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Scientific Reports volume 16, Article number: 10537 (2026) Cite this article
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In view of the significant volatility and randomness of photovoltaic power, traditional forecasting methods are unable to meet the requirements for high prediction accuracy. It is urgent to develop a prediction model with high accuracy and strong stability. Therefore, this study proposes a novel multi-step short-term photovoltaic power prediction model based on Variational Mode Decomposition (VMD), Improved Whale Migration Algorithm (IWMA), and Convolutional Neural Network-Kernel Extreme Learning Machine structure (CNN-KELM). Initially, VMD is employed to decompose the original power sequence to reduce its nonlinearity and complexity. Furthermore, we construct a CNN-KELM hybrid model, and the IWMA algorithm, which integrates chaotic mapping, dynamic inertia weight and dynamic factor adjustment with Lévy flight strategy, is introduced to optimize the model parameters, thereby enhancing the prediction performance. Moreover, for each component, a VMD-CNN-IWMA-KELM forecasting model is established, and the predicted results are reconstructed and superimposed to obtain the final prediction. Finally, the performance of the proposed model is validated using two datasets. The experimental results show that the proposed model in this paper shows significant advantages in accuracy and stability. Its goodness-of-fit values reach 96.71% and 92.33%, respectively, effectively improving the accuracy of photovoltaic power prediction.
Driven by the global “carbon neutrality” goal, renewable energy has become the core force in the transformation of the energy structure¹. Solar energy has the advantages of wide resource distribution, easy accessibility, high availability, and zero carbon emissions. It is one of the fastest-growing forms of renewable energy. It has now become an important direction for energy development and utilization in various countries. Photovoltaic (PV) technology, as a convenient means of utilizing solar energy, can directly convert solar energy into the most urgently needed and highly adaptable form of energy—electricity².
On the one hand, photovoltaic power generation is the most mainstream conversion pathway of solar energy. The accuracy of its power forecasting is of vital importance to the scheduling and operation of power systems. It directly affects the efficiency of optimized operation of the distribution network³. On the other hand, future population growth will drive a simultaneous increase in electricity demand, and society’s reliance on photovoltaic power generation will continue to rise^4,5. Moreover, the grid-connected power generation capacity of photovoltaic energy is closely related to weather conditions and is easily constrained by weather intermittency⁶. As a result, solar power generation exhibits nonlinear and nonstationary characteristics, which pose significant challenges to photovoltaic power forecasting^{7,8,9,10,11,12,13,14}.
Therefore, developing a reliable photovoltaic power forecasting model that reduces noise interference and achieves accurate predictions is of significant theoretical and practical importance for promoting the widespread adoption of photovoltaic power generation and ensuring the safe, efficient, and economical operation of the entire power system.
Current photovoltaic power generation forecasting methodologies are primarily categorized into three types: physical forecasting methods, statistical forecasting methods, and hybrid forecasting methods. Physical methods rely on the physical principles of photovoltaic systems to explore the relationship between power and the power generation characteristics of photovoltaic modules. For instance, literature¹⁵ adopts an equivalent circuit model to calculate the current-voltage (I-V) characteristics of photovoltaic panels, and adjusts the output voltage of the PV panels via an enhanced MPPT method to obtain photovoltaic electrical power. Ref¹⁶. proposes a Power Law Model (PLM) for photovoltaic module modeling, which dynamically updates the model using meteorological parameters. Literature¹⁷ constructs a physics-informed model using numerical weather prediction (NWP) data and obtains the expected power output through calculations, which achieves favorable performance. Although physical methods for prediction can deeply explore the physical characteristics of photovoltaic systems, when modeling with physical methods, assumptions and simplifications are required, making it difficult to accurately capture stable patterns in practical applications and thus not suitable for PV power prediction. To address the demand for short-term forecasting, statistical methods emerge. The core of this approach is to construct a mapping association between multi-variable past operational data and photovoltaic power generation by extracting correlations and change patterns from historical data, thereby achieving future power prediction. Among the common statistical prediction models are: Autoregressive Moving Average Model (ARMA)¹⁸, Autoregressive Integrated Moving Average Model (ARIMA)¹⁹, etc. In Ref²⁰. mines data from 8 photovoltaic (PV) power plants in South Korea and conducts a performance comparison between the vector autoregressive (VAR) and the ARIMA for the prediction of hourly regional power generation. Literature²¹ establishes ARIMA and comparative models for predictive performance analysis, introducing a solar radiation variability index to enhance