Enhancing Load Frequency Control of Interconnected Power System Using Hybrid PSO-AHA Optimizer

Abstract

The integration of nonconventional energy sources such as solar, wind, and fuel cells into electrical power networks introduces significant challenges in maintaining frequency stability and consistent tie-line power flows. These fluctuations can adversely affect the quality and reliability of power supplied to consumers. This paper addresses this issue by proposing a Proportional–Integral–Derivative (PID) controller optimized through a hybrid Particle Swarm Optimization–Artificial Hummingbird Algorithm (PSO-AHA) approach. The PID controller is tuned using the Integral Time Absolute Error (ITAE) as a fitness function to enhance control performance. The PSO-AHA-PID controller’s effectiveness is evaluated in two networks: a two-area thermal tie-line interconnected power system (IPS) and a one-area multi-source power network incorporating thermal, solar, wind, and fuel cell sources. Comparative analyses under various operational conditions, including parameter variations and load changes, demonstrate the superior performance of the PSO-AHA-PID controller over the conventional PSO-PID controller. Statistical results indicate that in the one-area multi-source network, the PSO-AHA-PID controller achieves a 76.6% reduction in overshoot, an 88.9% reduction in undershoot, and a 97.5% reduction in settling time compared to the PSO-PID controller. In the dual-area system, the PSO-AHA-PID controller reduces the overshoot by 75.2%, reduces the undershoot by 85.7%, and improves the fall time by 71.6%. These improvements provide a robust and reliable solution for enhancing the stability of interconnected power systems in the presence of diverse and variable energy sources.

1. Introduction

Current power systems are undergoing fast changes due to the extensive application of smart grid technologies as well as a range of nonconventional sources. Loads, distribution, transmission, and generation are components of an integrated power network. A tie line links several widely distributed service areas geographically in a large, linked power network. Modern power systems are projected to face more obstacles and difficulties as they keep growing in size in addition to complicacy [1,2]. To maintain the stability of system efficiency, power generation changes proportionately with consumer load. The constancy of any power unit in a network, or any connected systems, is impacted by sudden increases in power demand [3,4].

To maintain a planned power exchange between multiple areas, energy production in each area must be regulated. LFC methods in an interlinked power system (IPS) generate and also provide electrical power constantly as well as accurately while keeping to reasonable frequency controls. The load in a power system varies continuously. The difference between generating power and consumer demand causes the system’s frequency to shift in a highly unfavorable way, causing fluctuations and the equilibrium state to float. When this equilibrium varies, the frequency is restored to a different level. This results in severe electrical network instability in addition to serious harm to costly equipment. LFC control loop’s speed governor mechanism can be used to adjust the active power needs, which will control frequency fluctuation. Control is necessary to keep the equilibrium of active capabilities. Therefore, a control system is required to address the problem of unpredictable load alterations and to maintain frequency at desired levels. The precise tuning scheme design of the controller plays a major role in providing optimal control. In order to efficiently control system frequency for single-area multiple-source as well as two-area tie-line IPS, this study proposes to optimize PID gains using optimization approaches and PSO along with PSO-AHA [5,6,7,8,9,10]. MATLAB/Simulink 2022a was used to develop suggested power networks; by varying system parameters in both networks by ±75% for various scenarios and varying the load in distinct scenarios for double-area tie-line IPS, suggested networks with PIDs were tested. PID controller gains were tuned using PSO as well as PSO-AHA approaches, with ITAE being a fitness function. In suggested networks, frequency responses, along with tie-line power variations, were successfully optimized by a PID controller utilizing PSO and PSO-AHA techniques. Signal characteristics and frequency responses show that PSO-AHA-PID is superior to PSO-PID. The sections below discuss the results of the PID controller in addition to optimization techniques that we investigated and applied to the suggested networks.

1.1. Literature Review

The author of [11] solved the interconnected thermal power networks LFC problem by using the PSO technique to optimize the PID control’s parameters. When comparing HC in addition to the GA-optimized controller’s efficiency of suggested PSO-PID, we achieved superior results. In [12,13], the authors used the PID controller and also the ACO algorithm for the LFC of a self-sustaining nuclear power plant. A method of trial and error was applied in order to contrast responses. In [14], utilizing the ARA technique, the author tuned the PID controller’s parameters for double-area non-reheat multiple-area power systems (MAPSs). In [15,16], using wind farms along with a photovoltaic model, an LFC power network was designed. In [17], the frequency was adjusted using LFC techniques, along with multi-source electricity from wind, solar, and diesel-powered generators, all of which were regarded as parts of a power network with many sources. In [18], optimal PID parameter values for the LFC renewable power network were determined by employing the nonconventional quasi-oppositional dragonfly method. Long-term significant fluctuations in tie-line power interchange along with network frequency inside LFC power networks can harm equipment, cause operating insecurity, limit the life of connected devices, or even create energy system collapse. Using a differential evolution algorithm and an optimized parameter-based controller, an analysis of LFC within a multiple-source power network was carried out [19,20].

Stochastic PSO was used to optimize the gain value of the PI controller considered for a single-area energy network LFC. It intended to employ the PSO technique via BF-tuned PID controller to solve linked power network LFC, including thermal along with PV networks [21]. Grid-connected electrical power networks’ LFC was implemented utilizing the PSO-associated MARL approach [22]. In [23], the FPA technique was applied to address LFC problems with linked power grids by optimizing PID controller parameters, and by comparing the results of the suggested FPA technique with those of GA-PID along with PSO-PID controllers, the technique’s efficiency was verified. In [24], in order to solve the single-area energy network LFC crisis, the authors investigated a PSO-tuned PID controller. Using the PSO technique in addition to four different fitness functions, the PID was considered for LFC within an isolated energy network. In [4], for LFC, a FOFPID controller was designed. In this situation, the BBO approach was used to tune controller values. However, in a steady state, the integral term performed poorly in terms of oscillation as well as frequency variations. In [25,26], the author proposed a novel optimization technique to accomplish automatic LFC. The optimization method specifically controlled the frequency variation of a multi-source power network. However, the approach also led to slow convergence, and outcomes were not ideal.

The literature review clearly demonstrates how oscillations and power system performance are caused by unpredictable load demands inside a power-generating unit. This implies that the electrical system could experience LFC/AGC issues. These issues can be solved by employing various optimization approaches in different controllers to increase gains under different circumstances as well as requirements [4,11,12,15,17,18,19,21,22,23,24,25,27,28,29,30]. A review of the literature under study is given in

Table 1. Review of studied literature.

Optimization Approach/Secondary Controller	Energy Source	Work	Reference
PSO/PID	Thermal	Contrasted results GA in addition to HC	[11]
ARA/PID	Non-reheat thermal	Contrasted performance PSO, DE, JAYA	[14]
GA/PID	Thermal	Efficiency PI as well as PID with and without GRC	[31]
BESSO/PID	Hydro, gas, thermal	Examined controller performance along with contrasted finding with ordinary PID	[32]
DE/PID	Thermal, hydro, gas	Examined controller performance using I, PI, as well as PID	[19]
HBFOA/PID	Thermal, PV	PSO as well as BFO results investigated	[21]
GBO/PID	Gas, thermal, hydro, PV, wind	Results contrasted with GBO-I-PD, GBO-TID, and also GBO-I-P	[33]
ANN/PID	Distributed power sources (WTG, DEG, AE, FC)	Superiority of proposed technique achieved by employing GOA	[34]
PSO/PID	PV, nuclear, hydro, gas, thermal	Contrast PSO-PID along with ordinary PID	[24]
PSO/PID	Thermal, solar, wind	Efficiency analysis of standard I, PI, in addition to PID	[35]
PSO/PIDF	Thermal	Results compared with PIDF and PI controller	[36]
PSO/PID	Thermal, hydro, gas	Results compared with DE-PID and GA-PID	[37]
PSO-GA-ACO-CSA/PID	PV and wind	Performance of LFC evaluated under different cases by utilizing PSO, GA, ACO, and CSA	[38]
GWO/PID	Thermal, hydro, and nuclear	Results compared with GA, PSO and ACO algorithms	[13]
GTO/PI	Thermal and PV	Performance analyzed by changing the system parameters and load variations	[39]
GA/MPC	Wind, solar	Results compared with fuzzy-MPC, PSO-MPC, and conventional MPC	[16]
GA-PSO/PID	Thermal	Results compared with PSO-PID and GA-PID	[40]
PSO-DE/PID	Thermal two-area	Results compared with PSO-PID and DE-PID	[41]
PSO/PID	Thermal	By varying load frequency, model system dynamics were tested	[42]
AHA/PID	Thermal	Results compared with PSO-PID and AHA-PID	[9]
AHA/TFOIDFF	Thermal, wind, solar, and EV	Results compared tilt with filter and FOPID with filter	[43]
AHA/3DOF-PID	Wind, solar, and thermal	Results compared AHA/3DOF-PID and existing controllers	[44]
AHA/FOPI	Thermal	Results compared AHA/FOPI, GWO/FOPI, and PSO/FOPI	[45]

.

To maintain tie-line power as well as frequency at the rated value, many control strategies that make use of soft computing methodologies have been developed recently. Table 1 lists the relevant literature for the proposed work. In order to prove PSO-AHA-PID’s superiority, this paper compares the results with the PSO-PID controller in multiple cases.

