# The Electric Riverboat Charging Station Location Problem

Authors: Daniel Villa; Alejandro Montoya; Aura Maria Herrera
Journal: Journal of Advanced Transportation
 (2020)

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## Abstract

Nowadays, the electric mobility is mainly focused on urban areas. However, the use of Photovoltaic-assisted Charging Stations (PVCSs) can contribute to implement the electric mobility in rural areas disconnected from the national grid. Inspired by the new river operations with an Electric Boat (EB), we introduce a new location problem named the Electric Riverboat Charging Station Location Problem (ERCSLP). This problem estimates the necessary infrastructure for an EB to be able to perform a round trip. In this case, we decide the location of the PVCSs and the size of the EB battery aiming to minimize the sum of the PVCS and the EB battery costs. In this problem, we include the nonlinear behavior of the charging function and the variation of the solar radiation. For solving this problem, we propose a Mixed-Integer Linear Programming (MILP) formulation. For testing this MILP formulation, we build a set of instances based on real river transport operations that have the potential to migrate to the electric mobility. In our computational experiments, we show that our MILP formulation can find the optimal solution of the instances. Finally, we perform a sensitivity analysis and an economic viability analysis of the electric mobility in these operations.


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## Body

## 1. Introduction

Electric mobility is emerging as an attractive alternative to counteract the Greenhouse Gas (GHG) emissions and to reduce the dependency on fossil fuels. Different organizations from both the public and the private sectors have implemented the use of Electric Vehicles (EVs) for transport operations [1]. Most of these operations are found in urban areas, since the concentration of polluting gases is considerably higher compared to rural areas [2]. Additionally, the electric infrastructure is more developed in urban areas, which implies better energy availability, crucial for EVs’ driving range limitations [1]. However, this does not mean that the development of the electric mobility is not important in rural areas. The use of fossil fuels in rural areas still accounts for CO2 emission and can potentially contaminate natural resources such as water bodies. But, the implementation of transport operations with EVs in rural areas has been limited by the long distances that are normally greater than the driving range of the EVs. Besides, the grid infrastructure is often unreliable or it is even nonexistent.Nevertheless, recent technological developments aim to overcome the limitations of the EVs for transport operations in rural areas. Some of the most important technological improvements are addressed to bothBattery Energy Density (BED) and Photovoltaic-assisted Charging Stations (PVCSs). The BED is the quantity of energy that a battery is capable to store per unit of mass kWh/kg. Over the last few years, the BED has substantially increased through the improvement of the materials for both anode and cathode [3]. This improvement of the BED is fundamental for rural transport operations, as it increases the driving range of EVs. This allows them to perform routes that were not possible before. Furthermore, the PVCSs are the charging system that use solar panels for generating electric power, and some of them have a battery to store energy. Understanding that the rural areas are poorly connected to the national grid, the use of PVCSs rises as a good alternative for supporting the electric mobility in these areas. To help with the implementation of this technology, manufacturers have lowered the manufacturing cost, thanks to different improvements in conversion technologies and installation processes [4, 5].In rural areas, there are different types of transport operations that can be developed around electric mobility systems. One of these operations is the river transport for both passengers and freight in areas with no connection to the national grid, for instance, the river transport operation in the Amazon region. In this region, approximately 90% of the transport operations are carried out by river, due to its lack of road connectivity [6, 7]. A problem of river transport operations with Internal Combustion Boats (ICBs) is related to the unreliable and inconstant supply of fuel. This factor could also affect the quality of life of the inhabitants of these isolated areas. Another problem of using ICBs is linked to its impact on the water sources due to the increase of noise and fuel spillage. This can represent a threat to the species that dwell in those ecosystems, which have suffered dramatic reductions in their populations during the last few years [8]. As can be seen, Electric Boats (EBs) have several advantages compared to ICBs as the reduction of CO2 emissions and the reduction of noise affecting the fauna [9]. For these reasons, in some places of the Amazon basin, people are starting to use EBs as an alternative to ICBs. Some examples of river transport operations with EBs have been introduced by several authors. Jaimurzina et al. [7] presented a solution for the Putumayo River in Ecuador; Ponce [9] presented a study for the insertion of an electrosolar boat in the Napo River in Ecuador; and Miniguano and Elizabeth [10] described the Kara project, which intends to develop a transport system for native communities in two Ecuadorian rivers.Although EBs are an excellent alternative to perform river transport operations, it must be considered that the communities that inhabit these zones, near from the river, are isolated from each other. Therefore, the EBs would need to move across large distances, which occasionally could be larger than the autonomy of the EBs. For avoiding this problem, a possible solution would be the installation of a higher battery capacity; however, this solution requires a higher investment while reducing the efficiency of the EBs. The latter happens due to the energy consumption being directly related to the weight of the battery. A different solution for improving the autonomy would be the installations of Charging Stations (CSs) along the river. Nevertheless, it is important to be aware that this remote area is not connected into the national power grid. For example, in Colombia, 51% of the territory is not connected to the grid. Most of this area is located in the Amazon region, which represents 41.8% of the national territory [11]. For addressing this problem, PVCSs arise as a promising solution, as this type of CS does not need to be connected to the grid. Prior to implementation of this solution, there are two questions to be answered. Firstly, where should the PVCSs be located in such a way that the EB is capable of completing its route? For the second question, it is important to consider two possible scenarios. In the first scenario, the EB has enough battery capacity to complete the route (without using any intermediate PVCSs). In the second scenario, the EB has a smaller battery capacity and uses intermediate PVCSs. With this in mind, the second question is which scenario is more cost effective?For answering these questions, it is necessary to consider an optimization problem that allows determining the capacity of the EB battery and the location of the PVCSs minimizing the total investment cost, defined as the PVCS cost plus the EB battery cost. In this optimization problem, it is important to consider several components of the electric mobility system, such as the solar radiation and the energy generation of the PV system, nonlinear charging functions for the Li-ion batteries, and energy consumption estimations. Additionally, it is worth noting that there is a trade-off between the investment in the capacity of the EB battery and the amount of PVCSs to place. In this paper, we formally define this problem as the Electric Riverboat Charging Station Location Problem (ERCSLP).For solving this problem, we propose a Mixed-Integer Linear Programming (MILP) formulation. To test the performance of the MILP formulation, we used a set of instances built from ICB real river operations in Colombia that have the potential to operate with EBs. The results show that the MILP formulation is capable of solving the test instances in competitive CPU times. The results also show that, for some instances, even if it is possible to complete the route with a big battery without PVCSs, the optimal solution contemplates the installation of PVCSs while using small batteries. Additionally, we compare the investment costs of the electric mobility system against the fuel costs of the operation with an ICB, for a time horizon of 10 years (considering the battery lifespan). For several instances, it is possible to recover the investment of the electric mobility system in 10 years or less.The remainder of this paper is organized as follows. Section2 reviews the literature related to the problem. Section 3 briefly explains some technical issues about electric mobility, aiming to understand the formulation of the problem. Section 4 formally introduces the ERCSLP and presents a MILP formulation for solving the problem. Section 5 presents the computational experiments performed for the ERCSLP. Finally, Section 6 concludes the paper. For easy reference, Appendix A summarizes our notation.


## 2. Literature Review

In the last few years, the CS location problem literature has been gaining importance; however, most researchers have focused their attention on the urban context and applications where the CSs are connected to the grid. CS location problems have been addressed from two different perspectives: one from the CS user and the other from the CS owner. The former is related to optimizing objective functions that impact the user; for example, minimizing the traveled distance [12], reducing the total number of recharges [13, 14], and reducing the travel time [15]. On the other hand, the owner’s perspective is related to optimizing objective functions that impact the interest of the CSs’ owner, for example, minimizing the installation costs [16], maximizing the coverage [17, 18], minimizing new CSs deployment costs [19], and maximizing the number of EVs served [20].Recently, some researchers have integrated Renewable Energy Sources (RESs) to CS location problems. In this way, Moradi et al. [21] presented a model for locating and sizing CSs with RESs, where the CSs work on grid connected systems because their objectives are to reduce grid power losses, improve voltage stability, and minimize EV charging costs. In a different study, Mozafar et al. [22] designed a model that aims to the optimal location of RESs and CSs. Their objective function is to minimize a multiobjective function composed by voltage fluctuations, energy supplying costs, and power losses. Similarly, Amini et al. [23] proposed a model for the simultaneous location of EVs parking lots and distributed renewable resources in a smart power distribution network. Their approach has two phases. In the first phase, they simultaneously determined candidate locations for both EV parking lots and RES. In the second phase, they considered the distribution of RES and their penetration.The problem that we address in this paper can be seen in two levels: strategic and operative. On the strategic level, we determine the CS location and the EB battery capacity. On the operative level, we consider operational decisions such as where and how much to charge accounting for operative constraints. In the literature, we have found other related type of problems that also consider two levels on the decision-making process, known as location-routing problems with EVs. These problems consider the decisions involved in the CS location (strategic level) and the routing of the EVs (operative level). In this context, the studies in the literature consider the autonomy restrictions of the EVs and their charging times to complete their operations. The location-routing problems with EVs can be traced back to Yang and Sun [24], where they presented a model to determine battery swap station locations and, simultaneously, design a routing plan for a fleet of EVs for the delivering of goods. In this problem, the authors considered the EV driving range limitations and different demands from the customers that had to be satisfied. Li-ying and Yuan-bin [25] presented an extension of the work by Yang and Sun [24], in which they made decisions about CS infrastructure and their location. Charging infrastructure is referred to the technology of the CS, which can impact the charging times. Additionally, they decided the vehicle-routing plan considering Time Windows (TWs) for a set of customers. Schiffer and Walther [26] proposed a location-routing problem with TWs and partial recharging. Recently, Schiffer and Walther [27] presented an extension on their work, including the acquisition of the EVs. Another approach is presented by Cui et al. [28] where they present a location-routing problem; they present a location-routing problem for Mobile Charging Vehicles (MCVs) considering fixed candidate locations and, after locating the CSs and determining their charging rates, routing the MCVs to the EVs that are requiring the service.Considering that ERCSLP is in the context of a river operation, the EB has to state on a fixed route. To the best of our knowledge, there is no previous work on CS location in a fixed route context. Nonetheless, in the literature, there are works related to the charging decision problem for EVs in a fixed route. This problem was introduced by Montoya et al. [29]; in this study, they presented this problem as a subproblem of an electric vehicle-routing problem with nonlinear charging functions. To solve this problem, they proposed a MILP formulation and a heuristic approach. Recently, Villa et al. [30] introduced a problem of a fixed route for a river transport operation with EBs considering speed and charging decisions for minimizing charging costs and battery degradation costs.The previous studies on CS location have addressed the urban context and land applications, while rural areas have received scarce attention. Moreover, nobody has studied the CS location problem for a river transport operation, considering PV generation in standalone applications. In this regards, the closest work has been proposed by Zhang et al. [31]. This work presents a CSs location problem for Electric Ships (ESs). However, they did not consider fixed routes as their context is on a lake. Their problem is divided into three phases: finding the candidate sites to be CSs, selecting the CSs among the candidates, and finally, determining the capacity of the CSs. They present a MILP formulation for solving this problem.


