location design of electrification road in transportation

Research ArticleLocation Design of Electrification Road in TransportationNetworks for On-Way Charging

Yue Qiu1 Yuchuan Du 1 Shanchuan Yu2 and Shengchuan Jiang1

1Key Laboratory of Road and Traffic Engineering of the Ministry of Education Tongji University Shanghai 201804 China2China Merchants Chongqing Communications Research amp Design Institute Co Ltd Chongqing 400067 China

Correspondence should be addressed to Yuchuan Du ycdutongjieducn

Received 18 January 2020 Revised 14 March 2020 Accepted 26 June 2020 Published 23 July 2020

Academic Editor Wei (David) Fan

Copyright copy 2020 YueQiu et al-is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Electric vehicles tend to be a great mobility option for the potential benefits in energy consumption and emission reduction On-way charging (OWC) has been recognized to be a promising solution to extend driving range for electric vehicles Location of theelectrification road (ER) is a critical issue for future urban traffic management to accommodate the new mobility option -ispaper proposes a mathematical program with equilibrium constraint (MPEC) approach to solve this problem which minimizesthe total travel time with a limited construction budget To describe the driversrsquo routing choice a path-constrained networkequilibrium model is proposed to minimize their travel time and prevent running out of charge We develop a modified activeset algorithm to solve the MPEC model Numerical experiments are presented to demonstrate the performance of the model andthe solution algorithm and analyze the impact of charging efficiency battery size and comfortable range

1 Introduction

Electrification systems based on renewable energy powersources are introduced in the urban transportation systemfor positive environmental effect and carbon reduction[1 2] Recently with the increasing concern about sus-tainable transportation [3] electric vehicles (EVs) are widelyadopted in urban travel Electrification road (ER) is con-sidered as a promising domain for sustainable electricalenergy harvesting to support on-way charging (OWC) forEVs thanks to the advancement of application of photo-voltaic piezoelectric and pyroelectric materials convertingsolar energy physical pressure and thermal energy intoelectrical energy eg [4ndash8] OWC provides a new mode ofcharging for EV drivers to extend driving range who aresuffering from range anxiety of running out of power on theway [9 10] and long charging time ranging from 05 to 2hours for a full charge [11] OWC has raised the interests oforganizations and is being tested all over the world ERdeployed on a network has been tested in South Korea [12]UK has conducted a series of OWC recharging tested onhighways [13] In Sweden the Volvo Group and the Swedish

Transport Association planned to build 500-meter ER forwireless power transfer [14] Such tests and experimentsshow that OWC would come true in future Owing to thehigh investment in the ER which reaches $4 million per lanemile [15] it is imperative to find the optimal location

Most previous studies related to the location design ofcharging facilities are based on user equilibrium (UE)problems with EVs which investigate how the limiteddriving range of EVs and location of charging facilitiesaffect the routing choice and subsequently the flow equi-librium distribution on road networks Among thesestudies Jiang et al [16] first proposed a path-constrainedassignment model where lengths of selected routes arewithin the driving ranges of EVs -e model is furtherextended to consider mixed gasoline and electric vehicleflows and their combined routing choices under differentproblem settings [17 18] He et al investigated the optimalprices of electricity and the integrated prices of electricityand roads based on the multiclass spatial distribution ofelectric vehicles across the transportation network [19 20]All these studies assumed the energy consumption of EVs isflow-independent

HindawiJournal of Advanced TransportationVolume 2020 Article ID 7096767 13 pageshttpsdoiorg10115520207096767

-e aforementioned studies do not consider the locationof charging facilities in networks Problems related to theoptimal location of charging stations or battery swappingstations have received much attention He et al [21] firstproposed a UE-based framework for locating chargingstations to maximize social welfare considering routechoices of electric vehicles and further developed threemodels of different flow dependencies and energy con-sumption [22 23] Xu et al [24] defined a network withbattery swapping stations where swapping time is inde-pendent with charging capacity considering the impact ofqueuing to find the best charging strategy -ese existingrelevant studies are intended for charging stations or batteryswapping stations where EVs reach the station stop and getreplenished Such models cannot address issues for OWCdirectly

OWC unlike charging stations or battery swappingstations can charge EVs during the ride OWC technologyhas been implemented in a network in South Korea [12] Alimited number of research studies have been done on theOWC problem to select the optimal ER locations so farRiemann et al [25] proposed a mixed-integer nonlinearprogram to describe the flow-capturing location modelconsidering driverrsquos routing behavior Chen et al [26] in-vestigated the optimal deployment of charging lanes basedon the speed control of EV drivers on the charging lanesChen et al [27] proposed the integrated deployment ofcharging lanes and charging stations along a long trafficcorridor which cannot be applied in the transportationnetwork directly Liu and Song [28] adopted a robust op-timization methodology to provide robust location opti-mization of wireless charging facilities for the electric bussystem to reduce total cost of batteries and charging facilitieswith different uncertainty levels Liu et al [29] developed amixed-integer linear program for OWC location and batterysize optimization and battery size for an electric bus systemwith overlapping bus lines Ahmed et al [30] introduced amethod to find the best combination of battery capacity andwireless charger characteristics to solve the tradeoff betweenmaximizing charge sustaining minimum battery capacityand minimizing the initial investment Bi et al [31] adopteda genetic algorithm to optimize the rollout of OWC infra-structure both spatially and temporally in order to minimizelife cycle costs and energy burdens Zhao et al [32] proposeda biobjective optimization problem for integrated EV lo-cation and on-board battery size design of electric bussystems to minimize deployment cost and reduce energyconsumption of electrified systems

-e above research studies can solve the ER locationoptimization to some extent but they tend to assume thatcharging amount is only related to the charging time Undersuch assumption drivers tend to deliberately slow down toextend the charging time and even EV may revisit the samecharging lanes All the vehicles behind the EV that is movingslowly to get charged are delayed A road contains more thantwo lanes one of which is the ER Vehicles without chargingintention would perform lane-changing which reduces theroad capacity Relationship between the cost of building theER and width is seldom considered

In this paper charging amount is assumed to be relatedwith the driving distance while charging and the candidatesof the ER have to satisfy the minimum energy supply -ewidth of the ER is considered to affect ER building cost -elink performance function is set as the BPR function innumerical examples -erefore the route choice of driverswill be complicated and the UE flows of EVs in thetransportation network will be different

-is paper investigates the location design problem ofthe ER based on the analysis of the routing behavior of EVdrivers which is different from the previous studies withthe charging station or swapping station Our modelinherits from the paper written by He et al [21] butcharging behaviors and routing behaviors are differenttracking back to the specific situation of OWC in contrastto charging at station which leads to a totally differentlocation design Assuming that the energy consumptionrate and recharging rate are flow-independent this paperproposes a path-constrained network equilibrium model(PCNE) considering the routing and recharging behaviorof EV drivers as well as the constraint of driving range-en we develop an iterative solution framework to solvethe PCNE problem Lastly we investigate a mathematicalprogram with equilibrium constraint (MPEC) model tooptimize the ER location to minimize the system totaltravel time under a given construction budget limit Withthe proposed model discussions about the effect of dif-ferent recharging rates comfortable ranges and batterysizes on the location design and system performance areconducted

-e remainder of the paper is organized as follows -enotation assumption formulation of the PCNE problemand a solution framework are elaborated in Section 2Section 3 formulates the location design problem as a MPECprogram Section 31 proposes a modified active set methodto solve the problem Section 4 presents numerical casesbased on the Sioux Falls network and Section 5 offers theconclusions of this study and implications for future re-search on the transportation network of the ER

2 Path-Constrained Network EquilibriumModel for On-Way Charging

-e section proposes a PCNE model to describe the routingand charging behaviors of EV drivers under a given locationdesign which determine the network flow pattern Decisionvariables of themodel include equilibrium flow and chargingamount of the EV

21 Notation and Assumption Following assumptions aremade before the model formulation

Assumption 1 Drivers do not need to revisit the same ER toguarantee that the EV will not be out of charge

Assumption 2 If the EV is full of charge charging processstops while running on the ER

2 Journal of Advanced Transportation

Assumption 3 Given an ER location design a physical pathfrom the origin to the destination may not be feasible for EVdrivers due to the limit for the EV battery

Assumption 4 All vehicles in networks are EVs -is as-sumption can be relaxed as the model can be extended toaccommodate both EVs and regular vehicles

Assumption 5 Recharging rate is not affected by speed ofthe EV

When EV drivers travel from their origins to destina-tions they are assumed to select routes to minimize theirtravelling cost Travelling cost includes electricity cost andtravel time cost Electricity cost is much smaller than traveltime cost [23]-erefore we simply assumed that EV driversselect routes of the least travelling time While running onthe ER EV drivers can decide whether to charge andcharging amount Charging amount is not greater than themaximum charging amount that equals charging ratemultiplied by the distance of the ER

