# Optimization Approach for Long-Term Planning of Charging Infrastructure for Fixed-Route Transportation Systems

^{*}

*World Electric Vehicle Journal*in 2021)

## Abstract

**:**

## 1. Introduction

- A clear definition of the purpose of the analysis or optimization, which implies clarifying the use-case, its system boundaries, the stakeholder perspective, and the stakeholder objective.
- A careful a-priori selection of a reduced number of important influence factors—variables or parameters—depending on the use-case definition, the perspective, and the ultimate objective.

#### 1.1. Literature Overview of Stakeholder Perspectives and Objective Formulation

- converting different objectives to one via weighted sum of objectives,
- a hierarchical optimization model with layers, and
- a cross-entropy method.

#### 1.2. Literature Overview of Modeling Methodologies

**Table 2.**Overview of modeling methodologies for vehicle fleets with the purpose of identifying suitable charging infrastructure.

Methodology | Remarks | Reference |
---|---|---|

Facility Location Problem | Generic problem in transportation research | |

Estimation of Stationary Demand Density at System Nodes | Estimates charging demand at homes, stores, working places | [3] |

Estimation of Spatial Demand and Mobility of BEVs | Estimation is based on traffic flow models; demand can be covered along the routes | [3] |

Estimation of Spatial-temporal Demand | Real-world GPS data or fleet schedules extend demand estimation to the time domain | [3] |

Flow-Capturing Location Model | Captures as many routes as possible by placing charging points along them | [3] |

Multipath-Refueling Location Model | Allows drivers to deviate from their original path and to refuel more than once along the way | [3] |

Spatial-Temporal Model: Multistage Infrastructure Planning | Budgeted multistage planning | [7] |

Queuing Model | Implemented for a taxi fleet with waiting areas | [19] |

Bi Level Stochastic Queuing Models | ||

Graph Theoretic Model |

#### 1.3. Impact Factors for the Planning of Charging Infrastructure

- infrastructure,
- technology related,
- operational planning,
- bus network, and
- energy consumption.

- In their comprehensive comparative investigation of heavy-duty vehicles performance, Giakoumis et al. found that vehicle velocity was the most influential parameter affecting performance and the whole operation of it. Further, the indicators stops-per-kilometer and relative positive acceleration correlate very well with fuel or energy consumption [21] (p. 16). Eßer et al. confirmed that vehicle consumption profiles strongly depend on the driving profiles [22].
- Impact factors on the consumption were also sorted in a graph by Gallet et al. in their study regarding a bus fleet [23] (p. 14). The three most influential parameters in decreasing order are: curb mass, auxiliary power, and rolling resistance.
- Regarding the fleet charging strategy, if overnight charging is chosen, this requires bigger battery capacities for the electric vehicles, according to Gallet et al. [23].
- Short distances between bus stops favor the use of electric buses because they are better suited for driving profiles with frequent start and stop situations than conventional buses [24] (p. 191).
- Kunith et al. stated that, in general, as concluded by previous studies, maximum charging power has a significant influence on the number of charging stations needed [4] (p. 9).
- Kunith et al. found that the extension of dwell time requires the adaptation of the operational schedule or the increase of the number of vehicles but relaxes the infrastructure requirements [4] (p. 9).
- Battery aging over time becomes a more restrictive constraint for the fleet management because with it the range of the vehicles decreases. These stricter constraints result in a higher fleet management effort for the fleet operator. However, it is seldom considered.

#### 1.4. The Fixed-Route Transportation System Problem of Fleet Operators

## 2. Materials and Methods

#### 2.1. Optimization Framework Structure

#### 2.2. Preparing the Optimization: Pre-Processing

#### 2.3. Core Optimization Process

_{total}related to the operation of a fleet, including the cost for its charging infrastructure. The total cost is the sum of the costs that arise in each period j over all periods considered, as can be seen in Equation (1):

_{j}

_{,operation}and investment costs C

_{j}

_{,investment}.

_{p}, as can be seen in Equation (3). The potential costs are approximated a priori uniquely for every CPC p in the frTS. Since a detailed cost estimation considering the cost for charging modules, inverters, cables, etc., might be very time consuming, basing the cost on the maximum available charging power q is a reasonable first approach. In a real planning scenario, fleet operators and distribution system operators should be consulted because their knowledge in this regard is central for the estimation of potential costs. In Equation (3), α

_{p}represents the monetary cost for activating CPC p. Note that, in the following, the index of the period j is omitted.

