A Fractional Hybrid Strategy for Reliable and Cost-Optimal Economic Dispatch in Wind-Integrated Power Systems

Abstract

Economic dispatch in wind-integrated power systems is a critical challenge, yet many recent metaheuristics suffer from premature convergence, heavy parameter tuning, and limited ability to escape local optima in non-smooth valve-point landscapes. This study proposes a new hybrid optimization framework, the Fractional Grasshopper Optimization algorithm (FGOA), which integrates fractional-order calculus into the standard Grasshopper Optimization algorithm (GOA) to enhance its search efficiency. The FGOA method is applied to the economic load dispatch (ELD) problem, a nonlinear and nonconvex task that aims to minimize fuel and wind-generation costs while satisfying practical constraints such as valve-point loading effects (VPLEs), generator operating limits, and the stochastic behavior of renewable energy sources. Owing to the increasing role of wind energy, stochastic wind power is modeled through the incomplete gamma function (IGF). To further improve computational accuracy, FGOA is hybridized with Sequential Quadratic Programming (SQP), where FGOA provides global exploration and SQP performs local refinement. The proposed FGOA-SQP approach is validated on systems with 3, 13, and 40 generating units, including mixed thermal and wind sources. Comparative evaluations against recent metaheuristic algorithms demonstrate that FGOA-SQP achieves more accurate and reliable dispatch outcomes. Specifically, the proposed approach achieves fuel cost reductions ranging from 0.047% to 0.71% for the 3-unit system, 0.31% to 27.25% for the 13-unit system, and 0.69% to 12.55% for the 40-unit system when compared with state-of-the-art methods. Statistical results, particularly minimum fitness values, further confirm the superior performance of the FGOA-SQP framework in addressing the ELD problem under wind power uncertainty.

1. Introduction

1.1. Motivation

In recent decades, the demand for electrical energy has grown steadily, driven by expanding urban areas, rapid population growth, and the increasing dependence on electrical devices across residential, commercial, and industrial sectors. This surge in consumption, combined with the integration of renewable energy sources (RES) that fluctuate unpredictably, has made reliable and efficient power system operation increasingly challenging. Under such stressed operating conditions, utilities face higher generation, operational, and maintenance costs, along with amplified risks of system losses, instability, and outages. The ELD problem has become particularly important in modern power systems due to the escalating cost of electricity production and the diminishing availability of fossil fuels used in conventional thermal units. The prime aim of ELD is to find the most optimum way of allocating real power generations from the thermal plants, keeping in mind all the constraints. The goal of optimum dispatch is to minimize the overall generation cost, increase reliability, and optimize efficiency. In this study, wind power uncertainty is addressed within a stochastic steady-state ELD framework using expected overestimation and underestimation cost functions, rather than real-time or time-coupled day-ahead dispatch modeling.

The increasing focus of the scientific community is on deriving a solution to the ELD problem that considers actual operating conditions. In this regard, the fuel cost curves of thermal power plants are not convex, as they are impacted by multi-valve steam mechanisms. This brings VPLEs, which cause ripples in the fuel cost function, making it non-differentiable and nonlinear. This is typically expressed by including the absolute sine function to the standard quadratic cost function, which helps describe the cost curve with increased precision.

1.2. Related Work

In addition, in practice, power system operation of generating units is also subject to ramp-rate limits, where the rate at which their output can increase or decrease during operation is restricted, and prohibited operating zones arising from mechanical and structural constraints of the equipment [1,2]. These, together with the nonlinear, nonconvex and non-smooth characteristics inherent in generation cost curves, make the ELD problem highly constrained. Classical solution techniques such as Lagrangian relaxation [3], branch-and-bound [4], and linear and nonlinear programming [5,6] have historically been used to address ELD. Although these methods rely on systematic heuristics to sequentially coordinate generator outputs, they typically become inefficient for large-scale systems or problems with severe practical restrictions, as their computational burden grows rapidly. Effective constraint handling is an essential component of ELD optimization. Advanced variants including stochastic ranking and ε-constrained strategies handle constraint violations as independent objective components [7,8]. Penalty-based and feasibility-based mechanisms are two widely used constraint-handling techniques [9]. In penalty approaches, equality and inequality constraint violations are combined with the objective function using fixed or adaptive penalty factors. This may generate an unintended biased or unstable search process [10]. However, due to complex, nonlinear, and non-smooth natures, real-world ELD problems often cannot be handled properly with all these aforementioned traditional techniques. Therefore, soft computing and metaheuristic algorithms have recently come into the focus of research as a means of overcoming some of the serious limitations of classical optimization techniques.

The use of RES, especially wind energy and solar power, has been significant in minimizing the increasing cost of power production as well as the rising level of harmful emissions. The use of RES in the existing thermal power plants changes the existing quadratic fuel cost relationship with the aid of probability distribution functions such as Beta distribution, which represents the variability in available solar energy, and Weibull distribution, which represents the variability in available wind energy [11,12]. The use of fossil fuel-based thermal power plants, on the other hand, leads to high emissions of harmful gases such as nitrogen oxides (NOx), sulfur oxides (SO), and carbon monoxide (CO). In order to mitigate the environment-related problems, sophisticated computational models for optimizing power production with reduced emissions of pollutants have been introduced [13].

The incorporation of real-world constraints in the ELD problem gradually increases the difficulties, because such constraints result in highly nonlinear and nonconvex problems, which cannot be efficiently solved with existing numerical solution methods that remain confined to local solutions and lack efficiency to globally scout the solution space. In this regard, researchers are presently applying nature-inspired meta-heuristics, because such meta-heuristics are apparently more adaptable when handling tricky solution spaces. For instance, a modified ant colony optimization approach has been used to manage the handling of six hybrid generating units together [14], while the application of growth optimizer (GO) has been apparently justified when handling the nonlinear situation of ELD problems [15]. Use of three metaphor-less algorithms has been commonly reported when classical deterministic solution methods are incapable of seeking globally optimum solutions [16]. In addition, application of fractional-order fish migration optimization (FO-FMO) has been reported while handling multi-objective dispatch problems that take into account economic cost as well as reduced emissions together [17], while hybrid slime mold and genetic algorithm (GSMA) has been efficiently used when handling constrained ELD problems [18]. On the other hand, particle swarm optimization (PSO), which is apparently known for possessing robust globally intensive solution searching properties, has been commonly used when handling hybrid dispatch problems that include stochastic wind energy sources together [19,20,21,22]. In particular, the HIC-SQP optimization technique has been apparently salutary when used for handling thermal as well as wind power generation together, with goals aiming to minimize overall multiple cost components as well as reduce polluting emissions, and encouraging results have been apparently obtained within the literature [23].

More recent research on the use of fractional-order versions of optimizers such as the moth–flame optimizer (MFO) has proven very promising in handling more complex engineering optimization problems [24,25,26,27,28]. This further supports the overall observation that the coupling of fractional calculus (FC) with metaheuristic optimizers can significantly promote performance in power and energy-related problems of optimization. Following on from such a rationale, more diverse dynamic modeling hybrids, such as the GA and PSO-based lead-lag compensator for multiband power system stabilizers, further support such a combined approach with dynamic modeling, albeit with a clearer aim toward GA, PSO, or other metaheuristic optimizers [29]. In a parallel vein, FC research on robotic systems has clearly illustrated how FC-based variants of optimizers such as the classical PSO may effectively address a broad array of engineering tasks [30,31,32,33,34]. Such research has emerged because of a growing appreciation that particular mathematical instruments offered by FC, specifically non-integer order derivatives, may significantly add value to dynamic modeling frameworks with regard to overall control behavior. FC has been used with a view towards advances, for instance, in the realms of feature selections, fractional filters, and Kalman filters, as well as robotic motion control. Complementary research in nonlinear system identifications has, in turn, identified analogous potential when such fractional approaches are entwined with modeling toolkits [35]. Such overall research, therefore, very clearly supports a hybrids approach with FC modeling toolkits, such as meta-heuristics optimizers, toward efficiently handling constrained engineering-related tasks of optimization in the power sector.

Table 1. ELD and wind-integrated dispatch studies: objectives, key findings, and limitations.

References	Main Objective	Key Findings	Limitations
[15]	ELD cost minimization using growth Optimizer	Achieved competitive cost reduction and stable convergence	No fractional memory, limited escape from local optima
[16]	Develop an improved economic load dispatch algorithm integrating advanced optimization strategies for ELD under practical system constraints	Demonstrated improved solution quality and convergence over classical GA and basic heuristics	Limited evaluation on wind-integrated systems; local-optima escape under VPLE not robustly addressed
[17]	Fractional Fish Migration for renewable-gas dispatch	Lower cost when natural gas + RES integrated	Scalability not tested on large generator sets
[18]	Hybrid SMA + GA optimizer (GSMA)	Achieving lowest dispatch cost in microgrid ELD simulations.	No wind modeling via IGF, higher complexity, and scalability not validated on large real-time generator sets.
[19]	Economic load dispatch using metaheuristic techniques	Minimize ELD cost and transmission loss by comparing PSO, ACO, and HS on 6- and 15-unit thermal plants	Single-objective only, fixed demand assumptions
[24]	Blockchain-integrated fractional ELD	Demonstrated feasibility of FC-based dispatch in smart contracts	Overhead of blockchain not quantified
[25,26,27,28]	Fractional-order MFO/Whale hybrids for energy systems	Fractional variants improve convergence smoothness and stability	Runtime impact for large-scale not discussed
[36]	Quasi-oppositional political optimizer for valve-point ELD	Opposition-based learning improves cost under VPLEs	No FO memory, runtime not quantified
[37]	Knowledge-refined optimization for ELD	Refining past knowledge improves cost vs. classical heuristics	No FO memory, no SQP local polishing
[38]	Data-center decarbonization via computing-power migration	Power-aware spatial computing migration supports system-level decarbonization	Not coupled with generator dispatch or SQP
[39]	Review of option value in energy economics	Option value is key for flexibility valuation under uncertainty	No synergy with metaheuristics or SQP explored

provides a concise summary of recent ELD studies with wind-power uncertainty integration, outlining their objectives, main findings, and reported limitations.

1.3. Contribution and Paper Structure

Previous research has suggested that many metaheuristics are highly dependent on rigorous parameter tuning, with high-quality solutions often obtained at the cost of considerable computational expense. In this regard, FGOA is proposed in this research to address such challenges. The approach relies on the natural swarm behavior of grasshoppers, involving attraction, repulsion, and position adjustment, which constitutes the fundamental ideology of the conventional GOA. The employment of FC further improves the resilience of the approach strategy by making it more sensitive to changes in the environment with memory dynamics. The previous studies suggested that the fraction-order process might be capable of producing swifter convergence, enhanced exploitation, and diminished need for advanced parameter adjustment. Although the conventional GOA possesses scalability, it has been found that it converges towards a small region centered on the optimum solution in the latter stage of the optimization process. This premature convergence behavior limits the algorithm’s ability to effectively explore highly nonconvex and non-smooth search spaces, such as those encountered in practical ELD problems with valve-point effects and stochastic wind integration. By incorporating fractional-order calculus, the proposed FGOA introduces long-memory and history-dependent dynamics, enabling each agent’s movement to be influenced by its past trajectories. The fractional-order calculus embedded in the proposed FGOA framework introduces a memory-related behavior, whereby the search dynamics depend not only on the current state but also on the historical trajectory of each agent. This long-memory characteristic smooths abrupt oscillations in the search process caused by valve-point loading effects, which typically generate multiple local optima in the fuel cost landscape. As a result, the optimizer avoids being trapped in shallow local minima by retaining directional information from past iterations, enabling a more informed and persistent exploration of the search space. This mechanism significantly enhances the algorithm’s ability to escape valve-point-induced local optima and contributes to improved convergence reliability in non-smooth and highly nonconvex ELD problems. This mechanism enhances exploration diversity and mitigates stagnation around local optima. To overcome this limitation, an improved hybrid optimization scheme, termed FGOA-SQP, is developed by integrating fractional-order grasshopper dynamics with the local search precision of SQP. The incorporated SQP enables fast and accurate local refinements, while the FGOA component preserves strong global search capability. Based on this hybrid framework, the proposed FGOA-SQP achieves an improved exploration–exploitation balance and is well suited for solving constrained nonlinear ELD problems under chaotic and stochastic wind-power fluctuations.

SQP is selected as the local search component due to its proven effectiveness in solving constrained nonlinear optimization problems through successive quadratic approximations of the objective function and linearization of constraints. In the context of the ELD problem with valve-point loading effects, the cost function is non-smooth and highly nonconvex, which limits the effectiveness of standalone gradient-based methods. The fractional-order dynamics of FGOA provide robust global exploration and memory-driven search behavior, guiding the solution toward a high-quality neighborhood. Within this refined search region, SQP efficiently exploits local curvature information to perform precise constraint-aware refinements, allowing the hybrid FGOA-SQP framework to accurately handle non-smooth cost characteristics while ensuring fast and reliable convergence.

The incorporation of FC in the optimization process adds flexibility as well as several memory-related features that are proven to be valuable in dealing with complex and uncertain systems. The algorithm is capable of adapting itself by utilizing the concept of fractional derivatives, ensuring that it has favorable convergence properties in addition to producing improved results. The following are the major benefits that are achieved by combining FC with GOA:

(a)

Enhanced search dynamics:

Fractional derivatives allow for finer tuning of the trade-off between exploration and exploitation, enabling the algorithm to escape from local optima more effectively and thus explore the solution space more thoroughly.

(b)

Improved robustness to uncertainty:

Intrinsic memory property of FC strengthens the algorithm’s capability to cope with the irregular and unpredictable behavior of wind-power fluctuations, which causes more stable and resilient optimization performance.

