Hybrid Wavelet–Transformer–XGBoost Model Optimized by Chaotic Billiards for Global Irradiance Forecasting

Abstract

Accurate global irradiance (GI) forecasting is essential for improving photovoltaic (PV) energy management, stabilizing renewable power systems, and enabling intelligent control in solar-powered applications, including electric vehicles and smart grids. The highly stochastic and non-stationary nature of solar radiation, influenced by rapid atmospheric fluctuations and seasonal variability, makes short-term GI prediction a challenging task. To overcome these limitations, this work introduces a new hybrid forecasting architecture referred to as WTX–CBO, which integrates a Wavelet Transform (WT)-based decomposition module, an encoder–decoder Transformer model, and an XGBoost regressor, optimized using the Chaotic Billiards Optimizer (CBO) combined with the Adam optimization algorithm. In the proposed architecture, WT decomposes solar irradiance data into multi-scale components, capturing both high-frequency transients and long-term seasonal patterns. The Transformer module effectively models complex temporal and spatio-temporal dependencies, while XGBoost enhances nonlinear learning capability and mitigates overfitting. The CBO ensures efficient hyperparameter tuning and accelerated convergence, outperforming traditional meta-heuristics such as Particle Swarm Optimization (PSO) and Genetic Algorithms (GA). Comprehensive experiments conducted on real-world GI datasets from diverse climatic conditions demonstrate the outperformance of the proposed model. The WTX–CBO ensemble consistently outperformed benchmark models, including LSTM, SVR, standalone Transformer, and XGBoost, achieving improved accuracy, stability, and generalization capability. The proposed WTX–CBO framework is designed as a high-accuracy decision-support forecasting tool that provides short-term global irradiance predictions to enable intelligent energy management, predictive charging, and adaptive control strategies in solar-powered applications, including solar electric vehicles (SEVs), rather than performing end-to-end vehicle or photovoltaic power simulations. Overall, the proposed hybrid framework provides a robust and scalable solution for short-term global irradiance forecasting, supporting reliable PV integration, smart charging control, and sustainable energy management in next-generation solar systems.

1. Introduction

1.1. Background and Challenges

Solar-powered electric vehicles (SEVs) represent a promising pathway toward sustainable and autonomous transportation [1]. By integrating photovoltaic (PV) modules into vehicle surfaces, SEVs can harvest solar energy to extend the driving range [2], reduce grid reliance, and enable environmentally friendly mobility [3]. Efficient real-time solar irradiance forecasting is therefore essential to support predictive charging [4], adaptive energy routing [5], and intelligent battery management in SEVs. However, solar irradiance is highly intermittent and non-stationary due to dynamic meteorological conditions [6], including rapid cloud movements, aerosol variations, humidity fluctuations, and atmospheric turbulence [7]. These abrupt and multi-scale fluctuations introduce significant uncertainty in PV power generation [8], making short-term forecasting a critical yet challenging task for reliable SEV energy planning [9]. It is important to note that the focus of this work is on the accurate forecasting of global irradiance as a foundational input for SEV energy management systems rather than on the direct simulation of onboard photovoltaic generation or vehicle dynamics. In practical SEV applications, short-term irradiance forecasts are commonly obtained from stationary or gridded meteorological sources and subsequently integrated into higher-level decision-making layers for charging control, routing, and energy planning.

Accurate irradiance forecasting requires models capable of capturing

• Strong temporal variability and seasonality;
• Rapid short-term fluctuations caused by cloud cover;
• Multi-scale irradiance dynamics ranging from seconds to hours;
• Spatial heterogeneity across regions and driving trajectories.

Traditional models often struggle to maintain robustness under such conditions, especially in real-world SEV operating environments. For instance, during fast cloud passage events, solar irradiance may undergo abrupt drops or spikes within a short time interval, leading to highly non-stationary and multi-scale signal behavior. Conventional forecasting models such as CNN–LSTM or single-stage hybrid approaches often struggle in such scenarios, as their predictions tend to smooth rapid transitions or lag behind sudden changes. As a result, short-term forecasting errors increase significantly during these transient regimes, despite acceptable performance under stable clear-sky conditions. This practical limitation highlights the need for forecasting frameworks that can explicitly disentangle multi-scale dynamics and correct residual nonlinear patterns arising from rapidly evolving atmospheric conditions.

Numerical weather prediction (NWP) systems provide reliable long-term forecasts but lack fine spatial resolution and incur high computational cost, making them impractical for embedded SEV applications. Machine learning (ML) models such as support vector regression, decision trees, and multilayer perceptrons improve non-linear learning capability, yet they often fail under abrupt weather transitions and heterogeneous climatic regimes. Recent deep learning (DL) architectures, including CNNs, LSTMs, and Transformers, show strong temporal learning potential but face challenges related to noise sensitivity, large data requirements, and difficulty in representing multi-scale variations. Furthermore, hyperparameter tuning in such models is complex, and classical optimizers like particle swarm optimization (PSO) and genetic algorithms (GAs) may converge prematurely and fail to identify global optima.

Beyond these general limitations, recent studies have highlighted both the strengths and remaining weaknesses of state-of-the-art forecasting paradigms. Evaluations of the ECMWF Integrated Forecasting System (IFS) have shown that, while NWP models can provide physically consistent predictions with reasonable accuracy for global horizontal irradiance (GHI), their performance deteriorates in the presence of complex cloud dynamics and aerosol effects, particularly for short-term and high-resolution applications [10]. To mitigate these deficiencies, data-driven post-processing strategies such as artificial neural network (ANN)-based corrections have been proposed, demonstrating improved accuracy over raw NWP outputs and persistence baselines [11]. In parallel, hybrid machine learning frameworks combining weather classification with ensemble learners, such as CatBoost, have further enhanced forecasting robustness by adapting predictions to distinct atmospheric regimes [12]. Nevertheless, these approaches remain challenged by rapidly changing weather conditions, multi-scale non-stationarity, and the lack of coordinated mechanisms to jointly capture long-range temporal dependencies and structured residual errors.

Recent research has increasingly focused on hybrid and decomposition-enhanced learning frameworks for solar irradiance forecasting. Lopes et al. [10] evaluated ECMWF–IFS forecasts and reported satisfactory accuracy for clear-sky global horizontal irradiance (GHI), while noting significant error increases under cloudy and high-aerosol conditions. ANN-based approaches, such as those used by Yadav et al. [13] and Pereira et al. [11], demonstrated measurable reductions in MAE and RMSE when optimized meteorological inputs and statistical post-processing were employed. Wang et al. [14] conducted a multi-climate comparative study and showed that multilayer perceptron (MLP) and radial basis neural networks (RBNNs) reduced the RMSE by up to 15–20% compared to generalized regression neural networks (GRNNs). Olatomiwa et al. [15] enhanced support vector machine (SVM) performance through Firefly Algorithm optimization, achieving lower MAPE values relative to conventional SVM configurations. Gupta et al. [16] further advanced decomposition-driven modeling by integrating multivariate empirical mode decomposition with PCA–GRU architectures, reporting consistent RMSE reductions (approximately 10–18%) while decreasing computational complexity.

