
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12784-3
10.1016/j.heliyon.2024.e36753
e36753
Research Article
2DOF-PID-TD: A new hybrid control approach of load frequency control in an interconnected thermal-hydro power system
Shahi Md. Nazmush Shakib nazmushshakib@iut-dhaka.edu
⁎
Orka Nabil Anan nabilanan@iut-dhaka.edu

Ahmed Ashik ashik123@iut-dhaka.edu

Department of Electrical & Electronic Engineering, Islamic University of Technology, Gazipur, Dhaka, Bangladesh
⁎ Corresponding author. nazmushshakib@iut-dhaka.edu
23 8 2024
15 9 2024
23 8 2024
10 17 e367537 6 2024
31 7 2024
21 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
The Load Frequency Control (LFC) scheme, with its primary aim being the maintenance of uniform frequency, has been a heavily researched topic for decades. Achieving a consistent frequency necessitates a delicate balance between load demand and power generation. Researchers strive to find an optimal solution within the LFC domain—one that can effectively withstand drastic load fluctuations. Despite a plethora of efforts, the LFC dilemma remains unresolved, complicated by factors such as dwindling demand-supply and the rapid integration of renewables. Furthermore, the lack of innovation in controller structure design exacerbates the complexity of solving modern LFC problems. Consequently, a robust control approach capable of handling uncertainties while simultaneously regulating system frequency becomes crucial. In light of this, we propose a novel hybrid control architecture called 2DOF-PID-TD. This architecture combines Two Degrees Of Freedom Proportional-Integral-Derivative (2DOF-PID) and Tilt-Derivative (TD) controllers. To optimize the proposed controller, we employ a metaheuristic called the Artificial Gorilla Troops Optimizer (AGTO), which mimics the social behavior and intelligence of gorilla troops. The proposed approach is analyzed in a realistic multi-area multi-source hydro-thermal system, accounting for nonlinearities, random load perturbations, and system parametric uncertainties. Experimental results, when compared with current state-of-the-art optimization algorithms and traditional controller structures, demonstrate the prowess of our approach in terms of precision, robustness, and resilience.

Keywords

Load Frequency Control (LFC)
Hybrid control architecture
2DOF-PID-TD
Artificial Gorilla Troops Optimizer (AGTO)
Multi-area multi-source power system
==== Body
pmc1 Introduction

The power generation system requires modification to meet rapidly changing load demands. However, the dynamic nature of these loads results in frequent fluctuations in electrical power consumption, leading to instability between the electrical and mechanical torque of the generator. Consequently, the generator speed deviates from its nominal value. Since frequency is directly proportional to generator speed [1], any change in generator speed affects the system frequency. To address this issue, the Load Frequency Control (LFC) mechanism is employed.

In recent years, the industry has shifted toward hybrid power systems due to increased supply-demand dynamics, evolving lifestyles, rapid industrialization, and growing environmental concerns [2]. As a result, more and more renewable energy systems are being applied to LFC studies, even though conventional Interconnected Power Systems (IPS) are still dominant [2]. Renewable sources such as solar [3,4], hydro [5,6], and wind energy [4,7] —are frequently combined with thermal energy in both single and multi-area IPS. When it comes to renewable energy sources, hydroelectric power plants are by far the largest global renewable energy sources [8]. Along with hydroelectric power plants, thermal power plants provide most of the energies of the daily energy needs [8,9]. Therefore, considering the dominance of these energy sources, a multi-sourced model that combines thermal and hydropower is more feasible for the LFC study. Recent research trends indicate a strong preference for multi-sourced thermal-hydro systems in addressing the LFC dilemma [[10], [11], [12], [13]]. Furthermore, LFC studies have been addressed in one area [7], two areas [[14], [15], [16]], and even three areas [4,17]. However, most of the previously mentioned works exclude the inclusion of the dominant thermal-hydro multi-sourced model, except for the studies presented in Refs. [[10], [11], [12]], and [13].

LFC is a crucial mechanism in power systems that helps maintain the delicate balance between power generation and load demand. However, several challenges are associated with LFC. In interconnected power systems, the loads are inherently unpredictable and uncertain, indirectly causing deviations in system frequency and tie-line power from their nominal values [18]. Primarily, the deviated frequency is controlled by the governor [19]. Nevertheless, ensuring precise frequency and tie-line power control for a multi-area interconnected power system becomes a daunting task for the governor alone [19]. Renewable energy sources, such as solar and wind, further complicate the LFC process due to their intermittent nature and varying output power based on weather conditions. Consequently, increased fluctuations in load demand at primary generation stations, including thermal and hydro plants are observed. To mitigate this issue, Automatic Generation Control (AGC) must be applied to the system by incorporating optimal controllers. However, integrating an appropriate controller into the AGC system is a rigorous undertaking, involving the design and fine-tuning of controller parameters to achieve the desired performance in frequency and power regulation [20].

Regarding controller architecture, there are two main categories: integer and fractional [21]. Among these, Proportional-Integral-Derivative (PID) [22] controllers stand out as the most widely employed due to their simplicity and favorable cost-benefit ratio [23]. However, as electricity consumption increases and system complexity grows, PID controllers alone are no longer sufficient or robust [23]. Cascading architectures are also commonly used, where integer and/or fractional order controllers form a cascading structure. This arrangement aims to minimize frequency fluctuations more effectively [24]. Despite this, many existing works lack innovation in controller design. Recent research papers [11,20], and [25], emphasize the importance of innovative controller structures for Load Frequency Control (LFC) in multi-unit multi-source power systems. Additionally, incorporating extra Degrees Of Freedom (DOF) enhances the controller's ability to reject load perturbations [20,26].

In recent studies, researchers have employed various controllers to address the Load Frequency Control (LFC) problem. These controllers include, but are not limited to, Proportional-Integral-Derivative (PID) [27], Tilt-Integral-Derivative (TID) [28], Fractional Order PID (FOPID) [29], and Two Degrees Of Freedom PID (2DOF-PID) [30]. These controllers are the most traditional LFC controllers, as discussed in Ref. [31]. According to Refs. [27,32], PID controllers effectively minimize frequency deviations and tie-line power fluctuations in two-area power systems. In Refs. [28,33], the authors demonstrated the load perturbation rejection capabilities of TID controllers in their chosen LFC models. Meanwhile, the fractional parameters of FOPID controllers were harnessed for maintaining uniform frequency and tie-line power flow, as discussed in Refs. [29,34]. Furthermore, in Refs. [30,35], the 2DOF-PID controller was utilized to regulate frequency and ensure desired tie-line power flow in a two-area power system.

To ensure optimal controller performance, proper tuning of its parameters is essential. However, manual tuning becomes increasingly challenging when dealing with complex controller structures that involve numerous parameters. Consequently, soft computing techniques, such as metaheuristic optimization algorithms, have gained prominence in the LFC domain [2,36]. Even if the scope is narrowed down to only renewable energy-incorporated IPS, a plethora of optimizers come into play. These include, but are not limited to, Harris Hawks Optimization (HHO) [3], Dragonfly Algorithm (DA) [6], Hybrid of Firefly Algorithm and Pattern Search (hFA-PS) [37], Improved Ant Colony Optimization (IACO) [5], Adaptive JAYA Optimization Algorithm (AJOA) [14], Quasi-Oppositional Dragonfly Algorithm (QODA) [17], Grey Wolf Optimization (GWO) [15], JAYA Algorithm (JA) [16], Spotted Hyena Optimization Algorithm (SHOA) [7], and Grasshopper Optimization Algorithm (GOA) [4]. However, adhering to the “no free lunch” theorem, no single optimization technique universally excels for all problems [38]. Therefore, it is imperative to explore newer optimization algorithms and compare their outcomes with existing ones to achieve improved optimization performance.