the model. The comparative study underscores the superior efficiency of the ARIMA model. However, statistical methods are heavily reliant on historical data, and their advantages may diminish when dealing with highly nonlinear behaviors, unforeseen events, or poor-quality historical datasets. Therefore, many scholars turn their attention to artificial intelligence models, such as machine learning algorithms like Support Vector Machine (SVM)²² and Extreme Learning Machine (ELM)^23,24 and deep learning algorithms like Convolutional Neural Network (CNN)²⁵ and Long Short-Term Memory Artificial Neural Network (LSTM)^26,27, which have been widely used due to their robustness and high prediction accuracy. As a variant of SVM models, the Support Vector Regression (SVR) model plays a critical role in short-term photovoltaic power generation forecasting. Literature²⁸ builds a BP model and an SVR model. It uses different evaluation metrics to compare the two models. The result shows that the SVR model has higher forecasting precision and better generalization ability when there are only a small number of samples. Literature²⁹ develops the FWOA-SVR model, which integrates fractional calculus to enhance hyperparameter optimization, thereby improving the prediction performance of solar power generation. As a feedforward neural network, Extreme Learning Machine (ELM) features fast learning speed and strong generalization ability, and has demonstrated excellent application effects in multiple fields³⁰. Literature³¹ integrates the persistence method and the ELM model by leveraging the assumption of a high correlation between the current and future values of power data. This enables the P-ELM model to not only retain the advantages of ELM but also achieve stable storage of model parameters through persistence, thereby reducing the training cost of the model. The Kernel Extreme Learning Machine (KELM) is a model improved upon the ELM by introducing kernel methods to address the issues of insufficient generalization ability and excessive dependence on the number of hidden layer nodes when the ELM handles nonlinear problems. It has better prediction accuracy and robustness. Literature³² mines the internal features of data through multi-scale similar day clustering and fast iterative filtering decomposition, then uses the KELM model for prediction, and demonstrates the model’s advantages through multi-fusion strategies.
The above-mentioned physical and statistical methods enhance the forecasting precision of short-term PV power generation to a notable degree. However, due to the non-linear and non-stationary characteristics of photovoltaic data, the prediction effect for the segments with strong fluctuations is still not satisfactory. Therefore, seeking effective data processing and feature extraction methods to alleviate noise interference and improve data quality becomes a focus of attention. Researchers introduced signal processing methods to convert power signals into components with periodic characteristics, providing the model with higher-quality feature inputs. Widely adopted signal processing approaches involve wavelet transform (WT)³³, empirical mode decomposition (EMD)³⁴, and variational mode decomposition (VMD)³⁵, etc. Literature³⁶ proposes a novel power generation data prediction method based on Wavelet Transform (WT) and adaptive hybrid optimization, which not only effectively mitigates noise interference but also enhances feature extraction capability. Literature³⁴ proposes a dual decomposition model combining time-varying filtering and empirical mode decomposition (TVF-EMD), which can effectively alleviate the volatility of photovoltaic power and thereby improve the prediction accuracy of the model. WT and EMD can both break down the original waveform to make prediction results better. But they still have some limitations. The WT model has poor robustness, while EMD may have endpoint effects and over-enveloping problems in some situations. In contrast, VMD model can effectively solve the mode aliasing problem existing in EMD. This model not only has stronger anti-interference ability, but also can effectively reduce the impact of endpoint effects. Literature³⁷ integrates the VMD and KELM models to predict photovoltaic output under major extreme weather conditions. It is verified through four different extreme weather conditions that the combination with signal methods can enhance the adaptability of the model to extreme conditions. Signal decomposition methods can reduce the difficulty of predicting power signals affected by noise. But these methods still have some limits when extracting useful features. Convolutional neural networks (CNN), with their unique network structure, can effectively capture local correlations and hierarchical features in data, and perform exceptionally well in improving parameter efficiency and model robustness. Combining CNN with other predictive models can make up for the deficiency of traditional methods in learning sequence patterns. For example, in Ref³⁸., CNN is respectively combined with Bi-LSTM, Transformer structures, and multi-layer perceptions (MLP). Experiments demonstrate that the two hybrid models outperform the single baseline model in prediction tasks at the daily, weekly, and monthly scales. Literature³⁹ combines VMD, CNN, the IPSO algorithm, and the LSSVM. By using VMD to decompose photovoltaic electrical data into a series of intrinsic mode functions (IMFs), the capacity for extracting the temporal-frequency characteristics of the signals is significantly improved.
