## 3 Overvoltage Simulation Model Including Variable Capacitance
### 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:  
1. The radius of the arc column is constant;
2. The temperature of the arc column is uniformly distributed radially;
3. Conductivity and a radiative heat source are functions of temperature;
4. 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:  
\[g=F(Q)=F\bigg{[}\int{(P-P_{0})dt}\bigg{]} \tag{3}\]  
\[\frac{dg}{dt}=\frac{dQ}{dt}\cdot\frac{dg}{dQ}=(P-P_{0})\cdot\frac{dg}{dQ} \tag{4}\]  
where \(P\) represents the input power of the pantograph-catenary arc, \(P_{0}\) denotes the arc output power, and \(Q\) represents the energy accumulated during arc formation. Both \(P\) and \(P_{0}\) are functions [18] of the arc conductivity \(g\),  
\[P=\frac{i^{2}}{g} \tag{5}\]  
\[P_{0}=k_{1}g^{-\beta}L \tag{6}\]  
where \(i\) represents the arc current, \(L\) represents the arc length, and \(k_{1}\) and \(\beta\) are arc constants. By substituting (2) to (5), we derive  
\[\frac{dg}{dt}=\frac{dQ}{dt}\cdot\frac{dg}{dQ}=\bigg{(}\frac{i^{2}}{g}-k_{1}g^ {-\beta}L\bigg{)}\frac{dg}{dQ} \tag{7}\]  
The arc energy \(Q\) and arc conductivity \(g\) are given by:  
\[Q=q\pi^{2}L \tag{8}\]  
\[g=\frac{\pi r^{2}\sigma}{L} \tag{9}\]  
According to the theory of gas molecule motion, the accumulated energy per unit volume of the arc \(q\) is derived as [19]  
\[q=0.354p\bigg{(}1-\frac{T_{1}}{T_{0}}\bigg{)} \tag{10}\]  
where \(p\) represents atmospheric pressure, \(T_{1}\) is the ambient temperature, and \(T_{0}\) is the temperature of the generated arc. Assuming that atmospheric pressure \(p\) and pressure dependent specific heat capacity are constants, the arc conductivity \(\sigma\) is derived from the arc theory's Saha equation [16]:  
\[\sigma=\sigma_{0}e^{-\frac{m}{t_{0}}} \tag{11}\]  
where \(\sigma_{0}\) 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 \(T_{0}\). By solving Equations (7) and (11) simultaneously, we derive  
\[\frac{dg}{dQ}=\frac{\frac{dg}{dT_{0}}}{\frac{dQ}{dT_{0}}}=\frac{\frac{d\,\varpi _{0}^{2}}{dT_{0}}}{\frac{d(q\varpi^{2}L)}{dT_{0}}}=\frac{mg}{0.354 pT_{1}\varpi ^{2}L}=k_{2}\frac{g}{L} \tag{12}\]  
Substituting Equation (9) into Equation (12), the arc is derived as  
\[\frac{dg}{dt}=\frac{dQ}{dt}\cdot\frac{dg}{dQ}=(P-P_{0})\cdot\frac{dg}{dQ} \tag{4}\]0  
In the simulation model, the expression of arc resistance can be solved by designing the correlation function-solving module. The range of \(\beta\) is specified as \(-3<\beta<0\)[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.