Frequency Regulation for State-Space Model-Based Renewables Integrated to Multi-Area Microgrid Systems

Abstract

This paper focuses on effective frequency regulation of isolated multi-area microgrid structures using polynomial functions extracted from the modelling of renewable sources. This paper highlights the necessity of state-space (SS) formation for intermittent renewables such as solar and wind for specified quantities compared with the standard functions. The approximate gain and time constants for the variable generating sources used in the literature cannot be rationalized while carrying out frequency regulation since it does not epitomize the system dynamics. Thus, the article recommends non-linear modelling of renewables for executing frequency regulation in MATLAB/Simulink. The supremacy of the proposed work in the presence of a 3-degree of freedom (DOF)-based fractional PID controller aided with a recently evolved Artificial Hummingbird Algorithm (AHA) is determined through the transient and sensitivity analysis. The augmented results proved the outstanding response of AHA-optimized 3DOF-FOPID-based systems when compared with the existing controllers under variable operating conditions. Stringent evaluation is executed in the presence of commonly used heuristic approaches for witnessing the robustness of the optimized controller. The proposed system’s hardware-in-loop (HIL)-based implementation is also presented in the paper for a significant characterization of its working in the physical environment.

1. Introduction

The foremost motive of energy sectors is to generate, transmit, and distribute energy to all ranges of loads such as primary, secondary, and tertiary consumers. Load frequency control is the crucial issue that has been addressed for conventional, deregulated, and microgrid systems. The effective balance of power mismatch between the generation and load is maintained to deliver quality power to the consumers. Conventional setups such as thermal and hydropower plants are widely used for carrying LFC [1,2,3,4,5], which acts as the superior control stratagem for system stabilization. The standard systems are compared with different controllers, and their performance is evaluated with cost functions such as integral square error (ISE), integral time square error (ITSE), integral absolute error (IAE), integral time absolute error (ITAE), integral weighted square error (IWSE), and mean integral square error (MISE). The vertically integrated monopoly in generation, transmission, and distribution systems was less competitive and inefficient, resulting in unbundling of services to a private forum. This enhanced the way towards deregulation and instigated extensive research [6,7]. The system was exposed to uncertainties such as generation rate constraint (GRC), governor dead band (GDB), and thermal limits [8,9,10,11,12,13,14,15,16] with comparison charts for optimal solutions.

The notion of privatization has taken over the place of governmental bodies in power sectors. The essential need for independent energy resources caused by the exhaustion of fossil fuels paved the way for microgrid systems. In general, it comprises solar, wind, battery, and fuel cells, which are extensively used in load frequency control that illustrates the real-time systems for performance evaluation [17,18,19,20,21]. However, the transfer function representation for these renewables is not accurately derived based on the system ratings, instead of which, predefined gain and time constants are used globally. In this circumstance, the frequency response characteristics obtained from these systems cannot be truly accepted and taken into consideration for real-time system study. The concept of higher-order functions was introduced with the sliding mode controller [22,23,24,25,26] for LFC, but the representations were in the form of standard functions. Numerous research has been applied in LFC-like design and modelling of different controllers, contracts-based market operations, optimization algorithms for fine-tuning of gain constants [27], etc., where first-order transfer functions have been executed. Even though the author [25] has derived a linearized model for a solar thermal type for solar and wind, the representation of the system is confined to first-order lag transfer function form. These ideal stratagems without the inclusion of its dynamical effect resulted in regular behavior of the system characteristics under all operating conditions. The process of addressing the drawback led to the motivation of introducing derived state space formations for the intermittent supplies in the frequency control of microgrids.

Modest controllers such as I, PI, PID etc., were mainly used in order to minimalize the deviations in frequency, tie-line, and realize null change between the overall supply and load. However, in the recent era, fractional order (FO), model predictive, fuzzy and neural-based controllers are extensively used in solving all frequency-related issues [28,29] with minimal analytical periods and reduced complexity in the system dynamics. Among all, the classified and cascaded FO controllers have been leveraged in different applications [30] for performance evaluation. The 2DOF-FOPID [31] has greater damping ability in terms of peak over and under shoots, but the secondary control loop along with the demand change as the feedback input in the 3 DOF-FOPID controller makes it more suitable and viable for the dynamic response of highly complicated and sophisticated systems. The tuning of gain constants of the controllers is done either manually or optimally. The GA [32] is widely used for optimal tuning of controller gains because of its simplicity and convergence characteristics. The systems investigated for frequency-based issues are subjected to step load and random load fluctuations [33], so as to study its effectiveness on a real-time basis. In all the above-mentioned types, first-order transfer functions are used, which results in simplicity of structure and fast computational time while carrying out frequency regulation for interconnected systems. The main drawback of these first-order systems is that, on evaluating its performance, the exact dynamic response of the system cannot be measured.

Recently, the application of hybrid algorithms such as PSO-GSA, HHOPSO along with cascaded controllers [24,25], played a vital role in load frequency control. The response of microgrid during regulation of frequency in the presence of latest and hybrid algorithms are clearly mentioned and justified. The most topical approach is the Seagull optimization Algorithm (SOA), which is proven to be more suitable for solving computationally classy problems [34]. The realization of the system configuration is explicitly addressed in algorithms such as Moth flame optimization (MFO) [35], Multiverse Optimization (MVO) [36], Marine Predator algorithm (MPA) [37], Generalized Normal Distribution Optimization (GNDO), Gorilla Troops Optimizer (GTO) [38], and several more. However, amongst all the recently evolved algorithms, Artificial Hummingbird Algorithm’s (AHA) capability of solving highly complex systems is very high since it has befitted and proved its efficiency in twenty-three and forty-four benchmark test functions [39]. Such a promising algorithm has provided remarkable solutions to all cost functions. The algorithm has the capability to capture the local minima without altering the control variables exclusively and suppressing the oscillations even in the presence of non-linearities across thermal generators. This latest process has still not been implemented in frequency regulation-related problems.

On summarizing the literature, it is crucial to recognize the significant controller and the optimal approach for estimating the control parameters required in the execution of frequency control in the suggested microgrid system. The main scope of the controller should be highly confined to the overall improvement of the time domain specifications. For the aforementioned reasons, the AHA-based 3DOF-FOPID controller in the improved microgrid network representation fits in close proximity to the requirement.