Consider a transportation network containing ER run-ning EVs denoted by G (N A) where N and A are sets ofnodes and link respectively -e link (i j) isin A is directionaland emanates from the starting node i isin N to the endingnode j isin N with the length Dij and the road gradeGij Traveldemands are between a set of O-D pairs W qod representsthe travel demand between the O-D pairs od isinW and Rod

represents the set of paths between the O-D pairs

22 Formulation Following definitions are introduced hereto describe the PCNE problem

Definition 1 A path is usable if an EV is able to complete thepath with the remaining charge state never falling below thecomfortable range

EV drivers are assumed to select routes of the least traveltime among all the usable paths At least one usable pathexists for each O-D pair here for model generality As theenergy consumption rate and recharging rate are assumed tobe flow-independent the usability of a path is independentof the driversrsquo routing choice which can be predeterminedbefore flow distribution given the location design of the ER-e following path-constrained network equilibrium pro-posed by He et al [23] is adopted here

Definition 2 At the path-constrained network equilibriumall utilized paths are usable and travel time of all utilizedpaths of a given O-D pair is the same which is not more thanthat of any unutilized usable paths of the same O-D pair

-e formulation of the path-constrained networkequilibrium is as follows

PCNE

minimizef

1113944(ij)isinA

11139461113936odisinW1113936

risin1113954Rod fod

r χodijr

0tij(z)dz

subject to

(1)

1113944

risinRod

fodr q

od forallod isinW (2)

fodr ge 0 forallod isinW r isin 1113954R

od (3)

where 1113954Rod represents the set of all usable paths between a

given O-D pair od isinW Energy consumption rate andcharging rate depend on the travel time which are flow-independent -erefore the usability of any path is flow-independent which can be calculated f ( fod

r ) is avector of path flow fod

r represents the traffic flow on pathr isin Rod of O-D pair od isinW Constraint (2) demonstratesthe flow balance for each O-D pair Constraint (3) ensuresnonnegativity of flow for each usable path

Here χodijr is a binary variable to represent whether path r

traverses link (i j) isin A which equals 1 if path r traverseslink (i j) isin A and 0 otherwise vij represents the traffic flowon the link (i j) isin A -e travel time of link tij(vij)(i j) isin A is a strictly increasing function of the flow on link(i j) isin A Here assume that link travel time takes the fol-lowing form of Bureau of Public Roads (BPR)

tij t0ij 1 + 015

vij

cij

1113888 1113889

4⎡⎣ ⎤⎦ forall(i j) isin A (4)

where t0ij represents the free-flow travel time of link (i j) andcij represents the capacity of link (i j)

23 SolutionProcedure If we can enumerate all usable pathsbeforehand the proposed PCNE problem is a regularnonlinear optimization which can be solved easily bycommercial nonlinear solvers such as CONOPT Howeverenumeration of usable paths is a time-consuming workHere we adopt the solution procedure proposed by He et al[22] to obtain the solution -e procedure starts with asubset of 1113954R

od od isinW and solves a restricted version ofPCNE defined upon the subset -en the feasibility of thesolution to the restricted PCNE is tested If not a new usablepath will be generated and added to the subset Iterationproceeds until termination

Some new variables are introduced in the subproblemHere lmax and l0 are the battery size and the initial state ofcharge For an EV travelling between the O-D pair od isinWthe state of charge at node i is lod

i δij is a binary parameterrepresenting whether link (i j) isin A is equipped with the ERto support OWC -e variable equals 1 if the street (i j) isequipped with the ER and 0 otherwise ApsubeA denotes the setof all links equipped with the ER Let ωij denote the energyconsumption rate on link (i j) isin A and ϖ denote therecharging rate of the ER -erefore link energy con-sumption is denoted as cij (ωij minus δijϖ)Dij mod is theminimum charge state within the comfortable range fordrivers of the O-D pair od isinW Nt(n) is the set of tails ofthose links heading to node n and Nh(n) is the set of headsof those links emanating from node n K andM are sufficientlarge constants xod

ij is a binary variable indicating utilizationincidence which equals 1 if link (i j) is utilized for traveldemands between the O-D pair od isinW and 0 otherwise

Journal of Advanced Transportation 3

Accordingly ρodij is a variable that equals 0 if link (i j) is

utilized for travel demands between the O-D pair od isinW and is unrestricted otherwise ξn d is a binary variable whichequals 1 when n d and 0 otherwise ξon is a binary variablewhich equals 1 when n o and 0 otherwise and eod

ij rep-resents the recharging amount at link (i j) isin A

For an O-D pair od isinW the link flow obtained( v

[k]ij ) at the kth iteration of PCNE is adopted at the

shortest usable path solution framework (denoted as SUPF-k) -e formulation of SUPF-k is shown as follows

SUPF-k

x[k]

e[k]

1113872 1113873 isin argminxeisinΩ

1113944(ij)isinA

tij v[k]ij1113872 1113873x

odij (5)

where Ω consists of the following conditions

1113944iisinNt(n)

xodin minus 1113944

jisinNh(n)

xodnj minus δn d + δon 0 foralln isin N (6)

lodj minus l

odi + ωijDij minus e

odij ρod

ij forall(i j) isin A (7)

0le eodij leDijδijϖ forall(i j) isin A (8)

0le lodn le lmax foralln isin N (9)

minus K 1 minus xodij1113872 1113873le ρod

ij leK 1 minus xodij1113872 1113873 forall(i j) isin A (10)

lodi minus ωijDij minus e

odij1113872 1113873ge minus M 1 minus x

odij1113872 1113873 + m

od forall(i j) isin A

(11)

lodo l0 (12)

ωij φ Gij1113872 1113873 forall(i j) isin A (13)

xodij isin 0 1 forall(i j) isin A (14)

In the above the objective function is to minimize thetravel time Constraint (6) guarantees the balance of trafficflow constraint (7) specifies the relation between the statesof the charge of EV batteries travelling from node i to node jlink (i j) isin A Constraint (8) ensures the recharging amountof electricity on the ER at link (i j) isin AP does not exceed themaximum charging quantity that equals charging ratemultiplied by the distance of the link Constraint (9) sets theupper and lower bounds of the states of the charge of the EVConstraint (10) suggests that the EV driver can only rechargewhen travelling on the ER Constraint (11) sets the lowerbounds of the states of charge of the EV during the trip tosatisfy comfortable range for drivers Constraint (12)specifies the initial state of charge Constraint (13) is a givenelectricity consumption function with respect to the roadgrade Constraint (14) requires xod

ij to be binary-e formulation assumes that EV drivers can control the

recharging process on the ER -ey can determine the

quantity of recharging on the ER SUPF-k is a mixed-integerlinear program which can be easily solved by commercialsolvers like CPLEX 128 for small- or medium-sizedproblems

-e solution to SUPF-k for each O-D pair denoted as( xod[k]

ij e[k]ij ) is adopted in the set con-

struction of the shortest usable path -e iterative steps ofPCNE is shown as follows

Step 0 set the iteration variable k 0 and traffic flow( v

[k]ij ) ( 0 ) for each O-D pair

od isinW Solve SUPF-k to obtain the optimal solution( x

od[k]ij e

[k]ij ) Construct 1113954R

od 1113954R

od[k]

S Step 1 solve the restricted NE upon 1113954R

od Denote( v

[k+1]ij ) and ( μod ) as the optimal so-

lutions and multipliers associated with constraint (2)Step 2 for each O-D pair od isinW solve SP-(k + 1) andobtain the optimal solution ( x

od[k+1]ij

e[k+1]ij ) For 1113957o1113957d isinW if μ1113957o1113957d gt1113936(ij)isinAtij(v

[k+1]ij )

x1113957o1113957d[k+1]ij 1113954R

od 1113954R

odcup1113954R1113957o1113957d[k+1]

S and go to Step 1 k k + 1If μod le1113936(ij)isinAtij(v

[k+1]ij )x

od[k+1]ij for all O-D pairs

terminate and ( v[k+1]ij ) is the equilibrium link

flow distribution

-e above procedure terminates in a finite number of steps

3 Location Design of the ER

-is section investigates the location design under a givenbudget limit to maximize social welfare (ie minimize thetotal travel time of all EVs in our paper) -e followingassumptions are introduced here

Assumption 5 Construction cost of the ER is related tolength width of the street

Assumption 6 When a street is designed to be equippedwith the ER vehicles driving on any lane of the street can getcharged -at is streets in the network are not mixed withcharging lanes and regular lanes

When the ER location is determined the equilibriumtraffic flow can be derived as well as the recharging androuting behavior -e problem of the location design forelectrification road (LDER) is formulated as follows

