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CASPT 2015

The Electric Vehicle Scheduling Problem

A study on time-space network based and heuristic solution

approaches

Josephine Reuer ·Natalia Kliewer ·

Lena Wolbeck

Abstract In the last years, many public transport companies have launched

pilot projects testing the operation of electric buses. Therewith new chal-

lenges arise in the planning process. In this work, we deﬁne the electric vehicle

scheduling problem (EVSP) and multi-vehicle-type vehicle scheduling problem

with electric vehicles (MVT-(E)VSP). For both problems we present solution

approaches and results. Therefore, we extend the traditional deﬁnition of the

vehicle scheduling problem (VSP) and consider the limited battery capac-

ity restriction as well as the vehicles’ possibility to recharge their batteries.

First, we use an existing time-space network based exact solution approach

for the VSP and provide an algorithm that adds chargings to the schedule

when necessary in order to build vehicle blocks for electric vehicles. For this,

we tested diﬀerent strategies for the network ﬂow decomposition. Second, we

use two heuristic approaches in order to obtain feasible vehicle schedules for

the EVSP and upper bounds for the required number of vehicles.

Keywords electric vehicle ·public transport ·time-space network ·vehicle

scheduling

1 Introduction

The scarcity of fossil fuels as well as many countries’ current climate targets in

regard to the reduction of CO2emission require a paradigm shift to renewable

energies. Albeit, electric vehicles (EVs) still only represent a small share of

Josephine Reuer

Freie Universit¨at Berlin, Garystr. 21, 14195 Berlin, Germany

Tel.: +49-30-838-65495

Fax: +49-30-838-53692

E-mail: josephine.reuer@fu-berlin.de

Natalia Kliewer ·Lena Wolbeck

Freie Universit¨at Berlin, Garystr. 21, 14195 Berlin, Germany

the vehicle market, we can observe an increasing electro-mobility particularly

in urban areas. Also many public transport companies have launched pilot

projects testing the operation of electric buses. In Germany, for instance, the

public transport operator in Brunswick already introduced two electric buses1

in 2014 and is preparing three more for line operation.2This only accounts for

a share of about 3 %3of the total ﬂeet, but still entails additional restrictions

for vehicle scheduling. The traditional vehicle scheduling problem (VSP) is

the task of assigning a given set of timetabled trips to a set of vehicles while

considering operational restrictions and minimizing total costs. While this is

a well studied problem (cf. Bunte and Kliewer, 2009), new challenges arise

due to the use of EVs: (1) The vehicles have a much smaller range due to

battery capacity than traditional diesel vehicles and (2) they can be recharged

at speciﬁed stations only.

Thus, the transit operator faces several decision problems of which the two

most signiﬁcant will be considered in the following. First, he has to decide

on the EV type. In general one distinguishes between hybrid electric vehicles

(HEVs), fuel cell vehicles (FCVs) and battery electric vehicles (BEVs). HEVs

use an electric drive as well as a combustion engine and thus do not belong

to the zero emission vehicles. FCVs generate electricity, that is stored in the

battery, in fuel cells mostly from hydrogen. BEVs store energy in a battery or

ultracapacitor and of all EVs have the smallest range. For further descriptions

on the technology we refer to Chan (2007). Second, the chosen vehicle type

determines greatly the range of a bus and in conclusion the necessary charging

(or refueling) infrastructure. If either HEVs or BEVs are used the operator has

to choose a charging technology. Yilmaz and Krein (2013) distinguish three

power levels: Level 1 slow charging overnight, Level 2 semifast charging and

Level 3 fast charging. In Brunswick diﬀerent charging technologies are used.

While the buses are primary recharged overnight in the depot, the battery can

also be charged inductively during passenger boarding and alighting as well

as about 11 minutes at the ﬁnal stop. Additionally one can choose battery

swapping instead of charging, as used e.g. during Shanghai World Expo (Li,

2014).

Right now, it is diﬃcult to forecast which technology will prevail. In this

paper we focus on BEVs as they set the greatest range restrictions. Further-

more, for the ﬁrst we assume a given charging infrastructure where battery

recharging is possible at certain ﬁnal stops only and takes about the time of

battery swapping. For reason of convenience we will still talk about charging.