The literature survey reveals that the performance and characteristics of Automatic Generation Control (AGC) in interconnected power systems significantly depend on the structural design of the controller [11]. Therefore, considering the benefits of added an extra Degree Of Freedom (DOF) and acknowledging the lack of innovation in controller structure design, this work combines a Two Degrees Of Freedom PID (2DOF-PID) structure with a Tilt-Derivative (TD) controller to address the modern LFC problem. To the best of the authors’ knowledge, the implementation of a 2DOF-PID-TD control framework represents a pioneering effort in the LFC field.

In accordance with the “no free lunch” theorem [38], we employ a nature-inspired metaheuristic algorithm known as the Artificial Gorilla Troops Optimizer (AGTO) [12,39] to determine the optimal parameters for this novel controller. AGTO is a relatively recent optimization algorithm within LFC studies [12,40]. To validate its excellence, we compare the performance of the AGTO-tuned 2DOF-PID-TD controller against several other optimization algorithms, including the Marine Predators Algorithm (MPA) [41], Multi-Verse Optimizer (MVO) [42], Enhanced Gradient Based Optimizer (EGBO) [43], and Flow Direction Algorithm (FDA) [44]. Among these, MPA [41], MVO [42], and EGBO [43] have been tested extensively in the LFC field. Here, Integral of Time multiplied Absolute Error (ITAE) [45] is applied as the performance index. The reason behind using ITAE [45] as performance index is illustrated in section II(C).

Furthermore, considering the prevalence of thermal and hydro-electric power plants, as well as the recent trend toward using multi-sourced thermal-hydro systems for LFC studies, this work adopts a two-area multi-sourced thermal-hydro system for the proposed controller architecture and algorithm.

The PID [32], TID [28], FOPID [34], and 2DOF-PID [35] controllers are well-tested in the LFC field. In various recently published works [27,29,33], and [30], these controllers have been employed to address modern LFC challenges. Consequently, we verify the results of our proposed approach against these established controllers to emphasize the novelty and superiority of our work.

The objective of this study is to propose a novel 2DOF-PID-TD controller within the Load Frequency Control (LFC) field, substantiating its dynamic performance and robustness when compared to existing controllers, including PID [27], TID [33], FOPID [29], and 2DOF-PID [30]. The key contributions of this work are outlined below.• Novel Hybrid Controller (2DOF-PID-TD): We introduce an innovative architecture for LFC—a hybrid controller that combines Two Degrees Of Freedom Proportional-Integral-Derivative (2DOF-PID) with a Tilt-Derivative (TD) structure.

• Performance Evaluation: We rigorously evaluate the effectiveness of the novel 2DOF-PID-TD controller in the LFC domain. By comparing its performance against the traditional PID [32], TID [28], FOPID [34], and 2DOF-PID [35] controllers, we establish its superiority.

• System Development: We model a realistic two area four source thermal-hydro system for the case study.

• Optimization Algorithm Selection: Through statistical analysis, we determine the optimal optimization algorithm for controller comparison in a two-area multi-sourced thermal-hydro system. We consider AGTO [39], Marine Predators Algorithm (MPA) [41], Multi-Verse Optimizer (MVO) [42], Enhanced Gradient Based Optimizer (EGBO) [43], and Flow Direction Algorithm (FDA) [44].

• AGTO Implementation: We adapt the Artificial Gorilla Troops Optimizer (AGTO) [12] to work seamlessly with the novel hybrid controller and the specified thermal-hydro model.

• Testing and Validation: Our proposed approach undergoes rigorous testing against fixed and random step load perturbations, system nonlinearities, and parametric uncertainties.

The rest of the paper includes the system under analysis, optimization technique, results & discussions, and conclusion. Section II delves the system under analysis including the in-depth information of the LFC model, controller structure and objective function. Moving on to Section III, the optimization technique is presented here which includes a comprehensive overview of the AGTO algorithm. In Section IV, four different scenarios are showcased, each of which illustrates an extensive analysis regarding the performance of the controllers in terms of LFC. Finally, the work is concluded in section V.

2 System under analysis

2.1 LFC model

In this study, we simplify the analysis by employing a two-area interconnected power system as the simulation model, as illustrated in Fig. 1. Each area in this model comprises a non-reheat thermal power plant, a hydro power plant, and an alternator with a load (Rotating Mass & Load). Within each area, two dedicated controllers regulate the frequency and tie-line power deviations for the two power plants (Thermal & Hydro). Specifically, a thermal power plant consists of a governor and a turbine unit, while each hydro power plant includes a mechanical hydraulic governor and a hydro turbine unit. We choose the non-reheat type thermal power plant due to its compact structure [46], faster response, and improved stability [47].Fig. 1 Block diagram of the LFC model.

Fig. 1

For the non-reheat thermal power plant, transfer function for the governor can be represented as [48],(1) Gsg(s)=11+sTsg

Here, Tsg represents the time constant of governor in thermal plant.

Now the turbine of thermal power plant can be represented by the following transfer function [48],(2) Gt(s)=11+sTt

Here, Tt represents the time constant of steam turbine.

The mechanical hydraulic governor consists of two parts: hydro governor and droop. Fig. 2 represents the mechanical hydraulic governor. The overall transfer function of the mechanical hydraulic governor can be represented as [49,50],(3) Ggh(s)=1+sTrs(1+sTgh)(1+sTrh)

Here, Trs, Tgh, and Trh represent droop resetting time, time constant for hydro turbine governor, and droop time constant, respectively [49,51].Fig. 2 Mechanical hydraulic governor.

Fig. 2

Now the transfer function of the hydro turbine can be represented as [49],(4) Gw(s)=1−0.5sTw1+0.5sTw

Here, Tw represents water starting time.

The transfer function for the rotating mass & load can be represented as [48],(5) Gps(s)=Kps1+sTps

Here, Tps and Kps represent the time constant and gain of the rotating mass & load, respectively, where Kps=1D and Tps=2HfD [5]. Here, the load frequency dependency parameter (D) and the inertia constant (H) are 0.01 p.u. MW/Hz and 5 p.u. MW/Hz, respectively. The parameter D represents the ratio of the nominal load of each area (PL) to the nominal frequency (f) [5]. For reference, the nominal values of f and PL are 50 Hz and 0.5 p.u. MW, respectively. Additionally, each area has a rated load of 2000 MW [5].

The nominal values of the constants and other system parameters that are used in this model are illustrated in Appendix A.

2.2 Overview of the proposed controller

The system comprises two separate controllers in each area, responsible for controlling thermal and hydro units independently. To maintain the desired frequency and tie-line power, we propose a novel 2DOF-PID-TD controller, as depicted in Fig. 3.Fig. 3 Block diagram of the 2DOF-PID-TD controller.

Fig. 3

One of the inputs to the novel hybrid controller is the Area Control Error (ACE), which can be represented as follows [5],(6) ACE1=BΔf1+ΔPtie

(7) ACE2=BΔf2+a12ΔPtie

Here, Δf1 and Δf2 represent the frequency deviation in area 1 and area 2, respectively, while ΔPtie denotes the tie-line power deviation. Furthermore, B is the frequency bias coefficient and a12 is the area size ratio. The specific values of B and a12 are depicted in Appendix A.

