Energy efficient route planning for electric vehicles with ......Yen algorithm helps solving a multi-objective opti-mization problem with three optimization variables representing

/ : 25 September 2020/ Published online

Energy Efficiency (2020) 13:1705–1726https://doi.org/10.1007/s12053-020-09900-5

ORIGINAL ARTICLE

Energy efficient route planning for electric vehicleswith special consideration of the topographyand battery lifetime

Theresia Perger ·Hans Auer

Received: 22 July 2019 / Accepted: 28 August 2020© The Author(s) 2020

Abstract In contrast to conventional routing sys-tems, which determine the shortest distance or thefastest path to a destination, this work designs aroute planning specifically for electric vehicles byfinding an energy-optimal solution while simul-taneously considering stress on the battery. Afterfinding a physical model of the energy consump-tion of the electric vehicle including heating, airconditioning, and other additional loads, the streetnetwork is modeled as a network with nodes andweighted edges in order to apply a shortest pathalgorithm that finds the route with the smallest edgecosts. A variation of the Bellman-Ford algorithm,the Yen algorithm, is modified such that batteryconstraints can be included. Thus, the modifiedYen algorithm helps solving a multi-objective opti-mization problem with three optimization variablesrepresenting the energy consumption with (vehiclereaching the destination with the highest state ofcharge possible), the journey time, and the cycliclifetime of the battery (minimizing the number of

T. Perger (�) · H. AuerInstitute of Energy Systems and Electrical Drives, EnergyEconomics Group (EEG), TU Wien, Gusshausstrasse 25-29E370-3, 1040, Wien, Austriae-mail: [email protected]

H. Auere-mail: [email protected]

charging/discharging cycles by minimizing theamount of energy consumed or regenerated). For theoptimization problem, weights are assigned to eachvariable in order to put emphasis on one or the other.The route planning system is tested for a suburbanarea in Austria and for the city of San Francisco,CA. Topography has a strong influence on energyconsumption and battery operation and therefore thechoice of route. The algorithm finds different resultsconsidering different preferences, putting weightson the decision variable of the multi-objective opti-mization. Also, the tests are conducted for differentoutside temperatures and weather conditions, as wellas for different vehicle types.

Keywords Energy efficiency · Electric vehicles ·Route planning · Yen algorithm · Multi-objectiveoptimization · Battery lifetime

Nomenclature

Vehicle parametersm Masscw Drag coefficientA Cross-sectional areaSoC State of charge of the batteryw/h Width/heightv Velocity

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Road parameterss Length of a road sectionq Slope of a road section in %α Slope of a road section in rad

External parametersT Outside temperatureg Gravitational constantρ Air densityfR Rolling resistance coefficient

Efficienciesηd Final driveηm/ηg Motor/generatorηinv DC-AC inverterηacc Accessoriesηhc Heating and coolingηcha/ηdis Battery charging/discharging

Topography dataRearth Earth radiusφ/ψ Coordinates in longitude/latitude�x/�y Distances�z Height difference

Energy consumptionFroll Rolling resistanceFair Air resistanceFgrad Gradient resistanceFdrive Driving forcePdrive Power for drivingPheat Power for heatingEdrive Energy for drivingEregen Regenerative energy from drivingEacc Energy consumption for the

accessoriesEhc Energy for heating and coolingEout/Ein Energy from/to the battery

Multi-objective optimizationa Minimum battery capacity factorγ /δ Weights for multi-obj.

optimization

f(k)i /g

(k)i /h

(k)i Costs to node i (kth iteration)

di,j /ti,j /ai,j Edge costs from i to jEmin/Tmin/ Amin Optimization results

Introduction

Combustion engine driven cars have been dominat-ing our world for more than a century. With globalwarming concerns (Gota et al. 2019) and increasinglystringent environmental policies ahead, vehicles withelectric motors could be part of the solution, a precon-dition being that electrical power comes from renew-able sources. In addition, further challenges come upwith this new technology option in transportation.Despite higher efficiency than combustion engines,the task to store energy in the vehicle is still chal-lenging. The battery offers a much more limited rangecompared with conventional cars. Moreover, recharg-ing an electric vehicle is more time-consuming thanrefilling a tank. This leads to a more complex tripplanning with an electric vehicle. A good advice is toidentify the location of charging stations beforehandand to plan the route accordingly. Energy consumptioncan vary significantly depending on the path chosen.This work elaborates on optimal route planning to adesired destination while considering the special char-acteristics of electric vehicles. The main focus lies onimproving the battery lifetime, as well as minimiz-ing energy consumption and journey time while takinginto account impacts of topography.

The first objective of this work is to find a routefrom a starting to a destination point with the leastamount of energy used. This task is expanded suchthat it is possible to calculate the shortest journeytime as well as the best route to maximize batterylifetime. Those route planning options are combinedinto a weighted multi-objective optimization problem.Shortest path algorithms are used in networks con-sisting of nodes, edges, and edge costs to solve theoptimization problem. With the help of these algo-rithms, it is possible to find the path from one nodeto another with the smallest edge costs. Based ona model of the electric vehicle, which describes thephysics of different driving modes, the energy con-sumption and travel time are approximated. It includesthe energy required for driving as well as additionalloads for air conditioning and other accessories. Themodel also considers engine operation as either amotor or as a generator and thus respects regenerativeenergy.

1706 Energy Efficiency (2020) 13:1705–1726

The following “State of the art” section givesan overview of relevant work covering this topic.The “Methodology” section explains the model, whichis used to calculate the vehicle’s energy consump-tion, and the impact additional loads and efficiencieshave on the vehicle. Furthermore, the street networkmodel, the shortest path algorithm, and the multi-objective optimization are explained. The “Results”section presents the results of the optimization algo-rithm tested for an urban as well as a suburban area. Inaddition, a sensitivity analysis investigates the effectsof temperature and different vehicle models from dif-ferent manufacturers. The “Conclusion and outlook”section presents conclusions and elaborates on stillopen questions in this field of research.

