Optimizing PID controller parameters for robust automatic voltage regulator system through indirect design approach-2

0 Introduction

Synchronous generators are the most widely used equipment for power generation owing to their ability to produce electricity across a broad operating range.The terminal voltages of these synchronous generators can be regulated to a set-point value under fluctuating load conditions by controlling the exciter input voltage.Maintaining a steady terminal voltage is critical in power system networks because the proper functioning and reliability of connected electrical devices depend on preserving the nominal system voltage.Therefore,the primary goal of control in power systems is to stabilize the voltage magnitude at a consistent nominal level.By preventing the voltage from deviating,the sensitive equipment can continue to operate as intended.Effective voltage control enables the integration of synchronous generators across expansive and variable power grids [1].Employing an automatic voltage regulator (AVR) enables the generator’s output voltage to be maintained at the set point.This improves the stability,reliability,and power quality of the power system amid voltage fluctuations.However,achieving rapid and stable AVR control is challenging due to load variations and high inductance in the alternator field coils.Hence,enhancing the control performance of AVR systems is critical for ensuring stable and reliable power distribution across the grid.Therefore,research efforts continue to focus on advancing voltagecontrol responsiveness and transient stability to augment power-system distribution capabilities.

A considerable amount of literature has been published on the development of accurate controllers for AVR systems.These studies include fuzzy logic controllers (FLCs) [2],sliding mode controllers [3],adaptive neurofuzzy inference systems [4],and brain emotional learningbased intelligent controllers [5].In recent decades,proportional-integral-derivative (PID) controllers have become the most widely used controllers in research.PID controllers offer robust performance,a simple structure,and low computational demand [6].Therefore,they have been extensively utilized across several engineering domains,including applications in piezoelectric ultrasonic motors [7],doubly fed induction motors [8],2nd-order nonlinear time-invariant plants [9],vehicle steer-by-wire systems [10],robotic manipulators [11],and wind turbine generators [12].However,traditional PID controllers have certain limitations.For example,they are sensitive to noise,particularly in the derivative component,which can cause significant fluctuations in control signals.Additionally,they exhibit integrator windup,in which the integral term can accumulate a large value in cases of substantial errors,leading to overshooting and prolonged stabilization times.To address this,additional anti-windup schemes are required that add complexity to the control system.

Therefore,researchers and practitioners have developed variants of PID controllers to improve the performance of AVR systems.These variants comprised proportionalintegral-derivative-acceleration (PIDA) [13],PID with additional second-order derivative terms (PIDD2) [14-17],sigmoid PID [18],2DOF PI controller [19],real PID with derivative filter (PIDN) [20],fractional-order PID (FOPID) controller [21],fractional order (FO) proportional-integralderivative plus second-order derivative (FOPIDD2) [22],cascaded real proportional-integral-derivative with second order derivative and fractional-order proportional-integral (FOPI) controller [23],one plus proportional derivative with filter-fractional order proportional-integral controller [24],fractional-order proportional derivative -(one+fractional order integrator) cascade controller [25],and fuzzy tilt integral derivative with filter -one plus integral controller [26].However,these PID variants introduce additional parameters,which increase the complexity of parameter optimization.

Therefore,researchers continue employing the PID controller for AVR systems,because it only has three parameters that need to be tuned.To achieve excellent PID controller performance,the proportional Kp,integral Ki,and derivative Kd gains must first be optimally tuned.However,tuning these parameters has posed challenges to operators and researchers,especially when using classical techniques such as Ziegler-Nichols [27],Cohen-Coon [28],and gain-phase margin [29].These classical techniques operate by manually attempting different gains until suitable settings are obtained.However,this trial-anderror approach requires considerable effort,particularly for complicated and time-varying dynamics.Additionally,AVR systems make PID tuning even more difficult because of variable operating points and nonlinear loads.The manual tuning process struggles to find gains that work under all conditions.Overall,classical techniques are slow and often fail to find the best-fit PID gains for AVR control.Hence,better automated PID tuning is required rather than arduous manual trial-and-error techniques,particularly for systems such as the AVR,which has numerous parameter uncertainties.

