Event-Driven Real-Time Multi-Objective Charging Schedule Optimization For Electric Vehicle Fleets

$E_{a}$ and $R$ are the activation energy and gas constant. $T_{i,v}$ and $T_{amb}$ are the battery and ambient temperatures. The battery’s cyclic capacity fading model coefficients $k_{1}$, $k_{2}$, $k_{3}$ and $k_{4}$ was obtained by fitting (1) to the Stanford battery degradation dataset [22] using the Matlab surface fitting tool. The cyclic capacity fade prediction was tested on cells and it had a root mean squared percentage error of 3.7%.

The non-linear cyclic capacity fading model, $C_{loss,cyc,i,v}$ as a function of the charging current $I_{i,v}$ and the initial SoC, $SoC_{init}$ is given in (4). $SoC_{init}$ is the initial SoC at each time slot. $\Delta t$ is the length of a time slot.

From visualizing the plot in Fig. 1, two disjoint pieces of the surface plot can be distinguished by splitting the domain on either side of the line $I_{i,v}=480\cdot SoC_{init}$. A second order polynomial function in (5) with two different set of coefficients given in table 1 are used to approximate the two regions. The goodness of fit parameter, R-square on the domain $I<480*SoC_{init}$ was 0.9992 and 0.9983 on the domain $I\geq 480*SoC_{init}$. This approximation is necessary to ensure real-time performance in optimization calculations as the branch and bound algorithm in Gurobi takes too long to converge with the original non-linear model.

The calendric capacity fading model was obtained from [21] and is shown in (6).

In (6) $K(T_{amb},SoC_{i,v})$ and $(1+\alpha(T))$ are the kinetic dependence and temperature dependent alpha parameter. $t$ is the time for which the operating conditions are maintained. Calendric capacity fade, $C_{loss,cal}$ versus $SoC$ has a nearly linear relationship so it is approximated by (7). Coefficients for $p_{1}$ and $p_{2}$ are 0.0001347 and 0.00005356 respectively.

The exact total capacity fading model is the sum of cyclic and calendric capacity fading models given in (4) and (6) while the approximate total capacity fading model used in the simulation is the sum of (5) and (7).

The weight $\alpha_{1}$ in (8) multiplies the charging cost objective which is the sum of products of forecasted wholesale electricity price, $WEP_{i}$ with energy supplied during the charging period. The charging period is the elapsed number of time slots, $TT_{v}$ between the starting time, $t_{s}$ of the simulation and the departure time, $T^{dep}_{v}$. With this information, lower current will be consumed during high electricity prices and vice versa. The total charging cost of all the vehicles over their charging periods is considered as an objective function that has to be minimized. The constraints pertaining to all objectives is given in (9) - (13).

$WEP_{i}$ in (16) is the dollar cost per kilowatt hour (KWh) given in [23]. The constraints in (9) and (10) are current limits in each individual time slot and CS current limits respectively. (11) calculates the number of time slots in a charging period. (12) and (13) ensure that departure SoC is reached within the deadline.

The weight $\alpha_{2}$ multiplies the total battery capacity fading objective, $C_{loss,tot,i,v}$ which is the sum of approximate $C_{loss,cyc,i,v}$ and $C_{loss,cal,i,v}$ given by (5) and (7). Battery capacity fade is minimized by considering the stress factors in the battery capacity fading models [8] and [21]. In general, high SoC levels, $SoC_{avg,i,v}$ high depth of discharge, $SoC_{dev,i,v}$ and high charging currents, $I_{i,v}$ lead to higher fading. The battery should only be charged with enough energy for the upcoming trip. Therefore, charging the battery gradually later on near the end of the charging period reduces the SoC levels, depth of discharge and charging currents throughout the charging period. Minimizing the calendric and cyclic capacity fading models achieves this objective. The capacity losses are linked to the charging current through the state of charge and charge processed equality constraints in (14) - (17).

The weight $\alpha_{3}$ multiplies the ride availability objective, which is the negative product of decaying weights, $w_{i,v}$ multiplied to the corresponding power allocated. When rides are abound and EVs need to be charged quickly to service them in the order of earliest to latest deadlines the power allocated to the different vehicles are weighted using monotonically decreasing weights, $w_{i,v}$ as given in equality constraint (18).

Minimizing the total negative power allocated to all the vehicles in all their time slots is equivalent to minimizing the total charging time. The decreasing weights are distributed throughout the time slots for each vehicle. The size of the decreasing weights are influenced by the charging period $TT_{v}$. The charging period is divided into uniform time slots, $i$ for each vehicle, $v$ and the cost of each of the three objective is computed for each vehicle at each time slot. The total cost for each objective is summed over all the time slots and vehicles in the schedule. These individual objectives are normalized using utopia and nadir points and then summed together as weighted sum of objectives as given in (8). The solution of the optimization results in the current allocation, $I_{i,v}$ in each time slot for each vehicle. The power allocated in each time slot is simply the product of the current allocation, $I_{i,v}$ to the constant charging voltage, $V$.

