 Original research
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Design of a 2DOFPID controller using an improved sine–cosine algorithm for load frequency control of a threearea system with nonlinearities
Protection and Control of Modern Power Systems volume 7, Article number: 33 (2022)
Abstract
This paper proposes an improved sine–cosine algorithm (ISCA) based 2DOFPID controller for load frequency control. A threearea test system is built for study, while some physical constraints (nonlinearities) are considered for the investigation of a realistic power system. The proposed method is used as the parameter optimizer of the LFC controller in different scenarios. The 2DOFPID controllers are used because of their capability of fast disturbance rejection without significant increase of overshoot in setpoint tracking. The 2DOFPID controllers’ efficacy is observed by examining the responses with the outcomes obtained with PID and FOPID controllers. The simulation results with the suggested scheme are correlated with some of the existing algorithms, such as SCA, SSA, ALO, and PSO in three different scenarios, i.e., a disturbance in two areas, in three areas, and in the presence of physical constraints. In addition, the study is extended to a fourarea power system. Statistical analysis is performed using the Wilcoxon Sign Rank Test (WSRT) on 20 independent runs. This confirms the supremacy of the proposed method.
1 Introduction
1.1 Literature survey
The pivotal aspiration of the modern power system is to provide a reliable power supply without any interruption. This can be achieved when equilibrium between demand and generation of power is maintained. Frequency is one of the vital parameters which indicates the balance between demand and supply or generation. As frequency and load are inversely proportional to each other, frequency increases when the load is less than the generation, while frequency reduces when the load on the system is more than the supply. Maintaining the frequency at its standard value is an essential task and can be achieved by a technique known as load frequency control (LFC) [1]. LFC has the significant responsibility to maintain the drift in frequency within the permissible limit. In addition, it also maintains the drift in tieline power between multiarea systems to its permissible value. The mechanical input to the power generator is provided to have a balance between generation and demand of power and thereby, LFC controls the input to the generator according to the requirements. LFC basically perform the following tasks:

Nullifies the steadystate error in frequency that is due to step load changes and thereby minimizes transient response and time error.

Reduces the static change in tieline power to zero due to step load retribution.

Supplies the power to any area during emergency from the rest areas.
A detailed review of LFC using PID controller based on soft computing, IMC techniques, and robust control schemes is presented in [2]. In [3], a PID controller with a sliding mode control scheme based on the ALO method for a fourarea system is discussed, while a comparative analysis of a backtracking search algorithm and fruit fly optimizerbased PID controller for a twoarea system considering nonlinearities is explained in [4]. Reference [5] uses a PID controller based on the ALO method for a twoarea and a threearea system with a nonreheated thermal power system for analyzing different performance indices. In [6], a differential evolution (DE) method based on a PID controller is provided for a twoarea thermal system with GRC and a twoarea thermal system with diverse generating