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Optimizing the Deployment of Electric VehicleCharging Stations Using Pervasive Mobility Data

Mohammad M.Vazifeh, Hongmou Zhang, Paolo Santi and Carlo Ratti

Abstract—With recent advances in battery technology and theresulting decrease in the charging times, public charging stationsare becoming a viable option for Electric Vehicle (EV) drivers.Concurrently, wide-spread use of location-tracking devices inmobile phones and wearable devices makes it possible to trackindividual-level human movements to an unprecedented spatialand temporal grain. Motivated by these developments, we pro-pose a novel methodology to perform data-driven optimizationof EV charging stations location. We formulate the problem asa discrete optimization problem on a geographical grid, with theobjective of covering the entire demand region while minimizinga measure of drivers’ discomfort. Since optimally solving theproblem is computationally infeasible, we present computation-ally efficient, near-optimal solutions based on greedy and geneticalgorithms. We then apply the proposed methodology to optimizeEV charging stations location in the city of Boston, starting froma massive cellular phone data sets covering 1 million users over 4months. Results show that genetic algorithm based optimizationprovides the best solutions in terms of drivers’ discomfort and thenumber of charging stations required, which are both reducedabout 10% as compared to a randomized solution. We furtherinvestigate robustness of the proposed data-driven methodology,showing that, building upon well-known regularity of aggregatehuman mobility patterns, the near-optimal solution computedusing single day movements preserves its properties also in latermonths. When collectively considered, the results presented inthis paper clearly indicate the potential of data-driven approachesfor optimally locating public charging facilities at the urban scale.

Index Terms—Intelligent Transportation System (ITS), ElectricVehicle (EV), Network of Public Charging Stations (NPCS),Evolutionary Optimization (EA).

I. INTRODUCTION

The level of discomfort drivers experiences is, along sidehigh costs, one of the factors contributing to the slow adop-tion of plug-in all-electric vehicles (PEVs) [1], [2]. Driverdiscomfort depends on a number of factors ranging fromthe limitations imposed by the short vehicle range to thesignificant amount of time required to recharge batteries atthe charging stations. Current technological advances haveincreased mileage and reduced the time required for a singlefull recharging to around 30 minutes [3]–[6]. These trends,coupled with the increased push for environment friendlytransportation modes, signal the rapidly growing importanceof public charging stations as a viable and alternative optionto home-stations for charging EVs. An important question toaddress in this regard is how a network of public charging

M. Vazifeh and C. Ratti are with Senseable City Lab, MassachusettsInstitute of Technology, Cambridge, MA, USA. H. Zhang is with SenseableCity Lab and the Department of Urban Studies and Planning, MIT. P. Santiis with Senseable City Lab, MIT, and the Istituto di Informatica e Telematicadel CNR, Pisa, Italy. (email correspondence: mvazifeh@mit.edu).

stations should be deployed in order to minimize the driversdiscomfort in an urban environment.

As discussed in detail in the next section, there has beena number of studies aimed at introducing optimization frame-works for the deployment of charging stations. These studiesuse traditional methodologies developed in the field of facilitylocations optimization, which are based on computer simu-lations of the traffic flows and travel surveys [7]–[15]. Bothcomputer simulations and travel surveys are known to havemajor limitations in terms of scalability, accuracy, and/or cost.This explains the high interest in the possibility of accessingand analyzing real anonymized individual trajectories to feedthe optimization process. This is possible nowadays given theavailability of massive data sets of digital traces recordingindividual-level human movements, such as those provided bycell phone records. These and other data sets collected byurban sensing devices have been used in a number of recentpapers [16]–[20], [20]–[22]. However, to our best knowledgedata-driven approaches for optimally locating EV chargingstations have not been explored so far.

This motivated us to develop the data-driven optimizationmethodology presented in this paper. The methodology isbased on first constructing a model for charging station energydemand that takes into account the mobility patterns of realindividual movements recorded in the region surroundingBoston. More specifically, we have used a massive data setof over 1 million cell phone users whose activity has beenrecorded over a 4 months period. The second step of themethodology consists in formulating a discrete optimizationproblem to find the optimal configuration for a network ofcharging stations. To this end, the region of interest is dividedinto a geographical grid, and an objective function is definedto simultaneously minimize the overall number of chargingstations required and the aggregate distance EV drivers needto travel to reach the closest charging station. Since optimallysolving the problem at hand is computationally infeasible, wethen present computational efficient, near-optimal solutionsbased on greedy and genetic algorithms. The third and finalstep is assessing the performance and robustness of the pre-sented approach, which is done by: i) comparing quality ofthe obtained solutions vs. those provided by a randomized,but locally optimized approach; and ii) showing that thenear-optimal solution computed using single day movementspreserves 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 for-mulate the problem and introduce the data-driven optimizationframework. The optimization framework is then described in

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sections IV through VII: we describe the mobile phone datasetused in the analysis, and how it has been processed to feedthe optimization framework. In Section VIII, we present anddiscuss the results of the proposed optimization methodology,including a thorough robustness analysis. Finally, Section IXpresents conclusions and discusses future research directions.

