# Research on Multifractal Characteristics of Vehicle Driving Cycles

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## Abstract

**:**

## 1. Introduction

#### 1.1. Research Background

- -
- check the compliance of vehicle pollutant emissions with respect to the applicable emissions limits;
- -
- establish the reference vehicle fuel consumption and CO
_{2}performance; - -
- reduce the gap between type approval values and real world emissions.

#### 1.2. Introduction of the MF-DFA Method

#### 1.3. Statement of the Design Approach

## 2. Description of Different Driving Cycles

## 3. Description of the MF-DFA Method

#### 3.1. Calculation Steps of the MF-DFA Method

- (1)
- Constructing a new series $y(i)$

- (2)
- Dividing segments at equal intervals

- (3)
- Detrending

- (1)
- Determining the ${q}^{th}$ order fluctuation function for the full series:

- (2)
- Calculating the ${q}^{th}$ order generalized Hurst exponent and the multifractal singularity spectrum

#### 3.2. Extraction of Multifractal Characteristic Parameters

- (1)
- The generalized Hurst exponent parameters

- (2)
- The mass exponent spectrum ($\tau (q)~q$) parameter

- (3)
- The multifractal singularity spectrum ($f\left(\alpha \right)~\alpha $) parameter

## 4. Calculation of Multifractal Parameters for the Driving Cycles and Analysis of Results

#### 4.1. Calculation Modeling

#### 4.2. Calculation Results and Analysis

## 5. Conclusions

- (1)
- From Figure 6, the fluctuation functions of the four driving cycles satisfy a power-law relationship with scale s, which indicate that they are scale-free within the specified scale variation, have fractal characteristics. Meanwhile, it can be seen from Figure 6a–d that overall, the log-log curves of URRDC have the best linear relationship and the log-log curves of WLTC have the largest fluctuation.
- (2)
- From Figure 7, the generalized Hurst exponents of the four driving cycles decrease with the increase of $q$, which indicate the existence of irregular multifractal characteristics of each series. And by calculating the generalized Hurst exponent parameters, it is concluded that these exponents are all between 0 and 0.5, which indicate that the driving cycles have long-range anticorrelations. By comparing the values of $\Delta h(q)$, $\overline{h(q)}$ and ${S}_{h(q)}$, the multifractal strength in order from weak to strong is URRDC, NEDC, CLTC-P, and WLTC.
- (3)
- From Figure 8, the mass exponent spectrum curves of the four driving cycles are upwardly convex, further indicating that these cycles have multifractal characteristics. As mentioned above, a larger value of $\left|{K}_{\tau (q)}\right|$ indicates a more inhomogeneous distribution of the probability measures over the entire fractal structure of the time series, and a greater degree of nonlinearity. By comparing the magnitude of the $\left|{K}_{\tau (q)}\right|$, the inhomogeneity in the distribution of the driving cycles is concluded to be URRDC, NEDC, CLTC-P, and WLTC from lowest to highest, and this conclusion is the same as that of the generalized Hurst exponent curves.
- (4)
- From Figure 9, the multifractal singularity spectra of the four driving cycles are downward opening parabolas, also verifying that these cycles have multifractal characteristics. The width of the opening of each multifractal singular spectrum differs, indicating that the multifractal strength varies. By comparing the values of the width $\Delta \alpha $ of the multifractal singularity spectrum, it is concluded that the inhomogeneity of the probability measure distribution from the lowest to the highest is URRDC, NEDC, CLTC-P, and WLTC, which is the same conclusion as for the mass exponent spectrum parameters and the generalized Hurst exponent parameters.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