units such as thermal, hydro, and diesel. Reference [7] uses bacterial foraging optimization for LFC of an unequal threearea system, whereas a flower pollination algorithmbased PID controller is designed in [8] for frequency control of a twoarea system considering GDB. A fuzzy PID controller based on a sine cosine algorithm for LFC of a hybrid renewable system is presented in [9], and [10] uses a sine cosine algorithmbased fuzzy PID controller for the LFC of a threearea system having nonlinearities. An imperialist competitive algorithmbased fuzzy PI controller for LFC of twoarea systems is presented in [11], while [12] carries out a detailed review on the different types of fractional order controllers used for LFC where various controllers are used, including FOPI, FOPID, PIFOD, TID, FOPIDN, PFOID, etc. It has been observed from different literature surveys that FOPID is the most commonly used controller for LFC. The DEbased FOPID controller is used for LFC of a threearea system in [13], while a sine cosine algorithmbased TID controller is implemented for a twoarea hybrid source system in [14]. FOPID based on gases Brownian motion optimization [15] is proposed for LFC of a twoarea system having GDB, while a FOPID controller is designed in [16] for a single area system using Kharitonov’s theorem. A Salp Swarm Algorithm (SSA) based TID controller is suggested in [17] for LFC for the systems incorporating FACTS devices, while a Grey wolf optimized multidegree of freedom PID controller is proposed for the LFC of a twoarea system in [18]. A 2DOFPID controller based on the quasioppositional Jaya algorithm is applied for LFC of multisource threearea system in [19]. The 2DOFPID controller based on MFO is implemented for LFC of a twoarea system [20], and it is also suggested for a twoarea system with some nonlinearities in [21]. The SSAbased 2DOFPID controller is used in [22] for LFC of a twoarea multisource system. Some hybrid algorithms such as the hybrid gravitational search and pattern search algorithm [23], and the hybrid fireflypattern search algorithm [24] have been proposed for LFC. Recently, ALObased adaptive neuro fuzzy interference system (ANFIS) [25], PSO [26, 27], twodimensional sinelogistic mappedbased SCA [28], and SSA [29, 30] have been used for LFC.
It is observed from the literature survey that in the study of LFC, the researchers have mainly focused on three things, viz. designing new controllers, proposing new optimization techniques, and modeling different types of power system. SCA is a recently developed technique that has been implemented to solve various engineering problems [31]. However, SCA suffers from slow convergence, and getting trapped in local optima. For better performance, SCA is improved in this work, and the Improved SCA (ISCA) method is applied to tune the controller of LFC. The 2DOFPID controller based on ISCA is designed and implemented for LFC of a threearea system with and without nonlinearities, and then further applied to a fourarea system. The objective function used is ITAE, and by minimizing the ITAE, the different performance parameters such as peak undershoot, settling time of frequency and tieline power are improved.
To further confirm the superiority of the proposed method, a statistical analysis is performed. For that analysis, each method is simulated for 20 independent runs, and the Wilcoxon Sign Rank Test (WSRT) is used. This is a significance test where \(+, , \approx\) show ‘superior’, ‘inferior’, or ‘equivalent’ with respect to the compared ones.
1.2 Contribution and organization of the paper
The main contributions of the work are as follows:

