Advanced optimization models for the location of charging stations in e-mobility

Abstract

For a reduction in environmental pollution and dependency on petroleum, electric vehicles (EV) present an advantageous alternative to traditionally fossil-fuel powered automobiles. Rapid growth in the number of EVs requires an urgent need to develop an adequate charging station infrastructure to stimulate and facilitate their usage. Due to restricted investments in the development of a sufficient infrastructure, locations have to be chosen deliberately. In this paper, two extensions considering different objectives and further constraints to the deterministic flow refuelling location problem (DFRLP), described by Vries and Duijzer (Omega 69:102–114, 2017), are introduced. First, our research shows that location-dependent construction costs significantly influence the charging infrastructure obtained. Tests for different cost scenarios are carried out and policy implications are discussed. The original DFRLP assumes an unlimited capacity, meaning it is always possible to refuel all EVs at all charging stations, where they stop. Hence, for a more realistic modelling, we assume a limited capacity of charging stations in our second extension. Finally, both extensions are evaluated using benchmark instances based on test instances from the literature.

1 Introduction

In the 1970s, developed countries began to conduct research and development in the electric vehicle (EV) industry responding to the oil crisis and environmental pollution. As political pressure to reduce environmental pollution and dependency on petroleum became factors of great significance, the interest in looking for alternatives to traditionally fossil-fuel powered automobiles increased rapidly (Wu and Zhang 2017).

The growing use of such vehicles demands more focus on the impact of these alternative technologies, because they influence people’s driving behaviour and their acceptance of alternative mobility, particularly e-mobility. Most important challenges arising from the widespread use of EVs are posed by vehicles’ recharging needs (Mehar and Senouci 2013). To support the attractiveness of e-mobility and to trigger a change towards it, the need for a sufficient charging station (CS) infrastructure arises. One of the demanding goals is to set tax and subsidy policies so that infrastructure development matches the interests of particular states and regions. For an efficient implementation of such policies, various optimal mathematical location models for installing charging stations can be beneficial. Our paper enhances the deterministic flow refuelling location problem (DFRLP) by de Vries and Duijzer (2017) in two different ways:

• 1. to refine the modelling, i.e. to get rid of various simplifications, and

• 2. to illustrate the impact of price-controlling measures to policymakers, such as taxes and subsidies. These have an effect on construction costs of the particular CSs and thus on the future development of refuelling infrastructure.

The remainder of this paper is organized as follows: Sect. 1.1 gives an overview about related literature and our contribution. In Sect. 1.2 the DFRLP is formally introduced. Section 2 presents our two extensions to this model, afterwards Sect. 3 analyses the numerical results. The final Sect. 4 offers conclusions and opportunities for future research.

1.1 Related literature

Many scientific papers deal with optimization problems arising from the rapid development of electromobility. In general, there is a distinction between two different ways of expressing the demand of a network:

• Node-based demand is expressed at fixed points in the network and facilities are located centrally to fulfil demand. Well-known problems are the p-median problem (see Hakimi 1964; ReVelle and Swain 1970) and the location set-covering problem (see Toregas et al. 1971). These models are based on the assumption that extra charging trips are made instead of charging en-route.

• Flow-based Hodgson (1990) complains about the assumption of node-based demand, because customers usually satisfy their refuelling demand on their usual routes, i.e. along traffic flows.

It is not possible to give an exhaustive overview over this whole area. Consequently, we just focus on research papers dealing with CS location problems in flow-based context and refer to Sanguesa et al. (2021) and Kchaou-Boujelben (2021) for further information. To the best of our knowledge, the first paper dealing with so-called “flow-capturing” problems was written by Hodgson (1990). In his model, the flow capturing location problem (FCLP), a flow is simply defined by a given path from an origin to a destination node and covered if just a single CS exists on this path. Since it is often necessary to stop more than once along a path for “flow-refuelling”, the flow refuelling location problem (FRLP) appears to be more convenient. Compared to FCLPs, FRLPs generally model flows as round trips from an origin to a destination point and back to the origin, because cyclic models embed a much higher universality than models based on paths (Kuby and Lim 2005).

In order to explicitely take the driving range of an EV into account, Kuby and Lim developed a novel model formulated as a two-stage approach: it determines the maximum flow volume covered by using pre-generated combinations of facilities feasible with respect to EVs’ driving range. These exogenously generated combinations serve as an input for a mixed integer linear program (MILP) solved in the second stage. This formulation is, however, limited in its applicability, because the generation of feasible combinations for all paths is computationally burdensome, even for small instances. Thus, Lim and Kuby (2010) provide some heuristic algorithms, including greedy and genetic algorithms, to overcome the two-stage approach of the time-consuming pre-generation in the first stage and solving a MILP to locate a number of refuelling stations in the second stage. A radically new MILP formulation by Capar and Kuby (2012) does not require pre-generated combinations of possible facility locations, but incorporates the pre-generation in the original first stage into the MILP-constraints. This results in a larger and more complex model, but also equal or better running times. Nevertheless, the driving range is still not considered explicitly in their new model formulation.

Another issues with the model of Capar and Kuby is the assumption of an infinite CS-capacity. This assumption becomes impractical as the number of EVs increases. Upchurch et al. (2009) were the first researchers addressing this concern in their model. They extended the original FRLP by Kuby and Lim (2005), which requires pre-generated combinations of facilities as input, by limiting the capacity of EVs rechargeable at a refuelling station. Hosseini and Mirhassani (2017) defined capacity more accurately by estimating the quantity of consumed fuel at each refuelling station. For large-scale networks the authors propose an effective heuristic algorithm to obtain good quality solutions in appropriate time. Another FRLP-advancement by Wang and Lin (2013) considers a budget constraint and multiple types of charging stations.

Finally, de Vries and Duijzer (2017) worked out a MILP formulation, which takes the driving range of an EV explicitly as an input parameter into account. The authors introduce two FRLPs in their paper: one deterministic and one stochastic. Since our research is based on the deterministic one, we call it the deterministic flow refuelling location problem (DFRLP) in the remainder of this paper.