Thus, the vital role of the article is to propose the state-space modelling of renewable stratagem for the specific quantities and extract the non-linear transfer functions for the frequency regulation of the microgrid system. These variable sources need a rigorous evaluation of system dynamics so as to represent the real-time frequency and tie-line changes. The microgrid system is an interconnected two-area identical network that is inclusive of generating units such as Diesel Engine Generator (DEG) with Reheat and Non-Reheat Turbines, Photovoltaic (PV) system, Wind Turbine Generator (WTG), Bio-Gas plant (BGP) system [40], and Battery Energy Storage System (BESS), as presented in Figure 1; the system specifications are seen in Figure 2.

The efficacy of the AHA-based 3DOF-FOPID controller is analyzed with other algorithms in transient conditions under the effect of non-linearities such as Generation Rate Constraint (GRC) and Governor dead band (GDB). Abbreviations extends the comprehensive nomenclature of the article.

The foremost highlights/contributions of the work are listed as:

• Derivation of SS models for solar and wind systems for the given specifications.
• Development of identical two area microgrid model fed with standard functions for DEG, BGP, and BESS.
• The proposed higher degree polynomials of PV and wind are inserted into the system for rigorous testing of its behavior and impact.
• Simultaneous operation of different controllers such as PID, FOPID, 2DOF-FOPID, and 3DOF-FOPID controller in both the areas.
• Proposed network subjected to step perturbations and compared with PSO, PSO-GSA, SOA, MFO, MVO, MPA, GNDO, and GTO. The evident simulation results justify the proposed system to be more preferable when compared with the existing stigma.
• Procuring transient and sensitivity analysis under thermal constraints (GRC, GDB) and variable operating conditions (±25%, ±50% tolerance), respectively, which confines the feasibility and compatibility of the proposed network.

The remaining sequence of the article is enlisted as: detailed state-space modelling of solar-wind system and description of the conventional generating sources in Section 2, a report on the 2 stage and 3 stage FOPID controllers in Section 3, skeletal of the anticipated heuristic approach along with its pseudocode in Section 4, simulation outcomes with different loading conditions in Section 5, hardware implementation of the structure in the OPAL-RT platform and conclusion in Section 6 and Section 7, respectively.

2. Microgrid System Model

The basic illustration of the proposed microgrid system comprises two DEG units (both non-reheat and reheat type) and one BGP, PV, WTG, and BESS framing the single area (Figure 3).

The identical single areas are interconnected to form the two-area network subjected to comprehensive analysis and application. The components are summated to form the overall change in generation. The load change is given as the small disturbance pattern to estimate the change in frequency and tie-line errors. The controller is more specific in minimizing the error because of the divergence that occurs between the generation and load. The percentile proportion of energy contribution in Figure 4 describes the overall generation of 60 KW and demand of 55 KW. Non-conventional sources provide a major contribution in generation and transmission of power. These sources have been applied in most of the areas, especially LFC along with energy storage and FACTS devices [17,19,39].

2.1. Discrete Representation of the Components

2.1.1. Diesel Engine Generator (DEG)

The equilibrium between the total generation and load is maintained by the governor and turbine constants of the generator [17,19,20]. The deviation is approximated to zero using the speed control action in an autonomous microgrid system. The standard form of non-reheat (NR) turbine and two-stage turbines (R) used for the generator model is displayed in Equations (1) and (2) [34].

$$\Delta P_{DEG,NR}=\left(\frac{K_g}{1+S{T}_g}\right)\left(\frac{K_t}{1+S{T}_t}\right)$$

(1)

$$\Delta P_{DEG,R}=\left(\frac{1}{1+S{T}_t}\right)\left(\frac{K_r{T}_r}{1+S{T}_r}\right)$$

(2)

2.1.2. Biogas Plant (BGP) Model

The usage of biogas for the generation of power has been substantially increasing in the production market, so that growing demands across the globe can be addressed without depending on the natural sources [40]. Biogas is the resultant of organic waste that is decomposed to form methane, carbon dioxide, and hydrogen sulphide. The biogas plant comprises a governor, valve position, fuel system, and turbine unit. The linearized model is suggested below,

$$\Delta P_{BGP}=\frac{\left(1+s{X}_g\right)\left(1+s{T}_{cr}\right)}{\left(1+S{Y}_g\right)\left(1+S{b}_v\right)\left(1+s{T}_f\right)\left(1+s{T}_{cd}\right)}.$$

(3)

2.1.3. Battery Energy Storage Systems (BESS)

In general, the modelling of BESS comprises the batteries, power conditioning circuits, and control strategies. The batteries are stacked up in series/parallel combinations where chemical to electrical energy conversion takes place [23]. For long-term energy management and storage, battery backup is most essential. Electrical energy derived from solar is usually stored in battery systems. The response is given by,

$$\Delta P_{BESS}=\frac{K_{BESS}}{1+S{T}_{BESS}}.$$

(4)

2.1.4. Load Model

The change in load demand ∆P_D is given to the system in the form of single and multi-step load (case 1) and random fluctuations (case 2) [27,31] for estimating the mismatch between supply and demand so that the feedback in the form of ∆f is given to the controllers.

2.1.5. Power Model

The linear form of the power system model is given by

$$\frac{\Delta f}{\Delta P_g-\Delta P_L}=\frac{K_{ps}}{1+s{T}_{ps}}$$

(5)

The variance $\left(\Delta P_G-\Delta P_D\right)$ between the total change in power generation and demand is given as input to the transfer function model in order to obtain the change in frequency as the output.

$$\Delta P_G=\Delta P_{DEG,NR}+\Delta P_{DEG,R}+\Delta P_{BGP}+\Delta P_{PV}+\Delta P_{WTG}\pm \Delta P_{BESS}$$

(6)

2.1.6. Renewable Integration Models

Among the existing renewable energy resources, the most abundant form of energy available is solar and wind which is a prerequisite and is sustainable throughout the globe. In recent years, PV and WTG systems have been at the forefront in supplying power generation to local loads. Extensive application-oriented research has been carried out using PV systems [20,23,26,27]. The linearized form of PV and wind are given as

$$\Delta P_{pv}=\frac{K_{PV}}{1+S{T}_{PV}}$$

(7)

$$\Delta P_{WTG}=\frac{K_{WTG}}{1+S{T}_{WTG}}$$

(8)

However, these typical forms do not precisely replicate the dynamics of the PV and WTG systems in real scenarios and represent its equivalent system rating as in Figure 2. In order to implicate its system features, this article proposes the extraction of the SS model from the equivalent circuit of the single diode PV model and wind turbine generator.

