Research on arc length threshold of pantograph-catenary interaction leading to excessive disturbances in traction drive systems

1.1. Background description

The electrified railway catenary is a suspended system installed along the railway, which supplies electrical energy to trains through sliding contact with the pantograph, serving as the sole source of continuous power for electric locomotives. However, current collection issues remain a major constraint on the stable operation of electrified railways, with pantograph-catenary detachment identified as a critical factor affecting current collection stability. During high-speed train operation, the contact between the pantograph and the catenary is often disturbed by vehicle vibrations, track irregularities^[1], and structural hard points^[2], resulting in transient separations and electrical arcing^[3-5] [Figure 1]. Stability of current collection remains one of the major constraints on the maximum operating speed and reliability of electrified railways, with pantograph-catenary (PC) off-line events recognized as a dominant source of power instability^[6,7]. Arcing during these events causes voltage drops at the rectifier side of the traction converter, leading to torque fluctuations in the traction motor^[8]. In addition, the resulting arcs accelerate wear of the collector strip^[9] and generate harmonic and electromagnetic interference that disrupts traction communication systems^[10-12]. Understanding and mitigating PC arcing is therefore essential for improving current collection performance and ensuring the safe, stable operation of high-speed trains.

Figure 1. Schematic diagram of arc generated by pantograph-catenary off-line.

1.2. Literature review

Extensive studies worldwide have focused on traction network modeling, PC arc phenomena, and arc detection based on their characteristic features. For traction network modeling, Chen et al.^[13] proposed a Thévenin-Thévenin equivalent circuit model that incorporates distributed parameters while simplifying the traction network into a two-port circuit with controlled sources. Brenna et al. and Dolara et al.^[14,15] developed a general electromagnetic model for 2 × 25 kV systems, in which geometric and electrical parameters can vary with time and line conditions. Huang et al.^[16] established a traction network link model that includes the impedance-frequency characteristics of an integrated grounding system within tunnel sections. Traditional arc detection techniques primarily rely on analyzing the current derivative and its high-frequency components to identify key features such as peaks, shoulders, and rise rates^[17-19]. In recent years, image-recognition-based methods have advanced rapidly with the development of artificial intelligence and pattern recognition technologies^[20,21]. The optical radiation emitted by PC arcs can be captured using dedicated imaging equipment and analyzed through spectral decomposition, morphological analysis, and color feature extraction to achieve a multi-dimensional representation of arc characteristics. Although deep neural networks offer high accuracy and recall in real-time arc image recognition, their performance remains strongly dependent on ambient illumination and visibility, limiting effectiveness under occluded or low-light conditions. Gao et al.^[22] employed a self-developed fourth-order Hilbert curve fractal antenna to detect arcs based on their electromagnetic radiation signals. Yan et al.^[23] proposed a low-voltage direct-current (DC) arc detection approach using characteristic frequency analysis, applying a 90-110 kHz band-pass filter for feature extraction and an improved convolutional neural network (CNN) to achieve over 99.88% classification accuracy with real-time performance. Li et al.^[24] used a dedicated arc generator to observe current zero-crossing cycles, extracting five temporal parameters for quantitative characterization and employing an optimized support vector machine (SVM) classifier to achieve detection accuracy exceeding 96%. Du et al.^[25] enhanced arc detection robustness and accuracy through the fusion of current and electromagnetic signals combined with ensemble empirical mode decomposition (EEMD)-based feature extraction. In addition, several railway companies, such as the French National Railway Company (SNCF), have explored diagnosing arcs through their disturbances on electrical equipment^[26,27], though the corresponding arc threshold issue has not been addressed.