An improved form of SCA is proposed, and the superiority of the proposed ISCA method is justified using unimodal and multimodal benchmark functions.

The proposed ISCA is used to optimize the LFC controller variables, and then the performance is compared with some other recently developed algorithms such as SCA, ALO, SSA, and PSO.

A threearea test system and a fourarea test system are modeled for the case study.

Three types of controllers, viz. PID, 2DOFPID, and FOPID are implemented to test the systems, and the effectiveness of the 2DOFPID controller is validated.

For the first time, statistical analysis and WSRT are performed in LFC study to draw conclusions.

The impact of the presence of physical constraints on system performance is investigated.
The work is organized as follows: the proposed power system is discussed in Sect. 2, while Sect. 3 describes the proposed controller. The proposed optimization strategy is detailed in Sect. 4, and the problem formulation is defined in Sect. 5. The results are discussed in Sect. 6, and the conclusions are summarized in Sect. 7.
2 Proposed power system
A threearea system is considered for the case study. The system is an unequal system that consists of three thermal generators having different parameter values. In each area, there are governor, turbine, generation and load sections. The complete system is shown in Fig. 1 with mathematical models in transfer function forms. As shown in Fig. 1, the time constants of each area are denoted as: (1) T_{g1}, T_{g2,} T_{g3} for the governors; (2) T_{t1}, T_{t2}, T_{t3} for the turbines; and (3) T_{ps1,} T_{ps2}, T_{ps3} for the generation and load section. The gains for generation and load sections are denoted as K_{p1}, K_{p2}, K_{p3} for each area. The other parameters of the system are B_{1}, B_{2}, B_{3} for the frequency biased, R_{1}, R_{2}, R_{3} for the droops, and T_{12}, T_{23,} T_{31} for the synchronizing coefficients. Drifts in frequency for each area are represented by \(\Delta F_{1}\), \(\Delta F_{2}\), \(\Delta F_{3}\), while \(\Delta P_{tie12}\), \(\Delta P_{tie23}\), \(\Delta P_{tie31}\) are the incremental changes in tieline power. The numerical values of the parameters are shown in “Appendix 1”.
3 The proposed controller
Numerous variants of PID controller have been used for many years, because of its simplicity and ability to provide reliable results. In this paper, a variation of PID controller known as 2DOF PID controller is used as the LFC controller because of its capability of fast disturbance rejection without significant increase of overshoot in setpoint tracking. DOF stands for the degree of freedom, which means the extent of closedloop transfer function that can be handled distinctly in a control system. The basic arrangement of this controller is shown in Fig. 2, where two separated loops can be seen. Two inputs are applied to the controller, of which one is a reference, and the other is the output of the system. The error signal generated because of difference in these two signals is used by the controller for the generation of the controller output signal which consists of the proportional, integral, and derivate portions according to their weight. The mathematical expression for the 2DOFPID is:
where r and y are two input signals, r is a reference and y is the output of the system. K_{p}, K_{i}, K_{d} are the proportional, integral, and derivative weights, respectively. N is the filter coefficient, and u is the controller output. PW and DW are the setpoint weights on proportional and derivative sections, respectively.
4 The projected optimization techniques
4.1 Sine–cosine algorithm (SCA)
The SCA algorithm is a stochastic populationbased optimization technique inspired by the mathematical functions of sine and cosine. It was recently developed in [32]. Because of the use of the sine and cosine mathematical functions, this algorithm provides cyclic space for exploitation, in which search agents can update their position as per a position changing equation, as:
where \(a_{1}\), \(a_{2}\), \(a_{3}\) are the arbitrary numbers and are the main parameters of this algorithm, and \(Y_{j}^{n + 1}\) and \(Y_{j}^{n}\) are the next and current positions of the solution at the time of the nth iteration in the jth dimension, respectively. \(P_{j}^{n}\) is the terminus point in the jth dimension. Equations (2) and (3) can be combined by another parameter \(a_{4}\). Depending upon the value of \(a_{4}\) which is an arbitrary numeral in the array of [0, 1], the algorithm will choose the equation for renovating the position of the investigating agent, given as:
The parameter \(a_{1}\) will decide the next location province, which can be between the destination and another location. It has the objective of harmonizing the exploitation and exploration of this optimizer, and its value can be given by:
where N is the maximum number of iterations, b is a constant, and n represents the current iteration.The direction of movement of the search agent, whether towards the global optima or away from it, is decided by \(a_{2}\). Better results are obtained by considering the range of \(a_{2}\) between [− 2 to 2], while sine and cosine functions are between 0 to 2π. The objective of \(a_{3}\) is to emphasize the destination and is implemented by choosing a random value. If it is greater than 1 it will stochastically emphasize the destination while it will deemphasize if less than 1.
4.2 Improved sine–cosine algorithm (ISCA)
Nevertheless, SCA is very capable of handling the realtime problem, though there is scope to improve the algorithm to improve the rate of convergence, the ability not to be trapped in nearby optima, and to maintain a balance between exploration and exploitation. The above limitations of traditional SCA are due to the updating scheme of its search agents. In SCA, most of the search agents run towards the global optima, but sometimes can get trapped in local optima and thus converge to that premature local optima. To overcome this, a new scheme for updating the location of the search agent is introduced in this paper. This scheme mainly consists of the SCA/besttarget shown in (6) and (7), and the SCA/randtarget shown in (8) and (9). The besttarget search agent of the SCA assists the search agents in moving towards the best position obtained so far and searching locally around the best search agent, which results in the intensification of the solution. On the other hand, the randtarget search agent of SCA moves the search agents towards the arbitrary position, which results in more search space exploration. In the next step, the exploring capability of both schemes is combined by taking the mean as shown in (10), and the resultant is set as the new search agent. The characteristics of the proposed ISCA are as follows:

1.
It maintains balance between exploration and exploitation.

2.
It has fewer parameters, i.e., the number of parameters of the proposed ISCA is 3 while it is 4 in the original SCA.

3.
It has a better convergence rate than the SCA.

4.
It avoids getting trapped in local optima.
The values of the three parameters are decided in accordance with (11), (12), and (13), respectively.
where b is a constant which is set to 2, N is the maximum number of iterations, n is the current iteration, and \(rand\left( {0, 1} \right)\) denotes a random number that will be generated in the range of 0–1.
The flow chart of the ISCA is shown in Fig. 3. The algorithm mainly has three steps, i.e., initialization, iteration, and termination. In the first step, the algorithm initializes the parameters, such as the maximum number of iterations (N), search agent number (c), number of variables to be tuned (d) with their upper (ub) and lower (lb) bound, first set of search agents (solution). In the second step, it generates a single new search agent by taking the average of four search agents which are being generated by the proposed search schemes. In the last step, the best agent so far obtained is selected as the solution to the optimization problem.
4.3 Performance estimation of the suggested method
The superiority of the proposed technique is tested against ALO, SCA, SSA, and PSO using the 13 standard unimodal and multimodal benchmark functions. Every single algorithm is run 20 times for each benchmark function. The average and standard deviations of different benchmark functions for the ISCA, SCA, ALO, SSA, and PSO algorithms are shown in Table 1. The statistical Wilcoxon signedrank test is carried out on the results. This is shown in Table 2 to confirm the superiority of the ISCA. From Tables 1 and 2, it is found that the ISCA outperforms other methods in eight functions \(f_{1} ,f_{2} ,f_{3} ,f_{4} ,f_{5} ,f_{7} ,f_{10} ,f_{11}\), while PSO outperforms other methods for \(f_{6}\), \(f_{8} , f_{12}\) functions, and SCA and ALO outperform other methods for \(f_{9}\) and \(f_{13}\) functions, respectively. Hence, the proposed method achieves better performance than the existing methods.
5 Problem formulation
Whenever there is a disturbance in the system, the prime objectives of LFC are: (1) nullifying the drift in frequency; and (2) keeping the exchange power of the tieline at its schedule value. For any optimization problem in LFC, the objective function needs to be defined to achieve the objectives. Various criteria have been included in the literature to accumulate the frequency deviation and tieline power deviation in the objective function. From the literature survey, it is found that ITAE (integral of time multiplied absolute error) is a promising criterion and is generally preferred over others like IAE, ISE, and ISTE. Hence, ITAE is used as the objective function for the test system shown in Fig. 1. In LFC, ITAE of the drift in each area frequency and incremental change in tieline power is taken as the objective function:
where \(\Delta f_{i}\) is the drift in each area frequency, \(\Delta P_{{tie_{i  j} }}\) is the incremental change in tieline power, and \(t_{sim}\) is the simulation time period. The controller parameter boundary is the problem constraint. As a result, the design challenge can be expressed as an optimization problem:
For the PID controller
For the FOPID controller:
For the 2DOFPID controller:
6 Results and discussions
On the test system of Fig. 1, many simulation studies have been carried out in order to determine the optimal combination of the suggested algorithm and the controller in order to reach a better outcome. For this purpose, various types of studies have been considered. In Sect. 6.1, the performance of various controllers is compared to identify the better one for further study, whereas in Sect. 6.2, disturbances are applied in two areas, while various algorithms are compared and statistical analysis is carried out. Similarly, in Sect. 6.3, disturbances are applied to all three areas, various algorithms are compared, and statistical investigation is carried out to find the better algorithm. In Sect. 6.4, nonlinearities of GRC, GDB, and commutation delay are considered in the test system, with various algorithms applied to find the controller parameters, while statistical analysis is carried out to find the better algorithm. For statistical assessment of these techniques, WSRT is performed to demonstrate the superior (\(+\)), equivalent (\(\approx\)) or inferior (\(\)) schemes in contrast to the proposed ISCA method. The study is then extended to a fourarea power system shown in Fig. 8 in Sect. 6.5 and again various algorithms are compared for this scenario.
6.1 Examination of controllers
From the literature survey, it is found that some variants of PID controllers have been considered to implement the proposed algorithm. The most common controllers are the PID controller with filter (PIDF), 2DOFPID controller (2DOFPID), and Fractional order PID controller (FOPID). In this work, these controllers are initially compared for the considered test system (Fig. 1) to identify the better one for the present study. A disturbance of 2% is applied in area1 and area2. Figure 4 illustrates the convergence curve of the different controllers, the drift in frequency and tieline power of each area for this case, while Table 3 shows the performance parameter of LFC for these controllers. From Table 3 and Fig. 4, it can be seen that the 2DOF–PID controller outperforms the other two as it converges faster and attains the least objective function value. It also has a lower settling time and undershoot for the frequency deviation as well as for the tieline power. Given these advantages of the 2DOF PID controller, it will be used as the LFC controller for further study as discussed in the following subsections.
6.2 Examination of optimization techniques when disturbances are in two areas