The VSP with limited range is similar to the vehicle scheduling problem

with route constraints (VSP-RC). Bunte and Kliewer (2009) give an overview

on these models as well as on modeling and solution approaches for the clas-

sical VSP. Yet, there exist only few approaches for considering refueling or

recharging. Wang and Shen (2007) ﬁrst deﬁned the vehicle scheduling problem

1BEV (Solaris Urbino 12/18 electric) with Li-Ion-battery 60kW/90kW and 600 V.

2http://www.verkehr-bs.de/unternehmen/forschungsprojekt-emil.html

3http://www.verkehr-bs.de/unternehmen/fuhrpark.html

with route and fueling time constraints (VSPRFTC). They develop a heuris-

tic that incorporates route time constrictions and ﬁnds vehicle blocks starting

and ending at the depot. After that they use a bipartite graphic model to

connect these blocks regarding fuel time restrictions. Zhu and Chen (2013)

propose a heuristic approach for the EVSP which they tested on a real-world

instance with 119 service trips. They aim at minimizing vehicle costs as well

as total charging demand. Li (2013) models the VSP with limited energy using

time-expanded station nodes, thus considering the possibility to recharge and

the capacity of charging stations. The author presents a construction heuris-

tic producing schedules which serve as initial solutions for diﬀerent column

generation based approaches. To generate a feasible solution, Adler (2014)

enhances the concurrent scheduler algorithm (cf. Bodin et al, 1978) using an

algorithm that regards fueling constraints. The heuristic is tested on real-world

instances with up to 4,000 service trips. Besides he develops and evaluates an

exact column generation algorithm for small instances.

All of these approaches use connection based network models. In contrast,

time-space networks (TSN) used for solving the traditional VSP have proven

to build signiﬁcantly more compact models and thus are able to generate op-

timal solutions applying directly MIP-solver even for large instances.4That

is, because a TSN does not consider every possible connection between service

trips but aggregates these to groups of compatible connections. As a solution

of TSN based MIP-optimization one derives a network ﬂow. Next, this ﬂow

is decomposed into paths, i. e. vehicle blocks. Because of the aggregation of

connections many diﬀerent schedules may be constructed depending on the

decomposition strategy used. Such strategies have successfully been used be-

fore for diﬀerent restrictions such as line change considerations (Kliewer et al,

2008). In this paper we want to transfer those ﬁndings to vehicle scheduling

with EVs and evaluate its applicability.

In this paper another diﬀerence to previous research is that we support

a mixed ﬂeet composed of both internal combustion engine vehicles (ICEVs)

and electric buses. Given that electric buses induce high ﬁxed costs and ICEVs

will be replaced only slowly, this is a reasonable assumption. We deﬁne this

problem as multi-vehicle-type vehicle scheduling with EVs (MVT-(E)VSP).

Despite, we also consider the electric vehicle scheduling problem (EVSP) that

uses only EVs.

Hence, the contributions of this paper are twofold: First, we apply the TSN

approach and we extend it by proposing diﬀerent strategies for ﬂow decom-

position that support, for instance, the creation of waiting times at charging

stations. Therefore, we provide an algorithm that adds chargings to the sched-

ule if necessary. This way, we test the adaptability of the optimal schedules to

the new requirements due to electric vehicles and receive a feasible schedule

for the MVT-(E)VSP. That gives us a bound on the minimal percentage of

EVs in a mixed ﬂeet that can be used without making major changes to the

optimal schedule and requiring more buses. Second, we present two heuristics

4For a detailed description of the TSN model we refer to Kliewer et al (2006).

in order to construct feasible schedules for the EVSP from scratch. Therefore,

we follow the ideas of Adler (2014) by extending the concurrent scheduler

algorithm (cf. Bodin et al, 1978).

In the following, we ﬁrst deﬁne the EVSP and the MVT-(E)VSP as exten-

sions of the standard VSP for one depot. Note that although we use the con-

cepts for BEVs, i. e. battery capacity and charging instead of battery swapping

or refueling, the concepts can easily be adapted to other electric or alternative-

fuel vehicles. Section 3 describes the methods used for solving those problems

and section 4 presents our test results for several real-world instances. In or-

der to gain further knowledge on the problems, many experiments can be

conducted by changing the input parameters, e.g. charging technology, con-

sumption or battery capacity. Here, we will give one extension of the model

where we drop the restriction of limited charging facilities (cf. section 5). This

enables further insights into the necessary charging infrastructure. We sum-

marize our contributions and give a brief outlook on further extensions in

section 6.

