Optimizing the Deployment of Electric Vehicle Charging Stations Using Pervasive Mobility Data

The level of discomfort drivers experiences is, along side high costs, one of the factors contributing to the slow adoption of plug-in all-electric vehicles (PEVs) [1, 2]. Driver discomfort depends on a number of factors ranging from the limitations imposed by the short vehicle range to the significant amount of time required to recharge batteries at the charging stations. Current technological advances have increased mileage and reduced the time required for a single full recharging to around 30 minutes [3, 4, 5, 6]. These trends, coupled with the increased push for environment friendly transportation modes, signal the rapidly growing importance of public charging stations as a viable and alternative option to home-stations for charging EVs. An important question to address in this regard is how a network of public charging stations should be deployed in order to minimize the drivers discomfort in an urban environment.

As discussed in detail in the next section, there has been a number of studies aimed at introducing optimization frameworks for the deployment of charging stations. These studies use traditional methodologies developed in the field of facility locations optimization, which are based on computer simulations of the traffic flows and travel surveys [7, 8, 9, 10, 11, 12, 13, 14, 15]. Both computer simulations and travel surveys are known to have major limitations in terms of scalability, accuracy, and/or cost. This explains the high interest in the possibility of accessing and analyzing real anonymized individual trajectories to feed the optimization process. This is possible nowadays given the availability of massive data sets of digital traces recording individual-level human movements, such as those provided by cell phone records. These and other data sets collected by urban sensing devices have been used in a number of recent papers [16, 17, 18, 19, 20, 21, 20, 22]. However, to our best knowledge data-driven approaches for optimally locating EV charging stations have not been explored so far.

This motivated us to develop the data-driven optimization methodology presented in this paper. The methodology is based on first constructing a model for charging station energy demand that takes into account the mobility patterns of real individual movements recorded in the region surrounding Boston. More specifically, we have used a massive data set of over 1 million cell phone users whose activity has been recorded over a 4 months period. The second step of the methodology consists in formulating a discrete optimization problem to find the optimal configuration for a network of charging stations. To this end, the region of interest is divided into a geographical grid, and an objective function is defined to simultaneously minimize the overall number of charging stations required and the aggregate distance EV drivers need to travel to reach the closest charging station. Since optimally solving the problem at hand is computationally infeasible, we then present computational efficient, near-optimal solutions based on greedy and genetic algorithms. The third and final step is assessing the performance and robustness of the presented approach, which is done by: $i)$ comparing quality of the obtained solutions vs. those provided by a randomized, but locally optimized approach; and $ii)$ showing that the near-optimal solution computed using single day movements preserves its properties also in later months.

The rest of the paper is organized as follows: in Section II, we present an overview of related work. In Section III, we formulate the problem and introduce the data-driven optimization framework. The optimization framework is then described in sections IV through VII: we describe the mobile phone dataset used in the analysis, and how it has been processed to feed the optimization framework. In Section VIII, we present and discuss the results of the proposed optimization methodology, including a thorough robustness analysis. Finally, Section IX presents conclusions and discusses future research directions.

Facility Location Optimization (FLO) problems have been extensively studied in the field of logistics and transportation planning over the past few decades. With the exception of a single work mentioned below, to our best knowledge the methodologies used for FLO in the context of EV charging station were based on theoretical models, simulations, and/or aggregate transportation data obtained from census tracts or travel surveys. Some of these approaches were based on individual trip trajectory analysis which required theoretical modelling of the behavior of the service seeker agents or travel-surveys. A parking-based assignment introduced in [7] is based on minimizing EV users’ stations access costs while penalizing the unsatisfied demand. The case study relied on 30,000 personal trip records based on household travel surveys in Seattle, US. In [10], a multi-objective optimization problem is introduced considering various costs associated with charging stations (including construction, operational costs, charging related costs). The aim of this study was to minimize the costs with consideration of charging demand distribution obtained from aggregate transportation data for Beijing.