State of the art

Creating an optimal route planning system for electricvehicles is multi-disciplinary and requires profoundknowledge of electric vehicles, batteries, route plan-ning algorithms, and dynamic optimization.

Modeling electric vehicles Energy efficiency and bat-tery conservation are the main goals of the proposedoptimal route planning system. Therefore, the elec-tric vehicle is described based on a physical model inorder to enable calculation of the energy consumption.In this context, one special characteristic of electricvehicles is the ability to regenerate energy from brak-ing or driving downhill. In Lv et al. (2015) the energyflow from regenerative braking is modeled in detail.Another detailed model of the vehicle’s energy con-sumption is presented in Yi and Bauer (2017). Thegoal is to find a high-resolution powertrain efficiencyestimation. The energy consumption is described bythe physical forces acting on the moving vehicle. Asimilar concept can be found in Maia et al. (2011),where an energy consumption simulator is presented.An alternative approach to estimate the energy con-sumption of an electric vehicle is found in Hayeset al. (2011), where a range estimator is created byusing different drive cycles, which are originally basedon combustion engines and then adapted for elec-tric vehicles. However, the approach in Hayes et al.(2011) avoided modeling the physics of an electricvehicle.

Battery lifetime in electric vehicles Having batteriesas an energy storage in vehicles comes along withmany new challenges (see, e.g., Rizoug et al. (2018))compared with cars with conventional combustionengines. Apart from their costly production, batterieshave a limited lifetime. According to Li et al. (2017),by 2020 nearly one million electric vehicle batterypacks will be “retired” in the USA alone, meaning thatthey cannot be used in electric vehicles anymore. Thishappens when (i) the battery only provides less than80% of the original capacity and maximum power,or (ii) there are functional failures occurring. Sev-eral models to describe the battery degradation canbe found in Pelletier et al. (2017), where cycle agingas well as calendar aging mechanisms are considered.Fernández et al. (2013) propose an optimal chargingenergy management that minimizes the degradationof the battery as a function of temperature and depthof discharge. In Wang et al. (2016) a battery degra-dation model of the cycle losses is used, which islinear dependent in cumulative current throughput,but quadratic in temperature. One interesting result isfound in Peterson et al. (2010), where capacity fadeof commercial Li-ion batteries caused by driving andvehicle-to-grid usage is studied. Battery degradationwas found to be related to the amount of energy pro-cessed in the battery, while the depth of discharge hasless effects on the battery lifetime. The importance ofmethods to increase the battery lifetime correspondswith the work in Hawkins et al. (2012), where it isshown that replacing a battery frequently may harmthe environment and increases the risk of toxicity(e.g., for humans and freshwater) and metal depletionimpacts.

Route planning algorithms Route planning problemsare typically solved by applying shortest path algo-rithms. The Dijkstra algorithm (Dijkstra 1959) is avery efficient method to work with weighted graphsand is widely used in network theory. A Dijkstra-based speed-up technique for constrained shortest pathproblems is applied in Storandt (2012), where routeplanning is conducted for bicycles. The Dijkstra algo-rithm is also applied in Song et al. (2015). In thatwork, route planning in urban areas with influencesof traffic is considered. The Bellman-Ford (Bellman1958) algorithm has a higher complexity than Dijk-stra, but can be used in networks with negative edgecosts, which is useful for route planning for electric

1707Energy Efficiency (2020) 13:1705–1726

vehicles. Despite its high complexity, the Bellman-Ford algorithm works well with street networks. TheYen algorithm (Yen 1970) is based on Bellman-Ford,but it is generally able to solve the shortest pathproblem faster than Bellman (1958).

Route planning for electric vehicles It is mentioned inNeaimeh et al. (2013) that many people, who testedelectric vehicles, experience so-called range anxiety.It turns out that many drivers would change their driv-ing behavior and especially their route-choices to thedestination. In order to find an optimal route that canextend the range of the vehicle, Dijkstra’s shortestpath algorithm is applied. The goal is not to sim-ply find the energy-minimal route, but also to helpwith range anxiety, making drivers feel more comfort-able with e-mobility. In Nunzio and Thibault (2017)a range estimation for online use is created, whichis based on calculating the energy-optimal route. Thevehicle’s energy consumption is modeled includinginfluences of traffic and with a shortest path algo-rithm (Bellman-Ford) the range can be estimated. InStorandt and Funke (2012) battery switch station areincluded in the network and a modification of Dijk-stra’s algorithm is used. This modification is done byusing Johnson’s shifting technique (see Johnson 1977)in order to include regenerated energy of the elec-tric vehicle. In practice, battery switch stations arenot established, but charging stations become increas-ingly visible. In Storandt et al. (2013), the approachis similar to Storandt and Funke (2012), but includingcharging stations instead of battery switch stations. Itis also proposed to work on a multi-criteria optimiza-tion that includes the journey time and a maximumnumber of recharging events.

The present work implements a multi-objectiveoptimization approach determining a path that seeks tomeet best the drivers’ requirements. In detail, the opti-mization algorithm uses weighting factors on thesethree optimization variables:

(i) Energy consumption: The electric vehicleshould reach the destination with the highestpossible state of charge of the battery.

(ii) Time: The journey time should be as short aspossible.

(iii) Battery lifetime: In order to increase the bat-tery lifetime of the electric vehicle, the number

of charging and discharging cycles should beas small as possible, therefore the cumulatedenergy flow should be minimized.

The goal of this paper is to include the drivers’ prefer-ences in a very flexible way. For example, the driverscan choose a single-objective optimization of either ofthe above-mentioned variables, or give equal weightsto all three. Charging stations for electric vehicles arenot included in this work.

Methodology

Optimization problem

The objective of this work is to design a route plan-ning system for electric vehicles in order to optimizethe battery lifetime, energy consumption, and jour-ney time. A weighted multi-objective optimizationapproach is used to prioritize one aspect or the other.