2 Problem deﬁnition

We deﬁne the EVSP as an extension of the standard VSP (cf. Bertossi et al,

1987). The objective is to ﬁnd an assignment for a given set of timetabled trips

to a homogeneous set of vehicles which minimizes the number of vehicles and

operational costs such that:

–each service trip is assigned exactly once,

–each vehicle starts and ends its schedule at the same depot,

–each vehicle block contains a feasible sequence of trips,

–a vehicle’s battery capacity cannot fall below zero and,

–a vehicle can only be charged at deﬁned stop points.

We assume a constant consumption in kWh per kilometer which diﬀers on

service and deadhead trips. That is due to the higher weight and consump-

tion when passengers are transported. In addition, a vehicle’s battery capacity

is assumed to be constant as well. A vehicle always leaves the depot with a

full battery and, when recharged, it is charged to full capacity. The charging

time and costs do not depend on the capacity left when arriving at a charg-

ing station. Besides we do not presume any capacity restrictions at stations.

Furthermore, chargings and deadhead trips start initially on arrival of the pre-

vious trip. However, we do not consider chains of chargings, i.e. driving from

one charging station to another without serving a passenger trip. Hence, any

trip can be served with a fully charged battery.

Furthermore, we also deﬁne the multi-vehicle-type vehicle scheduling prob-

lem with electric vehicles (MVT-(E)VSP). Therefor, we assume a heteroge-

neous ﬂeet consisting of both internal combustion engine vehicles (ICEVs)

and electric vehicles. For ICEVs the above described restrictions regarding

electric engines do not apply and we assume an unlimited range for those

vehicles. Moreover, no changes are made opposite to the standard VSP.

3 Solution approaches

In the following, we extend a TSN based approach using six diﬀerent strate-

gies for ﬂow decomposition and develop an algorithm that inserts necessary

chargings to a given vehicle schedule. That way, we compute a solution for the

MVT-(E)VSP that uses the maximum possible number of EVs based on the

solution of a standard VSP. In addition, we test two heuristics that construct

feasible vehicle schedules for the EVSP following the ideas of Adler (2014).

3.1 TSN based approach

To consider electric vehicles in vehicle scheduling, we take up the TSN based

solution method from Kliewer et al (2006) and propose the following additional

six strategies for ﬂow decomposition:

– MaxChargings (MaxCh) connects in- and out-going activities at each

charging station node in such a way that potential charging times are

maximized.

– MinConsumption (MinCon) solves a bottleneck assignment problem

at each node with the goal to minimize the maximum consumption.

– MaxChargingsMinConsumption (MaxChMinCon) uses MaxCh at

charging station nodes and MinCon at the other nodes.

– Extended MaxChargingsMinConsumption (XMaxChMinCon) con-

nects MaxCh and MinCon in one weighting function at each node.

– Extended MinConsumption (XMinCon) extends MinCon in such a

way that the consumption between two chargings is regarded.

– MaxChargings Extended MinConsumption (MaxChXMinCon)

uses MaxCh at each charging station node and XMinCon at all other nodes.

The presented strategies were developed in order to promote the possibility

to build vehicle blocks for electric vehicles, for instance by increasing possible

charging times or by minimizing the total consumption for a vehicle block.

The strategies all act locally and are more or less myopic as they solve an

optimization problem at each node. The nodes that represent stop points at

diﬀerent times are handled sequentially. In addition, we used simple ﬁrst-in,

ﬁrst-out (FIFO) and last-in, ﬁrst-out (LIFO) ﬂow decomposition.

After decomposition we use the vehicle schedule without chargings and de-

velop an algorithm that identiﬁes when a vehicle needs to be recharged and, if

possible, inserts chargings into the schedule. The algorithm incorporates the

fact that deadhead trips may be shifted forwards or backwards in scope of the

attached buﬀered times if necessary to add chargings. By this means, we re-

ceive a vehicle schedule using a limited percentage of EVs as well as standard

ICEVs. Thus, by slightly modifying well-studied solution methods incorpo-

rated by diﬀerent strategies for ﬂow decomposition, we estimate a bound on

the maximum number of EVs in a MVT-(E)VSP based on the optimal solution

of a standard VSP.

