Optimal Transient Search Algorithm-Based PI Controllers for Enhancing Low Voltage Ride-Through Ability of Grid-Linked PMSG-Based Wind Turbine

: This paper depicts a new attempt to apply a novel transient search optimization (TSO) algorithm to optimally design the proportional-integral (PI) controllers. Optimal PI controllers are utilized in all converters of a grid-linked permanent magnet synchronous generator (PMSG) powered by a variable-speed wind turbine. The converters of such wind energy systems contain a generator-side converter (GSC) and a grid-side inverter (GSI). Both of these converters are optimally controlled by the proposed TSO-based PI controllers using a vector control scheme. The GSC is responsible for regulating the maximum power point, the reactive generator power, and the generator currents. In addition, the GSI is essentially controlled to control the point of common coupling (PCC) voltage, DC link voltage, and the grid currents. The TSO is applied to minimize the ﬁtness function, which has the sum of these variables’ squared error. The optimization problem’s constraints include the range of the proportional and integral gains of the PI controllers. All the simulation studies, including the TSO code, are implemented using PSCAD software. This represents a salient and new contribution of this study, where the TSO is coded using Fortran language within PSCAD software. The TSO-PI control scheme’s e ﬀ ectiveness is compared with that achieved by using a recent grey wolf optimization (GWO) algorithm–PI control scheme. The validity of the proposed TSO–PI controllers is tested under several network disturbances, such as subjecting the system to balanced and unbalanced faults. With the optimal TSO–PI controller, the low voltage ride-through ability of the grid-linked PMSG can be further improved.


Introduction
The remarkable development of wind power has become a pressing modern energy subject all over the world. This is due to the need to obtain new electrical power sources, avoid the need for fossil fuels, and the target to obtain clean energy, considering the environmental concerns [1]. Wind power represents the second-largest source after hydropower sources worldwide [2]. The wind power industry is extensively developed, resulting in a bright future of renewable energy sources. Based on the Global Wind Energy Council's statistical analysis and annual report, the last year (2019) was an outstanding year for the wind power industry, where 60.4 GW was globally installed by the end of 2019 [3]. The total cumulative wind power capacity reached 651 GW. An increase in wind power installation by 17% was achieved compared with that obtained in 2018. It is expected that this type of renewable energy source shall increase in the future and will rigorously contribute to the inductors and capacitors. The TSO has been successfully applied to twenty-three optimization problems, and its results have proven its high performance over other conventional and meta-heuristic algorithms. Many performance tests have been carried out for this issue like non-parametric sign test and p-value test. Furthermore, the TSO has been used to optimally design engineering applications like coil spring, welded beam, and pressure vessel. Recently, it was applied to determine the electrical parameters of the photovoltaic model [31].
In this paper, a new attempt in the application of the TSO algorithm is proposed to fully design the PI controllers of the GSC and GSI of grid-linked VSWT-PMSG systems. The GSC is optimally controlled by the TSO-PI controllers to efficiently control the maximum power point, the reactive generator power, and the generator currents. In addition, the GSI is optimally controlled by the TSO-PI controllers to control the PCC voltage, DC link voltage, and the grid currents. All these control strategies are built using cascaded or vector control schemes to produce the firing pulses of the converters' electronic switches. The TSO is applied to minimize the fitness function, which has the sum of these variables' squared error. The optimization problem's constraints include the range of the proportional and integral gains of the PI controllers. All the simulation studies, including the TSO code, are implemented using PSCAD software [32]. This represents a salient and new contribution of this study, where the TSO is coded using Fortran language within PSCAD software. The TSO-PI control scheme's effectiveness is compared with that achieved by using a recent grey wolf optimization (GWO) algorithm-PI control scheme. The proposed TSO-PI controllers' validity is tested under several network disturbances, such as subjecting the system to symmetrical and unsymmetrical faults. With the optimal TSO-PI controller, the LVRT ability of the grid-linked VSWT-PMSG can be further improved.
The paper's detailed structure is as follows: Section 2 points out the power system model components in detail. In Section 3, the frequency converter, GSC, and GSI models are presented. Section 4 presents the optimization methodology of using TSO algorithm. Section 5 demonstrates the optimization and simulation results under various network disturbances with many analyses. Lastly, a conclusion of this paper is formed in Section 6.

