Cascaded Fractional-Order Controller-Based Load Frequency Regulation for Diverse Multigeneration Sources Incorporated with Nuclear Power Plant

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Introduction
The modern interconnected power system (IPS) is designed with several control regions linked by tie lines.The daily energy demand is increasing as the world's population grows.Inconsistencies in load demand cause irregularities in both frequency and tie line power flow within the IPS's control zones.In order to overcome this challenge within the existing power system framework, load frequency regu-lators play an important role in balancing power consumption and generation [1,2].Both primary and secondary mechanisms play an important role in controlling frequency fluctuations.A governor uses a control method to alter the speed and frequency of basic control processes [3].However, after considerable deviations, supplementary control is required to stabilize frequency.The secondary controller, therefore, is of greater importance to the overall performance of the system [4].
1.1.Literature Survey.The principal aim of load frequency control (LFC)/automatic generation control (AGC) is to regulate the power deviation between adjacent regions linked by tie lines within defined boundaries and maintain the system frequency within predetermined limits [5].In the early stages of control scheme development for AGC, conventional controllers such as integral (I), proportional integral (PI), and PI derivative (D) were utilized.Researchers preferred these controllers due to their adaptability and simplicity; nevertheless, the outcomes frequently failed to meet expectations in terms of performance [6,7].For instance, in [8] proposed PID controller optimized with ant colony optimization (ACO) for single-area nuclear power system.The authors in reference [9] proposed PI controller to address the LFC problem in a single-area scenario incorporated with renewable energy resources.Researchers extended their investigations to tackle the LFC challenges arising in interconnected PSs.The researchers in [10] introduced an ACO algorithm for the purpose of optimizing the PID controller of a two-area interconnected nonreheat thermal power system with nonlinearity caused by governor dead band (GDB).Jagatheesan et al. in [11] proposed PI-based fuzzy logic controller for the frequency regulation of interconnected hydropower system incorporated with GDB as well as GRC nonlinearities.Apart from the conventional power systems, various researchers have worked in a deregulated power system by designing various types of controllers to regulate the frequency [12,13].
Various control mechanisms have been introduced in PSs to address the issue of load frequency management.A few examples include model predictive control [14], robust sliding mode controllers [15], artificial intelligence-based LFC approach [16], linear matrix inequality [17], resilient control methodologies [18], data-driven controllers [19], fuzzy logic control (FLC) [20,21], and robust virtual inertia control [22].To control the frequency of connected PSs, the classic PID controller has been the principal focus of academic research because of its ease of use and low cost.Even yet, the PID controller has a difficult time adapting to nonlinear properties and interruptions in the system by trial and error.To determine the best PID values, a considerable amount of effort has been expended.Due to advances in computing power that enable simulation and accurate implementation of the fractional-order controller, the use of fractional-based controllers has recently drawn more attention in power engineering issues, specifically for power optimization control.The FOPID and their modification have been utilized in [23][24][25] as a secondary LFC to improve the frequency stability of the interconnected two region power systems.Numerous cascaded controller forms [26][27][28] have been used to improve frequency persistence in PSs.Recent research [29,30] has investigated a second method for studying LFC that emphasizes the integration of two controllers.The frequency variations of a two-area coupled PS can be diminished using a PIDD2 controller framework, which was also proposed by authors in [31].Moreover, references [32,33] recommended the integral-(I-) tilt derivative (TD) and fractional-order (FO) I-TD controller for system frequency adaptation, respectively.The frequency effectiveness of the ID-T controller is superior to that of the TID controller [34].
There is numerous research in the literature that use fractional calculus to solve classical PID controller [35,36].According to previous research, FOPID and PIDD2 controllers can surpass PID controllers in numerous engineering applications in addition to the LFC system.To improve the control performance of LFC systems, an upgraded controller must be developed.As a result, for the first time, we proposed a cascaded based FOI-FOPIDD2 controller to enhance LFC transient and dynamic performance.The suggested controller can be viewed as a hybrid of fractional calculus and PIDD2.
This study is aimed at obtaining the right knobs for the recommended controller by developing a new metaheuristic algorithm called the squid game optimizer (SGO) technique, which is inspired by the fundamentals of a traditional Korean sport.During the squid game, attackers strive to reach their target, whereas players attempt to eradicate one another.It is typically acted on broad, open grounds with no predetermined extent and dimension limitations.Based on historical records, the playing area for this sport is commonly devised in a design resembling a squid and appears to be approximately half the dimensions of a standard basketball court.First, the numerical model of this approach is built by selecting the best nominee solutions and selecting an initialization method at random.In two groups, solution candidates move among defensive players, initiating a fight that is replicated by random movement towards defensive players.The position update procedure is completed, and the current position vectors are formed by judging the winner declarations of the players on opposite sides.These states are estimated based on the cost function.Twenty-five (25) unrestricted mathematical assessment functions are applied to examine the performance of the presented SGO algorithm, along with six others that regularly used metaheuristics for assessment 2 International Journal of Energy Research [53].Furthermore, the suggested SGO's capability is evaluated using advanced real-real-life challenges on the latest CEC, such as CEC 2020, with the SGO demonstrating remarkable results in haggling with these inspiring optimization challenges [53].
In addition, the primary conclusion drawn from the available literature is that LFC methods, such as FLC, Hinfinite approaches, and model predictive control, which rely on the controller designer's imagination to achieve the desired performance, accomplish that despite numerous design flaws and a lengthy setup process.In addition, conventional PD/PI/PID controllers struggle with responding to uncertainty in the system.The impact of system nonlinearities and boundary fluctuations on robustness evaluations has been underexplored in several earlier works.Most previous evaluations also ignored the significant integration of energy storage devices (ESD) without adjusting system parameters due to the included system nonlinearities/uncertainties and immediate demand variations.The list of nomenclature and notations is shown in Table 1.