In our paper, which is based on the master’s thesis of the third author (Kastner 2019), we enhance the DFRLP by considering different objectives and other constraints to make it more applicable to real-world situations.

• 1. The first model extension, introduced in Sect. 2.1, deals with location-dependent costs while maximizing the total flow volume covered (CFV). It returns useful information on the effects of cost differences concerning the construction costs for charging stations. This allows governments to plan their subsidy measures accordingly to control and support investments in installing adequate infrastructures in certain areas.

• 2. Due to the significantly rising number of EVs, it becomes necessary to take capacity limits at charging stations into account and therefore not only the location, but also the size of charging stations.

• 3. As mentioned, the remarkable rising number of EVs—besides location—also triggers the necessity of taking CS-capacities and -sizes into account.

  We deal with these enhancements to the DFRLP in Sect. 2.2, while again maximizing the total flow covered. Contrary to the idea of Upchurch et al. (2009), who define capacity as the number of vehicles “refuelable” at a station, we build on the ideas of Hosseini and Mirhassani (2017) and assume the energy consumption to be linearly proportional to the distance travelled since the last refuelling process and that batteries are always filled to entire capacity.

1.2 Formal problem definition and the basic model

The objective of the DFRLP (see de Vries and Duijzer 2017) is to maximize the total number of EVs which can complete their trip without running out of fuel by optimally locating an exogenously given number of charging stations. Given an undirected graph G(L, E), L is a set of nodes, i. e. locations, and E is a set of edges, i. e. streets, between these locations. The set of locations L is the union of three disjunct subsets: the set of driving origins O, the set of driving destinations D and the set of potential facility locations K for charging stations (CSs) with no capacity limitations.

The overall amount in the traffic network is defined by set of flows F. Each flow $f \in F$ is defined by its origin $O_f \in O$, its integer flow volume $v_f \in {\mathbb {N}}$, its destination $D_f \in D$, and by the desired path between the origin–destination pair in G. The nodes on this path are potential facility locations and define the set $K_f \subseteq K$. All in all, $L_f = \{O_f\} \cup K_f \cup \{D_f\}$. We say a flow is covered if the driving distance between all consecutively used CSs along a round-trip does not exceed the driving range of the EV. Like G and F, the fixed driving range R is a parametric input.

Based on de Vries and Duijzer (2017), we define sub-trips from one CS to another as so-called cycle segments:

Definition 1

A cycle segment of the flow f is identified by two nodes k and l and has corresponding distances $t_{k l} \in {\mathbb {R}}$ as defined below:

• If $k = O_f$ and $l \in K_f$, the cycle segment defined by these two nodes is the path $l \rightarrow O_f \rightarrow l$ and its distance $t_{k l}$ is given by the distance from l via $O_f$ to l, both along f.

• For all $k, l \in K_f$, where k occurs before l in the flow f on the way from $O_f$ to $D_f$, the cycle segment defined by these two nodes is the path $k \rightarrow l$ and its distance $t_{k l}$ is given by the distance from k to l, both along f.

• If $k \in K_f$ and $l = D_f$, the cycle segment defined by these two nodes is the path $k \rightarrow D_f \rightarrow k$ and its distance $t_{k l}$ is given by the distance from k via $D_f$ to k, both along f.

• For $k = O_f$ and $l = D_f$ we define no cycle segment, but set $t_{k l} = \infty$.

This definition of cycle segments represents the usual approach in literature (see Kuby and Lim 2005; Upchurch et al. 2009; Kuby et al. 2009; Capar and Kuby 2012; Hosseini and Mirhassani 2017; de Vries and Duijzer 2017). The modelling aspect of using cycle segments, and herewith corresponding loading locations, arises from the wish to optimally solve a MILP model that approximates a given reality, because more elaborated solutions are prevented directly by the given current PC-hardware available.

Example 1

Consider a flow f corresponding to the desired path along the following nodes $10 \rightarrow 7 \rightarrow 12 \rightarrow 21$, i. e. having the origin $O_f = 10$ and the destination $D_f = 21$. Assume that the distances of the connecting edges are 7, 2, and 3, respectively. Then the six possible cycle segments have distances: $t_{10, 7} = 14$, $t_{10, 12} = 18$, $t_{7, 12} = 2$, $t_{7, 21} = 10$, and $t_{12, 21} = 6$. Further, we have $t_{10, 21} = \infty$, but do not define a cycle segment for $k = 10$ and $l = 21$.

It is important to mention that an origin or destination node cannot hold a CS in the DFRLP. However, to enable this property, one must only add a dummy facility location to the original $O_f$ and/or $D_f$, which is connected to the original $O_f$/$D_f$ nodes by a zero-distance edge.

We adopt the DFRLP, notation and model, from de Vries and Duijzer (2017).

Parameters

F:

Set of flows

O ($O_f \in O$):

Set of origins (origin of flow f)

K ($K_f \in K$):

Set of potential facility locations (along flow f)

D ($D_f \in D$):

Set of destinations (destination of flow f)

$v_f \in {\mathbb {N}}$:

Volume of flow f

L ($L_f \in L$):

Set of locations, i. e. $L = O \cup K \cup D$ (set of locations along flow f, i. e. $L_f = \{O_f\} \cup K_f \cup \{D_f\}$)

E:

Set of edges between locations

$L_{k f}^- \subsetneq L$ ($L_{k f}^+ \subsetneq L$):

Set of locations along flow f passed before (after) location k on a trip from $O_f$ to $D_f$

$p \in {\mathbb {N}}$:

Number of new facilities to locate

$R \in {\mathbb {N}}$:

Driving range

$t_{k l} \ge 0$:

Length of the cycle segment identified by locations k and l

Decision variables

$x_k \in \{0, 1\}$:

1 if a facility is placed at location k and 0 otherwise

$y_f \in \{0, 1\}$:

1 if flow f is covered and 0 otherwise

$i_{k l f} \in \{0, 1\}$:

1 if cycle segment k, l is used in flow f and 0 otherwise

DFRLP

$$ \max \sum\limits_{f \in F} v_f y_f $$

(1)

$$\text{s.t.} \sum\limits_{k \in K} x_k = p$$

(2)

$$\sum\limits_{l \in L_{k f}^+} i_{k l f} t_{k l} - (1 - y_f) M \le R f \in F,\ k \in \{O_f \cup K_f\} $$

(3)

$$\begin{aligned}{} & {} \sum\limits_{l \in L_{k f}^+} i_{k l f} =x_k \;{} & {} f \in F,\ k \in K_f \end{aligned}$$

(4)

$$\begin{aligned}{} & {} \sum\limits_{l \in L_{O_f f}^+} i_{O_f l f} =1\;\;\;f \in F \end{aligned}$$

(5)

$$\begin{aligned}{} & {} \sum\limits_{k \in L_{l f}^-} i_{k l f} = x_l \;{} & {} f \in F,\ l \in K_f \end{aligned}$$

(6)

$$\begin{aligned}{} & {} \sum\limits_{k \in L_{D_f f}^-} i_{k D_f f} = 1 \;{} & {} f \in F \end{aligned}$$

(7)

$$ \begin{gathered} i_{{klf}} \in \{ 0,1\} \quad f \in F,\;k \in \{ O_{f} \cup K_{f} \} , \;l \in L_{{kf}}^{ + } \end{gathered} $$

(8)

$$ x_{k} ,\;y_{f} \in \{ 0,1\} \quad k \in K,\;f \in F $$

(9)

The objective (1) of the DFRLP is to maximize the total number of EVs covered by optimally locating an exogenously given number of charging stations stated by constraint (2). Constraint (3) ensures that a flow is covered if the length of each cycle segment used along the path of flow f does not exceed the driving range of an EV. Constraints (4)–(7) are flow constraints, which also link variables x and i. Constraints (8)–(9) define the decision variables of the model.

2 Enhanced models

Below we investigate the DFRLP and take further constraints and parameters into account. We start with the consideration of CS-construction costs.

2.1 Location-dependent costs per charging station (LC_DFRLP)

While planning a network of CSs, one has to consider different one-time construction costs depending on the CS location. This results mainly from different land costs in urban, sub-urban and rural areas. Thus, this extension, LC_DFRLP, involves location-dependent construction costs as an additional input parameter in the optimization model.

Additional parameters

$c_k \ge 0$:

Construction costs per charging station at location k

$B > 0$:

Available budget for all stations

LC_DFRLP Model

$$ \max \sum\limits_{f \in F} v_f y_f $$

(1)

$$ \begin{aligned} {\text{s}}.{\text{t}}.\;\;\;\;\;\;\;\;\;\sum\limits_{{k \in K}} {c_{k} } x_{k} & \le \;B \hfill \\ (3) & - (9). \end{aligned} $$

(10)

The objective function (1) and constraints (3)–(9) remain the same as in the DFRLP. The number of new facilities to locate is no longer exogenously given, i. e. constraint (2) is not used. In fact, this restriction is replaced by constraint (10), which takes into account that there is only a limited budget to build the CS infrastructure.

2.2 Capacitated DFRLPs: determination of the station size

The objective of this extension is to simultaneously decide upon the placement and the size of CSs in a network in order to maximize the flow volume covered. In the C_DFRLP we assume that each charging pole has a limited capacity. Like Hosseini and Mirhassani (2017), we assume the energy consumption to be proportional to the travelled distance and that batteries are always filled to entire capacity. The pole capacity is then defined as energy output in terms of total driving range per pole and observation period. A positive side effect of considering the CS size, i. e. the number of charging poles per CS, as decision variable, is the avoidance of idle charging facilities due to a low or zero utilisation at charging points (Wu and Niu 2017).

In this section, we will discuss two different approaches: first, we maximize the flow coverage under an exogenous number of CSs, later on, we minimize the number of installed CSs while satisfying a given coverage level.

2.2.1 Exogenous given number of charging poles (C_DFRLP)

To begin with, we assume that the number of charging poles is exogenous. As construction costs are not explicitly concerned in this model, the exogenous number of CSs (and thus poles as well) can be seen as a proxy for the budget available for deploying an infrastructure (Giménez-Gaydou et al. 2017). Moreover, the assumption from Hosseini and Mirhassani (2017) is taken, stating that flows are assumed to be divisible, i. e. flow coverage may be lower than 100%.

Additional decision variables

$n_k \in {\mathbb {N}}_0$:

Number of charging poles at location k

$z_f \in [0, 1]$:

Proportion of flow f that is covered

$w_{k l f} \in [0, 1]$:

Auxiliary variable for linearisation

Additional parameters

$\text{ Cap } \in {\mathbb {N}}$:

Capacity of charging pole given as the amount of available energy in distance units

$M_k \in {\mathbb {N}}$:

Location-dependent maximum number of charging poles at a charging station (if $M_k = M$ for all $k \in K$, use M)

$S \in {\mathbb {N}}$:

Total number of charging poles to locate

$e_f \in (0, 1]$:

Positive range-based refuelling proportion $\le 1$ of flow f to be covered per observation period for flows with $2 {\tilde{t}}_{O_f D_f} \le R$; otherwise, $e_f = 1$:

$$\begin{aligned}e_f = \frac{1}{\max \left\{ 1,\Bigl \lfloor \frac{R}{2 {\tilde{t}}_{O_f D_f}}\Bigr \rfloor \right\} }, \end{aligned}$$

where ${\tilde{t}}_{O_f D_f} > 0$ stays for the real distance between origin and destination (note that ${\tilde{t}}_{O_f D_f} \ne t_{O_f D_f}$, because we defined $t_{O_f D_f} = \infty$ in Definition 1); as explained later, this parameter is needed to model short trips properly

