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.

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 [

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 both

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 [

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 [

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. Section

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 [

Recently, some researchers have integrated Renewable Energy Sources (RESs) to CS location problems. In this way, Moradi et al. [

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 [

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. [

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. [

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.

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 [

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 [

Piecewise linear approximation for charging a battery of 16 kWh with a charging rate of 22 kW. Source: [

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 [

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. Figure

General scheme of a PVCS. Source: [

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. Figure

Solar radiation curve for the municipality of Sincelejo, Colombia. Source: data from IDEAM [

Let

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.

In this section, we present a numerical example intending to give the reader a better perspective of our problem. Figure

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.

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 of

Figure

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,

We now provide a MILP formulation for the ERCSLP. The MILP formulation uses the following decision variables:

The MILP formulation of the ERCSLP is as follows:

The objective function (

Subject to

Constraint (

Constraints (

Constraints (

Constraint (

Constraints (

Constraints (

Finally, constraints (

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.

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 [

For building the instances, we make the following considerations. For the discretization of the EB battery capacity, we use a delta of

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 values

For solving the MILP formulation of the ERCSLP, we used the Gurobi Optimizer (version 8.1.0) with a CPU time limit of

Initially, we solve the 12 original instances, and Table

Results of the test instances for the ERCSLP.

Instance | Length (km) | # nodes | # CS | % of nodes with CS | Bat. capacity (kWh) | % of battery used | Obj. function | % of cost due to CS | % of cost due to battery | CPU time (S) | Cost with fuel ($) | SCC ($) | Total cost fuel ($) | Relative difference (%) | Required price for fuel^{3} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Caqueta3 | 103 | 14 | 1 | 7.1 | 52 | 34.7 | 40,925 | 61.9 | 38.1 | 130.5 | 41,806 | 8,734 | 50,540 | 19.0 | NA |

Magdalena3 | 114 | 9 | 1 | 11.1 | 55 | 36.7 | 41,861 | 60.6 | 39.4 | 226.8 | 39,642 | 8,282 | 47,924 | 12.7 | NA |

Magdalena4 | 116 | 11 | 1 | 9.1 | 55 | 36.7 | 41,859 | 60.6 | 39.4 | 757.1 | 41,612 | 8,693 | 50,305 | 16.8 | NA |

Guaviare2 | 121 | 11 | 1 | 9.1 | 52 | 34.7 | 42,414 | 63.2 | 36.8 | 52.7 | 41,649 | 8,701 | 50,350 | 15.8 | NA |

Magdalena2 | 145 | 11 | 1 | 9.1 | 72 | 48.0 | 48,570 | 55.5 | 44.5 | 361.6 | 43,314 | 9,049 | 52,363 | 7.2 | NA |

Caqueta2 | 172 | 21 | 1 | 4.8 | 144 | 96.0 | 70,211 | 38.5 | 61.5 | 552.5 | 40,565 | 8,474 | 49,039 | −43.2 | 4.3 |

Magdalena1 | 174 | 17 | 1 | 5.9 | 105 | 70.0 | 58,447 | 46.1 | 53.9 | 291.8 | 49,977 | 10,441 | 60,418 | 3.3 | NA |

Caqueta1 | 210 | 6 | 3 | 50.0 | 64 | 42.7 | 98,245 | 80.5 | 19.5 | 47.9 | 82,649 | 17,266 | 99,915 | 1.7 | NA |

Guaviare | 273 | 15 | 3 | 20.0 | 71 | 47.3 | 97,186 | 78.1 | 21.9 | 547.1 | 57,854 | 12,086 | 69,941 | −39.0 | 4.2 |

Amazonas | 482 | 9 | 4 | 44.4 | 73 | 48.7 | 126,307 | 82.7 | 17.3 | 72.7 | 78,966 | 16,497 | 95,463 | −32.3 | 4.0 |