## 3. Technical Issues about the ERCSLP

In the problem that we are facing in this study, we include different components related to electric mobility such as energy consumption models, PV generation, and nonlinear charging times. Aiming to give the reader a better understanding of these components, we present a short explanation for each one.


### 3.1. Energy Consumption Models

When planning a transport operation system, in the context of the electric mobility, it is necessary to estimate the energy consumption of the EV, or in this case the EB. For the case of EVs, Villa and Montoya [32] made taxonomy of some energy consumption models in the literature. In this study, they found 25 models to estimate the energy consumption of the EVs. However, in the case of the EBs, the number of models to estimate the energy consumption is more limited. Minami and Yamachika [33] pioneered the study of the models that estimate the energy consumption of the EBs. In their work, they reviewed some previous models and proposed a new model. Additionally, they compared the accuracy of the model using experimental data. In our case, the energy consumption is an input parameter; therefore, any model could be used for the estimating the consumption. However, it is important to note that there are some key features for the estimation such as EB and river speed, EB dimensions, and EB weight. The latter is important, because the EB weight affects the power needed to propel the EB. The weight of the EB itself depends on its feature and the EB size. Considering that the battery size is a decision variable for this problem, the energy consumption parameter will be in terms of the EB battery size. Although, in our model, the energy consumption is an input parameter, it is important to mention that, in the case of restrictive waters (e.g., the rivers), the propulsive power of a boat is sensitive to the width, depth, and the current of the river. One of the first authors to analyze the impact of the depth on the propulsion quality was Schwanecke [34]. This author proposed a model for analyzing the hydrodynamic and the propulsion of a boat on restricted water depth. Recently, Thi Ngoc Hoa et al. [35] investigated the impact of the depth of the waterway on the ship’s resistance and propulsion using a computational fluid dynamic method.


### 3.2. Nonlinear Charging Function

The battery charging function is nonlinear, which seeks to avoid overcharging degradation. This function represents the relationship between the energy level of the battery and its charging time. The charging process is represented in a Constant Current (CC)-Constant Voltage (CV) scheme. Initially, a CC phase is performed, in which the current is held constant while the battery’s terminal voltage is below a maximum value. Then, the CV phase begins. In this phase, the current starts to decrease to hold the terminal voltage constant. This process causes a change in the linearity of the charging time [36]. Montoya et al. [29] used a piecewise linear approximation to represent the nonlinear charging function. Their approximation obtains an average relative absolute error of less than 2% for the data that they used in their experiments. Figure 1 shows a representation of a piecewise linear approximation. The data were obtained from a battery capacity of 16kWh, using a charger of 22kW of power. The breakpoints are represented in the coordinates shown in Figure 1. Now, the values of the breaking points depend on the size of the battery and the charging power supplied.Figure 1
Piecewise linear approximation for charging a battery of 16 kWh with a charging rate of 22 kW. Source: [30].


### 3.3. Photovoltaic-Assisted Charging Stations Behavior

As mentioned above, the increasing interest on the implementation of green renewable energies has impacted the development of energies related to CS systems. These developments have focused not only on the energy capture stages but also on the storage, tracking systems, and energy conversion. All these elements have a direct or indirect impact on the generation capacity of the PVCS. In this section, we intend to give a brief and simple description of a PVCS operation considering the relevant factors that affect the strategic decisions of this problem. For further description of the topics mentioned here, refer to [37] and [38].A PVCS consists of several components: a set of photovoltaic modules converting solar radiation into electric power; a DC-DC power converter with an algorithm that optimizes the power curve of the PV module; an internal battery for energy storage, called the Energy Storage System (ESS); and a DC charger that controls the input current to the EB. Figure2 depicts a general scheme for a standalone PVCS with a battery connected from an EV. Some standalone PVCSs could work without an ESS; however, this is impractical as the incoming radiation fluctuates; and thus, the generation is insufficient for charging the EB in short times [38]. Our study considers standalone PVCSs with an ESS.Figure 2
General scheme of a PVCS. Source: [38].For the PVCSs, the input power may vary depending on environmental factors such as temperature and geographical position of the photovoltaic modules. Considering this variation of the input energy, we present three possible scenarios for the PVCS operation. In the first scenario, we have a PVCS without any EB connected to it. In this scenario, the energy generated by the PV module flows directly to the ESS until it is fully charged. With standalone PVCSs, once the ESS capacity is full, the additional generated energy is lost. For the remaining two scenarios, we will consider an energy lower bound for the ESS, to avoid degradation due to the overdischarging. With this in mind, for the second scenario, the energy within the ESS is above the lower bound, and an EB is connected to the PVCS. In this case, the energy for charging the EB comes from the PV modules and simultaneously from the ESS, in order to ensure the charging power selected to charge the EB. It is important to mention that, in this scenario, the charging time is determined by the nonlinear charging function. Finally, in the last scenario, the energy within the ESS is below the lower bound, and an EB is connected to the PVCS. For this scenario, we consider that there is a control system that ensures that the energy for charging the EB comes only from the PV modules. In this case, the charging time will only depend on the PV generation.For the simplification of the modeling, we consider that PV modules are always in a horizontal position (the PVCS does not have a tracking system). Additionally, solar radiation is discretized in a finite number of time intervals, and the panel efficiency is considered constant for the whole day. Figure3 shows the average solar radiation in W/m2 per hour, in the month of June, for the municipality of Sincelejo, in Colombia.Figure 3
Solar radiation curve for the municipality of Sincelejo, Colombia. Source: data from IDEAM [39].


## 3.1. Energy Consumption Models

When planning a transport operation system, in the context of the electric mobility, it is necessary to estimate the energy consumption of the EV, or in this case the EB. For the case of EVs, Villa and Montoya [32] made taxonomy of some energy consumption models in the literature. In this study, they found 25 models to estimate the energy consumption of the EVs. However, in the case of the EBs, the number of models to estimate the energy consumption is more limited. Minami and Yamachika [33] pioneered the study of the models that estimate the energy consumption of the EBs. In their work, they reviewed some previous models and proposed a new model. Additionally, they compared the accuracy of the model using experimental data. In our case, the energy consumption is an input parameter; therefore, any model could be used for the estimating the consumption. However, it is important to note that there are some key features for the estimation such as EB and river speed, EB dimensions, and EB weight. The latter is important, because the EB weight affects the power needed to propel the EB. The weight of the EB itself depends on its feature and the EB size. Considering that the battery size is a decision variable for this problem, the energy consumption parameter will be in terms of the EB battery size. Although, in our model, the energy consumption is an input parameter, it is important to mention that, in the case of restrictive waters (e.g., the rivers), the propulsive power of a boat is sensitive to the width, depth, and the current of the river. One of the first authors to analyze the impact of the depth on the propulsion quality was Schwanecke [34]. This author proposed a model for analyzing the hydrodynamic and the propulsion of a boat on restricted water depth. Recently, Thi Ngoc Hoa et al. [35] investigated the impact of the depth of the waterway on the ship’s resistance and propulsion using a computational fluid dynamic method.


## 3.2. Nonlinear Charging Function

The battery charging function is nonlinear, which seeks to avoid overcharging degradation. This function represents the relationship between the energy level of the battery and its charging time. The charging process is represented in a Constant Current (CC)-Constant Voltage (CV) scheme. Initially, a CC phase is performed, in which the current is held constant while the battery’s terminal voltage is below a maximum value. Then, the CV phase begins. In this phase, the current starts to decrease to hold the terminal voltage constant. This process causes a change in the linearity of the charging time [36]. Montoya et al. [29] used a piecewise linear approximation to represent the nonlinear charging function. Their approximation obtains an average relative absolute error of less than 2% for the data that they used in their experiments. Figure 1 shows a representation of a piecewise linear approximation. The data were obtained from a battery capacity of 16kWh, using a charger of 22kW of power. The breakpoints are represented in the coordinates shown in Figure 1. Now, the values of the breaking points depend on the size of the battery and the charging power supplied.Figure 1
Piecewise linear approximation for charging a battery of 16 kWh with a charging rate of 22 kW. Source: [30].


## 3.3. Photovoltaic-Assisted Charging Stations Behavior

As mentioned above, the increasing interest on the implementation of green renewable energies has impacted the development of energies related to CS systems. These developments have focused not only on the energy capture stages but also on the storage, tracking systems, and energy conversion. All these elements have a direct or indirect impact on the generation capacity of the PVCS. In this section, we intend to give a brief and simple description of a PVCS operation considering the relevant factors that affect the strategic decisions of this problem. For further description of the topics mentioned here, refer to [37] and [38].A PVCS consists of several components: a set of photovoltaic modules converting solar radiation into electric power; a DC-DC power converter with an algorithm that optimizes the power curve of the PV module; an internal battery for energy storage, called the Energy Storage System (ESS); and a DC charger that controls the input current to the EB. Figure2 depicts a general scheme for a standalone PVCS with a battery connected from an EV. Some standalone PVCSs could work without an ESS; however, this is impractical as the incoming radiation fluctuates; and thus, the generation is insufficient for charging the EB in short times [38]. Our study considers standalone PVCSs with an ESS.Figure 2
General scheme of a PVCS. Source: [38].For the PVCSs, the input power may vary depending on environmental factors such as temperature and geographical position of the photovoltaic modules. Considering this variation of the input energy, we present three possible scenarios for the PVCS operation. In the first scenario, we have a PVCS without any EB connected to it. In this scenario, the energy generated by the PV module flows directly to the ESS until it is fully charged. With standalone PVCSs, once the ESS capacity is full, the additional generated energy is lost. For the remaining two scenarios, we will consider an energy lower bound for the ESS, to avoid degradation due to the overdischarging. With this in mind, for the second scenario, the energy within the ESS is above the lower bound, and an EB is connected to the PVCS. In this case, the energy for charging the EB comes from the PV modules and simultaneously from the ESS, in order to ensure the charging power selected to charge the EB. It is important to mention that, in this scenario, the charging time is determined by the nonlinear charging function. Finally, in the last scenario, the energy within the ESS is below the lower bound, and an EB is connected to the PVCS. For this scenario, we consider that there is a control system that ensures that the energy for charging the EB comes only from the PV modules. In this case, the charging time will only depend on the PV generation.For the simplification of the modeling, we consider that PV modules are always in a horizontal position (the PVCS does not have a tracking system). Additionally, solar radiation is discretized in a finite number of time intervals, and the panel efficiency is considered constant for the whole day. Figure3 shows the average solar radiation in W/m2 per hour, in the month of June, for the municipality of Sincelejo, in Colombia.Figure 3
Solar radiation curve for the municipality of Sincelejo, Colombia. Source: data from IDEAM [39].


