Optimal Routing and Scheduling of Charge for Electric

Research ArticleOptimal Routing and Scheduling of Charge for Electric VehiclesA Case Study

J Barco12 A Guerra3 L Muntildeoz3 and N Quijano2

1Engineering Department Electronics Engineering Program Institucion Universitaria CESMAG Carrera 20A No 14ndash54San Juan de Pasto Colombia2Electrical and Electronics Engineering Department Universidad de Los Andes Carrera 1 Este No 19 A 40 Bogota Colombia3Mechanical Engineering Department Universidad de Los Andes Carrera 1 Este No 19 A 40 Bogota Colombia

Correspondence should be addressed to L Munoz lui-munouniandeseduco

Received 8 April 2017 Revised 4 September 2017 Accepted 3 October 2017 Published 14 November 2017

Academic Editor Domenico Quagliarella

Copyright copy 2017 J Barco et al This 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

There are increasing interests in improving public transportation systems One of the proposed strategies for this improvement isthe use of Battery Electric Vehicles (BEVs)This approach leads to a new challenge as the BEVsrsquo routing is exposed to the traditionalrouting problems of conventional vehicles as well as the particular requirements of the electrical technologies of BEVs Examples ofBEVsrsquo routing problems include the autonomy battery degradation and charge processThis work presents a differential evolutionalgorithm for solving an electric vehicle routing problem (EVRP) The formulation of the EVRP to be solved is based on a schemeto coordinate the BEVsrsquo routing and recharge scheduling considering operation and battery degradation costs A model based onthe longitudinal dynamics equation of motion estimates the energy consumption of each BEV A case study consisting of an airportshuttle service scenario is used to illustrate the proposed methodology For this transport service the BEV energy consumption isestimated based on experimentally measured driving patterns

1 Introduction

The integration of Battery Electric Vehicles (BEVs) to thepublic transportation system has been encouraged by thefavorable efficiency in the use of energy and the reductionof CO2 emissions [1] From the energy consumption per-spective BEVs are driven by high-efficiency motors with thepossibility of implementing a regenerative braking systemAdditionally charging a BEV is less expensive than refuelinga conventional vehicle because the electric energy is cheaperthan its equivalent in fossil fuel (eg gasoline and diesel)From the emissions point of view when a BEV is used incombination with renewable sources for the electricity gen-eration the outcome is a reduction in emissions associatedwith fossil fuel combustion therefore BEVs are one of thebest alternatives to be integrated into cities as part of a publictransportation system

The use of BEVs in public transportation systems facesseveral challenges mainly related to the combination of

conventional-fuel-service characteristics with those of elec-tric vehicles Examples of these challenges are the routing ofelectric vehicles used in public transportation the rechargescheduling and the battery state of health (SOH) [2] Thesethree challenges are treated in this work using amethodologydeveloped for the optimal routing and scheduling of chargefor BEVs

The first challenge is the routing of BEVs In addition tothe usual routing issues the BEVs should be routed takingparticular attention to minimizing energy consumption Forthe routing of BEVs two steps are considered The first stepconsists in finding theminimum consumption paths to travelbetween two points In this step it is necessary to consider thetechnical characteristics of BEVs The second step consistsin determining optimal routes to satisfy the transportationdemand in different places and at different schedules whileminimizing energy consumption Similar to the first stepthe calculation of optimal routes is performed consideringthe BEVsrsquo characteristics For this case the vehiclersquos traveling

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 8509783 16 pageshttpsdoiorg10115520178509783

2 Mathematical Problems in Engineering

range is considered mainly defined by the battery technologyavailable [3] This consideration could require intermediaterecharge stages to extend the traveling range of the vehicles

The second challenge is the scheduling for rechargingThe recharge scheduling is a challenge as the energy ratevariation during peak and valley hours must be consideredThis process should be coordinated with the routing layoutto guarantee a reliable operation while the recharge costsare minimized Additionally for the recharge scheduling it isnecessary to consider the amount of energy required to per-form the subsequent trip and the time required to recharge

The third challenge is the SOH of the battery The aimis to increase the battery lifespan to reduce the long-termoperational cost It must be considered that the useful lifeof the battery depends on the charge and discharge cycles[4] This aspect is relevant because the battery is the mostexpensive component of the BEVs [5]

The three challenges for the implementation of BEVsfor public transportation systems described in this sectionhave been previously discussed in the literature Neverthelessmost of those studies discuss one topic at a time In [67] the energy consumption during trips is considered Theauthors propose a reduced order model for the energyconsumption which considers the BEVsrsquo characteristics (egweight rolling resistance and drag coefficient) and the roadcharacteristics (eg grade length and traffic) From thismodel an energy graph can be constructed to find the mini-mum energy consumption path between two locations usingclassic algorithms (eg Dijkstra [8] and Johnson [9])

On the other hand in [7 10] the formulated EVRPconsists of finding a set of minimum consumption routes tosatisfy the demand of the users Furthermore some addi-tional restrictions are included to consider the battery capac-ity of the BEVs In [11] the vehicle routing problem is studiedfor the case of alternative fueled vehicles considering theirfueling time In [12] a realistic energy consumption model isused for a vehicle routing problem in which the fleet includesboth BEVs and diesel vehicles In [13] the EVRP consid-ers a heterogeneous fleet composition In [14] an electrictraveling salesman problem is studied In [15] exact algo-rithms are presented for different charging conditions thatgenerate variants of the EVRP In [16 17] the EVRP model isformulated considering the BEVsrsquo recharge time and a set ofrecharge stations Nonetheless the battery degradation costsare not considered

Other works such as [18ndash21] discuss the recharge controlto minimize the energy recharge cost Some papers [19 21]study the recharge control for large fleets of BEVs The con-trol strategy proposed in those works minimizes the costsof energy generation and the BEVs fleet recharge This stra-tegy is not centralized allowing the vehiclersquos autonomy pre-servation nevertheless these articles do not consider theenergy consumed by each BEV nor the optimal rechargeprofiles obtained could increase the battery degradation

In [22ndash24] anothermethod to obtain recharge profiles forBEVs is discussed In these studies an optimization problemis formulated to minimize power losses and maximize theload factor The formulation considers the network topologycharacteristics Thereby the obtained charge profiles contain

information about the schedule and the vehicle that needs tobe recharged depending on the network connection pointwhereby the vehicles connected in the locations that inducehigher losses are recharged in off-peak hours However thesestudies do not consider the effect of the recharge profile on thebattery life nor the recharge cost in the objective function

In [25 26] the battery SOH is studied through a degrada-tionmodel for the lithium-ion batteriesThemodel considersthe main features that influence battery degradation such asthe mean state of charge (SOC) depth of discharge (DOD)and battery temperature In [25] some models to minimizethe battery degradation and the cost of energy rechargeare proposed The recharge profile obtained is considerablydifferent to those obtained in [19 21] where it is calculatedconsidering only the recharge cost

In [27 28] the EVRP is combined with infrastructureproblems In [27] the combination of the EVRP and thelocation-routing problem is presented including the deter-mination of battery swap stations and also the possibilityof intermediate stops In [28] different scenarios for theuse of artificial intelligence for the management of BEVsare presented those scenarios include issues for the grid-to-vehicle interaction as load balancing energy pricing placingof recharging places and rerouting of BEVs to charge points

This work proposes a methodology that relates the threechallenges presented before to determine minimum con-sumption paths a set of optimal routes the route assignmentfor each vehicle and the recharge scheduling for electricvehicles for public transportation A centralized controllerbased on a program that minimizes the costs of electric vehi-clesrsquo operation is proposed The objective function to mini-mize considers both the recharge cost and the cost associatedwith the battery degradation caused by route assignation andrecharge cycles

As a case study an airport shuttle service attended by afleet of BEVs is used to analyze the EVRP while consideringthe topics exposed previously This service consists of trans-porting passengers from a hotel to a nearby airport Compar-ative results between the recharge scheduling obtained with areduced order model and real energy consumption data of anelectric vehicle are presented Finally findings on the BEVsrsquocharge patterns routing and operational costs are presentedThis information could be useful for public transportationcompanies interested in upgrading their fleet to BEVs

This paper is organized as follows Section 2 presents theenergy consumption and optimal routing model the energyconsumption of the BEVs is estimated using a model basedon the longitudinal dynamics equation of motion Section 3introduces the recharge scheduling problem comprised of aroute assignation and the charge model Section 4 describesthe battery degradation model Section 5 presents an evo-lutionary algorithm to solve the proposed EVRP Section 6describes the case study and presents simulation resultsSection 7 presents the conclusion

2 Optimal Routing for BEVs

Theconsidered scenario takes into account the presence of anoperation center (OC) which coordinates both the transport

Mathematical Problems in Engineering 3

Clients

Operations CENTER (OC)

ElectricVEHICLES (BEVs)

Charge STATIONS (RSs)Utility

Transportation demand

Ener

gy ra

tes Availability

State of charge (SOC)

Route assignment and

CHARGE SCHEDULING

Figure 1 Operation center scheme

service and the charge scheduling The transport serviceconsists of a BEV fleet that offers an airport shuttle service

The charge scheduling seeks tominimize the cost for thisreason the OC assigns routes to the BEVs that satisfy thetransportation demand and establishes the charge schedulesin the available charge stations (RSs) according to the energyrate To coordinate the operation of the BEV fleet the OC cancommunicate with BEVs clients (ie potential passengers)RSs and the public utility as presented in Figure 1 In thisway the OC receives information regarding requests fromcustomers one day in advance availability of RSs state ofcharge of BEVs and energy rates from the public utility Theroute assignment and the charge scheduling are calculatedand transmitted to the electric vehicles

In the next subsections the models of the systemsin which the OC is based are described These modelsare presented in the following order estimation of energyconsumption on the road determination of optimal routesassignment of routes scheduling of a specific charge problemand the solution method based on differential evolution

21 Energy Consumption on the Road For a BEV the ener-gy consumption on the road is sensitive to environment roadand vehicle characteristics Factors as the road grade andthe travel speed can significantly influence the energy con-sumption Additionally the traffic conditions and other envi-ronmental factors can cause accelerationdeceleration ratesthat have an impact on the energy consumption of the vehicleConsequently a model that considers a vehicle moving at aconstant speed is not sufficient

The energy consumption model presented in this sectionis based on the longitudinal dynamics equations of motionpresented in [29] This model allows determining the instan-taneous power consumption of the vehicle during its opera-tion considering the resistive loads imposed on the vehicle(ie aerodynamic drag force rolling resistance force due tocontact between tires and road and longitudinal componentof the gravitational force due to driving on hilly roads) Forthe implementation the rolling resistance force was mod-eled by considering a constant rolling coefficient hence therolling resistance force changes linearly with the changesof the normal force between the vehicle and the road[29]

The energy consumption of the vehicle is determinedby integrating the instantaneous power consumption of the

0 50 100 150 200 250 300 350 40002468

101214161820

Time (s)

Spee

d (m

s)

Figure 2 Example of a speed profile for a route

vehicle which is a function of the environment road andvehicle characteristics For the total energy computation theenergy consumed by a BEV while accelerating or in motionwith a constant speed is considered the energy recovered bythe system with regenerative braking is not considered