(c)

Distinction from existing hybrid metaheuristic approaches:

Contrary to typical classical hybridized metaheuristic approaches, which involve the integration of two different integer-order-based optimization methods or a local search algorithm as a post-processing tool for refinement, the innovative contribution of the proposed FGOA-SQP lies in a novel way of applying a different type of hybridization. Within this manuscript, the application of FC is incorporated at the core motion dynamics of the proposed GOA in a way that goes beyond the mere auxiliary role of facilitating supplementary search refinement. By doing so, the optimizer is capable of incorporating the effects of memory and long-range dependencies in search path behavior, which play a significantly important role in shaping the search path over multiple iterations. Additionally, the hybridized algorithm with SQP is applied in a tightly integrated manner, in which the globally optimal result derived from the proposed FGOA is used for initializing SQP. Hence, unlike classical methods, which involve using local solvers in an independent manner, the proposed FGOA-SQP provides a better-balanced exploration and exploitation capability with guaranteed faster convergence and improved solution quality for nonlinear and nonconvex wind-integrated ELD problems.

The main contributions are as follows:

• This work introduces FGOA-SQP by first integrating FC into GOA and then hybridizing the resulting FGOA with SQP to enhance adaptability, memory effects, and the balance between global exploration and precise local refinement.
• A realistic ELD model is formulated, incorporating valve-point effects, transmission losses, generator limits, and chaotic stochastic wind power modeled through the incomplete gamma function.
• The proposed FGOA-SQP is evaluated on multiple benchmark power systems, including 3-unit, 13-unit, and 40-unit test systems, demonstrating its effectiveness across diverse dispatch scenarios
• Comparative analyses against state-of-the-art metaheuristic methods show that FGOA-SQP achieves superior solution quality, faster convergence, and improved handling of complex nonlinear constraints.
• Statistical analyses further confirm the reliability of FGOA-SQP, demonstrating consistent improvements in minimum fitness values and enhanced robustness under stochastic wind-power uncertainty.

The manuscript is organized as follows: Section 2 formulates the ELD model, including VPLEs and wind uncertainty using IGF. Section 3 details the proposed FGOA-SQP framework and its pseudocode. Section 4 presents simulation results and comparative analyses for Cases 1, 2, and 3, including cost reductions and statistical validation. Section 5 concludes the study and outlines future research directions.

2. System Model: ELD Problem

The ELD problem can be modeled through an input–output characteristic relationship, where the generated power serves as the independent variable and the corresponding production cost is treated as the dependent variable. This relationship is commonly described using a mathematical cost function, which forms the basis for economic dispatch analysis and is expressed as follows [40,41].

$${\mathit{F}}_{\mathit{cost}}\mathit{\left(}\mathit{P}\mathit{\right)}\mathit{=}\mathit{\sum }_{\mathit{j}\mathit{=}\mathit{1}}^{{\mathit{N}}_{\mathit{g}}}\left({\alpha }_{\mathit{j}}{\mathit{P}}_{\mathit{j}}^{\mathit{2}}\mathit{+}{\beta }_{\mathit{j}}{\mathit{P}}_{\mathit{j}}\mathit{+}{\gamma }_{\mathit{j}}\right)$$

(1)

In this formulation, ${\mathit{N}}_{\mathit{g}}$ denotes the total number of thermal generating units and ${\mathit{P}}_{\mathit{j}}$ represents the real power output of the j^th generator (MW). The coefficients ${\alpha }_{\mathit{j}}$, ${\beta }_{\mathit{j}}$, and ${\gamma }_{\mathit{j}}$ are the quadratic fuel cost coefficients of generator j, accounting for its operational and economic characteristics. The objective function ${\mathit{F}}_{\mathit{cost}}\left(\mathit{P}\right)$ represents the total generation fuel cost of all thermal units, expressed in monetary units (e.g., USD/h).

When the valve-point loading effect is taken into account, the cost function is modified and can be expressed as follows [35,40].

$$F_{\cos \mathrm{t}}^{\mathrm{V}\mathrm{P}\mathrm{L}\mathrm{E}}\left(P\right)=\sum _{j=1}^{N_g}\left({\alpha }_j{P}_j^2+{\beta }_j{P}_j+{\gamma }_j+\left|{\eta }_j\mathrm{s}\mathrm{i}\mathrm{n}\left({\xi }_j\left(P_j^{\mathrm{m}\mathrm{i}\mathrm{n}}-P_j\right)\right)\right|\right)$$

(2)

In Equation (2), $P_j^{\mathrm{m}\mathrm{i}\mathrm{n}}$ denotes the minimum power output limit of the j^th generating unit. The coefficients ${\eta }_j$ and ${\xi }_j$ represent the amplitude and frequency of the valve-point loading effect, respectively, which model the rippling behavior introduced by steam valve operations. The absolute sinusoidal term makes the fuel cost function non-smooth and highly nonconvex, thereby increasing the complexity of the ELD problem.

2.1. Power Balance Constraint

The primary and most fundamental constraint in a power system is to ensure that the total generated electrical power meets the load demand of end users in an optimal manner.

$$\sum _{j=1}^{N_g}{P}_j=P_D+P_L$$

(3)

In Equation (3), $P_D$ denotes the total consumer load demand, while $P_L$ represents the total transmission power losses (MW). This equality constraint ensures that the total generated power from all thermal units satisfies the system demand while accounting for network losses.

2.2. Generator’s Power Capacity Checks

When formulating the ELD problem, each generating unit is required to operate within its specified minimum and maximum power output limits.

$$P_j^{\mathrm{m}\mathrm{i}\mathrm{n}}\le P_j\le P_j^{\mathrm{m}\mathrm{a}\mathrm{x}},j=1,2,\dots ,N_g$$

(4)

In Equation (4), $P_j^{\mathrm{m}\mathrm{i}\mathrm{n}}$ and $P_j^{\mathrm{m}\mathrm{a}\mathrm{x}}$ denote the minimum and maximum allowable power output limits (MW) of the j^th generating unit, respectively. This inequality constraint ensures that each generator operates within its physical and operational capacity limits

2.3. Reducing Fuel Price for Integrated Power Plant Systems

The combined utilization of thermal and wind power generation is widely adopted to reduce operating costs in integrated power systems. However, wind availability is inherently uncertain due to the difficulty in accurately forecasting wind speed at a given time, which may lead system operators to overestimate or underestimate the available wind power. To address this uncertainty, wind turbines are incorporated into multi-objective dispatch formulations, enabling a more cost-effective coordination between wind and thermal generation sources [23].

$$F_{\mathrm{cost}}^{wind}(P)=\sum _{i=1}^n\left[\begin{array}{c}\left({C_{WP}}^{dir}\left(i\right)\right)+\left({C_{WP}}^{oe}\left(i\right)\right)+\left({C_{WP}}^{ue}\left(i\right)\right)\end{array}\right]$$

(5)

Here, $F_{\mathrm{cost}}^{wind}\left(P\right)$ represents the total wind power generation cost, and n denotes the number of wind power generating units. The cost components associated with wind power include the direct generation cost, overestimation cost, and underestimation cost, denoted by ${C_{WP}}^{dir}\left(i\right)$, ${C_{WP}}^{oe}\left(i\right)$, and $\left({C_{WP}}^{ue}\left(i\right)\right)$, respectively. The direct cost component is directly related to the wind power output and is expressed for the ith wind unit as follows.