II. RELATED WORK

Facility Location Optimization (FLO) problems have beenextensively studied in the field of logistics and transportationplanning over the past few decades. With the exception ofa single work mentioned below, to our best knowledge themethodologies used for FLO in the context of EV chargingstation were based on theoretical models, simulations, and/oraggregate transportation data obtained from census tracts ortravel surveys. Some of these approaches were based onindividual trip trajectory analysis which required theoreticalmodelling of the behavior of the service seeker agents ortravel-surveys. A parking-based assignment introduced in [7]is based on minimizing EV users’ stations access costs whilepenalizing the unsatisfied demand. The case study relied on30,000 personal trip records based on household travel surveysin Seattle, US. In [10], a multi-objective optimization prob-lem is introduced considering various costs associated withcharging stations (including construction, operational costs,charging related costs). The aim of this study was to minimizethe costs with consideration of charging demand distributionobtained from aggregate transportation data for Beijing.

In another work, the authors of [11] proposed an opti-mization problem from the EV driver’s perspective, wherethe objective is to minimize user discomfort measured as thedistance between the charging demand spots and the locationof the stations. This study used static population and income-level in Hong Kong as an input to model the spatial distributionof charging demand. In [13], EV traffic and drivers’ behaviourhave been simulated on the real road network of Vienna. Basedon a simple linear battery depletion model, an approach isproposed to optimize the location of charging stations for alimited number of EVs with the objective of minimizing theoverall travel time.

In [9], a study on optimizing the location of EV chargingstations for a neighborhood in Lisbon is presented based onestimation of the refuelling demand through the applicationof a maximal covering model. In this study, the refuellingdemand estimation was based on 2001 static census data. Inanother work, an optimization framework was introduced todeploy charging stations and to assign optimized number ofplugs based on real EV taxi-trajectory data, with the objectiveof minimizing the average time to find a charging stationand waiting time before charging [8]. Using LP-Rounding,the authors provided approximate solutions to the NP-hardoptimization problem. To our best knowledge, this is the onlystudy in the field which is based on sensor-based individualtravel trajectories data analytics. However, taxi data is knownto represent only a small fraction of the actual mobilitydemand, which can be better characterized by means of cellphone data [23] as done in this paper.

Fig. 1: The sampled individual location traces across spaceshown for two different users. The vertical dimension repre-sents time (an entire week in July 2009) in increasing orderstarting from the plane. Relatively more dispersed mobility ofthe red vs. the green user is clearly visible.

Other studies have proposed multi-objective optimizationproblems to locate charging stations. For instance, in [24]the authors proposed a model based on set cover with dualobjectives of minimizing the overall costs of opening newcharging stations and maximizing their overall coverage onTaiwan road network based on static population density data.In [15], a linear model is used to estimate EV penetrationin various sub-regions to determine the volume of EV flowsbetween the sub-regions based on demographic data. Thissimulation model is then fed into an optimization problemto obtain the location of charging stations considering anobjective function which maximizes the EVs that charge atthe new public stations.

The aforementioned works consider various aspects and areimportant first steps towards constructing a comprehensiveoptimization framework for efficient planning of EV chargingstations. In this work, we further grow the body of knowledgein this important topic by proposing the first optimizationframework based on cell-phone data. The importance of re-lying on cell phone records lies on the fact that a recent studyhas shown their very good accuracy in modelling individualtrip origin/destination flows in urban settings [23]. We thusbelieve that cell phone data fed optimization will become aprominent approach for many urban facility location problems,of which EV charging station is a first example.

III. INTRODUCING THE FRAMEWORK

Our data-driven optimization framework relies on analyzingthe movement patterns of individuals obtained through cell-phone data over the span of 4 months. Starting from that, weformulate a discrete optimization problem equivalent to thewell-known set cover problem [25] by dividing the region ofinterest into a geographical grid. The objective is to minimizethe total aggregate distance travelled by drivers on their route

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Fig. 2: The home-location population obtained using thehome-detection method described in the text. Higher popu-lation density in the Metro Boston area is clearly visible.

from the end of their intended trip to the closest availablecharging 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 chargingstation locations.

The overall optimization framework can be divided intothe following stages:

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

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

C) Analyzing the Data: Processing the raw data to estimateEV energy demand based on a simple assumption for EVadoption rates

D) Optimization Methods: Performing a number ofoptimization methods including Chvatal greedy searchand GA meta-heuristic search to find near-optimal locationsof EV charging stations.

