Application of Neuro-Fuzzy Controller to Replace SMIB and Interconnected Multi-Machine Power System Stabilizers

In this research, an effective application and performance assessment of the Neuro-Fuzzy Controller (NFC) damping controller is designed to replace a single machine infinite bus (SMIB) power system stabilizer (PSS), and coordinated multi PSSs in large interconnected power systems are presented. The limitation of the conventional PSSs on SMIB and interconnected multi-machine test power systems are exposed and disclosed by the proposed NFC stabilizer. The NFC is a nonlinear robust controller which does not require a mathematical model of the test power system to be controlled, unlike the conventional PSSs’ damping controller. The Proposed NFC is designed to improve the stability of SMIB, an interconnected IEEE 3-machine, 9-bus power system, and an interconnected two-area 10-machine system of 39-bus New England IEEE test power system under multiple operating conditions. The proposed NFC damping controller performance is compared with the conventional PSS damping controller to confirm the capability of the proposed stabilizer and realize an improved system stability enhancement. The conventional PSSs’ design problem is transformed into an optimization problem where an eigenvalue-based objective function is developed and applied to design the SMIB-PSS and the interconnected multi-machine PSSs. The time-domain phasor simulation was done in the SIMULINK domain, and the simulation results show that the transient responses of the system rise time, settling time, peak time, and peak magnitude were all impressively improved by an acceptable amount for all the test system with the proposed NFC stabilizer. Thus, the NFC was able to effectively control the LFOs and produce an enhanced performance compared to the conventional PSS damping controller. Similarly, the result validates the effectiveness of the proposed NFC damping controller for LFO control, which demonstrates more robustness and efficiency than the classical PSS damping controller. Therefore, the application and performance of the NFC has appeared as a promising method and can be considered as a remarkable method for the optimal design damping stabilizer for small and large power systems.

T Ai where the generator rotor angle is δ, the generator speed is ω, the synchronous speed is ω s , the internal transient voltage is E d and E q . They are behind X d and X q , which are the d-axis and q-axis transient reactances, respectively. I d and I q are the d-axis and q-axis constituents of the generator stator currents, respectively, the damping coefficient is D, the mechanical power input is T M , the generator inertia constant is H, the d-axis and q-axis reactance is X d and X q , respectively, while T d0 and T q0 are the open circuit d-axis and q-axis time variable constants, respectively. V is the terminal voltage for the synchronous generator, while the exciter voltage is E f d , the regulator gain is K A , the regulator time constant is T A and the regulator reference voltage is V re f . In Equations (1) and (4), i represents the ith synchronous generator.
To simplify the excitation controller design, we consider the mechanical input torque T Mi as a constant term, meaning we make an assumption here that the governor action is so slow that its impact is insignificant on the system dynamics [31]. Now the electrical torque is expressed as: T Ei = E di I di + E qi I qi + X qi − X di I di I qi (6) The electrical torque is introduced in Equation (4). For a power system having n buses and m generators, it provides m − n load buses. Then, the algebraic power system equations can be represented by: 0 = V i e jθi + (R si + jX di ) I di + jI qi e j(δi− π 2 ) − E di + X qi − X di I qi + jE qi e j(δi− π 2 ) i = 1, · · · , m (7) V i e jθi I di + jI qi + P Li (V i ) + jQ Li V i e jθi I di + jI qi + P Li (V i ) + jQ Li (V i ) = n k=1 V i V k Y ik e j(θi−θk−αik) , i = 1, · · · , m (10) where the active and reactive powers are P L and Q L , respectively, the admittance matrix is Ye jα , and the bus voltage angle is θ. Constant impedances in a power system having m generators represent the loads. The basic terms and definitions related to linear system analysis, along with the linearization method of nonlinear systems out of the DAEs from Equations (1)- (9), are explained below: Using a set of first-order nonlinear DAEs, system behavior can be represented as follows: .
x = f (x, z, u) 0 = g(x, z, u) y = h(x, z, u) (12) where f is the set of the first-order nonlinear differential equations that represent the system and the controller dynamics, x is the vector of the state variables, z represents the vector of the algebraic variables, u is the vector of the input variables, g is the set of nonlinear algebraic equations that describes the network power flow equations, y is a set of output variables, and h is the set of equations that represents the output variables. In this study, z represents the vector of the bus voltage magnitudes and phase angles, u represents the PSS output signals. x = δ, ω, E q , E f d T , where, δ and ω are the rotor angles and rotor speed, respectively, while E q and E f d are the internal and field voltages, respectively.
The design of the PSS is based on the linear incremental model and on [40,41], while the linearized power system model with n machines and m PSSs around an equilibrium point is given as follows: Now, eliminating ∆z and assuming that the power flow Jacobian g z is non-singular, the state space equation of the system can be readily written as follows: where A and B are 4n × 4n and 4n × m matrices, respectively, given by Equation (15).
The output variables can be written as Equation (16).
where C and D are This method of presenting the power system with DAEs is extensively used for power system small signal stability analysis.

PSS Design Procedure
A wide speed based shown in Figure 1 is utilized in this research [42]. The conventional PSS can be represented by the transfer function of the ith system in Equation (18), as shown in Figure 1.
This method of presenting the power system with DAEs is extensively used for power system small signal stability analysis.

PSS Design Procedure
A wide speed based shown in Figure 1 is utilized in this research [42]. The conventional PSS can be represented by the transfer function of the th system in Equation (18), as shown in Figure 1. Here, V PSSi is the stabilizing signal from the conventional PSS output at the ith machine, T w is called the time constant from the washout block, dω i is the mechanical speed deviation signal from the synchronous speed of the ith machine. The PSS parameters to be determined here are the stabilizer gain KG i and the T 1i , T 2i , T 3i and T 4i , respectively. These PSS parameters are tuned using a Farmland Fertility Algorithm (FFA). Figure 2 shows the conventional PSS system connected to the IEEE-type-ST1 excitation system. Here, is the stabilizing signal from the conventional PSS output at the th machine, is called the time constant from the washout block, is the mechanical speed deviation signal from the synchronous speed of the th machine. The PSS parameters to be determined here are the stabilizer gain and the , , and , respectively. These PSS parameters are tuned using a Farmland Fertility Algorithm (FFA). Figure 2 shows the conventional PSS system connected to the IEEE-type-ST1 excitation system.

Design and Implementation of the Neuro-Fuzzy Controller
NFC was designed and successfully applied to many industrial applications that work on the principle of employing Artificial Neural Network (ANN) constructively to execute the function of the Fuzzy Logic Controller (FLC) [34]. The FLC is capable of making inferences from expert knowledge, and the ANN has the capability of learning, generalizing, and adapting. The NFC is an intelligent control system combining the qualities of the FLC and ANN. Therefore, NFCs show characteristics of a nonlinear controller that is capable of learning, adapting, and making inferences. The NFC is considered to be one of the controllers suitable for model-free design. These properties make the NFC robust when associated with other conventional controllers. Consequently, NFCs are commonly used in non-linear systems with parameter variation and uncertainty. Consider the Fuzzy rules of the Sugeno type Fuzzy logic defined below: : X , . . . , then y = = + + + (19) where Xi and y are the input and output variables, respectively, and Ai j ϵ R are the coefficients of the linear fi = (x1, x2… xn) function. The schematic structure of the NFC that is employed in the control algorithm is shown in Figure 3.

Design and Implementation of the Neuro-Fuzzy Controller
NFC was designed and successfully applied to many industrial applications that work on the principle of employing Artificial Neural Network (ANN) constructively to execute the function of the Fuzzy Logic Controller (FLC) [34]. The FLC is capable of making inferences from expert knowledge, and the ANN has the capability of learning, generalizing, and adapting. The NFC is an intelligent control system combining the qualities of the FLC and ANN. Therefore, NFCs show characteristics of a nonlinear controller that is capable of learning, adapting, and making inferences. The NFC is considered to be one of the controllers suitable for model-free design. These properties make the NFC robust when associated with other conventional controllers. Consequently, NFCs are commonly used in non-linear systems with parameter variation and uncertainty. Consider the Fuzzy rules of the Sugeno type Fuzzy logic defined below: R j : I f X 1 , A j 1 . . . X n , A j n then y = f i = a j 0 + a j 1 X 1 + a j 2 X 2 + a j n X n (19) where X i and y are the input and output variables, respectively, and A i j R are the coefficients of the linear f i = (x 1 , x 2 . . . x n ) function. The schematic structure of the NFC that is employed in the control algorithm is shown in Figure 3. The inputs of the NFC are error (e) and change in error (∆e). The five membership functions were chosen for each input [43]. It can be seen that membership functions are executed in the second layer. In this case, the membership function is substituted by an activation function for each artificial neuron. The output of this layer is as follows: where σ and m are also input parameters that represent the parameters of membership functions to be adapted. Xi is the input of the ith cell of the 2 nd layer. Similar to FLC, the third layer of NFC consists of a rule base, and Fuzzy rules are determined in this layer.
where Xj 3 is the input of the jth cell of the third layer. The output of the system is defined using central clarification for Mamdani Fuzzy logic, as shown below: where layer-4 is called the normalization layer, in which the accuracy of Fuzzy rules is calculated. Layer-5 is called the firing size of a rule. The firing degree of normalized rules is multiplied by the linear f function in this layer. This layer generates the output values required for the interconnected test system.

