## 3 Methods
### Power Flow Formulation: A Linearized Approach
Then, the current \(I_{w}^{p^{*}}\) at node \(w\) with constant power load \(p\) is reflected in (40):  
\[I_{w}^{p}=\frac{S_{w}^{p^{*}}}{V_{w}^{*}} \tag{40}\]  
For the constant current load (\(\alpha=1\)), \(S_{w}^{i}\) is expressed as follows:  
\[S_{w}=S_{w}^{i}\cdot\left(\frac{V_{w}}{V_{nom}}\right)^{1}\] \[V_{w}\cdot I_{w}^{i^{*}}=\frac{S_{w}^{i}\cdot V_{w}}{V_{nom}}\] \[I_{w}^{i^{*}}=\frac{S_{w}^{i}}{V_{nom}}\]  
Subsequently, the current \(I_{w}^{i^{*}}\) at node \(w\) with constant current load \(i\) is reflected in (41):  
\[I_{w}^{i}=\frac{S_{w}^{i^{*}}}{V_{nom}} \tag{41}\]  
With regard to the constant impedance load (\(\alpha=2\)), \(S_{w}^{2}\) is treated as shown below:\[S_{w}=S_{w}^{z}\cdot\left(\frac{V_{w}}{V_{nom}}\right)^{2}\] \[S_{w}=S_{w}^{p}\cdot\left(\frac{V_{w}}{V_{nom}}\right)^{0}\]0 \[S_{w}=S_{w}^{p}\cdot\left(\frac{V_{w}}{V_{nom}}\right)^{0}\]1 \[S_{w}=S_{w}^{p}\cdot\left(\frac{V_{w}}{V_{nom}}\right)^{0}\]2  
Later, the current \(I_{w}^{z^{*}}\) at node \(w\) with constant impedance load \(z\) is shown in (42):  
\[S_{w}=S_{w}^{p}\cdot\left(\frac{V_{w}}{V_{nom}}\right)^{0}\]3  
Considering Equations (40)-(42), the currents flowing with the phases \(a\), \(b\), and \(c\) at node \(w\) are shown in (43)-(45), taking into account that \(h\) is the inverse of \(V_{nom}\):  
\[S_{w}=S_{w}^{p}\cdot\left(\frac{V_{w}}{V_{nom}}\right)^{0}\]4  
\[S_{w}=S_{w}^{p}\cdot\left(\frac{V_{w}}{V_{nom}}\right)^{0}\]5  
\[S_{w}=S_{w}^{p}\cdot\left(\frac{V_{w}}{V_{nom}}\right)^{0}\]6  
Notice in (43)-(45) that the component \(p\) of the \(ZIP\) model is the only nonlinear term in the equation for current. Based on the Laurent series expansion [55] of the function \(f(V)=\frac{1}{V}\) within a closed region in the range of \(|1-V|<e_{max}<1\), the nonlinear term in the \(ZIP\) model can be approximated for each phase \(a\), \(b\), and \(c\), considering that the angle between the voltage at each phase is \(120^{\circ}\) due to the nature of a three-phase system. This is represented by the vector \(T_{w}= \begin{bmatrix}1\angle 0^{\circ}&1\angle-120^{\circ}&1\angle 120^{ \circ}\end{bmatrix} ^{T}= \begin{bmatrix}T_{wa}&T_{wb}&T_{wc}\end{bmatrix} ^{T}\), and it is treated as shown below first for the current at phase \(a\):  
\[\frac{S_{wa}^{p^{*}}}{V_{wa}^{*}}=\frac{S_{wa}^{p^{*}}}{V_{nom}^{*}}\cdot \frac{T_{wa}}{V_{nom}^{*}}=h\cdot S_{wa}^{p^{*}}\cdot(2-h\cdot V_{wa}^{*})=2 \cdot h\cdot S_{wa}^{p^{*}}\cdot V_{wa}^{*}\]  
According to the above treatment, the current through phase \(a\) at node \(w\) is given by (46). \[I_{wa}=2\cdot h\cdot S_{wa}^{p^{*}}-h^{2}\cdot S_{wa}^{p^{*}}\cdot V_{wa}^{* }+h\cdot S_{wa}^{z^{*}}+h^{2}\cdot S_{wa}^{p^{*}}\cdot V_{wa} \tag{46}\]  
Likewise, for phase \(b\), it proceeds as follows:  
\[\frac{S_{wb}^{p^{*}}}{V_{wb}^{*}}=\frac{S_{wa}^{p^{*}}}{V_{nom}^{*}}\cdot \frac{T_{wb}}{T_{wb}}=h\cdot S_{wb}^{p^{*}}\cdot(2-h\cdot V_{wb}^{*}\cdot T_ {wb})\cdot T_{wb}\]Then, the current through phase \(b\) at node \(w\) is shown in (47):  
\[I_{wb}=h\cdot S_{wb}^{p^{*}}\cdot T_{wb}\cdot(2-h\cdot V_{wb}^{*}\cdot T_{wb})+h \cdot S_{wb}^{i^{*}}+h^{2}\cdot S_{wb}^{i^{*}}\cdot V_{wb} \tag{47}\]  
For phase \(c\), the nonlinear component is managed as follows:  
\[\frac{S_{\text{inv}}^{p^{*}}}{V_{\text{inv}}^{*}}=\frac{\frac{S_{\text{inv}}^{p^ {*}}}{V_{\text{inv}}}}{\frac{V_{\text{inv}}^{*}}{V_{\text{inv}}}}\cdot\frac{T_{ \text{inv}}}{T_{\text{inv}}}=h\cdot S_{\text{inv}}^{p^{*}}\cdot(2-h\cdot V_{ \text{inv}}^{*}\cdot T_{\text{inv}})\cdot T_{\text{vec}}\]  
In this sense, the current through phase \(c\) at node \(w\) is shown in (48):  
\[I_{\text{acc}}=h\cdot S_{\text{acc}}^{p^{*}}\cdot T_{\text{vec}}\cdot(2-h \cdot V_{\text{acc}}^{*}\cdot T_{\text{vec}})+h\cdot S_{\text{inv}}^{i^{*}}+ h^{2}\cdot S_{\text{inv}}^{i^{*}}\cdot V_{\text{vec}} \tag{48}\]  
Considering the expressions in (46)-(48), the representation of the currents \(I_{\text{inv}}\), \(I_{wb}\) and \(I_{\text{vec}}\) in matrix form is shown in (49):  
\[