Optimal forecasting of wind and solar PV power generation using enhanced HHO algorithm

Abstract

Accurate forecasting of renewable energy is crucial for grid stability, efficient scheduling, and reliable power system operation. However, photovoltaic (PV) and wind systems are highly nonlinear and intermittent due to changing environmental conditions, making prediction challenging. This study proposes an optimization-driven framework for PV parameter estimation and wind power curve modeling using real-world datasets, including data from the 226.8 MW Anantapur Wind Farm and experimental PV measurements. It highlights that the effectiveness of metaheuristic algorithms is problem-specific, with performance varying based on model and data characteristics. A comparative analysis using metrics such as RMSE, convergence behavior, and reliability shows that Leader Harris Hawks Optimization performs best for PV parameter estimation, while Harris Hawks Optimization is superior for wind modeling. The framework improves prediction accuracy and provides a scalable solution for integrating renewable energy into modern power systems.

Keywords

PV forecasting wind energy estimation metaheuristics optimization

Introduction

Background

Concerns about global climate change, rising greenhouse gas emissions, and the depletion of fossil fuel resources have accelerated the adoption of renewable energy technology in recent years. Among other renewable energy sources, photovoltaic and wind energy systems have attracted a lot of attention due to their technological maturity, environmental friendliness, and continually declining installation costs. Their extensive integration into current power grids is being further encouraged by government incentives, global environmental policies, and improved power electronic interfaces.

The effective use of wind and photovoltaic resources faces substantial challenges despite their apparent advantages. Both sources are sporadic by nature and depend on erratic natural events. While solar energy generation varies with sun irradiance, temperature, and seasonal oscillations, wind turbines rely on stochastic wind speeds driven by weather patterns and terrain, which vary considerably across regions such as the wind-rich districts of Andhra Pradesh (Satyanarayana et al., 2019). When large quantities of such erratic energy sources are introduced into the system, it becomes more challenging to maintain frequency stability, voltage regulation, and dependable power dispatch. Accurate short-term forecasting of renewable output has become essential for grid operators, distribution system planners, and energy storage schedulers. Conventional physical and statistical forecasting models are computationally efficient, but they often fail to capture highly nonlinear behaviors in weather-driven generation scenarios. The efficiency of machine-learning techniques is mostly dependent on model generalization and suitable parameter tweaking, notwithstanding their enhanced predictive capacity. In recent years, forecasting frameworks based on model optimization have demonstrated significant potential for improving prediction accuracy, convergence speed, and resilience to uncertainties. Metaheuristic optimization techniques are highly beneficial because they can explore multi-dimensional search spaces without requiring gradient information.

Advanced population-based optimization methods such as Particle Swarm Optimization, Grey Wolf Optimizer, and Harris Hawks Optimization have demonstrated potential for parameter estimation, feature selection, and meta-model calibration (Heidari et al., 2019; Kennedy and Eberhart, 1995; Mirjalili et al., 2014). These techniques increase the precision of PV and wind power prediction models by lowering error metrics like Root Mean Square Error (RMSE). Furthermore, energy efficiency, optimal load planning, and smart grid decision-making are enhanced by hybrid frameworks that integrate optimization techniques with data-driven forecasting models.

Literature review

For precise forecasting of renewable output, especially for PV and wind, machine-learning, hybrid, and optimizer-assisted ensembles have replaced deterministic and conventional statistical approaches. Simultaneously, multi-horizon, multi-asset, and probabilistic formulations have surfaced (Inman et al., 2013; Zendehboudi et al., 2018; Zhang et al., 2014). Metaheuristic tuning and ensemble approaches are key components of contemporary systems for both sources (Mohsin et al., 2022; Ren et al., 2015; Wang et al., 2022).

Traditional and statistical methods

Early research focused on univariate and multivariate time-series models such as persistence, AR(M)A, ARIMA, exponential smoothing, and state-space Kalman techniques, as well as physical (deterministic) models including NWP downscaling and device-level physics (Hill et al., 2012; Inman et al., 2013; Tascikaraoglu and Uzunoglu, 2014). These baselines remain useful but deteriorate under non-stationarity, regime changes, and mesoscale variability typical of wind shear, wakes, and cloud transients. Physical models require dense, high-quality inputs and thorough bias correction, while purely statistical models struggle with abrupt ramps and nonlinear interactions among drivers such as irradiance, temperature, and wind coupling (Inman et al., 2013; Ren et al., 2015; Tascikaraoglu and Uzunoglu, 2014; Wang et al., 2022). For PV systems, clear-sky transformers and NWP-guided ARIMA with Kalman filters are commonly applied; for wind, regime-stratified ARIMA and vector-autoregressive approaches are helpful but still underperform during ramp events and cut-in/cut-out regimes (Foley et al., 2012; Gallego-Castillo et al., 2015; Hill et al., 2012; Inman et al., 2013; Lydia et al., 2014; Tascikaraoglu and Uzunoglu, 2014; Voyant et al., 2017; Wang et al., 2016).