2.2. SS Formation of Renewables

2.2.1. PV System

For the Single Diode Equivalent Circuit [41,42] in Figure 5, the output current across the PV array is

$$I_{pv}=N_{ph}\ast I_{ph}-N_P\ast I_o\left[exp\left\{\frac{q\ast \left(V_{PV}+I_{PV}{R}_s\right)}{N_{s}AKT}\right\}-1\right]$$

(9)

where $I_{ph}$ and $I_o$ are the photovoltaic and saturation currents of the PV ideal cell, K Boltzmann constant, q Coulomb constant, T is the cell temperature, and A is the ideality factor. For the constant irradiance G of 1000 W/m² and a temperature of 25 °C, the $N_p$ and $N_s$ cells decide the output PV voltage $V_{pv}$ and output PV current $I_{pv}$. The output voltage $V_o$ is measured across the capacitance C. For the quantified details in datasheet of PV (

Table 1. Datasheet of PV.

Type	Spec. Variables	Parametric Values
PV panel	Voc, Isc VMPP, IMPP Power Pmax (W)	49.7 V, 13.82 A 41.9 V, 13.13 A 550 W
PV Array (3 × 10) {3 modules in series and 10 modules in parallel}	VOC, ISC VMPP, IMPP Power Pmax (W)	149.1 V, 152.02 A 141.3 V, 151.33 A 16 KW
Passive Elements	Rs, RP C Ns × Np	0.1685, 103 4700 μF 3 × 10

), the PV panel and array is simulated and the I-V and V-P characteristics are extracted at maximum power point (Figure 6).

On solving the equations, the state-space form is expressed as

$$\frac{d}{dt}\left[V_{pv}\right]=\left[\frac{-1}{C\left(R_p+R_s\right)}\right] \left[V_{pv}\right]+\left[\frac{R_p}{c\left(R_p+R_s\right)}\right] \left[I_{pv}\right]$$

(10)

$$\left[V_o\right]=\left[1\right] \left[V_{pv}\right]+\left[0\right] \left[I_{pv}\right]$$

(11)

where $\left[A\right]=\left[\frac{-1}{C\left(R_p+R_s\right)}\right], \left[B\right]=\left[\frac{R_p}{c\left(R_p+R_s\right)}\right], \left[C\right]=\left[1\right], \left[D\right]=\left[0\right]$.

2.2.2. Wind System

The wind turbine is widely used for extracting the dynamic energy from wind energy and converting it into mechanical energy [20,21,22,23,24,25]. It comprises a wind turbine along with a driver circuit, generator, and power converter circuit. The kinetic energy is extracted from the swept area of the blades of the turbine. The mechanical power extracted from the wind is given by [28,29,30],

$$P_m=0.5\ast \rho \ast A\ast {\upsilon }^3\ast C_P\left(\lambda ,\beta \right)$$

(12)

where the mathematical estimation of the power coefficient and tip speed ratio is given by

$C_P\left(\lambda ,\beta \right)$ = $0.22\left(\frac{116}{\gamma }-0.4\ast \beta -5\right).\exp \left(\frac{-12.5}{\gamma }\right)$ with $\frac{1}{\gamma }=\frac{1}{\lambda +0.08\beta }-\frac{0.035}{{\beta }^3+1}$, $\lambda =\left({\omega }_{t}R\right)/\upsilon$.

The mechanical torque

$$T_m=\frac{P_m}{\omega }$$

(13)

The modelling and computation of the PMSG- based wind turbine generator [43,44] is usually not presented in the abc reference frame because of its complicated and tedious representation. However, in this proposal, the equivalent circuit of the three-phase winding of the generator (Figure 7) with an LC filter and three-phase parallel RLC load is considered for deriving the state-space form. For the given standards of wind speed and pitch angle (

Table 2. Datasheet of WTG system.

Spec. Variables of 4 KW	Parametric Values
Pitch Angle β	0
Rotor Radius R	3.266 m
Wind speed, Vm	7.7 m/s
Blade angular velocity, ωm	15 rad/s
Armature Resistance Ra	1.5 Ω
Load resistance RL	7.5 Ω
Synchronous inductance Ls	0.115 H
Load impedance, ZL	326.66 + j 5.4358

), mechanical power can be extracted for 4 KW (Figure 8).

For equivalent circuit, the state-space form for the ‘a’ phase winding of the generator is attained as

$$\left[V_{0,a}\right]=\left[\begin{array}{ccccc}0& 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{i}_a\\ V_{c{f}_a}\\ i_{R_{La}}\\ i_{L_{La}}\\ V_{C_{La}}\end{array}\right] \mathrm{Output} \mathrm{equation}$$

(14)

$$\begin{gathered} \frac{d}{dt}\left[\begin{array}{c}{i}_a\\ V_{c{f}_a}\\ i_{R_{La}}\\ i_{L_{La}}\\ V_{C_{La}}\end{array}\right]=\left[\begin{array}{ccccc}-\left(\frac{R_{aa}+R_L}{L_{sa}+L_{fa}}\right)& -\left(\frac{1}{L_{sa}+L_{fa}}\right)& 0 & 0 & 0\\ 0& \left(\frac{1}{C_{fa}{R}_L}\right)& 0& 0& 0\\ 0& \left(\frac{1}{L_{fa}}\right)& 0& 0& 0\\ 0& 0& 0& 0& \left(\frac{1}{L_{La}}\right)\\ 0& 0& \left(\frac{1}{C_{La}}\right)& 0& 0\end{array}\right] \left[\begin{array}{c}{i}_a\\ V_{c{f}_a}\\ i_{R_{La}}\\ i_{L_{La}}\\ V_{C_{La}}\end{array}\right] \\ +\left[\begin{array}{c}\left(\frac{1}{L_{sa}+L_{fa}}\right)\\ 0\\ 0\\ 0\\ 0\end{array}\right] \left[\begin{array}{ccccc}{e}_a& 0& 0& 0& 0\end{array}\right] \\ \mathrm{State} \mathrm{equation} \end{gathered}$$

(15)

Similarly, for the ‘b’ phase and ‘c’ phase, state and output equations can be considered with the resultant of 15 state variables. The frequency regulation is executed, upon converting these SS into transfer functions.

The above components are organized together to form a two-area system for carrying LFC, and the value of AFRC is considered for maintaining a higher frequency bias and contributing adequate frequency control [34].

