# Statistical Study on the Time Characteristics of the Transient EMD Excitation Current from the Pantograph–Catenary Arcing Discharge

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Setup and Measured Time Characteristics

- Peak amplitudes (A
_{+}and A_{−}), the positive and negative peak amplitudes of a single current pulse; - Rise time (t
_{r}), the time interval between the instants at which the instantaneous current value first reaches 0.1 A_{+}and then 0.9 A_{−}; - Pulse width (t
_{w}), i.e., halfwave duration, the period between the instants at which the instantaneous value reaches 50% of its peak amplitude; - Pulse repetition interval (t
_{in}), the time interval between two successive pulses in a pulse train.

## 3. Experimental Results and Analysis

#### 3.1. Statistical Distribution of Transient Current Waveform Parameters

_{+}and A

_{−}), rise time (t

_{r}), pulse width at half maximum (t

_{w}), and interval time (t

_{in}) of current pulses, as well as their probability density functions (PDF), are calculated with the distribution fitting tool from MATLAB software. As depicted in Figure 5, the statistical distributions of the waveform parameters with a sample size of 1000 are graphically described in the form of distribution histograms associated with various theoretical reference distributions. The characteristics of individual pulses are quite different, which is mainly reflected in the randomness of pulse amplitude, rise time, and interval time.

_{+}as an example) is between 4 and 17 A, which accounts for 96.2% of the total number of samples, and the right-biased distribution of the pulse amplitude A

_{+}essentially follows the Burr distribution law. The rise time is predominantly distributed in the region of 23 to 27 ns, and there is less distribution in the range of 27 to 39 ns. The t Location–Scale distribution can better describe the data distribution with heavier tails (more prone to outliers). The pulse width corresponds more closely to the Birnbaum–Saunders distribution model, which results in a better fit at the distribution peak. The repetition interval time distribution of the pulse presents a left-skewed Extreme Value distribution, and the smaller the pulse interval time, the higher the repetition frequency, so the fitting accuracy on the left side of the peak is very important.

^{2}) and the root mean squared error (RMSE). A smaller RMSE value or an R

^{2}value closer to 1 is employed to choose a suitable model among the candidate theoretical distributions.

#### 3.2. Statistical Analysis on the Peak Amplitudes of Transient Current Pulses

_{+}and A

_{−}) as they vary with the applied voltage and the gap spacing distance. The probability density function $f\left(A\right)$ has the following expression:

_{+}and A

_{−}) increase with higher applied voltages, primarily due to the occurrence of multi-point discharges caused by the strengthening electric field. Positive and negative peak amplitudes increase significantly more in the lower voltage range of 20 to 25 kV, whereas the distribution of peak amplitudes varies little and stabilizes gradually in the higher voltage range of 25 to 35 kV. Figure 7d–f shows that, despite a constant applied voltage, the peak value of the current pulse increases considerably as the distance of the contact gap increases. The influence of the gap spacing on the peak amplitudes of the discharge current can be explained as follows: a greater d allows a space charge to travel a greater distance, resulting in the accumulation of more charges and a reduction in the total electric field. The decrease of the total electric field weakens the discharge at the conductor’s surface, hence reducing the number of seed charges at the beginning of the discharge and the peak amplitudes of the resulting discharge pulse.

_{−}is more compact than that of A

_{+}, indicating that the variance of A

_{−}is significantly smaller. Comparing the average values of A

_{+}and A

_{−}with various applied voltages reveals that the changes of A

_{+}with the applied voltages are comparable to those of A

_{−}, and that the average values of the two are nearly twice as close. Figure 8 depicts scatter plots for A

_{+}versus A

_{−}under different settings, showing the value relationship between A

_{+}and A

_{−}from a single current pulse. From the plots, we can observe a generally tight positive correlation between the positive peak amplitudes of the pulsed current and its negative peak amplitudes, which indicates a nearly identical ratio between A

_{+}and A

_{−}under the same condition. The ratio of the positive peak amplitude A

_{+}and the following negative peak amplitude A

_{−}in one single current pulse can be expressed as

_{A}, the variation of R

_{A}at different voltages from 20 to 40 kV with gap spacing distance changes from 5 to 15 mm is shown in Figure 9. The average values and standard deviations (STD) are used to represent the fluctuations of R

_{A}in this figure.

