Leveraging the Performance of Integrated Power Systems with Wind Uncertainty Using Fractional Computing-Based Hybrid Method

Abstract

Reactive power dispatch (RPD) in electric power systems, integrated with renewable energy sources, is gaining popularity among power engineers because of its vital importance in the planning, designing, and operation of advanced power systems. The goal of RPD is to upgrade the power system performance by minimizing the transmission line losses, enhancing voltage profiles, and reducing the total operating costs by tuning the decision variables such as transformer tap setting, generator’s terminal voltages, and capacitor size. But the complex, non-linear, and dynamic characteristics of the power networks, as well as the presence of power demand uncertainties and non-stationary behavior of wind generation, pose a challenging problem that cannot be solved efficiently with traditional numerical techniques. In this study, a new fractional computing strategy, namely, fractional hybrid particle swarm optimization (FHPSO), is proposed to handle RPD issues in electric networks integrated with wind power plants (WPPs) while incorporating the power demand uncertainties. To improve the convergence characteristics of the Particle Swarm Optimization and Gravitational Search Algorithm (PSOGSA), the proposed FHPSO incorporates the concepts of Shannon entropy inside the mathematical model of traditional PSOGSA. Extensive experimentation validates FHPSO effectiveness by computing the best value of objective functions, namely, voltage deviation index and line loss minimization in standard power systems. The proposed FHPSO shows an improvement in percentage of 61.62%, 85.44%, 86.51%, 93.15%, 84.37%, 67.31%, 61.64%, 61.13%, 8.44%, and 1.899%, respectively, over ALC_PSO, FAHLCPSO, OGSA, ABC, SGA, CKHA, NGBWCA, KHA, PSOGSA, and FPSOGSA in case of traditional optimal reactive power dispatch(ORPD) for IEEE 30 bus system. Furthermore, the stability, robustness, and precision of the designed FHPSO are determined using statistical interpretations such as cumulative distribution function graphs, quantile-quantile plots, boxplot illustrations, and histograms.

1. Introduction

The research community of the energy and power sector has paid close attention to reactive power dispatch (RPD) due to its critical importance in the global energy management system with the goals of enhancing voltage profile, minimizing electrical transmission losses, and overall operating cost while fulfilling the network system power demand and network constraints [1]. The said aims are met by tuning a specified set of decision parameters that include the rating of the capacitor bank, transformer tap setting, and generator terminal voltage that assure both the suitability and optimality of the planned operational state. Nevertheless, because of the uncertain nature of the load, the existence of discrete and continuous constraints and the non-linear and dynamic nature of the power system leads the RPD towards a difficult optimization issue. As a result, a plethora of optimization methods have been developed, including evolutionary programming [2], genetic algorithm [3], whale optimization algorithm [4], gravitational search algorithms [5], adaptive particle swarm optimization [3], fuzzy genetic algorithm [6], improved gravitational search algorithm [7], population-based methods like artificial bee colony with firefly [8], ant lion optimizer [9], chaotic krill herd algorithm [10], differential evolution [11], quasi-oppositional chemical reaction optimization [12], and others, as listed in Table 1, to solve the optimal reactive power dispatch (ORPD) problems. These already proposed methods have several advantages, such as improved convergence rates, robustness, effective handling of constraints and multi-objective optimization, increased flexibility and adaptability to different system conditions, enhanced ability to handle complex, solution quality, and non-linear problems, and stability in solving large-scale power systems. Even some algorithms, such as differential evolution, adaptive genetic algorithms, and evolutionary programming, have shown exceptional performance in solving ORPD problems. The incorporation of meta-heuristic techniques, like ant colony optimization, particle swarm optimization, and the firefly algorithm, has also demonstrated promising results; however, despite their several advantages, these techniques also have some limitations, like difficulty in handling high-dimensional problems, risk of convergence to suboptimal solutions, sensitivity to parameter settings and initial conditions, computational complexity, and limited ability to handle dynamic and uncertain systems. Among several others, some methods, such as the chaotic krill herd and enhanced firefly algorithm may involve extensive parameter tuning, while others, like the whale optimization and oppositional krill herd, may suffer from convergence issues. Additionally, the optimization strength of these algorithms can be affected by the complexity and size of the power network being optimized. Keeping this in mind, the recent development of fractional calculus inside the mathematical model of traditional metaheuristic techniques is revealing promising results such as improved stability, reduced computational complexity, handling dynamic enhanced exploration, flexibility and adaptability, avoiding suboptimal solutions, improved convergence, uncertain systems, ability to handle high-dimensional problems, and robustness to parameter settings. However, the synergy of fractional calculus (FC) and entropy diversity inside the methodology of traditional algorithms has yet to be explored in the energy and power industry, notably in the realm of RPD considering uncertainties associated with the power demand and wind power [13,14,15]. Recent developments in fractional variants of the swarm and evolutionary-based computing methods that embrace the principle concept of FC inside the mathematical model of the algorithm have yielded encouraging results. PSO with fractional order (FO) velocity, FO robotic PSO, and FO Darwinian PSO (FO-DPSO) are examples [16,17,18,19]. Furthermore, these methods are effectively adopted for solving optimization problems in robot path controller design [20], electromagnetic plane wave [21], power system and parameter estimation [22,23,24,25,26,27,28], classification of hyperspectral images [29,30], image processing [31], and localization, design of discretized fractional order filters [32], segmentation of optic disc [33], land-cover monitoring [34], and swarm formation control for UAV [35]. Additionally, we may also cite the coupled tank systems [36], continuous slide mode-based nonlinear observer [37], implementation of PID controllers [38,39], emission prediction [40], non-linear model identification [41], and multi-band power stabilizer based on integrated GAPSO [42]. These advancements motivate the incorporation of FC into a mathematical model of algorithms for tackling RPD issues while taking wind power plants into consideration [43]. At first, the voltage collapse proximity index (VCPI) approach is used to detect the weak buses of electric networks for optimal allocation of the WPPs. Subsequently, the minimization of the voltage deviation index (VDI) and line losses is employed as the fitness assessment function for enhancing the capability of the network, taking into account the wind generation and power demand uncertainties. In the given analysis, therefore, wind generation and variable power demand are used as factors of uncertainty. For wind speed, the Rayleigh probability distribution function (PDF), while for the load uncertainty estimation, the normal PDF, is employed. The following are the salient features of the proposed work, which summarize the main contributions and novel insights:

• The Gaussian probability distribution function is used in scenario-based optimum RPD to provide an effective characterization of wind power output and load uncertainty.
• The modeling of a novel integrated optimization mechanism, namely FHPSO, based on the collaboration of Shanon entropy, fractional calculus, PSO, and GSA, is shown as an alternate method for solving probabilistic RPD problems.
• The computational intelligence and rich pedigree of FHPSO versions based on various fractional orders are demonstrated by minimizing the voltage deviation index and line losses while fulfilling operational constraints and scenario power demand in electric networks.
• A detailed statistical comparison of the proposed FHPSO and original FPSOGSA-EE based on cumulative distribution function graph, boxplot illustration, quantile-quantile plot, and histograms to validate the consistency, stability, efficacy, and scalability of the FHPSO. The remainder of this research document is organized as follows.

Table 1. A summary of designed schemes for RPD (f₁ = P_loss and f₂ = VDI minimization) [43].

Ref.	Methods	Objectives	Year	Ref.	Methods	Objectives	Year
[44]	Evolutionary programming	f1	1995	[45]	Enhanced firefly	f1, f2	2015
[46]	Adaptive genetic algorithm	f1	1998	[47]	Differential evolution	f1	2015
[48]	PSO	f1	2000	[49]	Backtracking search	f1, f2	2016
[50]	Multi-agent PSO	f1, f1	2005	[10]	Chaotic krill herd	f1, f2	2016
[51]	Improved GA	f1	2005	[52]	Exchange market algorithm	f1, f2, stability	2016
[53]	GA-interior point method	f1	2006	[54]	Quasi-oppositional DE	f1, f2, stability	2016
[55]	Modified PSO	Stability	2007	[56]	Oppositional krill herd	f1, f2	2016
[57]	Turbulent crazy PSO	f1, f2	2009	[58]	Two-point estimate method	f1, f2, stability	2016
[59]	Self-adaptive real-coded GA	f1	2009	[60]	Moth-flame optimization	f1	2017
[61]	Comprehensive learning PSO	f1	2010	[62]	GBWC	f1, f2	2017
[63]	MNSGA-II	f1, stability	2011	[12]	Chemical reaction	f1, f2, stability	2018
[64]	Ant colony optimization	f1	2011	[8]	ABC-FF	f1, f2, stability	2018
[65]	Biogeography optimization	f1, f2	2011	[66]	Whale optimization	f1	2018
[67]	Harmony search algorithm	f1, f2, f3	2011	[68]	Sine cosine algorithm	f1, f2	2019
[69]	Adaptive approaches	f1, f2	2012	[70]	Moth Swarm Algorithm	f1, f2	2019
[71]	HFMOEA	f1, stability	2013	[72]	ALC-PSO algorithm	f1, f2	2019
[73]	Opposition-based GSA	f1, f2, stability	2013	[74]	Lightning Attachment	f1	2019
[75]	MICA-IWO	f1	2014	[76]	Enhanced GWO	f1, f2	2019
[77]	Teaching learning	f1	2015	[78]	Artificial bee colony	f1, f2, stability	2020
[79]	Hybrid firefly algorithm	f1, f2	2015	[80]	Chaotic Bat Algorithm	f1, f2, stability	2020
[81]	Gray wolf optimizer (GWO)	f1, f2	2015

Table 1. A summary of designed schemes for RPD (f₁ = P_loss and f₂ = VDI minimization) [43].