In another work, the authors of [11] proposed an optimization problem from the EV driver’s perspective, where the objective is to minimize user discomfort measured as the distance between the charging demand spots and the location of the stations. This study used static population and income-level in Hong Kong as an input to model the spatial distribution of charging demand. In [13], EV traffic and drivers’ behaviour have been simulated on the real road network of Vienna. Based on a simple linear battery depletion model, an approach is proposed to optimize the location of charging stations for a limited number of EVs with the objective of minimizing the overall travel time.

In [9], a study on optimizing the location of EV charging stations for a neighborhood in Lisbon is presented based on estimation of the refuelling demand through the application of a maximal covering model. In this study, the refuelling demand estimation was based on 2001 static census data. In another work, an optimization framework was introduced to deploy charging stations and to assign optimized number of plugs based on real EV taxi-trajectory data, with the objective of minimizing the average time to find a charging station and waiting time before charging [8]. Using LP-Rounding, the authors provided approximate solutions to the NP-hard optimization problem. To our best knowledge, this is the only study in the field which is based on sensor-based individual travel trajectories data analytics. However, taxi data is known to represent only a small fraction of the actual mobility demand, which can be better characterized by means of cell phone data [23] as done in this paper.

Other studies have proposed multi-objective optimization problems to locate charging stations. For instance, in [24] the authors proposed a model based on set cover with dual objectives of minimizing the overall costs of opening new charging stations and maximizing their overall coverage on Taiwan road network based on static population density data. In [15], a linear model is used to estimate EV penetration in various sub-regions to determine the volume of EV flows between the sub-regions based on demographic data. This simulation model is then fed into an optimization problem to obtain the location of charging stations considering an objective function which maximizes the EVs that charge at the new public stations.

The aforementioned works consider various aspects and are important first steps towards constructing a comprehensive optimization framework for efficient planning of EV charging stations. In this work, we further grow the body of knowledge in this important topic by proposing the first optimization framework based on cell-phone data. The importance of relying on cell phone records lies on the fact that a recent study has shown their very good accuracy in modelling individual trip origin/destination flows in urban settings [23]. We thus believe that cell phone data fed optimization will become a prominent approach for many urban facility location problems, of which EV charging station is a first example.

Our data-driven optimization framework relies on analyzing the movement patterns of individuals obtained through cell-phone data over the span of 4 months. Starting from that, we formulate a discrete optimization problem equivalent to the well-known set cover problem [25] by dividing the region of interest into a geographical grid. The objective is to minimize the total aggregate distance travelled by drivers on their route from the end of their intended trip to the closest available charging station. Since the set cover problem is NP-hard [25], we use Chvatal’s greedy [26] and a Genetic Algorithm (GA) [27] approach to approximate near-optimal EV charging station locations.

The overall optimization framework can be divided into the following stages:

A) Demand Modelling: Constructing an energy demand model for EVs based on individual trip trajectories

B) Formulating the Optimization Model: Formulating the location optimization problem as a set cover problem

C) Analyzing the Data: Processing the raw data to estimate EV energy demand based on a simple assumption for EV adoption rates

D) Optimization Methods: Performing a number of optimization methods including Chvatal greedy search and GA meta-heuristic search to find near-optimal locations of EV charging stations.

Each step will be described in a separate section.

We assume the urban area is partitioned into a number of non-overlapping square cells $\mathcal{C}=\{C_{1},C_{2},...,C_{N}\}$, e.g., corresponding to a square grid partitioning. Similar to space, time is also divided into a number of non-overlapping intervals, corresponding to, e.g., $\Delta t=30~min$ slots. In the following, we use $i$ as a generic cell index and $t$ as a generic time slot index. Equipped with the above definitions, we now define the following quantity for each spatio-temporal slot $(i,t)$:

We concatenate consecutive trip segments if the stay time between them is less than $\tau_{min}$, where $\tau_{min}$ represents the minimum time interval needed for completely charging an EV, and is set to $30~min$ in the following. The trip segments counted in $IN_{i,t}$ are then those at which ends it is possible to completely charge the EV.