Based on the above analysis, integrating data processing methods with prediction models to construct hybrid models becomes the main trend in recent studies on PV power prediction. The Kernel Extreme Learning Machine (KELM), due to its strong nonlinear fitting ability and high computational efficiency, demonstrates an excellent adaptability to the task of photovoltaic electrical output forecasting. However, its predictive accuracy and generalization ability largely relies on the selection of hyperparameters. Therefore, most studies adopt heuristic optimization algorithms to optimize the hyperparameters of the KELM model. Literature⁴⁰ employs the Improved Dung Beetle Optimization (IDBO) algorithm, literature⁴¹ adopts the Improved Moth-Flame Optimization (IMFO) algorithm, etc. These algorithms automatically determine the selection of hyperparameters through their own excellent optimization capabilities, solving the problem that manual parameter tuning struggles to balance underfitting and overfitting, and ultimately improving the prediction accuracy. A whale migration algorithm (WMA) proposed in Literature⁴² effectively balances the global exploration and local exploitation processes by integrating the leader-follower mechanism with an adaptive migration strategy, demonstrating superior optimization capability compared to traditional algorithms. Therefore, this paper will improve the WMA algorithm to enhance the model’s requirements for parameter optimization.
Based on the relevant models and specific methods discussed in the literature review and Table 1, current photovoltaic short-term power forecasting has established a mainstream framework and improved performance through technological integration. However, existing research still faces significant limitations in addressing core issues such as data nonlinearity and non-stationarity:The adaptability and predictive accuracy of both single-model forecasting and traditional hybrid models still exhibit shortcomings.
The adaptability and predictive accuracy of both single-model forecasting and traditional hybrid models still exhibit shortcomings.
Physical methods are not suitable for short-term forecasting, and traditional statistical methods are constrained by the quality of historical data. Single and simple hybrid models have limitations and cannot extract features in depth. Traditional KELM hyperparameter tuning struggles to balance global and local optimization. The above shortcomings limit the prediction accuracy. This paper constructs a hybrid VMD-CNN-IWMA-KELM model. By employing VMD for denoising and CNN-KELM for deep feature extraction and fitting, the model’s adaptability and prediction accuracy are enhanced.
The performance of heuristic optimization algorithms in model hyperparameter tuning requires improvement.
Existing heuristic algorithms for optimizing KELM hyperparameters suffer from slow convergence, low accuracy, susceptibility to local optima, and poor stability, making them ill-suited for complex photovoltaic data. This paper proposes the IWMA algorithm, which employs chaotic mapping, dynamic inertial weighting, and dynamic factor-adjustd Lévy flight to optimize CNN-KELM hyperparameters. This approach avoids manual intervention and maximizes the model’s predictive potential.
Existing mixed models lack comprehensive validity verification and targeted analysis.
The current hybrid model validation lacks systematic rigor, featuring insufficient comparative models and inadequate assessment of generalization capabilities and robustness. The absence of ablation studies and complexity analysis hinders practical engineering applications. Based on actual datasets from two power plants, this paper establishes nine comparative models. Through multi-indicator analysis, ablation experiments, and complexity analysis, it comprehensively validates the superiority of the proposed model across multiple dimensions, providing robust support for engineering applications.
In summary, existing research exhibits gaps in feature extraction, model adaptability, hyperparameter tuning, and effectiveness validation. This paper addresses the aforementioned issues by constructing hybrid models, improving optimization algorithms, and conducting comprehensive validation, thereby enhancing the accuracy and robustness of short-term photovoltaic power forecasting.
