# Optimal Scheduling for Electric Vehicle Charging under Variable Maximum Charging Power

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## Abstract

**:**

## 1. Introduction

- We consider the aggregator’s EV charging-scheduling problem under the assumption of variable maximum charging power.
- We propose mathematical formulations for two different charging schemes: non-preemptive and preemptive charging.
- We also introduce an adaptation of the proposed formulation as a way of preventing frequent interruptions in the charging process.
- Our numerical simulations compare the different charging schemes and demonstrate that preemptive charging with limited interruptions is an attractive alternative in terms of both cost and practicality.
- We also show that our formulation is computationally efficient in solving practical, large-scale charging-scheduling problems.

## 2. Scenario Description

#### 2.1. Aggregator Settings

#### 2.2. Variable Maximum Charging Power

#### 2.3. Charging Schemes

## 3. Problem Formulation

#### 3.1. Non-Preemptive Charging

#### 3.2. Preemptive Charging

- Charging transition $(t,k)\to (t+1,k+1)$: EV v with SOC level k at the outset of time t is charged during time t,
- Idle transition $(t,k)\to (t+1,k)$: EV v with SOC level k at the outset of time t is not charged during time t.

**Proposition**

**1.**

#### 3.3. Controlling Frequency of Interruption

## 4. Numerical Simulations

#### 4.1. Simulation Settings

#### 4.2. Results and Discussion

- NP: Non-preemptive charging,
- P60: Preemptive charging with $\Delta $ = 60 min,
- P30: Preemptive charging with $\Delta $ = 30 min,
- P15: Preemptive charging with $\Delta $ = 15 min.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 6.**Examples of charging schedules corresponding to different charging schemes: (

**a**) non-preemptive charging; (

**b**) preemptive charging with $\Delta $ = 60 min; (

**c**) preemptive charging with $\Delta $ = 30 min; and (

**d**) preemptive charging with $\Delta $ = 15 min.

Parameter | PCP-E | PCP | |||||
---|---|---|---|---|---|---|---|

$\left|\mathit{V}\right|$ | ${\mathit{L}}^{\mathit{t}}$ | avg | min | max | avg | min | max |

500 | 600 | 1.1 | 0.8 | 1.5 | 4.2 | 1.9 | 19.4 |

750 | 0.9 | 0.8 | 1.3 | 33.4 | 1.6 | 300* | |

900 | 0.9 | 0.8 | 1.0 | 2.0 | 1.0 | 2.5 | |

1000 | 1200 | 2.2 | 1.9 | 3.6 | 11.9 | 5.5 | 58.0 |

1500 | 2.0 | 1.8 | 3.5 | 5.8 | 2.9 | 11.5 | |

1800 | 2.0 | 1.9 | 2.1 | 5.3 | 3.6 | 8.6 | |

1500 | 1800 | 3.4 | 3.3 | 3.5 | 9.3 | 7.5 | 13.2 |

2250 | 2.9 | 2.8 | 3.0 | 8.1 | 6.5 | 10.0 | |

2700 | 3.1 | 2.9 | 3.3 | 7.7 | 7.0 | 9.4 | |

2000 | 2400 | 4.8 | 4.6 | 5.0 | 14.1 | 7.6 | 27.1 |

3000 | 4.0 | 3.8 | 4.2 | 21.5 | 4.5 | 122.5 | |

3600 | 4.2 | 3.9 | 4.5 | 10.4 | 4.6 | 23.1 | |

2500 | 3000 | 8.4 | 6.7 | 10.0 | 16.6 | 7.5 | 22.6 |

3750 | 5.4 | 5.2 | 5.6 | 11.8 | 6.0 | 17.1 | |

4500 | 5.9 | 5.3 | 7.1 | 15.5 | 8.0 | 24.1 | |

3000 | 3600 | 11.7 | 8.4 | 15.1 | 24.1 | 8.4 | 28.9 |

4500 | 6.9 | 6.6 | 7.4 | 19.2 | 7.2 | 49.8 | |

5400 | 7.6 | 6.7 | 8.4 | 17.6 | 7.3 | 24.8 |

**Table 2.**Comparison of total charging costs (in thousand dollars) between non-preemptive and preemptive charging schemes.