$${C_{WP}}^{dir}(i)=\sum _{i=1}^n(d_i\ast W{P}_i)$$

(6)

As indicated in Equation (6), $W{P}_i$ denotes the actual power output of the ith wind generating unit, while $d_i$ represents its corresponding direct energy cost coefficient, expressed in USD/MWh. ${C_{WP}}^{oe}\left(i\right)$ denotes the imbalance cost caused by overestimating wind power availability, which leads to power deficits that must be compensated by procuring additional real power, as expressed mathematically below

$${C_{WP}}^{oe}\left(i\right)=\sum _{i=1}^n(S_{rw,t}\ast T(U_{oe,i}))$$

(7)

Equation (7) defines the expected wind power overestimation for the ith wind generating unit, denoted by $T\left(U_{oe,i}\right)$. The corresponding overestimation cost coefficient is represented by $S_{rw,t}$ and is expressed in USD/MWh.

$$\begin{gathered} T\left(U_{OE,i}\right)=W{P}_i\left[\begin{array}{c}1-\mathit{exp}\left(-{\displaystyle \frac{V_{in,i}^i}{S_i^{K_i}}}\right)+\mathit{exp}\left(-{\displaystyle \frac{V_{out,i}^{E_i}}{S_i^{E_i}}}\right)\end{array}\right]+\left(\begin{array}{c}{\displaystyle \frac{W{P}_{r,i}\ast V_{in,i}}{V_{r,i}-V_{in,i}}}+W{P}_i\end{array}\right).\left[\begin{array}{c}\mathit{exp}\left(-{\displaystyle \frac{V_{in,i}^{E_i}}{S_i^{E_i}}}\right)-\mathit{exp}\left(-{\displaystyle \frac{V_{1,i}^{E_i}}{S_i^{E_m}}}\right)\end{array}\right] \\ +\left({\displaystyle \frac{W{P}_{r,i}\ast S_i}{V_{r,i}-V_{in,i}}}\right)\left\{\begin{array}{c}\Gamma \left[1+{\displaystyle \frac{1}{E_i}},{\left({\displaystyle \frac{V_{1,i}}{S_i}}\right)}^{E_i}\right]-\Gamma \left[1+{\displaystyle \frac{1}{E_i}},{\left({\displaystyle \frac{V_{in,i}}{S_i}}\right)}^{E_i}\right]\end{array}\right\}. \end{gathered}$$

(8)

V_in, V_out, and V_r denote the cut-in, cut-out, and rated wind speeds (m/s), respectively. An intermediate term P_r is introduced through the relation V₁ = V_in + (V_r − V_in) (WP_i/WP_r). The Weibull distribution parameters S_i and E_i represent the scale and shape factors of the ith wind generating unit, respectively. Here, WP_i and WP_r denote the actual and rated wind power outputs (MW). The incomplete gamma function, defined by two parameters, is mathematically expressed as follows

$$\Gamma \left(\omega ,\xi \right)=1/\Gamma \left(\xi \right)\ast {\int }_0^{\omega }{t}^{\xi -1}{e}^{-t}dt$$

(9)

A conventional gamma function includes a single parameter.

$$\Gamma \left(\omega \right)={\int }_0^{\omega }{t}^{\omega -1}{e}^{-\omega }dt$$

(10)

${C_{WP}}^{oe}\left(i\right),$ represents the penalty cost arising from underestimating wind power availability, which occurs when the actual wind power output exceeds the scheduled value, requiring compensation to wind power producers.

$${C_{WP}}^{oe}\left(i\right),={\sum }_{i=1}^n(C_{ew,i}\ast T(U_{UE,i}\left)\right).$$

(11)

For the ith wind generating unit, $C_{ew,i}$ denotes the underestimation cost coefficient in USD/MWh, while $T\left(U_{UE,i}\right)$ represents the expected value of wind power underestimation.

The overall total generating cost (Tgc) function will become

$${T_{gc}=F}_{\mathrm{cost}}\left(P\right)+F_{\mathrm{cost}}^{\mathrm{V}\mathrm{P}\mathrm{L}\mathrm{E}}\left(P\right)+F_{\mathrm{cost}}^{wind}\left(P\right)$$

(12)

3. Design Methodology

The proposed hybrid framework is utilized to solve constrained optimization problems evident in modern power generation systems. The global exploration capability of the FGOA is combined with the precise local refinement provided by SQP to achieve fast convergence toward high-quality solutions in the proposed scheme. In order to evaluate the performance of the hybrid FGOA-SQP approach, three case studies have been considered, including 3, 13, and 37 thermal generating units along with 3 wind powered units. The overall procedure and flow of the proposed methodology are illustrated in Figure 1.

3.1. Grasshopper Optimization Algorithm

The GOA is a metaheuristic created by Mirjalili as presented in [42], which is inspired by the interesting swarm behavior shown by grasshoppers. While individual grasshoppers can normally live independently, they also have the ability to create massive groups that can cause considerable loss to agricultural areas, deeming grasshoppers among the most feared insects on the global stage [42]. Grasshoppers also have an interesting behavior characteristic which involves swarming during both the young and adult stages. While grasshoppers exhibit behaviors characteristic of the young stage by moving from one place to another in the form of rolling cylinders, their counterparts, or adult grasshoppers, fly and move over very long distances in search of food.

Spreading, grouping, movement, and specific behaviors of foraging comprise the conceptual foundation of GOA. In optimization terms, these behaviors translate to the concepts of “exploration” or “global search” and “exploitation” or “refinement through local movement,” which naturally occur to grasshoppers in the process of finding resources. “Exploration/Refinement through Local Movement” in optimization is modeled in a mathematical formulation presented in [42].

$$X_i=S_i+G_i+A_i$$

(13)

From the structure of Equation (13), it is evident that the proposed algorithm models the movement of locusts toward a food source through three primary influences: the social interaction term $S_i$ reflecting the forces between individuals, the gravitational component $S_i$ representing environmental pull, and the wind advection term $A_i$ accounting for atmospheric effects. In this formulation, $X_i$ denotes the position of the ith locust within the search space

$$S_i={\sum }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N}s\left(d_{ij}\right)\widehat{d_{ij}}$$

(14)

$$d_{ij}=\left|x_j-x_i\right|$$

(15)

$$\widehat{d_{ij}}=\left(x_j-x_i\right)/d_{ij}$$

(16)

$$s\left(r\right)=f{e}^{-r/l}-e^{-r}$$

(17)

In this formulation, $d_{ij}$ represents the Euclidean distance between two grasshoppers, while $\widehat{d_{ij}}$ denotes the corresponding unit direction vector from one individual to the other. The interaction function $s$ governs the social behavior between grasshoppers, producing attraction when its output is positive and repulsion when negative. The parameter $f$ controls the magnitude of this attraction, whereas the parameter $l$ influences the interaction range. It is also essential that the function s does not generate excessively strong forces when grasshoppers are far apart. For effective swarm behavior, the inter-grasshopper distance is typically maintained within a practical interval of [1,4]. The gravitational influence and wind advection acting on each grasshopper can be expressed mathematically as follows:

$$G_i=-g\widehat{e_g}$$

(18)

$$A_i=u\widehat{e_w}$$

(19)

Building on the earlier definitions, the gravitational component G is determined by the gravity coefficient g together with the unit vector $\widehat{e_g}$, which points toward the center of the Earth. Similarly, the wind effect is characterized by the wind magnitude u and the unit vector $\widehat{e_w}$, indicating the direction of the wind.