To further boost the accuracy of PV power prediction, this study improves the WMA algorithm by introducing chaotic mapping, dynamic inertia weight, and flight strategy based on dynamic factor adjustment, forming the Improved Whale Migration Algorithm (IWMA). Compared with the WMA algorithm, the IWMA is enhanced in terms of convergent precision, rate of convergence, and the local optima avoidance capability. To address the volatility of the photovoltaic power series, Variational Mode Decomposition (VMD) is employed to decompose the sequence, thereby mitigating the influence of noise. Then, a CNN-KELM hybrid model is constructed to enhance the model’s feature extraction capability. Subsequently, the IWMA algorithm is used to optimize the hyperparameters of the CNN-KELM model, aiming to effectively enhance the model’s fitting ability and generalization performance. Therefore, this paper constructs a hybrid CNN-KELM forecasting model optimized by VMD and IWMA. Compared to the single KELM model, the proposed model in this paper demonstrates higher prediction accuracy and greater stability in short-term photovoltaic power forecasting tasks. The main contributions and novelties are summarized as follows:
By introducing chaotic mapping, dynamic inertia weight and a dynamic factor-adjusted Lévy flight strategy, an Improved Whale Migration Algorithm (IWMA) is proposed to enhance the algorithm’s exploration and exploitation capabilities, thereby improving the optimization efficiency.
To obtain an improved prediction model, a CNN-KELM hybrid prediction model is constructed, and the hyperparameters of the hybrid model are optimized using the IWMA algorithm, thereby enhancing the prediction accuracy of the model.
A photovoltaic power prediction model constructed on VMD-CNN-IWMA-KELM is established. By using the VMD signal decomposition method, the instability of the original power data is mitigated, providing higher-quality input data for the model and thereby enhancing the prediction accuracy of the model.
Nine comparative models are selected to conduct multiple sets of simulation experiments on the actual data set of a power plant in China. The effectiveness of the proposed model is verified from multiple perspectives, including true-predicted curve comparison, linear regression, model evaluation index, ablation experiment and model complexity analysis.
This paper has six parts, and the main content of the subsequent parts is arranged as follows: The “Methodology” part explains the basic theories behind VMD, CNN, KELM and the CNN-KELM structure; The “Improved Whale Migration Algorithm” part presents the basic theory of WMA and conducts evaluation experiments on the improvement strategies and performance of the IWMA; The “Prediction of PV power generation based on VMD-CNN-IWMA-KELM” part presents the IWMA-optimized CNN-KELM model and the VMD-CNN-IWAM-KELM photovoltaic power generation power prediction model, and introduces the prediction process; The “Experimental design” part describes the specific details of the experiments as well as the software and solvers used to conduct them; The “Results and analysis of the experiment”part presents the relevant experimental analysis. The “Conclusion” part sums up the whole paper with key findings and extends the model; The “Research limitations” part discusses the limitations of the study and future work.
Variational Mode Decomposition (VMD)⁴³ is an adaptive signal processing method, and it does not use recursion. By iteratively searching the optimal solution of the variational model, the center frequency and bandwidth of each IMF are adaptively determined, and the corresponding modes are estimated, so as to properly balance the error between the modes, so as to realize the effective separation of the signal frequency domain components. By appropriately balancing the errors among modes, it achieves effective separation of the signal’s frequency-domain components. When compared with the classical empirical mode decomposition, VMD can effectively alleviate the modal aliasing effect. The definition of the IMF is as follows:
where, ({A_k}left( t right)) denotes the envelope assignment, and ({varphi _k}left( t right)) represents the instantaneous phase. In fact, the specific implementation process of VMD decomposition is to transform the constrained variational problem into an unconstrained variational problem for solution. The decomposition problem of any signal can be described by Eq. (2).
In Eq. (2), ({u_k}=left{ {{u_1},{u_2}, cdot cdot cdot {u_k}} right}) denotes the k-th modal component function, and ({omega _k}=left{ {{omega _1},{omega _2}, cdot cdot cdot {omega _k}} right}) stands for the center frequency of the k-th modal mode. (*) stands for the operator symbol for convolution. (delta left( t right)) is the Dirac delta function, “s.t.” indicates the constraint condition, and (fleft( t right)) denotes the initial input data.