LDER

minimizeδ

1113936(ij)isinA

tij vlowastij1113872 1113873vlowastij

subject to (15)

Dijδijϖge minus G 1 minus δij1113872 1113873 + lmax (i j) isin A (16)

1113944(ij)isinA

WijDijδijBleΘ(17)


δij isin 0 1 (i j) isin A (18)

flowast vlowast

( 1113857 isin argminfisinΨ(δ)

1113944(ij)isinA

11139461113936odisinW1113936

risin1113954Rod

(δ)fod

r χodijr

0tij(z)dz

(19)

where Ψ(δ) f | 1113936risinRod fodr qodforallod isinW fod

r ge 01113864 forallod

isinW r isin 1113954Rod

(δ) Wij is the width of link (i j) isin A B is theconstruction cost of the unit-area ER which is the functionof the recharging rate ϖ Θ is the budget limit and G is asufficiently large constant

In the above the objective function represents the totaldriving time Constraint (16) presents the shortest length ofthe street chosen as the ER guaranteeing that Assumption 1always holds Constraint (17) specifies the budget limit forthe ER deployment Constraint (18) requires δij to be binaryConstraint (19) states the traffic flow following the resultpredicted by the PCNE model

31 Active Set Method LDER is a typical bilevel discretenetwork design problem with NE at lower level Many existingsolutions from the literature can be explored for examplebranch and bound [33] simulated annealing [34] and SOrelaxation [35] with consensus of reformulating an equivalentsingle-level model with equivalent optimality conditions of thelower-level problem like KKTconditions to find the solution ofthe bilevel program Problems with the nonconvex feasibleregion generally contain many local optimal solutions andtypically require a time-consuming branch-and-bound schemeto search for a globally optimal solution Such algorithms are notreadily applicable considering the nonlinear and complemen-tary constraints adding enormous amount of calculation

To solve the LDEP we utilize the active set method [36]which terminates after a finite number of iterations ending witha strongly stationary solution -e active set method has beenproved to have the potential for solving larger network designproblems-e active setmethod assigns nonnegative variables ineach pair as zero to construct the location design problem as aregular nonlinear programAdoptingmultipliers associatedwiththe constraints forcing nonnegative variables to be zero theproposed algorithm formulates the binary knapsack problem tosolve the zero-value assignments to decrease the total travel time

-e active set algorithm starts by setting an initial fea-sible solution U δ0ij |(i j) isin A1113966 1113967 and considering two sets

Ω0 (i j) δij 01113966 1113967 (20)

Ω1 (i j) δij 11113966 1113967 (21)

where Ω0cupΩ1 A and Ω0capΩ1 ϕ Under the locationdesign for the ER corresponding to the initial feasible so-lution U0 the NE problem can be solved easily by the path-based iteration procedure proposed in Section 2 Let Vlowast

vlowastij (i j) isin A1113966 1113967 be the solution set of NE which is uniquebecause of the monotonically increasing BPR function

-e basic principle of the active set algorithm is to ex-change elements between Ω0 and Ω1 in order to reduce theoverall travel time in each iteration until the system optimalsolution is found In each iteration the active sets are ad-justed to meet the following theorems proved by Zhang et al[36]

Theorem 1 Given that Ω0 and Ω1 are feasible solution setsif λij lt 0 for some (i j) isin Ω0 switching (i j) from Ω0 to Ω1yields less system travel time If μij gt 0 for some (i j) isin Ω1switching (i j) from Ω1 to Ω0 yields less system travel time

Theorem 2 Ae active set method converges after a finitenumber of iterations

λij and μij are the Lagrangian multipliers associated withequations (20) and (21) respectively

To ensure that budget limit and shortest-length limit aresatisfied in each iteration according to constraints (16) and(17) an embedded program is proposed as follows

minimize 1113936(ij)isinΩ0

λijgij minus 1113936(ij)isinΩ1

μijhij

subject to(22)

1113944(ij)isinΩ0

Dijgijϖ + 1113944(ij)isinΩ1

Dij 1 minus hij1113872 1113873ϖge 1113944(ij)isinΩ0

minus G 1 minus gij1113872 11138731113872 1113873

+ 1113944(ij)isinΩ0

minus Ghij + lmax1113872 1113873 forall(i j) isin A

(23)

1113944(ij)isinΩ0

WijDijgijB + 1113944(ij)isinΩ1

WijDij 1 minus hij1113872 1113873B forall(i j) isin A

(24)

1113944(ij)isinΩ0


μijhij ge θ (i j) isin A(25)

gij hij isin 0 1 forall(i j) isin A (26)

where gij 1 records a shift in (i j) isin Ω0 to Ω1 and hij 1records a shift in (i j) isin Ω1 to Ω0 -e objective of program-ming (22) is to minimize the estimated decrease the negativevalue ofwhich implies a potential reduction of the objective value-e process terminates when equation (22) is zero

Constraint (23) is consistent with constraint (16)Constraint (24) is consistent with constraint (17) Parameterθ is set to guarantee a decrease in the objective function Inthe first iteration θ minus infin In the next iteration θ can becalculated by the following equation

θ ε + 1113944(ij)isinΩ0


μijhij (27)

where ε is a sufficiently small constant Equation (27) pre-vents the solution from degenerating back to the previousiteration by increasing θ by ε


Steps of the algorithm are shown in the following Here SOdenotes the objective value of programming (15) and TD de-notes the objective value of programming (22) (Algorithm 1)

4 Numerical Examples

41 Basic Settings In this section numerical examples onthe Sioux Falls network in South Dakota are presented todemonstrate the performance of the proposed model asshown in Figure 1 which consists of 24 nodes 76 links and14 OD pairs All links are available to be equipped with theER Table 1 reports the link characteristics including free-flow travel time (FFTT) width length and capacity-e ODdemand is given in Table 2

We assume that the battery size lmax is 25 kWh -einitial state of the charge l0 is set as 25 lmax For simplicityit is assumed that energy consumption rate ω at all links inthe network is set as 03(kWhmile) and the recharging rateϖ is set as 25(kWhmile)-e comfortable range for alldrivers mod is set as zero Assume that the cost of buildingunit-area ER is (($2 times 105)(mile times meter)) at the recharg-ing rate ϖ 25 (kWhmile) -e budget limit Θ is set as$200 000 000 All the above values are chosen for illustrativepurpose We adopt CPLEX128 to solve the PCNE SP andLDER

42 BaseCaseResult -e LDER problem is solved under theaforementioned basic setting as shown in Figure 2(a)Figure 2(b) depicts the convergence performance of the SP-kalgorithm indicating the potential of the SP-k algorithm insolving the PCNE problem -e equilibrium link flows arereported in Table 3 with the total travel time of 920 times 104 h middot

veh and the average travel time 0575 h per vehicle

43 Strategy Comparison under Different Budget LimitsIn this section we consider different location designs of theproposed model under different budget limits to demonstratethe system performance as shown in Figure 3(a) Figure 3compares the total travel time cost of the ER and the numberof ERs under different budget limits As expected total traveltime of the network decreases with an increase in the budgetMore ERs are built with an increasing budget when the budget

ranges from $13 times 108 to $2 times 108 generating more usableroute choice for the EV to complete the tour which promotesaverage distribution of traffic flow leading to reduction in thesystem travel time Nevertheless more ERs are built under thebudget of $12 times 108 comparing with the budget of $13 times 108and $14 times 108 It can be observed from Figure 3(a) that whenthe budget exceeds $14 times 108 link (2018) is always chosen tobe equipped with ER -e usability of link (2018) has asignificant impact on EV driversrsquo routing behavior due to itsspecial location However ER on link (2018) of the 15mwidth is an expensive project which has to be weighedaccording to the budget limit Figure 3(d) compares theequilibrium link flow of different location designs underdifferent budgets It can be observed that under a relativelylow budget the routing choice is limited which leads to

Set the iteration variable η 1 SO0 +infin TD0 minus infin and the initial feasiblefixed solutions (Ωη0 Ωη1)

Solve (1)ndash(14) with (Ωη0 Ωη1) and derive ληij and μηij Calculate system totaltravel time SOη If SOη ge SOη then η η minus 1 go to step 3

Solve (22)ndash(26) and θ ε + TDηminus 1

If the optimal objective value is zero (Ωη0 Ωη1) is the best solution and theiteration ends

If not derive (gij hij)) and the objective value TDη and go to step 4Obtain a new solution to (Ωη+1

0 Ωη+11 )

Ωη+10 Ωη0 minus (i j) isin Ωη0 gij 11113966 1113967 + (i j) isin Ωη1 hij 11113966 1113967

Ωη+11 Ωη1 minus (i j) isin Ωη1 hij 11113966 1113967 + (i j) isin Ωη0 gij 11113966 1113967