Using these elements, and following the interaction rules and formulations discussed previously, the updated position of each locust in the search space can be derived, resulting in the expression given in Equation (20).

$$X_i={\sum }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^{N}s\left(\left|x_j-x_i\right|\right){\displaystyle \frac{x_j-x_i}{d_{ij}}}-g\widehat{e_g}+u\widehat{e_w}$$

(20)

Consequently, the overall mathematical model can be expressed as follows

$$X_i^d=\beta \left({\sum }_{\begin{array}{c}j=1\\ j\ne i\end{array}}^N\beta {\displaystyle \frac{u{b}_d-l{b}_d}{2}}s\left(\left|x_j^d-x_i^d\right|\right){\displaystyle \frac{x_j-x_i}{d_{ij}}}\right)+\widehat{T_d}$$

(21)

In Equation (21), $u{b}_d$ and $l{b}_d$ denote the upper and lower bounds of the dth dimension, respectively. The variable N represents the total number of grasshoppers in the swarm, while $\widehat{T_d}$ refers to the best position found so far in the d-dimensional search space. The coefficient β functions as a constriction factor that gradually decreases over iterations, thereby reducing global exploration and strengthening the algorithm’s capability for precise local searching.

$$\beta =\beta max-p{\displaystyle \frac{\beta max-\beta min}{P}}$$

(22)

Here, $\beta max$ is equal to max $\beta$ and $\beta min$ is equal to min $\beta$. The variable P denotes the current iteration index, while P_max represents the total number of iterations allowed for the algorithm.

3.2. Sequential Quadratic Programming

SQP is a well-established iterative optimization technique for solving constrained nonlinear programming problems, in which a sequence of quadratic programming subproblems approximates the original nonlinear problem. SQP has long been regarded as one of the most effective methods for handling both equality and inequality constraints with strong theoretical convergence properties. Comprehensive formulations and algorithmic details are provided in seminal works such as Boggs and Tolle’s survey on SQP methodologies [43] and the widely used large-scale SQP implementation presented in the SNOPT algorithm by Gill et al. [44].

3.3. Fractional Grasshopper Optimization (FGOA)

FC has become increasingly prominent across engineering, physics, and applied mathematics due to its ability to model systems with memory and hereditary effects—capabilities not offered by classical integer-order calculus. FC extends traditional derivatives and integrals to non-integer orders, providing a more flexible framework for describing complex dynamic processes. Various definitions for fractional derivatives exist in the literature [45,46], with the Grünwald–Letnikov formulation being particularly useful for numerical implementation. Using this definition, the fractional expressions required for developing the algorithm can be derived. For any signal s(t), its Grünwald–Letnikov fractional derivative given in Equation (23) forms the basis for incorporating fractional behavior into the proposed optimization method.

$$D_{GL}^{\alpha }[f(t)]=\underset{\mathsf{\Delta }\mathrm{t}\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\left[{\displaystyle \frac{1}{{\left(\mathsf{\Delta }\mathrm{t}\right)}^{\alpha }}}\sum _{j=0}^{\infty }\left[{\displaystyle \frac{{\left(-1\right)}^j\Gamma \left(\alpha +1\right)f\left(t-j\mathsf{\Delta }\mathrm{t}\right)}{\Gamma \left(\alpha -j+1\right)\Gamma \left(j+1\right)}}f\left(t-j∆t\right)\right]\right]$$

(23)

where $D_{GL}^{\alpha }$ denotes the Grünwald–Letnikov fractional derivative of order α of the function f(t) with respect to time t. f(t) is a real-valued signal or function of time, α is a fractional derivative order usually in the range of [0 1]. $\Gamma$ is the amma function, which generalizes the factorial. The discrete time consumption can be expressed as

$${D_t}^{\delta }s(t)\approx {\displaystyle \frac{1}{T^{\delta }}}\sum _{k=0}^N{(-1)}^k\left(\begin{array}{c}\delta \\ k\end{array}\right)s(t-kT)$$

(24)

where

$$\left(\begin{array}{c}\delta \\ k\end{array}\right)=\left({\displaystyle \frac{\Gamma (\delta +1)}{\Gamma (k+1)\Gamma (\delta -k+1)}}\right)$$

s(t) is the signal (or function) evaluated at time t. T > 0 is the sampling/step size (time increment). k is the summation index (integer). N is the truncation length, Γ is the Gamma function, and $\left(\begin{array}{c}\delta \\ k\end{array}\right)$ is the generalized binomial coefficient defined via Gamma functions. The term $s(t-kT)$ is the k-step delayed sample of the signal. The forward difference operator can be expressed as

$$\mathsf{\Delta }s\left(t\right)=s(t+\mathsf{\Delta }\mathrm{t})-s\left(t\right)$$

(25)

The forward-difference operator provides a discrete approximation of the first derivative by evaluating the change in the signal between s(t) and $s(t+\mathsf{\Delta }\mathrm{t})$, forming the basis for fractional-order derivative models. The position of each grasshopper is updated using its velocity component, as shown in Equation (26), where the previously defined fractional-calculus operator is incorporated to enhance the local search performance of the standard GOA.

$$G_p^n=G_p^{n-1}+{\mathsf{\Delta }}_H^{p,n}$$

(26)

In Equation (26), $G_p^n$ denotes the position update (movement increment) of the pth grasshopper at iteration n, while $G_p^{n-1}$ represents its value at the previous iteration. The term ${\Delta }_H^{p,n}$ captures the social-interaction-driven displacement arising from grasshopper swarming behavior. The superscript n indicates the iteration index, and the subscript p identifies the corresponding agent. Unlike PSO-based formulations, the proposed model does not employ velocity vectors or personal/global best concepts; instead, agent movements are governed by attraction–repulsion dynamics and collective behavior, consistent with the GOA framework

$$G_p^n={\alpha \rho }_1(X_{best}^n-X_p^{n-1})$$

(27)

$G_p^n$ is computed using the acceleration coefficient α and a random weight ${\rho }_1\in (0,1$). The displacement depends solely on the difference between the current and previous positions $(X_{best}^n-X_p^{n-1})$, reflecting GOA’s attraction–repulsion-based motion dynamics. When δ = 1, the model reduces to a conventional first-order update without memory, while 0 < δ < 1 enables fractional-memory influence as expanded in Equation (30).

$$D^{\delta }{G}_p^n=G_p^{n-1}+{\alpha \rho }_1(X_{best}^n-X_p^{n-1})$$

(28)

