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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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