η η minus 1 go to step 2

ALGORITHM 1

1

3 4

12 11

5 6

9 8

2

7

10 16 18

17

14 15

23 22

19

202413 21

Figure 1 Sioux Falls network


extremely high traffic flow on some links accompanyingconsiderable travel time like link 7 and link 35 As budget isincreasing EV drivers have more usable path to selectEquilibrium flow on such links decreases

44 Strategy Comparison with Different Recharging RatesIn order to investigate the impact of the charging rate onlocation design and equilibrium flow (see Figure 4) we solvethe DLEP and NE with charging rate ϖ set as 2kWhmile(LEVEL1) 25kWhmile (LEVEL2) and 35kWhmile(LEVEL3) Cost of the ER is set as $1 times 105 $2 times 105 and$3 times 105 respectively Budget limit is set as $19 times 108 In-terestingly the minimum total travel time can be obtainedunder this budget in LEVEL 1 and LEVEL 2 cases -at iseven when the budget limit increases the total travel timewill not be reduced To explore the minimum travel time inLEVEL 3 we repeat to solve the DLEP and NE in the LEVEL3 case with the budget limit increasing gradually and find theminimum travel time of 913 times 104h middot veh at the budget of$273 times 108 which is less than the travel time of 920 times 104h middot

veh in the LEVEL 2 case Figure 4(b) compares the total

Table 1 Link characteristics FFTT (min) width (meter) length(mile) and capacity (103 vehh)

Link FFTT Width Length Capacity1-2 6 15 15 25901-3 4 15 10 23402-1 6 15 15 25902-6 5 35 125 4963-1 4 15 10 23403-4 4 1125 10 17113-12 4 15 10 23404-3 4 1125 10 17114-5 2 1125 5 17784-11 6 35 15 4915-4 2 1125 5 17785-6 4 35 10 4955-9 5 75 125 10006-2 5 35 125 4966-5 4 35 10 4956-8 2 35 5 4907-8 3 75 75 7847-18 2 15 5 23408-6 2 35 5 4908-7 3 75 75 7848-9 10 35 25 5058-16 5 35 125 5059-5 5 75 125 10009-8 10 35 25 5059-10 3 75 75 139210-9 3 75 75 139210-11 5 75 125 100010-15 6 1125 15 135110-16 4 35 10 48510-17 8 35 20 49911-4 6 35 15 49111-10 5 75 125 100011-12 6 35 15 49111-14 4 35 10 48812-3 4 15 10 234012-11 6 35 15 49112-13 3 15 75 259013-12 3 15 75 259013-24 4 34 10 50914-11 4 35 10 48814-15 5 35 125 51314-23 4 35 10 49215-10 6 1125 15 135115-14 5 35 125 51315-19 3 1125 75 145615-22 3 75 75 96016-8 5 35 125 50516-10 4 35 10 48516-17 2 35 5 52316-18 3 15 75 196817-10 8 35 20 49917-16 2 35 5 52317-19 2 35 5 48218-7 2 15 5 234018-16 3 15 75 196818-20 4 15 10 234019-15 3 1125 75 145619-17 2 35 5 48219-20 4 35 10 500

Table 1 Continued

Link FFTT Width Length Capacity20-18 4 15 10 234020-19 4 35 10 50020-21 6 35 15 50620-22 5 35 125 50821-20 6 35 15 50621-22 2 35 5 52321-24 3 35 75 48922-15 3 75 75 96022-20 5 35 125 50822-21 2 35 5 52322-23 4 35 10 50023-14 4 35 10 49223-22 4 35 10 50023-24 2 35 5 50824-13 4 35 10 50924-21 3 35 75 48924-23 2 35 5 508

Table 2 Network O-D demand (103 vehh)

O D Demand1 20 1620 1 161 13 813 1 81 7 127 1 122 13 1013 2 102 20 820 2 87 13 1413 7 147 20 1220 7 12


travel time and cost in these four cases It can be observedthat under the budget limit of $19 times 108 the ER of LEVEL 2gets the best performance which is much better than the

other two cases -ree explanations are offered here Firstdue to constraint (16) shortest length of the ER is different asrecharging rate changes In the case of LEVEL 1 only 26

1 2

3 4 5 6

12

13

11 10 16

14 15

23 22

19

24 21 20

8

18

79

17

Nodes of transportation network

LinksLinks equipped with electrification road

(a)

Rela

tive g

ap

006005004003002001

01 50 100 150

Number of iterations200 250

(b)

Figure 2 Base case result in the Sioux Falls network (a) Location design of the ER (b) Performance of the SP algorithm

Table 3 Equilibrium link flow (103 vehh)

Link Flow Link Flow Link Flow Link Flow1-2 1015 8-7 1205 13-12 2668 19-15 6581-3 4604 8-9 599 13-24 1072 19-17 2562-1 1136 8-16 675 14-11 759 19-20 7782-6 796 9-5 1375 14-15 534 20-18 31253-1 4482 9-8 610 14-23 013 20-19 6623-4 2277 9-10 653 15-10 000 20-21 6093-12 2333 Link Flow 15-14 759 20-22 5364-3 2345 10-9 776 15-19 1030 21-20 5334-5 1725 10-11 539 15-22 391 21-22 0004-11 552 10-15 635 16-8 567 21-24 6465-4 1747 10-16 858 16-10 893 22-15 3525-6 585 10-17 025 16-17 422 22-20 6915-9 1262 11-4 598 16-18 1793 22-21 0386-2 918 11-10 866 17-10 422 22-23 3986-5 494 11-12 699 17-16 281 23-14 0006-8 1253 11-14 547 17-19 000 23-22 5527-8 1333 12-3 2144 18-7 2968 23-24 3987-18 2840 12-11 861 18-16 1860 24-13 10448-6 1284 12-13 2696 18-20 2930 24-21 533

24-23 539




Budget (times108 USD)

(A) Budget = 12 (B) Budget = 13 (C) Budget = 14 (D) Budget = 15

(E) Budget = 16 (F) Budget = 17 (G) Budget = 18 (H) Budget ge 19

(a)28

12 13 14 15Budget (times108 USD)

Total travel timeCost of ER

16 17 18 19

2426

22

Tota

l tra

vel t

ime (

times104

h)

201816141210

8

19

17

18

16

15

14

13

12

11

(b)


16 17 18 19

Num

ber o

f ERs

12

8

10

6

4

2

0

(c)

Figure 3 Continued


links of the length not less than 125 miles are qualified to beequipped with the ER In the case of LEVEL 3 62 links can bechosen as the candidate location of the ER -us usablepaths of ER drivers and total travel time are affected Secondthe cost per unit ER increases with the rise of the rechargingrate resulting in less ER constructed giving a budget con-straint -ird the state of charge and charging behavior areaffected by the recharging rate bringing about further im-pact on routing behavior When the budget exceeds

$273 times 108 total travel time of the network equipped withthe ER of LEVEL 3 attains the minimum As expected moreusable paths are generated as recharging rate increaseswhich may reduce the total travel time with more probablelinks sharing the traffic loads

45 Impacts of Comfortable Range on Location Design andSystemPerformance Variations in comfortable range changethe usable path sets and thereby hinder routing choice and

Link 7

Link no1 10 20 30 41 51 61 71 76

Link 35Budget (times108 USD)

Equi

libriu

m li

nk fl

ow (1

03 veh

h)

50

40

30

20

10

0

Capacity

Budget = 12Budget = 15




(d)

Figure 3 Comparison under different budget limits (a) Comparison of location designs under different budgets (b) Total travel time andER cost (c) -e number of ERs under different budgets (d) Equilibrium flow of different location designs under different budgets


LinksLinks equipped with ER

LEVEL 1 LEVEL 2 LEVEL 3

LEVEL 3

Budget = $19 times 108

Budget ge $273 times 108

(a)

3111

920

1480

913

3300

2800

2300

1800

1300

800

Tota

l tra

vel t

ime (

times104 h

)

3

25

2

15

1

05

0

Cos

t of E

R (times

108 U

SD)

Recharging level of ERLEVEL 1 LEVEL 2 LEVEL 3 LEVEL 3

Budget = $19 times 108 Budget ge $273 times 108

Cost of ERTotal travel time

(b)

Figure 4 Comparison with the ER of different recharging rates (a) Location designs (b) Total travel time and cost of the ER


the ER location design In this section the impact of com-fortable range on location design and system performance istested with DLEP of different comfortable ranges mod 15kWh and mod 2kWh under the budget limit of $21 times 108and $27 times 108 solved respectively comparing with mod 0Location designs of the ER are shown in Figure 5(a)Figure 5(b) compares the total travel time and cost in these sixcases Under the budget limit of $21 times 108 change of mod

from 0 to 2kWh increases the total travel time by 78 Tofurther explore the impacts of comfortable range we raise thebudget limit to $27 times 108 and find that the variations of thetotal travel time with the change in comfortable range are notobvious -at is to say under a given budget of $21 times 108increased comfortable range leads to the increment in thetotal travel time When the budget limit is set as $27 times 108the location design of mod 0 is the same as that under thebudget limit of $21 times 108 with the construction cost of$1865 times 108 -e location designs of mod 15 kWh andmod 2kWh change to yield the minimum travel time which

is close to the number when mod 0 with the constructioncost increasing to $263 times 108 and $25 times 108 respectively