1.3. Contribution of this work

The aforementioned traction network models typically simplify the messenger wire, droppers, and contact wire into a single equivalent conductor, making it impossible to accurately measure current distribution parameters or detect electrical connection signals. Consequently, these models exhibit certain discrepancies compared with real operating conditions. Existing studies on arc detection exhibit inherent limitations. Image recognition approaches are highly dependent on environmental conditions, while feature extraction methods are often affected by spectral overlap from operational interference sources - particularly harmonic distortions introduced by onboard alternating current (AC)-DC conversion modules - compromising the accuracy of arc feature identification. In summary, this study establishes an indirect detection framework based on the disturbance tolerance of electrical parameters in the traction drive system. By analyzing the impact of pantograph-catenary arcing on system voltage and current fluctuations, the occurrence of arcs is inferred, and the maximum tolerable arc length within allowable disturbance limits is determined. This study constructs an extended AT traction network model that accounts for components such as the messenger wire, introduces a parameter-extended Habedank black-box arc model, and develops a complete traction drive system simulation model incorporating a pulse rectifier circuit and traction motor. The main contributions of this work are as follows:

(1) An equivalent AT traction network impedance model incorporating components such as the messenger wire and droppers was established.

(2) A coupled arc-locomotive model is developed that accounts for the influence of locomotive power on arc dynamics, offering higher physical realism

(3) Based on the permissible fluctuation ranges of input and rectifier-side DC voltages, the study defines a critical arc length threshold beyond which the traction drive system experiences unacceptable disturbances.

2.1. Construction of the extended AT traction network model

In high-speed railway traction networks, the parallel transmission conductors conform to a multiconductor transmission line (MTL) model. However, the presence of autotransformers (ATs), electric locomotives, and cross-connection lines introduces current branches at regular intervals, dividing the network into multiple uniform subsections. This structure - comprising MTL segments as the backbone irregularly segmented by parallel branches - forms a chain-type circuit topology. The model consists of longitudinal series components and transverse shunt components: the longitudinal part represents the parallel MTL segments of the traction network, while the transverse branches mainly include traction substations, ATs, locomotives, and cross-connection lines, as illustrated in Figure 3.

Figure 3. Chain circuit.

The electrical parameters of the traction network - including series impedance and shunt admittance - form the foundation of electrified railway system modeling, as they are directly linked to the mathematical representation of the network. Accurate parameterization is essential for validating voltage levels, calculating voltage drops, and designing and setting relay protection strategies. Based on the conductor configuration shown in Figure 4 and the primary conductor properties listed in Table 1, the electrical parameters of each conductor - specifically the impedance and admittance matrices - are computed. For computational simplicity, the catenary structure is appropriately simplified: since droppers and electrical connectors are arranged perpendicularly between the contact and messenger wires, their mutual inductance with other conductors is neglected. Additionally, due to their short lengths relative to the contact wire, mutual inductance among these short conductors is also omitted. The calculation results are shown in Table 2.

Figure 4. Spatial distribution of conductors in the traction network.

2.1.1. Impedance calculation

The AT traction network primarily comprises the T, F, R, and PW conductors, with the overhead conductors including T, F, and PW. The equivalent self-impedance to ground for a single overhead conductor, as well as the mutual impedance between any two conductors, can be calculated using Carson’s theory. Under the assumption of Kirchhoff’s laws and equal voltage drops along the lines, a theoretical ground return conductor is introduced deep beneath the earth. Each overhead conductor forms a closed loop with this fictitious return path, establishing an overhead line model with the earth serving as the return medium. Figures 5 and 6 illustrate the equivalent modeling of single- and two-conductor configurations, respectively.

Figure 5. Equivalent model of a single-conductor-to-ground loop.

Figure 6. Equivalent model of a two-conductor-to-ground loop.

The self- and mutual impedances shown in the Figure 6 are calculated according to Equation (1)

(1) $$ \begin{equation} \begin{aligned} \begin{array}{l}Z_{i i}=r_{i}+r_{g}+j 0.1466 \lg \frac{D_{g}}{R_{i}}(\Omega / \mathrm{km}) \\Z_{i j}=r_{g}+j 0.1466 \lg \frac{D_{g}}{d_{i j}}(\Omega / \mathrm{km}) \\D_{g}=\frac{0.2085}{\sqrt{f \sigma \times 10^{-9}}}(\mathrm{~cm}) \end{array}\end{aligned} \end{equation} $$

where r_i and r_g denote the internal resistance of the conductor and the ground, Ω/km, respectively; R_i is the equivalent radius of the conductor, cm; d_ij represents the distance between conductors i and j, m; f is the current frequency, Hz; σ is the soil resistivity, 1/(Ω/cm); and Z_ii, Z_ij are the self- and mutual impedances of the conductors, Ω/km, respectively.