_{t}in that period. Additionally, the costs for the electricity contract for every charging point are added. Considering that a fleet operator is likely to sign a general agreement with a distribution system operator rather than separately doing so for each charging point, we consider the maximum aggregate power which is drawn in a certain period as the proxy for this share of the cost C

_{power contract}.

_{power contract}is determined via the decision variable Z

_{c}that chooses exclusively one power contract from a set of available ones. Z

_{c}determines the cost as in Equation (5) based on the price for a respective power contract δ

_{c}. Further, Z

_{c}constrains the maximum aggregate power that can be drawn in a single time step.

_{t}on representative day t can be written as shown in Equations (6)–(8). The trip-dependent fuel consumption is given by Ω

_{k,v,r}, where k references a trip, v a specific vehicle, and r the respective fuel consumed, e.g., diesel. The fuel price β

_{r}is different for each fuel r. Y

_{v}

_{,k}is a binary decision variable indicating if vehicle v serves trip k. It is worth noting that hybrid vehicles can be considered as well. In such a case, Ω

_{k}

_{,v,r}will only represent a share of the total energy required for the vehicle operation. The electric energy is considered separately in Equation (8). Note that the index of the representative day t is omitted in Equations (7) and (8).

_{p}

_{,ts,v}stands for the amount of energy charged to vehicle v at charging point p in time step ts. Summing up over all vehicles, time steps, and charging points, one gets the total amount of electric energy necessary for the fleet operation. The electricity price is assumed to remain constant over one representative day and is given by γ

_{t}. The starting SoC is set by the fleet operator, and a constraint makes sure that the cumulative amount of electric energy stored in all the vehicles at the end of a representative day is equal to the one at the start of the day. This makes sure that vehicles are not completely drawn of their power, which would render them unprepared for the next day. One alternative is the implementation of vehicle-specific SoC constraints at the end of a representative day. However, this is too restrictive. We illustrate this with the following example: It might be the case that some vehicles arrive late and with a low SoC at the depot. Nonetheless, the fleet operation of the next day is not jeopardized because other sufficient vehicles are ready and charged.

_{min}and SoC

_{max}, respectively. Additionally, the maximum SoC in each period is influenced by the battery aging. In every period, a percentage capacity loss is implemented to account for the battery aging. This effect propagates over the whole optimization time horizon and has a direct impact on the fleet operation. Depending on the battery cell technology, the user can set a suitable aging factor. Farmann et al. suggested coulomb counting as a simple method for estimating battery aging [31]. Implementing this renders the optimization problem very time consuming for the use-case of Darmstadt. Darmstadt represents a rather small problem compared to bigger cities. In fact, when implementing coulomb counting, the solving process had to be stopped because the RAM on the mid-end solving computer was exhausted. In order to not completely omit the influence of battery aging—especially long-term—a constant aging factor of 2% capacity loss per year is implemented. Equation (9) shows the SoC constraint for an electric vehicle ev in time step ts. This is valid on all representative days t and for all time-steps belonging to t, which we summarize in the set I

_{ts}. The set of electric vehicles is denoted by I

_{ev}.

_{min}and SoC

_{max}are percentage values entered prior to optimizing. The SoC

_{ts}at time-step ts for an electric vehicle ev on a specific day is computed as follows:

_{k}

_{,ev,cts}denotes the electric consumption that vehicle ev would have on trip k in each time-step cts, and N

_{cum. time steps}denotes the highest number in the set of time-steps I

_{cts}. The latter is comprised of ts and all previous time-steps on that representative day; e.g., in time step 3, the set becomes I

_{cts}= [1, 2, 3]. As can be seen, the electric consumption is only considered if Y

_{ev}

_{,k}is set to 1. As above, θ

_{p}

_{,cts,ev}is the variable representing the electric energy throughput from charging point p to vehicle ev in time step cts.

#### 2.4. Evaluation of Optimization Results: Post Processing

- at what point in time does there have to be an infrastructure expansion,
- with what maximum charging capacity should charging points be equipped, and
- what is the fleet management like on every single representative day?

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Overview of key micro- and macroscopic impact factors of the energy and transportation sector for planning charging infrastructure.

**Figure 3.**Fleet management for a selection of vehicles on one representative day as determined by an exemplary optimization process. Each trip has a distinctive color. The colored trips are assigned to the vehicle numbers 1 to 20.