In this part, to validate the effectiveness of the proposed method, the ISCA is applied for the tuning of the LFC controller parameters. The 2DOFPID controlled test system is simulated when disturbances of 2% are applied in area1 and area2. The comparison is carried out with recently used algorithms of PSO, SSA, SCA, and ALO. Figure 5 illustrates the convergence curves and the transient responses of the test system. The convergence curves show that the ISCA achieves the lowest objective function (ITAE value) and converges faster than the other methods. Table 4 gives the performance parameters of the different algorithms for this case. From Fig. 5 and Tables 4, it can be seen that the ISCA performs much better than the other optimization techniques.
To further highlight the efficacy of the ISCA in optimizing the LFC controller parameters, simulation is carried out for 20 independent times by ISCA, PSO, SSA, ALO, and SCA. Table 5 presents the mean and standard deviation of each technique. The ISCA gives the minimum mean value of the objective function. Further, for statistical assessment of these techniques, WSRT is performed on the test system for this scenario, and the results are shown in Table 5.
6.3 Examination of optimization techniques when the disturbance is in each area
In this section, the ISCA is used as the optimizer for the controller parameters. The 2DOFPID controlled test system is simulated when a disturbance of 2% is applied in each area of the test system, and the results are compared with the PSO, SSA, SCA, and ALO. Figure 6 compares the convergence curves and the transient responses of the test system. As seen, the convergence curve of the ISCA shows that the ISCA achieves the lowest objective function (ITAE value) and converges faster than the other methods. Table 6 gives the performance parameters of different algorithms for this case.
To further show the efficacy of the ISCA in optimizing the LFC controller parameters, simulation is done for 20 independent times by ISCA, PSO, SSA, ALO, and SCA. Table 7 presents the mean and standard deviation of each technique. The ISCA gives the minimum mean value of the objective function. Further, for statistical assessment of these techniques, WSRT is performed, and the results are given in Table 7. It is evident that the proposed method is superior.
6.4 Examination of optimization techniques considering physical constraints of GRC, GDB, and communication delay
In this section, the test system is modified and some physical constraints of GRC, GDB, communication delay, and reheated turbine are taken into consideration. Since practical power systems have nonlinearity as mentioned above, the modified system with consideration of these constraints gives a better representation of the practical systems. The communication delay of 40 ms, GDB of 0.036 pu, and GRC of 3% pu are considered in each area of the system, while a disturbance of 1% is applied to area1. The proposed and other algorithms are compared for the 2DOFPID controlled system. The dynamic responses and convergence curves for this case are given in Fig. 7, and the performance parameters are shown in Table 8. It is concluded from the convergence curve, dynamic response, and performance parameters that the proposed ISCA method has much better tuning efficiency than other optimization methods.
To further highlight the efficacy of the ISCA in optimizing the LFC controller parameters, simulation is done for 20 independent times by ISCA, PSO, SSA, ALO, and SCA. Table 9 shows the mean and standard deviation of each technique. The ISCA gives the minimum mean value of the objective function. Further, for statistical assessment of these techniques, WSRT is performed and the results are given in Table 9.
6.5 Extension to fourarea power system
To further examine the capability of the proposed algorithm the study is extended to a fourarea power system. The schematic diagram of this system is shown in Fig. 8 while the values of relevant parameters are given in “Appendix 1”. The 2DOFPID controlled system is simulated for the 2% disturbance in area1. Figure 9 shows the comparative analysis with various techniques for this test system while Table 10 gives the corresponding performance parameters. From Fig. 9 and Table 10, it can be seen that the proposed ISCA algorithm performs better than other techniques.
To further demonstrate the efficacy of ISCA in optimizing the LFC controller parameters, simulation is done for 20 independent times by ISCA, PSO, SSA, ALO, and SCA. Table 11 shows the mean and standard deviation of each technique. The ISCA again gives the minimum mean value of the objective function. Again, WSRT is performed and the results are given in Table 11.
7 Conclusion
In this work, an improved version of the SCA, i.e. the ISCA, is proposed as the tuning tool for the load frequency controller of a multiarea unequal system. The balance between exploitation and exploration is maintained as the proposed method combines the exploitation proficiency of the SCA/besttarget search agent and the exploration proficiency of the SCA/randtarget search agent. First, some controllers are compared for the considered test system, and the 2DOFPID controller emerges as the best. This is then further used as the load frequency controller for different analyses. The performance of the ISCA tuned LFC controller is examined for the test system for various schemes, such as disturbances in two areas, disturbance in each area, disturbance in the presence of various nonlinearities in the test system, and in an extended fourarea power system. The performance of the ISCA is compared with some of the existing algorithms, while the effectiveness of the proposed method is tested by statistical analysis for different scenarios. After considering all the analyses, it is observed that the proposed method is better than the other techniques, such as PSO, SCA, SSA, and ALO, in terms of better convergence rate, least objective function value, minimum settling time, and undershoot of deviation in frequency and tie line power. Therefore, it can be concluded that the proposed method can be applied to solve realworld engineering problems.
Availability of data and materials
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 ISCA:

Improved sine–cosine algorithm
 ALO:

Antlion optimization
 LFC:

Load frequency control
 DOF:

Degree of freedom
 T_{ps} :

Time constant of the power system
 FOPID:

Fractional order proportionalintegralderivative
 SCA:

Sine–cosine algorithm
 T_{sg} :

Time constant of speed governor
 R_{i} :

Speed regulation constant
 K_{ps} :

Gain of the power system
 PID:

Proportionalintegralderivative
 β:

Frequency bias constant
 SSA:

Salp swarm algorithm
 PW, DW:

Proportional and derivative setpoint weights respectively
 T_{t} :

Time constant of the turbine
 ΔP_{tie} :

Tieline power deviation
 T_{ij} :

Synchronization time constant
 \(\Delta P_{ref}\) :

Reference input to the controller
 PSO:

Particle swarm optimization
 \(\Delta P_{D}\) :

Load disturbance
 Δf:

Frequency deviation
 WRST:

Wilcoxon sign rank test
 SLD:

Step load disturbance
 GRC:

Generation Rate Constraint
 GDB:

Governor dead band
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NKG: Formulation of the method, Simulation and Result, Paper writing. MKK: Formulation of the method. AKS: Supervisions and Paper correction. All authors read and approved the final manuscript.
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Neelesh Kumar Gupta born in 1991. He received B. Tech degree form UPTU Lucknow, Utter Pradesh, India in electrical and electronics engineering in 2014, M. Tech in Power System Engineering from NIT Jamshedpur, Jharkhand, India, in 2018. Currently he is the Research Scholar in Electrical Engineering department of NIT Jamshedpur Jharkhand, India.
Manoj Kumar Kar born in 1988. He received B. Tech degree from B.P.U.T, Odisha India, in 2010, M. Tech (Power Electronics and Control Drives) from VSSUT, Burla, Odisha, India, in 2012. He is having five years of teaching experience. Currently, he is the Research Scholar in the Electrical Engineering Department, NIT Jamshedpur Jharkhand, India.
Arun Kumar Singh born in 1956. He received his B. Sc. (Engg.) degree from Kurkshetra University, India in 1982, M Tech from IIT BHU, India in 1985, Ph.D. from IIT Kharagpr, India in 1994. He is having more than thirty years of teaching experience. Currently, he is working as Professor in the Electrical Engineering Department, NIT Jamshedpur Jharkhand, India.
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Appendix 1
Appendix 1
1.1 Three area unequal power system
T_{g1} = 0.08; T_{g2} = 0.06; T_{g3} = 0.07; T_{t1} = 0.4 T_{t2} = 0.44; T_{t3} = 0.3; K_{p1} = 105; K_{p2} = 100; K_{p3} = 120; T_{ps1} = 20; T_{PS2} = 22; T_{ps3} = 20; B_{1} = 0.3483; B_{2} = 0.3827; B_{3} = 0.3692; R_{1} = 3; R_{2} = 2.73; R_{3} = 2.82; T_{12} = 0.2 T_{23} = 0.44 T_{31} = 0.3.
1.2 Four area unequal power system
T_{g1} = 0.08; T_{g2} = 0.06; T_{g3} = 0.07, T_{g4} = 0.06; T_{t1} = 0.4 T_{t2} = 0.44; T_{t3} = 0.3, T_{t4} = 0.44; K_{p1} = 105; K_{p2} = 100; K_{p3} = 120, K_{p4} = 100; T_{ps1} = 20; T_{ps2} = 22; T_{ps3} = 20; T_{ps1} = 22; B_{1} = 0.3483; B_{2} = 0.3827; B_{3} = 0.3692; B_{4} = 0.3827; R_{1} = 3; R_{2} = 2.73; R_{3} = 2.82; R_{4} = 2.73; T_{12} = 0.2 T_{23} = 0.44 T_{31} = 0.3; T_{14} = 0.2.
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Gupta, N.K., Kar, M.K. & Singh, A.K. Design of a 2DOFPID controller using an improved sine–cosine algorithm for load frequency control of a threearea system with nonlinearities. Prot Control Mod Power Syst 7, 33 (2022). https://doi.org/10.1186/s4160102200255w
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DOI: https://doi.org/10.1186/s4160102200255w