Fractional-Order African Vulture Optimization for Optimal Power Flow and Global Engineering Optimization

Abstract

This paper proposes a novel fractional-order African vulture optimization algorithm (FO-AVOA) for solving the optimal reactive power dispatch (ORPD) problem. By integrating fractional calculus into the conventional AVOA framework, the proposed method enhances the exploration–exploitation balance, accelerates convergence, and improves solution robustness. The ORPD problem is formulated as a constrained optimization task with the objective of minimizing real power losses while satisfying generator voltage limits, transformer tap ratios, and reactive power compensator constraints. The general optimization capability of the FO-AVOA is verified using the CEC 2017, 2020, and 2022 benchmark functions. In addition, the method is applied to the IEEE 30-bus and IEEE 57-bus test systems. The results demonstrate significant power loss reductions of up to 15.888% and 24.39% for the IEEE 30-bus and IEEE 57-bus systems, respectively, compared with the conventional AVOA and other state-of-the-art optimization algorithms, along with strong robustness and stability across independent runs. These findings confirm the effectiveness of the FO-AVOA as a reliable optimization tool for modern power system applications.

1. Introduction

1.1. Problem Background

The ORPD issue has become a central area of study in power system engineering due to its significant impact on modern-day energy management policies. Its primary objective is to reduce P_loss throughout transmission networks without compromising a superior voltage profile, as well as satisfying a vast range of equality and inequality operating constraints. Thus, RP allocation is regarded as a critical requirement of ensuring electric power system reliability, efficiency, and sustainability, which explains the rising academic attention on this area [1,2,3]. For these objectives to be achieved, adjustment of principal decision variables is crucial, e.g., generator voltage magnitudes, transformer tap levels, shunt capacitor/reactor installations, flexible AC transmission system (FACTS) devices, and active power generation levels. Because of the nonlinearity and high complexity of ORPD, gradient-free algorithms are commonly applied in real applications.

1.2. Related Work

All metaheuristic-based optimization techniques have proven to be helpful, with particle swarm optimization (PSO) is a resilient approach grounded in collective intelligence in nature. PSO has been vastly utilized in economic load dispatch (ELD) studies and, in combination with differential evolution (DE), has shown strong potential to optimize hybrid energy generation units [4]. Genetic algorithms (GAs) have been widely utilized for the ELD problem, particularly when conventional numerical algorithms are unable to identify globally optimal solutions within manageable system constraints. Through well-designed active power output allocation, enhanced versions of GA have shown higher accuracy and robustness in delivering credible results [5]. More recently, the energy and power sectors have adopted a broad range of metaheuristic search techniques, as highlighted in [6]. Among these, Mirjalili’s [7] nature-inspired grey wolf optimization (GWO) method has emerged as a powerful tool capable of handling complex, real-world ELD scenarios. Beyond ELD, GWO has also been applied to diverse optimization tasks across engineering and computational domains. Traditional optimization techniques, such as sequential, quadratic, linear, dynamic programming, and the Newton–Raphson method, have been employed in power system analysis for many years. However, their performance is very much compromised by early convergence and sensitivity to initial conditions [8], rendering them incapable of reproducibly reaching global optima. For this, hybrid optimization frameworks that use complementary algorithms have gained popularity, as they are able to leverage the positive aspects of different methods to produce improved global solutions [9].

Nowadays, power system optimization is very challenging owing to many realistic constraints such as demand–supply balance, generator capacity, emissions, prohibited operating regions (POZs), ramp rate limits, random variability from renewable sources, bus voltage controls, transformer tap controls, RP compensation, and the presence of continuous, discrete, and integer decision variables. All these together render power system optimization extremely nonlinear and nonconvex [10]. Most traditional optimization methods tend to become trapped in local optima and hence are not so effective for power system planning, operation, and management. But it has been found that bio-inspired algorithms can serve as effective alternatives in solving complex optimization problems. Their population-based and stochastic nature allows them to explore larger solution spaces and reduce the probability of entrapment in suboptimal regions, as opposed to deterministic derivative-based algorithms relying mainly on derivative information rather than fitness evaluations [11]. Bio-inspired approaches are often classified as evolutionary, ecological, and swarm-based algorithms, while hybridized versions offer even more potential by combining complementary features to achieve global solutions without premature convergence.

A particularly critical application in this area is the ELD problem with emission constraint, which minimizes fuel costs in relation to poisonous gas emissions [12]. This has been addressed by hybrid solutions such as a fuzzy-based PSO–DE method (FB-PSO-DE) taking into account valve-point loading effects (VPLEs), POZs, and multi-fuel sources for test systems of 10, 40, and 160 generation units. The hybrid solution was also more stable and convergent than the standalone PSO in more than seven test functions and 100 independent runs [13]. Similarly, adaptive differential evolution combined with a continuous greedy randomized adaptive search heuristic in the CGRASP–SADE algorithm was used to optimize generation cost under VPLE constraints [14]. Integration of renewable materials has also been explored for cost minimization encompassing the wind energy embedded in ELD using the moth–flame optimization (MFO) algorithm [15]. Further advancements witnessed the hybrid modified imperialist competitive algorithm (MICA), and invasive weed optimization (IWO) was explored for optimal power flow, where the VAR compensator’s decision variables, generator voltages, and transformer taps were taken into account [16].

A few other contributions include the use of multiagent-based PSO for loss minimization under nonlinear cost functions [17], the harmony search algorithm (HSA) for RP dispatch in the IEEE 30-bus system [18], and GWO for managing inequality constraints using penalty functions [19]. Hybridized methods, i.e., fractional order and the PSO algorithm for the IEEE 30- and 57-bus systems [20] and MFO algorithms for loss minimization in IEEE bus networks [21], have also provided promising results. However, the conventional MFO has the tendency to converge prematurely during the later optimization stages, with most of the individuals clustering around a nearby local area of the search space. Apart from that, GWO is also utilized for parameter tuning in the case of the IEEE 57-bus system [22], and seeker optimization (SO) is utilized for RP dispatch in IEEE systems [23]. In [24], a novel fractional swarming method was explored for optimum RP dis-patch. Hybrid PSO and the imperialist competitive algorithm (ICA) were revisited for the ORPD issue in [25]. Similarly, swarm intelligence for optimization [26], accurate power system modeling, and load flow analysis [27,28,29], as well as control design and nonlinear system performance [30,31], have been explored. Recent studies have demonstrated that the application of fractional calculus (FC) and fractional derivatives in modeling systems can significantly enhance performance in a wide range of scientific and engineering applications. Feature selection, image and hyperspectral data processing, robotic path planning, Kalman filter, and fractional-order filter design are just some of the applications where these methods have been used successfully. In this context, researchers suggest that FC techniques, along with evolutionary optimization algorithms, have immense potential to provide solutions to complex problems in energy-related fields. A few noteworthy examples include fractional-order robotic PSO, FPSO with fractional-order velocity terms [32,33,34,35,36], optical disc localization and segmentation [37], parameter estimation for Kalman filtering [38], land cover monitoring [39], feature selection applications [40], hyperspectral image classification [41], and fractional-order controllers for robot path planning [42]. Other contributions in this context are the multiband stabilizer design based on a hybrid GA–PSO lead–lag compensator dynamics for power systems [43], nonlinear system identification [44], and the power system and global engineering problem [45,46,47,48]. Tempered fractional gradient descent [49], Caputo fractional optimization [50], fractional gradient descent theory [51], convergence analysis [52], and fractional optimization survey [53]. Altogether, these developments have established the efficacy of FC-based solutions and demonstrated very clearly that FC will be amalgamated with metaheuristic algorithms to solve optimization problems in energy and power systems.

Table 1. Feature-based comparison of existing ORPD optimization approaches and the proposed algorithm.

Features	Conventional OPF Methods	Classical Metaheuristics	Hybrid Optimization Methods	Fractional-Order Optimization Methods	Proposed Algorithm
Fractional-order modeling	✗	✗	✗	✓	✓
Real power loss minimization	✓	✓	✓	✓	✓
Handling nonlinear ORPD constraints	Limited	✓	✓	✓	✓
Adaptive exploration–exploitation balance	Limited	✓	✓	✓	✓
Avoidance of premature convergence	✗	Limited	✓	✓	✓
Scalability of large power systems	Limited	Limited	✓	✓	✓
Robust convergence behavior	✗	Limited	✓	✓	✓
Global optimization capability	✗	✓	✓	✓	✓
Sensitivity to control parameter tunning	High	High	Medium	Medium	Medium
Computational efficiency	High	Medium	Medium	Medium	Medium
Simultaneous control of OPF method	Limited	✓	✓	✓	✓

highlights the main limitations of existing ORPD optimization approaches and clearly positions the proposed method among recent techniques. The table emphasizes how the proposed approach improves the coordination of generator voltages, transformer tap settings, and reactive power compensators under both radial and meshed network conditions. By effectively handling the nonlinear, constrained, and multimodal nature of the ORPD problem, the proposed method avoids premature convergence and ensures stable operation under varying loading conditions. This distinctive capability differentiates the proposed approach from conventional methods and enables more reliable and efficient minimization of real power loss.

Despite the strong performance of existing metaheuristic, hybrid, and fractional-order optimization approaches for ORPD, several limitations persist. Many methods suffer from premature convergence in later iterations, weak exploration–exploitation balance, high sensitivity to parameter tuning, and reduced scalability when applied to large-scale and highly constrained power systems. In addition, most existing fractional-order optimizers are either computationally intensive or lack robust convergence control. These weaknesses motivate the need for a more balanced and memory-driven optimization framework. Accordingly, this study introduces the FO-AVOA to effectively address these gaps by combining the strong global search capability of the AVOA with the nonlocal memory characteristics of fractional-order calculus.