## 4. Problem Description and Formulation

### 4.1. Problem Description

LetN=0,…,r,…,n be a set of nodes that compose a round trip on a river (i.e., the EB should visit these nodes). The nodes 0 and n represent the same node, where 0 is the starting point and n is the final destination. The node r represents the return node. The subsets No=1,…,r−1 and Nr=r+1,…,n−1 represent the nodes to visit on the outbound and return trip. Considering that we assume that the EB visits the same nodes on the on the outbound and return trip, for this problem, the nodes in Nr are a copy of the nodes in No. For instance, the node n−1 is a copy of node 1, and the node r+1 is a copy of r−1. Therefore, a PVCS located at node i∈No could serve to charge the EB also on the return trip. It is important to note that if node j∈Nr is a copy of node i∈No, their service times and TWs could be different. Each node i∈N∖0,n has a service time oi (e.g., load and unload times, lunch breaks, and sleep times). Each node i∈N∖0 has a TW si,li, where si and li are the earliest and latest possible arrival times. The travel time between nodes i−1 and i∈N∖0 is defined as ti. The EB must return to the final destination n before a time limit Tmax. Let M=Qmin,…,Qmax be a set of battery capacities for the EB. Each battery capacity m∈M has associated an acquisition cost cm (in $). It is assumed that the EB leaves the origin with a fully charged battery. The energy consumption between nodes i−1 and i∈N∖0 using a EB with a battery capacity of m∈M is defined as eim. Let H be a set of different types of PVCSs’ technologies. Each technology is related with the charging rate (e.g., slow, moderate, and fast). The cost of installing a PVCS at the node i∈N∖0,n of type h∈H is defined as fih. For modeling the nonlinear charging behavior, we use a piecewise linear approximation with a set of break points B=0,1,…,b. Each segment of the piecewise linear between breakpoints k−1 and k∈B∖0 has a slope ρkh according to a charging rate h∈H, and each segment is bounded by the battery levels αk−1,mh and αkmh for a battery capacity of m∈M. For modeling the PV generation, let T=1,2,…,Tmax be the set of time intervals and Δ be the length of each interval. Each time interval j∈T has an upper bound uj (in hours) and a power generation gj (in kW). The latter is considered constant during each time interval. Finally, the PVCSs have an ESS capacity of size a (in kWh) and a lower energy limit v (in kWh) for preventing the overdischarge of the ESS.In the ERCSLP, the objective is to find the EB’s battery capacity and the number of located PVCSs such that the total cost of the system is minimized. This problem considers TWs and operation times for the EB at each node, the nonlinear behavior of the charging function, the variation of the energy generation of the PVCSs, the autonomy of the EB (defined by the battery capacity), and the time limit of the route.


### 4.2. Illustrative Example

In this section, we present a numerical example intending to give the reader a better perspective of our problem. Figure4 shows a numerical example illustrating the ERCSLP. In Figure 4(a), we depict a round trip route where the EB departs from an origin node (0), which will also be the final destination of the route; two intermediate nodes; and a return node where the EB arrives and, afterwards, starts the return trip. This means that the EB sequentially visits nodes 1, 2, and 3. Once it completes the outbound trip, it starts with the return trip visiting, once more, nodes 2 and 1 finally arriving to node 0.Figure 4
Fixed route for an EB in a river transport operation. (a) Instance information. (b) Power generated in terms of solar radiation. (c) Piecewise linear approximation. (d) Instance solution.
(a)(b)(c)(d)Each pair of nodes has associated a travel time (in hours) and energy consumption (in kWh) for both the outbound and the return trips. Considering that the weight of the EB battery affects the energy consumption, for each pair of nodes, we have an energy consumption value associated to each possible battery size. For this example, the battery capacity is discretized in modules of30kWh each, i.e., 30,60,…,150kWh, and the cost of each module will be 9,000$/module300$/kWh. In Figure 4(a), above the arrows, we show the energy consumption for each possible battery capacity, and at the bottom, we show the travel time. Each node in the route has the costs for opening different types of PVCSs (slow or fast); its time window; and its service time. Each PVCS that is candidate to be opened has a battery capacity of 60kWh and an energy lower bound 20% of its capacity. For this example, the maximum trip duration (Tmax) is 69h. Assuming that the trip starts at sunrise, Figure 4(b) shows the solar value radiation per hour until Tmax. Figure 4(c) presents the piecewise linear charging function for each of the battery capacities, as well as the two considered charging technologies (slow and fast).Figure4(d) presents a feasible solution for this instance. In this solution, the selected EB battery capacity is 60kWh. Each node has associated a table that shows both time and energy levels when the EB arrives to and departs from the node (for both the outbound and the return trips). One PVCS of slow charging technology is opened at node 2. For this node in particular, the table also shows the charged energy by the EB and the energy level of the ESS when the EB arrives to and departs from the PVCS. The objective function for this feasible solution is $43,373; this result comprises the PVCS’ cost and the EB battery’s cost. For the sake of comparison, another feasible solution considers an EB with a battery capacity of 150kWh, and no PVCS is opened. However, in this solution, the objective function is $45,000, which is higher than the solution with a lower EB battery capacity and a PVCS opened.Considering that the PVCS is opened at node 2, we will only detail how the system behaves at this node. As we mentioned above, when the EB departs from the origin, the energy level of the ESS corresponds to the lower bound, for example,12kWh. For this example, the EB departs from the origin at 2.1h, and it remains 3.2h at node 1. Then, the EB arrives at node 2 after 11h2.1+3.6+3.2+2.1, with an energy level of 33.5kWh. For this 11h, the PV modules have generated 29.3kWh. This energy has flowed into the ESS reaching a level of 41.3kWh. The charging power of the PVCS is 11kW. Considering that the EB needs 18.5kWh, the charging time will be 1.8h using this charging power. For computing the charging time, we use the piecewise linear charging function with a charging power of 11kW and the selected battery of 60kWh. During the charging time, the PV modules generated 0.3kWh. The battery level of the ESS when the EB finishes its charging process is 23.1kWh. This is computed as the ESS battery level when the EB arrives at PVCS 41.3kWh plus the generated energy during the charging time 0.3kWh minus the charged energy by the EB 18.5kWh. In the return trip, when the EB arrives at node 2, after visiting node 3, its energy level is 0kWh. At this moment, the available energy at the ESS is 52.7kWh (it has generated an additional 29.6kWh since the departure). At this point, the EB requires to charge 42.8kWh, for completing the route. The charging process begins at 41h, which is night time; Figure 4(b). Under this condition, the EB only takes energy from the ESS until it reaches the lower bound. At that moment, the EB has not completed its required charge (missing 2.1kWh). This first charging process ends at 44.8h (still night time). Therefore, the EB needs to wait that the PV modules resume generation of the remaining 2.1kWh. The PV modules start the generation at 48h, and the 2.1kWh are generated at 50.4h. At this moment, the EB is ready to depart and complete the route.