Each step will be described in a separate section.

IV. STEP 1: DEMAND MODELLING

We assume the urban area is partitioned into a number ofnon-overlapping square cells C = C1, C2, ..., CN, e.g., cor-responding to a square grid partitioning. Similar to space, timeis also divided into a number of non-overlapping intervals,corresponding to, e.g., ∆t = 30 min slots. In the following,we use i as a generic cell index and t as a generic time slotindex. Equipped with the above definitions, we now define thefollowing quantity for each spatio-temporal slot (i, t):

INi,t = number of trip segments ending (1)in cell Ci during time interval [t, t+ ∆t) (2)

We concatenate consecutive trip segments if the stay timebetween them is less than τmin, where τmin represents the

minimum time interval needed for completely charging anEV, and is set to 30 min in the following. The trip segmentscounted in INi,t are then those at which ends it is possible tocompletely charge the EV.

Note that, if we assume that the duration of a time slot∆t is 6 τmin, all segments contributing to INi,t belongs todifferent vehicles. A fraction 0 < δ ≤ 1 of INi,t, i.e., δ ·INi,t,can be defined as the number of candidate EVs arrivingand potentially demanding fast refuelling their batteries in(i, t). We construct this refuelling demand model based onthe durations of the stay at various cells, and by assuming alinear energy consumption model with the trip length. Morespecifically, uE(t) = (uE(0) − c · l)Θ(uE(0) − c · l) whereuE is 1 when the EV is fully charged and 0 when the batteryis empty of charge and Θ(x) is a step function which is zerofor x 6 0 and 1 when x > 0. The constant c depends on anumber of parameters including road conditions, traffic flow,driver behavior, and vehicle type. For simplicity, we assumethat the battery depletion is linear on average with distancetravelled, and that the range is 150km, which is equivalent tosetting c = 1/150km−1. In order to have a better estimate ofrefuelling demand at each cell, at each time slot we only counttrip segment end points with trip lengths equal to or largerthan lmin ∼ 100km. We obtain the number INi,t of needed”chargeable” parking spots at each cell according to theseconsiderations. It is important to note that cell phone data doesnot allow immediately distinguishing between transportationmodes. For this reason, we simply assume that the shares ofEVs are uniformly distributed in space proportionally to theenvisioned penetration ratio δ of electric vehicles. A betterapproximation would be to consider a spatial inhomogeneityin the penetration rate as it has been proposed in [15], whichis left for future work. In the rest of the paper we assumeδ = 0.25.

V. STEP 2: FORMULATING THE OPTIMIZATION MODEL

Based on the topology of the cell partitioning, we build acell network CN as follows. We add a node for each cell Ci ∈C, and we add a link between Ci and Cj if the two cells arespatially adjacent to each other. For instance, if C correspondsto a square grid partitioning, the node corresponding to cellCi in CN is connected to the nodes corresponding to the cellsadjacent to Ci in the North, East, South, and West direction.We now introduce a parameter h, where h is an integer > 0,which models driver discomfort in terms of how far a driver iswilling to drive to reach a charging station. Having set h, wesay that a charging station located at Ci covers a cell Cj if therespective nodes in CN have hop-distance at most h. In turn,we say that Cj is covered if there exists at least one cell athop-distance at most h from Cj that hosts a charging station.

The tradeoff between drivers discomfort and the coverageis now clear: if h = 0, driver discomfort is minimum, but acharging station located at Ci covers only the cell itself. Withincreasing values of h, we have a higher driver discomfort,but coverage of charging stations increases as well, and lesscharging stations are required to cover the entire city. Eachnode in CN is then weighted as follows. Let N j

i be the set of

4

nodes at hop-distance j from node Ci1 in CN . The weight wi

of node Ci is computed as

wi =1

k

∑s=1,...,k

∑j=0,...,h

(j − h) ·∑z∈Nj

i

δ · INz,ts

(3)

Weight wi is designed to model the discomfort imposed ondrivers by locating charging station at node i. The discomfortis assumed to be linearly proportional to the hop distance tothe charging station, hence it is 0 for drivers arriving in Ci, 1for drivers arriving in a cell Cz at hop distance 1 from Ci, andso on. So, the total discomfort is obtained by summing up thenumber of drivers arriving in cells at a certain hop distancefrom Ci, up to the maximum value of h. Furthermore, thisquantity is averaged across the total number of time slots kconsidered in the analysis. Note that as we only consider cellsin the coverage range of Ci in the computation of wi, all theweights are negative as j 6 h.

