Multi‐physics analysis and optimisation of high‐speed train pantograph–catenary systems allowing for velocity skin effect

A pantograph–catenary system (PCS) – an essential component to supply a high-speed train (HST) – faces a variety of new challenges due to the continuously increasing train speeds. The HST traction system receives power via an electrical contact between the pantograph strip and the high-voltage contact wire. This electrical contact is subject to serious mechanical shocks and significant electrochemical corrosion, making the modelling of the dynamic processes complicated, especially under high-speed and heavy-load conditions. The damage to the PCS – which is particularly noticeable at the edges of the pantograph strip – may become severe as the speed of the train rises. Moreover, as the speed increases, the distribution of the electrical current in the strip becomes uneven due to the velocity skin effect (VSE). To assess the impact of the VSE on the performance of PCSs, a multi-physics model has been created and is reported in this study. The model has been validated through experiments and the main aspects of its functionality – such as the VSE, friction, and air convection – have been identified and analysed at different speeds. The impact of speed on the traction current and the behaviour of thermal sources have been explored. With the increasing speed, the phenomenon of current clustering at the trailing edge of the strip becomes quite dramatic, resulting in a thermal surge in the region of the strip with high current density. To mitigate the negative impact caused by VSE in the PCSs, an improved kriging optimisation methodology has been utilised to optimise the parameters of the PCS. Recommendations regarding the optimal design of the PCS are put forward to improve the current-carrying performance and reduce the local temperature rise in the strip.


Introduction
The pantograph-catenary contact, fluid-structure interaction, and wheel-rail performance may be regarded as the three fundamental relationships for high-speed trains (HSTs), which determine their safety and reliability [1]. Being responsible for power transfer to HSTs, the pantograph-catenary system (PCS) plays a critical role in maintaining an uninterruptable and stable power supply for traction and control systems. The traction current is transferred from the overhead high-voltage catenary to the pantograph through an electrical contact. Unlike normal electrical connections, the sliding contact between the pantograph strip and the overhead wire is exposed to serious mechanical shocks, substantial electrochemical corrosion, and generated high joule heat, especially under high-speed and heavy-load conditions. Moreover, because of the high maintenance cost of the catenary -and to protect the contact wire -the pantograph strips are designed as consumable components to abrade preferentially (Fig. 1).
The wear of the pantograph strip has always been of crucial concern in relation to the sliding electrical contact of the PCS. To understand the factors influencing the wear of the strip, research studies have analysed the effects of the sliding speed, contact force, electrical current, friction coefficient, and arc energy [2][3][4], while models for wear prediction have been constructed to describe the relationship between those factors [5]. With the drive to pursue higher and higher speeds, improvements to the structure and material of the pantograph strip are essential to satisfy the growing requirements in terms of the impact resistance, abrasion resistance, and electrochemical erosion resistance. The research has mainly focused on the sliding electrical contact of the PCS at low and medium speeds. Recently, the fastest HST named China Standard EMU (the electric multiple unit) was put into operation with a maximum speed reaching 350 km/h; the new generation of HSTs are expected to be even faster. As the speed of HST exceeds 350 km/h, new technical challenges emerge [6]. The increase of relative motion velocity exacerbates vibration and friction of the PCS, resulting in pantograph-catenary arcing [7] and severe ablation [8].
A test rig to simulate the sliding electrical contact of the PCS at speeds of 350 km/h [6] has been used, and it has been reported that although the upward tendency of the wear rate of the pantograph strip has further increased with the increasing speed, the upward tendency of arc energy has slowed down, while the friction coefficient has decreased. It has also been reported in [6] that the upward tendency of the temperature of the pantograph strip has further increased with the increase of speed. Bearing in mind that the wear rate of the strip is positively related to its temperature [9], an analysis of the factors that can increase the temperature of the strip and thus further increase the wear of pantograph strip in addition to arc, vibration, and friction, appears to be crucial. As speed increases, more traction power is needed to overcome air resistance; the convection process is affected, and traction current increases. Moreover, due to the motion between the pantograph strip and the catenary, a velocity skin effect (VSE) becomes pronounced [10,11] and may become quite dramatic as the speed increases. The VSE is a phenomenon of current clustering observed in a region near the contact interface between two objects [12]. Owing to the VSE, the current tends to cluster at the trailing edge of the pantograph strip. The unevenly distributed current density inside the pantograph strip may directly lead to a local thermal surge. This local thermal surge at the trailing edge may trigger thermal damage in the region. Moreover, when pantograph strips pass under the rigid spots (the hitch points for hanging the contact wire) at high speed, the mechanical shock will exacerbate the damage, thus significantly reducing the efficiency and reliability of power transmission.
The damage to the pantograph is further aggravated by the already mentioned phenomenon of VSE, well known and thoroughly researched in the context of electromagnetic launchers, which has some similarities to the performance of PCSs, although the speeds here are not so high. Initially, VSE was observed as an asymmetric wear of recovered armatures [12]. Then, to explore its origins and properties, several VSE models were established for medium and low speeds [13,14]. The characteristics of the VSE were found to be affected by several factors, such as armature geometry, material resistivity, and contact surface. A scheme for mitigating the current clustering phenomenon caused by VSE was also considered [15,16]. The latest studies of VSE mainly focus on the multi-physics aspects, including vibration, gouging, transition, and melting erosion [17][18][19][20], which take place in EMLs at very high speeds.
Despite some similarities, there are also important differences between the operation of EMLs and PCSs; in the latter, the sliding electrical contact lasts a long time and the pantograph strip is continuously exposed to mechanical shocks, electrochemical corrosion, and high joule heat. The contact between the strip and the wire may be considered to be a line contact rather than a surface contact. To control the wear of the surface of the pantograph strip, the wire above the train body is arranged in a zigzag formation, thus the friction between the strip and contacting wire takes place not only in the direction of the train motion but also 'sideways'. In this study, the VSE in a PCS is examined. A multi-physics model -including electrical, magnetic, and thermal fields -has been created. Based on the model, the impact of the HST velocity on the traction current and other thermal sources can be assessed so that recommendations regarding the optimal design of the PCS may be put forward.