### 4.3. Mixed-Integer Linear Programming Formulation

We now provide a MILP formulation for the ERCSLP. The MILP formulation uses the following decision variables:(i)
wm: 1 if the EB battery capacity takes a value of m∈M, and 0 otherwise(ii)
yih: 1 if a PVCS of type h∈H is opened at node i∈N∖0,n, and 0 otherwise(iii)
xi: battery level when the EB arrives at node i∈N∖0(iv)
τi: arrival time of the EB at node i∈N∖0(v)
Ωikh: charged energy when the EB charges at the PVCS of type h∈H located at node i∈N∖0,n between breakpoints k−1 and k∈B∖0(vi)
μikh: battery level when the EB charges at the PVCS of type h∈H located at node i∈N∖0,n between breakpoints k−1 and k∈B∖0(vii)
θikmh: 1 if the EB charges on the segment between breakpoint k−1 and k∈B∖0 using a battery capacity of m∈M at the node i∈N∖0,n, and 0 otherwise(viii)
ϕi: charging time at the PVCS located at node i∈N∖0,n(ix)
δi: stay time at the PVCS located at node i∈N∖0,n(x)
γij: Fraction of the time interval j∈T that is being used to generate energy before the arrival of EB at the PVCS located at node i∈N∖0,n(xi)
pij: fraction of the time interval j∈T that is being used to generate energy during the charging process of EB at the PVCS located at node i∈N∖0,n(xii)
zij: 1 if the PVCS located at node i∈N∖0,n is generating energy before the arrival of the EB at the time interval j∈T(xiii)
dij: 1 if the PVCS located at node i∈N∖0,n is generating energy during the charging process of the EB at the time interval j∈T(xiv)
λi: total energy generated by the PVCS located at node i∈N∖0,n before the arrival time of the EB(xv)
ψi: available energy at the PVCS located at node i∈N∖0,n before the arrival time of the EB(xvi)
φi: energy generated while the EB is charging at the PVCS located at node i∈N∖0,nThe MILP formulation of the ERCSLP is as follows:(1)min∑m∈Mcmwm+∑i∈N∖0,n∑h∈Hfihyih.The objective function (1) seeks to minimize the total cost of the electric mobility system, which is composed of the PVCSs’ installation cost and the cost of the EB battery.Subject to(2)∑m∈Mwm=1.Constraint (2) ensures that exactly one battery capacity is selected for the EB.(3)x1=∑m∈Mmwm−∑m∈Me1mwm,(4)xi=xi−1+∑k∈B∖0∑h∈HΩi−1,kh−∑m∈Meimwm,∀i∈N∖0,1,(5)xi+∑k∈B∖0∑h∈HΩikh≤∑m∈Mmwm,∀i∈N∖0,n.Constraints (3) and (4) track the energy level when the EB arrives to node i∈N∖0 considering the charged energy at the located PVCSs. Constraint (5) ensures that the energy level of the EB does not exceed the maximum selected battery capacity.(6)μi1h=xi+Ωi1h,∀i∈N∖0,n,∀h∈H,(7)μikh=μi,k−1,h+Ωikh,∀i∈N∖0,n,∀k∈B∖0,1,∀h∈H,(8)μikh≥∑m∈Mαk−1,mhθikmh,∀i∈N∖0,n,∀k∈B∖0,1,∀h∈H,(9)μikh≤∑m∈Mαkmhθikmh+1−∑m∈MθikmhQmax,∀i∈N∖0,n,∀k∈B∖0,∀h∈H,(10)μikh−Ωikh≥∑m∈Mαk−1,mhθikmh,∀i∈N∖0,n,∀k∈B∖0,∀h∈H,(11)ϕi≥∑k∈B∖0∑h∈HΩikhρkh,∀i∈N∖0,n.Constraints (6) and (7) define the battery level of the EB when it finishes the charging process at the PVCS located at node i∈N∖0,n on the segment between breakpoints k−1 and k∈B∖0. Constraints (8)–(10) ensure that when the EB charges on a segment of the piecewise linear function, the energy level is between the corresponding energy levels αk−1,mh and αkmh. Constraint (11) determines that the charging time at the PVCS located at node i∈N∖0,n.(12)Ωikh≤∑m∈MθikmhQmax,∀i∈N∖0,n,∀k∈B∖0,∀h∈H,(13)θikmh≤wm,∀i∈N∖0,n,∀k∈B∖0,∀m∈M,∀h∈H,(14)θikmh≤yih+yn−i,h,∀i∈N∖0,r,n,∀k∈B∖0,∀m∈M,∀h∈H,(15)θr,kmh≤yr,h,∀k∈B∖0,∀m∈M,∀h∈H,(16)∑h∈Hyih+yn−i,h≤1,∀i∈No,(17)∑h∈Hyr,h≤1.Constraint (12) ensures that charging is allowed only on the selected segments of the piecewise linear function. Constraint (13) determines that charging is possible only for the battery capacity selected. Constraints (14) and (15) ensure that charging is allowed only at the located PVCSs. Constraints (16) and (17) ensure that, at most, one type h∈H of PVCS is located at each node.(18)δi≥oi,∀i∈N∖0,n,(19)δi≥ϕi,∀i∈N∖0,n,(20)τ1≥t1,(21)τi=τi−1+δi−1+ti,∀i∈N∖0,1,(22)si≤τi≤li,∀i∈N∖0.Constraints (18) and (19) define the stay time at each node, which is the maximum value between the operation time and the charging time. Constraints (20) and (21) determine the arrival time at each node. Constraints (22) ensure that the EB has to meet the TW of each node.(23)zij≤∑h∈Hyih+yn−i,h,∀i∈N∖0,r,n,∀j∈T,(24)zr,j≤∑h∈Hyr,h,∀j∈T,(25)γij≤Δzij,∀i∈N∖0,n,∀j∈T,(26)γij≤τi−uj−1+1−zijuj,∀i∈N∖0,n,∀j∈T∖1,(27)∑j∈Tγij+1−∑h∈Hyih+yn−i,hTmax≥τi,∀i∈N∖0,r,n,(28)∑j∈Tγr,j+1−∑h∈Hyr,hTmax≥τr,(29)λi=gjγij,∀i∈N∖0,n,(30)dij≤∑h∈Hyih+yn−i,h,∀i∈N∖0,r,n,∀j∈T,(31)dr,j≤∑h∈Hyr,h,∀j∈T,(32)pij≤Δdij,∀i∈N∖0,n,∀j∈T,(33)pij≤uj−τi+1−dijTmax,∀i∈N∖0,n,∀j∈T,(34)pij≤τi+ϕi−uj+Δ+1−dijTmax,∀i∈N∖0,n,∀j∈T,(35)∑j∈Tpij=ϕi,∀i∈N∖0,n,(36)φi=∑j∈Tgjpij,∀i∈N∖0,n,(37)ψi≤a−v,∀i∈N∖0,n,(38)ψi≤λi,∀i∈N:0<i≤r,(39)ψi≤λi−λn−i+ψn−i−∑k∈B∖0∑h∈HΩn−i,kh,∀i∈N:i>r,(40)∑k∈B∖0Ωikh≤ψi+φi,∀i∈N∖0,n.Constraints (23)–(25) ensure that there is generation only in the nodes where a PVCS was located. Constraints (26)–(28) control the time of generation to be equal to the arrival time of the EB to the PVCS. Constraint (29) calculates the energy generation until the arrival of the EB to the PVCS. This considers that the energy generated starts from the lower energy level. Constraints (30)–(32) control that the time of generation during the charging process is only for the nodes with a located PVCS. Constraints (33)–(35) estimate the time of generation to fulfill the energy requirements of the EB at its arrival. Constraint (36) calculates the energy generation during the time that the EB is charging. Constraint (37) limits the available energy for charging the EB at each PVCS to the maximum capacity, considering the lower energy limit. Constraint (38) ensures that the available energy at the PVCS in the outbound trip is, at most, the generated energy until the arrival. Constraint (39) limits the available energy at the PVCS, in the return trip, considering the energy retrieved by the EB in the outbound trip. Constraint (40) ensures that the energy taken by the EB at each PVCS has been already generated at the end of the charging process.(41)wm∈0,1,∀m∈M,(42)xi≥0,τi≥0,∀i∈N∖0,(43)yih∈0,1,∀i∈N∖0,n,∀h∈H,(44)Ωikh≥0,μikh≥0,∀i∈N∖0,n,∀k∈B∖0,∀h∈H,(45)θikmh∈0,1,∀i∈N∖n,∀b∈B∖0,∀m∈M,∀h∈H,(46)ϕi≥0,δi≥0,λi≥0,φi≥0,ψi≥0,∀i∈N∖0,n,(47)dij∈0,1,zij∈0,1,∀i∈N∖0,n,∀j∈T,(48)γij≥0,pij≥0,∀i∈N∖0,n,∀j∈T.Finally, constraints (41)–(48) define the domain of the decision variables.


## 4.1. Problem Description

LetN=0,…,r,…,n be a set of nodes that compose a round trip on a river (i.e., the EB should visit these nodes). The nodes 0 and n represent the same node, where 0 is the starting point and n is the final destination. The node r represents the return node. The subsets No=1,…,r−1 and Nr=r+1,…,n−1 represent the nodes to visit on the outbound and return trip. Considering that we assume that the EB visits the same nodes on the on the outbound and return trip, for this problem, the nodes in Nr are a copy of the nodes in No. For instance, the node n−1 is a copy of node 1, and the node r+1 is a copy of r−1. Therefore, a PVCS located at node i∈No could serve to charge the EB also on the return trip. It is important to note that if node j∈Nr is a copy of node i∈No, their service times and TWs could be different. Each node i∈N∖0,n has a service time oi (e.g., load and unload times, lunch breaks, and sleep times). Each node i∈N∖0 has a TW si,li, where si and li are the earliest and latest possible arrival times. The travel time between nodes i−1 and i∈N∖0 is defined as ti. The EB must return to the final destination n before a time limit Tmax. Let M=Qmin,…,Qmax be a set of battery capacities for the EB. Each battery capacity m∈M has associated an acquisition cost cm (in $). It is assumed that the EB leaves the origin with a fully charged battery. The energy consumption between nodes i−1 and i∈N∖0 using a EB with a battery capacity of m∈M is defined as eim. Let H be a set of different types of PVCSs’ technologies. Each technology is related with the charging rate (e.g., slow, moderate, and fast). The cost of installing a PVCS at the node i∈N∖0,n of type h∈H is defined as fih. For modeling the nonlinear charging behavior, we use a piecewise linear approximation with a set of break points B=0,1,…,b. Each segment of the piecewise linear between breakpoints k−1 and k∈B∖0 has a slope ρkh according to a charging rate h∈H, and each segment is bounded by the battery levels αk−1,mh and αkmh for a battery capacity of m∈M. For modeling the PV generation, let T=1,2,…,Tmax be the set of time intervals and Δ be the length of each interval. Each time interval j∈T has an upper bound uj (in hours) and a power generation gj (in kW). The latter is considered constant during each time interval. Finally, the PVCSs have an ESS capacity of size a (in kWh) and a lower energy limit v (in kWh) for preventing the overdischarge of the ESS.In the ERCSLP, the objective is to find the EB’s battery capacity and the number of located PVCSs such that the total cost of the system is minimized. This problem considers TWs and operation times for the EB at each node, the nonlinear behavior of the charging function, the variation of the energy generation of the PVCSs, the autonomy of the EB (defined by the battery capacity), and the time limit of the route.


## 4.2. Illustrative Example

In this section, we present a numerical example intending to give the reader a better perspective of our problem. Figure4 shows a numerical example illustrating the ERCSLP. In Figure 4(a), we depict a round trip route where the EB departs from an origin node (0), which will also be the final destination of the route; two intermediate nodes; and a return node where the EB arrives and, afterwards, starts the return trip. This means that the EB sequentially visits nodes 1, 2, and 3. Once it completes the outbound trip, it starts with the return trip visiting, once more, nodes 2 and 1 finally arriving to node 0.Figure 4
Fixed route for an EB in a river transport operation. (a) Instance information. (b) Power generated in terms of solar radiation. (c) Piecewise linear approximation. (d) Instance solution.
(a)(b)(c)(d)Each pair of nodes has associated a travel time (in hours) and energy consumption (in kWh) for both the outbound and the return trips. Considering that the weight of the EB battery affects the energy consumption, for each pair of nodes, we have an energy consumption value associated to each possible battery size. For this example, the battery capacity is discretized in modules of30kWh each, i.e., 30,60,…,150kWh, and the cost of each module will be 9,000$/module300$/kWh. In Figure 4(a), above the arrows, we show the energy consumption for each possible battery capacity, and at the bottom, we show the travel time. Each node in the route has the costs for opening different types of PVCSs (slow or fast); its time window; and its service time. Each PVCS that is candidate to be opened has a battery capacity of 60kWh and an energy lower bound 20% of its capacity. For this example, the maximum trip duration (Tmax) is 69h. Assuming that the trip starts at sunrise, Figure 4(b) shows the solar value radiation per hour until Tmax. Figure 4(c) presents the piecewise linear charging function for each of the battery capacities, as well as the two considered charging technologies (slow and fast).Figure4(d) presents a feasible solution for this instance. In this solution, the selected EB battery capacity is 60kWh. Each node has associated a table that shows both time and energy levels when the EB arrives to and departs from the node (for both the outbound and the return trips). One PVCS of slow charging technology is opened at node 2. For this node in particular, the table also shows the charged energy by the EB and the energy level of the ESS when the EB arrives to and departs from the PVCS. The objective function for this feasible solution is $43,373; this result comprises the PVCS’ cost and the EB battery’s cost. For the sake of comparison, another feasible solution considers an EB with a battery capacity of 150kWh, and no PVCS is opened. However, in this solution, the objective function is $45,000, which is higher than the solution with a lower EB battery capacity and a PVCS opened.Considering that the PVCS is opened at node 2, we will only detail how the system behaves at this node. As we mentioned above, when the EB departs from the origin, the energy level of the ESS corresponds to the lower bound, for example,12kWh. For this example, the EB departs from the origin at 2.1h, and it remains 3.2h at node 1. Then, the EB arrives at node 2 after 11h2.1+3.6+3.2+2.1, with an energy level of 33.5kWh. For this 11h, the PV modules have generated 29.3kWh. This energy has flowed into the ESS reaching a level of 41.3kWh. The charging power of the PVCS is 11kW. Considering that the EB needs 18.5kWh, the charging time will be 1.8h using this charging power. For computing the charging time, we use the piecewise linear charging function with a charging power of 11kW and the selected battery of 60kWh. During the charging time, the PV modules generated 0.3kWh. The battery level of the ESS when the EB finishes its charging process is 23.1kWh. This is computed as the ESS battery level when the EB arrives at PVCS 41.3kWh plus the generated energy during the charging time 0.3kWh minus the charged energy by the EB 18.5kWh. In the return trip, when the EB arrives at node 2, after visiting node 3, its energy level is 0kWh. At this moment, the available energy at the ESS is 52.7kWh (it has generated an additional 29.6kWh since the departure). At this point, the EB requires to charge 42.8kWh, for completing the route. The charging process begins at 41h, which is night time; Figure 4(b). Under this condition, the EB only takes energy from the ESS until it reaches the lower bound. At that moment, the EB has not completed its required charge (missing 2.1kWh). This first charging process ends at 44.8h (still night time). Therefore, the EB needs to wait that the PV modules resume generation of the remaining 2.1kWh. The PV modules start the generation at 48h, and the 2.1kWh are generated at 50.4h. At this moment, the EB is ready to depart and complete the route.