Note that, if we assume that the duration of a time slot $\Delta t$ is $\leqslant\tau_{min}$, all segments contributing to $IN_{i,t}$ belongs to different vehicles. A fraction $0<\delta\leq 1$ of $IN_{i,t}$, i.e., $\delta\cdot IN_{i,t}$, can be defined as the number of candidate EVs arriving and potentially demanding fast refuelling their batteries in $(i,t)$. We construct this refuelling demand model based on the durations of the stay at various cells, and by assuming a linear energy consumption model with the trip length. More specifically, $u_{E}(t)=(u_{E}(0)-c\cdot l)\Theta(u_{E}(0)-c\cdot l)$ where $u_{E}$ is $1$ when the EV is fully charged and $0$ when the battery is empty of charge and $\Theta(x)$ is a step function which is zero for $x\leqslant 0$ and $1$ when $x>0$. The constant $c$ depends on a number of parameters including road conditions, traffic flow, driver behavior, and vehicle type. For simplicity, we assume that the battery depletion is linear on average with distance travelled, and that the range is $150km$, which is equivalent to setting $c=1/150km^{-1}$. In order to have a better estimate of refuelling demand at each cell, at each time slot we only count trip segment end points with trip lengths equal to or larger than $l_{min}\sim 100km$. We obtain the number $IN_{i,t}$ of needed ”chargeable” parking spots at each cell according to these considerations. It is important to note that cell phone data does not allow immediately distinguishing between transportation modes. For this reason, we simply assume that the shares of EVs are uniformly distributed in space proportionally to the envisioned penetration ratio $\delta$ of electric vehicles. A better approximation would be to consider a spatial inhomogeneity in the penetration rate as it has been proposed in [15], which is left for future work. In the rest of the paper we assume $\delta=0.25$.

Based on the topology of the cell partitioning, we build a cell network $\mathcal{C}_{N}$ as follows. We add a node for each cell $C_{i}\in\mathcal{C}$, and we add a link between $C_{i}$ and $C_{j}$ if the two cells are spatially adjacent to each other. For instance, if $\mathcal{C}$ corresponds to a square grid partitioning, the node corresponding to cell $C_{i}$ in $\mathcal{C}_{N}$ is connected to the nodes corresponding to the cells adjacent to $C_{i}$ in the North, East, South, and West direction. We now introduce a parameter $h$, where $h$ is an integer $\geqslant 0$, which models driver discomfort in terms of how far a driver is willing to drive to reach a charging station. Having set $h$, we say that a charging station located at $C_{i}$ covers a cell $C_{j}$ if the respective nodes in $\mathcal{C}_{N}$ have hop-distance at most $h$. In turn, we say that $C_{j}$ is covered if there exists at least one cell at hop-distance at most $h$ from $C_{j}$ that hosts a charging station.

The tradeoff between drivers discomfort and the coverage is now clear: if $h=0$, driver discomfort is minimum, but a charging station located at $C_{i}$ covers only the cell itself. With increasing values of $h$, we have a higher driver discomfort, but coverage of charging stations increases as well, and less charging stations are required to cover the entire city. Each node in $\mathcal{C}_{N}$ is then weighted as follows. Let $N^{j}_{i}$ be the set of nodes at hop-distance $j$ from node $C_{i}$¹¹1To simplify notation, in the following $C_{i}$ denotes both a cell and its corresponding node in the cell network $\mathcal{C}_{N}$. in $\mathcal{C}_{N}$. The weight $w_{i}$ of node $C_{i}$ is computed as

Weight $w_{i}$ is designed to model the discomfort imposed on drivers by locating charging station at node $i$. The discomfort is assumed to be linearly proportional to the hop distance to the charging station, hence it is $0$ for drivers arriving in $C_{i}$, 1 for drivers arriving in a cell $C_{z}$ at hop distance 1 from $C_{i}$, and so on. So, the total discomfort is obtained by summing up the number of drivers arriving in cells at a certain hop distance from $C_{i}$, up to the maximum value of $h$. Furthermore, this quantity is averaged across the total number of time slots $k$ considered in the analysis. Note that as we only consider cells in the coverage range of $C_{i}$ in the computation of $w_{i}$, all the weights are negative as $j\leqslant h$.