PCS model structure
The PCS configuration is shown in Fig. 2. The traction current from a substation is transmitted to the train through an electric contact from a 25 kV wire to the pantograph strip.
To facilitate the pursuit of higher speeds and larger traction power, advanced pantographs of higher electrical conductivity (e.g. Ti 3 SiC 2 /Cu) have been considered. Owing to a larger traction current, higher speed, and better conductivity of the pantograph strip, the VSE in the PCS can no longer be ignored. A model of the pantograph strip and contact wire has been established to describe induced currents [21][22][23]. The geometry parameters of the pantograph strip (Type number: CS0039) and contact wire (Type number: CTMH-150) are assumed as for the currently used systems as listed in Table 1 (the labels of parameters are consistent with the ones mentioned in Fig. 2b). The structure of the model and experimental setup for the PCS is presented in Fig. 3; the cooling source in Fig. 3a is used to maintain the contact wire at ambient temperature. Material properties of the pantograph strip and contact wire are listed in Table 2, with a friction coefficient of 0.2 selected based on experimental results.

Magnetic effect
As mentioned before, with the rising operating speeds and increasing traction power, there is a requirement for advanced pantograph strips. Compared with original strips, which had poor conductivity (<10 5 S/m), the conductivity of advanced strips (10 6 -10 7 S/m) is generally much higher. Consequently, the currents induced by the magnetic field of the PCS cannot be omitted [11], which is consistent with where B is the magnetic flux density, v is the velocity of HST, µ is the magnetic permeability, and σ m is the electric conductivity of materials. Higher train speeds and increased currents both lead to a greater variation of the magnetic field.

Heat sources
Several heat sources contribute to temperature distribution within a PCS. The total heat Q T includes Joule heat Q J and friction Q f , so that where the total equivalent resistance R J is made up of the volume resistance R V and contact resistance R c , which in turn is the sum of a shrinkage resistance R s and film resistance R f [3,24]. The film resistance may be ignored when relative motion takes place between the pantograph and the catenary; as a result, the contact resistance R c is defined as where ρ 1 and ρ 2 are resistivities of the pantograph strip and contact wire, respectively, and α is the equivalent radius of the actual contact area. If the shape of the contact spot is assumed to be a circle, then the actual contact area of a rough surface is where F is the pantograph-catenary contact force, n is the number of contact spots per unit area, ξ is the modification factor of elastic deformation, and H is the Brinell hardness of materials. Finally, the heat source related to friction is described by where μ k is the friction coefficient of the PCS.

Heat diffusion
Cooling of the PCS is primarily achieved by convection at speeds of around or above 350 km/h. Following [25][26][27], the external forced convection Q c is described by where h c is the externally forced convection coefficient (taking account of the shape of the strip), T atm is the temperature of the airflow surrounding the HST, T is the surface temperature of the strip, ɛ is the surface emittance coefficient of materials, and σ is the Stefan-Boltzmann constant. Heat radiation effect Q r is included above in (7).