Recently,researchers explored the use of modern heuristic optimization techniques to automatically tune the optimal PID gains for AVR systems.These techniques minimize the objective function to derive PID parameters.Gaing [30] pioneered the implementation of heuristic optimization algorithms to tune AVR system controllers.Gaing proposed the use of particle swarm optimization (PSO) for PID tuning of an AVR system and compared its performance with that of a genetic algorithm (GA).More recent studies have focused on the PID tuning of AVR systems using state-of-the-art metaheuristic optimization algorithms.These studies include the Lévy flight-based reptile search algorithm [31],Runge-Kutta optimizer [32],zebra optimization algorithm [33],cuckoo search algorithm [34],sine–cosine algorithm (SCA) [35],nonlinear SCA [18],improved kidney-inspired algorithm (IKA) [36],water cycle optimization (WCO) [37],tree seed algorithm (TSA) [38],enhanced Harris hawks algorithm (EHHA) [39],equilibrium optimizer algorithm [40],hybrid simulated annealing-manta ray foraging optimization [20],artificial rabbits optimization (ARO) [41],and marine predators algorithm [21,42].

In the aforementioned literature,metaheuristic optimization algorithms have shown excellent performance in fine-tuning PID controllers in AVR systems to maintain and stabilize the nominal voltage level of the synchronous generator.These algorithms define an objective function with criteria such as the figure of demerit (FOD),which depends on time-response specifications,including overshoot,rise time,settling time,and steady-state error [43,44].Additionally,a few studies utilized error-based functions,including the integral square error (ISE),integral absolute error (IAE),integral time-square error (ITSE),and integral time absolute error (ITAE) [45,46].However,few have incorporated robustness constraints into the objective function when tuning optimal PID controllers for AVR systems [47].In particular,studies have incorporated robustness measures,such as the GM and PM,into the objective function by applying weights.This allows the optimization to be constrained to guarantee the desired minimum robustness.However,there is a tradeoff between robustness and performance;fixed constraints may worsen other objective function metrics.In unconstrained control optimization,robustness can be adjusted by tuning the weights of those terms.Weighting coefficients balance robustness with performance.However,selecting appropriate weights for each criterion poses challenges.

Although metaheuristic optimization algorithms can tune PID controllers for satisfactory AVR system response and robustness,parameter uncertainties are inevitable.Parameter uncertainties stemming from temperature,pressure,and other physical factors cause the actual system behavior to deviate from the theoretical results.Online manual finetuning of PID gains can help mitigate such issues.However,these risks compromise the robustness of the controller at its optimal value,and do not guarantee process stability.Moreover,the PID controller requires reoptimization for any new robustness constraint.

This study adopted indirect design approach-2 (IDA-2) from [48] to optimize the PID controller by adjusting a single parameter,thereby ensuring the robustness of the AVR system.IDA-2 involves the frequency-shifting of the PID controller based on a frequency-shifting constant ψ and previously optimized PID parameters.Updating ψ modifies all PID gains simultaneously,which in turn improves robustness and performance over the previously optimized PID controller.We validated the proposed approach by simulating an AVR system with a PID controller tuned via sine-cosine optimization (SCA) [35].The relation between frequency-shifting constant ψ and robustness measures (gain margin and phase margin) was demonstrated,along with impacts on maximum sensitivity,overshoot,rise time,settling time,and FOD.The key advantage of IDA-2 is guaranteed AVR system robustness while preserving the optimized time response performance.Notably,this study presents an expanded analysis of the preliminary work in [49].The novelty and contributions of this study are as follows:

1) The proposed method is the first to apply IDA-2 to fine-tune the optimized PID controller of an AVR system.Moreover,this study shows that the proposed PID-IDA-2 method can ensure the robustness of the AVR system during the fine-tuning process.

