Modified Frequency Regulator Based on TI^λ-TD^μFF Controller for Interconnected Microgrids with Incorporating Hybrid Renewable Energy Sources

Abstract

Reducing the emissions of greenhouse gases has directed energy sectors toward using renewable energy sources (RESs) and decreasing the dependency on conventional energy sources. Recently, developing efficient load frequency control (LFC) schemes has become essential to face the reduced inertia due to RESs installations. This paper presents a modified tilt fractional order (FO) integral–tilt FO derivative with a fractional filter (TFOI-TFODFF or namely TI${}^{\lambda }$-TD${}^{\mu }$FF) LFC method. Although the proposed controller uses the same elements of standard controllers, it adopts FO control capabilities and flexibilities, including the tilt, FO integral, FO derivative, and FO filter. Thence, a new control structure is obtained, merging the advantages of both controllers. Moreover, the proposed TFOI-TFODFF controller employs two control loops to be able to mitigate low-frequency as well as high-frequency disturbances in power grids. Additionally, a new modified marine predator algorithm (MMPA) is proposed for optimally tuning the parameters of the proposed TFOI-TFODFF LFC method. The performance of the MMPA is enhanced in terms of initialization and exploitation phases using the chaotic maps and weighting factor. A two-area interconnected power system case study is implemented with wind and photovoltaic RESs and electric vehicles (EVs) contribution. The proposed TFOI-TFODFF LFC is compared with the FOPID, TID, TI-DF, and FOTPID controllers, wherein the proposed TFOI-TFODFF has offered superior performance of the proposed controller. Moreover, the proposed modified MPA is compared with the original MPA and other competitive optimization algorithms, and statistical analyses are carried out through parametric and nonparametric tests.

1. Introduction

1.1. General Overview

Modern power systems are mainly characterized by the decreased mechanical power stored in system inertia. This stored mechanical power has a significant role in retaining power system stability during system oscillations, faults and load variations. Therefore, the increased penetration levels of RESs increase the stability problems that face such systems [1,2]. In this context, the wide use of converter-based RESs, such as solar power generation and wind energy, makes the situation even worse, since the system inertia constant is decreased and the capability of the system to preserve its stability becomes fragile. Therefore, the load frequency control (LFC) is playing a significant role in regulating system frequency during normal/abnormal conditions and controlling tie-line power flow among system areas [3]. The LFC is responsible for controlling the output power from various generation sources and load demands to achieve the smooth regulation of frequency signals and tie-line power between interconnected areas.

Several studies have discussed frequency stability issues in single-area, multi-area and deregulated power systems [4,5,6]. As well, different control techniques have been adopted for LFC problems, such as conventional control techniques [7], robust control techniques [8,9], sliding mode techniques [10], model predictive control techniques [11,12], intelligent control techniques [13], deep neural networks techniques [14], etc. The recently presented intelligent controllers have achieved enhanced stability and control performance. However, their wide applications are facing several limiting factors, such as the high computational requirements, huge training data volumes, requirements of powerful computing processors, and requiring experts for their tuning.

1.2. Literature Review

Conventional integer order (IO) and fractional order (FO) control techniques have been met with great concerns for LFC applications [15,16]. The conventional integral (I), proportional–integral (PI), and proportional–integral–derivatives (PID) controllers have found wide applications in the literature. In [17], an optimized PI controller has been presented based on the hybrid gravitational searching and firefly algorithms (hGFA) optimizer. Another optimized PI controller for deregulated power systems has been proposed in [18] using the binary moth flame optimizer (MFO) algorithm. Moreover, PID based has been proposed in [19] and optimized using an improved hybrid gravitational searching algorithm with the binary particle swarm optimizers (IGSA-BPSO). They have provided several advantages, such as their simple design and ease of implementation on low-cost digital processors, etc. However, they fail when considering the nonlinearities and parameters’ uncertainties of power systems. The FO-based techniques provide more degrees of freedom due to the extra included FO operators in the tilt (T), integral ($I^{\lambda }$), derivative ($D^{\mu }$), and FO filters as well. The benefits of the increased flexibilities have been discussed in [20], which compares their performance with their IO counterparts. The FOPID controller has achieved better damping and faster transients than the PID control technique.

In the literature, many metaheuristic optimization algorithms have been adopted to ensure the optimal design of LFC systems. The various control techniques and optimization algorithms are jointly presented for optimizing the frequency regulation response of power grids. A PI-based technique has been presented with the Harris-Hawks optimization algorithm (HHO) in [21]. In [22], the PI has been optimized using the gray wolf optimizer (GWO) algorithm for controlling hydrogen electrolysis systems with RES grids. The PID and artificial bee colony (ABC) algorithm have been proposed in [23]. Another stability boundary locus (SBL)-based PID technique has been presented in [24]. In addition, the adaptive integral optimized with elder scrolls online (ESO) and the balloon effects modulation (BE) have been introduced in [25] with featuring parameters uncertainty problem’s mitigation. These techniques have shown that IO LFC methods represent simple design, decreased complexities in implementations, and a fast convergence rate. However, they fail in mitigating system nonlinearities, RES generated disturbances, and uncertainty in power system parameters.

Regarding the existing FO LFC methods, the FOPID and the movable damped wave optimization (MDWA) have been presented in [26] with multi-area interconnected microgrids. Another sine–cosine optimizer algorithm (SCA)-based FOPID has been provided in [27]. Additionally, tilt-based LFC techniques have been introduced in the literature [28]. The ABC optimizer-based TID technique has been proposed in [29] with EV-based microgrids. Another TID design using the pathfinder optimizer algorithm (PFA) has been provided in [30]. Enhanced performance obtained using TIDF with differential–evolution (DE) optimization has been presented in [31]. The recent eagle strategy arithmetic optimizer algorithm (ESAOA) has been introduced for optimizing the PID and FOPID controllers in [32]. Moreover, some modified combinations of FO control techniques have been presented in the literature to benefit from multiple controllers simultaneously. The TID is jointed with FOPID in [33] for forming the hybrid FO controller structure, and an optimization process has been made using an artificial ecosystem optimization (AEO) algorithm. Another modified hybrid including the fractional filter and the artificial hummingbird optimizer algorithm (AHA) have been provided in [34].

Combined as well as cascaded arrangements for LFC have been developed in the literature. Cascaded FOPID FLC using the imperialist competitive optimization algorithm (ICA) has been developed in [35]. As well, a cascaded FO-IDF has been introduced in [36]. Furthermore, combined fuzzy with FO LFC techniques have been presented in [37] using FLC-FOPI-FOPD as in [38] using the FL-FOPIDF and in [39] using the FLC-PIDF-FOI LFC scheme. Likewise, an ICA-based optimized version of the FPIDN-FOPIDN control scheme has been introduced in [36] for two-areas power grids to combine the merits of the fuzzy logic and fractional controllers. Dual stage-based LFC schemes tuned by butterfly optimizer algorithms (BOA) have been adopted in [40]. In addition, PI-TDF-based LFCs with the slap swarm optimizer algorithm (SSA) has been proposed in [41]. The dual stage controllers present an outstanding performance; however, extreme computational calculations are required with an effective optimization technique. As well, different LFC techniques have been presented using PIDF in [42,43], two degrees-of-freedom (2DoF)-based PID (namely 2DoF-PID) in [44], cascaded PD with PID in [45], PID with second-order-based derivatives (PID2D) in [46], FLC with the PID scheme (namely FLC-PID) in [47], and neuro-fuzzy based LFCs in [48]. From another side, PI with PID and the slap swarm optimizer algorithm (SSA) has been presented in [49]. The 2DoF-PID-based LFC has been proposed in [50] and has been optimized using flower pollination optimization algorithms (FPAs). The presented controller has shown superior performance metrics compared to PI LFC, PID LFC, and 2DoF-PI LFC schemes.

1.3. Article Contributions

It has become apparent that several LFC techniques have been jointly proposed with optimizations algorithms for RES-based power system grids in the literature, in which FO-based LFC techniques have a proven better mitigation of frequency deviation, reduced complexity, and improved disturbance rejection. Furthermore, the MPA optimizer has enhanced the performance of several optimization problems in the literature. This, in turn, has encouraged the authors to integrate a new improved MPA optimizer with a modified FO LFC structure for a two-area EV and RES-based power grid system. The main contributions of this paper are summarized as the following:

• An improved version of the FO control technique is proposed based on the modified tilt fractional order (FO) integral-tilt FO derivative with fractional filter (TFOI-TFODFF or namely TI${}^{\lambda }$-TD${}^{\mu }$FF) controller. The proposed TFOI-TFODFF controller merges the benefits of several FO control capabilities, such as the tilt, FO integral, FO derivative, and FO filter. Thence, the proposed TFOI-TFODFF controller is capable of reducing peak overshoot/undershoot transients and steady-state error compared to existing LFC techniques.
• The proposed TFOI-TFODFF controller employs two control loops compared to some single-loop-based LFC structure. As a consequence, the proposed TFOI-TFODFF controller is capable of mitigating low-frequency as well as high-frequency disturbances in power grids.
• A coordinated participation of EVs with existing generation power plants is proposed through the proposed TFOI-TFODFF centralized controller. Therefore, EVs can participate effectively in damping systems’ disturbances and fluctuations thanks to their inherent batteries. This can lead to the better utilization of a future high number of connected EVs in power grids.
• A new modified marine predator algorithm (MMPA) is proposed for optimally tuning the controller’s parameters in the proposed TFOI-TFODFF LFC method. The proposed MMPA is modified using a chaotic map and weighting factors to boost the performance of the MMPA. The employed chaotic map can generate a high diverse-based initial population, whereas the weighting factor enhances the exploitation capabilities of the algorithm and accelerates the convergence in the subsequent iterations. The MMPA algorithm is being adopted for tuning the optimal parameters in the new proposed TFOI-TFODFF LFC method.
• A comprehensive statistical analysis is being performed to investigate the effectiveness of the MMPA using parametric and nonparametric tests.

