Social-Aware Long-Distance Trip Planner for Electric Vehicles Using Genetic Algorithm¹

Battery electric vehicles (BEVs) are a promising endeavor striving for the reduction of fossil fuel consumption and global greenhouse gas emissions. The average driving range of commercial BEVs increased from 127 km in 2010 to 336 km in 2020 due to the development of battery technology. One of the major issues for BEV adoption is the lack of charging infrastructure. There were 48,253 charging stations across the USA by the end of 2022, but only 13.5% of those are DC fast chargers, according to the U.S. Department of Energy. Compared with the number of gas stations and refueling time for internal combustion engine vehicles, BEV owners need to plan their schedules more carefully before long trips due to limited charging infrastructures. The optimization of the recharging schedule on a fixed trip is classified as a fixed route vehicle charging problem (FRVCP) in the literature, which is frequently studied as a subproblem of the electric vehicle (EV) routing problem. The FRVCP problem is challenging due to vehicle and environmental uncertainties such as the nonlinearity brought by battery charging and energy consumption. Reference [1] simplifies the problem by assuming that the vehicle gets fully charged whenever leaving a charging station. The author of Ref. [2] assumes a constant charging rate to get a solution that allows partial recharging. Furthermore, Montoya et al. [3] solve the FRVCP with nonlinear charging and heterogenous charging stations by linearizing the nonlinear charging curve. The solution is obtained by both a greedy heuristic and a mixed-integer programming formulation. In Ref. [4], the author builds a computationally efficient open-source solver for large-scale FRVCP based on the labeling algorithm. Some literature studies the FRVCP independently. For example, the author of Ref. [5] proposes a solution that addresses eco-driving, optimal charging, and thermal management of BEVs simultaneously during long, hilly trips using a model-based method. The vehicle dynamics, battery circuits, and thermodynamics are modeled and the optimization problem is solved using nonlinear programming. In Ref. [6], the author utilizes the local information of an EV and global charging station selection information through a mobile telecommunications network to develop two optimization algorithms. In Ref. [7], the author proposes a new variant of FRVCP called FRVCP with nonlinear energy management (FRVCP-NLEM). The author considers the nonlinearity of the energy consumption function caused by variable speed at each route segment. A mixed-integer programming model is built to solve the problem.

The charging schedules proposed in the aforementioned studies only consider the physical attributes of the vehicle. However, the potential stops caused by the will of the driver including the purpose of the trip (e.g., travel and delivery) and basic human needs (e.g., dining and lodging) are ignored. The social activities along the trips can greatly affect the optimization results for a BEV charging schedule. For example, during a driving tour, an individual will likely overlap dining time at a restaurant with charging time to reduce their overall stopping period. Only a few studies target the minimization of BEV users’ traveling time by taking advantage of mandatory stops due to social activities. Reference [8] uses an iterated local search method to solve FRVCP that allows the driver to serve customers by an alternative mode of transportation while leaving the truck at charging station. Reference [9] integrates the selection of restaurants to optimize the en-route waiting time during a multi-dimensional tour trip for an EV driver; however, the algorithm is not flexible enough to consider the dining needs of the driver on an individual basis—the meals are planned for a specific period and a restaurant stop is chosen within that time instead of letting the driver determine when a stop is necessary.

The main contributions of this paper include: (1) the co-optimization of speed, charging schedules, and social activities to minimize travel time for BEV users on a long trip; (2) modeling of temporal characteristics of human energy and incorporating it as a dynamic state into the co-optimization problem; (3) the co-optimization problem is formulated as a mixed-integer nonlinear programming model and solved using genetic algorithm (GA); and (4) simulation and real-world experiments are conducted to validate the model and solution.

The paper is organized as follows: Sec. 2 introduces models and formulates the optimization problem. Section 3 conducts case studies and analyzes the simulation results. Section 4 presents the experiment design and results. Section 5 concludes the paper.

For the battery charging rate, a look-up table in Fig. 1 is used. The charging data were collected by charging a 2019 SV+ from 18% to 95% using a 62.5-kW DC fast charger.