2) The PID-IDA-2 controller not only increases the robustness but also improves the time response of the AVR system compared to PID-SCA [35],PID-TSA [38],PIDARO [41],PID-IKA [36],PID-EHHA [39],and PID-WCO [37].

3) This study is the first to consider the disturbance rejection and measurement noise during the parameter uncertainty of the amplifier.

The remainder of this paper is structured as follows.Section 2 explains the implementation of a PID controller in the AVR structure.Section 3 describes the procedure for applying IDA-2 to tune the PID controller.Section 4 presents the results,and Section 5 concludes the study with final remarks.

1 PID controller of AVR system

This section illustrates the overall structure of the PID controller employed in the closed-loop control of the AVR system to maintain the terminal voltage at the correct level.A comprehensive understanding of the PID controller requires a proper mathematical model of the AVR system with linearization of all components,such as the exciter,sensor,amplifier,and generator.Fig.1 depicts the closed-loop block diagram of the AVR system with a PID controller,where Kp,Ki,and Kd represent the proportional,integral,and derivative gains,respectively.The gains of the amplifier,exciter,generator,and sensor are denoted by Ka,Ke,Kg,and Ks,respectively.Additionally,the time constants for all these components are denoted as Ta,Te,Tg,and Ts,respectively.The specific values of the gains and time constants were set as K s = 1,Ts = 0.01,K g = 1,Tg = 1,K e = 1,Te = 0.4,Ka=10,and Ta=0.1.It is crucial to highlight that the parameter values employed in this study are aligned with those used in [35].To mitigate issues such as saturation or other nonlinear characteristics,each component was linearized using a time constant and gain.

Fig.1 Closed-loop block diagram of the automatic voltage regulator (AVR) system with the proportional–integral–derivative (PID) controller

Fig.2 Three-dimensional (3D) plot of Ta vs.ψ vs.GM

Fundamentally,a controlled AVR system operates based on error detection.The voltage sensor transforms the output signal Vo ut(t ) into a measurement signal Vm (t).The difference between Vm (t ) and Vr ef(t) is the error voltage,ev (t ).The PID controller modulates the error voltage to enhance the dynamic response and eliminate steady-state errors.The amplifier then receives the output of the PID controller,which is responsible for adjusting the voltage of the exciter and operating the generator.Notably,the PID controller is distinguished by its simplicity,as it relies on only three parameters,Kp gain,Ki gain,and Kd gains,as exemplified in (1):

Notably,the lower and upper bounds for all gains were set to 0.2 and 2.0,respectively,based on [35].Notably,although the performance of the PID controller may be considered satisfactory,the current PID control design for AVR systems lacks robustness to parameter uncertainty.This disadvantage warrants upgrading the design to a robust PID controller,specifically one based on the IDA-2 methodology.

2 Robust PID controller tuning based on indirect design approach-2 (IDA-2)

The proposed IDA-2 emphasizes the controller rather than process shifting,which is the focal point of the indirect design approach (IDA) theory,as described in [50].This implies that the proposed PID tuning rules rely exclusively on the parameters of the optimized PID controller.Theorem 1 summarizes the concept of IDA-2 as described in [48].Assuming a stable closed-loop transfer function C (s ) P (s)/(1 +C (s ) P (s)) for process plant P (s) and controller C(s),Theorem 1 is formulated as follows:

Theorem 1 (IDA-2) [48]:The maximum sensitivity of the process decreases as the frequency-shifting constant ψ increases and vice versa.Meanwhile,the PM and GM of the open-loop transfer function Ci (s )C p(s -ψ) P (s) increase with an increase in the frequency-shifting constant ψ and vice versa,where Ci (s) is the controller transfer function and C p(s -ψ) is a controller transfer function with frequency-shifted version of the process plant P(s).This relationship can be mathematically expressed as follows:

where ψ ∈R, GMψ1 ,PMψ1,and Ms,ψ1 represents the value of the GM,PM,and maximum sensitivity at ψi,respectively.