PSO-AHA-PID is proposed in addition to the PSO-PID controller using the objective function ITAE proposed in this paper. The results from PSO-AHA-PID in the suggested networks are contrasted with PSO-PID in order to investigate frequency responses by considering different scenarios in the system with respect to variations in load along with system constraints in a two-area IPS. The main goal of this study is to increase the proposed system’s effectiveness and also to maintain network balance under stressful situations so that every consumer may get good quality power.

1.2. Major Contributions and Highlights

Dual-area tie-line IPS, along with a single-area multiple-source energy network consisting of PVwind, thermal, and fuel cells, are proposed systems in this article. By evaluating the effectiveness of the suggested controller in addition to optimization approaches in examined networks under system constraint alterations as well as load variations, the ITAE objective function is taken into consideration in the design and study for the suggested work. To identify the reliable supremacy of the proposed tuned controller, a thorough examination was carried out.

• The proposed study uses a variable, sustainable power network model that accounts for fluctuation and the effects of renewable energy sources. With detailed explanations of multiple mathematical models of nonconventional energy, the LFC issue can be resolved quite rapidly.
• This research includes a fuel cell model, fuzzy-based MPPT PV system, and PMSG-based wind energy system based upon the P & O MPPT approach in order to set up a single-area power network with multiple sources.
• A tie-line-connected double-area power network model is created.
• To optimize the PID controller’s parameter values, the PSO and PSO-AHA techniques are applied.
• Upon load variations, steady-state error is reduced to zero.
• In numerous scenarios, the effectiveness of PSO-AHA-PID for multiple-source single-area as well as double-area tie-line IPS is examined by adjusting system constraint values within the range of approximately ±75 and also by examining fluctuations in load for dual-area tie-line IPS.
• Compared to the PSO-PID control system, it produces superior outcomes and is more dependable. The suggested control system method has a lower undershot, overshoot, and settling time than the PSO-PID controller in every scenario this study investigates.

1.3. Structure of Article

The following is the arrangement of the manuscript: Section 1 reviews the literature on the topic of the current study as well as a range of controller optimization methods. The state space functions and mathematical models that support LFC power systems are described in Section 2. A possible control strategy employing PSO-PID and PSO-AHA-PID controllers, together with data, is covered in Section 3. The achieved results are covered in Section 4. Section 5 includes recommendations for more research in addition to a summary of the study’s main results.

2. Suggested Power Framework Modeling

The recommended power structure’s mathematical modeling is presented in the section that follows.

2.1. Components of the Thermal Power Network

The primary components of a thermal power network include the governor, turbine, and load. Each of these components plays a crucial role in the network’s operation and stability.

2.1.1. Governor

The governor is a mechanical device used to monitor and regulate an engine’s speed. Its primary function is to adjust the engine’s median speed when the load fluctuates. The block diagram of the speed governor for a power system is shown in Figure 1. In this figure, the governor constant time is denoted as T_G, and the speed-governing mechanism is demonstrated with an unstable fall correction G_c(s). Other parameters include the mechanical generator opening time T_M, the damping ratio D, the droop gain R, the turbine constant time T_r, the variation in frequency Δf, the change in generator power output ΔP, the turbine’s mechanical power output change ΔP_m, and the power load change ΔP_L.

2.1.2. Turbine

A turbine is a rotating mechanical machine that produces useful work by converting energy from the flow of a liquid, such as steam vapor, water, or air, into rotation. The transfer function model of the turbine is depicted in Figure 2. The time lag between the valve’s switching position and the turbine torque is represented by T_ch.

2.1.3. Load

The electrical grid is subject to various types of loads. Figure 3 illustrates the block structure of the load mathematical model. In this model, H represents the generator inertia constant, and ΔP_e denotes the change in electrical power.

$$\mathrm{Governor} {\displaystyle \frac{1}{1+s{T}_{sg}}}$$

(1)

$$\mathrm{Reheater}={\displaystyle \frac{1+s{K}_r{T}_r}{1+s{T}_r}}$$

(2)

$$\mathrm{Steam} \mathrm{generator}={\displaystyle \frac{1}{1+s{T}_t}}$$

(3)

While time constants for the steam turbine, reheater, as well as governor are denoted, T_sg, T_r, and T_t. Figure 4 shows a mathematical model representation for a single-area thermal electrical network.

The governor, turbine, and load components are interconnected to form the thermal power network. The governor adjusts the turbine’s mechanical power output ΔP_m in response to changes in frequency Δf and load ΔP_L. The turbine then converts this mechanical power into electrical power, which is supplied to the grid. The load component models the variations in electrical power demand, impacting the overall stability and performance of the power network.

2.2. Numerical Modeling of Wind Energy Network

The explanation for wind power system modeling is as follows:

2.2.1. Wind Speed Modeling

The wind resource of a wind energy system is a critical component that determines how much electricity can be generated, under ideal conditions, that wind turbine power can capture proportional to the cube of wind speed. The kinetic energy of wind can be computed using the following expression:

$$E={\displaystyle \frac{1}{2}}m{v}^2$$

(4)

where m is the moving mass of air and v motion speed. At that time, wind speed is reported as

$$P_{wind}={\displaystyle \frac{E}{\Delta t}}$$

(5)

Typically, a scalar function that changes over time to represent wind speed v is V = f(t). It can alternatively divide into two distinct elements, which stand for variations in wind: a part that varies slowly, marked as V₀, and a part that varies randomly, denoted as V_t. Consequently, wind speed can be expressed as below:

$$V(t)=V_0+V_t(t)$$

(6)

The literature provides three methods for modeling wind speed profiles mathematically:

The first technique is a white noise filtering technique, in which a low-pass filter with subsequent transfer function is utilized to correct the impact of turbulence:

$$F(s)={\displaystyle \frac{1}{1+\tau \times s}}$$

(7)

τ filters constant time, which depends upon the average wind speed, rotor diameter, as well as the intensity of wind turbulence.

The second way to create a wind speed profile uses meteorologist I. Van der Hoven’s established spectral density to characterize fluctuations in wind speed. As a result, wind speed variation V(t) is expressed as a harmonic sum:

$$V_v(t)=A+{\displaystyle \sum _{i=1}^n{\alpha }_k·\sin ({\omega }_k·t)}$$

(8)

i is the final harmonic rank retained in wind profile computation, α_k amplitude of K-order modulation, ω_k vibration of K-order modulation, and A average wind speed.

The third technique, the Weibull distribution, measures the average wind speed at average periods to determine the wind potential for a given site. A histogram is then used to divide acquired data into numerical values according to wind speed classifications. Given the chosen time instant and Weibull distribution, the wind profile is given as

$$V_v(t)=(1+{\xi }_v(t)-{\xi }_v)·V_v$$

(9)

where ${\xi }_v$ means the value of disturbance and V_v is the average wind speed, given by

$${\xi }_v(t)={(-{\displaystyle \frac{\ln (rand(t))}{C_v}})}^{{\displaystyle \frac{1}{k_v}}}$$

(10)

where (C_v, k_v) is a duplet of parameters found in analyzing the wind class histogram, and rand(t) is a function that generates random values between 0 and 1 in a uniform distribution. In general, C_v is a scaling factor larger than 5. If the histogram resembles a normal distribution, which is defined by a regular division throughout a mean value, then the shape factor k_v is larger than 3.

2.2.2. Modeling of Wind Turbine

Wind energy is a primary nonconventional energy source. Through the wind power network, the capacity of the wind turbine generator (WTG) is combined with the current energy network [46]. The dynamic model for WTG is as follows:

$$\Delta P_{WTG}={\displaystyle \frac{1}{T_{WTG}}}\Delta P\omega -{\displaystyle \frac{1}{T_{WTG}}}\Delta P_{WTG}$$

(11)

Pw denotes wind power, T_WTG is the WTG time constant, and ΔP_WTG denotes the WTG output power change. The blade-equipped rotor of a turbine converts wind energy into mechanical energy. The following formulas can be used to numerically express the rotor’s extracted wind power [47]:

$$P_{rotor}={\displaystyle \frac{1}{2}}\rho A{V}^3{C}_p$$

(12)

where A stands for the sweeping area, V is the wind speed, C_p is the power coefficient, and ρ is the wind density. The relationship between the input wind speed as well as active power is shown in the following way:

$$P_{GW}={\displaystyle \frac{\rho a^2{V}_{\omega }^3{C}_p}{2}}(T_{SR},\beta )$$

(13)

β is the blade angle, T_SR is the tip speed ratio, α is the area density, and V_ω is the wind velocity. Rotor efficiency C_p:

$$\begin{array}{l}{C}_p={\displaystyle \frac{T_{SR}-0.022{\beta }^2-5.6}{2}}{e}^{-0.17T_{SR}}\\ T_{SR}={\displaystyle \frac{r_{pm}\pi D}{60V}}\end{array}$$

(14)

Turbine-generated torque is expressed by

$$T_t={\displaystyle \frac{P_{GW}}{\omega t}}$$

(15)

ωt represents the wind turbine rotor’s angular rotational speed.

