Simulation on Operating Overvoltage of Dropping Pantograph Based on Pantograph–Catenary Arc and Variable Capacitance Model

Abstract

When the electric locomotive pantograph is dropping, the interruption of pantograph catenary contact causes electromagnetic oscillation and arcing. The frequent arc burning that occurs due to charge accumulation results in the amplitude of overvoltage increasing gradually, posing a threat to locomotive high-voltage equipment. However, the physical mechanisms and characteristics of overvoltage are still unclear. This paper proposes a simulation model of operating overvoltage due to a dropping pantograph based on the pantograph–catenary arc and variable capacitance. Distributed RLC electromagnetic oscillation is considered, which allows the real-time calculation of arc resistance and capacitance. Under the same working conditions, the error between the simulation and test results is less than 4.0%, which proves the credibility of the model. The variation law of overvoltage under different dropping speeds or catenary phases was investigated, which shows the max amplitude is 298.20 kV and steepness is 2096.80 kV/μs at 0.30 m/s speed. The waveform shows the characteristics of high amplitude and high steepness, similar to very fast transient overvoltage (VFTO). There is a sinusoidal relationship between the catenary phase and overvoltage amplitude. The closer the catenary phase to 90°, the higher the overvoltage amplitude. The research has important guiding significance for the overvoltage formation mechanism of a traction power supply system and the insulation coordination design of high-voltage equipment.

1. Introduction

When the locomotive pantograph is dropping, the interruption of the pantograph–catenary contact causes a sudden change in circuit, triggering electromagnetic oscillation. During this process, an ultra-fast transient overvoltage with an amplitude and steepness far higher than normal levels (AC 27.5 kV 50 Hz) is generated, which can be detected on the high-voltage equipment on the roof of the electric locomotive. This type of overvoltage is similar to the VFTO, which shows high amplitude and high steepness. This extreme electromagnetic phenomenon poses a threat to the electrical system of electric locomotives, resulting in frequent overvoltage shocks, insulation breakdown in high-voltage equipment, and train outages [1]. Accurately simulating ultra-fast transient operating overvoltage is crucial for revealing its generation mechanisms and evolution law.

In order to replicate the overvoltage wave by simulation, domestic and foreign scholars have proposed a variety of overvoltage simulation methods based on the arc model. Yuan utilized a Habedank arc model combined with the Cassie and Mayr model to study the pantograph dropping of a CR400BF high-speed train under different operating speeds, revealing the arc voltage peak was up to 30.00 kV at a speed of 350 km/h [2]. Li improved the dissipative power expression of the Mayr arc model and integrated the optimized model with a traction substation–catenary–train model to simulate the dropping pantograph overvoltage of the CR400BF train. The results show the amplitude of overvoltage around 70.00 kV [3]. Shi combined the Cassie arc model with an equivalent circuit model of the CRH2 train and a simulated operating overvoltage amplitude ranging from 60.00 to 75.00 kV, where the oscillation frequency of the voltage was around 60 kHz [4]. Liu proposed a pantograph arc modeling method by combining the Mayr and Cassie arc models, which achieved state switching through boundary length determination. In this method, the transient processes of arc extinction and reignition were considered, and it was observed that the voltage peak was 72.00 kV on the catenary at the moment of arc extinction during pantograph dropping [5]. Lv et al. studied the influence of electromagnetic voltage transformer iron core resonance on the dropping of operating overvoltage in the locomotive equivalent circuit, using the Mayr arc model. The results indicate that the overvoltage amplitude ranged from 39.00 to 77.00 kV [6]. He et al. simulated arc reignition by analyzing the multiple transient processes of the circuit. The results show that when the phase angle of the catenary voltage was 90°, the maximum catenary overvoltage was 66.45 kV [7]. He et al. studied the characteristics and generation mechanism of very fast transient overvoltage caused by disconnector operation. The simulation results show that the maximum amplitude of VFTO at the circuit breaker reached 1.45 p.u. [8]. Wang et al. established a monitoring system: by paralleling a voltage transformer next to the arrester and transmitting the low-voltage side signal to the data acquisition card through a coaxial cable, the roof overvoltage waveform data during train operation were collected. The maximum overvoltage amplitude of a CRH2A train during pantograph dropping was 67.10 kV; the maximum peak value of harmonic overvoltage on the CR400BF train was 52.80 kV [9].