Ref.	Methods	Objectives	Year	Ref.	Methods	Objectives	Year
[44]	Evolutionary programming	f1	1995	[45]	Enhanced firefly	f1, f2	2015
[46]	Adaptive genetic algorithm	f1	1998	[47]	Differential evolution	f1	2015
[48]	PSO	f1	2000	[49]	Backtracking search	f1, f2	2016
[50]	Multi-agent PSO	f1, f1	2005	[10]	Chaotic krill herd	f1, f2	2016
[51]	Improved GA	f1	2005	[52]	Exchange market algorithm	f1, f2, stability	2016
[53]	GA-interior point method	f1	2006	[54]	Quasi-oppositional DE	f1, f2, stability	2016
[55]	Modified PSO	Stability	2007	[56]	Oppositional krill herd	f1, f2	2016
[57]	Turbulent crazy PSO	f1, f2	2009	[58]	Two-point estimate method	f1, f2, stability	2016
[59]	Self-adaptive real-coded GA	f1	2009	[60]	Moth-flame optimization	f1	2017
[61]	Comprehensive learning PSO	f1	2010	[62]	GBWC	f1, f2	2017
[63]	MNSGA-II	f1, stability	2011	[12]	Chemical reaction	f1, f2, stability	2018
[64]	Ant colony optimization	f1	2011	[8]	ABC-FF	f1, f2, stability	2018
[65]	Biogeography optimization	f1, f2	2011	[66]	Whale optimization	f1	2018
[67]	Harmony search algorithm	f1, f2, f3	2011	[68]	Sine cosine algorithm	f1, f2	2019
[69]	Adaptive approaches	f1, f2	2012	[70]	Moth Swarm Algorithm	f1, f2	2019
[71]	HFMOEA	f1, stability	2013	[72]	ALC-PSO algorithm	f1, f2	2019
[73]	Opposition-based GSA	f1, f2, stability	2013	[74]	Lightning Attachment	f1	2019
[75]	MICA-IWO	f1	2014	[76]	Enhanced GWO	f1, f2	2019
[77]	Teaching learning	f1	2015	[78]	Artificial bee colony	f1, f2, stability	2020
[79]	Hybrid firefly algorithm	f1, f2	2015	[80]	Chaotic Bat Algorithm	f1, f2, stability	2020
[81]	Gray wolf optimizer (GWO)	f1, f2	2015

Section 2 contains the problem formulation for RPD, stochastic wind generation, and power demand uncertainty characterization. Section 3 explains the proposed methodology, general flow diagram, and pseudocode. In addition to providing numerical data, Section 4 and Section 5 include a thorough comparison with other methodologies and statistical analysis, respectively. Finally, Section 6 summarizes the key findings.

2. System Model

2.1. Uncertainty Characterization

(1) Modelling of demand variabilities: The Gaussian probability distribution formed on the known standard deviation (σ_D) and mean (μ_D) of power demand PDF is used to represent the probabilistic behavior of power demand in an electric network. The probability of occurrence πD for any dth load scenario is defined as follows:

$${ \pi }_D={\int }_{P_{D_d}^{\mathrm{m}\mathrm{i}\mathrm{n}}}^{P_{D_d}^{\mathrm{m}\mathrm{a}\mathrm{x}}} {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\displaystyle \frac{{\left(P_D-{\mu }_D\right)}^2}{2{\sigma }^2}}\right]d{P}_D$$

(1)

where, $P_{D_d}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $P_{D_d}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ defines the limits (minimum/maximum) of load scenario dth, whereas

$${ P}_{D_d}={\displaystyle \frac{1}{{\pi }_d}}{\int }_{P_{D,d}^{\mathrm{m}\mathrm{i}\mathrm{n}}}^{P_{D=d}^{\mathrm{m}\mathrm{a}\mathrm{x}}} \left(P_D\times {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\displaystyle \frac{{\left(P_D-{\mu }_D\right)}^2}{2{\sigma }^2}}\right]\right)d{P}_D$$

(2)

(2) Uncertainty modeling of wind generation: The Weibul or Rayleigh PDF is used to represent the uncertainty in wind speed ν (in m/s), mathematically, as follows:

$$\mathrm{P}\mathrm{D}\mathrm{F}(\nu )=\left({\displaystyle \frac{\nu }{c^2}}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left({\displaystyle \frac{\nu }{\sqrt{2}c}}\right)}^2\right]$$

(3)

The likelihood of all the adopted wind scenarios is shown below.

$${ \pi }_w={\int }_{v_{t,w}}^{v_{f,w}} \left({\displaystyle \frac{\nu }{c^2}}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left({\displaystyle \frac{\nu }{\sqrt{2}c}}\right)}^2\right]dv$$

(4)

$${ v}_w={\displaystyle \frac{1}{{\pi }_w}}{\int }_{v_{t,w}}^{v_{f,w}} \left(v\times \left({\displaystyle \frac{\nu }{c^2}}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left[-{\left({\displaystyle \frac{\nu }{\sqrt{2}c}}\right)}^2\right]\right)dv$$

(5)

Hereafter, c defines the scaling parameter, which is dependent on previous wind data, and v_{i, w} and v_{f, w} are the boundaries of the wind speed at the wth scenario. For different wind speeds, the anticipated available power output (P^ravl) of the WPPs may be quantified.

$${ P}_w^{avl}=\left\{\begin{array}{ll}{P}_r^w& v_{\mathrm{rated} }\le v_w\le v_{\mathrm{out} }^c\\ {\displaystyle \frac{v_w-v_{\mathrm{on} }^c}{v_{\mathrm{rated} }-v_{\mathrm{in} }^c}}{P}_r^w& v_{\mathrm{rated} }\le v_w\le v_{\mathrm{out} }^c\\ 0& v_{\mathrm{rated} }\le v_w\le v_{\mathrm{out} }^c\end{array}\right.$$

(6)

Here, v_in, v_rated, and v_out represents the cut-in, rated, and cut out speed. It is notable that, the individual scenarios are added to find the aggregate load-wind scenarios. The probability of overall scenario πs obtained by using the dth and wth scenarios may be quantified as follows:

$${ \pi }_s={\pi }_w\times {\pi }_d$$

(7)

For (1)–(7) the reference source is [43].

2.2. Problem Formulation

The voltage deviation index and transmission line loss minimization are used as objective functions during RPD in electric networks while taking WPPs into account in this study. The mathematical representation of these fitness functions and accompanying restrictions is summarized in the next section.

2.2.1. Voltage Deviation Index (VDI)

It is essential for the safe operation of the power system to maintain a constant voltage profile at all network nodes (buses). VDI is used as a fitness in the optimization framework to assess voltage deviation w.r.t. reference voltage. The reference voltage is set to 1 p.u. (per unit) while considering VDI as a fitness evaluation function [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. At standard nominal bus voltage, the per unit, i.e., p.u. value, is always 1 because p.u. = actual value/base value. A voltage deviation index of zero indicates that the voltage level is precisely at its reference value, representing no deviation from the standard voltage. Numerically,

$$V_D={\sum }_{i=1}^{N_{\mathrm{bus} }} \left|V_i-1.0\right|$$

(8)

hereafter, V_i defines the bus voltage and N_bus represents the total load of buses.