2.1.2. Admittance calculation

Let conductors i and j, as shown in Figure 7, be defined. Using methods from power system analysis, the self-potential coefficient P_ii of the conductor and the mutual potential coefficient P_ij between two conductors can be derived as follows:

(2) $$ \begin{equation} \begin{aligned}P_{i i}=\frac{1}{2 \pi \varepsilon_{0}} \ln \frac{2 h_{i}}{r_{i}}(\mathrm{~km} / \mathrm{F}) \\P_{i j}=\frac{1}{2 \pi \varepsilon_{0}} \ln \frac{D_{i j}}{d_{i j}}(\mathrm{~km} / \mathrm{F}) \end{aligned} \end{equation} $$

where ɛ₀ is the dielectric constant of air, ɛ₀ = 8.854 × 10^-9 (F/km); D_ij is the image distance between conductors i and j, m; d_ij is the physical distance between conductors i and j, m; r_i is the equivalent radius of conductor i, cm; and h_i is the height of conductor i above the ground, m.

Figure 7. Conductor and its image.

2.2. PC off-line arc modeling

Under varying operating conditions, Cassie and Mayr each derived mathematical formulations for off-line arcs based on the principle of energy conservation. The two models differ in applicability: the Mayr model effectively represents arc behavior near the current zero-crossing, whereas the Cassie model better captures arcs sustained away from this region. To achieve a more comprehensive representation of arc dynamics, Habedank integrated the two models in series, forming a unified black-box arc model. The corresponding nonlinear governing equation is expressed as follows:

where g_c represents the arc conductance of the Cassie model, S; g_m represents the arc conductance of the Mayr model, S; and g represents the arc conductance of the Habedank model, S. This model is centered around arc conductance, where the arc is embedded in the circuit topology in the form of conductance, thereby influencing the changes in electrical parameters.

The parameters of the arc model, such as the voltage gradient P₀ and dissipated power E₀, vary dynamically and cannot be regarded as constants. To more accurately capture the electrical behavior of PC off-line arcing, this study incorporates the dynamic separation process of the PC system into the Habedank black-box model. For the arc voltage gradient, following the Cassie model assumption - which neglects energy dissipation at the electrodes and assumes dependence solely on arc length - the dissipated power E₀ is reformulated as a function proportional to the arc length^[28]. Moreover, previous experimental investigations have shown that dissipated power exhibits an exponential dependence on arc conductance. further demonstrated, through repeated AC switching arc experiments, that the dissipated power also varies with the arc length L_arc. Accordingly, the extended expressions for both the voltage gradient and dissipated power are derived as follows:

By combining Equation (3) and Equation (4), the black-box arc model, considering the dynamic off-line process of the PC, can be expressed as

where τ_c and τ_m are the time constants of the Cassie arc model and the Mayr arc model, respectively. According to Liu et al.^[28], k₁ = 13, k₂ = 10¹⁰, and α = β = 1.

2.3. Traction drive system modeling

The main circuit of a high-speed train AC traction drive system primarily consists of a pantograph, main circuit breaker, onboard transformer, traction converter, traction motor, and transmission system, as illustrated in Figure 8. The traction converter comprises a four-quadrant pulse rectifier, an intermediate DC link, and a traction inverter. During operation, the roof-mounted pantograph collects 25 kV, 50 Hz single-phase AC power from the traction network, which is transmitted through a vacuum circuit breaker (VCB) and high-voltage cables to the onboard transformer. The transformer steps the voltage down to 1.55 kV, 50 Hz AC, supplying the four-quadrant rectifier. The rectifier converts this input into 3 kV DC, which is regulated by the intermediate DC link before being delivered to the traction inverter. The inverter generates three-phase AC with variable voltage and frequency, providing controlled torque and speed for the traction motor.

Figure 8. Schematic diagram of the traction drive system.

