Steady State Stability Analysis and Improvement using Eigenvalues and PSS A Case Study of a Thermal Power Plant in Jamshoro, Pakistan

The efficient handling and distribution of electrical power consist one of the most complex and appealing research problems. Due to the interconnection of different power plants and intensity of load, which gradually changes due to continuous change in load on generating units, careful treatment of the small disturbances in a power system which may lead to severe disturbance is necessary. Stability is an essential part in electrical power system operation and control. The stability problem is related with the behavior of synchronous machine after the power system is subjected to trouble. This work presents Steady State Stability (SSS) analysis of a Jamshoro Thermal Power Plant (JTPP) by using eigenvalue analysis of the different cases by varying load at three different positions. A mathematical model has been used for the JTPP with real data in order to examine the behavior of the system and to find the eigenvalues. A Simulink model of the JTTP for waveform analysis in MATLAB/Simulink has been used without and with Power System Stabilizer (PSS). Numerical quantification of the eigenvalues under the examined cases categorizes the stability of the system. The waveforms of the system are analyzed, and in cases of instability, the proposed procedure utilizing PSS helps in maintaining the system’s usual working conditions. The eigenvalue analysis and simulation results show the behavior of synchronous machines when loading changes gradually. The existing system becomes stable after more swings, whereas by using PSS in the existing system, stable regimes are attained in less time. The obtained results demonstrate the effectiveness of the proposed solution for SSS examination and securing of the disturbances of the JTPP. Keywords-steady state stability; power plant; eigenvalues; power system stabilizer; MATLAB


INTRODUCTION
In an electrical power system, it is essential to transmit reliable, secure and free from disturbances power, but load and external faults are independent which may affect voltage and frequency values, or even cause loss of synchronism [1]. The ability of a power system to remain stable after facing a small or large disturbance, which occurs in the existing system, is called power system stability. Stability problems are divided into two categories: Transient Stability (TS) and Steady State Stability (SSS) [2]. In TS, the power system remains in normal condition when a large disturbance occurs such as a loss of excitation, fault on the system, a sudden outage of line, removal of large load, etc. [3]. On the other hand, in SSS, the power system remains in normal condition when small disturbances occur, such as gradual variation in electrical load [4]. Nowadays, electrical power systems grow day by day, with increase of interconnection, installation of very large units, and extra high voltage (EHV) tie-lines [5]. TS is usually analyzed through Euler method, modified Euler method, and Runge-Kutta (RK) methods whereas SSS of the power system is usually examined using eigenvalues [6]. Eigenvalues and wave-forms of synchronous machines were analyzed in [7]. The examination was carried out by varying faults at three different locations: generating point, middle of transmission line, and an infinite position. The fault near the machine appeared to be most severe. In [8], by varying instantaneous power disturbance, eigenvalues and wave forms were analyzed in power system generating units, disturbance of waveforms dependence on the amplitude and location of disturbance. According to [9] the SSS of power systems by increasing loads is complementary to the usual TS investigations. To study the SSS of a complex power system, eigenvalue analysis was used. In [10], analysis of SSS on the near connection between the already-connected Luzon-Visayas grid with the isolated Mindanao power grid was investigated in free and open-source software. There are not publicly available studies regarding this inter-connection. In [11], small signal stability analysis for a hybrid renewable energy system was conducted and an integrated hybrid system was analyzed with the help of Lyapurav's stability criteria and small signal stability disturbance with power system stabilizer (PSS) through MATLAB/Simulink.
Power flow and SSS of interconnected three isolated Philippine power grids was studied in [12]. Eigenvalues analysis was employed to find the SSS of the main power grid and a model was developed in MATLAB/Simulink. According to [13], if the quantity of perturbation is very small, in the initial stage when there is small perturbation in the load, the system remains in stable condition (SSS negative real parts, the system is under stable condition. In case the system becomes unstable, this means that the eigenvalues have positive real parts, while complex eigenvalues, produce oscillations in the system. In [14], the TS of a power system was verified by using time domain (TD) analysis, and eigenvalues analysis when the system was brought under small disturbances. The use of FACTS devices for TS was investigated in [15]. Other approaches like the use of fuzzy logic controllers [16] and harmonic function [17] have also been used to improve PSS under different conditions [18].
In this paper, stable and efficient operation of a JTPP have been considered. Stable operation, voltage stability, frequency stability, and rotor angle stability were studied. Voltage stability considered voltage control through an Automatic Voltage Regulator (AVR) for frequency stability. Focus was given on Primary Frequency Control (PFC) through governor, and rotor angle stability focus is on the deviation of rotor speed. This work analyzes small signal stability which can be modified by using PSS. In that way, the system becomes stable and will remain in stable condition for a long period of time. The proposed procedure assures the secure and reliable operation of the JTPP even in conditions when disturbance, either small or large, occurs in the system, by changing different loads. The objective of the current work is to analyze the stability of a power system under varying load at alternator side. By varying load at alternator side, it has been shown that the system has more swings when large load is connected on a power system. So, in order to remove more swings a PSS was used. By using PSS, the system became stable in less time. The main contribution of this work lies on the use of eigenvalues as indicators of the small signal stability in usual operation so that the unstable regimen may be analyzed and taken care of in order to retain stability. The use of PSS on usual operation has also been tested and found to improve the stability regimen.