2.2.3. Gearbox Modeling

A wind turbine’s mechanical components are made up of a turbine shaft, which rotates slowly at a speed of Ω_t, and a gearbox, which has a multiplication gain of G and powers the generator Ω_g through a fast secondary shaft.

By using a gearbox, the turbine speed of Ω_t can be multiplied by a multiplication gain G in order to adjust to the generator’s higher speed of Ω_g. The elasticity, friction, and energy loss of the gearbox are deemed insignificant, making this device efficient. The following two equations represent the mathematical model of this device’s operation:

$$\begin{array}{l}{T}_g={\displaystyle \frac{T_{aer}}{G}}\\ {\Omega }_t={\displaystyle \frac{{\Omega }_g}{G}}\end{array}$$

(16)

where T_aer is the wind turbine’s aerodynamic torque; Ω_g is the speed of the generator shaft; T_g is the torque on the generator shaft; G is the multiplication gain, which is defined as G = N₁/N₂; and Ω_t is the turbine speed shaft.

Generator inertia J_g and turbine inertia J_t make up total inertia J, which can be expressed using the following equation:

$$J={\displaystyle \frac{J_t}{G^2}}+J_g$$

(17)

The generator friction constant f_g and turbine friction constant f_t add up to the total viscous friction constant f_v. Following is an expression for constant f_v:

$$f_v={\displaystyle \frac{f_t}{G^2}}+f_g$$

(18)

T_mec is the total mechanical torque, which determines generator speed Ω_g. This torque is the sum of torques supplied to generator shaft T_g, viscid friction torque T_v, and electromagnetic torque of generator T_em.

$$T_{mec}=J\times {\displaystyle \frac{d{\Omega }_g}{dt}}$$

(19)

$$T_{mec}=T_g-T_{em}-T_v$$

(20)

$$T_v=f\times {\Omega }_g$$

(21)

Consequently, the following represents the differential equation for mechanical network dynamics:

$$J\times {\displaystyle \frac{d{\Omega }_g}{dt}}=T_g-T_{em}-T_{vis}$$

(22)

Figure 5 displays a block diagram for the mechanical part of the wind turbine, and different power coefficient (C_p) numerical formulas are given in

Table 2. Different power coefficient (C_p) numerical formulas.

Coefficient Type, Cp	Formula
Exponential	$0.22\times \left[{\displaystyle \frac{116}{{\lambda }_i}}-0.4\times \beta -5\right]\times e^{{\displaystyle \frac{12.5}{{\lambda }_i}}}\mathrm{avec}{\displaystyle \frac{1}{{\lambda }_i}}={\displaystyle \frac{1}{\lambda +0.08\times \beta }}-{\displaystyle \frac{0.035}{{\beta }^3+1}}$
	$\begin{array}{l}0.5\times \left[{\displaystyle \frac{116}{{\lambda }_i}}-0.4\times \beta -5\right]\times e^{-{\displaystyle \frac{21}{{\lambda }_i}}}+0.068\times \lambda \\ \mathrm{Avec}:{\displaystyle \frac{1}{{\lambda }_i}}={\displaystyle \frac{1}{\lambda +0.08\times \beta }}-{\displaystyle \frac{0.035}{{\beta }^3+1}}\end{array}$
	$\begin{array}{l}0.5176\times \left[{\displaystyle \frac{116}{{\lambda }_i}}-0.4\times \beta -5\right]\times e^{-{\displaystyle \frac{21}{{\lambda }_i}}}+0.0068\times \lambda \\ \mathrm{Avec}:{\displaystyle \frac{1}{{\lambda }_i}}={\displaystyle \frac{1}{\lambda +0.08\times \beta }}-{\displaystyle \frac{0.035}{{\beta }^3+1}}\end{array}$
	$\begin{array}{l}0.5109\times \left[{\displaystyle \frac{116}{{\lambda }_i}}-0.4\times \beta -5\right]\times e^{-{\displaystyle \frac{21}{{\lambda }_i}}}+0.068\times \lambda \\ \mathrm{Avec}:{\displaystyle \frac{1}{{\lambda }_i}}={\displaystyle \frac{1}{\lambda +0.08\times \beta }}-{\displaystyle \frac{0.035}{{\beta }^3+1}}\end{array}$
	$\begin{array}{l}0.44\times \left[{\displaystyle \frac{125}{{\lambda }_i}}-6.94\right]\times e^{-{\displaystyle \frac{16.5}{{\lambda }_i}}}\\ \mathrm{Avec}:{\lambda }_i={\displaystyle \frac{1}{\frac{1}{\lambda }+0.002}}\end{array}$
Sinusoidal	$0.5-0.167\times (\beta -2)\times \sin \left[{\displaystyle \frac{\pi \times (\lambda -3)}{18.9-0.3\times (\beta -2)}}\right]-0.00184\times (\lambda -3)\times (\beta -2)$
	$0.3-0.00167\times (\beta -2)\times \sin \left[{\displaystyle \frac{\pi \times (\lambda +0.1)}{10-0.3\times (\beta -2)}}\right]-0.00184\times (\lambda -3)\times \beta$
Polynomial	$6\times {10}^{-7}\times {\lambda }^5+{10}^{-5}\times {\lambda }^4-65\times {10}^{-5}\times {\lambda }^3+2\times {10}^{-5}\times {\lambda }^2+76\times {10}^{-3}\times \lambda +0.007$
	$79.5633\times {10}^{-5}\times {\lambda }^{-5}-17.375\times {10}^{-4}\times {\lambda }^4+9.86\times {10}^{-3}\times {\lambda }^3-9.4\times {10}^{-3}\times {\lambda }^2+6.38\times {10}^{-2}\times \lambda +0.001$

.

In this article, the wind generator’s generating output is PMSG. Figure 6 and Figure 7 show block schematics of PMSG-wind turbines employing P & O MPPT. In MATLAB/Simulink, 2022a a control scheme for wind turbines with PMSG has been created, and it will be covered in Section 3. The following equations represent the PMSG wind turbine mathematically [48,49]. The output of the mathematical model, which is the rotation speed, is produced by the electromechanical torque of the generator and rotor torque acting as inputs.

$${\displaystyle \frac{d{w}_{gen}}{dt}}={\displaystyle \frac{1}{2H_{gen}}}\left[\left.-{\displaystyle \frac{P_{elec}}{w_{gen}+w_0}}-D_{tg}(w_{gen}-w_{rot})-k_{tg}\Delta {\theta }_m\right]\right.$$

(23)

$${\displaystyle \frac{d{w}_{rot}}{dt}}={\displaystyle \frac{1}{2H_{rot}}}\left[\left.-{\displaystyle \frac{P_{mech}}{w_{rot}+w_0}}+D_{tg}(w_{gen}-w_{rot})+k_{tg}\Delta {\theta }_m\right]\right.$$

(24)

$${\displaystyle \frac{d(\Delta {\theta }_m)}{dt}}=w_{base}(w_g-w_t)$$

(25)

w_gen denotes the speed of the generator, P_elec is the electric power, w₀ is the initial velocity, w_rot is the turbine speed, and P_mech is the mechanized power; a stable state appears when w_gen = w_rot, so d(Δθ)/dt = 0, and P_elec = P_mech. D_tg, K_tg, and w_base are constants. Mass’s geometrical dissipation demonstrates the inertial constant; the inertial moment was evaluated utilizing

$$H_{rotor}={\displaystyle \frac{J_{rotor}{w}_{rotor}^2}{2P_n}};H_{gen}={\displaystyle \frac{J_{gen}{w}_{gen}^2}{2P_n}}$$

(26)

The wind rotor’s moment of inertia can be computed generally using

$$J_{rotor}={\displaystyle \frac{1}{8}}{m}_r{R}^2$$

(27)

The rotor’s mass and radius are indicated by the letters m_r and R, respectively. The stator output voltages of the generator, d-q, are therefore stated as follows:

$$V_d=R_{d}I+L_d{\displaystyle \frac{\mathrm{d}{I}_d}{\mathrm{d}t}}-{\omega }_{gen}{L}_q{I}_q$$

(28)

$$V_q=R_q{I}_q+L_q{\displaystyle \frac{\mathrm{d}{I}_q}{\mathrm{d}t}}+{\omega }_{gen}(L_d{I}_d+{\phi }_f)$$

(29)

L is the generator’s inductance, R is the resistance, I is the current within d-q axes, φ_f is the permanent magnetic flux, and ω_gen is the PMSG’s rotational speed.

$${\omega }_{gen}=P_p{\omega }_{ref}$$

(30)

P_p stands for the number of pole pairs. The formula for electromechanical torque, or T_gen, is

$$T_{gen}={\displaystyle \frac{3}{2}}{P}_p{\omega }_{ref}(L_q-L_d)i_d{i}_q+{\phi }_f{i}_q$$

(31)

2.3. Mathematical Modeling of PV Energy System

Photovoltaic (PV) energy is an additional fundamental as well as widespread nonconventional energy source. PV panels are used to convert this energy into a current, which is then sent over conductors. To study PV power used in power networks, a basic model of PV panels can be created by considering connections between various electronic components [15,50].