46 Impacts of the Battery Size on LocationDesign and SystemPerformance Different battery sizes affect the routing choiceand location design in two ways First the shortest length ofthe ER is different according to constraint (16) Second thestate of charge and charging behavior are different accordingto constraints (7) and (9) In this section budget limit is set as$4 times 108 to explore the location design and system perfor-mance under different battery sizes lmax 20 25 and30 kWh Initial state of charge is set as l0 25 lmax Asrevealed in Figure 6 location design with the battery size oflmax 25 kWh has the least total travel time and constructioncost of the ER On the one hand as commonly considered alarger battery size allows more charging amount on the ERwhich may generate more usable path and reduce the totaltravel time On the other hand constraint (16) sets the

mod = 0 mod = 15 mod = 2

mod = 0 mod = 15 mod = 2Bu

dget

= $

21

times 10

8Bu

dget

= $

27

times 10

8

(a)

17

16

15

14

13

12

11

10

9

8

27

26

25

24

23

22

21

2

19

18mod = 0 mod = 15 mod = 2 mod = 0 mod = 15 mod = 2

Budget = $21 times 108 Budget = $27 times 108


(b)

Figure 5 Comparison with different comfortable ranges (mod kWh) (a) Location designs (mo d kWh) (b) Total travel time and cost ofthe ER

lmax = 20kWh lmax = 25kWh lmax = 30kWh

(a)



2422201816141210

8

(b)

Figure 6 Comparison of the ER with different battery sizes (a) Location designs (b) Total travel time and cost of the ER


minimum length of the ER which is related to the battery sizeWhen lmax 3 kWh only 26 links whose length is more than12 miles can be the candidate of the ER which affects the setof usable paths and the total travel time

From the above discussion we can find that

(1) System total travel time of the network decreaseswith an increase in the budget More ERs are builtwith an increasing budget generating more usableroute choice for the EV to complete the tour whichpromotes average distribution of traffic flow leadingto reduction in the system travel time

(2) Recharging rate sets the bottom boundary for thelength of the candidate of the ER which affects theEV driversrsquo routing choice Besides recharging rate isrelated to the construction cost which should bebalanced with the budget limit

(3) Increase in comfortable range changes the sets ofusable path -erefore system travel time increases

(4) Battery sizes affect the shortest length of the ER aswell as the initial state of charge and charging be-havior which should be balanced under differentbudgets

5 Conclusion

To enhance the transportation network performance as ERis equipped this paper proposes a LDER model to optimizethe location of the ER considering the routing andcharging behavior of EV drivers by a PCNE model EVdrivers are assumed to decide their routing and rechargingplan to minimize their travel time and prevent running outof charge We develop a modified active set algorithm toobtain the optimal location of the ER Numerical examplesare conducted to demonstrate the performance of themodel in comparison with the location designs in differentbudget limits recharging rates comfortable range andbattery sizes

-e energy consumption rate and the recharging rate ofthe proposed path-constrained network equilibrium areassumed to be flow-independent Our future study willextend our model to investigate the effect of the trafficcongestion on the energy consumption rate and therecharging rate and try to find the solution algorithm Be-sides the transportation network can be extended to ac-commodate both EVs and regular vehicles with impact ofthe mixed traffic flow investigated

Data Availability

-e data used to support the findings of this study areavailable from the corresponding author upon request

Disclosure

-e authors are responsible for all views and opinionsexpressed in this paper

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is study was supported by the Joint Science Foundationof Ministry of Education of China amp CMCC (Grant No2018202004) -e first author was supported by the Pro-gram for Changjiang Scholars and Innovative ResearchTeam in University -e third author was supported by theNational Natural Science Foundation for Young Scholars ofChina (Grant No 71901190)

References

[1] C W Yang and Y-L Ho ldquoAssessing carbon reduction effectstoward the mode shift of green transportation systemrdquoJournal of Advanced Transportation vol 50 no 5pp 669ndash628

[2] P Vorobiev and Y Vorobiev ldquoAbout the possibilities of usingthe renewable energy power sources on railway transportrdquoJournal of Advanced Transportation vol 47 no 8 pp 681ndash691 2013

[3] M D Meyer ldquoTransport planning for urban areas a retro-spective look and future prospectsrdquo Journal of AdvancedTransportation vol 34 no 1 pp 143ndash171 2000

[4] A Shukla and S A Ansari ldquoEnergy harvesting from roadpavement a cleaner and greener alternativerdquo InternationalResearch Journal of Engineering and Technology (IRJET)vol 5 no 2 2018

[5] A Northmore and S Tighe ldquoInnovative pavement design aresolar roads feasiblerdquo in Proceedings of the Conference of theTransportation Association of Canada Fredericton CanadaOctober 2012

[6] A S Dezfooli F M Nejad H Zakeri and S KazemifardldquoSolar pavement a new emerging technologyrdquo Solar Energyvol 149 pp 272ndash284 2017

[7] X Yu S Sandhu S Beiker R Sassoon and S Fan ldquoWirelessenergy transfer with the presence of metallic planesrdquo AppliedPhysics Letters vol 99 no 21 pp 1230ndash1610 2011

[8] P Ning J M Miller O C Onar C P White andL D Marlino ldquoA compact wireless charging system devel-opmentrdquo in Proceedings of the Twenty-Eighth Annual IEEEApplied Power Electronics Conference and Exposition (APEC)IEEE Long Beach CA USA March 2013

[9] D U Eberle andD R VonHelmolt ldquoSustainable transportationbased on electric vehicle concepts a brief overviewrdquo Energy ampEnvironmental Science vol 3 no 6 pp 689ndash699 2010

[10] N S Pearre W Kempton R L Guensler and V V ElangoldquoElectric vehicles how much range is required for a dayrsquosdrivingrdquo Transportation Research Part C Emerging Tech-nologies vol 19 no 6 pp 1171ndash1184 2011

[11] I E Agency and E V Global ldquoGlobal ev outlook under-standing the electric vehicle landscape to 2020rdquo 2013

[12] P Bansal ldquoCharging of electric vehicles technology andpolicy implicationsrdquo Ae Journal of Science Policy amp Gover-nance vol 6 no 1 p 38 2015

[13] J U K Tidy ldquoTest wireless charging for electric car 2015rdquohttpnewsskycomstory

[14] J Ramsey ldquoVolvo studying test of electric roads in swedenrdquo2014 httpwwwautoblogcom


[15] M Fuller ldquoWireless charging in California range rechargeand vehicle electrificationrdquo Transportation Research Part CEmerging Technologies vol 67 pp 343ndash356 2016

[16] N Jiang C Xie and T Waller ldquoPath-constrained trafficassignment model and algorithmrdquo Transportation ResearchRecord Journal of the Transportation Research Boardvol 2283 no 1 pp 25ndash33 2012

[17] N Jiang and C Xie ldquoComputing and analyzing equilibriumnetwork flows of gasoline and electric vehiclesrdquo in Proceedingsof the Transportation Research Board 92nd Annual MeetingWashington DC USA January 2013

[18] N Jiang C Xie J C Duthie and S T Waller ldquoA networkequilibrium analysis on destination route and parkingchoices with mixed gasoline and electric vehicular flowsrdquoEURO Journal on Transportation and Logistics vol 3 no 1pp 55ndash92 2014

[19] F He Y Yin and J Zhou ldquoIntegrated pricing of roads andelectricity enabled by wireless power transferrdquo TransportationResearch Part C Emerging Technologies vol 34 no 9pp 1ndash15 2013

[20] F He Y Yin J Wang and Y Yang ldquoSustainability SI optimalprices of electricity at public charging stations for plug-inelectric vehiclesrdquo Networks and Spatial Economics vol 16no 1 pp 131ndash154 2016

[21] F He D Wu Y Yin and Y Guan ldquoOptimal deployment ofpublic charging stations for plug-in hybrid electric vehiclesrdquoTransportation Research Part B Methodological vol 47 no 1pp 87ndash101 2013

[22] F He Y Yin and J Zhou ldquoDeploying public charging sta-tions for electric vehicles on urban road networksrdquo Trans-portation Research Part C Emerging Technologies vol 60pp 227ndash240 2015

