 Original research
 Open Access
 Published:
Small signal stability enhancement of a large scale power system using a bioinspired whale optimization algorithm
Protection and Control of Modern Power Systems volume 6, Article number: 35 (2021)
Abstract
A whale optimization algorithm (WOA)based power system stabilizer (PSS) design methodology on modified single machine infinite bus (MSMIB) and multimachine systems to enhance the smallsignal stability (SSS) of the power system is presented. The PSS design methodology is implemented using an eigenvalue (EV)based objective function. The performance of the WOA is tested with several CEC14 and CEC17 test functions to investigate its potential in optimizing the complex mathematical equations. The New England 10generator 39bus system and the MSMIB system operating at various loading conditions are considered as the test systems to examine the proposed method. Extensive simulation results are obtained which validate the effectiveness of the proposed WOA method when compared with other algorithms.
Introduction
It is wellknown that a power system is a practical example of a highly nonlinear interconnected system formed by a number of transmission lines, switchgear equipment, various generators with diversified configurations, and a variety of loads. Increasing electrical energy demand has led almost all power grids to operate at their stability limits. Under such highly stressed conditions, the nonlinear behavior of the system is more pronounced by small disturbances. The Power System Stabilizer (PSS) has been developed as a supplementary controller to produce the necessary damping torque to suppress electromechanical oscillations in the range of 0.1–1.5 Hz originated from small disturbances [1].
The role of the PSS in coordination with the excitation system in a generator is illustrated in [2,3,4,5,6,7,8], while many classical PSS design techniques on various scenarios have been developed after the realization of the role PSS role in the power system. Such techniques include the pole placement technique [9, 10], sliding mode approach [11,12,13,14], linear quadratic regulator [15, 16], H − ∞ technique [17, 18], and fuzzy logic [19, 20]. The design of a PSS using the abovementioned conventional methods for single machine and multimachine interconnected systems of highly varying loading conditions is tedious, requires extensive expertise, and consumes much computational time. Many mathematical calculations and several parameters are required to design the PSS parameters for such a highly nonlinear, timevarying, and complex large scale power system. In addition, the design of a PSS using conventional techniques for the multimachine power system, one that always operates at variable loading conditions is a complex process. In recent decades, heuristic search algorithms have evolved and have proven their significance in solving complex problems, optimizing design parameters, and minimizing the cost of fuel. PSS design using heuristic algorithms can reduce the above mentioned drawbacks.
Extensive literature is available on the methods of PSS design for the single machine and multimachine power systems includingTabu search [21, 22], and algorithms such as genetic [23, 24], particle swarm optimization (PSO) [25,26,27,28,29], differential evolution (DE) [30, 31], hybrid differential evolution [32, 33], chaotic swarm optimization [34], firefly [35], harmony search [36], bacterial foraging search [37], cuckoo search [38], bat [39, 40], gravitational search [41], and grey wolf [42] algorithms. The above PSS design techniques, implemented on single and multimachine systems have proven their efficacy in improving the smallsignal stability of the power system. However, some drawbacks and limitations have also been observed including lack of fast convergence, the need for large numbers of design parameters, and not maintaining a good balance between the exploitation and exploration stages. To address these issues, a Whale Optimization Algorithm (WOA) is implemented in this paper on single machine and large scale multimachine systems to design the PSS parameters. Recently, WOA [43, 44] has been developed based on the hunting strategy of the humpback whale. The advantage of the WOA is the maintenance of a good balance between exploitation and exploration stages to identify the best global solution, while very few numbers of control parameters are required to. Compared to other heuristic search algorithms the number of steps involved in the evolving process and the rate of the convergence in the WOA are also improved. To validate the performance of the WOA, it is tested on various CEC 14 and CEC 17 test functions before designing the PSS parameters. Two test cases working with various loading conditions under several disturbance conditions are considered in the design of the PSS parameters using WOA. The first test case is the recently developed MSMIB system which is the modified version of the SMIB system. This takes the transformer secondary bus voltage of the generator side, instead of an infinite bus voltage. The details of this model are illustrated in [44]. The wellknown IEEE New England 10generator 39bus MM test system is considered as the second test case to design the PSS parameters using the WOA.
The remainder of the paper is as follows: the test systems considered are described in Sect. 2, while the structure of the PSS and formulation of the objective function for the PSS design are provided in Sect. 3. PSS design using the WOA is described in Sect. 4, and in Sect. 5, the performance analysis of the WOA on CEC 14 and CEC 17 test functions is presented. Simulation results are presented in Sect. 6 and finally, Sect. 7 draws the conclusion.
Systems under consideration
Case 1: MSMIB system
