# Multi-Objective Electric Vehicles Scheduling Using Elitist Non-Dominated Sorting Genetic Algorithm

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## Abstract

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## Featured Application

**The proposed methodology can be used in the electric vehicles charging/discharging scheduling considering different multi-objective functions.**

## Abstract

## 1. Introduction

_{2}emissions costs are very significant. In this case, the goal is to obtain the Pareto front that contains all nondominated solutions [6], then EV aggregators can select the solution that best suits their interests using different techniques such as the fuzzy-based method proposed in the present paper. We can use a linear combination of objective functions with weight factors to determine the Pareto front as a single objective function. This is the most known classic method for handling multi-objective problems [4].

_{EV_max}, the search interval in NSGA-II is reduced to an interval around this value, instead of the normal interval (−P

_{EV_max}to P

_{EV_max}). In [10], several tools were tested to determine a good initial solution to be used as a starting point in simulated annealing (SA) heuristic. SA is a local optimization model that considers only one objective function. In this paper is proposed the use of a relaxed deterministic approach, like the one used in [10], but considering a multi-objective function. Moreover, this initial solution is combined with NSGA-II heuristic allowing the definition of nondominated solutions in the Pareto-front curve.

## 2. Optimization Problem and Methodology

#### 2.1. Objective Functions

#### 2.2. Constraints

_{Trip}) is always zero. It is also important to mention that the variable P

_{(EV)}is positive when the vehicle is charging and negative when the vehicle is injecting energy to the network. The variables P

_{Ch}

_{(EV,t)}and P

_{Dch}

_{(EV,t)}are used instead of P

_{(EV)}to facilitate the comprehension of the constraints.

#### 2.3. Hybrid Evolutionary Algorithm

_{POP}, the number of generations N

_{GEN}, the crossover probability p

_{Cross}and the mutation probability p

_{Mut}. Then, the parent population P

_{t}is generated based on the deterministic technique, which is further explained below. Each individual of P

_{t}is evaluated for the two objective functions that compose the multi-objective function. The next step consists in ranking all individuals of P

_{t}and determining the crowding distance among them using the methods described in [6]. The rank mechanism ranks all individuals (or solutions) based on nondomination levels: level 1 is the best one, level 2 is the second-best one, and so on. The crowding distance metric determines an estimate of the perimeter of the cuboid formed by using the nearest neighbours of the solution as the vertices. The idea is to assign a high crowding distance value for solutions located in a less crowded region.

_{t}. The first criterion is the selection of the individual with the lower rank (meaning it is better). Otherwise, if both individuals have the same rank, then the solution with a higher crowding distance is selected. The objective of this operator is to obtain a well-spread population for the next generation t + 1. Next, the crossover and mutation are performed to generate an offspring population Q

_{t}with same size N

_{POP}. Once again, all individuals of Q

_{t}are evaluated for each objective function.

_{t}and Q

_{t}(i.e., parent and offspring, respectively) are combined creating a new population R

_{t}with twice the size of the two populations; then, the combined population is ranked, and the crowding distance is also determined. The next step selects the best N

_{POP}individuals of R

_{t}based on the ranking and crowding distance, which creates the parent population (P

_{t}+ 1) for the next generation t + 1. The NSGA-II algorithm returns to the tournament selection block and continues the rest of the flowchart until t is equal to the number of generations N

_{GEN}. However, the NSGA-II algorithm can also stop after some iterations when no significant improvement in the population occurs.

#### 2.3.1. Decision Variables

- V
_{i}_{(t)}and q_{ij}_{(t)}—Voltage and power flows defined by AC optimal power flow (AC OPF) - P
_{(ES,t)}and Q_{(ES,t)}—Active and reactive power supplied by external suppliers - P
_{(DG,t)}, Q_{(DG,t)}and P_{GCP}_{(DG,t)}—Active and reactive power produced by DG units - P
_{(SNS,t)}—Active power of service not supplied. The SNS can be positive in case of load curtailment or negative in case of generation curtailment - P
_{(ESS,t)}—Active power charged in electric storage systems (negative values represent power discharge) - P
_{(EV,t)}—Active power charged in electric vehicles (negative values represent power discharge) - P
_{DR}_{(L,t)}—Load demand curtailment under demand response programs

_{i}

_{(t)}, q

_{ij}

_{(t)}and power losses from the knowledge of the other variables. This process is run before each individual being evaluated for the two functions composing the multi-objective function.

