## 3 Field-Circuit Coupling Analysis Method
### Electric Field and Stray Parameters Calculation  
In the offshore wind power MMC-HVDC transmission system, severe station faults lead to the operation of control and protection systems, which in turn cause the MMC valve to be blocked. Following the fault described in Figure 2, the voltage of the MMC valve is shown in Figure 7.  
Figure 6: Field–circuit coupling analysis model for overcurrent calculation. Figure 7: Voltage of MMC valve end during fault. As shown in Figure 7, following a severe system fault, the voltage variation can be divided into four stages. It can be observed that the terminal voltage of the MMC valve experiences the largest fluctuation and the most severe impulse during stage T\({}_{2}\), with the voltage falling edge at the terminal occurring approximately 100 us after the fault occurs. After estimation, the high-frequency components are expected to exceed 1 MHz. As shown in Figure 5b, the \(f_{CI}\) will exceed 1.5%, indicating that the stray parameters of the MMC valve will have a significant influence on the overcurrent. From Figure 7, it is evident that during system faults, the terminal voltage of the MMC valve is subjected to impulse excitation, characterized by relatively steep voltage rise and fall edges. In summary, under the influence of this impulse voltage, the following characteristics appear at the terminal location of the converter valve:  
1. Under impulse excitation, the stray parameters of the valve cause more pronounced changes in the fault loop. Therefore, it is necessary to calculate the stray parameters of the MMC valve under impulse excitation, primarily the stray capacitance. 2. When the voltage peak is high, it may lead to a higher electric field strength at the terminal of the MMC valve. If partial discharge occurs, it will also cause changes in the fault loop. 3. Under impulse excitation, when the voltage peak is large, the surge arrester at the terminal location of the MMC valve may be activated, leading to changes in the fault loop. The MMC valve is composed of multiple conductors, and all conductors need to be processed in the capacitance calculation. The capacitances of the valve shields are shown in Figure 8.  
As shown in Figure 8, \(C_{1 iG}\) is the capacitance between conductor \(i\) in the layer 1 and ground. \(C_{1 ij}\) is the capacitance between conductor \(i\) and conductor \(j\) in layer 1. \(C_{2 li1}\) is the capacitance between conductor \(i\) in layer 1 and conductor \(i\) in layer 2. From Equation (2), a change in \(Z_{\text{g}}\) will lead to alterations in the characteristics of the fault current. According to the physical definition, capacitance is a measure that describes the ability to store electric field energy. The structure of the MMC valve is relatively complex, with a large number of conductors of various shapes. The accurate calculation method of stray capacitance based on electric field energy calculation is defined as follows:  
Figure 8: Diagram of MMC valve stray capacitance. \[ \begin{cases}W_{\rm c}=\frac{1}{2}C(\Delta U)^{2}\\ \Delta U=\varphi_{1}-\varphi_{2}\\ C_{\rm gij}=f_{\rm C}(E_{ij})=\frac{\epsilon_{r}\epsilon_{0}\int E^{2}dV}{( \Delta U)^{2}}\end{cases}  \tag{6}\]  
In Equation (6), \(E_{ij}\) is the electric field between conductors \(i\) and \(j\), \(\varphi_{ij}\) are the potential distributions of the conductors, \(\epsilon_{r}\) is the relative permittivity of the insulating structure between the conductors, \(\epsilon_{0}\) is the vacuum permittivity, \(W_{\rm c}\) is the electric field energy stored between the conductors, \(C_{\rm gij}\) is the capacitance between the conductors, and \(\Delta U\) is the voltage between the conductors. Due to the influence of factors such as terminal connectors and bus-bars, the electric field at the terminal of the MMC valve is often large and unevenly distributed. Therefore, the electric field characteristics at the terminal of the converter valve are described through ground impedance.