## 2 Description of the Stability Problem for Three-Phase Unbalanced Systems  
Reference [18] showed how an EPS can be described through a set of differential and algebraic non-linear equations. This equation system contains dynamic and algebraic variables of the system:  
\[\dot{x}=f(x,y,u) \tag{1}\]  
\[0=g(x,y) \tag{2}\]  
where:  
* \(x\in\mathbb{R}^{n}\) represents the state variables of the machines;
* \(y\in\mathbb{R}^{m}\) represents the the algebraic variables of the system;
* \(u\in\mathbb{R}^{d}\) represents the input variables, in this case the parameter associated with the control systems of the generators;
* \(f(x,y,u)\) represents the differential equations of the generators;
* \(g(x,y)\) represents the algebraic equations associated with the generators and traditional load flow equations;
* \(n,m\) represent the number of state variables and algebraic equations. The calculation needed to directly and simultaneously solve the set of algebraic and differential equations, defined in Equations (1) and (2), is numerically very complex because of the non-linearity associated with the equations of the system [18]. As an alternative path, the procedure used linearizes the equation system around an operation point defined through the power flow. Thus, the following linear equivalent is obtained:  
\[ \begin{array}{c}\Delta\dot{x}=A\Delta x+B\Delta y+E\Delta u\\ 0=C\Delta x+D\Delta y\end{array} \right\}\Longrightarrow \begin{array}{c} \Delta\dot{x}=(A-BD^{-1}C)\Delta x+E\Delta u\\ \Delta\dot{x}=A_{sys}\Delta x+E\Delta u\end{array}  \tag{3}\]  
where:  
* \(\Delta x\) represents linearized state variables;
* \(\Delta y\) represents linearized algebraic variables. In classical small signal stability studies, it is considered that the input vector \(\Delta u=0\), and therefore, the eigenvalues of matrix \(A_{sys}\) are evaluated. In this context, the present article proposes a model that allows generalizing the structure of the \(A\), \(B\), \(C\), and \(D\) matrices for the cases in which the system operates under unbalanced conditions. The approach used by the authors is based on what was proposed in [2]. A sequence component methodology is used, which consists of representing an unbalanced three-phase system, through three three-phase balanced systems denominated as: positive sequence, negative sequence, and zero sequence. The description of each system considers the following:  
* Positive sequence: this is the real system considering that it operates in balanced conditions. * Negative sequence: this has the same topology as the positive sequence, but the equivalent voltage sources of the generators are not represented. * Zero sequence: this has the same topology as the negative system, but only here, the grounding connection and the existence of neutrals are modeled. The use of sequence components allows representing in the bases \(A\), \(B\), and \(C\) the variables associated with the algebraic equations. In general terms, matrix \(A\) does no suffer any modifications in the quantity of variables that define it. However, matrices \(B\), \(C\), and \(D\) are almost completely modified since each of the variables that define them are tripled in quantity. Once the new \(A\), \(B\), \(C\), and \(D\) matrices are obtained, the equivalent matrix \(A_{sys}(p)\) is redefined, and a small signal stability analysis is carried out. This study intends to determine the eigenvalues of the system (3), which are obtained by solving the following equation:  
\[det(\lambda\cdot I-A_{sys}(p))=0 \tag{4}\]  
where:  
* \(I\) is the identity matrix;
* \(\lambda\) represents the eigenvalue vector of the system (3). From the classical approach of permanent regime stability, the system of Equation (3) will be stable, only if all the real parts of the eigenvalues obtained from Equation (4) have a negative real part [18]. A commonly used indicator for stability is the damping ratio, which indicates how much each eigenvalue contributes to the stability of the system. It can be calculated with Equation (5).