Under the FC framework, the velocity derivative order can be extended to a real value 0 < δ < 1, which promotes smoother updates and introduces a long-term memory effect. Accordingly, Equation (28) can be reformulated using a discrete-time fractional difference operator as follows:

$${{\mathit{G}}_{\mathit{p}}}^{\mathit{n}}\mathit{=}\mathit{-}\frac{\mathit{1}}{{\tau }^{\delta }}\mathit{\sum }_{\mathit{j}\mathit{=}\mathit{0}}^{\mathit{n}}{\left(\mathit{-}\mathit{1}\right)}^{\mathit{j}}\frac{\Gamma \left(\delta \mathit{+}\mathit{1}\right)}{\Gamma \left(\mathit{j}\mathit{+}\mathit{1}\right)\Gamma \left(\delta \mathit{-}\mathit{j}\mathit{+}\mathit{1}\right)}{\mathit{G}}_{\mathit{p}}^{\mathit{n}\mathit{-}\mathit{j}}\mathit{+}{\alpha \rho }_{\mathit{1}}\mathit{(}{\mathit{X}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}^{\mathit{n}}\mathit{-}{\mathit{X}}_{\mathit{p}}^{\mathit{n}\mathit{-}\mathit{1}}\mathit{)}$$

(29)

Equation (30) can be reformulated by restricting the expansion to four terms, as expressed below

$${\mathit{G}}_{\mathit{p}}^{\mathit{n}}\mathit{=}{\mathit{G}}_{\mathit{p}}^{\mathit{n}\mathit{-}\mathit{1}}\mathit{+}\frac{\mathit{1}}{\mathit{2}}\delta \mathit{(}\mathit{1}\mathit{-}\delta \mathit{)}{\mathit{G}}_{\mathit{p}}^{\mathit{n}\mathit{-}\mathit{2}}\mathit{+}\frac{\mathit{1}}{\mathit{6}}\delta \mathit{(}\mathit{1}\mathit{-}\delta \mathit{)}\mathit{(}\mathit{2}\mathit{-}\delta \mathit{)}{\mathit{G}}_{\mathit{p}}^{\mathit{n}\mathit{-}\mathit{3}}\mathit{+}\frac{\mathit{1}}{\mathit{24}}\delta \mathit{(}\mathit{1}\mathit{-}\delta \mathit{)}\mathit{(}\mathit{2}\mathit{-}\delta \mathit{)}\mathit{(}\mathit{3}\mathit{-}\delta \mathit{)}{\mathit{G}}_{\mathit{p}}^{\mathit{n}\mathit{-}\mathit{4}}$$

(30)

In Equation (30), $G_p^{n-1}$, $G_p^{n-2}$, and $G_p^{n-3}$ represent its previous updates. The parameter δ ∈ (0,1) is the fractional order controlling memory depth, where lower δ values increase historical influence, helping agents maintain smoother motion and escape local optima induced by valve-point ripples. The coefficients in Equation (30) arise from the finite-memory fractional expansion applied to grasshopper displacement dynamics, preserving GOA’s attraction–repulsion behavior.

3.4. Impact of Fractional-Order Dynamics on Search Behavior and Convergence

The incorporation of fractional-order dynamics into the GOA plays a crucial role in enhancing the exploration–exploitation balance and improving convergence behavior. Unlike integer-order updates, fractional-order derivatives introduce non-local memory effects, enabling each grasshopper to exploit information from multiple past iterations rather than relying solely on the current state. This memory-aware mechanism allows the algorithm to explore the search space more effectively during early iterations, thereby reducing the likelihood of premature convergence to local optima.

As the optimization process progresses, the fractional-order formulation contributes to smoother and more stable position updates, which enhances exploitation capability and ensures steady convergence toward high-quality solutions. This behavior is particularly advantageous for ELD problems characterized by highly nonlinear, nonconvex, and non-smooth cost functions, such as those arising from VPLEs and stochastic wind-power integration. Consequently, the fractional-order GOA demonstrates faster convergence, improved solution stability, and enhanced robustness compared to its integer-order counterpart.

3.5. The FGOA-SQP Framework for Solving the ELD Problem

The proposed FGOA-SQP framework is employed to address the ELD optimization problem described in Section 2. In this hybrid approach, the FGOA performs the global exploration phase, and the best solution identified by FGOA is subsequently used as the initial point for the SQP procedure. In this formulation, each grasshopper represents a candidate solution vector containing N_var decision variables, where N_var corresponds to the number of generating units considered in the dispatch problem. Thus, the goal of the hybrid method is to determine the optimal active power outputs of all generators while satisfying system constraints and minimizing the total cost. FGOA performs global search for 100 iterations, and SQP is applied only for local refinement, where the SQP solution is accepted only if it reduces the total cost and satisfies feasibility; otherwise, the best FGOA solution is retained. The FGOA-SQP method proceeds through the following steps, and its pseudocode is presented in Algorithm 1:

Algorithm 1. Pseudocode of FGOA-SQP

1. Initialize the swarm size (NP), the number of decision variables, termination criteria, and the maximum number of function evaluations. 2. Evaluate the objective function for all candidate solutions and record the best position identified. 3. Update FGOA parameters, including the fractional-order operators that control interaction dynamics among grasshoppers. 4. Generate a random value p ∈ [0,1]. Depending on the parameter conditions, update the grasshopper positions using the appropriate motion model derived in the fractional GOA formulation. 5. Check stopping criteria. If satisfied, terminate the FGOA loop; otherwise, return to Step 2. 6. Store the best grasshopper position obtained as the initial solution for the SQP stage. 7. Execute the SQP algorithm, utilizing the FGOA-derived solution as the starting point to achieve rapid and accurate local convergence. 8. Apply SQP only for local refinement. Accept the SQP-updated solution only if it reduces total cost and satisfies feasibility; otherwise, retain the FGOA best solution.

4. Result and Discussion

In order to test the performance of the proposed FGOA-SQP framework, a number of extensive simulations are carried out on various ELD benchmark test systems. First, analyses are performed on systems with 3 and 13 thermal generating units, tested under VPLE and non-VPLE operating scenarios as well. A large-scale configuration featuring 40 units and integrated renewable resources is also tested, comprising 37 thermal generators as well as three wind-power units. For the 3-unit and 13-unit systems, standard operating limits and fuel-cost coefficients are used with a load demand of 850 MW and 1800 MW, respectively. In the 40-unit hybrid system, typical wind-generation parameters, fuel-cost characteristics, and permissible generation boundaries are considered, and a total system demand of 10,500 MW is applied for evaluation. These test systems collectively provide a diverse platform for validating the robustness and scalability of the FGOA-SQP approach.

4.1. Case 1: Three-Unit Test System

This case study investigates the performance of the proposed FGOA-SQP method for a three-unit generation system supplying an 850 MW load while incorporating the valve-point loading effect. The generator limits and fuel-cost coefficients used in this system configuration are taken from established benchmark studies reported in the literature [36,37,47,48,49,50]. A comparative summary of the results obtained using FGOA-SQP and the method is provided in

Table 2. Comparative optimal results for Case 1 using different optimization techniques.