Forecasting using machine learning

Machine-learning models, including MLP/ANNs, SVR, ELM, tree ensembles, and deep RNN/LSTM/GRU architectures, have become widely adopted for both wind and PV forecasting due to their ability to capture nonlinearities and complex interactions (Ahmed et al., 2020; Jung and Broadwater, 2014; Ren et al., 2015; Soman et al., 2010; Wang et al., 2022). For wind applications, ANN family models are among the most frequently used, mapping wind speed, direction, turbulence, atmospheric stability, and temperature to power output. For PV systems, analogous models relate GHI/DNI, ambient and module temperature, wind speed, and sky conditions to DC/AC output (Ahmed et al., 2020; Jung and Broadwater, 2014; Mellit and Kalogirou, 2008; Pedro and Coimbra, 2012; Ren et al., 2015; Soman et al., 2010; Wang et al., 2022). When combined with bias-corrected NWP inputs and exogenous features such as cloud motion vectors and sky imager cues, deep learning architectures including CNN, LSTM, and attention-based models further improve forecasting skill (Ahmed et al., 2020; Jung and Broadwater, 2014; Zendehboudi et al., 2018). However, ML and deep learning approaches can be data-hungry, sensitive to hyperparameter choices, and prone to overfitting and covariate shift unless carefully regularized and validated on independent test sets (Ahmed et al., 2020; Jung and Broadwater, 2014; Mohsin et al., 2022; Ren et al., 2015).

Hybrid and optimization-assisted models

Recognizing complementary strengths, hybrids combine physical/statistical predictors with ML learners or fuse heterogeneous experts via stacking/bagging/boosting (Inman et al., 2013; Ren et al., 2015; Wang et al., 2022). Metaheuristics (GA, PSO, GWO, WOA, HHO, etc.) are widely used for feature subset selection and hyperparameter search and parameter estimation (e.g., PV I-V model fitting), improving convergence and accuracy for non-convex objectives (Holland, 1992; Kennedy and Eberhart, 1995; Wang et al., 2022; Zendehboudi et al., 2018). For PV, swarm-trained neural ensembles and optimizer-tuned regressors often outperform standalone learners in day-ahead horizons; for wind, optimizer-tuned hybrid ANN/SVR/LSTM models reduce RMSE and improve ramp detection (Jung and Broadwater, 2014; Ren et al., 2015; Wang et al., 2022). A representative example uses harmony-search optimized ANN for both PV irradiance and wind speed, reporting improved MSE/RMSE against pure ANN/SVR baselines (Mohsin et al., 2022).

Metaheuristics in renewable forecasting

PSO/GA remain popular for their simplicity and robust performance across search landscapes, while newer nature-inspired algorithms (GWO, WOA, HHO, and variants) can better balance exploration-exploitation and escape local minima on high-dimensional surfaces (Heidari et al., 2019; Jung and Broadwater, 2014; Mirjalili et al., 2014; Mirjalili and Lewis, 2016; Ren et al., 2015; Wang et al., 2022; Zendehboudi et al., 2018). In wind applications, metaheuristics tune deep architectures (e.g., LSTM weights/structure and SVR kernels) and select multi-site spatiotemporal features; in PV, they accelerate and stabilize parameter extraction and ensemble fusion (Mohsin et al., 2022; Ren et al., 2015; Wang et al., 2022; Zendehboudi et al., 2018). Comparative studies generally show hybrid “ML + metaheuristic” models outperform pure ML or pure statistical baselines on RMSE/MAE/MAPE and reliability metrics (e.g., CRPS/coverage for probabilistic forecasts) (Jung and Broadwater, 2014; Ren et al., 2015; Wang et al., 2016, 2022).

Cross-cutting advances and gaps

Two methodological advances stand out:

• Ensemble design: Competitive (data/parameter diversity) and cooperative (pre/post-processing decomposition) ensembles systematically improve skill for wind speed/power and solar irradiance/power forecasts across horizons (Ren et al., 2015).

• NWP correction: Bias-correction and spatiotemporal downscaling of NWP (GFS/ECMWF, WRF) via ML (e.g., RF/GBM/LSTM) or filters (e.g., Kalman and PCA-DBN) are critical, given cubic wind-speed-power sensitivity and irradiance-power linearity (Wang et al., 2022).

Despite progress, gaps persist: (i) few unified frameworks compare PV and wind under identical data/metrics; (ii) algorithm selection and metaheuristic hyper parameterization remain ad hoc; (iii) computational cost and real-time deploy ability (latency and memory) are under-reported; (iv) the interaction between optimizer dynamics (exploration/exploitation) and model architecture (e.g., depth and attention) is not systematically mapped; and (v) integrated multi-asset (wind PV and often load) forecasting with shared uncertainty quantification is still limited (Ahmed et al., 2020; Jung and Broadwater, 2014; Ren et al., 2015; Wang et al., 2022; Zendehboudi et al., 2018). These motivate a single-study comparison across multiple metaheuristics for both sources, with emphasis on RMSE/CRPS, convergence profiles, and robustness.