1.3. Motivation and Contribution

The integration of fractional-order calculus into the African vulture optimization algorithm, referred to as the FO-AVOA, represents the core of advanced optimization in this study. By incorporating fractional dynamics, the proposed algorithm improves the efficiency of the exploration and exploitation processes for solving the ORPD problem, as fractional operators provide additional flexibility in controlling the search behavior compared to conventional integer-order operators. The inclusion of fractional calculus enhances the efficiency, scalability, and adaptability of the AVOA; consequently, the ORPD problem can be handled in a systematic manner to optimize power dispatch operations and minimize transmission losses. In addition, the FO-AVOA can contribute to improved energy market performance and grid management through real-time optimization and adaptive adjustment of load profiles, which promotes better resource utilization, reduced generation cost, lower system losses, and enhanced overall grid stability.

Although the use of fractional-order operators improves optimization efficiency and adaptability, it may introduce additional computational overhead due to the evaluation of fractional terms. However, this overhead is acceptable in view of the achieved improvement in solution quality and convergence behavior. Therefore, the main objective of this work is to provide an effective solution for the constrained ORPD optimization problem. To this end, this research proposes a novel application of the FO-AVOA as a new variant of the AVOA based on fractional calculus. The proposed algorithm provides an efficient computational solution for ORPD by determining optimal control parameters such as generator bus voltages, transformer tap settings, and reactive power compensators to satisfy load demands. In this study, the minimization of power loss is considered as the fitness function, while all equality and inequality constraints are maintained within acceptable limits. This proposed technique demonstrates novelty through the integration of fractional-order dynamics with a metaheuristic optimizer to achieve higher accuracy, reliability, and scalability in ORPD solutions.

The original contributions of the current work can be enumerated as:

• A novel fractional-order reformulation of the African vulture optimization algorithm (AVOA) is developed by embedding Grünwald–Letnikov fractional dynamics into the velocity update mechanism, introducing multi-iteration memory into the search process.
• A memory-driven exploration–exploitation balancing strategy is established through fractional-order control, enabling smoother convergence and significantly reducing premature stagnation compared to the conventional integer-order AVOA.
• A computationally efficient truncated fractional velocity model is derived for practical implementation, ensuring enhanced performance without introducing excessive computational overhead.
• A fractional-order ORPD optimization framework is constructed, enabling stable and simultaneous coordination of generator voltages, transformer tap settings, and reactive power compensators under nonlinear equality and inequality constraints.
• A rigorous comparative statistical validation framework is established using convergence analysis and probabilistic performance indicators to objectively quantify robustness, stability, and consistency of the proposed optimizer.
• A combined global–local validation strategy is employed using both large-scale CEC benchmark suites (2017–2022) and standard ORPD test systems to demonstrate the generalization capability and superiority of the proposed FO-AVOA.

The remainder of this paper is structured as follows: Section 2 introduces the formulation of the OPF problem, while Section 3 details the solution methodologies employed. The results, along with their statistical evaluation, are presented and discussed in Section 4, and conclusions are drawn in Section 5.

2. Problem Formulation

The ORPD problem plays a fundamental role in power system operation and voltage stability enhancement. Its main objective is to determine optimal control variable settings in order to minimize real power transmission losses, while satisfying both equality (power balance) and inequality (operational) constraints. In this study, ORPD is formulated based on standard IEEE test systems.

2.1. Real Power Loss Objective Function

The total active power loss in the transmission network is mathematically expressed as:

$$P_{loss}=\sum _{i=1}^{N_L}{G}_l\left(\left.U_i^2+U_j^2-2\ast U_i{U}_{j}cos({\delta }_i-{\delta }_j\right)\right)$$

(1)

where $N_L$ is the number of transmission lines, G is the conductance of line l, $U_i{ \mathrm{a}\mathrm{n}\mathrm{d} U}_j$ are the voltage magnitudes at buses i and j, and ${\delta }_i$ and ${\delta }_j$ are the voltage phase angles. This expression shows that transmission losses are governed by bus voltage magnitudes, phase angle differences, and line conductance.

2.2. Penalized Objective Function

To ensure that operating constraints are strictly satisfied, a quadratic penalty approach is used. The augmented objective function is defined as:

$$F_{obj}=P_{loss}+\sum _{k=1}^{N_v}{\mathsf{\lambda }}_v{\left(U_k-U_k^{lim}\right)}^2+\sum _{k=1}^{N_T}{\mathsf{\lambda }}_T{\left(T_k-T_k^{lim}\right)}^2+\sum _{k=1}^{N_G}{\mathsf{\lambda }}_{\mathrm{Q}}{\left(Q_m-Q_m^{lim}\right)}^2$$

(2)

where ${\mathsf{\lambda }}_v$, ${\mathsf{\lambda }}_T$, and ${\mathsf{\lambda }}_{\mathrm{Q}}$ are penalty multipliers, while $N_v$, $N_T$, and $N_G$ denote the numbers of voltage-controlled buses, transformers, and reactive generators, respectively. This penalty term becomes zero within permissible limits and positive upon violation, ensuring solution feasibility.

2.3. Equality Constraints

The equality constraints enforce system power balance at each bus:

$$P_{Gk}-P_{Dk}-U_k\sum _{m=1}^N{U}_m\left[C_{km}\mathrm{s}\mathrm{i}\mathrm{n}\left({\delta }_k-{\delta }_m\right)+S_{km}\mathrm{c}\mathrm{o}\mathrm{s}\left({\delta }_k-{\delta }_m\right)\right]=0$$

(3)

$$Q_{Gk}-Q_{Dk}-U_k\sum _{m=1}^N{U}_m\left[C_{km}\mathrm{c}\mathrm{o}\mathrm{s}\right({\delta }_k-{\delta }_m)+S_{km}\mathrm{s}\mathrm{i}\mathrm{n}({\delta }_k-{\delta }_m\left)\right]=0$$

(4)

where $P_{Gk}$ and $Q_{Gk}$ are generated real and reactive powers, $P_{Dk}$ and $Q_{Dk}$ are demanded real and reactive power, and $C_{km}$ and $S_{km}$ are the conductance and susceptance elements of the bus admittance matrix.

2.4. Inequality Constraints

(a) 

Tap Limits

The transformer tap settings are expressed as:

$$T_k^{\mathrm{m}\mathrm{i}\mathrm{n}}\le T_k\le T_k^{\mathrm{m}\mathrm{a}\mathrm{x}}, \hspace{1em} k=1,2,3,\dots ,N_T$$

(5)

(b) 

Generator Voltage Constraints

The generation voltage and RP limits are defined by their upper and lower bounds as follows:

$$U_{Gk}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le U_{Gk}\le U_{Gk}^{\mathrm{m}\mathrm{a}\mathrm{x}}, \hspace{1em} k=1,2,\dots ,N_G$$

(6)

$$Q_{Gk}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le Q_{Gk}\le Q_{Gk}^{\mathrm{m}\mathrm{a}\mathrm{x}}, \hspace{1em}k=1,2,\dots ,N_G$$

(7)

(c) 

Reactor Limits

The bus compensators are subject to the following operational limits:

$$Q_{Ck}^{\mathrm{m}\mathrm{i}\mathrm{n}}\le Q_{Ck}\le Q_{Ck}^{\mathrm{m}\mathrm{a}\mathrm{x}}, \hspace{1em} k=1,2,\dots ,N_C$$

(8)

2.5. Compact OPF Representation

The ORPD optimization problem is finally expressed in compact form as:

$$\mathit{min}{\mathit{F}}_{\mathit{o}\mathit{b}\mathit{j}}(\mathit{x},\mathit{u})$$

(9)

subject to $\mathit{g}\left(\mathit{x},\mathit{u}\right)=0$, $\mathit{h}(\mathit{x},\mathit{u})\le 0$; where $\mathit{x}=\left[\mathit{U},\mathit{\delta }\right]$ represents the state variables, and $\mathit{u}=[\mathit{T},\mathit{Q}]$ represents the control variables.

To enhance the global search dynamics and convergence behavior, the classical OPF formulation is generalized using fractional calculus as:

$${\mathit{D}}^{\mathit{\alpha }}\mathit{x}\left(\mathit{t}\right)=\mathit{F}\left(\mathit{x}\right(\mathit{t}),\mathit{u}(\mathit{t}\left)\right)$$

(10)

where ${\mathit{D}}^{\mathit{\alpha }}$ is the Caputo fractional derivative of order α. The Caputo definition is selected due to its physical interpretability and compatibility with standard initial conditions:

$${\mathit{D}}^{\mathit{\alpha }}\mathit{x}(\mathit{t})={\displaystyle \frac{1}{\Gamma (1-\mathit{\alpha })}}{\int }_0^{\mathit{t}} {\displaystyle \frac{\dot{\mathit{x}}\left(\mathit{\tau }\right)}{(\mathit{t}-\mathit{\tau }{)}^{\mathit{\alpha }}}}\mathit{d}\mathit{\tau }$$

(11)

This formulation introduces memory and hereditary characteristics, enabling the optimizer to account for long-term system behavior rather than only instantaneous gradients.

3. Design Methodology

3.1. African Vulture Optimization Algorithm (AVOA)

The AVOA, introduced by Abdollahzadeh et al. [54] in 2021, is a new addition to the bio-inspired metaheuristics family. The algorithm is motivated by the survival strategies and adaptive sagacity of African vultures, which are typified by their flexibility to adjust to different conditions of the environment. One of the salient features of these birds is their hunger-rate-driven decision-making process that regulates their competitive and navigation behaviors.

