Moving horizon-based optimal scheduling of EV charging: A power system-cognizant approach

The PEV charging setup considered in this work comprises of an aggregator providing charging service to a residential community. During the entire operating period, the aggregator is supposed to control the total real-time load of the station such that the power distribution system constraints for the entire feeder are satisfied. This arrangement minimizes the burden on utility for charging schedule and also reduces the dependency on high-end communication technology. It is assumed that the aggregator receives a day-ahead electricity prices, forecasted load, and forecasted generations at all buses. A deterministic model is considered for simplicity. However, it is assumed that the aggregator is not aware of the number of PEV’s arriving at different times across the day. Hence, a moving horizon based scheduling algorithm is developed that periodically updates the schedule based on real-time arrival of PEVs at the charging station.

The main novelty of the proposed algorithm is that if a PEV arrives at the charging station and a contract is established, the scheduling process guarantees that the charging commitments are met despite of uncertain future schedule revisions. The charging commitments comprise of two components: i) time of return after charging; and ii) final charging level of PEV on return. While the final charging level is determined by customers’ needs, the time of return is determined as described next. When a PEV arrives with a charging request, the aggregator gives the PEV owner the flexibility of choosing a charging cost option. For every cost option, the charging time required by the aggregator for charging the PEV to its required level will be provided. Based on this information, the PEV owner decides the charging cost. The charging time limit is decided by dividing the required charging power with an average charging rate for each cost option. In this setup, two charging cost options are given to the PEV owner.

Two charging prices $C_{1}$ ($/kWh) and $C_{2}$ ($/kWh) are considered here. This can be scaled for more number of charging options. Here, $C_{1}$ is greater than $C_{2}$. So, the costumers choosing $C_{1}$ as charging cost have higher priority than that of $C_{2}$. The reason for a customer opting for $C_{1}$ is less charging time guaranteed by the aggregator. The PEVs selecting the cost option $C_{1}$ are grouped as one category and PEVs selecting $C_{2}$ are grouped as another. The aggregator solves the optimization problem of maximizing its charging profit by generating optimal charging schedule at the beginning of each 1 hour time interval.

With new PEVs requesting charging, the aggregator can revise the previous charging schedule facilitating maximal PEV charging at all times, the aggregator can alter the previous charging schedule, based on the number of new PEVs requesting charging. Depending on the available charging capacity, power system load levels, and existing charging commitments from previous hours, the aggregator selects the PEVs from the two groups for providing the charging power.

In this setup, PEVs arriving for charging between two intervals are assumed available for charging from the next time step. Also, the charging for PEV is not necessarily continuous, i.e., a PEV might sometimes remain idle. The schematic diagram of the scheduling arrangement is summarized in Figure 1.

In a residential area, peak load hours are usually 6 pm-8 pm. During a weekday evening, residents come home from work and basic residential loads are in use, along with which PEV load is plugged in under uncoordinated charging. This practice leads to higher loading conditions. The off-peak hours in a residential setup is from 11 pm- 6 am and 8 am- 2 pm. So, the primary idea behind scheduling the PEV charging is to spread out the charging start times causing less steep peaking.

We consider Level 2 charging that uses a 240V AC setup and can be configured for variable charging power of 3.3- 19.2 kW. Depending on the make, model, distance range and affordability, PEVs can have varying range of battery capacities. The charging efficiency ($\eta$) is used as with increase in usage and losses incurred by the charging setup, the battery charged is less than the input grid energy.

Consider a single phase radial distribution system with $N$ nodes. Let $p_{i,t}$ and $q_{i,t}$ represent the net active and reactive power injection at node $i$ during time $t$. It is assumed that the active and reactive power generations $(p_{i,t}^{g},q_{i,t}^{g})$, and loads $(p_{i,t}^{\ell},q_{i,t}^{\ell})$ are known a priori. The nodes hosting the aggregator charging facility will have additional active power load $p_{i,t}^{EV}$. Thus power balance for any network node $i$ entails

Oftentimes there are maximum apparent power limits $\bar{s}_{i}$ based on feeder rating, transformer rating or contracted capacity. These constraints are quadratic in general. However, with known reactive power, the quadratic constraints $p_{i,t}^{2}+q_{i,t}^{2}\leq\overline{s}_{i}^{2}$ can be written as linear constraints