Optimization Techniques	P1 (MW)	P2 (MW)	P3 (MW)	Total Generation	Total Cost (USD/h)
GA-PS-SQP [49]	300.30	400	149.70	850	8234.10
PSO-SQP [49]	300.30	400	149.70	850	8234.10
QOPO [50]	300.25	400	149.75	850	8234.07
GA [49]	398.7	399.6	50.1	848.4	8222.1
GSA [48]	300.210	149.795	399.995	850	8234.1
PSO [49]	300.3	400	149.7	850	8234.1
MMFO [50]	396.769	328.4747	124.7563	850	8194.4800
FGOA-SQP	393.1698	334.6038	122.2264	850	8194.36

, alongside values published in earlier works. The corresponding optimal power distribution among the three generating units meets the load demand of 850 MW as depicted in Table 2. The FGOA-SQP framework achieves a minimum production cost of 8194.3561 USD/h, demonstrating a clear improvement over other recent optimization methods. Figure 2 shows the total generation cost produced by the FGOA-SQP algorithm across 100 independent runs of the ELD problem. Each run begins with different random initial conditions, and the vertical axis reflects the resulting cost in USD/h.

The plot illustrates the algorithm’s stability and robustness, as most runs yield consistently low costs with only minor variations. This demonstrates that FGOA-SQP reliably converges to high-quality solutions and effectively avoids poor local minima in repeated trials. The results show that the proposed method attains a lower total fuel cost than competing state-of-the-art algorithms, confirming its superior effectiveness in achieving optimal economic performance (USD/h). The convergence behavior for the FGOA-SQP refinement process for the three-generator ELD system is shown in Figure 3. The convergence plot depicts the final power output of each generator, associated with a minimum objective cost of 8194.36 USD/h achieved by the proposed FGOA-SQP method at convergence. The function-value demonstrates fast improvement during the first few iterations, stabilizing quickly as the solution approaches optimality.

Table 3. A comparative evaluation of the total generation fuel cost and percentage improvement of FGOA-SQP for Case 1 against state-of-the-art optimization algorithms.

Algorithms	Total Generation Fuel Cost (USD/h)	Improvement Achieved Against Other Algorithm
CPSO [49]	8234.07	0.48%
NSS [51]	8234.08	0.48%
NDS [52]	8234.07	0.48%
iBA [53]	8234.08	0.48%
EP [52]	8234.1357	0.48%
CPSO-SQP [49]	8234.07	0.48%
GWO [55]	8253.11	0.71%
MFEP [55]	8234.08	0.48%
GSA [48]	8234.1	0.48%
QOPO [50]	8234.07	0.48%
MFO [50]	8198.23141	0.047%
GA-PS-SQP [47]	8234.1	0.48%
GAB [55]	8234.08	0.48%
FGOA-SQP	8194.36

summarizes the optimal distribution of power among the generating units for a system load demand of 850 MW. The results clearly indicate that the proposed FGOA-SQP algorithm delivers enhanced economic performance by achieving a minimum total fuel cost of 8194.36 USD/h and achieves fuel-cost reductions ranging from 0.047% to 0.71%. This outcome is lower than those obtained using other recently published state-of-the-art optimization algorithms, as reported in the comparative studies available in the literature [48,49,50,51,52,53,54,55], thereby confirming the effectiveness of the proposed hybrid approach for solving the ELD problem.

4.2. Case 2: Thirteen-Unit Test System

This case study evaluates the performance of the proposed FGOA-SQP method for a 13-unit thermal generation system supplying an 1800 MW load while accounting for valve-point loading effects. Standard generator limits and fuel-cost parameters for this configuration are used in accordance with established benchmark data [36,54,55,56,57]. The results obtained using FGOA-SQP for the 1800 MW demand are summarized in

Table 4. Comparative optimal results for Case 2 using different optimization techniques.

Generating Units	MFO [50]	NN-EPSO [54]	GWO [54]	MMFO [50]	QOPO [49]	FGOA-SQP
1	807.1247	490	807.1247	481.7726	628.3183	506.9118
2	144.869	189	144.869	194.1905	298.1864	253.4559
3	297.9434	214	297.9434	244.7307	223.7622	253.4559
4	60	160	60	116.1982	60.00008	99.3628
5	60	90	60	117.4941	60	99.3628
6	60	120	60	132.1647	60	99.3627
7	60	103	60	77.94045	159.7331	99.3627
8	60	88	60	125.2659	60	99.3628
9	60.0362	104	60.0362	92.16435	60	99.3627
10	40	13	40	40	40	40.0000
11	40.0267	58	40.0267	43.26936	40	40.0000
12	55	66	55	78.6438	55	55.0000
13	55	55	55	56.16537	55.00001	55.0000
Total Cost (USD/h)	18,008.89	18,442.59	18,051.11	17,960.14	17,988.99	17,932.4741

, together with comparative results reported in earlier studies, while

Table 5. Comparison of total fuel cost against other algorithms for Case 2.

Algorithms	Total Fuel Cost	Improvement Achieved Against Other Algorithm (%)
GA-SQP [24]	24,650.96	27.25%
FA [24]	24,650.50	27.2%
CSA [24]	24,190.26	25.86%
FO_FA [24]	24,280.13	26.14%
FMFO [24]	23,938.71	25.09%
FWOA [58]	23,313.53	23.08%
CEP [55]	18,048.21	0.64%
FEP [55]	18,018.00	0.47%
MFEP [55]	18,028.09	0.53%
PSO [36]	18,030.72	0.54%
MFO [50]	18,008.89	0.42%
IFEP [55]	17,994.07	0.34%
PSO [57]	18,030.72	0.54
EP-SQP [57]	17,991.03	0.32%
QOPO [49]	17,988.99	0.31%
FGOA-SQP	17,932.4741

presents the corresponding minimum total fuel cost generation obtained by the proposed optimizer with other optimization techniques reported in the literature. The hybrid FGOA-SQP method achieves the lowest total cost of 17,932.4741 USD/h and yields fuel-cost improvements ranging from 0.31% to 27.25%, outperforming several well-known optimization techniques, including FEP, CEP, MFEP, IFEP, and PSO, GA-SQP, FA, CSA, FO-FA, and FMFO. These improvements become more evident as the system size increases. For the 13-unit case, FGOA-SQP consistently yields a substantial reduction in generation cost relative to competing algorithms. The number of independent trial and convergence behaviors is illustrated in Figure 4 and Figure 5. Figure 4 shows the total generation cost obtained by FGOA-SQP over 100 independent runs of the 13-unit ELD problem. Each point corresponds to a separate trial with different initial conditions. The narrow range of cost values indicates stable and repeatable performance, demonstrating that the algorithm consistently converges to high-quality solutions with minimal variability, while Figure 5 shows the convergence behavior.

FGOA-SQP for the 13-unit dispatch case. The total cost rapidly decreases during the early iterations and stabilizes at the optimal value, confirming fast and reliable convergence. The flat region after approximately 20 iterations indicates that the algorithm reaches its optimal solution efficiently. Overall, the results demonstrate that the proposed hybrid approach offers superior fuel-cost minimization and enhanced convergence performance for the ELD problem.