3. Summary of Two and Three Stage FOPID Controllers

The preliminary purpose of a primary controller in the outer loop of a control scheme is to estimate the error difference and nullify it to zero. Recently, the PID controller performance has been widely updated by introducing non-integer type integrating and derivative orders such as λ and μ [28,29] with the transfer function given by

$$G_{c,FOPID,i}\left(s\right)=K_{pi}\left(s\right)+\frac{K_{Ii}}{s^{\lambda i}}\left(s\right)+K_{Di}{s}^{\mu i}\left(s\right)0\le \lambda ,\mu \le 0)$$

(16)

The choice of FOPID controller for solving complex and non-linear system is because of its additional degree of freedom that results in improved control performance. The integrating and differentiating integer powers suppress the classical effects because of overshoot and resonance for any chaotic performance in the mathematical models. The dominance nature of such a controller has resulted in the expansion of the classical to stage controllers. The two-stage FOPID (Figure 9) has reference signal R(s) and the control error signal Y(s) fed as inputs to the proportional $(P_W)$ and derivative set point weights $(P_D)$ [30,31]. The implementation of 2DOF-FOPID in the investigation of frequency change at microgrids is widely acknowledged because of its robust performance enhancement at critical and variable operating conditions.

$$\begin{gathered} G_{c,2DOF-FOPIDi}\left(s\right)=K_{Pi}.\left[R\left(s\right)\ast P_{wi}-Y\left(s\right)\right]+\frac{K_{Ii}\left[R\left(s\right)-Y\left(s\right)\right]}{s^{\lambda i}\left(s\right)}+K_{D.}{S}^{\mu i}\left[R\left(s\right)\ast P_{Di}-Y\left(s\right)\right] \\ \left(0\le \lambda i,\mu i\le 1\right); \left(0\le P_{wi},P_{Di}\le 5\right); i=1,2; \end{gathered}$$

(17)

An additional feedforward signal ff(s) and the disturbance signal D(s) is considered in the formation of the three-stage FOPID (Figure 10) [30,31].

$$\begin{gathered} G_{c,3DOF-FOPIDi}\left(s\right)=K_{Pi}.\left[R\left(s\right)\ast P_{wi}-Y\left(s\right)+D\left(s\right)\ast ff\left(s\right)\right] \\ +\frac{K_{Ii}\left[R\left(s\right)-Y\left(s\right)+D\left(s\right)\ast ff\left(s\right)\right]}{s^{\lambda i}\left(s\right)} \\ +K_{D.}{S}^{\mu i} \left[R\left(s\right)\ast P_{Di}-Y\left(s\right)+D\left(s\right)\ast ff\left(s\right)\right] \\ \left(0\le \lambda i,\mu i\le 1\right); \left(0\le P_{wi},P_{Di}\le 5\right); i=1,2; \end{gathered}$$

(18)

Cumulatively, 14 variables are to be optimized for the stage controllers in the two-area network. The frequency change and power flow at tie-line are given as inputs for tuning the gains of the controller. ITAE is one such fitness function that has minimum overshoots/undershoots and disturbances [20]. The expression along with its bounds are given below

$$\mathrm{Fitness} \mathrm{function} \mathrm{J}=\mathrm{Minimize} \left\{\mathrm{ITAE}\right\}=\mathrm{Min} \{{{\displaystyle \int }}_0^{T}t. (|\left(\Delta f_1\left|+\left|\Delta f_2\right|+\right|\Delta P_{tie}|\right).dt\}$$

(19)

Sub to FOPID limits

$$\left\{\begin{array}{c}{K}_{min}\le K_{P_i}\le K_{max}\\ K_{min}\le K_{I_i}\le K_{max}\\ K_{min}\le K_{D_i}\le K_{max}\\ {\lambda }_{min}\le {\lambda }_i\le {\lambda }_{max}\\ {\mu }_{min}\le {\mu }_i\le {\mu }_{max}\\ P_{w,min}\le P_{wi}\le P_{w,max}\\ P_{D,min}\le P_{Di}\le P_{D,max}\end{array}\right.$$

(20)

ACE (area control error) is given as the input signal to the controller, and is formulated as

$$AC{E}_i=B_i\Delta f_i+\Delta P_{tie,ij}$$

(21)

The importance of ITAE [5] is enhanced because of its sensitivity and compatibility in producing minimal overshoots. The scenarios of examination for the proposed network are listed in

Table 3. Scenarios for validating the proposed system.

Cases	Components of the Microgrid System	Simulation Time (Secs)	Range 1	Load Pattern
1	DEG, PV, WTG, BESS, PD1	180	$P_{PV}=\left\{\begin{array}{c}0.01 p.u., 0\le t\le 120\\ 0.009 p.u., 120\le t\le 180\end{array}\right.$ $P_{WTG}=\left\{\begin{array}{c}0.03 p.u. , 0\le t\le 120\\ 0.01 p.u.,\hspace{1em}120\le t\le 180\end{array}\right.$ $P_{D1}=\left\{\begin{array}{c}0.01 p.u. , 0\le t\le 120\\ 0.015 p.u.,\hspace{1em}120\le t\le 180\end{array}\right.$	Figure 11a
2	DEG, PV, WTG, BESS, PD1	180	Multi-step disturbances	Figure 11b

¹ Ranges are calculated based on the system specifications in p.u.

. The single and multi-step load pattern [27,31] for the distributed generations and demand is displayed in Figure 11.

4. Skeletal of Anticipated Heuristic Approach

The Artificial Hummingbird Algorithm is a bio-inspired algorithm that captures the flighting skills and foraging techniques of hummingbirds in the environment. The flight skills include axial, diagonal, and omnidirectional methods. Further guided foraging, territorial foraging, and migrating foraging are realized along with the visit table for modelling the memory function of hummingbirds for collecting food particles [39]. The effectiveness of the proposed algorithm is proven when it is subjected to computational burdens and the precision of the solution results in the most suitable optimization algorithm for the load frequency control at microgrids with variable generating resources. The steps involved in the AHA are listed below:

4.1. Initialization

A randomly initialized huge population of n hummingbirds seated on n food particles is given by

$$x_i=Less+r.\left(More-less\right)\hspace{1em}i=1,2\dots \dots n$$

(22)

where More and less are the boundary limits for the d-dimensional problem, r is the random vector [0–1], and x_i is the location of the i^th food particle. The visit tabulation is listed below for the initialization of the food particles.

$$V{T}_{i,j}=\left\{\begin{array}{c}0\hspace{1em}if\hspace{1em}i\ne j\\ null\hspace{1em}i\ne j\end{array}i=1,2\dots \dots n;j=1,2\dots \dots n\right\}$$

(23)

4.2. Guided Foraging

Each hummingbird visits the food particle with a huge nectar volume so that the visiting level is more for the guided foraging behaviour. Once the target is fixed, the hummingbird flies for the feeding process. The different flight skills are included in the algorithm by using a direction switch vector for estimating the directions available in the d-dimension space. The hummingbirds master the axial and diagonal flights rather than the omnidirectional flight. The axial is given by

$$D^{\left(i\right)}=\left\{\begin{array}{c}1\hspace{1em}if\hspace{1em}i=randi\left(\left[1,d\right]\right)\\ 0\hspace{1em}else \end{array}\right.\hspace{1em}i=1,2,\dots \dots d$$