For a stability investigation, power system components, for example, synchronous generators, exciters, and transmission networks are required to structure proper dynamic models. A unique model of the power system, one which consolidates all the elements of the system, is represented by the SMIB system, which is modelled as the Kconstant model or HP model. This model has become very familiar to those studying SSS issues in a power system. Numerous controllers have also been developed for this model to investigate the smallsignal stability behavior [45]. A, modified version of this model is developed which gives similar performance to the Heffron Phillip's (HP) model and is called the Modified HP model (MHP). In this study, the PSS designed on the MHP model is considered as MPSS and on the multimachine system is considered as PSS. It is created by taking the generator side transformer secondary bus voltage as the reference rather than considering an infinite bus voltage. One line diagram for the MSMIB system is shown in Fig. 1 and the block diagram of the MHP model is represented in Fig. 2. Table 1 depicts the loading conditions considered for case 1.
Case 2: The New England 10generator 39bus system
This test system [46] is the second case study for the design of PSS. Optimal tuning of PSS parameters for the test case is carried out using the proposed techniques. Various disturbances are created on the system to investigate the performance of the proposed technique. The block diagram of the New England 10generator 39bus system is shown in Fig. 3.
PSS structure
It is wellknown that a complex interconnected power system is always subjected to various types of disturbances. Maintaining system stability under such conditions is a crucial task for the operation engineers. PSS has developed as the supplementary controller to the excitation system to provide the required damping to mitigate the oscillations originatig from the disturbances. Typically, the PSS is formulated by three important blocks namely, the phase compensation, washout, and gain blocks. The damping performance of the system is affected by the contribution of each block. The phase compensation block acts as a leadlag compensator, the wash out block acts as the high pass filter, and the gain block provides necessary gain value to damp the oscillations. The transfer function of the PSS is represented as
here V_{s} is the output from the PSS, K_{pssi} is the gain value of the PSS, subscript 'i' represents the ith machine, T_{wi} is the time constant of the washout block, T_{1i}, T_{2i}, T_{3i}, and T_{4i} are the phase compensation blocks' time constants and Δω_{i} is the speed deviation of the ith machine. It has been shown in the literature that the performance of the PSS for any test system can be significantly affected by its parameters. Thus the proposed algorithm is developed to properly design the parameters using the mentioned objective function for all the case studies.
Formulation of the EVbased objective function for PSS design
The single objective function for the design of the PSS is formed by combining two independent objective functions to place the lightly damped and unstable EVs of the system into the required splane regions. Figure 4 represents eigenvalue regions of the EVbased objective function. The first objective function is represented as
where N_{p} is the total number of loading conditions considered for the optimization process, σ_{i} is the real part of the ith eigenvalue of the system and σ_{0} is a constant value of the expected damping factor and is chosen as − 1.5. The real parts of poorly damped EVs of the system will be improved when the PSS is designed using the objective function. Hence the EVs will move to the left region of the imaginary axis as shown in Fig. 4a. The damping ratios of the lightly damped oscillating modes of the system will be improved, if the second objective function is considered alone. This is represented as
where \(\zeta_{i}\) is the damping ratio of the ith EV of the system, \(\zeta_{0}\) is a constant value of the expected damping ratio and is considered to be greater than 0.3. The EVs will be pushed into the wedge shape sector of Fig. 4b, when the PSS parameters are optimized based on this objective function. The advantage of this objective function is that the damping ratio of the low damped EVs of the system can be improved. Finally, the desired objective function is constructed by combining the two objective functions into one by assigning a weighting factorC, used to offset the weights of J_{a} and J_{b} as the value of c is chosen as 10 [22].
Thus, all the poorly damped eigenvalues will be shifted in the marked regions as shown in Fig. 4c, when \(J_{I}\) is used.
PSS design using WOA
The WOA was developed by Seyedali Mirjalili in 2016, by observing the hunting strategy of Humpback whales. Humpback whales, when attacking targets, follow a special type of hunting technique which is known as the bubblenet feeding method. In this technique, the whales create two types of mechanisms to reach the target, i.e.,: a shrinking encircle mechanism and a spiral mechanism. The WOA has been developed based on these two hunting mechanisms and it has several advantages over other methods. The biggest advantage of the WOA is that only one design parameter (a) is required for the optimization process [43]. This is fewer than other PSO and DE algorithms. The convergence rate of the WOA is superior and the number of steps required in the WOA is also fewer than PSO and DE. After initialization of the design parameters, PSO requires an update of the position and velocity for each particle. For DE algorithm, crossover and mutation steps are required for the optimization process. For any optimization algorithm, exploration and exploitation are the two important stages upon which the convergence of the optimal solution depends. The exploration stage is to get the global best solution and the exploitation stage is required to get the local best solution. Achieving a good balance between the two stages is the most challenging task for an optimization algorithm because of the stochastic nature of the algorithm. If an algorithm can succeeded in obtaining a good balance between the two, the optimal solutions can be found. As the WOA has been more successful in maintaining a good balance between the exploitation and the exploration stages than the other algorithms [43], it is used here for the design of the parameters of the PSS for the test cases.