2.2.2. Power Loss Minimization

The second objective function adopted in this study is power loss minimization which plays a vital role in improving power system efficiency, performance and for techno-economic reasons. It may be stated mathematically as follows:

$$\begin{array}{c}\mathrm{Minimize} f\left(x_1,x_2\right)=P_{\mathrm{loss} }\\ ={\displaystyle \sum _{r=1}^R} g_r\left[V_i^2+V_j^2-2\times V_i\times V_j\mathrm{c}\mathrm{o}\mathrm{s}\left({\delta }_i-{\delta }_j\right)\right]\end{array}$$

(9)

where $x_1$ and $x_2$ are defined as follows:

$$x_2=\left[\begin{array}{l}{Q}_{C1},Q_{C2},\dots ,Q_{CN},\\ V_{G1},V_{G2},\dots ,V_{GN},\\ T_1,T_2,\dots ,T_N,\end{array}\right]$$

(10)

$$x_1=\left[\begin{array}{l}{S}_{L1},S_{L2},\dots ,S_{LN},\\ V_{L1},V_{L2},\dots ,V_{LN},\\ Q_{G1},Q_{G2},\dots ,Q_{GN}\end{array}\right]$$

(11)

The variables and symbols in the above equations are defined below:

•

x₂ represents vector of the controlling parameters such as the transformers tap setting (T₁, T₂, …, T_NT), generators voltage magnitude (V_G1, V_G2, …, V_GNPV), reactive power compensators (Q_C1, Q_C2, …, Q_CNC).

•

x₁ represents the dependent parameters such as the reactive power of the generator (Q_G1, Q_G2, …, Q_GNPV), load voltages (V_L1, V_L2, …, V_LNL) and line loading (S_L1, S_L2, …, S_LNL).

•

R represents the total branches in power system.

•

f (x₁, x₂) is power loss minimization function.

•

g_r represents the line conductance.

•

V_j and V_i are, respectively, the receiving and sending end voltages.

•

δ_i and δ_j are the sending and receiving end voltage angles, respectively.

Power balancing equations describe the equality constraints as follows:

$${ P}_{Gi}-P_{Di}-V_i\sum _{j=1}^{N_{Bus}} V_j\left[\begin{array}{l}{B}_{ij}\mathrm{s}\mathrm{i}\mathrm{n}\left({\delta }_i-{\delta }_j\right)\\ +G_{ij}\mathrm{c}\mathrm{o}\mathrm{s}\left({\delta }_i-{\delta }_j\right)\end{array}\right]=0$$

(12)

$${ Q}_{Gi}-Q_{Di}-V_i\sum _{j=1}^{N_{Bus}} V_j\left[\begin{array}{l}{B}_{ij}\mathrm{c}\mathrm{o}\mathrm{s}\left({\delta }_i-{\delta }_j\right)\\ +G_{ij}\mathrm{s}\mathrm{i}\mathrm{n}\left({\delta }_i-{\delta }_j\right)\end{array}\right]=0$$

(13)

Here, P_Gi and P_Di are, respectively, the ith bus injected and demanded real powers, whereas Q_Gi and Q_Di defines, respectively, the ith bus injected and demanded reactive powers. The inequality limitations (min and max) include the reactive power of generator and output voltage, the tap setting of the transformer, and the capacitor bank size:

$$Q_{Gi}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le Q_{Gi}\le Q_{Gi}^{\mathrm{m}\mathrm{a}\mathrm{x}},i=\mathrm{1,2},\dots ,N_{PV}$$

(14)

$$V_{Gi}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le V_{Gi}\le V_{Gi}^{\mathrm{m}\mathrm{a}\mathrm{x}},i=\mathrm{1,2},\dots ,N_{PV}$$

(15)

$$T_i^{\mathrm{m}\mathrm{i}\mathrm{n}}\le T_i\le T_i^{\mathrm{m}\mathrm{a}\mathrm{x}},i=\mathrm{1,2},\dots ,N_T$$

(16)

$$Q_{ci}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le Q_{ci}\le Q_{ci}^{\mathrm{m}\mathrm{a}\mathrm{x}},i=\mathrm{1,2},\dots ,N_c$$

(17)

Hereafter, the variables N_pv, N_c, and N_T, respectively, denote the total number of voltage-controlled (PV) buses, shunt capacitors, and tap-changing transformers, respectively, whereas Q_Gi represents the ith generator’s reactive power, V_Gi is the output voltage of the ith generator bus, T_i is the ith tap-changing transformer, and Q_ci is ith fixed capacitor bank. For IEEE 30 bus system, the min limits in per unit (p.u.), that is $Q_{Gi}^{\mathrm{m}\mathrm{i}\mathrm{n}}$, $V_{Gi}^{\mathrm{m}\mathrm{i}\mathrm{n}}$, $T_i^{\mathrm{m}\mathrm{i}\mathrm{n}}$, and $Q_{ci}^{\mathrm{m}\mathrm{i}\mathrm{n}}$ are −1, 0.95, 0.9, and 0, while, maximum limits $Q_{Gi}^{\mathrm{m}\mathrm{a}\mathrm{x}}$, $V_{Gi}^{\mathrm{m}\mathrm{a}\mathrm{x}}$, $T_i^{\mathrm{m}\mathrm{a}\mathrm{x}}$, and $Q_{ci}^{\mathrm{m}\mathrm{a}\mathrm{x}}$ are 1, 1.1, 1.05, and 5, respectively.

3. Design Methodology

The model theory and mathematical structure behind the FHPSO components, which include the gravitational search algorithm, conventional PSO, hybrid PSOGSA, FC, and Shannon entropy, are presented in this section. Subsequently, a graphical explanation of the RPD optimization framework design is provided.

3.1. Conventional GSA

In Ref. [82], E. Rashedi et al. presented the GSA, an advanced heuristic approach, in 2009. The GSA idea relies on Newton’s law, which says that “everything in the universe attracts everything else with a force that is inversely proportional to the square of their distance apart and directly correlative to the product of their masses”. In the gravitational search method, a collection of alternative solutions, known as agents, have masses that are proportional to the value of the relevant fitness function. During the algorithm’s execution, the agents’ gravitational forces attract one another. The attraction force is proportionate to the masses magnitude. As an outcome, the one with huge size and probably nearest to the ideal worth draws different masses relying upon individual distances. The mathematical model of standard GSA is as presented below: Create an N agents model. After that, assign a random number to each agent in the search space. The force of gravity between agents j and i at any interval of time t is defined as follows:

$$F_{ij}^d(t)=G\left(t\right){\displaystyle \frac{M_{pi}\left(t\right)\times M_{aj}\left(t\right)}{R_{ij}\left(t\right)+\epsilon }}\left(x_j^d\left(t\right)-x_i^d\left(t\right)\right)$$

(18)

In the following, G_(t) is the gravitational constant, ε is a small coefficient, R_ij is the Euclidian distance between ith and jth agents, Maj is the active gravitational mass of the j_th agent, and M_pi is the passive gravitational mass of the ith agent. The mathematical illustration of the gravitational constant is given below:

$$G(t)=G_o\times exp\left(-\alpha \times {\displaystyle \frac{ \mathrm{iter} }{ \mathrm{maxiter} }}\right)$$

(19)

The current iteration is indicated by iter in (19), the maximum number of iterations is indicated by maxiter, and the initial value and descending factor are denoted by α and G_o, respectively. In d dimensional search space the commutative force on agent i is as follows:

$$F_i^d(t)=\sum _{j\ne i,j=1}^N {\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}_j{F}_{ij}^d(t)$$

(20)

where ϵ [0, 1] = rand_j. According to the law of motion, the acceleration of an agent is inversely proportional to its mass and directly proportional to the resultant force.

$$a{c}_i^d(t)={\displaystyle \frac{F_i^d(t)}{M_{ii}(t)}}$$

(21)

The time t, the mass Mii of the ith item, and the resultant force $F_i^d$ are shown below. The agent’s location and velocity are calculated as follows:

$$v_i^d(t+1)={\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}_i\times v_i^d\left(t\right)+a{c}_i^d\left(t\right)$$

(22)

$$x_i^d(t+1)=x_i^d\left(t\right)\times v_i^d(t+1)$$

(23)

3.2. Traditional PSO

The canonical PSO approach, developed in 1995 by Eberhart and Kennedy, is a meta-heuristic inspired by particle growth in a search space, specifically swarm, and may be applied both defensively and to identify the optimal solution. The velocity v and location x are the two numbers that immediately depict the particle’s motion. The local searching is governed by measuring the difference between the particle’s best location in a given iteration, or LBⁿ _t and its current position, or xⁿ_t. The global search is determined by measuring the difference between the position of the global best attained thus far, or GBⁿ_t, and the particle’s current position, or xⁿ_t. A weight ρ₁ and ρ₂ are randomly allocated to the local and global searching processes, respectively, as shown in Equations (24) and (25), as follows:

$$v_{t+1}^n=v_t^n,+{\rho }_1{r}_1\left(L{B}_t^n-x_t^n\right)+{\rho }_2{r}_2\left(G{B}_t^n-x_t^n\right)$$

(24)

$$x_{t+1}^n=x_t^n+v_{t+1}^n$$

(25)

3.3. The Hybrid PSOGSA

Syedali presented PSOGSA, a co-evolutionary optimization approach, in [83]. This strategy combines the global search potential g_best in PSO with the robustness of local searching in GSA.

$$\begin{array}{ll}& V_i(t+1)=w\times V_i\left(t\right)+c_1\times \mathrm{rand} \times a{c}_i\left(t\right)\\ & +c_2\times \mathrm{rand} \times \left( \mathrm{gest} -X_i\left(t\right)\right)\end{array}$$

(26)

In this case, the weighting function is defined by w, the acceleration coefficient is c_j, the best solution is g_best, a random number [01] is represented by rand, the acceleration of the ith agent is a_ci(t), and the velocity is V_i during iteration t. During each iteration, the following changes are made to each particle’s location:

$$X_i(t+1)=X_i\left(t\right)+V_i(t+1)$$

(27)

3.4. Fractional Calculus

The conventional integer differentiation and integration to a non-integer sequence is explained in FC. Notable mathematicians, including Weyl, Riemann, and Liouville, made significant contributions to the rise of FC as a research field in recent years. With applications in chaos, signal processing, irreversibility, modeling, diffusion, percolation, biology, wave propagation, viscoelasticity, control, physics, and fractals, fractional calculus has piqued the interest of the research community. The most popular interpretation of fractional derivative (FC) is the Grunwald–Letnikov (GL) explanation, which gives the following definition for the differential of a generic signal f(z) with fractional order α [84,85]:

$$D^{\alpha }[f(z)]=\underset{h\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}} \left[{\displaystyle \frac{1}{h^{\alpha }}}\sum _{w=0}^{\mathrm{\infty }} {\displaystyle \frac{(-1{)}^w\mathsf{\Gamma }(\alpha +1\left)f\right(z-wh)}{\mathsf{\Gamma }(w+1)\mathsf{\Gamma }(\alpha -w+1)}}\right]$$

(28)

The Eular gamma function is defined as Γ(), and the incremental step time is represented by h. This GL mathematics example clarifies a crucial distinction: the fractional derivative comprises infinite terms, which implies that it implicitly carries information from earlier trials, whereas the integer derivative only integrates a finite series. A discrete time generalization of the GL definition with sample period T and truncation order r is given below:

$$D^{\alpha }[f(z)]={\displaystyle \frac{1}{T^{\alpha }}}\sum _{w=0}^r {\displaystyle \frac{(-1{)}^m\mathsf{\Gamma }(\alpha +1\left)f\right(z-wT)}{\mathsf{\Gamma }(w+1)\mathsf{\Gamma }(\alpha -w+1)}}$$

(29)

Fractional order models are perfect for representing concepts like chaos and irreversibility because of their built-in memory. Construct an instance where the fractional calculus tool fits pertinently during the search process, given the dynamic behavior of particle advancement.

3.5. Fractional PSO

In 2010, Machado and associates [84] presented a revolutionary method to improve the properties of classical PSO, particularly its convergence, by including the fundamental fractional calculus hypothesis into the velocity update mechanism of conventional PSO. First, by reorganizing the initial velocity, the velocity derivative order is changed as follows:

$$v_{t+1}^n=\omega v_t^n,+{\rho }_1{r}_1\left(L{B}_t^n-x_t^n\right)+{\rho }_2{r}_2\left(G{B}_t^n-x_t^n\right)$$

(30)

In this case, ω represents the inertial weight, and x represents the particle’s location. Equation (30) can be shuffled using ω as follows:

$${ v}_{t+1}^n-v_t^n={\rho }_1{r}_1\left(L{B}_t^n-x_t^n\right)+{\rho }_2{r}_2\left(G{B}_t^n-x_t^n\right)=\nabla \left(v_{t+1}^n\right)$$

(31)

The following connection results when T = 1, since the term vⁿ _t+1–vⁿ_t is a simple derivative with order α = 1.

$${ D}^{\alpha }\left(v_{t+1}^n\right)={\rho }_1{r}_1\left(L{B}_t^n-x_t^n\right)+{\rho }_2{r}_2\left(G{B}_t^n-x_t^n\right)$$

(32)

Taking into account (29) with r = 4 (the first four terms), (32) can be reorganized as follows:

$$\begin{array}{cc}{v}_{t+1}^n-\hfill & \hfill \alpha v_t^n-{\displaystyle \frac{1}{2}}\alpha (1-\alpha )v_{t-1}^n-{\displaystyle \frac{1}{6}}\alpha (1-\alpha )(2-\alpha )v_{t-2}^n\\ & -{\displaystyle \frac{1}{24}}\alpha (1-\alpha )(2-\alpha )(3-\alpha )v_{t-3}^n\hfill \\ & ={\varphi }_1{r}_1\left(L{B}_t^n-s_t^n\right)+{\varphi }_2{r}_2\left(G{B}_t^n-s_t^n\right)\hfill \end{array}$$

(33)

or

$$\begin{array}{cc}\hfill v_{t+1}^n=& \alpha v_t^n+{\displaystyle \frac{1}{2}}\alpha (1-\alpha )v_{t-1}^n+{\displaystyle \frac{1}{6}}\alpha (1-\alpha )(2-\alpha )v_{t-2}^n\hfill \\ & +{\displaystyle \frac{1}{24}}\alpha (1-\alpha )(2-\alpha )(3-\alpha )v_{t-3}^n\hfill \\ & +{\varphi }_1{r}_1\left(L{B}_t^n-s_t^n\right)+{\varphi }_2{r}_2\left(G{B}_t^n-s_t^n\right)\hfill \end{array}$$

(34)

The fractional order α may be approximated to a number between [0, 1] if the fractional calculus viewpoint is applied, showing an enlarged memory attribute and reasonable fluctuation. The fractional PSO (FPSO) with order α = 1 is a subset of the standard PSO, according to (32). Since the FPSO uses the FC concept to affect particle features like convergence rate, it is necessary to classify the coefficient alpha in order to guarantee a significant amount of exploration during algorithm development. A comprehensive and extensive literature on FPSO can be found in [19,85,86,87,88].

3.6. Entropy

Throughout the years, a number of interpretations of entropy, including freedom, chaos, mixing, diversification, disorder, uncertainty, unpredictability, and information, have been developed. The basic idea of entropy, which measures a system’s disorder during a change from one state to another, was introduced by Boltzmann. The quantitative measure of energy distribution in a thermodynamic system at a specific temperature was demonstrated by Guggenheim using the concept of spreading. Lewis found in an isolated system that when a gas expands impulsively, there is an increase in variability or data loss and a weakening of the data related to particle position. Shannon then put forth information theory, focusing on the statistical and physical limitations that restrict signal transmission, to list the lost information lost during message transmission in a network. Shannon created H as a measure of uncertainty, chaos, and information. Regarding random parameters (x, y) ϵ (X, Y):

$$H(X)=-K\sum _{x\in X} p_i(x)\mathrm{l}\mathrm{o}\mathrm{g}{p}_i(x)$$

(35)

$$H(X,Y)=-K\sum _{x\in X} \sum _{y\in Y} \mathrm{l}\mathrm{o}\mathrm{g}{p}_i(x,y)p_i(x,y)$$

(36)

hereafter, x ϵ X is the discrete variable, p(x) defines the probability distribution, whereas, K represents the constant parameter.

3.7. Fractional Hybrid PSO

The optimization framework for this research was developed with the need to use and examine the entropy evolution throughout the evolution of fractional hybrid PSO. In order to increase algorithm optimization characteristics by avoiding sub-optimal solutions and boosting convergence rate while searching for the best solution, the FHPSO combines the computational power of Shanon’s entropy PSO, GSA, and FC. In the meantime, each iteration of the FHPSO implementation directly tracks the influence of the entropy measure throughout the algorithm’s evolution through acceleration updates. Entropy, in fact, quantifies the tendency of a system energy to change over time, or the distribution of particles across the search space. In light of this, the distance di between the ideal global particle and the position of the ith particle is calculated. The probability pi of each particle is then determined by dividing the distance di by the greatest probable distance dmax, as follows:

$$p_i={\displaystyle \frac{d_i}{d_{\mathrm{m}\mathrm{a}\mathrm{x}}}}$$

(37)

Mathematically, with $k=1$, the particle’s diversity index is stated as follows:

$$H(X)=-\sum _{i=1}^n p_i\mathrm{l}\mathrm{o}\mathrm{g}{p}_i$$

(38)

Here, $n$ represents the population size.