[23] F He Y Yin and S Lawphongpanich ldquoNetwork equilibriummodels with battery electric vehiclesrdquo Transportation Re-search Part B Methodological vol 67 no 3 pp 306ndash3192014

[24] M Xu Q Meng and K Liu ldquoNetwork user equilibriumproblems for the mixed battery electric vehicles and gasolinevehicles subject to battery swapping stations and road gradeconstraintsrdquo Transportation Research Part B Methodologicalvol 99 pp 138ndash166 2017

[25] R Riemann D Z W Wang and F Busch ldquoOptimal locationof wireless charging facilities for electric vehicles flow-cap-turing location model with stochastic user equilibriumrdquoTransportation Research Part C Emerging Technologiesvol 58 pp 1ndash12 2015

[26] Z Chen F He and Y Yin ldquoOptimal deployment of charginglanes for electric vehicles in transportation networksrdquoTransportation Research Part B Methodological vol 91pp 344ndash365 2016

[27] Z Chen W Liu and Y Yin ldquoDeployment of stationary anddynamic charging infrastructure for electric vehicles alongtraffic corridorsrdquo Transportation Research Part C EmergingTechnologies vol 77 pp 185ndash206 2017

[28] Z Liu and Z Song ldquoRobust planning of dynamic wirelesscharging infrastructure for battery electric busesrdquo Trans-portation Research Part C Emerging Technologies vol 83pp 77ndash103 2017

[29] Z Liu Z Song and Y He ldquoOptimal deployment of dynamicwireless charging facilities for an electric bus systemrdquoTransportation Research Record Journal of the TransportationResearch Board vol 2647 no 1 pp 100ndash108 2017

[30] A S Ahmed A M Mohamed and Z H U Lei ldquoSystemdesign and optimization of in-route wireless charging

infrastructure for shared automated electric vehiclesrdquo IEEEAccess vol 7 pp 79968ndash79979 2019

[31] Z Bi G A Keoleian Z Lin et al ldquoLife cycle assessment andtempo-spatial optimization of deploying dynamic wirelesscharging technology for electric carsrdquo Transportation Re-search Part C Emerging Technologies vol 100 pp 53ndash672019

[32] T Zhao Z Wu and Y Zhang ldquoBi-objective optimization ofelectric bus dynamic wireless charging facilities location andonboard battery size considering both facility cost and energyconsumptionrdquo in Proceedings of the Transportation ResearchBoard 98th Annual Meeting Washington DC USA January2019

[33] L J Leblanc ldquoAn algorithm for the discrete network designproblemrdquo Transportation Science vol 9 no 3 pp 183ndash1991975

[34] T L Friesz H-J Cho N J Mehta R L Tobin andG A Nandalingam ldquoA simulated annealing approach to thenetwork design problem with variational inequality con-straintsrdquo Informs vol 26 no 1 pp 18ndash26 1992

[35] S Wang Q Meng and H Yang ldquoGlobal optimizationmethods for the discrete network design problemrdquo Trans-portation Research Part B Methodological vol 50 pp 42ndash60

[36] L Zhang S Lawphongpanich and Y Yin An Active-SetAlgorithm for Discrete Network Design Problems SpringerScience+Business Media Berlin Germany 2009


-e aforementioned studies do not consider the locationof charging facilities in networks Problems related to theoptimal location of charging stations or battery swappingstations have received much attention He et al [21] firstproposed a UE-based framework for locating chargingstations to maximize social welfare considering routechoices of electric vehicles and further developed threemodels of different flow dependencies and energy con-sumption [22 23] Xu et al [24] defined a network withbattery swapping stations where swapping time is inde-pendent with charging capacity considering the impact ofqueuing to find the best charging strategy -ese existingrelevant studies are intended for charging stations or batteryswapping stations where EVs reach the station stop and getreplenished Such models cannot address issues for OWCdirectly

OWC unlike charging stations or battery swappingstations can charge EVs during the ride OWC technologyhas been implemented in a network in South Korea [12] Alimited number of research studies have been done on theOWC problem to select the optimal ER locations so farRiemann et al [25] proposed a mixed-integer nonlinearprogram to describe the flow-capturing location modelconsidering driverrsquos routing behavior Chen et al [26] in-vestigated the optimal deployment of charging lanes basedon the speed control of EV drivers on the charging lanesChen et al [27] proposed the integrated deployment ofcharging lanes and charging stations along a long trafficcorridor which cannot be applied in the transportationnetwork directly Liu and Song [28] adopted a robust op-timization methodology to provide robust location opti-mization of wireless charging facilities for the electric bussystem to reduce total cost of batteries and charging facilitieswith different uncertainty levels Liu et al [29] developed amixed-integer linear program for OWC location and batterysize optimization and battery size for an electric bus systemwith overlapping bus lines Ahmed et al [30] introduced amethod to find the best combination of battery capacity andwireless charger characteristics to solve the tradeoff betweenmaximizing charge sustaining minimum battery capacityand minimizing the initial investment Bi et al [31] adopteda genetic algorithm to optimize the rollout of OWC infra-structure both spatially and temporally in order to minimizelife cycle costs and energy burdens Zhao et al [32] proposeda biobjective optimization problem for integrated EV lo-cation and on-board battery size design of electric bussystems to minimize deployment cost and reduce energyconsumption of electrified systems

-e above research studies can solve the ER locationoptimization to some extent but they tend to assume thatcharging amount is only related to the charging time Undersuch assumption drivers tend to deliberately slow down toextend the charging time and even EV may revisit the samecharging lanes All the vehicles behind the EV that is movingslowly to get charged are delayed A road contains more thantwo lanes one of which is the ER Vehicles without chargingintention would perform lane-changing which reduces theroad capacity Relationship between the cost of building theER and width is seldom considered

In this paper charging amount is assumed to be relatedwith the driving distance while charging and the candidatesof the ER have to satisfy the minimum energy supply -ewidth of the ER is considered to affect ER building cost -elink performance function is set as the BPR function innumerical examples -erefore the route choice of driverswill be complicated and the UE flows of EVs in thetransportation network will be different

-is paper investigates the location design problem ofthe ER based on the analysis of the routing behavior of EVdrivers which is different from the previous studies withthe charging station or swapping station Our modelinherits from the paper written by He et al [21] butcharging behaviors and routing behaviors are differenttracking back to the specific situation of OWC in contrastto charging at station which leads to a totally differentlocation design Assuming that the energy consumptionrate and recharging rate are flow-independent this paperproposes a path-constrained network equilibrium model(PCNE) considering the routing and recharging behaviorof EV drivers as well as the constraint of driving range-en we develop an iterative solution framework to solvethe PCNE problem Lastly we investigate a mathematicalprogram with equilibrium constraint (MPEC) model tooptimize the ER location to minimize the system totaltravel time under a given construction budget limit Withthe proposed model discussions about the effect of dif-ferent recharging rates comfortable ranges and batterysizes on the location design and system performance areconducted

-e remainder of the paper is organized as follows -enotation assumption formulation of the PCNE problemand a solution framework are elaborated in Section 2Section 3 formulates the location design problem as a MPECprogram Section 31 proposes a modified active set methodto solve the problem Section 4 presents numerical casesbased on the Sioux Falls network and Section 5 offers theconclusions of this study and implications for future re-search on the transportation network of the ER

2 Path-Constrained Network EquilibriumModel for On-Way Charging

-e section proposes a PCNE model to describe the routingand charging behaviors of EV drivers under a given locationdesign which determine the network flow pattern Decisionvariables of themodel include equilibrium flow and chargingamount of the EV

21 Notation and Assumption Following assumptions aremade before the model formulation

Assumption 1 Drivers do not need to revisit the same ER toguarantee that the EV will not be out of charge

Assumption 2 If the EV is full of charge charging processstops while running on the ER













PCNE

minimizef

1113944(ij)isinA

11139461113936odisinW1113936

risin1113954Rod fod

r χodijr

0tij(z)dz

subject to

(1)

1113944

risinRod

fodr q



od (3)







tij t0ij 1 + 015

vij

cij

1113888 1113889















SUPF-k

x[k]

e[k]


1113944(ij)isinA

tij v[k]ij1113872 1113873x

odij (5)


1113944iisinNt(n)

xodin minus 1113944

jisinNh(n)


lodj minus l


odij ρod








odij1113872 1113873 + m


(11)

lodo l0 (12)













od[k]ij e


od 1113954R

od[k]


od Denote( v



od[k+1]ij


[k+1]ij )

x1113957o1113957d[k+1]ij 1113954R

od 1113954R

odcup1113954R1113957o1113957d[k+1]


[k+1]ij )x



flow distribution







LDER

minimizeδ

1113936(ij)isinA


subject to (15)