EMS | Energy Management Strategies |

MF-DFA | Multifractal Detrended Fluctuation Analysis |

NEDC | New European Driving Cycle |

WLTC | World-wide harmonized Light duty Test Cycle |

CLTC-P | China Light-duty Vehicle Test Cycle for Passenger Car |

URRDC | Urban Road Real Driving Cycle |

DFA UDC | Detrended Fluctuation Analysis Urban Driving Cycles |

EUDC | Extra Urban Driving Cycle |

EUE | Economic Commission of Europe |

CATC | China Automotive Test Cycle |

CLTC | China Light-duty Vehicle Test Cycle |

CLTC-C | China Light-duty Vehicle Test Cycle for Commercial Car |

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**Figure 3.**The speed trace for the CLTC test cycles. (

**a**) The speed trace for the CLTC-P test cycle; (

**b**) The speed trace for the CLTC-C test cycle.

**Figure 6.**Log-log curves of fluctuation function ${F}_{q}(s)$ versus scale s for each driving cycle. (

**a**) NEDC; (

**b**) WLTC; (

**c**) CLTC-P; (

**d**) URRDC.

$\mathit{h}(2)$ | ${\mathit{h}(\mathit{q})}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{h}(\mathit{q})}_{\mathit{m}\mathit{a}\mathit{x}}$ | $\mathbf{\Delta}\mathit{h}(\mathit{q})$ | $\overline{\mathit{h}(\mathit{q})}$ | ${\mathit{S}}_{\mathit{h}(\mathit{q})}$ | |
---|---|---|---|---|---|---|

NEDC | 0.042 | −0.039 | 1.911 | 1.951 | 0.814 | 0.860 |

WLTC | 0.036 | −0.015 | 2.182 | 2.197 | 0.950 | 0.993 |

CLTC-P | 0.036 | −0.015 | 2.068 | 2.083 | 0.891 | 0.935 |

URRDC | 0.024 | −0.015 | 1.843 | 1.858 | 0.782 | 0.831 |

NEDC | WLTC | CLTC-P | URRDC | |
---|---|---|---|---|

${K}_{\tau (q)}$ | 0.918 | 1.064 | 1.006 | 0.893 |

${\mathit{\alpha}}_{0}$ | ${\mathit{\alpha}}_{\mathit{m}\mathit{i}\mathit{n}}$ | ${\mathit{\alpha}}_{\mathit{m}\mathit{a}\mathit{x}}$ | $\mathbf{\Delta}\mathit{\alpha}$ | $\mathit{f}({\mathit{\alpha}}_{\mathit{m}\mathit{i}\mathit{n}})$ | $\mathit{f}({\mathit{\alpha}}_{\mathit{m}\mathit{a}\mathit{x}})$ | $\mathbf{\Delta}\mathit{f}$ | $\mathit{Z}$ | |
---|---|---|---|---|---|---|---|---|

NEDC | 0.087 | −0.091 | 2.024 | 2.115 | 0.488 | −0.127 | 0.615 | 0.188 |

WLTC | 0.048 | −0.049 | 2.289 | 2.338 | 0.653 | −0.070 | 0.723 | 0.043 |

CLTC-P | 0.047 | −0.049 | 2.179 | 2.229 | 0.652 | −0.110 | 0.763 | 0.045 |

URRDC | 0.030 | −0.044 | 1.950 | 1.994 | 0.714 | −0.070 | 0.785 | 0.038 |

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**MDPI and ACS Style**

Yuan, M.; Luo, W.; Lan, H.; Qin, Y.
Research on Multifractal Characteristics of Vehicle Driving Cycles. *Machines* **2023**, *11*, 423.
https://doi.org/10.3390/machines11040423

**AMA Style**

Yuan M, Luo W, Lan H, Qin Y.
Research on Multifractal Characteristics of Vehicle Driving Cycles. *Machines*. 2023; 11(4):423.
https://doi.org/10.3390/machines11040423

**Chicago/Turabian Style**

Yuan, Mengting, Wenguang Luo, Hongli Lan, and Yongxin Qin.
2023. "Research on Multifractal Characteristics of Vehicle Driving Cycles" *Machines* 11, no. 4: 423.
https://doi.org/10.3390/machines11040423