C_DFRLP Model

$$\begin{aligned}&\text{ max } \ \sum\limits _{f \in F} v_f z_f \end{aligned}$$

(11)

$$\begin{aligned}&\quad \text{ s.t. } \ \sum\limits_{k \in K} n_k \ = \ S \end{aligned}$$

(12)

$$\begin{aligned}&\quad x_k \ \le \ n_k \quad \quad k \in K \end{aligned}$$

(13)

$$\begin{aligned}&\quad n_k \ \le \ M_k x_k \quad k \in K\end{aligned}$$

(14)

$$\begin{aligned}&\quad z_f \le \ y_f \quad \quad f \in F \end{aligned}$$

(15)

$$\begin{aligned}&\quad \sum\limits_{f \in F:k \in K_f} \left( \sum\limits_{l \in L_{k f}^-} t_{l k} v_f e_f w_{l k f} + \sum\limits_{l \in L_{k f}^+} t_{k l} v_f e_f w_{k l f} \right) \ \nonumber \\&\quad \hspace{1.65cm} \le \ Cap \cdot n_k \; \; \; \; \; \; \; \; \; \; k \in K \end{aligned}$$

(16)

$$\begin{aligned}&\quad w_{k l f} \le \ i_{k l f} \; \; \; \; \;\; \; \; \; \;f \in F, k \in K_f, l \in L_{k f}^+ \end{aligned}$$

(17)

$$\begin{aligned}&\quad w_{k l f} \le \ z_f \;\; \; \; \;\; \; \; \; \;f \in F, k \in K_f, l \in L_{k f}^+ \end{aligned}$$

(18)

$$\begin{aligned}&\quad w_{k l f} l \ge z_f - (1 - i_{k l f}) \; \; \; \; \; f \in F, k \in K_f, l \in L_{k f}^+ \end{aligned}$$

(19)

$$\begin{aligned}&\quad z_f \in \ [0, 1] \;\; \; \; \;\; \; \; \; \;f \in F \end{aligned}$$

(20)

$$\begin{aligned}&\quad n_k \in \ {\mathbb {N}}_0 \;\; \; \; \;\; \; \; \; \;k \in K \end{aligned}$$

(21)

$$ \begin{gathered} w_{{klf}} \in [0,1] \;\; \; \; \;\; \; \; \; \;f \in F,k \in K_{f} ,l \in L_{{kf}}^{ + } \hfill \\ \;\; \; \; \;\; \; \; \; \; \quad \quad\; \; \;\; \; \; \; \; (3) - \;(9).\; \hfill \\ \end{gathered} $$

(22)

The model above covers two elemental changes from the basic DFRLP: first, we make use of a new integer variable $n_k$, defining the number of charging poles at a potential facility location. Second, by postulating a limited capacity at refuelling stations, it might be impossible for one or more CSs to satisfy the total flow volume (TFV) of the flows using said stations. Thus the continuous variable $z_f$, which indicates the proportion of flow f that can be covered, is introduced.

The objective function (11) maximizes the flow volume covered, CFV. Note that it is the real-valued equivalent of objective function (1), obtained by substituting the binary variable $y_f$ for the proportion $z_f$. While the DFRLP parametrises the total number of CSs in (2), here the total number of charging poles is ensured by (12). Constraint (13) states that there is at least one charging pole installed when a location is intended to be a charging station. The maximum number of charging poles allowed at a charging station is defined in constraint (14). (15) ensures that the proportional coverage of flow f is zero if the flow is not covered at all, i.e. if $y_f = 0$. The limited capacity of charging poles is considered with constraint (16). It is important to understand that (16) is already a linearisation of an originally quadratic context as explained in the following.

To guarantee that capacity is not exceeded at any charging station, it is necessary to know which flows are recharging at a station. This information can be obtained from variable $i_{k l f}$, indicating at which locations (k and l) a flow f stops for refuelling. The limited capacity at charging poles is expressed by the following non-linear constraint:

$$\begin{aligned}&\sum\limits_{f \in F:k \in K_f} \left( \sum\limits_{l \in L_{k f}^-} t_{l k} i_{l k f} + \sum\limits_{l \in L_{k f}^+} t_{k l} i_{k l f} \right) v_f e_f \cdot z_f \nonumber \;\le Cap \cdot n_k \quad k \in K \end{aligned}$$

(23)

The left-hand side of this constraint totals the amount of energy demand at location k. As discussed, the EVs stop at CS k twice, once on their forward and a second time on their return journey. Consequently, the energy consumption of an EV at location k is linearly proportional to the distance travelled since the last charging stop on the forward trip plus the distance from the last CS used before k on the backward trip (which is expressed by means of forward distance $t_{k l}$ in our symmetric case). The right-hand side of the equation defines the total capacity (in units of distance) available at the potential location site k.

The linearisation is done by introducing a new variable $w_{l k f} :=i_{l k f} z_f$ and adding linking constraints (17)–(19). If the right-hand side is 0, i. e. no CS is opened at location k, on the first view, there are four possibilities how to fulfil (23): Either $z_f$, $e_f$, $v_f$ or $i_{k l f}$ equals 0 for all $f \in F$, $k \in L_f$ and $l \in L_{k f}^+$. $z_f = 0$ corresponds to zero flow coverage. $e_f$ cannot be 0 following its definition. In the special case $v_f = 0$, $z_f$ might take a value greater than zero, but this has no influence on the objective function value, which maximizes the total flow volume covered (CFV), i. e. the product of $v_f$ and $z_f$. If $i_{k l f} = 0$, the flow f is not covered at all according to the variable definition, because no CS is used. Finally, if $z_f = 0$ or $i_{k l f} = 0$, $w_{l k f}$ is forced to zero by (17) and (18).