Power System Modeling
In this study, the wind power harvesting, conversion, and supply to the electrical grid are modeled as depicted in Figure 1. This model consists of a three-bladed horizontal-axis wind turbine (WT), which harvests the kinetic energy of wind and converts it into mechanical power. Then, the WT is interconnected directly (without gearbox) to the shaft of the PMSG, which will convert the mechanical power to electrical power. However, the rotor side's rotation speed is deficient, so the frequency converter links the generator and the grid sides. The GSC converts the AC power into DC power, where the DC link capacitor (C dc ) and a chopper resistor (R ch ) exists. The GSI converts the DC power to AC power, which is delivered to the electrical grid through inductance filter (L f ), a step-up transformer, and double-circuit overhead transmission lines (TLs). The control system synchronizes the rectifier and the inverter with the voltage and frequency on each side. The data of the power system are listed in the Appendix A.

Horizontal-Axis WT Modeling
A horizontal-axis three-bladed wind turbine is the most widespread type of wind turbine, where the generator is placed on the top of the tower. The airflow is with density ρ a = 1.225 kg/m 3 . v a is the velocity and the blades have radius (R b ). The kinetic power (P k ) is converted into rotating power (P r ) with efficacy called power coefficient (C p ) as in Equation (1). However, the efficacy of WT is limited by the Betz constant (0.593) [33].
Electronics 2020, 9, 1807 4 of 20 where T r is the rotor torque, and ω r is the rotor speed. C p is expressed in Equation (4), where its variables are the tip-speed ratio (TSR λ) and pitch angle (β) of three-blades. C p is plotted against λ at different values of β, as depicted in Figure 2, where the peak value of C p occurs at β = 0. Additionally, the rotor power and maximum power (P max ) are expressed in Equations (7) and (8). Figure 3 shows the rotor power variations and the locus of maximum power versus the rotor speed at different wind speed values [34].

Horizontal-Axis WT Modeling
A horizontal-axis three-bladed wind turbine is the most widespread type of wind turbine, where the generator is placed on the top of the tower. The airflow is with density ρa=1.225 kg/m 3 . va is the velocity and the blades have radius (Rb). The kinetic power (Pk) is converted into rotating power (Pr) with efficacy called power coefficient (Cp) as in Equation (1). However, the efficacy of WT is limited by the Betz constant (0.593) [33].
where Tr is the rotor torque, and ωr is the rotor speed. Cp is expressed in Equation (4), where its variables are the tip-speed ratio (TSR λ) and pitch angle (β) of three-blades. Cp is plotted against λ at different values of β, as depicted in Figure 2, where the peak value of Cp occurs at β = 0. Additionally, the rotor power and maximum power (Pmax) are expressed in Equations (7) and (8). Figure 3 shows the rotor power variations and the locus of maximum power versus the rotor speed at different wind speed values [34].

Gearless Shaft Modeling
The shaft of PMSG is interconnected to the wind turbine's shaft without a gearbox, so the rotor speed is low, resulting in a low frequency in the electrical side of PMSG. Therefore, the frequency converter is applied to convert the low frequency to the grid frequency. On the other hand, the frequency converter reduces the grid stresses to the PMSG. A single shaft model can then be used to model the shafts of wind turbines and PMSG, as in Equation (9) where Ds is the shaft damping, js is the shaft inertia, and Tg is the generated torque by the PMSG.

Gearless Shaft Modeling
The shaft of PMSG is interconnected to the wind turbine's shaft without a gearbox, so the rotor speed is low, resulting in a low frequency in the electrical side of PMSG. Therefore, the frequency converter is applied to convert the low frequency to the grid frequency. On the other hand, the frequency converter reduces the grid stresses to the PMSG. A single shaft model can then be used to model the shafts of wind turbines and PMSG, as in Equation (9) [35].
where D s is the shaft damping, j s is the shaft inertia, and T g is the generated torque by the PMSG.