## 4.3. Mixed-Integer Linear Programming Formulation

We now provide a MILP formulation for the ERCSLP. The MILP formulation uses the following decision variables:(i)
wm: 1 if the EB battery capacity takes a value of m∈M, and 0 otherwise(ii)
yih: 1 if a PVCS of type h∈H is opened at node i∈N∖0,n, and 0 otherwise(iii)
xi: battery level when the EB arrives at node i∈N∖0(iv)
τi: arrival time of the EB at node i∈N∖0(v)
Ωikh: charged energy when the EB charges at the PVCS of type h∈H located at node i∈N∖0,n between breakpoints k−1 and k∈B∖0(vi)
μikh: battery level when the EB charges at the PVCS of type h∈H located at node i∈N∖0,n between breakpoints k−1 and k∈B∖0(vii)
θikmh: 1 if the EB charges on the segment between breakpoint k−1 and k∈B∖0 using a battery capacity of m∈M at the node i∈N∖0,n, and 0 otherwise(viii)
ϕi: charging time at the PVCS located at node i∈N∖0,n(ix)
δi: stay time at the PVCS located at node i∈N∖0,n(x)
γij: Fraction of the time interval j∈T that is being used to generate energy before the arrival of EB at the PVCS located at node i∈N∖0,n(xi)
pij: fraction of the time interval j∈T that is being used to generate energy during the charging process of EB at the PVCS located at node i∈N∖0,n(xii)
zij: 1 if the PVCS located at node i∈N∖0,n is generating energy before the arrival of the EB at the time interval j∈T(xiii)
dij: 1 if the PVCS located at node i∈N∖0,n is generating energy during the charging process of the EB at the time interval j∈T(xiv)
λi: total energy generated by the PVCS located at node i∈N∖0,n before the arrival time of the EB(xv)
ψi: available energy at the PVCS located at node i∈N∖0,n before the arrival time of the EB(xvi)
φi: energy generated while the EB is charging at the PVCS located at node i∈N∖0,nThe MILP formulation of the ERCSLP is as follows:(1)min∑m∈Mcmwm+∑i∈N∖0,n∑h∈Hfihyih.The objective function (1) seeks to minimize the total cost of the electric mobility system, which is composed of the PVCSs’ installation cost and the cost of the EB battery.Subject to(2)∑m∈Mwm=1.Constraint (2) ensures that exactly one battery capacity is selected for the EB.(3)x1=∑m∈Mmwm−∑m∈Me1mwm,(4)xi=xi−1+∑k∈B∖0∑h∈HΩi−1,kh−∑m∈Meimwm,∀i∈N∖0,1,(5)xi+∑k∈B∖0∑h∈HΩikh≤∑m∈Mmwm,∀i∈N∖0,n.Constraints (3) and (4) track the energy level when the EB arrives to node i∈N∖0 considering the charged energy at the located PVCSs. Constraint (5) ensures that the energy level of the EB does not exceed the maximum selected battery capacity.(6)μi1h=xi+Ωi1h,∀i∈N∖0,n,∀h∈H,(7)μikh=μi,k−1,h+Ωikh,∀i∈N∖0,n,∀k∈B∖0,1,∀h∈H,(8)μikh≥∑m∈Mαk−1,mhθikmh,∀i∈N∖0,n,∀k∈B∖0,1,∀h∈H,(9)μikh≤∑m∈Mαkmhθikmh+1−∑m∈MθikmhQmax,∀i∈N∖0,n,∀k∈B∖0,∀h∈H,(10)μikh−Ωikh≥∑m∈Mαk−1,mhθikmh,∀i∈N∖0,n,∀k∈B∖0,∀h∈H,(11)ϕi≥∑k∈B∖0∑h∈HΩikhρkh,∀i∈N∖0,n.Constraints (6) and (7) define the battery level of the EB when it finishes the charging process at the PVCS located at node i∈N∖0,n on the segment between breakpoints k−1 and k∈B∖0. Constraints (8)–(10) ensure that when the EB charges on a segment of the piecewise linear function, the energy level is between the corresponding energy levels αk−1,mh and αkmh. Constraint (11) determines that the charging time at the PVCS located at node i∈N∖0,n.(12)Ωikh≤∑m∈MθikmhQmax,∀i∈N∖0,n,∀k∈B∖0,∀h∈H,(13)θikmh≤wm,∀i∈N∖0,n,∀k∈B∖0,∀m∈M,∀h∈H,(14)θikmh≤yih+yn−i,h,∀i∈N∖0,r,n,∀k∈B∖0,∀m∈M,∀h∈H,(15)θr,kmh≤yr,h,∀k∈B∖0,∀m∈M,∀h∈H,(16)∑h∈Hyih+yn−i,h≤1,∀i∈No,(17)∑h∈Hyr,h≤1.Constraint (12) ensures that charging is allowed only on the selected segments of the piecewise linear function. Constraint (13) determines that charging is possible only for the battery capacity selected. Constraints (14) and (15) ensure that charging is allowed only at the located PVCSs. Constraints (16) and (17) ensure that, at most, one type h∈H of PVCS is located at each node.(18)δi≥oi,∀i∈N∖0,n,(19)δi≥ϕi,∀i∈N∖0,n,(20)τ1≥t1,(21)τi=τi−1+δi−1+ti,∀i∈N∖0,1,(22)si≤τi≤li,∀i∈N∖0.Constraints (18) and (19) define the stay time at each node, which is the maximum value between the operation time and the charging time. Constraints (20) and (21) determine the arrival time at each node. Constraints (22) ensure that the EB has to meet the TW of each node.(23)zij≤∑h∈Hyih+yn−i,h,∀i∈N∖0,r,n,∀j∈T,(24)zr,j≤∑h∈Hyr,h,∀j∈T,(25)γij≤Δzij,∀i∈N∖0,n,∀j∈T,(26)γij≤τi−uj−1+1−zijuj,∀i∈N∖0,n,∀j∈T∖1,(27)∑j∈Tγij+1−∑h∈Hyih+yn−i,hTmax≥τi,∀i∈N∖0,r,n,(28)∑j∈Tγr,j+1−∑h∈Hyr,hTmax≥τr,(29)λi=gjγij,∀i∈N∖0,n,(30)dij≤∑h∈Hyih+yn−i,h,∀i∈N∖0,r,n,∀j∈T,(31)dr,j≤∑h∈Hyr,h,∀j∈T,(32)pij≤Δdij,∀i∈N∖0,n,∀j∈T,(33)pij≤uj−τi+1−dijTmax,∀i∈N∖0,n,∀j∈T,(34)pij≤τi+ϕi−uj+Δ+1−dijTmax,∀i∈N∖0,n,∀j∈T,(35)∑j∈Tpij=ϕi,∀i∈N∖0,n,(36)φi=∑j∈Tgjpij,∀i∈N∖0,n,(37)ψi≤a−v,∀i∈N∖0,n,(38)ψi≤λi,∀i∈N:0<i≤r,(39)ψi≤λi−λn−i+ψn−i−∑k∈B∖0∑h∈HΩn−i,kh,∀i∈N:i>r,(40)∑k∈B∖0Ωikh≤ψi+φi,∀i∈N∖0,n.Constraints (23)–(25) ensure that there is generation only in the nodes where a PVCS was located. Constraints (26)–(28) control the time of generation to be equal to the arrival time of the EB to the PVCS. Constraint (29) calculates the energy generation until the arrival of the EB to the PVCS. This considers that the energy generated starts from the lower energy level. Constraints (30)–(32) control that the time of generation during the charging process is only for the nodes with a located PVCS. Constraints (33)–(35) estimate the time of generation to fulfill the energy requirements of the EB at its arrival. Constraint (36) calculates the energy generation during the time that the EB is charging. Constraint (37) limits the available energy for charging the EB at each PVCS to the maximum capacity, considering the lower energy limit. Constraint (38) ensures that the available energy at the PVCS in the outbound trip is, at most, the generated energy until the arrival. Constraint (39) limits the available energy at the PVCS, in the return trip, considering the energy retrieved by the EB in the outbound trip. Constraint (40) ensures that the energy taken by the EB at each PVCS has been already generated at the end of the charging process.(41)wm∈0,1,∀m∈M,(42)xi≥0,τi≥0,∀i∈N∖0,(43)yih∈0,1,∀i∈N∖0,n,∀h∈H,(44)Ωikh≥0,μikh≥0,∀i∈N∖0,n,∀k∈B∖0,∀h∈H,(45)θikmh∈0,1,∀i∈N∖n,∀b∈B∖0,∀m∈M,∀h∈H,(46)ϕi≥0,δi≥0,λi≥0,φi≥0,ψi≥0,∀i∈N∖0,n,(47)dij∈0,1,zij∈0,1,∀i∈N∖0,n,∀j∈T,(48)γij≥0,pij≥0,∀i∈N∖0,n,∀j∈T.Finally, constraints (41)–(48) define the domain of the decision variables.


## 5. Computational Experiments

Aiming to test the performance of our MILP formulation for the ERCSLP, we used a set of instances based on real river transport operations, from different regions in Colombia. In these experiments, we initially focus on the capability of the MILP formulation to solve different types of instances in reasonable CPU times. Additionally, we show the features of the solution in terms of the number of opened PVCSs and the selected battery size for the EB. Also, for evaluating the economical feasibility of the solutions, we compare their objective functions, which correspond to the investment costs, against the total cost of a 10-year operation using an ICB. In some cases, the investment could have a return. Furthermore, we perform a sensitivity analysis to test the robustness of the solutions when variations occur in key features, such as the EB speed, the average radiation, and the discretization of the battery capacity.