At present, the arc model considering a locomotive equivalent circuit is primarily adopted in simulations performed to research operational overvoltage, and the simulation results are generally in the range of 50.00~77.00 kV. GB/T 11032 standard [10] stipulates that the power-frequency flashover voltage of the arrester is generally 2.5 to 4.0 times the rated voltage (100 kV). The IEC60850 standard [11] points out that when the overvoltage exceeds three times the rated voltage (>75.00 kV), the risk of high-voltage equipment (especially high-voltage circuit breakers) massively increases. Existing simulation models show low amplitude, but these fail to reveal the mechanisms of severe faults in high-voltage equipment.

This paper proposes a simulation model of operating overvoltage due to a dropping pantograph based on the pantograph–catenary arc and variable capacitance considering the influence of pantograph–catenary capacitance on the operational overvoltage’s amplitude and steepness. The credibility of the model is proven by making a comparison between simulation and test results. The variation law of overvoltage under different dropping speeds or catenary phases is investigated.

2. Formation Mechanism of Pantograph–Catenary Arc and Arc Reignition Overvoltage

2.1. Pantograph–Catenary Arc Development Process

The pantograph–catenary arc is an electric spark phenomenon that occurs when the carbon contact strip of an electric locomotive disconnects from the catenary [12]. As shown in Figure 1, there are three stages during the arc development process. At the moment of the pantograph dropping, the voltage difference between the pantograph and catenary increases rapidly. The air gap between them is broken and then enters the arc burning stage. A bright channel can be seen between two contacts. The temperature of the channel is extremely high, reaching approximately 4000~5000 K within microseconds [13]. As the pantograph dropping process continues, the arc channel elongates and narrows. Simultaneously, the arc resistance increases and the conductivity decreases. The arc enters an unstable state and is easily interrupted by external interference. Finally, the insulation state between the two contacts is restored and leads to arc extinction.

2.2. Mechanism of Arc Reignition Overvoltage

Before the arc burning, the resistance of the pantograph–catenary gap is large (approximately 1 MΩ). At the moment of arc burning, the gap resistance exponentially decreases to around 0.5 Ω within a few microseconds [14]. In this process, the sudden change of arc current and voltage will cause high-frequency electromagnetic oscillation in the system, and then form operating overvoltage, which is generally 2~3 times the size of the system voltage. When considering arc reignition during pantograph dropping, high-amplitude and high-steepness overvoltage, which is up to 9 times the system voltage, will occur in the later stage of arc development process [15]. Figure 2 shows the results of a test of the operating overvoltage of a dropping pantograph waveform during an HX_D1 electric locomotive’s operation (test system used a high-precision oscilloscope with a minimum sampling rate of 500 MHZ and is linear), where multiple arc reignitions are observed. The amplitude of overvoltage increases with the number of arc burning steps, reaching peak during the final burning.

The condition of AC arc reignition is shown in Figure 3. It illustrates two situations when the arc crosses the zero point: reignition or permanent extinction. u₁ represents the dielectric strength and u₂ denotes the recovery voltage; both units are kV. The arc reignites depending on whether the recovery rate of the dielectric strength u₁ is greater than the recovery rate of the gap voltage. At point A in Figure 3a, u₂ is greater than u₁, which can cause arc reignition, and the gap voltage transitions to arc voltage. Figure 3b shows that there is no intersection between u₁ and u₂, meaning it fails to meet the conditions for arc reignition, so the arc undergoes permanent extinction [16].

By introducing the variable capacitance of the pantograph–catenary arc, the repeated breakdown of the pantograph–catenary gap during the pantograph dropping process can be simulated more accurately [17]. The model can simulate a situation of arc reignition after extinguishing and complex RC oscillation between the pantograph and the catenary.

3. Overvoltage Simulation Model Including Variable Capacitance

The existing references show that the Cassie and Mayr arc models are generally used to simulate operating overvoltage. These arc models can be introduced as follows.

3.1. Arc Models

The Mayr and Cassie arc models, based on the principle of energy conservation, are often employed for simulating arc process of power system.