The road characteristics are modeled in a directed graphG = (VE) The vertices (ie key points or nodes) V isin V

represent the points of special interest on the street mapsTheedges 120576119894119895 isin E where 120576119894119895 = (V119894 V119895) represent road sectionsbetween key points The model assumes that for each edgethe representative driving pattern associated with the givenedge is known and it is represented by its speed profile (seeFigure 2) Hence there is a function 119904 E rarr 120581(R) where119904119894119895 = 119904(120576119894119895) corresponds to the representative speed profileassociated with the road section that connects the vertices V119894and V119895 where 120581(R) is the space of continuous real functionswith compact support

The model also assumes that the key point elevation 119911 V rarr R is known On the other hand (1) presents the termsinvolved into the power consumption of the vehicle

119875119894119895 (119905)= 119875aerodynamic (119905) + 119875rolling (119905) + 119875specific (119905) + 119875gravitational (119905)120578 (1)

Thefirst termon the right side of (1) corresponds to the powerdissipated by aerodynamic drag The second term corre-sponds to the rolling resistance between tires and asphaltThethird term stands for the specific power required to overcome


the vehicle inertia and the fourth term corresponds to thepower consumedgained due to the height of the vehicleoriginated by the road grade represented by angle 120574 Thevehicle characteristics that influence the power consumption

are mass (119898) frontal area (119860) drag and rolling resistancecoefficients (119862119863 119891119903) and the powertrain efficiency (120578) Giventhese characteristics the power consumption at instant 119905 ofthe trip between vertex 119894 and vertex 119895 is given by

119875119894119895 (119905) = (12) 120588119860119862119863 [119904119894119895 (119905)]3 + 119898119892119891119903119904119894119895 (119905) + 119898 (119889119904119894119895119889119905) (119905) 119904119894119895 (119905) + 119898119892 tan (120574) 119904119894119895 (119905)120578 (2)

The energy consumption can be calculated by integrating (2)over the total time of each displacement between verticesto obtain the energy consumption associated with the trip

Under the assumption of constant powertrain efficiency thegravitational component can be analytically integrated Thisleads to the following expression

119888119894119895 = int119905119891

0((12) 120588119860119862119863 [119904119894119895 (119905)]3 + 119898119892119891119903119904119894119895 (119905) + 119898 (119889119904119894119895119889119905) (119905) 119904119894119895 (119905)120578 ) 119889119905 minus 119898119892 [119911 (V119895) minus 119911 (V119894)]120578 (3)

Finally energy consumption can be computed for an arbi-trary path composed of different edges of the graph A path119875ℎ is defined as a sequence of 119897 vertices (V1 V2 V119897) with(V119894 V119894+1) isin E for 119894 = 1 119897 minus 1 The energy necessary totravel along the path 119875ℎ in the road network is the sum of theenergy consumed to complete each one of the road sectionsthat compose the path as follows

119862 (119875ℎ) = 119897minus1sum119894=1

119888119894119895 (4)

22 Determination of Routes Similar to the classical vehiclerouting problem formulated in [30] an EVRP is proposed tomodel the airport shuttle service It consists of searching aset of minimum consumption routes that satisfy the trans-portation demand and the operational constraints Figure 1presents the symbols used to explain the routing model

The EVRPmodel is formulated on the energy graphG119904 =(V119904E119904) which is a simplification of the road network Thegraph nodes are composed ofV119904 = V1198891 V1198892 cup 119862 cup 119877119904 where119862 is the set of nodes within a transportation demand and119877119904 is the set of RSs The expression 2 sdot |119862| is the number ofrequestsmade by customers where the operator |sdot| representsthe set module The depot nodes are denoted by V1198891 andV1198892 which represent the routersquos starting and ending nodesThe graph edges defined by E119904 = (V119894 V119895) | V119894 V119895 isin V119878where (V119894 V119895) isin E119904 are paths of minimum consumptionthat can be found through routing algorithms applied to theroad network model Therefore the transportation demandimplies 119898 requests to pick up passengers and 119898 requeststo drop off passengers each one with its associated numberof passengers The requests are identified by two nodes 119894and 119898 + 119894 corresponding to the pick-up and drop-off stopsrespectively The set of pick-up nodes is denoted by 119875 =1 119898 and the set of drop-off nodes is denoted by 119863 =119898 + 1 2119898 Therefore it is possible to define 119862 = 119875 cup 119863

and if the request 119894 consists of transporting 119902119894 passengers from119894 to 119898 + 119894 then 119902(119898+119894) = minus119902119894The energy consumed during the trip from 119894 to 119895 is given

by 119888119894119895 and the traveling time is 119905119894119895 where 119894 119895 isin V119904 Eachcustomer 119894 isin 119862 has a time window [119886119894 119887119894] in which the servicemust take place Electric vehicles have a maximum passengercapacity 119876 and a maximum battery capacity 119861 The objectivefunctionminimizes the energy consumption for all routes 119867meaning

min sumℎisin119867

sum(119894119895)isinE119904

119888119894119895119909ℎ119894119895 (5)

where ℎ isin 119867 is the set of routes to satisfy all the transporta-tion demand and 119909ℎ119894119895 are the flow variables which are equal to1 if the arc (119894 119895) is used on route ℎ or 0 otherwise In additiontwo constraints are defined to guarantee that all the passengerdemands are satisfied which are

sumℎisin119867

sum119895isinV119904V1198891

119909ℎ119894119895 = 1 forall119894 isin 119875 (6)

sum119894isinV119904V1198892

119909ℎ119894119895 minus sum119894isinV119904V1198891

119909ℎ119895119898+1 = 0forallℎ isin 119867 119895 isin V119904 V1198891 V1198892 (7)

Equation (6) guarantees that each node with passengerdemand is attended or visited by one route Moreover (7)forces passengers picked up at node 119894 to be dropped off atnode 119898 + 119894 Three constraints are also defined to satisfy theflow through vertices these are

sum119894isinV119904V1198892


119909ℎ119895119894 = 0forallℎ isin 119867 119895 isin V119904 V1198891 V1198892 119894 = 119895 (8)


sum119895isinV 119904V1198892

119909ℎV1198891 119895 le 1 forallℎ isin 119867 (9)

sum119894isinV 119904V1198892

119909ℎ119894V1198892 le 1 forallℎ isin 119867 (10)

Equation (8) indicates that if a route ℎ arrives at node 119895then the same route must depart from node 119895 keeping theflow equilibrium Also (9) indicates that a route ℎ can exitone time of the departing node V1198891 On the other hand (10)indicates that route ℎ can enter once to the arriving nodeV1198892 Furthermore three sets of constraints are defined (i)time constraints (ii) capacity constraints and (iii) energyconstraints The capacity constraints are

119910ℎ119895 ge 119910ℎ119894 + 119902119895 minus 119876 + 119876119909ℎ119894119895 forallℎ isin 119867 119894 119895 isin 119875 119894 = 119895 (11)

119902119895 le 119910ℎ119895 le 119876 forallℎ isin 119867 119895 isin V119904 (12)

0 le 119910ℎ119895+119898 le 119876 minus 119902119895 forallℎ isin 119867 119895 isin 119875 (13)

where 119910ℎ119895 is the capacity variable representing the amountof passengers picked up or dropped off during a trip alongroute ℎ to node 119895 Therefore (11) counts the number ofpassengers traveling along route ℎ with constraints (12) and(13) guaranteeing that the capacity variable does not exceedthe maximum capacity 119876The time constraints are

119908ℎ119895 ge 119908ℎ

119894 + 119905119894119895 minus 119872 + 119872119909ℎ119894119895forallℎ isin 119867 119894 119895 isin V119904 119894 = 119895 (14)


where 119908ℎ119894 is the time variable specifying the instant at which

passenger 119894 is picked up or dropped off and 119872 is a constanthigher than any value of119908ℎ

119894 Therefore (14) counts the elapsedtime until passenger 119894 is attended by route ℎ In addition theconstraint in (15) guarantees that passenger 119894 is picked upor dropped off within the time window Finally the energyconstraints are

119890ℎ119895 le 119890ℎ119894 minus 119888119894119895119909ℎ119894119895 + 119861 minus 119861119909ℎ119894119895forallℎ isin 119867 119894 119895 isin V119904 119894 = 119895 (16)

119890min le 119890ℎ119895 forallℎ isin 119867 119895 isin V119904 (17)

where 119890ℎ119894 is the energy variable specifying the remaining bat-tery charge level when route ℎ arrives to vertex 119894Additionally(16) allows computing the battery level based on the traveledvertices sequenceMoreover (17) guarantees that at the end ofroute ℎ the battery level will never drop below the minimumlevel 119890min

The solution of proposed EVPR model is a set of optimalroutes 119867 which minimizes the energy consumed whilesatisfying the passenger demand indirectly reducing thecharge costs

[ ]1 N3 4 5

ts

te

2

Figure 3 Timeline in the programming horizon

3 Scheduling of Charge Problem

In this section the scheduling of charge for BEVs is presentedFirst the assignment of routes for the BEV fleet is definedThen the model for the charge control is established Finallythe problem formulation is presented

31 Assignment of Routes The assignment of routes consistsof the distribution of all the optimal routes 119878 within the BEVfleet It is considered that the assignment of a set of routes inthe vehicle fleet 119870 is performed in a programming horizon119873 The programming horizon is illustrated in Figure 3

The assignment of one route to a specific BEV is denotedby 119886119896119904 with 119896 isin 119870 a binary variable which is equal to 1 if route119904 is assigned to an EV or 0 otherwise A binary variable forldquounavailabilityrdquo is defined This variable indicates that someperiods of 119894 isin 119873 are used to travel along route s assigned to aBEV The unavailability variable is defined as

119889119896119904 (119894) = 1 if 119905119904119904 le 119894 le 119905119890119904 0 otherwise

(18)

1198891198961199041 (119894) + 1198891198961199042 (119894) le 1 forall119894 isin 119873 forall119896 isin 119870 1199041 1199042 isin 119878 (19)

where [119905119904119904 119905119904119890] is the interval of duration of each route 119904with starting time 119905119904119904 and ending time 119905119904119890 This way variable119889119896119904 (119894) allows locating each route 119904 on the horizon 119873 while(19) guarantees that two different routes (119904119894 1199042) with similarduration intervals are not assigned at the same time to thesame BEV On the other hand the energy consumed by theBEV 119896 in route 119904 for interval 119894 of the horizon 119873 is defined as119890119896119904 (119894)

Finally an assignation profile is defined with vector alefsym =119886119896119904 sum119896isin119870 sum119904isin119878 119886119896119904 = |119878| where the double sum guaranteesthat all the routes of 119878 are assigned to the electric vehiclescomposing the fleet Considering this an assignation profileis valid if the constraint on (19) is satisfied for all the BEVs inthe fleet

32 Model for Charge Control A charge control is proposedfor a BEV fleet 119870 over a programming horizon 119873 basedon the work presented in [19] The SOC is defined for eachvehicle 119896 isin 119870 at instant 119894 isin 119873 as SOC119896(119894) with the followingconsiderations