Generator-Side Modeling
The generator converts the rotational power to electrical power, where the input of the generator model is ω r, and its outputs are the stator voltages, as in Equation (10). The generator voltages (v d , v q ), currents (i d , i q ), and inductances (L d , L q ) are expressed in the direct and quadrature axis (d-q axis) [36].
where R s is the stator resistor, ω g is the electric generator speed, and the ψ f is the rotor flux linked the stator.

Grid Side Modeling
In this study, the grid is an extensive power system, which can be modeled by inductance in series with a controlled voltage source. The equivalent inductance (L g ) represents the inductances of the filter, transformer, and TLs. The terminal voltages at the GSI (v id , v iq ) are expressed in terms of the grid currents (i gd , i gq ) and grid voltages (v gd , v gq ) that are measured at the point of common coupling (PCC), as in Equation (11).

Frequency Converter Modeling
The frequencies in the generator and grid sides are not synchronized, so the frequency converter is utilized in this study. The frequency converter is constructed from two voltage source converters connected in a back-to-back manner. These converters are based on the insulated gates bipolar transistors (IGBTs) paralleled with diodes. The converter at the generator side, called the GSC, converts the generated AC power to DC power. However, the converter at the grid side, which is called GSI, converts the DC power to AC power that is synchronized with the grid frequency. The IGBTs are switched on and off using pulse width modulation (PWM) signals produced by cascaded control systems of GSC and GSI. The control system consists of four proportional-integral (PI) controllers at each side, so eight PI controllers are used in total. Each PI controller has two parameters (proportional gain (k p ) and integral time constant (T i )).

GSC Control Modeling
The GSC vector control objectives are to obtain i) maximum power point tracking (MPPT) and ii) unity power factor at the generator side. Since T g is proportional to the i q , as in Equation (12), then the vector control of GSC is a field-oriented control, and the generated real power (P g ) is proportionate with i q , as in Equation (13) [33,36].
where L d ≈ L q , then So, PI1 regulates the P g with respect to P max to generate the reference i q * , as in Equation (14), while PI2 regulates the reactive power (Q g ) of generator with respect to the reference Q ref to generate the reference i d * , as in Equation (15). Consequently, PI3 and PI4 regulate the currents i q , i d with respect to i q * , i d * to generate the references v q * and v d * , as in Equations (16) and (17). The reference voltage is then transformed using dq-abc transformer, which is synchronized using the phase angle (θ) of PMSG by measuring ω r . So, the cascaded control system of the GSC is depicted in Figure 4.

GSI Control Modeling
The objectives of the control system of the GSI are: i) low voltage ride-through capability enhancement during short circuits by producing reactive power to support stability restoration of the grid, and ii) regulating the DC link voltage to deliver smooth power to the grid. Hence, the DC link voltage (V dc ) is proportionate to the i gd , as in Equation (18), PI5 regulates the V dc with respect to the reference V dc * to generate the reference i gd * , as in Equation (19). Furthermore, PI6 regulates the PCC voltage V rms relative to the reference voltage V rms * to create the reference i gq * , as in Equation (20).
Consequently, PI7 and PI8 regulate the currents i gd , i gq relative to i gd * , i gq * to generate the references v iq * and v id * , as in Equations (21) and (22), as shown in Figure 5a. Then, the reference voltages are transformed to abc form using dq-abc transformer. The phase-locked loop (PLL) is utilized to produce the phase angle (θ g ) of the grid for phase synchronization. PLL is based on the dq-frame, where the q-axis voltage is regulated with zero reference, as shown in Figure 5b. In addition, the angular frequency of grid is ω s = 2π × 60 = 377 rad/s.