II. MATERIALS AND METHODS
The JTPP has four units, one with 16.5KV rated voltage and three with 15.75KV rated voltage. Two transformers are used which step up from the rated voltage to 220KV transmission line voltage. A generating unit was developed in the mathematical model, at alternator side, at excitation side and at prime mover governing system side. The eigenvalues were found and the analyzed system's stability was improved by PSS.

A. Plant Overview
The plant was visited and all the necessary data, parameters, and specifications mentioned in Tables I-II were collected.

B. Small Signal Stability Analysis
The ability of the running power system to maintain synchronism after the occurrence of small interruption is known as Steady State Stability [1]. As variations occur in the system, they cause small unbalances. A dynamic equation model of the system will be developed and will be linearized to understand SSS. Under balanced condition, we may define the system as: It is defined as the rate of change of each state variable with the linear composition of all thee states available in the system. To inscribe the small signal analysis problem, the modified state vector z is described in equation (2): The variable matrix Φ is the right modal matrix of vector A. A is consisting of n right eigenvectors φᵢ;i∈n . Equation (3) (5) The resolved form of (5) is: zᵢሺtሻ = zᵢሺ0ሻ݁ ᵢ୲ (6) To obtain z i (0), (7), can be written as: zሺtሻ = Φ¯ଵ∆xሺtሻ = Ѱ∆xሺtሻ (7) where Ψ is the left matrix of vector A, which consist of n left eigenvectors ψ i , hence every variable of state and its starting condition are given as: zᵢሺtሻ = Ѱᵢ∆xሺtሻ ; zᵢሺ0ሻ = Ѱᵢ∆xሺ0ሻ (8) In accordance with (6), the actual state vector ∆x(t) is written as: ∆xሺtሻ = ∑ Ѱᵢcᵢ݁ ᵢ୲ ୀଵ with cᵢ as zᵢ(0). The actual state variable ∆xᵢ(0) can be extracted from (2) and (9): ∆xᵢሺtሻ = Ѱᵢ₁c₁݁ భ ୲ + Ѱᵢ₂c₂݁ మ ୲ + ⋯ + Ѱ ୧୬ c ୬ ݁ ୲ (10) Free motion quickness of the power system is determined by the linear composition of eigenvalues, dynamic modes and left/right eigenvectors. The frequency behavior of the system can be successfully examined and evaluated by using the SSA in terms of the time-domain simulation (TDS). Linear analysis can be obtained by using DIgSILENT software and the nonlinear TDS of the dynamic system, which makes it easy by software. The major benefits of the linear model are that it gives the opportunity for using the eigenvalues to adjust the frequency handling controllers and the use of eigenvectors to trace the contents of the frequency regulation mode. Generators' frequency shows the condition of the systems which may be scripted in the forms of entire eigenvalues and conditions of the system by using (10) [19]: ωᵢሺtሻ = Ѱᵢ ଵ c ଵ ಓ భ ౪ + Ѱᵢ ଶ c ଶ ಓ మ ౪ + ⋯ + Ѱᵢ ୬ c ୬ ݁ ୲ (11)

C. General Mathematical Model
The electrical system exists on synchronous machines connected to an infinite bus bar. There are four conditions to sink the alternator with the infinite bus. It must have the same value of voltage, frequency, phase sequence, and phase angle difference. Every alternator has two main controllers, one for speed, called governor and another for voltage, the AVR. By varying load gradually at the alternator side through MATLAB/Simulink, we analyzed the small disturbances without PSS and with PSS. In the two axis model, the synchronous alternator consists of the transient and sub transient effect. The transient effects are taken into consideration and sub transient effects are considered as negligible, so the effects of transients are overcome with the rotor circuit which is the field circuit in the d-axis and q-axis formed by solid rotor [1].