SOC119896 (119894) = SOC119896 (119894 minus 1)+ sum119911isin119877

[120578119911119903119911119906119896119911 (119894) minus 120578119889119911119903119889119911 V119896119911 (119894) minus 119890119896119911 (119894)]minus sum119904isin119878

119890119896119904 (119894) forall119894 isin 119873 forall119896 isin 119870(20)


where 119906119896119911(119894) is a binary variable equal to 1 when the vehicle119896 performs a charge action at instant 119894 119903119911 is the charge rateand 120578119911 is the efficiency of the charge process V119896119911(119894) is a binaryvariable which allows us to integrate the vehicle to building(V2B) energy discharge action this variable is equal to 1 whenthe vehicle 119896 performs a discharge action at instant 119894 119903119889119911 isthe discharge rate and 120578119889119911 is the efficiency of the dischargeprocess Subscript 119911 isin 119877 indicates the place where the chargeor discharge is performed Variable 119890119896119911(119894) indicates the energyconsumed during the displacement towards a RS 119911 explainedin the next section Also a restriction for SOC of the vehiclesmust be considered as

SOCmin le SOC119896 (119894) le SOCmax 119894 isin 119873 forall119896 isin 119870 (21)

where SOCmin and SOCmax are the lower andupper bounds ofSOC respectively Both bounds can be established accordingto the battery manufacturer recommendation consideringthe battery technology For this study the initial and finalSOC are equal to the SOCmax value to guarantee that thevehicle begins and ends in the programming horizonwith thesame battery energy level meaning

SOC119896 (0) = SOCmaxSOC119896 (119873) = SOCmax

forall119896 isin 119870(22)

where SOC119896(0) and SOC119896(119873) are the initial and final statesof charge of a BEV 119896 Finally a charge profile for a vehicle119896 is defined as a set of charge actions 119906119896119911(119894) A charge profilefor the BEV 119896 is denoted u119896 and defined as u119896 = 119906119896119911(119894) 120578119911119903119911 sum119894isin119873 119906119896119911(119894) = sum119894isin119873(119890119896119904 (119894)+119890119896119911(119894)) where the sum of all thecharge actions must be equal to the energy consumed duringall the trips The profile u119896 is valid if the constraints on (20)to (22) are satisfied

33 Reroute of Charge Stations In themodel for determiningroutes it is assumed that when a vehicle 119896 is not travelingalong any of them then the vehicle is parked on the depart-ingarriving node (V1198891 V1198892) where the vehicles can be chargedin a private station denoted by 1198771 isin 119877 It is also consideredthat the vehicle can be charged in public stations identified inthe operation zones these charge actions on public stationsmight be necessary when the vehicles do not have enoughenergy to return to the arrivingdeparting node or when theprivate stations are occupied by other vehicles

When considering that the vehicles can be charged inpublic stations it must also be considered that their trajectoryhas to be reprogrammed or their initially assigned routemust be modified which is known as rerouting Reroutingis defined as the redirection of the vehicle towards a stationto perform a charge action 119906119896119911 119911 = 1198771 without altering thevehiclersquos destination The redirecting or rerouting is denotedwith subscript 119911 in the charge action 119906119896119911 If subscript 119911indicates a place different than 1198771 then BEV 119896 must travelto another station 1198772 isin 119877 to perform the charge action

It must be considered that redirecting towards a stationimplies that the BEV must have available time to travel backand forth to the RS Similarly the BEV must have some timeintervals available to charge the energy that it needs this isassured using two constraints The first constraint is

119889119896119904 (119895) + 119906119896119910 (119895) = 0119894119911 minus Δ119894119911 le 119895 lt 119894119911 forall119910 = 119911 119910 isin 119877 (23)

where 119894119911 isin 119873 is the interval at the beginning of the charge and119906119896119911 Δ119894119911 is the number of time intervals that a BEV 119896 needs totravel from V1198891 to 119911 If the result of (23) is zero the BEV 119896 canbe rerouted towards a public recharge station because it hasenough time to perform the action The second constraint is

119889119896119904 (119895) + 119906119896119910 (119895) = 0119894119911 + Δ119894119911 le 119895 lt 119894119911 + Δ119894119906119911 + Δ119894V1198891 forall119910 = 119911 (24)

whereΔ119894119906119911 is the number of time intervals inwhich the chargeaction needs to be completed and Δ119894V1198891 is the number oftime intervals that a BEV 119896 needs to travel from 119911 to V1198891 Equations (23) and (24) guarantee that BEV 119896 can be reroutedwhen there is enough time available On the other hand theenergy consumed in the rerouted trip is denoted as 119890119896119911 whichcorresponds to the energy consumed traveling from V1198891 to 119911and back This is considered in (20) assuring that the vehiclehas the energy necessary to perform the displacement neededwhen redirected

34 Optimization Problem of Charge for BEVs The proposedoptimization problem seeks to minimize the operation costof the electric vehicle fleet defined as

min sum119911isin119877

sum119896isin119870

sum119894isin119873

119901119911 (119894) 119906119896119911 (119894) + sum119896isin119870

119888119896deg (25)

The first term is the charge energy cost for all the BEVs inthe fleet where 119901119909(119894) is the energy rate in the recharge station119909 at instant 119894 the last term of the objective function is thecost of battery degradation for all the BEVs in the fleet Theobjective function in (25) is subjected to a valid assignmentprofilealefsym and a valid charge profile u119896 for all the BEVs 119896 isin 119870Besides if there is redirecting towards public stations thenthe constraints in (23) and (24) should be satisfied

Likewise a constraint related to the recharge stationsrsquoavailability is defined as sum119896isin119870 119906119896119911(119894) le 119860119911(119894) forall119894 isin 119873 119911 isin119877 to guarantee that the charge actions are programmed inavailable charge points where 119860119911(119894) is the number of spacesavailable in recharge station 1199114 Battery Degradation Model

In this section a simplified version of the lithium-ion batterydegradation model proposed in [25] is presented

The model used in this work estimates the cost of batterydegradation 119888deg in terms of the reduction of the lifespan ofa battery which is a function of three factors temperature


SOC and DOD These factors are related to the cost as fol-lows

119888deg = 119888bat (119871119876119879 + 119871119876SOC + 119871119876DOD) (26)

where 119888bat is the initial cost of the battery and 119871119876119879 119871119876SOCand 119871119876DOD are the percentage terms of battery degradationdue to temperature SOC and DOD respectively

Each of the terms 119871119876119879 119871119876SOC and 119871119876DOD can beinterpreted as a ratio (Δ119871119909119871119909) where Δ119871119909 is the batterylifetime degradation due to a complete charge cycle during aday and 119871119883 is the total lifetime of the battery if the evaluatedcharge cycle is repeated until the end of the battery life thatis when the energy capacity of the battery is lower than 80

The term 119871119876119879 refers to the battery degradation due totemperature and charge time This term is proportional tothe charge power since a high operation temperature corre-sponds to a high charge power This term is defined as

119871119876119879 = int119905ch

1119899ℎ119910119897119910 (119879amb + 119877th1003816100381610038161003816119875119905 (119905)1003816100381610038161003816)119889119905

+ 119905max minus 119905ch119899ℎ119910119897 (119879amb) (27)

where 119905ch is the charge time in hours 119905max is the maximumtime available to perform a charge in hours and 119899ℎ119910 is thenumber of hours in a year 119879amb is the ambient temperatureestablished with a constant value of 25∘C and 119877th is thethermal resistance of the battery established with a constantvalue of 2∘CkW 119875119905 corresponds to the charge or dischargepower in (kW) and 119897119910(119879) is a function that estimates the totalnumber of years that the battery will last at a temperature 119879defined as 119897119910(119879) = 119886119890119887119879 where 119886 and 119887 are the parameters ofthe model of a lithium battery fixed as a = 373 times 10minus4 andb = 636 according to [25]

The term 119871119876SOC describes the battery lifetime degrada-tion due to average SOC defined as

119871119876SOC = 119898119886SOCavg minus 119889119886CFmax119910119901119899ℎ119910 (28)

where 119898119886 = 16times10minus5 and 119889119886 = 64times10minus6 are the parametersof the model for the lithium-ion batteries defined in [25]while 119910119901 = 15 corresponds to the number of years whichthe battery is expected to last and CFmax = 080 representsthe fading capacity at the end of the battery life From (28) itcan be noticed that 119871119876SOC is proportional to SOCavg thus ahigh SOCavg reduces the useful life of the battery more thana moderated one

The term 119871119876DOD describes the degradation of the batterylifetime due to the mean DOD This term is calculated usingthe energy performance concept [31] hence 119871119876DOD is theratio between the energy throughput used in a completecharge cycle and the energy throughput used during the bat-tery lifetime defined as

119871119876DOD = 119861 sum119873119888119894=1DOD119894119873119897 (DODavg) lowast DODavg lowast 119861 (29)

where DOD119894 is the 119894th subcycle in a complete charge cycle119873119888 is the number of subcycles DODavg is the average DODcycle and 119873119897(DOD) is the life time of the battery in cyclesfor a given DOD This is defined as 119873119897(DOD) = (DOD14571)minus106844 which is fitted for lithium-ion battery tech-nology

Summarizing the model has two attributes that allowits integration to the optimization problem First it allowscomputing the battery degradation cost of a BEV due tothe actions of charge and discharge previously programmedSecond the simplicity of the model allows performing thecalculations in a fast way reducing the use of computationalresources embedding it with the optimization problem andits solution method explained in the next section

5 Differential Evolution

A differential evolution (DE) algorithm is presented to solvethe problem formulated in previous sections The DE is adirect stochastic search algorithm based on the evolution ofa population which was developed by Storn et al in 1996[32 33] The DE operates through steps that are similar tothose used in a standard evolutionary algorithm (EA)

The DE employs the differences between members of thepopulation to explore the objective function For this reasonit does not use a probability function to generate the offspringHence theDEuses an approach that is less stochasticmakingit more efficient for solving several kinds of problems [34]

In the next subsections the format of a solution and thesteps composing the DE algorithm are presented

51 Format of a Solution Before solving the problem apossible solution must be represented in a format that theDE can optimize For this case the possible solution of theelectric vehicle charge problem is represented by the validassignment profilealefsym and the collection of the charge profilesu119896 associated with each vehicle The assignment profile con-tains each route assignation variable 119886ℎ119896 forall119896 isin 119870 ℎ isin 119867 Foreach vehicle 119896 isin 119870 the associated charge profile containseach charge action variable for that vehicle 119906119896119911(119899) forall119911 isin 119877119899 isin 119873

119883119906 = 11990611 (1) 11990611 (119873) 119906|119870|1 (1) 119906|119870|1 (119873) 1199061|119877| (1) 1199061|119877| (119873) 119906|119870||119877| (1) 119906|119870||119877| (119873)

119883119886 = 11988611 119886|119870|1 1198861|119867| 119886|119870||119867| (30)

where 119883119906 and 119883119886 maintaining the notation of [32] aredenoted as vectors of charge and assignment parameters res-pectively Considering this 119883 = 119883119906 cup 119883119886 is defined asa feasible solution for the electric vehicle charge problemdenoted as the parameter vector119883 in the development of eachstep of the DE algorithm which are described next

52 Population Initialization Similar to the EA the DE stepsare (i) population initialization (ii) mutation (iii) crossingover and (iv) selection A population is composed of anumber 119873119901 of vectors 119883 where each vector is known as an


individual The following notation is adopted to represent avector of parameters 119894 isin 119873119875 of the population in the currentgeneration 119866119890

119883119894119866119890= [1199091119894119866119890 1199092119894119866119890 119909119898V 119894119866119890

] (31)

where 119866119890 = 0 1 119866max denotes the future generations119866max the last generation and 119898V the size or dimension of theparameter vector

The population of vectors is produced through a randomgenerator where each vector must meet the constraintsotherwise it will be rejected and will not be part of thepopulation This rejection step ensures the feasibility ofthe individuals of the population given that the constraintswere defined specifically for that purpose The size of thepopulation 119873119875 is considered as a control variable of the DEmethod In this case 119873119875 = 10119898V is used according to [34]

53 Mutation The mutation allows generating a donor vec-tor 119881119894119866119890 by means of a differential operation This operationis defined as

119881119894119866119890 = 1198831199031 119866119890OR (1198831199032 119866119890

XOR1198831199033 119866119890) (32)

where 1198831199031 119866119890 1198831199032 119866119890

and 1198831199033 119866119890are three individuals of the

population generation 119866119890 and the indices 1199031 1199032 and 1199033 arerandomly selected from the range [1 119873119875]These indicesmustbe mutually exclusive to diversify the results of the mutationoperation

The differential operation consists of adding the differ-ence of two vectors to another vector of the population [33]In this case the differential operation is implemented withthe exclusive disjunction (XOR) By using the logic operatorXOR the algorithm identifies the differences between twovectors After that the sum operation is implemented usingthe disjunction operator (OR) which permits combining thedifference between the two vectors with another vector of thepopulation

54 Crossover The crossover operation allows the exchangeof components between the donor vector 119881119894119866119890 and the targetvector119883119894119866119890

The result of the crossover is a trial vector119880119894119866119890=[1199061119894119866119890 1199062119894119866119890 119906119898V 119894119866119890

] This operation is defined as

119906119895119894119866119890 = V119895119894119866119890 for 119899V + 1 le 119895 le 119899V + 119871 minus 1119906119895119894119866119890 = 119909119895119894119866119890 otherwise (33)

where 119899V is randomly selected among [1 119898V] with uniformprobability and 119871 is obtained from [1 119898V] with probabilityequal to 119862119903 isin [0 1] The variable 119862119903 isin [0 1] is known asthe crossover rate and constitutes a control variable of the DEmethod 119862119903 must be considerably lower than one (eg 03) Ifthe convergence towards a solution is not achieved then 119862119903

can be chosen in the interval [05 1] according to [34]55 Selection The selection operation allows choosing be-tween the target vector 119883119894119866119890

and the vector 119880119894119866119890(obtained

in the crossover operation) to be part of the next generation119866119890 + 1 The selection operation is defined as

119883119894119866119890+1= 119880119894119866119890

if 119891 (119880119894119866119890) le 119891 (119883119894119866119890

) 119883119894119866119890+1

= 119883119894119866119890otherwise (34)

where 119891(sdot) is the objective function presented in (25) to beminimized This operation indicates that if the trial vector119880119894119866119890

has a value 119891(sdot) equal to or lower than the value 119891(sdot) ofthe target vector then the trial vector will be part of the nextgeneration otherwise the target vector119883119894119866119890

is retained in thepopulation of generation 119866119890 +1Therefore the steps of muta-tion crossover and selection are cycled until the maximumgeneration 119866max is reached A summary of this process ispresented in Algorithm 1

56 Genetic Algorithm Benchmark For the analysis of theperformance of the DE algorithm a standard Genetic Algo-rithm (GA) was used as a benchmark The performance wasanalyzed in terms of convergence and computational costFor the GA formulation the chromosomes that describe eachindividual are binary strings equal to those used in the DE

The GA used has the typical stages of an evolutionaryalgorithmThe first stage is the generation of the initial popu-lation with the population size (119875119866) as a parameter of adjust-ment Population size was chosen as 10 times the quantity ofchromosomersquos genes (119873119866) The second stage is the selectionby tournament in this stage two individuals with the highestfitness are chosen from a randomly selected subpopulationThe third stage is crossover A two-point crossover operationwith a crossover probability of 119875119888 = 08 was performed Thefourth stage is mutation which is intended to maintain aportion of random search Mutation provides the possibilityof changing one of the genes of the chromosomes of thedescendants Mutation probability was chosen as 119875119898 = 001Finally if the ending criterion (ie generation is equal tomaximum generation 119866 = 119866max) is not reached then agenerational change is performed In the generational changethe descendants obtained through the last stages replace theirparents by initiating a new cycle On the other hand if theending criterion is reached the GA finishes Figure 4 presentsa block diagram of the GA used as a benchmark

6 Case Study

The case study consists of an airport shuttle service using aBEV fleet For this service it is required to determine theroute assignment and the charge scheduling in a coordinatedmannerThe location defined for this case study is El Doradoairport in Bogota Colombia In this section the case studyis described in detail through four aspects (i) the operationzone (ii) the BEVsrsquo characteristics and data logging (iii) thedata processing and (iv) the demand for transportation

61 Operation Zone The operation zone used for this casestudy is defined by three nodes El Dorado airport definedas the north-western node a mall located in the south nodeand a public recharge station (RS) in the eastern node (seeFigure 5) Within the area defined by the nodes a total of 54


Inputs Read values of the control parameters 119862119903 119873119875

Initialization of population (PGe) 119866119890 = 0 119875119866 = 1198831119866119890

119883119873119875119866119890

for 119866119890 = 1 to 119866max dofor 119894 = 1 to 119873119875 do

Mutation Generate a donor vector 119881119894119866119881119894119866119890 = 1198831199032119866119890OR (1198831199032 119866119890

XOR 1198831199033 119866119890)

Crossover Generate a trial vector 119880119894119866119890119906119895119894119866119890 = V119895119894119866119890 for 119899V + 1 le 119895 le 119899V + 119871 minus 1119906119895119894119866119890 = 119909119895119894119866119890 119900119905ℎ119890119903119908119894119904119890Selection Evaluate the trial vector 119880119894119866119890119883119894119866119890+1

= 119880119894119866119890if 119891(119880119894119866119890

) le 119891(119883119894119866119890)119883119894119866119890+1

= 119883119894119866119890otherwise

end forend for

Algorithm 1 DE algorithm

Find FITNESS

Selection byTOURNAMENT(TWO individualare selected)

POPULATIONNew

End

InitialPOPULATIONPG = 10NG

G = GＧＲ

Crossover

(TWO descendantsare created)

Pc = 08

MutationPm = 001

Figure 4 Block diagram of the GA used as a benchmark

nodes are used as references to determine the routes Fromthese nodes six are identified as main nodes and are used tobuild the graphics (see Figure 5 and Table 1) The main nodesare (1) airport whereabouts 1 (2) airport whereabouts 2 (3)hotel operation center (OC) private RS with 1199032 = 3 kW 1205782 =09 (4) public RS with 1199033 = 6 kW 1205783 = 09 (5) mall and (6)bus terminal

62 Test Vehicle Characteristics The aim of the case study isto analyze the feasibility of the implementation of a fleet ofBEVs for an airport shuttle service In order to do this theresults of scheduling and routing are used

The kinematical information about the displacements isneeded to estimate the energy consumption of each pathThisinformation is measured by using a different vehicle that hasbeen selected to be as similar as possible to the BEV understudyThe characteristics of the vehicle used in the case studyare summarized in Table 2

The most relevant characteristics considered to estimatethe energy consumption properly are those related to resistiveforces and traction acting on the vehicle The characteristicsinclude the dimensions (ie frontal area and drag coefficient)

and weight to power relation of the vehicle The test vehicleselected is a saloon type It has been necessary to add weightto the vehicle to obtain a similar weight to power ratio

63 Data Logging and Data Processing The characteristicsof the GPS unit used to measure and register the data arepresented in Table 3 With the data obtained from the speedprofile (see Figure 6) it has been determined that a signalconditioning stage must be implemented to handle signalnoise and atypical points due to any interruptions of com-munication between GPS and satellitesThese issues are orig-inated by the buildings located near the streets and by bridgeson the paths Figure 6 presents an example of a lost-of-signalissue which generates a signal spike on the raw speed profile

The data processing considers the use of filters to dealwith the data logging issues Two filtering methods wereconsidered to obtain a smooth speed profile the Kalmanfilter described in [35] and the Savitzky-Golay filter describedin [36 37] Figure 7 presents a comparison between theprocessed data obtained by using these filters The resultobtained by using each filter is compared with the originaldata The goal is to remove the signal spikes originated by


El Dorado Airport(AIRPORT whereabout 2)

Airport whereabout 1

Public RS

Hotel OC private RS

Transport TERMINAL

Shopping MALL

Figure 5 Main nodes of reference located in the operation zone

Table 1 Distance matrix

Distance [km]Departarrive Airport 1 Airport 2 Hotel Public RS Mall TerminalAirport 1 mdash 094 592 955 762 669Airport 2 294 mdash 586 954 76 666Hotel 62 711 mdash 55 179 333Public RS 699 787 421 mdash 421 59Mall 774 862 261 49 mdash 459Terminal 748 837 21 603 325 mdash

Table 2 Characteristics of the test vehicle

Parameter Value119898 [kg] 1312119860 [m2] 186119862119863 032119891119903 00117120578 09Length [m] 432Width [m] 169

the issues of the data logging process keeping the rest of thesignal unaltered It can be observed that for this case studythe Savitzky-Golay filter (Figure 7(a)) is more effective thanthe Kalman filter (Figure 7(b))

Table 3 Characteristics of the GPS unit

Parameter ValueSampling frequency [Hz] 100Speed resolution [kmh] 001Speed accuracy [kmh] 01Distance resolution [m] 001Distance accuracy [] 005

64 Transport Demand An illustrative transport demand isdefined It consists of a case where one passenger is departingfrom the airport whereabouts 1 and two passengers departingfrom the airport whereabouts 2 This scenario is repeatedevery hour from 800 to 1600 Each passenger must pay a5USD ticket and has to be picked up within a time window of


0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

Figure 6 Raw speed profile data

30minutes It is assumed that the passengersmay be travelingto themall the terminal or the hotel according to their needs

65 Energy Consumption The results of energy consumed bythe vehicle during a trip along each path are calculated thisis achieved by the introduction of the filtered speed profile(see Figure 7) in the longitudinal dynamics model describedin Section 21 The results are presented in the energy matrixshown in Table 4

66 Simulation Results This section presents the results ofthe implementation of the proposed methodology for thesolution of the case study The comparison of the proposedDE metaheuristic with a Genetic Algorithm (GA) is alsopresented