We are now ready to formulate the charging station opti-mization problem. The problem of finding the optimal locationof charging stations can be formulated as a weighted set-covering problem [25] in which the universe set, U , and theset of subsets, S, are given as follows

U = C1, C2, ..., CN, (4)

S = s1, s2, ..., sN, (5)

where si is defined as

si = Cj |Cj ∈h⋃

z=0

N iz, (6)

i.e., the set of all cells within hop-distance h from Ci in CN .Each si is weighted with w(si) = wi, which is a measure ofthe total discomfort experienced by the drivers in cells coveredby Ci. The optimization problem is to find a subset Sopt of Ssuch that all elements of U are covered, and the sum of theweights in Sopt is minimized. Note that when h = ∞, eachof the si’s coincides with the universe set, so the optimal Cis the si that has the lowest wi. When h = 0, si = Ci andall the weights are zero, therefore optimal Sopt = S itself,implying that there must be a charging station at every cell.We are then interested in investigating the non-trivial case inwhich 0 < h <∞.

The weighted set covering problem described above can beformulated in terms of the following constrained integer linearoptimization problem (ILP):

MinimizeN∑i=1

(wi · xi), (7)

subject to: (8)

pi =∑

j=1,...,h

∑z∈Ni

j

xz > 1 (i = 1, ..., N), (9)

xz ∈ 0, 1

1To simplify notation, in the following Ci denotes both a cell and itscorresponding node in the cell network CN .

Note that pi is the number of available stations in the h-proximity of Ci. In practice, there is a finite capacity on thenumber of EVs a charging station can handle at a time, i.e.,nc. If we require that the capacity is not exceeded, then oneneed to make sure that the following condition is satisfied forevery cell:

pi > ceiling[

max(INi,t)

nc

]= ki (10)

where max(INi,t) = maxt INi,t is the maximum value ofINi,t during the observed time interval. Interestingly, this mod-ifies the inequality constraint in a simple way. The differenceis that instead of > 1, we have > ki. Therefore, the problembecomes:

MinimizeN∑i=1

(wi · xi), (11)

subject to: (12)

pi =∑

j=1,...,h

∑z∈Ni

j

xz > ki (i = 1, ..., N), (13)

xz ∈ 0, 1In the following, we assume that the capacity of charging

stations is large enough to address all energy demand inthe coverage range, which is equivalent to setting ki = 1for all Cis. As discussed in Appendix A, the model can beeasily generalized to consider capacitated case for the chargingstations. This way, the number of EV charging plugs requiredin each cell can be optimized to maximize the coverage andto distribute the load evenly between the charging stations.

The set cover problem can be represented also in binarymatrix form – useful for GA implementation – as follows. AnN×N binary matrix is formed where rows are associated withcells that need to be covered, and columns are associated withstations at various cell locations. Now the (i, j)-element ofthis matrix is either 1 or 0 depending on whether the stationat Cj covers Ci or not. This binary matrix, which we callSCP, along with a weights array W uniquely represents theset cover problem:

SCPhij =

1 Cj ∈

⋃hz=0N

iz

0 otherwise. (14)

Each configuration of charging stations in this binary repre-sentation is a binary vector in which the indices of nonzeroelements correspond to indices of the cells which have charg-ing stations in them.

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

VI. STEP 3: ANALYZING THE DATA

A. DatasetThe dataset consists of estimated anonymous location traces

from approximately a million users in the Boston Metropolitan

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Fig. 3: Layout of charging stations as computed by the GA for different values of stations coverage h = 0, 1, 2, 3, 4, 5.

Areas, Massachusetts, USA collected by Airsage over the spanof 4 months in 2009. The data set contains connection events,including when a text message is sent or received, when acall is placed or received, and when the user connects to theInternet. The location estimation is generated through triangu-lation by means of the AirSage’s Wireless Signal ExtractionTechnology. The average inter-event time measured for thewhole population is approximately 260 min. However, thedistribution of the inter-event times has an arithmetic averageof medians which is around 84 mins, and the geometricmedian is around 10.3 min. This relatively high temporalresolution enables us to detect significant portion of the eventsassociated with location changes of the users. The data setis then suitable to characterize individuals’ mobility behavior,which is needed to feed the proposed optimization framework.

B. Processing call data records

The first data processing step is to identify mobility from thedetailed records of cellphone usage. If a continuous sequenceof records occurring in the same cell are observed, the useris assumed to stay in that cell from the earliest time in thesequence to the latest. This implies that the records in themiddle of the sequence are not relevant for our analysis. Basedon this observation, we reduce the list of records from

ci0t0 , . . . , ci0tl, . . . , c

iptk, . . . , c

iptk+m

, . . . , ciqtn−s

, . . . , ciqtn

where ciptk represents a cellphone usage recorded at time pointtk in cell Cip , to a shorter list

ci0t0 , ci0tl, . . . , c

iptk, c

iptk+m

, . . . , ciqtn−s

, ciqtn

obtained removing in-cell records. The list is then transformedinto the following list of tuples

(i0, t0, tl), . . . , (ip, tk, tk+m), . . . , (iq, tn−s, tn) .