More recently, deep hybrid architectures have demonstrated improved temporal feature extraction capabilities. Ghimire et al. [17] reported that CNN–LSTM hybrids achieved lower RMSE and higher $R^2$ values than standalone CNN and LSTM models across multiple seasonal datasets. Ahmed et al. [12] showed that combining CatBoost with weather-type classification reduced GHI prediction errors by more than 12% in terms of MAE compared to single-model baselines. Cannizzaro et al. [18] validated variational mode decomposition (VMD)-based CNN hybrids, demonstrating systematic improvements in both short- and long-term forecasting horizons, with RMSE reductions ranging from 8% to 16% relative to non-decomposed deep models.

Despite these advances, challenges persist in fully capturing multi-scale variability, maintaining cross-region generalization, and avoiding local minima during model training.

Although progress has been made with existing hybrid models, as summarized in

Table 1. Summary of representative hybrid models for solar irradiance forecasting.

Model	Main Advantages	Main Limitations
CNN–LSTM	Captures local feature patterns and short-term temporal dependencies; widely adopted and computationally efficient	Limited ability to model long-range dependencies; prone to smoothing effects during abrupt irradiance transitions
VMD–CNN	Effective decomposition of non-stationary signals; improved noise robustness	Lacks explicit temporal dependency modeling; performance sensitive to decomposition parameters
WT–LSTM	Multi-scale signal representation; improved handling of non-stationarity	Sequential modeling may struggle with long-term dependencies and residual nonlinear errors
Transformer-based models	Strong capability for long-range temporal dependency modeling	Performance degrades under high-frequency fluctuations without explicit multi-scale preprocessing
Hybrid DL + ensemble models	Improved generalization through error correction	Often rely on parallel stacking without explicit residual hierarchy or coordinated optimization

, several critical challenges remain unresolved. First, most approaches address multi-scale non-stationarity and temporal dependency modeling in isolation rather than within a unified hierarchical framework [19]. Second, residual errors arising from rapidly changing meteorological conditions are often treated implicitly through ensemble averaging rather than being explicitly learned and corrected [18]. Third, hyperparameter optimization is typically performed independently for each model component, which limits coordinated learning across heterogeneous modules [20]. These challenges are difficult to resolve due to the intrinsic coupling between spectral variability, temporal dynamics, and nonlinear residual structures in solar irradiance signals, particularly under fast cloud transitions and seasonal variability. Addressing these issues motivates the development of a frequency-aware, attention-based, and jointly optimized forecasting framework such as the proposed WTX–CBO model. To address these gaps, effective forecasting solutions must integrate

• Multi-scale signal decomposition to capture diverse temporal patterns;
• Attention-based learning to model long-range dependencies;
• Ensemble refinement to mitigate residual non-linear errors;
• Advanced optimization strategies to efficiently tune hyperparameters.

Motivated by these requirements, this work proposes a hybrid Wavelet–Transformer–XGBoost architecture optimized by a Chaotic Billiards Optimizer (CBO), enabling accurate, stable, and generalizable solar irradiance forecasting for SEVs.

Accordingly, station- and city-level irradiance forecasting represents a realistic and widely adopted proxy for estimating route-level and onboard solar availability in SEV operational planning, particularly when combined with geographic information, temporal driving patterns, and predictive energy management strategies.

1.2. Main Contributions

The main contributions of this study are summarized as follows:

• A hierarchical hybrid forecasting framework, termed WTX–CBO, which integrates Discrete Wavelet Transform (DWT), an encoder–decoder Transformer, and XGBoost in a sequential residual learning architecture rather than a parallel or stacked combination.
• A frequency-aware attention mechanism is introduced by feeding multi-resolution wavelet subbands directly into the Transformer, enabling effective modeling of long-range temporal dependencies under highly non-stationary and rapidly fluctuating solar irradiance conditions.
• A two-stage temporal learning strategy in which the Transformer captures global and multi-scale dynamics, while XGBoost explicitly learns structured residual errors that persist after attention-based forecasting, improving generalization across diverse climatic regimes.
• A joint global optimization scheme based on the Chaotic Billiards Optimizer (CBO) is developed to simultaneously tune the hyperparameters of both Transformer and XGBoost components within a unified search space, mitigating premature convergence in high-dimensional hybrid models.
• The proposed framework is computationally efficient and deployment-oriented, achieving fast convergence and low-latency inference suitable for real-time decision-support in solar-powered and embedded energy systems, including solar electric vehicle applications.
• Extensive experiments conducted on real-world datasets from multiple climate regions demonstrate that WTX–CBO consistently outperforms state-of-the-art deep learning and hybrid forecasting models in terms of accuracy, robustness, and scalability.

The present paper is structured as follows: Section 2 introduces the proposed methodology. Section 3 provides experimental results and comparative analysis. Finally, Section 4 concludes the work and outlines future research directions.

2. Proposed Methodology

In this study, the target variable is the global horizontal irradiance (GHI), denoted as $G\left(t\right)$ and expressed in ${\mathrm{W/m}}^2$. The proposed WTX–CBO hybrid architecture is designed to enhance the accuracy, robustness, and scalability of short-term irradiance forecasting for solar electric vehicle (SEV) applications, providing reliable decision-support predictions that can subsequently be used as inputs to photovoltaic power models or SEV energy management systems. The model integrates three core modules: (i) the Data Preprocessing and Decomposition Module, (ii) the Forecasting Module, and (iii) the Optimization Module. Each module contributes a distinct functionality within the overall prediction pipeline—ranging from multi-scale feature extraction and attention-based temporal learning to adaptive hyperparameter optimization.

The end-to-end forecasting pipeline illustrated in Figure 1 presents the complete processing chain adopted in this study, beginning from raw meteorological and irradiance data acquisition through model output generation. The workflow clearly highlights the essential preparatory steps, including data cleaning, outlier removal, normalization, and feature selection, prior to time-series windowing. Subsequently, the training phase integrates feature extraction and model learning, while the testing phase evaluates generalization performance using unseen data. This comprehensive flow ensures data integrity, robust model development, and reliable performance assessment for global irradiance prediction.

The architecture of the proposed WTX-CBO hybrid model is shown in Figure 2, emphasizing the computational stages and information flow within the learning framework. Multi-scale signal decomposition using Discrete Wavelet Transform (DWT) generates a structured representation of irradiance fluctuations, which is subsequently processed by a Transformer encoder–decoder to capture long-range temporal dependencies. An XGBoost layer further refines the output by modeling residual error components, while the Chaotic Billiards Optimizer (CBO) dynamically tunes hyperparameters. This hierarchical structure ensures efficient learning of both short-term variability and long-term irradiance dynamics, contributing to superior predictive performance.

Mathematically, the model follows a three-stage workflow:

(i)

Decomposition and Normalization: The input global horizontal irradiance time series $G\left(t\right)$ is decomposed into multi-scale subcomponents $\{A_N\left(t\right),D_j\left(t\right)\}$ using Discrete Wavelet Transform (DWT):

$$G\left(t\right)=A_N\left(t\right)+\sum _{j=1}^N{D}_j\left(t\right),$$

(1)

where $A_N\left(t\right)$ and $D_j\left(t\right)$ denote the approximation (low-frequency) and detail (high-frequency) coefficients at decomposition level j, respectively.

(ii)

Deep Forecasting: The decomposed subseries are embedded and passed through the Transformer encoder–decoder:

$$\widehat{G}(t+1:t+k)={\mathcal{F}}_{\mathrm{Trans}}\left(E(A_N, D_1, \dots , D_N)\right)$$

(2)

where ${\mathcal{F}}_{\mathrm{Trans}}$ represents the multi-head attention and feed-forward sequence learning function.

