## 3 Small-Signal Stability Analysis
\[k_{px}=\left(\frac{\partial p}{\partial X_{V}}\right)=\frac{(EV\sin\delta) \Big{(}R^{2}+X^{2}\Big{)}-\Big{(}RE^{2}-REV\cos\delta+XEV\sin\delta\Big{)}(2X) }{\big{(}R^{2}+X^{2}\big{)}^{2}}. \tag{12}\]  
The same partial derivative of the active power based on virtual impedance can be expressed as shown in Equation (13):  
\[k_{px}=\left(\frac{\partial p}{\partial X_{V}}\right)=\frac{R^{2}EV\sin\delta- 2 XRE^{2}}{\big{(}R^{2}+X^{2}\big{)}^{2}}-\frac{X^{2}EV\sin\delta-2 XREV\cos \delta}{\big{(}R^{2}+X^{2}\big{)}^{2}}. \tag{13}\]  
Now, by considering the partial derivative of the active power with respect to the angle, the term shown in Equation (14) is obtained:  
\[\Delta P(S)=k_{px}\Delta X_{V}(S)+k_{p\delta}\Delta\delta(S). \tag{5}\]0  
From the partial derivative of the reactive power with respect to the virtual impedance, the expression shown in Equation (15) is obtained:  
\[\Delta P(S)=k_{px}\Delta X_{V}(S)+k_{p\delta}\Delta\delta(S). \tag{5}\]1  
Furthermore, from the partial derivative of the reactive power with respect to angle, the following expression shown in Equation (16) is obtained:  
\[\Delta P(S)=k_{px}\Delta X_{V}(S)+k_{p\delta}\Delta\delta(S). \tag{5}\]2  
By considering that active and reactive power are commonly measured using a first-order low-pass filter, the cut-off frequency \(W_{f}\) of the active power controller can be linearized around the same point:  
\[\Delta P(S)=k_{px}\Delta X_{V}(S)+k_{p\delta}\Delta\delta(S). \tag{5}\]3  
\[\Delta P(S)=k_{px}\Delta X_{V}(S)+k_{p\delta}\Delta\delta(S). \tag{5}\]4  
After replacing \(\Delta P(S)\), as defined in Equations (17) and (18), the following equation is obtained:  
\[\Delta P(S)=k_{px}\Delta X_{V}(S)+k_{p\delta}\Delta\delta(S). \tag{5}\]5  
Thus, the virtual impedance is defined by considering Equation (19):  
\[\Delta P(S)=k_{px}\Delta X_{V}(S)+k_{p\delta}\Delta\delta(S). \tag{5}\]6  
After replacing \(\Delta X_{V}\) in Equation (20), Equation (21) is obtained, and considering that \(\Delta P(S)=\frac{S\Delta\delta(S)}{-k_{p}}\):  
\[\Delta P(S)=k_{px}\Delta X_{V}(S)+k_{p\delta}\Delta\delta(S). \tag{5}\]7By solving Equation (21), the large expression shown in Equation (22) is obtained. \[ \begin{split}&\Delta W(S)\\ &=\frac{-S\Delta\delta(S)k_{px}w_{f}{}^{2}-\Delta\delta(S)k_{px}k_{ p}k_{p0}w_{f}{}^{2}+S\Delta\delta(S)k_{p0}k_{p}k_{px}w_{f}+\Delta\delta(S)k_{p0}k_{p}k_{px}w_{f} {}^{2}}{-S^{2}k_{px}-2 Sw_{f}k_{px}-w_{f}{}^{2}k_{px}}.\end{split}  \tag{22}\]  
After considering that \(\Delta W(S)=S\Delta\delta(S)\) and replacing this in Equation (22), the expression shown in Equation (23) is obtained. \[S\Delta\delta(S)=\frac{-S\Delta\delta(S)k_{px}w_{f}{}^{2}-\Delta\delta(S)k_{ px}k_{p}k_{p0}w_{f}{}^{2}+S\Delta\delta(S)k_{p0}k_{p}k_{px}w_{f}+\Delta\delta(S)k_{ p0}k_{p}k_{px}w_{f}{}^{2}}{-S^{2}k_{px}-2 Sw_{f}k_{px}-w_{f}{}^{2}k_{px}}. \tag{23}\]  
Then, after solving Equation (23) and expressing the result in a simplified form, the expression shown in Equation (24) is obtained. \[S^{3}\Delta\delta(S)+2S^{2}\Delta\delta(S)w_{f}+S\Delta\delta(S)\Big{(}w_{f}{} ^{2}+k_{p0}k_{p}w_{f}-w_{f}{}^{2}\Big{)}+\Delta\delta(S)\Big{(}-2\ k_{p}w_{f}{ }^{2}k_{p0}\Big{)}=0. \tag{24}\]  
This homogeneous equation describes the free movement of the system for small disturbances around the equilibrium point (\(\delta\), E, V), and it can be reduced when considering some constant parameters, as the expression presented in Equation (25):  
\[S^{3}\Delta\delta(S)+aS^{2}\Delta\delta(S)+bS\Delta\delta(S)+c\ \Delta\delta(S)=0, \tag{25}\]  
where the parameters \(a\), \(b\), and \(c\) are defined in Equations (26)-(28) as follows. \[a=2w_{f}, \tag{26}\]  
\[b=\Big{(}w_{f}{}^{2}+k_{p0}k_{p}w_{f}-w_{f}{}^{2}\Big{)}, \tag{27}\]  
\[c=\Big{(}-2\ k_{p}w_{f}{}^{2}k_{p0}\Big{)}. \tag{28}\]  
Then, the small-signal stability can be studied based on the previous equations. They are implemented in this work for each control system used in the inverters to respond to the load variations at different times and nodes in the network.