### 5.1. Test Instances for the ERCSLP

Within the frame of the research project “Energética 2030,” a river transport operation with EBs is planned to be implemented on the Magdalena River, in Colombia [40]. This project aims to start with a pilot model of a river transport operation using EBs including intermediate PVCSs. It is expected that this pilot serves as the starting point for launching future projects of electric mobility on other rivers in Colombia. Therefore, for the construction of the test instances, we consider different rivers that could have a river transport operation with EBs. For the test instances, we consider 6 of the most important rivers in Colombia: Amazonas, Caquetá, Guaviare, Magdalena, Orinoco, and Putumayo. For the Caquetá River, we build 3 instances; for the Magdalena River, we build 4 instances; for Guaviare River, we build 2 instances; and for the remaining rivers, we build one instance each. Then, the total number of instances is 12. For each instance, we locate the set of nodes in places where there are towns or settlements. This is performed aiming to supply the excess of energy to the community and reducing the impact on the river (caused for the installation of the PVCSs). It is important to remark that most of the nodes are disconnected from the national grid because they are located in isolated areas.For building the instances, we make the following considerations. For the discretization of the EB battery capacity, we use a delta of1kWh. For the energy consumption estimation, we use Crouch’s formula presented by Minami and Yamachika [33]. This estimation was made for each EB possible battery capacity. We selected Crouch’s formula because it is easy to implement and has the potential to produce reliable results. However, any consumption model could be used given that the energy consumption is a parameter. For estimating the weight associated to each EB possible battery capacity, we consider an energy density of 0.15kWh/kg for a Li-ion battery, following the values presented by Epec [41]. The maximum battery capacity is bounded to a maximum weight value of a ton, which is an adequate value for the type of EB considered. For computing the travel time between two nodes, we use a constant speed of 10km/h, which agrees with the data presented by Jaimurzina et al. [7]. It is important to remark that both the energy consumption and travel time are also affected by the direction of the river flow (i.e., with or against the river flow). For generating the TWs and service times, we consider the operation conditions of the route, for example, eating, sleeping, loading, and unloading times. For estimating the breakpoints of the piecewise linear charging function, we use the data presented by Martin et al. [42]. For determining the battery cost, we use the data of a battery pack manufacturing costs per kWh by Bloomberg New Energy Finance [43] and Battery University [44]. From this information, we set the cost as 300USD/kWh. For establishing the standalone PVCS cost, we took the information from a PVCS commercial supplier [45]. We added an additional factor to the PVCS cost for including the installation costs. For estimating the PV generation, we use the hourly averages presented by IDEAM [39]. For the total generation, we assumed a panel efficiency of 14% since this value is in the rank of commercial PV modules [46]. The lower energy level for the ESS is set at 20% of the EB battery capacity following the recommendation by Venu et al. [47].Moreover, for evaluating how some parameters affect the structure of the solution (i.e., the number of charging stations and size of the EB’s battery) and the objective function, we varied some parameters in each instance. For each of the instances, we varied the radiation, the average speed, and the delta of the discretization. These variations were performed one parameter at a time. For the radiation, we set the values to the historical maximum and minimum. For the average speed, we evaluated the set of values6,7,8,9,11,12,13,14, and 15km/h. Finally, for the delta of the discretization, we evaluated the set of values 0.5,0.75,1.25, and 1.5kWh. After applying this variation, we had 15 additional instances for each of the original instances, for a total of 180 instances.


### 5.2. Experimental Environment

For solving the MILP formulation of the ERCSLP, we used the Gurobi Optimizer (version 8.1.0) with a CPU time limit of10,800s and a relative MIP optimality gap of 1e−4. When the optimality gap (the optimality gap is calculated as Gap=Dual Solution−Primal Solution/Dual Solution) is lesser than the relative MIP optimality gap, the solver finishes and reports the best primal bound as the optimal solution. We ran all the experiments on a computer with an Intel Xeon E5-2670 processor (with 16 cores at 2.6 GHz) and 64 GB of RAM running on Linux CentOS 6.6. The feasibility of the solutions reported by the Gurobi Optimizer was validated in an external application implemented on Java (jdk-1.8.0_162-b12).


### 5.3. Results of the Computational Experiments

Initially, we solve the 12 original instances, and Table1 presents the results in detail. In this table, the results are sorted based on the length of each route in a river with the shortest on top. Hereafter, we explain the information contained in each column of Table 1. Column 1 lists the instances names. Column 2 shows the total length (in km) of each route. Column 3 presents the number of nodes that compose each route. Column 4 shows the number of PVCSs that were open for each instance. Column 5 presents the percentage of nodes that were opened as PVCS in each route. Column 6 contains the battery capacity (in kWh) that was selected for the EB in each route. Column 7 presents the percentage of the selected battery, in terms of the maximum battery capacity. Column 8 presents the objective function value, which is the total cost. Column 9 presents the percentage for the PVCS cost, in terms of the total cost. Column 10 shows the percentage of the selected battery cost, in terms of the total cost. Finally, column 11 shows the CPU time. In Table 1, we also included the economic comparison of the operation of the route using EB against ICB. For this comparison, we computed the total cost of the route operation using ICB during 10 years. This period was selected due to the estimated lifespan of the EB battery [48]. It is important to mention that the PVCS has a lifespan of 20 years, which is twice the lifespan of the EB battery. For comparing both operations, we needed to estimate the amount of fuel to perform each of the routes. For this estimation, we converted the total energy needed for completing each of the routes (in kWh) to the amount of fuel (in gallons) using the information presented by Lee [49]. For the total cost using an ICB, we considered two elements, the fuel cost and the Social Cost of Carbon (SCC). For estimating the SCC, we first computed the CO2 emission using an average value for marine Diesel [50], and then, we transformed the CO2 emission into cost considering the value presented by Nordhaus [51]. Also, for the routes where the ICB operation cost was under the investment cost of the EB operation, we estimated what would be the minimum fuel cost for both operations to have the same cost. From this point up, the operation with the EB will be the best economic option. From these considerations, Column 12 shows the fuel cost of performing each route using an ICB. Column 13 presents the SCC. Column 14 shows the total cost of the fuel operation (fuel cost + SCC). Column 15 shows the relative difference (Relative Difference=Cost With Fuel−Objective Function/Cost With Fuel) between the investment cost of the electric mobility system and the operation cost with an ICB. Finally, Column 16 presents the required fuel price (in USD/gallon) for making both operation cost equal.Table 1
Results of the test instances for the ERCSLP.
InstanceLength (km)# nodes# CS% of nodes with CSBat. capacity (kWh)% of battery usedObj. function% of cost due to CS% of cost due to batteryCPU time (S)Cost with fuel ($)SCC ($)Total cost fuel ($)Relative difference (%)Required price for fuel3Caqueta31031417.15234.740,92561.938.1130.541,8068,73450,54019.0NAMagdalena31149111.15536.741,86160.639.4226.839,6428,28247,92412.7NAMagdalena41161119.15536.741,85960.639.4757.141,6128,69350,30516.8NAGuaviare21211119.15234.742,41463.236.852.741,6498,70150,35015.8NAMagdalena21451119.17248.048,57055.544.5361.643,3149,04952,3637.2NACaqueta21722114.814496.070,21138.561.5552.540,5658,47449,039−43.24.3Magdalena11741715.910570.058,44746.153.9291.849,97710,44160,4183.3NACaqueta12106350.06442.798,24580.519.547.982,64917,26699,9151.7NAGuaviare27315320.07147.397,18678.121.9547.157,85412,08669,941−39.04.2Amazonas4829444.47348.7126,30782.717.372.778,96616,49795,463−32.34.0Putumayo57615746.77550.0201,11788.811.21010.773,14715,28188,428−127.46.8Orinoco65313861.512180.7248,36985.414.6166.481,85617,10098,956−151.07.5Min103614.85234.740,92538.511.247.939,6428,28247,924−151.04.0Average26212.72.723.278.352.292,95966.833.2351.556,08611,71767,803−26.45.4Max65321861.514496.0248,36988.861.51010.782,64917,26699,91519.07.53NA indicates that the EB investment cost is lower than the ICB operation cost.For all the instances shown in Table1, the optimality gap of the solution was smaller than the relative MIP optimality gap, which allows us to conclude that the instances are solved optimally. The average CPU time is around 6 minutes, which is a competitive CPU time for a strategic planning problem. In average, 23.2% of the nodes are selected to open a PVCS. It is important to mention that the 4 instances with the shortest lengths (Caqueta; 3, Magdalena; 3, Magdalena; 4, and Guaviare; 2) can be operated without a PVCS if the EB has a large battery size (respecting the maximum battery size). However, in the optimal solution, one PVCS is opened in all 4 routes. These results prove that, in some cases, it is better to select a small battery size and open a PVCS. As we can expect, the longest routes (Amazonas, Orinoco, and Putumayo) require a significant number of PVCSs. From the results in Table 1, we can also see that most of the solutions required a battery size that is less than the 50% of the maximum battery size. When analyzing the distribution of the cost, in average, 66.8% of the cost is devoted to opening PVCSs. Therefore, when we consider the EB operation, the most representative cost is related to the PVCSs. When comparing the investment cost against the cost of performing the same operation with an ICB during 10 years, we notice that in 7/12 instances, the electric mobility option performs better than the ICB operation. For the remaining instances, there is a minimum of 7.5USD/gallon to ensure that the electric mobility which is the best option (see Column 17). Even though this value seems very high, considering the isolations of these areas, the fuel transportation impacts the final fuel costs. Under this premises, the fuel cost could surpass the 7.5USD/gallon threshold.


### 5.4. Sensitivity Analysis

We now provide a sensitivity analysis to show how certain modifications on some parameters can impact the objective function. To do this, we used the 192 total instances, comprised of the 12 original instances and the 180 additional instances. Firstly, we proposed a variation on the radiation level. The radiation being an input for the PVCSs, we suppose that different radiation conditions can affect the decision-making when locating the PVCSs. Secondly, considering that the average EB speed impacts the energy consumption and the travel time, we hypothesize that this variable could affect the feasibility of the solutions and the value of the objective functions. Finally, we wanted to evaluate how different deltas of the discretized battery capacity can affect the objective function value and the CPU time.