In the AVOA, this biological mechanism is converted mathematically into search dynamics to allow the algorithm to change its exploration and exploitation balance according to the optimization state. The step-by-step execution mechanism of the African vulture optimization algorithm is presented in Figure 1. The mathematical model of the hunger rate mechanism is as follows.

$$F_t(t)=(2\times rand+1)\times z\times \left(1-{\displaystyle \frac{t}{T}}\right)+dt$$

(12)

$$dt=h\times \left({\mathrm{s}\mathrm{i}\mathrm{n}}^w\left({\displaystyle \frac{\pi }{2}}\times {\displaystyle \frac{t}{T}}\right)+\mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{\pi }{2}}\times {\displaystyle \frac{t}{T}}\right)-1\right)$$

(13)

In the hunger rate formulation, $F_t\left(t\right)$ represents the hunger level of the kth vulture at iteration t. The term $dt$ denotes a fixed parameter that is predefined prior to the execution of the algorithm. The variable $t$ indicates the current iteration, while $T$ specifies the maximum number of iterations. The random term $rand$ is uniformly distributed in the interval [0, 1] whereas $h$ is a random variable within the range [−2, 2], and $z$ is randomly selected from [−1, 1]. The coefficient $w$ is assigned a fixed value of 2.5 in the AVOA. A negative hunger rate $(F_t\left(t\right)<0)$ indicates that the vulture is in a hungry state, whereas values of $(z\ge 0)$ correspond to a satiated condition. To model leadership dynamics, either the best or second-best vulture in the population is chosen as the leader, whose mathematical representation is provided in the subsequent equation:

$$R_k(t)=\left\{\begin{array}{cc}BV1& if p>rand\\ BV2& Otherwise\end{array}\right.$$

(14)

Here, $R_k\left(t\right)$ denotes a randomly chosen vulture, while $BV1$ and $BV2$ correspond to the best and second-best vultures in the population, respectively. The parameter $p$ is a control constant, fixed at 0.8.

3.1.1. Exploration Phase

When $F_t\left(t\right)$, the vultures begin searching for food across different regions, indicating that the AVOA has transitioned into the exploration phase. In this stage, the foraging behavior is modeled through two distinct strategies that describe the vultures’ movement patterns in safeguarding potential food sources. The corresponding mathematical formulation is expressed as follows:

$$P_k(t+1)=\left\{\begin{array}{c}Equation \left(16\right) if p_1\ge ran{d}_{p1}\\ Equation \left(17\right) if p_1<ran{d}_{p1}\end{array}\right.$$

(15)

$$P_k(t+1)=R_k\left(t\right)-D_k\left(t\right)\times F_k\left(t\right)$$

(16)

$$P_k(t+1)=R_k\left(t\right)-F_k\left(t\right)+rand\times \left((ub-lb)\times rand\times lb\right)$$

(17)

$$D_k\left(t\right)=|X\times R_k(t)-P_k(t\left)\right|$$

(18)

In this model, $P_k(t+1)$ denotes the updated position of the vulture in the subsequent iteration, while $p_1$ is a constant parameter with a fixed value of 0.6. The term $ran{d}_{p1}$ and $ran{d}_{p1}$ represents a random number uniformly distributed in the interval [0, 1]. The variable $D_k\left(t\right)$ corresponds to the distance between the current vulture and the selected leader vulture. The symbol $X$ is a random number generated within the range [−2, 2], and $ub$ and $lb$ indicate the upper and lower bounds of the search space, respectively.

3.1.2. Exploitation Phase

When $F_t\left(t\right)$, the vultures focus on food sources within a limited region, indicating that the AVOA has transitioned into the exploitation phase. In this stage, the search process is modeled using two distinct strategies that characterize the vultures’ movements while competing for and protecting food. The mathematical representation of these strategies is provided as follows:

$$P_k(t+1)=\left\{\begin{array}{c}Equation \left(20\right) if p_2\ge r_2\\ Equation \left(21\right) if p_2<r_2\end{array}\right.$$

(19)

$$P_k(t+1)=D_k\left(t\right)-\left(F_k\right(t)+r_1)-d_k\left(t\right)$$

(20)

$$P_k(t+1)=R_k\left(t\right)-(S_1-S_2)$$

(21)

$$d_k\left(t\right)=R_k\left(t\right)-P_k\left(t\right)$$

(22)

$$\left\{\begin{array}{c}{S}_{1k}=R_k(t)\times \left({\displaystyle \frac{r_3\times P_k(t)}{2\times \pi }}\right)\times \cos (P_k(t))\\ S_{2k}=R_k(t)\times \left({\displaystyle \frac{r_4\times P_k(t)}{2\times \pi }}\right)\times \sin (P_k(t))\end{array}\right.$$

(23)

In this formulation, $p_2$ is a constant parameter with a fixed value of 0.4, and $R_k\left(t\right)$ denotes one of the top-performing vultures in the population. The variable $rand$ is a uniformly distributed random number within the interval [0, 1]. During the second phase, when the value of F falls below 1, the algorithm models the vultures’ behavior of food accumulation and their intense competition for resources. In the above equations, the subscript k denotes the index of the vulture (search agent), while $S_{1k}$ and $S_{2k}$ are computed independently for each agent at iteration t.

$$P_k(t+1)=\left\{\begin{array}{c}Equation (25) if p_3\ge ran{d}_{p_3}\\ Equation (26) if p_3<ran{d}_{p_3}\end{array}\right.$$

(24)

$$P_k(t+1)={\displaystyle \frac{A_1+A_2}{2}}$$

(25)

$$P_k\left(t+1\right)=R_k\left(t\right)-\left|d_k\left(t\right)\right|\times F_k\left(t\right)\times Levy\left(d\right)$$

(26)

$$\left\{\begin{array}{c}{A}_{1k}=BV1(t)-{\displaystyle \frac{BV1(t)-P_k{(t)}^2}{BV1{(t)}^2-P_k{(t)}^2}}\times F_k(t)\\ A_{2k}=BV1(t)-{\displaystyle \frac{BV2(t)-P_k{(t)}^2}{BV2(t)-P_k{(t)}^2}}\times F_k(t)\end{array}\right.$$

(27)

$$Levy(d)=0.01\times {\displaystyle \frac{u}{|v{|}^{\frac{1}{\beta }}}},u~(0,{\sigma }_u^2),v~(0,{\sigma }_v^2)$$

(28)

$${\sigma }_{\mathrm{u}}={\left({\displaystyle \frac{\mathrm{\Gamma }(1+\beta )\times \mathrm{s}\mathrm{i}\mathrm{n}\left(\frac{\pi \beta }{2}\right)}{\mathrm{\Gamma }\left(\frac{1+\beta }{2}\right)\times \beta \times 2^{\left(\frac{\beta -1}{2}\right)}}}\right)}^{{\displaystyle \frac{1}{\beta }}}$$

(29)

The parameters $A_{1k}$, $A_{2k}$, and $F_k\left(t\right)$ are computed independently for each agent at iteration t. Here, $p_3$ is a constant parameter assigned a value of 0.4, while $ran{d}_{p_3}$ represents a uniformly distributed random number in the interval [0, 1]. The variables $u$ and $v$ are random numbers generated according to a Gaussian distribution. The parameters $\sigma$ and $\beta$ are fixed at 1 and 1.5, respectively, and $\mathrm{\Gamma }$ denotes the standard gamma function.

3.2. Fractional-Order African Vulture Optimization Algorithm (FO-AVOA)

The work introduces a new notion where FC is incorporated into the traditional AVOA and constructs a fractional-order AVOA (FO-AVOA). The motivation for such unification lies in the fact that it attempts to overcome the issue of premature convergence and to improve the quality of solutions in general. The AVOA has a good global search capability, but local exploitation capacity could be weak, which may adversely affect convergence in complex problems. In order to overcome this shortcoming, FC is incorporated in the AVOA model. With its built-in memory and retention characteristics, FC enables information sharing between candidate solutions during the exploitation step. This adjustment is likely to boost accuracy and convergence efficiency.