Flowchart

To achieve the objective of this work—finding theweighted optimal path to a desired destination withan electric vehicle—it is important to have a modeldescribing the energy consumption of the electricvehicle and the street network. Then, shortest pathalgorithms can be used for optimization. A flow chart(see Fig. 1) presents an overview of this optimizationproblem. At first, a starting and a destination pointhave to be defined. Then, it is necessary to have aroad network that includes topography information.Each road section has a defined length s, a velocityv, and a slope q in %, calculated from the topogra-phy information. The next step is to define the vehicleparameters. The initial state of charge of the batterySoCinit and its maximum capacity SoCmax are nec-essary, as well as the mass m, the drag coefficientcw, and the cross-sectional area A of the vehicle.When having information on the outside temperatureT , all parameters are available for the calculation ofthe energy consumption model of the electric vehicleand the journey time on all the possible roads of thenetwork.

As this work formulates an optimization problemwith constraints, the boundary conditions have to be


Fig. 1 Flowchart of theoptimization problem

defined. Naturally, the state of charge of the batterySoC cannot be negative or above the maximum capac-ity. In the interest of the vehicle owner, the SoC shouldnot fall below a certain minimum capacity, becausedeep discharge can decrease battery lifetime. A factora, 0 ≤ a < 1, is added to the optimization problem,such that the constraint equation becomes

a SoCmax ≤ SoC ≤ SoCmax. (1)The task will be extended to a multi-objective opti-mization with the variables energy, time, and batterylifetime. The factors γ and δ, with γ +δ ≤ 1, are usedto give weights to the optimization variables. The laststep is to apply a shortest path algorithm.

Modeling the energy consumption

Energy consumption for driving

We start with the calculation of the energy consump-tion of an electric vehicle for driving. The basicprinciples of the calculations are similar to conven-tional vehicles. The driving force Fdrive is calculated

as the sum of the rolling resistance Froll, the air resis-tance Fair, and the gradient resistance Fgrad, whileother factors are neglected for simplicity:

Fdrive = Froll + Fair + Fgrad (2)= mgfR cos(α) + 1

2ρcwAv

2 + mg sin(α). (3)The next step is the calculation of the power resultingfrom the driving force

Pdrive(t) = Fdrive(t) · v(t), (4)and then finally

Edrive =∫ t2

t1

Pdrive(t) dt, (5)

for the energy required over a certain period of timet1 ≤ t ≤ t2.

Total energy consumption of the electric vehicle

Apart from regenerative energy from braking or driv-ing downhill, the battery is the only source of energyin electric vehicles. Other loads than driving must


Fig. 2 Energyflow—engine operates as amotor

be covered by the battery as well. The major con-sumers are the heating and cooling, which are powereddirectly by the high voltage battery (around 400 V),without any DC-DC converter (see Jeschke 2016).Other so-called accessory loads are headlights, fan,windshield wipers, rear window heating, and radio.These need low voltage (12 V) and therefore a DC-DCconverter.

Figures 2 and 3 show the energy flow from the bat-tery to the components of the vehicle with the engineoperating as a motor or as a generator, respectively.Eout is the energy the battery provides. Eacc and Ehcrepresent the low voltage accessories, and the heatingand cooling, respectively. Edrive is used for driving, asin (5). The amount of energy E that decreases the stateof charge of the battery considering efficiency factorsis

E = 1ηdis

(1

ηdηmηinvEdrive+ 1

ηaccEacc+ 1

ηhcEhc

),(6)

if the engine operates as a motor (Fig. 2).With electric vehicles, recharging the battery with

regenerated energy from braking or driving downhillis possible. In this case, the driving power Pdrive in(4) is negative and the engine operates as a genera-tor (see Fig. 3). There are two cases to distinguish: (i)

the battery charges, only if the regenerated energy issufficient to cover the accessory loads; (ii) the batterydischarges, if the regenerated energy cannot cover allthe accessory loads. The energy Eout = −Ein can becalculated as

Eout = −ηdηgηinvEregen + 1ηacc

Eacc + 1ηhc

Ehc, (7)

and the total energy including charging and discharg-ing efficiencies is

E =⎧⎨⎩

1ηdis

Eout, if Eout > 0ηcharge Eout, if Eout < 00, if Eout = 0.

(8)

If the efficiency of the engine operating as a generatoris the same as for a motor (ηm = ηg), Eqs. (6) and (7)become

Eout =(

1

ηdηmηinv

)sign(Edrive)Edrive+ 1

ηaccEacc+ 1

ηhcEhc.

(9)

If the charging and discharging efficiencies of the bat-tery are equal (ηcha = ηdis), Eqs. (6) and (8) merge toone new equation

E =(

1

ηdis

)sign(Eout)Eout, (10)

Fig. 3 Energyflow—engine operates as agenerator


with Eout as in Eq. (9).

Modeling the street network

The next task of the route planning is to model thereal-life street network with its properties such asslope, distances, speed limits, and road type. In orderto solve a shortest path problem, a specific structure ofthe street network is needed.

Networks

Networks consist of so-called nodes and edges. Thenodes are connected with each other by edges, whichcan have assigned values called “edge weights” or“edge costs.”

Nodes Nodes are the decision points of the network.Each node is connected to different nodes via edges.Some nodes are directly connected, others are onlyreachable by passing other nodes on the way. In thereal-world street network, nodes are intersections ofroads.

Edges Edges are the roads connecting the nodes. Inthis network model, the edges are split into straightsections with constant slope and constant speed limit.

Edge costs Networks have the purpose to expressrelations between the nodes. There are different waysto characterize such relations. The most elementaryones are a binary relations, describing whether thereis a direct connection (an edge) or not. Edge costs canhave physical meanings, such as the distance betweennodes. A shortest path algorithm finds the path thatcumulates the least amount of edge costs on its way.When the physical meaning of the edge costs is theenergy consumption of an electric vehicle, the net-work becomes bidirectional. The main reason is thetopography, as two nodes can be on different altitudesand, therefore, the required energy is different for bothdirections.