Proof: Here is the expression for a PID controller with proportional gain kc,integral time constant τi,and derivative time constant τd.

This can be further simplified as

where p1 and p2 are the roots of C p(s).These poles can be real or complex.Thus,a generalized expression for p1 and p2 can be written as

Substituting C p(s) with C p(s-ψ) in (5) and (6),we obtain:

The GM and PM of the open-loop transfer function G (jω ) C I DA2 (jω) are defined as follows:

By examining Eqs.(10) and (11) for

the following conclusions can be drawn:

1) In (10),as the value of ψ increases,the denominator value decreases,leading to an increase in the GM,and vise versa.This relationship can be expressed mathematically as

2) In (11),the PM of the process increases with an increase in ψ and decrease when ψ decreases.This relationship can be expressed as

3) According to the relationship between maximum sensitivity,PM,and GM [50] is

A relationship between Ms and ψ,derived from (13)– (16) can be expressed as

This completes the proof [48].

According to Theorem 1 [48],the PID controllers in (1) is expressed as follows:

where

The PID controllers derived in (18) was subsequently employed in the AVR system illustrated in Fig.1.The objective is to assess the robustness of the proposed PID controller for the AVR system by systematically varying the frequency-shifting constant ψ along with the optimized PID parameters.In this research,the frequency-shifting constant ψ,utilized in the PID controller of the AVR system,is defined with lower and upper limits,denoted as ψL=–4.61 and ψU=47,737.27,respectively.More importantly,these values were derived from initial investigations,ensuring stability through consistent positive values for both GM and PM.

3 Simulation results

In this section,the validity of the proposed PID tuning method is established for the PID control design of an AVR system.Its effectiveness is evaluated and compared with the optimized PID controller for the AVR system,which was tuned using the SCA [35].Furthermore,the robustness of the proposed design method is assessed using key indicators such as the maximum sensitivity (Ms),GM,and PM of the process.In addition,a performance index based on the time-response specifications,including the maximum overshoot (Mp),rise time (Tr),settling time (Ts),and FOD,is considered.The FOD equation can be expressed as

where M p,Ess,Ts and Tr are the maximum overshoot,steady-state error,settling time,and rise time,respectively.Notably,the FOD was chosen as the objective function because it met the designer’s requirements through the weighting factor β.By setting β greater than 0.7,we can reduce the overshoot and steady-state error.Conversely,setting β less than 0.7 helps reduce the rise time and settling time.Notably,the value of β in (20) is fixed to 1.0 according to [35].In this study,the settling time was defined as the duration required for the output response to reach 5% of the final steady-state output.The procedure for assessing the robustness of a PID controller for an AVR system is outlined as follows:

(1) The PID controller was initially designed using established optimal tuning techniques,that is,SCA,as outlined in [35].The optimized PID controller gains obtained using the SCA were Kp=0.9826,Ki=0.8337,and Kd=0.4982.

(2) Subsequently,the PID controller optimized in [35] was fine-tuned using the proposed IDA-2 method to enhance robustness.

(3) The effectiveness of IDA-2 was demonstrated in the context of diverse parameter uncertainties.

(4) The effectiveness of the proposed PID-IDA-2 method was further tested by introducing disturbances and measurement noise amid parameter uncertainty (Ka=15) and comparing the results with those of PID-SCA [35],PIDTSA [38],PID-ARO [41],PID-IKA [36],PID-EHHA [39],and PID-WCO [37].

(5) A frequency response analysis was conducted to evaluate the robustness of the PID-IDA-2 method against existing methods.

Table 1 presents the results of PID tuning based on IDA-2 for uncertainties in the amplifier parameters,considering various values of Ta (0.097,0.12,0.13,and 0.15) and Ka (9.7,12,13,and 15).The performance indices in (19) are examined by varying ψ,where the results for ψ=0 refer to the existing optimized PID parameters in [35].Table 1 also lists the values of PID parameters (Kp,Ki,and Kd) for each variation of ψ,which are recalculated based on (19).The table demonstrates that the online tuning of the PID controller enables users to adjust the controller’s robustness in real time,resulting in a superior transient response of the AVR system compared to the optimized PID controller,especially under parameter uncertainties.