Putumayo | 576 | 15 | 7 | 46.7 | 75 | 50.0 | 201,117 | 88.8 | 11.2 | 1010.7 | 73,147 | 15,281 | 88,428 | −127.4 | 6.8 |

Orinoco | 653 | 13 | 8 | 61.5 | 121 | 80.7 | 248,369 | 85.4 | 14.6 | 166.4 | 81,856 | 17,100 | 98,956 | −151.0 | 7.5 |

Min | 103 | 6 | 1 | 4.8 | 52 | 34.7 | 40,925 | 38.5 | 11.2 | 47.9 | 39,642 | 8,282 | 47,924 | −151.0 | 4.0 |

Average | 262 | 12.7 | 2.7 | 23.2 | 78.3 | 52.2 | 92,959 | 66.8 | 33.2 | 351.5 | 56,086 | 11,717 | 67,803 | −26.4 | 5.4 |

Max | 653 | 21 | 8 | 61.5 | 144 | 96.0 | 248,369 | 88.8 | 61.5 | 1010.7 | 82,649 | 17,266 | 99,915 | 19.0 | 7.5 |

^{3}NA indicates that the EB investment cost is lower than the ICB operation cost.

For all the instances shown in Table

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.

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 Figure

Variation of the objective function in terms of the solar radiation.

Caquetá 1 instance solution for the three radiation scenarios.

Node | |||||||
---|---|---|---|---|---|---|---|

Scenario | 1 | 2 | 3 | 4 | 5 | 6 | Battery |

Maximum | — | 22 kW | 11 kW | — | 22 kW | — | 64 kWh |

Average | — | 22 kW | 11 kW | — | 22 kW | — | 64 kWh |

Minimum | — | 22 kW | 11 kW | — | 22 kW | — | 73 kWh |

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 the

Changes in the objective function according to the average EB speed.

As we mention in the problem description, the battery capacity is a decision variable. This variable takes discrete values from the set

Changes in the objective function according to the battery capacity discretization.

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.

Set of nodes

Starting point

Final destination

Return node

Nodes to visit on the outbound trip

Nodes to visit on the return trip

Service time at node

Earliest possible arrival time at node

Latest possible arrival time at node

Travel time between nodes

Time limit

Set of battery capacities

Acquisition cost of battery capacity

Energy consumption between nodes

Set of different types of PVCSs’ technologies

Cost of installing a PVCS at node

Set of breakpoints of the piecewise linear approximation

Slope of the segment defined between breakpoints

Upper bound of the battery level of the segment defined between breakpoints

Set of time intervals

Length of the time intervals

Upper bound (in hours) of interval

Power generation of interval

Capacity of the ESS

Lower energy limit of the ESS

Binary variable equal to 1 if the EB battery capacity takes a value of

Binary variable equal to 1 if a PVCS of type

Battery level when the EB arrives at node

Arrival time of the EB at node

Charged energy when the EB charges at the PVCS of type

Battery level when the EB charges at the PVCS of type

Binary variable equal to 1 if the EB charges on the segment between breakpoint

Charging time at the PVCS located at node

Stay time at the PVCS located at node

Fraction of the time interval

Fraction of the time interval

Binary variable equal to 1 if the PVCS located at node

Binary variable equal to 1 if the PVCS located at node

Total energy generated by the PVCS located at node

Available energy at the PVCS located at node

Energy generated while the EB is charging at the PVCS located at node

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors would like to thank Universidad EAFIT for supporting this research through the Research Assistantship grant from project 828-000068. This research has also been developed in the framework of the “ENERGETICA 2030” Research Program, with code 58667 in the “Scientific Colombia” initiative, funded by The World Bank through the call “778-2017 Scientific Ecosystems,” managed by the Colombian Administrative Department of Science, Technology, and Innovation (COLCIENCIAS). The authors would like to thank Jorge Lozano for his assistance in the instances building automation and his contribution in the literature review and Simón Polanía for his help in the construction of the images. Additionally, the authors would like to thank Universidad EAFIT scientific computing center (APOLO) for its support for the computational experiments.