In addition, FC can readily model complex dynamics, such as irreversibility and chaotic processes, due to its long-memory nature. Combining these features with the AVOA’s flexibility, the FO-AVOA provides a novel optimization framework that integrates the strengths of both methods with enhanced performance. The use of fractional calculus (FC) has garnered a lot of interest from researchers working in a broad array of disciplines from engineering, computational mathematics, and applied physics. Unlike conventional integer-order calculus, FC is a more general mathematical formalism that is able to resolve the problems the classical ones cannot. Fractional derivatives unify derivatives of non-integer order, and fractional derivatives are a powerful means of modeling memory-type systems with hereditary features, particularly for simulating complex dynamic phenomena. Various definitions have been offered for fractional-order derivatives [55,56]. Among them, the GL definition is employed extensively to derive fractional calculus-based mathematical models. The Grünwald–Letnikov fractional derivative for an arbitrary signal s(t) is provided by:

$$D^{\delta }(s\left(t\right))=\underset{h\to 0}{\lim }\left[\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(-1\right)}^k\tau \left(\delta +1\right)s(t-kh)}{\tau (\delta -k+1)\tau (k+1)}}\right]$$

(30)

Equation (30) represents the continuous Grünwald–Letnikov definition of the fractional derivative of order δ, formulated as a limit when the step size h → 0. This definition models fractional differentiation through a weighted infinite summation, naturally capturing the memory effect of the system. For numerical implementation, the limit operation is approximated by fixing the step size as the sampling interval T, which yields the discrete-time realization provided in Equation (31). In this case, the infinite summation is truncated to a finite memory length τ where k = 0, 1, …,τ. The fractional order satisfies 0 < δ ≤ 1, where δ = 1 reduces the formulation to the classical first-order derivative. The sampling step satisfies T = h > 0, and a sufficiently small step is selected to ensure numerical stability and accuracy.

$$D^{\delta }(s\left(t\right))={\displaystyle \frac{1}{T^{\delta }}}\left[\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(-1\right)}^k\tau \left(\delta +1\right)s\left(t-kh\right)}{\tau \left(\delta -k+1\right)\tau \left(k+1\right)}}\right]$$

(31)

In this expression, $T$ denotes the sampling interval, while $\tau$ specifies the truncation order. The term $\left(s\left(t\right)\right)$ refers to the discretized form of the signal. In the special case where $\delta$, the formulation reduces to the conventional integer-order derivative, which corresponds to the standard first-order derivative and can be expressed as follows:

$$D^1\left[s\left(t\right)\right]=s\left(t+1\right)-s(t)$$

(32)

To enhance the local exploitation of the conventional AVOA, the position of each vulture is updated as a function of its velocity, which is defined in terms of the fractional derivative.

$$P_k(t)=P_k(t-1)+V_k(t)$$

(33)

where $V_k^n(t)$ denotes the velocity of the kth vulture at iteration t. The fractional-order modification allows the velocity update to include both present and past states, thereby creating a memory-driven search mechanism. This approach balances exploration and exploitation more effectively, prevents stagnation in local optima, and provides smoother convergence toward the global optimum.

$$V_k(t)-V_k(t-1)=C_1\cdot r_1\cdot \left(LB-P_k(t-1)\right)+C_2{r}_2\left(GB-P_k(t-1)\right)$$

(34)

where $V_k(t)$ and $V_k(t-1)$ denote the velocity at the current and previous iterations, $P_k(t-1)$ is the previous position of the vulture, $LB$ and $GB$ are the local and global best positions, $C_1$ and $C_2$ are acceleration coefficients, and $r_1 \mathrm{a}\mathrm{n}\mathrm{d} r_2$ are uniformly distributed random variables in the range of $[0,1]$. This form represents the first-order integer difference of the velocity. By introducing fractional calculus, the model generalizes this into a fractional-order derivative of velocity:

$$D^{\delta }\left[V_k(t)\right]=C_1\cdot r_1\cdot \left(LB-P_k(t-1)\right)+C_2\cdot r_2\cdot \left(GB-P_k(t-1)\right)$$

(35)

where $D^{\delta }$ denotes the fractional derivative operator of order $\delta \in (0,1]$.

$$\begin{array}{c}{V}_k^{\delta }(t)=-\left[{\displaystyle \sum _{n=0}^r} {\displaystyle \frac{(-1{)}^n\tau (\delta +1)V_k(t-nh)}{\tau (\delta -n+1)\tau (n+1)}}\right]+C_1{r}_1\left(LB-P_k(t-1)\right)\\ +C_2{r}_2\left(GB-P_k(t-1)\right.\end{array}$$

(36)

The fractional velocity of the kth vulture at iteration t, with r-term expansion, is provided as:

$$\begin{array}{l}{V}_k^{\delta }\left(t\right)={\sum }_{i=1}^{\infty }\left\{\right[\delta \left(\delta -1\right)\left(\delta -2\right)\dots (\delta -m+1)/m!].V_k(t-m)\}+\\ C_1{r}_1\left(LB-P_k\left(t-1\right)C_2{r}_2\left(GB-P_k\left(t-1\right)\right)\right.\end{array}$$

(37)

Equation (38) is obtained by truncating the infinite Grünwald–Letnikov fractional series in Equation (37) to the first four memory terms to achieve a computationally efficient implementation:

$$\begin{array}{c}{V}_k^{\delta }(t)=\delta V_k(t-1)+{\displaystyle \frac{1}{2}}\delta (1-\delta )V_k(t-2)+{\displaystyle \frac{1}{6}}\delta (1-\delta )(2-\delta )V_k(t-3)+\\ {\displaystyle \frac{1}{24}}\delta (1-\delta )(2-\delta )(3-\delta )V_k(t-4)+C_1{r}_1\left(LB-P_k(t-1)C_2{r}_2\left(GB-P_k(t-1)\right)\right.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\end{array}$$

(38)

In practical implementation, the infinite Grünwald–Letnikov fractional series is approximated by truncating it to a finite number of past velocity terms r, as shown in Equation (38), to maintain computational efficiency. The fractional order δ ∈ (0, 1] acts as a tuning parameter that controls the influence of historical information; smaller values of δ emphasize global exploration, while values closer to unity enhance local exploitation. Unlike the integer-order AVOA, which updates velocity based only on the most recent iteration, the proposed fractional update exploits multi-iteration memory, leading to smoother convergence and reduced premature stagnation.

The pseudocode of the proposed FO-AVOA for solving the ORDP is presented in Algorithm 1 and illustrated in Figure 2. Figure 2 illustrates the complete workflow of the proposed FO-AVOA for solving the ORPD problem. The flowchart presents the main stages of the algorithm, including initialization of the vulture population using IEEE benchmark system parameters, evaluation of fitness based on real power loss minimization, selection of optimal and suboptimal vultures, and adaptive switching between exploration and exploitation strategies. The fractional-order updating mechanism is incorporated within the exploitation phase to introduce memory-based search dynamics. This visual representation complements the mathematical formulation and clearly demonstrates how the proposed FO-AVOA iteratively updates control variables until convergence is achieved.

Algorithm 1. Pseudocode of FO-AVOA

Input:     Population size N     Maximum iterations MaxIter     Problem dimension D     Search space bounds [LB, UB]     Objective function f(x)     Fractional-order parameter α Output:     Best solution xbest     Best fitness fbest 1. Initialization 1.1.  Initialize population X = {x1, x2, …, xN} randomly within [LB, UB] 1.2. Evaluate fitness f (xi) for all xi ∈ X 1.3. Determine leader vultures:     Leader1 = best solution, Leader2 = second-best solution 2. Main loop For t = 1 to T do   For each vulture k = 1 to N do 2.1 Update control parameters:     Compute exploitation factor Fk(t) using Equation (12).     Compute probabilities P1, P2, P3      Generate random numbers rand1, rand2, rand3∈ [0, 1] 2.2. Select search strategy If ∣Fk(t)∣ ≥ 1 then   If P1 ≥ rand1 then Update the position of vulture k using the fractional velocity model, Equation (36). Else Update the position of vulture k using the truncated fractional model, Equation (38). Else if ∣Fk(t)∣ ≥ 0.5 then   If P2 ≥ rand2 then Update Pk(t+1) using the AVOA position update in Equation (20). Else Update Pk(t + 1) using the AVOA position update in Equation (21). Else (i.e., ∣Fk(t)∣ < 0.5|) If P3 ≥ rand3 using Equation (25). Else Update Pk(t + 1) using Equation (26). 2.3 Boundary control If any component of Pk(t + 1) violates [LB, UB] project it to the nearest bound. 2.4 Fitness evaluation and greedy selection Evaluate f (Pk(t + 1)) If f(Pk(t+1)) < f(Pk(t)) then set Pk(t) = Pk(t + 1) End For 3. Return result Set xbest = Leader1 and fbest = f(xbest).

4. Results and Discussion

The performance of the new FO-AVOA was initially tested on the classical CEC 2017, 2020, and 2022 benchmark functions, which provide a good test bed for assessing constraint-handling optimization techniques. Details about the benchmark functions can be found in [57,58,59]. The test on multimodal objective functions (MMOFs) confirms the robustness and versatility of the FO-AVOA to different optimization tasks. As seen from

Table 2. Comparative analysis of FAVOA vs. AVOA for CEC 2020 MMOF.

Function	Best		Median		Mean		Standard Deviation		Worst
	FAVOA	AVOA	FAVOA	AVOA	FAVOA	AVOA	FAVOA	AVOA	FAVOA	AVOA
F1	7.5000 × 10−1	7.5000 × 10−1	7.5735 × 10−1	7.5735 × 10−1	7.503 × 10−1	7.8706 × 10−1	1.47 × 10−3	5.8796 × 10−3	9.476 × 10−1	2.4051 × 100
F2	7.5000 × 10−1	7.5000 × 10−1	7.5735 × 10−1	8.1770 × 10−1	7.5153 × 10−1	8.6194 × 10−1	1.419 × 10−2	4.0313 × 10−3	2.639 × 100	3.0681 × 100
F3	1.000 × 100	1.000 × 100	1.000 × 100	1.000 × 100	1.000 × 100	1.000 × 100	1.040 × 10−3	1.2826 × 10−3	1.1345 × 100	1.1943 × 100
F4	7.5000 × 10−1	7.5000 × 10−1	7.5002 × 10−1	7.8984 × 10−1	7.5154 × 10−1	8.2661 × 10−1	2.97 × 10−3	1.6749 × 10−2	1.2061 × 100	3.000 × 100
F5	7.5000 × 10−1	7.5000 × 10−1	7.5002 × 10−1	8.0330 × 10−1	7.5090 × 10−1	8.5371 × 10−1	4.764 × 10−3	2.4652 × 10−2	1.397 × 100	4.2679 × 100
F6	−3.972 × 10−1	−3.972 × 10−1	−3.972 × 10−1	−2.8041 × 10−1	−3.932 × 10−1	−1.277 × 10−1	2.338 × 10−2	3.088 × 10−2	6.794 × 10−1	1.215 × 100
F7	−9.9865 × 10−1	−1.000 × 100	−9.9829 × 10−1	−8.9967 × 10−1	−9.93 × 10−1	−7.5918 × 10−1	2.2303 × 10−2	3.4269 × 10−2	1.4015 × 100	5.9659 × 100
F8	1.3602 × 100	1.3603 × 100	1.3602 × 100	1.3781 × 100	1.360 × 100	1.4370 × 100	9.088 × 10−3	8.9089 × 10−3	2.2731 × 100	2.5289 × 100
F9	2.0373 × 100	2.0373 × 100	2.0373 × 100	2.0452 × 100	2.0378 × 100	2.0743 × 100	7.494 × 10−3	5.1947 × 10−3	2.9116 × 100	2.9116 × 100
F10	9.5950 × 10−1	9.5950 × 10−1	9.5950 × 10−1	9.6699 × 10−1	9.60823 × 10−1	9.9958 × 10−1	5.217 × 10−3	7.7669 × 10−3	1.5835 × 100	1.8824 × 100