Recent advances in optimization-based renewable forecasting

Recent studies have shown a growing interest in optimization-driven approaches for renewable energy forecasting, particularly for photovoltaic and wind systems. Advanced metaheuristic techniques and their improved variants have been widely applied for parameter estimation and model tuning, demonstrating improved accuracy and reduced RMSE under nonlinear operating conditions (Heidari et al., 2019; Mirjalili et al., 2014; Mirjalili and Lewis, 2016). Several recent works emphasize hybrid optimization frameworks that enhance convergence stability and robustness compared to conventional methods, especially in wind power curve modeling and PV parameter extraction (Antonanzas et al., 2016; Mohsin et al., 2022). Although deep learning-based approaches are increasingly explored, their dependence on large datasets and computational complexity limits their practical applicability in certain scenarios (Ahmed et al., 2020; Aslam et al., 2021). In contrast, optimization-based techniques continue to offer a computationally efficient and reliable alternative, particularly for limited or real-time data environments, thereby reinforcing the motivation for the present study.

Contributions

As sustainable and clean alternatives to fossil fuels, photovoltaic (PV) and wind energy technologies have become significant contributors to renewable power generation. Every source has different characteristics. For instance, wind energy relies on changes in wind speed, whereas photovoltaic systems rely on solar radiation. Despite their independence, both exhibit nonlinear and intermittent behaviors that are influenced by environmental influences, making accurate power prediction a difficult problem. Accurate forecasting of these renewable sources is essential for efficient load management, grid stability, and optimal energy consumption.

Recently, researchers have focused more on mathematical modeling and optimization-based methods to increase the precision and reliability of forecasts for renewable energy. Metaheuristic optimization algorithms have become a powerful alternative to classic deterministic models since real-world data often involves nonlinearities and changing conditions. These algorithms, which draw inspiration from natural processes like evolution, swarming, or predation, can efficiently navigate challenging search regions and identify optimal or nearly ideal answers with less computational effort. Unlike gradient-based methods, which rely on derivative information, metaheuristics are highly adaptable in solving non-convex and multi-dimensional optimization problems that are commonly seen in PV and wind parameter estimates.

Algorithms such as Genetic Algorithm, Particle Swarm Optimization, Whale Optimization Algorithm, Grey Wolf Optimizer, and Harris Hawks Optimization have been widely used to minimize objective functions like Root Mean Square Error in order to increase the accuracy and convergence efficiency of predicted outputs.

Recent research have shown that the Leader Harris Hawks Optimization technique performs better for PV parameter estimation, providing greater accuracy and faster convergence. Similarly, the strong global search behavior and adaptive exploration-exploitation balance of the fundamental Harris Hawks Optimization algorithm make it an excellent tool for optimizing wind power models. Applying these two methods separately to PV and wind forecasts ensures better estimation accuracy and system modeling efficiency, all of which are necessary for robust renewable energy analysis.

This work primarily focuses on following contributions summarized in below given points:

(1) creation of independent forecasting frameworks based on optimization for wind and photovoltaic systems inside a single hybrid study.

(2) The best metaheuristic algorithms for PV and wind forecasting, respectively, were determined by comparing and implementing several of them.

(3) RMSE performance, convergence behavior, and overall forecast dependability for both renewable sources are highlighted in this thorough examination.

Problem formulation

Single diode photovoltaic system

The Single Diode Model (SDM), whose equivalent circuit is shown in Figure 1, accurately depicts the electrical behavior of a photovoltaic (PV) cell or module given in equation (1) (Mellit and Kalogirou, 2008; Pedro and Coimbra, 2012).

Figure 1.

Equivalent circuit diagram of the Single Diode Model (SDM) for a photovoltaic (PV) cell.

The circuit includes a current source representing the photo-generated current, a diode with current following the Shockley diode equation (2), a shunt resistance with current (Equation (3)), and a series resistance.

Applying Kirchhoff’s current law at the output node yields the fundamental current balance equation of the SDM.

I = I_{L} - I_{D} - I_{s h}

(1)

The diode current component follows the Shockley diode equation, given explicitly in equation (2).

I_{D} = I_{s} (\exp (\frac{q (V + I R_{s})}{n k T}) - 1)

(2)

The shunt current is represented by equation (3).

I_{s h} = \frac{V + I R_{s}}{R_{s h}}

(3)

In equation (4), a number of parameters are given such as the diode saturation current, the diode ideality factor, electronic charge, Boltzmann constant, cell temperature in Kelvin, and series and shunt resistances.

Combining these elements, the implicit nonlinear equation for the terminal current is given in equation (4):

I = I_{L} - I_{s} (\exp (\frac{q (V + I R_{s})}{n k T}) - 1) - \frac{V + I R_{s}}{R_{s h}}

(4)

The unknown parameters ${I_{L}, I_{s}, R_{s}, R_{s h}, n}$ in equation (4) must be accurately estimated to reliably simulate PV behavior (Jordehi, 2016) Parameter estimation is formulated as minimizing the root mean square error (RMSE) between measured currents and calculated currents over all data points in equation (5).

RMSE = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(I_{meas, i} - I_{calc, i})}^{2}}

(5)

The Single Diode Model captures important nonlinearities including recombination and resistive losses, offering a well-known and physically significant approximation of PV cells that is necessary for precise simulation and system design.

Logistic function model for wind

The logistic curve model is an excellent tool for demonstrating the nonlinear relationship between wind speed and the power output of existing wind turbines. The logistic function successfully illustrates the naturally sigmoidal version of this relationship due to its mathematical characteristics and link to turbine performance. Because the logistic function is continuous and differentiable, it can smoothly transition from a zero output at very low wind speeds to a saturation point at the rated power at higher wind speeds, reflecting the operational reality of turbine aerodynamics, mechanics, and control systems. This model is essential for the operational and design optimization of the wind energy sector because it shows how power output progressively increases with wind speed before plateauing owing to safety measures and physical constraints.