For the optimization problem, the goal is to minimize the overall trip time by optimizing the driving speed and charging/dining choices simultaneously. Assume there are N stops for a trip that includes start and destination, define v(i) ∈ [v_lb, v_ub] as the speed while driving on the ith road segment whose length is L_i. The driving power can be found as P_batt(i) from (3). The charging time is defined as a continuous variable T_c(i) ∈ [T_clb, T_cub], and choices for dining are defined as an integer variable S_d(i) ∈ {0, 1} (i = 1, 2, …, N − 1). The reason that i only counts to N − 1 is that stop N is the destination, once the driver arrives at the destination, the trip is over. The actual dining time T_e is a constant. The driver energy will recover for a constant value h_c after dining. Every time the driver stops for charging/dining, a constant T_o is added to account for the time the driver spent from highway to the stop. Additionally, another integer variable z_i ∈ {0, 1} is used as an indicator for stop events. z_i is 1 whenever the stop time at ith stop is greater than 0. The stop time is identified by the maximum between T_c(i) and S_d(i)T_e. Note that the energy spent on finding the charger/restaurant is small compared with the energy spent on the whole trip, therefore it is negligible. For vehicle states, q_i ∈ [q_min, q_max] and h_i ∈ [h_min, h_max] represent the battery state of charge and human energy level when the vehicle arrives at ith stop, respectively. Then, the problem can be formulated as

Equation (10) is the objective of the problem which includes the overall stop time, driving time, and time spent on finding the stops. Equations (11) and (12) guarantee the battery SOC when the vehicle arrives/leaves the stop, stays in a range such that the battery will neither be overcharged nor over-consumed. Note that the energy consumption while driving is calculated by multiplying the driving power to the driving time and then dividing the energy by the battery capacity following Eq. (6). f₁ and f₂ here represent the interpolation of the look-up table to find charging time according to SOC and SOC according to charging time, respectively. Equations (13) and (14) enforce the human energy constraints to guarantee the driver comfort. Similar to SOC constraints, the human energy when the driver arrives/leaves the stop is limited within a range. Since the human energy is assumed to be changing with time, if the charging takes longer than dining, the human energy will start to deplete even though the driver is not driving. Equations (15)–(17) restrict the range of charging time, vehicle speed, and dining choices. Equation (18) shows the values of z_i.

GA is a well-developed metaheuristic algorithm inspired by Darwinian's theory of evolution. The nature of “Survival of the fittest” and their genes is simulated [13]. The algorithm starts with generating random populations within the searching space, where the populations represent chromosomes of individuals, and each chromosome contains a set of variables that represent the genes. Then by evaluating the fitness function, the genes of individuals with higher fitness will be more likely to contribute to the next generation following certain selection mechanics like roulette wheel selection. Some of them are selected as elite members and are guaranteed to survive to the next generations, and the rest follow the sorting mechanics above to crossover. The genes from parents will be split and recombined together randomly and be passed to their children, thus different potential solutions can be exploited. However, this mechanics is a greedy strategy and it possibly leads to a locally optimal solution. Therefore, mutation mechanics is used to randomly change genes after crossover. GA is widely used in optimization problems and shows promising performance. This paper implements it using the toolbox in matlab.

For comparative purposes, a logic-based algorithm is used to simulate the driver's choices. To minimize the total travel time, the driver will drive at the max speed that is allowed. At each stop, the driver will roughly estimate if the current SOC is enough to drive the vehicle to the next stop without charging at the current stop. If it is not enough, the driver will charge the vehicle SOC to the maximum value. If the current SOC is enough to reach the next stop but not enough to reach the stop after next stop, and the next stop happens to not have a charging station, then the driver will still charge the vehicle SOC at the current stop as well. After the driver makes sure the vehicle will not run out of battery, the dining is the next concern. The dining choices are made following similar logic to charging. This logic-based planner will try to minimize the total travel time while strictly satisfying input and SOC constraints, but not strictly enforce human energy constraints.

In this section, two case studies are conducted, as shown in Figs. 2 and 3. The vehicle parameters used in Table 1 are from the model in Ref. [14]. The dining time is assumed to be 30 min which is considered the minimum expected dining time of people at a restaurant from a survey [15]. The offset time T_o is selected as 8 min which represents the time required to get off the highway, reach the charging station, and return to highway. q_min and q_max are bounded between 0.2 and 0.9 to ensure battery health. The slope k is chosen as 4.6 × 10⁻⁴ such that the human energy will decrease by around 70% in 5 h.

Figures 4 and 5 show results of the two trips with different initial conditions. The labels (SxHy) represent the initial q₀ at x% and human energy h₀ at y%. In Fig. 4, GA outperforms the logic-based planner in all cases on trip 1. The time saving ranges from 15% to 27%. For trip 2, Fig. 5 shows that the total travel time of GA in two cases is slightly worse than baseline. Since the nature of GA is a search method, it only provides a fast and good solution. For the low initial SOC but high human energy case, the GA planner takes 2% more time. To find out the reason, Fig. 6 shows the states including battery SOC and human energy when arriving/leaving each stop. It is clear that the human energy of the logic-based planner violates the pre-defined constraint. This indicates that GA-based planner sacrifices total time slightly to strictly follow the human energy constraints. For the rest of the two cases, the GA-based planner saves 10% and 7% more time than baseline.