GSI Control Modeling
The objectives of the control system of the GSI are: i) low voltage ride-through capability enhancement during short circuits by producing reactive power to support stability restoration of the grid, and ii) regulating the DC link voltage to deliver smooth power to the grid. Hence, the DC link voltage (Vdc) is proportionate to the igd, as in Equation (18), PI5 regulates the Vdc with respect to the reference Vdc * to generate the reference igd * , as in Equation (19). Furthermore, PI6 regulates the PCC voltage Vrms relative to the reference voltage Vrms * to create the reference igq * , as in Equation (20). Consequently, PI7 and PI8 regulate the currents igd, igq relative to igd * , igq * to generate the references viq * and vid * , as in Equations (21) and (22), as shown in Figure 5a. Then, the reference voltages are transformed to abc form using dq-abc transformer. The phase-locked loop (PLL) is utilized to produce the phase angle (θg) of the grid for phase synchronization. PLL is based on the dq-frame, where the q-axis voltage is regulated with zero reference, as shown in Figure 5b. In addition, the angular frequency of grid is ωs = 2π × 60 = 377 rad/s.

Optimization Methodology
This study's main objective is to improve the low voltage ride-through ability of the grid-linked PMSG-based WT. This objective can be achieved by designing optimal PI controllers that exist in the control system. However, as shown in Figure 1, the power system is a big nonlinear dynamic system, which cannot be represented by a transfer function. Therefore, the objective function is considered the sum of the integral of PI controllers' squared errors (Equation (23)). The stochastic optimization algorithms are applied to design PI controllers' optimal parameters (k p and T i ) by executing the modeled power system using PSCAD/EMTDC software, as depicted in the flowchart in Figure 6. The power system model's simulation time is 10 s, and a balanced fault is applied at t = 5 s, with a duration of 0.15 s. In this study, the TSO and GWO algorithms are utilized in minimizing the objective function. where

Transient Search Optimization (TSO)
The TSO algorithm is inspired by the transient response of switched electrical circuits that consists of one and two energy storage elements (inductors (L) and capacitors (C)) besides resistors (R), as depicted in Figure 7. The exploration of the TSO algorithm imitates the underdamped transient response of RLC circuits. However, the exploitation of TSO imitates the exponential decay of the transient response of RC circuits. So, the voltage responses across the capacitors in RC and RLC circuits are expressed in Equations (24) and (25).
where fd is the damping frequency, and B1 and B2 are constant numbers.
Therefore, the main frame of the Equations (24) and (25) is exploited to mode the TSO algorithm mathematically, as in Equation (26). However, the resistors, capacitors and inductor (R1, R2, C1, C2, and L) in Equations (24) and (25) are replaced with random numbers, such as (W and A), that are suitable for metaheuristic algorithms. The variables of TSO algorithm are the search agent Xl+1 and

Transient Search Optimization (TSO)
The TSO algorithm is inspired by the transient response of switched electrical circuits that consists of one and two energy storage elements (inductors (L) and capacitors (C)) besides resistors (R), as depicted in Figure 7. The exploration of the TSO algorithm imitates the underdamped transient response of RLC circuits. However, the exploitation of TSO imitates the exponential decay of the transient response of RC circuits. So, the voltage responses across the capacitors in RC and RLC circuits are expressed in Equations (24) and (25).
where f d is the damping frequency, and B 1 and B 2 are constant numbers.

Grey Wolf Optimizer (GWO)
The GWO is inspired by the leader and followers' behavior in the group of the grey wolves [37]. There are main leaders (called α) and secondary leaders (called β), and followers (called δ). Grey wolves start the hunting procedure searching for the food, then surround the prey, and then attack the target. So, the mathematical expression of surrounding the food is: where X is the grey wolf position, X * is the food position, a = 2 − 2l/Lmax, A = 2ar1 − a, C = 2r2, r1, and r2 are real numbers that are randomly distributed between 0 and 1. The variables of the GWO algorithm are the search agent Xl and Xl+1 and the best position Xα, however, the constants are a, C and A. The leaders' Xα start attacking the prey and are supported by the secondary leaders' Xβ and the followers' Xδ, where the attacking process is modeled (Equations (29)-(31)). The pseudo-code of the GWO algorithm is explained in Algorithm 2.  Therefore, the main frame of the Equations (24) and (25) is exploited to mode the TSO algorithm mathematically, as in Equation (26). However, the resistors, capacitors and inductor (R 1 , R 2 , C 1 , C 2 , and L) in Equations (24) and (25) are replaced with random numbers, such as (W and A), that are suitable for metaheuristic algorithms. The variables of TSO algorithm are the search agent X l+1 and X l , which are equivalent to the variables v(t) and v(0) in Equations (24) and (25). Additionally, the variable X * is the best agent, which is equivalent to the v(∞). In addition, B 1 = B 2 = |X l − W.X l *|, where W is expressed as W = k × r 2 × a + 1, k is a real number, a = 2 − 2 × l/L max , L max is the total number of iterations, and r 2 is a random number between 0 and 1. The exploration and exploitation of TSO are balanced by using a random number r 1 .
where, A = 2 × a × r 3 − a, and r 3 are real random numbers between 0 and 1. The pseudo-code of the TSO algorithm is illustrated in Algorithm 1.