1.

Cassie Model

The Cassie arc model establishes the relationship between arc current and voltage. The equation for the Cassie arc model is

$${\displaystyle \frac{1}{R_{\mathrm{c}}}}{\displaystyle \frac{d{R}_{\mathrm{c}}}{dt}}={\displaystyle \frac{1}{{\tau }_{\mathrm{c}}}}(1-{\displaystyle \frac{u^2}{u_{\mathrm{c}}^2}})$$

(1)

where R_c is the dynamic resistance, τ_c is the arc time constant, u_c is the arc voltage constant, and u is the arc voltage. The model indicates that the arc conductivity changes with the arc current. It is suitable for when there is a high current before current zero crossing.

2.

Mayr Model

The Mayr arc model considers the thermal equilibrium of the arc conductor and gas dynamic characteristics. The equation for the Mayr arc model is

$${\displaystyle \frac{1}{R_{\mathrm{m}}}}{\displaystyle \frac{d{R}_{\mathrm{m}}}{dt}}={\displaystyle \frac{1}{{\tau }_{\mathrm{m}}}}(1-{\displaystyle \frac{Ei}{P_{\mathrm{loss}}}})$$

(2)

where R_m is the dynamic resistance, τ_m is the arc time constant, P_loss is the dissipative power during arc burning, E is the electric field strength in the arc column, and i is the arc current. Equation (2) illustrates the relationship between arc conductivity and arc dissipation power.

According to Section 2.1 and Section 2.2, the arc extinction and reignition processes, as well as the impact of circuit voltage and current on arc resistance, should be taken into account in the pantograph dropping simulation. The Mayr model is suitable for when there is a small current around the current crossing zero, consistent with the characteristics of the pantograph dropping arc. Additionally, the model can describe the relationship between the dissipated power and the arc resistance during arc burning. In summary, this paper chooses the Mayr arc combined with pantograph–catenary variable capacitance as the simulation model to research the operating overvoltage of a dropping pantograph.

3.2. Calculation of Arc Resistance and Variable Capacitance

Due to the difficulties met in directly obtaining arc dissipation power and the arc time constant in the Mayr arc formula, an expression is derived for arc resistance based on the principle of energy conservation in this paper. The expression takes pantograph–catenary distance and arc current as variables, making the calculation of arc resistance more straightforward and less prone to convergence issues in simulations. For simplicity, the following assumptions are made:

• The radius of the arc column is constant;
• The temperature of the arc column is uniformly distributed radially;
• Conductivity and a radiative heat source are functions of temperature;
• The arc column plasma does not contain a space charge.

Based on the above assumptions, the expression for arc conductivity in terms of total arc energy is derived as follows:

$$g=F(Q)=F\left[{\displaystyle \int \left(P-P_0\right)dt}\right]$$

(3)

$${\displaystyle \frac{dg}{dt}}={\displaystyle \frac{dQ}{dt}}\cdot {\displaystyle \frac{dg}{dQ}}=(P-P_0)\cdot {\displaystyle \frac{dg}{dQ}}$$

(4)

where P represents the input power of the pantograph–catenary arc, P₀ denotes the arc output power, and Q represents the energy accumulated during arc formation. Both P and P₀ are functions [18] of the arc conductivity g,

$$P={\displaystyle \frac{i^2}{g}}$$

(5)

$$P_0=k_1{g}^{-\beta }L$$

(6)

where i represents the arc current, L represents the arc length, and k₁ and β are arc constants. By substituting (2) to (5), we derive

$${\displaystyle \frac{dg}{dt}}={\displaystyle \frac{dQ}{dt}}\cdot {\displaystyle \frac{dg}{dQ}}=\left({\displaystyle \frac{i^2}{g}}-k_1{g}^{-\beta }L\right){\displaystyle \frac{dg}{dQ}}$$

(7)

The arc energy Q and arc conductivity g are given by:

$$Q=q\mathsf{\pi }{r}^{2}L$$

(8)

$$g={\displaystyle \frac{\mathsf{\pi }{r}^2\mathsf{\sigma }}{L}}$$

(9)

