**1. Introduction**

Developing technology, ever-growing urbanization and environmental conditions have forced energy systems to work close to the stability limit. This has increased the importance given to the subject of voltage stability and it has become much more important. The stability of a power system is the ability to keep the amplitudes of continuous or transient load bus bars within certain limits. In addition, voltage stability is the ability of these power systems to return to their former stable state when confronted with a disturbing effect and to retain the voltage in all bus bars within a certain level. One of the criteria for qualifying a system as a stable system is if the power given to a bus bar increases numerically, the amplitude of the voltage in that bus bar increases and this process proceeds similarly for all other bus bars in the system. If the reactive power given to that bus bar is increased in any of the bus bars in the system, and the voltage amplitude does not increase, it can then be said that there is voltage instability in this system [1]. Also, if the V-Q sensitivity for each bus bar is directly proportional or positive, the system is stable in terms of voltage. Thus, if the V-Q sensitivity is negative

for at least one bus bar, the system voltage is unstable. Failure of power systems to reach a voltage stability state is referred to as a "voltage failure", which occurs in the event of overload, failure, or insu fficient reactive power. In recent years, the voltage stability problem called voltage collapse has been experienced quite a lot. This has led to an increase in voltage stability studies [2,3]. In some studies, solutions are made by providing a methodology aimed at maintaining frequency stability by taking into account the latency associated with the frequency measurement process, while acquiring virtually equal inertia limitations from virtual operational generators [4]. Voltage stability and voltage instability are described depending on the size of static faults that may occur [5]. The oscillations in power systems raise voltage-related problems along with a small signal stability problem, which is one of the most important factors that limit power transmission capacity and jeopardize safe operation [6,7]. Analytical solutions and mathematical models were used to analyze the e ffect of stochastic continuous disturbances on the power system small signal stability [8,9].

This study aims to estimate the modal characteristics of the system including modal frequency, damping and shape. Most of the signal processing algorithms described in this section are the basis of developing several software tools. The majority of these tools are used to perform an engineering analysis on the grid in an o ffline or post-degradation environment [10]. Online real-time software tools and applications have recently been developed [11] and will continue to be the focus of research for the power system community. Voltage stability is sometimes referred to as load balancing [12]. The terms voltage instability and voltage collapse are often used interchangeably. Voltage instability is a dynamic process involving voltage dynamics, as opposed to rotor angle (synchronous) stability. Voltage collapse is defined as a process in which voltage instability in a significant portion of the system leads to a very low voltage profile. The voltage instability limit is not directly connected to the maximum power transmission limit of the grid [5,11,12]. Generally, local modes are in the range of 1–2 Hz, while in-field modes can range from 0.2 to 1.0 Hz [13]. Typically, in-field modes cause a little more trouble. Consistent with the dynamic system of a power system, it can be linearized at an operating point of the power system [10,11]. The proposed method o ffers an advantage for the di fficulty of stability analysis of nonlinear systems. In addition, it is impossible to practically analyze very large powerful complex systems. The systems examined in this article are not actually of a linear nature. Since the deviations occurring at the equilibrium are small (small signal), they can be approximated to the linear system. Therefore, instead of analyzing the nonlinear system, we can analyze the system approximated to being linear, which is easier.

Stability in a power system means that the system normally has the desired parameters. In other words, it can be defined as the ability of the system to return to its nominal state in a short period in case of a failure. In some studies, a small signal and a large signal stability analysis were performed using the Lyapunov linearization method. A combined stability criterion was then proposed to predict small and large signal stability problems [14–16]. In a power system, when the system is in a stable state, there is equilibrium between the incoming mechanical momentum and the outgoing electrical momentum. This equilibrium causes the velocity to remain constant. When short-circuit faults are also included, the equilibrium may be explained in terms of static stability. If an error occurs between power systems, the balance between the incoming mechanical momentum and the outgoing electrical momentum is relatively eliminated. According to the law of motion of rotating objects, the synchronous machine rotor will have a positive or negative speed, so it will rotate at high or low speed compared to other generators. This uncoordinated rotation will alter the stability of the system by changing the rotor angle. Furthermore, this problem is solved by using an approach based on the largest Lyapunov base for online transient rotor angle stability assessment using data from large area measurement systems only [17,18].