( )
The output of layer-6 is as follows: To update input and output parameters by using the analog teaching method with the backpropagation algorithm, the squared error (E) which minimizes tracking error (e) is determined as follows [44]: The inputs of the NFC are error (e) and change in error (∆e). The five membership functions were chosen for each input [43]. It can be seen that membership functions are executed in the second layer. In this case, the membership function is substituted by an activation function for each artificial neuron. The output of this layer is as follows: where σ and m are also input parameters that represent the parameters of membership functions to be adapted. X i is the input of the ith cell of the 2nd layer. Similar to FLC, the third layer of NFC consists of a rule base, and Fuzzy rules are determined in this layer.
where X j 3 is the input of the jth cell of the third layer. The output of the system is defined using central clarification for Mamdani Fuzzy logic, as shown below: where layer-4 is called the normalization layer, in which the accuracy of Fuzzy rules is calculated. Layer-5 is called the firing size of a rule. The firing degree of normalized rules is multiplied by the linear f function in this layer. This layer generates the output values required for the interconnected test system.
The output of layer-6 is as follows: To update input and output parameters by using the analog teaching method with the backpropagation algorithm, the squared error (E) which minimizes tracking error (e) is determined as follows [44]: The parameters to be adapted can be updated by the following equation: Here, ϑ is the parameter to be adapted and η is the learning rate. The chain rule is used to obtain the partial derivative. The derivative chain up to the output of the NFC is obtained from the following equation: (27) where the local gradient is δ 1 . Updating the result parameters of the NFC used in the interconnected test system was achieved as follows: In the NFC structure, precondition parameters of the membership layer have been trained in the simulation model. During the simulation studies, outcome parameters have been trained using a back-propagation learning algorithm. The complete structure of the NFC damping controller-based interconnected test power system is shown in Figure 4. To control the LFO in the system, the synchronous rotor speed deviation dω is applied to the NFC controller. A reference value for NFC signal V n f cs is obtained from the output of the NFC controller. These voltage signals are sent to the excitation system block, which generates the required signals for damping the LFOs in the test system. back-propagation learning algorithm. The complete structure of the NFC damping controller-based interconnected test power system is shown in Figure 4. To control the LFO in the system, the synchronous rotor speed deviation is applied to the NFC controller. A reference value for NFC signal is obtained from the output of the NFC controller. These voltage signals are sent to the excitation system block, which generates the required signals for damping the LFOs in the test system.

NEURO-FUZZY CONTROLLER (NFC)
NFC Stabilizer signal π π π π π π π π π ∑ IEEE-type-ST1 Exciter New England 10-machine, 39-bus power system  Table 1 shows the suitable choice of parameters used for the FFA algorithms, which help the algorithms achieve optimal convergence speed and a fast computational cost for the PSS design [45]. The optimization process was terminated by a pre-specified number of iterations for all the three algorithms, and it is worth mentioning that the PSS design method was initialized and run several times before the optimal parameters of the PSS were chosen [46].   Table 1 shows the suitable choice of parameters used for the FFA algorithms, which help the algorithms achieve optimal convergence speed and a fast computational cost for the PSS design [45]. The optimization process was terminated by a pre-specified number of iterations for all the three algorithms, and it is worth mentioning that the PSS design method was initialized and run several times before the optimal parameters of the PSS were chosen [46].

Results and Discussion
This section discusses the eigenvalue, time-domain, and transient response simulation results of the three test power systems under multiple operating conditions.

SMIB Power System Results and Discussion under Operating Condition 1
The SMIB power system structure used was modeled in the SIMULINK domain with the following system parameters, and the complete system data can be found in [40]. The SMIB synchronous generator is represented by fourth-order models of the DAEs explained by the test system modeling equations described earlier. In this research, MATPOWER software was utilized to execute the system power flow which computes the system's initial condition states. The solution of the DAE Equations (1)-(9) shows the power system's nonlinear dynamic behavior and the DAEs are solved via an ODE solver in MATLAB/SIMULINK.