Contribution of the
Paper.This research proposes a novel FOI-FOPIDD2 controller that improves system frequency steadiness as accounting for perturbations affected by diverse power sources.In accord with the SGO, the settings for the suggested FOI-FOPIDD2 controller have also been established to guarantee frequency and system constancy under atypical conditions.The main contribution of the paper is summarized as follows, in contrast to earlier studies on related subjects: (i) Employing a reliable FOI-FOPIDD2 controller to enhance frequency reliability for dual zone coupled

Modelling of Hybrid Power System
In this portion, a dual area PS that is combined with reheat thermal, nuclear, gas, hydro, capacitor energy storage, and redox flow battery is mathematically modeled which is shown in Figure 1.The schematic diagram is shown in Figure 2. The distribution of all the included power generations between the two regions is assumed to be equal.The PS material from [24,41,54] is used to construct the system in Simulink/MATLAB and is available in Table 2.For the thermal system, the GRC of 10%/min is considered for both rising and dropping rates.Growing generation is utilized into consideration for the hydro portion at a typical GRC of 270%/min, while decreasing generation is considered into action at a regular GRC of

Gas governor
Valve positioner Fuel system and combusture

Gas turbine
Gas participation Factor

Hydro participation factor
Termal participation factor 1 R h

Hydro governor
Penstock Hydro turbine Low Pressure Turbine-2 with Reheater Termal participation factor Hydro participation factor Area-2  International Journal of Energy Research 360%/min [12].The mathematical formulation for GDB is described below [55].