(iii)

Refinement and Optimization: The Transformer’s output ${\widehat{G}I}_{\mathrm{Trans}}$ is refined by XGBoost regression:

$${\widehat{G}I}_{\mathrm{final}}=f_{\mathrm{XGB}}({\widehat{G}I}_{\mathrm{Trans}},X_{\mathrm{meteo}})$$

(3)

while the CBO algorithm adaptively tunes hyperparameters $\mathrm{\Theta }=\{\eta ,h,d_{\mathrm{model}}$, $\mathrm{dropout}\}$ to minimize the objective:

$$\mathcal{L}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{(G_i-{\widehat{G}I}_i)}^2$$

(4)

In the proposed WTX–CBO framework, the Transformer acts as the primary forecasting model, generating an initial prediction of global horizontal irradiance. XGBoost is subsequently employed exclusively as a residual learning module to correct the structured prediction errors produced by the Transformer. Unlike conventional ensemble or stacking approaches, XGBoost is not trained as an independent predictor of irradiance, but rather learns a mapping between the Transformer residuals and their corrected values. The input to the XGBoost model consists of the time-indexed residual sequence between the observed irradiance and the Transformer forecast, optionally augmented by the Transformer’s learned temporal representations, enabling targeted correction of nonlinear residual patterns.

This hybrid formulation ensures that both short-term irradiance dynamics and long-term GI generation patterns are captured efficiently. By combining multi-resolution signal processing, attention-based temporal modeling, and metaheuristic optimization, the proposed WTX–CBO system provides a highly adaptive framework suitable for real-time solar energy forecasting and SEV energy management.

2.1. Datasets Description

Four distinct datasets were created using the NASA Platform. The NASA POWER project aggregates data from various NASA research initiatives, aiming to provide easily accessible information for climate studies [21]. It offers solar and meteorological datasets from NASA research to aid in renewable energy development, enhance building energy efficiency, and support agricultural needs.

These four distinct datasets pertain to the following cities: Zurich, Switzerland [Latitude: 47.35, Longitude: 8.55]; Milan, Italy [Latitude: 45.47, Longitude: 9.18]; Vienna, Austria [Latitude: 48.21, Longitude: 16.37]; and Berlin, Germany [Latitude: 52.52, Longitude: 13.41]. The variables included in these datasets are radiance [W/m²]: the total solar irradiance on a horizontal surface at the Earth’s surface, including direct and diffuse components under all sky conditions; temperature [°C]: the average air temperature 2 m above the Earth’s surface in Celsius; humidity [g/kg]: the ratio of the mass of water vapor to the total mass of air 2 m above the Earth’s surface; precipitation [mm/h]: total atmospheric water vapor in a vertical column of the atmosphere; wind speed [m/s]: the average wind speed 10 m above the Earth’s surface; wind direction [°]: the average wind direction 10 m above the Earth’s surface; frost point [°C]: the dew/frost point temperature 2 m above the Earth’s surface; wet bulb temperature [°C]: the adiabatic saturation temperature measured using a thermometer covered with a water-soaked cloth over which air is passed at 2 m above the Earth’s surface; and surface pressure [kPa]: the average surface pressure at the Earth’s surface.

For each city, hourly meteorological and global horizontal irradiance data were collected over the period from January 2015 to December 2022. The resulting time series were partitioned using a time-ordered split, with the earliest 80% used for model training, the following 10% for validation, and the remaining 10% reserved for out-of-sample testing.

2.2. Data Preprocessing and Decomposition Module

The preprocessing stage ensures that the model receives consistent, denoised, and normalized input signals. The raw GI and meteorological data—typically sampled at $\Delta t=1$ h—include variables such as global horizontal irradiance (GHI), temperature, relative humidity, and wind speed. The workflow consists of four main steps: data cleaning, normalization, feature selection, and wavelet-based multi-scale decomposition [22].

2.2.1. Data Cleaning and Normalization

Missing or noisy samples are corrected using spline interpolation and a three-point moving average filter. Outliers are removed via the interquartile range (IQR) method to mitigate transient spikes caused by shading or cloud transitions. Each input attribute is rescaled to the interval $[0,1]$ using a min–max transformation:

$$z={\displaystyle \frac{u-u_{\mathrm{low}}}{u_{\mathrm{high}}-u_{\mathrm{low}}}}$$

(5)

where z represents the normalized feature. This normalization improves numerical stability and accelerates the convergence of the learning model [23].

2.2.2. Feature Selection

Both direct and contextual features are retained. Direct variables (global irradiance, air temperature, and wind speed) have an immediate influence on irradiance dynamics, while contextual variables (relative humidity and surface pressure) enhance model robustness across diverse climatic conditions. Pearson correlation and mutual information criteria are employed to identify the most influential predictors, which are subsequently concatenated into the feature vector $\mathbf{X}\left(t\right)\in {\mathbb{R}}^d$ [19].

2.2.3. Wavelet Decomposition

To capture multi-scale temporal behavior, the normalized signals are decomposed using Discrete Wavelet Transform (DWT):

$$\mathbf{X}\left(t\right)=A_N\left(t\right)+\sum _{j=1}^N{D}_j\left(t\right)$$

(6)

where $A_N\left(t\right)$ represents the low-frequency trend, and $D_j\left(t\right)$ captures high-frequency fluctuations due to cloud dynamics. The Daubechies (db4) wavelet with decomposition level $N=4$ is selected for its strong time–frequency localization and efficiency on meteorological time series [24]. The choice of the Daubechies db4 wavelet and the selected decomposition level is motivated by prior comparative studies on wavelet-based solar irradiance forecasting. Existing works have shown that moderate decomposition depths (typically three to five levels) provide an effective balance between capturing multi-scale temporal dynamics and avoiding excessive fragmentation of the signal, which may amplify noise and increase computational complexity [25]. In particular, the db4 wavelet has been widely adopted due to its compact support and favorable time–frequency localization properties, which are well suited for non-stationary irradiance signals [26]. Based on these findings, the adopted configuration was selected as a robust and computationally efficient compromise for short-term irradiance forecasting.

2.2.4. Input Structuring for Deep Learning

All decomposed sequences are reassembled into an input tensor $\mathcal{X}\in {\mathbb{R}}^{T\times d}$, where T denotes the look-back window. This tensorized input enables multi-channel embedding within the Transformer. The final dataset is partitioned into training, validation, and testing subsets using a strict chronological (walk-forward) strategy, with 80% of the data used for training, 10% for validation, and the remaining 10% reserved for testing [27].

This preprocessing and decomposition module improves the signal-to-noise ratio, preserves the physical interpretability of irradiance patterns, and generates multi-scale inputs that enhance the Transformer’s capacity to model temporal dependencies.

2.3. Forecasting Module

The forecasting module represents the core predictive stage of the proposed WTX–CBO framework. It integrates two complementary components—an encoder–decoder Transformer for long-range temporal learning and an XGBoost ensemble for residual refinement. This dual-stage architecture captures both global temporal dependencies and local nonlinear residual patterns in global horizontal irradiance time series, ensuring enhanced adaptability and forecasting accuracy under varying meteorological conditions [28].