It has become clear from the literature review that more proper design and control methods are needed for future power grids incorporating hybrid renewable energy resources. In addition, the Introduction section has shown the detailed contribution of this paper to the existing work in the literature. Accordingly, the remaining sections of the paper are organized as follows: The selected system structure is introduced in Section 2 with a detailed modeling of each element. In addition, the mathematical modeling with state-space models is developed in this section. Section 3 provides the existing LFC methods and the proposed proposed TFOI-TFODFF controller structure. This section also introduces the fractional order calculus and the proposed design optimization process and constraints. Detailed representation of the new proposed MMPA algorithm is provided in Section 4. It also includes the mathematical representation of the proposed algorithm and pseudocodes. Section 5 presents the obtained verification results of the new proposed MMPA optimizer using 23 benchmark functions. Section 6 shows the simulation results of the studied power grid with the new proposed TFOI-TFODFF controller and proposed modified optimization algorithms. Finally, Section 7 concludes the paper’s findings.

2. Mathematical Representations of Selected Power Grid

2.1. System Construction

The construction of the proposed TFOI-TFODFF controller with the selected two-area interconnected power grids with RESs and EVs is shown in Figure 1. The RESs are distributed between areas, with installing PV energy in area a and wind energy in area b. Meanwhile, the thermal unit exists in area a, whereas a hydro-power plant exists in area b. Moreover, it is assumed that EVs are distributed equally between the two areas. The system is built in Matlab/Simulink based on the system data from [33,51], which are provided in

Table 1. Parameters of the modeled grid ($x\in \{a,b\}$), [52].

Parameters	Symbols	Value
		Area a	Area b
Rated capacities	$P_{rx}$ (MW)	1200	1200
Droop constants	$R_x$ (Hz/MW)	2.4	2.4
Frequency biases	$B_x$ (MW/Hz)	0.4249	0.4249
Valve gate limit (minimum)	$V_{vlx}$ (p.u.MW)	−0.5	−0.5
Valve gate limit (maximum)	$V_{vux}$ (p.u.MW)	0.5	0.5
Time constant of thermal governor	$T_g$ (s)	0.08	-
Thermal turbines (time constants)	$T_t$ (s)	0.3	-
Hydraulic governor (time constants)	$T_1$ (s)	-	41.6
Transient droop for hydraulic governor (time constants)	$T_2$ (s)	-	0.513
Hydraulic governor (reset times)	$T_R$ (s)	-	5
Hydro turbine (water starting times)	$T_w$ (s)	-	1
Inertia constants	$H_x$ (p.u.s)	0.0833	0.0833
Damping coefficients	$D_x$ (p.u./Hz)	0.00833	0.00833
PV generation (time constants)	$T_{PV}$ (s)	-	1.3
PV generation (gains)	$K_{PV}$ (s)	-	1
Wind generation (time constants)	$T_{WT}$ (s)	1.5	-
Wind generation (gains)	$K_{WT}$ (s)	1	-
EV models
Penetrations level	-	5–10%	5–10%
Voltages (nominal value)	$V_{nom}$ (V)	364.8	364.8
Batteries capacities	$C_{nom}$ (Ah)	66.2	66.2
Series resistance	$R_s$ (ohms)	0.074	0.074
Transient resistance	$R_t$ (ohms)	0.047	0.047
Transient capacitance	$C_t$ (farad)	703.6	703.6
Constant values	$RT/F$	0.02612	0.02612
Battery’s SOC (maximum limits)	%	95	95
Battery’s energy capacities	$C_{batt}$ (kWh)	24.15	24.15

.

A Thevenin-based model of EV-installed batteries is developed in this paper using the EV model in [33]. Moreover, fluctuations in PV output power are modeled using the models in [51]. The output power variations of a wind plant are modeled using the model in [51]. The LFC objectives include controlling the frequency deviation in area a ($\Delta f_a$), controlling the frequency deviation in area b ($\Delta f_b$), and controlling the tie-line power deviations between areas ($\Delta P_{tie,ab}$). The proposed TFOI-TFODFF LFC is included in the system, wherein two signals are fed in each area ($AC{E}_a$ and $\Delta f_a$ for area a and $AC{E}_b$ and $\Delta f_b$ for area b). The system is modeled and linked with the proposed modified MPA optimizer to determine the optimal parameters of the proposed TFOI-TFODFF LFC method. Moreover, the models are used for obtaining the results and performance comparisons of both the proposed TFOI-TFODFF controller with other existing LFCs in the literature and the proposed modified MPA optimizer with the other existing optimizers in the literature.

2.2. Mathematical Models

The complete construction and models of each component in the selected interconnected power grids case study are shown in Figure 2. The used parameters of the selected grid are tabulated with their model definitions in Table 1. Additionally, the models include the Thevenin equivalent circuit of the installed EV in both areas. The state-space model of the system is constructed as follows:

$$\dot{x}=Ax+B_1\omega +B_{2}u$$

(1)

$$y=Cx$$

(2)

where x and y stand for the vectors representing state variables and output states, respectively. The disturbance and control variables are expressed by $\omega$ and u vectors, respectively. The state variables’ vector x and disturbances’ vector $\omega$ are constructed as follows:

$$x={\left[\begin{array}{cccccccccc}\Delta f_a& \Delta P_{ga}& \Delta P_{ga1}& \Delta P_{WT}& \Delta f_b& \Delta P_{gb}& \Delta P_{gb1}& \Delta P_{gb2}& \Delta P_{PV}& \Delta P_{tie,ab}\end{array}\right]}^T$$

(3)

$$\omega ={\left[\begin{array}{cccc}\Delta P_{la}& P_{WT}& \Delta P_{lb}& P_{PV}\end{array}\right]}^T$$

(4)

As explained above, the control variables comprised ACE signals ($AC{E}_a$ and $AC{E}_b$) and EV supplied power to the grid ($\Delta P_{EVa}$ and $\Delta P_{EVb}$). The control vector can be expressed as follows:

$$u={\left[\begin{array}{cccc}AC{E}_a& \Delta P_{E{V}_a}& AC{E}_b& \Delta P_{E{V}_b}\end{array}\right]}^T$$

(5)

The structure of matrices A, $B_1$, $B_2$, and C are based on the power system model parameters for state-space modeling. They are constructed as follows:

$$A=\left[\begin{array}{cccccccccc}-\frac{D_a}{2H_a}& \frac{1}{2H_a}& 0& \frac{1}{2H_a}& 0& 0& 0& 0& 0& -\frac{1}{2H_a}\\ 0& -\frac{1}{T_t}& \frac{1}{T_t}& 0& 0& 0& 0& 0& 0& 0\\ -\frac{1}{R_a{T}_g}& 0& -\frac{1}{T_g}& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -\frac{1}{T_{WT}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -\frac{D_b}{2H_b}& \frac{1}{2H_b}& 0& 0& \frac{1}{2H_b}& \frac{1}{2H_b}\\ 0& 0& 0& 0& \frac{2T_R}{R_b{T}_1{T}_2}& -\frac{2}{T_w}& \frac{2T_2+2T_w}{T_2{T}_w}& \frac{2T_R-2T_1}{T_1{T}_2}& 0& 0\\ 0& 0& 0& 0& -\frac{T_R}{R_b{T}_1{T}_2}& 0& -\frac{1}{T_2}& \frac{T_1-T_R}{T_1{T}_2}& 0& 0\\ 0& 0& 0& 0& -\frac{1}{R_b{T}_1}& 0& 0& -\frac{1}{T_1}& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& -\frac{1}{T_{PV}}& 0\\ 2\pi T_{\mathit{tie},\mathit{eq}}& 0& 0& 0& -2\pi T_{\mathit{tie},\mathit{eq}}& 0& 0& 0& 0& 0\end{array}\right]$$

(6)

$$B_1=\left[\begin{array}{cccc}-\frac{1}{2H_a}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& \frac{K_{WT}}{T_{WT}}& 0& 0\\ 0& 0& -\frac{1}{2H_b}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{K_{PV}}{T_{PV}}\\ 0& 0& 0& 0\end{array}\right],\hspace{1em}\hspace{1em}\mathrm{and}\hspace{1em}\hspace{1em}{B}_2=\left[\begin{array}{cccc}0& -\frac{1}{2H_a}& 0& 0\\ 0& 0& 0& 0\\ -\frac{1}{T_g}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -\frac{1}{2H_b}\\ 0& 0& \frac{2T_R}{T_1{T}_2}& 0\\ 0& 0& -\frac{T_R}{T_1{T}_2}& 0\\ 0& 0& -\frac{1}{T_1}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$$

(7)

$$C=\left[\begin{array}{cccccccccc}1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ B_a& 0& 0& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& B_b& 0& 0& 0& 0& -1\end{array}\right]$$

(8)

3. Proposed TFOI-TFODFF Controller

The integer-order control has found tremendous industrial applications, particularly in the LFC of power systems. The PI and PID and their families have been implemented and verified in the literature. Moreover, several optimizers algorithms have been addressed for optimum tuning of the parameters. The gains of each of the three terms in PID can be tuned to control transient time, rise time, peak overshoot, peak undershoot, stability, steady-state errors, etc. The prevalent LFC in the literature can be summarized as follows:

$$\begin{array}{c}{C}_{PI}\left(s\right)=\frac{Y\left(s\right)}{E\left(s\right)}=K_p+\frac{K_i}{s}\\ C_{PID}\left(s\right)=\frac{Y\left(s\right)}{E\left(s\right)}=K_p+\frac{K_i}{s}+K_{d}s\\ C_{PIDF}\left(s\right)=\frac{Y\left(s\right)}{E\left(s\right)}=K_p+\frac{K_i}{s}+K_{d}s\frac{N_f}{s+N_f}\end{array}$$

(9)

From the other side, FO control theory-based LFC methods have proven better performance compared with their non-integer counterparts. Generally, the FO theory-based calculus is commonly defined using the Grunwald–Letnikov methodology, Riemann–Liouville methodology, and Caputo methodology. The methodology of Grundwald–Letnikov’s expresses ${\alpha }_{th}$ FO-based derivative term for the function f in range a to t bounds as follows:

$$D^{\alpha }{|}_a^t=\underset{h\to 0}{lim}\frac{1}{h^{\alpha }}\sum _{r=0}^{\frac{t-a}{h}}{(-1)}^r\left(\begin{array}{c}n\\ r\end{array}\right)f(t-rh)$$