_{A}increases with d; however, at the same spacing, R

_{A}maintains a constant value while U changes from 20 kV to 40 kV, showing that the applied voltage has no influence over R

_{A}. This phenomenon is thought to be brought on by a variation in the arc length, or the distance between the electrodes, which changes the arc resistance. The arc resistance R

_{arc}can be expressed using the following formula with reference to [33] when the arc is thought of as a plasma cylinder.

#### 3.3. Statistical Analysis on the Pulse Repetition Intervals

_{0}is the coefficient that adjusts the applied voltage value to match the experimental data more closely. The physical significance of U

_{0}is the inception voltage of the arcing discharge. However, Equation (9) is insufficient to describe the changing law of the repetition frequency with varying gap spacing distances. Fortunately, close correlations between K

_{R}and d as well as U

_{0}and d were found in the experiments, as shown in Figure 10. The fitted relational expression between K

_{R}and d exhibits a power law:

_{0}and d exhibit a linear association because of the linear relationship between the breakdown field strength and the gap distance,

_{R}, U

_{0}, and d are expressed in kHz, kV, and mm, respectively.

#### 3.4. Statistical Study of Transient Current Pulse Rise Time and Pulse Width

_{r}follows the t Location–Scale model, represented as

_{r}under various experimental settings are determined using the ML estimation approach with a 95% confidence interval.

_{w}.

## 4. Deviation of the Stochastic Model of Arcing Currents

_{p}is the zero-crossing point of the positive pulse component, and t

_{e}is the end time of the single current pulse.

_{r}of several nanoseconds and t

_{w}of tens of nanoseconds, whereas the MDEF model was proposed to characterize pulses with low t

_{w}/t

_{r}ratios.

_{r}, pulse width at half maximum t

_{w}, and fall time t

_{f}(the time interval from 90% to 10% of the maximum value), and the function’s mathematical parameters, denoted as α and β, commonly need to be transformed into each other with high precision. Using the numerical solution and asymptotic formulations to make estimations, Ref. [36] established the correlations for βt

_{w}, t

_{w}/t

_{r}, and t

_{f}/t

_{r}with β/α for Equation (23), which are expressed as follows:

_{+}), rise time (t

_{r}), pulse width (t

_{w}), and separation intervals (t

_{in}) of current pulses, can be obtained through a pseudo-random number generation algorithm. The proportion ratio between the positive and negative amplitudes at different voltages, shown in Figure 9, can be used to calculate the negative amplitude (A

_{−}). The mathematical parameters α and β are calculated with Equations (24) and (25), which, in turn, can be used to estimate the fall time t

_{f}in Equation (26) to determine t

_{p}and t

_{e}in the following equation.

_{r}, t

_{w}) generated by the parameter distribution functions. Assuming N to be the total number of single pulses in a pulse train, the N single pulses can be represented by the notation ${I}^{\mathrm{i}}\left(t\right)$, $1\text{}\text{}\mathrm{i}\text{}\text{}N$. Then, we can produce N groups of random t

_{in}based on the Extreme Value distribution stated in Equation (15) and apply the following formula to splice the pulse train sequence:

_{in}has a mean value of 0.185 and a standard deviation of 0.064, consistent with statistical calculations. The pulse train simulated by this method is very similar to the measured one. The stochastic model of the discharge waveform is effective based on the description of the statistical distribution of the waveform.

_{p}, t

_{e}) and physical parameters (A

_{+}, A

_{−}, t

_{r}, t

_{w}, t

_{in}, t

_{f}), and the establishment of the time-domain waveform of the transient current for both single pulse and pulse train. The detailed description of the calculation process has been given in previous sections.

_{+}, t

_{r}, t

_{w}, and t

_{in}with statistically typical characteristics can be obtained from the identified distribution function and the selected quantiles for different severity levels. Using the relationship between the analytic parameters (α and β) and the waveform parameters in Equations (24)–(27), the MDEF model (in the form of Equation (21)) representing the single pulse waveform can be finally derived from the triplet (A

_{+}, t

_{r}, t

_{w}).

