## 3 Theoretical Analysis of Undersea DC Power System Stability
### Mixed Potential Function Method  
[PERSON] and [PERSON] proposed three mixed potential function theorems for analyzing nonlinear RLC networks [21; 22; 28]. Based on the state equation of the power system, a mixed potential function should be constructed to judge the system stability under large disturbances. The CSO undersea DC power system is a topological complete RLC circuit, and its dynamic performance can be described by a mixed potential function. The mixed potential function \(P(i,u)\) can be uniformly expressed as  
\[P(i,u)=-A(i)+B(u)+D(i,u) \tag{13}\]  
where, \(-A(i)\) represents the current potential function of some non-energy storage elements in the circuit, \(B(u)\) represents the voltage potential function of non-energy storage elements in the circuit and \(D(i,u)\) represents the energy of capacitors and part of non-energy storage elements in the circuit [29; 30]. Figure 6: The medium-voltage experimental prototype of a single-node CSO undersea DC power system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & \(U_{S}\) & \(n\) & \(R_{0}\) & \(L_{0}\) & \(C_{0}\) & \(P_{L}\) \\ \hline Values & 400 V & 36 & 10 \(\Omega\) & 4 mH & 2 \(\mu\)F & 84 W \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The parameters of each part in the medium-voltage experimental prototype. In this paper, the mixed potential function third theorem is used to analyze the large-signal stability of the CSO undersea DC power system with long-distance submarine cables [21]. For the convenience of application, the third theorem is introduced as follows:  
\[P_{u}=\frac{\partial P(i,u)}{\partial u} \tag{14}\]  
\[P_{i}=\frac{\partial P(i,u)}{\partial i} \tag{15}\]  
\[A_{ii}(i)=\frac{\partial^{2}A(i)}{\partial i^{2}} \tag{16}\]  
\[B_{uu}(u)=\frac{\partial^{2}B(u)}{\partial u^{2}} \tag{17}\]  
\(L_{n}\) and \(C_{n}\) are diagonal arrays of branch inductance and branch capacitance, respectively. So,  
\[\mu_{1}=\min\biggl{\{}\lambda\biggl{(}L_{n}{}^{-\frac{1}{2}}A_{ii}(i)L_{n}{}^{- \frac{1}{2}}\biggr{)}\biggr{\}} \tag{18}\]  
\[\mu_{2}=\min\biggl{\{}\lambda\biggl{(}C_{n}{}^{-\frac{1}{2}}B_{uu}(u)C_{n}{}^{ -\frac{1}{2}}\biggr{)}\biggr{\}} \tag{19}\]  
where \(\lambda(M)\) represents the eigenvalue of the matrix M. In case  
\[\mu_{1}+\mu_{2}>0 \tag{20}\]  
and when \(|i|+|u|\rightarrow\infty\),  
\[P^{*}(i,u)=\frac{\mu_{1}-\mu_{2}}{2}P(i,u)+\frac{1}{2}\Bigl{(}P_{i},L_{n}{}^{ -1}P_{i}\Bigr{)}+\frac{1}{2}\Bigl{(}P_{u},C_{n}{}^{-1}P_{u}\Bigr{)}\rightarrow\infty. \tag{21}\]  
When \(t\rightarrow\infty\), the system is gradually stable over a wide range. It should be pointed out that this conclusion is only a sufficient condition, not a necessary condition. For the circuit shown in Figure 5, assuming that \(u_{0}=U_{S}\), the following mixed potential function can be derived:  
\[P(i,u)=\sum_{k=1}^{n}\left[-\frac{1}{2}R_{0}i_{k}{}^{2}+i_{k}(u_{k-1}-u_{k}) \right]+\int_{0}^{u_{n}}\frac{P_{L}}{u}du. \tag{22}\]  
It satisfies the following equation:  
\[\left\{ \begin{array}{ll}L_{0}\frac{d{\rm{i}}_{k}}{dt}=\frac{\partial P}{d{ \rm{i}}_{k}},k=1,2,\cdots,n\\ -C_{0}\frac{d{u_{k}}}{dt}=\frac{\partial P}{d{\rm{i}}_{k}},k=1,2,\cdots,n \end{array} \right.. \tag{23}\]  
From Equation (22), it can be derived that:  
\[A_{ii}(i)=\left[ \begin{array}{ccccc}R_{0}&0&\cdots&0&0\\ 0&R_{0}&\cdots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&R_{0}&0\\ 0&0&\cdots&0&R_{0}\end{array}