Thanks to the studies conducted on maintaining small signal stability, a few advantages in large power systems will be addressed. One of the benefits of small signal stability is that each synchronous generator present in the system can be linearized around the operating point. However, di fferential equations need to be solved systematically in transient stability, where the di fferential equations that dominate the pre-fault system, the during-fault power system, and the equations that dominate the post-fault system must be solved. Furthermore, if the protection relays do not work in the system during this time, the system will lose its synchronized state. It is not possible to find di fferential equations, even when solved in a systematic manner, that determine the cleaning time and security index in the relays after the fault. In order to find the critical time, the system needs to be simulated several times in all error-occurring situations, which will result in a grea<sup>t</sup> loss of time, and the fact that the system inspects its behavior for errors will reduce the time to intervene and cause even more time loss. However, the time spent in inspection of small signal stability is between 10 and 20 s. In the case of small signal stability, the stability of the system can be determined very easily through the master data, and only the positive or negative data will be su fficient to examine the stability of the system, without the need to inspect the data even after the calculations.

This article provides an overview of the challenges applied to the prediction of time-synchronized data of more successful analysis techniques of electromechanical mode. The theoretical basis, applications and performance characteristics of these methods are explained. When inter-zone modes are studied in a power system, several generators fluctuate in the opposite direction compared to other generators due to other failures. This is caused by a connection of two groups of generators over a weak line. The frequency of these fluctuations is between 0.2 and 1 Hz. Regional and comprehensive modes are today's most modern modes and are studied in stability studies of power systems. As modern power systems are directly connected to each other, the connection lines are often outdated and due to their high costs, the renewal of the lines is avoided. Despite the construction of new power plants, inter-regional fluctuations often occur because these power plants are connected to power systems through weak lines. Many PSS studies today are focused on power systems. The PSS system should not use local signals. However, it can use signals from other regions as input signals. In this case, too, a delay may occur when the signal is sent from other regions, which may impair the small signal stability. When there are multiple synchronous machines, the variable parameters will increase, and the mathematical analysis of the system will become more di fficult. It is very di fficult to find the determinant of a large matrix. Therefore, the straightforward method can only be discussed theoretically, but it does not have much use in practice. In the determination of the main quantities by means of numerical methods such as the square method, inverse square method and the Arnold method, the largest main amount is obtained in terms of absolute magnitude. Consequently, di fferent measures should be taken to find other main quantities. For this reason, the method of similar transformations is one of the practical methods. In the Gionesis Rotation method, which is based on orthogonal similarities, only one element in the given matrix is reset at each stage after transformation. Because of repeated transformations, the given matrix is orthogonally homologized to an upper triangle or Hasenberg Matrix. Thus, the given matrix can be decomposed as QR (Q, an orthogonal matrix; R, an upper triangular matrix). The main advantage of the Householder method presented in this article is that each column of the given matrix is transformed into the column of an upper-triangular matrix at each transformation stage. This method is important when compared particularly to the Gionesis method because, in the Gionesis method, for the transformation of an (n × n) matrix to an upper-triangular matrix, a conversion (matrix multiplication) operation must be performed n (n – 1/2) times. However, the conversion operation must be performed, at most, n times in the Householder method. In numerical terms of the data, the number of computational operations is important, because rounded errors can accumulate due to the large number of operations and ultimately have an impact on the result. In particular, when the actual specific quantities are close to zero, these rounded errors may not be able to determine the sign of that main quantity. Therefore, the method presented is of grea<sup>t</sup> importance—both in terms of the number of computational operations and in terms of obtaining all the main quantities—and has undeniable superiority over similar methods.