(24)

The diagonal flight is

$$D^{\left(i\right)}=\left\{\begin{array}{c}1\hspace{1em}if\hspace{1em}i=P_j,\hspace{1em}j \in \left[1,k\right], P=randperm\left( k\right),\\ 0\hspace{1em}else \end{array}\right.i=1,2,\dots \dots d$$

(25)

The omnidirectional flight is

$$D^{\left(i\right)}=1\hspace{1em}\hspace{1em}\hspace{1em}i=1,2,\dots \dots d$$

(26)

The expression for the guided foraging behavior is

$$v_i\left(t+1\right)=x_{i,tar}\left(t\right)+a.D.(x_i\left(t\right)-x_{i,tar}\left(t\right), \mathrm{a} ~ \mathrm{N} (0,1)$$

(27)

where x_i(t) gives the i^th food position at t time, and x_i,tar(t) gives the target food particle position that is visited by the i^th hummingbird with guided factor ‘a’.

The i^th food particle position is updated position by

$$x_i\left(t+1\right)=\left\{\begin{array}{c}{x}_i\left(t\right)\hspace{1em}\hspace{1em}\hspace{1em}f(x_i\left(t\right))\le f(v_i\left(t+1\right))\\ v_i\left(t+1\right)\hspace{1em}\hspace{1em}f(x_i\left(t\right))\le f(v_i\left(t+1\right))\end{array}\right.$$

(28)

4.3. Territorial Foraging

When the flower nectar has been eaten by the hummingbird on visiting the target food particle, it reaches the neighboring region within its boundary for a new food source. The equation that describes the local search of the hummingbird in the above strategy with the desired food particle is given as

$$v_i\left(t+1\right)=x_i\left(t\right)+b.D.x_i\left(t\right)$$

(29)

where b~N(0,1) and b is the territorial factor subjected to mean and standard deviation of 0 and 1.

4.4. Migration Foraging

The migration process begins when the hummingbird lacks food, and so the search is expanded to a distant region. In this algorithm, the migration coefficient is defined as constant and when the number of iterations exceeds the coefficient, the hummingbird migrates to the new location by leaving behind the old food source, resulting in the revision of the visit table. The foraging process from the source with the poor refilling rate of nectar to the new place randomly is given by

$$x_{wor}\left(t+1\right)=Less+r.\left(More-Less\right)$$

(30)

where x_wor is the poor nectar-filling rate-based food particle.

The flow diagram (Figure 12) commences with the initialization of parameters essential for the optimization. The variables of frequency regulation are taken into account for approximating the fitness function across the best food particle. The decision of finding the new food source is based on the types of foraging technique. On consecutive inspection of the predetermined value for migration foraging, the termination criterion is satisfied. The iterative process is executed for the maximum number of iterations so as to obtain the best optimal solutions for the required parameters.

5. Investigations and Discussions

The acceptable function of any control system is based on faster response and stability. The time-domain specifications, such as settling time and peak overshoot/undershoot with minimal performance indices, are generally used to determine the performance of any system. The functional operation of the proposed higher-order system is analyzed by exposing it to stringent cases mentioned in Table 3 and executing its dynamic evaluation under different meta-heuristic approaches.

5.1. Transient Analysis

The dominance of the system is analyzed in the condition where the system is connected as isolated two-area multi-microgrid systems interconnected in the presence of DEG, BGP, PV, WTG, and BESS. The comparative analysis is done with different controllers such as PID, FOPID, 2DOF-FOPID, and 3 DOF-FOPID. The performance index, ITAE, is calculated for these controllers in different scenarios, and feasible and acceptable solutions are derived. Further, the performance of the proposed system in the presence of advanced 3DOF-FOPID controller against different algorithms (Figure 13) is compared and justifiable results prove the effective action of the AHA-based 3DOF-FOPID controller for the presented work. All experimentations are implemented in MATLAB R2018 b, Microsoft windows 8 (64 bit), with an i5 core processor with 3 GHz and 8 GB RAM.

5.1.1. Assessment of Multi-Microgrid with Single-Step Load Disturbance in Area 1

In this scenario, two microgrids are interconnected using a tie-line and subjected to step load disturbance in area 1 (Figure 11). The deviations in frequency and tie-line of the higher-order system in the presence of the PSO-optimized 3DOF-FOPID controller is compared against PID, FOPID, and 2DOF-FOPID, and the fallouts are displayed in Figure 14. The settling time (ts), maximum peak overshoot (MPo+) and optimum gains are tabulated in

Table 4. ts, MPo+, and optimal gains of controllers for comparison.

Parametric Constants	Controllers—Single Step Disturbance in Area 1 @ 0 Secs
	PID		FOPID		2DOF-FOPID		3DOF-FOPID (Proposed)
	ts (secs)	MPo+	ts (secs)	MPo+	ts (secs)	MPo+	ts (secs)	MPo+
$\Delta f_1\left(Hz\right)$	14.35	0.0593	3	0.0602	18.9	0.0537	7.9	0.0140
$\Delta f_2\left(Hz\right)$	12.4	0.0611	12.65	0.0613	6.6	0.0464	15.95	0.0128
$\Delta P_{tie12}\left(p.u.\right)$	16.65	NA	27.8	NA	9.85	0.0502	25.3	0.0066
FODITAE	52.131		57.110		57.57		46.415
	A1	A2	A1	A2	A1	A2	A1	A2
KP	1	1	2	0.510	0.512	0.5124	0.0649	0.0623
KI	0.1937	0.1311	0.502	1.300	0.123	0.235	0.0218	0.0538
KD	1	1	1.837	1.496	0.245	0.334	0.1000	0.1000
$K_{\lambda }$	-	-	1	0.3328	0.951	0.875	0.8651	1
$K_{\mu }$	-	-	0.664	0.698	0.864	1	0.7539	0.7751
PW	-	-	-	-	4.46	5	3.657	4.7590
PD	-	-	-	-	4.583	2.322	4.435	5

. It is evident that ∆f₁, ∆f₂, and ∆P_tie of the 3DOF-FOPID based system have lesser ts and FODITAE when compared to the most prominent controllers available in the literature. The best and optimal solutions are obtained when a versatile and highly convergent PSO optimized 3DOF-FOPID controller is used because of its feed forward ability and input disturbance signal. The peak over shoots epitomizes the actual response of the system. For the FOD expression in (19), the cost evaluation is implemented and results in 46.415, which is minimal amongst the existing controllers. At the 50th iteration with initial disturbance at 0 sec, the Mpo+ for the ∆f₁, ∆f₂, and ∆P_tie are 0.0140, 0.0128, and 0.0066, respectively. In spite of deduced overshoots, the period taken for the deviations to saturate is little longer when compared to preceding controllers. Thus, 3DOF-FOPID is exposed to promising algorithms of the literature to prove the befitting capability of the proposed algorithm.