3.8. VCPI Method

The work outlined below employs the maximum power transfer (MPT) theory-based VCPI approach to identify weak buses in order to carry out the optimal allocation of WPPs [4]. Consider a voltage source Vs that is powering a load with an impedance Z_L∠φ and has a correlated impedance Z_s∠θ. The MPT theorem states that power flow peaks at impedances Z_L/Z_s = 1, producing a voltage collapse predictor. Accuracy and job simplification can be achieved by taking a fluctuating load impedance and maintaining a constant φ. In the meantime, when the power demand rises, the load current rises as well and Z_L falls, which causes a large line drop and a low receiver side voltage level, mathematically.

$$\begin{array}{c}I={\displaystyle \frac{V_s}{\sqrt{\left[{\left(Z_s\mathrm{c}\mathrm{o}\mathrm{s}\theta +Z_L\mathrm{c}\mathrm{o}\mathrm{s}\phi \right)}^2+{\left(Z_s\mathrm{s}\mathrm{i}\mathrm{n}\theta +Z_L\mathrm{s}\mathrm{i}\mathrm{n}\phi \right)}^2\right]}}}\\ \end{array}$$

(39)

$$V_r=Z_{L}I$$

(40)

For

$$V_r={\displaystyle \frac{Z_r}{Z_s}}{\displaystyle \frac{V_s}{\sqrt{1+{\left(\frac{Z_r}{Z_s}\right)}^2+2\mathrm{c}\mathrm{o}\mathrm{s}(\theta -\phi )\left(\frac{Z_r}{Z_s}\right)}}}$$

(41)

the line loss is given by

$$P_l={\displaystyle \frac{V_s^2/Z_s}{\sqrt{1+{\left(\frac{Z_r}{Z_s}\right)}^2+2\mathrm{c}\mathrm{o}\mathrm{s}(\theta -\phi )\left(\frac{Z_r}{Z_s}\right)}}}\mathrm{c}\mathrm{o}\mathrm{s}\phi$$

(42)

whereas receiving end power by

$$P_r=V_{r}I\mathrm{c}\mathrm{o}\mathrm{s}\varphi$$

(43)

$$\begin{array}{c}\\ P_r={\displaystyle \frac{V_s^2/Z_s}{\sqrt{1+{\left(\frac{Z_r}{Z_s}\right)}^2+2\left(\frac{Z_r}{Z_s}\right)\mathrm{c}\mathrm{o}\mathrm{s}(\theta -\phi )}}}{\displaystyle \frac{Z_r}{Z_s}}\mathrm{c}\mathrm{o}\mathrm{s}\phi \end{array}$$

(44)

By applying the boundary conditions $\left(\partial P_r/\partial Z_L\right)=0$, which translates to $Z_L/Z_s=1$, the maximum power $P_r$ is found. Given this circumstance, (44) can be expressed as follows:

$$P_r={\displaystyle \frac{\mathrm{c}\mathrm{o}\mathrm{s}\phi }{4{\mathrm{c}\mathrm{o}\mathrm{s}}^2\left(\frac{\theta -\phi }{2}\right)}}{\displaystyle \frac{V_s^2}{Z_s}}$$

(45)

Since the VCPI concept is derived from MPT, it can be stated as follows:

$$VCPI={\displaystyle \frac{P_r}{P_{r(\max )}}}$$

(46)

In order for the system to be stable, the VCPI indication needs to be less than one. If a bus’s VCPI is approaching unity, it is considered weak and is close to the instability limit. Subsequently, it is also suggested that the optimal possible location for WPP integration is this assessed weak bus. The FHPSO execution procedure and its graphical abstract are depicted in Figure 1. The key contribution of Figure 1 is the introduction of fractional calculus into the mathematical model of canonical PSOGSA. These changes updated mass, force, and acceleration of particles with the help of Fractional Order Derivative to give them better convergence behavior and exploration capabilities. The new additions have been highlighted in the updated portions of the flowchart, demonstrating where and how this improvement has impacted the performed algorithm steps, respectively.

4. Case Study and Simulation Results

To show the optimization efficacy of the proposed FHPSO in solving stochastic RPD problems considering wind power generation, IEEE 30 and 57 bus networks are adopted as test cases. The input parameters, including the power density function, cumulative Weibull PDF, and speed time series of wind power, are taken from [89]. Using the VCPI method, weak buses in the aforementioned power networks are identified for the installation of WPPs. Following that, VDI and Ploss minimization, being adopted as fitness assessment functions, are minimized by tuning network constraints such as generator’s terminal voltage, transformer tap setting, and capacitor bank size by using the FHPSO. To demonstrate the efficacy of FHPSO, the following scenarios are adopted:

4.1. Reactive Power Dispatch under Wind/Load Uncertainties

In this scenario, the RPD problems in an electric network integrated with WPPs are handled by taking into account the variabilities associated with system wind power and power demand, which are defined by a scenario-based approach. In the presented research work, the power demand, comprising the real and reactive loads, is defined by normal PDF. Here, we have adopted five different load scenarios, so the load PDF is divided into five sections. The distribution function’s mean gauge is adopted as the nominal power demand, whereas 2% of the mean load is selected as a standard deviation. A 12 MW WPP is installed at bus 4, which has been identified as a weak bus of the IEEE 30 bus system. Using the VCPI approach with [90], 25 load-wind scenarios (including 5 generation scenarios for WPPs) are considered with their relevant probabilities, including πd, πw, and πs, as listed in

Table 2. Wind-load scenarios, likelihood, and global best solutions.

Inputs [43]			Probabilities [43]			Outputs
Scenario	Wind%	Load%	πd	πw	πs	EPloss	EVDI
S-25	100	105	0.025	0.1227	0.0031	0.0181	0.0006
S-24	86.83	105	0.025	0.1992	0.005	0.0281	0.0015
S-23	49.37	105	0.025	0.4048	0.0101	0.0588	0.0018
S-22	12.87	105	0.025	0.2044	0.0051	0.0291	0.0008
S-21	0	105	0.025	0.0689	0.0017	0.0097	0.0006
S-20	100	103	0.135	0.1227	0.0166	0.0973	0.0033
S-19	86.83	103	0.135	0.1992	0.0269	0.1516	0.0097
S-18	49.37	103	0.135	0.4048	0.0546	0.3094	0.0176
S-17	12.87	103	0.135	0.2044	0.0276	0.1580	0.0048
S-16	0	103	0.135	0.0689	0.0093	0.0535	0.0044
S-15	100	100	0.680	0.1227	0.0834	0.4752	0.0144
S-14	86.83	100	0.680	0.1992	0.1355	0.7928	0.0548
S-13	49.37	100	0.680	0.4048	0.2753	0.002811	0.0028
S-12	12.87	100	0.680	0.2044	0.139	0.0166	0.0028
S-11	0	100	0.680	0.0689	0.0469	0.2386	0.0160
S-10	100	97	0.135	0.1227	0.0166	0.0992	0.0060
S-09	86.83	97	0.135	0.1992	0.0269	0.0048	0.0079
S-08	49.37	97	0.135	0.4048	0.0546	0.3125	0.0187
S-07	12.87	97	0.135	0.2044	0.0276	0.0564	0.0017
S-06	0	97	0.135	0.0689	0.0093	0.0527	0.0020
S-05	100	95	0.025	0.1227	0.0031	0.0178	0.0012
S-04	86.83	95	0.025	0.1992	0.005	0.0291	0.0024
S-03	49.37	95	0.025	0.4048	0.0101	0.0592	0.0046
S-02	12.87	95	0.025	0.2044	0.0051	0.0291	0.0012
S-01	0	95	0.025	0.0689	0.0017	0.0098	0.0004

and are being taken from [43]. The obtained best solution for expected voltage deviation index EVPI, i.e., πs × V PI, and expected line loss reduction E_Ploss, i.e., πs × P_loss, yielded by FHPSO throughout each scenario can also be seen in Table 2. The minimum EVPI is always lower than 0.0564 p.u. while the minimum E_Ploss is lower than 1.39 MW [43] In all of the 25 different scenarios, demonstrating a superior performance of FHPSO in evaluating the fitness function at fractional order.

4.2. V DI Minimization in the IEEE 30 Bus

In this test case, the IEEE 30 bus system is adopted without the integration of WPPs, and the voltage deviation index is minimized considering the standard case data settings as cited [43,91]. This IEEE 30 bus system includes 6 generators, 9 capacitor banks, and 41 branches that serve as operational constraints in the optimal RPD. Following that, the designed fractional versions of FHPSO are used to determine the best tuning of control variables such as generator terminal voltages, tap setting, and capacitor bank size while satisfying the power demand. Figure 2 depicts the learning curves during VDI minimization based on FHPSO using fractional order α = [0, 0.1, 0.2,…, 0.9]. As shown from the convergence graph, the proposed algorithm achieved faster convergence in less number of iterations.

Table 3. Optimized control variables of IEEE 30 bus during VDI minimization without WPPs.