The routing problem proposed is solved using the opti-mization tool XPRESSThe results show that nine routesmustbe used to satisfy the total transport demand of the case studyEach route is optimal and has an energy consumption ofapproximately 166 kWh Traveling along each route takes lessthan one hourThe routesmust be followed on an hourly basisduring the period from 0730 to 1530 according to the trans-portation demand The charge and route assignment actionsare planned with a 24-hour programming horizon starting at700 The programming horizon is discretized in half hourintervals Four scenarios (SC) are considered with differentworking conditions to illustrate possible circumstances

In SC1 the charge is allowed only in the private station(RS2) with a two-level energy rate 020USDkWh between[700ndash2200) and 015USDkWh between [2200ndash700) ForSC1 the battery degradation cost is not considered In SC2the charge is also allowed only in the private station (RS2)with the same two levels energy rate of SC1 and the batterydegradation cost is considered

In SC 3 the charge is allowed in both the private station(RS2) and the public station (RS3) In SC3 the energy rateof RS2 is the same one used in SC1 and RS3 has a two-level energy rate 010 USDkWh between [700ndash2200) and005USDkWh between [2200ndash700) In SC3 the batterydegradation cost is considered and the V2B operation isallowed (ie energy discharge of BEVs towards the hotel)In SC4 the conditions are similar to those in SC3 exceptfor the difference in energy rate where both RS2 and RS3

have single-level energy rates 050USDkWh for RS2 and005USDkWh for RS3 In SC4 the energy rate for the privatestation is artificially incremented to investigate a scenariowhere V2B operations are intuitively profitable

The results obtained with the scenarios previouslydescribed are presented in Figures 8 and 9 For each scenariothe rate evolution and the SOC for two vehicles of the fleetand their charge actions on each station are presented acrossthe programming horizon

Figure 8(a) presents the charge and route assignmentobtained for SC1 using DE It is observed that the chargeactions were programmed at the end of the horizon takingadvantage of the low energy rate during the night

Figure 8(b) presents the charge and route assignmentobtained for SC2 using DE It is found that the SOC of theBEVs describes a travel-charge pattern meaning that theBEVs travel along one or two routes and immediately haveto charge When analyzing this behavior it is found that thetravel-charge pattern reduces the battery degradation cost bymaintaining a low DOD This is consistent with the batterymodel since the cycles with a large DOD are the main sourceof battery degradation

SC1 and SC2were used to compare the impact of scenarioconditions on battery lifespan When the charge and roadassignments obtained for SC1 are implemented the expectedbattery lifespan is approximately 6400 cycles On the otherhand when the SC2 assignments are used the expectedbattery lifespan is approximately 8400 cycles This representsa difference of 2000 cycles which is comparable with fiveyears of operation of a BEV in transportation service

Figure 9(a) presents the charge and route scheduling forSC3 obtained with DE It was obtained that the algorithmdoes not schedule the V2B operation although the energydischarges are allowed It is observed that the energy rate forSC3 is not profitable when performing the energy dischargeactions because the economic benefit to afford a V2B opera-tion is surpassed by the battery degradation cost

Figure 9(b) presents the charge and route scheduling forSC4 obtained with DE It was obtained that several chargeactions in RS3 are programmed as well as the operation V2Bthrough discharge actions This behavior can be explaineddue to the high difference in the energy rate in the rechargestations RS2 and RS3 For this reason the V2B operationbecomes profitable However this difference in rates is noteasily accomplished in real scenarios

67 Benchmark Results To analyze the convergence of thesolution and the computational cost of the DE algorithma standard GA algorithm was used as a benchmark asdescribed in Section 56 The algorithms were comparedunder the conditions of scenarios SC1 and SC2 Given the factthat both algorithms depend on pseudorandom variablesseveral iterations were considered The results are presentedindicating the average values and the standard deviationobtained for each observed variable

Table 5 presents the results obtained for the comparisonunder SC1 conditions Both DE and GA converged to thesame value of the daily recharge cost of a vehicle (ie142USD) The recharge schedules obtained were essentially


400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)

Figure 7 Comparison between filtered data Savitzky-Golay filter (a) Kalman filter (b)

0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)

Figure 8 Charge scheduling for two BEVs operating in SC 1 (a) and SC2 (b)


0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)

Figure 9 Charge scheduling for two BEVs operating in SC3 (a) and SC4 (b)

Table 4 Energy consumption matrix

Energy consumption [kWh]Departarrive Airport 1 Airport 2 Hotel Public RS Mall TerminalAirport 1 mdash 0128 0574 1073 079 0719Airport 2 0285 mdash 0567 1076 0634 0656Hotel 0593 0773 mdash 0658 0183 0429Public RS 0688 0828 051 mdash 0443 0727Mall 0805 0876 0321 0497 mdash 0536Terminal 0839 0925 028 072 0407 mdash

the same Consequently the recharging schedules producea similar battery lifespan for both cases 6434 cycles for DEand 6322 cycles for GA Regarding the computational costthe simulation times obtained were 1457 for the DE and 1791for the GA For SC1 the use of DE instead of GA led to areduction of 186 in the computational cost The computa-tional cost difference obtained can be important when a largefleet is considered

Figure 10 shows the convergence curves obtainedwithDEand GA Both algorithms presented expected behaviors TheDE convergence curve is monotonously decreasing The GA

convergence curve presents some oscillations before converg-ing The advantage of DE is the selection process performedbetween generations The choice of GA as benchmark takesadvantage of its capability to reach global solutions From thispoint of view the results of the benchmark performed suggestthat the DE is converging to the global minimum

Table 6 presents the results obtained for the comparisonunder SC2 conditions For this scenario the total daily costwas computed The total cost considers the recharging costand the battery degradation cost The difference between theaverage convergence values obtained with DE and GA for the


0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248

14248Recharging cost

Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)

Figure 10 Convergence curves of DE (a) and GA (b) in SC1

Table 5 Results of the simulation obtained with the DE and GAalgorithms in SC1

Average (Std Dev)DE GA

Recharge cost [USD] 142 (sect) 142 (sect)Battery lifespan [cycles] 6434 (16) 6322 (137)Simulation time [s] 1457 (0201) 1791 (0126)sect represents a standard deviation under 001 USD



Total cost [USD](recharging + batterydegradation)

521 (005) 529 (007)

Battery lifespan [cycles] 8368 (185) 8042 (281)Simulation time [s] 1513 (023) 1742 (0357)total cost is 14 The difference between the average con-vergence values obtained with DE and GA for the batterylifespan is 39

7 Conclusions and Future Research

A methodology to plan the energy charge and route assig-nation for a fleet of BEVs providing a passenger transportservice has been presented This methodology considers thesearch of optimal routes and the minimization of operationalcosts It has been found that the charge scheduling and routeassignment have effects on the battery lifetime The resultobtained for the charge scheduling allows increasing thebattery lifespan Additionally it is observed that the consid-eration of battery degradation modifies the charge patterns

In this work some conditions to perform aV2Boperationhave been reported These conditions are related to thebattery technology the battery degradation model and theenergy rate It was found that batteries with lithium-iontechnology studied in this work do notmeet the requirementsto provide the V2B operation as their degradation cost ishigh Therefore the V2B operation is profitable only for theBEV owner when the difference in energy rate between therecharge station and the discharge station is approximately05USDkW Nonetheless this difference in energy rates isdifficult to achieve in a real scenario

For future work both models for the estimation of theenergy consumption and for battery degradation should berefined The energy consumption model should considerregenerative braking The battery degradation model shouldconsider the detailed behavior of different types of batteriesAlso the performance of additional metaheuristics should beinvestigated

Notations

Notation Used in the BEV Routing Problem

G119904 Simplified energy graphV119904 Vertices or nodes of the energy graphG119904

E119904 Edges of the energy graphG119904

V1198891 V1198892 Depot nodes119862 Set of customer nodes119877119904 Set of nodes on recharge stations119875 Set of pick-up nodes119863 Set of drop-off nodes119888119894119895 Energy consumed by traveling from node 119894to node 119895119905119894119895 Time elapsed during the trip from node 119894to node 119895[119886119894 119887119894] Time window to pick up a passenger atnode 119894


119902119894 Number of passengers at node 119894119876 Maximum passenger capacity of the BEV119861 Battery capacity119872 Constant higher than any value of 119908ℎ119895119890min Lower bound of battery level119867 Set of routes119909ℎ119894119895 Binary flow variable to specify that route ℎ

travels between nodes 119894 and 119895119910ℎ119895 Number of passengers picked up ordropped off at node 119895 when traveling onroute ℎ119908ℎ

119895 Instant at which a passenger 119894 has to bepicked up or dropped off along route ℎ119890ℎ119895 Remaining level of battery charge at theend of node 119894 on route ℎ

Notation Used in the Description of the ChargeScheduling Problem

N Programming horizon119873 Final time slot of programming horizon119886ℎ119896 Route assignation variable119889119896ℎ(119899) Unavailability variable119905119904ℎ Starting time for a trip along route ℎ119905119890ℎ Ending time for a trip along route ℎalefsym Assignation profileSOC119896(119899) State of charge of BEV 119896 battery at instant119899120578119911 Charge efficiency of a charge station 119911119903119911 Charge rate of charge station 119911119906119896119911(119899) Charge action variable of a BEV 119896 at

instant 119899 in charge station 119911119890119896ℎ(119899) Variable of energy consumed by BEV 119896 atinstant 119899 in route ℎ

u119896 Charge profile of BEV 119896119901119911(119899) Energy cost in charge station 119911 at instant 119899119888119896deg Battery degradation cost of BEV 119896Notation Used in the Description of the BatteryDegradation Model

119888bat Battery cost119871119876119879 Battery degradation due to temperature119871119876SOC Battery degradation due to state of charge119871119876DOD Battery degradation due to depth ofdischarge119899ℎ119910 Number of hours in a year119897119910(sdot) Function of the lifespan of the battery inyears119879amb Ambient temperature119877th Thermal resistance119875119905(sdot) Charge power119905max Time available to perform a charge inhours119905ch Charge time in hours

CFmax Battery charge capacity

119910119901 Battery lifespan estimated in years119873119897(sdot) Battery lifespan estimated in cycles

Notation Used in the Description of the DifferentialEvolution Algorithm

XU Vector containing the charge parametersor the information about the chargeactions

Xa Vector containing the route assignationparameters119883119894119866119890

Vector containing the parameters ofsubject 119894 of generation 119866119890119881119894119866119890 Donor vector that contains the parametersof subject 119894 of generation 119866119890119880119894119866119890

Trial vector that contains the parametersof subject 119894 of generation 119866119890119873119901 Number of parameters of the vector119862119903 Crossover rate119898V Number of elements in a parameter vector

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was partially supported by three sources ProjectCIFI 2012 Facultad de Ingenierıa Universidad de LosAndes Codensa SA ESP through Silice III project underGrant P122454220062012 and the Colombian Adminis-trative Department of Science Technology and Innovation(Colciencias) Asoingenierıa Ltda and Universidad de LosAndes under Grant 256-2013

References

[1] A Emadi ldquoTransportation 20rdquo in IEEE Power and EnergyMag-azine vol 9 pp 18ndash29 IEEE 2011