The Adaptive Charging Network, ACN [25] dataset contains real-world EV charging patterns at CSs across California Institute of Technology. The task information obtained from this data were connection time, disconnection time, energy requested in KWh, minutes available for charging and charging slot ID. Low and high vehicle charging density scenarios from two different weeks were used to compare the baseline solution with our proposed multi-objective optimization solution. The low density charging scenario was between 2021-05-22 to 2021-05-28 and the high density charging scenario was between 2021-05-01 to 2021-05-07. The maximum number of vehicles to optimize at one time for the low density scenario was 4 and for high density scenario was 14. The electricity price during the two weeks was obtained from [23]. The simulation code can be found here ¹¹1https://github.com/JosePeeterson/ACN_caltech.

The simulation starts on the first day with the first arrival event of an EV. The task information is recorded and a schedule is generated with power allocated over the different time slots of the EV’s charging period using either the baseline solution or the proposed solution. The schedule is then implemented and the states of the EV such as the remaining charging time and current SoC level are updated until another event such as an arrival or departure of an EV occurs. When this happens, the new task information is recorded together with the updated states of the existing EVs and a new schedule is generated. This process repeats until the end of the week.

The Performance metrics given in table 2 are used to evaluate and compare the results from the proposed solution with the baseline solution for all the vehicles in the two weeks of low and high vehicle charging density.

Each of the performance metrics is calculated from all EVs in the entire week with some vehicles charging simultaneously. Unity weights represent equal weights for all 3 objectives, i.e. $\alpha_{1},\alpha_{2},\alpha_{3}=1$. Overall, the proposed solution outperforms the baseline solution on all metrics except total charging time and maximum optimization time. The increase in total charging time is acceptable for the proposed solution as the schedule generated was feasible. The optimization times are less than 1 second so real-time performance is still guaranteed. The total charging cost and total battery capacity fading cost is lower for the proposed solution in both cases of low and high density. In the low density case, the proposed solution has 53.2% lower charging cost and 16% lower capacity fade for the proposed solution. Total Amortized Battery Value Loss is computed for a 86 KWh battery that has a cost of $135/KWh. The cost of the battery multiplied to the ratio of total capacity faded to the nominal capacity gives the value lost in dollars. The impact of the baseline charging strategy on the grid during the high density case is measured by the total peak power period metric which is the total time of peak power usage for all vehicles. Peak power period is defined as a time slot allocated with greater than 75% of maximum power. For the low density case, this leads to a 33% reduction in total peak power period. Fig. 2 shows the total peak power period for all the charging EVs at each optimization time for the high charging density week. The proposed solution shown in blue uses unity weights and the baseline solution is shown in red. The average of the total peak power period over all optimization time for the proposed solution is 1.8 hours. This is lower compared to 4.3 hours for the baseline solution.

Fig. 3 shows one instance of the optimization using baseline and proposed solution taking place on 2021-05-07 at 15:44 for an arrival event of an EV. The top subplot shows the electricity price followed by the subplots for the charging current allocated to each vehicle at each time slot by the baseline solution and the proposed solution using unity weights. Each vehicle is given a distinct colour and their charging current values are represented as a stacked bar plot. For the baseline solution the schedule lasts only till 19:14 compared to the proposed solution that lasts till 01:59 on the next day. The maximum number of vehicles in this optimization are only 5 for baseline compared to 13 for the proposed solution. These observations are due to the maximum power allocation in successive optimization events for baseline solution whereas for the proposed solution charging is allocated closer to the departure and this results in aggregation for charging tasks that spill over into future optimization events. The baseline solution comes at the cost of charging during higher electricity prices, causing higher capacity fade and higher number of peak power periods. In contrast, the proposed solution has a monotonically decaying trend which can be attributed to the urgency created by the ride availability objective. Current peaking between 17:14 to 17:44 is the result of a drop in electricity price. The distribution of currents throughout the time slots till the end of the charging period of each vehicle and the two small current allocations near the end of the schedule for vehicle 0 are due to the battery capacity fade objective that tries to maintain lower SoC levels till the end of the charging period of each vehicle in the schedule.

Although setting unity weights in conflicting objectives for every optimization event in the entire week leads to the deterioration of some objectives in some optimization events, overall the combined objectives is still better than the BAU policy of charging at the maximum rate as shown by the results in table 2. The lower part of table 2 also shows the effect of prioritizing one objective over another by setting higher weights. Clearly, setting a higher weight for the charging cost (0.6,0.3,0.1) has the most significant reduction in the charging cost compared to the other two weight combinations. Similarly, setting a higher weight for ride availability reduces the total charging time. The cost savings on battery value loss is not as significant for a single week of operation but in the long run it helps to mitigate the range anxiety problem and improve the resale value of the EV.