(10)

where h stands for step-time, whereas operator $[·]$ takes only integer parts in (15). Whereas, n is used to achieve the condition of ($n-1$$<\alpha <n$. The definition of coefficients of the binomial is made as follows:

$$\left(\begin{array}{c}n\\ r\end{array}\right)=\frac{\Gamma (n+1)}{\Gamma (r+1)\Gamma {(n-r+1)}^{\prime }}$$

(11)

where the Gamma function definition is made through using the following:

$$\Gamma (n+1)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}dt$$

(12)

Liouville and Riemann proposed the FO derivative definitions, which can avoid using the sums and the limits. Instead, the definition uses an IO-based derivative, and the integral representations, as follows:

$$D^{\alpha }{|}_a^t=\frac{1}{\Gamma (n-\alpha )}(\frac{d}{dt}{)}^n{\int }_a^t\frac{f\left(\tau \right)}{{(t-\tau )}^{\alpha -n+1}}d\tau$$

(13)

Another methodology for the definition of FO-based derivative has been presented by Caputo, which is represented as follows:

$$D^{\alpha }{|}_a^t=\frac{1}{\Gamma (n-\alpha )}{\int }_a^t\frac{f^{\left(n\right)}\left(\tau \right)}{{(t-\tau )}^{\alpha -n+1}}d\tau$$

(14)

Whereas, $D^{\alpha }{|}_a^t$ represents various forms as follows:

$$D^{\alpha }{|}_a^t=\left\{\begin{array}{cc}\alpha >0\to \frac{{\mathrm{d}}^{\alpha }}{\mathrm{d}{t}^{\alpha }}\hfill & \mathrm{FO}\mathrm{derivative}\hfill \\ \alpha <0\to {\int }_{t_0}^{t_{\mathrm{f}}}\mathrm{d}{t}^{\alpha }\hfill & \mathrm{FO}\mathrm{integral}\hfill \\ \alpha =0\to 1\hfill \end{array}\right.$$

(15)

Furthermore, Oustaloup-recursive approximation (ORA) for the implementation process for FO-based derivatives has been widely utilized as a suitable real-time digital-based control implementation. It has become very familiar in the optimized tuning of FO control. Based on the domination of ORA implementation, it is selected in this article for the modeling process of a FO-based integral term and derivative term. The approximated mathematical-based representation for the ${\alpha }^{th}$ FO-based derivative ($s^{\alpha }$) is as follows:

$$s^{\alpha }\approx {\omega }_h^{\alpha }\prod _{k=-N}^N\frac{s+{\omega }_k^z}{s+{\omega }_k^p}$$

(16)

where ${\omega }_k^p$ and ${\omega }_k^z$ denote the poles locations and zeros locations for ${\omega }_h$ sequence, respectively. Their calculations are as follows:

$${\omega }_k^z={\omega }_b{\left(\frac{{\omega }_h}{{\omega }_b}\right)}^{\frac{k+N+\frac{1-\alpha }{2}}{2N+1}}$$

(17)

$${\omega }_k^p={\omega }_b{\left(\frac{{\omega }_h}{{\omega }_b}\right)}^{\frac{k+N+\frac{1+\alpha }{2}}{2N+1}}$$

(18)

$${\omega }_h^{\alpha }={\left(\frac{{\omega }_h}{{\omega }_b}\right)}^{\frac{-\alpha }{2}}\prod _{k=-N}^N\frac{{\omega }_k^p}{{\omega }_k^z}$$

(19)

where the approximated function contains $(2N+1)$ poles/zeros locations. Thence, N defines the order for ORA’s filter mode with an order of $(2N+1)$. The ORA methodology is utilized in this article with $(M=5)$ within frequency ranges ($\omega \in [{\omega }_{\mathrm{b}},{\omega }_{\mathrm{h}}]$), and they are selected within $[{10}^{-3},{10}^3]$ rad/s.

The widely applied FO LFCs are as follows:

$$\begin{array}{c}{C}_{FOPI}\left(s\right)=\frac{Y\left(s\right)}{E\left(s\right)}=K_p+\frac{K_i}{s^{\lambda }}\\ C_{FOPID}\left(s\right)=\frac{Y\left(s\right)}{E\left(s\right)}=K_p+\frac{K_i}{s^{\lambda }}+K_d{s}^{\mu }\\ C_{FOPIDF}\left(s\right)=\frac{Y\left(s\right)}{E\left(s\right)}=K_p+\frac{K_i}{s^{\lambda }}+K_d{s}^{\mu }\frac{N_f}{s+N_f}\\ C_{TID}\left(s\right)=\frac{Y\left(s\right)}{E\left(s\right)}=K_t{s}^{-\left(\frac{1}{n}\right)}+\frac{K_i}{s}+K_{d}s\\ C_{TIDF}\left(s\right)=\frac{Y\left(s\right)}{E\left(s\right)}=K_t{s}^{-\left(\frac{1}{n}\right)}+\frac{K_i}{s}+K_{d}s\frac{N_f}{s+N_f}\end{array}$$

(20)

Based on (9) and (20), FO-based LFCs involve more parameters in the tuning procedure than their counterparts of IO-based LFCs. For example, the FOPID ($P{I}^{\lambda }{D}^{\mu }$ includes five parameters for tuning compared with existing three parameters for PID tuning. Figure 3 indicates widely utilized IO-based and FO-based LFCs.

It is clear that FOPID and TID controllers represent the principal types of FO-based LFCs. In addition, TID LFCs permits an easier parameter tuning process, disturbance rejections capability, and improved sensitivity against parametric variation in the controlled systems. The extra parameters included by FO add more flexibility for their tuning, which increase their ability for facing disturbances and the parameters’ uncertainty in the system. As illustrated in Figure 3, three parallel branches are included in the $AC{E}_x$ loop to control frequency regulation. Other modified versions of the TID LFC scheme have been provided based on the I-TD LFC in [53,54] and the ID-T LFC in [55], as shown in Figure 4.

The proposed TFOI-TFODFF controller is shown in Figure 5. The proposed TFOI-TFODFF LFC uses two input signals in each controlled area ($AC{E}_a$ and $\Delta f_a$ in area a and $AC{E}_b$ and $\Delta f_b$ in area b), whereas the controller output is used to control the generation systems in addition to the connected EVs in each area. It is considered as a modified version of TID while adding extra flexibility from the FOPID controller in addition to the flexibility of the fractional filter. The FO operators in the integral and derivative terms increase the number of tunable parameters in the proposed TFOI-TFODFF LFC method. Furthermore, adding the fractional filter to the derivative term leads to the better mitigation of high-frequency noise resulting from derivative term operation. The complete transfer function model of the proposed TFOI-TFODFF is expressed as follows:

$$\begin{array}{c}{Y}_a\left(s\right)=(K_{t1}{s}^{-\left(\frac{1}{n1}\right)}+\frac{K_{i1}}{s^{{\lambda }_1}})\left(AC{E}_a\right)+(K_{t2}{s}^{-\left(\frac{1}{n2}\right)}+K_{d1}{s}^{{\mu }_1}\frac{N_{f1}}{s^{{\lambda }_{f1}}+N_{f1}})(\Delta f_a)\\ Y_b\left(s\right)=(K_{t3}{s}^{-\left(\frac{1}{n3}\right)}+\frac{K_{i2}}{s^{{\lambda }_2}})\left(AC{E}_b\right)+(K_{t4}{s}^{-\left(\frac{1}{n4}\right)}+K_{d2}{s}^{{\mu }_2}\frac{N_{f2}}{s^{{\lambda }_{f2}}+N_{f2}})(\Delta f_b)\end{array}$$

(21)

In this article, the proposed TFOI-TFODFF LFC method is proposed as an upgraded version of TID and FOPID LFCs. The proposed TFOI-TFODFF LFC can be applied to various structures of power grids. The main improvements and benefits of the proposed TFOI-TFODFF LFC method are its simplicity, flexibility, enhanced transient response, reduction of overshoot amplitude and settling time, and mitigation of low/high-frequency fluctuations.

Figure 6 shows the parameters’ optimization of the proposed TFOI-TFODFF LFC using the proposed modified MPA algorithm. The optimization problem is formulated using the integral squared-error (ISE) objective function. The ISE objective function represents the desired performance of the proposed optimization process, including the minimization of frequency deviations ($\Delta f_a$ and $\Delta f_b$) and the tie-line power fluctuation ($\Delta P_{tie}$). The mathematical representation of the ISE in the proposed optimized parameters’ process is expressed as follows:

$$ISE=\underset{0}{\overset{t_s}{\int }}({(\Delta f_a)}^2+{(\Delta f_b)}^2+{(\Delta P_{tie})}^2)\mathrm{d}t$$

(22)

Based on (21), the proposed TFOI-TFODFF LFC includes 10 parameters in the tuning process in each studied area. There are a 20 of total parameters for the tuning process in the proposed optimized parameters determination. The constraints for each parameter during the optimization process are expressed as follows:

$$\begin{array}{c}{K}_t^{min}\le K_{t1},K_{t2},K_{t3},K_{t4}\le K_t^{max}\\ K_i^{min}\le K_{i1},K_{i2}\le K_i^{max}\\ K_d^{min}\le K_{d1},K_{d2}\le K_d^{max}\\ n^{min}\le n_1,n_2,n_3,n_4\le n^{max}\\ {\lambda }^{min}\le {\lambda }_1,{\lambda }_2\le {\lambda }^{max}\\ {\mu }^{min}\le {\mu }_1,{\mu }_2\le {\mu }^{max}\\ N_f^{min}\le N_{f1},N_{f2}\le N_f^{max}\\ {\lambda }_f^{min}\le {\lambda }_{f1},{\lambda }_{f2}\le {\lambda }_f^{max}\end{array}$$

(23)

where ${\left(f\right)}^{min}$ and ${\left(f\right)}^{max}$ are the upper and lower bounds of the tuned parameters within the optimization process. The set constraints in the optimization process are as follows: ($K_t^{min}$, $K_i^{min}$, $K_d^{min}$ = 0), ($K_p^{max}$, $K_t^{max}$, $K_i^{max}$, $K_d^{max}$ = 2), ($n^{min}$ = 2), ($n^{max}$=10), (${\mu }^{min}$, ${\lambda }^{min}$, ${\lambda }_f^{min}$ = 0), (${\mu }^{max}$, ${\lambda }^{max}$, ${\lambda }_f^{max}$ = 1), ($N_f^{min}$ = 5), and ($N_f^{max}$) = 400).