The PV current is defined by the following formulas.

$$I=I_{Ph}-I_d-I_{R_{sh}}$$

(32)

I_ph, I_d, and I_Rsh show the photocurrent, current across the diode, and current across the shunt resistance. The PV produces a photocurrent upon the stated temperature when exposed to solar. I_ph is expressed as follows:

$$I_{Ph}=I_{SC}({\displaystyle \frac{S}{1000}})+K_t(T-T_R)$$

(33)

S is the solar radiance, I_SC is the current across the short circuit, K_t is the temperature co-efficient, T is the PV temperature, and T_R is the reference temperature. I_d is expressed as follows:

$$I_d=I_o\left[\left.\exp ({\displaystyle \frac{V+I{R}_s}{nJT}})-1\right]\right.$$

(34)

The voltage across terminal is represented by V, R_s is the resistance across series, n is the diode ideal factor, and J is Boltzmann’s constant. I_o is the saturation current, expressed as follows:

$$I_o=I_d{({\displaystyle \frac{T}{T_R}})}^3\exp \left[\left.{\displaystyle \frac{q{E}_g}{nJ}}({\displaystyle \frac{T-T_R}{T_{R}T}})\right]\right.$$

(35)

I_d is the reverse current across the diode, E_g is the cell bandgap energy, and q is the electron charge. I_R is expressed as follows:

$$I_R={\displaystyle \frac{V+I{R}_s}{R_h}}$$

(36)

The PV grid’s 1st-order transfer function is as follows [50,51]:

$$\Delta P_v={\displaystyle \frac{K_{pv}}{1+s{T}_{pv}}}\Delta P_{solar}$$

(37)

T_pv is the photovoltaic temperature, and K_pv is the specified PV gain (one). The majority of the mathematical connections between two- and one-diode photovoltaic systems are identical [52]. The fundamental circuits for single- and two-diode PV systems are shown in Figure 8 and Figure 9.

This paper describes a fuzzy-based MPPT PV system that uses MATLAB/Simulink. Figure 10 and Figure 11 display this control approach’s block illustration, including the fuzzy MPPT technique. In Figure 11, I_in(k) is the current, V_in(k) is the voltage, P_in(k) is the power, P_in (k − 1) is the preceding power, V_in (k − 1) is the preceding voltage, and E is the instant as well as the preceding value of voltage as well as power.

2.4. Fuel Cell System Modeling

This is a complicated system consisting of hydrodynamic, thermal, and electrochemical variables. The proportional relationship between a gas’s partial pressure and flow through a valve can be expressed as follows:

$${\displaystyle \frac{Q_{H_2}}{p_{H_2}}}={\displaystyle \frac{L_{an}}{\sqrt{M_{H_2}}}}=L_{H_2}$$

(38)

Q_H2 is the hydrogen molar fluid, P_H2 is the partial pressure of hydrogen, L_an is the anode valve constant, M_H2 is the hydrogen molar mass, and L_H2 is the hydrogen valve molar constant.

$${\displaystyle \frac{Q_{H_{2}O}}{p_{H_{2}O}}}={\displaystyle \frac{L_{an}}{\sqrt{M_{H_{2}O}}}}=L_{H_{2}O}$$

(39)

Q_H2O is the water molar fluid, P_H2O is the water vapor partial pressure, M_H2O is the water molar mass, and L_H2O is the molar constancy water valve.

$${\displaystyle \frac{d}{dt}}{p}_{H_2}={\displaystyle \frac{RT}{V_{an}}}(Q_{H_2}^{in}-Q_{H_2}^{out}-Q_{H_2}^r)$$

(40)

The following represents the basic electrochemical relationship between the stack current and hydrogen flow:

$$Q_{H_2}^r={\displaystyle \frac{NI}{2F}}=2K_r{I}_{fc}$$

(41)

N is the stack’s number, I_fc is the current of the fuel cell, F is Faraday’s constant, and K_r is the modeling constant. Using Laplace transformation upon (23) and (27), the hydrogen partial pressure is written in the s domain as follows:

$$p_{H_2}={\displaystyle \frac{1/L_{H_2}}{1+{\tau }_{H_2}s}}(Q_{H_2}^{in}-2K_r{I}_{fc})$$

(42)

$${\tau }_{H_2}={\displaystyle \frac{V_{an}}{L_{H_{2}RT}}}$$

(43)

τ_H2 is the hydrogen constant time, V_an is the volume across the anode, T is the temperature across the stack, and R is the universal gas constant. The output voltage is given below:

$$V_{cell}=E+{\eta }_{act}+{\eta }_{ohm}$$

(44)

Nernst voltage, when given in terms of gas molarities, is as follows:

$$E=N\left[\left.E_0+{\displaystyle \frac{RT}{2F}}\log \left[\left.{\displaystyle \frac{p_{H_2}{p}_{O_2}^{0.5}}{p_{H_{2}O}}}\right]\right.\right]\right.$$

(45)

E is Nerst’s voltage, E₀ is the cell open voltage, V_cell is the voltage across the cell, and η_act is the ohm overvoltage.

2.5. Double-Area Tie-Line Modeling

The electrical network proposed in this article is the dual-area thermal power network, shown in Figure 12. The primary components of each area consist of a turbine, generator, and speed-regulating device with two outputs and three inputs. The dual-area tie-line IPS block illustration is displayed in Figure 12. Consequently, both model’s components are equipped with a PID controller. Control signals (Figure 12) are represented by u₁ and u₂, power demand variations are represented by ΔP_D1 and ΔP_D2, power variation in the generator is represented by ΔP_g, power variation in the turbine is represented by ΔP_t, output area control errors are represented by ACE₁ and ACE₂, the frequency bias co-efficient is represented by B₁ and B₂, and fluctuations in system frequency are represented by Δf₁ and Δf₂.

B₁ and B₂ of the dual-area power network are calculated as follows:

$$B_1={\displaystyle \frac{1}{R_1}}+D_1$$

(46)

$$B_2={\displaystyle \frac{1}{R_2}}+D_2$$

(47)

$$AC{E}_1=B_1\Delta f_1+\Delta P_{tie1}$$

(48)

$$AC{E}_2=B_2\Delta f_2+\Delta P_{tie2}$$

(49)

Utilizing specified schemes along with transfer functions, suggested networks were designed; Simulink models were designed for analysis within MATLAB/Simulink for LFC. Using PSO and PSO-AHA approaches, PID controller gains were tuned.

Table 3. One-area thermal power network constraint values.

Constraint	Constraint Value
Turbine constant time	0.5 s
Governor constant time	0.2 s
Generator inertia constant	5 s
Governor’s speed regulation	0.05 pu
Turbine power rating	250 MW
Frequency	60 Hz
Duty ratio	0.8

and

Table 4. Constraint values along with electrical features of wind, fuel cell, and solar.

Characteristics of Advanced Renewable Energy AREi-230W-M6-G PV Module
Strings in parallel	75
Module connected in series	40
Maximal output power	213.15 W
Voltage across open circuit	36.3 V
Voltage at maximal power	29 V
Temperature co-efficient across open circuit voltage	−0.36099
Every module’s cell	60
Short circuit current	7.84 A
Current at maximal power	7.35 A
Temperature co-efficient	0.102
Stated temperature	25 °C, 1000 (W/m2)
Diode Ideal factor	0.98117
Resistance across shunt	313.3991 ohm
Resistance across series	0.39383 ohm
PMSG wind turbine features
Mechanized output power	800 Kw
Generator’s base power	800 Kw/0.9
Maximal power (Pu)	0.90
Generator’s rotational speed (Pu)	1.2
PMSG’s stator phase resistance	0.0485 ohms
Wind speed	12 m/s
Fuel cell constraint
Function 1: 1.36908036 atomic mass units	60,000 × 8.3145 × (273 + 95) × 400 × u (1)/(2 × 96,485 × (3 × 101,325) × 0.919 × 0.995)
Function 2: 5.89420573 atomic mass units	60,000 × 8.3145 × (273 + 95) × 400 × u (1)/(4 × 96,485 × (3 × 101,325) × 0.5057 × 0.21)
Stack power (Watts)	50,000
Fuel cell resistance (ohms)	0.66404
Nerst voltage of one cell (V)	1.1342
Exchange current (A)	0.91636
Exchange coefficient	0.26402
Temperature across system (K)	338
Fuel supply pressure (bar)	1.5
Air supply pressure (bar)	1
Nominal stack efficiency (%)	55
Number of cells	900
Operating temperature (Celsius)	65
Nominal air flow rate (lpm)	2100

.

3. Suggested Control Scheme

This section explains the suggested study, a single-area energy network linked with a PMSG wind turbine based upon the P & O MPPT technique, a fuzzy-based MPPT solar PV, as well as a fuel cell, in addition to double-area tie-line IPS. PID controller gains are adjusted using PSO-AHA and also PSO approaches. MATLAB/Simulink is used to simulate suggested control research in order to investigate and regulate the frequency. Figure 12 and Figure 13 display block illustrations for dual-area as well as single-area multiple-source energy networks.