Topography data

Topography data is obtained from the “USGS EarthExplorer” website (https://earthexplorer.usgs.gov/),where geographical data is available for download.The file obtained from the USGS is a TIFF-file with

3601 × 3601 pixel, containing a digital elevationmap from the NASA Shuttle Radar Topography Mis-sion (SRTM). It has a high spatial resolution of onearc-second for longitude (east-west) as well as for lati-tude (north-south). The height resolution is one meter.Since two adjacent pixels are one arc-second apart, theTIFF-file covers an area of one degree in longitudeand one degree in latitude. For the calculations of theenergy consumption of the electric vehicle, it is nec-essary to convert the distances given in degrees intodistances in meters. The conversion is achieved by anapproximation of the earth as a perfect sphere1 withradius of Rearth = 6371 km = 6371000 m. φ1 andψ1 are the coordinates of a point in degree longitudeand degree latitude, respectively. φ2 and ψ2 are thecoordinates of another point, and �φ = φ1 − φ2 and�ψ = ψ1 − ψ2 are the distances between point 1 andpoint 2 (in degrees). The distances �x for longitudeand �y for latitude in meters are converted using

�y = 2πRearth360

�ψ, (11)

and

�x = 2πRearth360

cos(ψ1π/180)�φ. (12)

It can be noticed that Eq. (12) depends on the coor-dinates of the latitude. At the equator, we havecos(ψ1π/180) = 1. The coordinate lines of the lon-gitude converge when moving further away from theequator. Two places with a �φ of 1◦ at the equato-rial line have a higher �x than two places with thesame �φ at another degree of latitude. The resolutionof one arc-second of the map obtained from NASAmeans having a resolution of about 31 m north-south.It has to be mentioned that Eqs. (11) and (12) are onlyapproximations of the real distances.

The next task is to calculate the slope of a roadsection that is straight and has a constant gradient.The difference in altitude �z between the start andthe end point of the section, which is derived from theTIFF-file mentioned in the beginning of this section,is divided by the Euclidean distance between thosepoints and multiplied by 100 in order to get the slopeq in %:

q = 100 �z√(�x)2 + (�y)2 . (13)

1Another approach to calculate the distance between two pointson the surface of a sphere is, e.g., Haversine formula.


https://earthexplorer.usgs.gov/

Street network

After obtaining topographical data, the next step is toget an adequate network of roads containing the regionof interest. One option is using “OpenStreetMaps,” abig database for streets, roads, and paths. The use of“OpenStreetMaps” is problematic because many roadsare missing labels with no possibility of distinguishingif they are footpaths, waterways, or streets. The samegoes for data obtained from official government web-sites. Thus, a more simple approach is chosen. First, itis decided which roads should be part of the network.Only the main roads in the area of interest are includedby taking coordinates of road sections. Some of theroads (edges) are a few kilometers long and split upinto straight consecutive sections with a length varyingfrom 50 m to a few hundred meters. For the purposeof testing the algorithm, this resolution is satisfying.For an advanced route planning system, a higher res-olution would be needed. The coordinates of the startand end points of the road segments and the matchingtopography information obtained, the distances andheight differences are calculated as explained in the“Topography data” section. This ensures each segmentis straight and has a constant slope.

Optimization with a shortest path algorithm

The multi-objective optimization is solved by apply-ing a shortest path algorithm. Shortest path algorithmsuse networks made of nodes and edges. The goal ofshortest path algorithms is to find the path with theleast amount of edge costs. In this case the edge costsrepresent the energy consumption, journey time, orthe battery degradation. The “Yen algorithm” sectionexplains the shortest path algorithm for one optimiza-tion variable. Then, the multi-objective optimizationbased on the shortest path algorithm is described in the“Multi-objective optimization” section.

Yen algorithm

A well-known shortest path algorithm is the Bellman-Ford algorithm (see Bellman 1958). An improved ver-sion of Bellman-Ford is the algorithm proposed by JinY. Yen (see Yen 1970). Both algorithms find the short-est paths from all nodes of the network to one specificdestination node. The iterative search goes backwards,always starting from the destination node. In contrast,

the state of charge of the battery (SoC) is calculatedforward, starting from the starting point. The con-straints are violated if the SoC exceeds the boundaries(see Eq. (1)). In addition, the goal for energy-optimalroute planning is to arrive at the destination with thehighest state of charge possible. The backwards itera-tions of the Yen algorithm are, therefore, not suitablefor the optimization problem with SoC constraints.Hence, the following solution is introduced. Insteadof computing all minimum paths to the destination,all minimum paths from the start are calculated. Theprinciples of the Yen algorithm remain the same, butthere are a few modifications. The algorithm is appliedto find the energy-optimal path in a network with Nnodes in the following manner, starting with iteration0,

f(0)i = d1i , (14)

where d1i are the edge costs from node 1 to nodei = 2, . . . , N , representing the energy consumptionrequired to travel from node 1 to i. The edge costsare calculated adding up the energy consumption El(derived from Eq. (10)) of M adjacent road sections:

dij =M∑l=1

El. (15)

f(k)i are the costs from node 1, the start node, to node

i at iteration k. A distinction is made between oddand even numbered iterations. For odd iterations, theminimum is found using

f(2k−1)i = min1≤ji(f(2k)j + dji, f (2k−1)i ), (18)

f(2k)N = f (2k−1)N , (19)

with i = N − 1, N − 1, . . . , 2. For all iterations, thebattery constraints are checked. If SoC > SoCmax, thepath is still feasible, but the current state of charge isset to the maximum capacity of the battery (SoC =SoCmax), because no further charging is possible. Theenergy consumption f (k)j +dji is adapted accordingly.If SoC < a SoCmax, that path is considered unfea-sible and f (k)i is set to infinity, such that this path isneglected in all further calculations. If all minimum


paths fi of iteration k remain the same compared tothe previous iteration k − 1, the optimum is found:

f (k) =

⎡⎢⎢⎢⎢⎣

f(k)1

f(k)2...

f(k)N

⎤⎥⎥⎥⎥⎦ ≡

⎡⎢⎢⎢⎢⎣

f(k−1)1

f(k−1)2...

f(k−1)N

⎤⎥⎥⎥⎥⎦ = f

(k−1). (20)

For time-optimal route planning, the optimum g(k)i ofeach iteration k is calculated having the edge coststj i (see Eq. (A1.1))), the journey time from node jto node i (details in Appendix 1). Similarly, the opti-mum for the battery lifetime h(k)i works with the edgecosts aji (see Appendix 2). aji represents the abso-lute energy flow to and from the battery. The maindifference between dij and aij concerns regenerative

energy, which decreases the value f (k)i , but increases

h(k)i , and therefore the Yen algorithm finds different

minima.