4. Proposed Modified MPA Optimizer

4.1. Marine Predators Algorithm

The Lévy and Brownian motions of ocean predators, together with the optimal encounter rate policy in the biological interplay between predator and prey, are the key sources of inspiration for MPA. When it comes to encounter rates between predators and prey in marine habitats, MPA abides by the natural laws that determine the best foraging strategy. The next part in this section outlines the major phases of the MPA algorithm. In a population-based method known as MPA, the initial answer, $X_0$, is evenly dispersed over the search region as follows:

$$X_0=X_{min}+rand(X_{max}-X_{min})$$

(24)

where $X_{min}$ denotes the lower limit, $X_{max}$ denotes the upper bound for variables, and rand denotes a uniform random vector with values between 0 and 1.

The hypothesis of the survival of the fittest states that top predators in nature are more skilled hunters. In order to create the $Elite$ matrix, the top predator is chosen as the fittest answer. Arrays in this matrix search for and locate prey based on information about the prey’s location.

$$Elite=\left[\begin{array}{ccc}{X}_{1.1}^t& \cdots & X_{1.d}^t\\ ⋮& \ddots & ⋮\\ X_{n.1}^t& \cdots & X_{n.d}^t\end{array}\right]$$

(25)

where $\overrightarrow{X}$ stands for the vector of the top predator, which is pretended n times to produce the Elite matrix. n stands for the quantity of search agents, whereas d stands for the quantity of dimensions. Prey is represented by a matrix with the same size as Elite that predators employ to update their positions.

$$Prey=\left[\begin{array}{ccc}{X}_{1.1}& \cdots & X_{1.d}\\ ⋮& \ddots & ⋮\\ X_{n.1}& \cdots & X_{n.d}\end{array}\right]$$

(26)

The MPA optimization approach is divided into three basic phases, each of which considers a certain velocity ratio while modeling the whole life of a predator and prey as follows:

4.1.1. High Velocity Ratio

This circumstance arises when the predator is outpacing the prey or early in the optimization process when exploration is crucial. The best predator strategy in a high-velocity ratio $(v\ge 10)$ scenario is to remain still. The following is the mathematical model of this rule:

$$\mathrm{iter}<\frac{1}{3}{\mathrm{iter}}_{\max }$$

(27)

$$\overrightarrow{\mathrm{step}{}_l}=\overrightarrow{R_B}\otimes \left(\overrightarrow{\mathrm{Elte}{}_l}-\overrightarrow{R_B}\otimes \overrightarrow{\mathrm{Prey}{}_l}\right)i=1,\dots ,n$$

(28)

$$\overrightarrow{\mathrm{Prey}{}_l}=\overrightarrow{\mathrm{Prey}{}_l}+P·\overrightarrow{R_B}\otimes \overrightarrow{\mathrm{step}{}_l}$$

(29)

where $R_B$ is a collection of random numbers that follow Brownian motion, and it comes from the normal distribution. Multiplications of entries one by one are shown by the symbol ⊗. Multiplying $R_B$ by prey simulates prey movement. $P=0.5$ and R is a vector of uniformly distributed random values between [0, 1]. When the step size or movement speed is large throughout the first third of iterations, it indicates a high level of exploration abilities. $iter$ is the abbreviation for iteration, and $ite{r}_{max}$ stands for the maximal iteration.

4.1.2. Unit Velocity Ratio

It demonstrates that both the predator and the prey are searching for their prey while traveling at approximately the same speed. This stage happens early in the optimization process, when exploratory attempts are temporarily transformed into effective exploitation. Exploration and exploitation are crucial at this stage. As a result, half of the population is assigned to exploration and the other half is assigned to exploitation. The predator is in charge of exploration at this time, while the prey is in charge of exploitation. As a result, if prey travel in a Lévy way, the best predator approach is Brownian.

$$\frac{1}{3}\mathrm{iter}{}_{\max }<\mathrm{iter}<\frac{2}{3}\mathrm{iter}{}_{\max }$$

(30)

In the case of the first half of the population

$$\overrightarrow{\mathrm{step}{}_l}=\overrightarrow{R_L}\otimes \left(\overrightarrow{\mathrm{Eltte}{}_l}-\overrightarrow{R_L}\otimes \overrightarrow{\mathrm{Prey}{}_l}\right)i=1,\dots ,n/2$$

(31)

$$\overrightarrow{\mathrm{Prey}{}_l}=\overrightarrow{\mathrm{Prey}{}_l}+P.\overrightarrow{R}\otimes \overrightarrow{\mathrm{step}{}_l}$$

(32)

In the case of the second half of the population

$$\overrightarrow{\mathrm{step}{}_l}=\overrightarrow{R_B}\otimes \left(\overrightarrow{R_B}\otimes \overrightarrow{\mathrm{Prey}{}_l}-\overrightarrow{\mathrm{Eltete}{}_l}\right)i=1,\dots ,n/2$$

(33)

$$\overrightarrow{{Prey}_l}=\overrightarrow{{Prey}_l}+P.CF\otimes \overrightarrow{{step}_l}$$

(34)

$$CF={\left(1-\frac{iter}{ite{r}_{\max }}\right)}^{\left(\frac{2iter}{ite{r}_{\max }}\right)}$$

(35)

In order to track the step size, CF is a variable parameter.

4.1.3. Low Velocity Ratio

For a low velocity ratio, the predator moves faster than the prey. This condition takes place close to the final point of the optimization procedure, which is often connected to a high exploitation capability. For low velocity ratios $(v=0.1)$, Lévy is the most effective predator strategy. This phase is represented as:

$$\mathrm{iter}>\frac{2}{3}\mathrm{iter}{}_{\max }$$

(36)

$$\overrightarrow{\mathrm{step}{}_l}=\overrightarrow{R_L}\otimes \left(\overrightarrow{R_L}\otimes \overrightarrow{\mathrm{Eltte}{}_l}-\overrightarrow{\mathrm{Prey}{}_l}\right)i=1,\dots ,n$$

(37)

$$\overrightarrow{\mathrm{Prey}{}_l}=\overrightarrow{\mathrm{Elite}{}_l}+P.\mathit{CF}\otimes \overrightarrow{\mathrm{step}{}_{ı}}$$

(38)

By multiplying $R_L$ and Elite, the Lévy approach replicates predator movement, whereas by adding the step size to the Elite position, it simulates predator movement to help with the updating of the prey location. The ability of MPA to replicate predator behavior increases the likelihood of escaping from local optimums. This benefit results from the fact that variables such as eddy formation and fish-aggregating devices (FADs) can affect the behavior of predators. Because of this, $20\%$ of the time, the predators would flee to areas with an abundance of prey, while the other $80\%$ was spent hunting for prey in the immediate region. The following are some methods for creating FADs:

$$\overrightarrow{\mathrm{Prey}{}_l}=\left\{\begin{array}{cc}\overrightarrow{\mathrm{Prey}{}_l}+CF\left[\overrightarrow{X_{min}}+\overrightarrow{R_L}\otimes \left(\overrightarrow{X_{\max }}-\overrightarrow{X_{min}}\right)\right]\otimes \overrightarrow{U},& r\le \mathit{FADs}\hfill \\ \overrightarrow{\mathrm{Prey}{}_l}+[(1-r)FADs+r]\left(\overrightarrow{\mathrm{Prey}{}_{r1}}-\overrightarrow{\mathrm{Prey}{}_{r2}}\right),& r>FADs\hfill \end{array}\right.$$

(39)

where $FADs=0.2$ stands for the likelihood that FADs may affect the optimization process. The binary vector U is built by constructing a random vector ranging from [0, 1] and setting its array to zero if it is less than 0.2 and one if it is more than 0.2. The uniform random number, r, falls between [0, 1]. Random prey matrix indices are represented by the subscripts $r_1$ and $r_2$. Algorithm 1 provides an overview of the MPA’s implementation. The flowchart of the proposed MPA technique is shown in Figure 7.

Algorithm 1 Pseudocode of MPA.

1: Create the initial Prey population ($i=1,2,..,n$) 2:   Randomly create a set of random search Prey $X_0$ using (24) within the bounds $X_{max}$ and $X_{min}$ 3:   while ($iter<ite{r}_{max}$) do 4:         Calculate the fitness function and create the $Elite$ matrix 5:         if $\mathrm{iter}<\frac{1}{3}{\mathrm{iter}}_{\max }$ then                                                                                                    ▹ High velocity ratio 6:             Use (28) and (29) to update the Prey position 7:         else if $\mathrm{iter}>\frac{1}{3}\mathrm{iter}{}_{\max }\mathrm{AND}\mathrm{iter}<\frac{2}{3}\mathrm{iter}{}_{\max }$ then                               ▹ Unit velocity ratio 8:             Update the first half of the population using (31) and (32) 9:             Update the second half of the population using (33) to (35) 10:       else if $\mathrm{iter}>\frac{2}{3}\mathrm{iter}{}_{\max }$ then                                                                                       ▹ Low velocity ratio 11:           Update the Prey position using (37) and (38) 12:         end if 13:         Apply FADs effect and update the position using (39) 14: end while 15:   Return the best solution obtained so far

4.2. Modified MPA

Two strategies are used in this study to increase the MPA’s performance. The following part is explaining how the initial population is generated using a sine chaotic map:

$$\begin{array}{c}{y}_{i+1}=sin\left(\frac{b\pi }{y_i}\right)\hfill \\ X_0=X_{min}+y_{i+1}\ast (X_{max}-X_{min})\hfill \end{array}$$

(40)

where b set to 0.7 in the proposed method. The exploitation phase of the optimization process has been increased by using weighting factors. Consequently, in this work, the weighting factor is used to modify the best predator as follows:

$$\begin{array}{c}w\left(t\right)=(w_{max}-w_{min})\ast e^{\left(\frac{-t}{T}\right)}+w_{min}\hfill \\ Eltt{e}_l=w\left(t\right)\ast {Prey}_l\hfill \end{array}$$

(41)

where $w_{max}=0.09$ and $w_{min}=0.04$. The proposed WCMPA’s pseudocode, based on the chaotic and the weighting factor, is presented in Algorithm 2. However, the flow chart for the proposed MMPA technique is shown in Figure 8.