Verification of the PCS model
To verify the effectiveness of the proposed model, tests using a PCS experimental platform (see Fig. 3) have been undertaken and the results are presented in Fig. 4. The thermal distributions locating on the strip's surface are observed by an infrared thermal imager (Type number: NEC InfRec R300 series) with the measuring range between 0 and 2000°C and with a temperature resolution as 0.05°C. Experiments at speeds of 0 and 20 m/s have been carried out with a test current of 100 A. Comparison between experimental and modelling results reveals that the deviation of stabilised temperature is within 4%. It may therefore be argued that the model reflects with acceptable accuracy the thermal processes of a PCS at low speeds. The measurements were constrained by the limitation of the speed of the test device of 72 km/h. The temperature distributions around the strip's surface during the tests are presented in Fig. 5. Over time, the temperature rise in the strip is higher than the temperature rise in the contact wire, while the maximum temperature appears at the centre of the contact surface.
The current distribution of the PCS is affected by the VSE when the HST operates at 350 km/h; the velocity skin depth δ v [28,29] is defined by where k e indicates the comparability between the saw tooth wave and the sine form current [28] (in our case k e = 1), w is the interface length of the contact region (here the width of the pantograph strip), and v is the pantograph speed with respect to the catenary. For the purpose of verification, a finite difference method (FDM) was also applied and the results compared with the analytical method and finite element simulations (FEMs). In the FDM model of the PCS, the pantograph strip was assumed to be stationary with the straight contact wire moving relative to this static strip. Otherwise, the remaining geometrical parameters in the FDM model were the same as for the FEM. Following (1), the magnetic diffusion equations governing the process in the contact wire and in the pantograph strip may be described as where σ w is the electric conductivity of the contact wire and σ s is the electric conductivity of the pantograph strip. The boundary conditions suggested in [30] have been adapted here for the geometry of the PCS. The uneven ratio of current r δ is defined as the ratio of maximum current density j m to current density at velocity skin depth j δ on the contact surface of the pantograph strip, is utilised to evaluate the uneven current caused by VSE. The uneven ratio r δ = e according to [29]. Assuming electric parameters of the PCS as in Table 2 and the speed of 350 km/h, several conditions with different widths of pantograph strips have been attempted by the analytical functions to observe the variation tendency of the velocity skin depth. Since the uneven ratio of current r δ is the same at the velocity skin depth, the velocity skin depth can also be obtained by the PCS model. The results of δ v obtained by the analytical method, the FDM model of PCS, and the FEM model of PCS are presented in Fig. 6.
Comparing the results obtained from the FEM model, FDM model and analytical equations, the deviation between the FEM model and analytic model is ∼7.4% on average, while the deviation between the FEM model and FDM model is ∼9.1%, which is considered to be acceptable.

Modelling results
The contact wire of a PCS is static, while the pantograph strip slides forward against the bottom of the wire. The current in the strip is affected by the magnetic field of the current in the wire. The sliding strip is exposed to a VSE; the consequences of such influence are depicted in Fig. 7, where the cases without and with the effect are shown for a Ti 3 SiC 2 /Cu strip (assuming current of 1 kA and speed of 350 km/h).
It can be seen that the VSE makes the current cluster at one side of the strip. The operating current is normally selected under the assumption of uniform current distribution, thus localised increases above such value will inevitably lead to undesirable thermal expansion effects and may accelerate the damage to the strip. Therefore, when pantograph strips have higher conductivity, the magnetic effects must not be neglected.

Velocity skin effect (VSE)
The speed of the train directly influences the VSE and consequently the current distribution. Assuming the traction current of 1 kA and strip conductivity of 2.5 × 10 6 S/m, the current density distributions along the width of the strip at different speeds are shown in Fig. 8a as a clear demonstration that the VSE becomes stronger with rising speed.
Conductivity also contributes to the VSE, although the current is hardly affected by the magnetic field in low conductivity materials, but strongly if conductivity is high, as seen in Fig. 8b, assuming electric parameters of four typical strip materials as in Table 3 and the speed of 350 km/h. Table 3 presents the velocity skin depth of different potential materials for the pantograph strips, which is calculated based on (8).
There is another, perhaps less obvious, influence related to the movement of the train in relation to the direction of the supply current. As the train moves along the track, it is supplied by one of the substations, either behind or in front, while passing by a substation also results in a change (from 'left line' to 'right line'), as explained by a simple sketch in Fig. 9. Thus, the pantograph strip either moves 'away' from the supply current or 'towards' it, as indicated at the bottom of Fig. 9; this results in two distinct conditions for the strip.
The effect is elucidated by the results of Fig. 10. The two distributions, on the left and on the right, may appear to be a mirror image, but in reality there is a noticeable difference in the actual values of both the flux density and, more importantly, the current density, as clearly marked in relevant boxes. Thus, when the train pulls away from the supplying substation the current density may be higher by as much as 10% compared with the case when it is moving towards a substation.

