## 2 The Equivalent Circuits of a Long-distance Submarine Cable  
In scientific CSOs, submarine electro-optic cables are used to transmit electrical power to the seafloor, and mainly consist of the insulating sheath, the composite conductor, the steel wires strand, the steel tube, the thixotropic jelly and optical fibers [12]. According to the electrical conductivity and radius of different materials in submarine cables, the resistance, inductance and capacitance of per unit length submarine cable can be obtained by theoretical calculation, shown in Table 1 for a typical submarine cable [12; 27]. In Table 1, R\({}_{1}\), L\({}_{1}\) and C\({}_{1}\) represent the resistance, the inductance and the capacitance of 1 km submarine cable, respectively. The lumped parameter equivalent circuits of long-distance submarine cables are the approximation of the actual distributed characteristics within a certain range. In our studies, multiple \(\pi\)-type lumped parameter models are cascaded to simulate a long-distance submarine cable. The length of each cable segment represented by a \(\pi\)-type lumped parameter model, as shown in Figure 2, will affect the accuracy of the equivalent circuits of a submarine cable. The two-port network of a submarine cable can be expressed as  
\[\left\{ \begin{array}{l}Z=Z_{\mathrm{C}}\mathrm{sinh}(\gamma l)\\ Y=\frac{1}{Z_{\mathrm{C}}}\frac{2(\mathrm{cosh}\gamma l-1)}{\mathrm{sinh} \gamma l}\end{array} \right. \tag{1}\]  
\[\omega=2\pi f \tag{2}\]  
\[z=R+j\omega L \tag{3}\]  
\[y=G+j\omega\mathrm{C} \tag{4}\]  
\[Z_{\mathrm{C}}=\sqrt{\frac{z}{y}} \tag{5}\]  
\[\gamma=\sqrt{yz} \tag{6}\]  
where \(f\) is the characteristic frequency of the undersea DC power system, \(f\) = 1 kHz in our study. \(\omega\) is the angular velocity of the undersea DC power system. \(l\) is the overall length of the submarine cable. \(R\), \(L\), \(G\) and \(C\) are corresponding to the equivalent resistance, inductance, conductance and capacitance of the submarine cable, respectively. The insulation resistance of submarine cables is very high, so that the conductance between the copper conductor and the seawater can be negligible, that is, \(G\) = 0. \(z\) and \(y\) are the impedance and admittance of per unit length submarine cable, respectively. \(Z_{\mathrm{C}}\) is the wave impedance. \(\gamma\) is the propagation coefficient. When \(\gamma l<<1\), \(Z\) and \(Y\) can be expressed approximately as  
\[Z\approx zl \tag{7}\]  
\[Y\approx yl. \tag{8}\]  
In order to study the influence of the cable segment length on the model accuracy, the two-port network model of the submarine cable is regarded to be accurate as a reference, and the \(\pi\)-type lumped parameter models of different cable segment lengths are taken as approximations. The comparison of the two-port network and lumped parameter model under different cable segment lengths are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **R1** & **L1** & **C1** \\ \hline Values & 1 \(\Omega\) & 0.4 mH & 0.2 \(\mathrm{\SIUnitSymbolMicro F}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of 1 km typical submarine cable. Figure 2: The \(\pi\)-type lumped parameter model of a submarine cable segment. shown in Table 2. When multiple \(\pi\)-type lumped parameter models are cascaded to simulate a long-distance submarine cable, the longer the cable segment represented by each \(\pi\)-type equivalent circuit, the greater the model parameter error is between the actual submarine cable characteristics and the equivalent circuits. Computer simulation is performed by using MATLAB. The simulation file is shown in Figure 3. The output voltage of the shore station power feeding equipment (PFE) is 10 kV, and the rising time of the output voltage is 1 ms. The length of the submarine cable is 200 km, and the load is 200 \(\Omega\). The simulation results are shown in Figure 4. The red line indicates the distributed parameter model of the submarine cable, closest to the actual submarine cable characteristics.