Algorithm 2 Pseudocode of WCMPA.

1: Create the initial Prey population ($i=1,2,..,n$) 2:   Randomly create a set of random search Prey $X_0$ using a Sine chaotic map (40) within the bounds $X_{max}$ and $X_{min}$   ▹ Chaotic initialization 3:   while ($iter<ite{r}_{max}$) do 4:         Calculate the fitness function and create the $Elite$ matrix 5:         Update Weighting factor $w\left(iter\right)$ and the $Elite$ matrix using (41)                            ▹ Weighting factor 6:         if $\mathrm{iter}<\frac{1}{3}{\mathrm{iter}}_{\max }$ then                                             ▹ High velocity ratio 7:                 Use (28) and (29) to update the Prey position 8:         else if $\mathrm{iter}>\frac{1}{3}\mathrm{iter}{}_{\max }\mathrm{AND}\mathrm{iter}<\frac{2}{3}\mathrm{iter}{}_{\max }$ then                            ▹ Unit velocity ratio 9:                 Update the first half of the population using (31) and (32) 10:               Update the second half of the population using (33) to (35) 11:       else if $\mathrm{iter}>\frac{2}{3}\mathrm{iter}{}_{\max }$ then                                          ▹ Low velocity ratio 12:               Update the Prey position using (37) and (38) 13:       end if 14:       Apply FADs effect and update the position using (39) 15: end while 16:   Return the best solution obtained so far

5. Optimizer Verification Results

This section evaluates the performance of the modified MPA using 23 benchmark functions. The benchmarks, which range from F1 to F7, are unimodal and have a single global optimum solution. Furthermore, multimodal components are included in the benchmarks (F8-F14). Nevertheless, (F15-F23) are composite functions, and the performance and effectiveness of the modified (WCMPA) algorithm is evaluated by comparison with the original MPA and other new optimization algorithms such as artificial ecosystem-based optimization (AEO), Slime Mold Algorithm (SMA), and Honey Badger Algorithm (HPA). Parametric tests in terms of average, worst, and best values and nonparametric tests using a Wilcoxon signed-rank test (WSRT) are carried out for each benchmark function. In total, 30 runs are simulated with 200 iterations, while the number of populations set to 30 for each optimization algorithm.

5.1. Parametric Test

In this test, the average, worst, and best values are arranged in

Table 2. Parametric test values for unimodal benchmark functions.

		WCMPA	MPA	AEO	HPA	SMA
F1	Best	5.33 × 10${}^{-299}$	4.23 × ${10}^{-12}$	1.06 × ${10}^{-85}$	7.14 × 10${}^{-77}$	2.02 × ${10}^{-264}$
	Average	2.47 × ${10}^{-283}$	5.86 × ${10}^{-11}$	5.89 × ${10}^{-73}$	1.49 × ${10}^{-70}$	2.91 × ${10}^{-220}$
	Worst	6.80 × ${10}^{-282}$	2.61 × ${10}^{-10}$	1.76 × ${10}^{-71}$	3.00 × ${10}^{-69}$	8.74 × ${10}^{-219}$
F2	Best	2.24 × ${10}^{-150}$	7.91 × ${10}^{-08}$	7.16 × ${10}^{-44}$	9.27 × ${10}^{-41}$	5.89 × ${10}^{-140}$
	Average	7.06 × ${10}^{-144}$	7.33 × ${10}^{-07}$	5.17 × ${10}^{-38}$	2.97 × ${10}^{-38}$	3.14 × ${10}^{-95}$
	Worst	1.29 × ${10}^{-142}$	2.05 × ${10}^{-06}$	6.05 × ${10}^{-37}$	4.00 × ${10}^{-37}$	9.42 × ${10}^{-94}$
F3	Best	2.10 × ${10}^{-290}$	1.49 × ${10}^{-06}$	7.09 × ${10}^{-80}$	1.35 × ${10}^{-61}$	5.30 × ${10}^{-269}$
	Average	1.01 × ${10}^{-271}$	8.96 × ${10}^{-05}$	4.57 × ${10}^{-69}$	9.37 × ${10}^{-57}$	4.59 × ${10}^{-206}$
	Worst	2.16 × ${10}^{-270}$	4.44 × ${10}^{-04}$	1.36 × ${10}^{-67}$	2.23 × ${10}^{-55}$	8.37 × ${10}^{-205}$
F4	Best	1.60 × ${10}^{-147}$	2.94 × ${10}^{-05}$	2.33 × ${10}^{-44}$	2.97 × ${10}^{-35}$	2.29 × ${10}^{-134}$
	Average	8.53 × ${10}^{-141}$	8.69 × ${10}^{-05}$	3.81 × ${10}^{-35}$	2.99 × ${10}^{-32}$	4.43 × ${10}^{-112}$
	Worst	6.90 × ${10}^{-140}$	2.23 × ${10}^{-04}$	1.03 × ${10}^{-33}$	4.72 × ${10}^{-31}$	1.04 × ${10}^{-110}$
F5	Best	2.83 × ${10}^{-09}$	3.37 × ${10}^{+00}$	2.14 × ${10}^{+00}$	3.01 × ${10}^{+00}$	6.85 × ${10}^{-05}$
	Average	2.47 × ${10}^{-06}$	4.49 × ${10}^{+00}$	3.12 × ${10}^{+00}$	3.56 × ${10}^{+00}$	2.39 × ${10}^{+00}$
	Worst	1.33 × ${10}^{-05}$	5.81 × ${10}^{+00}$	4.66 × ${10}^{+00}$	4.33 × ${10}^{+00}$	7.42 × ${10}^{+00}$
F6	Best	1.05 × ${10}^{-20}$	9.38 × ${10}^{-11}$	1.01 × ${10}^{-15}$	1.52 × ${10}^{-15}$	3.21 × ${10}^{-05}$
	Average	4.78 × ${10}^{-19}$	2.50 × ${10}^{-10}$	5.53 × ${10}^{-12}$	2.66 × ${10}^{-10}$	1.20 × ${10}^{-04}$
	Worst	2.26 × ${10}^{-18}$	8.01 × ${10}^{-10}$	6.03 × ${10}^{-11}$	5.66 × ${10}^{-09}$	2.66 × ${10}^{-04}$
F7	Best	3.05 × ${10}^{-07}$	2.86 × ${10}^{-04}$	1.01 × ${10}^{-04}$	2.59 × ${10}^{-05}$	5.23 × ${10}^{-06}$
	Average	5.87 × ${10}^{-05}$	1.67 × ${10}^{-03}$	8.67 × ${10}^{-04}$	4.13 × ${10}^{-04}$	1.84 × ${10}^{-04}$
	Worst	1.58 × ${10}^{-04}$	4.91 × ${10}^{-03}$	2.94 × ${10}^{-03}$	1.34 × ${10}^{-03}$	6.16 × ${10}^{-04}$

,

Table 3. Parametric test values for multimodal benchmark functions.

		WCMPA	MPA	AEO	HPA	SMA
F8	Best	−4.19 ×${10}^{+03}$	−4.07 ×${10}^{+03}$	−4.19 ×${10}^{+03}$	−4.07 ×${10}^{+03}$	−4.19 ×${10}^{+03}$
	Average	−4.19 ×${10}^{+03}$	−3.51 ×${10}^{+03}$	−3.88 ×${10}^{+03}$	−3.46 ×${10}^{+03}$	−4.19 ×${10}^{+03}$
	Worst	−4.19 ×${10}^{+03}$	−3.16 ×${10}^{+03}$	−3.48 ×${10}^{+03}$	−2.76 ×${10}^{+03}$	−4.19 ×${10}^{+03}$
F9	Best	0.0000	0.0000	0.0000	0.0000	0.0000
	Average	0.0000	0.0717	0.0000	0.0000	0.0000
	Worst	0.0000	1.0126	0.0000	0.0000	0.0000
F10	Best	8.88 × ${10}^{-16}$	3.64 × ${10}^{-07}$	8.88 × ${10}^{-16}$	8.88 × ${10}^{-16}$	8.88 × ${10}^{-16}$
	Average	8.88 × ${10}^{-16}$	3.36 × ${10}^{-06}$	8.88 × ${10}^{-16}$	8.88 × ${10}^{-16}$	8.88 × ${10}^{-16}$
	Worst	8.88 × ${10}^{-16}$	7.33 × ${10}^{-06}$	8.88 × ${10}^{-16}$	8.88 × ${10}^{-16}$	8.88 × ${10}^{-16}$
F11	Best	0.00 × ${10}^{+00}$	4.68 × ${10}^{-10}$	0.00 × ${10}^{+00}$	0.00 × ${10}^{+00}$	0.00 × ${10}^{+00}$
	Average	0.00 × ${10}^{+00}$	1.97 × ${10}^{-03}$	0.00 × ${10}^{+00}$	0.00 × ${10}^{+00}$	0.00 × ${10}^{+00}$
	Worst	0.00 × ${10}^{+00}$	4.37 × ${10}^{-02}$	0.00 × ${10}^{+00}$	0.00 × ${10}^{+00}$	0.00 × ${10}^{+00}$
F12	Best	1.11 × ${10}^{-21}$	1.46 × ${10}^{-11}$	1.49 × ${10}^{-15}$	6.38 × ${10}^{-15}$	1.21 × ${10}^{-06}$
	Average	3.96 × ${10}^{-19}$	2.28 × ${10}^{-10}$	1.40 × ${10}^{-12}$	1.58 × ${10}^{-11}$	1.89 × ${10}^{-05}$
	Worst	4.45 × ${10}^{-18}$	1.00 × ${10}^{-09}$	2.70 × ${10}^{-11}$	1.95 × ${10}^{-10}$	5.30 × ${10}^{-05}$
F13	Best	1.16 × ${10}^{-20}$	1.30 × ${10}^{-10}$	1.50 × ${10}^{-13}$	1.63 × ${10}^{-14}$	4.30 × ${10}^{-04}$
	Average	4.61 × ${10}^{-19}$	1.01 × ${10}^{-09}$	1.28 × ${10}^{-03}$	9.06 × ${10}^{-03}$	3.42 × ${10}^{-03}$
	Worst	2.33 × ${10}^{-18}$	2.75 × ${10}^{-09}$	1.10 × ${10}^{-02}$	9.74 × ${10}^{-02}$	1.49 × ${10}^{-02}$
F14	Best	0.99800	0.99800	0.99800	0.99800	0.99800
	Average	0.99800	0.99800	0.99800	1.16341	0.99800
	Worst	0.99800	0.99800	0.99800	2.98211	0.99800