3.1. PID Controller

The primary goal of the controller is to determine the optimal set of controls that will enable the system to continuously reach the intended state with the fewest possible variations. PID controllers are extensively used in control as well as automation domains due to enhanced functionalities along with well-proven design methodologies [53]. The sum of derivatives, along with the proportionate integral of derivatives and the proportionately integral controller outputs, is the output it generates. Figure 14 shows the fundamental configuration of the PID controller. R(s) is the input signal, the error signal is represented by E(s), the control signal is represented by U(s), and the output signal is represented by Y(s) in Figure 14.

Numerically,

$$u(t)\propto e(t)+{\displaystyle \int e(t)+\frac{d}{dt}}e(t)$$

(50)

The transfer function is expressed below:

$${\displaystyle \frac{U(s)}{E(s)}}=K_P+{\displaystyle \frac{K_I}{s}}+K_{D}s$$

(51)

K_P, K_I, and K_D represent controller gains. The transfer function is expressed as

$$\begin{array}{l}{K}_P\left[1+{\displaystyle \frac{K_I}{K_{Ps}}}+{\displaystyle \frac{K_D}{K_P}}s\right]\\ K_P\left[1+{\displaystyle \frac{1}{T_{is}}}+T_{D}s\right]\\ T_i={\displaystyle \frac{K_P}{K_I}}\&T_D={\displaystyle \frac{K_D}{K_p}}\end{array}$$

(52)

The PID controller determines its control value by taking errors from the past, present, as well as future into account.

It is used in conjunction with the one-area multiple-source network in addition to dual-area tie-line IPS to maintain the system’s intended frequency and also to decrease power variations.

3.2. Objective Function

A scale called objective function can be used to optimize controller gains. In this article, the ITAE objective function is used to modify the PID controller’s gains. It has the following mathematical representation [12]:

$$ITAE={\displaystyle \underset{t=0}{\overset{t=final}{\int }}\Delta f\times t\times dt}$$

(53)

Suggested approach tunings utilizing PSO along with PSO-AHA algorithms, with ITAE being the cost function, are covered in the sections that follow.

3.3. One-Area Multi-Source Power Network

A RESs-connected single-area energy network model was created within the Simulink environment. Table 3 displays the constraint values of the single-area thermal power network; Table 4 displays the electrical properties along with the parameter values of fuel cell, solar, and wind power sources.

In Section 2, an illustration showing a wind turbine along with the PMSG-based P & O MPPT technique and also the solar PV fuzzy-based MPPT technique is provided. Fuzzy rule ranges, membership functions, as well as types for solar PV MPPT, in addition to fuzzy rules, respectively, are displayed in

Table 5. PV MPPT fuzzy rules.

ΔVPV × (o/p)	ΔVPV(i/p)
ΔPPV(i/p)		NB	NS	ZE	PS	PB
	NB	PS	PB	NB	NB	NS
	NS	PS	PS	NS	NS	NS
	ZE	ZE	ZE	ZE	ZE	ZE
	PS	NS	NS	PS	PS	PS
	PB	NS	NB	PB	PB	PS

and

Table 6. Fuzzy rule ranges, membership function (MF), and kinds.

Input/Output	No. of MF	MF Range	MF Type
Input 1	10	−12.5 to 14.5	Triangular
Input 2	10	−4.6 to 2.3	Triangular
Output	10	−7.2 to 3.8	Triangular

. Table 5 shows negative big NB, negative small NS, zero ZE, positive small PS, and positive big PB. To determine PID optimal gains, PSO and PSO-AHA strategy operators were used. PSO, as well as AHA operators, are shown in

Table 7. PSO and AHA algorithm operators for one-area multiple-source energy network.

Parameters	Assigned Value
AHA
Fitness function	ITAE
The population’s size	70
Maximum iteration count	200
No. of variables	3
Dimension size	100
Lower bounds	0
Upper bounds	1
PSO
Fitness function	ITAE
The population’s size	70
Maximum iteration count	200
No. of variables	3
Minimum weight of inertia	0.4
Maximum inertia of weight	0.9
The cognitive component	1.85
The social component	1.85
Random numbers count	U (0, 1)

. The optimal gains of the PID controller are shown in

Table 8. PID gains for one-area multiple-source energy network.

PID Gains	KP	KI	KD
PSO-PID	178.0278	129.2391	113.3715
PSO-AHA-PID	103.9671	55.7804	27.4175

.

3.4. Dual-Area Tie-Line IPS LFC

The need for power in commercial, industrial, and residential sectors is increasing. Since it reduces reliance, integrating renewable energy is a superior option. Standard operating conditions can be changed by deregulating the electrical grid. Distributed generations in dual-region energy systems have attracted a lot of attention. Most modern power networks are made up of different control zones along with different generating sources and non-identical capacities to handle an excessive load. Any variation in generating and consumer load causes the area frequency to accidentally fluctuate from the set level, which, in turn, causes power-sharing with nearby control zones. Two thermal units make up the two-area power network tie-line IPS used in this study. Tie-lines provide the transmission of electricity between connected sectors. When adverse circumstances transpire, like extreme load disruptions, the control unit monitors changes in tie-line power and makes periodic attempts to return the system to its typical state. As a result, ITAE is represented as follows and is considered a goal function:

$$ITAE={\displaystyle \underset{0}{\overset{t}{\int }}(\left|\Delta f_1\right|}+\left|\Delta f_2\right|+\left|\Delta P_{tie}\right|)\times t\times dt$$

(54)

where ΔP_tie represents the tie-line frequency variation, and Δf₁ and Δf₂ denote the frequency variation in areas 1 and 2, respectively. In Section 2, a schematic illustrating this control approach, along with the mathematical model, is provided. The relevant model values for two-area IPS are given in

Table 9. Dual-area tie-line IPS constraint.

Constraint	Area 1	Area 2
R	0.05	0.0625
D	0.6	0.9
H	5	4
Base power	1000 MVA	1000 MVA
Gt	0.2 s	0.3 s
Tt	0.5 s	0.6 s

.

Table 10. PSO and AHA operators for dual-area tie-line IPS.

Parameters	Assigned Value
AHA
Fitness function	ITAE
The population’s size	55
Maximum iteration count	150
No. of variables	9
Dimension size	100
Lower bounds	0
Upper bounds	1
PSO
Fitness function	ITAE
The population’s size	55
Maximum iteration count	150
No. of variables	9
Minimum weight of inertia	0.45
Maximum inertia of weight	0.92
The cognitive component	1.85
The social component	1.85
Random numbers count	U (0, 1)

displays the configuration of the PSO and AHA parameters, while

Table 11. PID gains for dual-area tie-line IPS.

Controller	Area 1			Area 2			Tie-Line
	P	I	D	P	I	D	P	I	D
PSO-AHA-PID	0.49724	0.39757	−0.48765	0.24581	0.908370	−0.83745	0.49563	9.43 × 10−2	−0.06856
PSO-PID	0.75821	0.8453	0.4149	0.27183	−0.3218	0.55273	−0.8367	0.7691	0.51874

displays the PID controller’s optimum values.

3.5. PSO Algorithm

The PSO algorithm was created by Dr. Kennedy and Eberhart in 1995, with influence from collective nature observed in fish-schooling and bird-swarming processes. Every solution in the search field is referred to as a “particle” in the PSO approach. The optimizing process is determined by the fitness values of the objective function. The particle searches space for best values, both generally as well as individually, and its momentum directs its path. Initially, the swarm is put together using a set of randomly distributed particles; then, iterations are made in order to look for optimization. To achieve optimal gain values, principles of both personal as well as global are best applied to each particle. After every cycle, the best response, referred to as the local best, is determined. The term “global best” refers to the final best value, which is found after tuning each local optimal value [11].

There are N particles in population within the D-dimensional search space, and the position of the ith particle is presented as X_i = (x_i1, x_i2…x_iD) in the D-dimensional space, with the velocity V_i = (v_i1, v_i2…v_iD), i = 1,2…N. An iteration process is used to update the locations and velocities of the particles while determining the ideal values for each individual and collective particle creation.

The following equation gives the velocity of particles:

$$V_{k+1}^i=\omega V_k^i+c_1{r}_1(P_k^i-X_k^i)+c_2{r}_2(P_k^g-X_k^i)$$

(55)

The positions of the particles are constantly being updated by

$$X_{k+1}^i=X_k^i+V_{k+1}^i$$

(56)

$X_k^i$ is the position of the particle, $V_k^i$ is the velocity of the particle, $P_k^i$ is the best position of the individual particle, $P_k^g$ is the best position for a swarm, c₁ and c₂ are the cognitive and social parameters, and r₁ and r₂ are the random numbers between zero and one. The algorithm for PSO is depicted in the flow chart that can be found in Figure 15. Due to its simple structure and effectiveness, the PSO algorithm is gaining popularity. Furthermore, in terms of performance, it outperforms gradient-based optimization techniques, which necessitate the differentiation of the optimization issue. But if a particle gets trapped in a nearby optimal state, the PSO algorithm may produce suboptimal solutions.