Temperature distributions
The variation of the maximum strip temperature with the speed is shown in Fig. 11a, with the assumed traction current of 0.5 kA and the strip conductivity of 2.0 × 10 6 S/m. The temperature rises as the speed increases, and the strip endures higher temperatures at speeds above 350 km/h.
The variation of the maximum strip temperature with the strip conductivity is shown in Fig. 11b. Lower conductivity means larger volume and contact resistance of the strip, resulting in a sharp rise of the temperature. Consequently, manufacturers impose a rigid limit on current density to restrict the use of low conductivity strips; to obtain larger current capacity, high conductivity is necessary. With the increase of strip conductivity, the contribution of Joule heat loss to the temperature rise of the strip is noticeably reduced, thus the effects of friction and thermal dissipation of convection become dominant. However, higher conductivity results in enhanced VSE in the strip, causing a local increase of current density. Overall, due to the thermal effects of the contact resistance, the temperature of the contact area slightly increases.
The degree of wear of the surface of the contact wire also affects the temperature distribution in the pantograph strip. Compared with the strip, the contact wire has a longer service lifespan of 10-15 years. The bottom section of a new contact wire is arc-shaped, as shown in Fig. 12a. However, the contact wire is exposed to wear, gradually causing a flat bottom as in Fig. 12b; the contact area between a worn wire and the pantograph strip is slightly increased compared with the case of a brand new wire.
Accounting for the wear condition, the temperature distribution of the pantograph strip varies, as shown in Fig. 11c, ultimately causing a reduction of the maximum temperature of the pantograph strip.

Initialisation of the PCS structural optimisation
The wear of the pantograph strip and the resulting damage to the PCS are described in [7,31] and are clearly aggravated as the speed of the train goes up, as explained before. The temperature rise may, however, be limited somewhat by careful choice of the shape and dimensions of the strip [32]; this observation gave the motivation to conduct proper optimisation of the PCS. Hence the focus of this stage of the study was to achieve some optimal parameters of the pantograph strip and the contact wire to improve the thermal dissipation performance. Many parameters can influence the maximum temperature and current distributions. After careful consideration, the following three were identified as the most important: the strip width w strip , the thickness of the strip h strip , and the cross-sectional radius of the contact wire r wire , as shown in Table 4, while the remaining parameters were fixed as already given in Tables 1 and 2.

Objective functions
The thermal performance and current-carrying capability are the performance indices when designing pantograph strips. According to Fig. 11b, it can be found that the maximal temperature in the pantograph strip varies non-linearly with the unevenness ratio of current distribution. Hence when optimising the structure of pantograph strips, the design objectives of the optimisation problem need to involve the uneven ratio of current r a [33] and surficial temperature of pantograph strip T max separately. r a is defined as the ratio of maximum current density j m to average current density j a on the contact surface of pantograph strip. The two objectives are defined as follows: The initial values of the critical design variables are shown in Table 4, while the remaining parameters are fixed as shown in Table 2. To control the emphasis on these two objectives, a pair of weighting parameters is imposed on them. The objective function including two sub-objectives is defined as