Control Variables	Fractional Order									Integer Order
	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 1
V1	1.009997	1.01	1.00996	1.009357	1.008535	1.009993	1.01	1.01	1.009943	1.01
V2	1.01	1.01	1.009976	1.009766	1.01	1.01	1.01	1.009989	1.009999	1.01
V5	1.01	1.009526	1.01	1.009884	1.009977	1.01	1.01	1.01	1.01	1.01
V8	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.009263
V11	0.997338	0.979931	1.01	1.009941	1.009756	1.00507	1.01	1.009878	0.998376	1.009335
V13	1.007964	1.009966	1.008925	1.01	1.009993	1.009386	1.01	1.00106	1.01	1.00638
T11	0.968603	0.958173	0.974194	0.963291	0.993197	0.951334	0.97888	0.994752	0.959817	0.980802
T12	0.950104	0.95	0.950052	0.950978	0.950601	0.975811	0.95	0.951426	0.95	0.95
T15	0.963189	0.970253	0.973144	0.969639	0.952203	0.973865	0.96685	0.962683	0.976754	0.978289
T36	0.971012	0.964969	0.950336	0.952066	0.950982	0.951319	0.953682	0.962722	0.960634	0.973938
Qc10	3.897704	4.437704	4.501827	4.873459	2.345744	3.217345	2.705423	4.786584	4.999846	2.378371
Qc12	4.954354	3.819101	4.069773	0.090632	0.60293	1.636259	0.079362	4.566777	2.296087	3.541337
Qc15	1.490825	4.079146	4.128829	2.989181	1.899343	2.995714	4.520463	4.855639	4.717713	4.999899
Qc17	4.287732	2.549029	2.310929	1.523325	3.706118	3.27968	1.87727	3.362823	0.000793	4.997936
Qc20	4.76726	3.66755	4.716818	4.989153	4.908928	4.696337	3.475392	4.92343	5	3.721863
Qc21	4.62632	3.132631	1.729241	4.117648	4.537567	4.841364	4.828549	4.96083	5	5
Qc23	4.931755	4.861373	4.949557	4.041004	4.448722	4.914961	4.503106	3.958634	5	5
Qc24	2.723139	4.944161	2.525881	3.056695	3.419034	4.99979	4.659123	3.936695	3.35988	4.560676
Qc29	4.930604	3.955492	1.165535	1.229564	1.326223	0.941414	2.153882	2.634725	2.576249	4.41817
VDI (p.u.)	0.130052	0.132484	0.128501	0.131554	0.133731	0.126458	0.124852	0.11517	0.117782	0.1174

lists the best control variable settings for each order α, as well as the related minimum value of the V DI. One may infer that the α = 0.8 generated the smallest fitness value, 0.11517 p.u., which implies a 90.007% V DI reduction w.r.t. the initial value, i.e., 1.1606 per unit [43]. Whereas, the PSOGSA at α = 1 has yielded 0.1174 p.u. of V DI, which implies 89.88% V DI reduction w.r.t. the base case value. Additionally, almost all the fractional coefficients yielded a lower voltage deviation index w.r.t. the base case value. This reveals that the FHPSO has shown a highly successful rate in evaluating the objective function while solving the RPD problem. For this experimental setup, the archive size is set to 10 and the number of iterations is set to 80.

4.3. VDI Minimization Considering WPPs in IEEE 30 Bus

In this scenario, the VDI is minimized considering wind power plants installed at weak buses of the IEEE 30 bus system. The equality and inequality constraints, iteration number archive size, and initial conditions are the same as in case 1, while the number of wind power plants is 4, placed at weak buses based on the VCPI method. The base case estimate of the VDI, which corresponds to the initial setting of the decision parameters, is 1.1606 per unit. Figure 3 depicts the learning curves obtained while minimizing the voltage deviation index adopting FHPSO with α = [0, 0.1, 0.2,…, 0.9], which shows a very fast convergence rate in proceeding towards the final global solution for all the αs.

Table 4. Optimized control variables of IEEE 30 bus during VDI minimization with WPPs.

Control Variables	Fractional Order									Integer Order
	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 1
V1	1.009598	0.992727	1.000642	0.998938	1.003732	1.009926	1.000825	1.009964	0.984796	0.998802
V2	0.980401	1.005783	1.006349	1.009831	0.994001	1.000287	1.01	1.008954	1.01	1.01
V5	1.009361	1.009923	1.008598	1.007585	1.009853	1.01	1.01	1.009888	1.01	1.01
V8	1.002166	1.008009	1.000658	0.998546	1.009409	1.001073	0.999756	0.995423	0.994774	1.005358
V11	1.008156	0.952376	0.99613	0.993479	0.974025	1.01	0.950909	1.001683	1.01	0.95
V13	1.007451	1.009774	0.989689	0.996711	0.994486	1.003881	1.006022	1.005176	1.00953	0.999609
T11	1.017491	0.959128	0.994153	0.992835	0.967558	1.011877	0.952593	0.996736	1.012935	0.950322
T12	1.038584	1.008402	1.044878	1.037886	1.04449	1.026261	1.001931	1.049976	1.05	1.030283
T15	0.966919	0.969242	0.952456	0.957599	0.958898	0.95	0.972916	0.95	0.981771	0.95
T36	0.959815	0.954373	0.953198	0.954614	0.971034	0.950509	0.963596	0.965718	0.950554	0.952637
Qc10	4.997096	2.773556	2.026303	0.187836	0.015959	2.767962	1.270832	2.910785	3.441165	0.931014
Qc12	1.709004	3.638932	4.197716	3.198421	3.007224	2.958404	4.723731	0	1.644165	4.789313
Qc15	2.729187	0.026141	3.850031	1.14113	4.992537	0.354864	2.589197	3.58932	3.008114	2.762203
Qc17	2.918021	4.584324	4.732506	4.926041	0.866615	2.621195	0.279706	0.601639	3.70388	3.496858
Qc20	0.965048	0.096781	0.016877	3.120113	0.659116	0.566124	1.213956	0.212836	1.804096	0
Qc21	3.708417	0.864623	4.381865	3.699563	4.28813	2.309885	3.33615	4.076081	3.018913	4.385511
Qc23	4.376691	4.730329	4.632192	4.011251	4.975529	1.5858	4.999824	2.219595	4.992568	1.881586
Qc24	2.400727	2.130312	2.166979	4.266552	4.879632	3.842694	2.741557	4.48471	3.8594	1.640443
Qc29	1.786826	0.882162	1.627655	1.005825	2.563779	0.638681	2.544383	3.030669	0.942991	1.086004
VDI (p.u.)	3.18 × 10−4	3.08 × 10−4	3.05 × 10−4	3.18 × 10−4	3.16 × 10−4	3.26 × 10−4	3.22 × 10−4	3.18 × 10−4	3.08 × 10−4	3.23 × 10−4

lists the optimum control variable settings for various α values, as well as the associated value of the VDI. According to the tabulated data, α = 0.3 generated the smallest value of V DI, that is 3.05 × 10⁻⁴ p.u., which is less than that yielded by the integer counterpart (PSOGSA), i.e., 3.23× 10⁻⁴ p.u., and far less than the initial value. Additionally, in comparison with its integer equivalent, most fractional orders offered a lower value of VDI. As a result, the efficacy of FHPSO as an alternative solution has been reaffirmed.

4.4. Ploss Minimization Considering WPPs

In third scenario, the strength of FHPSO is endorsed by considering line loss minimization in the IEEE 30 bus as an objective function. The convergence graphs obtained during Ploss minimization in a given power system incorporated with WPPs at weak buses are depicted in Figure 4, while

Table 5. Optimized control variables of IEEE 30 bus during Ploss minimization with WPPs.