Given the nodal power injections, the voltages may be determined using the power flow equations. In this formulation the linearized distribution flow (LDF) model is employed for computational tractability [15]. Let $v_{i,t}$ be the squared voltage at node $i$ at time $t$, and $\mathbf{v}_{t}$ be the $N-1$-length vector collecting all nodal squared voltages other than the substation node voltage $v_{0}$ for time $t$. Similarly, let $\mathbf{p}_{t}$ and $\mathbf{q}_{t}$ be the collection of active and reactive power injections at non-substation nodes, respectively. Then, the linear power flow model dictates

where $\mathbf{R}$ and $\mathbf{X}$ are derived from the network topology and impedances as detailed in [15]. Next, any stipulated limits on power injections and voltages may be imposed as

where ($\underline{p},~{}\underline{q}$ and ($\overline{p},~{}\overline{q}$) are the limits on active and reactive power respectively. The voltage limits $\underline{v}$ and $\overline{v}$ are the minimum and maximum permissible squared voltages of the distribution system. As a voltage deviation between a distribution transformer and the service voltage is expected, the voltages at the distribution transformers level is maintained within $\pm 3\%$ pu, resulting in the squared voltage limits as [$0.97^{2}~{}1.03^{2}$].

For a given day, the entire time horizon may be divided into intervals of length $\Delta t$, say $\Delta t=1$ hr. Let us index the intervals as $t=1,2,\cdots,T$. Let $M_{k}$ denote the number of PEVs that have entered a charging contract by the $k$-th interval and have not yet reached the desired charging level. Denote the number of new PEVs requesting charging at the beginning of $k$-th interval as $n_{k}$. Thus, at the beginning of $k$-th interval, the aggregator would try to prepare a charging schedule for $N_{k}:=M_{k-1}+n_{k}$ over the remaining $T_{k}=(T-k+1)$ intervals $t=k,k+1,\cdots,T$. An optimal schedule shall first ensure that the charging requirements for the previously contracted $M_{k-1}$ PEVs is fulfilled in the remaining $T_{k}$ intervals. Next, a subset of the new $n_{k}$ PEVs shall be selected to enter a charging contract such that the optimal schedule can successfully fulfill their charging requirements. Indexing the $N_{k}$ PEVs by ${n=1,\dots,N_{k}}$, let the binary variable $u_{n}$ denote whether a contract is established with PEV $n$ ($u_{n}=1$); or otherwise $(u_{n}=0)$. Since the first $M_{k-1}$ PEVs are already contracted from previous binary instances, we have

Note that while $M_{k}$ and $n_{k}$ are problem parameters, the decision for establishing a contract $u_{n}$ is an optimization variable.

Define matrices $\mathbf{D}^{(k)}\in\{0,1\}^{N_{k}\times T_{k}}$, and $\mathbf{P}^{(k)}\in\mathbb{R}^{N_{k}\times T_{k}}$ to represent the overall schedule prepared at the beginning of $k$-th interval, as detailed next. The binary entry $D_{nt}=1$ represents that PEV $n$ is scheduled to receive a charging power $P_{nt}$ during the $t$-th time interval. Otherwise, entry $D_{nt}=0$ implies PEV $n$ remains idle during interval $t$, and receives no charging. Since only the PEVs entering a contract shall participate in the schedule, the following constraints hold

where $[\underline{p}^{EV},~{}\overline{p}^{EV}]$ represent the limits on charging power of PEVs. The assumption of common limits $[\underline{p}^{EV},~{}\overline{p}^{EV}]$ is without loss of generality.

Let $a_{n}^{k}$ denote the number of time intervals starting from the $k$-th interval within which PEV $n$ must receive $s_{n}^{k}$ units of energy. The values for parameters $a_{n}^{k}$ and $s_{n}^{k}$ are derived based on the contract established between the aggregator and PEV owner, and is updated after every time interval as detailed in Section III-D. The following constraints impose the requirements for charging times and total charge needed

where $\mathbf{u}$ and $\mathbf{s}^{(k)}$ collect the contract statuses $u_{n}$’s and charging requirement $s_{n}^{k}$’s for the $N_{k}$ PEVs, and $\odot$ represents the entry-wise product of the two vectors. Constraint (7a) ensures that in the prepared schedule at the beginning of interval $k$, PEV $n$ receives no charging power after time $a_{n}^{k}$. Constraint (7b) ensures that the row-sum of $\mathbf{P}^{k}$, representing the total charge received by PEV $n$, considering $\Delta t=1$, matches with the needed charge $s_{n}^{k}$.