When calculating the active power production of wind turbines, the logistic curve model is crucial for illustrating the S-shaped relationship between wind speed and power generation.

The basic mathematical formula is provided in equation (6) (Foley et al., 2012; Lydia et al., 2014).

P (v) = \frac{Pmax}{1 + \exp [- C \cdot (ν - d)]}

(6)

Here, P(v) denotes the predicted power output at wind speed v, P_max represents the turbine’s rated maximum power capacity, C is the steepness parameter that controls how quickly the power approaches the rated output as the wind speed increases, and d indicates the midpoint wind speed where the power output reaches half of its rated value.

The logistic function is used within the limitations imposed by the turbine’s cut-in and cut-out wind speeds to guarantee accurate modeling that complies with turbine operating regulations, identified as $v_{c i}$ and $v_{c o}$ , respectively, as follows: the turbine starts generating power only after the wind speed exceeds $v_{c i}$ , and it stops operating beyond $v_{c o}$ for protection.

This operational behavior is mathematically expressed as

\tilde{P} (v) = {\begin{cases} 0, {v < v}_{ci} \\ \frac{Pmax}{1 + \exp [- C \cdot (ν - d)]}, v_{ci} \leq {v \leq v}_{co} \\ 0, {v > v}_{co} \end{cases}

(7)

This improves the model’s physical realism and sets precise limits with the permitted power output. This increases the physical realism of the model and establishes exact restrictions on the amount of power that can be produced. The phases of regulated shutdown, active power generation, and gradual start-up that are typical of operating turbines within these parameters are accurately captured by the logistic model.

For analytical convenience and parameter estimation, the normalized logistic power curve is commonly utilized. It is expressed as

\tilde{P} (v) = \frac{1}{1 + \exp [- c \cdot (v - d)]}

(8)

As a result, the ramp-up curve is scaled from zero to one, independent of turbine rating. Two uses for this normalized form are fitting empirical data across turbine sizes and developing generic models that might be applied at the fleet or regional levels. By optimizing the parameters c and d through nonlinear regression techniques using historical wind speed and power output data, the model can be closely aligned with the observed behaviors, thereby improving the prediction accuracy for both performance assessment and forecasting.

One of the main features of the logistic model is its ability to be differentiable. Examining the power ramp rate the speed at which a turbine’s power output varies in response to changes in wind speed becomes simpler as a result.

Equation (9) is used to differentiate the logistic function.

\frac{dP}{dv} = \frac{c P_{\max} \exp [- c (v - d)]}{{[1 + \exp [- c (v - d)]]}^{2}}

(9)

The max slope is reached at the midpoint wind speed $v = d$ , as expressed in equation (10).

{\frac{dP}{dv} |}_{v = d} = \frac{c P_{\max}}{4}

(10)

This slope measures the turbine’s responsiveness to wind speed fluctuations near d, providing crucial insights for designing control systems, analyzing grid stability, and forecasting power in real-time wind farm operations. A steeper slope affects how grid operators and turbine controllers regulate variability and guarantee dependability since it shows a quick rise or fall in power with slight wind fluctuations.

Beyond its formal formulation, the logistic curve model’s practical utility is demonstrated by its extensive use in wind resource assessments, turbine certification, and grid integration studies (Lydia et al., 2015; Sohoni et al., 2016). It combines theoretical modeling with empirical fitting to ensure that power forecasts accurately reflect actual turbine performance and adapt to site-specific wind conditions. Additionally, the model allows system designers to optimize turbine performance parameters, assess the expected energy production, and standardize interconnection studies in accordance with regulatory frameworks.

Power curve modeling for wind

The physical power curve model provides a fundamental explanation of the active power output behavior of renewable energy generators, such as wind turbines and solar photovoltaic systems, dependent on environmental input variables, such as wind speed or irradiance. This model provides a physically grounded and continuous mapping between the incident resource and the electrical power generated, reflecting the limitations imposed by the system design, control strategies, and operational constraints given in equation (11).

Similar to the logistic model, the physical power curve shows the change from negligible power at low resource levels to rated power output at higher inputs. It does this by incorporating the natural saturation characteristics that prevent the output from exceeding the equipment ratings and ensure safe and reliable operation. The following basic equation can be used to mathematically express a wind turbine’s physical power production in the aerodynamic regime:

P = \frac{1}{2} ρ C_{p} A v^{3}

(11)

Here, $P$ is the mechanical power extracted from the wind, $ρ$ is the air density, $C_{p}$ is the power coefficient representing the efficiency of converting wind energy into mechanical energy, $A$ is the rotor blade-swept area, and $v$ is the wind speed. This cubic relationship demonstrates that the power available in the wind increases dramatically with wind speed, which results in the steep ramp-up characteristic observed in turbine power curves at low to moderate wind speeds (Manwell et al., 2009).

