HAL Id: hal-01331293
https://hal.archives-ouvertes.fr/hal-01331293v3
Preprint submitted on 28 Nov 2016
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The electric vehicle routing problem with nonlinear
charging function
Alejandro Montoya, Christelle Guéret, Jorge E. Mendoza, Juan G. Villegas
To cite this version:
Alejandro Montoya, Christelle Guéret, Jorge E. Mendoza, Juan G. Villegas. The electric vehicle
routing problem with nonlinear charging function. 2016. �hal-01331293v3�
The electric vehicle routing problem with nonlinear charging
function
Alejandro Montoya
a,b
, Christelle Gu´eret
a
, Jorge E. Mendoza
c,d
, Juan G. Villegas
e,∗
a
Universit´e d’Angers, LARIS (EA 7315), 62 avenue Notre Dame du Lac, 49000 Angers, France
b
Departamento de Ingenier´ıa de Producci´on, Universidad EAFIT, Carrera 49 No. 7 Sur - 50, Medell´ın,
Colombia
c
Universit´e Fran¸cois-Rabelais de Tours, CNRS, LI (EA 6300), OC (ERL CNRS 6305), 64 avenue Jean
Portalis, 37200 Tours, France
d
Centre de Recherches Math´ematiques (UMI 3457 CNRS), Montr´eal, Canada.
e
Departamento de Ingenier´ıa Industrial, Facultad de Ingenier´ıa, Universidad de Antioquia, Calle 70 No.
52-21, Medell´ın, Colombia
Abstract
Electric vehicle routing problems (E-VRPs) extend classical routing problems to consider
the limited driving range of electric vehicles. In general, this limitation is overcome by intro-
ducing planned detours to battery charging stations. Most existing E-VRP models assume
that the battery-charge level is a linear function of the charging time, but in reality the
function is nonlinear. In this paper we extend current E-VRP models to consider nonlinear
charging functions. We propose a hybrid metaheuristic that combines simple components
from the literature and components specifically designed for this problem. To assess the im-
portance of nonlinear charging functions, we present a computational study comparing our
assumptions with those commonly made in the literature. Our results suggest that neglect-
ing nonlinear charging may lead to infeasible or overly expensive solutions. Furthermore,
to test our hybrid metaheuristic we propose a new 120-instance testbed. The results show
that our method performs well on these instances.
Keywords: Vehicle routing problem, Electric vehicle routing problem with nonlinear
charging function, Iterated local search (ILS), Matheuristic
1. Introduction
In the last few years several companies have started to use electric vehicles (EVs) in
their operations. For example, La Poste operates at least 250 EVs and has signed orders
for an additional 10,000 (Kleindorfer et al. 2012); and the French electricity distribution
∗
Corresponding author
Email addresses: jmonto36@eafit.edu.co (Alejandro Montoya),
christelle.gueret@univ-angers.fr (Christelle Gu´eret), jorge.mendoza@univ-tours.fr (Jorge E.
Mendoza), juan.villegas@udea.edu.co (Juan G. Villegas)
Preprint submitted to Elsevier November 28, 2016
company ENEDIS runs 2,000 EVs, accounting for 10% of their fleet in 2016
1
. Despite these
developments, the large-scale adoption of EVs for service and distribution operations is
still hampered by technical constraints such as battery charging times and limited battery
capacity. For the most common EVs used in service operations, the minimum charging time
is 0.5 h and the battery capacity is around 22 kWh. The latter leads to a nominal driving
range of 142 km (Pelletier et al. 2014). In reality, the driving range could be significantly
lower because the energy consumption increases with the slope of the road, the speed, and the
use of peripherals (De Cauwer et al. 2015). For instance, Restrepo et al. (2014) documented
that the heating and air conditioning respectively reduce the driving range of an EV by
about 30% and 8% per hour of use.
Automakers and battery manufacturers are investing significant amounts of capital and
effort into the development of new technology to improve EV autonomy and charging time.
For instance, General Motors (GM) reinvested USD 20 million into the GM Global Battery
Systems Lab to help the company developing new battery technology for their vehicles (Mar-
cacci 2013). The results of these efforts, however, are transferred only slowly to commercially
available EVs. In the meantime, companies using EVs in their daily operations need fleet
management tools that can take into account limited driving ranges and slow charging times
(Felipe et al. 2014). To respond to this challenge, around 2012 the operations research com-
munity started to study a new family of vehicle routing problems (VRPs): the so-called
electric VRPs (E-VRPs) (Afroditi et al. 2014, Pelletier et al. 2016). These problems con-
sider the technical limitations of EVs. Because of the short driving range, E-VRP solutions
frequently include routes with planned detours to charging stations (CSs). The need to
detour usually arises in rural and semi-urban operations, where the distance covered by the
routes on a single day is often higher than the driving range.