3.2 Heuristic approach

We develop two versions of a heuristic method inspired by the conception in

Adler (2014). The approach focuses on both simple and fast generation of a

vehicle schedule as well as reducing the number of vehicles and the operating

costs. Both algorithms construct solutions for the EVSP iteratively starting

with an ordered list of service trips. The fundamental procedure is as follows:

1. Set i= 1. Assign the uncovered trip tito vehicle 1.

2. Increment i. Find vehicle vthat minimizes total operating costs resulting

from the assignment. Consider that deadhead trip to nearest charging sta-

tion after trip timust be possible. If no existing vehicle can take the trip

add new vehicle. If charging before trip tiis necessary go to step 2a, else

go to step 2b.

(a) Insert charging and necessary deadhead trips.

(b) Insert deadhead trip if necessary.

3. Assign trip tito vehicle v.

4. If unassigned trips are left go to step 2, otherwise return vehicle schedule.

Both heuristic versions follow this schema but diﬀer in the number of vehi-

cles at initialization. The ﬁrst heuristic H1 starts with only one vehicle while

heuristic H2 starts with the number of vehicles which corresponds to the max-

imum number of trips at times of peak load.

4 Results

The provided approaches are implemented in C# under .Net using the opti-

mization library of IBM ILOG CPLEX 12.5. Our computational experiments

are conducted on ten real-world instances with up to 10,000 service trips. These

instances are characterized by diﬀerent kinds of distributions of the timetabled

trips over the day as shown in ﬁgure 1. The names of the instances contain

the total number of service trips. We distinguish three groups: (1) small-size

instances with up to 1,000 service trips, (2) medium-size instances with be-

tween 1,000 and 5,000 service trips and (3) large instances with more than

5,000 service trips.

For this study the instances were adjusted in order to capture the require-

ments of electric vehicles in the following way:

–Depending on the size of the instance we added between 5 and 26 charging

stations at highly frequented stop points. For reasons of comparability,

the charging stations were chosen to cover about 50 % of all service trip

departure and arrival stations. Thus, the probability of passing a charging

station is nearly the same for all instances.

–Battery capacity is set to 120 kWh.

–We assume a consumption of 1 kWh/km on service and 0.8 kWh/km on

deadhead trips.

–We consider a charging time of 10 minutes.

06:00 12:00 18:00 00:00

0

10

20

30

40

t424

# service trips

06:00 12:00 18:00 00:00

0

10

20

30

40

t426

06:00 12:00 18:00 00:00

0

20

40

60

80

t867

# service trips

06:00 12:00 18:00 00:00

0

20

40

60

80

t1135

06:00 12:00 18:00 00:00

0

20

40

60

80

t1296

# service trips

06:00 12:00 18:00 00:00

0

40

80

120

t2452

06:00 12:00 18:00 00:00

0

40

80

120

t2633

# service trips

06:00 12:00 18:00 00:00

0

40

80

120

t2825

06:00 12:00 18:00 00:00

0

40

80

120

t3067

# service trips

06:00 12:00 18:00 00:00

0

100

200

300

t10710

Fig. 1 Proﬁle of active service trips for each instance

These ﬁgures chosen for battery capacity, consumption and charging time

were motivated by research as well as practice. For instance, Li (2013) and

Adler (2014) assume ranges between 120 km to 150 km and 10 minutes for ser-

vice at charging stations. Another example are the buses used during Shanghai

World Expo. They had a range of 150 km and needed 10 minutes for battery

swapping (Li, 2014). We confer to Li (2014) for a review on recent devel-

opments of electric buses where more ﬁgures concerning range and charging

technology is given.

In the following, we ﬁrst present our results regarding the TSN based so-

lution approach for the MVT-(E)VSP. Second, we discuss our ﬁndings for the

heuristic solution of the EVSP.