, the FO-AVOA consistently demonstrates superior performance than the baseline AVOA in best, median, mean, worst, and standard deviation values; hence, it demonstrates superior convergence accuracy and stability. The same is confirmed by the statistical analysis reported in

Table 3. Statistical comparison of FO-AVOA and AVOA performance using Wilcoxon signed-rank test.

Test Metric	p-Value	Test Statistic (W)	Conclusions (Reject H0)
Best value	0.00000	238,461.000	Yes
Median value	0.00000	252,010.000	Yes
Mean value	0.00000	260,212.000	Yes
Std	0.00000	259,431.000	Yes
Worst value	0.00000	187,934.000	Yes

. In all the performance indicators, the FO-AVOA and AVOA are significantly different from each other as indicated by the Wilcoxon signed-rank test p-values all being below 0.05. To further validate the effectiveness of the proposed FO-AVOA, extensive experiments were conducted on benchmark functions from the CEC 2017 and CEC 2022 suites.

Table A1. Experimental setup and benchmark configuration.

Category	Parameter	Value/Description
Benchmark suite	Test functions	CEC 2017/2022
	Function types	Unimodal, multimodal, hybrid composition
	Dimensionality (D)	30
	Search bounds	[−100, 100]
	Number of runs	30 independent runs
	Performance metrics	Best, mean, median, worst, standard deviation
General settings	Population size (N)	30
	Maximum iterations	500
	Simulation platform	MATLAB R2025b
Compared algorithms	Optimization methods	FO-AVOA, AVOA, PSO, MFO, DE, GA, HSA, SOA

summarizes the experimental setup and benchmark configuration, while

Table A2. Mathematical model of different benchmark suites.

Function	Formula	n	Range	Minimum
F1	$F_1={\sum }_{i=1}^n{x}_i^2$	30	[−100, 100]	0
F2	$F_2={\sum }_{i=1}^n\left|x_i\right|+{\prod }_{i=1}^n\left|x_i\right|$	30	[−100, 100]	0
F3	$F_3={\sum }_{i=1}^n{\left({\sum }_{j-1}^i{x}_j\right)}^2$	30	[−10, 10]	0
F4	$F_4={\sum }_{i=1}^n{\left({\sum }_{j-1}^i{x}_j\right)}^2$	30	[−100, 100]	0
F5	$F_5={\mathit{max}}_i\{\left|x_i\right|,1\le i\le n\}$	30	[−100, 100]	0
F6	$F_6={\sum }_{i=1}^n([x_i+0.5]{)}^2$	30	[−100, 100]	0
F7	$F_7={\sum }_{i=1}^{n}i{x}_i^4+random[\mathrm{0,1})$	30	[−100, 100]	0
F8	$F_6={10}^6{x}_1^2+{\displaystyle \sum _{i=2}^n}{x}_i^2$	30	[−100, 100]	0
F10	$F_8={\sum }_{i=1}^n-x_i\mathit{sin}(\sqrt{\left|x_i\right|})$	30	[−500, 500]	−418.9829 × 5

and

Table A4. Mathematical model of different benchmark suites.

Function	Formula	n	Range	Minimum
F1	$f_1(x)=x_0^2+{10}^6\cdot {\displaystyle \sum _{i=1}^{n-1}}{x}^2$	30	[−100, 100]	0
F2	$f_2(x,y)=2A+x^2-A\cdot \mathrm{c}\mathrm{o}\mathrm{s}(2\pi x)+y^2-A\cdot \mathrm{c}\mathrm{o}\mathrm{s}(2\pi y)$	30	[−5.12, 5.12]	0
F3	$f_3(x)={\displaystyle \sum _{i=1}^n}{a}^{{\displaystyle \frac{i-1}{n-1}}}{x_i}^2$	30	[−2, 2, 100]	0
F4	$f(x_1,x_2)={\displaystyle \sum _{i=1}^2}{{(x_i-a_i)}^2)}^2+{\displaystyle \frac{({\sum }_{i=1}^2{(x_i-a_i)}^2)}{{10}^6}}$	30	[−20, 20, 40]	0
F5	$f(x,y)=100{\left(y-x^2\right)}^2+(1-x{)}^2$	30	[−2, 2, 0.15]	0
F6	$1+\left({\displaystyle \frac{1}{4000}}\right)\ast {\displaystyle \sum _{i=1}^{D-1}} x{i}^2-{\displaystyle \prod _{i=1}^D} \mathrm{c}\mathrm{o}\mathrm{s}\left({\displaystyle \frac{xi}{\sqrt{(i)}}}\right)$	30	[−2, 2, 100]
F7	$f(x,y)=-20e^{-0.2\sqrt{0.5\left(x^2+y^2\right)}}-e^{0.5(\mathrm{c}\mathrm{o}\mathrm{s}(2\pi x)+\mathrm{c}\mathrm{o}\mathrm{s}(2\pi y))}+e+20$	30	[−5, 5, 100]	0

present the mathematical formulations of the considered benchmark functions. The corresponding statistical performance results are reported in

Table A3. Summary statistics of benchmark functions.