Let us discuss the parameter $e_f$ in more detail. If $e_f \in (0, 1)$, then the range of the EV allows for a longer round-trip than the flow f demands. The setting $e_f = 1$ reflects two real situations. In the first case, the EV’s range exactly corresponds to the length of the round-trip. In the second case, more than one charging stop is needed to finish the whole round-trip. The latter case is upper-bounded with the value $e_f = 1$, because it is not possible to charge more energy then the battery capacity allows and more than one charging stops per round-trip are necessary.

Example 2

Imagine an EV with driving range $R = 200$. First, we assume that the round-trip to be covered has length 100. Then two such trips are possible before recharging is needed; consequently only half of the flow volume needs to be recharged per round-trip, i. e. $e_f = 0.5$. Next, assume the round-trip length to be 200. Now exactly one round-trip can be done without recharging, i. e. $e_f = 1$. Finally, assume that our round-trip has length 400. Then obviously one recharging process per trip is not enough and more than one CS on the trip is needed since the charging volume is bounded the battery capacity, i. e. $e_f = 1$. Note, obviously a minimum of 2 CSs is needed in this case.

2.2.2 Number of charging poles as decision variable (C+MC_DFRLP)¹

Allowing the number of charging poles to be no longer exogenously given, the objective is to install a minimum number of charging poles while guaranteeing a certain level of coverage. This objective seems to be reasonable as an incentive to increase consumer acceptance of EVs.

Additional variables

$n_k$, $z_f$:

Described in Sect. 2.2.1

Additional set and parameters

$C \in [0,1]$:

Minimum required flow volume coverage level

${{ Cap}}$, $e_f$, $M_k$:

Described in Sect. 2.2.1

C+MC_DFRLP Model

$$\text{ min }\sum\limits _{k \in K} n_k{} $$

(24)

$$ \begin{aligned} {\text{s}}{\text{.t}}{\text{.}}\;\;\;\;\;\;{\text{ }}\frac{{\sum\nolimits_{{f \in F}} {v_{f} } z_{f} }}{{\sum\limits_{{f \in F}} {v_{f} } }} &\ge C \hfill \\ (13) & - (22) \hfill \\ (3) & - (9). \hfill \\ \end{aligned} $$

(25)

The objective function (24) minimizes the number of charging poles located within the network. Constraint (25) forces the model to install enough charging poles to cover at least a certain proportion of all EVs driving within the network. Furthermore, constraints (13)–(22) are borrowed from the model described in Sect. 2.2.1. Constraints (3)–(9) take the driving range explicitly into account and define the decision variables.

Note that the C+MC_DFRLP can be used in a combined approach. First, the C+MC_DFRLP calculates the maximum achievable CFV and then the C_DFRLP takes it as input to minimize the construction costs, i.e. the number of CSs.

3 Evaluation

In this section, results of various numerical tests are described. We used the AMPL-IDE² together with Gurobi³ for modelling and calculation, respectively. All tests were run on a macOS Sierra⁴ computer with a 2.3 GHz Intel Core i5⁵ processor and an 8 GB/2133 MHz memory. The solve time always reports the overall running time, i. e. not only the running time of the solver, but also the time needed for starting the AMPL process and for generating the model.⁶

3.1 Benchmark instances

All models were tested on four instances randomly generated by de Vries and Duijzer (2017). In addition, for model LC_DFRLP, we partition the nodes in severals cost classes as outlined below. In their names, “sXwY”, “X” stays for the number of potential facility locations |K| and “Y” corresponds to the number of origin and destination nodes $|OD |$. We also use the same driving range of an EV, 250, for the basic model and all extensions.

Table 1 shows a summary reflecting the characteristics of all test instances.

Table 1 Test instance characteristics

3.2 Computational results and performance analysis

This section discusses parameters which are required in some model extensions in more detail, followed by a numerical analysis of the performance of the model extensions described in previous sections. As the baseline case we present computational results for the instance s60w30 (see e. g. Fig. 1) in more detail. For the other test instances, the general results and conclusions from the solutions are very similar, so we refer to our accompanying technical report, Kastner et al. (2023), which is an extended version of this paper. This report contains further pretests, remarks, and also a third enhanced model, the MC_DFRLP (minimum flow volume coverage), of the DFRLP.

Fig. 1

Model LC_DFRLP, test instance s60w30—cost categories: each of the nodes is representing an origin or destination or/and a pure potential facility location. OD nodes are in bold font and the size of the OD nodes represents how many EVs are starting/ending at these locations. The thickness of the road segments between the nodes represents the flow volume travelling along with these nodes. Black rectangles represent urban potential facility locations, red points denote locations in sub-urban areas and rural potential locations are marked with blue circles (colour figure online)

3.2.1 Numerical analysis: location-dependent costs per charging station (LC_DFRLP)

In the following, we define and discuss the parameters required for the LC_DFRLP: location-dependent costs per charging location and a limiting factor, a budget constraint.

Table 2 Number of potential facility locations assigned to particular cost categories

Fig. 2

Different charging cost scenarios for charging stations. Black filled square stays for urban, red circle for suburban, and blue circle for rural areas (colour figure online)

Test set-up

• 1. Partitioning facility locations into cost classes:

  The LC_DFRLP extends the DFRLP by taking costs of installing a charging station for EVs into consideration. This assumption is based on the idea that construction costs for building a charging station located along rural areas are lower than construction costs for locations in dense urban areas due to the scarity of land in urban areas. Nevertheless, the chosen costs are parameters in the LC_DFRLP, so our model can also be used for other cost structures. For simplicity and representational purposes, the individual nodes are classified into different cost categories and weighted with category specific costs. Therefore, a density-based cluster algorithm is used to classify the nodes into three different categories, which can be understood as urban, sub-urban and rural. Urban nodes are defined by either being closely located to each other in terms of radial distance or if they are affected with a considerably high proportion of the TFV starting or ending at that node. Rural nodes are defined in an opposite way and sub-urban ones in between. For more details and exact parameter settings see our accompanying technical report, Kastner et al. (2023). The resulting partition into the three categories (urban, sub-urban, and rural) is statistically summarized in Table 2. Figure 1 visualises the partitioning of the potential facility locations into cost categories for the test instance s60w30. In order to avoid results of only limited informative value, we tested 16 different cost structures, i.e. scenarios, for each instance by systematically changing the proportions of the construction costs in urban, sub-urban and rural areas (see Fig. 2). Due to the fact, that construction costs can vary significantly from country to county or even from region to region, one of these scenarios might represent the “real-world case” in a specific area. Moreover, this set of scenarios can be interpreted as possible set of options for subsidy systems a government can choose from. The public sector can grant subsidies in order to change a given cost structure and therefore enhance investments in the development of an adequate infrastructure. Depending on their extent, subsidies support the deployment of charging stations in certain areas. In the test cases, location-dependent construction costs are chosen within an interval of [1, 7]. This is based on the idea to enable special cases, like installing a charging station in a sub-urban region with construction costs more than twice as in a rural area and having construction costs in urban areas, which are more than twice as expensive as in a sub-urban area. Scenario 16 corresponds to the extreme case, where every CS costs 7 units.

• 2. Definition of the budget:

  The model takes a limited budget into account. Based on preliminary tests, in the scenarios for the LC_DFRLP, the available budget B for installing a charging station infrastructure is a quarter of the costs of installing all facilities

  $$\begin{aligned} \begin{aligned} B =&(\# urban\ nodes \cdot c_u + \# sub\text {-}urban\ nodes \cdot c_{su} +\# rural\ nodes \cdot c_r) / 4, \end{aligned} \end{aligned}$$

  (26)

  where $c_u, c_{su}, c_r$ describe the construction costs in urban, sub-urban and rural areas, respectively.

• 3. Testing for different cost scenarios:

  We want to highlight the following interesting results, obtained from scenario categories, S.1, S.2 and S.3, of specific scenario sequences:

  • First, all pairs differing in costs of charging stations (CS) in only one cost category can give insights into the behaviour of the LC_DFRLP. Thus we define the following two scenario categories, S.1 and S.2, each of them containing four scenario sequences.

    S.1::

    1-2-3-4-5, 6-7-8-9, 10-11-12, and 13-14: in these sequences, the costs of urban and rural CSs are constant while the costs of sub-urban CSs gradually increase by 1.

    S.2::

    5-9-12-14-15, 4-8-11-13, 3-7-10, and 2-6: here, the costs of urban and sub-urban CSs do not change, but the costs of rural CSs gradually increase.

  • Sometimes, it is informative to compare situations which are “similar” with respect to the general cost structure, but differ in more than one cost category.

  S.3::

  1-6-10-13-15: all these scenarios have one thing common: the costs of CSs in sub-urban and rural locations are similar while the costs of those in urban areas are much higher. Note that this property gets less significant when moving from scenario 1 towards scenario 15.

  Apart from the sequences described above, some particular scenarios and situations are of special interest.

  • Scenario 1 depicts the extreme situation where the sub-urban and rural CS costs are very small compared to those of the urban ones.

  • In scenario 5 the costs of CSs in urban and sub-urban areas are relatively similar, but the costs in rural locations are much smaller.

  • Scenario 15 versuss 16: 15 corresponds more or less to the case where all CSs have relatively similar costs; thus a comparison with scenario 16 makes it possible to evaluate whether even small cost differences between urban, sub-urban, and rural locations are significant. We will outline this situation in more detail below.

  Finally, scenarios, which lead to the same total number of opened charging stations, are of special interest.

Analysing baseline case s60w30

Before dealing with the particular results, one important property should be outlined. Since, e.g., in the first scenario sequence of S.1, 1-2-3-4-5, budget increases from one scenario to the next scenario (compare (26)), one might assume that an optimal solution from a scenario is also feasible for the next scenario. However, this need not be the case if for each cost category the proportion of possible CS locations in total possible locations does not correspond to the proportion of optimal CS locations in total optimal locations. An illustrative example can be found in the accompanying technical report, Kastner et al. (2023).

Table 3 Results of LC_DFRLP for all scenarios of s60w30: budget, optimal CFV and total number of opened CSs followed by relative (rel.) and absolute (#) number of CSs per cost category

Fig. 3

Proportion of optimal CS locations in total optimal locations in different scenarios for the test instance s60w30. The left bar (black) filled bar stays for urban, the middle bar (red) bar for uburban, and the right bar (blue) bar for rural areas (colour figure online)

The optimal flow coverage and proportion of opened urban/suburban/rural stations for all s60w30 scenarios are described in Table 3 and depicted in Fig. 3. Moreover, as an example, the results for scenario 13 are visualised in Fig. 4.

Fig. 4

Model LC_DFRLP, test instance s60w30—scenario 13: black rectangles represent urban potential facility locations, red points denote locations in sub-urban areas and rural potential locations are marked with blue circles. Surrounded nodes indicate opened charging stations and dark marked paths indicate covered flows, where the dashed lines represent the proportional coverage. Flows that cannot be covered are depicted as dashed lines in light colour (colour figure online)

S.1 scenario sequences

As sub-urban construction costs increase under the assumption that urban and rural costs remain the same, the proportion of sub-urban CSs is decreasing while the proportion of both, urban and rural CSs is increasing. Reason for it is the enlarged budget (which increases by 25% of the number of sub-urban locations in every move towards the sequence end; compare (26)).

As sub-urban costs increase, urban locations become relatively less expensive. Therefore, the leftover budget, resulting from a bigger budget or from the failure of building a more expensive sub-urban CS, can be used to build urban and/or rural CSs.

Moreover, it might become optimal to give up a sub-urban location that was built in a previous scenario and use the leftover budget to build one or more urban or rural stations, which are relatively cheaper due to the increasing budget.

Finally, compared to the other test instances, s60w30 is characterized by longer average travel distances between origins and destinations. Thus, sometimes it can be optimal to prefer particular rural stations over urban and sub-urban ones, even if they are relatively expensive, because they are essential to cover long distance flows.