#### 2.3.2. Initial Solution

#### 2.3.3. Crossover and Mutation Operators

_{t}that were previously selected by the tournament selection. Next, it is decided if the two-parent individuals will be crossed or not to generate two offspring individuals. This is accomplished if a random number is lower or equal to the defined crossover probability p

_{Cross}, otherwise the two selected individuals will be part of the offspring population Q

_{t}. The next step is applying the SBX operator to cross the variables regarding EVs and storage units scheduling (P

_{ST}

_{(ST,t)}and P

_{EV}

_{(EV,t)}) from the two selected individuals. The SBX starts with the first EV and if the new solutions generated from the SBX violate the EV constraints they are rejected. These steps are applied to all EVs and storage units. After the SBX operator generates two offspring individuals, the heuristics implemented in [20] for neighbourhood solutions in metaheuristics are applied to improve the two offspring individuals.

_{Mut}). The polynomial mutation operator [21] is used to each EV and storage system because it is one of the most recommended operators to handle with a continuous variable. If the new solution generated from the polynomial mutation operator violates the EV or storage constraint it is rejected.

#### 2.3.4. Stopping Criteria

^{−4}and 20, respectively.

#### 2.4. Fuzzy-Based Method

- Definition of the maximum and minimum values for each objective function.
- Definition of the membership function for each objective function (see Equation (16)).
- Definition of the normalized membership function (see Equation (17)).
- Order the Pareto solutions according to the normalized membership function.

## 3. Case Study

_{2}emissions. The case study is composed of a distribution grid with 37 buses that are connected to the upstream network by two 10 MVA power transformers [24]. This case study contains 1908 consumers: 1850 domestic houses, two industries, 50 commerce stores, and six service buildings. These consumers are aggregated by buses resulting in 22 aggregated loads. All consumers have two direct load control DR programs: the maximum power reduction of DR_A and DR_B correspond to 3% and 2% of the power consumption in each period. The energy roadmap for 2050 proposed in [25], which takes into account the EU targets [26], was used to establish the scenario for the DG units’ capacity. This resulted in a DG penetration composed of photovoltaic (PV) panels with a total installed power of 8MW, three Combined Heat and Power (CHP) units with a total installed power of 1.5MW, and four storage systems with the capability of storing 750 kWh. The penetration of EVs is around 50% and has been established based on the study developed in [27], resulting in 2000 EVs connected to this distribution network. Figure 4 shows the scheme of the 37-bus distribution network considering the DER’s penetration, where the substation (bus 1) connects this network to the system. The substation is composed of two transformers, with a power capacity of 10 MVA each, in which the energy from the 10 external suppliers flows. The input data related to the distribution network and the distributed energy resources are attached to the paper, as Supplementary Materials.

^{®}Xeon

^{®}E5-2620 v2 210 GHz, each one with two cores, 16 GB of random-access-memory.

#### 3.1. Pareto front Result

#### 3.2. Best Compromise Solution

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Parameters | |

$\alpha $ | Greenhouse gas emission fixed coefficient |

$\beta $ | Greenhouse gas emission linear coefficient |

$\gamma $ | Greenhouse gas emission quadratic coefficient |

${\eta}_{c}$ | Grid-to-vehicle efficiency |

${\eta}_{d}$ | Vehicle-to-grid efficiency |

$B$ | Imaginary part in admittance matrix [pu] |

$c$ | Resource cost in period t [m.u./kWh] |

$E$ | Stored energy in the battery of vehicle at the end of period t [kWh] |

${E}_{Trip}$ | Energy consumption in the battery during a trip that occurs in period t [kWh] |