Table 1 Outcomes of proportional–integral–derivative (PID) tuning,utilizing the proposed IDA-2 method,considering uncertainties in the amplifier’s parameters (Ta and Ka) for the automatic voltage regulator (AVR) system

As evident in Table 1,changing the value of ψ influences the robustness of the controller,as proposed in Theorem 1.When ψ increases from–0.05 to 0.05 for both Ta and Ka uncertainties,the values of GM and PM increase.In the case of Ta=0.097,GM and PM increase by approximately 0.4% and 1.3%,respectively,as ψ increases from 0 to 0.05.Similar trends are observed for other parameter uncertainties,such as for Ta=0.12,where GM and PM increase by approximately 0.48% and 1.45%,respectively.Furthermore,the positive change in Ta contributes to a more stable AVR system,resulting in a smaller maximum sensitivity Ms and a higher GM for the nominal case (ψ=0).As observed in Table 1,the GM increases by approximately 0.5%,0.8%,and 1.5% for Ta=0.12,Ta=0.13,and Ta=0.15,respectively.Meanwhile,Ms decreases by approximately 2.4%,3.4%,and 5.1% for Ta= 0.12,Ta=0.13,and Ta=0.15,respectively.Additionally,marginal improvements in the values of FOD and Mp can be observed at larger ψ for every positive change of Ta.In the case of Ta=0.13,the FOD and Mp decrease by approximately 1% as ψ increases from 0 to 0.05.However,Ts and Ts gradually increase with the increase of ψ.In the case of Ta=0.15,Ts and Ts increase by approximately 1.3% and 1.1%,respectively,as ψ increases from 0 to 0.05.

Conversely,an increase in Ka makes the AVR system less stable,leading to a higher Ms and smaller GM and PM for the nominal case (ψ=0).As observed in Table 1,GM decreases by approximately 9%,0.8%,and 1.5% for Ka=12,Ka=13,and Ka=15,respectively.Meanwhile,Ms increases by approximately 3.9%,5.7%,and 8.9% for Ka=12,Ka=13,and Ka=15,respectively.Despite this,an increase in ψ leads to slight improvements in FOD and Mp for every positive change in Ka.In the case of Ka= 13,FOD and Mp decrease by about 1.3% and 0.8%,respectively,as ψ increases from 0 to 0.05.Nonetheless,Ts and Tr also gradually increase with the increment of ψ.In the case of Ka=15,Ts and Tr increase by approximately 0.8% and 0.5%,respectively,as ψ increases from 0 to 0.05.

Table 2 presents the results of PID tuning based on IDA-2 for uncertainties in the generator parameters,considering a varying Tg (0.8.0.97,1.1,and 1.5) and Kg (0.97,1.2,1.3,and 1.5).Notably,the values of GM and PM increase as ψ increases from–0.05 to 0.05 amid parameter uncertainties of both Tg and Kg.In the case of Tg=0.8,GM and PM increase by approximately 0.4% and 1.2%,respectively,as ψ increases from 0 to 0.05.Similar patterns are observed for other parameter uncertainties,such as for Tg=0.97,where GM and PM increase by approximately 0.41% and 1.3%,respectively.Additionally,a positive change in Tg contributes to a more stable AVR system,resulting in a smaller maximum sensitivity Ms and a higher GM and PM for the nominal case (ψ=0).Specifically,as evident in Table 2,the GM increased by approximately 8%,13.4%,and 26.8% for Tg=0.97,Tg=1.1,and Tg=1.5,respectively.Meanwhile,Ms decreased by approximately 3.4%,5.2%,and 9.5% at Tg=0.97,Tg=1.1,and Tg=1.5,respectively.Additionally,there are slight improvements in the values of FOD and Mp when a larger ψ is applied for every positive change of Tg.In the case of Tg=1.5,FOD and Mp decrease by approximately 13.9% and 0.1%,respectively,as ψ increases from 0 to 0.05.However,Ts and Tr gradually increase with ψ.In the case of Tg =1.1,the Ts and Tr increase by approximately 2.9% and 1.1%,respectively,as ψ increases from 0 to 0.05.

Table 2 Outcomes of PID tuning,utilizing the IDA-2 method,considering uncertainties in the generator’s parameters (Tg and Kg) for the AVR system

Conversely,an increase in Kg reduces the stability of the AVR system,leading to a higher Ms and smaller GM and PM for the nominal case (ψ=0).As shown in Table 2,GM decreases by approximately 9%,12.4%,and 18.4% for Kg=1.2,Kg=1.3,and Kg=1.5,respectively.Meanwhile,Ms increases by approximately 3.9%,5.7%,and 8.9% for Kg=1.2,Kg=1.3,and Kg=1.5,respectively.Nonetheless,with an increment in ψ,the FOD and Mp slightly improve for every positive change in Kg.In the case of Kg=1.3,FOD and Mp decrease by approximately 1.3% and 0.8%,respectively,as ψ increases from 0 to 0.05.Nevertheless,Ts and Tr gradually increase with an increase in ψ.In the case of Kg=1.5,Ts and Tr increase by approximately 0.7% and 0.5%,respectively,as ψ increases from 0 to 0.05.

The results in Table 1 and Table 2 collectively demonstrate that a larger positive value of ψ significantly improves the robustness of the PID controller for the AVR system,as proven by the GM and PM results.This observation aligns with Theorem 1 of [48].

Figure 2 presents a three-dimensional (3D) plot of the relationship between Ta,ψ,and GM.As evident in the plot,increments in Ta and ψ results in an increase in the GM.Simultaneously,Fig.3 displays a 3D plot of the relationship between Ta,ψ,and PM.The plot clearly illustrates that PM is directly proportional to ψ and inversely proportional to Ta.In simpler terms,as ψ increases,PM increases,while an increase in Ta leads to a decrease in PM.

Fig.3 3D plot of Ta vs.ψ vs.PM

Fig.4 3D plot of Ka vs.ψ vs.GM

Figure 4 depicts a 3D plot of the relationship between Ka,ψ,and GM.As evident in the figure,the GM is directly proportional to ψ and inversely proportional to Ka.To elaborate,GM increases as ψ increases,while GM decreases with an increase in Ka.Simultaneously,Fig.5 exhibits a 3D plot of the relationship between Ka,ψ,and PM.This plot demonstrates that PM is directly proportional to ψ and inversely proportional to Ka.In simpler terms,as ψ increases,the PM increases,while an increase in Ka results in a decrease in the PM.

Fig.5 3D plot of Ka vs.ψ vs.PM

Additionally,Fig.6 presents a 3D plot of the relationship between Tg,ψ,and GM.In this figure,the GM is directly proportional to both ψ and Tg.Meanwhile,Fig.7 displays a 3D plot of the relationship between Tg,ψ,and PM.Here,PM is directly proportional to both ψ and Tg.

Fig.6 3-D plot of Tg vs.ψ vs.GM

Fig.7 3D plot of Tg vs.ψ vs.PM

Figure 8 displays the AVR system responses generated by the PID-SCA [35] and PID-IDA-2 controller at a reference of 1 per unit (p.u.) and a simulation time of 5 s.The AVR responses from both controllers were recorded under parameter uncertainties of Ta=0.15,Ka=15,Tg=1.5,and Kg=1.5.Specifically,for PID-IDA-2,the ψ values are configured as [–0.05,–0.01,0.01,0.05].Notably,the PID-SCA response is replicated by directly using the PID parameters from the respective studies.Examining the time response in Fig.8 reveals that,with each parameter uncertainty,increasing ψ leads to a reduction in Mp.This result shows that the ψ=0.05 yields the best results for all parameter uncertainty cases by scoring the lowest value of Mp.Although slight increments in Ts and Tr occur in all parameter uncertainty cases,an overall improvement in the objective function FOD was evident,as indicated in Table 1 and Table 2.

Fig.8 AVR system responses produced by PID-SCA [35] and PID-IDA-2 (ψ=[-0.05,-0.01,0.01,0.05]) for various parameter uncertainties (a) Ta=0.15,(b) Ka=15,(c) Tg=1.5,and (d) Kg=1.5

As evidenced by improvements in terms of Ms,GM,and PM,augmenting the frequency-shifting constant ψ appears to enhance the robustness of the AVR system.The inclusion of ψ also proves effective in preserving a robust AVR response even amid system parameter uncertainties.Overall,the proposed IDA-2 method emerges as a suitable tool for PID controller tuning,demonstrating commendable attributes in terms of both robustness and simplicity.

Moreover,to compare the effectiveness of the proposed method with a state-of-the-art controller,we considered [3] and [51],who proposed a sliding mode controller (SMC) and FLC,respectively.Notably,the comparison was performed without subjecting the AVR to any parameter uncertainty.Table 3 lists the time-response specifications in terms of percentage overshoot (%OS),rise time (Tr),and settling time (Ts) for PID-IDA-2 (ψ=0.05),SMC [3],and FLC [51].The results show that the proposed PIDIDA-2 method achieves faster Tr and Ts than SMC and FLC.However,in terms of %OS,PID-IDA-2 ranked third behind FLC and SMC.Overall,PID-IDA-2 demonstrates competitive performance compared with state-of-the-art controllers.

Table 3 Time-response specification results for PID-IDA-2 (ψ=0.05),SMC [3],and FLC [51]

Furthermore,the effectiveness of the PID-IDA-2 method was tested by introducing disturbances during the parameter uncertainty (Ka=15).To test the rapid reactions to disturbances,a step input was applied to the AVR system.Then,at t=1 and 3 s,the disturbances with amplitudes 0.5 and–0.5,respectively,are applied to the output voltage.Fig.9 shows the AVR system responses with disturbance rejection amid parameter uncertainties (Ka =15) produced by PID-IDA-2 (ψ=0.05),PID-SCA [35],PID-TSA [38],PID-ARO [41],PID-IKA [36],PID-EHHA [39],and PIDWCO [37].Notably,the PID controller parameters for PID-SCA,PID-TSA,PID-ARO,PID-IKA,PID-EHHA,and PID-WCO were directly adopted from their respective studies,as listed in Table 4.As shown in Fig.9,PID-IDA-2 can react to disturbances better than the other methods.This can be proven by the performance indices in terms of the ISE,IAE,ITSE,and ITAE,as listed in Table 5.As shown in the table,PID-IDA-2 produces competitive results by obtaining the lowest ISE value alongside PID-EHHA,followed by PID-IKA,PID-SCA,PID-TSA,PID-ARO,and PID-WCO.It also obtained the lowest IAE value compared to the other methods.Meanwhile,for ITSE and ITAE,PIDIDA-2 achieved the second lowest value,following PIDARO.This was followed by PID-SCA,PID-TSA,PID-IKA,PID-EHHA,and PID-WCO.

Table 4 Comparison of PID controller parameters for AVR system

Table 5 ISE,IAE,ITSE,and ITAE produced by all methods for disturbance rejection amid parameter uncertainties (Ka=15)

Fig.9 AVR system responses with disturbance rejection amid parameter uncertainties (Ka=15)