Multi-objective optimization

The multi-objective optimization has three optimiza-tion variables: energy consumption of the vehicle,journey time, and cyclic lifetime of the battery. Thealgorithm applied for this task is based on the modifiedversion of the Yen algorithm from the “Yen algorithm”section. The optimal path from the start node number1 to all nodes i is calculated in each iteration, takinginto account all three variables and their weights, aswell as the battery constraints. We start with iteration0, setting all three variables in the same manner:

f(0)i = d1i , (21)g

(0)i = t1i , (22)

h(0)i = a1i . (23)

The next step is to go into the details of the odditerations. At first,

f(2k−1)1 = f (2k−2)1 , (24)g

(2k−1)1 = g(2k−2)1 , (25)h

(2k−1)1 = h(2k−2)1 (26)

are set. Then, for the other nodes i, the energy-optimalpath is calculated following Eq. (16). If at least onepath is feasible, Emin = |f (2k−1)i | is set and Tmin,the minimum in time, and Amin, the smallest abso-lute energy consumption, are calculated according to

Eqs. (A1.4) and (A2.3), respectively. All three resultsare combined to multi-objective optimization with theweights γ for energy, δ for time, and 1 − (γ + δ) forbattery lifetime. The optimization problem solved fornode i = 2, 3, . . . , N in iteration 2k − 1 is:

x(2k−1)i = min1≤ji

(γ

f(2k)j +djiEmin

+ δ g(2k)j +tj iTmin

+(1 − (γ + δ))h(2k)j +ajiAmin

,

γf

(2k−1)j

Emin+ δ g

(2k−1)j

Tmin

+(1 − (γ + δ))h(2k−1)j

Amin

). (31)

Reference values and assumptions

The proposed multi-objective optimization algorithmis implemented in MATLAB2 (MATLAB 2019)and two vehicles selected for testing are the Nissan

2The solving time for the tests in the “Suburban area: nearVienna, Austria” section is around 0.15 s.


Leaf, see data sheet (https://www.nissan.co.uk), andthe Mitsubishi i-MiEV, see data sheet (https://www.mitsubishi-motors.com/en/products/#search-models).All parameters necessary for the calculations are sum-marized in Table 1. For the mass m, the curb weightof the vehicle, is combined with the weight of oneperson, the driver. The driver’s mass is assumed to be75 kg. To calculate the air resistance Fair accordingto Eq. (3), the cross-sectional area A and the dragcoefficient cw are necessary. If A is not specified inthe official data sheet, it may be estimated accordingto Haken (2013), knowing the width w and height hof the vehicle:

A = 0.81 · w · h. (32)

Accessory loads and efficiencies are obtained from themeasurements in Geringer and Tober (2012). Someefficiency factors cannot be found in Geringer andTober (2012), therefore they are assumed to be 100%,except for the engine efficiency, which is assumed to

be 90%. The main differences between the two vehicletypes are mass and the dimensions of height, weight,and drag coefficient.

Results

Suburban area: near Vienna, Austria

The street network covers the main roads of theWienerwald area West of Vienna. The city of Viennais excluded. The network has a total of 31 nodes and41 edges. Different routes were tested with the multi-objective optimization algorithm. Planning a trip fromPassauerhof (a in Fig. 4) to Maria Gugging (b) pro-vides interesting results (see Fig. 4). There are twodifferent solutions of the multi-objective optimization.The routes are:

(a) The first path, where the electric vehicle needsthe least amount of energy, starts at Passauerhof,and passes through Katzlsdorf, Königstetten,

Table 1 Parameters of the two different electric vehicles

Nissan leaf Mitsubishi i-MiEV

Dimensions

Curb weight 1516 1090 kg

Mass with driver m 1591 1165 kg

Drag coefficient cw 0.28 0.33 –

Width with mirrors 1967 1792 mm

Width w/o mirrors w 1770 1475 mm

Height h 1550 1610 mm

Cross-sectional area A 2.22 2.14 m2

Accessory loads

Heating Pheat 90 95 W/◦CCooling Pcool 40 30 W/◦CLights Plight 48 38/127 W

Fan Pair 62 48 W

Battery

Capacity SoCmax 24 16 kWh

Efficiencies

Drive ηd 1 1

Motor ηm 0.90 0.90

Inverter DC-AC ηinv 0.96 0.91

Battery ηdis 0.90–0.96 0.88–0.95

Accessories DC-DC ηacc 1 0.83

Heating/cooling ηhc 1 1


https://www.nissan.co.ukhttps://www.mitsubishi-motors.com/en/products/#search-modelshttps://www.mitsubishi-motors.com/en/products/#search-models

Fig. 4 Results from Passauerhof to Maria Gugging (edited screen-shot from Google Maps; Google and the Google logo are registeredtrademarks of Google LLC, used with permission)

and St. Andrä before it reaches the destinationMaria Gugging (see Fig. 4).

(b) The other route (b) is both the fastest and theshortest route, but presents more variation intopography than (a), passing small villages suchas Unterkirchen and Hintersdorf before reachingMaria Gugging (see Fig. 4).

Table 2 presents the solutions of the multi-objectiveoptimization for various combinations of the weights.