Time-Domain Simulation of the SMIB Test Power System without PSS and NFC Damping Controllers under Operating Condition 1
In this section, the system undergoes a disturbance without a damping controller (PSSs and NFC) for LFO mitigation, and hence system stability enhancement. Eigenvalues simulation analysis and nonlinear time-domain simulations should be conducted for comprehensive evaluation. The SMIB system was subjected to a three-phase fault without a PSS damping controller. A symmetrical 100 ms three-phase fault was observed at t = 1 s, and a nonlinear time-domain simulation was performed. After the fault was cleared, at 0.2 s, the system stable condition was restored. Table 2 shows the eigenvalues results with their associated damping ratio and the frequency for the power system case without the PSS and NFC installed. From Table 2, Mode 1 produces a weak damping ratio for the no PSS case condition with 0.0588 ± j8.3601 eigenvalues. Similarly, the worst damping ratio of EM was found to be −0.0070. In addition, a nonlinear time-domain simulation was performed, and Figure 5 shows the unstable generator relative power angles δ in radian for the system without PSS and NFC, while Figure 6 shows the unstable generator speed deviation dω in radian per second for the system without PSS and NFC. Similarly, Figure 7 shows the unstable system rotor speed response ω in radian per second for the system without PSS and NFC.
In addition, a nonlinear time-domain simulation was performed, and Figure 5 shows the unstable generator relative power angles δ in radian for the system without PSS and NFC, while Figure 6 shows the unstable generator speed deviation in radian per second for the system without PSS and NFC. Similarly, Figure 7 shows the unstable system rotor speed response in radian per second for the system without PSS and NFC.   For the PSS stabilizer design, a symmetrical 100 ms three-phase fault was observed at = 1 s, and a nonlinear time-domain simulation was performed. To increase the damping properties of the electromechanical modes (EMs), there are two types of objective function used for the optimization process: the eigenvalue-based objective function Equation (29), which is employed to prevent unstable modes formation that will lead the system's eigenvalues to move towards the left-hand side (LHS) of the complex plane, and the time-domain-based objective function. In this study, the damping properties of the electromechanical modes (EMs) were improved using an eigenvaluebased objective function in Equation (30), where the PSS control parameters can be determined by  For the PSS stabilizer design, a symmetrical 100 ms three-phase fault was observed at = 1 s, and a nonlinear time-domain simulation was performed. To increase the damping properties of the electromechanical modes (EMs), there are two types of objective function used for the optimization process: the eigenvalue-based objective function Equation (29), which is employed to prevent unstable modes formation that will lead the system's eigenvalues to move towards the left-hand side (LHS) of the complex plane, and the time-domain-based objective function. In this study, the damping properties of the electromechanical modes (EMs) were improved using an eigenvaluebased objective function in Equation (30), where the PSS control parameters can be determined by solving the system state matrix from Equation (14). For the PSS stabilizer design, a symmetrical 100 ms three-phase fault was observed at t = 1 s, and a nonlinear time-domain simulation was performed. To increase the damping properties of the electromechanical modes (EMs), there are two types of objective function used for the optimization process: the eigenvalue-based objective function Equation (29), which is employed to prevent unstable modes formation that will lead the system's eigenvalues to move towards the left-hand side (LHS) of the complex plane, and the time-domain-based objective function. In this study, the damping properties of the electromechanical modes (EMs) were improved using an eigenvalue-based objective function in Equation (30), where the PSS control parameters can be determined by solving the system state matrix A from Equation (14).
where λ i represents the ith eigenvalues of the power system state matrix A from Equation (14), P F is a constant called the penalty constant, that is applied in forming the positive eigenvalues (and can also improve the slow eigenvalues) and was considered to be 50 in this study. The objective function J egn minimizes the inter-regional and EM system damping, while at the same time preventing unstable modes formation that will lead the system's eigenvalues to move towards the left-hand side ( The rate at which the PSS design index converges using the FFA search algorithm is shown in Figure 8, while the optimal parameters of the designed PSSs are shown in Table 3. From the convergence characteristics, the FFA method shows a good convergence rate at eight iterations. The rate at which the PSS design index converges using the FFA search algorithm is shown in Figure 8, while the optimal parameters of the designed PSSs are shown in Table 3. From the convergence characteristics, the FFA method shows a good convergence rate at eight iterations.   Table 4 shows the eigenvalues results with their associated damping ratio and the frequency for the power system case with PSS installed in the system, respectively. From Table 2, Mode 2 produces a weak damping ratio for the no PSS case condition. After the optimal PSS design, these modes were impressively enhanced from 0.0588 ± 8.3601 up to a stable mode of −0.4772 ± 4.3029, respectively, using the FFA design method. Similarly, the worst damping ratio EM was improved from − 0.0070 to 0.1102 using the FFA design method. Similarly, Figure 9 illustrates the eigenvalues plot for the system with the FFA-PSS controller based on the numerical simulation results of Tables 2 and 4.   Table 4 shows the eigenvalues results with their associated damping ratio and the frequency for the power system case with PSS installed in the system, respectively. From Table 2, Mode 2 produces a weak damping ratio for the no PSS case condition. After the optimal PSS design, these modes were impressively enhanced from 0.0588 ± j8.3601 up to a stable mode of −0.4772 ± j4.3029, respectively, using the FFA design method. Similarly, the worst damping ratio EM was improved from −0.0070 to 0.1102 using the FFA design method. Similarly, Figure 9 illustrates the eigenvalues plot for the system with the FFA-PSS controller based on the numerical simulation results of Tables 2 and 4.  Figure 8. Convergence characteristics of FFA in finding optimal design of PSS.  Table 4 shows the eigenvalues results with their associated damping ratio and the frequency for the power system case with PSS installed in the system, respectively. From Table 2, Mode 2 produces a weak damping ratio for the no PSS case condition. After the optimal PSS design, these modes were impressively enhanced from 0.0588 ± 8.3601 up to a stable mode of −0.4772 ± 4.3029, respectively, using the FFA design method. Similarly, the worst damping ratio EM was improved from − 0.0070 to 0.1102 using the FFA design method. Similarly, Figure 9 illustrates the eigenvalues plot for the system with the FFA-PSS controller based on the numerical simulation results of Tables 2 and 4.  Moreover, a nonlinear time-domain simulation was performed, and Figure 10 shows the stable generator relative power angles δ in radian for the system with PSS, while Figure 11 shows the stable generator speed deviation dω in radian per second for the system with PSS. Similarly, Figure 12 shows the stable system rotor speed response ω in radian per second for the system with PSS.  Moreover, a nonlinear time-domain simulation was performed, and Figure 10 shows the stable generator relative power angles δ in radian for the system with PSS, while Figure 11 shows the stable generator speed deviation in radian per second for the system with PSS. Similarly, Figure 12 shows the stable system rotor speed response in radian per second for the system with PSS.    Moreover, a nonlinear time-domain simulation was performed, and Figure 10 shows the stable generator relative power angles δ in radian for the system with PSS, while Figure 11 shows the stable generator speed deviation in radian per second for the system with PSS. Similarly, Figure 12 shows the stable system rotor speed response in radian per second for the system with PSS.    In this case, the test power system is equipped with an NFC stabilizer, and a symmetrical three-phase fault is subjected to the system. Table 5 shows the eigenvalues results with their associated damping ratio and frequency for the power system case with NFC installed in the system, respectively. From Table 2, Mode 2 produces a weak damping ratio for the no PSS case condition. After the NFC design, these modes were impressively enhanced from 0.0588 ± j8.3601 up to a stable mode of −0.3421 ± j3.3027, respectively, using the NFC stabilizer. Similarly, the worst damping ratio EM was improved from −0.0070 to 0.2103 using the NFC. Similarly, Figure 13 illustrates the eigenvalues plot for the system with the NFC stabilizer controller based on the numerical simulation results of Tables 2 and 5. In this case, the test power system is equipped with an NFC stabilizer, and a symmetrical threephase fault is subjected to the system. Table 5 shows the eigenvalues results with their associated damping ratio and frequency for the power system case with NFC installed in the system, respectively. From Table 2, Mode 2 produces a weak damping ratio for the no PSS case condition. After the NFC design, these modes were impressively enhanced from 0.0588 ± 8.3601 up to a stable mode of −0.3421 ± 3.3027, respectively, using the NFC stabilizer. Similarly, the worst damping ratio EM was improved from −0.0070 to 0.2103 using the NFC. Similarly, Figure 13 illustrates the eigenvalues plot for the system with the NFC stabilizer controller based on the numerical simulation results of Tables 2 and 5.   Moreover, a nonlinear time-domain simulation was performed with NFC installed in the system, and Figure 14 shows the stable generator relative power angles δ in radian for the system with NFC, while Figure 15 shows stable the generator speed deviation in radian per second for the system with NFC. Similarly, Figure 16 shows the stable system rotor speed response in radian per second for the system with NFC. Moreover, a nonlinear time-domain simulation was performed with NFC installed in the system, and Figure 14 shows the stable generator relative power angles δ in radian for the system with NFC, while Figure 15 shows stable the generator speed deviation dω in radian per second for the system with NFC. Similarly, Figure 16 shows the stable system rotor speed response ω in radian per second for the system with NFC.

Comparison between the Proposed NFC and PSS Damping Controllers under Operating Condition 1
This section presents comparative simulation results between the proposed NFC and the PSS damping controllers for LFO's robustness. The generator power angles δ in radian, the generator rotor speed deviation in radian per second for the SMIB system using PSS, and the proposed NFC damping controllers are shown in Figures 17 and 18 respectively.

Comparison between the Proposed NFC and PSS Damping Controllers under Operating Condition 1
This section presents comparative simulation results between the proposed NFC and the PSS damping controllers for LFO's robustness. The generator power angles δ in radian, the generator rotor speed deviation in radian per second for the SMIB system using PSS, and the proposed NFC damping controllers are shown in Figures 17 and 18 respectively.

Comparison between the Proposed NFC and PSS Damping Controllers under Operating Condition 1
This section presents comparative simulation results between the proposed NFC and the PSS damping controllers for LFO's robustness. The generator power angles δ in radian, the generator rotor speed deviation in radian per second for the SMIB system using PSS, and the proposed NFC damping controllers are shown in Figures 17 and 18 respectively.

Comparison between the Proposed NFC and PSS Damping Controllers under Operating Condition 1
This section presents comparative simulation results between the proposed NFC and the PSS damping controllers for LFO's robustness. The generator power angles δ in radian, the generator rotor speed deviation dω in radian per second for the SMIB system using PSS, and the proposed NFC damping controllers are shown in Figures 17 and 18 respectively.  Similarly, Figure 19 shows the variation in the generator speed in radian per second for the SMIB system for PSS and the proposed NFC stabilizer. From the time-domain simulation results, it was seen that the stabilizer using the proposed NFC was effectively able to damp out the LFOs of the test system under severe system disturbances. Thus, the NFC damping controller can be applied as a general damping controller for the robust design stabilizer and other similar science and engineering controller applications.

Response Scenario for Controller Performance Analysis under Operating Condition 1
This section explains the transient response simulation which describes the performance analysis of the SMIB power system when the system has no controller (NC), with a PSS controller, and with the proposed NFC damping controller for the SMIB system, respectively. Table 6 considers how the SMIB system responds to the transient situation with respect to settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM) for NC, PSS, and NFC damping controller cases, respectively. The rotor speed (ω) SMIB transient response in terms of settling time and rise time were all remarkably improved by an amount of 73.21% and 87.02%, respectively, by the proposed NFC stabilizer as compared with the NC in the system.  Similarly, Figure 19 shows the variation in the generator speed in radian per second for the SMIB system for PSS and the proposed NFC stabilizer. From the time-domain simulation results, it was seen that the stabilizer using the proposed NFC was effectively able to damp out the LFOs of the test system under severe system disturbances. Thus, the NFC damping controller can be applied as a general damping controller for the robust design stabilizer and other similar science and engineering controller applications.

Response Scenario for Controller Performance Analysis under Operating Condition 1
This section explains the transient response simulation which describes the performance analysis of the SMIB power system when the system has no controller (NC), with a PSS controller, and with the proposed NFC damping controller for the SMIB system, respectively. Table 6 considers how the SMIB system responds to the transient situation with respect to settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM) for NC, PSS, and NFC damping controller cases, respectively. The rotor speed (ω) SMIB transient response in terms of settling time and rise time were all remarkably improved by an amount of 73.21% and 87.02%, respectively, by the proposed NFC stabilizer as compared with the NC in the system. Similarly, Figure 19 shows the variation in the generator speed ω in radian per second for the SMIB system for PSS and the proposed NFC stabilizer.  Similarly, Figure 19 shows the variation in the generator speed in radian per second for the SMIB system for PSS and the proposed NFC stabilizer. From the time-domain simulation results, it was seen that the stabilizer using the proposed NFC was effectively able to damp out the LFOs of the test system under severe system disturbances. Thus, the NFC damping controller can be applied as a general damping controller for the robust design stabilizer and other similar science and engineering controller applications.