4.1 Results for the MVT-(E)VSP using the TSN based solution method

The MVT-(E)VSP is solved using the TSN based solution approach described

in section 3.1. As this paper does not focus on the comparison of diﬀerent

decomposition strategies, we choose the best strategy based on the maximal

possible number of EVs for each instance. If two strategies produce the same

number then we select the best one on the basis of the total cost, so that the

schedule with the fewest chargings is chosen. Note, that ﬂow decomposition

only takes a few seconds for all instances and thus represents a small part of

the overall runtime. Hence, all decomposition strategies can be run one after

another without major eﬀect on the runtime.

Table 1 depicts diﬀerent ﬁgures of the best solution found for the MVT-

(E)VSP. It shows the number of vehicles needed in the optimal solution of the

standard VSP and the maximal percentage of blocks served by electric vehicles

as well as those that can be served without needing to recharge. Besides, the

average number of trips before an EV needs to recharge is given.

Table 1 Results for the TSN based approach

instance no. of

vehicles

max. pct.

of EVs

pct. of EV

blocks w/o

charging

avg. service

trips before

charging

t424 29 51.72 % 6.90 % 5.53

t426 32 100.00 % 56.25 % 30.43

t867 69 57.97 % 37.68 % 20.87

t1135 75 72.00 % 28.00 % 16.74

t1296 47 82.98 % 4.26 % 23.38

t2452 101 35.64 % 10.89 % 22.57

t2633 125 18.40 % 5.60 % 21.00

t2825 92 45.65 % 6.52 % 18.09

t3067 165 60.61 % 30.91 % 25.62

t10710 349 28.94 % 9.74 % 23.78

The small instances produce diverse results. For t426 we obtain a vehicle

schedule that can be served using only EVs and is thus a solution to the EVSP.

This is also due to the high percentage of vehicle blocks that do not exceed

the capacity restriction of the battery. In comparison, t424 has almost the

same number of service trips but only uses half as much EVs. This might be

a result of the high energy consumption for this plan due to long deadhead

distances. The ﬁgures suggest that for t426 six times as much service trips can

be served before a vehicle needs to be recharged. It can also be assumed that

the distribution of timetabled trips for t426 supports the insertion of chargings

as it has long oﬀ-peak hours with far less need of vehicles. Nevertheless, if we

only consider the absolute number of vehicles for which chargings were added,

it is almost the same for both schedules.

For the medium-size instances there is a slight negative correlation between

the number of vehicles and the percentage of EVs: the more vehicle blocks a

solution the less the percentage of EVs. One exception is the schedule with

3067 service trips which holds more than 60 % of EVs. The smallest percentage

of EVs appears for t2633 which also has the highest average consumption of

the medium-size instances. In comparison, the vehicle schedule for the large

instance uses about 29 % of EVs which is a lot if we consider the total number

of vehicle blocks.

Our results show that there is not only one criteria that is responsible for

the performance of the TSN based approach but several inﬂuencing factors.

One is the average consumption of a vehicle block: naturally, if it is low, there

are more blocks that are feasible for EVs without needing to recharge. Another

factor is the distribution of timetabled trips as there is more time to recharge

a vehicle during oﬀ-peak hours. Other parameters are the number of vehicle

blocks in total and the number of service trips.

Nevertheless, considering the slow shift towards the operation of electric

buses in public transport our results suggest that for the ﬁrst years in which

only few EVs are used, the proposed modiﬁed TSN based approach is suﬃ-

cient. Note that the solution we get is not optimal for the MVT-(E)VSP but

feasible. As it only made slight changes to the optimal solution of the VSP it

is interesting to look at the new cost component, namely the costs for charg-

ings and charging infrastructure. Table 2 shows ﬁgures associated with these

costs. Here, we only show the charging stations that are actually used in the

solution.

The schedules distinguish greatly in the number of charging stations needed.

There is no signiﬁcant correlation between the number of charging stations and

neither the percentage of EVs nor the size of the instance. For instance, t867

uses as many charging stations as t2825 but charges four times less on average.

Although, we did not incorporate charging station capacities in our model

we evaluated those afterwards. Recognize that the capacity of a charging sta-

tion seldom exceeds 2 buses. For the largest instance the exceeding is only 13

minutes in total and less for the other instances. Even exceeding 1 bus occurs

seldom in our results. In most cases for less than 4 % of the total charging time

a charging station is occupied by 2 or more buses.