In conventional AC drive systems, rectifier control strategies typically include DC voltage control, AC control, indirect current control, and direct current control. In this study, a three-level four-quadrant rectifier adopts a transient current control strategy that integrates an outer voltage loop and an inner current loop, forming a dual closed-loop architecture. This configuration offers a simplified design, reduced DC-link voltage ripple, and enhanced dynamic response. The specific control function is expressed in^[28]

where K_p and T_i denote the parameters of the PI controller; K represents the amplifier gain; I_d and U_d are the DC-link current and voltage, respectively; U_N(t) and U_N denote the instantaneous and nominal values of the grid-side voltage; i_N is the grid-side current; and U_d^* and i_N^* are the reference values of the DC voltage and grid-side current, respectively.

For the inverter stage, SVPWM is implemented, with coordinated control by a dual closed-loop regulator and proportional-integral (PI) controller. The voltage outer loop regulates the rectifier input, while the current inner loop governs the inverter output. The PI controller accelerates transient response, minimizes steady-state error, and improves overall system stability and performance^[28].

3.1. The traction network model validation

To validate the proposed model, this study conducts a comparative analysis between simulation results and measured current distribution data from a high-speed railway traction network. As shown in Figure 9, a high-current generator was connected to the messenger wire at a fixed feeding point, and the current values of current-carrying droppers were measured at various positions. A simulation model replicating the same structural configuration (three consecutive spans) was constructed under conditions matching the field test. The accuracy of the contact suspension model and its parameters was verified by comparing simulated and measured current values.

Figure 9. Schematic diagram of current distribution test for the contact suspension.

The relative error (Re) between the measured and simulated current data, calculated using Equation (7), is shown in Table 3. The current variation trend obtained from the simulation closely matches the measured results, with the deviation consistently maintained within 10%. Furthermore, as the current magnitude increases, the reliability of the contact suspension model improves, validating the accuracy and validity of the model.

where I_S represents the measured current data, A, and I_E represents the simulated current data, A.

3.2. Arc model validation

This study validates the PC arc dynamic model through a combination of simulation and experimental measurements. If the simulation waveform and the measured waveform exhibit consistency in qualitative analysis, and the electrical external characteristics of the arc are closely matched, it indicates that the constructed PC arc dynamic model and the associated parameter settings can effectively describe the PC arc phenomenon. The model is reasonable, and the results are valid. Gao et al.^[29] developed a PC arc testing system based on the arc generation mechanism. The system captures voltage and current signals during the pantograph-raising process through its integrated measurement modules. The schematic structure of the experimental setup is shown in Figure 10.

Figure 10. Schematic structure of the experimental setup.

A comparison between Figure 11 and the measured waveform by Gao et al.^[29] shows that the simulated arc voltage waveform exhibits excellent agreement with the measured results in the overall trend. Within each cycle, the amplitudes of the arc initiation, sustaining, and extinction voltages gradually decrease with increasing arcing duration. These findings confirm that the extended formulations of the voltage gradient E₀ and dissipated power P₀, together with the selected parameters, provide an accurate representation of the dynamic behavior of PC off-line arcing.

Figure 11. The arc waveform obtained from the simulation.

3.3. Traction drive system model validation

The traction drive system parameters are configured according to Table 4. The simulated waveforms obtained from the numerical model are presented in Figure 12.

Figure 12. (A) represents the DC-side voltage of the rectifier, while (B) represents the output voltage of the inverter.

As shown in Figure 12A, the output of the numerical model stabilizes at approximately 3,000 V, fully consistent with the rectifier parameters of the China Railway High-Speed 3 (CRH3) high-speed train traction converter. The inverter output waveform for one phase, presented in Figure 12B, exhibits a well-formed sinusoidal shape with an amplitude of 3,000 V and a frequency of 138 Hz, closely matching the actual operating conditions of the CRH3 train.