4.3. Case 3: Forty-Unit Test System with Wind Power Units

This case study examines a large-scale ELD-VPLE system composed of 40 generating units, including 37 thermal units and 3 wind-power units. Benchmark data for both thermal and wind generators are adopted from established studies [24,55,57,58,59,60]. The optimal generation schedule for meeting a total demand of 10,500 MW is presented in

Table 6. Optimal result for forty generators with wind power units.

Power Units	MMFO [50]	MFO [50]	FGOA-SQP
1	112.2146	114	36.0000
2	85.7714	110.782043	114.0000
3	88.2117	97.35768193	97.4002
4	180.9641	179.853732	190.0000
5	82.4790	47	97.0000
6	139.9986	140	68.0000
7	300	300	259.6000
8	289.7228	300	300.0000
9	288.4185	285.1041	300.0000
10	200.5044	130	130.0000
11	289.2551	318.0878	94.0000
12	243.7934	94	94.0000
13	304.4608	216.8874	394.2788
14	390.7212	484.0405941	394.2789
15	500	500	484.0380
16	353.3224	500	304.5192
17	313.0460	500	489.2799
18	421.2108	220	500.0000
19	495.8544	511.4687	421.5199
20	518.3697	550	511.2795
21	534.9080	523.2265	523.2804
22	519.7360	345.1678	523.2801
23	461.0149	523.2798	523.2798
24	532.9676	550	523.2799
25	532.8027	523.2365	523.2800
26	541.2884	522.6056	523.2799
27	80.9368	47	47.0000
28	112.6556	163.3979	190.0000
29	126.9149	169.6291	190.0000
30	158.8551	190	159.7338
31	199.9890	172.465	200.0000
32	172.3346	166.535	200.0000
33	90	90	164.8002
34	86.84495	65.63347	89.1143
35	57.10207	110	110.0000
36	72.98398	110	101.1978
37	500.4913	511.2403	511.2794
38	19.85508	18	18.0000
39	46.0001	46	46.0000
40	54	54	54.0000
Total Cost	138,155.7853	139,576.3965	137,208.4176

, where the results of the proposed method are compared with the MFO and MMFO solutions reported in [50], which achieve the minimum fuel cost.

Table 7. Comparison of total fuel cost and improvement for the 40-unit ELD system with wind-power integration.

Optimization Techniques	Minimum Fuel Cost	Improvement Achieved Against Other Algorithm (%)
FWOA-I [58]	149,598.9	8.28%
FWOA-II [58]	145,750.4	5.86%
FWOA-III [58]	142,413.4	3.65%
FWOA-IV [58]	141,195.7	2.82%
PWTED1 [58]	137,984.38	0.562%
COOT [24]	139,000.63	1.28%
GAEPSO [60]	146,035.00	6.04%
DWTED2 [59]	154,993.00	11.47%
EMA [60]	144,356.00	4.95%
PWTED2 [54]	156,878.97	12.53%
Best Compromise [59]	143,587.90	4.44%
PSO [60]	142,068.00	3.42%
MFO [50]	139,576.3965	1.69%
MMFO [50]	138,155.7853	0.68%
FGOA-SQP	137,208.4176

compares the total operating cost achieved by the proposed FGOA-SQP method with those obtained using alternative optimization techniques for Case 3, highlighting fuel-cost improvements ranging from 0.69% to 12.55%. The results confirm that the bi-objective dispatch problem has strong performance with FGOA-SQP, showing a significantly reduced total cost while maintaining convergence behavior reliably. It follows that hybrid methods produce efficient and dependable solutions consistently, out-performing conventional algorithms in accuracy and speed.

Figure 6 indicates the total cost obtained by FGOA-SQP on a total of 100 runs for the 40-unit system. Every point shown on the figure represents a different run with varying initial conditions. The small range of fluctuation establishes that the technique is robust with respect to convergence to high-quality solutions on the large-scale ELD problem. Across these 100 independent runs, the proposed FGOA-SQP achieves a standard deviation of 1004.4614 USD/h, quantitatively confirming its robustness and stable convergence behavior under wind power uncertainty. For a population size of 3500 search agents and 500 iterations, the proposed FGOA-SQP completes execution in 2.31 s with 7970 function evaluations, indicating that the fractional-order memory term does not introduce a significant computational burden. A sensitivity sweep over fractional orders δ ∈ [0.1–1.0] confirms that δ = 0.3 maintains stable convergence and cost consistency, while fairness of comparison is preserved by using identical system configuration, constraints, and cost coefficients across all benchmarked algorithms.

The statistical analysis in Figure 7 illustrates the performance consistency of the FGOA-SQP algorithm for the 40-unit system. The histogram (left) displays the total-cost distribution over 100 independent runs, showing that most solutions concentrate around the lower-cost region. The box plot (center) reinforces this observation, exhibiting a narrow interquartile range and minimal dispersion, which indicates low variability in the obtained results. The empirical cumulative distribution function (right) further demonstrates that a high proportion of runs converge to near-optimal cost values, confirming the robustness and reliability of the proposed FGOA-SQP approach.

5. Conclusions

This study introduced a novel hybrid optimization framework, FGOA-SQP, which combines the global search capability of the FGOA with the high-precision local refinement of SQP. The proposed approach was rigorously evaluated on small, medium, and large-scale ELD test systems, including VPLEs and integrated wind-power units. In all cases, the proposed FGOA-SQP showed lower total generation costs, better convergence characteristics, and higher stability than many established methods. Specifically, the proposed approach achieved fuel-cost reductions ranging from 0.047% to 0.71% for the small-scale system, 0.31% to 27.25% for the medium-scale system, and 0.69% to 12.55% for the large-scale system when compared with state-of-the-art algorithms. These results reveal that the major improvement of the exploration behavior is due to the fractional order dynamics while the SQP stage ensures high-quality solutions will be converged fast and precisely. In addition, statistical evaluation over 100 independent runs for the large-scale system yields a standard deviation of 1004.4614 USD/h, quantitatively confirming the robustness and solution stability of the proposed FGOA-SQP under wind power uncertainty. Further confirmation on the robustness and repeatability of the proposed method was proven by the statistical analyses, and especially in big-scale systems where uncertainty in renewable energy creates more complexity. Overall, the proposed FGOA-SQP method offers an efficient, reliable, and scalable solution for modern ELD problems. Its strong performance in handling nonlinear, nonconvex, and non-smooth characteristics together with renewable integration positions it as a promising tool for real-world power-system operation and future intelligent energy-management systems. Although the SQP component involves matrix-based operations that may increase computational cost for very large-scale systems, its use is confined to local refinement, which mitigates potential scalability bottlenecks. The proposed FGOA-SQP framework is limited to stochastic steady-state ELD with fixed load demand and does not explicitly consider time-varying loads, transmission uncertainties, or real-time market constraints. Future research may extend the proposed framework to multi-objective ELD formulations, higher renewable penetration scenarios, real-time dispatch applications under increased uncertainty, and emerging paradigms such as flexible loads, computing power migration, and option-value-based decision-making in power systems [38,39].