To simplify notation, we relabel this list with continuoussubscripts according to their orders in the list, and label thestarting and ending times of a location with superscripts ‘start’and ‘end’

(i0, tstart0 , tend

0 ), (i1, tstart1 , tend

1 ), . . . , (iw, tstartw , tend

w ) .

C. Home Detection

For each user in the data set, we then identify a home cellamongst the list of visited cells. This step is necessary becausewe assume that a user does not need to charge her EV ata public charging station if she is at home. Therefore, thearrivals at home location cells will not be counted in INi,t.Fig. 1 illustrates two sample users’ location traces during aweek using the vertical axis as time. Each cellphone recordis a point on the map based on its location ((x, y) axes) andtime of occurrence (z axis).

Following the methodology proposed in [22], the homelocation H(uj) of a user uj is determined by finding the cellwhere she cumulatively spends most of the time from 8p.m.to 6a.m. in a week, i.e.

H(uj) = Ck, k = argmaxp

∑s:is=ip

(tends − tstart

s )

where tstarts , tend

s are only considered if they are between 8p.m.and 6a.m. In Fig. 2 we show the spatial distribution of theidentified home locations with a sample of a week’s data.

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July-07-2009 Oct-05-2009

d d

< d > (km) < d > (km)

h = 1km

h = 2km

h = 3km

h = 4km

Fig. 4: The scatter plot of the average driving distance to the closest station, and its variance for a population of the solutionsbefore (larger cluster) and after (smaller cluster) GA. Each point represents a charging station layout.

D. Counting the Number of Trips Ending in a Cell

The process of counting INi,t is implemented with a linearsearch. For each user, starting from the beginning of thereduced list of records, we track the total distance the userhas traveled from the last charging cell to the current cellthrough a path along all the intermediate cells without acharging opportunity, i.e., with stays shorter than 30 minutes.For example, if the last charging cell is Cis , since then theuser has gone through Cis+1 . . . , Cis+k−1

, while the currentcell is Cis+k

, the total tracked distance is

Ds+k =

s+k−1∑p=s

Dip,ip+1

where Dip,ip+1 is the Manhattan distance from Cip to Cip+1 .If during the tracking process, the user stays for longer

than 30 minutes at a non-home cell Ci and the total traveleddistance is at least lmin = 100km, the trip is counted in INi,t

for the respective time period t. Then the tracking distancewill start from 0 again, until next charging cell is found.

This process is repeated for all the users in the data set,so that the INi,t values of all cells for all time periods aredetermined.

VII. STEP 4: OPTIMIZATION METHODS

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

A. Greedy search

The first approach is based on a greedy search [26], whichis used to find a set of column indices that cover all the rows inthe binary matrix representation. The algorithm is summarizedin the pseudo-code reported in Algorithm 1. In this greedysearch method, at each optimization step the location of acharging station is chosen by sorting all the available cellsaccording to their average weight per uncovered cell in theircoverage range, and a station is placed in a cell corresponding

time

<d

>(k

m)

Fig. 5: The average distance to the charging stations for GAand a randomly selected layout obtained based on a single dayin July 2009 computed for the demand on various days in aweek selected from the months July, September, and October2009. The line with darker color correspond to GA and thelighter lines represent a randomly selected solution from therandom population of size 1000. The shaded area representsthe deviation in distance from the average.

to the lowest value of the average weight per uncovered cell.This process is repeated until all cells are covered. Once thegreedy coverage phase is terminate, the algorithm then checkwhether cells hosting redundant charging stations exist, and,in case they exist, remove them. By eliminating redundantcells, the number of stations will be subsequently minimized.Later on in the paper, we discuss how the objective functioncan be modified to reduce the number of stations ever furtherby simultaneously minimizing the number of stations requiredand the average distance to the closest charging station.

B. Generating initial GA population

While providing a provable approximation bound ofO(logN), the solution provided by the greedy search methodis known to be inferior to those provided by more sophisticatedheuristic/meta-heuristic methods. In this section, we present aGenetic Algorithm (GA) approach.