and

Table 4. Parametric test values for composite benchmark functions.

		WCMPA	MPA	AEO	HPA	SMA
F15	Best	0.00031	0.00031	0.00031	0.00031	0.00031
	Average	0.00031	0.00031	0.00042	0.00249	0.00066
	Worst	0.00033	0.00033	0.00159	0.02255	0.00125
F16	Best	−1.03163	−1.03163	−1.03163	−1.03163	−1.03163
	Average	−1.03163	−1.03163	−1.03163	−1.03163	−1.03163
	Worst	−1.03163	−1.03163	−1.03163	−1.03163	−1.03163
F17	Best	0.39789	0.39789	0.39789	0.39789	0.39789
	Average	0.39789	0.39789	0.39789	0.39789	0.39789
	Worst	0.39789	0.39789	0.39789	0.39789	0.39789
F18	Best	3.00000	3.00000	3.00000	3.00000	3.00000
	Average	3.00000	3.00000	3.00000	3.00000	3.00000
	Worst	3.00000	3.00000	3.00000	3.00000	3.00000
F19	Best	−3.86278	−3.86278	−3.86278	−3.86278	−3.86278
	Average	−3.86278	−3.86278	−3.86278	−3.86226	−3.86272
	Worst	−3.86278	−3.86278	−3.86278	−3.85490	−3.86091
F20	Best	−3.32194	−3.32200	−3.32200	−3.32200	−3.32198
	Average	−3.32091	−3.32200	−3.27840	−3.25151	−3.26175
	Worst	−3.31672	−3.32200	−3.20310	−2.84042	−3.19766
F21	Best	−10.15274	−10.15320	−10.15320	−10.15320	−10.15319
	Average	−10.15097	−10.15320	−9.15017	−9.65168	−10.15240
	Worst	−10.14359	−10.15320	−2.63047	−2.63047	−10.14966
F22	Best	−10.40259	−10.40294	−10.40294	−10.40294	−10.40291
	Average	−10.39883	−10.40294	−9.70313	−9.00332	−10.40207
	Worst	−10.38432	−10.40294	−2.76590	−2.76590	−10.39884
F23	Best	−10.53613	−10.53641	−10.53641	−10.53641	−10.53635
	Average	−10.52902	−10.53641	−10.31304	−9.81918	−10.53558
	Worst	−10.50445	−10.53641	−3.83543	−2.42173	−10.53370

for unimodal, multimodal, and composite functions, respectively. The result in Table 2 approves the effectiveness of the modified MMPA (WCMPA) compared to traditional MPA and the other optimization algorithms. However, in the case of multimodal functions that are characterized by the existence of numerous local minimum, the WCMPA is still giving the best solutions. Finally, to check the balance of the exploration and exploitation phases, composite functions are used to demonstrate the efficiency of the WCMPA.

5.2. Nonparametric Test

A nonparametric statistical test using the Wilcoxon signed-rank test (WSRT) is used to show the superiority of the MMPA technique compared to the classic MPA. To compare two approaches using WSRT, it is required to compute the R+ and R− values together with the corresponding p values. For the situation where the first method performs better than the second, the sum of rankings is represented by $\sum R^{+}$. When the second method is superior to the first, the sum of rankings in this case is represented by $\sum R^{-}$. Therefore, a WCMPA of 375 has been utilized in this work as a reference for the compared algorithms (MPA, AEO, HPA, and SMA). The $\sum R^{+}$ value in

Table 5. Nonparametric test using WSRT values for unimodal benchmark functions.

		MPA	AEO	HPA	SMA
F1	$\sum R^{+}$	465	465	465	465
	$\sum R^{-}$	0	0	0	0
	p-Value	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$
F2	$\sum R^{+}$	465	465	465	465
	$\sum R^{-}$	0	0	0	0
	p-Value	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$
F3	$\sum R^{+}$	465	465	465	465
	$\sum R^{-}$	0	0	0	0
	p-Value	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$
F4	$\sum R^{+}$	465	465	465	465
	$\sum R^{-}$	0	0	0	0
	p-Value	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$
F5	$\sum R^{+}$	465	465	465	465
	$\sum R^{-}$	0	0	0	0
	p-Value	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$
F6	$\sum R^{+}$	465	465	465	465
	$\sum R^{-}$	0	0	0	0
	p-Value	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$
F7	$\sum R^{+}$	465	465	453	439
	$\sum R^{-}$	0	0	−12	−26
	p-Value	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	5.89 × ${10}^{-06}$	2.21 × ${10}^{-05}$

,

Table 6. Nonparametric test using WSRT values for multimodal benchmark functions.

		MPA	AEO	HPA	SMA
F8	$\sum R^{+}$	465	465	465	465
	$\sum R^{-}$	0	0	0	0
	p-Value	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$
F9	$\sum R^{+}$	465	$N{V}^{\ast }$	$N{V}^{\ast }$	$N{V}^{\ast }$
	$\sum R^{-}$	0
	p-Value	1.78 × ${10}^{-06}$
F10	$\sum R^{+}$	$N{V}^{\ast }$	$N{V}^{\ast }$	$N{V}^{\ast }$	$N{V}^{\ast }$
	$\sum R^{-}$
	p-Value
F11	$\sum R^{+}$	465	$N{V}^{\ast }$	$N{V}^{\ast }$	$N{V}^{\ast }$
	$\sum R^{-}$	0
	p-Value	1.78 × ${10}^{-06}$
F12	$\sum R^{+}$	465	465	465	465
	$\sum R^{-}$	0	0	0	0
	p-Value	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$
F13	$\sum R^{+}$	465	465	465	465
	$\sum R^{-}$	0	0	0	0
	p-Value	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$	1.78 × ${10}^{-06}$
F14	$\sum R^{+}$	0	87	3	0
	$\sum R^{-}$	−465	−378	−462	−465
	p-Value	1.78 × ${10}^{-06}$	0.002812	2.41 × ${10}^{-06}$	1.78 × ${10}^{-06}$

$\mathit{N}{\mathit{V}}^{\ast }$ = There are no variations, so a Wilcoxon test cannot be performed

and

Table 7. Nonparametric test using WSRT values for composite benchmark functions.

		MPA	AEO	HPA	SMA
F15	$\sum R^{+}$	29	87	192	464
	$\sum R^{-}$	−436	−378	−273	−1
	p-Value	2.91 × ${10}^{-05}$	0.002812	4.08 × ${10}^{-01}$	1.97 × ${10}^{-06}$
F16	$\sum R^{+}$	$N{V}^{\ast }$	$N{V}^{\ast }$	$N{V}^{\ast }$	$N{V}^{\ast }$
	$\sum R^{-}$
	p-Value
F17	$\sum R^{+}$	$N{V}^{\ast }$	$N{V}^{\ast }$	$N{V}^{\ast }$	$N{V}^{\ast }$
	$\sum R^{-}$
	p-Value
F18	$\sum R^{+}$	$N{V}^{\ast }$	$N{V}^{\ast }$	$N{V}^{\ast }$	$N{V}^{\ast }$
	$\sum R^{-}$
	p-Value
F19	$\sum R^{+}$	$N{V}^{\ast }$	59	465	$N{V}^{\ast }$
	$\sum R^{-}$		−28	0
	p-Value		0.753774	1.78 × ${10}^{-06}$
F20	$\sum R^{+}$	0	275	312	345
	$\sum R^{-}$	−465	−190	−153	−120
	p-Value	1.78 × ${10}^{-06}$	3.85 × ${10}^{-01}$	1.03 × ${10}^{-01}$	2.10 × ${10}^{-02}$
F21	$\sum R^{+}$	0	114	59	85
	$\sum R^{-}$	−465	−351	−406	−380
	p-Value	1.78 × ${10}^{-06}$	0.015007	3.66 × ${10}^{-04}$	2.46 × ${10}^{-03}$
F22	$\sum R^{+}$	0	87	165	48
	$\sum R^{-}$	−465	−378	−300	−417
	p-Value	1.78 × ${10}^{-06}$	0.002812	1.67 × ${10}^{-01}$	1.51 × ${10}^{-04}$
F23	$\sum R^{+}$	0	30	87	23
	$\sum R^{-}$	−465	−435	−378	−442
	p-Value	1.78 × ${10}^{-06}$	3.18 × ${10}^{-05}$	2.81 × ${10}^{-03}$	1.68 × ${10}^{-05}$

$\mathit{N}{\mathit{V}}^{\ast }$ = There are no variations, so a Wilcoxon test cannot be performed.

is high, demonstrating that the improved algorithm performs better than the benchmark for all examined methods. However, WSRT cannot be used in the cases of F9, F10, F11, F16, F17, F18, and F19, since there are no variances in the data. Furthermore, the p-value is determined. It can be observed that the modified MPA gives the best solution for a variety of optimization problems.