Algorithm 1 Transient search optimization (TSO)
Input the random initial values of search agents X l = {k p1 , k p2 , . . . , k p8 , T i1 , T i2 . . . ., T i8 } between 0.01 ≤ k p ≤ 10 and 0.001 ≤ T i ≤ 1 Calculate the ISE(X l ) while l < L max Modify A and W do all search agents X l Update the search agents by Equation (26) end do Compute the ISE(X l+1 ) Compare the ISE(X l ) with ISE(X l+1 ) and find the new best search agents X* l = l + 1 end while return the best search agents X *

Grey Wolf Optimizer (GWO)
The GWO is inspired by the leader and followers' behavior in the group of the grey wolves [37]. There are main leaders (called α) and secondary leaders (called β), and followers (called δ). Grey wolves start the hunting procedure searching for the food, then surround the prey, and then attack the target. So, the mathematical expression of surrounding the food is: where X is the grey wolf position, X * is the food position, a = 2 − 2l/L max , A = 2ar 1 − a, C = 2r 2 , r 1, and r 2 are real numbers that are randomly distributed between 0 and 1. The variables of the GWO algorithm are the search agent X l and X l+1 and the best position X α , however, the constants are a, C and A. The leaders' X α start attacking the prey and are supported by the secondary leaders' X β and the followers' X δ , where the attacking process is modeled (Equations (29)-(31)). The pseudo-code of the GWO algorithm is explained in Algorithm 2.

Algorithm 2 Grey wolf optimizer (GWO)
Input the random initial values of search agents X l = {k p1 , k p2 , . . . , k p8 , T i1 , T i2 . . . , T i8 } between 0.01 ≤ k p ≤ 10 and 0.001 ≤ T i ≤ 1 Compute the ISE(X l ) while l < L max Modify a, A, and C for All search agents Update the search agent (X l+1 ) by Equation (31) end for Compute the ISE(X l+1 ) Compare ISE(X l+1 ) with ISE(X l )and update X α , X β , and X δ l = l + 1 end while Output best search agent X α

Optimization Results
The TSO and GWO are applied to find the PI controllers' optimal variables by minimizing the objective function in Equation (23). These algorithms are coded using FORTRAN and implemented in PSCAD to design the optimal PI controllers. The parameters of TSO and GWO are set as shown in Table 1, where the number of optimization iterations is 1000 for each algorithm and the simulation time of each iteration is 10 s. Additionally, the balanced fault is considered during optimization to emphasize enhancing the low voltage ride-through (LVRT) ability of grid-linked PMSG-based WT. Hence, Figure 8 illustrates the convergence curves of the integral squared-error (ISE) minimization by using TSO and GWO, where the TSO algorithm has achieved a lower ISE value (0.25953) than the ISE value (0.27269) of the GWO algorithm. In addition, Tables 2 and 3 show the optimal 16 variables of eight PI controllers that are optimally based on TSO and GWO.