7

The first step is randomly constructing a population ofpossible layouts by stochastically and uniformly distributingcharging stations until all the cells are covered, and thenremoving the redundant cells that contribute the most to theoverall objective function. This way, we can generate a pop-ulation of locally optimized layouts for charging stations thatcan be used as the starting point of the GA. Another approachto generate a population of the solutions is a generalization ofChvatal’s search method by adding stochasticity to the searchscheme. In this probabilistic version of Chvatal’s algorithm,deterministic choice of the best possible cell at each stageis replaced by random choice of a cell from a small subsetof k elite locations, which are the best k possibilities out ofall available cells at each step according to how locationsare sorted in Chvatal’s search. The required change in thealgorithm is explained in pseudo-code reported in Algorithm2. This way, the solution is no longer deterministic, and eachexecution of the algorithm results in a different solution. Werepeat this process until a population of a desired size isproduced. As one increases k, the search space becomes morediverse; however, the average quality of the solutions degrades.On the other hand, decreasing k reduces the size of thesearch space; however, the average quality of the populationis higher. In the following, we set k = 5, which represents agood compromise between size of the search space and thepopulation quality.

After the initial population is generated, we need to definerules to replace less fit members of the population withnew offsprings. Towards this purpose, we closely follow thestrategy presented in [27], which uses a binary tournamentselection and a fusion crossover operator to evolve the popu-lation. The relevant pseudo-code is reported in Algorithm 3,4. This way the average weight of the population improvesand finally the population converges to a high quality set ofsolutions, which has a significantly lower average distance tothe charging stations than that of the best member of the initialpopulation in the most cases – see next section for details. It isimportant to note that new binary vectors produced using thefusion crossover operation are not necessarily feasible layoutsas they may contain redundant cells or uncovered cells. Weneed to perform a local search to find a close feasible solutionin which all the demand cells are covered and there is noredundancy in the layout. This part is the most computationallyexpensive part of the algorithm, and it is considered as anecessary step to make sure that the minimum number ofstations is used to cover the demand. What we do in this step isvery simple: we find all the uncovered cells, and then performa greedy Chvatal’s approximation to cover these remainingcells as we described in the previous section. Then, we removeredundant stations in exactly the same fashion as described inthe redundant cell removal stage of Algorithm 1. To diversifyour search space further, we use constant mutation operatorwhich mutates one of the components in the newly generatedvector (corresponding to a charging station configuration) inthe binary vector representation. The component is chosenamong the ones that two parents share with each other. We runthe GA by generating 40, 000 children and replacing them withthe less fit members of the population. The initial population

evolves to a new population after each iteration, till the totalnumber of 40, 000 iterations is reached.

Algorithm 1 Chvatal greedy approximation

InitializationW ← get the weights arrayS ← set of all cell indices at which the stations can belocatedU ← set of all cell indices which must be coveredΓ(s) ← set of cell indices in the coverage range of theelement s in S, i.e., row indices corresponding to thenonzero elements on the i-th column of SCPΩ(u) ← set of allowed cell indices which can provide cov-erage for element u in U , i.e., column indices correspondingto the nonzero components on the i-th row of SCPUc ← set of the so far covered cell indices ← ∅Sc ← set of so far picked columns ← ∅Greedy coverage phasewhile not Uc = U do

pick an item s in S that has the minimum weight perits still uncovered rows

S ← S − sSc ← Sc ∪ sUc ← Uc ∪ previously uncovered rows covered by s

end whileSc ← list(Sc) list of picked column indices sorted accord-ing to their weights in decreasing orderQ← ScRedundant cell removalfor s in Sc do

counter ← 0for c in Γ(s) do

if Ω(c)⋂Q− s = ∅ then

counter ← counter + 1end if

end forif counter = 0 then

then Q← Q− send if

end forreturn Q

VIII. RESULTS

Fig. 3 shows the best configuration obtained for variousvalues of the coverage range h, where the INi,t values arecomputed from the movement data of a single day in July2009. The h = 0 case is the trivial case where there is acharging station at each demand spot, and it just reported toshow cells with non-zero energy demand. As one expects,by increasing the value of h, a lower number of chargingstations is required to cover all the demand. It is interestingto observe that, while the solutions obtained for a relativelylarger value of h contains relatively less stations as expected,the computed charging station location sets are not subsets ofthose representing previous solutions, indicating the non-trivialcombinatorial structure of the problem at hand.

8

Algorithm 2 Stochastic Chvatal

Replace the while loop in the Chvatal greedy approximationwith the followingwhile not Uc = U do

Sk ← pick k item from S that has the least weight pertheir uncovered rows

s← choose one item randomly from Sk

S ← S − sSc ← Sc + sUc← Uc + previously uncovered rows covered by s

end while

We can also compute the average distance to a chargingstation from each demand spot, and compare the initial andfinal populations of configuration layouts. In Fig. 4 a we reporta scatter plot where each point represents a charging stationconfiguration, and the horizontal and vertical axes representsthe average distance and the variance of the distances to thecharging stations from the demand spots. Different colors cor-responds to different values of h. As one can see, for any valueof h the final population (smaller clusters) has significantlylower average distance to charging stations compared to initialpopulations (larger clusters of points).