Response Scenario for Controller Performance Analysis under Operating Condition 1
This section explains the transient response simulation which describes the performance analysis of the SMIB power system when the system has no controller (NC), with a PSS controller, and with the proposed NFC damping controller for the SMIB system, respectively. Table 6 considers how the SMIB system responds to the transient situation with respect to settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM) for NC, PSS, and NFC damping controller cases, respectively. The rotor speed (ω) SMIB transient response in terms of settling time and rise time were all remarkably improved by an amount of 73.21% and 87.02%, respectively, by the proposed NFC stabilizer as compared with the NC in the system. From the time-domain simulation results, it was seen that the stabilizer using the proposed NFC was effectively able to damp out the LFOs of the test system under severe system disturbances. Thus, the NFC damping controller can be applied as a general damping controller for the robust design stabilizer and other similar science and engineering controller applications.

Response Scenario for Controller Performance Analysis under Operating Condition 1
This section explains the transient response simulation which describes the performance analysis of the SMIB power system when the system has no controller (NC), with a PSS controller, and with the proposed NFC damping controller for the SMIB system, respectively. Table 6 considers how the SMIB system responds to the transient situation with respect to settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM) for NC, PSS, and NFC damping controller cases, respectively. The rotor speed (ω) SMIB transient response in terms of settling time and rise time were all remarkably improved by an amount of 73.21% and 87.02%, respectively, by the proposed NFC stabilizer as compared with the NC in the system. The NFC controller's transient simulation analysis was found to be able to improve the system stability in terms of ST, RT, PT, and PM by an acceptable amount, when compared with the PSS, and thus to damp out the LFOs under credible contingency. Nonetheless, the minimum control effort of the NFC damping controller showed its effectiveness to control LFOs and thereby enhance the overall dynamic stability of the SMIB system, as compared to the application of the PSS damping controller.

WSCC 3-Machine Test Power System Results and Discussion under Operating Condition 1
This section of the work considered the familiar Western System Coordinated Council (WSCC) 3-machine 9-bus power system. The single line configurations of the power system are as shown in Figure 20, and the complete system data can be found in [31]. The dynamic stability of the WSCC test system can be obtained considering generator G1 (which is the slack bus) as the reference generator, which is not equipped with the PSSs. The interconnected synchronous generators are represented by fourth-order models of the DAEs explained by the test system modeling equations described earlier.  The NFC controller's transient simulation analysis was found to be able to improve the system stability in terms of ST, RT, PT, and PM by an acceptable amount, when compared with the PSS, and thus to damp out the LFOs under credible contingency. Nonetheless, the minimum control effort of the NFC damping controller showed its effectiveness to control LFOs and thereby enhance the overall dynamic stability of the SMIB system, as compared to the application of the PSS damping controller.

WSCC 3-Machine Test Power System Results and Discussion under Operating Condition 1
This section of the work considered the familiar Western System Coordinated Council (WSCC) 3-machine 9-bus power system. The single line configurations of the power system are as shown in Figure 20, and the complete system data can be found in [31]. The dynamic stability of the WSCC test system can be obtained considering generator G1 (which is the slack bus) as the reference generator, which is not equipped with the PSSs. The interconnected synchronous generators are represented by fourth-order models of the DAEs explained by the test system modeling equations described earlier.   A complete simulation model of all systems was designed in MATLAB/Simulink software. The Simulink model of the complete system is depicted in Figure 21. Furthermore, the simulation parameters related to this model can be found in [26]. A complete simulation model of all systems was designed in MATLAB/Simulink software. The Simulink model of the complete system is depicted in Figure 21. Furthermore, the simulation parameters related to this model can be found in [26]. Sustainability 2020, 12, x FOR PEER REVIEW 17 of 43 The performance of the proposed NFC stabilizer was assessed for three different operating conditions, taking into account criteria such as settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM). The operating conditions are given below.
 Condition-1: contingency at bus 9 with no change in the system loading.  Condition-2: contingency at bus 9 with a 120% increase in active power on generator 2 and generator 3.  Condition-3: contingency at bus 9 with an 80% decrease in active power on generator 2 and generator 3.

Time-Domain Simulation of the 3-Machine Test Power System without PSS and NFC Damping Controllers under Operating Condition 1
In this section, the system undergoes a disturbance without a damping controller (PSSs and NFC) for LFOs mitigation, and hence system stability enhancement. An eigenvalues simulation analysis and nonlinear time-domain simulations should be carried out for comprehensive evaluation. The test system was subjected to a three-phase fault without a PSS damping controller. A symmetrical 100 ms three-phase fault was observed at = 1 s, and a nonlinear time-domain simulation was performed. After the fault was cleared, at 0.2 s, the system's stable condition was restored. Table 7 shows the eigenvalues results with their associated damping ratio and frequency for the power system case without the PSS and NFC installed. From Table 7, Mode 1 and Mode 2 produce a weak damping ratio for the no PSS case condition with − 0.6856 ± 12.7756 and − 0.1229 ± 8.2867 eigenvalues. Similarly, the worst damping ratio EM was found to be 0.0536 and 0.0148, respectively. The performance of the proposed NFC stabilizer was assessed for three different operating conditions, taking into account criteria such as settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM). The operating conditions are given below.
Condition-1: contingency at bus 9 with no change in the system loading. Condition-2: contingency at bus 9 with a 120% increase in active power on generator 2 and generator 3. Condition-3: contingency at bus 9 with an 80% decrease in active power on generator 2 and generator 3.

Time-Domain Simulation of the 3-Machine Test Power System without PSS and NFC Damping Controllers under Operating Condition 1
In this section, the system undergoes a disturbance without a damping controller (PSSs and NFC) for LFOs mitigation, and hence system stability enhancement. An eigenvalues simulation analysis and nonlinear time-domain simulations should be carried out for comprehensive evaluation. The test system was subjected to a three-phase fault without a PSS damping controller. A symmetrical 100 ms three-phase fault was observed at t = 1 s, and a nonlinear time-domain simulation was performed. After the fault was cleared, at 0.2 s, the system's stable condition was restored. Table 7 shows the eigenvalues results with their associated damping ratio and frequency for the power system case without the PSS and NFC installed. From Table 7, Mode 1 and Mode 2 produce a weak damping ratio for the no PSS case condition with −0.6856 ± j12.7756 and −0.1229 ± j8.2867 eigenvalues. Similarly, the worst damping ratio EM was found to be 0.0536 and 0.0148, respectively. In this section, the system undergoes a disturbance without PSSs and NFC damping controller for LFOs mitigation, and hence system stability enhancement. A nonlinear time-domain simulation was performed, and Figure 22 shows the unstable generator power angles δ relative to δ 1 in radian for the interconnected test power system for the operating condition 1 while Figure 23 shows the unstable generator speed deviation dω in radian per second for the system without PSS and NFC for the operating condition 1. Similarly, Figure 24 shows the unstable system rotor speed response ω in radian per second for the system without PSS and NFC for the interconnected test power system. Sustainability 2020, 12, x FOR PEER REVIEW 18 of 43 In this section, the system undergoes a disturbance without PSSs and NFC damping controller for LFOs mitigation, and hence system stability enhancement. A nonlinear time-domain simulation was performed, and Figure 22 shows the unstable generator power angles δ relative to in radian for the interconnected test power system for the operating condition 1 while Figure 23 shows the unstable generator speed deviation in radian per second for the system without PSS and NFC for the operating condition 1. Similarly, Figure 24 shows the unstable system rotor speed response in radian per second for the system without PSS and NFC for the interconnected test power system.   In this section, the system undergoes a disturbance without PSSs and NFC damping controller for LFOs mitigation, and hence system stability enhancement. A nonlinear time-domain simulation was performed, and Figure 22 shows the unstable generator power angles δ relative to in radian for the interconnected test power system for the operating condition 1 while Figure 23 shows the unstable generator speed deviation in radian per second for the system without PSS and NFC for the operating condition 1. Similarly, Figure 24 shows the unstable system rotor speed response in radian per second for the system without PSS and NFC for the interconnected test power system.   In this section, the system undergoes a disturbance without PSSs and NFC damping controller for LFOs mitigation, and hence system stability enhancement. A nonlinear time-domain simulation was performed, and Figure 22 shows the unstable generator power angles δ relative to in radian for the interconnected test power system for the operating condition 1 while Figure 23 shows the unstable generator speed deviation in radian per second for the system without PSS and NFC for the operating condition 1. Similarly, Figure 24 shows the unstable system rotor speed response in radian per second for the system without PSS and NFC for the interconnected test power system.