According to the theory of gas molecule motion, the accumulated energy per unit volume of the arc q is derived as [19]

$$q=0.354p\left(1-{\displaystyle \frac{T_1}{T_0}}\right)$$

(10)

where p represents atmospheric pressure, T₁ is the ambient temperature, and T₀ is the temperature of the generated arc. Assuming that atmospheric pressure p and pressure-dependent specific heat capacity are constants, the arc conductivity σ is derived from the arc theory’s Saha equation [16]:

$$\mathsf{\sigma }={\mathsf{\sigma }}_0{e}^{-{\displaystyle \frac{m}{T_0}}}$$

(11)

where σ₀ represents the conductivity per unit length of the arc, and m is the coefficient. It is observed that the arc energy Q and arc conductivity g are functions of the arc temperature T₀. By solving Equations (7) and (11) simultaneously, we derive

$${\displaystyle \frac{dg}{dQ}}={\displaystyle \frac{\frac{dg}{d{T}_0}}{\frac{dQ}{d{T}_0}}}={\displaystyle \frac{\frac{d\frac{\mathsf{\sigma }\mathsf{\pi }{r}^2}{L}}{d{T}_0}}{\frac{d\left(q\mathsf{\pi }{r}^{2}L\right)}{d{T}_0}}}={\displaystyle \frac{mg}{0.354p{T}_1\mathsf{\pi }{r}^{2}L}}=k_2{\displaystyle \frac{g}{L}}$$

(12)

Substituting Equation (9) into Equation (12), the arc is derived as

$${\displaystyle \frac{dg}{dt}}={\displaystyle \frac{k_2}{L}}{i}^2-k_1{k}_2{g}^{1-\beta }$$

(13)

In the simulation model, the expression of arc resistance can be solved by designing the correlation function-solving module. The range of β is specified as −3 < β < 0 [19].

In the process of arc development, the voltage between the pantograph and catenary contact changes with time, resulting in the corresponding fluctuation of the voltage difference between the contact gaps. This phenomenon can be explained by capacitance effects between the contacts, indicating the existence of variable capacitance in the gap. Gap capacitance causes the voltage to fluctuate sharply in a very short time, thereby inducing overvoltage, which helps to simulate the arc transition process between burning and extinction. The variable capacitance is a physical quantity that changes dynamically with the distance between the pantograph and catenary, which is considered in the simulation model.

ANSYS software (version:2022-R1)was used to simulate the capacitance between the pantograph and catenary. The pantograph model was built. Figure 4a is the actual pantograph structure of the HX_D1 electric locomotive, and Figure 4b is the simulated pantograph with equal proportions.

Since the pantograph dropping process is a process of two contacts being separated, the electric field change between them can be approximately regarded as the electric field change of the plate capacitor, and the gap capacitance decreases with the increase in the pantograph–catenary distance. However, with regard to the process of pantograph dropping, the distance between the pantograph and the catenary begins at 0 m, but it cannot be used as a denominator in the variable capacitance formula. Therefore, in order to ensure the numerical stability of the model, it is necessary to set the limit of the pantograph–catenary capacitance. After setting the grid and boundary conditions in ANSYS, the equivalent capacitance of 250 pF was simulated with a pantograph–catenary distance of 1 μm. When the pantograph–catenary distance is less than or equal to 1 μm, the gap capacitance is fixed at 250 pF. With the increase in pantograph–catenary distance, the capacitance between pantograph and catenary becomes smaller. The variable capacitance expression for the pantograph–catenary is as follows:

$$C=\left\{\begin{array}{llll}{\displaystyle \frac{\epsilon S}{L}}={\displaystyle \frac{250}{L}}& & & (L>1 \mathsf{\mu }\mathrm{m})\\ 250& & & (L\le 1 \mathsf{\mu }\mathrm{m})\end{array}\right.$$

(14)

The equivalent circuit of the pantograph–catenary gap is shown in Figure 5, where R and C represent the arc resistance and variable capacitance between the pantograph and catenary (because there is no variable conductance module in the simulation software(ANSYS 2022-R1), there is only a variable resistance module, so variable resistance was used instead of variable conductance in the simulation model).