To demonstrate the efficiency of the improved algorithm, a ranking is developed for each benchmark function of all addressed algorithms. The mean ranking is then computed and given in

Table 8. Mean absolute error ranking.

	WCMPA	MPA	AEO	HBA	SMA
F1	1	5	3	4	2
F2	1	5	4	3	2
F3	1	5	3	4	2
F4	1	5	3	4	2
F5	1	5	3	4	2
F6	1	3	2	4	5
F7	1	5	4	3	2
F8	1	4	3	5	2
F9	1	5	2	3	4
F10	1	2	3	4	5
F11	1	5	2	3	4
F12	1	4	2	3	5
F13	1	2	3	5	4
F14	4	1	5	3	2
F15	2	1	3	5	4
F16	1	2	3	4	5
F17	1	2	3	4	5
F18	4	2	1	5	3
F19	2	3	5	1	4
F20	2	1	3	5	4
F21	3	1	5	4	2
F22	3	1	4	5	2
F23	3	1	4	5	2
Mean Rank	1.6522	3.0435	3.1739	3.9130	3.2174
Rank	1	2	3	5	4

. This table illustrates the dominance of WCMPA over other algorithms by showing how it is rated first using the mean ranking.

5.3. MMPA Convergence Characteristics

The convergence characteristics of the modified MPA compared to the other optimization algorithms are shown in Figure 9, Figure 10 and Figure 11 for unimodal, multimodal, and composite benchmark functions, respectively. Figure 10 shows the superiority of the WCMPA over the MPA, AEO, HPA, and SMA in achieving the optimal solutions. However, Figure 10 and Figure 11 illustrate that the WCMPA finds the optimal solution in fewer iterations. Thus, based on these figures, the modified WCMPA has the highest rate of convergence.

6. Proposed TFOI-TFODFF LFC Verification Results

This section highlights the validity and efficiency of the proposed control strategy of the new TFOI-TFODFF controller and EV system in the LFC loop for improving the performance of the multi-area interconnected power systems. The proposed control strategy and the other comparable techniques are optimized utilizing the new modified MPA. The proposed method is verified via the environment of MATALB/Simulink by interfacing the Simulink of the two-area system with the modified MPA m-file code to achieve the objective function of the LFC. The overall code of the multi-area microgrid system with the modified MPA optimization technique is implemented using a PC with a processor Intel Core i7 CPU of 2.9 GHz, 64-bit version. In order to assess the robustness of the proposed TFOI-TFODFF concept, it is compared with more recent control techniques such as TID, ID-T, I-TD, and TI-TD. Many scenarios are used to investigate the performance of the proposed controller as follows:

• Scenario 1: Applying a 1% step load disturbance (SLD);
• Scenario 2: Applying 10% multi-step disturbances (MSD);
• Scenario 3: Applying a random load profile (RLP);
• Scenario 4: Applying a variation of RESs;
• Scenario 5: Applying a high penetration of RESs.

6.1. Scenario 1

In the first test case, to validate the new proposed combination TFOI-TFODFF controller and its coordination with the EV system, area a is subjected to a 12 MW step load disturbance (SLD) at the first instant of simulation. The tuned parameters values of all utilized controllers are represented in

Table 9. The obtained optimal controllers parameters with the step load scenario.

Control	Area	Coefficients
		${\mathit{K}}_{\mathit{t}1}$	${\mathit{K}}_{\mathit{t}2}$	${\mathit{K}}_{\mathit{i}}$	${\mathit{K}}_{\mathit{d}}$	${\mathit{\lambda }}_1$	${\mathit{\mu }}_1$	${\mathit{n}}_1$	${\mathit{n}}_2$	${\mathit{N}}_{\mathit{f}}$	${\mathit{\lambda }}_{\mathit{f}}$
TID	Area a	1.8022	―	1.9561	1.8213	―	―	3.01	―	―	―
	Area b	1.9763	―	1.2044	1.3331	―	―	2.98		―	―
ID-T	Area a	1.8184	―	1.5674	0.9969	―	―	4.95	―	―	―
	Area b	1.8909	―	1.1891	1.9492	―	―	4.91		―	―
I-TD	Area a	1.8726	―	1.8859	1.7689	―	―	2.65	―	―	―
	Area b	1.1364	―	0.6997	0.4305	―	―	3.66	―	―	―
TI-TD	Area a	4.3749	4.9837	3.1151	1.6403	―	―	4.58	4.37	―	―
	Area b	0.5839	4.7702	0.7011	3.3158	―	―	2.01	2.62	―	―
Proposed	Area a	4.8285	4.8577	4.7162	4.6441	0.456	0.136	4.40	3.24	207	0.745
	Area b	2.7803	4.4415	0.5121	4.9637	0.641	0.788	2.81	4.01	120	0.851

. The proposed coordination strategy has been compared to other individual and cascaded controllers that are deliberated in the literature as means of evaluation. The system dynamic responses of ($\Delta f_1$, $\Delta f_2$, and $\Delta P_{tie}$), are shown in Figure 12. This figure includes the performance of the proposed approach and other techniques such as TID, ID-T, I-TD, and TI-TD on the same system under the same conditions to find the best one. It is clear that the proposed TFOI-TFODFF controller can damp the system fluctuations of $\Delta f_1$, $\Delta f_2$, and $\Delta P_{tie}$, to 1 × 10${}^{-3}$ Hz, 5 × 10${}^{-4}$ Hz, and 4.8 × 10${}^{-4}$ p.u, respectively. It shows the best results of the proposed controller compared to 2 × 10${}^{-3}$ Hz, 1.03 × 10${}^{-3}$ Hz, and 7.5 × 10${}^{-4}$ p.u for the TI-TD controller, 2.7 × 10${}^{-3}$ Hz, 1.3 × 10${}^{-3}$ Hz, and 1.22 × 10${}^{-3}$ p.u for the I-TD controller, and 3.3 × 10${}^{-3}$ Hz, 1.7 × 10${}^{-3}$ Hz, and 1.28 × 10${}^{-3}$ p.u for the ID-T controller, while the TID controller comes in the last order with 5.7 × 10${}^{-3}$ Hz, 4.3 × 10${}^{-3}$ Hz, and 1.4 × 10${}^{-3}$ p.u as noted in

Table 10. Performance comparisons of the tested scenarios.

No.	Controller	$\mathbf{\Delta }{\mathit{f}}_{\mathit{a}}$			$\mathbf{\Delta }{\mathit{f}}_{\mathit{b}}$			$\mathbf{\Delta }{\mathit{P}}_{\mathit{tie}}$
		PO	PU	ST	PO	PU	ST	PO	PU	ST
	TID	0.0007	0.0057	12	0.001	0.0041	13	0.0001	0.0014	19
	ID-T	0.0008	0.0032	14	0.0011	0.0017	16	0.0001	0.0013	19
No.1	I-TD	0.0002	0.0027	19	0.0002	0.0013	17	0.00001	0.0012	16
	TI-TD	0.00007	0.0019	17	0.0001	0.0011	18	0.00002	0.0007	15
	Proposed	0	0.0011	5	0	0.0005	6	0	0.0004	7
	TID	0.003	0.107	OS	0.0121	0.1438	OS	0.0442	0.0113	OS
	ID-T	0.0025	0.0856	OS	0.0097	0.1152	OS	0.0427	0.002	OS
No.2	I-TD	0.0071	0.0738	14	0.0083	0.1108	15	0.034	0.0015	OS
	TI-TD	0.0067	0.0375	13	0.0008	0.0485	12	0.0289	0.0013	OS
	Proposed	0	0.009	4	0.0004	0.0251	7	0.0046	0	9
	TID	0.0652	0.0616	OS	0.0661	0.0769	OS	0.0179	0.0191	OS
	ID-T	0.0454	0.0522	OS	0.0322	0.0398	OS	0.0141	0.0122	OS
No.3	I-TD	0.0231	0.0272	OS	0.0212	0.0222	OS	0.0096	0.0113	OS
	TI-TD	0.0097	0.0094	OS	0.0078	0.0092	OS	0.0093	0.0089	OS
	Proposed	0.0029	0.0026	OS	0.0024	0.0021	OS	0.0009	0.0011	OS
	TID	0.1624	0.1052	27	0.1439	0.1444	OS	0.0484	0.0496	OS
	ID-T	0.0878	0.0711	21	0.1111	0.1104	39	0.0442	0.0443	OS
No.4	I-TD	0.0573	0.0464	19	0.0622	0.0605	35	0.0431	0.0426	OS
	TI-TD	0.0465	0.0254	18	0.0438	0.0442	30	0.0302	0.0301	OS
	Proposed	0.0245	0.0105	9	0.0174	0.0175	11	0.0088	0.0082	10
	TID	0.0782	0.0341	OS	0.1035	0.0231	OS	0.0126	0.0321	OS
	ID-T	0.0548	0.0181	OS	0.0814	0.0085	OS	0.0135	0.0304	OS
No.5	I-TD	0.0254	0.0177	OS	0.0422	0.0112	OS	0.0075	0.0288	OS
	TI-TD	0.0185	0.0098	15	0.0235	0.0032	OS	0.0048	0.0185	OS
	Proposed	0.0111	0.0022	11	0.0169	0.0008	14	0.0024	0.0084	18

OS: Oscillation.

, which summarizes the maximum overshoot (MO), undershoot (MU) and settling time (ST) of frequencies and tie-line power for all addressed controllers. These observations proved the sovereignty of the new proposed combination TFOI-TFODFF controller over the other controllers.