Table 2 Charging station statistics for the TSN based approach

instance no. of

charg-

ings

avg.

charg-

ings per

EV

no. of

charging

stations

used (in %)

max. ca-

pacity of

charging

station

pct. of total

charging time

with

occupancy ≥2

avg.

chargings

per

charging

station

t424 32 2.13 5 (100 %) 2 0.63 % 6.40

t426 14 0.44 4 (67 %) 1 0.00 % 3.50

t867 15 0.38 9 (75 %) 1 0.00 % 1.67

t1135 43 0.80 9 (82 %) 2 1.42 % 4.78

t1296 48 1.23 11 (65 %) 1 0.00 % 4.36

t2452 28 0.78 5 (56 %) 3 2.74 % 5.60

t2633 22 0.96 6 (55 %) 1 0.00 % 3.67

t2825 58 1.38 9 (75 %) 2 0.87 % 6.44

t3067 53 0.53 12 (46 %) 2 3.52 % 4.42

t10710 93 0.92 18 (82 %) 4 8.19 % 5.17

4.2 Results for the EVSP using the heuristic method

The EVSP is solved with the heuristic approach described in section 3.2.5The

focus of this approach is on ﬁnding an upper bound on the number of necessary

EVs for the EVSP. A comparison with the results of the TSN based solution

method does not take place because of the diﬀerent underlying vehicle set for

the MVT-(E)VSP. Comparing the two versions of the heuristic H1 and H2

with each other shows that two vehicle schedules contain one respectively two

vehicles less if they are produced with H2. For t426 and t1296 the initialization

with the minimum number of vehicles leads to a slightly better distribution of

service trips on the EVs.

Due to the rather similar results for both heuristic versions, we will not dis-

tinguish between them in the following. Instead, we look at the most economic

solution for each instance. Table 3 depicts selected ﬁgures of the best solution

found for the EVSP and the optimal number of vehicles for the VSP. To also

examine the inﬂuence of the heuristic method irrespective of the consideration

of EVs, we use a similar heuristic approach to solve the VSP. Furthermore,

the table shows the percentage of EVs that can be served without charging

and the average number of trips before an EV needs to recharge.

The number of vehicles is slightly higher in ﬁve vehicle schedules due to

the heuristic. For instance, the results of t3067 show four and of t10710 two

vehicles more than the optimal solution. Otherwise, the increased number of

vehicles of three other instances is due to the use of electric vehicles. t424,

t1296 and t2633 produce vehicle schedules with 12 to 17 % more vehicles

than the optimal and the heuristic solution for the VSP. We assume that

the distribution of the timetabled trips (shown in ﬁgure 1) is responsible for

these outcomes. Their services trips are relatively evenly spread throughout

the day, which causes an accumulation of trips and chargings at the same time

5Note that all schedules were computed in less than ﬁve seconds.

Table 3 Results for the heuristic approach

instance opt. no. of

vehicles

for VSP

no. of

vehicles

for VSP

(heuristic)

no. of

vehicles for

EVSP

(heuristic)

pct. of EV

w/o

charging

avg. service

trips before

charging

t424 29 29 34 8.82 % 7.31

t426 32 32 32 75 % 53.25

t867 69 70 70 90 % 123.86

t1135 75 75 75 60 % 37.83

t1296 47 48 53 9.43 % 19.06

t2452 101 102 102 11.76 % 25.28

t2633 125 125 140 2.86 % 9.37

t2825 92 92 92 3.26 % 18.59

t3067 165 169 169 27.81 % 20.72

t10710 349 351 351 2.85 % 19.76

and thus increases the number of vehicles needed. Compared to other instances

of similar sizes, they have a much lower percentage of electric vehicles without

charging and the number of executable service trips before a recharge is lower.

These are also indications for an increased number of vehicles. Whereas we do

not know the optimal number of vehicles for the EVSP, we cannot assess the

results to their full extent.

Table 4 depicts ﬁgures of the heuristic method relating to the costs for

chargings and charging infrastructure. The results for the small instances diﬀer

greatly. For t426 and t867 we obtain a vehicle schedule containing very few

chargings and with low occupancy of the charging stations. This is due to the

fact that only a few of the vehicles need to load at all and many service trips

can be served before recharging (see table 3). In contrast, the schedule for t424

contains seven times more chargings and the charging stations’ occupancy is

in every respect higher. This might be an outcome of the heuristic approach

which does not operate well in this case or it might be due to the distribution

of the timetabled trips.