1113944(ij)isinA

WijDijδijBleΘ(17)



flowast vlowast


1113944(ij)isinA

11139461113936odisinW1113936

risin1113954Rod

(δ)fod

r χodijr

0tij(z)dz

(19)









Ω0 (i j) δij 01113966 1113967 (20)

Ω1 (i j) δij 11113966 1113967 (21)










μijhij

subject to(22)

1113944(ij)isinΩ0



minus G 1 minus gij1113872 11138731113872 1113873

+ 1113944(ij)isinΩ0


(23)

1113944(ij)isinΩ0



(24)

1113944(ij)isinΩ0






θ ε + 1113944(ij)isinΩ0


μijhij (27)
















0 Ωη+11 )




ALGORITHM 1

1

3 4

12 11

5 6

9 8

2

7

10 16 18

17

14 15

23 22

19

202413 21








Table 1 Continued



O D Demand1 20 1620 1 161 13 813 1 81 7 127 1 122 13 1013 2 102 20 820 2 87 13 1413 7 147 20 1220 7 12




1 2

3 4 5 6

12

13

11 10 16

14 15

23 22

19

24 21 20

8

18

79

17



(a)

Rela

tive g

ap

006005004003002001

01 50 100 150


(b)




24-23 539







(a)28



16 17 18 19

2426

22

Tota

l tra

vel t

ime (

times104

h)

201816141210

8

19

17

18

16

15

14

13

12

11

(b)


16 17 18 19

Num

ber o

f ERs

12

8

10

6

4

2

0

(c)

Figure 3 Continued





Link 7

Link no1 10 20 30 41 51 61 71 76


Equi

libriu

m li

nk fl

ow (1

03 veh

h)

50

40

30

20

10

0

Capacity





(d)





LEVEL 3



(a)

3111

920

1480

913

3300

2800

2300

1800

1300

800

Tota

l tra

vel t

ime (

times104 h

)

3

25

2

15

1

05

0

Cos

t of E

R (times

108 U

SD)




(b)







mod = 0 mod = 15 mod = 2


dget

= $

21

times 10

8Bu

dget

= $

27

times 10

8

(a)

17

16

15

14

13

12

11

10

9

8

27

26

25

24

23

22

21

2

19




(b)



(a)



2422201816141210

8

(b)









5 Conclusion



Data Availability


Disclosure




Acknowledgments


References



















































PCNE

minimizef

1113944(ij)isinA

11139461113936odisinW1113936

risin1113954Rod fod

r χodijr

0tij(z)dz

subject to

(1)

1113944

risinRod

fodr q



od (3)







tij t0ij 1 + 015

vij

cij

1113888 1113889















SUPF-k

x[k]

e[k]


1113944(ij)isinA

tij v[k]ij1113872 1113873x

odij (5)


1113944iisinNt(n)

xodin minus 1113944

jisinNh(n)


lodj minus l


odij ρod








odij1113872 1113873 + m


(11)

lodo l0 (12)













od[k]ij e


od 1113954R

od[k]


od Denote( v



od[k+1]ij


[k+1]ij )

x1113957o1113957d[k+1]ij 1113954R

od 1113954R

odcup1113954R1113957o1113957d[k+1]


[k+1]ij )x



flow distribution







LDER

minimizeδ

1113936(ij)isinA


subject to (15)


1113944(ij)isinA

WijDijδijBleΘ(17)



flowast vlowast


1113944(ij)isinA

11139461113936odisinW1113936

risin1113954Rod

(δ)fod

r χodijr

0tij(z)dz

(19)









Ω0 (i j) δij 01113966 1113967 (20)

Ω1 (i j) δij 11113966 1113967 (21)










μijhij

subject to(22)

1113944(ij)isinΩ0



minus G 1 minus gij1113872 11138731113872 1113873

+ 1113944(ij)isinΩ0


(23)

1113944(ij)isinΩ0



(24)

1113944(ij)isinΩ0






θ ε + 1113944(ij)isinΩ0


μijhij (27)
















0 Ωη+11 )




ALGORITHM 1

1

3 4

12 11

5 6

9 8

2

7

10 16 18

17

14 15

23 22

19

202413 21








Table 1 Continued



O D Demand1 20 1620 1 161 13 813 1 81 7 127 1 122 13 1013 2 102 20 820 2 87 13 1413 7 147 20 1220 7 12




1 2

3 4 5 6

12

13

11 10 16

14 15

23 22

19

24 21 20

8

18

79

17



(a)

Rela

tive g

ap

006005004003002001

01 50 100 150


(b)




24-23 539







(a)28



16 17 18 19

2426

22

Tota

l tra

vel t

ime (

times104

h)

201816141210

8

19

17

18

16

15

14

13

12

11

(b)


16 17 18 19

Num

ber o

f ERs

12

8

10

6

4

2

0

(c)

Figure 3 Continued





Link 7

Link no1 10 20 30 41 51 61 71 76


Equi

libriu

m li

nk fl

ow (1

03 veh

h)

50

40

30

20

10

0

Capacity





(d)





LEVEL 3



(a)

3111

920

1480

913

3300

2800

2300

1800

1300

800

Tota

l tra

vel t

ime (

times104 h

)

3

25

2

15

1

05

0

Cos

t of E

R (times

108 U

SD)




(b)







mod = 0 mod = 15 mod = 2


dget

= $

21

times 10

8Bu

dget

= $

27

times 10

8

(a)

17

16

15

14

13

12

11

10

9

8

27

26

25

24

23

22

21

2

19




(b)



(a)



2422201816141210

8

(b)









5 Conclusion



Data Availability


Disclosure




Acknowledgments


References














































SUPF-k

x[k]

e[k]


1113944(ij)isinA

tij v[k]ij1113872 1113873x

odij (5)


1113944iisinNt(n)

xodin minus 1113944

jisinNh(n)


lodj minus l


odij ρod








odij1113872 1113873 + m


(11)

lodo l0 (12)













od[k]ij e


od 1113954R

od[k]


od Denote( v



od[k+1]ij


[k+1]ij )

x1113957o1113957d[k+1]ij 1113954R

od 1113954R

odcup1113954R1113957o1113957d[k+1]


[k+1]ij )x



flow distribution







LDER

minimizeδ

1113936(ij)isinA


subject to (15)


1113944(ij)isinA

WijDijδijBleΘ(17)



flowast vlowast


1113944(ij)isinA

11139461113936odisinW1113936

risin1113954Rod

(δ)fod

r χodijr

0tij(z)dz

(19)









Ω0 (i j) δij 01113966 1113967 (20)

Ω1 (i j) δij 11113966 1113967 (21)










μijhij

subject to(22)

1113944(ij)isinΩ0



minus G 1 minus gij1113872 11138731113872 1113873

+ 1113944(ij)isinΩ0


(23)

1113944(ij)isinΩ0



(24)

1113944(ij)isinΩ0






θ ε + 1113944(ij)isinΩ0


μijhij (27)
















0 Ωη+11 )




ALGORITHM 1

1

3 4

12 11

5 6

9 8

2

7

10 16 18

17

14 15

23 22

19

202413 21








Table 1 Continued



O D Demand1 20 1620 1 161 13 813 1 81 7 127 1 122 13 1013 2 102 20 820 2 87 13 1413 7 147 20 1220 7 12




1 2

3 4 5 6

12

13

11 10 16

14 15

23 22

19

24 21 20

8

18

79

17



(a)

Rela

tive g

ap

006005004003002001

01 50 100 150


(b)




24-23 539







(a)28



16 17 18 19

2426

22

Tota

l tra

vel t

ime (

times104

h)

201816141210

8

19

17

18

16

15

14

13

12

11

(b)


16 17 18 19

Num

ber o

f ERs

12

8

10

6

4

2

0

(c)

Figure 3 Continued





Link 7

Link no1 10 20 30 41 51 61 71 76


Equi

libriu

m li

nk fl

ow (1

03 veh

h)

50

40

30

20

10

0

Capacity





(d)





LEVEL 3



(a)

3111

920

1480

913

3300

2800

2300

1800

1300

800

Tota

l tra

vel t

ime (

times104 h

)

3

25

2

15

1

05

0

Cos

t of E

R (times

108 U

SD)




(b)







mod = 0 mod = 15 mod = 2


dget

= $

21

times 10

8Bu

dget

= $

27

times 10

8

(a)

17

16

15

14

13

12

11

10

9

8

27

26

25

24

23

22

21

2

19




(b)



(a)



2422201816141210

8

(b)









5 Conclusion



Data Availability


Disclosure




Acknowledgments


References









































flowast vlowast


1113944(ij)isinA

11139461113936odisinW1113936

risin1113954Rod

(δ)fod

r χodijr

0tij(z)dz

(19)









Ω0 (i j) δij 01113966 1113967 (20)

Ω1 (i j) δij 11113966 1113967 (21)