3.3. Overvoltage Simulation Model Including Variable Capacitance

In this paper, PSCAD software (version:PSCAD-v46)has been used to simulate the operating overvoltage. This software is suitable for the electromagnetic transient analysis of the power system. It can better simulate the transient process in the power system and has the ability to deal with the complex power system model and high frequency response. The simulation model of the HX_D1 electric locomotive’s roof high-voltage system is shown in Figure 6. The high-voltage equipment mainly includes a pantograph, a high-voltage transformer, a vacuum circuit breaker, and an arrester, where C_ij (i = 1,2; j = 1,2,3,4) represents the capacitance to ground of the wires and high-voltage equipment, R_mn (m = 1,2; n = 1,2,3) represents the equivalent resistance of the connecting wires, Z₁ and Z₂ are surge arresters, Z₃ represents the high-voltage connector between two cars, and S₁ and S₂ represent the locomotive load.

When overvoltage occurs during pantograph dropping, arcs are generated between the catenary-side contact and the vehicle-side contact (Figure 6). The RC element in the model represents the equivalent distributed parameter line of the connecting wire within the high-voltage equipment on the roof.

The simulation software (PSCAD-v46)provides modular functions, allowing users to design circuits with different functions as independent modules and encapsulate these circuits into simulation model to achieve a certain function. The three functional modules designed in this paper are the pantograph–catenary distance control module, the arc burning detection module and the arc resistance and variable capacitance calculation module. This combination of three packaged modules can finally realize the numerical calculation of output arc resistance and variable capacitance. Then, the results are converted into resistance and capacitance types of signals input into the arc between the catenary side and the vehicle side.

3.3.1. Pantograph–Catenary Distance Control Module

The function of the pantograph–catenary distance control module is to simulate the gap distance variation during the pantograph dropping process. As shown in Figure 7, the module’s input parameters include dropping speed V, time t, and delay time Δt. The real-time calculation formula for the pantograph–catenary distance is L = Vt. The actual situation is that pantograph–catenary distance does not infinitely increase, but remains at a constant value after the pantograph dropping process ends.

Therefore, a selector is designed to implement the above-mentioned logic: when L_t > L₀, L_t is the pantograph–catenary distance; otherwise, L₀ is output indicating the end state of the pantograph dropping process. Another output of this module is the signal C₁, which indicates whether arc calculation can begin. Since the pantograph–catenary distance starts from 0, there is a risk of simulation divergence when calculating the arc at very small distances. A threshold is set to avoid this situation: the arc calculation proceeds only when the pantograph–catenary distance exceeds the threshold value (1 μm).

3.3.2. Arc Burning Detection Module

The function of the arc burning detection module is to output the logical signal C₂ that indicates gap breakdown and power frequency current zero crossing. As shown in Figure 8, the module’s input parameters include pantograph–catenary distance L, pantograph–catenary voltage U, and arc current I. The breakdown threshold voltage U₀ is calculated based on L. When U > U₀, the breakdown condition is satisfied and the comparator module outputs a logic signal of “1”. Then the arc current I is filtered to I₁ (50 Hz), which is put into the zero-crossing detection module. When the power frequency current crosses zero, the zero detector module outputs a logic signal of “1”. These conditions for voltage breakdown and current crosses zero are combined through an AND gate. Gap breakdown occurs only when both the voltage breakdown condition is satisfied and the power frequency current has not crossed zero. Otherwise, no breakdown occurs under any other conditions. The output signal is denoted as C₂.

3.3.3. Arc Resistance and Variable Capacitance Calculation Module

The arc resistance and capacitance calculation module is designed to simulate the impedance variation during the arcing process. As shown in Figure 9, by inputting the parameters of the arc differential formula, the differential expression of the arc conductance with respect to time can be obtained. The signal C₁, which is output from the pantograph–catenary distance control module, is the first integration control signal. When the pantograph dropping process is completed, C₁ is set to zero to indicate that the conductance is infinitely small and enters the arc extinction state. C₂ is output from the arc burning detection module, which is the second integration control signal. When the condition of arc extinction is satisfied, the output result is equal to the calculated value of the real-time pantograph–catenary impedance. If the condition is not met, the differential calculation module will output a resistance value (10²⁰ Ω). In order to reduce the amount of calculation in the simulation process, a selector is set to limit the arc resistance to 2.5 × 10¹² Ω when arc extinction occurs.