S.2 scenario sequences

In these scenario sequences, cost differences between sub-urban and urban locations are constant. In contrast, rural costs are significantly lower but increase towards the sequence end. A trend for a decreasing proportion of rural CS can be observed when looking at absolute and proportional numbers. This behaviour emerges from the fact that building CSs in rural locations becomes more expensive and that at rural locations, which are often situated on the shortest path of long distance flows, less frequented flows are usually passing by.

There is another general phenomenon that is specific to S.2 scenario sequences: CFV is increasing continuously when moving towards the sequence end. A continuous increase is possible due to the fact that rural costs are the only changing component and therefore the urban and sub-urban CSs can be taken over from one scenario into the next one.

Scenario sequence S.3

The general trend is an increasing proportion of urban locations and decreasing proportions of sub-urban and rural locations. The reason for this stems from the decreasing cost differences between sub-urban and urban locations, resulting in a trade-off between rural/sub-urban locations and urban locations, as it becomes relatively cheaper to build a CS in an urban location. Due to the fact that more frequented flows are passing by urban locations, CFV will usually increase by installing a CS in an urban area.

There are only small changes between scenarios 1 and 6 and scenarios 10 and 13. The comparison of scenarios 6 and 10, however, shows a far more strongly change in the structure of optimal CS placement. Contrary to scenario 6, in scenario 10 building two sub-urban CSs no longer incurs lower costs than building one urban CS, which can be understood as an effect of a cost threshold. This threshold, when exceeded, motivates more urban CS locations, which are usually stronger frequented.

Scenarios leading to the same total number of opened stations

The scenario pairs 5-14, 7-10, 8-9, 9-12, 10-11, 11-13, and 13-14 result in the same total number of opened stations for this test instance. Apart from the pair 5-14, these scenarios differ in the costs of CSs in only one category and in the usual case, the increasing budget is used to move a CS into a better location, i. e. usually for moving a CS from a rural location to a sub-urban one or from a sub-urban location to an urban one. E.g., consider scenarios 8 and 9 more closely: the increased budget in scenario 9 (compare (26)) allows for substituting a sub-urban CS for an urban one. This results in a better CFV compared to the coverage of the solution for scenario 8, which would also be possible for scenario 9.

Scenario 15 versus 16

Scenario 15 is characterized by similar cost differences between rural, sub-urban and urban locations. Moreover, the difference between rural and urban construction costs is small. Expectedly, this should result in a CS placement similar to a placement obtained for scenario 16 with no cost differences. Indeed, this is not the case, because the additional CFV from scenario 15 to 16 is considerably high (see Table 3). This demonstrates that already small construction cost differences may have a significant influence on the optimal placement of CSs as well as on the CFV.

Finally note that scenarios 16, 15, and 1 return the highest CFV.

The analysis of the results of the LC_DFRLP provides main insight that the cost structure of urban, sub-urban and rural costs has a strong influence to the optimal charging station infrastructure. As a result, policymakers are able to enhance investments in deployment of an adequate CS infrastructure by developing and designing efficient subsidy systems. Governments have the possibility to use subsidies to influence the current cost structure of urban, sub-urban and rural location and subsequently change the structure of CS locations.

3.2.2 Numerical analysis: determination of the station size (C_DFRLP and C+MC_DFRLP)

This section focuses on the analysis of the C_DFRLP and the C+MC_DFRLP, while repeating the testing process for different exogenously given numbers of charging poles to locate.

Test set-up

• Step 1 Define the parameter capacity per charging pole, Cap:

  In order to estimate the instance-specific CS capacity, we use a consumer-oriented ideal feasible solution, in which a CS is built at each possible locations. Then we calculate the median energy demand per CS, excluding those CSs that have a zero energy demand (note that in a time-dependent, i. e. dynamic model, it could be of interest to model a capacity restriction rather on peak than on median demand). Afterwards, to define the capacity of a charging pole, the median capacity per CS is divided by the maximum number of charging poles per location M, for which we assume $M = 4$. The pole capacities generated are shown in Table 4 (for a better representation, a scaling factor of 0.001 is used).

• Step 2 Estimating the maximum possible flow volume coverage level C for C+MC_DFRLP:

  To obtain information about the maximum possible CFV, given the capacity limitation of charging poles, the C_DFRLP with a maximum S, called $S_{max}$, is solved, requiring that all possible locations are used with a maximum number of poles. Consequently, this yields the number of charging poles to be located is M times the number of potential facility locations, $S_{max}=M\cdot |K|$.

• Step 3: Minimizing the number of charging poles, S, given the maximum possible C:

  Since the CFV from Step 2 could be guaranteed with a lower number of charging poles, the C+MC_DFRLP is applied to find the minimum number of poles needed to cover the maximum possible flow volume estimated in Step 2.

  In the C+MC_DFRLP the number of charging poles to locate is now no longer exogenously given, but it becomes an optimized minimal value to cover a certain proportion of EVs (in our case C from Step 2). Note that the obtained objective value, S, can be interpreted as a so-called maximum economically viable number of charging poles (EVCP), because locating more charging poles, would not increase the CFV.

Table 4 Capacity per charging pole

Evaluation

The exogenously given number of charging poles to be placed within a network can be interpreted as a budget constraint. For more insight, we solve the C_DFRLP and the C+MC_DFRLP for different levels of the parameter S, namely 25%, 50%, 75% and 100% of the EVCP. Unfortunately, the solve time for the C_DFRLP increases rapidly when the number of charging poles to locate decreases. Thus we sometimes stopped the solver before the optimality of the found solution could be proven. This effect will be exemplified in the next paragraph.

Analysing the baseline case s60w30

Testing the C_DFRLP for $S=108=0.75\cdot \text{ EVCP }$ with the Gurobi solver, we found that the incumbent objective value only improved within the first 1000 seconds, while a running time limit of 4600 seconds was set. We observed the same behaviour for the settings $S=72=0.50\cdot \text{ EVCP }$ and $S=36=0.25\cdot \text{ EVCP }$ within the first 1600 and 500 seconds, respectively. Consequently, we used a time limit of 1800 seconds for these two settings.