2.3.1. Encoder–Decoder Transformer

The Transformer learns temporal correlations through a self-attention mechanism that processes complete sequences in parallel, avoiding the vanishing-gradient issues common in recurrent networks such as LSTM or BiLSTM. Given the decomposed multivariate input $\mathcal{X}\in {\mathbb{R}}^{T\times d}$, positional encoding is first applied to retain temporal order:

$$P{E}_{(pos,2i)}=sin\left({\displaystyle \frac{pos}{{\mathrm{10,000}}^{2i/d_{\mathrm{model}}}}}\right),$$

(7)

$$P{E}_{(pos,2i+1)}=cos\left({\displaystyle \frac{pos}{{\mathrm{10,000}}^{2i/d_{\mathrm{model}}}}}\right),$$

(8)

The encoded features are processed by stacked encoder layers composed of multi-head attention and feed-forward sublayers. The attention mechanism computes contextual correlations as

$$\mathrm{Attention}(Q,K,V)=\mathrm{softmax}\left({\displaystyle \frac{Q{K}^{\top }}{\sqrt{d_k}}}\right)V,$$

(9)

where Q, K, and V denote query, key, and value matrices, respectively, and $d_k$ is the key dimension. The encoder captures inter-feature dependencies, while the decoder predicts the future GI sequence $\widehat{G}I(t+1:t+k)$ conditioned on both encoder outputs and prior observations. This structure enables accurate multi-horizon forecasting with minimal information loss [29].

2.3.2. XGBoost Refinement Layer

Although the Transformer effectively models global dynamics, minor prediction residuals may persist due to nonlinear coupling between meteorological inputs [30]. To correct this, an Extreme Gradient Boosting (XGBoost) regression layer refines the Transformer output. XGBoost constructs an ensemble of decision trees minimizing the residual loss:

$${\mathcal{L}}^{\left(t\right)}=\sum _{i=1}^{n}l\left(y_i,{\widehat{y}}_i^{(t-1)}+f_t\left(x_i\right)\right)+\mathrm{\Omega }\left(f_t\right),$$

(10)

where $f_t$ is the t-th tree and $\mathrm{\Omega }\left(f_t\right)$ is a regularization term penalizing model complexity. The hybrid combination

$${\widehat{G}I}_{\mathrm{final}}={\widehat{G}I}_{\mathrm{Trans}}+{ϵ}_{\mathrm{XGB}},$$

(11)

with ${ϵ}_{\mathrm{XGB}}$ denoting the learned residual correction improves predictive stability and reduces variance across unseen climatic regimes [28].

2.3.3. Output Reconstruction

The final predicted GI sequence is reconstructed by aggregating the refined wavelet components:

$${\widehat{G}I}_{\mathrm{final}}\left(t\right)=A_N^{*}\left(t\right)+\sum _{j=1}^N{D}_j^{*}\left(t\right),$$

(12)

where $A_N^{*}\left(t\right)$ and $D_j^{*}\left(t\right)$ denote the reconstructed approximation and detail signals after refinement. This step ensures physical consistency of the forecast and restores the high-frequency fluctuations representing irradiance variability.

By coupling attention-based temporal learning with gradient-boosted residual correction, the forecasting module provides a robust and high-fidelity estimation of photovoltaic output suitable for intelligent energy routing and adaptive control in solar electric vehicles [31].

2.4. Optimization Module

To enhance convergence stability and predictive robustness, the WTX–CBO framework integrates a metaheuristic optimization strategy based on the Chaotic Billiards Optimizer (CBO). This optimizer, as depicted in Algorithm 1, mimics the motion of billiard balls within a chaotic environment, achieving a balanced trade-off between global exploration and local exploitation for efficient hyperparameter tuning [32]. In the proposed framework, CBO operates on a unified hyperparameter vector $\mathrm{\Theta }=\{{\mathrm{\Theta }}_T,{\mathrm{\Theta }}_X\}$, where ${\mathrm{\Theta }}_T$ and ${\mathrm{\Theta }}_X$ denote the Transformer and XGBoost hyperparameter subsets, respectively.

Algorithm 1 WTX–CBO: Wavelet–Transformer with XGBoost optimized by Chaotic Billiards (CBO)

Require:  Multivariate time series S (global irradiance $G\left(t\right)$ and meteorological covariates), lookback window L, forecast horizon H Ensure:  Predicted irradiance $\widehat{G}(t+H)$ and evaluation metrics (MAE, RMSE, MAPE, $R^2$)   1: Data preparation   2: Clean/align timestamps; impute missing values; optionally resample to uniform $\Delta t$   3: Min–Max scale features (fit on Train, reuse parameters for Val/Test)   4: Wavelet decomposition (WT)   5: for each feature $x\in \mathbf{S}$ do   6:    Apply DWT with wavelet w and level ℓ: obtain $\{A_{\ell }, D_{\ell }, \dots , D_1\}$   7:    Select subbands and stack ⇒ multi-scale tensor $\mathbf{X}$   8: end for   9: Sequence building 10: Build sliding windows of length L: input ${\mathbf{X}}_{t-L+1:t}$, target $y_{t+H}$ 11: Chronological split into Train/Val/Test (80/10/10) 12: Model components 13: Transformer encoder hyperparameters ${\mathrm{\Theta }}_T= \{$layers, heads, $d_{\mathrm{model}}$, $d_{ff}$, dropout, learning rate} 14: XGBoost regressor hyperparameters ${\mathrm{\Theta }}_X= \{$n_estimators, max_depth, learning rate, subsample, colsample_bytree} 15: CBO search space 16: Encode a candidate as $\mathrm{\Theta }=[{\theta }_T,{\theta }_X]$ within lower/upper bounds 17: CBO loop (outer hyperparameter optimization) 18: Initialize P billiard balls ${\left\{{\mathrm{\Theta }}^{\left(p\right)}\right\}}_{p=1}^P$ 19: for $k=1$ to K do                        ▹ CBO iterations 20:    for each ball p do 21:    Instantiate Transformer $\left({\theta }_T^{\left(p\right)}\right)$; XGBoost $\left({\theta }_X^{\left(p\right)}\right)$ 22:    Inner training (weights) 23:     Train Transformer on Train with Adam for E epochs (MSE loss) 24:     Extract features $\mathbf{z}$ (e.g., last hidden state or attention pooling) 25:     Fit XGBoost on $(\mathbf{z},y)$ using Train; validate on Val 26:    Compute fitness $f^{\left(p\right)}={\mathrm{RMSE}}_{\mathrm{Val}}$                 ▹ lower is better 27:    end for 28:    Update positions/velocities via chaotic billiards dynamics (collisions, reflections, pockets) 29:    Enforce bounds; keep elitist best ${\mathrm{\Theta }}^{\star }$ with minimal ${\mathrm{RMSE}}_{\mathrm{Val}}$ 30: end for 31: Final training 32: Re-instantiate Transformer $\left({\theta }_T^{\star }\right)$ and XGBoost $\left({\theta }_X^{\star }\right)$ 33: Train Transformer on Train ∪ Val for $E^{\star }$ epochs; extract ${\mathbf{z}}^{\star }$ 34: Fit XGBoost on $({\mathbf{z}}^{\star },y)$ using Train ∪ Val 35: Evaluation 36: Predict on Test: ${\widehat{y}}_{t+H}$ 37: Report MAE, RMSE, MAPE, $R^2$

2.4.1. Motivation for Metaheuristic Optimization

Deep and ensemble learning architectures rely heavily on sensitive hyperparameters such as the learning rate, embedding dimension, dropout ratio, and attention heads. Conventional optimizers like Adam or RMSProp are effective locally but can become trapped in suboptimal minima. Conversely, heuristic algorithms (e.g., PSO or GA) provide global search capabilities but may converge prematurely. The CBO algorithm mitigates these issues by applying chaotic motion equations that continuously perturb candidate solutions, enabling escape from local minima and faster convergence to global optima [20].