Table 4 displays the finest and best suited gain constants of the controllers for a step load variation, and the performance index is calculated based on the FOD, showing that the derived SS model can provide actual, optimal, and acceptable performance in LFC. Using AHA, the controller gain constants are measured, and ITAE is estimated in the upcoming section.

5.1.2. Assessment with the Modern Heuristic Method in the Absence of Constraints

The newest artificial hummingbird algorithm is yet to be applied in the field of testing frequency-related issues. Thus, this article delivers the application of AHA in the 3DOF-FOPID controller-based microgrid system for realizing load frequency control. The study of its effectiveness is well established in Figure 15, where the MPo+ of the anticipating algorithm is minimal and the ts is far saturated in advance of the remaining algorithms.

Concerning AHA in the 3DOF-FOPID controller-based microgrid system for realizing load frequency control, the study of its effectiveness is well established in Figure 15, where the MPo+ of the anticipating algorithm is minimal and the ts is far saturated in advance of the remaining algorithms.

Table 5. ts, MPo+, and optimal gains of controllers with AHA against prevailing algorithms for comparison.

Parmetrizes/Algorithms	Controllers—Single Step Disturbance in Area 1 @ 0 Secs for 3DOF-FOPID
	PSO		PSOGSA		SOA		MFO		MVO		MPA		GNDO		GTO		AHA (Proposed)
	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+
$\Delta f_1\left(Hz\right)$	7.9	0.014	8.3	0.057	6.4	0.091	12.65	0.067	11.3	0.001	11.25	0.057	9.4	0.074	9.35	0.064	6.7	0.062
$\Delta f_2\left(Hz\right)$	15.95	0.012	12.6	0.073	3.45	0.069	9.45	0.072	9.5	0.066	9.5	0.056	6.2	0.066	6.45	0.064	6	0.063
$\Delta P_{tie12}$ $\left(p.u\right)$	25.3	0.006	0.007	NA	15.35	0.011	41.25	0.002	14.75	0.067	15.55	NA	15.25	NA	22.45	0.001	12.3	0.001
FODITAE	46.415		64.380		93.1354		50.454		48.479		43.9902		73.325		71.322		40.9703
Gains/Areas	A1	A2	A1	A2	A1	A2	A1	A2	A1	A2	A1	A2	A1	A2	A1	A2	A1	A2
KP	0.064	0.062	0.1	0.096	0.040	0	0.089	0.059	0.041	0.064	0.100	0.100	0.094	0.070	0.100	0.099	0.097	0.086
KI	0.021	0.053	0	0.023	0	0.050	0.078	0.080	0.019	0.097	0.100	0.099	0	0.027	0	0.021	0.087	0.044
KD	0.100	0.100	0.1	0.090	0.100	0.100	0.100	0.100	0.096	0.073	0.100	1.100	0.081	0.070	0.046	0.057	0.093	0.088
$K_{\lambda }$	0.865	1	0.858	0.999	0.934	1	0.974	0.989	0.769	0.991	0.946	0.100	0.978	0.984	0.951	0.949	0.894	0.907
$K_{\mu }$	0.753	0.775	0.744	0.707	0.580	0.622	0.688	0.757	0.769	0.738	0.789	0.788	0.748	0.703	0.694	0.678	0.758	0.741
PW	3.657	4.759	5	4.169	5	0	1.331	4.730	0.645	2.500	5	4.954	4.192	2.258	4.878	4.981	3.817	4.002
PD	4.435	5	4.677	3.472	0	0	1.762	4.764	4.927	4.784	5	5	5	4.866	5	4.987	4.494	3.410

showcases the time domain specifications and gain constants of the control plant for the suggested two-area structure. The trails are carried out in the absence of thermal constraints such as GRC and GDB, where its limiting effect is not considered.

5.1.3. Valuation with the Modern Heuristic Method in the Presence of GRC and GDB

The decentralized microgrid structure has two thermal units in each area with reheat and non-reheat turbines. The effect of non-linearities confines the ramp rate of the turbines, resulting in a weaker dynamic response with larger overshoots and slower ts. Nominally, the GRC ranges between 3%/min to 10%/min [44] where the system deterioration occurs when it exceeds beyond 10%/min. Thus, the GRC for both the turbines is fixed at 3%/min. The GDB is the defined scale of constant change in speed within which no valve position change is acceptable and the droop for the thermal plates is fixed at 4%. Thus, the validation is extracted for the suggested structure against the aforementioned algorithms in the presence of system non-linearities.

Figure 16 demonstrates the impact of GRC and GDB in the thermal turbines resulting in restrained frequency and tie-line fluctuations. These limiters control the system oscillations caused by load disturbances, making it trivial against various odds. On witnessing the measurements in

Table 6. ts, MPo+, and optimal gains of controllers with AHA against system non-linearities.

Parameters/Algorithms	Controllers—Single Step Disturbance in Area 1 @ 0 Secs for 3DOF-FOPID
	PSO		PSOGSA		SOA		MFO		MVO		MPA		GNDO		GTO		AHA
	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+	ts	MPo+
$\Delta f_1\left(Hz\right)$	17.60	0.276	18.84	0.248	15.85	0.233	18.15	0.281	17.3	0.272	15.95	0.260	15.1	0.297	15.2	0.279	14.90	0.278
$\Delta f_2\left(Hz\right)$	18.50	0.240	18.28	0.240	15.65	0.247	20.4	0.241	15.65	0.263	15.95	0.024	12.03	0.280	12.55	0.252	12.75	0.270
$\Delta P_{tie12}$ $\left(p.u\right)$	17.62	0.014	17.62	0.014	17.70	0.009	22.05	0.023	14.51	0.295	26.12	0.016	12.05	0.001	7.24	0.024	6.23	0.014
Gains/Areas	A1	A2	A1	A2	A1	A2	A1	A2	A1	A2	A1	A2	A1	A2	A1	A2	A1	A2
KP	0.100	0.044	0.091	0.084	0.074	0.057	0.088	0.100	0.099	0.031	0.036	0.099	0.042	0.048	0.002	0.041	0.100	0.087
KI	0.037	0.063	0.030	0.036	0.095	0.091	0.003	0.052	0.044	0.083	0.024	0.078	0.043	0.070	0	0.091	0.098	0.099
KD	0.100	0.100	0.089	0.079	0.100	0.100	0.065	0.100	0.068	0.090	0.071	0.094	0.053	0.072	0.099	0.100	0.099	0.100
$K_{\lambda }$	0.696	0.636	0.569	0.557	0.636	0.639	0.776	0.585	0.720	0.739	0.722	0.923	0.071	0.812	0.997	0.686	0.8103	0.697
$K_{\mu }$	0.520	0.771	0.798	0.698	0.785	0.757	0.662	0.797	0.682	0.481	0.491	0.649	0.519	0.628	0.777	0.704	0.796	0.739
PW	4.991	4.480	4.017	4.177	2.040	5	0.776	5	2.628	2.386	2.991	4.540	1.849	3.243	3.022	0.009	5	0
PD	0	5	4.910	4.431	5	4.137	4.101	4.988	4.92	0.002	4.045	4.320	1.642	3.082	4.421	3.972	4.96	5
FODITAE	241.75		239.26		241.557		228.89		255.84		235.97		360.60		245.40		225.11
FODITSE	7.654		7.824		7.524		7.641		7.827		7.450		8.205		7.448		6.899
FODIAE	0.180		0.202		0.180		0.190		0.197		0.184		0.226		0.210		0.172
FODISE	0.224		0.2005		0.198		0.182		0.191		0.180		0.209		0.188		0.203