4.1. Considering the allowable disturbance range of the input voltage

To investigate the acceptable disturbance range of the PC off-line arcing threshold under different U_in, simulation calculations were conducted for five train speeds: 200, 225, 250, 275, and 300 km/h. For each scenario, arc lengths (L) were set to 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 cm. The input voltage at the secondary side of the traction drive system’s onboard transformer was measured for each of the six arc lengths at the same train speed. By comparing these measurements, the arc threshold for the PC system, considering the allowable disturbance range of the input voltage, was determined. Figure 13 illustrates the schematic of the U_in position. Simulation results are shown in Figures 14 and 15.

Figure 13. Schematic diagram of the onboard transformer secondary-side input voltage.

Figure 14. Disturbance of different arc lengths at V = 200 km/h.

Figure 15. Corresponding arcing threshold under different vehicle speeds.

Figure 14 illustrates the input voltage waveforms under a train speed of 200 km/h, with arc lengths ranging from 0.5 cm to 1.0 cm. Figure 15 presents the corresponding input voltage values under five operating speeds. As shown in Figure 14, when PC separation occurs and arcing is initiated, the amplitude of U_in experiences varying degrees of attenuation. During recontact, U_in first drops sharply and then recovers, forming a transient voltage spike. According to the work by Zhang^[30], the acceptable voltage range of the traction system is 1,085-1,922 V. From the enlarged view in Figure 14, the disturbance arc-length threshold at 200 km/h is approximately 0.8 cm. Figure 15 summarizes the L_Tin across different speeds. The inset shows that U_in decreases with increasing train speed for a given arc length, and L_T decreases from 0.887 cm to 0.826 cm as the speed rises from 200 km/h to 300 km/h.

The phenomena observed in Figures 14 and 15 result from the transformation of the initially stable “contact wire-pantograph-vacuum circuit breaker (VCB)-high-voltage cable-onboard transformer” path into a strongly nonlinear, time-varying circuit due to PC arcing. When the pantograph briefly loses contact with the catenary, an arc is self-generated to maintain current continuity, and the arc voltage increases with train speed, leading to a reduction in the transformer input voltage. Upon recontact, the sudden extinction of the arc and abrupt restoration of a metallic conduction path cause a transient current surge, producing a voltage spike. Moreover, as the train speed increases, aerodynamic loading, vehicle vibration, and catenary dynamics intensify, increasing the arc’s dissipated power. To sustain energy transfer across the arc channel, higher energy is required, resulting in greater voltage attenuation at the same arc length and a corresponding decrease in L_Tin.

4.2. Considering the fluctuation range of the DC side voltage

According to the EN50163:2020 standard and relevant manufacturer specifications, when the DC-link voltage fluctuation of the traction converter exceeds 5%, the traction inverter can still operate, but its performance begins to deteriorate. When the fluctuation exceeds 10%, significant torque and current pulsations occur, leading to degraded inverter performance and potentially compromising the long-term safe and stable operation of the train.

Therefore, this study defines the allowable fluctuation range of the DC-link voltage as 5%. Simulations are conducted under five train speeds with arc lengths set to 0.4, 0.6, 0.8, 1.0, 1.2, and 1.4 cm. The amplitude of DC voltage fluctuation (ΔU_dc) during arcing is calculated to determine L_Tdc within the permissible DC-link voltage fluctuation range, as given in

where U_dcN denotes the nominal DC-link voltage amplitude under normal operating conditions, which is 3,000V, and U_dcarc represents the DC-link voltage during arcing, V.

Figure 16 shows the schematic diagram of the U_dc measurement location. The simulation results are presented in Figures 17 and 18.

Figure 16. Schematic diagram of DC-side voltage.

Figure 17. Disturbance of different arc lengths at V = 200 km/h.

Figure 18. Corresponding arcing threshold under different vehicle speeds.