6.2. Scenario 2

The capability of the new coordination between the novel combination of a TFOI-TFODFF controller based on the modified MPA technique is tested under a rarely worst-case scenario, which may occur when an MSD is applied for both areas. Hence, 120 MW is shed from area a at t = 10 s and then followed by an enforced 120 MW in area b at t = 30 s. The obtained results of $\Delta f_1$, $\Delta f_2$, and $\Delta P_{tie}$ for the proposed approach and other employed controllers under such conditions are presented in Figure 13. It is concluded from this figure that the TID controller achieved the worst performance as the frequency deviated to more than 0.08 Hz and 0.1 Hz at 10 s and 30 s for area a and 0.06 Hz and 0.14 Hz at the same time for area b with a long oscillation time far from zero. Meanwhile, the combination of ID-T and I-TD gives a frequency and power deviation less than that of the TID controller for 0.06 and 0.04 Hz at 10 s and 0.08 Hz and 0.06 Hz at 30 s in area a and 0.04 Hz and 0.02 Hz at 10 s and around 0.12 Hz at 30 s in area b for both controllers, respectively. However, they suffer from protracted damped oscillations especially at the instant of t = 30 s, as they could not restore the frequency to its nominal value. Meanwhile, the other combination of TI-TD gave satisfactory results compared with previous controllers as it can reduce the frequency oscillations to less than 0.02 Hz at 10 s and 0.03 Hz at 30 s in area a and also 0.01 Hz at 10 s and 0.04 Hz at 30 s in area b with acceptable settling times at both instants as illustrated in Table 10. On the other hand, the proposed TFOI-TFODFF controller is the best in damping the frequency and tie-line power deviations to 0.008 Hz and 0.009 Hz in area a at t = 10 and 30 s, respectively, and for 0.005 Hz and 0.025 Hz at the same instants for area b being quickly driven back to zero error again.

6.3. Scenario 3

This scenario is used to verify the performance of the proposed TFOI-TFODFF technique coordinated with the EV system to enhance the LFC loop of the two-area interconnected power system. Hence, a random load change is applied to area a of the studied multi-area system as shown in the load profile in Figure 14 to monitor the performance of the TFOI-TFODFF controller-based modified MPA. This case is very close to a real-world situation as the demand load is varying throughout the day depending on each power system area. The simulation results of the frequency and exchange power fluctuations of the proposed two-area power system for this case are depicted in Figure 15 with different state-of-the-art controllers such as TID, ID-T, I-TD, and TI-TD. It is quite clear that the proposed coordination technique performs significantly better than the other suggested individual and combination controllers, as it has the ability to mitigate the high frequency and power deviations due to the severely random load change with regard to fast frequency and tie-line power recovery and non-oscillations restraints. In contrast, it can damp the frequency deviation within ±0.0025 Hz for area a and ±0.002 Hz for area b, and it can keep power fluctuations within ±0.001 p.u. Meanwhile, the TI-TD controller comes second in reducing the oscillations to ±0.009 Hz for area a, ±0.0075 Hz for area b, and ±0.0093 p.u for system frequencies and tie-line exchange power, respectively. Furthermore, the combination of I-TD and ID-T controllers comes third as they uphold the frequency deviation within ±0.04 Hz and ±0.02 Hz for area a and ±0.03 Hz and ±0.02 Hz for area b and keep the tie-line power change within ±0.011 p.u and 0.014 p.u, respectively. Lastly, there is the TID controller with ±0.06 Hz for area a, ±0.08 Hz for area b, and ±0.018 p.u. It is concluded that the TFOI-TFODFF controller with EVs participation using MMPA has the lowest overshoots and settling times compared to the other controllers as summarized in Table 10 under the impact of uncertainties of load demand.

6.4. Scenario 4

In this scenario, the competing performance of the proposed coordination strategy of the TFOI-TFODFF controller beside the participation of EVs percentage is confirmed by adding the impact of the variation and intermittency of RESs to the studied two-area microgrid power system. Therefore, wind and PV power plants are employed in both areas as shown in Figure 2. The changes in wind, PV, and load power are depicted in Figure 16. The optimal parameters of controllers in this case are summarized in

Table 11. The obtained optimal controllers parameters with RESs scenario.

Control	Area	Coefficients
		${\mathit{K}}_{\mathit{t}1}$	${\mathit{K}}_{\mathit{t}2}$	${\mathit{K}}_{\mathit{i}}$	${\mathit{K}}_{\mathit{d}}$	${\mathit{\lambda }}_1$	${\mathit{\mu }}_1$	${\mathit{n}}_1$	${\mathit{n}}_2$	${\mathit{N}}_{\mathit{f}}$	${\mathit{\lambda }}_{\mathit{f}}$
TID	Area a	2.5123	―	2.6055	1.9822	―	―	3.63	―	―	―
	Area b	2.6011	―	2.0034	1.671	―	―	3.77	―	―	―
ID-T	Area a	2.9164	―	0.7622	1.9969	―	―	3.90	―	―	―
	Area b	1.8909	―	1.8055	2.3588	―	―	361	―	―	―
I-TD	Area a	3.9542	―	0.8566	2.4329	―	―	3.74	―	―	―
	Area b	1.7929	―	1.2055	2.4532	―	―	2.79	―	―	―
TI-TD	Area a	4.1099	3.2298	4.2241	3.7303	―	―	2.68	4.37	―	―
	Area b	3.6645	4.0332	3.2121	2.9058	―	―	3.19	2.52	―	―
Proposed	Area a	4.1088	4.2633	3.9701	4.5545	0.613	0.936	3.55	3.22	311	0.665
	Area b	3.9831	3.9801	0.9047	4.1998	0.744	0.988	3.03	2.49	178	0.921

. The PV generation is inserted at the initial time and disconnected at t = 70 s, while the wind farm is connected at t = 40 s. Figure 17 shows the two-area responses at different disturbance instances. It can be noticed that during the start of the simulation, the frequency variations increase provisionally in response to the extra generation output from the PV generation at t = 0 s, especially with the conventional TID controller. In addition, more deviations at area b at around 0.14 Hz at zero seconds and more than 0.1 Hz at t = 40 s then dropped significantly to −0.145 Hz when the PV plant is disconnected at t = 70 s. Meanwhile, the ID-T combination controller reduces the deviation levels to nearly 0.075 Hz, 0.087 Hz and −0.07 Hz in area a at 0 s, 40 s, and 70 s respectively, and less than 0.11 Hz, 0.04 Hz, and −0.1 Hz at area b at the same instants of PV and wind power connection and disconnection. However, the I-TD gives an acceptable performance by maintaining the system frequency at 0.045 Hz, 0.055 Hz and −0.045 Hz at 0 s, 40 s, and 70 s in area a, respectively, which is less than that of the TID and ID-T controllers. When using the combination TI-TD controller, the frequency deviation is maintained around acceptable values such as 0.026 Hz at 0 s, 0.04 Hz at 40 s, and −0.025 Hz at 70 s in area a and 0.04 Hz at 0 s, 0.034 Hz at 40 s, and −0.043 Hz at 70 s in area b with faster performance than the previous controllers. However, Figure 17 and Table 10 confirm that the proposed TFOI-TFODFF controller is the most efficient and the fastest technique in restoring the multi-MG system frequency in terms of overshoot, undershoot, rise time, and settling time.

6.5. Scenario 5

This scenario investigates the performance of the proposed new coordination of TFOI-TFODFF controller performance and its cooperation with EVs in the LFC loop. A drastic case for the capability of the multi-area MG system is tested under the impact of uncertainty of RESs by increasing their penetration level in the studied system. This case includes a 120 MW SLD at the initial time of simulation in addition to connecting the PV1 plant at t = 20 s, PV2 plant at t = 60 s, wind farm 1 at t = 40 s, and wind farm 2 at t = 80 s. The changes in wind, PV, and load power are depicted in Figure 18. Meanwhile, Figure 19 depicts the obtained performance comparisons of the proposed and suggested controllers. It is shown that the TID controller has poor frequency damping for frequency and power fluctuations during the disturbances of this case, especially at instants of wind connection at 40 s and 80 s, respectively. Meanwhile, the ID-T controller gives better damping characteristics than the TID controller. However, it takes a long time to settle down as noted in the obtained simulation results. When using the I-TD controller, it succeeded to keep the frequency fluctuations within lower values than the previous controllers. However, it suffers from protracted damped oscillations especially at the instants of 40 s, 60 s, and 80 s, respectively. Furthermore, the TI-TD and TFOI-TFODFF controllers have a strong dynamic response in restoring the frequency deviations in both areas but with small deviations for TI-TD compared to TFOI-TFODFF, which offers superior performance by effectively handling this contingency over all five steps of disturbances in this case. Therefore, this severe scenario represents the effectiveness and robustness of the proposed coordination technique over the other methods.

7. Conclusions

A novel modified TFOI-TFODF is proposed in this paper. The developed controller exhibits the merits of the tilt controller with the features of FO integral, FO derivative, and FO filter. Furthermore, a new optimization algorithm based on MMPA is also proposed as well. Chaotic map and weighting factor techniques are used in this paper to boost the MPA’s performance. They are used to enhance the initialization and exploitation processes of the traditional MPA. The performance of the MMPA has been assessed using standard functions and parametric/nonparametric statistical analysis and has been compared with recent techniques such as AEO, HPA, and SMA. The dominance of the proposed MMPA over the conventional techniques has been verified using the mean ranking method. On the other hand, the proposed controller tuned by MMPA coordinates the participation of the EVs in joint with the generation power plants to overcome inter-area frequency oscillations and tie-line power fluctuations as well. The performance of the proposed controller with MMPA has been investigated on a two-area power system. Through the simulation results with different perturbation scenarios and varying generations, the proposed controller outperforms the performance of the conventional controllers such as TID, ID-T, I-TD, and TI-TD. For instance, the settling time using the proposed controller has been reduced to be less than 50% compared to the conventional controllers. Moreover, the percentage overshoot/undershoot is almost eliminated from the dynamic response using the proposed controller. It is concluded from the simulation results that the proposed controller with MMPA has demonstrated its capability to regulate the system performance and mitigate frequency oscillations. Future research of this work includes the performance study of the proposed controller and optimizer over a higher number of interconnected power system areas. Moreover, more analysis can be conducted for studying the complexity of hardware implementation of the new proposed controller compared with existing controllers in the literature.