, the outcome of the system’s reaction is obvious in terms of MPo+ and ts. The setup is also exposed to typical FODs (Figure 17) for professing the effectiveness of ITAE over ISE, whereby IAE and ITSE are dominant and active in terms of oscillations and settling periods. The numerical solutions conclude that ITAE is widely acceptable for stringent evaluation on system settings. Thus, the convergence characteristics of ITAE for the test system with recommended PSO, PSO-GSA, SOA, MFO, MVO, MPA, GNDO, GTO, and AHA is presented in Figure 18. It is viable that AHA performs very well in terms of computational capability and faster convergence rate, so that the global solution is attained at 20 iterations whereas the other explicit algorithms settle down only after 30 to 40 iterations.

5.1.4. Effect of Multi-step Disturbance in Realization of Response

The expected variation in frequency at area 1 is obtained when the skeletal is proposed to multi-step load disturbances. The response curve is displayed in Figure 19.

5.2. Sensitivity Analysis

To examine the robustness and effectiveness of the anticipated system with the 3DOF-FOPID based AHA controller, the formulated SS-based two-area network is subjected to parametric uncertainties such as ±25% and ±50% of the loading condition at Tt, Tg, Tr, Tps, Kr, Kps, Ri, and Bi (Figure 20). The proposed test system is exposed to step disturbance and uncertainties within the tolerance band at area-1. In the above constraints, the globally used AHA resolves the fluctuations in frequency and tie-line deviations with minimal response characteristics, optimal gain constants, and precise fitness function. The system responds to the changes that take place and prove AHA is preferable for the application.

Table 7. Sensitivity Analysis for percentile variation in parameters.

Quantity	% Change	FODITAE	Settling Period (Secs)			Controller Gains
			∆f1	∆f2	∆Ptie12	KP1	KI1	KD1	Kλ1	Kμ1	PW1	PD1	KP2	KI2	KD2	Kλ2	Kμ2	PW2	PD2
Standard load	Nil	225.11	14.90	12.75	16	0.100	0.098	0.099	0.810	0.796	5	4.96	0.087	0.099	0.100	0.697	0.739	0	5
Tt	+50	275.82	14.7	15.40	16.13	0.013	0	0.091	0.794	0.618	3.459	1.610	0.046	0.090	0.078	0.732	0.549	3.737	1.267
	+25	300.17	12.5	14.11	16.03	0.004	0	0.084	0.862	0.681	3.776	3.827	0	0.079	0.024	0.766	0.505	0.548	3.626
	−25	291.43	9.61	15.69	13.75	0.074	0.001	0.091	0.904	0.635	2.750	2.683	0.053	0.069	0.092	0.799	0.739	3.702	3.206
	−50	269.07	9.61	17.71	16.15	0.066	0	0.065	0.973	0.650	4.203	3.316	0.027	0.089	0.090	0.725	0.559	3.471	0.802
Tg	+50	239.7	9.45	10.58	16.00	0.051	0.057	0.095	0.729	0.693	0.989	3.924	0.092	0.042	0.095	0.632	0.738	3.717	3.941
	+25	244.29	9.16	15.80	18.24	0.038	0.052	0.081	0.600	0.704	4.898	4.736	0.023	0.097	0.092	0.661	0.763	1.395	4.941
	−25	241.44	12.8	24.9	29.15	0.038	0.063	0.099	0.770	0.787	3.438	4.619	0.082	0.053	0.080	0.663	0.663	2.482	3.899
	−50	357.29	15.9	24.69	26.70	0.056	0.072	0.087	0.825	0.681	3.361	3.408	0.024	0.095	0.062	0.889	0.526	3.923	1.765
Tr	+50	283.44	9.6	15.65	7.9	0.016	0	0.072	0.967	0.624	0.304	4.187	0.049	0.089	0.088	0.749	0.531	3.386	0.089
	+25	294.10	9.55	14.95	9.24	0.073	0	0.047	0.980	0.567	3.519	3.127	0.083	0.037	0.041	0.627	0.601	2.745	3.66
	−25	353.29	12.25	21.58	26.11	0.021	0.078	0.087	0.780	0.627	3.077	2.858	0.049	0.013	0.097	0.661	0.577	1.496	0.932
	−50	282.88	11.88	24.65	20.76	0.070	0	0.068	0.984	0.729	2.916	3.822	0.057	0.064	0.080	0.812	0.691	0.243	4.738
Tps	+50	286.53	11.88	15.43	10.64	0.037	0.014	0.066	0.796	0.542	2.534	3.859	0.077	0.075	0.087	0.713	0.599	4.405	3.418
	+25	236.50	9.047	12.35	8.50	0.064	0	0.094	0.949	0.627	2.869	3.918	0.039	0.033	0.090	0.540	0.528	0.410	1.971
	−25	385.63	12.00	16.02	26.78	0.071	0.073	0.076	0.828	0.730	3.118	4.005	0.081	0.066	0.094	0.860	0.861	3.924	4.406
	−50	398.07	14.60	24.18	28.78	0.058	0	0.058	0.993	0.772	3.636	2.992	0.077	0.095	0.061	0.671	0.898	3.601	4.220
Kr	+50	350.489	9.619	17.71	16.15	0.073	0.051	0.071	0.801	0.687	2.464	3.783	0.083	0.054	0.084	0.713	0.674	4.250	4.012
	+25	280.101	12.54	18.73	18.30	0.081	0.041	0.068	0.782	0.632	2.414	3.124	0.864	0.042	0.074	0.712	0.512	3.564	4.089
	−25	363.35	15.69	20.19	19.21	0.083	0.042	0.064	0.690	0.656	3.964	3.769	0.087	0.036	0.059	0.898	0.530	3.046	1.807
	−50	368.95	12.88	24.69	16.79	0.041	0.001	0.084	0.653	0.570	0.790	2.120	0.087	0.058	0.094	0.728	0.627	2.199	2.430
Kps	+50	310.014	12.12	21.95	25.25	0.071	0.065	0.086	0.792	0.851	4.456	3.613	0.069	0.087	0.099	0.738	0.828	2.896	4.662
	+25	286.46	8.3	NA	NA	0.068	0.043	0.086	0.772	0.794	2.776	4.932	0.026	0.046	0.084	0.724	0.749	4.142	3.952
	−25	189.021	9.16	15.80	18.24	0.066	0.097	0.095	0.703	0.721	3.021	4.917	0.036	0.027	0.090	0.545	0.629	3.567	3.259
	−50	140.370	6.46	18.61	21.53	0.084	0.052	0.080	0.595	0.576	3.356	4.336	0.090	0.056	0.099	0.615	0.529	4.903	1.937
Ri	+50	335.42	11.30	15.46	16.76	0.025	0.003	0.074	0.980	0.452	3.283	0.002	0.098	0.062	0.079	0.740	0.509	3.311	0.696
	+25	247.107	17.71	14.45	16.31	0.057	0	0.099	0.853	0.774	3.730	4.264	0.044	0.059	0.087	0.644	0.693	0.504	4.265
	−25	271.543	NA	NA	NA	0.090	0	0.043	0.943	0.485	2.750	0.088	0.060	0.048	0.086	0.676	0.676	2.589	2.933
	−50	269.124	NA	NA	NA	0.058	0	0.084	0.846	0.573	2.524	1.487	0.098	0.058	0.085	0.742	0.751	4.890	3.713
Bi	+50	244.714	20.04	14.45	NA	0.071	0	0.079	0.870	0.778	4.829	3.818	0.029	0.094	0.074	0.691	0.751	4.021	4.693
	+25	223.05	9.25	22.05	18.95	0.055	0.058	0.090	0.683	0.810	3.052	4.481	0.029	0.005	0.095	0.468	0.768	2.672	4.602
	−25	279.05	14.28	9.50	9.88	0.086	0	0.084	0.970	0.691	3.065	4.734	0.059	0.096	0.087	0.781	0.581	2.762	0.782
	−50	283.842	17.45	12.25	17.85	0.041	0	0.094	0.755	0.569	1.724	1.946	0.037	0.071	0.096	0.697	0.698	1.375	4.769