Figure 17 presents the rectifier-side DC voltage (U_dc) waveforms for train operation at 200 km/h with arc lengths ranging from 0.4 cm to 1.4 cm, while Figure 18 shows the corresponding U_dc values under five speed conditions. As shown in Figure 17, during the arcing period (2.1-2.18 s), U_dc exhibits varying degrees of attenuation. The voltage drop reaches its maximum around t = 2.15 s for all six arc length conditions. Based on the 5% allowable fluctuation criterion for U_dc, the enlarged view in Figure 17 indicates that the L_Tdc at 200 km/h is approximately 1.2 cm. In Figure 18, as the arc length increases, the corresponding ΔU_dc rises across all train speeds. When V = 200 km/h, increasing the arc length from 0.4 cm to 1.4 cm raises the voltage fluctuation amplitude from 2.15%-5.41%. Moreover, for the same arc length, the fluctuation amplitude increases with train speed - from 2.15% at 200 km/h to 6.59% at 300 km/h for L = 0.4 cm. Within the 5% permissible range, the corresponding L_Tdc decreases from 1.28 cm at 200 km/h to 0.75 cm at 300 km/h.

The observed phenomena in Figures 17 and 18 can be attributed to the nonlinear load characteristics of the PC arc, which introduce transient voltage oscillations and high-frequency noise into the system. During arcing, the discontinuous arc current and the repeated extinction-reignition process generate overvoltages and current spikes, which propagate through the rectifier and cause fluctuations in U_dc. As the arc length increases, arc impedance and energy dissipation both rise, leading to enhanced ΔU_dc. Meanwhile, higher train speeds amplify PC instability and arc intensity, inducing stronger transient disturbances and greater voltage drops. Consequently, for a fixed arc length, the DC voltage fluctuation becomes more pronounced at higher speeds. Furthermore, as the train speed increases from 200 km/h to 300 km/h, the L_Tdc corresponding to the 5% allowable voltage fluctuation decreases from 1.28 cm to 0.75 cm. This indicates that the traction converter becomes more sensitive to voltage disturbances at higher speeds. The combined effects of increased arc impedance, intensified dynamic interactions, and reduced voltage tolerance at high speeds lead to the nonlinear rise in ΔU_dc and the decrease in L_Tdc.

Figure 19 summarizes the critical arc length thresholds corresponding to excessive voltage disturbances under different operating speeds. When the train speeds are 200, 225, 250, 275, and 300 km/h, the arc length thresholds corresponding to the L_Tin are 1.28, 1.25, 1.23, 1.07, and 0.75 cm, respectively, while those for the L_Tdc are 0.887, 0.878, 0.848, 0.839 and 0.826 cm. Notably, L_Tin drops sharply by 29.9% as the speed increases from 275 km/h to 300 km/h. Moreover, at speeds below 290 km/h, L_Tin remains consistently higher than L_Tdc. Both the numerical results and Figure 19 reveal a clear decreasing trend in arc length thresholds with increasing train speed. This indicates that as the operating speed rises, the traction drive system becomes more sensitive to PC arcing, reducing its tolerance to arc length. The primary reason lies in the intensified dynamic interaction between the pantograph and the catenary at higher speeds, which increases the arc voltage gradient and dissipated power, thereby amplifying voltage disturbances even for identical arc lengths. The sharp drop in L_Tin between 275 and 300 km/h suggests that beyond a critical speed, the voltage stability of the system deteriorates rapidly, and the impact of arc energy on the input voltage exhibits nonlinear amplification. Furthermore, the consistently higher L_Tin than L_Tdc below 290 km/h implies that the AC-side input voltage has a slightly greater tolerance to arc disturbances compared to the DC side, owing to the rectifier’s filtering effect and the dynamic response characteristics of the DC link. Overall, the formation and maintenance of an arc essentially represent a transient energy transfer process, in which the dissipated power directly reflects the intensity of energy injection from the arc into the traction system. Experimental observations indicate that as the arc length increases, the arc voltage rises while the arc current slightly decreases. The combined effect of these variations leads to a significant increase in the rate of energy injection per unit time, causing over-limit voltage fluctuations at the input terminal of the traction system. This phenomenon reveals a pronounced nonlinear coupling relationship between arc energy and the energy dynamics of the traction system. Specifically, under low-speed conditions, the system’s filtering and inductive energy storage components can absorb part of the arc energy, resulting in relatively small voltage fluctuations. In contrast, under high-speed conditions, the arc length varies more dramatically, and the rate of change in arc power is higher, preventing the input voltage disturbance from attenuating within a short time and thereby producing larger transient overvoltage responses.

Figure 19. Arc length threshold.