In the last subsection, it is found that the proposed planner has different performance depending on the initial conditions. In order to eliminate the initial condition's effect, Monte Carlo simulation is performed on both trips. Five hundred and 300 samples with random initial conditions are tested for the two routes, respectively, and the results are presented in Table 2. In the shorter trip, GA saves overall 21.98% trip time comparing with the baseline. Meanwhile, the GA follows the constraints well while baseline shows a significant violation rate of constraints. In the case of trip 2, the performance of GA is 4.67% better than the baseline. Although the constraint violation rate is higher than case 1, it is still much smaller than baseline.

Next, to investigate the impact of initial conditions on the results, the cost with respect to initial SOC and human energy level of trip1 is plotted in Fig. 7. The first observation is that the cost increases as the initial SOC decreases for both algorithms. The distribution of the cost for baseline has a clear division when initial SOC is at around 53%. However, for GA, the cost increases smoothly as initial SOC decreases. For initial human energy, the high cost of the baseline mostly occurs when the initial value is between 0.4 and 0.6. GA did not show similar pattern. The average cost of GA at high initial human energy region (0.4–0.9) is lower than the low initial human energy region (0.1–0.4) and the cost does not have an obvious peak like baseline. A common phenomenon can be found that the cost of GA changes smoothly with different initial conditions while the cost of baseline contains sudden change. This is because the baseline follows a simple logic and does not have the ability to coordinate the charging and dining events. It simply adds the number of stops when the states are about to violate the constraints, and this causes the clear division on the cost pattern. For GA, it is capable of coordinating the charging and dining schedule, and thus will not have to increase the number of stops at low initial condition cases. For example, if GA decides to stop for two times to guarantee the SOC constraints at high human energy level, it can still guarantee the human energy constraints when it is low by eating during charging. Therefore, the overall number of stops and stop time will not significantly increase. However, for the baseline, the driver may need to stop for another time to eat.

Since the battery and vehicle models used in this paper are simplified mathematical models, the model's accuracy is questionable. To validate the model accuracy, a real-world experiment is conducted. The selected route is a 140-mile route on I-40 near Cookeville, TN. The DC fast charging station is located at Exit 287. During the experiment, the driver will drive the vehicle back and forth between the different exits marked in Fig. 8 to simulate the driving on the longer route. The information about the stops is shown in Table 3. Note that the “CS” and “R” in the stop type column represent charging stations and restaurants, respectively. The detailed information of the route (including the road segments and the stops (bolded)) are shown in the route detail column in Table 3. For example, the driver started route at Exit 287, then traveled to Exit 280, and then returned to Exit 287. The second visit of Exit 287 is counted as the second stop.

The case with starting SOC and human energy both equal to 80% had been tested in the simulation before the experiment, then the driver follows the results by GA in the real-world driving. The simulation and experimental results for the proposed route are shown in Fig. 9. The first 1000 s of the experiment are considered as the preparation phase which took place in the local route. Note that, in the experiment, the initial SOC when the vehicle entered highway is around 85%, which is 5% higher than the initial SOC in the simulation. To account for the difference of initial SOC between the simulation and experiment, the SOC profile in the experiment was offset by 5% before it is compared with simulation result as shown in Fig. 9. Additionally, the test was conducted right after lunch (around 1 p.m.) to simulate an 80% initial human energy level as shown in the second subfigure. In the third subfigure, the experimental speed is presented. The dashed line circles represent the six stops. By following the schedule generated by the planner, the actual SOC fits the simulated SOC well, which validates the models used in the trip planner. The total trip time (on-highway + off-highway portion) is 9447 s.

This paper formulates the driver-friendly fixed route charging problem into a mixed-integer nonlinear programming model and generates optimal solutions using GA. The algorithm minimizes the total travel time of a given trip by optimizing vehicle speed and the stopping schedule for charging and social activities at the same time. The human energy is modeled as a dynamic state similar to SOC. The efficacy of the algorithm is proven by comparing the proposed method to a logic-based human-like planner under two different trips. The proposed method is able to minimize the total travel time while satisfying SOC and human energy constraints, and it outperforms the baseline in Monte Carlo simulations. In the experiment, the accuracy of the models used is validated.

This work was supported by U.S. Department of Energy under Award No. DE-EE0009223.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.