Time-Domain Simulation of the Interconnected Test Power System with PSS Damping Controller for Operating Condition 1
For the PSS stabilizer design, a symmetrical 100 ms three-phase fault was observed at t = 1 s, and a nonlinear time-domain simulation was performed. The rate at which the PSS design index converges using the FFA search algorithm is shown in Figure 25, while the optimal parameters of the designed PSSs are shown in Table 8. From the convergence characteristics, the FFA method shows a good convergence rate at 41 iterations. For the PSS stabilizer design, a symmetrical 100 ms three-phase fault was observed at = 1 s, and a nonlinear time-domain simulation was performed. The rate at which the PSS design index converges using the FFA search algorithm is shown in Figure 25, while the optimal parameters of the designed PSSs are shown in Table 8. From the convergence characteristics, the FFA method shows a good convergence rate at 41 iterations.  Figure 25. Convergence characteristics of FFA in finding the optimal design of PSS. Table 9 shows the eigenvalues results with their associated damping ratio and frequency for the power system case with PSS installed in the system, respectively. From Table 7, Mode 1 and Mode 2 produce a weak damping ratio for the system without a controller. After the optimal PSS design, these modes were impressively enhanced from −0.6856 ± 12.7756 and −0.1229 ± 8.2867 to −4.0720 ± 13.1963 and −4.4351 ± 7.3550, respectively, using the FFA design method. Similarly, the worst damping ratio EM was improved from 0.0148 to 0.4442 using the FFA design method. Similarly, Figure 26 illustrates the eigenvalues plot for the system with the FFA-PSS controller based on the numerical simulation results of Tables 7 and 9.  Figure 25. Convergence characteristics of FFA in finding the optimal design of PSS.  Table 9 shows the eigenvalues results with their associated damping ratio and frequency for the power system case with PSS installed in the system, respectively. From Table 7, Mode 1 and Mode 2 produce a weak damping ratio for the system without a controller. After the optimal PSS design, these modes were impressively enhanced from −0.6856 ± j12.7756 and −0.1229 ± j8.2867 to −4.0720 ± j13.1963 and −4.4351 ± j7.3550, respectively, using the FFA design method. Similarly, the worst damping ratio EM was improved from 0.0148 to 0.4442 using the FFA design method. Similarly, Figure 26 illustrates the eigenvalues plot for the system with the FFA-PSS controller based on the numerical simulation results of Tables 7 and 9.  Moreover, a nonlinear time-domain simulation was performed and Figure 27 shows the stable generator power angles δ relative to in radian for the interconnected test power system for FFA-PSS, while Figure 28 shows the stable generator speed deviations response in radian per seconds for and , respectively. Similarly, Figure 29 shows the stable system rotor speed response in radian per seconds for and , respectively, for the FFA-PSS.   Moreover, a nonlinear time-domain simulation was performed and Figure 27 shows the stable generator power angles δ relative to δ 1 in radian for the interconnected test power system for FFA-PSS, while Figure 28 shows the stable generator speed deviations response dω in radian per seconds for G 2 and G 3 , respectively. Similarly, Figure 29 shows the stable system rotor speed response ω in radian per seconds for G 2 and G 3 , respectively, for the FFA-PSS.  Moreover, a nonlinear time-domain simulation was performed and Figure 27 shows the stable generator power angles δ relative to in radian for the interconnected test power system for FFA-PSS, while Figure 28 shows the stable generator speed deviations response in radian per seconds for and , respectively. Similarly, Figure 29 shows the stable system rotor speed response in radian per seconds for and , respectively, for the FFA-PSS.    Moreover, a nonlinear time-domain simulation was performed and Figure 27 shows the stable generator power angles δ relative to in radian for the interconnected test power system for FFA-PSS, while Figure 28 shows the stable generator speed deviations response in radian per seconds for and , respectively. Similarly, Figure 29 shows the stable system rotor speed response in radian per seconds for and , respectively, for the FFA-PSS.    . Stable rotor speed response of the interconnected power system for a three-phase contingency at bus 9 with FFA-PSS for operating condition 1.

Time-Domain Simulation of the Interconnected Test Power System with NFC Damping Controller for Operating Condition 1
In this case, the test power system is equipped with an NFC stabilizer, and a symmetrical threephase fault is subjected to the system. Table 10 shows the eigenvalues results with their associated damping ratio and frequency for the power system case with NFC installed in the system, respectively. From Table 7, Mode 1 and Mode 2 produce a weak damping ratio for the system without a controller. After the NFC design, these modes were impressively enhanced from −0.6856 ± 12.7756 and −0.1229 ± 8.2867 to −2.0665 ± 10.2851 and −2.4242 ± 5.2661, respectively, using the NFC stabilizer. Similarly, the worst damping ratio of EM was improved from 0.0148 to 0.7285 using the NFC. Similarly, Figure 30 illustrates the eigenvalues plot for the system with the NFC stabilizer controller based on the numerical simulation results of Tables 7 and 10.  Figure 30. Eigenvalues plot for the system with the NFC damping controller.
Moreover, a nonlinear time-domain simulation was performed with NFC installed in the system, and Figure 31 shows the stable generator relative power angles δ in radian for the system with NFC, Figure 29. Stable rotor speed response of the interconnected power system for a three-phase contingency at bus 9 with FFA-PSS for operating condition 1.

Time-Domain Simulation of the Interconnected Test Power System with NFC Damping Controller for Operating Condition 1
In this case, the test power system is equipped with an NFC stabilizer, and a symmetrical three-phase fault is subjected to the system. Table 10 shows the eigenvalues results with their associated damping ratio and frequency for the power system case with NFC installed in the system, respectively. From Table 7, Mode 1 and Mode 2 produce a weak damping ratio for the system without a controller. After the NFC design, these modes were impressively enhanced from −0.6856 ± j12.7756 and −0.1229 ± j8.2867 to −2.0665 ± j10.2851 and −2.4242 ± j5.2661, respectively, using the NFC stabilizer. Similarly, the worst damping ratio of EM was improved from 0.0148 to 0.7285 using the NFC. Similarly, Figure 30 illustrates the eigenvalues plot for the system with the NFC stabilizer controller based on the numerical simulation results of Tables 7 and 10.  Figure 29. Stable rotor speed response of the interconnected power system for a three-phase contingency at bus 9 with FFA-PSS for operating condition 1.

Time-Domain Simulation of the Interconnected Test Power System with NFC Damping Controller for Operating Condition 1
In this case, the test power system is equipped with an NFC stabilizer, and a symmetrical threephase fault is subjected to the system. Table 10 shows the eigenvalues results with their associated damping ratio and frequency for the power system case with NFC installed in the system, respectively. From Table 7, Mode 1 and Mode 2 produce a weak damping ratio for the system without a controller. After the NFC design, these modes were impressively enhanced from −0.6856 ± 12.7756 and −0.1229 ± 8.2867 to −2.0665 ± 10.2851 and −2.4242 ± 5.2661, respectively, using the NFC stabilizer. Similarly, the worst damping ratio of EM was improved from 0.0148 to 0.7285 using the NFC. Similarly, Figure 30 illustrates the eigenvalues plot for the system with the NFC stabilizer controller based on the numerical simulation results of Tables 7 and 10.  Figure 30. Eigenvalues plot for the system with the NFC damping controller.
Moreover, a nonlinear time-domain simulation was performed with NFC installed in the system, and Figure 31 shows the stable generator relative power angles δ in radian for the system with NFC, Moreover, a nonlinear time-domain simulation was performed with NFC installed in the system, and Figure 31 shows the stable generator relative power angles δ in radian for the system with NFC, while Figure 32 shows the stable generator speed deviation dω in radian per second for the system with NFC. Similarly, Figure 33 shows the stable system rotor speed response ω in radian per second for the system with NFC. while Figure 32 shows the stable generator speed deviation in radian per second for the system with NFC. Similarly, Figure 33 shows the stable system rotor speed response in radian per second for the system with NFC. of the interconnected power system for a three-phase contingency at bus 9 with NFC for operating condition 1.    while Figure 32 shows the stable generator speed deviation in radian per second for the system with NFC. Similarly, Figure 33 shows the stable system rotor speed response in radian per second for the system with NFC. of the interconnected power system for a three-phase contingency at bus 9 with NFC for operating condition 1.   while Figure 32 shows the stable generator speed deviation in radian per second for the system with NFC. Similarly, Figure 33 shows the stable system rotor speed response in radian per second for the system with NFC. of the interconnected power system for a three-phase contingency at bus 9 with NFC for operating condition 1.

Figure 32.
Stable rotor speed deviation response of the interconnected power system for a three-phase contingency at bus 9 with NFC for operating condition 1.

Figure 33.
Stable rotor speed response of the interconnected power system for a three-phase contingency at bus 9 with NFC for operating condition 1. Figure 33. Stable rotor speed response of the interconnected power system for a three-phase contingency at bus 9 with NFC for operating condition 1.