In Fig. 3, Pw and Dw are the proportional set point weight and derivative set point weight, respectively. The values of these two parameters are constrained to the range of 1–5 [49]. Additionally, Kp, Ki, and Kd denote the proportional, integral, and derivative gains of the controller, while Tt and Td correspond to the tilt gain and derivative gain, respectively. For the parameter n, the preferred value lies between 2 and 3 [52].

The proposed controller structure combines 2DOF-PID [49] and TD [53] controllers. The output equation of the proposed controller is formulated as expressed in Ref. [49]. Considering the two inputs, ACE and Δf, the output of the 2DOF-PID-TD controller in Laplace domain can be represented as follows:(8) U(s)=Kp×M+Kis×N+Kds×S+Tts−(1n)×Q+Tds×Q

Where:(9) M={ACE(s)×Pw−Δf(s)}

(10) N={ACE(s)−Δf(s)}

(11) S={ACE(s)×Dw−Δf(s)}

(12) Q=ACE(s)

These variables (M, N, S, and Q) are introduced for the simplicity of Eq. (8).

Eqs. (9), (10), (11), (12)) define the M, N, S, and Q variables, respectively. Specifically.• M represents the subtraction of Δf(s) from the product of ACE(s) and Pw.

• N represents the subtraction of Δf(s) from ACE(s).

• S represents the subtraction of Δf(s) from the product of ACE(s) and Dw.

• Q corresponds to ACE(s).

For the remaining parameters of Eq. (8) and the ranges of the controller parameters, please refer to Appendix B.

2.3 Objective function

The objective function plays a crucial role in justifying the performance of a system. A system is considered optimal when the controller parameters are adjusted to minimize the objective function [5].

For optimizing the 2DOF-PID-TD controller, we first need to define the objective function. The following conditions must be satisfied for Load Frequency Control (LFC).1. Fast Restoration: Both the frequency and the tie-line power should be restored to their nominal values as quickly as possible after applying Step Load Perturbations (SLPs).

2. Minimal Peaks: The peak value of tie-line power deviation and frequency deviation during the transient should be as small as possible.

To meet these conditions, we adopt the Integral of Time multiplied Absolute Error (ITAE) [45] as the objective function. According to various sources [[54], [55], [56]], the ITAE gives more weight to errors, occurring later in the transient response. This means that the ITAE objective function can effectively minimize frequency and tie-line power deviations compared to other objective functions [55,56] such as Integral of Absolute Error (IAE) [57,58], Integral of Squared Error (ISE) [57,58], and Integral of Time multiplied Squared Error (ITSE) [57]. Additionally, the ITAE promotes system robustness and stability by penalizing larger errors [54]. The ITAE can be mathematically represented as follows [59]:(13) ITAE=∫0tsim(|Δf1|+|Δf2|+|ΔPtie|)×t×dt

Here, Δf1, Δf2 are FD in area 1 and area 2, respectively. ΔPtie is the tie-line power deviation and tsim is the maximum simulation time.

For the full forms of the acronyms, please refer to Appendix B.

3 Optimization technique

3.1 Artificial Gorilla Troops Optimizer

Artificial Gorilla Troops Optimizer (AGTO) is a metaheuristic algorithm inspired by gorilla troops’ social intelligence, i.e., their group life and how they collect food [39]. For exploration and exploitation, AGTO mimics the following five strategies [39,60].• Strategy 1 - Traversing the previously unexplored areas

• Strategy 2 - Migration to other gorilla troops

• Strategy 3 - Migration to a familiar territory

• Strategy 4 - Following the leader

• Strategy 5 - Competition for female partners

The first three strategies comprise the exploration phase. In this phase, the strategies increase the exploration capability, strike balance between the two optimization operations and boost searching in different optimization spaces, respectively. The exploitation phase includes the latter two strategies of AGTO, which ensures a systematic and perennial exploration.

3.1.1 Exploration mechanism

The mathematical formulae of the exploration parts [39]:(14) xG(t+1)={(UB−LB)r1+LBrand<p(r2−P)xr(t)+QRrand≥0.5x(i)−Q(Q(x(t)−XGr(t))+r3(x(t)−XGr(t)))rand<0.5

(15) P=cos(2r4+1)×(1−ItiItmax)

(16) Q=PI

(17) R=Zx(t)

Here, xG(t+1) is the gorilla candidate position vector next to t-th iteration, x(t) is the current gorilla position vector, xr(t) is a randomly selected gorilla from the group at t-th iteration, x(i) is the initial gorilla position vector, and XGr(t) is a randomly selected gorilla from the gorilla candidate position vector. In addition, p denotes the probability of selecting the migration mechanism to a hitherto unexplored location, and UBandLB constitute upper and lower bounds of variables, respectively. The r1,r2,r3, and rand variables are random numbers in [01] updated each iteration. P, Q, R represent the intermediate variables, calculated with the help of Eq. (15), Eq. (16), and Eq. (17). In these equations, Iti represents the current iteration number and Itmax represents the maximum iteration number. Moreover, Z,I, and r4 are random numbers in the range of [−PP], [−11], and [01], respectively.

3.1.2 Exploitation mechanism

In the exploitation phase, depending on the P value of Eq. (15), two mechanisms occur. If P≥W, the group follows the leader silverback using Eq. (18), Eq. (19), and Eq. (20). But, if P<W, the competition of attracting adult females commences using Eq. (21), Eq. (22), Eq. (23), and Eq. (24). It should be noted that the value of W is user-defined and must be selected before the optimization initialization.(18) xG(t+1)=QM(x(t)−xsb)+x(t)

(19) M=(|1N∑i=1NxGi(t)|g)1g

(20) g=2Q

(21) xG(i)=xsb−aF(xsb−x(t))

(22) F=2r5−1

(23) a=bVE

(24) VE={N1rand≥0.5N2rand<0.5

Here, F, a, b, VE, and N describe the impact force, the degree of violence, a specific value selected before the optimization operation, the violence effect on the solutions' dimension, and the total number of gorillas, respectively. N1 and N2 are random values from the normal distribution, whereas r5 is a random number in [0 1]. Moreover, xsb represents silverback gorilla position vector and xGi(t) denotes the candidate position vector of each gorilla at t-th iteration. Mandg serve as intermediate variables in the exploitation stage. After the comparison among all xG solutions and x(t), the algorithm finds the silverback gorilla, i.e., the best solution among the population set.

Fig. 4 illustrates the implementation flowchart of the AGTO algorithm. First, a population of gorillas is initialized with random positions or values, and the fitness values corresponding to these positions are calculated. The algorithm runs for a set number of iterations, referred to as the maximum iteration number, Itmax. After exploring the solution space using different heuristics and updating the positions, the fitness value of each gorilla is recalculated. Depending on the intermediate P and the threshold value W, the algorithm applies either Strategy 4 or Strategy 5 to further update the positions. The fitness value is then recalculated. Once the iteration reaches Itmax, the algorithm terminates and returns the best solution found.Fig. 4 Flowchart of the AGTO algorithm.

Fig. 4

4 Results & discussions

The model depicted in Fig. 1 is developed using the MATLAB-Simulink [61] environment. We have tested a total of five controllers (PID [27], TID [28], FOPID [29], 2DOF-PID [30], and 2DOF-PID-TD) in the same model to compare the effectiveness of the controllers. The 2DOF-PID-TD controller, a novel hybrid design, is compared against the PID [27], TID [28], FOPID [29], and 2DOF-PID [30] controllers to showcase its superior performance.