Control Variables	Fractional Order									Integer Order
	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 1
V1	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01
V2	1.009139	1.008497	1.008984	1.01	1.008663	1.009636	1.008175	1.009719	1.009871	1.01
V5	0.991447	0.991898	0.991027	0.991812	0.991838	0.993039	0.990261	0.989737	0.991319	0.993865
V8	0.997199	0.997331	0.997572	0.997876	0.997914	0.999895	0.995167	0.996172	0.999264	0.998556
V11	1.00973	1.00532	1.009439	1.00381	1.009616	1.009975	1.01	1.009219	1.01	1.01
V13	1.009999	1.01	1.009974	1.01	1.009523	1.01	1.01	1.00982	1.01	1.01
T11	1.016388	0.977609	0.96998	1.007841	0.952357	1.016355	0.994707	0.988959	0.993752	1.003823
T12	0.950022	1.040717	1.016363	0.95	1.04808	0.971971	0.958049	0.95	0.951116	0.95
T15	0.952916	0.979169	0.956783	0.950151	0.950167	0.952517	0.952476	0.950043	0.95	0.95
T36	0.952993	0.967388	0.955043	0.964451	0.960877	0.964672	0.950321	0.953925	0.95	0.964321
Qc10	1.30842	3.039407	2.498887	1.555517	2.800039	1.654898	2.300174	1.436268	3.06738	4.531357
Qc12	2.615445	1.235816	3.616906	4.946201	2.364442	4.952389	1.625983	4.763014	4.155678	4.119073
Qc15	3.412006	0.497456	0.023099	0.062106	0.412725	2.592019	2.132699	3.850767	1.129889	0.563277
Qc17	4.467295	3.882846	4.646732	3.051384	3.0589	4.987051	1.646051	1.685549	2.839364	1.796974
Qc20	0.030214	0.149475	1.461066	0.081318	1.6133.4	1.182738	1.04643	0.325587	1.111193	0.001759
Qc21	4.09634	0.366882	2.671941	2.722601	2.976372	2.691452	1.728655	3.369179	1.938579	1.365593
Qc23	1.136409	4.274631	3.394322	0.07818	2.136938	2.679618	4.984786	1.087084	2.861772	2.223829
Qc24	1.584408	4.784188	2.545824	4.507205	3.78813	4.872162	4.340718	2.614785	4.340777	4.421699
Qc29	0.00929	1.466873	1.008741	2.111019	1.079646	2.444771	1.523108	2.092179	1.129839	3.253818
Ploss	2.858	2.8631	2.858735	2.855258	2.864563	2.850616	2.856476	2.850347	2.845852	2.849088

reports the optimal settings of the operational constraints, corresponding to different α, as yielded by FHPSO. In this scenario, again, the FHPSO outperforms its integer counterpart by yielding minimum Ploss at α = 0.9, that is, 2.845852 MW, implying a 51.10% lesser w.r.t. the initial setting, i.e., 5.82 MW. One may infer better optimization brilliance of FHPSO in this scenario as well.

4.5. Ploss Minimization Considering WPPs in IEEE 57 Bus

The usefulness of FHPSO is further demonstrated on a large-scale electric network, namely, IEEE 57 bus, adopting Ploss minimization as the performance evaluation function. The FHPSO time evolution by means of iterative updates during minimization of the fitness function is represented in Figure 5, for different order α and depicting the best, average, and worst behaviors. From the learning graph, it has been observed that the convergence is faster and attains global value in less number of iterations.

Table 6. Optimized control variables of IEEE 57 bus during Ploss minimization with WPPs.

Control Variables	Fractional Order									Integer Order
	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9	α = 1
V_G1	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01	1.01
V_G2	1.007645	1.009223	1.007051	1.009654	1.01	1.01	1.009446	1.01	1.01	1.00946
V_G3	0.998833	0.999628	0.998998	1.002971	1.000687	1.000615	0.998295	1.001342	1.002205	0.996966
V_G6	0.992719	0.995446	0.99835	1.003004	0.992987	0.996591	0.991579	0.992051	0.993074	0.984087
V_G8	1.009946	1.01	1.01	1.01	1.01	1.01	1.01	1.009723	1.009934	1.003579
V_G9	0.997128	0.994634	0.995134	0.997333	0.998467	0.999352	0.994094	0.995522	0.996622	0.98164
V_G12	0.990462	0.989106	0.988193	0.9942	0.993211	0.98976	0.990659	0.991376	0.995006	0.980991
T_4-18	0.97723	0.960907	0.988011	1.022341	1.002013	1.030412	1.031282	1.045078	0.95	0.95
T_4-18	1.030279	0.970857	1.0337	1.043736	1.0281	0.95	0.985417	0.964339	1.05	1.013221
T_21-20	1.024093	1.007874	0.98204	1.047049	1.049585	1.039553	1.025576	1.05	1.05	1.005138
T_24-26	0.983092	1.047792	0.988998	1.004138	0.991509	0.998622	0.977219	1.020306	0.994999	1.05
T_7-29	0.962191	0.995391	0.959904	0.959586	0.953988	0.955238	0.95	0.961194	0.95	0.999998
T_34-32	0.989965	0.97816	1.04373	1.011779	0.951606	0.967364	0.961342	1.027341	0.950826	1.05
T_11-41	1.017831	0.952886	0.957979	1.043545	1.043317	1.049826	1.045441	0.951355	0.961136	0.973287
T_15-45	0.973424	0.950198	0.991595	0.95	0.952318	0.966384	0.966524	0.95	0.950071	0.95
T_14-46	0.98882	0.97591	0.984263	0.951325	0.979581	0.951056	0.970844	0.951524	0.95	0.953992
T_10-51	0.980632	0.966671	0.976927	0.969647	0.979036	0.953734	0.965839	0.95	0.950288	0.95
T_13-49	0.958235	0.950137	0.980628	0.950435	0.950164	0.95007	0.959208	0.95	0.95	0.95
T_11-43	0.955294	1.046944	0.953916	0.952371	0.960695	0.95	0.950002	1.036775	0.95	0.95
T_40-56	0.981962	1.049014	1.05	1.013506	1.05	0.95	1.018456	0.95	0.977272	0.95
T_39-57	0.959673	0.972857	0.953979	0.952371	0.968534	1.013002	1.031243	1.005092	0.992674	0.95
T_9-55	0.964146	0.990958	0.957799	0.963741	0.954701	0.968277	0.950565	0.95	0.95	0.992709
Q_C18	9.183823	4.919385	1.200194	0.404448	2.517985	1.488137	3.61981	0.004937	3.707711	4.043883
Q_C25	13.89894	4.966403	3.745585	6.312066	4.020983	9.6559	10.68565	9.398614	13.0572	9.531845
Q_C53	11.59676	10.84459	10.61797	7.870676	11.99991	13.02508	7.088589	11.87762	11.46538	10.52381
Ploss	20.60795	20.55402	20.56932	20.49348	20.45659	20.3324	20.41443	20.39718	20.27219	20.63867

shows the optimal configuration of the regulating parameters with corresponding transmission line losses. Based on tabular interpretations, it is possible to conclude that FHPSO has outperformed its integer counter art by yielding far better results. One may infer that the minimum transmission loss is achieved at α = 0.9, i.e., 20.27219 MW, whereas the PSOGSA has achieved 20.63867 MW, which is far better than the standard line losses, that is, 27.86 MW [43] in said network. Summarizing, the synergy of entropy metric and FC adds to enhanced computational behavior by allowing better convergence of FHPSO towards the final solution of the optimization problem in large-scale networks.

4.6. Traditional Optimal RPD Problems (without Considering Wind Power Plants)

Till now, the optimization efficacy of FHPSO has been established by solving RPD problems in electric networks equipped with WPPs being placed at weak buses based on the VCPI technique. Since this approach is applied for the first time, the results yielded by the FHPSO are compared with those produced by traditional PSOGSA. Keeping this in mind, the wide range of FHPSO efficacy is further validated by addressing traditional RPD problems such as VDI minimization in a standard power system, namely, IEEE 30 bus, using standard data, as reported in the literature. The results as obtained by FHPSO execution while minimizing the fitness function are documented in

Table 7. Optimized variables of IEEE 30 bus system yielded by ALC-PSO, FAHCLPSO, PSO, OGSA, GSA, ABC, CKHA, SGA (1), NGBWCA, KHA, SGA (2), and ABS, during VDI minimization (traditional RPD) [43].