Through the introduction of a quadratic penalty term (theta) and Lagrangian multiplier (lambda left( t right)) into Eq. (1), the restricted formulation is transformed into an un restricted one through the method of Lagrange multipliers (LMM) and the Alternating Direction Method of Multipliers (ADMM), thereby converting Eq. (2) into Eq. (3).
where, (L({text{ }} cdot {text{ }})) represents the Lagrangian function, (alpha) denotes the penalty factor, and (lambda left( t right)) signifies the Lagrange multiplier.
The Convolutional Neural Network (CNN)⁴⁴ is a deep learning model motivated by the structure of the biological visual cortex. Its core is to extract local features of data and realize feature hierarchical abstraction through convolution operation, weight sharing and pooling operation, which can be widely used in data processing and analysis.
One-dimensional Convolutional Neural Network (1D CNN)⁴⁵ is an important variant of Convolu-tional Neural Network (CNN) and is a type of model specifically designed for processing sequential data. Its structure typically consists of an input layer, a convolutional layer, an activation function, a pooling layer, and an output layer.
The data in the input layer is processed by the convolutional kernel through its receptive field to complete feature extraction and mapping. We can express this process as:
In Eq. (4), (a_{j}^{l}) denotes the convolution output data, and (a_{i}^{{l – 1}}) represents the previous convolution input feature data. (k_{j}^{l}) is the convolution kernel, (b_{j}^{l}) denotes the bias parameter, (x( * )) is the activation.
To introduce nonlinearity into the network, we use the piecewise linear ReLU function as the activation function, with its expression given below:
The pooling layer filters the convolutional feature data according to Eq. (6) and constructs the feature sequence.
The output layer corresponds to the target variable, and the CNN structure is illustrated in Fig. 1.
Structure diagram of 1DCNN.
Kernel Extreme Learning Machine (KELM)⁴⁶ is a machine learning model developed by integrating the kernel technique into the conventional Extreme Learning Machine (ELM). It retains the high efficiency of ELM while using a kernel function to overcome the issue of output fluctuations caused by random initialization in ELM, thereby demonstrating superior learning ability and generalization capability. The kernel matrix of KELM is defined based on Mercer’s condition, with its expression presented below:
where, ({H^T}) denotes the Moore-Penrose inverse of H, ({x_i}) and ({x_j}) are the input values of the samples, (Kleft( {{x_i},{x_j}} right)) is the kernel function. Considering the fitting ability and robustness of the comprehensive model, this paper selects the Gaussian kernel function, which has fewer parameters and stronger universality, as its kernel function. Its expression is:
In Eq. (8), (delta) denotes the kernel parameter. After ELM incorporates the above kernel function matrix, the model output of KELM is:
where, I denotes the identity matrix, which is used to preserve matrix structure. (lambda) is the regularization coefficient, (hleft( x right)) represents the hidden-layer feature mapping, (fleft( x right)) is the model’s output term. The specific network structure of KELM is illustrated in Fig. 2.
Kernel extreme learning machine structure diagram.
In Fig. 2, ({x_i}) denotes the input variable, ({y_i}) the output variable, while ({a_{ij}}) and ({v_{ij}}) stand for the connection weights between the i-th neuron and the j-th neuron in different layers.
CNN-KELM is a hybrid deep learning model that combines the strengths of 1D CNN and KELM. The structure selection is based on the core requirements of photovoltaic power prediction, taking into account both the accuracy of feature extraction and the efficiency of prediction. Compared with the common CNN-LSTM, CNN-GRU, and Transformer-based structures, although they can capture temporal dependencies, they have the drawbacks of numerous training parameters, slow convergence, high risk of overfitting, and high requirements for computing power. The combination of CNN and KELM can effectively remedy the above deficiency.
The CNN-KELM model first uses the convolutional and pooling layers of 1D CNN to complete the extraction of core features. Among them, the convolutional layer extracts local temporal characteristics by sliding the convolution kernel across the series data. Pooling layers perform down-sampling on the convolved features, reducing data redundancy while enhancing the model’s robustness against minor signal variations. The 1D CNN model processes the original signal. It turns the signal into a representative set of feature vector.