Simulation Results
In this section, the LVRT ability of the optimal TSO-PI and GWO-PI controllers is tested under balanced and unbalanced short circuits. These abnormal conditions are simulated and applied to the power system model in PSCAD/EMTDC, where the time step is 20 µs.
Case 1, balanced short circuits: This means that TL's three-phase lines are short-circuited together with ground, which is called three-phase to ground fault (3PGF). This fault occurs at the PCC bus bar at the end of one circuit of the double-circuit TLs, as depicted in Figure 1. The power system model is operating under steady state before the fault occurs at t = 0 s, the circuit breakers (CBs) trip the short-circuited TL after 0.15 s.
Firstly, a temporary balanced short circuit is assumed (e.g., flashover, arc, . . . , etc.), so the CBs are assumed to reclose at t = 1 s, as depicted in Figure 9. The transient response of V rms at the PCC bus bar is depicted in Figure 9a, which changes from 1 (pu) at steady state to 0 during the effect of the 3PGF, then returns to the steady-state value once a fault is cleared. During CBs reclosing, the voltage recovery is combined with adverse overshoot and oscillates around the steady-state value until it reaches this value. Therefore, the TSO-PI controllers have attained a lower overshoot than that of the GWO-PI controllers; however, they have the same steady-state error. Figure 9b depicts the real power response during 3PGF, which decreases to 0 during fault at PCC, then recovers to the steady state after fault clearance. Figure 9b shows that the TSO-PI recovers the power faster than the GWO-PI.
Furthermore, the DC link voltage increases during fault due to the excessive power in the DC link, so an adverse overvoltage occurs across the DC capacitors and can be damaged. These capacitors are protected using chopper circuits, as in Figure 1. Additionally, the control system regulates the DC voltage across the capacitors. Figure 9c illustrates that the TSO-PI controller has attained an overvoltage at DC link lower than that of the GWO-PI controller. Moreover, the reactive power at PCC is near the zero value at steady state for a unity power factor correction. Then, after fault clearance, the GSI supports the grid with reactive power to restore the grid stability. Figure 9d shows that reactive power supplied by the TSO-PI controllers is lower than that supplied by the GWO-PI controllers.
Secondly, a permanent balanced short circuit is assumed, so the CBs fail to reclose at t = 1 s, as depicted in Figure 10, then the short-circuited TLs are tripped permanently for maintenance. Figure 10a shows the response of the Vrms measured at PCC during permanent fault and failure of CBs to reclose the tripped TL. The overshoot of TSO-PI controllers is lower than that of GWO-PI during voltage recovery in the first and the second operations of CBs. Moreover, Figure 10b demonstrates the response of real power at PCC during both operations of CBs, where the power response of GWO-PI is much slower in the second act of CBs than the first operation of CBs. However, TSO-PI controllers have attained a fast power response during both operations of CBs. In addition, Figure 10c reveals that the overvoltage and oscillations of DC voltage for TSO-PI are lower than that of GWO-PI. Finally, the reactive power supply during voltage recovery by GWO-PI is higher than that of TSO-PI, as depicted in Figure 10d.
Case 2, unbalanced short circuits: single-phase to ground fault (1PGF) and phase to phase fault (PPF) are considered unbalanced short circuits in the TLs of the power system. Temporary unbalanced short circuits are applied at t = 0, and then the CBs isolate the short-circuited TL at t = 0.15 s; finally, the CBs are assumed to reclose the tripped TL at t = 1 s. Firstly, the PCC voltage responses, real power, DC voltage, and reactive power are measured during the applied 1PGF at the end of one TL, as shown in Figure 11. So, the response of V rms of TSO-PI is slightly lower than that of the GWO-PI, as displayed in Figure 11a. However, the overshoot of real power of GWO-PI is much higher than that of the TSO-PI, as depicted in Figure 11b. Furthermore, the DC voltage response of the GWO-PI is higher than that of the TSO-PI, as depicted in Figure 11c. In addition, the reactive power supply using GWO-PI after fault clearing is much higher than that of the TSO-PI, as depicted in Figure 11d. Secondly, a temporary PPF is applied, and the response of TSO-PI is compared with the response of GWO-PI, as depicted in Figure 12. The Vrms' response using TSO-PI has a lower rise time than that of the GWO-PI, as shown in Figure 12a. However, the real power response of the TSO-PI is faster than that of the GWO-PI, as depicted in Figure 12b. Additionally, the DC voltage response of the TSO-PI has better damped response, as illustrated in Figure 12c. finally, the reactive response of the TSO-PI is nearly similar to that of the GWO-PI, as depicted in Figure 12d.
Electronics 2020, 9, x FOR PEER REVIEW 13 of 21 the response of real power at PCC during both operations of CBs, where the power response of GWO-PI is much slower in the second act of CBs than the first operation of CBs. However, TSO-PI controllers have attained a fast power response during both operations of CBs. In addition, Figure  10c reveals that the overvoltage and oscillations of DC voltage for TSO-PI are lower than that of GWO-PI. Finally, the reactive power supply during voltage recovery by GWO-PI is higher than that of TSO-PI, as depicted in Figure 10d. Case 2, unbalanced short circuits: single-phase to ground fault (1PGF) and phase to phase fault (PPF) are considered unbalanced short circuits in the TLs of the power system. Temporary unbalanced short circuits are applied at t = 0, and then the CBs isolate the short-circuited TL at t = 0.15 s; finally, the CBs are assumed to reclose the tripped TL at t = 1 s. Firstly, the PCC voltage responses, real power, DC voltage, and reactive power are measured during the applied 1PGF at the end of one TL, as shown in Figure 11. So, the response of Vrms of TSO-PI is slightly lower than that of the GWO-PI, as displayed in Figure 11a. However, the overshoot of real power of GWO-PI is much higher than that of the TSO-PI, as depicted in Figure 11b. Furthermore, the DC voltage response of the GWO-PI is higher than that of the TSO-PI, as depicted in Figure 11c. In addition, the reactive power supply using GWO-PI after fault clearing is much higher than that of the TSO-PI, as depicted in Figure 11d. Secondly, a temporary PPF is applied, and the response of TSO-PI is compared with the response of GWO-PI, as depicted in Figure 12. The Vrms' response using TSO-PI has a lower rise time than that of the GWO-PI, as shown in Figure 12a. However, the real power response of the TSO-PI is faster than that of the GWO-PI, as depicted in Figure 12b. Additionally, the DC voltage response of the TSO-PI has better damped response, as illustrated in Figure 12c. finally, the reactive response of the TSO-PI is nearly similar to that of the GWO-PI, as depicted in Figure 12d.