To represent the actual turbine operation, the physical power curve is often modeled as a piecewise function:

P (v) = {\begin{cases} 0, {v < v}_{c i} \\ f (v), v_{c i} \leq {v \leq v}_{r} \\ P m a x, v_{r} {< v \leq v}_{c o} \\ 0, {v > v}_{c o} \end{cases}

(12)

In this context, $v$ represents the wind speed, $v_{c i}$ is the cut-in wind speed below which no power is produced, $v_{r}$ is the rated wind speed at which the turbine reaches its peak power output $P_{\max}$ , and $v_{c o}$ is the cut-out wind speed at which the turbine ceases operation for safety reasons. The function $f (v)$ describes the nonlinear rise in power from the cut-in to the rated wind speed, often derived from the aerodynamic power equation and adjusted with empirical or manufacturer-provided data to reflect the mechanical and control system efficiencies in equation (12).

A similar physical power curve in solar photovoltaic systems connects sun irradiance to power output; the curve’s characteristics are influenced by other factors like temperature effects and inverter restrictions. The major characteristic is still a nonlinear rise in output power with increasing input resources up to the rated capacity, after which the output levels out, even with different physical inputs.

The physical power curve is important because it may combine turbine design, environmental physics, and control constraints into a single descriptive model. System planning, grid integration studies, and accurate performance modeling are made easier with this approach. Analyzing the derivative of the power curve, particularly the slope $\frac{d P}{d v}$ within the ramp region, offers insights into control systems and stability analysis, showing how sensitively the power output reacts to small changes in the resource availability.

Mathematical formulation of the optimization problem

To ensure clarity and reproducibility of the proposed framework, the forecasting task is formulated as a parameter estimation problem for both photovoltaic and wind power models. The objective is to determine a set of optimal model parameters that closely match the measured data obtained from real-world observations.

In the case of the photovoltaic system, the Single Diode Model represents the nonlinear current–voltage behavior of the PV cell. The model parameters, including diode characteristics and resistive components, are not directly measurable and must be estimated. Similarly, for the wind energy system, the relationship between wind speed and power output is modeled using a logistic function and a physical power curve, both of which depend on a set of unknown parameters.

To evaluate the accuracy of these models, an objective function is defined based on the Root Mean Square Error (RMSE), which quantifies the difference between measured values and model predictions (Calasan et al., 2020). A lower RMSE indicates a better fit between the model and actual system behavior. Therefore, the optimization problem is defined as the minimization of RMSE over all available data samples.

The parameter estimation process is inherently nonlinear and involves multiple interacting variables, making it difficult to solve using conventional analytical or gradient-based optimization techniques. To overcome this challenge, metaheuristic optimization algorithms are employed (Li et al., 2021; Pillai and Rajasekar, 2018). These algorithms are well-suited for complex search spaces, as they do not require derivative information and are capable of escaping local minima through stochastic search mechanisms.

During the optimization process, each candidate solution represents a possible set of model parameters. These solutions are iteratively updated based on the search strategy of the respective algorithm, aiming to reduce the RMSE at each step. The process continues until a stopping criterion is satisfied, such as reaching the maximum number of iterations or achieving convergence.

This formulation ensures that the developed models accurately capture the nonlinear characteristics of both photovoltaic and wind power systems, while maintaining computational efficiency and robustness under varying operating conditions.

Proposed forecasting algorithm

Harris Hawks Optimization (HHO) is a nature-inspired, population-based computational intelligence technique that replicates the strategic group-hunting behavior observed in Harris’ Hawks (Heidari et al., 2019). In this framework, each individual in the population encodes a candidate solution to the problem, while the prey represents the current global best solution. A core feature governing the algorithm’s behavior is an energy scalar that models the prey’s capacity to evade capture, expressed as $E = 2 E_{0} (1 - \frac{t}{T_{\max}}) .$ As this energy diminishes with each iteration, the algorithm transitions progressively from wide-area global search to focused local convergence, enabling an effective balance between exploration and exploitation. The current study employs both the standard HHO for wind power curve fitting and an enhanced variant, the Leader Harris Hawks Optimization (LHHO), for photovoltaic parameter extraction. The performance of HHO and LHHO is further benchmarked against the Hippopotamus Optimization Algorithm (HOA) and the weighted-mean-based Improved Novel Flexible Optimization (INFO) algorithm (Ahmadianfar et al., 2022; Amiri et al., 2024). In LHHO, an elite-guided perturbation mechanism is integrated to enrich solution diversity and guard against premature stagnation. The algorithmic phases and their governing mathematical expressions are detailed in the subsections that follow.

Exploration phase (|E| ≥ 1)

When the normalized escape energy magnitude satisfies |E| ≥ 1, the algorithm operates in its global search mode. During this phase, individuals scan the solution space by selecting between two alternative repositioning strategies. Each agent chooses its movement pattern according to an adaptive probability threshold F^ad, which is derived from a positional relationship between the highest-performing and lowest-performing members of the current population. Specifically, when the random variable q falls below F^ad, a stochastic repositioning strategy is activated using population boundary information; otherwise, the agent adjusts its position based on the mean swarm location combined with a random offset drawn from an arbitrarily chosen member. This dual-path mechanism maintains population spread and prevents premature clustering in suboptimal regions. The positional update governing this exploratory behavior is formulated in equation (13):

(X_{r a b b i t} (t) - P_{m} (t)) - r_{3} (L B + r_{4} (U B - L B)) q < F_{a d}

(13)

Here, is the best solution found at iteration t, P(t) denotes the current population matrix, and P^e(t) is the mean position computed across all agents. The scalars r₃ and r₄ are uniformly distributed random values in [0, 1], while LB and UB represent the lower and upper bounds of the search domain, respectively. The binary variable q controls the branching between the two repositioning strategies, and Fm^e is the adaptive probability function calibrated from elite and weak population members. This formulation ensures diversified coverage of the search space throughout the early phases of optimization, preventing the algorithm from committing prematurely to any local region.