Structural optimisation of pantograph strips
As a kind of regression model, kriging [34] is able to exploit the spatial correlation of data to predict the shape of the objective function and pursue the global optimum based on limited known information. Moreover, it can estimate the accuracy of this prediction throughout the iterative process, which may be very helpful in guiding the optimiser to locate the next evaluation point (vector). As the critical step in the kriging model, the correlation function is defined as follows (it relies on the linear regression model (13) and the Gaussian correlation model (14)) The sum ∑ k = 1 m β k f k x includes the linear combination of the values of initial sampled points x and may be viewed as a global approximation to the true function. The coefficients β k are regression parameters, while ε is an additive Gaussian noise, representing the uncertainty. The model is statistical by assuming that the errors ε x i and ε x j are correlated. θ k determines how fast the correlation between design vectors drops away in the kth coordinate direction, meanwhile p k determines the function smoothness in the kth coordinate direction. The whole optimisation process made by the kriging surrogate model may be viewed as a Gaussian process γ, while a number of updating schemes can be adopted in this process. This allows the concept of improvement to be defined: for a single objective to be minimised, the improvement may be measured by comparing the value realised by the objective function with the current minimum of prediction, as presented below The expectation of improvement, which may also be called the expected improvement (EI) [35], is based on a potential error -the 'mean square error' produced by the kriging model -and is commonly used to select multiple designs for further evaluation.
Here the structural optimisation of pantograph strips may be viewed as minimising the objective function (13). The EI function is defined as where ŷ(x) is a value of the objective function predicted by kriging; f min is the minimum value of y at the existed samples; s(x) is the root mean square error produced by kriging; Φ and φ are the normal density and normal distribution functions, respectively. The maximum of EI indicates the position to select new sampling points when proceeding with iterations.
To reduce the computational cost of kriging in finding the next sampling point by constructing a full field EI over the whole design space based on very large correlation matrices, the kriging optimiser [36], developed previously, offers an effective methodology for searching a global optimum and was selected to perform this three-variable optimisation task. The 'direct optimisation' approach, motivated by a Lipschitzian algorithm, can address difficult global optimisation problems with constraints. The Lipschitzian optimisation process may be described as If the function f is Lipschitz continuous with a constant α, then this information can be utilised to seek the optimum of f iteratively. The optimisation algorithm used in this study is a hybrid kriging optimiser, which is a combination of the direct algorithm and kriging [37]. The optimising procedures for kriging and hybrid kriging are shown in Fig. 13. The direct algorithm was used to assist kriging in selecting the next sampling point with an optimal value of EI, rather than building up a full-scale EI over the whole design space based on very large correlation matrices. The number of sampling points selected by kriging was increasing while the data produced by the EI remained almost constant.

Multi-objective optimisation
The optimisation goal of the pantograph-catenary structure contains two sub-objectives, which are minimising the uneven ratio of current distribution r a and minimising the maximal temperature T max appearing in the strips, when considering the impact brought from the Joule heat and friction heat. To adjust the balance between these two sub-objectives, two weighting parameters w 1 and w 2 are imposed on them based on (12) and (13). Some optimisation results with a different combination of weights are presented in Table 5.
When w 1 = w 2 = 0.5, the kriging optimiser required 429 calls of the FEM software before converging at the optimum solution (w strip = 35.2 mm, h strip = 30.5 mm, r wire = 7.4 mm), which provides the maximum temperature of 106.3°C, hence a noticeable improvement of 11.9% compared with the maximum temperature in a prototype, which was 120.7°C. The optimum solution above is to reduce the max temperature as much as possible while limiting the maximum current density. When w 1 = 0.3 and w 2 = 0.7, the best result in terms of the maximum temperature of the pantograph strip is achieved amongst the three cases. However, the best result  related to the current evenness is achieved when w 1 = 0.7 and w 2 = 0.3, which provides the max current density of 1.57 × 10 7 A/m 2 , while the maximum current density in the prototype was 1.63 × 10 7 A/m 2 . Therefore, in practice the weights on these two subobjectives can be adjusted based on the preference of different performance requirements. Finally, to fully validate the thermal distribution of the PCS under the high-speed scenario, a field test will be developed, as the next step, to capture the VSE.

Conclusion
The PCS is a critical component for transmitting constant and stable power to HSTs. In contrast to stationary contacts, a sliding electric contact in a PCS experiences continuous mechanical shocks, arc erosion and high joule heat generation, especially under high-speed and heavy-load conditions. As the speed increases, the strips are prone to more damage, especially at the edges. The VSE phenomenon causes the electric current to cluster at the edges; this effect cannot be neglected at higher speeds and results in local thermal surges leading to thermal damage. To investigate the impact of VSE on the electrical contact of PCS, a multi-physics FEM model of the PCS has been built and its validity verified by a series of experiments and analytical models. Pantograph strips made of different materials have been considered and their performance evaluated in the context of VSE. The relationship between the conductivity and the maximum temperature of the pantograph strip exhibits non-linearly. When the conductivity of the pantograph strip rises, the maximum temperature in the strip initially decreases due to the decreasing volume resistance; this is then followed by an increase in the temperature because of the current clustering in a local region. The optimal thermal performance of the pantograph strips of different conductivity was found at the inflection point of the maximum temperature in the strip.
Some parameters of the PCS have influence on the electrical and thermal performances. The VSE effect is mitigated somewhat when the width of the pantograph strip is increased. A thinner pantograph strip exhibits better thermal performance, but aggravates the VSE effect. The radius of the contact wire also influences the current distribution. However, the possibility of adjusting geometry parameters is constrained by many factors, such as fixed weight and aerodynamic performance.
Overall, because of the VSE, the current clusters and heat accumulates, therefore the quality of the electrical contact in the PCS deteriorates, thus accelerating the ageing and damage to the region near the strip edges. To alleviate these problems a krigingassisted optimisation model has been developed to improve the current-carrying and thermal dissipation performance of pantograph strips.