The total power consumed by the PEVs getting charged at time interval $t$ appears in the power flow equations of the distribution network as $p_{i,t}^{EV}$ where $i$ is the power system node hosting the aggregator EV charging facility. The aforementioned coupling is captured by

Additionally, the maximum number of PEVs getting charged at a time interval is limited by the maximum number of available charging spots $\overline{\mathsf{N}}$

At the beginning of $k$-th interval, the aggregator would try to prepare a schedule over the next $T_{k}$ intervals that maximizes the profit. Let $\mathcal{N}_{1}^{k}$ and $\mathcal{N}_{2}^{k}$ be the set of PEV owners willing to pay the charging prices $C_{1}$ ($/kWh) and $C_{2}$ ($/kWh) respectively. Thus, the total number of new PEVs is $n_{k}=|\mathcal{N}_{1}^{k}\cup\mathcal{N}_{2}^{k}|$. Based on the prepared schedule, the aggregator would have to pay the electricity prices to the utility. With $\mathbf{c}_{e}\in\mathbb{R}^{T_{k}\times 1}$ representing the electricity prices for the next $T_{k}$ instances, the anticipated electricity cost is given by $\mathbf{1}^{\top}\mathbf{P}^{(k)}\mathbf{c}_{e}$. Therefore, the scheduling problem solved by the aggregator at the start of $k$-th time interval can be formulated as an MILP

Problem (P1) if feasible yields a possible schedule such that the PEVs selected for establishing a charging contract ${(u_{n}=1)}$ receive their requested charging $s_{n}^{k}$ within the agreed time $a_{n}^{k}$. We next delineate the algorithm for periodically solving (P1) in a moving horizon basis while ensuring feasibility and profitability.

In a realistic setup for local aggregators, the number of PEVs $n_{k}$, arriving at time $k$ is not known a priori. Therefore, an optimal schedule can not be generated at the start of the day. Rather, based on the initial $n_{1}$, an initial candidate schedule may be prepared, which is subsequently updated. In detail, at the start of any interval $k$, the aggregator shall solve (P1) and obtain a schedule for the next $T_{k}$ intervals; implement the schedule for $k$-th interval; discard the remaining schedule and resolve (P1) at the begining of $k+1$-th interval.

The proposed framework gurantees that the chrging commitments $(s_{n},a_{n})$ are fulfilled for PEVs with contract ($u_{n}=1$). To see this, note that for the first instance $t=1$, contracts are established for PEVs for which there exists a feasible schedule $\mathbf{P}^{(1)}$ over the next $T$ intervals. Thus, after implementing the first interval, the remaining $T-1$ columns of $\mathbf{P}^{(1)}$ are still a candidate solution to (P1) for $t=2$ with no new contracts established; hence guaranteeing feasibility of (P1). Similarly, since the truncated schedule from $\mathbf{P}^{(1)}$ is still feasible, any new schedule generated as $\mathbf{P}^{(2)}$ must yield a higher profit by optimality. Continuing the argument for all intervals, it is evident that the novel approach of enforcing (5) and (7) in a moving horizon basis guarantees fulfillment of commitment towards PEV owners while maximizing profit.

The initialization and updation of parameters $s_{n}^{k}$’s and $a_{n}^{k}$’s are explained next. When a new PEV arrives, its charging energy requirement is computed as

where $\mathsf{SOC_{plugout}}$ is the final SOC of an PEV to be reached and $\mathsf{SOC_{plugin}}$ is the SOC at the PEV’s time of arrival. $\mathsf{Bcap_{n}}$ is the battery capacity of $n^{th}$ PEV. Once the total energy requirement $s_{n}$ is computed, the guranteed time of return $a_{n}$ may be computed based on the price option $C_{1}$ or $C_{2}$ opted by the PEV owner. In detail, let $P_{C_{1}}$ and $P_{C_{2}}$ be the average charging power for the two cost options with $P_{C_{1}}>P_{C_{2}}$ for $C_{1}>C_{2}$. Then the time of return is given by

Once the optimal schedule is obtained for $k^{th}$ time, the charging power requirement and time availability of the selected $M_{k}$ PEVs are updated for the $(k+1)^{th}$ time interval.