KELM demonstrates excellent applicability in multi-step prediction. It maps the low-dimensional features to a high-dimensional space through a kernel function to solve nonlinear regression problems. Then, the extracted feature vectors are directly input into the KELM. Compared with models such as LSTM and GRU, KELM requires no complex iterative training with higher computational efficiency. At the same time, its structure is simple and the parameters have low redundancy. It can effectively reduce the cumulative error in multi-step prediction, and improve the generalization ability and prediction stability of the model. In addition, KELM does not require a large number of samples for support and is suitable for the sample scenario of this study. It can make up for the drawbacks of long training time and insufficient generalization ability in traditional deep learning models. The integration of CNN and KELM enables the collaborative optimization from automatic feature learning to efficient and accurate prediction.
The Whale Migration Algorithm (WMA)⁴² is a meta-heuristic optimization algorithm that simulates the collaborative migration behavior of humpback whale populations. This algorithm classifies humpback whales into two distinct groups: experienced leader whales and inexperienced follower whales (or calves). It achieves a balance between global exploration and local exploitation by simulating the collective coordination and social interaction behaviors of whale groups during migration. The WMA algorithm incorporates the leader-follower mechanism and adaptive migration strategy. These two components work in tandem to establish an efficient mathematical model for problem-solving. Compared with the Whale Optimization Algorithm (WOA), although both are inspired by whale behavior, WOA focuses on the bubble-net foraging mechanism, featuring strong local exploitation but weak global exploration, whereas WMA emphasizes population migration and delivers superior stability in global search. Compared with classic algorithms such as Particle Swarm Optimization (PSO) and Grey Wolf Optimizer (GWO), WMA circumvents the problems of high computational complexity and longtime consumption via its dual mechanisms, featuring a simpler iterative mechanism and more stable structure.
The algorithm starts by creating search agents. It sets their evolutionary direction clearly. The population then optimizes step by step along this direction. Specifically, WMA simulates the initial distribution of the whale population by randomly generating an initial population within the defined search space, where each individual’s position is expressed as:
where, L and U are the search space’s lower and upper bounds. ({text{rand }}(1,D)) is a D-dimensional random vector generated within the interval [0,1], and “(odot)” denotes the Hadamard product of two vectors.
Furthermore, the algorithm sorts all population individuals in descending order of fitness. The individual with the highest fitness is marked as ({X_1}), which is the current optimal solution ({X_{Best}}). The sorted population can be expressed by formula (11).
In the WMA algorithm, the leader refers to the individual in the population that is more experienced, has a better position, and achieves a higher fitness. Half the population size (({N_{pop}})) is designated as the number of leading whales (({N_L})). To characterize the overall position of the whale pod during migration, the average value ({X_{Mean}}) of all leaders’ positions is calculated, which reflects the concentration trend of the population within the current search domain. The calculation formula is as follows:
Assuming the objective function value is analogous to the age of a whale, each juvenile whale emulates and follows the individual in the population whose “age” most closely approximates its own. For instance, the movement of individual whale ({X_i}(i={N_L}+1, cdot cdot cdot ,{N_{pop}})) within the whale group is influenced by the nearest whale ({X_{i – 1}}) that is ranked higher in terms of fitness, and the degree of influence is quantified by ({text{rand }}(1,D) odot ({X_{i – 1}} – {X_i})).
If the distance between ({X_{Mean}}) and ({X_{Best}}) gradually shortens, it indicates that the leader population is approaching ({X_{Best}}). At the same time, the less-experienced whale must also move synchronously in the direction of ({text{rand }}(1,D) odot ({X_{Best}} – {X_{Mean}})). If the objective function value of the new position is superior, that is, when the condition (f(X_{i}^{{new}})) < (f({X_i})) is satisfied, the algorithm updates the position of the calf whale. The position update Equation for the i-th young whale is as follows:
During whale migration, the leader draws upon its experience to identify and select the optimal route. If the objective function value at the new position is superior, i.e., when the specified condition (f(X_{i}^{{new}}) ) < (f({X_i})) is met, the algorithm updates the position of the i-th leader whale using formula (15).
where, ({r_1}) and ({r_2}) are D-dimensional random vectors, L is the lower bound vector of position, (U – L) specifies the search direction and range. The algorithm sorts the entire whale population by fitness from best to worst, then selects the top ({N_L}) optimal individuals as leaders for the next generation.