**Figure 4.**Comparison of the status-quo scenario and scenarios 1 to 7 with varying charging point candidates and the consequential implication for the problem size in terms of number of variables and constraints. The lower part displays the objective value after the optimization (

**left**) and the relative mixed integer problem gap (

**right**).

**Table 1.**Overview of stakeholders in the energy and transportation sectors and their different optimization objectives from economic, environmental, and social welfare perspectives.

Sector | Stakeholder | Perspective | Objective | Reference |
---|---|---|---|---|

Energy | Electricity Producers | Economical | Minimize electricity production costs | Authors. |

Electricity Producers | Environmental | Minimize greenhouse-gas emissions | Authors. | |

Electricity Producers | Environmental | Minimize resource utilization | Authors. | |

Grid Operators | Economical | Maximize voltage stability | [3] | |

Grid Operators | Economical | Minimize cost of infrastructure | Authors. | |

Grid Operators | Economical | Minimize transmission losses | [3] | |

Grid Operators | Economical | Minimize resource utilization | Authors. | |

Transportation | End-user/Client | Social Welfare | Minimize individual agent’s travel time | Authors. |

End-user/Client | Social Welfare | Minimize cumulative travel time | Authors. | |

End-user/Client | Social Welfare | Minimize delivery times | Authors. | |

End-user/Client | Social Welfare | Minimize discomfort of drivers | [8] | |

End-user/Client | Social Welfare | Maximize social welfare | [9] | |

End-user/Client | Social Welfare | Maximize number of reached households | [10] | |

End-user/Client | Environmental | Minimize energy consumption | Authors. | |

End-user/Client | Environmental | Minimize greenhouse-gas emissions | Authors. | |

End-user/Client | Environmental | Maximize share of electric vehicle-kilometers traveled | [11] (p. 166) | |

Fleet Operator | Economical | Minimize infrastructure cost | Authors. | |

Fleet Operator | Economical | Minimize investment cost including vehicles | Authors. | |

Fleet Operator | Economical | Minimize number of charging stations | Authors. | |

Fleet Operator | Economical | Maximize schedule length of buses | [12] | |

Fleet Operator | Economical | Maximize the share of distance traveled electrically | [13] | |

Fleet Operator | Economical | Minimize operation cost | Authors. | |

Fleet Operator | Economical | Minimize total cost of ownership | [12] | |

Fleet Operator | Environmental | Minimize greenhouse-gas emissions | Authors. | |

Fleet Operator | Environmental | Minimize noise pollution | Authors. | |

Fleet Operator | Environmental | Maximize electrically traveled range | [11] | |

Vehicle Manufacturer | Economical | Minimize production costs | Authors. | |

Vehicle Manufacturer | Environmental | Minimize resource utilization | Authors. |

**Table 3.**Reference data for the optimization of the use-case of the public transport operator of Darmstadt.

Description | Unit | Value |
---|---|---|

Time horizon | Years | 2 |

Number of reference days | - | 4 |

Total number of vehicles | - | 76 |

Number of unique trips | - | 240 |

Underlying time discretization for the optimization | Minutes | 30 |

Cost for electricity | € per kWh | 0.12 |

Cost for diesel fuel | € per liter | 1.05 |

Cost for activation of a CPC with 50 kW maximum charging power | € | 7500 |

Cost for activation of a CPC with 150 kW maximum charging power | € | 11,500 |

Annual cost for a power contract with 1 MW aggregate power | € | 10,000 |

Annual cost for a power contract with 2 MW aggregate power | € | 20,000 |

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## Share and Cite

**MDPI and ACS Style**

Blat Belmonte, B.D.; Rinderknecht, S.
Optimization Approach for Long-Term Planning of Charging Infrastructure for Fixed-Route Transportation Systems. *World Electr. Veh. J.* **2021**, *12*, 258.
https://doi.org/10.3390/wevj12040258

**AMA Style**

Blat Belmonte BD, Rinderknecht S.
Optimization Approach for Long-Term Planning of Charging Infrastructure for Fixed-Route Transportation Systems. *World Electric Vehicle Journal*. 2021; 12(4):258.
https://doi.org/10.3390/wevj12040258

**Chicago/Turabian Style**

Blat Belmonte, Benjamin Daniel, and Stephan Rinderknecht.
2021. "Optimization Approach for Long-Term Planning of Charging Infrastructure for Fixed-Route Transportation Systems" *World Electric Vehicle Journal* 12, no. 4: 258.
https://doi.org/10.3390/wevj12040258