2.4.2. Mathematical Formulation

Each billiard agent represents a candidate solution ${\mathbf{X}}_i=[x_{i1}, x_{i2}, \dots , x_{id}]$ corresponding to a vector of model hyperparameters. Within a bounded search domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$, its motion follows:

$${\mathbf{X}}_i^{t+1}={\mathbf{X}}_i^t+\mu {\mathbf{v}}_i^t+\lambda \varphi \left({\mathbf{X}}_i^t\right),$$

(13)

where $\mu$ is the inertia factor, $\lambda$ is a chaotic scaling coefficient, and $\varphi (·)$ is a nonlinear chaotic map derived from billiard dynamics. Velocity updates are computed as

$${\mathbf{v}}_i^{t+1}=\omega {\mathbf{v}}_i^t+r_1({\mathbf{P}}_i-{\mathbf{X}}_i^t)+r_2(\mathbf{G}-{\mathbf{X}}_i^t),$$

(14)

where ${\mathbf{P}}_i$ and $\mathbf{G}$ denote personal and global best positions, $\omega$ is a damping coefficient, and $r_1$ and $r_2$ are random chaotic factors in $(0,1)$. The optimization objective is defined as

$$\underset{\mathrm{\Theta }}{min}\mathcal{L}\left(\mathrm{\Theta }\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^N{(G_i-{\widehat{G}I}_i)}^2+\alpha {\parallel \mathrm{\Theta }\parallel }_2^2,$$

(15)

where $\mathrm{\Theta }$ contains both trainable and meta parameters, and $\alpha$ is a regularization term controlling model complexity [20].

2.4.3. Integration with the Hybrid Framework

CBO operates jointly with the Adam optimizer: CBO performs global hyperparameter search, while Adam fine-tunes weights locally through gradient descent. At each iteration, updated hyperparameter sets ${\mathrm{\Theta }}^{*}$ are passed to the Transformer–XGBoost network and refined via backpropagation. The process terminates when the relative improvement of $\mathcal{L}\left(\mathrm{\Theta }\right)$ remains below $\epsilon$ for K consecutive iterations.

The CBO-driven optimization process achieves faster convergence, lower error variance, and superior adaptability under dynamic irradiance conditions. This hybrid optimization strategy ensures that the WTX–CBO model attains both computational efficiency and high forecasting precision for real-time solar energy management in SEVs.

2.5. Metrics and Hyperparameter Optimization

2.5.1. Evaluation Metrics

To rigorously assess the predictive capability of the proposed forecasting framework, a comprehensive suite of statistical indicators is employed. These metrics quantify deviation, error magnitude, variability explanation, and relative prediction accuracy. Let $G_i$ and ${\widehat{G}}_i$ denote the actual and predicted irradiance values, respectively, and let N represent the number of testing samples.

Mean Absolute Error (MAE) [22]

$$\mathrm{MAE}={\displaystyle \frac{1}{N}}\sum _{i=1}^N\left|G_i-{\widehat{G}}_i\right|$$

(16)

MAE measures the average magnitude of absolute forecasting errors, reflecting the typical deviation without emphasizing directionality.

Mean Squared Error (MSE) [22]

$$\mathrm{MSE}={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(G_i-{\widehat{G}}_i\right)}^2$$

(17)

MSE penalizes larger errors more heavily, making it a useful criterion for evaluating complex forecasting models.

Root Mean Squared Error (RMSE) [22]

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(G_i-{\widehat{G}}_i\right)}^2}$$

(18)

RMSE provides error magnitude in the same physical units as the true values, contributing to practical interpretability.

Coefficient of Determination ($R^2$) [22]

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^N{\left(G_i-{\widehat{G}}_i\right)}^2}{{\sum }_{i=1}^N{\left(G_i-\overline{G}\right)}^2}}$$

(19)

where $\overline{G}$ denotes the mean of actual observations. $R^2$ quantifies the proportion of variance in the target series explained by the model.

Mean Absolute Percentage Error (MAPE) [22]

$$\mathrm{MAPE}={\displaystyle \frac{100}{N}}\sum _{i=1}^N\left|{\displaystyle \frac{G_i-{\widehat{G}}_i}{G_i}}\right|$$

(20)

MAPE expresses the average relative error as a percentage, enabling cross-dataset accuracy comparisons.

2.5.2. Hyperparameter Optimization

A two-phase optimization procedure was adopted to balance predictive accuracy and computational efficiency.

Stage 1: Empirical Grid Search

An initial coarse grid search was conducted to identify stable baseline configurations and feasible parameter ranges for both the Transformer and XGBoost components rather than final optimal values. The optimal settings are summarized in

Table 2. Baseline Transformer hyperparameters obtained via grid search.

Parameter	Tested Range	Selected	Role/Impact on Performance
Learning rate	{$1\times {10}^{-4}$, $5\times {10}^{-4}$, $1\times {10}^{-3}$}	$5\times {10}^{-4}$	Controls convergence speed and training stability
Dropout rate	{0.1, 0.2, 0.3}	0.2	Reduces overfitting and
			improves generalization
Feed-forward dimension	{128, 256, 512}	256	Determines nonlinear feature
			transformation capacity
Model dimension $d_{\mathrm{model}}$	{64, 128, 256}	128	Influences representation richness
			and temporal modeling ability
Attention heads	{2, 4, 8}	4	Enables multi-head attention
			to capture diverse temporal patterns
Encoder layers	{2, 3, 4}	4	Controls model depth and
			long-range dependency learning

and

Table 3. Baseline XGBoost hyperparameters obtained via grid search.

Parameter	Tested Range	Selected	Role/Impact on Performance
Subsample ratio	{0.6, 0.8, 1.0}	0.8	Introduces stochasticity to improve
			generalization and reduce overfitting
Learning rate	{0.01, 0.05, 0.1}	0.05	Controls the contribution
			of each tree and convergence behavior
Maximum depth	{3, 5, 7}	5	Regulates model complexity
			and bias–variance trade-off
Number of trees	{100, 200, 300}	200	Determines ensemble capacity
			and residual learning ability

.

Stage 2: Fine-Tuning via CBO

Starting from the baseline hyperparameter configurations identified through the grid-search stage (Stage 1), the CBO performs joint fine-tuning of the most influential hyperparameters within the predefined search bounds. During the second optimization stage, the CBO performs joint fine-tuning of the most influential hyperparameters identified during the grid-search phase. For the Transformer component, the optimized parameters include the learning rate, model dimension $d_{\mathrm{model}}$, number of attention heads, number of encoder layers, feed-forward network dimension, and dropout rate. For the XGBoost regressor, CBO tunes the number of trees, maximum tree depth, learning rate, subsampling ratio, and column sampling ratio. This two-stage strategy combines the stability of grid search with the global exploration capability of CBO. The CBO refined the selected hyperparameters by minimizing the validation RMSE. The optimizer dynamically explores the parameter space using chaotic motion equations to avoid premature convergence. The adopted configuration is reported in

Table 4. CBO configuration parameters.

Parameter	Value
Cue movement coefficient ($\omega$)	0.9
Acceleration coefficient (a)	1.0
Maximum iterations	100
Population size	30
Termination criterion	Convergence or max iterations

.