Steps involved in the WOA algorithm
In the WOA, the steps of the hunting strategy of the whales are listed as:
Step 1: Initialization
To start the optimization process, PSS parameters are selected as control variables and their ranges with minimum and maximum values are listed in Table 2. The number of iterations taken as 100, and the population size is selected to be 40. The initial solutions are randomly generated by using the expression given as:
where 'X' is the control variable, and X_{j}^{min} and X_{j}^{max} are its minimum and maximum values, respectively. j = 1, 2,…, N, where N is the number of control variables, and i = 1, 2, 3, …, N_{P}, where N_{P} is the population size, rand €[0, 1] is a random number that changes between 0 and 1. All the tuneable parameters are selected with their ranges and randomly generated using (16).
Step 2: Determination of EVbased objective function
To place the EVs in predetermined regions, the EVbased objective function described in Sect. 3 is used. After initial solutions of the PSS are randomly generated, the Simulink models of the test cases in the MATLAB program are called and then the EVs of the test systems are determined. The EVbased objective function for the given population size s then found, and the EVbased objective function is determined for both test cases working with various operating conditions.
Step 3: Updating hunting agent path using shrinking encircling mechanism
After determining the objective function, updating the hunting agent process using the WOA is imitated. All the PSS parameters are treated as hunting agents in the optimization process. When the location of the target is identified, the hunting agent updates its posture by using this mechanism to reach the prey. Since the best solution is not obvious, WOA assumes that the current solution is the optimum solution and updates its position using:
where \(\vec{S}\) is the distance between the whale and target, \(\vec{R}\) and, \(\vec{A}\) are the coefficient vectors, t is the current iteration, P* corresponds to the optimum solution attained up to now, and \(\vec{P}\) stands for the position vector.;  represents the absolute value, and '.' is the multiplication of elements to elements. The vectors \(\vec{A}\), and \(\vec{R}\) are characterized as:
where \(\vec{A}\) is a random value that decreases from 2 to 0 during the optimization process and \(\vec{r}\) is an arbitrary number that varies between (0and,1). In every iteration process, the values of a, A, R are updated for each hunting agent. The present hunting agents update their locations by (7), but if the value of \(\vec{A}\) is less than 1, the following are used instead:
where \(\Delta \omega\) is a position of the whale selected randomly from the present population.
Step 4: Updation of hunting agent path using a spiral mechanism
In this step, the hunting agents follow in a spiralshaped path to reach the target. To simulate the spiralshaped track between the whale and target, a spiral equation is formulated. All the hunting agents update their position based on the formulated equations as:
where
Here 'l' is a stochastic limit that diverges between 0 and 1.
The shrinking encircle itinerary and spiral itinerary are merged by giving 50% probability to each of them to update the positions of the hunting agents to reach the prey as:
where δ is an arbitrary number that varies between 0 to 1.
The flowchart for the WOA to design the PSS parameters is depicted in Fig. 5, which describes various steps involved in the WOA. It starts with the initialization of the PSS parameters with their minimum and maximum values, and then leads to the generation of initial solutions, determination of the objective function, optimization of the PSS parameters according to the mechanisms mentioned above, and stopping criteria of the algorithm when the number of generations is completed. Table 3 shows the design parameters of the optimization algorithms.
Performance analysis of WOA
Many complex test functions are tested with the proposed WOA to prove its performance in finding the global solution, and the test results for CEC 14 and CEC 17 functions are displayed in Tables 4 and 5 respectively. CEC14 test functions,e.g., High Conditioned Elliptic, Discuss, Rosenbrocks, Weierstrass, Griewanks, Rastrigins's, Katsuura, Happy Cat, and Griewanks plus Rosenbrocks Functions are with the proposed WOA. The test results are also compared with PSO and, DE algorithms. As shown in Table 4, the results show that the WOA is succeeded in finding the global best fitness values for all test functions better than the other algorithms.
Similarly, the CEC17 test functions are such as Rastrigins's, Lunaeek biRastrigin, Ackley's Function, Griewanks, HG Bat, Griewanks plus Rosenbrocks and Schffer's functions are tested with the proposed WOA to derive the best fitness value. The test results prove that the WOA can derive the best i.e. better than the other algorithms. The convergence plots of CEC14 and CEC 17 functions are shown in Fig. 6. From these results, it can be seen that WOA performed better in deriving the global best fitness values than the other algorithms.
Simulation results and discussions
Case 1
The WOA and DE algorithms are run several times considering the new objective function. Evolved PSS parameters for four loading conditions are listed in Table 6. The following disturbances are considered to test the efficacy and vigor of the proposed WOAMPSS.