To identify the most suitable optimizer for the system, we evaluate the performance of the AGTO [39,40] tuned 2DOF-PID-TD controller against other optimization algorithms, including the Marine Predators Algorithm (MPA) [41], Multi-Verse Optimizer (MVO) [42], Enhanced Gradient Based Optimizer (EGBO) [43], and Flow Direction Algorithm (FDA) [44], using the Integral of Time multiplied Absolute Error (ITAE) [45] as the performance metric.

Table 1 presents a comparative analysis of the performance of these five different optimizers for the 2DOF-PID-TD controller in terms of ITAE [45]. The controller parameters are optimized for the mentioned two-area power system with Step Load Perturbations (SLPs) of 0.1 p.u. in both areas. All optimizers use a population size of 20 and a maximum of 50 iterations. To ensure fairness, we maintain consistent controller parameter ranges across all optimization methods. Details of the parameter ranges are provided in Appendix B.Table 1 Comparative results of the five different optimizers.

Table 1Iterations	Optimizers	Controller	ITAE	
50	Artificial Gorilla Troops Optimizer (AGTO) [40]	2DOF-PID-TD	0.003755173	
50	Multi-Verse Optimizer (MVO) [42]	0.003856133	
50	Marine Predators Algorithm (MPA) [41]	0.004244198	
50	Enhanced Gradient Based Optimizer (EGBO) [43]	0.003970269	
50	Flow Direction Algorithm (FDA) [44]	0.012661477	

Table 1 unequivocally demonstrates that AGTO [40] reigns supreme in terms of ITAE [45], outperforming the other optimizers in this system (specifically, ITAEs are 0.003755173 for AGTO [40], 0.003856133 for MVO [42], 0.004244198 for MPA [41], 0.003970269 for EGBO [43], and 0.012661477 for FDA [44]). The convergence curves of all five optimizers are illustrated in Fig. A1.Fig. A1 Convergence curves of the optimizers.

Fig. A1

From Fig. A1, we can further verify the excellence and robustness of AGTO [40] in terms of ITAE [45]. AGTO [40] exhibits the fastest convergence characteristics with the lowest ITAE [45] among the five optimizers.

Table X1 presents the statistical data for the mentioned five optimizers. For statistical analysis, each optimizer is run 20 times to determine the best, worst, and mean ITAE [45] values. During this analysis, the Step Load Perturbations (SLPs) are set to 0.1 p.u. in both areas. For each optimizer run, we maintain a population size of 20 and a maximum of 50 iterations.Table X1 Comparative statistical study of the five optimizers

Table X1Runs	Optimizers	ITAE values	
Best	Worst	Mean	
20	Artificial Gorilla Troops Optimizer (AGTO) [40]	0.003755	0.004664	0.004028	
20	Multi-Verse Optimizer (MVO) [42]	0.003856	0.004968	0.004439	
20	Marine Predators Algorithm (MPA) [41]	0.004244	0.005165	0.004676	
20	Enhanced Gradient Based Optimizer (EGBO) [43]	0.00397	0.005051	0.004493	
20	Flow Direction Algorithm (FDA) [44]	0.012661	0.028194	0.023155	

In Table X1, AGTO [40] stands out as the best performer among the listed five optimizers, with ITAE values of 0.003755 (best), 0.004664 (worst), and 0.004028 (mean). In contrast, the best values for MVO [42], MPA [41], EGBO [43], and FDA [44] are 0.003856, 0.004244, 0.00397, and 0.012661, respectively. The worst values for these optimizers are 0.004968, 0.005165, 0.005051, and 0.028194, while the mean values are 0.004439, 0.004676, 0.004493, and 0.023155, respectively.

Therefore, considering the supremacy of AGTO [40] in terms of ITAE [45], we optimize the parameters of the mentioned controllers using the AGTO [40] algorithm based on ITAE [45] criteria. The goal of this work is to evaluate the dynamic performance of the novel hybrid controller (2DOF-PID-TD controller) and compare it with the other mentioned controllers. To achieve this, we consider the following four scenarios.⁃ Scenario 1: 0.15 p.u. SLP in area 1 and 0.1 p.u. SLP in area 2.

⁃ Scenario 2: Random SLP in both areas.

⁃ Scenario 3: Sensitivity analysis with added parametric uncertainties.

⁃ Scenario 4: System performance with nonlinearity.

SLPs are applied at t = 1 s for Scenarios 1, 3, and 4. To implement AGTO [40] in the system, we set the maximum number of iterations to 50 and the population size to 20.

4.1 Scenario 1

In this scenario, we apply a 0.15 p.u. Step Load Perturbation (SLP) in area 1 and a 0.1 p.u. SLP in area 2. Based on these disturbances the controllers are optimized for the system. The optimal gains of the controllers of area 1 for the thermal and hydro units are depicted in Table 2 (b), Table 2 (a), respectively.Table 2 (a) Optimal gains of the controllers for thermal unit (area 1) in scenario 1.

Table 2 (a)AGTO Controllers	Thermal unit (Area 1)	
Kp1	Ki1	λ1	Kd1	μ1	Pw1	Dw1	Tt1	1n1	Td1	
2DOF-PID-TD (Proposed)	45	75		22.761		5	5	75	0.5	20	
2DOF-PID [30]	30	60		14.54		4.741	2.271				
FOPID [29]	45	75	1	35	0.863						
TID [28]		75						75	0.33	35	
PID [27]	17.576	74.547		15.929							

Table 2 (b) Optimal gains of the controllers for hydro unit (area 1) in scenario 1.

Table 2 (b)AGTO Controllers	Hydro unit (Area 1)	
Kp1	Ki1	λ1	Kd1	μ1	Pw1	Dw1	Tt1	1n1	Td1	
2DOF-PID-TD (Proposed)	30	1.002		45		5	4.809	20	0.333	35	
2DOF-PID [30]	30	1		11.295		1.077	1.867				
FOPID [29]	30	40	0.75	45	1						
TID [28]		1						1	0.33	1	
PID [27]	1.0002	1		1							

In Fig. 5 (a), we observe the frequency deviation in area 1 (Δf1) for each controller. Notably, the 2DOF-PID-TD controller stands out, demonstrating superior performance in managing the frequency of area 1. Delving into the details from Table 4, the extracted information is as follows:1. Settling Time: The 2DOF-PID-TD controller achieves the fastest settling time for the Δf1 curve, clocking in at 1.93076 s. In comparison, the settling times for the other controllers are as follows:o 2DOF-PID [30]: 4.08886 s

o FOPID [29]: 4.2378 s

o TID [28]: 3.92581 s

o PID [27]: 5.65047 s

2. Peak Overshoot and Undershoot: The 2DOF-PID-TD controller outperforms other controllers in terms of both peak overshoot and peak undershoot for Δf1:o Peak Overshoot: 0 Hz (compared to 0.0001973 Hz for 2DOF-PID [30], 0.005671 Hz for FOPID [29], 0.02395 Hz for TID [28], and 0.04131 Hz for PID [27])

o Peak Undershoot: 0.0056153 Hz (compared to 0.0068303 Hz for 2DOF-PID [30], 0.02 Hz for FOPID [29], 0.04923 Hz for TID [28], and 0.07933 Hz for PID [27])

Fig. 5 (a) FD in area 1, (b) FD in area 2, for the scenario 1.