Exploitation phase (|E| < 1)

Once the escape energy falls below the threshold (|E| < 1), the algorithm switches to its local refinement mode. At this stage, agents converge upon the identified prey location using targeted encirclement maneuvers. The intensity and style of the attack are conditioned on both the remaining escape energy and an escape probability parameter J, which introduces stochasticity into the convergence trajectory. A key attack strategy, employed when the prey retains sufficient energy and escape probability exceeds 50%, involves a soft encirclement maneuver. In this configuration, agents update their positions by referencing the difference vector between the current population position and the best solution, scaled by both E and J to introduce controlled randomness. The governing update equation for this convergence behavior is given in equation (14):

P (t + 1) = Δ P (t) - E | J X_{r a b b i t} (t) - P (t) |

(14)

In this expression, P(t + 1) is the updated agent position, is the current best solution (prey location), ΔP(t) captures the positional gap between the prey and the current agent, E is the prey escape energy, and J = 2(1 − r₅) is a random leap factor governed by the stochastic variable r₅ ∈ [0, 1] (Heidari et al., 2019). As iterations advance and E continues to diminish, agents close in on the prey with increasing precision, enabling fine-grained local search. The interplay between E and J ensures that the convergence trajectory retains diversity, reducing sensitivity to local optima. In the LHHO variant, an additional elite-guided mutation step is applied at this stage, wherein a perturbation vector constructed from the two highest-ranked population members is used to generate a candidate update, accepted only if it improves upon the current position. This targeted mutation substantially broadens the effective search landscape during exploitation, yielding consistently superior convergence behavior over the standard HHO. The algorithm requires only two control parameters: population size (N_h = 50) and maximum iterations (Tᵐ^aᵋ = 100).

Overall Framework of the Proposed Methodology

The proposed optimization-based framework integrates photovoltaic parameter estimation and wind power curve modeling within a unified structure comprising Framework A and Framework B. In Framework A, experimental photovoltaic data are first acquired and preprocessed, after which the single diode model is employed to represent the nonlinear electrical behavior of the PV cell. Metaheuristic optimization algorithms are then used to estimate the five unknown parameters by minimizing the RMSE between measured and calculated current values, and the algorithms are compared based on accuracy and convergence to identify the most effective method. In Framework B, real-time wind speed–power data are collected and preprocessed, and both the logistic S-function model and the physical power curve model are utilized to capture the nonlinear wind speed–power relationship. Metaheuristic algorithms are applied to estimate optimal model parameters by minimizing RMSE between measured and predicted power outputs, with performance evaluated in terms of fitting accuracy and convergence behavior to select the best algorithm. Finally, the optimized models from both frameworks are validated by comparing predicted and measured characteristics, ensuring reliable and accurate renewable power forecasting, as illustrated in Figure 2.

Figure 2.

Overall framework of the proposed methodology.

Results and discussion

Solar PV forecasting analysis

The single diode model-based PV power forecasting process utilizes a variety of advanced metaheuristic parameter estimation algorithms to extract the most accurate model parameters for photovoltaic systems. Table 1 provides a detailed comparison of the Root Mean Square Error values achieved by each optimization method. Genetic Algorithm, Whale Optimization Algorithm, Harris Hawks Optimization, Leader Harris Hawks Optimization, Grey Wolf Optimizer, Improved Novel Flexible Optimization, and Particle Swarm Optimization are the algorithms represented by the letters A through G. According to the findings, the LHHO has the lowest RMSE of 0.000757, indicating that it is better at handling the nonlinear multimodal aspects of PV parameter estimation. While traditional algorithms like GA and GWO may suffer from local optima, INFO and HHO also show high accuracy with RMSE values near LHHO. This demonstrates the efficacy of hybrid and swarm-based metaheuristics for PV modeling; LHHO’s exceptional performance is attributed to its adaptive mutation method and strong exploration exploitation balance as depicted in Table 1. With the lowest RMSE of 0.000757, the Leader Harris Hawks Optimization approach is important for offering the most accurate parameter estimation in the single diode photovoltaic model. This demonstrates LHHO’s resilience in handling nonlinearities, attaining quick convergence, and avoiding local minima, highlighting its applicability for realistic PV forecasting applications and improving the reliability of renewable energy integration provided in Figure 3 and Table 1.

Table 1.

Parameters for single diode model.

Method	Best RMSE
LHHO (proposed)	0.000757
GA	0.000890
WOA	0.001062
HHO	0.000773
GWO	0.000873
INFO	0.000772
PSO	0.000773

Figure 3.

Estimated measured current & power curves of Leader Harris Hawks Optimization. (LHHO).