4. Credibility Verification of Simulation Model

In this paper, a test of the operating overvoltage of a dropping pantograph in an HX_D1 electric locomotive is carried out, and the credibility of the simulation model is verified by comparing the test overvoltage waveform with the amplitude and steepness of the simulated waveform.

4.1. Overvoltage Test

Figure 10 illustrates the structure of the testing system, with the test point located at “M”. Within the substation, electricity at 110 kV was transformed into a single-phase 27.5 kV catenary voltage. The overvoltage wave at the pantograph during the dropping operation was extracted from point M and transmitted through a high-voltage shielded cable to the high-voltage side of the voltage divider, and then the signal was taken from the low-voltage side and input into the high-speed digital storage oscilloscope. The sampling rate of the oscilloscope used in the test ranged from 500 MHz to 2 GHz. The test system could retain the high-steepness and high-amplitude part of the overvoltage wave to the greatest extent.

4.2. Comparison of Simulated and Test Waveforms

In this model, we set simulation parameters consistent with the test to ensure that the simulation accurately reflects the actual conditions. The test parameters are shown in

Table 1. Test and simulated results with parameters.

Test Times	Parameters	Pantograph Dropping Speed	Pantograph–Catenary Distance	Catenary Voltage Phase	Operating Overvoltage of Dropping Pantograph	Error of Simulation Data Compared to Test Data
1	Test Data	0.31 m/s	1.90 m	89°	305.10 kV	2.3%
	Simulation Data	0.31 m/s	1.90 m	89°	298.20 kV
2	Test Data	0.32 m/s	1.90 m	87°	273.90 kV	2.2%
	Simulation Data	0.32 m/s	1.90 m	87°	267.90 kV
3	Test Data	0.29 m/s	1.90 m	91°	253.70 kV	3.9%
	Simulation Data	0.29 m/s	1.90 m	91°	243.80 kV

.

Under the same operating conditions, the error in overvoltage amplitude between the test overvoltage data and the simulation data is less than 4.0%, which proves that the simulation model has high precision. In this paper, only the difference between the simulation and test amplitudes is compared. In Figure 11, there are some differences between the voltage amplitude before and after the arcing. This is because the simulation model cannot fully reflect the actual circuit structure, so there will be a little difference in voltage amplitude. It can be seen in the Figure 11 that the duration time of the arc is tens of nanoseconds, which is similar to the characteristics of VFTO. These conclusions indicate that the simulation model can sufficiently reflect the test conditions, and the error in the simulation results is relatively small.

5. Operating Overvoltage Variation Law of Dropping Pantograph under Different Conditions

5.1. Influence of Pantograph Dropping Speed on Dropping Pantograph Operating Overvoltage

Due to the elongation of the arc column in the process of pantograph dropping, the power injected into the arc and the arc voltage will increase [20]. With the gradual elongation of the pantograph arc, the arc column expands and the contact area between the arc and the air increases, which contributes to the dissipation of energy [21]. Especially when the pantograph and the catenary are separated at a fast speed, the expansion effect may be aggravated, which helps to accelerate the extinction of the arc. In order to study the influence of dropping speed on overvoltage, different conditions are simulated by changing the dropping speed in the model, and the results are shown in

Table 2. Operating overvoltage of dropping pantograph at different pantograph dropping speeds.

Pantograph Dropping Speed (m/s)	Operating Overvoltage of Dropping Pantograph (kV)	Arc Duration (μs)
0.30	298.20	2.35
0.40	201.39	1.96
0.50	122.95	1.90
0.60	94.12	2.34
0.70	78.24	1.96
0.80	78.85	2.22
0.90	39.48	2.02

.

It can be seen in Table 2 that the arc duration is generally a few microseconds. When the pantograph dropping speed is 0.30 m/s, the amplitude of overvoltage reaches its highest value of 298.20 kV, and the arc duration is the longest. When the pantograph dropping speed is 0.90 m/s, the overvoltage amplitude is the lowest (39.48 kV) and the operating overvoltage is almost unobservable in the system. This is due to the faster pantograph dropping speed compressing the space and time available for arc development, causing the arc energy to dissipate rapidly during the process. When the rate of energy dissipation exceeds the rate of energy accumulation, the temperature of the plasma in the arc decreases as the arc resistance increases. The capacitive discharge process is rapid, which induces a weak transient oscillation, and the overvoltage amplitude is relatively small. The law of the variation in the operating overvoltage of the dropping pantograph as dictated by pantograph dropping speed is shown in Figure 12, and it is evident that the amplitude of the operating overvoltage decreases with the increase in pantograph dropping speed.