[2] C Ozkurt F Camci V Atamuradov and C Odorry ldquoIntegra-tion of sampling based battery state of health estimationmethodin electric vehiclesrdquo Applied Energy vol 175 pp 356ndash367 2016

[3] S Vazquez S M Lukic E Galvan L G Franquelo and J MCarrasco ldquoEnergy storage systems for transport and grid appli-cationsrdquo IEEE Transactions on Industrial Electronics vol 57 no12 pp 3881ndash3895 2010

[4] A Millner ldquoModeling lithium ion battery degradation inelectric vehiclesrdquo in Proceedings of the 2010 IEEE Conferenceon Innovative Technologies for an Efficient and Reliable Elec-tricity Supply CITRES 2010 pp 349ndash356 Waltham MA USASeptember 2010

[5] L Gaines and R Cuenca ldquoCost of Lithium-Ion Batteries forVehiclesrdquo Center for Transportation Research Energy SystemsDivision Center for Transportation Research Energy SystemsDivision 2000

[6] M Sachenbacher M Leucker A Artmeier and J HaselmayrldquoEfficient Energy-Optimal Routing for Electric Vehiclesrdquo inProceedings of theTwenty-Fifth AAAI Conference on ArtificialIntelligence pp 1402ndash1407 2011


[7] N Touati-Moungla and V Jost ldquoCombinatorial optimizationfor electric vehicles managementrdquo Journal of Energy and PowerEngineering vol 6 pp 738ndash743 2012

[8] E W Dijkstra ldquoA note on two problems in connexion withgraphsrdquo Numerische Mathematik vol 1 pp 269ndash271 1959

[9] D B Johnson ldquoEfficient algorithms for shortest paths in sparsenetworksrdquo Journal of the ACM vol 24 no 1 pp 1ndash13 1977

[10] A Artmeier J Haselmayr M Leucker and M SachenbacherldquoTheOptimal Routing Problem in the Context of Battery- Pow-ered Electric Vehiclesrdquo in proceedings of the Workshop CROCSat CPAIOR-10 Second International Workshop on ConstraintReasoning and Optimization for Computational Sustainabilitypp 1ndash13 2010

[11] S Erdogan and E Miller-Hooks ldquoA Green Vehicle RoutingProblemrdquo Transportation Research Part E Logistics and Trans-portation Review vol 48 no 1 pp 100ndash114 2012

[12] D Goeke and M Schneider ldquoRouting a mixed fleet of electricand conventional vehiclesrdquo European Journal of OperationalResearch vol 245 no 1 pp 81ndash99 2015

[13] G Hiermann J Puchinger S Ropke and R F Hartl ldquoTheelectric fleet size and mix vehicle routing problem with timewindows and recharging stationsrdquo European Journal of Opera-tional Research vol 252 no 3 pp 995ndash1018 2016

[14] R Roberti and M Wen ldquoThe Electric Traveling SalesmanProblem with Time Windowsrdquo Transportation Research Part ELogistics and Transportation Review vol 89 pp 32ndash52 2016

[15] G Desaulniers F Errico S Irnich and M Schneider ldquoExactalgorithms for electric vehicle-routing problems with timewindowsrdquo Operations Research vol 64 no 6 pp 1388ndash14052016

[16] R G Conrad and M A Figliozzi ldquoThe Recharging VehicleRouting Problemrdquo inProceedings of the 2011 Industrial Engineer-ing Research Conference 2011

[17] M Schneider A Stenger and D Goeke ldquoThe electric vehicle-routing problem with time windows and recharging stationsrdquoTransportation Science vol 48 no 4 pp 500ndash520 2014

[18] Z Ma D Callaway and I Hiskens ldquoDecentralized chargingcontrol for large populations of plug-in electric vehicles Appli-cation of the Nash certainty equivalence principlerdquo in Proceed-ings of the 2010 IEEE International Conference on Control Appli-cations CCA 2010 pp 191ndash195 Yokohama Japan September2010

[19] Z Ma D S Callaway and I A Hiskens ldquoDecentralized charg-ing control of large populations of plug-in electric vehiclesrdquoIEEE Transactions on Control Systems Technology vol 21 no 1pp 67ndash78 2013

[20] L Gan U Topcu and S Low ldquoOptimal decentralized protocolfor electric vehicle chargingrdquo in Proceedings of the 2011 50thIEEE Conference on Decision and Control and European ControlConference CDC-ECC 2011 pp 5798ndash5804 Orlando FL USADecember 2011

[21] L Gan U Topcu and S H Low ldquoOptimal decentralized proto-col for electric vehicle chargingrdquo IEEE Transactions on PowerSystems vol 28 no 2 pp 940ndash951 2013

[22] K Clement E Haesen and J Driesen ldquoCoordinated chargingof multiple plug-in hybrid electric vehicles in residential dis-tribution gridsrdquo in Proceedings of the IEEEPES Power SystemsConference and Exposition (PSCE rsquo09) pp 1ndash7 Seattle WAUSA March 2009

[23] K Clement-Nyns E Haesen and J Driesen ldquoThe impactof charging plug-in hybrid electric vehicles on a residential

distribution gridrdquo IEEE Transactions on Power Systems vol 25no 1 pp 371ndash380 2010

[24] E Sortomme andM A El-Sharkawi ldquoOptimal charging strate-gies for unidirectional vehicle-to-gridrdquo IEEE Transactions onSmart Grid vol 2 no 1 pp 131ndash138 2011

[25] A Hoke A Brissette D Maksimovic A Pratt and K SmithldquoElectric vehicle charge optimization including effects of lithi-um-ion battery degradationrdquo in Proceedings of the 7th IEEEVehicle Power and Propulsion Conference VPPC 2011 ChicagoIL USA September 2011

[26] T Markel K Smith and A Pesaran ldquoImproving PetroleumDisplacement Potential of PHEVrsquos Using Enhanced ChargingScenariosrdquo in proceedings of the EVS-24 24th InternationalBattery Hybrid and Fuel Cell Electric Vehicle SymposiumNRELCP-540-45730 pp 1ndash9 2009

[27] J Hof M Schneider and D Goeke ldquoSolving the batteryswap station location-routing problem with capacitated electricvehicles using an AVNS algorithm for vehicle-routing prob-lems with intermediate stopsrdquo Transportation Research Part BMethodological vol 97 pp 102ndash112 2017

[28] E S Rigas S D Ramchurn and N Bassiliades ldquoManagingElectric Vehicles in the Smart Grid Using Artificial IntelligenceA Surveyrdquo IEEE Transactions on Intelligent Transportation Sys-tems vol 16 no 4 pp 1619ndash1635 2015

[29] L E Munoz J C Blanco J P Barreto N A Rincon and S DRoa ldquoConceptual design of a hybrid electric off-road vehiclerdquoin Proceedings of the 2012 IEEE International Electric VehicleConference IEVC 2012 Greenville SC USA March 2012

[30] P Toth and D Vigo Eds The vehicle routing problem vol 9 ofSIAM Monographs on Discrete Mathematics and ApplicationsSociety for Industrial and Applied Mathematics (SIAM) Phila-delphia PA 2002

[31] V Marano S Onori Y Guezennec G Rizzoni and NMadellaldquoLithium-ion batteries life estimation for plug-in hybrid electricvehiclesrdquo in Proceedings of the 5th IEEE Vehicle Power and Pro-pulsionConference VPPC rsquo09 pp 536ndash543DearbornMIUSASeptember 2009

[32] R Storn and K Price ldquoDifferential evolution a simple and effi-cient heuristic for global optimization over continuous spacesrdquoJournal of Global Optimization vol 11 no 4 Article ID 341359pp 341ndash359 1997 httpdxdoiorg101023A1008202821328

[33] S Das and P Suganthan ldquoDifferential evolution a survey ofthe state-of-the-artrdquo IEEE Transactions on Evolutionary Com-putation vol 15 no 1 pp 4ndash31 2011

[34] R Storn ldquoOn the Usage of Differential Evolution for FunctionOptimizationrdquo in Proceedings of North American Fuzzy Infor-mation Processing pp 519ndash523 1996

[35] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Journal of Basic Engineering vol 82 no 1 pp 35ndash451960

[36] S J Orfanidis Introduction to Signal Processing Prentice Hall2010

[37] A Savitzky and M J E Golay ldquoSmoothing and differentiationof data by simplified least squares proceduresrdquoAnalytical Chem-istry vol 36 no 8 pp 1627ndash1639 1964

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


range is considered mainly defined by the battery technologyavailable [3] This consideration could require intermediaterecharge stages to extend the traveling range of the vehicles

The second challenge is the scheduling for rechargingThe recharge scheduling is a challenge as the energy ratevariation during peak and valley hours must be consideredThis process should be coordinated with the routing layoutto guarantee a reliable operation while the recharge costsare minimized Additionally for the recharge scheduling it isnecessary to consider the amount of energy required to per-form the subsequent trip and the time required to recharge

The third challenge is the SOH of the battery The aimis to increase the battery lifespan to reduce the long-termoperational cost It must be considered that the useful lifeof the battery depends on the charge and discharge cycles[4] This aspect is relevant because the battery is the mostexpensive component of the BEVs [5]

The three challenges for the implementation of BEVsfor public transportation systems described in this sectionhave been previously discussed in the literature Neverthelessmost of those studies discuss one topic at a time In [67] the energy consumption during trips is considered Theauthors propose a reduced order model for the energyconsumption which considers the BEVsrsquo characteristics (egweight rolling resistance and drag coefficient) and the roadcharacteristics (eg grade length and traffic) From thismodel an energy graph can be constructed to find the mini-mum energy consumption path between two locations usingclassic algorithms (eg Dijkstra [8] and Johnson [9])

On the other hand in [7 10] the formulated EVRPconsists of finding a set of minimum consumption routes tosatisfy the demand of the users Furthermore some addi-tional restrictions are included to consider the battery capac-ity of the BEVs In [11] the vehicle routing problem is studiedfor the case of alternative fueled vehicles considering theirfueling time In [12] a realistic energy consumption model isused for a vehicle routing problem in which the fleet includesboth BEVs and diesel vehicles In [13] the EVRP consid-ers a heterogeneous fleet composition In [14] an electrictraveling salesman problem is studied In [15] exact algo-rithms are presented for different charging conditions thatgenerate variants of the EVRP In [16 17] the EVRP model isformulated considering the BEVsrsquo recharge time and a set ofrecharge stations Nonetheless the battery degradation costsare not considered

Other works such as [18ndash21] discuss the recharge controlto minimize the energy recharge cost Some papers [19 21]study the recharge control for large fleets of BEVs The con-trol strategy proposed in those works minimizes the costsof energy generation and the BEVs fleet recharge This stra-tegy is not centralized allowing the vehiclersquos autonomy pre-servation nevertheless these articles do not consider theenergy consumed by each BEV nor the optimal rechargeprofiles obtained could increase the battery degradation