Fig. 5

Now the optimal gains of the controllers of area 2 for the thermal and hydro units are illustrated in Table 3 (b), Table 3 (a), respectively.Table 3 (a) Optimal gains of the controllers for thermal unit (area 2) in scenario 1.

Table 3 (a)AGTO Controllers	Thermal unit (Area 2)	
Kp2	Ki2	λ2	Kd2	μ2	Pw2	Dw2	Tt2	1n2	Td2	
2DOF-PID-TD (Proposed)	50	75		25		3.719	1.023	1	0.406	30	
2DOF-PID [30]	30	41.48		35		5	1				
FOPID [29]	50	73.751	1	25	0.9						
TID [28]		41.753						40	0.33	25	
PID [27]	17.569	51.551		15.523							

Table 3 (b) Optimal gains of the controllers for hydro unit (area 2) in scenario 1.

Table 3 (b)AGTO Controllers	Hydro unit (Area 2)	
Kp2	Ki2	λ2	Kd2	μ2	Pw2	Dw2	Tt2	1n2	Td2	
2DOF-PID-TD (Proposed)	45	40		25		5	3.031	40	0.33	55	
2DOF-PID [30]	1.4	1		17.8		4.63	3.11				
FOPID [29]	1	40	0.75	25	0.75						
TID [28]		25.214						40	0.33	1	
PID [27]	1.066	1		1							

Table 4 Comparison of Settling Time, Peak Overshoot, and Peak Undershoot for FD in area 1 & 2.

Table 4Parameters	Controller	Scenario 1	
Settling Time (s)	Peak Overshoot (Hz)	Peak Undershoot (Hz)	
Δf1	2DOF-PID-TD (Proposed)	1.93076	0	0.0056153	
2DOF-PID [30]	4.08886	0.0001973	0.0068303	
FOPID [29]	4.2378	0.005671	0.02	
TID [28]	3.92581	0.02395	0.04923	
PID [27]	5.65047	0.04131	0.07933	
Δf2	2DOF-PID-TD (Proposed)	2.41039	0	0.008207	
2DOF-PID [30]	2.44425	0	0.008296	
FOPID [29]	3.37586	0.0004792	0.02456	
TID [28]	4.56404	0.01323	0.04025	
PID [27]	6.23988	0.02251	0.05358	

In Fig. 5(b), the frequency deviation in area 2 (Δf2) is depicted for each controller. Among the controllers considered, the 2DOF-PID-TD controller demonstrates superior performance in managing the frequency deviation of area 2. Specifically, the details from Table 4 are as follows:1. Settling Time: The 2DOF-PID-TD controller achieves the fastest settling time for the Δf2 curve, with a value of 2.41039 s. Comparatively, the settling times for the other controllers are as follows:o 2DOF-PID [30]: 2.44425 s

o FOPID [29]: 3.37586 s

o TID [28]: 4.56404 s

o PID [27]: 6.23988 s

2. Peak Overshoot and Undershoot: The 2DOF-PID-TD controller also excels in terms of peak overshoot and undershoot for the Δf2 curve:o Peak Overshoot: 0 Hz (compared to 0 Hz for 2DOF-PID [30], 0.0004792 Hz for FOPID [29], 0.01323 Hz for TID [28], and 0.02251 Hz for PID [27]).

o Peak Undershoot: 0.008207 Hz (compared to 0.008296 Hz for 2DOF-PID [30], 0.02456 Hz for FOPID [29], 0.04025 Hz for TID [28], and 0.05358 Hz for PID [27]).

These impressive results highlight the 2DOF-PID-TD controller's superior stability and robustness in maintaining consistent frequency compared to its counterparts.

Moving on to Fig. 6, we examine the tie-line power deviation (ΔPtie) for each controller in scenario 1.Fig. 6 ΔPtie for the scenario 1.

Fig. 6

According to Fig. 6, the 2DOF-PID-TD controller outperforms its counterparts in maintaining the tie-line power. The specific analysis from Table 5 is as follows.1. Settling Time: The 2DOF-PID-TD controller achieves the fastest settling time for the ΔPtie curve, with a value of 24.0845 s. In comparison, the settling times for the other controllers are as follows:o 2DOF-PID [30]: 28.7657 s

o FOPID [29]: 32.4442 s

o TID [28]: 25.1742 s

o PID [27]: 25.5629 s

2. Peak Overshoot and Undershoot: Although the 2DOF-PID-TD controller doesn't have the lowest peak overshoot, it outperforms its counterparts in terms of peak undershoot for the (ΔPtie) curve:o Peak Overshoot: 0.000008183 p.u. (compared to 0.000006469 p.u. for 2DOF-PID [30], 0.00003844 p.u. for FOPID [29], 0.00012943 p.u. for TID [28], and 0.0001845 p.u. for PID [27]).

o Peak Undershoot: 0.00001951 p.u. (compared to 0.00004729 p.u. for 2DOF-PID [30], 0.0002459 p.u. for FOPID [29], 0.00006021 p.u. for TID [28], and 0.0002582 p.u. for PID [27]).

Table 5 Comparison of settling time, peak overshoot, and peak undershoot for ΔPtie.

Table 5Parameter	Controller	Scenario 1	
Settling Time (s)	Peak Overshoot (p.u.)	Peak Undershoot (p.u.)	
ΔPtie	2DOF-PID-TD (Proposed)	24.0845	0.000008183	0.00001951	
2DOF-PID [30]	28.7657	0.000006469	0.00004729	
FOPID [29]	32.4442	0.00003844	0.0002459	
TID [28]	25.1742	0.00012943	0.00006021	
PID [27]	25.5629	0.0001845	0.0002582	

Finally, in Fig. 7, we analyze the convergence curves for all five controllers in scenario 1. The 2DOF-PID-TD controller stands out, demonstrating superior performance in managing ITAE. It exhibits the fastest convergence characteristics, significantly outperforming other controllers.Fig. 7 Convergence curves of the controllers for the scenario 1.

Fig. 7

The objective function (ITAE) confirms the superiority of the 2DOF-PID-TD controller. From Table 6, the best ITAE for the 2DOF-PID-TD controller is 0.01443844, whereas for other controllers.• 2DOF-PID [30]: 0.039021077

• FOPID [29]: 0.052369866

• TID [28]: 0.080131451

• PID [27]: 0.228159714

Table 6 Objective function (ITAE) of the controllers for scenario 1.

Table 6Optimizer	Controller	Best ITAE	
AGTO	2DOF-PID-TD (Proposed)	0.01443844	
2DOF-PID [30]	0.039021077	
FOPID [29]	0.052369866	
TID [28]	0.080131451	
PID [27]	0.228159714	

In summary, the 2DOF-PID-TD controller excels in managing frequency, tie-line power, and overall system convergence.

4.2 Scenario 2

In this scenario, random Step Load Perturbation (SLP) is applied to both areas as disturbances to analyze the performance of the controllers. Using the optimized parameters of the controllers, we generate frequency deviation curves for both areas and tie line power deviation curves, which are depicted in Fig. 9 (a), Fig. 9 (b), Fig. 9 (c)(a–c), respectively.

Fig. 8 shows the random SLP (ΔPL) in per unit (p.u.) versus time in seconds. The SLP changes from 0 to 0.1 p.u. at 1-s intervals from 1 to 20 s, then remains at 0.2 p.u. from 20 to 60 s. Subsequently, it drops to 0.05 p.u. at 60 s and remains there until 105 s. Further, it decreases to −0.1 p.u. at 105 s and remains constant until 140 s. Finally, it returns to 0 p.u. at 140 s and remains there until 160 s.Fig. 8 Random SLP pattern for both areas.