Now we consider the test set-up for s60w30. The results of Step 2, where $S_{max} = 4 \cdot 60 = 240$, show that with the capacity per charging pole, $Cap=$ 3 362.01 (see Table 4), it is not possible to cover more than 827 525 (82.75%) of the TFV. Using this proportion of TFV, $C =$ 0.827 525 as an input for the C+MC_DFRLP in Step 3, gives a result, in which EVCP = 144 charging poles are located. This value serves as an upper bound for the number of charging poles, because the CFV would not increase when further charging poles are installed.

Table 5 Results of C_DFRLP and C+MC_DFRLP for instance s60w30

For the computational results in Table 5 some general remarks apply: CF stands for all flows in the (optimal) solution, which are at least partially covered. Note that, opposite to CF, $\sum _{f \in f} y_f$ is the number of flows, which could be theoretically covered with the located CSs if there were no capacity restriction given. The solve time seconds address the whole running time of the combined model, C_DFRLP and C+MC_DFRLP, while the time limit only targets the C_DFRLP. Finally, ${\overline{n}}$ is built only over all open CS locations.

From Step 2 one can experience that it is not possible to cover more than 82.75% of the TFV and from Step 3 we get $\text{ EVCP }$ = 144 charging poles, where on average 3.00 poles are installed per station. Theoretically, assuming infinite CS capacities, 416 of 435 flows could be covered, however, given the limited capacity at charging poles, 188 flows can actually refuel and are therefore covered. Using only $S=108=0.75\cdot \text{ EVCP }$ charging poles, still covers 95.41% of the CFV for the maximal $S = 144$ under capacity limitation. The 108 charging poles are located in 45 facility locations. Compared to the previously described case, there are now on average 2.40 charging poles per station. When locating $S=72=0.50\cdot \text{ EVCP }$ charging poles, again limiting charging pole capacity, it is still possible to receive a CFV of 81.30% compared to the case with $S=144=1.00\cdot \text{ EVCP }$. Locating $S=36=0.25\cdot \text{ EVCP }$ charging poles (see Fig. 5), still covers 47.68% from the CFV of the 100% case given the capacity limit of charging poles. On average there are 1.7 charging poles per station. With this charging station placement it would be possible to theoretically cover 38 flows, but due to the capacity limitation at charging poles, it is just possible to cover 23 flows.

Fig. 5

Model C_DFRLP, test instance s60w30—$S=36=0.25\cdot \text{ EVCP }$: open charging stations are marked with a surrounding circle, where the size of the circle represents the number of installed charging poles per location (1, 2, 3 or 4). Covered paths are indicated with dark dashed lines representing the proportional coverage. Flows that cannot be covered given the charging station infrastructure are dotted in light colour

Finally, note that only a small amount of flows is saturated in Fig. 5. In particular, when $S=36=0.25\cdot \text{ EVCP }$, 36 charging poles are allocated, the respective numbers are: 10 flows (out of 23) fully covered ($z_f = 1.00$), 6 flows by less than 50% ($z_f < 0.50$) and 7 in between. In the case of $S=144=1.00\cdot \text{ EVCP }$ the respective numbers are: 176 (out of 188) flows fully covered ($z_f = 1.00$) and just a single flow by less than 50% ($z_f < 0.50$). A corresponding figure is depicted in Kastner et al. (2023). This shows too few poles lead not only to a smaller CFV, but also to a minor degree of coverage of the individual flows.

4 Conclusions and future research

Due to the rapid growth in the number of EVs in the last years, mathematical models dealing with the development of a proper CS infrastructure become increasingly important. In this paper, two extensions considering different objectives and various constraints to the deterministic flow refuelling location problem (DFRLP), described by de Vries and Duijzer (2017), are introduced and implemented. Furthermore, these extensions are analysed using problem instances from the literature, partially extended with CS price categories.

The first model extension, introduced in Sect. 2.1, deals with location-dependent costs and returns useful information on the effects of cost differences concerning the construction costs for charging stations. This research shows that when considering location-dependent costs, results heavily depend on their relation. Thus comparing solutions for different cost relations can be used by governments to influence them by developing proper subsidy and tax policies so that infrastructure development matches the interests of the state.

The second enhanced model, described in Sect. 2.2, considers the location and size of particular CSs simultaneously. In comparison to existing models, capacity is defined as the quantity of energy available at a charging pole per observation period. This model can be used especially by CS building companies for designing their CS infrastructure.

4.1 Further research

In the models C_DFRLP and C+MC_DFRLP, which deal with limited capacity per charging pole, on a given segment we allow only the sequence of all opened CSs to be used for each flow using that segment. E.g., it is not possible to split the flow volume of a certain flow f such that different flow proportions use different sequences of opened CSs. According to our model formulation and the definition of the cycle segment indicator $i_{k l f}$, substantial changes would be necessary to allow different combinations of charging stations within such a segment. This model variation could be the objective of further research.

Moreover, in these models, C_DFRLP and C+MC_DFRLP, the price of each charging pole is constant. However, based on economic considerations, it would be reasonable to assume linearly or even non-linearly decreasing costs per charging pole with increasing size of the CS. This is justified by sharing construction costs among several charging poles (Upchurch et al. 2009). Thus, installing more than one charging pole at a location leads to an increasing utilisation per monetary unit. Also different types of charging poles could be modelled: faster ones, which are more expensive, and slower ones, which are cheaper.

Finally, one could also target stochasticity, where the driving range can vary due to many factors, e.g., air temperature or age of the battery. Therefore, such models, taking this issue into account, could be one of the next steps towards reality.

Some further results and a third extension of the DFRLP, the MC_DFRLP, can be found in the accompanying technical report, Kastner et al. (2023), which is an extended version of this paper.