Current power curves for different algorithms are shown as estimated versus measured in Figures 4 and 5. GA, WOA, HHO, and GWO are compared in Figure 4, and the fits for INFO and PSO are displayed in Figure 5. The effectiveness of the suggested optimization strategy is confirmed by the LHHO, which shows visually closer alignment with measured data, supporting the quantitative superiority indicated in Table 1 with tighter fitting and lower error.

Figure 4.

Estimated measured current & power curves of GA, WOA, HHO and GWO.

Figure 5.

Estimated measured current & power curves of INFO and PSO.

Wind forecasting analysis

In this section, metaheuristic algorithms such as Harris Hawks Optimization, Hippopotamus Optimization Algorithm, Particle Swarm Optimization, and Genetic Algorithm are used to estimate the wind turbine power curve parameters using the logistic S-function model. With the lowest Root Mean Square Error of 0.0249, the HHO algorithm outperforms the other approaches in terms of fitting accuracy and convergence, as Table 2 illustrates.

Table 2.

Parameters for s-function model.

Technique	Pmax	c-parameter	d-parameter	Best RMSE	Actual RMSE (kW)
HHO [Proposed]	2495.64	0.556	8.826	0.0249	52.33
HOA	23808.31	0.608	8.484	0.0281	58.97
PSO	2310.00	0.608	8.488	0.0280	58.85
GA	2290.24	0.628	8.447	0.0290	90.97

The Harris Hawks Optimization (HHO) algorithm, shown in Figure (d), outperforms other metaheuristic methods in estimating the parameters of the logistic S-function model, with the lowest RMSE of 0.0249, followed by the Hippopotamus Optimization Algorithm (HOA), shown in Figure (c), with the next best RMSE of 0.0281, which is roughly 11.35% higher than that of HHO. The logistic power curve fits in Figures (a) and (b), respectively, show these differences, with HHO (Figure d) offering the closest alignment to the experimental wind turbine power output across all wind speed ranges. The improved convergence capabilities and accuracy of HHO in modeling the nonlinear relationship between wind speed and power generation are confirmed by this superior graphical and numerical performance, which greatly increases the efficacy of wind power forecasting. (Figure 6).

Figure 6.

Estimated wind turbine power curve-logistic fit of PSO, GA, HOA, and HHO.

As shown in Table 3, many metaheuristic optimization algorithms are systematically compared to validate the wind turbine power curve model under rated operating conditions. Important system parameters, including input and output voltages, rated power, power coefficient, and the associated RMSE values for each algorithm, are displayed in this table. The Genetic Algorithm (GA) has greater RMSE values than the others, which suggests that it is less accurate. On the other hand, the Hippopotamus Optimization Algorithm (HOA) and Harris Hawks Optimization (HHO) both attain the lowest RMSE of 0.1336. Regardless of the optimization technique used, all examined theoretical models consistently maintain a high-power coefficient close to 0.48, indicating sustained aerodynamic efficiency.

Table 3.

Parameters of theoretical power curve model.

Technique	V_in	V_r	V_out	P_r	C_P	Best RMSE	Actual RMSE (kW)
HHO [Proposed]	3.000	9.048	15.929	3200.00	0.480	0.1336	280.55
HOA	3.000	9.050	25.106	3200.00	0.480	0.1345	280.55
PSO	3.000	9.048	30.000	3200.00	0.480	0.1342	280.55
GA	3.018	9.008	18.211	3195.64	0.479	0.1346	282.74

The model fit quality for each algorithm is visually compared in Figure 7(a) through (d), where solid lines represent the projected logistic power curves and red circles represent measured wind power data. HHO provides the closest match with the fewest errors over the whole operational wind speed range, as shown in Figure 7. This graphic proof highlights the greater resilience and dependability of HHO in real-world wind power system modeling and confirms the quantitative results in Table 3. The graphical comparisons and numerical indicators closely match, confirming the usefulness of HHO for advanced forecasting applications as well as research.

Figure 7.

Estimated Wind Turbine Power Curve using PSO, GA, HOA and HHO.

In the comparison analysis, the suggested HHO method outperforms conventional metaheuristic algorithms with the lowest RMSE of 0.1336. In particular, compared to the HOA and GA, which have RMSE values of 0.1345 and 0.1346, respectively, HHO lowers the RMSE by roughly 0.67% and 0.74%. PSO achieves a substantially lower RMSE of 0.1342, but its practical usefulness is limited since its matching output voltage and system performance characteristics greatly differ from the anticipated operational levels. The HHO approach, on the other hand, ensures both low error and reliable power generation results by striking a balance between high precision and practical operating performance.

The graphical comparison of RMSE values across all tested algorithms is presented in Figures 8 –10. For PV parameter estimation (Figure 8), LHHO achieves the lowest RMSE of 0.000757, outperforming HHO (0.000773), INFO (0.000772), PSO (0.000773), GWO (0.000873), GA (0.000890), and WOA (0.001062). For wind power curve modeling using the S-function model (Figure 9), HHO achieves the best RMSE of 0.0249, compared to HOA (0.0281), PSO (0.0280), and GA (0.0290). Similarly, for the physical power curve model (Figure 10), HHO and HOA both achieve an RMSE of 0.1336 and 0.1345, respectively, maintaining superior accuracy over PSO and GA. These results confirm that algorithm effectiveness is problem-dependent: LHHO excels for PV due to its adaptive mutation mechanism, while HHO is best suited for wind power modeling owing to its balanced exploration-exploitation strategy.

