## 2 Formulation of the Power Flow Problem
### Non-Grounded Neutral Wire  
This subsection explores the extension of the power flow formulation for bipolar DC networks where the neutral wire is uniquely grounded in the substation bus. The main difference in this case is that the current per pole depends on the difference between the pole and load voltages. We define the vector of demanded current per node as \(i_{dk}^{pon}\):  
\[i_{dk}^{pon}= \begin{bmatrix}i_{dk}^{p}\\ i_{dk}^{p}\\ i_{dk}^{n}\end{bmatrix}  \tag{17}\]  
where \(i_{dk}^{p}\), \(i_{dk}^{o}\), and \(i_{dk}^{n}\) are the net current injections at node \(k\) in the positive, neutral, and negative poles, respectively. These currents are calculated as follows:  
\[i_{dk}^{p} =\frac{P_{dk}^{p}}{v_{k}^{p}-v_{k}^{o}}+\frac{P_{dk}^{p-n}}{v_{k}^ {p}-v_{k}^{n}}, \tag{18}\] \[i_{dk}^{o} =\frac{P_{dk}^{p}}{v_{k}^{o}-v_{k}^{p}}+\frac{P_{dk}^{n}}{v_{k}^{ o}-v_{k}^{n}},\] (19) \[i_{dk}^{n} =\frac{p_{dk}^{n}}{v_{k}^{n}-v_{k}^{o}}+\frac{P_{dk}^{n-p}}{v_{k}^ {n}-v_{k}^{p}}, \tag{20}\]  
Equations (18)-(20) have the same hyperbolic relation defined for the bipolar current \(i_{dk}^{n}\) in Equation (3). This implies that the currents for the positive, neutral, and negative poles can be approximated as presented in (7) by using the Taylor series expansion with its linear term. To obtain the power flow formulation for the non-grounded neutral bipolar DC case, we rewrite the matrix Equation (9), as presented in Equation (21). \[-i_{d}^{pon}=G_{ds}^{pon}V_{s}^{pon}+G_{dd}^{pon}V_{d}^{pon}, \tag{21}\]  
where \(i_{d}^{pon}\in\mathbb{R}^{(3l-3)\times 1}\) and \(V_{d}^{pon}\in\mathbb{R}^{(3l-3)\times 1}\) are the vectors of demanded currents and voltages in the demand nodes. \(V_{s}^{pon}\in\mathbb{R}^{3\times 1}\) is the voltage output in the slack node. \(G_{ds}^{pon}\in\mathbb{R}^{3\times(3l-3)}\) and \(G_{dd}^{pon}\in\mathbb{R}^{(3l-3)\times(3l-3)}\) are submatrices associated with the bipolar conductance matrix considering the neutral wire. To define the general power flow formula for a neutral non-solidly grounded wire, the following vectors are defined. \[i_{d}^{pon}= \begin{bmatrix}i_{2}^{p}\\ i_{2}^{p}\\ i_{2}^{p}\\ \vdots\\ i_{l}^{p}\\ i_{l}^{p}\end{bmatrix} ,\;V_{d}^{pon}= \begin{bmatrix}v_{2}^{p}\\ v_{2}^{p}\\ v_{2}^{p}\\ \vdots\\ v_{l}^{p}\\ v_{l}^{p}\\ \vdots\\ v_{l}^{p}\end{bmatrix} ,\;P_{d}^{p}= \begin{pmatrix}P_{2}^{p}\\ P_{2}^{p}\\ 0\\ \vdots\\ P_{l}^{p}\\ P_{l}^{p}\\ 0\end{pmatrix} ,\] \[P_{d}^{p}= \begin{pmatrix}0\\ P_{2}^{p}\\ P_{2}^{p}\\ \vdots\\ P_{s}^{p}\\ 0\end{pmatrix} ,\;P_{d}^{p-n}= \begin{pmatrix}P_{2}^{p-n}\\ 0\\ P_{2}^{n-p}\\ \vdots\\ P_{s}^{p-n}\\ 0\end{pmatrix}  \tag{22}\]  
in addition, we define the following auxiliary matrices:  
\[M_{x_{3\times 3}}= \begin{bmatrix}1&-1&0\\ -1&1&0\\ 0&0&1\end{bmatrix} ,\;M_{y_{3\times 3}}= \begin{bmatrix}1&0&0\\ 0&1&-1\\ 0&-1&1\end{bmatrix} ,\;M_{z_{3\times 3}}= \begin{bmatrix}1&0&-1\\ 1&0&0\\ -1&0&1\end{bmatrix}  \tag{23}\]  
matrices \(\mathbf{X}\in\mathbb{R}^{(3l-3)\times(3l-3)}\), \(\mathbf{Y}\in\mathbb{R}^{(3l-3)\times(3l-3)}\), and \(\mathbf{Z}\in\mathbb{R}^{(3l-3)\times(3l-3)}\), were constructed by the authors in order to vectorize the recursive formula, and they are defined as follows:  
\[\mathbf{X}= \begin{bmatrix}M_{x_{3\times 3}}&\cdots&0_{3\times 3}&\cdots&0_{3 \times 3}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ 0_{3\times 3}&\cdots&M_{x_{3\times 3}}&\cdots&0_{3\times 3}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ 0_{3\times 3}&\cdots&0_{3\times 3}&\cdots&M_{x_{3\times 3}}\end{bmatrix} ,\; \mathbf{Y}= \begin{bmatrix}M_{y_{3\times 3}}&\cdots&0_{3\times 3}&\cdots&0_{3 \times 3}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ 0_{3\times 3}&\cdots&M_{y_{3\times 3}}&\cdots&0_{3\times 3}\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ 0_{3\times 3}&\cdots&0_{3\times 3}&\cdots&M_{y_{3\times 3}}\end{bmatrix} ,\] \[\mathbf{Z}=