Comparison between the Proposed NFC and PSS Damping Controllers for Operating Condition 1
This section presents comparative simulation results between the proposed NFC and the PSS damping controllers for LFO's robustness. The generator power angles δ in radian for G 2 (δ 2 − δ 1 ) and G 3 (δ 3 − δ 1 ) with respect to G 1 illustrate the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 34 and 35 This section presents comparative simulation results between the proposed NFC and the PSS damping controllers for LFO's robustness. The generator power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 34 and 35, respectively.   Figure 34. Power angle response of G2 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 1. Figure 35. Power angle response of G3 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 1.
The variation in the generator speeds for ( − ) and ( − ) relative to corresponds to the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 36 and 37 respectively.

Comparison between the Proposed NFC and PSS Damping Controllers for Operating Condition 1
This section presents comparative simulation results between the proposed NFC and the PSS damping controllers for LFO's robustness. The generator power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 34 and 35, respectively.   Figure 34. Power angle response of G2 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 1. Figure 35. Power angle response of G3 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 1.
The variation in the generator speeds for ( − ) and ( − ) relative to corresponds to the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 36 and 37 respectively. The variation in the generator speeds for G 2 (ω 2 − ω 1 ) and G 3 (ω 3 − ω 1 ) relative to G 1 corresponds to the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 36 and 37 respectively.  From the time-domain simulation results, it was seen that the stabilizer using the proposed NFC was effectively able to damp out the LFOs of the test system under severe system disturbances. Thus, the NFC damping controller can be applied as a general damping controller for the robust design stabilizer and other similar science and engineering controller applications.

Proposed FFA Based NFC Design Results Comparison with FFA-PSS for Operating Condition 2
To verify the performance of the proposed NFC controller, this section considers a 120% increase in active power on generator 2 and generator 3. Relative power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system using FFA-PSS and the proposed NFC design methods, as shown in Figures 38 and 39 respectively.   Figure 36. Rotor speed response of G2 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 1.
Sustainability 2020, 12, x FOR PEER REVIEW 24 of 43 Figure 36. Rotor speed response of G2 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 1. Figure 37. Rotor speed response of G3 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 1.
From the time-domain simulation results, it was seen that the stabilizer using the proposed NFC was effectively able to damp out the LFOs of the test system under severe system disturbances. Thus, the NFC damping controller can be applied as a general damping controller for the robust design stabilizer and other similar science and engineering controller applications.

Proposed FFA Based NFC Design Results Comparison with FFA-PSS for Operating Condition 2
To verify the performance of the proposed NFC controller, this section considers a 120% increase in active power on generator 2 and generator 3. Relative power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system using FFA-PSS and the proposed NFC design methods, as shown in Figures 38 and 39 respectively. From the time-domain simulation results, it was seen that the stabilizer using the proposed NFC was effectively able to damp out the LFOs of the test system under severe system disturbances. Thus, the NFC damping controller can be applied as a general damping controller for the robust design stabilizer and other similar science and engineering controller applications.

Proposed FFA Based NFC Design Results Comparison with FFA-PSS for Operating Condition 2
To verify the performance of the proposed NFC controller, this section considers a 120% increase in active power on generator 2 and generator 3. Relative power angles δ in radian for G 2 (δ 2 − δ 1 ) and G 3 (δ 3 − δ 1 ) with respect to G 1 illustrate the system stability profile of the system using FFA-PSS and the proposed NFC design methods, as shown in Figures 38 and 39 respectively.      The variation in the generator speeds for G 2 (ω 2 − ω 1 ) and G 3 (ω 3 − ω 1 ) relative to G 1 that corresponds to the system stability profile of the system using FFA-PSS, and the proposed NFC design methods are shown in Figures 40 and 41 respectively.    The section on condition 3 considers an 80% decrease in active power on generator 2 and generator 3. Relative power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system using FFA-PSS, and the proposed NFC design methods as shown in Figures 42 and 43 respectively. The section on condition 3 considers an 80% decrease in active power on generator 2 and generator 3. Relative power angles δ in radian for G 2 (δ 2 − δ 1 ) and G 3 (δ 3 − δ 1 ) with respect to G 1 illustrate the system stability profile of the system using FFA-PSS, and the proposed NFC design methods as shown in Figures 42 and 43 respectively.   Figure 41. Rotor speed response of G3 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 2.
3.2.6. Proposed FFA Based NFC Design Results Comparison with FFA-PSS for Operating Condition 3 The section on condition 3 considers an 80% decrease in active power on generator 2 and generator 3. Relative power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system using FFA-PSS, and the proposed NFC design methods as shown in Figures 42 and 43 respectively.    . Rotor speed response of G3 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 2.
3.2.6. Proposed FFA Based NFC Design Results Comparison with FFA-PSS for Operating Condition 3 The section on condition 3 considers an 80% decrease in active power on generator 2 and generator 3. Relative power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system using FFA-PSS, and the proposed NFC design methods as shown in Figures 42 and 43 respectively.   Figure 42. Power angle response of G2 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 3. Figure 43. Power angle response of G3 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 3.
The variation in the generator speeds for ( − ) and ( − ) relative to corresponds to the system stability profile of the system using FFA-PSS, and the proposed NFC design methods are shown in Figures 44 and 45 respectively. The variation in the generator speeds for G 2 (ω 2 − ω 1 ) and G 3 (ω 3 − ω 1 ) relative to G 1 corresponds to the system stability profile of the system using FFA-PSS, and the proposed NFC design methods are shown in Figures 44 and 45 respectively. . Rotor speed response of G2 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 3. Figure 44. Rotor speed response of G2 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 3. Figure 44. Rotor speed response of G2 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 3. Figure 45. Rotor speed response of G3 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 3.

Transient Response Scenario for Controller Performance Analysis under Operating Condition 1
This section explains the transient response simulation which describes the performance analysis of the interconnected power system when the system has no controller (NC), with a PSS controller, and with the proposed NFC damping controller for the three machines, respectively. Table  11 considers how the Machine 1 plant responds to the transient situation with respect to settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM) for NC, PSS, and NFC damping controller cases, respectively. It is seen that the rotor speed for machine 1's transient response, , in terms of settling time and rise time, is remarkably improved by an amount of 66.57% and 64.70%, respectively, by the proposed NFC stabilizer, as compared with the NC in the system.  Figure 45. Rotor speed response of G3 relative to G1 for a contingency at bus 9 in the interconnected power system for operating condition 3.

Transient Response Scenario for Controller Performance Analysis under Operating Condition 1
This section explains the transient response simulation which describes the performance analysis of the interconnected power system when the system has no controller (NC), with a PSS controller, and with the proposed NFC damping controller for the three machines, respectively. Table 11 considers how the Machine 1 plant responds to the transient situation with respect to settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM) for NC, PSS, and NFC damping controller cases, respectively. It is seen that the rotor speed for machine 1's transient response, ω 1 , in terms of settling time and rise time, is remarkably improved by an amount of 66.57% and 64.70%, respectively, by the proposed NFC stabilizer, as compared with the NC in the system. Furthermore, Table 12 shows, for Machine 2's system, that the ST response for ω 2 was remarkably improved by an amount of 69.66% when the system had NC as compared to the proposed NFC stabilizer. Similarly, Table 13 explains, for Machine 3's system, how the ω 3 reacts to ST when the system has NC and when the NFC stabilizer is used. The FFA controller was able to improve the system's stability in terms of ST with an amount of 75.70% and thus damp out the LFOs under credible contingency.  The NFC damping controller in all the three machines' transient simulation analyses was found to be able to improve the system's stability in terms of ST, RT, PT, and PM by an acceptable amount, compared with the PSS, and thus to damp out the LFOs under credible contingency. Nonetheless, the minimum control effort of the NFC damping controller exhibited its effectiveness to control LFOs and thereby enhance the overall dynamic stability of the interconnected system compared to the application of the PSS damping controller.