Figure 8.

Rmse comparison of optimization algorithms for PV parameter estimation.

Figure 9.

Rmse comparison of optimization algorithms for wind S-function model.

Figure 10.

Rmse comparison of optimization algorithms for wind physical power curve model.

Sensitivity analysis of model parameters

To evaluate the robustness and reliability of the proposed forecasting framework, a sensitivity analysis was performed using the available hourly wind speed and power measurements. In real-world renewable energy systems, input data are inherently subject to uncertainty due to factors such as sensor inaccuracies, data acquisition noise, and short-term atmospheric variations. These uncertainties can influence model predictions and must be carefully examined to ensure the practical applicability of the proposed approach.

In this study, the sensitivity of the model was analyzed by introducing controlled perturbations to both the input wind speed data and key parameters of the logistic power curve model. Specifically, variations within a ±10% range were applied to the wind speed values as well as to the model parameters governing the shape and scaling of the power curve. For each perturbed case, the model was re-evaluated and the corresponding Root Mean Square Error (RMSE) was computed to quantify the impact on prediction accuracy, and the results are presented in Figure 11.

Figure 11.

Rmse under ±10% parameter perturbation (sensitivity analysis).

The results indicate that the variations in RMSE remain within a relatively narrow band, as shown in Figure 11, demonstrating that the proposed optimization-based framework exhibits stable performance under moderate levels of input uncertainty. Although slight deviations in wind speed and model parameters lead to minor changes in the predicted power output, the overall trend of the power curve and the convergence behavior of the optimization algorithms remain consistent across all test cases.

Furthermore, it was observed that the optimization algorithms are capable of adapting to variations in the input data, effectively recalibrating model parameters to maintain a close fit with the measured values. This adaptive behavior highlights the strength of metaheuristic optimization techniques in handling nonlinear and uncertain environments.

It is important to note that the present analysis is conducted using a short-duration dataset, representing practical field measurement conditions where long-term data may not always be available. Despite this limitation, the proposed method demonstrates reliable and consistent performance, indicating its suitability for preliminary analysis, real-time monitoring, and short-term forecasting applications.

Overall, the sensitivity analysis confirms that the proposed framework is robust, adaptable, and capable of maintaining prediction accuracy under realistic uncertainty conditions, as evidenced by Figure 11, thereby enhancing its applicability for renewable energy integration in modern power systems.

Limitations of the study

The proposed work has some limitations that should be noted. The analysis is based on short-duration wind and photovoltaic data, and therefore does not capture long-term seasonal variations. The model focuses on steady-state behavior and does not include dynamic effects such as rapid wind fluctuations or turbulence. In addition, the results are dependent on the available dataset and may require recalibration for different operating conditions or locations.

These limitations provide scope for future improvements and can be addressed in extended studies with larger datasets and advanced modeling approaches.

Conclusions

This study presents an optimization-based framework for photovoltaic parameter estimation and wind power curve modeling using real measured datasets, where multiple metaheuristic algorithms were evaluated based on Root Mean Square Error (RMSE) and convergence characteristics. The results show that the Leader Harris Hawks Optimization algorithm achieves the lowest RMSE of 0.000757 for photovoltaic modeling, while for wind energy systems, the Harris Hawks Optimization algorithm provides the best performance for both the logistic model and the physical power curve model, with RMSE values of 0.0249 and 0.1336, respectively. These findings demonstrate that the effectiveness of optimization algorithms is inherently problem-dependent, and selecting an appropriate algorithm significantly improves prediction accuracy and convergence behavior. The proposed framework also maintains stable performance under input variations and provides reliable results even with short-duration datasets, highlighting its suitability for practical applications. Overall, the study confirms that optimization-based approaches offer a computationally efficient and reliable solution for renewable energy forecasting, and future work will focus on extending the framework to long-term datasets and incorporating additional environmental factors to further enhance model performance.

Footnotes

Acknowledgments

The authors acknowledge the institutional support provided by the Department of Electrical and Electronics Engineering, Madanapalle Institute of Technology & Science, Madanapalle, India, and the Department of Electrical Engineering, Netaji Subhas University of Technology, Dwarka, New Delhi, India.

ORCID iDs

Vineet Kumar

Ethical considerations

This research does not involve human participants, animals, or any data collected from social media platforms. No ethical approval was required for this study.

Author contributions

Lokesh Vadla: Conceptualization, Methodology, Software, Writing – Original Draft. Vineet Kumar (MITS): Conceptualization, Methodology, Supervision, Writing – Review & Editing. Ramesh Kumar Ravi: Validation, Software, Data Curation. A V Pavan Kumar: Validation, Formal Analysis. Bhumireddy Poojitha Reddy: Investigation, Writing – Original Draft. Thupakula Pradeep: Investigation, Visualization. Vineet Kumar (NSUT): Supervision, Writing – Review & Editing.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

The data supporting the findings of this study are available from the corresponding author upon reasonable request.*

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