Fig. 8

Fig. 9 (a) FD in area 1 for scenario 2.

Fig. 9 (a)

Fig. 9 (b) FD in area 2 for scenario 2.

Fig. 9 (b)

Fig. 9 (c) Tie-line Power Deviation for random SLP.

Fig. 9 (c)

In Fig. 9 (a), the Frequency Deviations (FD) in area 1 (Δf1) for each controller are depicted. These curves are generated based on the random Step Load Perturbation (SLP) pattern shown in Fig. 8. Notably, the 2DOF-PID-TD controller exhibits significantly better frequency control capability compared to the other controllers during the 1–20 s timeline. Table 7 provides further insights: the Settling Time, Peak Undershoot, and Peak Overshoot of the FD curve for the 2DOF-PID-TD controller during this period are 1.5527 s, 0.0066 Hz, and 0 Hz, respectively. These values represent the fastest Settling Time and the smallest Peak Undershoot and Overshoot within this timeline.Table 7 Comparison of Settling Time, Peak Undershoot, and Peak Overshoot for FD in area 1.

Table 7Timeline (s)	Controller	Δf1	
Settling Time (s)	Peak Undershoot (Hz)
	Peak Overshoot (Hz)	
1–20	2DOF-PID-TD (Proposed)	1.5527	0.0066	0	
2DOF-PID [30]	2.5793	0.0119	0.0002172	
FOPID [29]	5.9854	0.0113	0.0027	
TID [28]	4.2726	0.0329	0.0157	
PID [27]	6.45	0.0856	0.0244	
20–60	2DOF-PID-TD (Proposed)	20.3868	0.0174	0.0012	
2DOF-PID [30]	20.631	0.01674	0.000149	
FOPID [29]	23.5413	0.0456	0.0132	
TID [28]	23.355	0.0498	0.0189	
PID [27]	34.6461	0.0759	0.0353	
60–105	2DOF-PID-TD (Proposed)	60.5004	0.0118	0.0031	
2DOF-PID [30]	60.5877	0.0134	0.0013	
FOPID [29]	61.111	0.0361	0.0033	
TID [28]	64.0697	0.1044	0.035	
PID [27]	64.6494	0.2009	0.0856	
105–140	2DOF-PID-TD (Proposed)	105.384	0.0038	0.0109	
2DOF-PID [30]	105.5875	0.0054	0.0136	
FOPID [29]	107.9057	0.0064	0.0139	
TID [28]	108.0373	0.0253	0.0475	
PID [27]	111.3493	0.0278	0.0653	
140–160	2DOF-PID-TD (Proposed)	140.2912	0.0021	0.00439	
2DOF-PID [30]	140.3101	0.0079	0.00984	
FOPID [29]	140.8982	0.0044	0.0173	
TID [28]	142.4582	0.0322	0.0379	
PID [27]	145.9094	0.0474	0.1168	

The 2DOF-PID-TD controller exhibits the fastest settling capability for the Frequency Deviation (FD) curve in area 1 during 20–60 s timeline. Specifically, the Settling Time for the 2DOF-PID-TD controller is 20.3868 s, while the Settling Times for the 2DOF-PID [30], FOPID [29], TID [28], and PID [27] controllers are 20.631, 23.5413, 23.355, and 34.6461 s, respectively.

In the 60–105 s timeline, the 2DOF-PID-TD controller achieves a Settling Time of 60.5004 s and a Peak Undershoot of 0.0118 Hz. These values represent the fastest Settling Time and smallest Peak Undershoot within this timeline.

The 2DOF-PID-TD controller achieves the fastest Settling Times of 105.384 s and 140.2912 s in the 105–140 s and 140–160 s timelines, respectively. Additionally, it exhibits the smallest Peak Undershoot and Peak Overshoot of the Δf1 curve in both timelines. Specifically, the values are 0.0038 Hz and 0.0109 Hz in the 105–140 s timeline, and 0.0021 Hz and 0.00439 Hz in the 140–160 s timeline.

Fig. 9 (b) depicts the FD curves for each controller in area 2. Notably, the FD curve for the 2DOF-PID-TD controller settles down much faster than the other controllers across all timelines. This observation can be verified using Table 8.Table 8 Comparison of Settling Time, Peak Undershoot, and Peak Overshoot for FD in area 2.

Table 8Timeline (s)	Controller	Δf2	
Settling Time (s)	Peak Undershoot (Hz)	Peak Overshoot (Hz)	
1–20	2DOF-PID-TD (Proposed)	1.54	0.0086	0	
2DOF-PID [30]	2.2209	0.0081	0.0001629	
FOPID [29]	3.7797	0.0175	0.0028	
TID [28]	4.1605	0.0401	0.0158	
PID [27]	6.2301	0.0614	0.038	
20–60	2DOF-PID-TD (Proposed)	20.5994	0.0129	0	
2DOF-PID [30]	24.6959	0.0057	0.0003878	
FOPID [29]	21.1849	0.0458	0.0071	
TID [28]	23.256	0.0669	0.0231	
PID [27]	25.3459	0.0626	0.0382	
60–105	2DOF-PID-TD (Proposed)	60.1566	0.0093	0.0024	
2DOF-PID [30]	60.4365	0.0143	0.0026	
FOPID [29]	61.2226	0.0363	0.0043	
TID [28]	65.096	0.1007	0.0136	
PID [27]	64.464	0.2096	0.0642	
105–140	2DOF-PID-TD (Proposed)	105.844	0.0050	0.014	
2DOF-PID [30]	105.9815	0.0057	0.0151	
FOPID [29]	107.2403	0.0123	0.0278	
TID [28]	108.2425	0.0235	0.053	
PID [27]	113.2481	0.0249	0.0669	
140–160	2DOF-PID-TD (Proposed)	140.453	0.0043	0.0062	
2DOF-PID [30]	140.7961	0.0021	0.0068	
FOPID [29]	140.7807	0.0092	0.0231	
TID [28]	142.427	0.0325	0.0376	
PID [27]	146.3522	0.0428	0.1032	

Table 8 reveals that the 2DOF-PID-TD controller achieves the fastest Settling Times in various timelines: 1.54, 20.5994, 60.1566, 105.844, and 140.453 s for the 1–20, 20–60, 60–105, 105–140, and 140–160 s intervals, respectively. Furthermore, the 2DOF-PID-TD controller exhibits the smallest Peak Overshoots (0 Hz, 0 Hz, 0.0024 Hz, 0.014 Hz, and 0.0062 Hz) in the respective timelines. Additionally, it generates the smallest Peak Undershoots for the Δf2 curve in the 60–105 s and 105–140 s timelines, with values of 0.0093 Hz and 0.0050 Hz, respectively.

In Fig. 9 (c), Tie-line Power Deviations (ΔPtie) for each controller are depicted. These deviations are generated based on the random Step Load Perturbation (SLP) pattern. Notably, the 2DOF-PID-TD controller exhibits significantly better ΔPtie control capability compared to the other controllers. Specifically, the ΔPtie curve for the 2DOF-PID-TD controller settles down more quickly in most of the timelines. This observation can be further verified by referring to Table 9.Table 9 Comparison of settling time, peak undershoot, and peak overshoot for ΔPtie.