IEEE 10-Machine Test Power System Results and Discussion under Operating Condition 1
This section of the work considered the well-known IEEE 10-machine, 39-bus New England power system as the test system. The single line configurations of the power system are shown in Figure 46, and the complete system data can be found in [31]. The dynamic stability of the test system can be obtained considering generator G2, which is an equivalent power source representing parts of the USA-Canadian interconnection system as the reference generator, which is not equipped with the PSSs. The interconnected synchronous generators are represented by fourth-order models of the DAEs explained by the test system modeling equations described earlier.  This section discusses the eigenvalue, time-domain, and transient response simulation results of the three test power systems under multiple operating conditions. The performance of the proposed NFC stabilizer was assessed for three different operating conditions, taking into account criteria such as settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM). The operating conditions are given below.
Condition-1: contingency at bus 39 with no change in the system loading. Condition-2: contingency at bus 39 with a 120% increase in active power on generator 5 and generator 8. Condition-3: contingency at bus 39 with an 80% decrease in active power on generator 5 and generator 8. In this section, the system undergoes a disturbance without a damping controller (PSSs and NFC) for LFOs mitigation, and hence system stability enhancement. An eigenvalues simulation analysis and nonlinear time-domain simulations should be carried out for comprehensive evaluation. The test system was subjected to a three-phase fault without a PSS damping controller. A symmetrical 100 ms three-phase fault was observed at t = 1 s, and a nonlinear time-domain simulation was performed. After the fault was cleared, at 0.2 s, the system's stable condition was restored. Table 14 shows the eigenvalues results with their associated damping ratio and frequency for the power system case without the PSS and NFC installed. From Table 14, Mode 1 and Mode 2 produce a weak damping ratio for the no PSS case condition with −1.8689 ± j10.9970 and −0.4579 ± j9.4704 eigenvalues. Similarly, the worst damping ratio EM was found to be 0.1675 and 0.0483, respectively. In this section, the system undergoes a disturbance without PSSs and NFC damping controller for LFOs mitigation, and hence system stability enhancement. A nonlinear time-domain simulation was performed, and Figure 47 shows the unstable generator power angles δ relative to δ 2 in radian for the interconnected test power system for the operating condition 1, while Figure 48 shows the unstable generator speed deviation dω in radian per second for the system without PSS and NFC for the operating condition 1. Similarly, Figure 49 shows the unstable system rotor speed response ω in radian per second for the system without PSS and NFC for the interconnected test power system.
for LFOs mitigation, and hence system stability enhancement. A nonlinear time-domain simulation was performed, and Figure 47 shows the unstable generator power angles δ relative to in radian for the interconnected test power system for the operating condition 1, while Figure 48 shows the unstable generator speed deviation in radian per second for the system without PSS and NFC for the operating condition 1. Similarly, Figure 49 shows the unstable system rotor speed response in radian per second for the system without PSS and NFC for the interconnected test power system.  was performed, and Figure 47 shows the unstable generator power angles δ relative to in radian for the interconnected test power system for the operating condition 1, while Figure 48 shows the unstable generator speed deviation in radian per second for the system without PSS and NFC for the operating condition 1. Similarly, Figure 49 shows the unstable system rotor speed response in radian per second for the system without PSS and NFC for the interconnected test power system.  For the PSS stabilizer design, a symmetrical 100 ms three-phase fault was observed at = 1 s, and a nonlinear time-domain simulation was performed. The rate at which the PSS design index converges using the FFA search algorithm is shown in Figure 50, while the optimal parameters of the designed PSSs are shown in Table 15. From the convergence characteristics, the FFA method shows a good convergence rate at 92 iterations. For the PSS stabilizer design, a symmetrical 100 ms three-phase fault was observed at t = 1 s, and a nonlinear time-domain simulation was performed. The rate at which the PSS design index converges using the FFA search algorithm is shown in Figure 50, while the optimal parameters of the designed PSSs are shown in Table 15. From the convergence characteristics, the FFA method shows a good convergence rate at 92 iterations.  Table 16 shows the eigenvalues results with their associated damping ratio and frequency for the power system case with PSS installed in the system, respectively. From Table 14, Mode 1 and   Table 16 shows the eigenvalues results with their associated damping ratio and frequency for the power system case with PSS installed in the system, respectively. From Table 14, Mode 1 and Mode 2 produce a weak damping ratio for the system without a controller. After the optimal PSS design, these modes were impressively enhanced from −1.8689 ± j10.9970 and −0.4579 ± j9.4704 to −0.1524 ± j0.2106 and −0.1284 ± j0.1863, respectively, using the FFA design method. Similarly, the worst damping ratio EM was improved from 0.0483 to 0.5676 using the FFA design method. Similarly, Figure 51 illustrates the eigenvalues plot for the system with the FFA-PSS controller based on the numerical simulation results of Tables 14 and 16.  Moreover, a nonlinear time-domain simulation was performed, and Figure 52 shows the stable generator power angles δ relative to in radian for the interconnected test power system for FFA-PSS, while Figure 53 shows the stable generator speed deviations response in radian per seconds for to , respectively. Similarly, Figure 54 shows the stable system rotor speed response in radian per seconds for to , respectively, for the FFA-PSS. Moreover, a nonlinear time-domain simulation was performed, and Figure 52 shows the stable generator power angles δ relative to δ 2 in radian for the interconnected test power system for FFA-PSS, while Figure 53 shows the stable generator speed deviations response dω in radian per seconds for G 1 to G 10 , respectively. Similarly, Figure 54 shows the stable system rotor speed response ω in radian per seconds for G 1 to G 10 , respectively, for the FFA-PSS. Moreover, a nonlinear time-domain simulation was performed, and Figure 52 shows the stable generator power angles δ relative to in radian for the interconnected test power system for FFA-PSS, while Figure 53 shows the stable generator speed deviations response in radian per seconds for to , respectively. Similarly, Figure 54 shows the stable system rotor speed response in radian per seconds for to , respectively, for the FFA-PSS.     In this case, the test power system was equipped with an NFC stabilizer, and a symmetrical three-phase fault was subjected to the system. Table 17 shows the eigenvalues results with their associated damping ratio and frequency for the power system case with NFC installed in the system, respectively. From Table 14, Mode 1 and Mode 2 produce a weak damping ratio for the system without a controller. After the NFC design, these modes were impressively enhanced from−1.8689 ± 10.9970 and −0.4579 ± 9.4704 to −0.1025 ± 0.1107 and −0.1193 ± 0.1773, respectively, using the NFC stabilizer. Similarly, the worst damping ratio of EM was improved from 0.0483 to 0.4676 using the NFC.

Time-Domain Simulation of the Interconnected Test Power System with NFC Damping Controller for Operating Condition 1
In this case, the test power system was equipped with an NFC stabilizer, and a symmetrical three-phase fault was subjected to the system. Table 17 shows the eigenvalues results with their associated damping ratio and frequency for the power system case with NFC installed in the system, respectively. From Table 14, Mode 1 and Mode 2 produce a weak damping ratio for the system without a controller. After the NFC design, these modes were impressively enhanced from −1.8689 ± j10.9970 and −0.4579 ± j9.4704 to −0.1025 ± j0.1107 and −0.1193 ± j0.1773, respectively, using the NFC stabilizer. Similarly, the worst damping ratio of EM was improved from 0.0483 to 0.4676 using the NFC. Moreover, a nonlinear time-domain simulation was performed with NFC installed in the system, and Figure 55 shows the stable generator relative power angles δ in radian for the system with NFC, while Figure 56 shows the stable generator speed deviation dω in radian per second for the system with NFC. Similarly, Figure 57 shows the stable system rotor speed response ω in radian per second for the system with NFC. −0.0150 ± 0.0680, 0.3002 Moreover, a nonlinear time-domain simulation was performed with NFC installed in the system, and Figure 55 shows the stable generator relative power angles δ in radian for the system with NFC, while Figure 56 shows the stable generator speed deviation in radian per second for the system with NFC. Similarly, Figure 57 shows the stable system rotor speed response in radian per second for the system with NFC.   −0.0150 ± 0.0680, 0.3002 Moreover, a nonlinear time-domain simulation was performed with NFC installed in the system, and Figure 55 shows the stable generator relative power angles δ in radian for the system with NFC, while Figure 56 shows the stable generator speed deviation in radian per second for the system with NFC. Similarly, Figure 57 shows the stable system rotor speed response in radian per second for the system with NFC.   This section presents comparative simulation results between the proposed NFC and the PSS damping controllers for LFO's robustness. The generator power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 58 and 59 respectively.

Comparison between the Proposed NFC and PSS Damping Controllers for Operating Condition 1
This section presents comparative simulation results between the proposed NFC and the PSS damping controllers for LFO's robustness. The generator power angles δ in radian for G 5 (δ 5 − δ 2 ) and G 8 (δ 8 − δ 2 ) with respect to G 2 illustrate the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 58 and 59 respectively. contingency at bus 39 with NFC for operating condition 1.

Comparison between the Proposed NFC and PSS Damping Controllers for Operating Condition 1
This section presents comparative simulation results between the proposed NFC and the PSS damping controllers for LFO's robustness. The generator power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 58 and 59 respectively.   contingency at bus 39 with NFC for operating condition 1.

Comparison between the Proposed NFC and PSS Damping Controllers for Operating Condition 1
This section presents comparative simulation results between the proposed NFC and the PSS damping controllers for LFO's robustness. The generator power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 58 and 59 respectively.   The variation in the generator speeds for G 5 (ω 5 − ω 2 ) and G 8 (ω 8 − ω 2 ) relative to G 2 corresponds to the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 60 and 61  The variation in the generator speeds for ( − ) and ( − ) relative to corresponds to the system stability profile of the system without the controller, with PSS, and with the proposed NFC, as shown in Figures 60 and 61 respectively.   Figure 60. Rotor speed response of G5 relative to G2 for a contingency at bus 39 in the interconnected power system for operating condition 1. Figure 60. Rotor speed response of G5 relative to G2 for a contingency at bus 39 in the interconnected power system for operating condition 1. Figure 60. Rotor speed response of G5 relative to G2 for a contingency at bus 39 in the interconnected power system for operating condition 1. Figure 61. Rotor speed response of G8 relative to G2 for a contingency at bus 39 in the interconnected power system for operating condition 1.
From the time-domain simulation results, it was seen that the stabilizer using the proposed NFC was effectively able to damp out the LFOs of the test system under severe system disturbances. Thus, the NFC damping controller can be applied as a general damping controller for the robust design stabilizer and other similar science and engineering controller applications.