3.6. AHA Algorithm

AHA is the newest biologically inspired optimization approach employed in engineering applications, and it was unveiled by year’s end in 2021. The concept is derived from the intelligence exhibited by hummingbirds [54]. Similar optimization techniques are employed in the development of the exploration and exploitation phases, as well as the AHA. Three elements—food sources, optimization, and a visit table—are used to characterize the development procedure for addressing various optimization challenges. The general attributes of sources are evaluated by hummingbirds, including the nectar content and grade of particular flowers, the rate of nectar production, and the most recent visitation date of the flower. Hummingbirds may communicate with their fellow hummingbirds about the location and pace of nectar replenishment at a certain food source. The hummingbird visiting table is utilized to document the count of hummingbirds, which consume food from each source. The visit table is typically modified at the end of each iteration. Development is influenced by three distinct foraging techniques: directed, territorial, and migratory foraging. The following mathematical representations depict the three distinct foraging strategies.

3.6.1. Initialization

A population of n hummingbirds is assigned to n food sources at random [1].

$$x_i=Low+r\times (U_p-U_L)i=1,\dots ,n$$

(57)

Here, U_L and U_P correspond to the lower and upper limits; x_i is the place that contains the ith food source, which is the answer to a certain problem; r is a random vector in the range of [0, 1]; and the food source visit table is set up as follows:

$$V{T}_{i,j}=\left\{\begin{array}{ll}0\hfill & \mathrm{if} \mathrm{i}\ne \mathrm{j}\hfill \\ \mathrm{null}\hfill & \mathrm{i}=\mathrm{j}\hfill \end{array}\right. \mathrm{i}=1,\dots ,\mathrm{n};\mathrm{j}=1,\dots ,\mathrm{n}$$

(58)

In the given scenario, it can be observed that when i = j, the value of VT_i,j is null, indicating that a hummingbird is actively consuming food from a specific source. Specifically, this signifies that the hummingbird indexed by i has recently visited the food source. On the other hand, when i ≠ j, the value of VT_i,j is 0, indicating that the hummingbird indexed by j is currently engaged in the iteration.

3.6.2. Guided Foraging

Every individual hummingbird selects the nectar source that offers the most quantity of nectar. Hummingbirds exhibit three distinct modes of flying, including axial flight, diagonal flight, and omnidirectional flight. The term axial flight can be defined as follows:

$$D^{(i)}=\left\{\begin{array}{rr}1& \mathrm{if} \mathrm{i}=\mathrm{randi}([1,d])\\ 0\hfill & \mathrm{else}\hfill \end{array}\right.i=1,\dots ,d$$

(59)

The following is a description of a diagonal flight:

$$D^{(i)}=\left\{\begin{array}{rr}1& \mathrm{if} i=P(j),j\in [1,k],P= \mathrm{randperm} (k),k\in [2,[r_1\times (d-2)]+1]\hfill \\ 0\hfill & \mathrm{else}\hfill \end{array}\right. \mathrm{i}=1,\dots ,\mathrm{d}$$

(60)

The following is a description of an omnidirectional flight:

$$D^{(i)}=1i=1,\dots ,d$$

(61)

The function randperm (k) generates a permutation of integers ranging from 1 to k, whereas the function randi [1, d] selects a number at random between 1 and d. Additionally, r₁ represents a random number within the range of [0, 1]. The subsequent expression is the mathematical equation utilized for the purpose of modeling directed foraging behavior in the presence of a suitable food source:

$$v_i(t+1)=x_{i,tar}(t)+a\times D\times (x_i(t)-x_{i,tar}(t))$$

(62)

$$aN(0,1)$$

(63)

The position of the i_th hummingbird’s desired food source could be represented as x_i,tar(t), where a is a guiding factor. The location of the i_th food source at time t, denoted as x_i(t), follows a normal distribution N (0, 1) with a mean of 0 and a standard deviation of 1.

The following is the latest position update for the i_th food source:

$$x_i(t+1)=\left\{\begin{array}{ll}{x}_i(t)& f(x_i(t))\le f(v_i(t+1))\hfill \\ v_i(t+1)\hfill & f(x_i(t))>f(v_i(t+1))\hfill \end{array}\right.$$

(64)

The hummingbird discontinues its use of the present food source and instead consumes from the potential food source indicated by Equation (62) if the potential food source’s rate of nectar replenishment surpasses that of the current food source Equation (64).

3.6.3. Territorial Foraging

The hummingbird has a remarkable ability to efficiently navigate within its established territory, swiftly relocating to a neighboring area in search of an alternative food source that may offer more benefits compared to its present one. The above mathematical equation outlines the local foraging method employed by hummingbirds in conjunction with an appropriate food supply that aligns with their territorial foraging approach.

$$v_i(t+1)=x_i(t)+b\times D\times x_i(t)$$

(65)

$$bN(0,1)$$

(66)

The variable b represents a territorial component that follows a normal distribution with a mean of 0 and a standard deviation of 1, denoted as N (0, 1), using its unique flight abilities.

3.6.4. Migration Foraging

The migration of a hummingbird from the nectar source with the lowest nectar refilling rate to a random one might be characterized as follows:

$$x_{wor}(t+1)=Low+r\times (Up-Low)$$

(67)

where x_wor is the food for which the population’s rate of nectar replenishment is the lowest. In the absence of any other food sources, a hummingbird engaging in both directed foraging and territorial foraging would sequentially visit each food source as its target source throughout each iteration, as indicated by the visit table. Let us consider a scenario where the selection between the territorial and directed foraging stages is characterized by a probability of success of 50%. Additionally, in the context of guided foraging, the likelihood of success in visiting every alternate source is also 50%. In this scenario, the adoption of a migratory foraging strategy becomes necessary in order to disrupt the state of equilibrium and expand the scope of the search region. As a consequence, the proposed population size parameters for the migration coefficient are as follows:

M = 2n

(68)

The algorithm for AHA is depicted in the flow chart that can be found in Figure 15.

3.7. PSO-AHA Hybrid Algorithm

The hybrid PSO-AHA optimization algorithm is used to optimize the K_P, K_I, and K_D values for the PID controller in a single-area multi-source and two-area tie-line IPS to achieve the desired objectives, as mentioned in Figure 15. We will outline the steps involved in the AHA cycle and the PSO cycle and then explain the decision-making process for incorporating both into a hybrid PSO-AHA strategy.

3.7.1. AHA Algorithm Cycle

AHA algorithm process are given as:

3.7.2. Initialization

Specify the values for the number of agents NP, max_iter, d, n_vars, Low, and Up for K_P, K_I, and K_D. Set up the random positions of the hummingbirds (agents) within the boundaries.

3.7.3. Fitness Evaluation

Determine the best possible fitness value for each hummingbird using the objective function associated with PID controller performance.

3.7.4. Position Update Based on Foraging Strategy

Determine whether territorial foraging applies to each hummingbird (rand < 0.5). If so, adjust the position using the foraging strategy: (59) should be used to update the location if r < 1 = 3. Update the position with Equation (60) if r < 2 = 3. Otherwise, use Equation (61) to update the location. Regardless of the outcome of the territorial foraging, update the location using Equation (62).

3.7.5. Calculation of the Objective Function

Determine the objective function for the updated positions to evaluate the effectiveness of the new solutions.

3.7.6. Migration Foraging

If the number of iterations exceeds maxiter, execute migration foraging to possibly prevent local optima.

3.7.7. Record Best Position

At the end of each iteration, or when the termination criteria are met, note the best position identified and its fitness value.

3.7.8. PSO Algorithm Cycle

The PSO algorithm process is given as follows:

3.7.9. Initialize PSO Parameters

Define variables such as NP, w_min, w_max, c₁, and c₂. Specify the initial positions and velocities of the particles.

3.7.10. Evaluate Fitness

Compute the fitness of each particle by applying the objective function.

3.7.11. Update Individual and Global Best

Update the position of each particle according to its best-known position and the swarm’s best-known position after evaluating its fitness.

3.7.12. Update Velocity and Position

Equation (55) is used to update the velocity of each particle depending on its current velocity, the best position of the swarm, and its own position. Equation (56) should be used to update each particle’s position, taking into account the newly calculated velocity.

3.7.13. Termination Check

If the termination conditions are met (which may be based on the number of iterations, the degree of improvement, etc.), proceed to the hybrid decision to make.

3.7.14. Hybrid Decision of PSO-AHA Optimizer

The hybrid decision of the PSO-AHA Optimizer process is given as:

3.7.15. Evaluate Performance

Determine the performance of both AHA and PSO in optimizing K_P, K_I, and K_D values at each iteration. Determine the optimal position and fitness value achieved by both algorithms.

3.7.16. Decide Next Steps

Determine whether to proceed with the AHA cycle, the PSO cycle, or a hybrid of both based on their performance. This decision might be based on an enhancement in fitness.

3.7.17. Termination Check

Determine if the termination conditions have been satisfied (according to the fitness threshold). Once the termination requirements are satisfied, finalize the results. If not, proceed with the hybrid loop.

3.7.18. Finalize Results and Generate Output

Once the termination conditions are met, identify the optimal K_P, K_I, and K_D values derived from the hybrid algorithm. Determine the global best particle and its corresponding PID controller parameters. Output the optimal K_P, K_I, and K_D values for the PID controller to improve the LFC in the one-area multiple-source and two-area tie-line IPS.