Summary Statistics of F1
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	2.1643 × 107	3.7678 × 106	3.9437 × 107	2.1296 × 107	1.1656 × 108
AVOA	2.6410 × 107	4.1943 × 106	4.2199 × 107	7.7167 × 109	5.5973 × 1010
PSO	7.7468 × 109	2.7629 × 1010	2.8394 × 1010	1.1242 × 1010	5.0198 × 1010
MFO	1.5329 × 109	7.5519 × 109	9.8828 × 109	6.1536 × 109	2.3419 × 1010
DE	6.7345 × 103	3.7678 × 1010	5.6352 × 107	1.6853 × 108	8.6670 × 108
GA	3.8101 × 107	1.0112 × 106	5.9918 × 107	8.0726 × 109	5.7495 × 1010
HSA	1.2686 × 1010	2.0960 × 1010	2.1176 × 1010	4.4768 × 109	2.9085 × 1010
SOA	9.4121 × 1010	1.2441 × 1011	1.2845 × 1011	1.9753 × 1010	1.7095 × 1011
Summary statistics of F2
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	3.8059 × 104	6.1647 × 104	6.3729 × 104	1.7890 × 104	1.0322 × 105
AVOA	1.7036 × 105	2.7899 × 105	2.8391 × 105	6.5328 × 104	4.0269 × 105
PSO	1.5223 × 105	2.4027 × 105	2.5099 × 105	6.9494 × 104	4.2101 × 105
MFO	1.5838 × 105	2.4348 × 105	2.3890 × 105	4.2268 × 104	3.1742 × 105
DE	3.8130 × 104	1.0579 × 105	1.0764 × 105	2.2871 × 104	1.5937 × 105
GA	7.1930 × 104	7.3547 × 104	7.6951 × 104	2.5952 × 104	1.5473 × 105
HSA	1.3641 × 105	2.6622 × 105	2.6844 × 105	6.8340 × 104	4.5753 × 105
SOA	1.6362 × 105	4.4013 × 105	4.2977 × 105	7.2136 × 104	5.1866 × 105
Summary statistics of F3
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	4.1136 × 102	5.1795 × 102	5.1248 × 102	3.1127 × 101	6.5610 × 102
AVOA	5.9099 × 103	9.6558 × 103	9.9792 × 103	2.5597 × 103	1.6609 × 104
PSO	9.6264 × 102	3.8270 × 103	5.0011 × 103	3.4744 × 103	1.3753 × 104
MFO	6.4773 × 102	1.0780 × 103	1.2185 × 103	4.7337 × 102	2.9642 × 103
DE	5.8697 × 103	1.0061 × 104	1.1107 × 104	3.5890 × 101	5.7678 × 102
GA	5.0687 × 102	5.5667 × 102	5.5745 × 102	3.8336 × 103	2.0490 × 104
HSA	2.3397 × 103	3.7817 × 103	4.0272 × 103	1.1029 × 103	6.1275 × 103
SOA	2.3170 × 104	5.4235 × 104	5.2950 × 104	1.5530 × 104	8.1082 × 104
Summary statistics of F4
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	5.5844 × 102	5.9659 × 102	5.9846 × 102	3.6008 × 101	1.0058 × 103
AVOA	8.5610 × 102	9.5658 × 102	9.6119 × 102	4.2432 × 101	1.0632 × 103
PSO	6.4835 × 102	7.3517 × 102	7.4028 × 102	4.6965 × 101	8.4803 × 102
MFO	6.7193 × 102	7.5875 × 102	7.6998 × 102	4.8397 × 101	9.0918 × 102
DE	6.4205 × 102	7.0934 × 102	7.0705 × 102	1.6966 × 101	7.3671 × 102
GA	8.5257 × 102	9.4374 × 102	9.4164 × 102	2.0161 × 101	6.4002 × 102
HSA	7.8436 × 102	8.4916 × 102	8.4141 × 102	2.5291 × 101	8.7983 × 102
SOA	1.0773 × 103	1.2116 × 103	1.2019 × 103	5.4851 × 101	1.2998 × 103
Summary statistics of F5
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	6.0003 × 102	6.0012 × 102	6.0018 × 102	8.5883 × 10−1	7.1069 × 102
AVOA	6.6447 × 102	6.9258 × 102	6.9203 × 102	1.2391 × 101	7.2267 × 102
PSO	6.2581 × 102	6.4077 × 102	6.4041 × 102	9.3720 × 100	6.6352 × 102
MFO	6.2454 × 102	6.4053 × 102	6.4438 × 102	1.6875 × 101	7.0768 × 102
DE	6.7604 × 102	6.9346 × 102	6.9387 × 102	7.8507 × 100	6.0057 × 102
GA	6.0214 × 102	6.0317 × 102	6.0336 × 102	8.5883 × 10−1	6.0603 × 102
HSA	6.3126 × 102	6.4875 × 102	6.4731 × 102	6.1130 × 100	6.6008 × 102
SOA	7.2646 × 102	7.4720 × 102	7.4637 × 102	1.2376 × 101	7.6947 × 102
Summary statistics of F6
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	8.2392 × 102	8.7027 × 102	8.7198 × 102	7.5462 × 101	1.6819 × 103
AVOA	1.2281 × 103	1.4501 × 103	1.4433 × 103	7.5218 × 101	1.5977 × 103
PSO	8.8314 × 102	1.1726 × 103	1.2040 × 103	1.8685 × 102	1.6097 × 103
MFO	9.5995 × 102	1.2051 × 103	1.2243 × 103	1.5569 × 102	1.6827 × 103
DE	9.1760 × 102	9.5854 × 102	9.6136 × 102	1.6912 × 101	1.0020 × 103
GA	1.3301 × 103	1.5148 × 103	1.5149 × 103	2.2379 × 101	9.3043 × 102
HSA	1.3303 × 103	1.4823 × 103	1.5061 × 103	9.8022 × 101	1.7282 × 103
SOA	3.0706 × 103	3.6373 × 103	3.6287 × 103	2.8000 × 102	4.1253 × 103
Summary statistics of F7
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	8.4857 × 102	8.8762 × 102	8.8527 × 102	3.4001 × 101	1.2093 × 103
AVOA	1.0788 × 103	1.1450 × 103	1.1395 × 103	2.7180 × 101	1.1870 × 103
PSO	9.5756 × 102	1.0190 × 103	1.0204 × 103	4.0561 × 101	1.1052 × 103
MFO	9.8344 × 102	1.0742 × 103	1.0752 × 103	4.8218 × 101	1.1621 × 103
DE	9.5735 × 102	1.0102 × 103	1.0063 × 103	1.6865 × 101	1.0295 × 103
GA	1.0813 × 103	1.1419 × 103	1.1444 × 103	1.7906 × 101	9.1171 × 102
HSA	1.0944 × 103	1.1372 × 103	1.1360 × 103	2.3169 × 101	1.1827 × 103
SOA	1.3330 × 103	1.4285 × 103	1.4234 × 103	5.1259 × 101	1.5431 × 103
Summary statistics of F8
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	9.0111 × 102	9.4590 × 102	9.8417 × 102	1.9753 × 103	1.4648 × 104
AVOA	7.8359 × 103	1.1572 × 104	1.1579 × 104	1.9580 × 103	1.4861 × 104
PSO	5.3354 × 103	1.0484 × 104	1.1700 × 104	4.1828 × 103	2.2658 × 104
MFO	6.8544 × 103	1.1938 × 104	1.2276 × 104	4.0847 × 103	1.9158 × 104
DE	7.6450 × 103	1.1574 × 104	1.1388 × 104	1.2620 × 102	1.5417 × 103
GA	1.0389 × 103	1.4339 × 103	1.5065 × 103	3.3435 × 102	2.5254 × 103
HSA	6.9211 × 103	1.0340 × 104	1.0425 × 104	1.8707 × 103	1.3727 × 104
SOA	2.9518 × 104	4.0341 × 104	4.1177 × 104	5.6897 × 103	5.3855 × 104
Summary statistics of F9
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	3.2883 × 103	4.1963 × 103	4.2635 × 103	4.6337 × 102	9.7094 × 103
AVOA	8.1037 × 103	9.1269 × 103	9.0557 × 103	4.1959 × 102	9.8132 × 103
PSO	4.8721 × 103	5.8397 × 103	5.9544 × 103	7.2477 × 102	7.4915 × 103
MFO	5.6290 × 103	6.6465 × 103	6.6999 × 103	6.5365 × 102	7.9928 × 103
DE	6.7962 × 103	8.6159 × 103	8.5805 × 103	4.7900 × 102	9.1860 × 103
GA	8.1288 × 103	8.9356 × 103	8.9100 × 103	5.1435 × 102	5.4959 × 103
HSA	9.2560 × 103	9.8436 × 103	9.8392 × 103	3.8227 × 102	1.0690 × 104
SOA	8.7495 × 103	9.2512 × 103	9.2450 × 103	2.3934 × 102	9.8751 × 103

and

Table A5. Summary statistics of different benchmark suite functions.

Summary statistics of F1
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	2.606 × 10−115	8.152 × 10−108	9.407 × 10−104	3.606 × 10−103	1.700 × 10−102
AVOA	1.102 × 109	1.663 × 109	1.721 × 109	4.260 × 108	2.996 × 109
PSO	3.988 × 105	2.281 × 106	6.689 × 108	2.537 × 109	1.000 × 1010
MFO	7.840 × 108	2.677 × 109	3.034 × 109	1.821 × 109	9.057 × 109
DE	3.800 × 100	1.841 × 101	5.666 × 101	1.116 × 102	5.640 × 102
GA	3.409 × 106	1.004 × 107	1.008 × 107	3.855 × 106	1.998 × 107
HSA	1.207 × 1010	1.768 × 1010	1.764 × 1010	3.035 × 109	2.493 × 1010
SOA	4.790 × 1010	6.304 × 1010	6.252 × 1010	6.083 × 109	7.243 × 1010
Summary statistics of F2
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100
AVOA	1.079 × 103	2.234 × 103	2.310 × 103	5.982 × 102	4.206 × 103
PSO	1.172 × 102	1.958 × 102	1.187 × 103	3.038 × 103	1.019 × 104
MFO	1.121 × 103	2.993 × 103	3.978 × 103	3.648 × 103	2.012 × 104
DE	1.595 × 102	2.058 × 102	2.037 × 102	1.301 × 101	2.272 × 102
GA	7.664 × 101	1.032 × 102	1.072 × 102	2.236 × 101	1.643 × 102
HSA	1.106 × 104	1.676 × 104	1.671 × 104	2.599 × 103	2.228 × 104
SOA	4.821 × 104	6.495 × 104	6.514 × 104	6.619 × 103	7.482 × 104
Summary statistics of F3
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	4.008 × 10−122	5.005 × 10−112	1.183 × 10−105	4.795 × 10−105	2.497 × 10−104
AVOA	1.151 × 107	4.216 × 107	4.604 × 107	2.230 × 107	9.309 × 107
PSO	9.817 × 106	9.137 × 107	1.035 × 108	7.581 × 107	3.477 × 108
MFO	1.741 × 106	7.735 × 106	1.087 × 107	8.564 × 106	3.477 × 107
DE	7.058 × 10−3	4.500 × 10−2	8.869 × 10−2	1.436 × 10−1	6.917 × 10−1
GA	2.780 × 104	7.641 × 104	1.192 × 105	1.284 × 105	5.417 × 105
HSA	1.130 × 108	2.118 × 108	2.208 × 108	6.295 × 107	3.547 × 108
SOA	6.430 × 108	2.159 × 109	2.155 × 109	7.936 × 108	3.558 × 109
Summary statistics of F4
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	3.750 × 102	3.750 × 102	3.750 × 102	1.588 × 10−4	3.750 × 102
AVOA	4.320 × 102	5.492 × 102	5.941 × 102	1.117 × 102	8.789 × 102
PSO	3.761 × 102	3.817 × 102	2.097 × 103	5.220 × 103	1.750 × 104
MFO	1.583 × 103	6.399 × 103	7.596 × 103	4.869 × 103	2.113 × 104
DE	2.714 × 103	6.824 × 103	6.578 × 103	1.804 × 103	1.051 × 104
GA	3.878 × 102	4.016 × 102	4.013 × 102	7.539 × 100	4.162 × 102
HSA	2.630 × 104	3.512 × 104	3.523 × 104	6.385 × 103	4.821 × 104
SOA	9.571 × 104	1.303 × 105	1.305 × 105	1.362 × 104	1.541 × 105
Summary statistics of F5
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	2.780 × 101	2.852 × 101	2.848 × 101	2.780 × 10−1	2.878 × 101
AVOA	8.370 × 106	3.383 × 107	3.861 × 107	2.091 × 107	8.638 × 107
PSO	7.088 × 103	1.136 × 105	4.724 × 105	4.939 × 105	1.060 × 106
MFO	3.928 × 107	4.461 × 108	1.085 × 109	1.915 × 109	1.004 × 1010
DE	2.466 × 101	8.100 × 101	1.483 × 102	3.113 × 102	1.691 × 103
GA	1.386 × 103	7.321 × 103	7.119 × 103	2.980 × 103	1.554 × 104
HSA	1.550 × 109	3.902 × 109	3.894 × 109	1.123 × 109	5.893 × 109
SOA	2.039 × 1010	2.754 × 1010	2.784 × 1010	5.049 × 109	3.860 × 1010
Summary statistics of F6
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100	0.000 × 100
AVOA	1.306 × 100	1.518 × 100	1.554 × 100	1.525 × 10−1	1.861 × 100
PSO	4.426 × 10−2	1.831 × 10−1	4.843 × 10−1	8.843 × 10−1	3.493 × 100
MFO	1.130 × 100	1.737 × 100	1.764 × 100	3.127 × 10−1	2.695 × 100
DE	4.585 × 10−7	3.277 × 10−6	3.780 × 10−3	6.586 × 10−3	2.217 × 10−2
GA	2.448 × 10−1	3.398 × 10−1	3.593 × 10−1	8.358 × 10−2	5.433 × 10−1
HSA	3.849 × 100	5.284 × 100	5.387 × 100	7.846 × 10−1	6.990 × 100
SOA	1.384 × 101	1.697 × 101	1.716 × 101	1.571 × 100	2.026 × 101
Summary statistics of F7
Algorithm	Best	Median	Mean	Std	Worst
FO-AVOA	4.441 × 10−16	3.997 × 10−15	3.405 × 10−15	2.104 × 10−15	7.550 × 10−15
AVOA	9.295 × 100	1.620 × 101	1.609 × 101	1.593 × 100	1.828 × 101
PSO	2.000 × 101	2.000 × 101	2.000 × 101	4.867 × 10−9	2.000 × 101
MFO	2.000 × 101	2.000 × 101	2.002 × 101	1.976 × 10−2	2.006 × 101
DE	1.999 × 101	2.000 × 101	2.000 × 101	1.037 × 10−3	2.000 × 101
GA	1.083 × 101	2.002 × 101	1.972 × 101	1.678 × 100	2.004 × 101
HSA	2.032 × 101	2.053 × 101	2.053 × 101	9.133 × 10−2	2.073 × 101
SOA	2.000 × 101	2.000 × 101	2.000 × 101	2.988 × 10−9	2.000 × 101