Parameter		Algorithms
	ALC-PSO	FAHL CPSO	GSA	OGSA	PSO	ABC	SGA (2)	CKHA	NGB WCA	KHA	PSOGSA	FPSOGSA	FHPSO
V1	1.05	1.1	1.071652	1.05	1.1	1.05	1.05	1.05	1.0502	1.05	1.01	1.01	1.01
V2	1.0384	1.0387	1.022199	1.041	1.0425	1.0615	1.0445	1.0473	1.0382	1.0381	1.01	1.01	1.009989
V5	1.0108	1.0161	1.040094	1.0154	1.0871	1.0711	1.024	1.0293	1.0107	1.011	1.01	1.01	1.01
V8	1.021	1.029	1.050721	1.0267	1.0859	1.0849	1.026	1.035	1.0212	1.025	1.01	1.009263	1.01
V11	1.05	1.0123	0.977122	1.0082	1.0647	1.1	1.05	1.05	1.0503	1.05	1.01	1.009335	1.009878
V13	1.05	1.1	0.96765	1.05	1.081	1.0665	1.05	1.05	1.05	1.05	1.01	1.00638	1.00106
T11	0.9521	1.0223	1.09845	1.0585	1.0255	0.97	1.049	0.9916	0.952	0.9541	1.016833	0.980802	0.994752
T12	1.0299	0.9107	0.982481	0.9089	0.95	1.05	0.9	0.9538	1.0295	1.0412	0.95	0.95	0.951426
T15	0.9721	1.0098	1.095909	1.0141	1.0021	0.99	0.988	0.9603	0.972	0.9514	0.95	0.978289	0.962683
T36	0.9657	0.9744	1.059339	1.0182	1.0416	0.99	0.965	0.967	0.9661	0.9541	0.970105	0.973938	0.862722
Qc10	0.009	0.034125	1.6537	0.033	2.6425	5	0.05	0.0092	0.0097	0.0089	4.03815	2.378371	4.786584
Qc12	0.0126	0.05	4.372261	0.0249	4.7822	5	0.05	0	0.0125	0	4.948908	3.541337	4.566777
Qc15	0.0209	0.020981	0.119957	0.0177	4.4309	5	0.05	0.0153	0.0212	0.0141	2.870865	4.999899	4.855639
Qc17	0.05	0.05	2.087617	0.05	4.3706	5	0.05	0.0497	0.0541	0.04989	4.7608	4.997936	3.362823
Qc20	0.0031	0.035512	0.357729	0.0334	-	0.05	0.05	0.0302	0.0043	0.0314	4.981661	3.721863	4.92343
Qc21	0.0293	0.004001	0.260254	0.0403	-	0.05	0.05	0.05	0.0289	0.0345	3.335246	5	4.96083
Qc23	0.0226	5	0	0.0269	-	0.05	0.036	5	0.0229	0.0241	4.998355	4.998355	3.958634
Qc24	0.05	0.0488	1.383953	0.05	-	0.05	0.05	5	0.0498	0.05	4.560676	4.560676	3.936695
Qc29	0.0107	0.021	5.5372	0.000317	-	0.0256	0.0275	3.52052	0.0106	0.0107	4.41817	4.41817	2.634725
VDI (p.u.)	0.3001	0.7914	1.6552	0.854	-	1.6815	0.7372	0.3524	0.3003	0.2963	0.1258	0.1174	0.11517

and compared with several other state-of-the-art algorithms including ALC-PSO, FAHCLPSO, GSA, OGSA, PSO, ABC, SGA(2), CKHA, SGA (1), NGBWCA, KHA, PSOGSA, FPSOGSA [43], FHPSO. It is worth noting that the FHPSO generated much better results, in terms of minimum gauge of VDI, than all counterparts leading towards a best solver that can also tackle traditional RPD problems. Hence, the remarkable performance of FHPSO in all the aforementioned scenarios proves its efficacy and supremacy while suggesting it as a reliable alternative optimization mechanism for solving optimization problems in the energy and power sectors.

5. Discussion and Statistical Analysis

The FHPSO computations were utilized to address voltage deviation index and line losses while simultaneously meeting operational constraints and power demand scenarios for various IEEE benchmark systems, both with and without wind power plant requirements. The proposed approach successfully provided the best possible solution for addressing the voltage deviation index and minimizing power losses. It demonstrated swift convergence and required less time for computation to reach the ultimate global result. The results confirm that FHPSO has superior capabilities in terms of optimal reactive power dispatch and convergence rate when compared to other optimization approaches. After evaluating the FHPSO method in relation to other techniques described in the literature, it is clear that the suggested approach outperforms all other techniques. To ensure an accurate contrast with the other methods, identical limits and variables were employed. The convergence characteristic graphs depicted in Figure 2, Figure 3, Figure 4 and Figure 5 illustrate that convergence happens more swiftly and attains the best possible result via fewer number of iterations. For the scenario of reactive power dispatch under wind/load uncertainties, the obtained minimum value EVPI is less than 0.0564 in all scenarios, while the obtained minimum EPloss is lower than 1.39 MW for all test scenarios. In the first test case for the IEEE 30 bus system, the FHPSO obtained the advantage of 90.007% with respect to the initial value, i.e., 1.1606; furthermore, all fractional coefficients yielded lower voltage deviation with respect to the base case. In the second case study for VDI minimization with WPPs, the FHPSO obtained a minimum value of 3.05 × 10⁻⁴ as compared to PSOGSA; additionally, in this scenario as well from all fractional orders, the FHPSO obtained a lower value of VDI, which is 3.23 × 10⁻⁴, which reaffirmed the efficiency. In the third case study of Ploss minimization considering WPP the FHPSO yields a minimum Ploss of 2.84582 which implies a 51.10% lesser with respect to the base value of 5.82 MW. In the fourth scenario of Ploss minimization considering WPPs in the IEEE 57 bus system, the FHPSO has outperformed its integer counterpart PSOGSA by yielding a superior result. In this case, a minimum transmission loss of 20.27219 MW is achieved as compared to PSOGSA, which has achieved 20.63867 MW, which is far better than the standard line losses of 27.86 MW. In the case of traditional optimal RPD, the optimization efficiency of FHPSO is further validated by addressing traditional RPD such as VDI minimization in the case of IEEE 30 bus using standard data cited in the literature. It is noteworthy that the FHPSO yielded significantly superior outcomes. In this case, the comparison of optimal settings determined by FHPSO with ALC_PSO, FAHLCPSO, OGSA, ABC, SGA, CKHA, NGBWCA, KHA, PSOGSA, and FPSOGSA shows 61.62%, 85.44%, 86.51%, 93.15%, 84.37%, 67.31%, 61.64%, 61.13%, 8.44%, and 1.899% improvement, respectively.

The convergence figures observed throughout the simulation process for all IEEE scenarios indicate an impressive convergence rate, leading to a good solution with a minimal number of iterations. An extensive statistical analysis is conducted on the IEEE-57 bus system test case to assess the validity and uniformity of the FHPSO algorithm. The analysis involves determining the ideal fractional coefficient α, which ranges between 0 and 1. For precise inferences, a series of 100 independent executions is conducted, and the median of the performance assessment function is utilized as a reference point for indicating the α value. The statistical examination is carried out by viewing cumulative distribution function (CDF) plots, boxplot illustrations, quantile-quantile plots, histograms, and minimal fitness gauges in each independent execution, as shown in Figure 6. The probability charts for the CDF, as shown in Figure 6a, show that the chance of finding a global solution using FHPSO is higher than that of PSOGSA. The median of the final fitness value in autonomous executions is depicted by boxplots as depicted in Figure 6b, revealing that the median gauge generated by FHPSO is always on the lower side when compared to its integer counterpart, along with a tight data spread. Figure 6c demonstrates that the minimal gauge of fitness against the percentile of a standard normal distribution is considerably optimal. The histogram charts, as shown in Figure 6d, illustrate that a large number of trials offer a good indicator of fitness function. Figure 6e shows that the FHPSO assesses minimal fitness values, including Ploss, in most autonomous simulations w.r.t. the PSOGSA. Furthermore, a very small fluctuation in the terminal value of the fitness assessment function for all the adopted instances reveals that the proposed fractional processing paradigm is very accurate.

6. Conclusions

This paper proposed a new fractional computing strategy based on the integration of Shannon entropy and fractional calculus inside the mathematical model of the conventional PSOGSA, i.e., FHPSO. The optimization strength of the FHPSO is then tested by solving the reactive power dispatch problems, including the VDI and line loss minimization, in the IEEE 30 and 57 bus incorporated with the WPPs. The wind power plants are allocated at weak buses of the electric network being identified by the voltage collapse proximity index. Following that, the proposed FHPSO is used to calculate the optimal value of control variables such as the terminal voltage of generators, tap setting of the transformer, and capacitor bank size corresponding to the minimum value of the objective function while meeting the load requirement of the given electric network. The findings show that the FHPSO outperformed its integer correlative, i.e., PSOGSA in all the considered test cases by computing the best global solution, i.e., minimum voltage deviation index and line loss. Following that, the efficacy of FHPSO is also ascertained by optimally solving conventional RPD problems in standard power systems using base case data, where FHPSO performed much better than the well-known algorithms including ALC-PSO, FAHCLPSO, GSA, OGSA, PSO, ABC, SGA (2), CKHA, SGA (1), NGBWCA, KHA, PSOGSA, FPSOGSA, and EE-PSOGSA. The consistency, reliability, and robustness of the FHPSO are validated further by statistical investigations, which include CDF plots, boxplots, quantile-quantile plots, and histograms as measurements of diversity indices and central tendency.

In the future, the presented method can also be augmented by incorporating it with other cerebral methods [91,92,93,94,95,96] to propose new hybrid methods for deciphering the optimization issues in different fields of energy and power. An alternate solution that is gaining interest is the incorporation of entropy metric inside the fractional order meta-heuristic mathematical model to avoid suboptimality while finding global solutions during algorithm evolution.