Table 2 Results from passauerhof to maria gugging with nissanleaf at 20 ◦C

γ /δ 0 0.2 0.4 0.6 0.8 1

0 (a) (a) (a) (b) (b) (b)

0.2 (a) (a) (a) (b) (b) –

0.4 (a) (a) (b) (b) – –

0.6 (a) (a) (b) – – –

0.8 (a) (b) – – – –

1 (a) – – – – –

It can be noticed that combinations with small valuesof γ and δ result in option (a). This route has less vari-ation in topography than (b), as it can be seen in Figs. 5and 6. If γ and δ are small, the focus is on increas-ing the battery lifetime. In contrast to the optimizationvariable for energy consumption, where the regener-ated energy has a negative sign and, therefore, helpsminimizing the costs, the optimization variable rep-resenting the battery lifetime takes the absolute valueof the energy. In this case, the regenerated energycounts the same way as the consumed energy. There-fore, driving downhill or braking increases the costsand is avoided by the optimization algorithm. A routewith little elevation up and down is preferred. It canbe seen in Fig. 6 that the absolute energy flow to andfrom the battery for route (a) is smaller than for route(b).

Combinations with a high value of δ lead to thefastest route (b). The difference in time is almost fourminutes, which is a quarter of the total journey timeof route (b). The difference in energy consumption isless significant (see Fig. 5). Choosing route (a), only


Fig. 5 Topography andenergy consumption onroute (a) and route (b) fromPassauerhof to MariaGugging

a few Watt-hours are saved. If the battery lifetime isneglected (γ + δ = 1), a high value of γ and a smallδ lead to route (b), while only with γ = 1, route (a)is chosen, because the savings in energy consumptionare insignificant compared with the savings in journeytime.

Comparing under different temperatures

The route planning system is also tested for differentoutside temperatures. During winter, the temperaturesfrequently fall below zero in Austria. Therefore, heat-ing is necessary for the passengers in the vehicle.With the help of the measurements in Geringer andTober (2012), it is possible to find a linear functionof the Nissan Leaf’s power demand for heating, whichis Pheat = 90 W/◦�C. The energy consumption isexpected to rise with lower temperatures, which mayinfluence the results of the multi-objective optimiza-

tion problem. Assuming an outside temperature ofT = −10 ◦C, we havePhc = Pheat(T0 − T ) = 1800 W, (33)with T0 = 20 ◦C. For the relatively short trip of around20 min, 0.6 kWh are used only for heating. In addition,battery (dis-)charging efficiency ηdis is decreasing dueto low temperatures (see Table 1). Table 3 shows theresults for varying combinations of the optimizationweights.

These results are not surprising considering thatthe energy consumption of all accessories, includingheating and cooling, is time-dependent. This factormakes fast routes more attractive also when lookingat energy-optimal route planning. In this case, route(a), which has less energy consumption at 20 ◦C, nowrequires more energy than route (b), making route (b)the most energy and time efficient route at tempera-tures of − 10 ◦C outside (see Fig. 7). When focusing

Fig. 6 Topography andabsolute energyconsumption on route (a)and route (b) fromPassauerhof to MariaGugging


Table 3 Results from Passauerhof to Maria Gugging withdifferent outside temperatures (20 ◦C—top; − 10 ◦C—bottom)γ /δ 0 0.2 0.4 0.6 0.8 1

0 (a) (a) (a) (b) (b) (b)0.2 (a) (a) (a) (b) (b) -0.4 (a) (a) (b) (b) - -0.6 (a) (a) (b) - - -0.8 (a) (b) - - - -1 (a) - - - - -

0 (a) (a) (b) (b) (b) (b)0.2 (a) (a) (b) (b) (b) -0.4 (a) (b) (b) (b) - -0.6 (b) (b) (b) - - -0.8 (b) (b) - - - -1 (b) - - - - -

different routes highlighted in bold

on the cyclic lifetime of the battery only (low γ andδ), route (a) is still chosen because of the topogra-phy, as explained previously. Figure 8 shows that thecumulated absolute energy consumption and regener-ation of route (a) is still smaller than that of route (b),but with a narrower margin at − 10 ◦C compared with20 ◦C.

Comparing two different types of electric vehicles

So far, the tests are performed exclusively with the NissanLeaf. In the following part, the Mitsubishi i-MiEVvehicle is added, assuming its characteristics as given

in Table 1. The Mitsubishi vehicle is almost 30%lighter in weight than the Nissan, but its cw-value ishigher and its efficiencies are lower. In Table 4, theresults for the Nissan Leaf and Mitsubishi i-MiEV arecompared.

Figure 9 shows that the energy consumption onboth routes, (a) and (b), is higher for the Mitsubishidespite having less weight. Another finding is thaton both routes the cumulated absolute value (Fig. 10)of the consumed and regenerated energy of the Mit-subishi is smaller compared to Nissan. This resultshows that the Mitsubishi regenerates less energy.Only once does the result change compared to the Nis-san, which is for γ = 0.2 and δ = 0.4. The absoluteenergy has the weight of 1 − (γ + δ) = 0.4. In thiscase, the objective to be minimized in (27) is almostequal for both routes, but with route (b) being slightlymore efficient.

Urban area: San Francisco, CA

Further tests are conducted for the city of San Francisco,CA, because of its interesting topography. The streetnetwork of San Francisco is fundamentally different fromthe street network of Wienerwald, Austria. It is anurban area with a high road density. The network is agrid of perpendicular streets. Each intersection repre-sents a node of the network. Because of the high roaddensity, there is also a high number of nodes. There-fore, only a small part of downtown San Francisco isselected in order to keep the network more compact.