#### 5.4.1. Solar Radiation Variation

For testing the impact of the radiation on the objective function, we evaluated three different scenarios, where we consider the months with the highest, average, and lowest solar radiation per hour. In Figure5, we present the impact of these variations on the average value of the objective function from all the instances. This figure shows that the objective function increases when there is less solar radiation because more infrastructure is needed. However, there is not a significant variation after comparing all three scenarios. From a detailed analysis of the solutions, we found that, in most of the instances, the location and the number of PVCSs do not change in the three scenarios. The variation of the irradiation impacts the size of the battery. Considering that the PVCS cost is more expensive than the battery cost, there is not a significant variation on the average value of the objective function among the three scenarios. In Table 2, we show the solution of the instance Caquetá 1 for the three scenarios. Columns 2 to 7 present the nodes where a PVCS can be installed. The value of the cells represents the power of the selected PVCS; if a cell is empty, it is because no PVCS was installed at that associated node for that scenario. Furthermore, the last column shows the selected battery size. Table 2 also shows that the number and the location of the PVCSs are the same among the scenarios. On the other hand, the battery size changes between the scenarios with average and minimum radiation, and there is not a change between the scenarios with average and maximum radiation. In the latter, although both solutions have the same battery size the charging decisions are different. The EB must charge more energy for the average radiation scenario. Considering that variation of the radiation do not change the solutions significantly, it is adequate to use the average radiation level for this strategic problem.Figure 5
Variation of the objective function in terms of the solar radiation.Table 2
Caquetá 1 instance solution for the three radiation scenarios.
NodeScenario123456BatteryMaximum—22 kW11 kW—22 kW—64 kWhAverage—22 kW11 kW—22 kW—64 kWhMinimum—22 kW11 kW—22 kW—73 kWh


#### 5.4.2. Average Boat Speed Variation

As we mentioned before, the EB speed has a significant impact on the energy consumption. This impact has a direct effect on the number of PVCSs and EB battery capacity. Furthermore, the EB speed can also affect the solution feasibility by the TW at each node and theTmax for the operation. In Figure 6, we present how the variation of the EB speed affects the objective function. In this figure, we show the average value of the objective function from all the instances. It is important to note that none of the instances below 9km/h had a feasible solution. This figure also shows that there is a significant impact of the EB speed on the total infrastructure cost. It is important to notice that increasing the EB speed could improve the service in the operation; however, this increment on the speed will directly impact the infrastructure needed to support the operation.Figure 6
Changes in the objective function according to the average EB speed.


#### 5.4.3. Discretization of the Battery Capacity

As we mention in the problem description, the battery capacity is a decision variable. This variable takes discrete values from the setM that represents the possible values of the battery capacity. For building that set, we define a delta value of the battery capacity, which is interpreted as the step that increases the battery capacity in the set M. In this analysis, we want to test how the delta of the discretization affects the objective function. For the analysis, we choose battery capacity discretization values of 0.5, 0.75, 1, 1.25, and 1.5kWh being 1kWh the value for the discretization of the original instances. Figure 7 shows the average results of the objective function for the different delta values. For this variation of the delta, the MILP formulation was capable of optimally solving all the instances. When we analyze the impact of this variation on the objective function, we see that, for most of the delta variations, there is not a significant impact. The only delta variation that presented an impact on the cost was 1.5kWh. In this case, there is 0.34% increase in the objective function when compared to the results using a delta of 1kWh. From these results, we can conclude that using a 1kWh delta is an acceptable choice because we can obtain competitive solutions (in terms of the objective function) and reasonable CPU times for this kind of problems.Figure 7
Changes in the objective function according to the battery capacity discretization.


## 5.1. Test Instances for the ERCSLP

Within the frame of the research project “Energética 2030,” a river transport operation with EBs is planned to be implemented on the Magdalena River, in Colombia [40]. This project aims to start with a pilot model of a river transport operation using EBs including intermediate PVCSs. It is expected that this pilot serves as the starting point for launching future projects of electric mobility on other rivers in Colombia. Therefore, for the construction of the test instances, we consider different rivers that could have a river transport operation with EBs. For the test instances, we consider 6 of the most important rivers in Colombia: Amazonas, Caquetá, Guaviare, Magdalena, Orinoco, and Putumayo. For the Caquetá River, we build 3 instances; for the Magdalena River, we build 4 instances; for Guaviare River, we build 2 instances; and for the remaining rivers, we build one instance each. Then, the total number of instances is 12. For each instance, we locate the set of nodes in places where there are towns or settlements. This is performed aiming to supply the excess of energy to the community and reducing the impact on the river (caused for the installation of the PVCSs). It is important to remark that most of the nodes are disconnected from the national grid because they are located in isolated areas.For building the instances, we make the following considerations. For the discretization of the EB battery capacity, we use a delta of1kWh. For the energy consumption estimation, we use Crouch’s formula presented by Minami and Yamachika [33]. This estimation was made for each EB possible battery capacity. We selected Crouch’s formula because it is easy to implement and has the potential to produce reliable results. However, any consumption model could be used given that the energy consumption is a parameter. For estimating the weight associated to each EB possible battery capacity, we consider an energy density of 0.15kWh/kg for a Li-ion battery, following the values presented by Epec [41]. The maximum battery capacity is bounded to a maximum weight value of a ton, which is an adequate value for the type of EB considered. For computing the travel time between two nodes, we use a constant speed of 10km/h, which agrees with the data presented by Jaimurzina et al. [7]. It is important to remark that both the energy consumption and travel time are also affected by the direction of the river flow (i.e., with or against the river flow). For generating the TWs and service times, we consider the operation conditions of the route, for example, eating, sleeping, loading, and unloading times. For estimating the breakpoints of the piecewise linear charging function, we use the data presented by Martin et al. [42]. For determining the battery cost, we use the data of a battery pack manufacturing costs per kWh by Bloomberg New Energy Finance [43] and Battery University [44]. From this information, we set the cost as 300USD/kWh. For establishing the standalone PVCS cost, we took the information from a PVCS commercial supplier [45]. We added an additional factor to the PVCS cost for including the installation costs. For estimating the PV generation, we use the hourly averages presented by IDEAM [39]. For the total generation, we assumed a panel efficiency of 14% since this value is in the rank of commercial PV modules [46]. The lower energy level for the ESS is set at 20% of the EB battery capacity following the recommendation by Venu et al. [47].Moreover, for evaluating how some parameters affect the structure of the solution (i.e., the number of charging stations and size of the EB’s battery) and the objective function, we varied some parameters in each instance. For each of the instances, we varied the radiation, the average speed, and the delta of the discretization. These variations were performed one parameter at a time. For the radiation, we set the values to the historical maximum and minimum. For the average speed, we evaluated the set of values6,7,8,9,11,12,13,14, and 15km/h. Finally, for the delta of the discretization, we evaluated the set of values 0.5,0.75,1.25, and 1.5kWh. After applying this variation, we had 15 additional instances for each of the original instances, for a total of 180 instances.


## 5.2. Experimental Environment

For solving the MILP formulation of the ERCSLP, we used the Gurobi Optimizer (version 8.1.0) with a CPU time limit of10,800s and a relative MIP optimality gap of 1e−4. When the optimality gap (the optimality gap is calculated as Gap=Dual Solution−Primal Solution/Dual Solution) is lesser than the relative MIP optimality gap, the solver finishes and reports the best primal bound as the optimal solution. We ran all the experiments on a computer with an Intel Xeon E5-2670 processor (with 16 cores at 2.6 GHz) and 64 GB of RAM running on Linux CentOS 6.6. The feasibility of the solutions reported by the Gurobi Optimizer was validated in an external application implemented on Java (jdk-1.8.0_162-b12).


## 5.3. Results of the Computational Experiments

Initially, we solve the 12 original instances, and Table1 presents the results in detail. In this table, the results are sorted based on the length of each route in a river with the shortest on top. Hereafter, we explain the information contained in each column of Table 1. Column 1 lists the instances names. Column 2 shows the total length (in km) of each route. Column 3 presents the number of nodes that compose each route. Column 4 shows the number of PVCSs that were open for each instance. Column 5 presents the percentage of nodes that were opened as PVCS in each route. Column 6 contains the battery capacity (in kWh) that was selected for the EB in each route. Column 7 presents the percentage of the selected battery, in terms of the maximum battery capacity. Column 8 presents the objective function value, which is the total cost. Column 9 presents the percentage for the PVCS cost, in terms of the total cost. Column 10 shows the percentage of the selected battery cost, in terms of the total cost. Finally, column 11 shows the CPU time. In Table 1, we also included the economic comparison of the operation of the route using EB against ICB. For this comparison, we computed the total cost of the route operation using ICB during 10 years. This period was selected due to the estimated lifespan of the EB battery [48]. It is important to mention that the PVCS has a lifespan of 20 years, which is twice the lifespan of the EB battery. For comparing both operations, we needed to estimate the amount of fuel to perform each of the routes. For this estimation, we converted the total energy needed for completing each of the routes (in kWh) to the amount of fuel (in gallons) using the information presented by Lee [49]. For the total cost using an ICB, we considered two elements, the fuel cost and the Social Cost of Carbon (SCC). For estimating the SCC, we first computed the CO2 emission using an average value for marine Diesel [50], and then, we transformed the CO2 emission into cost considering the value presented by Nordhaus [51]. Also, for the routes where the ICB operation cost was under the investment cost of the EB operation, we estimated what would be the minimum fuel cost for both operations to have the same cost. From this point up, the operation with the EB will be the best economic option. From these considerations, Column 12 shows the fuel cost of performing each route using an ICB. Column 13 presents the SCC. Column 14 shows the total cost of the fuel operation (fuel cost + SCC). Column 15 shows the relative difference (Relative Difference=Cost With Fuel−Objective Function/Cost With Fuel) between the investment cost of the electric mobility system and the operation cost with an ICB. Finally, Column 16 presents the required fuel price (in USD/gallon) for making both operation cost equal.Table 1
Results of the test instances for the ERCSLP.
InstanceLength (km)# nodes# CS% of nodes with CSBat. capacity (kWh)% of battery usedObj. function% of cost due to CS% of cost due to batteryCPU time (S)Cost with fuel ($)SCC ($)Total cost fuel ($)Relative difference (%)Required price for fuel3Caqueta31031417.15234.740,92561.938.1130.541,8068,73450,54019.0NAMagdalena31149111.15536.741,86160.639.4226.839,6428,28247,92412.7NAMagdalena41161119.15536.741,85960.639.4757.141,6128,69350,30516.8NAGuaviare21211119.15234.742,41463.236.852.741,6498,70150,35015.8NAMagdalena21451119.17248.048,57055.544.5361.643,3149,04952,3637.2NACaqueta21722114.814496.070,21138.561.5552.540,5658,47449,039−43.24.3Magdalena11741715.910570.058,44746.153.9291.849,97710,44160,4183.3NACaqueta12106350.06442.798,24580.519.547.982,64917,26699,9151.7NAGuaviare27315320.07147.397,18678.121.9547.157,85412,08669,941−39.04.2Amazonas4829444.47348.7126,30782.717.372.778,96616,49795,463−32.34.0Putumayo57615746.77550.0201,11788.811.21010.773,14715,28188,428−127.46.8Orinoco65313861.512180.7248,36985.414.6166.481,85617,10098,956−151.07.5Min103614.85234.740,92538.511.247.939,6428,28247,924−151.04.0Average26212.72.723.278.352.292,95966.833.2351.556,08611,71767,803−26.45.4Max65321861.514496.0248,36988.861.51010.782,64917,26699,91519.07.53NA indicates that the EB investment cost is lower than the ICB operation cost.For all the instances shown in Table1, the optimality gap of the solution was smaller than the relative MIP optimality gap, which allows us to conclude that the instances are solved optimally. The average CPU time is around 6 minutes, which is a competitive CPU time for a strategic planning problem. In average, 23.2% of the nodes are selected to open a PVCS. It is important to mention that the 4 instances with the shortest lengths (Caqueta; 3, Magdalena; 3, Magdalena; 4, and Guaviare; 2) can be operated without a PVCS if the EB has a large battery size (respecting the maximum battery size). However, in the optimal solution, one PVCS is opened in all 4 routes. These results prove that, in some cases, it is better to select a small battery size and open a PVCS. As we can expect, the longest routes (Amazonas, Orinoco, and Putumayo) require a significant number of PVCSs. From the results in Table 1, we can also see that most of the solutions required a battery size that is less than the 50% of the maximum battery size. When analyzing the distribution of the cost, in average, 66.8% of the cost is devoted to opening PVCSs. Therefore, when we consider the EB operation, the most representative cost is related to the PVCSs. When comparing the investment cost against the cost of performing the same operation with an ICB during 10 years, we notice that in 7/12 instances, the electric mobility option performs better than the ICB operation. For the remaining instances, there is a minimum of 7.5USD/gallon to ensure that the electric mobility which is the best option (see Column 17). Even though this value seems very high, considering the isolations of these areas, the fuel transportation impacts the final fuel costs. Under this premises, the fuel cost could surpass the 7.5USD/gallon threshold.