This hybrid PSO-AHA optimizer strategy will repeatedly refine the K_P, K_I, and K_D settings in order to improve the PID controller capabilities within the proposed networks. The AHA enables exploration of the search space, whereas PSO concentrates on the exploitation of the best solutions identified. The hybrid technique obtains the strengths of both algorithms, conceivably resulting in a more robust and effective optimization process.

4. Simulation Results

By analyzing results using PSO-PID as well as PSO-AHA-PID and also contrasting them with the PSO-PID controller, this section shows the superiority of the suggested control approach. Based upon MATLAB 2022a results, the proposed power network was analyzed into two cases:

• Variations in system parameters in dual-area tie-line linked power systems and one-area multi-source power networks.
• Load changes within dual-area tie-line IPS.

4.1. Multi-Source Single-Area Power Network

Figure 16 displays the results achieved using PSO-AHA-PID and PSO-PID. The transient responses of the suggested power network substantially improved in terms of signal characteristics. Contrasted with the PSO-PID controller, the PSO-AHA algorithm successfully adjusted PID controllers’ gains, producing the best results in terms of incredibly low undershoot and quick response times.

Robustness Investigations (Case Study 1)

Continuous changes in system parameters can severely harm closed-loop system performance. Thus, to examine the effect of parametric variations on the system, experiments were carried out. To achieve this, the initial value of each system parameter was adjusted by approximately ±75%.

Table 12. Constraint variations in single area.

Cases	Constraint	Initial Value	Alteration Range	New Value
Case 1	Tg	0.2	+75%	0.35
	H	5	+75%	8.75
	D	0.8	−75%	0.2
	R	0.05	−75%	0.0125
Case 2	Tg	0.2	−75%	0.05
	H	5	−75%	1.25
	D	0.8	+75%	1.4
	R	0.05	+75%	0.0875

illustrates two potential situations of parameter uncertainty for a single-area power network.

Under parametric uncertainty, frequency responses of single-area multiple-source energy networks are displayed in Figure 17 and Figure 18. The frequency response of PSO-AHA-PID is superior to that of the PSO-PID controller, with less overshoot as well as undershoot and an exceptionally quick settling period. The frequency response signal characteristics of a one-area multiple-source power network are displayed in

Table 13. Frequency response one-area multiple-source signal values.

Controller	Frequency Responds
	Undershoot %	Overshoot %	Settling Time sec	Fall Time ms	Error	Pre-Shoot %	High/Low Hz
PSO-AHA-PID	0.118	−0.118	0.6	16.962	0.0006	0.505	60/59.98
PSO-PID	0.505	1.576	5.5	327.339	0.3236	0.241	60/59.65
Case 1
PSO-AHA-PID	0.341	13.068	0.9	17.886	0.0008	0.568	60/59.97
PSO-PID	0.505	0.098	4.5	414.607	0.4457	0.194	60/59.67
Case 2
PSO-AHA-PID	0.246	−0.246	1.5	11.329	0.0005	0.505	60/59.96
PSO-PID	6.989	0.464	6.6	113.586	0.3144	0.538	60/59.50

.

4.2. Two-Area Tie-Line IPS

The PSO-AHA-PID and PSO-PID controller results are displayed in Figure 19. Regarding actual frequency, tie-line power variations, variation of frequency in Pu, as well as alterations of frequency in Hz, the suggested dual-area tie-line IPS shows significantly improved transient responses. The PSO-AHA approach optimized gains of PID. When contrasted to PSO-PID, it provides optimal performance in terms of incredibly little undershooting and exceptionally quick response times.

4.2.1. Robustness Investigations (Case Study 1)

Variability affects a system’s stability. The LFC network required control action against constraint variations in the controlled network to achieve an acceptable level of resiliency. Thirteen cases for constraint variations were taken to analyze the resiliency of the proposed PSO-AHA-PID as well as the PSO-PID controller within double-area IPS. Each network constraint separately varied by ±75% from its initial value to start this analysis.

In scenarios 1–12, just one constraint was altered once, varying the initial value by +75%/−75%;

Table 14. Investigated cases under system constraint variations.

Case Number	Constraint	Initial Values		Alteration Range	New Values
		Area 1	Area 2		Area 1	Area 2
1	H	5	4	+75%	8.75	7
2	H	5	4	−75%	1.25	1
3	Tt	0.5	0.6	+75%	0.875	1.05
4	Tt	0.5	0.6	−75%	0.125	0.15
5	B	20.6	16.9	+75%	75.75	21.175
6	B	20.6	16.9	−75%	5.15	4.225
7	D	0.6	0.9	+75%	1.05	1.575
8	D	0.6	0.9	−75%	0.15	0.225
9	Tg	0.2	0.3	+75%	0.35	0.525
10	Tg	0.2	0.3	−75%	0.05	0.075
11	R	0.05	0.0625	+75%	0.0875	0.1093
12	R	0.05	0.0625	−75%	0.0125	0.0156
13	B	20.6	16.9	−75%	5.15	4.225
	H	5	4	+75%	8.75	7
	R	0.05	0.0625	+75%	0.0875	0.1093
	D	0.6	0.9	−75%	0.15	0.225
	Tt	0.5	0.6	+75%	0.875	1.05
	Tg	0.2	0.3	+75%	0.35	0.525

shows that constraints T_g, T_t, H, and R changed +75% in both areas in scenario 13. Concurrently, there was a −75% change in B and D. Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30, Figure 31 and Figure 32 depict the dynamic outcomes of double-area tie-line IPS with PSO-AHA-PID as well as PSO-PID under constraint-ambiguity conditions. The frequency response signal characteristics of dual-area tie-line IPS are listed in

Table 15. Frequency response of dual-area signal characteristics.

Frequency Response Area 1					Frequency Response Area 2				Tie-Line Power (Pu) Response
Controller	Ush%	Osh%	Psh%	H/L Hz	Ush%	Osh%	Psh%	H/L Hz	Ush%	Osh%	Psh%	Fall time ms
PSO-AHA-PID	1.913	0.505	0.505	60/59.84	−1.339	11.049	217.108	60/59.99	1.143	0.847	−0.129	208.088
PSO-PID	2.022	0.313	0.505	60/59.78	34.477	27.564	0.641	60/59.71	0.802	0.505	−0.270	1196
Case 1
PSO-AHA-PID	0.427	2.577	0.515	60/59.88	1.483	3.549	0.962	60/59.98	1.030	0.769	−0.262	316.360
PSO-PID	1.998	0.316	0.505	60/59.83	13.472	11.798	0.562	60/59.78	−0.055	71.552	0.862	1038
Case 2
PSO-AHA-PID	1.834	0.510	1.531	60/59.84	12.411	31.945	124.336	60/59.99	1.269	0.877	−0.380	190.147
PSO-PID	23.570	−0.035	0.505	60/59.77	40.694	30.921	0.658	60/59.69	1.658	21.599	3.512	880.243
Case 3
PSO-AHA-PID	1.918	0.505	0.505	60/59.81	−0.106	10.248	0.877	60/59.97	1.403	0.847	−0.526	274.872
PSO-PID	27.172	27.564	0.641	60/59.74	30.737	57.937	0.794	60/59.66	−12.085	148.893	115.792	329.824
Case 4
PSO-AHA-PID	1.919	0.595	18.452	60/59.91	−53.882	98.696	302.210	60/59.99	−42.006	84.552	165.379	21.547
PSO-PID	1.910	0.477	0.505	60/59.84	1.836	0.505	0.505	60/59.78	0.511	44.203	0.725	786.243
Case 5
PSOAHA-PID	1.707	1.531	0.510	60/59.86	−8.842	50.413	348.726	60/59.98	0.777	0.847	0.455	191.507
PSO-PID	17.723	6.989	0.538	60/59.78	33.303	38.406	6.522	60/59.73	−2.470	38.099	5.862	805.789
Case 6
PSO-AHA-PID	1.846	0.489	0.505	60/59.73	6.415	28.610	0.806	60/59.93	1.816	0.943	0.484	295.817
PSO-PID	0.505	1.985	0.505	60/59.77	0.505	1.359	0.505	60/59.70	1.688	0.556	−0.180	8154
Case 7
PSO-AHA-PID	1.605	0.505	0.505	60/59.85	1.519	6.964	254.111	60/59.99	1.012	0.847	0.516	206.415
PSO-PID	1.756	−0.229	0.505	60/59.79	31.426	19.880	0.602	60/59.72	78.036	63.115	0.820	8578
Case 8
PSO-AHA-PID	1.727	0.505	0.505	60/59.84	24.021	3.357	0.943	60/59.97	1.271	0.847	−0.785	209.712
PSO-PID	19.340	−1.032	0.510	60/59.78	6.742	71.609	79.217	60/59.88	1.790	0.562	−0.435	1063
Case 9
PSO-AHA-PID	1.899	0.505	0.505	60/59.82	4.711	61.479	110.905	60/59.99	1.483	0.877	−0.652	241.410
PSO-PID	24.506	10.556	0.556	60/59.76	14.998	72.399	63.669	60/59.86	1.576	0.602	−0.348	1078
Case 10
PSO-AHA-PID	1.583	0.575	14.368	60/59.90	0.247	214.728	1.211	60.1/60	1.011	0.735	0.628	131.202
PSO-PID	15.69	6.98	0.53	60/59.61	0.50	1.96	0.50	60/59.91	0.483	80.909	0.909	1034
Case 11
PSO-AHA-PID	1.657	10.185	75.00	60/59.91	246.560	−246.560	493.550	60/59.99	1.183	0.820	0.588	214.165
PSO-PID	12.827	24.375	0.625	60/59.72	19.935	44.203	0.725	60/59.65	0.478	37.476	28.826	925.219
Case 12
PSO-AHA-PID	1.805	0.595	18.452	60/59.87	−1.333	25.031	1333.15	60/59.98	1.160	0.847	0.740	208.194
PSO-PID	16.289	0.476	0.505	60/59.77	31.025	27.564	0.641	60/59.70	0.719	0.521	−0.347	1194
Case 13
PSO-AHA-PID	1.745	0.499	0.505	60/59.69	0.813	2.487	0.862	60/59.94	1.163	0.893	0.615	325.216
PSO-PID	0.505	1.095	0.505	60/59.68	0.50	1.532	38.01	60/59.60	1.916	85.000	−0.151	1447

, where high and low frequencies are represented by H/L, O_sh represents overshoot, P_sh represents pre-shoot, and U_sh represents undershoot.

