## 3 Modeling and Linearization of the System
### Internal Voltage And control Systems' Differential Equations  
The synchronous machine model has its origin in the application of the coordinates in the direct and quadrature axes (\"dq\"), as was described in [18]. The internal voltage shown in Figure 1 is decomposed into \(E_{d{q_{i}}}^{{}^{\prime}}\) and \(E_{q{q_{i}}}^{{}^{\prime}}\). The differential equations that define them are presented below:  
\[E_{q{i}}^{{}^{\prime}} = \frac{1}{T_{do{i}}^{{}^{\prime}}}(-E_{q{i}}^{{}^{\prime}}-(X_{d{i} }-X_{d{i}}^{{}^{\prime}})I_{d{i}}+E_{fd{i}}) \tag{7}\] \[E_{d{i}}^{{}^{\prime}} = \frac{1}{T_{q{i}}^{{}^{\prime}}}(-E_{d{i}}^{{}^{\prime}}+(X_{q{i} }-X_{q{i}}^{{}^{\prime}})I_{q{i}})\] (8) \[\dot{\delta_{i}} = w_{i}-w_{s} \tag{9}\]  
where:  
* \(T_{do{i}}^{{}^{\prime}}\) and \(T_{do{i}}^{{}^{\prime}}\) the direct transient time constants and of quadrature (s);
* \(X_{d{i}}^{{}^{\prime}}\) and \(X_{q{i}}^{{}^{\prime}}\) the transient direct reactance and of quadrature (pu);
* \(I_{d{i}}\) and \(I_{q{i}}\) the direct axis current and of quadrature (pu);
* \(E_{fd{i}}\) the equivalent field voltage (pu).  
Synchronous machines can adjust their internal voltage through what is known as field control or excitation current/power. We will use the standard IEEE Type 1 model to represent the voltage control monitoring dynamics:  
\[\dot{E_{fd{i}}} = \frac{-E_{fd{i}}}{T_{E_{i}}}[K_{E{i}}+K_{E_{i}}S_{E{i}}(E_{fd{i}} )]+\frac{1}{T_{E_{i}}}V_{R} \tag{10}\] \[\dot{R_{fi}} = \frac{1}{T_{F_{i}}}(-R_{fi}+\frac{K_{E_{i}}}{T_{F_{i}}}E_{fd{i}})\] (11) \[V_{R{i}} = \frac{1}{T_{A_{i}}}(-V_{R{i}}+K_{A{i}}R_{fi}-\frac{K_{A{i}}K_{Fi} }{T_{F_{i}}}E_{fd{i}}+K_{A{i}}(V_{ref{i}}-V_{Ti}+V_{PSS_{i}})) \tag{12}\]  
where:* \(V_{ref}\), the reference voltage of the field control (pu);
* \(K_{A_{i}}\) the gain of the voltage regulator;
* \(T_{A}\) the time constant of the voltage regulator (s);
* \(K_{Ei}\) the gain of the exciter;
* \(T_{Fi}\) the time constant of the field voltage feedback (s);
* \(V_{Ri}\) the regulator output;
* \(S_{Ei}(E_{fdi})\) the non-linear function of the voltage regulator; for this case, \(S_{Ei}(E_{fdi})=0.0039\cdot e^{1.555\cdot E_{fd}}\).  
In the differential equation for \(\dot{V}_{R_{i}}\) appears the concept \(V_{Ti}\), which corresponds to the voltage in the terminals/clampings of the machine and consists of an algebraic variable. Thus, in this case, we will consider that \(V_{Ti}=[V_{i}^{a}+V_{i}^{b}+V_{i}^{c}]/3\) in the same way as in [2].  
A method to reduce the oscillations that may occur in the EPS operations consists of introducing the power system stabilizer (PSS). This controller takes a speed signal and transforms it into a voltage signal, which powers/sustains/fuels the exciter system. This device is composed of three blocks: gain, washout, and phase compensators. The dynamic equations that characterize it are:  
\[\Delta X_{1i} = \bigg{[}\frac{K_{PSS_{i}}(T_{1i}+T_{3i})}{\omega_{s}T_{2i}T_{4i}} -\frac{K_{PSS_{i}}T_{1i}T_{3i}(T_{wi}T_{2i}+T_{wi}T_{4i}+T_{2i}T_{4i})}{\omega_ {s}T_{wi}T_{2i}^{2}T_{4i}^{2}}\bigg{]}\Delta\omega_{i} \tag{13}\] \[- \frac{T_{wi}T_{2i}+T_{w}T_{4i}+T_{2i}T_{4i}}{T_{wi}T_{2i}T_{4i}} \Delta X_{1i}+\Delta X_{2i}\] (14) \[\Delta X_{2i} = \bigg{[}\frac{K_{PSS_{i}}}{\omega_{s}T_{2i}T_{4i}}-\frac{K_{PSS_ {i}}T_{1i}T_{3i}(T_{2i}+T_{4i}+T_{wi})}{\omega_{s}T_{wi}T_{2i}^{2}T_{4i}^{2}} \bigg{]}\Delta\omega_{i}\] (15) \[- \frac{T_{2i}+T_{4i}+T_{wi}}{T_{wi}T_{2i}T_{4i}}\Delta X_{1i}+ \Delta X_{3i}\] (16) \[\Delta X_{3i} = -\frac{K_{PSS_{i}}T_{1i}T_{3i}}{\omega_{s}T_{wi}T_{2i}^{2}T_{4i}^ {2}}\Delta\omega_{i}-\frac{1}{T_{wi}T_{2i}T_{4i}}\Delta X_{1i}\] (17) \[\Delta V_{PSS_{i}} = \frac{K_{PSS_{i}}T_{1i}T_{3i}}{\omega_{s}T_{1i}T_{4i}}\Delta \omega_{i}+\Delta X_{1i} \tag{18}\]  
where:  
* \(X_{1i},X_{2i},X_{3i}\) the internal variables of the PSS controller;
* \(K_{PSS_{i}}\) the gain of the PSS controller;
* \(T_{1i},T_{2i},T_{3i},T_{4i}\) the time constants of the lead/lag blocks (s);
* \(T_{w}\) the time constants of the washout block (s).