in terms of best, median, mean, standard deviation, and worst values over multiple independent runs. Across the majority of test functions and evaluation metrics, the FO-AVOA consistently achieves superior or highly competitive results compared to the other optimization algorithms, demonstrating its strong robustness, stability, and convergence capability. These outcomes provide clear evidence that incorporating fractional-order dynamics significantly enhances the optimization performance. The inability to eliminate the null hypothesis in all cases verifies that the FO-AVOA achieves a statistically significant enhancement over the original AVOA, emphasizing the real-world advantage of the proposed adjustments. To further test the FO-AVOA, it was applied to the IEEE 30- and 57-bus test systems to optimize the power loss minimization problem. The obtained solutions were compared with existing and newly introduced metaheuristic approaches. In all instances, the FO-AVOA demonstrated strong global performance in discrete and continuous power system optimization models. These findings confirm that the FO-AVOA not only improves optimization accuracy but also offers greater reliability in solving real-world problems in reducing power loss in electrical networks.

4.1. ORPD with Thirteen Control Variables in the IEEE 30-Bus System (Case 1)

In the current case study, the proposed FO-AVOA is applied to the IEEE 30-bus power system, whose detailed single-line diagram is presented in Figure 3. The test system involves a total of thirteen control variables, including three shunt compensator units, four tap-changing transformers, and six generator voltage magnitudes, all of which are to be optimized with the aim of minimizing active P_loss and enhancing system performance. The related system description and parameter data are listed in

Table 4. Test system description.

Parameters	Variables
Total buses	30
Total load buses	24
Total branches	41
Total generators	6
Total transformers	4
Total reactors	0
Total capacitors	9

.

Optimization of these control variables is of utmost concern for successful ORPD, as they have direct influences on the voltage stability, RP balance, and transmission line losses. By making use of the fractional-order dynamics of the FO-AVOA, the algorithm can retain memory of past iterations, thereby improving exploration–exploitation trade-off during optimization. This assists in avoiding premature convergence of the algorithm and enhancing its capacity to discover high-quality solutions in big search spaces. In this paper, the real and RP requirements were obtained word for word from the IEEE base case, and boundary conditions of the optimization variables were defined according to the requirements in [16]. To provide a fair and qualitative comparison, all the simulations employed the MATPOWER load flow software, which provides a consistent computational environment for evaluating transmission line loss calculations on the basis of diverse optimization algorithms.

Table 5. Comparative analysis of the proposed FO-AVOA with the literature.

Control Variables	IWO [16]	MICA-IWO [16]	ICA [16]	PSO [17]	HSA [18]	SGA [18]	GA [21]	C-PSO [19]	DE [20]	AVOA	FO_AVOA
V1	1.06965	1.07972	1.0785	1.0725	1.0726	1.0512	1.0721	1.1000	1.095319	1.1000	1.1000
V2	1.06038	1.07055	1.06943	1.0633	1.0625	1.0421	1.063	1.1000	1.085946	1.0941	1.0937
V5	1.03692	1.04836	1.06943	1.0410	1.0399	1.0322	1.0377	1.0747	1.062628	1.0735	1.0713
V8	1.03864	1.04865	1.04714	1.410	1.0422	0.9815	1.0445	1.0867	1.065076	1.0757	1.0751
V11	1.02973	1.07518	1.03485	1.0648	1.0318	0.9766	1.0132	1.1000	1.0266	1.0421	1.0651
V13	1.05574	1.07072	1.07106	1.0597	1.0681	1.1	1.0898	1.1000	1.014253	1.0963	1.0674
T6-9	1.05	1.03	1.08	1.03	1.01	0.95	1.0221	0.99	1.017796	0.9695	0.9890
T6-10	0.96	0.99	0.95	0.95	1	0.98	0.9917	1.05	0.979277	0.9001	0.9990
T4-12	0.97	1	1	0.99	0.99	1.04	0.9964	0.99	0.977843	0.9083	0.9791
T28-27	0.97	0.98	0.97	0.97	0.97	1.02	0.971	0.96	1.008938	0.9398	0.9600
Qc3	8	−7	−6	0.00	34	12	5.3502	9	20.22359	10.7731	7.610
Qc10	35	23	36	16	12	−10	36	30	9.584327	8.9863	23.060
Qc24	11	12	11	12	10	30	12.4175	8	13.02992	4.7651	7.866
PLoss (MW)	4.934	4.859	4.863	4.926	4.9059	4.940	4.8775	4.6801	4.888081	4.750	4.611

represents the comparative performance results. The results suggest that the proposed FO-AVOA efficiently achieved quality solutions by significantly minimizing overall active P_loss in the system. Relative to other state-of-the-art optimization methods, the FO-AVOA consistently provided improved solutions, displaying effective convergence characteristics and effective handling of system constraints. The results point toward the capability of the FO-AVOA to provide nearly optimal solutions for ORPD, thereby improving power system efficiency and reliability.

As presented in Table 5, the application of the FO-AVOA results in a loss of 4.611 MW, reflecting a 15.88% reduction relative to the base case loss of 5.482 MW. In contrast, other methods, such as IWO, MICA-IWC, ICA, PSO, HSA, SGA, GA, C-PSO, DE, and AVOA, achieved reductions of 9.99%, 11.36%, 11.29%, 10.14%, 10.5%, 9.88%, 11.027%, 14.62%, 10.83%, and 13.35% with the base, respectively, when compared against recently established optimization techniques. These results confirm that in solving the ORPD problem, the FO-AVOA consistently delivers superior solutions compared to widely recognized alternatives. Furthermore, all optimization variables remained within their specified operating limits throughout the simulations. Figure 4 illustrates the convergence and learning behaviors of the FO-AVOA during the ORPD optimization process.

It is evident that the FO-AVOA converges significantly faster than the classical AVOA and other compared algorithms, reaching the optimal solution within a substantially lower number of iterations. The learning curve shows that after a relatively small number of iterations, the real power loss stabilizes around 4.611 MW, which confirms the strong robustness and numerical stability of the proposed algorithm. As the iteration count increases, all independent runs consistently converge to the same optimal solution, indicating reliable repeatability and strong convergence characteristics. Moreover, the convergence trend reveals that during the initial iterations, global exploration dominates the search process, while after nearly 100 iterations, the exploitation phase becomes dominant, leading to fine local refinement around the global optimum.

This smooth transition between exploration and exploitation is a direct outcome of the fractional-order memory mechanism, and it clearly explains the superior instantaneous performance and faster convergence behavior of the FO-AVOA compared with recently established optimization approaches. The percentage net improvement in P_loss reduction is shown in Figure 5 and Figure 6 with the base case, as well as with other optimization techniques by the proposed optimizer and with the rest of the optimization techniques cited in the literature, and the proposed FO-AVOA outperformed the other optimizers in achieving more reduction in P_loss in the case of the IEEE 30-bus system.