FOI-FOPIDD2
where N 1 = 0 8 and Furthermore, a time delay (TD) of 2 seconds is inserted after the controller design to make the arrangement more realistic, which is given in equation ( 3).Similarly, a boiler dynamic is added as a nonlinearity into the thermal reheat system, and their schematic type is shown in Figure 3.The mathematical representation for the proposed boiler dynamics is also given in equation ( 4) [12,55].The area control error (ACE) for multigeneration unit is given in equation ( 5).
2.1.Modelling of Reheat Thermal, Hydro, Nuclear, and Gas Power Plants.Conventional power systems comprised of reheat thermal (with submodel of governor/turbine/reheater) and hydropower generation (with submodel of governor/pen-stock/droop compensation).The mathematical descriptions for reheating thermal structure with their subsequent submodel including (governor/turbine/reheater) are shown in the below equations, respectively [24,28].
The transfer functions (TFs) of the gas generation system, which includes the fuel system with combustor (G GC s ), gas turbine (G GT s ), (GFC), gas governor (G GG s ), and valve positioner (G GV s ) are expressed by [49,57] 2. Capacitive Energy Storage (CES) Modeling.Due to its ability to rapidly charge and discharge with a substantial amount of power, capacitive energy storage devices are growing in favor of modern power systems [58].The benefit of CES is that it responds to a boost in demand by producing an abundance of electricity.It is affordable and easy to use.It has a long service life and does not perform worse for it.A supercapacitor [59] serves as the CES system's main energy storage factor.Capacitor plates are utilized for retaining energy in the form of static charge.CES emits energy back into the grid when demand is at where T 1 -T 4 stand for the two-stage phase compensation blocks' time constants.Both analyzed PSs have incorporated CES scoring into their repertoires.The input control signal for each CES unit is the phase shift in frequency between two areas of the PS.The TF model of CES is shown in Figure 5.

Modelling of Redox Flow Battery (RFB)
. RFB has become prevalent as a quick-rechargeable battery in recent times.An electrochemical transformation process is utilized in the redox progression, and a dual converter manages the rectifier and inverter functions.The benefit of the RFB is its rapid storage operation, which minimizes the impact on the environment by addressing the governor response delay and eliminating oscillations.RFB is made up of electrolyte, pumps, pipes, tanks, flow cells, and other modules that store energy during charging and release it under load demand [60,61].RFB can function at room temperature and is suitable for power ratings between kW and MW with storage durations of 2 to 10 hours [62].Its control response time to frequency variations is particularly quick [63].The key features that set RFB apart as an excellent ESD are its flexible power capability, low environmental effect, high efficiency, and versatility.Equation (14) shows the RBF transfer function [60,61].

Squid Game Optimize
The proposed SGO algorithm is offered as a unique metaheuristic approach prompted by the basic rules of a tradi-tional Korean sport.During the squid game, attackers strive to reach their target, whereas players attempt to eradicate one another.It is typically acted on broad, open grounds with no predetermined extent and dimension limitations.Based on historical records, the playing area for this sport is commonly devised in a design resembling a squid and appears to be approximately half the dimensions of a standard basketball court.First, the numerical model of this approach is built by selecting the best nominee solutions and selecting an initialization method at random.In two groups, solution candidates move among defensive players, initiating a fight that is replicated by random movement towards defensive players.The position update procedure is completed, and the current position vectors are formed by judging the winner declarations of the players on opposite sides.These states are estimated based on the cost function.Twenty-five (25) unrestricted mathematical assessment functions are applied to examine the performance of the presented SGO algorithm, along with six others that regularly used metaheuristics for assessment [53].Furthermore, the suggested SGO's capability is evaluated using advanced real-real-life challenges on the latest CEC, such as CEC 2020, with the SGO demonstrating remarkable results in haggling with these inspiring optimization challenges [53].SGO algorithms consist of the following steps [53].
3.1.Mathematical Formulation.In this section, the numerical description of the SGO method is considered employing the squid game strategy.The initialization technique is executed as follows in the initial step, with the search space treated as a specific area of the playing field and  7 International Journal of Energy Research the prospective contenders (X i ) supposed to be players [53]: Here, "n" signifies the overall count of players within the field, which corresponds to the search space."d" denotes the dimensionality of the problem at hand.The initial position of the ith candidate is influenced by the j th decision variable, represented as x j i .The upper and lower bounds of the jth variable are termed as x j i,Max and x j i,Min , respectively.The random number, denoted as "rand," follows a uniform distribution between 0 and 1.In the second phase of the algorithm, players are categorized into two equal-sized groups known as defensives (Def) and offensives (Off).Below is an algebraic depiction of these elements [53].
Here, "m" represents the total count of players within each game group.The ith offensive player is denoted as i , and the kth defensive player is represented as X Def i .Once the game starts, an offensive player maneuvers among the defensive players to initiate a confrontation.It is important to highlight that each attacking player is constrained to move and engage in battle using a single foot, whereas defensive players have the liberty to use both feet.In mathematical terms, these elements are represented as follows [53]: Here, "r 1 " and "r 2 " denote two random numbers within the range of [0, 1], signifying the ability of the offensive players."X Def r3 " is a random integer ranging from 1 to "m".X offNew1 i represents the position vector of the upcoming ith offensive player in the field, while "DG" stands for the defensive group.The subsequent step involves the evaluation of the objective function for each player, following a confrontation between the ith offensive player and a specific defensive player.The winning state (WS) of the players is then determined.If the offensive player emerges as the winner, based on the squid game's fundamental rules, they join the successful offensive group (SOG).The offensive player can use both feet for this purpose if the defensive player's winning state is lower than the offensive player's winning state.The mathematical representation of these aspects is articulated as [53] Defensive players are deemed the game's champions and are asked to join the SDG if their winning states are better than those of the offensive players.It is anticipated that the defensive players in this group will defend the bridge, the playground's pivotal feature.In preparation for starting a fresh fight, the thriving defensive players move among the attacking performers in the group.The mathematical appearance of these parts is given below [53].
The algorithm introduces an additional search loop, where offensive players within the successful offensive group (SOG) endeavor to navigate a bridge guarded by defensive players in the successful defensive group (SDG).This inclusion is aimed at intelligently adapting the exploration and exploitation phases of the proposed algorithm.To achieve this, a position-updating operation is executed for all offensive players in SOG, involving advancement towards the best-known solution candidate and a specific defensive player in SDG.This simulates the reward for an offensive player attempting to cross the bridge.The mathematical expression of these components is detailed as follows [53]:

Design of Cascaded Based Controllers and Expression of Fitness Function
4.1.Concept of Fractional Order.Fractional calculus is a field of mathematical analysis that expands traditional calculus to include integrators and differentiators with noninteger orders.These orders are represented as a D γ t , where "a" and "t" indicate the operation boundaries and γ is a real number (R).The formulas for fractional-order integrator and differentiator can be expressed as follows [64,65]: The field of fractional calculus is guided by three essential principles: the Riemann-Liouville (RL) principle, the Caputo principle, and the Grunwald-Letnikov (GL) principle.These principles are represented by equations ( 14)-( 16) correspondingly [64,65].

24
In this study, oustaloup-based recursive approximations are applied with a filter order of 5 and within the range of [10 −3 , 10 3 ] rad/s.The primary objective of the new design of the cascaded controller is to regulate and enhance the frequency response of a diverse power system coping with immediate load variations and variations.The controller has been recommended in both regions to reduce fluctuations in frequency and interconnected tie line power discrepancies between both regions.Conventional PID controllers are commonly utilized in manufacturing due to their straightforward design and efficient functioning.Like the standard PID structure, the PIDD2 assembly also incorporates a second-order derivative gain [66].The FOI-FOPIDD2 controller has not yet been utilized in a study, despite numerous approaches being tested to improve the control performance of LFC systems.Researchers have shown that PIDD2 and FOPIDD controllers outperform traditional PID controllers.Figure 7 displays the cascaded based FOI-FOPIDD2 controller that was established by fusing FOPIDD2 controller and fractionalorder integral controller.
Equations ( 25) and ( 26) describe the transfer function of the FOI and FOPIDD2 controllers.Similarly, equation (27) illustrates the connection between the system's output and the error signal.International Journal of Energy Research

ITSE objective function
where λ 1 , λ 2 , and μ are the integral-differentiator operators; N d and N dd denote the filter constants; and Kp, K d , K i1 , and K i2 indicate the proportional, derivative, and integral coefficients of the proposed controller.The FOI-FOPIDD2 controller gains have been established by reducing the cost function through the utilization of the SGO.The settling period is shortened, and high oscillations are swiftly suppressed using an ITSE-based cost function [25,33,37]:    12 International Journal of Energy Research  The FOI-FOPIDD2 controller gains are subject to the following restrictions.
The range of values for the parameters Kp, K i1 , K i2 , K d , N d , and N dd is from 0 to 10, while the values for λ 1 , λ 2 , and μ are from 0 to 2.