As has been the case for other optimization problems inspired by practical applications,
research in E-VRPs started with primarily theoretical variants and is slowly moving toward
problems that better capture reality. In general, E-VRP models make assumptions about
the EV energy consumption, the charging infrastructure ownership, the capacity of the
CSs, and the battery charging process. Most E-VRPs assume that energy consumption is
directly and exclusively related to the traveled distance. However, as mentioned before, the
consumption depends on a number of additional factors. To the best of our knowledge only
Goeke & Schneider (2015) and Lin et al. (2016) use consumptions computed over actual
road networks taking into account the EV parameters and their loads.
Similarly, most E-VRP models implicitly assume that the charging infrastructure is pri-
vate. In this context, the decision-maker controls access to the CSs, so they are always
available. However, in reality, mid-route charging is often performed at public stations and
so the availability is uncertain. To our knowledge only Sweda et al. (2015) and Kullman
et al. (2016) deal with public infrastructure and consider uncertainty in CS availability.
CS capacity is another area in which current E-VRP models are still a step behind reality.
All existing E-VRP research that we are aware of assumes that the CSs can simultaneously
handle an unlimited number of EVs. In practice, each CS is usually equipped with only
1
http://www.avere-france.org/Site/Article/?article_id=5644. Last accessed 11/16/2016.
2
a few chargers. In some settings this assumption may be mild (e.g., a few geographically
distant routes and private CSs). However, in most practical applications CS capacity plays
a restrictive role.
Finally, in terms of the battery charging process, E-VRP models make assumptions about
the charging policy and the charging function approximation. The former defines how much
of the battery capacity can (or must) be restored when an EV visits a CS, and the latter
models the relationship between battery charging time and battery level. In this paper, we
focus on these assumptions.
In terms of the charging policies, the E-VRP literature can be classified into two groups:
studies assuming full and partial charging policies. As the name suggests, in full charging
policies, the battery capacity is fully restored every time an EV reaches a CS. Some studies
in this group assume that the charging time is constant (Conrad & Figliozzi 2011, Erdo˘gan
& Miller-Hooks 2012, Montoya et al. 2015). This is a plausible assumption in applications
where the CSs replace a (partially) depleted battery with a fully charged one. Other re-
searchers, including Schneider et al. (2014), Goeke & Schneider (2015), Schneider et al.
(2015), Desaulniers et al. (2016), Hiermann et al. (2016), Lin et al. (2016), and Szeto &
Cheng (2016), consider full charging policies with a linear charging function approximation
(i.e., the battery level is assumed to be a linear function of the charging time). In their
models, the time spent at each CS depends on the battery level when the EV arrives and on
the (constant) charging rate of the CS. In partial charging policies, the level of charge (and
thus the time spent at each CS) is a decision variable. To the best of our knowledge, all
existing E-VRP models with partial charging consider linear function approximations (Fe-
lipe et al. 2014, Sassi et al. 2015, Bruglieri et al. 2015, Schiffer & Walther 2015, Desaulniers
et al. 2016, Keskin &
˘
Catay 2016).
In general, the charging functions are nonlinear, because the terminal voltage and current
change during the charging process. This process is divided into two phases. In the first
phase, the charging current is held constant, and thus the battery level increases linearly
with time. The first charging phase continues until the battery’s terminal voltage increases
to a specific maximum value (see Figure 1). In the second phase, the current decreases
exponentially and the terminal voltage is held constant to avoid battery damage. The
battery level then increases concavely with time(Pelletier et al. 2015).
3
Figure 1: Typical charging curve, where i, u, and SoC represent the current, terminal voltage, and state of
charge respectively. The SoC is equivalent to the battery level. (Source: H˜oimoja et al. 2012).
Although the shape of the charging functions is known, devising analytical expressions to
model them is complex because they depend on factors such as current, voltage, self-recovery,
and temperature (Wang et al. 2013). The battery level is then described by differential
equations. Since such equations are difficult to incorporate into E-VRP models, researchers
rely on approximations of the actual charging functions. Bruglieri et al. (2014) use a linear
approximation that considers only the linear segment of the charging function, i.e., between
0 and (around) 0.8Q, where Q represents the battery capacity. This approximation avoids
dealing with the nonlinear segment of the charging function (i.e., from (around) 0.8Q to Q).
He henceforth refer to this approximation as FS. Felipe et al. (2014), Sassi et al. (2014),
Bruglieri et al. (2015), Desaulniers et al. (2016), Schiffer & Walther (2015), and Keskin &
˘
Catay (2016) approximate the whole charging function using a linear expression. They do
not explain how the approximation is calculated, but two options can be considered. In the
first (L1) the charging rate of the function corresponds to the slope of its linear segment (see
Figure 2b). This approximation is optimistic: it assumes that batteries charge to the level
Q faster than they do in reality. In the second approximation (L2) the charging rate is the
slope of the line connecting the first and last observations (see Figure 2c) of the charging
curve. This approximation tends to be pessimistic: over a large portion of the curve, the
charging rate is slower than in reality.
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