Table 9Timeline (s)	Controller	ΔPtie	
Settling Time (s)	Peak Undershoot (p.u.)	Peak Overshoot (p.u.)	
1–20	2DOF-PID-TD (Proposed)	10.92	0.000004486	0.000003439	
2DOF-PID [30]	18.13	0.000023150	0.000002399	
FOPID [29]	11.01	0.00009279	0.000077810	
TID [28]	15.04	0.00004259	0.0001658	
PID [27]	11.1659	0.0007813	0.0003379	
20–60	2DOF-PID-TD (Proposed)	25.26	0.000006774	0.00003784	
2DOF-PID [30]	27.3945	0.00009494	0.00002867	
FOPID [29]	34.6558	0.0001574	0.000283	
TID [28]	28.53	0.00006436	0.000284	
PID [27]	38.9532	0.0006578	0.0002729	
60–105	2DOF-PID-TD (Proposed)	65.2852	0.00000917	0.000000705	
2DOF-PID [30]	102.6973	0.000004634	0.000005048	
FOPID [29]	67.7589	0.00001635	0.00002239	
TID [28]	68.2835	0.0001071	0.0007353	
PID [27]	73.1319	0.0001122	0.0003661	
105–140	2DOF-PID-TD (Proposed)	113.357	0.00002683	0.00001037	
2DOF-PID [30]	120.4288	0.00000416	0.00001333	
FOPID [29]	124.5053	0.00009709	0.00008022	
TID [28]	114.2	0.0001945	0.00003686	
PID [27]	122.0494	0.0001075	0.0005329	
140–160	2DOF-PID-TD (Proposed)	142.19	0.000003209	0.000003719	
2DOF-PID [30]	143.4582	0.00001283	0.000009208	
FOPID [29]	143.1627	0.00002714	0.00001102	
TID [28]	148.5715	0.000006607	0.000007775	
PID [27]	148.6901	0.0002932	0.0006445	

From Tables 9 and it becomes evident that the 2DOF-PID-TD controller provides the fastest Settling Times across the entire timeline. These Settling Times are as follows.• 10.92 s in the 1–20 s timeline

• 25.26 s in the 20–60 s timeline

• 65.2852 s in the 60–105 s timeline

• 113.357 s in the 105–140 s timeline

• 142.19 s in the 140–160 s timeline

2DOF-PID-TD controller generates the smallest Peak Undershoots for the ΔPtie curve in the 1–20 s, 20–60 s, and 140–160 s timelines, with values of 0.000004486 p.u., 0.000006774 p.u., and 0.000003209 p.u., respectively. Additionally, it exhibits the smallest Peak Overshoots (0.000000705 p.u. and 0.00001037 p.u.) in the 60–105 s and 105–140 s timelines, respectively.

4.3 Scenario 3

We perform sensitivity analysis by varying the power system parameters (Tps and Kps), which directly impact the operating load conditions [5]. These parameters are deviated from the nominal conditions (illustrated in Appendix A) in the range of +50 % to −50 %, with a step size of 25 %. The optimum gains of the novel 2DOF-PID-TD controller and the disturbances (SLPs) in both areas remain the same as in scenario 1. This consistency allows us to assess the robustness of the novel controller against parametric uncertainties.

Table 10 and Fig. 10 (a), Fig. 10 (b) and (c) reveal that, despite the changes in loading conditions, the deviations in Integral of Time-multiplied Absolute Error (ITAE) [45] and settling times for the Δf1, Δf2, and ΔPtie curves are almost negligible compared to the nominal values.Table 10 Sensitivity analysis due to the parametric uncertainties.

Table 10Conditions	Changes	ITAE	Settling time (s)	
Δf1	Δf2	ΔPtie	
Nominal	No change	0.01443844	1.93076	2.41039	24.0845	
Rotating mass & load (Kps & Tps)	+50 %	0.01444079	1.93087	2.41049	24.0998	
+25 %	0.01443927	1.93082	2.41042	24.0954	
−25 %	0.01443977	1.93085	2.41066	24.0995	
−50 %	0.01444003	1.93107	2.41099	24.0996	

Fig. 10 (a) Frequency deviation in area 1.

Fig. 10 (a)

Fig. 10 (b) Frequency deviation in area 2.

Fig. 10 (b)

Fig. 10 (c) Tie-line power deviation, due to the parametric uncertainties in loading condition.

Fig. 10 (c)

4.4 Scenario 4

Governor Dead Band (GDB) nonlinearity is incorporated into the system to further demonstrate the effectiveness of the novel 2DOF-PID-TD controller in managing the system response. The GDB causes the system to oscillate and due to this nonlinear situation, the system tends to generate a continuous sinusoidal oscillation [5]. The transfer function of governor with GDB for the thermal unit can be written as [5],(25) GsgGDB(s)=N1+s(N2ω0)1+sTsg

Here, N1=0.8, N2=−0.2, and ω0=π. These coefficients represent Fourier components.

Fig. 11 represents the system model with GDB nonlinearity. All nominal system parameter values are detailed in Appendix A. In this scenario, 0.1 per unit (p.u.) Step Load Perturbations (SLPs) are applied to both areas, and Artificial Gorilla Troops Optimizer (AGTO) [40] is used to optimize the controllers with Integral of Time-multiplied Absolute Error (ITAE) [45] objective function.Fig. 11 System model with GDB nonlinearity.

Fig. 11

Table 11 clearly indicates that the 2DOF-PID-TD controller achieves the lowest ITAE [45] value (0.006666656), outperforming the other controllers. Specifically, the 2DOF-PID [30], FOPID [29], TID [28], and PID [27] controllers have ITAE [45] values of 0.061275044, 0.061517555, 0.770123756, and 1.618335018, respectively.Table 11 Objective function (ITAE) of the controllers for scenario 4.

Table 11Optimizer	Controller	Best ITAE	
Artificial Gorilla Troops Optimizer (AGTO) [40]	2DOF-PID-TD (Proposed)	0.006666656	
2DOF-PID [30]	0.061275044	
FOPID [29]	0.061517555	
TID [28]	0.770123756	
PID [27]	1.618335018	

Fig. 12 (a), Fig. 12 (b) and 13, along with Table 12 (b), Table 12 (a), reinforce the superiority of the 2DOF-PID-TD controller in managing frequency deviation and tie line power deviation under nonlinear conditions. Notably.• The Δf1 and Δf2 curves exhibit faster settling times (2.409 s and 2.467 s, respectively) compared to the other controllers (as verified in Table 12 (a)).

• For the ΔPtie curve, the 2DOF-PID-TD controller showcases the best system response among all five controllers: settling time of 11.784 s, peak undershoot of 0.0000343 p.u., and peak overshoot of 0.000013 p.u (as justified by Table 12 (b)).

Fig. 12 (a) Frequency deviation in area 1.

Fig. 12 (a)

Fig. 12 (b) Frequency deviation in area 2, for scenario 4.

Fig. 12 (b)

Fig. 13 ΔPtie for the scenario 4.

Fig. 13

Table 12 (a) Comparison of settling time, peak undershoot, and peak overshoot for Δf1 and Δf2.