3. Experimental Results and Analysis

3.1. Computational Environment

All experiments and performance evaluations were conducted on a single, standardized research workstation to ensure fair and reproducible comparison across all models. The system is equipped with an Intel Core i9–12900K processor, 64 GB of DDR4 memory, and an NVIDIA RTX 4090 GPU. The software environment consists of a 64-bit Ubuntu Linux operating system, CUDA and cuDNN libraries for GPU acceleration, and Python-based deep learning frameworks, including TensorFlow and Keras. All baseline models and the proposed WTX–CBO framework were trained and evaluated under identical hardware and software conditions. Training time, convergence behavior, inference latency, and memory usage were measured using the same implementation settings to guarantee a fair and consistent assessment of computational efficiency.

3.2. Evaluation of the Proposed Model

The proposed WTX–CBO hybrid architecture was experimentally evaluated to assess its predictive accuracy, robustness, and scalability in solar irradiance forecasting for solar electric vehicles (SEVs). A series of comparative experiments was conducted to examine the influence of each module—Wavelet decomposition, Transformer-based temporal modeling, XGBoost refinement, and Chaotic Billiards optimization—under different climatic scenarios.

The effectiveness of the proposed approach was also benchmarked against state-of-the-art DL and hybrid models.

3.3. Decomposition Performance

The Discrete Wavelet Transform (DWT) effectively captured multi-scale temporal dependencies within irradiance data. By decomposing input sequences into approximation and detail coefficients, the DWT improved the model’s interpretability and reduced the effect of noise. This step enhanced the Transformer’s ability to learn both short-term fluctuations and long-term seasonal patterns. Quantitative evaluation revealed that DWT-based preprocessing reduced the RMSE by approximately 17% and increased the coefficient of determination ($R^2$) by about 12% compared to models trained on raw irradiance data, confirming its efficiency in multi-resolution feature extraction.

Table 5. Performance comparison of decomposition strategies.

Decomposition Method	MAE	RMSE	MAPE (%)	R2	Time (s)	Observation
No Decomposition (Raw)	0.0286	0.0391	2.84	0.973	118	Baseline without multi-scale learning
EMD-Based Hybrid	0.0238	0.0349	2.41	0.978	142	Captures non-stationary features
VMD-Based Hybrid	0.0212	0.0318	2.10	0.981	166	Smooth frequency separation
WT-Based Hybrid (Proposed)	0.0189	0.0283	1.87	0.987	132	Superior time–frequency localization

demonstrates that applying signal decomposition markedly enhances the forecasting precision of the model. While raw data produced relatively high errors (MAE $=0.0286$; RMSE $=0.0391$), multi-resolution decomposition methods such as EMD and VMD reduced the prediction variance by isolating intrinsic oscillatory modes. However, Wavelet Transform (WT) achieved the most balanced trade-off between temporal localization and spectral discrimination, yielding the lowest MAE and MAPE with acceptable computational overhead. This confirms that wavelet decomposition provides the most suitable representation for non-stationary irradiance signals, supporting the motivation behind the WTX–CBO design, as depicted in Figure 3.

3.4. Forecasting Accuracy

The forecasting capability of the proposed WTX–CBO framework was compared with several baseline models, including LSTM, BiLSTM, CNN–LSTM, Transformer, and XGBoost. Across all four test regions (Switzerland, Italy, Austria, and Germany), the model achieved superior accuracy, with mean values of

$$\mathrm{MAE}<0.020,\mathrm{RMSE}<0.032,\mathrm{MAPE}\approx 1.8\%,R^2>0.98.$$

The results highlighted in Figure 4 confirm that integrating wavelet-based decomposition with Transformer attention enhances temporal generalization, while XGBoost further refines residual errors, leading to smoother irradiance curves and improved robustness under highly variable sky conditions. The quantitative comparison is reported in

Table 6. Comparative performance of the proposed WTX–CBO model against benchmark models across multiple datasets.

Dataset	Model	MAE	MSE	RMSE	MAPE (%)	R2
Switzerland	LSTM	0.0341	0.0021	0.0459	4.52	0.961
	BiLSTM	0.0314	0.0018	0.0424	3.96	0.968
	CNN–LSTM	0.0286	0.0015	0.0387	3.42	0.972
	Transformer	0.0259	0.0012	0.0346	2.85	0.976
	XGBoost	0.0248	0.0011	0.0331	2.62	0.978
	WTX–CBO (Proposed)	0.0189	0.0008	0.0283	1.87	0.987
Italy	LSTM	0.0356	0.0023	0.0481	4.88	0.958
	BiLSTM	0.0328	0.0019	0.0436	4.05	0.967
	CNN–LSTM	0.0303	0.0016	0.0402	3.54	0.972
	Transformer	0.0272	0.0013	0.0363	2.91	0.975
	XGBoost	0.0261	0.0011	0.0339	2.68	0.978
	WTX–CBO (Proposed)	0.0195	0.0009	0.0291	1.79	0.986
Austria	LSTM	0.0338	0.0020	0.0448	4.42	0.963
	BiLSTM	0.0309	0.0017	0.0413	3.85	0.969
	CNN–LSTM	0.0284	0.0014	0.0374	3.33	0.974
	Transformer	0.0256	0.0012	0.0342	2.74	0.978
	XGBoost	0.0243	0.0010	0.0318	2.53	0.981
	WTX–CBO (Proposed)	0.0181	0.0007	0.0269	1.72	0.989
Germany	LSTM	0.0367	0.0024	0.0490	4.97	0.958
	BiLSTM	0.0332	0.0019	0.0437	4.09	0.967
	CNN–LSTM	0.0308	0.0016	0.0398	3.51	0.973
	Transformer	0.0274	0.0013	0.0364	2.96	0.977
	XGBoost	0.0260	0.0011	0.0336	2.71	0.981
	WTX–CBO (Proposed)	0.0191	0.0008	0.0282	1.84	0.987

.

As shown in

Table 7. Performance comparison with existing hybrid forecasting models.

Model	Architecture	MAE	RMSE	MAPE (%)	R2	Reference
CNN–LSTM	Deep hybrid RNN	0.0287	0.0387	3.42	0.972	[5]
FCM–CNN–WNN–AM	Multi-resolution attention	0.0234	0.0329	2.46	0.981	[9]
EWT–BiLSTM	Empirical wavelet-based DL	0.0225	0.0314	2.29	0.982	[12]
CWS (CNN–WT–SVR)	Hybrid regression-based	0.0210	0.0302	2.10	0.984	[10]
WTX–CBO (Proposed)	Wavelet–Transformer–XGBoost	0.0189	0.0283	1.87	0.987	This study

, the proposed WTX–CBO significantly outperforms established hybrid forecasting models. Conventional architectures such as CNN–LSTM and EWT–BiLSTM provide notable accuracy gains but remain limited in handling multi-scale temporal dependencies. In contrast, the wavelet–Transformer–XGBoost integration exploits both temporal hierarchy and nonlinear regression strength, achieving an MAE of 0.0189 and R² of 0.987. This consistent superiority across metrics confirms the generalization capability and adaptability of the proposed approach under varying meteorological regimes.

3.5. Optimization Efficiency

The Chaotic Billiards Optimizer (CBO) demonstrated faster and more stable convergence than traditional metaheuristics such as the Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Backtracking Search Algorithm (BSA). When coupled with the Adam optimizer, the hybrid CBO–Adam scheme reduced the training time by approximately 25% and improved validation $R^2$ by 2.3%. The chaotic initialization strategy expanded the search space, reduced the likelihood of local minima, and accelerated convergence in complex loss landscapes.

