**4. Conclusions**

This article demonstrates the formation of voltage stability in a system using conventional and small signal stability methods used to differentiate voltage stability. According to the literature, the non-linear system of the Household Method has been found to provide a linearization around the working point provided that the signal that changes the working point is small. Thus, it has been proved that easier linear systems can be analyzed instead of analysis of demanding nonlinear systems. However, it is not easy to obtain the Lyapunov Function to study the stability of a fixed point. This is because there is no general method for finding this function. The complex ambiance will be compared to non-linear functions with thousands of variables by using the concepts of coordinate functions. It has been shown that it is almost impossible to examine ultra-large systems with hundreds and thousands of variables without linearizing them. The only practical way of examining large systems is small signal analysis, since it can be linearized. The stability analysis of these systems is related to finding the master data of the mode matrix. Whether the real parts of these values are negative or positive determines whether the system is stable. Although it is highly challenging to find the master data of such a large matrix, it can be facilitated by making it linear so that the application capability of computer simulation can be improved.

The methods of finding the master data of large matrices are generally designed on homologous transformations (especially Householder transformations), because the successive transformations of the householder will result in many zeros in the columns of the matrix. The QR decomposition is made with the matrix obtained through the multiplication of the orthogonal matrix by the upper triangular matrix. The matrix obtained using the QR algorithm is then multiplied by the upper-triangular matrix (if this matrix does not exist, by the upper Hessenberg Matrix) and made homologous. Homologous transformations and degrees of the master data are retained so that, since the master data of the triangular matrix, the elements on the diagonal, the master data of the resulting matrix emerges from the triangular matrix. The QR algorithm can also generate the master data matrix (Modal Matrix). It is also designed to find the zero of functions in some other algorithms. These methods consider a particular polynomial as a function and try to find the zeros of the function by methods such as the Newton–Raphson Method. Other methods that are better suited than the Newton–Raphson Method are also used in finding the zero of different functions. Another method to find the master data is to use random algorithms. To achieve this goal, finding zeros is considered as minimizing, and then using complementary algorithms, such as genetic algorithms, solutions are provided for these issues. As a result, the most efficient method of finding main data in order to perform small signal stability analysis in large power systems is the Householder Method.

**Author Contributions:** Conceptualization, A.S. and M.R.T.; Methodology, A.S.; Software, E.H.; Validation, R.B., S.P. and A.S; Formal Analysis, M.R.T.; Investigation, A.S.; Resources, A.S.; Data Curation, M.R.T and E.H.; Writing-Original Draft Preparation, M.R.T.; Writing-Review & Editing, M.R.T.; Visualization, R.B.; Supervision, R.B and S.P.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest
