Modern PID/FOPID controllers for frequency regulation of interconnected power system by considering different cost functions

This article presents frequency regulation of an interconnected three-area power system (Thermal + Wind + Hydro). Fractional Order PID (FOPID) and Proportional-Integral-Derivative (PID) controllers are applied as subsidiary regulators to control the electrical power interconnected system at the time of sudden load variation. To accomplish this study, Genetic Algorithm (GA), Grey Wolf Optimizer (GWO), Sine Cosine Inspired Algorithm (SCIA) and Atom Search Inspired Algorithm (ASIA) are implemented to optimize the secondary regulators' gains (PID and FOPID) by considering various cost functions such as Integral Absolute Error (IAE), Integral Time Absolute Error (ITAE), Integral Square Error (ISE), and Integral Time Square Error (ITSE). Performance analysis in this work is conducted using various cost functions based on GA, GWO, SCIA and ASIA. The comparative analysis of the attained results reveals that GWO-PID and ASIA–PID settle at (83.83 s) and (30.31 s), respectively and ASIA-FOPID at (25.12 s). The controllers based on ITSE as a cost function outperform the comptrollers with other cost functions (ISE, IAE and ITAE). In addition, the ISE-based GA–PID and SCIA–PID settle at (113.92 s) and (35.1 s), respectively and SCIA-FOPID at (24.78 s). The ISE-based regulators yield improved response equated to other cost functions (ITSE, IAE and ITAE) optimized controllers. The robustness test also is carried out to validate the effectiveness of the proposed optimization techniques by changing the system parameters within ± 25% and ± 50% from their nominal values as well as changing the load pattern.

www.nature.com/scientificreports/ The key contributions provided by the authors of the proposed work are as follows: 1. Two modern optimization techniques namely, the Sine Cosine Inspired algorithm (SCIA) and Atom Search Inspired Algorithm (ASIA) are employed to design optimal LFC-based Proportional-Integral-Derivative (PID) and Fractional Order PID (FOPID) controllers with four different cost functions to obtain the most optimal controller gain values of the suggested interconnected power system.2. Robustness test is carried out to validate the efficiency of the suggested optimization framework and the developed controllers by changing the load pattern and system parameter values from their nominal values.3. A comprehensive comparative study of the applicable cost functions is synthesized to validate the efficiency of four different cost functions-based (GA-PID, GWO-PID, ASIA-PID and SCIA -PID, ASIA-FOPID and SCIA -FOPID) controllers response during sudden load demand in the proposed system.
The paper's organization is as follows: The introduction section gives crucial details about the present work and a literature review of various related works to resolve LFC/AGC crisis in multi-area connected power systems.The Proposed Simulink model that investigates the power generating system is given in the "Investigated power system model" section.The transfer function model and controller details and the objective function is given in "Design of controller and cost function".Comparisons of the simulated response of PID and FOPID controllers tuned using different algorithms and cost functions are compared and discussed in "Simulation result and discussion" section and at the end; the "Conclusion" section summarizes the planned work's performance in tuning controller gain values.

Investigated power system model
The proposed interconnected three-area power generating system is displayed in Fig. 1.It consists of thermal, hydro and wind power systems.That All three power-generating areas are organized with the support of a tieline and are implemented as a subordinate controller to regulate power-generating system performance at the time of the unexpected load-changing scenario.Nominal parameter values of each proposed power-generating network are dispatched in [12][13][14] and relevant abbreviations are given as follow: f = 60 Hz, T t = 0.The secondary PID controller input is Area Control Error (ACE) and it is defined as a linear grouping of errors in system frequency & errors in tie-line power flow changes.The output of the controller is u1, u2 and u3 control signals.The expression of the input signal is given in Eq. ( 1)-(3) 12,13 .where ACE 1 , ACE 2 , ACE 3 represent ACE in areas 1, 2 and 3. B 1 , B 2 and B 3 represent frequency bias constant in areas 1, 2 and 3 correspondingly.∆f is denoted as frequency deviation in areas 1,2 and 3 respectively.∆P tie represents the deviation of power flow in the tie-line between connected power system areas 1 to 2, area 2 to 3 and area 3 to1.

Design of controller and cost function
PID controller.The PID controller output is the combination of proportional, integral, and derivative control actions 61 .The proportional controller takes care of and reduces the steady state error in system response [62][63][64][65] .The integral controller reduces the settling time and the derivative controller is responsible for the stability of the entire system at the time of emergencies [66][67][68][69] .This combination reduces the consequence of a disturbance and shortens the time it takes for the frequency level to return to its set point in all critical situations 70 .Moreover, PID is the most simple industrial controller for solving critical issues 6,[71][72][73][74][75] .
PID controller configuration is shown in Fig. 2. It contains three controllers namely, proportional, integral and derivative.The input of the secondary controller is area control error (ACE) and based on the error signal it generates the required control signal (delP ref) to power generating system for frequency regulation.Therefore, the controller design with suitable gain values plays a vital part to achieve a healthier controller response during sudden load demand 13,15 .

FOPID controller.
The main modification between PID controller and FOPID controller is that the order of the FOPID controller is not an integer one [76][77][78] .Based on this characteristic, it provides an extra degree of freedom for tuning controller gain values and its performance is superior compared to conventional PID controller.Regarding the above characteristics of FOPID controller over conventional PID controller, it receives a considerable amount of attention over fast few years.Podlubny introduces the concept of FOPID controller in 76,77 .The transfer function of FOPID controller is given as follows: The gain values of proportional, integral and derivative controllers are represented as K p , K i and K d .λ, µ represents the order of integral (I) and derivative (D) controller, respectively.Figure 3 demonstrates the structure of the proposed FOPID controller.
The dynamic response of the controller should provide quick relaxation time with minimal values of peak over and undershoot during sudden load demand.From the literature review, it is undoubtedly evident that many nature-inspired algorithms are presented to obtain optimal gain values of the controller.In this work, SCIA and ASIA optimization techniques are proposed to optimize controller gain values with four dissimilar cost functions and are compared with well-mature optimizers namely GA and GWO.The cost functions used are ISE, IAE, ITAE and ITSE.To design the regulator, the cost function is initially defined based on the essential description and constraint.

Meta-heuristic optimization techniques (MOTs)
In AGC of the power system, area control error acts as the input for the controller to generate the required control signal that complies with the desired output response during the sudden load disturbance in the system.The simulation process ended after reaching maximum iteration and the system yield a better response at the time of sudden load demand situations in the power system.Different MOTs have been developed and applied in recent years to address numerous dynamic computational issues and provide solutions for LFC problems 79 .Moreover, MOTs have better flexibility compared to conventional optimization methods and it deals with various complex optimization problems effectively such as blade pitch control problem in wind energy field 71,[80][81][82][83][84] , maximum power point tracking in photovoltaic-based energy field [85][86][87][88][89][90][91][92][93][94][95] and energy management schemes in renewable energy-based electrical power systems [96][97][98][99][100] .Despite MOTs being utilized in various applications.However, there is no single MOT that can solve all optimization problems 24,25 .A population-based meta-heuristic approach, which is the actual GWO value, is suggested in 101 .It can be successfully extended to many functional implementations due to the GWO algorithm's simplicity, consistency, and efficiency 101 .Many improvements have subsequently been made to the original GWO algorithm.These modifications, however, use binary or decimal encoders to place the GW; information on the individual genes is also minimal.The simple GWO, therefore has recently been suggested as a modern MOT to solve numerous optimization problems 102,103 .
In the proposed research work, GWO technique is developed to obtain optimal controller gain values in LFC of a multi-area power-generating system network.GWO technique is used to solve nonconvex engineering optimization problems.In nature, it replicates the social stratum and chasing system of grey wolves.The grey wolves are split into four primary levels.Leaders are the first-level wolves and their duty is decision-making.Decisions or other actions are made by the second-tier wolves.Scouts, sentinels, elders, hunters, and caretakers are the third-level wolves who perform orders but can direct other underlying entities and their duties.In all the wolves, in the fourth stage wolves act as an executor.
Optimized gain values are collected by applying four types of cost functions which are IAE, ITAE, ISE and ITSE.The performance of GWO-PID controller is verified by comparing it against GA-PID controller's performance based on the same system 18 .The details of the proposed algorithm and optimized gain values are given in the following section.

Grey Wolf optimization. The Grey Wolf Optimization technique is a most recently developed MOTs based
on the hunting behavior and leadership hierarchy nature of grey wolves.GWO is the freshest metaheuristics www.nature.com/scientificreports/swarm intelligence computational method.Due to its attractive characteristics over other swarm computational intelligence techniques.In addition, it is simple to use, more flexible and scalable.Due to these reasons, GWO has nowadays gained an enormous research interest with tremendous listeners from numerous domains in engineering.There are a greater number of factors considered at the time of developing a genetic algorithm for optimization.Many normal parameters value can be customized to affect the performance.Such as variable specification, tight variable bounds, weighting strategies and constraints.which may slow down the optimization process.
The major role of MOT algorithms is to find global optima while avoiding being stuck in local optima.The proposed GWO technique is robust, simple and it has been developed to solve various complex optimization issues [101][102][103] .In GWO technique, the group of grey wolves is classified into four different groups such as alpha (α), beta (β), omega (ω) and delta (δ).
The hunting manners are divided into the following three steps: Step 1: Tracking, Hurtling and approaching the prey.
Step 2: Surrounding and distressing prey till it stops moving.
Step 3: Aggressive prey.The surrounding behavior is represented by the following mathematical expression 101 : In these above equations − → X P -indicates the prey position vector,- − → X signifies the grey wolf position vector, − → A and − → C vectors are coefficients and t represents the current iteration.
− → α Value decreases linearly from 2 to 0 during iteration and r 1 , and r 2 indicate the random numbers which lie in the range of 0 to 1.During the optimization process, values of ω wolves revise their positions around α , β and δ .Based on these values,ω wolves are repositioned.The updated mathematical models of wolves' positions are determined as per the following [20][21][22] : − → X δ are signifies the position of the, β, and δ respectively, indicates the current solution position and the number of iterations is represented by (t).Then, based on the position's values of α, β and δ the ω wolves update their positions regularly.
The − → A which represents the random vector and − → C which represents the adaptive vector used to assist the algorithm with the local optimization value (prey).The half of iterations are committed to investigation, if |A > 1|, the value of A is decreased from A > 1 over the course of the iteration.The value of C is in the iteration value range of 2 to 0. When the value of C is greater than 1, the vector C starts exploration.In between, the remaining other half of the iterations are dedicated to exploitation when |A|< 1 exists.When the condition C < 1 occurs, exploitation is confirmed 101,102 .
The steps followed by GWO algorithm during PID controller gain value optimization are as follows [102][103][104] .
Step 1: Start the process.
Step 3: Calculate the fitness function of each search agent.
Step 4: Update the position of the current agent a, A and C. www.nature.com/scientificreports/ Step 5: All search agents' fitness is calculated.
Step 6: Check if the satisfied condition is reached or not.
Step 7: Go to Step 4 if, No and repeat.
Step 8: Display optimal PID gain values, if Yes.
Sine cosine inspired algorithm.Mirjalili 104 developed a sine cosine-inspired algorithm based on the sine cosine laws.The random solutions are generated automatically in the first population (controller gain values in the form of a variable vector) by the software.The best locations are defined based on the fitness function with the help of search agents for the minimization of area control error.With the support of this new position P is obtained by assessing the entire population in search agent.
The mathematical modeling of SCIA technique is developed based on search agent updating 104 , as depicted in the below Eq. ( 20).
In the above equation k,R 2 & R 3 are randomly generated random values, in addition R 1 is calculated with the support of Eq. ( 21).
In the above t and t max is pointing out the iterations number and maximum value, whereas S represents constant value.
Figure 4 denotes SCIA technique and its methodology that depends on a circular pattern.In that center of the circle, it is denoted that the best solution and remaining feasible solutions are available outside.The limitations and constraints for operations are represented in the border of the circle.The variable vector upper and lower limits are denoted in the borders of each segment.In addition, two sub-areas are divided and assumed to discover potential areas of Xi solutions.
The motion direction of X_i^ ( P position is outward when R_1 value < 1 and inward when R_1 is > 1) is defined using control factor R1. Based on the R1 and R2 values the inward and outward movements of Xi are determined within the range of 0 to" 2π".Within those probabilistic weights, P is defined using R3.Also switching factor k between two parts (sine and cosine) is determined randomly as per Eq. ( 20) and the random number Ki is (0:1).
At the time of Xi movement, the boundaries are defined in the range of [-1, + 1] to achieve Pbest position and each search space exploited by the algorithm.The search space is exploited using Pbest position.The steps involved in SCIA methodology are clearly shown in the below flow chart Fig. 5.

Atom search inspired algorithm.
Based on the behavior of atomic dynamics, the atom search-inspired algorithm (ASIA) is developed in 2018 by Zhao et al. 105 .ASIA is developed to mimic the atomic motion model because it follows certain molecular mechanics standards.In the atmosphere, all substances include atoms.Atoms are bounded by bonds (covalent) and transformed into molecules.There are two types of forces generated within the atom, such as attractive and repulsive forces.The interactions and forces are developed because of the gap between atoms.Whenever the gap between atoms decreases equally repulsive force increases between them.The frequency of atoms increases when the gap between atoms increases [105][106][107][108] .By using the constraint force (CF), the flow of motion of the atom is controlled and with the support of interactive forces (IF), the motion action is induced in atoms.The value of acceleration is calculated by applying the second law of Newton and it is given by: (20) In the above equation α i signifies the acceleration of ith atom, F i printouts the interaction force, G i indicates constraint force and m i represents the mass of the ith atom.The motion of the atom due to forces is shown in Fig. 6 108 .
The velocity and position of ith atom at (t + 1) iteration are given by [105][106][107][108] : The enhancement of exploration and exploitation at the time of the initial and final process of iterations had been validated by the authors of [105][106][107][108]   Figure 7 shows the convergence curves for applying the optimization methodologies ASIA, SCIA, and GWO in addition to GA with the pre-specified constraints on the interconnected power system using ITSE cost function.From this figure, it is noted that for the same number of iterations (300) and the same number of search agents (100), ASIA converges faster than SCIA, GWO and GA methodologies in terms of minimum iterations number and low computation time.Moreover, ASIA needs only 33 iterations, whereas SCIA, GWO and GA require 41, 57 and 137 iterations to reach the optimal objective function value, respectively.
By applying ASIA technique, the controller gain values are optimized and the respective gain values are listed in Tables 4 and 6.The GA-PID, GWO-PID, ASIA-PID and SCIA-PID controllers with various objective functions are dispatched in Tables 1, 2, 3 and 4. The gain parameters of ASIA-FOPID and SCIA-FOPID controllers are given in Tables 5 and 6.

Simulation result and discussion
The model (Simulink) of the proposed power-generating network is designed and demonstrated by using MAT-LAB/SIMULINK environment by considering PID and FOPID controllers as secondary controllers.The PID controller's gain values are optimized using SCIA, ASIA, GWO and GA algorithms with four different cost functions by considering 1% sudden load pattern (SLP) in area 1 is given in Tables 1, 2   www.nature.com/scientificreports/Further investigation shows that ISE cost function with GA-tuned controller response settles faster compared to other cost functions.The system responses are yields minimum damping oscillation during sudden load demand.
Figure 9 shows a relaxation time comparison.It is noted that the GA-PID controller response based on ISE settles faster compared to the GA-PID controller response based on other cost functions.

Analysis of GWO-tuned PID controller behavior. Figure 10 represents the deviations in frequency by
considering GWO algorithm-tuned PID controller with four cost functions.Based on the results, it is detected that ISE-GWO-PID controller provides a fast response and settles with lesser peak overshoot and undershoot value compared to ITSE, IAE, and ITAE-based PID controllers.
Figure 11 displays the relaxation time comparison plot and it is apparent that the ITSE-based GWO-PID controller response settles faster than the ISE, IAE, and ITAE-PID-based controllers' response.
Table 8 provides a performance analysis of ITSE cost function tuned GWO-PID controller over the other three cost functions (ISE.IAE, ITAE) based GWO-PID controller response during unexpected load variation in the grid-connected power system.The values tabulated in Table 8 justify that ITSE-based GWO-PID controller gives superior results over ISE, IAE, ITAE based controllers.

Analysis of SCIA-tuned PID and FOPID controller behavior. The behavior of SCIA technique tuned
PID and FOPID controller is analyzed in this section under 1% SLP in area 1 with four different cost functions.
Analyzing the response comparisons in Fig. 12 and bar chart comparisons in Fig. 13 effectively depicted that ISE cost function utilized SCIA technique tuned controller provides better controller performance compared to additional cost functions (IAE, ITSE and ITAE).Moreover, Table 9 validate the Percentage of improvement of ISE-based SCIA-PID controller over other cost functions.www.nature.com/scientificreports/ The different functions created FOPID controller behavior in Fig. 14 and numerical values comparisons in bar chart Fig. 15 proved that ISE cost functions based FOPID controller get a more superior response over the other cost functions utilized controller response.Also, the improvement of ISE cost function-based response in terms of settling time is given in Table 10.The response assessments in Fig. 16 effectively show that ITSE-based ASIA PID controller provides more enhanced response remaining functions (cost) employing minimal settling time.Also, respective numerical values of improvement for ITSE-based ASIA-FOPID controller over other functions are highlighted in Fig. 17 and tabulated in Table 11.

Analysis of ASIA-tuned PID and FOPID controller behavior. ASIA optimization technique is applied to tune the gain values of PID and FOPID controllers in frequency regulation of interconnected power systems
The response comparisons of ASIA technique-tuned FOPID controller response with different cost function in area 1 is shown in Fig. 18.The numerical values of settling with different cost bar chart comparisons are shown in Fig. 19.
By examining the response comparisons in Fig. 18 and bar chart comparisons in Fig. 19, ITSE cost functionbased ASIA FOPID controller provides better controller action over IAE, IAE and ITAE functions-based ASIA FOPID controller.Also, the system performance is improved using ITSE cost function-based ASIA FOPID controller by achieving minimal settling time during sudden load demand.Moreover, Table 12 validate the Percentage of improvement of ITSE-based SCIA-PID controller over other cost functions.

Analysis of the proposed controllers' robustness by changing system parameters and load pattern.
In this section, the robustness of the suggested optimization techniques and controllers are examined by  www.nature.com/scientificreports/changing system parameters ± 25% and ± 50% from its nominal value and the load pattern change by 1%, 2%, 5% and 10%.The details are given in the following two sub-sections.

System parameter change
In this section governor and turbine time constants are changed from their nominal value.The behavior comparisons are given in Figs. 20, 21, 22, 23, 24, 25, 26, and 27.Figures 20, 21, 22, and 23 show response comparisons of GA, GWO, ASIA and SCIA techniques tuned controller performance when turbine time constant changes from its nominal value.It is effectively indicating that the proposed optimization techniques are efficient under system parameter change scenarios.Figures 24, 25, 26, and 27 show the response (are 1 frequency deviations) comparisons of GA, GWO, ASIA and SCIA techniques tuned PID controller performance when governor time constant values change in the analyzed power system.

Changing load pattern
The effectiveness of both proposed optimization techniques and controllers is examined by applying different load patterns for both PID and FOPID controllers.The performance comparisons are depicted in Figures 36,  37, 38, 39, 40, 41, 42, and 43.The response of the system equipped with PID controller tuned by the suggested optimization techniques subjected to different load patterns is elaborated in figures 36, 37, 38, and 39.The main observation of the above robustness analysis (Figs. 36, 37, 38, 39, 40, 41, 42, and 43) originates that the analyzed ASIA and SCIA optimization techniques based on FOPID controllers provide better control action under different load patterns situations.In this analysis, the suggested well tunned FOPID controllers provide and yield better controller action in the investigated system.The simulation results depict that ITSE cost function based ASIA optimized controller (PID) yields better performance over the remaining cost functions employing better time domain parameters and also it improves the response of the system by 53.44%, 52.71% and 29.18% over IAE, ISE and ITAE cost functions-based controllers, respectively.Moreover, ITSE-based ASIA-FOPID controller provides superior response over other cost functions (10.60%, 6.89% and 8.95% over IAE, ISE and ITAE).This proposed article indicates that the proposed ISE cost function-based GA-PID, SCIA -PID and FOPID controllers in addition to ITSE-based GWO-PID, ASIA-PID and FOPID controllers outperformed well than other cost functions based controllers for multi-area interlinked power system with renewable power plant even in the case of system parameters change and different loading conditions.

Figure 1 .
Figure 1.Three area interconnected power generating system Transfer function model with PID controller.
, 3, and 4. Similarly, FOPID controller parameters (gain values) are optimized by applying SCIA, ASIA technique with four dissimilar cost functions.The simulation results are compared and evaluated in the following sections: Analysis of GA-based PID controller behavior.Figure 8 illustrates the response of area 1 (thermal) frequency deviations by considering GA-optimized gain values with four different cost functions.Based on the comparison, it is undeniably observed that compared to other cost features, the PID controller with ISE provides a quicker settled response.

Figure 6 .
Figure 6.The motion of the atom due to CF and IF.

Figure 7 .
Figure 7. convergence curves of the applied optimization techniques for ITSE cost function.

Figure 10 .
Figure 10.Area 1 Frequency deviation comparisons of GWO-tuned PID controller response with different cost functions.

Figure 12 .Figure 13 .
Figure 12.Area 1 Frequency deviation comparisons of SCIA-tuned PID controller response with different cost functions.

Figure 14 .
Figure 14.Area 1 Frequency comparisons of SCIA-tuned FOPID controller response with different cost functions.

Figure 15 .
Figure 15.Settling time comparisons for SCIA-tuned FOPID controller with different cost functions.

Figure 16 .Figure 17 .
Figure 16.Area 1 Frequency deviation comparisons of ASIA-tuned PID controller response with different cost functions.

Figure 18 .
Figure 18.Area 1 Frequency deviation comparisons of ASIA-tuned FOPID controller response with different cost functions.

Figure 19 .
Figure 19.Settling time comparisons for ASIA-tuned FOPID controller with different cost functions.

Table 7
represents the percentage of improvement of ISE-based GA-PID controller over IAE, ITAE and ITSEbased GA-PID controller at the time unexpected load demand conditions.

Table 1 .
Parameters of PID controller by applying GA method with various cost functions.Significant values are in [bold].

Table 2 .
Parameters of PID controller by applying GWO method with various cost functions.Significant values are in [bold].

Table 3 .
Parameters of PID controller by applying SCIA method with various cost functions.Significant values are in [bold].Based on the values tabulated in Table7, it is evident that ISE constructed GA-PID controller improves system performance during emergency load demand conditions.

Table 4 .
Parameters of PID controller by applying ASIA method with various cost functions.Significant values are in [bold].

Table 5 .
Optimized gain values of FOPID controller by utilizing SCIA method with various cost functions.Significant values are in [bold].

Table 7 .
Percentage of improvement of ISE-based GA-PID controller over other cost functions.

Table 8 .
Enhancement of ITSE-based GWO-PID controller over other cost functions.

Table 9 .
Percentage of improvement of ISE-based SCIA-PID controller over other cost functions.

Table 10 .
Percentage of improvement of ISE-based SCIA-FOPID controller over other cost functions.

Table 11 .
Percentage of improvement of ISE-based GA-PID controller over other cost functions.

function Relaxation time (s) Percentage of improvement (%)
ConclusionIn this work, PID and FOPID controllers have been projected as secondary controllers for three area thermalhydro-wind interconnected power generating systems.The optimized gain values of PID controller are obtained by implementing GA, GWO, SCIA and ASIA methods with four types of cost functions namely, IAE, ITAE, ISE and ITSE.Based on the simulation results, it is observed that ITSE cost function gives a fast settled response in GWO-PID controller response and it improves the performance of the system by 46.01%, 37.37% and 71.64% over IAE, ISE and ITAE cost functions based GWO-PID controllers, respectively.Similarly, ISE-based GA-PID controller performance yields superior performance compared to other cost functions tuned controllers by 25.93%, 12.49% and 14.11% over IAE, ITAE and ITSE-based GA-PID controller performance, respectively.

Table 12 .
Percentage of improvement of ITSE-based ASIA-FOPID controller over other cost functions.