Table 12 (a)Parameters	Controller	Scenario 4	
Settling Time (s)	Peak Undershoot (Hz)	Peak Overshoot (Hz)	
Δf1	2DOF-PID-TD (Proposed)	2.409	0.008	0.00067	
2DOF-PID [30]	4.336	0.006	0.00038	
FOPID [29]	3.726	0.033	0.00198	
TID [28]	7.228	0.149	0.03676	
PID [27]	6.775	0.212	0.04387	
Δf2	2DOF-PID-TD (Proposed)	2.467	0.012	0.001	
2DOF-PID [30]	8.712	0.005	0.00037	
FOPID [29]	5.076	0.037	0.00361	
TID [28]	6.576	0.153	0.03789	
PID [27]	7.987	0.204	0.04709	

Table 12 (b) Comparison of settling time, peak undershoot, and peak overshoot for ΔPtie.

Table 12 (b)Parameter	Controller	Scenario 4	
Settling Time (s)	Peak Undershoot (p.u.)	Peak Overshoot (p.u.)	
ΔPtie	2DOF-PID-TD (Proposed)	11.784	0.0000343	0.000013	
2DOF-PID [30]	15.536	0.0000607	0.000034	
FOPID [29]	30.519	0.0000662	0.000085	
TID [28]	30.035	0.0003862	0.000159	
PID [27]	30.677	0.0008901	0.000113	

The optimal gains of the controllers for this scenario are illustrated in Table 13 (b), Table 13 (a) and Table 14 (b), Table 14 (a).Table 13 (a) Optimal gains of the controllers for thermal unit (area 1) in scenario 4.

Table 13 (a)AGTO Controllers	Thermal unit (Area 1)	
Kp1	Ki1	λ1	Kd1	μ1	Pw1	Dw1	Tt1	1n1	Td1	
2DOF-PID-TD (Proposed)	45	75		35		3.242	1	75	0.033	20	
2DOF-PID [30]	25	60		35		4.5	1				
FOPID [29]	45	75	1	35	0.8316						
TID [28]		4.6591						5.3629		4.8202	
PID [27]	2.3847	3.2894		2.2456							

Table 13 (b) Optimal gains of the controllers for hydro unit (area 1) in scenario 4.

Table 13 (b)AGTO Controllers	Hydro unit (Area 1)	
Kp1	Ki1	λ1	Kd1	μ1	Pw1	Dw1	Tt1	1n1	Td1	
2DOF-PID-TD (Proposed)	30	1		45		5	2.7098	20	0.5	35	
2DOF-PID [30]	25	40		25		4	5				
FOPID [29]	30	1	1	32.475	0.75						
TID [28]		1.0746						1		1.03742	
PID [27]	2.4905	1.7219		2.524							

Table 14 (a) Optimal gains of the controllers for thermal unit (area 2) in scenario 4.

Table 14 (a)AGTO Controllers	Thermal unit (Area 2)	
Kp2	Ki2	λ2	Kd2	μ2	Pw2	Dw2	Tt2	1n2	Td2	
2DOF-PID-TD (Proposed)	50	75		25		5	5	1	0.33	1	
2DOF-PID [30]	26	60		35		4.5	5				
FOPID [29]	50	75	1	25	1						
TID [28]		4.62						5.532		4.71	
PID [27]	2.8696	3.645		2.424							

Table 14 (b) Optimal gains of the controllers for hydro unit (area 2) in scenario 4.

Table 14 (b)AGTO Controllers	Hydro unit (Area 2)	
Kp2	Ki2	λ2	Kd2	μ2	Pw2	Dw2	Tt2	1n2	Td2	
2DOF-PID-TD (Proposed)	1	1		25		5	5	1	0.376	55	
2DOF-PID [30]	12	60		40		4	5				
FOPID [29]	45	40	0.75	25	0.75						
TID [28]		1.874						5.2332		1	
PID [27]	2.566	1		2.207							

5 Conclusion

This study introduced a novel hybrid controller that combined a 2DOF-PID and a TD controller for the load frequency control of a two-area four-source interconnected thermal-hydro power system. The proposed 2DOF-PID-TD controller was optimally designed using the ITAE objective function and AGTO algorithm. The authors compared its performance with four traditional LFC controllers: 2DOF-PID, FOPID, TID, and PID controllers. Results from scenarios 1 and 2 demonstrated that the novel 2DOF-PID-TD controller excelled in minimizing frequency deviations in both areas and tie-line power deviations, regardless of fixed or random step load perturbations. Under scenarios 3 and 4, when the system faced parametric uncertainties and GDB nonlinearities, the proposed 2DOF-PID-TD controller showcased superior stability and robustness compared to the other controllers. While this study focuses on a two-area power system, future work could extend the control scheme to include additional areas. Furthermore, integrating Electric Vehicles (EVs) and other Renewable Energy Sources (RES), such as wind and solar power, into the existing model would allow for analyzing system responses. Additionally, future studies could explore controller comparisons using MVO, EGBO, MPA, and FDA algorithms. Overall, this study presents a promising and effective solution to the load frequency control problem using a novel hybrid controller.

Funding

This research did not receive any specific funding.

Data availability statement

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CRediT authorship contribution statement

Md. Nazmush Shakib Shahi: Writing – review & editing, Visualization, Validation, Software, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Nabil Anan Orka: Writing – review & editing, Writing – original draft, Data curation. Ashik Ahmed: Writing – review & editing, Validation, Supervision.

Declaration of competing interest

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

Appendices Appendix A [5,48,49].Symbol	Name	Value	
Tsg	Time constant of governor in thermal plant	0.4 s	
Tt	Time constant of steam turbine	0.5 s	
Trs	Droop resetting time	4.9 s	
Tgh	Time constant for hydro turbine governor	0.2 s	
Trh	Droop time constant	28.749 s	
Tw	Water starting time	1.1 s	
Tps	Time constant of rotating mass & load	20 s	
Kps	Gain of rotating mass & load	100 Hz/p.u.MW	
RT	Governor speed regulation constant for thermal plant	3 Hz/p.u.MW	
RH	Governor speed regulation constant for hydro plant	3 Hz/p.u.MW	
B	Frequency bias coefficient	0.425 p.u.MW/Hz	
a12	Area size ratio	−1	
KT	Participation factor of thermal unit	0.7127	
KH	Participation factor of hydro unit	0.2873	
T12	Synchronization coefficient	0.00796	

Appendix B Ranges of the controller parameters which are used for simulation are as follows.Controller parameters	Kp	Ki	Kd	Pw	Dw	Tt	n	Td	
Values	min	1	1	1	1	1	1	2	1	
max	50	75	45	5	5	75	3	55	

Meaning of the controller parameters and full form of the controller acronyms are illustrated below.Controller Parameters/Acronyms	Meaning/Full form	
Ki	Integral gain.	
Kp	Proportional gain.	
λ	Order of the Integral controller.	
Kd	Derivative gain.	
μ	Order of the Derivative controller.	
Pw	Proportional set-point weight.	
Dw	Derivative set-point weight.	
Tt	Tilt gain of the Tilt controller.	
n	Order of the Tilt controller.	
Td	Derivative gain of the Tilt-Derivative controller.	
Δf	Frequency deviation.	
ΔPtie	Tie line power deviation.	
ACE	Area Control Error.	
SLP	Step Load Perturbation.	
2DOF-PID-TD	2 Degree Of Freedom Proportional Integral Derivative Tilt Derivative.	
2DOF-PID	2 Degree Of Freedom Proportional Integral Derivative.	
FOPID	Fractional Order Proportional Integral Derivative.	
TID	Tilt Integral Derivative.	
PID	Proportional Integral Derivative.	
ITAE	Integral of Time multiplied Absolute Error.	
IAE	Integral of Absolute Error	
ISE	Integral of Squared Error	
ITSE	Integral of Time multiplied Squared Error
==== Refs
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