In [22ndash24] anothermethod to obtain recharge profiles forBEVs is discussed In these studies an optimization problemis formulated to minimize power losses and maximize theload factor The formulation considers the network topologycharacteristics Thereby the obtained charge profiles contain

information about the schedule and the vehicle that needs tobe recharged depending on the network connection pointwhereby the vehicles connected in the locations that inducehigher losses are recharged in off-peak hours However thesestudies do not consider the effect of the recharge profile on thebattery life nor the recharge cost in the objective function

In [25 26] the battery SOH is studied through a degrada-tionmodel for the lithium-ion batteriesThemodel considersthe main features that influence battery degradation such asthe mean state of charge (SOC) depth of discharge (DOD)and battery temperature In [25] some models to minimizethe battery degradation and the cost of energy rechargeare proposed The recharge profile obtained is considerablydifferent to those obtained in [19 21] where it is calculatedconsidering only the recharge cost

In [27 28] the EVRP is combined with infrastructureproblems In [27] the combination of the EVRP and thelocation-routing problem is presented including the deter-mination of battery swap stations and also the possibilityof intermediate stops In [28] different scenarios for theuse of artificial intelligence for the management of BEVsare presented those scenarios include issues for the grid-to-vehicle interaction as load balancing energy pricing placingof recharging places and rerouting of BEVs to charge points

This work proposes a methodology that relates the threechallenges presented before to determine minimum con-sumption paths a set of optimal routes the route assignmentfor each vehicle and the recharge scheduling for electricvehicles for public transportation A centralized controllerbased on a program that minimizes the costs of electric vehi-clesrsquo operation is proposed The objective function to mini-mize considers both the recharge cost and the cost associatedwith the battery degradation caused by route assignation andrecharge cycles

As a case study an airport shuttle service attended by afleet of BEVs is used to analyze the EVRP while consideringthe topics exposed previously This service consists of trans-porting passengers from a hotel to a nearby airport Compar-ative results between the recharge scheduling obtained with areduced order model and real energy consumption data of anelectric vehicle are presented Finally findings on the BEVsrsquocharge patterns routing and operational costs are presentedThis information could be useful for public transportationcompanies interested in upgrading their fleet to BEVs

This paper is organized as follows Section 2 presents theenergy consumption and optimal routing model the energyconsumption of the BEVs is estimated using a model basedon the longitudinal dynamics equation of motion Section 3introduces the recharge scheduling problem comprised of aroute assignation and the charge model Section 4 describesthe battery degradation model Section 5 presents an evo-lutionary algorithm to solve the proposed EVRP Section 6describes the case study and presents simulation resultsSection 7 presents the conclusion

2 Optimal Routing for BEVs

Theconsidered scenario takes into account the presence of anoperation center (OC) which coordinates both the transport


Clients





Ener

gy ra

tes Availability



CHARGE SCHEDULING








0 50 100 150 200 250 300 350 40002468

101214161820

Time (s)

Spee

d (m

s)











119875119894119895 (119905) = (12) 120588119860119862119863 [119904119894119895 (119905)]3 + 119898119892119891119903119904119894119895 (119905) + 119898 (119889119904119894119895119889119905) (119905) 119904119894119895 (119905) + 119898119892 tan (120574) 119904119894119895 (119905)120578 (2)



119888119894119895 = int119905119891

0((12) 120588119860119862119863 [119904119894119895 (119905)]3 + 119898119892119891119903119904119894119895 (119905) + 119898 (119889119904119894119895119889119905) (119905) 119904119894119895 (119905)120578 ) 119889119905 minus 119898119892 [119911 (V119895) minus 119911 (V119894)]120578 (3)


119862 (119875ℎ) = 119897minus1sum119894=1

119888119894119895 (4)






sum(119894119895)isinE119904

119888119894119895119909ℎ119894119895 (5)


sumℎisin119867

sum119895isinV119904V1198891

119909ℎ119894119895 = 1 forall119894 isin 119875 (6)

sum119894isinV119904V1198892




sum119894isinV119904V1198892




sum119895isinV 119904V1198892


sum119894isinV 119904V1198892







119908ℎ119895 ge 119908ℎ










[ ]1 N3 4 5

ts

te

2







(18)













forall119896 isin 119870(22)







119889119896119904 (119895) + 119906119896119910 (119895) = 0119894119911 + Δ119894119911 le 119895 lt 119894119911 + Δ119894119906119911 + Δ119894V1198891 forall119910 = 119911 (24)




sum119896isin119870

sum119894isin119873

119901119911 (119894) 119906119896119911 (119894) + sum119896isin119870

119888119896deg (25)







119888deg = 119888bat (119871119876119879 + 119871119876SOC + 119871119876DOD) (26)




119871119876119879 = int119905ch

1119899ℎ119910119897119910 (119879amb + 119877th1003816100381610038161003816119875119905 (119905)1003816100381610038161003816)119889119905

+ 119905max minus 119905ch119899ℎ119910119897 (119879amb) (27)














119883119906 = 11990611 (1) 11990611 (119873) 119906|119870|1 (1) 119906|119870|1 (119873) 1199061|119877| (1) 1199061|119877| (119873) 119906|119870||119877| (1) 119906|119870||119877| (119873)

119883119886 = 11988611 119886|119870|1 1198861|119867| 119886|119870||119867| (30)





119883119894119866119890= [1199091119894119866119890 1199092119894119866119890 119909119898V 119894119866119890

] (31)




119881119894119866119890 = 1198831199031 119866119890OR (1198831199032 119866119890

XOR1198831199033 119866119890) (32)

where 1198831199031 119866119890 1198831199032 119866119890












119883119894119866119890+1= 119880119894119866119890

if 119891 (119880119894119866119890) le 119891 (119883119894119866119890

) 119883119894119866119890+1

= 119883119894119866119890otherwise (34)






6 Case Study






119883119873119875119866119890



XOR 1198831199033 119866119890)


= 119880119894119866119890if 119891(119880119894119866119890

) le 119891(119883119894119866119890)119883119894119866119890+1

= 119883119894119866119890otherwise

end forend for


Find FITNESS


POPULATIONNew

End


G = GＧＲ

Crossover


Pc = 08

MutationPm = 001












Public RS

Hotel OC private RS

Transport TERMINAL

Shopping MALL











0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)


















400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)


0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)



0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Clients





Ener

gy ra

tes Availability



CHARGE SCHEDULING








0 50 100 150 200 250 300 350 40002468

101214161820

Time (s)

Spee

d (m

s)











119875119894119895 (119905) = (12) 120588119860119862119863 [119904119894119895 (119905)]3 + 119898119892119891119903119904119894119895 (119905) + 119898 (119889119904119894119895119889119905) (119905) 119904119894119895 (119905) + 119898119892 tan (120574) 119904119894119895 (119905)120578 (2)



119888119894119895 = int119905119891

0((12) 120588119860119862119863 [119904119894119895 (119905)]3 + 119898119892119891119903119904119894119895 (119905) + 119898 (119889119904119894119895119889119905) (119905) 119904119894119895 (119905)120578 ) 119889119905 minus 119898119892 [119911 (V119895) minus 119911 (V119894)]120578 (3)


119862 (119875ℎ) = 119897minus1sum119894=1

119888119894119895 (4)






sum(119894119895)isinE119904

119888119894119895119909ℎ119894119895 (5)


sumℎisin119867

sum119895isinV119904V1198891

119909ℎ119894119895 = 1 forall119894 isin 119875 (6)

sum119894isinV119904V1198892




sum119894isinV119904V1198892




sum119895isinV 119904V1198892


sum119894isinV 119904V1198892







119908ℎ119895 ge 119908ℎ










[ ]1 N3 4 5

ts

te

2







(18)













forall119896 isin 119870(22)







119889119896119904 (119895) + 119906119896119910 (119895) = 0119894119911 + Δ119894119911 le 119895 lt 119894119911 + Δ119894119906119911 + Δ119894V1198891 forall119910 = 119911 (24)




sum119896isin119870

sum119894isin119873

119901119911 (119894) 119906119896119911 (119894) + sum119896isin119870

119888119896deg (25)







119888deg = 119888bat (119871119876119879 + 119871119876SOC + 119871119876DOD) (26)




119871119876119879 = int119905ch

1119899ℎ119910119897119910 (119879amb + 119877th1003816100381610038161003816119875119905 (119905)1003816100381610038161003816)119889119905

+ 119905max minus 119905ch119899ℎ119910119897 (119879amb) (27)














119883119906 = 11990611 (1) 11990611 (119873) 119906|119870|1 (1) 119906|119870|1 (119873) 1199061|119877| (1) 1199061|119877| (119873) 119906|119870||119877| (1) 119906|119870||119877| (119873)

119883119886 = 11988611 119886|119870|1 1198861|119867| 119886|119870||119867| (30)





119883119894119866119890= [1199091119894119866119890 1199092119894119866119890 119909119898V 119894119866119890

] (31)




119881119894119866119890 = 1198831199031 119866119890OR (1198831199032 119866119890

XOR1198831199033 119866119890) (32)

where 1198831199031 119866119890 1198831199032 119866119890












119883119894119866119890+1= 119880119894119866119890

if 119891 (119880119894119866119890) le 119891 (119883119894119866119890

) 119883119894119866119890+1

= 119883119894119866119890otherwise (34)






6 Case Study






119883119873119875119866119890



XOR 1198831199033 119866119890)


= 119880119894119866119890if 119891(119880119894119866119890

) le 119891(119883119894119866119890)119883119894119866119890+1

= 119883119894119866119890otherwise

end forend for


Find FITNESS


POPULATIONNew

End


G = GＧＲ

Crossover


Pc = 08

MutationPm = 001












Public RS

Hotel OC private RS

Transport TERMINAL

Shopping MALL











0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)


















400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)


0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)



0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







119875119894119895 (119905) = (12) 120588119860119862119863 [119904119894119895 (119905)]3 + 119898119892119891119903119904119894119895 (119905) + 119898 (119889119904119894119895119889119905) (119905) 119904119894119895 (119905) + 119898119892 tan (120574) 119904119894119895 (119905)120578 (2)



119888119894119895 = int119905119891

0((12) 120588119860119862119863 [119904119894119895 (119905)]3 + 119898119892119891119903119904119894119895 (119905) + 119898 (119889119904119894119895119889119905) (119905) 119904119894119895 (119905)120578 ) 119889119905 minus 119898119892 [119911 (V119895) minus 119911 (V119894)]120578 (3)


119862 (119875ℎ) = 119897minus1sum119894=1

119888119894119895 (4)






sum(119894119895)isinE119904

119888119894119895119909ℎ119894119895 (5)


sumℎisin119867

sum119895isinV119904V1198891

119909ℎ119894119895 = 1 forall119894 isin 119875 (6)

sum119894isinV119904V1198892




sum119894isinV119904V1198892




sum119895isinV 119904V1198892


sum119894isinV 119904V1198892







119908ℎ119895 ge 119908ℎ










[ ]1 N3 4 5

ts

te

2







(18)













forall119896 isin 119870(22)







119889119896119904 (119895) + 119906119896119910 (119895) = 0119894119911 + Δ119894119911 le 119895 lt 119894119911 + Δ119894119906119911 + Δ119894V1198891 forall119910 = 119911 (24)




sum119896isin119870

sum119894isin119873

119901119911 (119894) 119906119896119911 (119894) + sum119896isin119870

119888119896deg (25)