## 5. Conclusions

- The statistical distribution of pulse peak amplitude, rise time, pulse width at half maximum, and pulse interval time follows a specific reference distribution determined by the distribution fitting method. The chosen reference distribution model was validated through a Kolmogorov–Smirnov hypothesis test.
- Statistical analysis demonstrates that the distribution parameters of each fitted model vary with the applied voltage and the gap spacing distance between the electrodes. The fitting relation between the waveform parameters and the experimental conditions is established using a maximum likelihood estimation approach. This allows for the determination of statistically typical current waveform parameters representing different EMD severities and parameters in the later deduced current function.
- A new bipolar MDEF is introduced to describe the amplitude of the PC arcing current as a function of time, and is used to model the ratio between the positive and negative peak amplitudes, which increases with a larger gap spacing. The mathematical parameters of the bipolar MDEF are calculated from the waveform parameters generated by a pseudo-random number generation algorithm, based on the known time characteristics’ statistical distributions. The measured and calculated waveforms show a correlation coefficient of approximately 0.956, confirming the validity of the proposed generation method for a single pulse.
- The PC discharge current pulse train is characterized as the superposition of separated single current pulses generated by the stochastic model. The simulated pulse train closely matches the measured one, indicating that the stochastic model of the discharge waveform is effective in describing the statistical distribution of the waveform.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 5.**Distribution histogram of measured waveform parameters and theoretical reference distributions: (

**a**) pulse amplitude (A

_{+}); (

**b**) rise time (t

_{r}); (

**c**) pulse width (t

_{w}); (

**d**) repetition interval time (t

_{in}).

**Figure 7.**Statistical distributions of pulse amplitudes under different conditions: (

**a**) U = 20 kV, d = 5 mm; (

**b**) U = 25 kV, d = 5 mm; (

**c**) U = 30 kV, d = 5 mm; (

**d**) U = 35 kV, d = 5 mm; (

**e**) U = 35 kV, d = 10 mm; (

**f**) U = 35 kV, d = 15 mm.

**Figure 8.**A

_{+}versus A

_{−}from one single current pulse at different conditions: (

**a**) U = 20 kV, d = 5 mm; (

**b**) U = 25 kV, d = 10 mm; and (

**c**) U = 35 kV, d = 15 mm.

**Figure 9.**Ratio relationship between the positive amplitude and negative amplitude at different voltages: (

**a**) average values; (

**b**) STDs.

**Figure 10.**Measured and fitted results of K

_{R}and U

_{0}with varying gap spacing: (

**a**) K

_{R}; (

**b**) U

_{0}. The solid lines are the least squares fitting lines.

**Figure 11.**Measured waveform and simulated waveform of discharge current pulse. Measured physical characteristics: t

_{r}= 22.98, t

_{w}= 45.84, A

_{+}= 8.53, A

_{−}= 3.88; Calculated mathematical parameters: α = 0.0242, β = 0.0289, t

_{p}= 92.91, t

_{e}= 185.81; Fitted function parameters: k

_{1}= 8.064, k

_{2}= 8.427.

**Figure 12.**Schematic diagram of the random current pulse train. Single pulses are stretched to show their shapes in a pulse train.

**Figure 13.**Comparison of the simulated and measured current pulse train: (

**a**) measured pulse train with $\overline{{t}_{\mathrm{in}}}=0.185$ ms and ${s}_{\mathrm{in}}=0.064$ ms; (

**b**) simulated pulse train with $\mu $ = 0.214 and $\sigma $ = 0.050 for the Extreme Value distribution of t

_{i}.

**Figure 15.**Simulated excitation current for moderate and serious EMD situations: (

**a**) moderate situation with parameters: α = 0.0217, β = 0.0413, A

_{+}= 14.580, R

_{A}= 2.2, t

_{r}= 18.731 ns, t

_{w}= 40.949 ns; (

**b**) severe situation with parameters: α = 0.0196, β = 0.0489, A

_{+}= 19.218, R

_{A}= 2.2, t

_{r}= 17.522 ns, t

_{w}= 42.386 ns; (

**c**) critical situation with parameters: α = 0.0182, β = 0.0588, A

_{+}= 26.252, R

_{A}= 2.2, t

_{r}= 15.419 ns, t

_{w}= 43.805.

Parameter | Reference Distribution | ML Estimation | R^{2} | RMSE |
---|---|---|---|---|