There is a slight correlation between the number of chargings and the

percentage of charging stations needed as well as the size of the instance. In

four solutions all possible charging stations are used and in the other cases over

70 % are needed. An exception of this are t426 and t867 which only use three

or two chargings stations while having signiﬁcantly lower number of chargings.

Furthermore, we evaluate the charging station capacities. There is a corre-

lation between the chargings per station and the concurrent occupation of the

charging station. For instance, the vehicle schedule of t2825 has an average of

12.67 chargings per charging station and the capacity exceeds two vehicles for

more than 85 minutes. Accordingly, the result of t2633 show twice the number

of chargings per station and the capacity exceeds two vehicles for 420 minutes

of which four vehicles load simultaneously for ﬁve minutes only.

Table 4 Charging station statistics for the heuristic approach

instance no. of

charg-

ings

avg.

charg-

ings per

EV

no. of

charging

stations

used (in %)

max.

capacity

of

charging

station

pct. of total

charging time

with

occupancy ≥2

avg.

chargings

per

charging

station

t424 58 1.71 4 (80 %) 3 7.82 % 14.50

t426 8 0.25 3 (50 %) 1 0.00 % 2.67

t867 7 0.10 2 (17 %) 1 0.00 % 3.50

t1135 30 0.40 8 (73 %) 2 1.01 % 3.75

t1296 68 1.28 15 (88 %) 2 0.74 % 4.53

t2452 97 0.95 9 (100 %) 3 11.70 % 10.78

t2633 281 2.01 11 (100 %) 4 14.95 % 25.55

t2825 152 1.65 12 (100 %) 2 5.63 % 12.67

t3067 148 0.88 21 (81 %) 2 5.11 % 7.05

t10710 542 1.54 22 (100 %) 5 19.08 % 24.64

5 Problem extension: variation of charging infrastructure

So far we assumed a ﬁxed charging infrastructure, thus limiting the upper

bound for EVs in a schedule. Now, we will drop this restriction by allowing to

recharge at every start or end station, i.e. every station is a potential charging

station. By this means, we will also receive an upper bound on the necessary

charging infrastructure. Note that charging during a service trip is still not

allowed.

Table 5 displays the maximal percentage of EVs and the number of neces-

sary charging stations in the extended model as well as the percentage change

towards the model with a given charging infrastructure as shown in tables 1

and 2 for the MVT-(E)VSP. Furthermore, it gives the maximal capacity of a

charging station and the average number of chargings per charging station as

an indicator of occupancy rate.

Table 5 Results for the TSN based approach with all possible charging stations

instance max.

pct. of

EVs

– change no. of

charging

stations

– change max.

capacity of

charging

station

avg.

chargings

per charging

station

t424 82.76 % 60.00 % 13 160.00 % 2 4.31

t426 100.00 % 0.00 % 6 50.00 % 2 2.33

t867 73.91 % 27.50 % 19 111.11 % 1 1.79

t1135 80.00 % 11.11 % 18 100.00 % 1 3.17

t1296 89.36 % 7.69 % 23 109.09 % 2 2.04

t2452 55.45 % 55.56 % 17 240.00 % 1 4.53

t2633 30.40 % 65.22 % 16 166.67 % 2 3.06

t2825 72.83 % 59.52 % 23 155.56 % 2 4.48

t3067 87.27 % 44.00 % 41 241.67 % 2 3.22

t10710 41.26 % 42.57 % 43 138.89 % 2 3.93

As expected, we now receive a higher percentage of EVs for each instance.

At the same time, this entails a stronger increase on the number of charging

stations which in practice leads to higher ﬁxed costs for the construction of

charging infrastructure. Except for one instance the results also show a de-

creased occupancy rate on average for each charging station. The capacity of

a charging station never exceeds two vehicles.

These results imply interesting questions for practical use that can be ad-

dressed in future research. For instance, as there is a trade-oﬀ between the

number of EVs and the number of necessary charging stations – both entail-

ing ﬁxed costs for public transport companies – we could use multi-criteria

optimization to determine a good solution for the MVT-(E)VSP with a high

percentage of EVs and a small number of charging stations.