## 5.4. Sensitivity Analysis

We now provide a sensitivity analysis to show how certain modifications on some parameters can impact the objective function. To do this, we used the 192 total instances, comprised of the 12 original instances and the 180 additional instances. Firstly, we proposed a variation on the radiation level. The radiation being an input for the PVCSs, we suppose that different radiation conditions can affect the decision-making when locating the PVCSs. Secondly, considering that the average EB speed impacts the energy consumption and the travel time, we hypothesize that this variable could affect the feasibility of the solutions and the value of the objective functions. Finally, we wanted to evaluate how different deltas of the discretized battery capacity can affect the objective function value and the CPU time.


### 5.4.1. Solar Radiation Variation

For testing the impact of the radiation on the objective function, we evaluated three different scenarios, where we consider the months with the highest, average, and lowest solar radiation per hour. In Figure5, we present the impact of these variations on the average value of the objective function from all the instances. This figure shows that the objective function increases when there is less solar radiation because more infrastructure is needed. However, there is not a significant variation after comparing all three scenarios. From a detailed analysis of the solutions, we found that, in most of the instances, the location and the number of PVCSs do not change in the three scenarios. The variation of the irradiation impacts the size of the battery. Considering that the PVCS cost is more expensive than the battery cost, there is not a significant variation on the average value of the objective function among the three scenarios. In Table 2, we show the solution of the instance Caquetá 1 for the three scenarios. Columns 2 to 7 present the nodes where a PVCS can be installed. The value of the cells represents the power of the selected PVCS; if a cell is empty, it is because no PVCS was installed at that associated node for that scenario. Furthermore, the last column shows the selected battery size. Table 2 also shows that the number and the location of the PVCSs are the same among the scenarios. On the other hand, the battery size changes between the scenarios with average and minimum radiation, and there is not a change between the scenarios with average and maximum radiation. In the latter, although both solutions have the same battery size the charging decisions are different. The EB must charge more energy for the average radiation scenario. Considering that variation of the radiation do not change the solutions significantly, it is adequate to use the average radiation level for this strategic problem.Figure 5
Variation of the objective function in terms of the solar radiation.Table 2
Caquetá 1 instance solution for the three radiation scenarios.
NodeScenario123456BatteryMaximum—22 kW11 kW—22 kW—64 kWhAverage—22 kW11 kW—22 kW—64 kWhMinimum—22 kW11 kW—22 kW—73 kWh


### 5.4.2. Average Boat Speed Variation

As we mentioned before, the EB speed has a significant impact on the energy consumption. This impact has a direct effect on the number of PVCSs and EB battery capacity. Furthermore, the EB speed can also affect the solution feasibility by the TW at each node and theTmax for the operation. In Figure 6, we present how the variation of the EB speed affects the objective function. In this figure, we show the average value of the objective function from all the instances. It is important to note that none of the instances below 9km/h had a feasible solution. This figure also shows that there is a significant impact of the EB speed on the total infrastructure cost. It is important to notice that increasing the EB speed could improve the service in the operation; however, this increment on the speed will directly impact the infrastructure needed to support the operation.Figure 6
Changes in the objective function according to the average EB speed.


### 5.4.3. Discretization of the Battery Capacity

As we mention in the problem description, the battery capacity is a decision variable. This variable takes discrete values from the setM that represents the possible values of the battery capacity. For building that set, we define a delta value of the battery capacity, which is interpreted as the step that increases the battery capacity in the set M. In this analysis, we want to test how the delta of the discretization affects the objective function. For the analysis, we choose battery capacity discretization values of 0.5, 0.75, 1, 1.25, and 1.5kWh being 1kWh the value for the discretization of the original instances. Figure 7 shows the average results of the objective function for the different delta values. For this variation of the delta, the MILP formulation was capable of optimally solving all the instances. When we analyze the impact of this variation on the objective function, we see that, for most of the delta variations, there is not a significant impact. The only delta variation that presented an impact on the cost was 1.5kWh. In this case, there is 0.34% increase in the objective function when compared to the results using a delta of 1kWh. From these results, we can conclude that using a 1kWh delta is an acceptable choice because we can obtain competitive solutions (in terms of the objective function) and reasonable CPU times for this kind of problems.Figure 7
Changes in the objective function according to the battery capacity discretization.


## 5.4.1. Solar Radiation Variation

For testing the impact of the radiation on the objective function, we evaluated three different scenarios, where we consider the months with the highest, average, and lowest solar radiation per hour. In Figure5, we present the impact of these variations on the average value of the objective function from all the instances. This figure shows that the objective function increases when there is less solar radiation because more infrastructure is needed. However, there is not a significant variation after comparing all three scenarios. From a detailed analysis of the solutions, we found that, in most of the instances, the location and the number of PVCSs do not change in the three scenarios. The variation of the irradiation impacts the size of the battery. Considering that the PVCS cost is more expensive than the battery cost, there is not a significant variation on the average value of the objective function among the three scenarios. In Table 2, we show the solution of the instance Caquetá 1 for the three scenarios. Columns 2 to 7 present the nodes where a PVCS can be installed. The value of the cells represents the power of the selected PVCS; if a cell is empty, it is because no PVCS was installed at that associated node for that scenario. Furthermore, the last column shows the selected battery size. Table 2 also shows that the number and the location of the PVCSs are the same among the scenarios. On the other hand, the battery size changes between the scenarios with average and minimum radiation, and there is not a change between the scenarios with average and maximum radiation. In the latter, although both solutions have the same battery size the charging decisions are different. The EB must charge more energy for the average radiation scenario. Considering that variation of the radiation do not change the solutions significantly, it is adequate to use the average radiation level for this strategic problem.Figure 5
Variation of the objective function in terms of the solar radiation.Table 2
Caquetá 1 instance solution for the three radiation scenarios.
NodeScenario123456BatteryMaximum—22 kW11 kW—22 kW—64 kWhAverage—22 kW11 kW—22 kW—64 kWhMinimum—22 kW11 kW—22 kW—73 kWh


## 5.4.2. Average Boat Speed Variation

As we mentioned before, the EB speed has a significant impact on the energy consumption. This impact has a direct effect on the number of PVCSs and EB battery capacity. Furthermore, the EB speed can also affect the solution feasibility by the TW at each node and theTmax for the operation. In Figure 6, we present how the variation of the EB speed affects the objective function. In this figure, we show the average value of the objective function from all the instances. It is important to note that none of the instances below 9km/h had a feasible solution. This figure also shows that there is a significant impact of the EB speed on the total infrastructure cost. It is important to notice that increasing the EB speed could improve the service in the operation; however, this increment on the speed will directly impact the infrastructure needed to support the operation.Figure 6
Changes in the objective function according to the average EB speed.


## 5.4.3. Discretization of the Battery Capacity

As we mention in the problem description, the battery capacity is a decision variable. This variable takes discrete values from the setM that represents the possible values of the battery capacity. For building that set, we define a delta value of the battery capacity, which is interpreted as the step that increases the battery capacity in the set M. In this analysis, we want to test how the delta of the discretization affects the objective function. For the analysis, we choose battery capacity discretization values of 0.5, 0.75, 1, 1.25, and 1.5kWh being 1kWh the value for the discretization of the original instances. Figure 7 shows the average results of the objective function for the different delta values. For this variation of the delta, the MILP formulation was capable of optimally solving all the instances. When we analyze the impact of this variation on the objective function, we see that, for most of the delta variations, there is not a significant impact. The only delta variation that presented an impact on the cost was 1.5kWh. In this case, there is 0.34% increase in the objective function when compared to the results using a delta of 1kWh. From these results, we can conclude that using a 1kWh delta is an acceptable choice because we can obtain competitive solutions (in terms of the objective function) and reasonable CPU times for this kind of problems.Figure 7
Changes in the objective function according to the battery capacity discretization.


## 6. Conclusions

In this paper, we introduced a new optimization problem named the Electric Riverboat Charging Station Location Problem (ERCSLP). This problem consists in estimating the necessary infrastructure so that an EB can perform a round trip on a river visiting several nodes. The objective of this problem is to decide the location of the PVCSs and the size of the EB battery aiming to minimize the total investment cost. This cost is defined as the sum of the PVCSs and battery costs. The ERCSLP includes some components from real river transport operations such as TW, service times, and a time limit of the route. Furthermore, this problem includes some components of the electric mobility such as the behavior of the nonlinear charging function and the variation of the solar radiation during the day. For solving this problem, we propose a MILP formulation that is able to consider all the characteristics of the proposed problem.For testing the MILP formulation, we built a set of instances based on different river transport operations, in Colombia, including a river operation that will be a pilot for the use of an EBs. The results show that our MILP formulation is capable of optimally solving all the instances in competitive CPU times for a strategic problem. Moreover, it is important to note that there are some instances where it possible to perform the route without placing PVCSs; however, the MILP formulation shows that, in this case, it is better to place a PVCSs operating with a smaller battery. Furthermore, we found that, for most of the tested river operations, the electric mobility could be economic feasible, when compared against the same operation with internal combustion cost. Finally, we perform a sensitivity analysis evaluating the impact on the objective function when we vary some parameters. We determinate that the EBs speed has a significant impact on the objective function; conversely, the variation of solar radiation had a negligible impact.



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*Source: 6527924-2020-11-16.xml*