We are now ready to formulate the charging station optimization problem. The problem of finding the optimal location of charging stations can be formulated as a weighted set-covering problem [25] in which the universe set, $\mathcal{U}$, and the set of subsets, $\mathcal{S}$, are given as follows

where $s_{i}$ is defined as

i.e., the set of all cells within hop-distance $h$ from $C_{i}$ in $\mathcal{C}_{N}$. Each $s_{i}$ is weighted with $w(s_{i})=w_{i}$, which is a measure of the total discomfort experienced by the drivers in cells covered by $C_{i}$. The optimization problem is to find a subset $\mathcal{S}_{opt}$ of $\mathcal{S}$ such that all elements of $\mathcal{U}$ are covered, and the sum of the weights in $\mathcal{S}_{opt}$ is minimized. Note that when $h=\infty$, each of the $s_{i}$’s coincides with the universe set, so the optimal $\mathcal{C}$ is the $s_{i}$ that has the lowest $w_{i}$. When $h=0$, $s_{i}=\{C_{i}\}$ and all the weights are zero, therefore optimal $\mathcal{S}_{opt}=\mathcal{S}$ itself, implying that there must be a charging station at every cell. We are then interested in investigating the non-trivial case in which $0<h<\infty$.

The weighted set covering problem described above can be formulated in terms of the following constrained integer linear optimization problem (ILP):

Note that $p_{i}$ is the number of available stations in the $h$-proximity of $C_{i}$. In practice, there is a finite capacity on the number of EVs a charging station can handle at a time, i.e., $n_{c}$. If we require that the capacity is not exceeded, then one need to make sure that the following condition is satisfied for every cell:

where $\text{max}(\text{IN}_{i,t})=\max_{t}IN_{i,t}$ is the maximum value of $\text{IN}_{i,t}$ during the observed time interval. Interestingly, this modifies the inequality constraint in a simple way. The difference is that instead of $\geqslant 1$, we have $\geqslant k_{i}$. Therefore, the problem becomes:

In the following, we assume that the capacity of charging stations is large enough to address all energy demand in the coverage range, which is equivalent to setting $k_{i}=1$ for all $C_{i}$s. As discussed in Appendix A, the model can be easily generalized to consider capacitated case for the charging stations. This way, the number of EV charging plugs required in each cell can be optimized to maximize the coverage and to distribute the load evenly between the charging stations.

The set cover problem can be represented also in binary matrix form – useful for GA implementation – as follows. An $N\times N$ binary matrix is formed where rows are associated with cells that need to be covered, and columns are associated with stations at various cell locations. Now the $(i,j)$-element of this matrix is either 1 or 0 depending on whether the station at $C_{j}$ covers $C_{i}$ or not. This binary matrix, which we call SCP, along with a weights array $W$ uniquely represents the set cover problem:

Each configuration of charging stations in this binary representation is a binary vector in which the indices of nonzero elements correspond to indices of the cells which have charging stations in them.

It is also important to note some of the cells in the partitioning have zero energy demand in the observed time period. Therefore, we can reduce the problem size by removing the associated rows/columns.

Since optimally solving the set-covering problem is computationally unfeasible, in the following we present two computational efficient methods to produce near-optimal solutions.

Fig. 3 shows the best configuration obtained for various values of the coverage range $h$, where the $IN_{i,t}$ values are computed from the movement data of a single day in July 2009. The $h=0$ case is the trivial case where there is a charging station at each demand spot, and it just reported to show cells with non-zero energy demand. As one expects, by increasing the value of $h$, a lower number of charging stations is required to cover all the demand. It is interesting to observe that, while the solutions obtained for a relatively larger value of $h$ contains relatively less stations as expected, the computed charging station location sets are not subsets of those representing previous solutions, indicating the non-trivial combinatorial structure of the problem at hand.