Conclusions
This paper has proposed a novel application of the TSO algorithm to optimally fine-tune all PI controller gains of the frequency converters of a PMSG-based VSWT. The main purpose of this study is to achieve LVRT capability enhancement of such systems. The TSO is extensively applied to the fitness function, which possesses the sum of the system variables' squared error. The optimization problem constraints have taken into account all ranges of the controllers' parameters. The simulation results have proven the high performance and superiority of the proposed TSO-PI controllers to the GWO-PI controllers under balanced and unbalanced network fault conditions. The results using TSO-PI controllers have given an improvement of more than 5% of some quantities of the system under study. The proposed controller's high performance comes from the proper design of the TSO during the design procedure, which relies on the designer experience. These good and salient achievements shall encourage researchers to use the TSO-PI controllers to enhance several systems' responses, including renewable energy systems, microgrids, and smart grid systems.

Conclusions
This paper has proposed a novel application of the TSO algorithm to optimally fine-tune all PI controller gains of the frequency converters of a PMSG-based VSWT. The main purpose of this study is to achieve LVRT capability enhancement of such systems. The TSO is extensively applied to the fitness function, which possesses the sum of the system variables' squared error. The optimization problem constraints have taken into account all ranges of the controllers' parameters. The simulation results have proven the high performance and superiority of the proposed TSO-PI controllers to the GWO-PI controllers under balanced and unbalanced network fault conditions. The results using TSO-PI controllers have given an improvement of more than 5% of some quantities of the system under study. The proposed controller's high performance comes from the proper design of the TSO during the design procedure, which relies on the designer experience. These good and salient achievements shall encourage researchers to use the TSO-PI controllers to enhance several systems' responses, including renewable energy systems, microgrids, and smart grid systems.

Conflicts of Interest:
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A
The data of the power system under study are listed in Table A1.