Proposed NFC Design Results Comparison with FFA-PSS for Operating Condition 2
To verify the performance of the proposed NFC controller, this section considers a 120% increase in active power on generator 5 and generator 8. Relative power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system using FFA-PSS and the proposed NFC design methods, as shown in Figures 62 and 63 respectively. From the time-domain simulation results, it was seen that the stabilizer using the proposed NFC was effectively able to damp out the LFOs of the test system under severe system disturbances. Thus, the NFC damping controller can be applied as a general damping controller for the robust design stabilizer and other similar science and engineering controller applications.

Proposed NFC Design Results Comparison with FFA-PSS for Operating Condition 2
To verify the performance of the proposed NFC controller, this section considers a 120% increase in active power on generator 5 and generator 8. Relative power angles δ in radian for G 5 (δ 5 − δ 2 ) and G 8 (δ 8 − δ 2 ) with respect to G 2 illustrate the system stability profile of the system using FFA-PSS and the proposed NFC design methods, as shown in Figures 62 and 63 respectively.  The variation in the generator speeds for ( − ) and ( − ) relative to corresponds to the system stability profile of the system using FFA-PSS, and the proposed NFC design methods are shown in Figures 64 and 65 respectively.  The variation in the generator speeds for ( − ) and ( − ) relative to corresponds to the system stability profile of the system using FFA-PSS, and the proposed NFC design methods are shown in Figures 64 and 65 respectively.  The variation in the generator speeds for G 5 (ω 5 − ω 2 ) and G 8 (ω 8 − ω 2 ) relative to G 2 corresponds to the system stability profile of the system using FFA-PSS, and the proposed NFC design methods are shown in Figures 64 and 65 respectively. Figure 63. Power angle response of G8 relative to G2 for a contingency at bus 39 in the interconnected power system for operating condition 2.
The variation in the generator speeds for ( − ) and ( − ) relative to corresponds to the system stability profile of the system using FFA-PSS, and the proposed NFC design methods are shown in Figures 64 and 65 respectively.  The section on condition 3 considers an 80% decrease in active power on generator 5 and generator 8. Relative power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system using FFA-PSS and the proposed NFC design methods, as shown in Figures 66 and 67 respectively.  The section on condition 3 considers an 80% decrease in active power on generator 5 and generator 8. Relative power angles δ in radian for G 5 (δ 5 − δ 2 ) and G 8 (δ 8 − δ 2 ) with respect to G 2 illustrate the system stability profile of the system using FFA-PSS and the proposed NFC design methods, as shown in Figures 66 and 67 respectively. The section on condition 3 considers an 80% decrease in active power on generator 5 and generator 8. Relative power angles δ in radian for ( − ) and ( − ) with respect to illustrate the system stability profile of the system using FFA-PSS and the proposed NFC design methods, as shown in Figures 66 and 67 respectively.    The variation in the generator speeds for G 5 (ω 5 − ω 2 ) and G 8 (ω 8 − ω 2 ) relative to G 2 corresponds to the system stability profile of the system using FFA-PSS, and the proposed NFC design methods are shown in Figures 68 and 69 respectively. The variation in the generator speeds for ( − ) and ( − ) relative to corresponds to the system stability profile of the system using FFA-PSS, and the proposed NFC design methods are shown in Figures 68 and 69 respectively.  . Rotor speed response of G8 relative to G2 for a contingency at bus 39 in the interconnected power system for operating condition 3.

Transient Response Scenario for Controller Performance Analysis under Operating Condition 1
This section explains the transient response simulation which describes the performance analysis of the interconnected power system when the system has no controller (NC), with a PSS controller, and with the proposed NFC damping controller for the two machines, respectively. Tables  18 and 19 illustrate how Generator 5 and Generator 8 respond to the transient situation with respect to settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM) for NC, PSS, and NFC damping controller cases, respectively. It was seen that the rotor speed for machine 5's transient Figure 68. Rotor speed response of G5 relative to G2 for a contingency at bus 39 in the interconnected power system for operating condition 3.
Sustainability 2020, 12, x FOR PEER REVIEW 39 of 43 The variation in the generator speeds for ( − ) and ( − ) relative to corresponds to the system stability profile of the system using FFA-PSS, and the proposed NFC design methods are shown in Figures 68 and 69 respectively.  . Rotor speed response of G8 relative to G2 for a contingency at bus 39 in the interconnected power system for operating condition 3.

Transient Response Scenario for Controller Performance Analysis under Operating Condition 1
This section explains the transient response simulation which describes the performance analysis of the interconnected power system when the system has no controller (NC), with a PSS controller, and with the proposed NFC damping controller for the two machines, respectively. Tables  18 and 19 illustrate how Generator 5 and Generator 8 respond to the transient situation with respect to settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM) for NC, PSS, and NFC damping controller cases, respectively. It was seen that the rotor speed for machine 5's transient response in terms of settling time and rise time was remarkably improved, by 76.80% and 96.16%, Figure 69. Rotor speed response of G8 relative to G2 for a contingency at bus 39 in the interconnected power system for operating condition 3.

Transient Response Scenario for Controller Performance Analysis under Operating Condition 1
This section explains the transient response simulation which describes the performance analysis of the interconnected power system when the system has no controller (NC), with a PSS controller, and with the proposed NFC damping controller for the two machines, respectively. Tables 18 and 19 illustrate how Generator 5 and Generator 8 respond to the transient situation with respect to settling time (ST), rise time (RT), peak time (PT), and peak magnitude (PM) for NC, PSS, and NFC damping controller cases, respectively. It was seen that the rotor speed for machine 5's ω 5 transient response in terms of settling time and rise time was remarkably improved, by 76.80% and 96.16%, respectively, by the proposed NFC stabilizer, as compared with the NC in the system.  Furthermore, Table 19 shows that the rotor speed for machine 8's ω 8 transient response in terms of settling time was remarkably improved by 77.33% by the proposed NFC stabilizer, as compared with the NC in the system.
The NFC damping controller in all the three machines' transient simulation analyses was found to be able to improve the system stability in terms of ST, RT, PT, and PM by an acceptable amount, as compared with the PSS, and thus to damp out the LFOs under credible contingency. Nonetheless, the minimum control effort of the NFC damping controller exhibited its effectiveness to control LFOs and thereby enhance the overall dynamic stability of the interconnected system compared to the application of the PSS damping controller.

Conclusions
In this paper, the performance of the neuro-fuzzy controller (NFC) as a stabilizer to replace conventional single machine infinite bus (SMIB) and interconnected multi-machine large power system stabilizers (PSSs) design was assessed. The drawback of the conventional PSSs on SMIB and interconnected multi-machine test power systems were exposed and disclosed by the proposed NFC stabilizer. The proposed NFC stabilizer is a nonlinear robust controller which does not require a mathematical model of the test power system to be controlled, unlike the conventional PSS damping controller. The Proposed NFC is designed to improve the stability of SMIB, an interconnected IEEE 3-machine, 9-bus power system, and an interconnected two-area 10-machine system of 39-bus New England IEEE test power system under multiple operating conditions. Through the application of the proposed NFC stabilizer on the three-test systems, the damping ratio of the electromechanical modes (EMs) has increased from −0.0070 to 0.2103 for the SMIB test-system, 0.0148 to 0.7285 for the 3-machine test system, and 0.0483 to 0.4676 for the 10-machine test system, respectively. This damping ratio is an acceptable value for EMs because it has increased to a value greater than 0.1. Nonlinear simulations were carried out for the designed PSSs with MATLAB/SIMULINK software. The variations in different quantities of the three test power systems, due to no change in the system loading condition, with a 120% increase in active power on generator 5 and generator 8 and with an 80% decrease in active power on generator 5 and generator 8, respectively, revealed the effectiveness of the proposed NFC stabilizer. It was observed in the Figures from the nonlinear time-domain simulations that oscillations have become significantly damped for all three test systems under these system operating variations using the proposed method. Moreover, the phasor simulation results show that the transient responses of the system rise time, settling time, peak time, and peak magnitude were all impressively improved by an acceptable amount for the three test systems with the proposed NFC stabilizer. Thus, the application and performance of the proposed NFC stabilizer in replacing the conventional PSSs for SMIB and large multi-machine power systems have appeared robust and superior and can be considered as a very remarkable technique for the optimal design of the damping controller in large interconnected power systems.