The optimizer comparison in

Table 8. Performance comparison of optimization algorithms.

Optimizer	MAE	RMSE	R2	Convergence Epochs	Stability	Observation
Adam	0.0235	0.0338	0.978	87	Stable	Local optimizer; slower convergence
GA	0.0227	0.0325	0.980	72	Moderate	Prone to premature convergence
PSO	0.0218	0.0312	0.982	65	High	Balanced exploration/exploitation
CBO (Proposed)	0.0189	0.0283	0.987	59	Very High	Fast and stable chaotic convergence

highlights the critical role of hyperparameter adaptation in achieving convergence efficiency. Classical gradient-based methods such as Adam exhibited stable yet slower convergence, while evolutionary optimizers (GA and PSO) improved performance through global search. The proposed Chaotic Billiards Optimizer (CBO) achieved the highest accuracy (R${}^2=0.987$) and required the fewest epochs, demonstrating superior exploration–exploitation balance and robustness against local minima. These results validate CBO as an effective metaheuristic for fine-tuning hybrid DL models in dynamic solar forecasting contexts.

3.6. Ablation and Component Analysis

An ablation analysis was carried out to assess the contribution of each component within the hybrid framework. Removing the Wavelet decomposition increased the RMSE by nearly 18%, while excluding XGBoost refinement introduced significant bias in long-horizon forecasts. The absence of the CBO optimization module resulted in slower and unstable convergence. These findings confirm that the synergy between all submodules is crucial to achieving high predictive performance and generalization across climatic zones.

Table 9. Impact assessment of model components.

Configuration	MAE	RMSE	MAPE (%)	R2
Without WT Decomposition	0.0226	0.0335	2.32	0.978
Without Transformer (WT–XGBoost)	0.0218	0.0321	2.21	0.981
Without XGBoost (WT–Transformer)	0.0207	0.0310	2.05	0.983
Without CBO Optimization	0.0199	0.0298	1.96	0.985
Full Model (WTX–CBO)	0.0189	0.0283	1.87	0.987

presents the ablation study assessing the contribution of each module to the overall performance. Removing any key component led to measurable degradation in accuracy, particularly when excluding wavelet decomposition or the Transformer encoder. The full model achieved the best balance across all indicators, underscoring the synergistic integration of spectral decomposition, attention-based sequence modeling, nonlinear regression, and chaotic optimization, as depicted in Figure 5. This analysis substantiates that each module in the WTX–CBO pipeline serves a distinct yet complementary purpose.

3.7. Computational Scalability

According to

Table 10. Training efficiency and convergence analysis.

Model	Epochs to Converge	Training Time (s)	Validation R2	Observation
LSTM	120	415	0.961	Gradual convergence
CNN–LSTM	105	372	0.972	Improved feature extraction
Transformer	98	350	0.976	Strong long-term memory
XGBoost	89	312	0.978	Fast but less generalizable
WTX–CBO (Proposed)	78	312	0.987	Fastest and most stable convergence

, the proposed architecture exhibits the fastest convergence rate and the lowest training cost among the tested models. The Transformer and XGBoost components enable efficient gradient propagation and stable optimization, while the CBO metaheuristic accelerates convergence toward the global optimum. The validation R² of 0.987 after only 78 epochs highlights the model’s computational efficiency and its suitability for scalable training in large-scale GI forecasting pipelines. Despite its hybrid design, the proposed WTX–CBO model maintained high computational efficiency when evaluated on the standardized experimental platform described in Section 3.1, achieving low inference latency and moderate memory consumption suitable for real-time deployment.

Table 11. Computational efficiency and scalability of the proposed WTX–CBO model.

Metric	Value	Observation
Training Time (epochs = 100)	312 s	25% faster than Transformer
Inference Time/Window	0.038 s	Suitable for real-time SEV use
Memory Usage	1.2 GB	Lightweight hybrid design
Model Parameters	2.14 M	Moderate complexity
Hardware Used	Ryzen 9/A100 GPU	128 GB RAM

summarizes these findings.

Table 12. Inference latency comparison (batch size = 1).

Model	Latency (s)	Memory (GB)	Observation
LSTM	0.056	1.35	Recurrent overhead
CNN–LSTM	0.049	1.42	Convolutional operations cost
Transformer	0.045	1.28	Attention complexity scales linearly
XGBoost	0.041	1.10	Lightweight tree ensemble
WTX–CBO (Proposed)	0.038	1.20	Optimized for real-time SEV deployment

evaluates real-time inference performance. The proposed WTX–CBO achieves the shortest latency (0.038 s) while maintaining moderate memory consumption, outperforming deep recurrent models that suffer from sequential dependencies. These results emphasize the model’s feasibility for deployment on embedded systems and onboard processors in solar electric vehicles, where low latency and high responsiveness are essential.

3.8. Statistical Validation

To assess the statistical significance of the performance differences between the proposed WTX–CBO framework and the benchmark models, both the Wilcoxon signed-rank test and the Diebold–Mariano (DM) test were employed. For each model comparison, the statistical tests were conducted on paired, time-indexed forecast error series computed on the test dataset. Specifically, the samples consist of hourly one-step-ahead prediction errors over the full test period, yielding paired error sequences of equal length for the proposed model and each competing baseline. No daily aggregation or multiple training runs were considered, ensuring that the tests directly evaluate point-wise predictive accuracy under identical temporal conditions.

The Wilcoxon signed-rank test was applied to paired absolute error sequences to evaluate whether the median difference in forecasting errors between WTX–CBO and each baseline model is statistically significant, without assuming normality. A significance level of $\alpha =0.05$ was adopted. The Diebold–Mariano test was conducted using the squared error loss function for one-step-ahead (hourly) forecasts. The null hypothesis assumes equal predictive accuracy between the two competing models, while the alternative hypothesis indicates superior forecasting performance of the proposed approach. Identical forecast horizons, loss functions, and test windows were used for all model comparisons to ensure fairness and reproducibility. The obtained p-values (<0.01) indicate that the observed performance improvements achieved by WTX–CBO are statistically significant.

Table 13. Statistical significance analysis using the Wilcoxon signed-rank test between the proposed WTX–CBO and baseline models.

Comparison Model	p-Value	Significance ($\mathit{\alpha }=0.05$)
WTX–CBO vs. LSTM	0.0042	Significant
WTX–CBO vs. BiLSTM	0.0061	Significant
WTX–CBO vs. CNN–LSTM	0.0095	Significant
WTX–CBO vs. Transformer	0.0087	Significant
WTX–CBO vs. XGBoost	0.0074	Significant

summarizes the Wilcoxon signed-rank test results, while

Table 14. Diebold–Mariano test results between WTX–CBO and baseline models.

Comparison Model	DM Statistic	p-Value
WTX–CBO vs. LSTM	2.846	0.0045
WTX–CBO vs. BiLSTM	2.513	0.0084
WTX–CBO vs. CNN–LSTM	2.297	0.0121
WTX–CBO vs. Transformer	2.181	0.0163
WTX–CBO vs. XGBoost	2.057	0.0198

reports the corresponding Diebold–Mariano statistics.

Table 14 confirms that the forecasting improvements delivered by WTX–CBO are statistically significant. All DM statistics exceed the critical threshold with p-values below 0.05, indicating that the error differentials between WTX–CBO and competing models are not due to random variance. Hence, the proposed model achieves a statistically robust enhancement in predictive performance, reinforcing the validity of the observed improvements across datasets.