We can also compute the average distance to a charging station from each demand spot, and compare the initial and final populations of configuration layouts. In Fig. 4 a we report a scatter plot where each point represents a charging station configuration, and the horizontal and vertical axes represents the average distance and the variance of the distances to the charging stations from the demand spots. Different colors corresponds to different values of $h$. As one can see, for any value of $h$ the final population (smaller clusters) has significantly lower average distance to charging stations compared to initial populations (larger clusters of points).

An important aspect to investigate is whether the charging station configuration computed using data from a single day is still near-optimal also for the following days. In other words, we would like to assess the robustness of the computed solution in presence of variation of the input data, which is especially important in infrastructure planning processes given the high costs and long lifetime of infrastructure. To this purpose, we have used the near-optimal charging station configuration obtained using a single day in July, and assessed its performance when the $IN_{i,t}$ values are computed from data of various days in a week selected from the months of July, September, and October 2009. As reported in Fig. 4 b, using the movement data for an equivalent day two months later in October yields very similar results, indicating the robustness of the computed near-optimal configuration. Fig. 5 reports the average distance and the variance for the movement data for three weeks chosen from three months of July, September and October, using the same configuration obtained for the single day in July. This plot clearly shows how the achieved improvement in distance remains stable over time. The robustness of the proposed optimization approach is a consequence of the stationarity of the aggregate movement patterns, which have been repeatedly observed in the literature.

Another way to investigate the robustness of our optimization approach is to tune the parameters of the EV demand to see how sensitive the average driven distance is with respect to a change in these parameters. In Fig. 6, we take the $h=1$ case as a reference, consider the layout configuration obtained for $\tau_{min}=30~mins$ and $l_{min}=~100km$, and recompute the average distance based on the demand for different values of $\tau_{min}$ and $l_{min}$. Again, the gap in the average distance between a randomly obtained layout and the best layout obtained using GA shows significant robustness in the space of these parameters. This shows that the computed charging station configuration remains stable not only across time, but also in case technological development significantly change charging times and EV ranges.

Finally, we consider robustness to changes in the EV penetration ratio $\delta$. In this case, it is easy to observe that the structure of the computed solutions is completely independent of the penetration ratio, since $\delta$ is a parameter the uniformly multiplies all weights in the problem formulation.

So far, in the minimal objective function we have not considered the cost associated with building new charging stations. Therefore, it is important to compare the number of stations required in different layouts versus the average distance to make sure the number of charging stations is minimized properly. In Fig. 7, we present a scatter plot of the number of stations versus the average distance to the charging stations for each configuration for charging stations network. This plot shows that layouts with similar numbers of stations can lead to significantly different average distances. The GA population tends to have higher number of stations compared to the average of the initial population. This is consistent with the fact that no penalty term exists in the objective function to penalize addition of new charging stations reflecting the cost of building and operation. To address this issue, we modify the objective function by adding a positive constant $w_{0}$ to all cell weights. This offset value can in general reflect the cost associated with building a new charging station or other spatial restriction which may exist such as compatibility with the power grid. In the simplest case, we consider a spatially uniform offset and optimize its value to significantly decrease the number of charging stations without too much sacrificing the improvement gained in the average distance. In Fig. 8, we reproduce the result with this modification to the objective function. The number of charging stations is significantly reduced compared to the previous case; however, the improvement in average distance remains still significant in most cases.

Efficient deployment of the network of public charging stations is an important matter which will play a significant role in further increasing the market share of the EVs in the near future. Motivated by this, we have proposed a modelling and optimization framework to find efficient layout of charging stations to minimize overall EV drivers discomfort, which is reflected in the distance they need to travel to reach the closest charging station starting from their refuelling demand spots. Our minimalistic model can be easily generalized to include more objectives. For instance, spatial variations in the cost of constructing charging stations and the capacity of each station can be considered as an optimization variable without changing the problem structure and the required optimization method. We also showed that the computed near-optimal layout is robust to variation in input data, in EV penetration rate, as well as in EV and charging station technology. This aspect is particularly important given the high costs related to infrastructure deployment, and promote data-driven planning as a viable solution for optimal EV charging station location.