4.2.2. Load Variations (Case Study 2)

Four cases were considered against fluctuations in load, as indicated in

Table 16. Load alteration for dual area.

Case No.	Area 1	Area 2
1	250 MW increment	200 MW increment
2	100 MW decrement	300 MW decrement
3	400 MW increment	150 MW decrement
4	200 MW decrement	350 MW increment

, to test the flexibility of the system.

Figure 33, Figure 34, Figure 35 and Figure 36 display the results obtained for dual-area load alterations. The frequency responses for dual-area signal characteristics in load variations are displayed in four scenarios in

Table 17. Frequency response for dual-area signal characteristics in load alteration.

Area 1					Area 2				Tie-Line Power (pu)
Controller	Ush%	Osh%	Psh%	H/L Hz	Ush%	Osh%	Psh%	H/L Hz	Ush%	Osh%	Psh%	Fall time ms
Case 1
PSO-AHA-PID	1.775	0.505	0.505	60/59.51	0.812	8.610	169.786	60/59.98	1.787	0.847	−0.147	208.101
PSO-PID	18.901	9.341	0.549	60/59.17	24.648	5.803	51.751	60/59.83	0.505	1.682	0.505	1219
Case 2
PSO-AHA-PID	−0.398	91.346	0.213	60.10/60	−62.720	211.491	198.838	60/59.99	−7.401	68.644	0.847	26.065
PSO-PID	1.560	0.595	0.375	61.21/60	−6.502	184.362	110.787	60.08/60	1.921	44.203	0.661	703.60
Case 3
PSO-AHA-PID	1.742	0.505	0.505	60/59.22	−2.249	15.178	268.494	60/59.99	1.144	0.847	−0.145	208.033
PSO-PID	1.284	0.610	−0.205	60.60/60	−13.481	231.389	90.462	60/59.95	−1.563	50.758	0.644	703.60
Case 4
PSO-AHA-PID	0.398	0.505	0.109	60.39/60	−61.603	209.938	100.933	60/60	−13.041	68.644	0.847	26.061
PSO-PID	18.500	11.798	0.562	60/58.56	6.757	−1.713	0.505	60/59.56	0.505	0.818	0.390	1238

. The proposed study examined the effectiveness of two power system models, a two-area tie-line IPS and a single-area multi-source power network for LFC. We analyzed these systems’ responses to a range of scenarios, including changes in load and parameter values.

We assessed the system performance for the single-area multi-source power network by adjusting system parameters within ±75%. This system includes fuzzy-based MPPT solar power, PMSG wind power utilizing P&O MPPT, fuel cells, and isolated thermal power systems. These changes, as shown in Table 12, were implemented methodically to evaluate how robust the control measures were. The measured results for the frequency response characteristics undershoot, overshoot, and settling time are shown in Table 13 and Figure 16, Figure 17 and Figure 18. As shown in Table 14 and Table 16, we investigated the response of the two-area tie-line linked power system under 13 distinct scenarios of parameter adjustments and 2 scenarios of load variations in Area 1 and Area 2.

Dynamic Outcomes Under Constraint Ambiguity: Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30, Figure 31 and Figure 32 present the dynamic outcomes of the two-area tie-line IPS when subjected to different scenarios of parameter variations. These figures illustrate the system’s frequency response under conditions of constraint ambiguity, comparing the performance of the PSO-AHA-PID controller with that of the PSO-PID controller.

Stability and Frequency Response: The PSO-AHA-PID controller demonstrates superior stability and better frequency response characteristics compared to the PSO-PID controller across all scenarios. Specifically, the PSO-AHA-PID controller consistently shows lower overshoot, undershoot, and settling times, indicating a more robust performance.

Resilience to Parameter Variations: The figures highlight the resilience of the PSO-AHA-PID controller to significant variations in system parameters (±75%). The frequency deviations in both areas are minimized more effectively by the PSO-AHA-PID controller, showing a better ability to handle uncertainties and maintain system stability.

Frequency Response Characteristics: Table 15 provides detailed frequency response characteristics for both Area 1 and Area 2, as well as tie-line power responses under various parameter-variation scenarios.

Performance Metrics: The metrics listed in Table 15, including undershoot (U_sh%), overshoot (O_sh%), pre-shoot (P_sh%), high/low frequencies (H/L Hz), and fall time, clearly show that the PSO-AHA-PID controller outperforms the PSO-PID controller. For instance, the PSO-AHA-PID controller has lower overshoot and undershoot percentages and faster settling times, reflecting improved dynamic performance.

Consistency Across Scenarios: The consistency of the PSO-AHA-PID controller’s performance across different scenarios is evident. It maintains more stable and desirable frequency characteristics, regardless of the specific parameter changes applied, demonstrating its robustness and adaptability to varying operating conditions.

5. Conclusions and Future Directions

5.1. Conclusions

This study presents a novel approach to Load Frequency Control (LFC) for dual-area tie-line interconnected power systems (IPS) and single-area multiple-source power networks. The innovation lies in the application of Particle Swarm Optimization (PSO) combined with the Artificial Hummingbird Algorithm (AHA) to fine-tune Proportional–Integral–Derivative (PID) controller gains, resulting in the PSO-AHA-PID controller. This hybrid approach leverages the exploratory capabilities of AHA and the exploitative strengths of PSO, ensuring robust and optimized control settings for LFC.

Key Scientific Contributions and Their Validation are given as:

5.1.1. Enhanced Frequency Stability

• Contribution: The PSO-AHA-PID controller demonstrates superior performance in minimizing frequency deviations compared to the conventional PSO-PID controller.
• Validation: In the one-area multiple-source network, frequency variations are limited to 59.98 Hz to 60 Hz with the PSO-AHA-PID controller, compared to 59.65 Hz to 60 Hz with the PSO-PID controller (Table 12 and Table 13, Figure 17 and Figure 18). In the dual-area system, similar improvements in frequency stability are observed (Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30, Figure 31 and Figure 32, Table 15).

5.1.2. Robust Performance across Scenarios

• Contribution: The proposed controller maintains consistent performance across various scenarios, including load changes and parameter variations up to ±75%.
• Validation: The robustness of the PSO-AHA-PID controller is demonstrated by its consistent performance in scenarios with parameter variations (Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30, Figure 31 and Figure 32, Table 15) and load changes (Figure 33, Figure 34, Figure 35 and Figure 36, Table 17).

5.1.3. Reduction in Power Variations

• Contribution: The methodology effectively reduces tie-line power variations, a critical factor for maintaining system stability in interconnected power networks.
• Validation: The PSO-AHA-PID controller achieves lower overshoot, undershoot, and faster settling times compared to the PSO-PID controller. For example, in the one-area multi-source network, the settling time is reduced to 0.6 s with the PSO-AHA-PID controller compared to 5.5 s with the PSO-PID controller (Table 13, Figure 17 and Figure 18).

5.1.4. Scalability and Applicability

• Contribution: This study’s approach can be scaled and applied to other power systems, including those integrating renewable energy sources (RESs).
• Validation: The applicability and scalability are validated through the modeling and analysis of a single-area multi-source power network, which includes fuzzy-based MPPT solar power, PMSG wind power using P&O MPPT, fuel cells, and isolated thermal power systems (Table 12 and Table 16, Figure 17, Figure 18, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30, Figure 31 and Figure 32).

5.2. Future Directions

Stability issues with linked power networks for nonconventional energy sources remain a significant concern. Future research will focus on the integration and maximization of different renewable energy sources. Additionally, the emergence of innovative technologies such as artificial intelligence, cloud computing, and big data holds great potential for addressing LFC-related issues. Future research will address the following areas:

• Nonconventional Energy: Continued research into nonconventional energy sources is vital as they introduce variability and complexity into power generation. The integration of LFC systems with RESs presents challenges in stabilization, necessitating effective device matching and information transfer.
• Flexibility: A highly variable and secure linked power network is required due to the high penetration of RESs. Enhancing the flexibility of multi-area power systems will involve developing new optimization approaches for control, storage, and market strategies.