In addition,measurement noise was added to the AVR system to further evaluate the effectiveness of each controller during parameter uncertainty (Ka=15).White noise with a noise power of 0.001 was used as the measurement noise.Fig.10 illustrates the AVR response amid measurement noise obtained by PID-IDA-2 (ψ=0.05),PID-SCA [35],PID-TSA [38],PID-ARO [41],PID-IKA [36],PID-EHHA [39],and PID-WCO [37].As evident from this response,PID-IDA-2’s response can follow the reference signal satisfactorily.Table 6 lists the numerical results of all performance indices obtained by all algorithms.The table shows that the PID-IDA-2 controller produces highly competitive results by obtaining the second-lowest values for ITSE,second to PID-EHHA.For the IAE and ITAE results,PID-IDA-2 produces the thirdlowest values for PID-IKA and PID-EHHA.For ISE,PIDIDA-2 scored the fourth lowest value,followed by PIDEHHA in first place,PID-IKA in the second place,and PID-TSA in the third place.In essence,although the AVR system is subjected to measurement noise during the trajectory tracking simulation,the PID-IDA-2 controller can still cater to it,proving that this method is the most efficient and robust.

Table 6 ISE,IAE,ITSE,and ITAE produced by all controllers amid measurement noise and parameter uncertainties (Ka=15)

Fig.10 AVR system responses amid measurement noise and parameter uncertainties (Ka=15)

Finally,the frequency response analysis of PIDIDA-2 and other methods was conducted using a Bode plot with parameter uncertainties (Ka=15).The stability characteristics were analyzed from the Bode plot by analyzing the GM and PM.Generally,an AVR system is said to be stable if GM and PM are positive,whereas a system with larger GM and PM is considered more stable.In particular,the stability performance of the system using PID-IDA-2 was compared with PID-SCA [35],PID-TSA [38],PID-ARO [41],PID-IKA [36],PID-EHHA [39],and PID-WCO [37].Fig.11 shows the Bode plots of magnitude and phase for PID-IDA-2 and other methods,while Table 7 lists the GM and PM results obtained by all methods.According to the results,PID-IDA-2 obtained highly competitive results by scoring the second-highest GM and PM values of 16.8625 and 41.4511,respectively.These values were less than 10% of the highest value produced by the PID-ARO.The third highest GM value was produced by PID-SCA,followed by PID-WCO,PID-TSA,PID-IKA,and PID-EHHA.Overall,PID-IDA-2 has produced a good frequency response by considering the second highest GM and PM values.

Table 7 The GM and PM produced by all controllers during the parameter uncertainty (Ka=15)

Fig.11 Bode plots for AVR systems during the parameter uncertainty (Ka=15)

4 Conclusion

This study introduces a novel PID tuning approach for an AVR system that exploits IDA-2 to enhance its robustness and performance.The key concept involves single-parameter adjustment of the frequency-shifting constant in an optimized PID control method.This modification provides an additional capability for online robustness and control performance adjustments.The proposed fine-tuning rules are independent of the process model and are derived from a previously optimized PID controller for AVR systems.The effect of tuning the frequency-shifting constant on the PID controller of the AVR system is demonstrated through simulation results.These findings indicate that a large positive value of the frequency-shifting constant significantly improves the controller’s robustness.Moreover,the simulation results revealed that PID-IDA-2 can produce a good time response even when the AVR system is subject to disturbances and measurement noise amid parameter uncertainties.Additionally,the frequency response analysis showed that PID-IDA-2 obtained competitive results compared to the other methods.However,it is currently only applicable to specific frequency-shifting constants.Future studies should explore the implementation of an adaptive frequencyshifting constant within the proposed stability range to address various PID controller challenges.Furthermore,the proposed method can be applied to other energy conversion systems if the PID controller parameters for those systems are available.

Acknowledgments

The authors express their gratitude to the Malaysian Ministry of Higher Education (MOHE) for their support through the Fundamental Research Grant Scheme (FRGS/1/2021/ICT02/UMP/03/3) (UMPSA Reference:RDU 210117).Additionally,appreciation is extended to Universiti Malaysia Pahang Al-Sultan Abdullah for providing the necessary laboratory facilities.Additionally,heartfelt thanks are extended to Universiti Teknikal Malaysia Melaka for their invaluable assistance in completing this paper.

Declaration of competing interest

We declare that we have no conflict of interest.