## 2 Materials and Methods
### Modelling of Synchronous Generators in Large Scale Power System  
The analysis of the dynamic behavior of power systems is an important research topic, as it is considered a problem for secure system operation [2; 3]. The system performance is continuously changing due to the connected loads and it is possible for it to present disturbances of diverse nature and duration. Thus, the operation of installed controllers is vital to update the required reference values and respond to disturbances.  
Transient stability studies in the power system are recommended for proving the closed-loop stable nonlinear dynamic system operation under severe time-varying disturbances. In this sense, a nonlinear dynamic model of the synchronous generator can be used for design purposes of efficient reference trajectory tracking control. In this context, the well known transient model for a synchronous generator constitutes a very good alternative. Moreover, this nonlinear dynamic model can be extended to large scale power systems as follows [24].  
\[\dot{\delta_{i}} =\omega_{i}\] \[\dot{\omega_{i}} =(P_{m_{i}}-P_{e_{i}}-D_{i}\omega_{i})\sigma_{i}\] \[\dot{E_{i}} =-a_{i}E_{i}+b_{i}\sum_{j=1,j\
eq i}^{n_{g}}E_{j}Y_{ij}\cos(\delta_ {i}-\delta_{j}+\theta_{ij})+E_{fi}+u_{i} \tag{1}\]  
with  
\[P_{e_{i}}=G_{ii}E_{i}^{2}+E_{i}\sum_{j=1,j\
eq i}^{n_{g}}E_{j}Y_{ij}\sin(\delta _{i}-\delta_{j}+\theta_{ij}),\quad i=1,\ldots,n_{g} \tag{2}\]  
where \(B_{Mii}\) is the susceptance and \(G_{ii}\) is the conductance, both from the reduced admittance matrix. Here, the subscript \(i\) is used to denote the \(i\)-\(th\) machine under analysis and \(n_{g}\) represents the number of synchronous generators in the electrical grid. In addition, \(\delta_{i}\) is the power angle; \(\omega_{i}\) is the speed deviation from synchronous velocity; \(D_{i}\) is the damping coefficient; \(\sigma_{i}=\pi f/H_{i}\) with \(H_{i}\) as the inertia constant and \(f\) the grid nominal frequency; \(P_{m_{i}}\) is the mechanical input power, which could exhibit possible variations; \(P_{e_{i}}\) is the electric power; \(E_{fd}\) is the steady state value of the excitation voltage; thus, \(E_{f}=\frac{E_{fd}}{T_{d0_{i}}^{d}}\); \(u_{i}\) identifies the control input signal which defines the variation from steady state value of the automatic voltage regulator; \(Y_{ij}\) and \(\theta_{ij}\) are the magnitude and the angle of the \(ij\) component of reduced admittance matrix of \(Y_{bus}\), respectively,  
\[Y_{ij}\triangleq\sqrt{G_{ij}^{2}+B_{ij}^{2}},\quad\theta_{ij}\triangleq arctan \frac{G_{ij}}{B_{ij}} \tag{3}\]  
The values for \(a_{i}\) and \(b_{i}\) are determined by the generator parameters as  
\[a_{i}\triangleq\frac{1}{T_{d0_{i}}^{\prime}}[1-B_{Mii}(x_{di}-x_{di}^{\prime} )],\quad b_{i}\triangleq\frac{x_{di}-x_{di}^{\prime}}{T_{d0_{i}}^{\prime}} \tag{4}\]  
where \(T_{d0_{i}}^{\prime}\) is the direct-axis transient time constant; \(x_{di}\) and \(x_{di}^{\prime}\) are the direct-axis synchronous reactance and transient reactance, respectively.  
The SG is represented by a simplified transient model. This representation considers an internal voltage source behind the direct axis transient reactance in series with the armature resistance. Thus, using the reduced admittance matrix, the dynamics of the power system can be evaluated considering the internal voltages.  
The reduced admittance matrix is calculated after obtaining the power flow solution by applying the Kron reduction. Thus, all loads are determined by equivalent admittances to ground and are included in the bus admittance matrix. The power system dynamics is evaluated with this formulation, where only the generator internal nodes are used for each iteration utilizing the reduced admittance matrix of dimension \(n_{g}\times n_{g}\).