describes the detailed sensitivity analysis of the two-area network with step load disturbance and physical constraints.

The radial chart in Figure 21 illustrates the numerical estimation of different approaches in 3DOF-FOPID against the cost functions. The calculated cost functions (Table 6) of each algorithm under system uncertainties prove that the proposed AHB is versatile and adaptable for all operating conditions.

6. Experimental Authentication in Real-Time Simulator

The simulated outcomes require experimental substantiation for real-time implementation in further study and analysis. Thus, the proposed ideology is replicated in the HIL approach using OPAL-RT 11/OP4500. The HIL implicates the errors, delays, and overruns that are immeasurable in traditional off-line simulations. Figure 22 portrays the HIL structure that includes the real-time simulator (RTS) embedded through the FPGA processor for controller action and PC for the program host in which the simulation files are transformed into the codes required to execute in the OPAL-RT platform [21]. The plant model in Figure 23 is fed into the host PC and the FPGA processor acts as the controller where the optimized gains are specified as control parameters. The model in this target is grouped into the subsystem, namely the main and the required outputs such as $\Delta f_1$, $\Delta f_2$, $\Delta P_{tie,}$ and $\Delta P_{D1,}$ which are grouped to form the console. The ‘OpComm’ block is the medium for the transmission of the signals to the oscilloscope during the regulation process. The fallouts are captured in 4 channel DSO for better visibility and understanding of how the presented work behaves in real-time with the proposed and optimized controller design.

The frequency and tie-line deviations in area 1 and area 2 against the given step disturbance at 0 sec is apprehended at the DSO and displayed in Figure 24. The system performance of the proposed 3DOF-FOPID controller optimized with AHA with uncertainties against existing meta heuristic types is showcased, where the peak overshoots and settling time are extremely minimal. The ‘Opctrl’ block in the console carries gain values such that the MPo+ is maintained at nominal values. The visual results along with the analytical measurements in Table 6 describe the variations in peak overshoots/undershoots and settling period of each approach in factual spells. These measurements in the OPAL-RT platform are augmented based on the gain setting in the main system and voltage threshold of the ‘Analogout’ block. The results exhibited show the potential effect of the proposed structure with the recommended controller gains and settings.

7. Conclusions

Regarding the autonomous hybrid microgrid, when interconnected to form a two-area network, load frequency control is required to maintain null difference between the total generation and demand. This helps in reducing the frequency deviations in the multi-areas interconnected. The paper tends towards state-space modelling of PV and wind systems based on the system specifications, thereby considering the dynamic behavior and higher degree of freedom. The derived transfer functions are injected into the two-area network to execute load frequency control. The derived transfer functions are also verified by linear analysis in MATLAB/Simulink for the PV and WTG systems. The frequency and tie-line deviations are analyzed in the presence of AHA tuned 3DOF-FOPID controller and the results are compared with elementary controllers such as PID, FOPID, and 2DOF-FOPID. The fitness function and the convergence characteristics for various algorithms are evaluated as in the literature and AHA proves to be more feasible and acceptable by escalating its performance. The system is subjected to uncertainties, variable load disturbances, and percentile parameter variations for tolerance check. Hence, the paper suggests the necessity of the state-space form of renewables for definite system capacities, making it more convenient and suitable for real-time applications and study.

8. Future Recommendations

Owing to privatization of the power industry, the unbundling of services has led to the restructuring of power sectors with market forecasting and settlement processes. The suggested work can be implied in deregulated scenarios for instigating different market models that are reliable and feasible for automatic generation control. The fast transactive load frequency control is a recent technique in which effective research is being carried out. On framing the proposed model in restructured scenarios, the transactive frequency regulation can be executed for system testing and detailed analysis.