$G$ | Real part in admittance matrix [pu] |

$N$ | Total number of resources |

$S$ | Maximum apparent power [pu] |

$T$ | Total number of periods |

$\overline{V}$ | Complex amplitude of voltage [pu] |

$\overline{y}$ | Series admittance of line that connects two buses [pu] |

$\overline{{y}_{sh}}$ | Shunt admittance of line that connects two buses [pu] |

Variables | |

$\theta $ | Voltage angle |

$P$ | Active power [pu] |

$Q$ | Reactive power [pu] |

$V$ | Voltage magnitude [pu] |

$X$ | Binary variable |

Indices | |

$BatMax$ | Battery energy capacity |

$BatMin$ | Minimum stored energy to be guaranteed at the end of period t |

$Bus$ | Bus |

$Ch$ | Batteries charge |

$D$ | Power demand |

$Dch$ | Batteries discharge |

$DG$ | Distributed generation unit |

$DR$ | Load with demand response contract |

$ES$ | External supplier |

$ESS$ | Energy storage system |

$EV$ | Electric vehicle |

$G$ | Power generation |

$i,j$ | Bus i and Bus j |

$L$ | Load |

$Max$ | Upper bound limit |

$Min$ | Lower bound limit |

$SNS$ | Service not supplied |

$Stored$ | Stored energy in the batteries |

$t$ | Period |

$TL$ | Thermal limit |

## Appendix A. Input Data Used in the Case Study

Parameters | NSGA-II |
---|---|

Population number—N_{POP} | 30 |

Number of generations—N_{GEN} | 50 |

Crossover probability—p_{Cross} | 0.9 |

Simulated Binary Crossover (SBX) distribution index | 5 |

Mutation probability—p_{Mut} | 0.2 |

Polynomial mutation index | 2 |

Threshold | 10^{−2} |

Threshold iterations | 20 |

Resource | Coefficients | |||
---|---|---|---|---|

Type | Bus | α [ton/h] | β [ton/kWh] | γ [ton/kWh^{2}] |

CHP | 14 | 0.01 | 2.5 × 10^{−4} | 3.0 × 10^{−6} |

16 | 0.03 | 5.0 × 10^{−3} | 7.0 × 10^{−6} | |

24 | 0.02 | 3.3 × 10^{−4} | 4.1 × 10^{−6} | |

External Supplier | 1 | 0.008 | 3.0 × 10^{−3} | 3.8 × 10^{−6} |

0.01 | 3.2 × 10^{−3} | 4.0 × 10^{−6} | ||

0.012 | 3.4 × 10^{−3} | 4.2 × 10^{−6} | ||

0.014 | 3.6 × 10^{−3} | 4.4 × 10^{−6} | ||

0.016 | 3.8 × 10^{−3} | 4.6 × 10^{−6} | ||

0.018 | 4.0 × 10^{−3} | 5.0 × 10^{−6} | ||

0.02 | 4.2 × 10^{−3} | 5.2 × 10^{−6} | ||

0.022 | 4.4 × 10^{−3} | 5.4 × 10^{−6} | ||

0.024 | 4.6 × 10^{−3} | 5.6 × 10^{−6} | ||

0.026 | 4.8 × 10^{−3} | 5.8 × 10^{−6} |

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**Figure 3.**Membership function example presented in [22].

**Figure 4.**37-Bus distribution network with distributed energy resources (DERs) penetration for 2050.

**Table 1.**Limits in the objective functions allowing the virtual power plants (VPP) to select the best solution.

Objective Function | Limits | Scenario |
---|---|---|

Operation Cost | Max-F_{l}^{max} | 26,500 |

Min-F_{l}^{min} | 24,500 | |

GHG emissions | Max-F_{l}^{max} | 3400 |

Min-F_{l}^{min} | 2800 |

Best Solution | Cost (m.u) | GHG (ton) |
---|---|---|

1 | 25,649.69 | 2808.86 |

2 | 25,689.80 | 2774.97 |

3 | 25,580.27 | 2832.88 |

4 | 25,297.04 | 2925.31 |

5 | 25,265.53 | 2947.30 |

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**MDPI and ACS Style**

Morais, H.; Sousa, T.; Castro, R.; Vale, Z.
Multi-Objective Electric Vehicles Scheduling Using Elitist Non-Dominated Sorting Genetic Algorithm. *Appl. Sci.* **2020**, *10*, 7978.
https://doi.org/10.3390/app10227978

**AMA Style**

Morais H, Sousa T, Castro R, Vale Z.
Multi-Objective Electric Vehicles Scheduling Using Elitist Non-Dominated Sorting Genetic Algorithm. *Applied Sciences*. 2020; 10(22):7978.
https://doi.org/10.3390/app10227978

**Chicago/Turabian Style**

Morais, Hugo, Tiago Sousa, Rui Castro, and Zita Vale.
2020. "Multi-Objective Electric Vehicles Scheduling Using Elitist Non-Dominated Sorting Genetic Algorithm" *Applied Sciences* 10, no. 22: 7978.
https://doi.org/10.3390/app10227978