119888deg = 119888bat (119871119876119879 + 119871119876SOC + 119871119876DOD) (26)




119871119876119879 = int119905ch

1119899ℎ119910119897119910 (119879amb + 119877th1003816100381610038161003816119875119905 (119905)1003816100381610038161003816)119889119905

+ 119905max minus 119905ch119899ℎ119910119897 (119879amb) (27)














119883119906 = 11990611 (1) 11990611 (119873) 119906|119870|1 (1) 119906|119870|1 (119873) 1199061|119877| (1) 1199061|119877| (119873) 119906|119870||119877| (1) 119906|119870||119877| (119873)

119883119886 = 11988611 119886|119870|1 1198861|119867| 119886|119870||119867| (30)





119883119894119866119890= [1199091119894119866119890 1199092119894119866119890 119909119898V 119894119866119890

] (31)




119881119894119866119890 = 1198831199031 119866119890OR (1198831199032 119866119890

XOR1198831199033 119866119890) (32)

where 1198831199031 119866119890 1198831199032 119866119890












119883119894119866119890+1= 119880119894119866119890

if 119891 (119880119894119866119890) le 119891 (119883119894119866119890

) 119883119894119866119890+1

= 119883119894119866119890otherwise (34)






6 Case Study






119883119873119875119866119890



XOR 1198831199033 119866119890)


= 119880119894119866119890if 119891(119880119894119866119890

) le 119891(119883119894119866119890)119883119894119866119890+1

= 119883119894119866119890otherwise

end forend for


Find FITNESS


POPULATIONNew

End


G = GＧＲ

Crossover


Pc = 08

MutationPm = 001












Public RS

Hotel OC private RS

Transport TERMINAL

Shopping MALL











0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)


















400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)


0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)



0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





sum119895isinV 119904V1198892


sum119894isinV 119904V1198892







119908ℎ119895 ge 119908ℎ










[ ]1 N3 4 5

ts

te

2







(18)













forall119896 isin 119870(22)







119889119896119904 (119895) + 119906119896119910 (119895) = 0119894119911 + Δ119894119911 le 119895 lt 119894119911 + Δ119894119906119911 + Δ119894V1198891 forall119910 = 119911 (24)




sum119896isin119870

sum119894isin119873

119901119911 (119894) 119906119896119911 (119894) + sum119896isin119870

119888119896deg (25)







119888deg = 119888bat (119871119876119879 + 119871119876SOC + 119871119876DOD) (26)




119871119876119879 = int119905ch

1119899ℎ119910119897119910 (119879amb + 119877th1003816100381610038161003816119875119905 (119905)1003816100381610038161003816)119889119905

+ 119905max minus 119905ch119899ℎ119910119897 (119879amb) (27)














119883119906 = 11990611 (1) 11990611 (119873) 119906|119870|1 (1) 119906|119870|1 (119873) 1199061|119877| (1) 1199061|119877| (119873) 119906|119870||119877| (1) 119906|119870||119877| (119873)

119883119886 = 11988611 119886|119870|1 1198861|119867| 119886|119870||119867| (30)





119883119894119866119890= [1199091119894119866119890 1199092119894119866119890 119909119898V 119894119866119890

] (31)




119881119894119866119890 = 1198831199031 119866119890OR (1198831199032 119866119890

XOR1198831199033 119866119890) (32)

where 1198831199031 119866119890 1198831199032 119866119890












119883119894119866119890+1= 119880119894119866119890

if 119891 (119880119894119866119890) le 119891 (119883119894119866119890

) 119883119894119866119890+1

= 119883119894119866119890otherwise (34)






6 Case Study






119883119873119875119866119890



XOR 1198831199033 119866119890)


= 119880119894119866119890if 119891(119880119894119866119890

) le 119891(119883119894119866119890)119883119894119866119890+1

= 119883119894119866119890otherwise

end forend for


Find FITNESS


POPULATIONNew

End


G = GＧＲ

Crossover


Pc = 08

MutationPm = 001












Public RS

Hotel OC private RS

Transport TERMINAL

Shopping MALL











0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)


















400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)


0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)



0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of









forall119896 isin 119870(22)







119889119896119904 (119895) + 119906119896119910 (119895) = 0119894119911 + Δ119894119911 le 119895 lt 119894119911 + Δ119894119906119911 + Δ119894V1198891 forall119910 = 119911 (24)




sum119896isin119870

sum119894isin119873

119901119911 (119894) 119906119896119911 (119894) + sum119896isin119870

119888119896deg (25)







119888deg = 119888bat (119871119876119879 + 119871119876SOC + 119871119876DOD) (26)




119871119876119879 = int119905ch

1119899ℎ119910119897119910 (119879amb + 119877th1003816100381610038161003816119875119905 (119905)1003816100381610038161003816)119889119905

+ 119905max minus 119905ch119899ℎ119910119897 (119879amb) (27)














119883119906 = 11990611 (1) 11990611 (119873) 119906|119870|1 (1) 119906|119870|1 (119873) 1199061|119877| (1) 1199061|119877| (119873) 119906|119870||119877| (1) 119906|119870||119877| (119873)

119883119886 = 11988611 119886|119870|1 1198861|119867| 119886|119870||119867| (30)





119883119894119866119890= [1199091119894119866119890 1199092119894119866119890 119909119898V 119894119866119890

] (31)




119881119894119866119890 = 1198831199031 119866119890OR (1198831199032 119866119890

XOR1198831199033 119866119890) (32)

where 1198831199031 119866119890 1198831199032 119866119890












119883119894119866119890+1= 119880119894119866119890

if 119891 (119880119894119866119890) le 119891 (119883119894119866119890

) 119883119894119866119890+1

= 119883119894119866119890otherwise (34)






6 Case Study






119883119873119875119866119890



XOR 1198831199033 119866119890)


= 119880119894119866119890if 119891(119880119894119866119890

) le 119891(119883119894119866119890)119883119894119866119890+1

= 119883119894119866119890otherwise

end forend for


Find FITNESS


POPULATIONNew

End


G = GＧＲ

Crossover


Pc = 08

MutationPm = 001












Public RS

Hotel OC private RS

Transport TERMINAL

Shopping MALL











0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)


















400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)


0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)



0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119888deg = 119888bat (119871119876119879 + 119871119876SOC + 119871119876DOD) (26)




119871119876119879 = int119905ch

1119899ℎ119910119897119910 (119879amb + 119877th1003816100381610038161003816119875119905 (119905)1003816100381610038161003816)119889119905

+ 119905max minus 119905ch119899ℎ119910119897 (119879amb) (27)














119883119906 = 11990611 (1) 11990611 (119873) 119906|119870|1 (1) 119906|119870|1 (119873) 1199061|119877| (1) 1199061|119877| (119873) 119906|119870||119877| (1) 119906|119870||119877| (119873)

119883119886 = 11988611 119886|119870|1 1198861|119867| 119886|119870||119867| (30)





119883119894119866119890= [1199091119894119866119890 1199092119894119866119890 119909119898V 119894119866119890

] (31)




119881119894119866119890 = 1198831199031 119866119890OR (1198831199032 119866119890

XOR1198831199033 119866119890) (32)

where 1198831199031 119866119890 1198831199032 119866119890












119883119894119866119890+1= 119880119894119866119890

if 119891 (119880119894119866119890) le 119891 (119883119894119866119890

) 119883119894119866119890+1

= 119883119894119866119890otherwise (34)






6 Case Study






119883119873119875119866119890



XOR 1198831199033 119866119890)


= 119880119894119866119890if 119891(119880119894119866119890

) le 119891(119883119894119866119890)119883119894119866119890+1

= 119883119894119866119890otherwise

end forend for


Find FITNESS


POPULATIONNew

End


G = GＧＲ

Crossover


Pc = 08

MutationPm = 001












Public RS

Hotel OC private RS

Transport TERMINAL

Shopping MALL











0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)


















400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)


0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)



0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119883119894119866119890= [1199091119894119866119890 1199092119894119866119890 119909119898V 119894119866119890

] (31)




119881119894119866119890 = 1198831199031 119866119890OR (1198831199032 119866119890

XOR1198831199033 119866119890) (32)

where 1198831199031 119866119890 1198831199032 119866119890












119883119894119866119890+1= 119880119894119866119890

if 119891 (119880119894119866119890) le 119891 (119883119894119866119890

) 119883119894119866119890+1

= 119883119894119866119890otherwise (34)






6 Case Study






119883119873119875119866119890



XOR 1198831199033 119866119890)


= 119880119894119866119890if 119891(119880119894119866119890

) le 119891(119883119894119866119890)119883119894119866119890+1

= 119883119894119866119890otherwise

end forend for


Find FITNESS


POPULATIONNew

End


G = GＧＲ

Crossover


Pc = 08

MutationPm = 001












Public RS

Hotel OC private RS

Transport TERMINAL

Shopping MALL











0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)


















400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)


0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)



0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







119883119873119875119866119890



XOR 1198831199033 119866119890)


= 119880119894119866119890if 119891(119880119894119866119890

) le 119891(119883119894119866119890)119883119894119866119890+1

= 119883119894119866119890otherwise

end forend for


Find FITNESS


POPULATIONNew

End


G = GＧＲ

Crossover


Pc = 08

MutationPm = 001












Public RS

Hotel OC private RS

Transport TERMINAL

Shopping MALL











0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)


















400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)


0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)



0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







Public RS

Hotel OC private RS

Transport TERMINAL

Shopping MALL











0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)


















400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)


0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)



0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0 100 200 300 400 500 600 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)


















400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)


0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)



0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





400 450 500 550 600 650 7000

10

20

30

40

50

60

70

Time (s)

Spee

d (m

s)

OriginalSavitzky

(a)

0

10

20

30

40

50

60

70

Spee

d (m

s)

400 450 500 550 600 650 700Time (s)

OriginalKalman

(b)


0

02

04

$kW

h

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

hours

SOCRA in 2

(a)

0

02

04

$kW

h

0

02

04

06

08

1

EV1

0

02

04

06

08

1

EV2

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

hours

SOCRA in 2

(b)



0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0

05

1$

kWh

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV2

0

02

04

06

08

1

EV1

(a)

0

05

1

$kW

h

minus02

0700 1000 1300 1600 1900 2200 0100 0400

SOCRA in 2

RA in 3

0700 1000 1300 1600 1900 2200 0100 0400

0700 1000 1300 1600 1900 2200 0100 0400

HOURS

0

02

04

06

08

1

EV1

minus02

0

02

04

06

08

EV2

(b)









0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0 10 20 30 40 50 60 70 80 90 10014244

14244

14245

14246

14246

14246

14247

14248


Generations

Cos

t ($)

(a)

Recharging cost

0 10 20 30 40 50 60 70 80 90 100Generations

Cos

t ($)

14244

14244

14245

14246

14246

14246

14247

(b)








521 (005) 529 (007)






Notations























Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






















Acknowledgments


References















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Documents

Optimal Routing and Scheduling of Charge for Electric