Figure 13 presents the overvoltage waveforms for four different pantograph dropping speeds (0.30 m/s, 0.50 m/s, 0.70 m/s, 0.90 m/s) given by simulations. The overvoltage at the pantograph dropping speed of 0.30 m/s takes a long time to re-attain the stable state, whereas the overvoltage at a high pantograph dropping speed of 0.90 m/s can rapidly re-attain its steady-state value. This is because when the pantograph is dropping at a low speed, the energy stored in the circuit will oscillate for a long time in the RC components, such as the parasitic capacitance of the cable or the magnetizing inrush current of the voltage transformer, resulting in a large amplitude. On the contrary, the oscillation time is short and the amplitude is small when the pantograph is dropping at high speed.

5.2. Influence of the Catenary Voltage Phase on the Dropping Pantograph Operating Overvoltage

When the pantograph is dropping, the catenary phase is unknown. For this reason, the generation time of the operating overvoltage of the dropping pantograph is uncertain. In order to further study the influence of the phase of the catenary voltage on the operating overvoltage of a dropping pantograph at the time of pantograph dropping, we can change the phase of the catenary at the time of pantograph dropping in the model to simulate this case. The simulation results are shown in

Table 3. Amplitude of pantograph dropping overvoltage at different catenary phases.

The Phase of Catenary Voltage (°)	Operating Overvoltage of Dropping Pantograph (kV)	Arc Duration (μs)
30	79.33	4.98
45	100.32	5.12
60	130.60	2.58
75	176.85	3.33
90	298.20	3.02
105	141.56	2.82
120	99.63	5.57
135	59.50	3.34
150	46.06	2.83

.

As shown in Table 3, when the phase of the catenary voltage is 90°, the operating overvoltage of the dropping pantograph reaches its maximum value of 298.20 kV; when the phase is 150°, the operating overvoltage of the dropping pantograph reaches its minimum value of 46.06 kV. The variation law of the operating overvoltage of a dropping pantograph with the phase of the catenary voltage is illustrated in Figure 14.

When the phase of the catenary voltage is close to 90°, the condition of arc extinction is strengthened, because the pantograph–catenary voltage difference also reaches its maximum value. On the contrary, when the phase of catenary voltage is close to 0°, the condition of arc burning is not easily met due to the voltage of the pantograph, and the catenary gap is almost 0. In engineering practice, a device for the real-time monitoring of the phase of the catenary voltage can be installed to select the best time to drop the pantograph.

6. Conclusions

(1)

In this paper, a simulation model of the operating overvoltage of a dropping pantograph based on the pantograph–catenary arc and variable capacitance is proposed. The error between the simulation and test overvoltage amplitudes under the same conditions is less than 4.0%, which proves the credibility of the model.

(2)

The duration of the arc is tens of nanoseconds, which is similar to the duration that is characteristic of VFTO. This shows that the operating overvoltage of a dropping pantograph has ultra-fast transient characteristics.

(3)

The amplitude of the operating overvoltage of a dropping pantograph increases with the decrease in pantograph dropping speed. When the speed is 0.30 m/s, the amplitude of the operating overvoltage of the dropping pantograph reaches the maximum value of 298.20 kV. When the speed is 0.90 m/s, the operating overvoltage of the dropping pantograph reaches the minimum value of 39.48 kV, and almost no overvoltage is observed at this time.

(4)

The amplitude of the operating overvoltage of a dropping pantograph varies sinusoidally with the phase of catenary voltage. When the phase of catenary voltage is 90°, the amplitude of the operating overvoltage of the dropping pantograph reaches the maximum value of 298.20 kV. When the phase of the catenary voltage is 150°, the amplitude of the operating overvoltage of the dropping pantograph reaches the minimum value of 46.06 kV.