The statistical analysis of the FO-AVOA using a box plot, the cumulative distribution function, a histogram, and the number of independent runs is shown in Figure 7. Figure 7a illustrates that the FO-AVOA consistently achieves a lower median value of the final solution across 100 independent trials compared to the standard AVOA. Figure 7b further confirms that the FO-AVOA substantially enhances the likelihood of reaching the optimal solution relative to the traditional AVOA. The histogram in Figure 7c highlights the distribution of lowest fitness values acquired throughout multiple runs, and Figure 7d displays the FO-AVOA’s repeated superiority, as it maintains lower levels of fitness throughout multiple experiments compared to the AVOA. Collectively, these graphical results underscore that fractional calculus implies an influential mathematical paradigm for enhancing traditional optimization algorithms, thereby increasing their efficiency, robustness, and reliability.

4.2. ORPD with Twenty-Five Control Variables in IEEE 57-Bus System (Case 2)

In this study, the proposed FO-AVOA is implemented in the IEEE 57-bus test system, which is depicted in Figure 8. Twenty-five control variables, such as generator voltages, transformer tap settings, and RP compensation devices, are optimized here. The IEEE 57-bus benchmark’s precise system parameters and parameter descriptions can be seen in [16], and their related data are presented in

Table 6. Test system description.

Parameters	Variables
Total buses	57
Total load buses	45
Total branches	80
Total generators	7
Total transformers	17
Total reactors	15
Total capacitors	3

. The FO-AVOA proposed was subjected to a more complex optimization environment in the IEEE 57-bus test network, which reflects the greater dimensionality and operating constraints inherent in large-scale systems. Contrary to smaller cases, here, variable bounds and working conditions are specifically geared for the IEEE 57-bus benchmark, with the aim of performing realistic and close evaluations The comparative performance results, as in

Table 7. Comparative analysis of the proposed FO-AVOA with the literature.

Controlled Variables	Base Case [23]	IWO [16]	SOA [23]	CLPSO [23]	FO- DPSO [24]	Hybrid [25]	PSO [25]	ICA [25]	GWO [21]	AVOA	FOAVOA
VG(1)	1.040	1.06	1.060	1.054	1.04	1.0395	1.0284	1.06	1.060	1.0998	1.0999
VG(2)	1.011	1.059	1.058	1.052	1.029	1.0259	1.0044	1.0388	1.056	1.0952	1.0949
VG(3)	0.985	1.047	1.043	1.033	1.009	1.0077	0.9844	1.0078	1.037	1.0823	1.0844
VG(6)	0.980	1.038	1.035	1.031	0.977	0.9982	0.9872	0.9688	1.020	1.0773	1.0736
VG(8)	1.050	1.059	1.054	1.049	0.985	1.0158	1.0262	0.9715	1.044	1.0984	1.0921
VG(9)	0.980	1.027	1.036	1.030	0.967	0.985	0.9834	09556	1.029	1.0792	1.0765
VG(12)	1.015	1.037	1.033	1.034	0.908	0.9966	0.9844	0.9891	1.031	1.0751	1.0745
TTS(4–18)	0.970	1.05	1	0.990	0.9	0.9265	0.9743	0.9584	0.984	0.9000	0.9000
TTS(4–18)	0.978	1.0	0.96	0.980	0.920	0.9532	0.9716	0.9309	0.932	0.9000	0.9000
TTS(21–20)	1.043	1.07	1.01	0.990	1.026	1.0165	1.0286	1.0269	0.957	0.9069	0.9000
TTS(24–26)	1.043	1.02	1.01	1.010	1.007	1.0071	1.0183	1.0085	0.996	0.9526	0.9213
TTS(7–29)	0.967	0.97	0.97	0.990	0.907	0.9414	0.9401	0.9	0.963	0.9000	0.9000
TTS(34–32)	0.965	0.99	0.97	0.930	0.987	0.9555	0.94	0.9872	0.981	0.9000	0.9001
TTS(11–41)	0.955	0.9	0.9	0.910	0.901	0.9032	0.9761	0.9097	1.062	0.9000	0.9000
TTS(15–45)	0.955	0.96	0.97	0.9700	0.9	0.9356	0.9211	0.9377	0.9755	0.9000	0.9000
TTS(14–46)	0.900	0.95	0.95	0.9500	0.9	0.9172	0.9165	0.9166	0.9639	0.9000	0.9000
TTS(10–51)	0.930	0.98	0.96	0.9800	0.916	0.9337	0.9044	0.9057	0.9723	0.9000	0.9000
TTS(13–49)	0.895	0.93	0.92	0.9500	0.9	0.9	0.9118	0.9	0.9248	0.9000	0.9000
TTS(11–43)	0.958	0.99	0.96	0.9500	0.9	0.9206	0.92	0.9	0.9554	0.9000	0.9000
TTS(40–56)	0.958	1.01	1	1.000	0.998	1.0042	0.9891	0.9575	1.1000	0.9038	0.9016
TTS(39–57)	0.980	1.04	0.96	0.9600	0.994	1.0297	0.9430	1.0476	0.9976	0.9000	0.9000
TTS(9–55)	0.940	0.96	0.97	0.9700	0.9	0.9294	0.9998	0.900	0.9845	0.9000	0.9000
QSC(18)	0	0.0442	9.984	0.0988	4	9.846	9	0	1.8917	−29.9682	−24.2087
QSC(25)	0	0.0443	5.904	0.0542	15	10	7.0185	10	5.2489	2.3179	4.3524
QSC(53)	0	0.0615	6.288	0.0628	11.67	10	5.0387	9.5956	5.1513	1.8662	4.0841
Ploss(MW)	28.462	24.593	24.26	24.515	26.68	25.5856	27.5543	26.999	24.752	21.9103	21.5175

, strongly illustrate that the FO-AVOA attains a remarkable improvement in loss minimization and convergence efficiency. With the common simulation platform being MATPOWER, the FO-AVOA’s transmission loss is 21.5175 MW—a 24.399% reduction from the base case value of 28.462 MW. On the other hand, the other approaches, such as IWO, SOA, CLPSO, FO-DPSO, Hybrid, PSO, ICA, and GWO, yield comparatively modest improvements of 13.593%, 14.76%, 13.867%, 6.26%, 10.106%, 3.189%, 5.140%, 13.034%, and 23.01% by the AVOA, respectively. These results show that the FO-AVOA not only provides better solutions in terms of quality and convergence rate but also scales enormously to high-dimensional ORPD problems with all control variables kept strictly within operational ranges.

Figure 9 depicts the convergence characteristics of the FO-AVOA for the IEEE 57-bus system. With increasing iterations, all independent simulation runs steadily approach the same optimal solution, demonstrating strong convergence reliability and repeatability. During the early iterations, the algorithm exhibits dominant exploration behavior, which progressively diminishes as the search advances, while the exploitation phase becomes predominant after approximately 100 iterations. This smooth exploration-to-exploitation transition confirms the effectiveness of the fractional-order memory mechanism in guiding the search toward the global optimum. Overall, this behavior highlights the superior convergence capability, numerical stability, and optimization efficiency of the FO-AVOA when compared with recently developed alternative techniques.

Figure 10 and Figure 11 present the percentage improvement in power loss reduction and compare the base case with other state-of-the-art optimizers by the proposed FO-AVOA reported in the literature. The results demonstrate that the FO-AVOA consistently outperforms competing techniques, achieving greater loss minimization in the IEEE 57-bus system.

Figure 12 presents the statistical evaluation of the FO-AVOA through box plots, cumulative distribution functions, histograms, and multiple independent runs. As shown in Figure 12a, the FO-AVOA consistently achieves a lower median fitness value across 100 independent trials when compared with the standard AVOA. The cumulative distribution in Figure 12b further highlights its enhanced probability of attaining near-optimal solutions, surpassing the performance of the conventional variant.

Figure 12c shows the histogram of minimum fitness values, which depicts a focused distribution around better results, whereas Figure 12d validates the FO-AVOA’s consistent lead by sustaining lower levels of fitness throughout repeated experimentation. Collectively, these statistical evaluations depict how the incorporation of fractional calculus adequately fortifies the underlying optimization process, providing improvements in efficiency, robustness, and overall reliability.

The superior performance of the FO-AVOA can be mainly attributed to the intrinsic characteristics of the fractional-order velocity update. Unlike integer-order updates, which rely only on the most recent search step, the fractional-order mechanism introduces a multi-iteration memory effect, allowing the algorithm to exploit historical search information. This memory-driven behavior smooths the search trajectory, mitigates oscillations in later iterations, and significantly reduces the risk of premature convergence. In high-dimensional search spaces, this effect becomes particularly important, as the fractional-order operator enables a more balanced exploration–exploitation trade-off, allowing the FO-AVOA to escape local optima while maintaining strong local refinement capability. Consequently, the FO-AVOA achieves faster convergence with higher solution accuracy and greater robustness compared to conventional metaheuristic algorithms.

5. Conclusions

This study proposed a FO-AVOA to solve the ORPD problem by minimizing real power transmission losses under system operating constraints. By integrating fractional-order calculus into the classical AVOA framework, memory and nonlocal search characteristics were introduced, leading to improved exploration–exploitation balance, faster convergence, and enhanced robustness. The effectiveness of the FO-AVOA was validated through extensive testing on CEC benchmark functions and standard IEEE 30-bus and 57-bus power systems, where it consistently outperformed several state-of-the-art optimization techniques. These results demonstrate the scalability, stability, and strong optimization capability of the proposed approach. Future work will focus on extending the FO-AVOA to large-scale systems, multi-objective ORPD formulations, and power networks with high renewable energy penetration.