3) Excitation System
Every alternator has an AVR. Through the AVR we can control the terminal voltage of an alternator by varying field current and maintain the terminal voltage as per our needs. Usually loop gain KA must be as per command of AVR, if the loop gain is very high the system will be unstable, if it is very low the system performance of AVR step response will be unsatisfactory. Therefore, to enhance the related stability and steady state behavior, the transformer is used for stabilizing.
The transient voltage of armature on d and q-axes may be indicated like:

4) Governor System
When the load on alternator increases suddenly, it causes rapid change in the governor intake. The deficient power is delivered by the kinetic energy of the rotary system. So, the depletion in kinetic energy decreases the alternator speed, hence change in new operating speed is identified by the governor which will act to adjust the intake valve to generate the mechanical power output which will counterbalance the normal operating position. The equations that express the prime-mover governing system are: The standard action for small faults is basically that the disturbed system may be linearized near quiescent operating point.

D. Proposed Stability Snalysis of the JTPP
The flowchart given in Figure 1 represents the work division in two parts. One part consists of the mathematical model, in which stability is analyzed through eigenvalues. If the eigenvalues are negative, the system is in stable state. If the obtained eigenvalues are positive, the system is in state of instability, and it will be stabilized by the PSS. In the other part, the developed MATLAB/Simulink model, stability is analyzed through waveforms, and is also improved by PSS. The equation for a system under an instability state can be given in matrix form as follows, here we can determine eigenvalues.
ሾFሿሾxሿ ሶ ൌ ሾBሿሾXሿ ሾDሿሾI ୢ୯ ሿ (32) The values of A matrix are known. Finally, the eigenvalues are obtained with MATLAB R2017a. Eigenvalues play an important part in determining power system stability. Two real separate eigenvalues have the same symbols. Both may be positive or negative. Positive values will show that the system is unstable and negative will show that the system is stable. Nodal real eigenvalues with two eigenvectors are unstable if these are positive and the opposite if they are negative. Saddle points with two distinct real eigenvalues of opposite signs are always unstable [6]. On the spiral chart, there are complex eigenvalues with non-zero and real parts. The system is unstable if the real part is positive and stable if the real part is negative. On the other hand, improper node represents unstable condition if it is positive and asymptotically stable if it is negative [19]. Figure 2 shows an existing electrical power system with generating unit, circuit breakers, transformer, and infinite bus. Three different loads are applied at generating stations and the waveforms are monitored. Severe disturbance occurs when more loads are applied. Figure 3, similarly, shows the developed MATLAB/Simulink model modified by PSS. Stability is analyzed by changing load after modification. It has been concluded that the system becomes stable in a very short time after the modification when compared to the system of Figure 2. The data regarding the parameters of the simulation model defined in (11)-(20) were obtained from the JTPP and are given in p.u. in Table III. III. RESULTS AND DISCUSSION By varying reheat system prime mover gain K rh =0.3, 2, 2.5, and 10, we find that at K rh =10 the system is unstable. Figures  4-7   The behavior of a synchronous machine with small disturbances has been analyzed with the help of MALTAB/Simulink, when 240MW, 300MW, and 175MW load changes. The respective waveforms are shown in Figures  8, 10, and 12 with PSS and in Figures. 9, 11, and 13 without PSS. The instabilities were identified using eigenvalues, as shown in Figures 4-7 IV. CONCLUSION Nowadays, electrical power systems are more complex in nature and widely interconnected. Continuous changes in load cause deviation in frequency and voltage. These changes result in small disturbances in the power system posing instability problems. Hence, steady state stability is important for generating stations. This work proposes a mathematical model and a MATLAB/Simulink model. Eigenvalues are found and employed for finding whether the system is stable or unstable. When they are negative the system stable and vice versa. MATLAB/Simulink model demonstrates the variations in the waveform by varying load, and then improves by applying PSS. This work promotes fast recovery in stable condition when unstable conditions occur in a system when the load gradually changes. This method is helpful for improving the reliability and stability of the system and small disturbances can be rectified without further isolation.