An important aspect to investigate is whether the chargingstation configuration computed using data from a single dayis still near-optimal also for the following days. In otherwords, we would like to assess the robustness of the computedsolution in presence of variation of the input data, whichis especially important in infrastructure planning processesgiven the high costs and long lifetime of infrastructure. Tothis purpose, we have used the near-optimal charging stationconfiguration obtained using a single day in July, and assessedits performance when the INi,t values are computed fromdata of various days in a week selected from the monthsof July, September, and October 2009. As reported in Fig.4 b, using the movement data for an equivalent day twomonths later in October yields very similar results, indicatingthe robustness of the computed near-optimal configuration.Fig. 5 reports the average distance and the variance for themovement data for three weeks chosen from three months ofJuly, September and October, using the same configurationobtained for the single day in July. This plot clearly showshow the achieved improvement in distance remains stable overtime. The robustness of the proposed optimization approach isa consequence of the stationarity of the aggregate movementpatterns, which have been repeatedly observed in the literature.

Another way to investigate the robustness of our optimiza-tion approach is to tune the parameters of the EV demand tosee how sensitive the average driven distance is with respectto a change in these parameters. In Fig. 6, we take the h = 1case as a reference, consider the layout configuration obtainedfor τmin = 30 mins and lmin = 100km, and recompute theaverage distance based on the demand for different values ofτmin and lmin. Again, the gap in the average distance betweena randomly obtained layout and the best layout obtainedusing GA shows significant robustness in the space of these

Fig. 6: The average distance to the charging stations for alayout obtained using GA based on a single day in July 2009,computed for various demand parameters.

parameters. This shows that the computed charging stationconfiguration remains stable not only across time, but also incase technological development significantly change chargingtimes and EV ranges.

Finally, we consider robustness to changes in the EVpenetration ratio δ. In this case, it is easy to observe that thestructure of the computed solutions is completely independentof the penetration ratio, since δ is a parameter the uniformlymultiplies all weights in the problem formulation.

So far, in the minimal objective function we have notconsidered the cost associated with building new chargingstations. Therefore, it is important to compare the numberof stations required in different layouts versus the averagedistance to make sure the number of charging stations isminimized properly. In Fig. 7, we present a scatter plot of thenumber of stations versus the average distance to the chargingstations for each configuration for charging stations network.This plot shows that layouts with similar numbers of stationscan lead to significantly different average distances. The GApopulation tends to have higher number of stations comparedto the average of the initial population. This is consistent withthe fact that no penalty term exists in the objective functionto penalize addition of new charging stations reflecting thecost of building and operation. To address this issue, wemodify the objective function by adding a positive constantw0 to all cell weights. This offset value can in general reflectthe cost associated with building a new charging station orother spatial restriction which may exist such as compatibilitywith the power grid. In the simplest case, we consider aspatially uniform offset and optimize its value to significantlydecrease the number of charging stations without too muchsacrificing the improvement gained in the average distance.In Fig. 8, we reproduce the result with this modification tothe objective function. The number of charging stations issignificantly reduced compared to the previous case; however,the improvement in average distance remains still significantin most cases.

9

< d > (km)

Ns

h = 1km

h = 2km

h = 3km

h = 4km

Fig. 7: The total number of charging stations in each layoutversus the average distance to charging stations from thedemand spots for set of layouts before (large clusters) andafter (small clusters) GA.

< d > (km)

Ns

h = 1km

h = 2km

h = 3km

h = 4km

Fig. 8: The total number of charging stations in each layoutversus the average distance to charging stations from thedemand spots for set of layouts before (large clusters) and after(small clusters) GA after the objective function is modified toconsider the cost of building a new charging station by addinga uniform positive offset, w0 = 25w to all of the weights,where w is the average weight of the cells.

IX. CONCLUSION

Efficient deployment of the network of public chargingstations is an important matter which will play a significantrole in further increasing the market share of the EVs in thenear future. Motivated by this, we have proposed a modellingand optimization framework to find efficient layout of chargingstations to minimize overall EV drivers discomfort, which isreflected in the distance they need to travel to reach the closestcharging station starting from their refuelling demand spots.Our minimalistic model can be easily generalized to includemore objectives. For instance, spatial variations in the costof constructing charging stations and the capacity of eachstation can be considered as an optimization variable withoutchanging the problem structure and the required optimizationmethod. We also showed that the computed near-optimal

Algorithm 3 Binary Tournament Selection and FusionCrossoverN ← Dimension of the binary vectors in the population,which is the same as the number of cells allowed toaccommodate charging stations(v1,v2) ← Randomly select two subset of binary vectormembers each of size T = 2, and select the fittest in eachsubset.(w1, w2) ← Compute the corresponding weights for eachof the selected memberspc ← w2

w1+w2

vnew ← v1

for i in 1, 2, ..., N doif v1(i) 6= v2(i) then

p← draw uniformly from [0, 1]if p > pc then

vnew(i)← v2(i)end if

end ifend for

Algorithm 4 Genetic Algorithm

P ← generate initial population set of size M usingstochastic Chvatal’s algorithmcounter ← 0Mc number of iterations in the GAconverged← Falsewhile not converged do

vnew Generate a new member using Binary Tournament,fusion crossover, and mutation operator

vnew ← make the new vector feasible using a localChvatal algorithm.

if not vnew ∈ P thenvl ← select a member randomly from the population

with a weight above P’s average weightP ← P − vlP ← P + vnewcounter ← counter + 1

end ifif counter = Mc then

converged← Trueend if

end whilevbest ← fittest member in Preturn vbest

layout is robust to variation in input data, in EV penetrationrate, as well as in EV and charging station technology. Thisaspect is particularly important given the high costs related toinfrastructure deployment, and promote data-driven planningas a viable solution for optimal EV charging station location.

APPENDIX ACAPACITATED CASE

Assume that a charging station can handle l EVs at a time.Now if it turns out that, frequently, there are q > l EVs

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in a particular cell over a significant time period, then onestation is not enough for that cell and that cell needs to becovered by more than one stations. One way to incorporatethis optimization dimension in the context of the set-coveringproblem is to duplicate the corresponding Ci in the universeset. In other words, if it turns out that cell Ci needs to becovered by at least ki stations, we can redefine the universeset U by replacing the Ci element in it with C1

i , C2i , ..., C

kii .

The weight needs also to be modified accordingly: INi,t →INi,t/ki. With these modifications, the universe set grows,and more subsets are required to cover the whole set. Also,the discomfort contribution of cell Ci decreases for a stationlocated at distance j, because its load is now divided betweenthe other ki − 1 available stations2. The interesting point isthat the problem is still a set covering problem.

ACKNOWLEDGMENT

The authors thank ENEL Foundation, Accenture China,American Air Liquide, Emirates Integrated Telecommunica-tions Company (du), Ericsson, Kuwait-MIT Center for Nat-ural Resources and the Environment, Liberty Mutual Insti-tute, Singapore-MIT Alliance for Research and Technology(SMART), Regional Municipality of Wood Buffalo, Volkswa-gen Electronics Research Lab, and all the members of the MITSenseable City Lab Consortium for supporting this research.

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Mohammad Vazifeh received the Ph.D. degree inphysics from the University of British Columbia(UBC), Vancouver, Canada. He is currently aPostdoctoral Fellow at MIT Senseable City Lab.His background is in theoretical condensed matterphysics. Since 2015, his research focus has shiftedto machine learning, optimization algorithms designand urban computing. Before joining Senseable CityLab, he worked as a senior algorithm researcher,designing optimization algorithms for the quantumcomputing hardware with adiabatic quantum anneal-

ing architecture. At MIT, he is involved in a number of urban mobility projectsincluding modelling human mobility across urban and regional scale and data-driven optimization in urban infrastructure planning. Currently, his main topicof interest is modelling and predicting human mobility in cities.

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Hongmou Zhang received the Master of City Plan-ning degree from the University of Pennsylvania. Heis currently a Ph.D. student at the Department ofUrban Studies and Planning, Massachusetts Instituteof Technology. His current research interests are ur-ban modelling and simulation with massive mobilitydata, and behavioural modelling in transportation.

Paolo Santi Paolo Santi is a Research Scientist atMIT Senseable City Lab where he leads the MIT-Fraunhofer Ambient Mobility initiative, and he isa Senior Researcher at the Istituto di Informatica eTelematica del CNR. His research interest is in themodeling and analysis of complex systems rangingfrom wireless multi hop networks to sensor and ve-hicular networks and, more recently, smart mobilityand intelligent transportation systems. In these fields,he has contributed more than 100 scientific papersand two books. Dr Santi has been Guest Editor of

the Proceedings of the IEEE, and has served in the Editorial Board of IEEETransactions on Mobile Computing and IEEE Transactions on Parallel andDistributed Systems. Dr Santi has recently been recognized as DistinguishedScientist by the Association for Computing Machinery.

Carlo Ratti received the Ph.D. degree from theUniversity of Cambridge, Cambridge, U.K. He iscurrently the Director of the Senseable City Labo-ratory, Massachusetts Institute of Technology, Cam-bridge, and an Adjunct Professor with QueenslandUniversity of Technology, Brisbane, Qld., Australia.He is also a founding partner (with W. Nicolino)and the Director of the architectural firm Carloratti.Dr. Ratti is a member of the Ordine degli Ingegneridi Torino and the Association des Anciens Elves del’Ecole Nationale des Ponts et Chaussees.