μijhij

subject to(22)

1113944(ij)isinΩ0



minus G 1 minus gij1113872 11138731113872 1113873

+ 1113944(ij)isinΩ0


(23)

1113944(ij)isinΩ0



(24)

1113944(ij)isinΩ0






θ ε + 1113944(ij)isinΩ0


μijhij (27)
















0 Ωη+11 )




ALGORITHM 1

1

3 4

12 11

5 6

9 8

2

7

10 16 18

17

14 15

23 22

19

202413 21








Table 1 Continued



O D Demand1 20 1620 1 161 13 813 1 81 7 127 1 122 13 1013 2 102 20 820 2 87 13 1413 7 147 20 1220 7 12




1 2

3 4 5 6

12

13

11 10 16

14 15

23 22

19

24 21 20

8

18

79

17



(a)

Rela

tive g

ap

006005004003002001

01 50 100 150


(b)




24-23 539







(a)28



16 17 18 19

2426

22

Tota

l tra

vel t

ime (

times104

h)

201816141210

8

19

17

18

16

15

14

13

12

11

(b)


16 17 18 19

Num

ber o

f ERs

12

8

10

6

4

2

0

(c)

Figure 3 Continued





Link 7

Link no1 10 20 30 41 51 61 71 76


Equi

libriu

m li

nk fl

ow (1

03 veh

h)

50

40

30

20

10

0

Capacity





(d)





LEVEL 3



(a)

3111

920

1480

913

3300

2800

2300

1800

1300

800

Tota

l tra

vel t

ime (

times104 h

)

3

25

2

15

1

05

0

Cos

t of E

R (times

108 U

SD)




(b)







mod = 0 mod = 15 mod = 2


dget

= $

21

times 10

8Bu

dget

= $

27

times 10

8

(a)

17

16

15

14

13

12

11

10

9

8

27

26

25

24

23

22

21

2

19




(b)



(a)



2422201816141210

8

(b)









5 Conclusion



Data Availability


Disclosure




Acknowledgments


References





















































0 Ωη+11 )




ALGORITHM 1

1

3 4

12 11

5 6

9 8

2

7

10 16 18

17

14 15

23 22

19

202413 21








Table 1 Continued



O D Demand1 20 1620 1 161 13 813 1 81 7 127 1 122 13 1013 2 102 20 820 2 87 13 1413 7 147 20 1220 7 12




1 2

3 4 5 6

12

13

11 10 16

14 15

23 22

19

24 21 20

8

18

79

17



(a)

Rela

tive g

ap

006005004003002001

01 50 100 150


(b)




24-23 539







(a)28



16 17 18 19

2426

22

Tota

l tra

vel t

ime (

times104

h)

201816141210

8

19

17

18

16

15

14

13

12

11

(b)


16 17 18 19

Num

ber o

f ERs

12

8

10

6

4

2

0

(c)

Figure 3 Continued





Link 7

Link no1 10 20 30 41 51 61 71 76


Equi

libriu

m li

nk fl

ow (1

03 veh

h)

50

40

30

20

10

0

Capacity





(d)





LEVEL 3



(a)

3111

920

1480

913

3300

2800

2300

1800

1300

800

Tota

l tra

vel t

ime (

times104 h

)

3

25

2

15

1

05

0

Cos

t of E

R (times

108 U

SD)




(b)







mod = 0 mod = 15 mod = 2


dget

= $

21

times 10

8Bu

dget

= $

27

times 10

8

(a)

17

16

15

14

13

12

11

10

9

8

27

26

25

24

23

22

21

2

19




(b)



(a)



2422201816141210

8

(b)









5 Conclusion



Data Availability


Disclosure




Acknowledgments


References













































Table 1 Continued



O D Demand1 20 1620 1 161 13 813 1 81 7 127 1 122 13 1013 2 102 20 820 2 87 13 1413 7 147 20 1220 7 12




1 2

3 4 5 6

12

13

11 10 16

14 15

23 22

19

24 21 20

8

18

79

17



(a)

Rela

tive g

ap

006005004003002001

01 50 100 150


(b)




24-23 539







(a)28



16 17 18 19

2426

22

Tota

l tra

vel t

ime (

times104

h)

201816141210

8

19

17

18

16

15

14

13

12

11

(b)


16 17 18 19

Num

ber o

f ERs

12

8

10

6

4

2

0

(c)

Figure 3 Continued





Link 7

Link no1 10 20 30 41 51 61 71 76


Equi

libriu

m li

nk fl

ow (1

03 veh

h)

50

40

30

20

10

0

Capacity





(d)





LEVEL 3



(a)

3111

920

1480

913

3300

2800

2300

1800

1300

800

Tota

l tra

vel t

ime (

times104 h

)

3

25

2

15

1

05

0

Cos

t of E

R (times

108 U

SD)




(b)







mod = 0 mod = 15 mod = 2


dget

= $

21

times 10

8Bu

dget

= $

27

times 10

8

(a)

17

16

15

14

13

12

11

10

9

8

27

26

25

24

23

22

21

2

19




(b)



(a)



2422201816141210

8

(b)









5 Conclusion



Data Availability


Disclosure




Acknowledgments


References










































1 2

3 4 5 6

12

13

11 10 16

14 15

23 22

19

24 21 20

8

18

79

17



(a)

Rela

tive g

ap

006005004003002001

01 50 100 150


(b)




24-23 539







(a)28



16 17 18 19

2426

22

Tota

l tra

vel t

ime (

times104

h)

201816141210

8

19

17

18

16

15

14

13

12

11

(b)


16 17 18 19

Num

ber o

f ERs

12

8

10

6

4

2

0

(c)

Figure 3 Continued





Link 7

Link no1 10 20 30 41 51 61 71 76


Equi

libriu

m li

nk fl

ow (1

03 veh

h)

50

40

30

20

10

0

Capacity





(d)





LEVEL 3



(a)

3111

920

1480

913

3300

2800

2300

1800

1300

800

Tota

l tra

vel t

ime (

times104 h

)

3

25

2

15

1

05

0

Cos

t of E

R (times

108 U

SD)




(b)







mod = 0 mod = 15 mod = 2


dget

= $

21

times 10

8Bu

dget

= $

27

times 10

8

(a)

17

16

15

14

13

12

11

10

9

8

27

26

25

24

23

22

21

2

19




(b)



(a)



2422201816141210

8

(b)









5 Conclusion



Data Availability


Disclosure




Acknowledgments


References













































(a)28



16 17 18 19

2426

22

Tota

l tra

vel t

ime (

times104

h)

201816141210

8

19

17

18

16

15

14

13

12

11

(b)


16 17 18 19

Num

ber o

f ERs

12

8

10

6

4

2

0

(c)

Figure 3 Continued





Link 7

Link no1 10 20 30 41 51 61 71 76


Equi

libriu

m li

nk fl

ow (1

03 veh

h)

50

40

30

20

10

0

Capacity





(d)





LEVEL 3



(a)

3111

920

1480

913

3300

2800

2300

1800

1300

800

Tota

l tra

vel t

ime (

times104 h

)

3

25

2

15

1

05

0

Cos

t of E

R (times

108 U

SD)




(b)







mod = 0 mod = 15 mod = 2


dget

= $

21

times 10

8Bu

dget

= $

27

times 10

8

(a)

17

16

15

14

13

12

11

10

9

8

27

26

25

24

23

22

21

2

19




(b)



(a)



2422201816141210

8

(b)









5 Conclusion



Data Availability


Disclosure




Acknowledgments


References











































Link 7

Link no1 10 20 30 41 51 61 71 76


Equi

libriu

m li

nk fl

ow (1

03 veh

h)

50

40

30

20

10

0

Capacity





(d)





LEVEL 3



(a)

3111

920

1480

913

3300

2800

2300

1800

1300

800

Tota

l tra

vel t

ime (

times104 h

)

3

25

2

15

1

05

0

Cos

t of E

R (times

108 U

SD)




(b)







mod = 0 mod = 15 mod = 2


dget

= $

21

times 10

8Bu

dget

= $

27

times 10

8

(a)

17

16

15

14

13

12

11

10

9

8

27

26

25

24

23

22

21

2

19




(b)



(a)



2422201816141210

8

(b)









5 Conclusion



Data Availability


Disclosure




Acknowledgments


References












































mod = 0 mod = 15 mod = 2


dget

= $

21

times 10

8Bu

dget

= $

27

times 10

8

(a)

17

16

15

14

13

12

11

10

9

8

27

26

25

24

23

22

21

2

19




(b)



(a)



2422201816141210

8

(b)









5 Conclusion



Data Availability


Disclosure




Acknowledgments


References














































5 Conclusion



Data Availability


Disclosure




Acknowledgments


References








































location design of electrification road in transportation

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