Fig. 7 Energyconsumption (cumulated)on route (a) and route (b)from Passauerhof to MariaGugging for differentoutside temperatures, 20 ◦Cand − 10 ◦C


Fig. 8 Absolute value(cumulated) of the energyconsumed or regenerated onroute (a) and route (b) fromPassauerhof to MariaGugging for differentoutside temperatures, 20 ◦Cand − 10 ◦C

Modeling an urban street network may be challeng-ing because there are many factors that are hard topredict. Traffic flow is on top of the agenda in thiscontext. When searching for the fastest route, the cur-rent traffic situation and the timing of traffic lights aremain concerns, other factors among many others may

Table 4 Results from Passauerhof to Maria Gugging for twodifferent vehicle types (Nissan Leaf—top; Mitsubishi i-MiEV—bottom) at 20 ◦C

γ /δ 0 0.2 0.4 0.6 0.8 1

0 (a) (a) (a) (b) (b) (b)

0.2 (a) (a) (a) (b) (b) -

0.4 (a) (a) (b) (b) - -

0.6 (a) (a) (b) - - -

0.8 (a) (b) - - - -

1 (a) - - - - -

0 (a) (a) (a) (b) (b) (b)

0.2 (a) (a) (b) (b) (b) -

0.4 (a) (a) (b) (b) - -

0.6 (a) (a) (b) - - -

0.8 (a) (b) - - - -

1 (a) - - - - -

different routes highlighted in bold

be being zebra crossings, stop signs, and bus stops.They influence not only the journey time but also theenergy consumption. Nevertheless, the impacts of traf-fic and road signs are neglected for simplification inthe present work. The focus is on the energy consump-tion and battery lifetime, which are expected to beconsiderably impacted by the topography of the city.Multi-objective optimization is performed with δ = 0.

Route: from Pier 39 to Russian Hill

Different starting and destination points are tested.One journey starts close to Pier 39, at the cornerBeach Street/Grant Avenue. The destination pointis Union Street/Hyde Street, close to Russian Hill.All combinations of the multi-objective optimizationweights result in the same route (see Fig. 11). Goingin the opposite direction, from Russian Hill to Pier39, gives the result seen in Fig. 12, for all valuesof γ .

Figure 13 (dashed lines) shows the topography ofthe two scenarios: Going from Pier 39 to Russian Hill(Union/Hyde Street) and the return. The energy con-sumption on both trips (see Fig. 13) strongly dependson the topography. Since Pier 39 is almost at sea level,going to Russian Hill takes a lot of energy, while theelectric vehicle regenerates energy by going in the


Fig. 9 Energy consumption(cumulated) on route (a) androute (b) from Passauerhofto Maria Guggingcomparing Nissan Leaf andMitsubishi i-MiEV at 20 ◦C

opposite direction down to Pier 39. In Fig. 14, it canbe seen that the absolute energy flow of the return tripis a lot smaller.

Further results of San Francisco show that focusingon the battery lifetime often results in routes differingfrom those focusing on energy efficiency. Switchingthe destination point to another one close to the orig-inal results in the three routes represented in Fig. 15.

Figures 16 and 17 show the topography as well as theenergy consumption and absolute energy consumptionof the three results. Route (a) is the result for γ = 0and γ = 0.2. It is longer and not as energy efficientas route (b) and (c), but shows better performance interms of battery lifetime due to a different topography.Route (c) is the most energy efficient route (γ = 0.8and γ = 1).

Fig. 10 Absolute value(cumulated) of the energyconsumed or regenerated onroute (a) and route (b) fromPassauerhof to MariaGugging comparing NissanLeaf and Mitsubishii-MiEV at 20 ◦C


Fig. 11 Results from Pier 39 to Russian Hill (screen-shot from Google Maps—Google and the Google logo are registered trademarksof Google LLC, used with permission)

Conclusion and outlook

This work shows that route planning specificallydesigned for electric vehicles has more to offerthan conventional routing systems, which only con-sider time or distance to the destination. The multi-objective optimization has three aspects : energy con-sumption, journey time, and battery lifetime. Withdifferent weights on the optimization variables, dif-ferent solutions are obtained. Tests are performed forexisting street networks. In some cases, there is onlyone optimal route for all variables, while other casesshow very distinct results for one of the optimizationvariables.

The results demonstrate the influence of the topog-raphy on the routes. The energy consumption dependssignificantly on the topography. Especially with elec-tric vehicles, which may regenerate energy by drivingdownhill, there is a strong correlation between energy

consumption and topography. The third aspect of themulti-objective optimization is to increase the batterylifetime. The approach in this paper evaluates batterydegradation by minimizing the total energy flow of thebattery and therefore the number of charging cyclesover the battery’s lifetime, which is different to arriv-ing at the destination with the highest state of chargepossible, which is essentially the first aspect of themulti-objective optimization. Routes with high topo-graphical variation increase stress on the battery andare therefore avoided with focus on the battery life-time. The battery is an expensive part and key elementof the vehicle and, as such, should be protected fromdegradation as much as possible. The influence ofadditional loads such as heating, air condition, lights,and fan is significant. Those loads are powered by thebattery and therefore increase the total energy con-sumption, with an important factor being the durationof the journey, as more energy is used the longer these


Fig. 12 Results from Russian Hill to Pier 39 (screen-shot from Google Maps—Google and the Google logo are registered trademarksof Google LLC, used with permission)

Fig. 13 Topography andenergy consumption fromPier 39 to Russian Hill andback


Fig. 14 Topography andabsolute energyconsumption from Pier 39to Russian Hill and back

Fig. 15 Route options (a), (b), and (c) in San Francisco (edited screen-shot from Google Maps; Google and the Google logo areregistered trademarks of Google LLC, used with permission)


Fig. 16 Topography andenergy consumption forroute options (a), (b), and(c) in San Francisco

additional loads operate. Accordingly, the results ofthe multi-objective optimization may change, due tofast routes becoming more energy efficient.