i.
10% step change at \(V_{ref}\)

ii.
10% step change at \(\Delta T_{m}\)
The simulation plots are shown in Fig. 7 for disturbance of 10% step change at \(V_{ref}\) and Fig. 8 for the disturbance of 10% step change at \(\Delta T_{m}\) for the four operating conditions. The plots represent system responses for the conventional design of PSS (MPSS), PSObased MPSS (PSOMPSS), PSOPIDbased MPSS (PSOPIDMPSS), and WOAbased MPSS (WOAMPSS). The plots obtained using the ISE objective function by PSO and PSObased PID are compared with the WOA and DE based PSS which are tuned with the EVbased objective function. The reason behind using the EVbased objective function is that it can relocate the unstable and lightly damped poles into the desired locations. From the results, it can be seen that that the intensity of the oscillations has been minimized with the WOAMPSS. The time taken for the oscillations to settle down is also reduced with WOAMPSS when compared to other stabilizers in all cases. Hence it is concluded that the proposed WOAMPSS tuned with the EVbased objective function shows superior performance over the MPSS, PSOMPSS, DEMPSS, and PSOPIDMPSS for all the loading conditions under the typical disturbances.
To further, demonstrate the performance of the proposed PSS design technique, EV analysis (EVA) is carried out for all the loading conditions and is displayed in Table 7 for four loading conditions with NOPSS, MPSS, PSOMPSS, DEMPSS, PSOPIDMPSS, and WOAMPSS. It is seen from the EVs that some of the poles lie on the righthand side of the splane when the PSS is not established in the system and hence the system becomes unstable. It is known that, for any system, the complex conjugate or imaginary poles with low damping ratio are responsible for the oscillatory behavior and the oscillatory behavior is pronounced when they are located near the imaginary axis. On the other hand, the real poles do not cause any oscillatory behavior in the system as the damping ratio of the real root is unity. In the present work, the real roots of the system with the proposed and other methods for all the loading conditions are already located in the desired and stable regions. However, the damping ratio of the oscillation causing eigenvalues is less than unity and there is a need to improve the damping ratio of these eigenvalues to improve the damping performance of the system. Therefore, in this work, much attention is paid to these complex conjugate poles with low damping ratios which causes oscillations in the system. Therefore, the oscillation causing eigenvalues or complex conjugate poles with low damping ratios are considered to be shifted into the stable regions. Because of this reason, the EVbased objective function is constructed in such a way that only lightly damped oscillating modes are shifted into the desired locations.
For example, for loading condition 1, with MPSS, the pair of complex conjugate poles which causes the oscillatory behavior are located at − 1.34 ± 6.39i with the damping ratio (\(\varsigma_{0}\)) of 0.20. It is shifted to − 2.03 ± 6.44i with a damping ratio of 0.30 when MPSS is optimized with PSO. It means that the real value (σ) of the complex conjugate pole is shifted from − 1.34 to − 2.03. Then it is further shifted to − 6.16 ± 10.1i with a damping ratio of 0.52 when PSS is tuned with PSO PIDMPSS. Finally, with WOAPSS, it is shifted to − 2.06 ± 1.57i and the damping ratio is improved to 0.79. Thus the damping ratio has been improved from 0.20 to 0.79 with the proposed WOAMPSS. Similarly, for loading condition 2, the oscillation causing eigenvalue pair are located at − 1.24 ± 6.27i with the damping ratio of 0.19. When the PSOMPSS is established in the system the damping ratio is improved to 0.23 and, with DEMPSS it is improved to 0.26. After that with PSOPIDMPSS, the damping ratio of the oscillatory causing eigenvalue is improved to 0.43, and finally to 0.80 with the proposed WOAMPSS For the loading condition 3, the damping ratio is improved from 0.15 to 0.70 with the proposed WOAMPSS. A similar improvement has also been observed with the fourth operating condition. Hence, from all these results it is seen that when the PSS parameters are optimized with the proposed WOA, the damping ratios of the oscillation causing eigenvalues are improved and the damping performances of the system with the proposed optimized parameters of the WOA are better enhanced for all the loading conditions than other methods.
Case 2
To test the efficacy of the suggested WOAPSS, a second test case is considered. The optimal design of PSS is carried out by considering the EVbased objective function using the WOA, DE, and PSO algorithms. All the generators except generator one, are equipped with PSS and the parameters are optimized using the WOA. In total 45 parameters named K_{Pssi}, T_{1i}, T_{2i}, T_{3i} and T4i are optimized, as listed in Table 8. To test the effectiveness and robustness of the proposed WOAPSS various disturbances are created at various locations of the test system. These conditions are extremely harsh from a stability point of view with the following cases studied to investigate the efficacy of the WOAPSS:

Scenario 1 line outage between 21 and 22.

Scenario 2 line outage between 14 and 15.

Scenario 3 25% increase in loads at buses 16 and 21, 25% increase in the generation of G_{7,} and line outage between 21 and 22.
Figure 9 represents Scenario 1 of case 2. To create the disturbance, the line between buses 21 and 22 is opencircuited and the stability behavior of the generators with the proposed WOAPSS and other PSSs are studied. After that, another disturbance of line outage between buses 14 and 15 (scenario 2) is created and the results are shown in Fig. 10. Similarly, the third disturbance condition of the line outage between 21 and 22 is created under scenario 3 to effectiveness of the proposed PSS design technique, and Fig. 11 shows the corresponding speed deviation of the system. It can be observed from the results that the system becomes unstable when PSSs are not connected to the generators. When PSSs are incorporated (without proper tuning), more oscillations are observed in the system. This is because of the lack of sufficient damping torque. This is not desirable from the stability point of view. On the other hand, when PSS parameters are designed with the WOA and placed at respective generators, the oscillations are reduced. The intensity of the oscillations is considerably lower with WOAPSS compared to PSOPSS and DEPSS, and the simulation results show that WOAPSS provides better dynamic performance characteristics over DEPSS and PSOPSS.
The system behavior under the second disturbance of line outage between 14 and 15 is studied. Extensive simulation results on the speed deviations are obtained and presented in Fig. 12. The simulation results show that WOAPSS provides better damping performance than the other two methods. For the third scenario, the speed deviation under the disturbance of a 25% increase in loads at buses 16 and 21, a 25% increase in the generation of G_{7,} and the line outage between 21 and 22 are shown in Fig. 13. From these responses, it is noted that the oscillations are reduced and settled in quicker with the WOAPSS compared to DEPSS and PSOPSS at all the generators shown. Hence, WOAPSS can provide better damping performance than the other two stabilizers. Table 9 depicts the time response specifications of generators under scenario 2. As seen, the settling time is decreased from 9.8300 to 8.9371 s with WOAPSS for generator two. In the case of generator three, the settling time is reduced from 8.4502 to 8.4913 s, where as for generator five, oscillations are settled in 7.9892 s with the WOAPSS. At generator six the settling time is 7.0971 s when PSOPSS is used and is reduced to 6.1351 s with DEPSS and to 5.9735 s with WOAPSS. Similarly, the time taken for settling down the oscillations at generator seven with PSOPSS is 7.1417 and is reduced to 6.7459 s with WOAPSS. A similar decrement is also observed in the remaining generators. The peak overshoot is also decreased with WOAPSS when compared to PSOPSS and DEPSS. Hence it is shown that the WOAbased stabilizers give better transient responses than the other DE and PSO based stabilizers for all the generators under this disturbance with significant improvements in peak overshoot and settling time.
The strength of the WOAPSS is tested with the third disturbance condition. and Fig. 13 depicts the speed deviation plots under the disturbance of a 25% increase in loads at buses 16 and 21, a 25% increase in the generation of G_{7}, and the line outage between 21 and 22. The peak overshoots of the oscillations at generator two, seven, eight, and nine are reduced with the WOAPSS compared to other stabilizers designed with the DE and PSO algorithms. The oscillations are also settled quicker with the WOAPSS. Thus, it is concluded that WOAPSS can provide better damping performance than DEPSS and PSOPSS.
EVA is carried out for all the scenarios to test the effectiveness of the proposed design technique. The EVA of the system under the three disturbance conditions is depicted in Table 10 and it can be seen that the lightly damped EVs are shifted more towards the lefthand side of the splane with the proposed WOAbased design technique at all the disturbance conditions than the other methods. This reveals that the damping factor of the lightly damped EVs is better improved and system enhanced with the WOAPSS than with PSOPSS and DEPSS under all the disturbance conditions. From all the simulation results it can be concluded that the SSS of the system is improved with the proposed WOAbased PSS design technique, and better than with the other design approaches.
Conclusion
A PSS design technique on the MSMIB and the large scale New England 10generator 39bus multimachine systems using a WOA is proposed. The PSS design approach has been carried out by considering an EVbased objective function. The performance of the proposed WOA has been tested on standard CEC14 and CEC17 test functions. The results have shown that the proposed WOAPSS design technique is very effective, requires less time for tuning, and reduces computational complexity compared to PSOPSS, and DEPSS. Results reveal that the proposed WOAbased PSS can improve the dynamic performance of the MSMIB and the large scale IEEE New England 10generator 39bus systems operating with various loading conditions at several disturbance conditions, and do it more effectively than the other methods.
Availability of data and materials
Please contact author for data and material request.
Abbreviations
 WOA:

Whale optimization algorithm
 PSS:

Power system stabilizer
 MPSS:

Modified power system stabilizer
 MSMIB:

Modified single machine infinite bus
 MM:

Multimachine
 SSS:

Small signal stability
 EV:

Eigenvalue
 PSO:

Particle swarm optimization
 DE:

Differential evolution
 HP:

Heffron Phillips
 MHP:

Modified Heffron Phillips
 EVA:

Eigenvalue analysis
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BD designed the study and formulated the objective function. BD and MS performed the simulations on test systems. MS and RS as supervisors helped in pursing the work with constructive suggestions and edited the manuscript. All authors read and approved the final manuscript.
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B. Dasu: The author completed his M. Tech from JNTU Kakinada and pursuing his PhD in JNTU Kakinada. His research interests include Control application to Power Systems and Evolutionary Algorithms.
Mangipudi Siva Kumar: The author completed his M.E and PhD from Andhra University. He is presently working as Professor in EEE Department, Gudlavalleru Engineering College, Gudlavalleru. He has contributed more than 40 technical papers in various referred journals and conference. He is a life member of ISTE, member of IEEE and IAEng and Fellow of Institution of Engineers. His research interests include model order reduction, interval system analysis, design of PI/PID controllers for Interval systems, sliding mode control and Soft computing Techniques.
R. Srinivasa Rao: The author completed his M.E from IISC Bangalore and PhD from JNTU Hyderabad. He is currently working as professor in EEE Department of JNTU Kakinada. His research interests include Optimization Algorithms, State estimation, Modeling and control of Induction Generators. He has published papers in IEEE transactions, Elsevier Publications.
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Dasu, B., Mangipudi, S. & Rayapudi, S. Small signal stability enhancement of a large scale power system using a bioinspired whale optimization algorithm. Prot Control Mod Power Syst 6, 35 (2021). https://doi.org/10.1186/s4160102100215w
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DOI: https://doi.org/10.1186/s4160102100215w
Keywords
 Smallsignal stability
 Power system stabilizer
 Stability
 Modified single machine infinite bus system
 Multimachine system
 Whale optimization algorithm
 Eigenvalues