# Electric Vehicle and Photovoltaic Power Scenario Generation under Extreme High-Temperature Weather

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## Abstract

**:**

## 1. Introduction

## 2. Kernel Density Estimation and Model Testing Methods

#### 2.1. Kernel Density Estimation Method

#### 2.2. Model-Checking Method

#### 2.2.1. Goodness of Fit Test

- (1)
- Pearson ${\chi}^{2}$

- (2)
- Kolmogorov–Smirnov

#### 2.2.2. Test of Fitting Accuracy

## 3. Correlation Modeling and Scene Generation of PV and EV Power under High-Temperature Weather Based on Copula Theory

#### 3.1. Copula-Related Theory

#### 3.1.1. Copula Functions and Classification

#### 3.1.2. PV Systems and EV Modeling

**(1)****PV Modeling**

**(2)****EV Modeling**

#### 3.1.3. Optimal Choice of Copula Function

**(1)****Joint scenario generation and reduction of PV and EV power**

- ①
- Functional image discrimination is to compare the probability density function images of each copula function with the probability density function of the sample data, and the closest image is the optimal copula function.
- ②
- Correlation coefficient discrimination method is to judge the goodness of fit using the Kendall rank correlation coefficient and Spearman rank correlation coefficient. The rank correlation coefficients of various copula functions are compared with the rank correlation coefficients of sample data. The closer the data are, the better the goodness of fit is, and the corresponding copula function is the best.

- ③
- The Euclidean distance discrimination method is to compare the Euclidean distance of each copula function with the empirical copula function of the sample data. The smaller the Euclidean distance is, the better the goodness of fit of the copula function is.

**(2)****Scenario generation based on cubic spline interpolation**

**(3)****Scenario reduction based on k-means clustering algorithm**

## 4. Case Study

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Description | |
---|---|

Diurnal Variation Curve | The diurnal variation curve of a photovoltaic power station illustrates the change in power generation over the course of a day. Typically, power generation gradually increases at sunrise, may reach its peak around noon, and then gradually decreases until sunset. |

Seasonal Variation | The power generation of a photovoltaic power station is influenced by seasonal changes. Summer, with longer sunlight hours and a higher solar zenith angle, may result in higher peak power during this season. |

Weather Impact | Weather conditions, such as clear, cloudy, or overcast skies, directly affect the output of a photovoltaic power station. Cloudy weather can lead to fluctuations and a reduction in power generation. |

Shadow Effect | If the photovoltaic power station is affected by shadows from buildings, trees, or other objects, irregular fluctuations may appear on the curve, known as the shadow effect. |

Start and End Times | The times when a photovoltaic power station begins and ends its power generation, influenced by sunrise and sunset times. |

Peak Power | The highest power generation of the photovoltaic power station during the day, typically occurring at noon when the solar zenith angle is at its maximum. |

Power Fluctuations | The fluctuation in power on the power curve of the photovoltaic power station, representing instantaneous changes in power, potentially influenced by shadows, cloud cover, and other weather factors. |

Description | |
---|---|

Charging Peak Period | Electric vehicles may experience a charging peak at night or during specific time periods, indicating users’ tendency to charge during low electricity price periods. |

Driving Peak Period | Daytime may witness a driving peak, signifying higher usage demand for electric vehicles during the day. |

Charging Efficiency Variations | The charging curve may reflect variations in charging efficiency at different charging power levels, influenced by battery and charging equipment performance. |

Charging Time Distribution | Describes the distribution of time required for electric vehicle charging, including short “top-up” charges and longer “full-charge” durations. |

Load Fluctuations: | Reflects the variability in electric vehicle power demand, with certain periods exhibiting significant power fluctuations. |

Charging Behavior Response | Describes whether electric vehicles respond to power system demand signals or price signals, adjusting their charging behavior accordingly. |

Usage Patterns | Distinguishes between weekdays and weekends, as well as different usage patterns during daytime and nighttime. |

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**Figure 5.**Probability distribution plots after reduction of scenarios under different copula functions.

Name | Formula |
---|---|

Box (or uniform) | $\frac{1}{2}S(\left|r\right|\le 1)$ |

Cosinus | $\frac{\pi}{4}\mathrm{cos}(\frac{\pi}{2}r)S(\left|r\right|\le 1)$ |

Epanechnikov | $\frac{3}{4}(1-{r}^{2})S(\left|r\right|\le 1)$ |

Gaussian | $\frac{1}{\sqrt{2\pi}}{e}^{-{r}^{2}/2}(\left|r\right|\le 1)$ |

Quaritic | $\frac{15}{16}{(1-{r}^{2})}^{2}S(\left|r\right|\le 1)$ |

Triangle | $(1-\left|r\right|)S(\left|r\right|\le 1)$ |

Triweight | $\frac{35}{32}{(1-{r}^{2})}^{3}S(\left|r\right|\le 1)$ |

Kendall Rank Correlation Coefficient | Spearman Rank Correlation Coefficient | Square Euclidean Distance | |
---|---|---|---|

Gaussian Copula | −0.06664 | −0.09983 | 0.45874 |

Gumbel Copula | 1.35753 × 10^{−6} | 2.05096 × 10^{−6} | 2.06240 |

Clayton Copula | 7.25427 × 10^{−7} | 1.09220 × 10^{−6} | 2.06237 |

Frank Copula | −0.09334 | −0.13967 | 0.24974 |

Sample Data | −0.08268 | −0.12254 | 0 |

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**MDPI and ACS Style**

Li, X.; Li, C.; Jia, C.
Electric Vehicle and Photovoltaic Power Scenario Generation under Extreme High-Temperature Weather. *World Electr. Veh. J.* **2024**, *15*, 11.
https://doi.org/10.3390/wevj15010011

**AMA Style**

Li X, Li C, Jia C.
Electric Vehicle and Photovoltaic Power Scenario Generation under Extreme High-Temperature Weather. *World Electric Vehicle Journal*. 2024; 15(1):11.
https://doi.org/10.3390/wevj15010011

**Chicago/Turabian Style**

Li, Xiaofei, Chi Li, and Chen Jia.
2024. "Electric Vehicle and Photovoltaic Power Scenario Generation under Extreme High-Temperature Weather" *World Electric Vehicle Journal* 15, no. 1: 11.
https://doi.org/10.3390/wevj15010011