the electrical grid, while guaranteeing the minimum SoC requirement
at the departure time. The uncertainty in the EV arrival time and SoC at the arrival time are considered as in
Section 3.2.
According to Definition 1, the concept of (upward or downward) flexibility is determined with respect to a nominal charging profile , whereas implies that no ancillary service may be offered to the grid. Hence, it could be crucial to set-up a charging strategy that always guarantees a certain amount of flexibility capacity. To pursue such an objective, two parallel paths can be developed, involving the flexibility as either an optimisation constraints or part of the cost function.
In fact, on the one hand, the optimisation constraints in Equation (11b) can be properly rephrased, in order to impose a minimum flexibility capacity to the chargers. Specifically, such an approach envisages two possible strategies for the constraint reformulation. In the first strategy, the equation in Equation (11b) binding the vehicle charging power is adapted to guarantee a certain degree of flexibility ,
being a parameter defining the flexibility requested to charger
i at time slot
k, while
is the overall flexibility offered by the EVCS, at the
k time slot. It is worth noting that the constraint Equation (
17) implies the same upward and downward flexibility, achievable for
. Moreover, all the
I chargers provide the same flexibility level at each time slot.
The second strategy for the constraints reformulation mainly consists of adding a new constraint to Equation (11b), binding the aggregated power
, yet leaving unmodified the single-vehicle power limits. Such a further constraint reads:
Note that, by introducing Equation (
18), the flexibility of each charger can be different. Indeed, the idea is to impose a gap in the power requested to the grid.
Finally, in spite of its capability to grant some level of flexibility, and, in turn, some extra energy service to the grid, a solution of the EVCS optimisation problem including the constraints Equation (
17) or Equation (
18) might result in being infeasible. In fact, the maximum flexibility
achievable by the EVCS is not known in advance. Indeed, this capacity depends on the state behavior.
On the other hand, in a second path to approach the charging problem improving the flexibility capacity, the optimisation problem formulation is re-framed according to an optimal control strategy, aimed to simultaneously maximise the charging flexibility and minimise the operational cost of the EVCS. This formulation assumes that remuneration factors (upward) and (downward) for the flexibility offered by the station are driven by prices.
To sum up, the aggregator deals with an optimal control problem, i.e.:
In this case, the aggregator decision variables are the optimal profile and the flexibilities and of the ith charger. Concerning and , in Equation (19), two new constraints are introduced. Indeed, and are considered as lower-bounded by zero and upper-bounded by . Then, the charging power is always positive.
This formulation allows for finding solutions that maximise both upward and downward flexibilities. Furthermore, the resulting charging strategy may lead to non-symmetric flexibility capacities, consistently with Definition 1.
Finally, the dynamic constraint for the SoC in
being in line with Equation (11b), its evolution can be again consistently expressed as per Equation (
14). Taking into account the discussion after Equation (
15) about linearity in the problem Equation (11), note also that the cost function in Equation (
19a) is a linear function of
as Equation (11a) and the new constraints are linear as well. Then, like for the eMPC case, the aggregator faces an LP convex problem. Similarly, the feasibility condition presented in Equation (16) also holds for the problem formulation Equation (19). From this perspective, it can be noticed how the approach Equation (19), in the worst-case scenario, would not allow any flexibility, thus reducing to the same outcome of the eMPC model in Equation (11).
To conclude, it is expected that this innovative OCCF strategy, with flexibility maximisation, generates higher expenses for the EVCS with respect to the eMPC one because the energy value is not the only element of the cost function. Nonetheless, it can provide to the grid a significant flexibility capacity. In turn, such additional flexibility would allow the generation of relevant extra revenues for the EVCS, due to the aggregator service in maintaining the electrical grid balance.