Using a weighted multi-objective optimization maybe more reasonable than using a single-objective opti-mization. If the second fastest route were much moreenergy efficient, but only slightly slower than thefastest route, a time-only optimization leads to anunreasonable result, but when different aspects areconsidered at the same time, the most practical solu-tion is achieved. In order to use this type of routeplanning for real-world scenarios, more accurate street

data is necessary as well as some additional informa-tion: Traffic lights and stop signs should be included,as well as traffic flow, as a theoretically optimal routemay in practice prove to be infeasible due to trafficholdups. Integration of the multi-objective optimiza-tion algorithm (knowing vehicle type and ambientconditions) into existing route planning systems withbetter street data could be an addition appreciated byEV owners. Future work on this topic should alsoinclude charging stations such that route planning ispossible for a journey farther than the battery range ofthe vehicle.

Fig. 17 Topography andabsolute energyconsumption for routeoptions (a), (b), and (c) inSan Francisco


Lastly, another important aspect to be included infuture work is driving behavior. Speed and accelera-tion influence the energy consumption of the vehicle.In urban areas, the journey time is mainly dependenton the traffic situation rather than the speed, whilein non-urban areas the speed may have a significantimpact. Assuming a certain speed profile could helpachieving realistic results. Either a statistical approachto obtain a speed profile, or learning from previouslyobtained data of the driver’s preferences could beutilized.

Funding Open access funding provided by TU Wien (TUW).

Compliance with ethical standards

Conflict of interest The authors declare that they have noconflict of interest.

Open Access This article is licensed under a Creative Com-mons Attribution 4.0 International License, which permitsuse, sharing, adaptation, distribution and reproduction in anymedium or format, as long as you give appropriate credit tothe original author(s) and the source, provide a link to the Cre-ative Commons licence, and indicate if changes were made. Theimages or other third party material in this article are includedin the article’s Creative Commons licence, unless indicated oth-erwise in a credit line to the material. If material is not includedin the article’s Creative Commons licence and your intended useis not permitted by statutory regulation or exceeds the permit-ted use, you will need to obtain permission directly from thecopyright holder. To view a copy of this licence, visit http://creativecommonshorg/licenses/by/4.0/.

Appendix 1. Time-optimal route planning

Compared with the search for an energy-optimal path,finding a time-optimal path is a lot less complicated.On the one hand, there are no constraints assumedconcerning the journey time. It is just about finding thefastest itinerary. On the other hand, there are no nega-tive edge costs. In fact, the graph is undirected if it isassumed that the same road with the same speed limitis available for the way back. This optimization prob-lem could be solved with a less complex and fasteralgorithm than Bellman-Ford or Yen, because thereare no negative edge costs. Although, when energyand time are combined to multi-objective optimiza-tion, it is easier to just use the same algorithm, becausethe computation time is not crucial in this work. Theprocedure is the same as for finding an energy-optimal

path. In this case, the edge costs represent the journeytime. The journey time of an edge connecting node iand j is the sum of M adjacent road sections

tij =M∑l=1

tl , (A1.1)

with each road section having a designated speed levelvl and the length sl :

tl = vl/sl . (A1.2)Iteration 0 is given as

g(0)i = t1i , (A1.3)

where t1i are the edges costs from node 1 to node i =2, . . . , N . For odd iterations, the minimum is foundusing

g(2k−1)i = min1≤ji(g(2k)j + tj i , g(2k−1)i ), (A1.6)

g(2k)N = g(2k−1)N , (A1.7)

with i = N − 1, N − 1, . . . , 2.

Appendix 2. Energy-optimal route planning inorder to increase battery lifetime

The third criterion, which may be used for route plan-ning is increasing battery lifetime. Battery wear-offshould be avoided for as long as possible, as it candecrease the capacity as well as limit the range of thevehicle. The battery is also a very expensive part ofthe electric vehicle to replace. There are some actionsto be taken in order to increase the battery lifetime.First, avoiding deep discharge is helpful. This can beincluded in the factor a, which can be tuned to a valuethat is safe for the battery.

A lithium-ion battery, which is the power sourcefor most electric vehicles, usually has a specific cycliclifetime. Decreasing the total number of cycles ofthe battery helps to increase its lifetime. This is thethird criterion of the multi-objective optimization. Itis performed in a similar way as the energy-optimalplanning, but takes the absolute value of the energy


http://creativecommonshorg/licenses/by/4.0/http://creativecommonshorg/licenses/by/4.0/

as the edge costs. Power provided by the battery fordriving or for accessories is treated the same as regen-erated energy from braking and driving downhill, thusincreasing the costs instead of decreasing them. Theedge costs from node i to j is aij , the sum of the abso-lute energy values of all road sections belonging to thisedge. Similar to (A1.1), aij is calculated

aij =M∑l=1

al, (A2.1)

with al = |El |. hi are the costs from node 1 to node i.Iteration 0 is given as

h(0)i = a1i , (A2.2)

where a1i are the edges costs from node 1 to node i =2, . . . , N . For odd iterations, the minimum is foundusing

h(2k−1)i = min1≤ji(h(2k)j + aji, h(2k−1)i ), (A2.5)

h(2k)N = h(2k−1)N , (A2.6)

with i = N − 1, N − 1, . . . , 2.

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Energy efficient route planning for electric vehicles with special consideration of the topography and battery lifetimeAbstractIntroductionState of the artModeling electric vehiclesBattery lifetime in electric vehiclesRoute planning algorithmsRoute planning for electric vehicles

MethodologyOptimization problemFlowchartModeling the energy consumptionEnergy consumption for drivingTotal energy consumption of the electric vehicle

Modeling the street networkNetworksNodesEdgesEdge costs

Topography dataStreet network

Optimization with a shortest path algorithmYen algorithmMulti-objective optimization

Reference values and assumptions

ResultsSuburban area: near Vienna, AustriaComparing under different temperaturesComparing two different types of electric vehicles

Urban area: San Francisco, CARoute: from Pier 39 to Russian Hill

Conclusion and outlookAppendix Appendix 1. Time-optimal route planning Appendix 2. Energy-optimal route planning in order to increase battery lifetimeAppendix Appendix 2. Energy-optimal route planning in order to increase battery lifetimeReferences

Energy efficient route planning for electric vehicles with ......Yen algorithm helps solving a multi-objective opti-mization problem with three optimization variables representing

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