Results, Execution, and Discussion
This study uses a combination of several hybrid power sources coupled with redox flow battery (RBF) and capacitor energy storage to test the effectiveness and validity of a novel FOI-FOPIDD2 controller.To meet the LFC goal function, the proposed controller coefficients are tweaked with the squid game optimizer utilizing the MATLAB programming language and connected with the Simulink tool.The SGObased controller knobs for the specified case study are indicated in Table 3 following 80 iterations of the optimization techniques using the materials from Table 4.The statistical values foe various techniques are shown in Table 5.While employing the same alignment with the RFB system that uses the SGO technique, the suggested FOI-FOPIDD2 15 International Journal of Energy Research controller's robustness is compared to other regulators like PIDD2, PID, and FOPID.The load changes are fixed at 5% = 0 05 per unit, in all cases.Results from the examined multiarea IPS are rigorously examined in the subsequent case studies.

Case 1 (Analyses of Algorithm Performance
).The squid game algorithm was compared with various recent algorithms including JSO, PSO, GWO, and firefly algorithm to determine its efficacy in this scenario.Each algorithm response was measured in terms of area 2 (ΔF 2 ), tie line (ΔP tie ), and area 1 (ΔF 1 ) as shown in Figures 8(a)-8(c).In Table 6, the overall performance is contrasted for various approaches in respect of transient parameters like U sh (undershoot), O sh (overshoot), and Ts (settling time) for Δ P tie , ΔF 2 , and ΔF 1 .The SGO strategy offered quicker settling times than JSO, GWO, PSO, and FA-based optimization approaches in regions 1 and 2 and the linked tie line.The            , we can see that the system's response to RFB and CES unit effects is better than its response without RFB and CES component effects in respect of O sh , U sh , and Ts.Additionally, Table 8 highlights the phenomenal results obtained by combining RFB and CES in our proposed method.21 International Journal of Energy Research 5.4.Case 4 (Sensitivity and Stability Analysis).Sensitivity analysis has been employed to examine the sturdiness of the recommended SGO-FOI-FOPIDD2 controller.The system's stability may occasionally be negotiated if the proposed control scheme is incapable of accommodating system parameter fluctuations.Various metrics for parameters like T gr , T re , and T rh have been adjusted by approximately ±40% and then contrasted with their nominal parameter response to validate the robustness of the proposed controller.Figures 16 and Table 9 illustrate the dependability of the suggested controller when constraint uncertainty manifests, based on the results obtained when system parameter variations were employed.Figure 17 represents the bode diagram for the proposed FOI-FOPIDD2 controller which confirms the stability of the system due to their positive gain margins and phase margin values.In order to pretend real-time conditions, the performance of the SGO-FOI-FOPIDD2 controller is validated under varied load disturbances up to ±25% and ±50%, as depicted in Figure 18.Numerous parameters respond near to their supposed values, as shown in Table 9, indicating that the suggested SGO-FOI-FOPIDD2 controller provides reliable performance over a spectrum of approximately ±40% of the system's characteristics.Furthermore, the optimal values of the proposed controller have no need to reset controller for a wide range of parameters if employed with the real values at stated value.

Figure 1 :
Figure 1: Dynamic modelling of the proposed PS.

Figure 2 :
Figure 2: Schematic diagram of the proposed PS.

Figure 4 :
Figure 4: Transfer function model of nuclear power plant.

Figure 13 :
Figure 13: Power output for various generation units in area 1.

Figure 14 :Figure 15 :
Figure 14: Power output for various generation units in area 2.

Figure 17 :
Figure 17: Bode diagram for the proposed system.

Table 1 :
List of nomenclature and notations.

Table 5 :
Statistical parametric analysis for various techniques.

Table 4 :
Value of SGO parameters.

Table 3 :
Optimal values for the recommended methods.

Table 7 :
Comparison performance for case 2.

Table 6 :
Comparison performance for case 1.

Table 8 :
Comparison performance for case 3.

Table 9 :
Variations in PS parameters.