## 2 Formulation of the Power Flow Problem
### Solidly Grounded Case  
The voltage \(v_{o}\) is zero for solidly grounded systems. Therefore, the general power flow equation for a multimodal bipolar DC can be represented by (1) (in the following equations, \(h,b,w\) are entries of the set \(\mathcal{B}\)):  
\[\dot{t}^{h}_{gkE}-\dot{t}^{h}_{dke}-\dot{t}^{h}_{dkC}=\sum_{b\in\mathcal{B}} \sum_{j\in\mathcal{L}}g^{hb}_{kj}v^{b}_{k},\ \{\forall k\in\mathcal{L},\ \forall h\in \mathcal{B}\}, \tag{1}\]  
where \(\dot{t}^{h}_{gkE}\) is the current generation in the power source connected at node \(k\) with an E-type connection in the pole \(h\); \(\dot{t}^{h}_{dke}\) is the current demanded at node \(k\) for a constant-power terminal with an E-type connection in the pole \(h\); \(\dot{t}^{h}_{dkC}\) is the current demanded at node \(k\) for a constant-power terminal with a C-type connection in the pole \(h\); \(g^{hb}_{kj}\) is the component (entry) of the conductance matrix that relates nodes \(k\) and \(j\) and the poles \(h\) and \(b\), respectively; \(v^{b}_{k}\) is the voltage at node \(k\) for the pole \(b\). The definitions of the demanded current for the E-type and C-type connections are presented in Equations (2) and (3). \[\dot{t}^{h}_{dke}=\frac{p^{h}_{dk}}{v^{h}_{k}}, \tag{2}\] \[\dot{t}^{h}_{dkC}=\frac{p^{h-w}_{dk}}{v^{h}_{k}-v^{w}_{k}}, \tag{3}\]  
where \(p^{h}_{dk}\) is the power consumed at node \(k\) in the pole \(h\) with respect to the neutral wire; \(p^{h-w}_{dk}\) represents the power consumption at node \(k\) between poles \(h\) and \(w\). Both (2) and (3) have hyperbolic form. A linear approximation of (2) and (3) is proposed here. It consists of a Taylor expansion of the hyperbolic representations \(f(x)=x^{-1}\) and \(g(x,y)=(x-y)^{-1}\) around the operation point \((x_{0},y_{0})\), which is defined in (4) and (5). \[f(x)\approx\frac{2}{x_{0}}-\frac{x}{x_{0}^{2}}, \tag{4}\] \[g(x,y)\approx\frac{2}{x_{0}-y_{0}}-\frac{x-y}{\left(x_{0}-y_{0} \right)^{2}}. \tag{5}\]  
Equations (4) and (5) are replaced in (2) and (3), so the equivalent currents for the E-type and C-type connections are found:  
\[\dot{t}^{h}_{dke}=\left(\frac{2}{v^{h}_{k0}}-\frac{v^{h}_{k}}{ \left(v^{h}_{k0}\right)^{2}}\right)p^{h}_{dk}, \tag{6}\] \[\dot{t}^{h}_{dkC}=\left(\frac{2}{v^{h}_{k0}-v^{w}_{k0}}-\frac{v^{ h}_{k}-v^{w}_{k}}{\left(v^{h}_{k0}-v^{w}_{k0}\right)^{2}}\right)p^{h-w}_{dk}. \tag{7}\]  
Figure 2: Examples of C-type and E-type loads. We separate the general power flow Equation (1) into two terms: The first term is related to the slack node, and the second term is associated with the demand nodes. This separation is presented in Equations (8) and (9), respectively. \[i_{gE}^{pn} =g_{ss}^{pn}v_{s}^{pn}+g_{sd}^{pn}v_{d}^{pn}, \tag{8}\] \[-i_{dE}^{pn}-i_{dC}^{pn} =g_{ds}^{pn}v_{s}^{pn}+g_{dd}^{pn}v_{d}^{pn}, \tag{9}\]  
where \(i_{gE}^{pn}\in\mathbb{R}^{2\times 1}\) contains the power injections in the slack node for the positive and negative poles (\(pn\)); \(i_{dE}^{pn}\in\mathbb{R}^{(2l-2)\times 1}\) and \(i_{dC}^{pn}\in\mathbb{R}^{(2l-2)\times 1}\) are vectors that represent the demanded currents in the E-type and C-type connections. In this definition of the variables, \(l\) represents the number of nodes in the bipolar DC grid; \(v_{s}^{pn}\in\mathbb{R}^{2\times 1}\) and \(v_{d}^{pn}\in\mathbb{R}^{(2l-2)\times 1}\) are vectors that contain the voltage output in the slack node and the voltage variables in the demand nodes for poles \(pn\), respectively; \(g_{ss}^{pn}\in\mathbb{R}^{2\times 2}\), \(g_{sd}^{pn}\in\mathbb{R}^{2\times(2l-2)}\), \(g_{sd}^{pn}\in\mathbb{R}^{(2l-2)\times 2}\), and \(g_{dd}^{pn}\in\mathbb{R}^{(2l-2)\times(2l-2)}\) are submatrices of the general conductance matrix \(g^{pn}\), which relates the slack and demanded nodes. Equation (8) is linear, since the variables of the power flow problem are the current injections in the slack node, i.e., \(i_{gE}^{pn}\), and the voltages in the demand nodes \(v_{d}^{pn}\). Hence, its solution is reached when the solution of Equation (9) is found.