Parameter | Non-Preemptive | Preemptive | Difference(%) | |||||||
---|---|---|---|---|---|---|---|---|---|---|

$\left|\mathit{V}\right|$ | ${\mathit{L}}^{\mathit{t}}$ | avg | min | max | avg | min | max | avg | min | max |

500 | 600 | 232 | 144 | 287 | 226 | 140 | 275 | 2.63 | 0.76 | 4.06 |

750 | 212 | 130 | 254 | 208 | 127 | 248 | 1.95 | 1.00 | 2.55 | |

900 | 206 | 124 | 244 | 203 | 121 | 239 | 1.49 | 0.85 | 2.40 | |

1000 | 1200 | 476 | 293 | 593 | 458 | 284 | 557 | 3.58 | 0.89 | 6.15 |

1500 | 429 | 264 | 511 | 420 | 258 | 499 | 1.93 | 0.94 | 2.53 | |

1800 | 416 | 251 | 493 | 409 | 244 | 480 | 1.86 | 1.09 | 2.85 | |

1500 | 1800 | 696 | 430 | 858 | 676 | 420 | 818 | 2.72 | 0.65 | 4.66 |

2250 | 629 | 388 | 747 | 620 | 381 | 733 | 1.55 | 0.75 | 2.00 | |

2700 | 612 | 371 | 721 | 603 | 362 | 705 | 1.51 | 0.83 | 2.36 | |

2000 | 2400 | 934 | 576 | 1159 | 899 | 557 | 1088 | 3.72 | 0.96 | 6.17 |

3000 | 840 | 517 | 998 | 824 | 505 | 976 | 1.89 | 0.94 | 2.48 | |

3600 | 816 | 492 | 961 | 803 | 479 | 938 | 1.60 | 0.92 | 2.59 | |

2500 | 3000 | 1174 | 723 | 1456 | 1132 | 701 | 1371 | 3.47 | 0.87 | 5.82 |

3750 | 1058 | 651 | 1258 | 1038 | 637 | 1229 | 1.89 | 0.90 | 2.45 | |

4500 | 1026 | 619 | 1210 | 1008 | 603 | 1179 | 1.75 | 0.99 | 2.62 | |

3000 | 3600 | 1404 | 866 | 1740 | 1358 | 842 | 1645 | 3.22 | 0.80 | 5.48 |

4500 | 1266 | 780 | 1504 | 1244 | 764 | 1474 | 1.70 | 0.82 | 2.26 | |

5400 | 1229 | 743 | 1450 | 1208 | 724 | 1414 | 1.66 | 0.97 | 2.52 |

**Table 3.**Effect of penalty parameter on number of interruptions and total charging costs (in dollars): The penalty parameter ${\Delta}_{v}^{t}$ is given as $x\times {M}^{t}$.

Preemptive | Non-Preemptive | |||
---|---|---|---|---|

x = 0 | x = 0.5 | x = 1 | ||

# Interruptions (average) | 0.76 | 0.52 | 0.49 | 0 |

Total costs (average) | 19,629.2 | 19,631.9 | 19,637.5 | 20,113.5 |

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**MDPI and ACS Style**

Han, J.; Park, J.; Lee, K. Optimal Scheduling for Electric Vehicle Charging under Variable Maximum Charging Power. *Energies* **2017**, *10*, 933.
https://doi.org/10.3390/en10070933

**AMA Style**

Han J, Park J, Lee K. Optimal Scheduling for Electric Vehicle Charging under Variable Maximum Charging Power. *Energies*. 2017; 10(7):933.
https://doi.org/10.3390/en10070933

**Chicago/Turabian Style**

Han, Jinil, Jongyoon Park, and Kyungsik Lee. 2017. "Optimal Scheduling for Electric Vehicle Charging under Variable Maximum Charging Power" *Energies* 10, no. 7: 933.
https://doi.org/10.3390/en10070933