A_{+} | Burr | α = 9.519, c = 6.259, k = 0.983 | 0.894 | 0.018 |

Birnbaum–Saunders | β = 9.593, γ = 0.277 | 0.873 | 0.020 | |

Gamma | a = 13.424, b = 0.742 | 0.856 | 0.021 | |

t_{r} | t Location–Scale | μ = 19.017, σ = 1.243, ν = 2.024 | 0.967 | 0.016 |

Burr | α = 17.518, c = 228.853, k = 0.037 | 0.932 | 0.023 | |

Lognormal | μ = 2.981, σ = 0.117 | 0.743 | 0.044 | |

t_{w} | Birnbaum–Saunders | β = 40.949, γ = 0.041 | 0.943 | 0.019 |

Burr | α = 40.416, c = 50.860, k = 0.696 | 0.942 | 0.021 | |

Log-Logistic | μ = 3.710, σ = 0.022 | 0.920 | 0.025 | |

t_{in} | Extreme Value | μ = 0.911, σ = 0.025 | 0.934 | 0.012 |

Rician | s = 0.896, σ = 0.029 | 0.894 | 0.020 | |

Burr | α = 0.945, c = 40.489, k = 5.229 | 0.923 | 0.015 |

Parameter | Distribution Law | D_{n} |
---|---|---|

A_{+} | Burr | 0.047 |

t_{r} | t Location–Scale | 0.023 |

t_{w} | Birnbaum–Saunders | 0.031 |

t_{in} | Extreme Value | 0.034 |

Significance level: $\alpha =0.01$ | ||

Critical value: $D(n,\alpha )=0.051$ |

Cumulative Probability | Peak Amplitude (A) | Rise Time (ns) | Pulse Width (ns) | Repetition Interval (ms) |
---|---|---|---|---|

5% | 5.96 | 15.41 | 38.27 | 0.835 |

50% | 9.55 | 19.01 | 40.95 | 0.902 |

80% | 11.94 | 20.33 | 42.39 | 0.923 |

95% | 15.36 | 22.61 | 43.81 | 0.939 |

Experimental Condition | Mean Value (ms) | Standard Deviation (ms) | CV (%) |
---|---|---|---|

U = 20 kV, d = 5 mm | 0.905 | 0.310 | 34.3 |

U = 25 kV, d = 5 mm | 0.453 | 0.157 | 34.7 |

U = 30 kV, d = 5 mm | 0.243 | 0.083 | 34.1 |

U = 35 kV, d = 5 mm | 0.185 | 0.064 | 34.8 |

U = 35 kV, d = 10 mm | 0.312 | 0.109 | 35.1 |

U = 35 kV, d = 15 mm | 0.571 | 0.196 | 34.4 |

Experimental Condition | $\mathbf{Location}\text{}\mathbf{Parameter}\text{}\mathbf{\mu}$ | $\mathbf{Scale}\text{}\mathbf{Parameter}\text{}\mathbf{\sigma}$ | $\mathbf{Shape}\text{}\mathbf{Parameter}\text{}\mathbf{\nu}$ |
---|---|---|---|

U = 20 kV, d = 5 mm | 19.017 | 1.243 | 2.024 |

U = 25 kV, d = 5 mm | 18.681 | 1.229 | 2.019 |

U = 30 kV, d = 5 mm | 19.118 | 1.235 | 2.014 |

U = 35 kV, d = 5 mm | 18.731 | 1.143 | 2.023 |

U = 35 kV, d = 10 mm | 20.032 | 1.361 | 2.026 |

U = 35 kV, d = 15 mm | 20.889 | 1.585 | 2.024 |

Experimental Condition | Scale Parameter β | Shape Parameter γ |
---|---|---|

U = 20 kV, d = 5 mm | 40. 949 | 0.041 |

U = 25 kV, d = 5 mm | 39.681 | 0.038 |

U = 30 kV, d = 5 mm | 39.562 | 0.041 |

U = 35 kV, d = 5 mm | 38.169 | 0.039 |

U = 35 kV, d = 10 mm | 37.124 | 0.038 |

U = 35 kV, d = 15 mm | 38.758 | 0.039 |

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## Share and Cite

**MDPI and ACS Style**

Jin, M.; Wang, S.; Liu, S.; Fang, Q.; Liu, W.
Statistical Study on the Time Characteristics of the Transient EMD Excitation Current from the Pantograph–Catenary Arcing Discharge. *Electronics* **2023**, *12*, 1262.
https://doi.org/10.3390/electronics12051262

**AMA Style**

Jin M, Wang S, Liu S, Fang Q, Liu W.
Statistical Study on the Time Characteristics of the Transient EMD Excitation Current from the Pantograph–Catenary Arcing Discharge. *Electronics*. 2023; 12(5):1262.
https://doi.org/10.3390/electronics12051262

**Chicago/Turabian Style**

Jin, Mengzhe, Shaoqian Wang, Shanghe Liu, Qingyuan Fang, and Weidong Liu.
2023. "Statistical Study on the Time Characteristics of the Transient EMD Excitation Current from the Pantograph–Catenary Arcing Discharge" *Electronics* 12, no. 5: 1262.
https://doi.org/10.3390/electronics12051262