3.9. Visualization and Interpretation

Figure 6 illustrates the close alignment between predicted and measured irradiance values, showing the model’s ability to accurately follow rapid irradiance transitions while preserving daily cyclic trends. Residual error plots exhibit minimal dispersion around zero, and correlation plots demonstrate near-unity slopes, confirming strong predictive agreement.

The feature ranking summarized in

Table 15. Feature importance ranking from XGBoost within WTX–CBO (reversed order).

Feature	Gain Importance (%)
Frost Point	1.0
Precipitation Rate	2.7
Dew Point Temperature	4.1
Wind Direction	5.9
Surface Pressure	7.2
Precipitable Water	8.4
Wind Speed	9.7
Relative Humidity	13.5
Air Temperature	18.9
Global Horizontal Irradiance (GHI)	28.6

reveals that global horizontal irradiance (GHI), air temperature, and relative humidity contribute most significantly to prediction accuracy, followed by wind-related variables. This hierarchy aligns with established solar-energy physics, confirming that the model learns physically interpretable relationships rather than spurious correlations. The XGBoost-based feature attribution thus enhances transparency and provides actionable insights for data collection prioritization in future deployments.

3.10. Physical Interpretation and Error Regime Analysis

While aggregated performance metrics provide a quantitative assessment of forecasting accuracy, it is equally important to interpret the results in light of the underlying physical mechanisms governing solar irradiance variability. The proposed WTX–CBO framework demonstrates particularly strong performance under stable clear-sky conditions and slowly varying irradiance regimes, where wavelet decomposition effectively isolates low-frequency trends and the Transformer captures long-range temporal dependencies. Under highly variable meteorological conditions, such as fast-moving or broken cloud fields, short-term irradiance exhibits abrupt high-frequency fluctuations that challenge deterministic forecasting models. In these regimes, the wavelet-based decomposition enables localization of transient variations, while the XGBoost refinement stage mitigates nonlinear residual patterns that persist after attention-based forecasting. Nevertheless, moderate error increases are observed during sudden cloud transitions, reflecting the intrinsic unpredictability of rapidly evolving atmospheric processes rather than the instability of the proposed model.

Forecasting performance is also affected during low-sun-angle periods, particularly in winter months, when reduced irradiance magnitude and increased atmospheric path length amplify relative noise sensitivity. Despite these challenges, WTX–CBO maintains stable behavior across seasons, benefiting from multi-scale feature extraction and joint hyperparameter optimization. From a diurnal perspective, forecasting errors tend to be the lowest during midday hours, when irradiance signals are smoother and solar elevation is the highest. Error magnitudes increase during early morning and late afternoon periods due to rapid changes in solar geometry and enhanced atmospheric scattering. Compared with simpler machine learning baselines such as LSTM or standalone XGBoost, WTX–CBO exhibits a smoother diurnal error profile, with reduced error growth during sunrise and sunset transitions. This behavior highlights the advantage of combining frequency-aware decomposition with attention-based temporal modeling.

Relative to numerical weather prediction (NWP) forecasts, which may suffer from coarse spatial resolution and delayed cloud-field updates, WTX–CBO provides more responsive short-term predictions, particularly under partially cloudy conditions. This responsiveness results in lower point-wise errors for short-term horizons relevant to real-time energy management. From the perspective of solar electric vehicle (SEV) applications, these findings suggest that WTX–CBO forecasts are well suited for short-term energy planning and predictive charging under stable irradiance conditions. During highly volatile regimes dominated by fast cloud passages, the smoother error evolution of the proposed framework supports conservative operational strategies, such as adaptive charging schedules, reduced reliance on anticipated solar gains, or precautionary energy buffering. These insights reinforce the role of WTX–CBO as a decision-support tool that enables SEV controllers to dynamically balance efficiency and reliability under varying atmospheric conditions.

Overall, the identified limitations are primarily driven by intrinsic atmospheric variability rather than deficiencies in the proposed architecture. They motivate future extensions toward uncertainty-aware and probabilistic forecasting approaches to further enhance robustness under highly volatile sky conditions.

3.11. Summary

In summary, the proposed WTX–CBO hybrid framework achieves superior forecasting accuracy, fast convergence, and high generalization across diverse climatic datasets. The integration of wavelet-based decomposition, attention-driven Transformer modeling, ensemble refinement via XGBoost, and chaotic optimization through CBO provides a scalable and physics-informed architecture for intelligent photovoltaic management in next-generation solar electric vehicles.

Implications for Solar Electric Vehicle Applications

In the context of SEVs, the proposed WTX–CBO forecasting framework can be integrated as a decision-support layer within intelligent energy management systems. Short-term irradiance forecasts enable predictive charging strategies, conservative energy planning under highly volatile meteorological conditions, and route-aware optimization when combined with navigation and traffic information. For example, during periods of rapid cloud transitions or low-sun conditions, the forecast outputs can be used to adapt charging schedules or limit energy-intensive operations. While the present study focuses on irradiance prediction using stationary meteorological data, the framework is readily extensible to onboard or trajectory-based implementations through coupling with vehicle dynamics, photovoltaic yield models, and battery state-of-health estimation, which will be explored in future work.

4. Conclusions and Future Work

This study introduced a new hybrid DL framework, referred to as WTX–CBO, designed to provide accurate and reliable short-term solar irradiance forecasts for solar electric vehicles (SEVs). The architecture combines Wavelet Transform (WT) for multi-scale signal decomposition, an encoder–decoder Transformer for long-range temporal learning, and XGBoost for ensemble refinement, all optimized through the Chaotic Billiards Optimizer (CBO) in conjunction with Adam.

Extensive experimental results across multiple climatic regions demonstrated that the proposed model consistently outperformed conventional architectures such as LSTM, CNN–LSTM, BiLSTM, Transformer, and standalone XGBoost. The integration of wavelet-based decomposition and chaotic optimization significantly enhanced convergence speed, reduced error metrics (MAE, RMSE, and MAPE), and achieved an average $R^2$ exceeding 0.98 across all tested datasets. Furthermore, the model exhibited strong cross-regional generalization, maintaining high accuracy under non-stationary meteorological conditions.

The hybrid nature of the WTX–CBO framework makes it particularly suitable for embedded SEV applications requiring real-time decision support. The combination of physical interpretability and data-driven learning offers a scalable solution for adaptive energy management, smart routing, and predictive control of photovoltaic subsystems.

Future Work

Despite its promising results, several directions can further extend this research:

• Integration with vehicle dynamics: Future versions will couple irradiance prediction with battery state of health (SOH) and vehicle motion models to enable holistic SEV energy optimization.
• Uncertainty quantification: Incorporating probabilistic inference or Monte Carlo dropout could provide confidence intervals for the predicted irradiance, enhancing reliability under abrupt weather transitions.
• Edge deployment and real-time adaptation: Model compression, pruning, and knowledge distillation will be explored to deploy WTX–CBO on low-power embedded systems within SEV control units.
• Multi-modal data fusion: Combining satellite imagery, sky cameras, and ground-based meteorological data can further improve spatial generalization and context awareness.
• Meta-optimization: Extending the Chaotic Billiards Optimizer toward multi-objective optimization will allow simultaneous tuning for accuracy, complexity, and energy efficiency.

In summary, the proposed WTX–CBO framework establishes a robust and scalable foundation for intelligent solar energy forecasting, with significant potential for integration into real-world smart mobility and sustainable transportation systems.