For the EVSP the heuristic results for the extended model are given in

table 6. It depicts the number of vehicles, charging stations and chargings as

well as the percentage change towards the results described in section 4.2.

Table 6 Results for the heuristic approach with all possible charging stations

instance no. of

EVs

– change no. of

charging

stations

– change no. of

chargings

– change

t424 33 -2.94 % 18 350.00 % 67 15.52 %

t426 32 0.00 % 8 166.67 % 15 87.50 %

t867 70 0.00 % 7 250.00 % 16 128.57 %

t1135 75 0.00 % 21 162.50 % 33 10.00 %

t1296 52 -1.89 % 37 146.67 % 70 2.94 %

t2452 102 0.00 % 25 177.78 % 100 3.09 %

t2633 133 -5.00 % 36 227.27 % 277 -1.42 %

t2825 95 3.26 % 41 241.67 % 168 10.53 %

t3067 169 0.00 % 64 204.76 % 167 12.84 %

t10710 351 0.00 % 87 295.45 % 563 64.62 %

For four instances the number of vehicles was reduced due to the newly won

charging possibilities. But it is striking that the number of charging stations

highly increases for all plans, i. e. for each instance it has more than doubled.

This is also apparent for the number of chargings needed (except for one

instance), although the change is not as high for most plans. As a result, for

most instances we receive a schedule with higher ﬁxed costs for infrastructure

without reducing neither the ﬁxed costs for EVs nor the operational costs

for charging. This implies that the heuristics perform better if less degrees of

freedom are given. Hence, it is necessary to either determine a good charging

infrastructure in advance or to incorporate the reduction of necessary charging

stations in the heuristic.

6 Summary and further research

In this contribution we deﬁne both the MVT-(E)VSP and the EVSP as an

extension of the standard VSP considering a limited battery capacity as well

as the possibility of recharging. For each problem we developed a solution

method: (1) a TSN based approach enhanced by six decomposition strategies

that is used to solve the MVT-(E)VSP and give bounds on the percentage

of EVs and (2) two heuristics to construct feasible vehicle schedules for the

EVSP. The approaches were tested on ten real-world instances with up to

10,000 service trips.

The results show that the performance of the presented methods depends

on the problem size as well as on the distribution of timetabled trips. Further

inﬂuencing factors of the TSN based method are for instance the average

consumption of a vehicle block and the number of vehicle blocks in total. For

the MVT-(E)VSP we got schedules containing between 18 and 100 % EVs,

which is a reasonable number for the ﬁrst applications with EVs considering

the slow shift towards their use in public transport. The computation of these

plans is fast due to the small changes to the optimal schedule for the VSP,

although the schedules themselves are not optimal. The analysis of charging

statistics has shown no signiﬁcant correlation between the number of charging

stations and the percentage of EVs or the problem size.

For the heuristics used to solve the EVSP we have shown that an even

spread distribution of timetabled trips is responsible for a higher increase in

the total number of vehicles towards the standard VSP. Regarding the number

of charging stations we have found a slight correlation with the number of

chargings as well as the size of the instance.

In an extension of our study we dropped the assumption of a given charg-

ing infrastructure, thus analyzing the change in performance and necessary

charging infrastructure. For the MVT-(E)VSP we received a higher number

of EVs for all plans but at the same time a decreased occupancy rate for

each charging station. In contrast, the heuristics for the EVSP have proven to

perform better if the charging infrastructure is determined in advance.

This study remains only the ﬁrst step towards more realistic concepts and

solution approaches for the MVT-(E)VSP. For subsequent research, on the one

hand, further analysis should be conducted varying the input parameters as

well as the underlying assumptions about the vehicle and charging technology

in order to evaluate our ﬁndings and gain further insights into the problem. On

the other hand, one could extend the solution methods used. For instance, new

heuristics or metaheuristics can be implemented to improve the solutions found

for the EVSP. Next, the focus should be on ﬁnding the optimal solution to

the EVSP using exact methods. In addition, the positioning and the amount

of charging infrastructure can be considered – as a stand-alone problem or

integrated in the scheduling process using multi-criteria optimization.

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