This paper addresses the shortcomings of the Whale Migration Algorithm (WMA), such as the unsatisfactory initial solution quality, low convergence accuracy, and weak global and local search capabilities, and proposes three improvement strategies to comprehensively enhance the optimization ability of the algorithm. Initially, we introduce the Tent chaotic map for initial population generation, leveraging its uniform distribution properties to enhance solution quality. Secondly, dynamic inertia weights are introduced in the global search stage to expedite convergence and enhance search precision. Finally, the Lévy flight strategy with dynamically adjusted factors is incorporated into the calves’ position update to boost the algorithm’s capacity to break free from local optima. Through the three strategies mentioned above, we propose the Improved Whale Migration Algorithm (IWMA).
The quality of the initial population directly determines the algorithm’s exploration capability and convergence rate. The more evenly the initial solutions are distributed across the search domain, the greater the likelihood that the algorithm will locate the global optimal solution. As stated in Ref⁴⁷., chaotic maps generate random sequences via deterministic systems. Due to their high sensitivity to initial conditions, these maps produce significantly different outputs, a characteristic that can be leveraged to improve population diversity in optimization algorithms. This paper selects the Tent chaotic function for population initialization because of its superior uniform distribution and good ergodicity, which helps to generate more comprehensive initial solutions within the search domain and thereby improve the global exploration ability⁴⁸. Its specific expression is:
where, (beta) is a positive real number defined on the interval (0,1). Extensive experiments show that the system exhibits optimal chaotic characteristics when (beta =0.4999.) (X_{i}^{{New}}) is a function of (X(X in [0,1])) with parameter (beta). The distribution of Tent chaotic mapping in the solution space is shown in Fig. 3. Its point distribution and numerical frequency both exhibit excellent uniformity, meeting the requirements for enhancing initial population quality.
Chaotic mapping spatial distribution diagram.
Balancing global exploration and local exploitation is a core element in enhancing the optimization accuracy and convergence speed of meta-heuristic algorithms. It is indicated in Literature⁴⁹ that a larger inertia weight value corresponds to stronger global exploration capability, while a smaller inertia weight value corresponds to stronger local search capability. To resolve the limitations of slow convergence speed and low precision during the early iterations of the WMA algorithm, this paper introduces the dynamic inertia weight factor adopted in Ref⁵⁰. into the position updating formula of the leader whale in the WMA algorithm. This factor incorporates a nonlinear decreasing mechanism allowing the algorithm to perform extensive global exploration in the early iterations and gradually concentrate on local search regions in the later stages. Thus, while enhancing the convergence speed, premature convergence can be effectively avoided. Its expression is as follows:
where, t and T denote the current iteration number and the maximum number of iterations, respectively. ({w_{hbox{max} }}=0.9,{mkern 1mu})and ({w_{hbox{min} }}=0.2). Additionally, we introduce a random term (rand) to the inertia weight, giving it a certain randomness in each iteration. This randomness helps improve the algorithm’s ability to escape local optimal solutions.
To address the limitation of the original WMA algorithm, which relies solely on the leader whale’s own position and random terms for updates during the early exploration phase. In this paper, we introduce a position update mechanism, which integrates individual and group experience, by introducing the global optimal position ({X_{Best}}), the average position of the population ({X_{Mean}}), and the dynamic inertia weight factor. This mechanism strengthens information exchange and guides the population to collaboratively search for the optimal solution. During the iterative process, if condition (f(X_{i}^{{new}})) < (f({X_i})) is satisfied, the position of the i-th leading whale is updated according to Eq. (18).
where, ({X_i}) denotes the current position, ({X_{Mean}}) represents the average of the leading whales, and ({X_{Best}}) refers to the current global optimal solution.
Then, perform boundary constraint processing on the updated position.
where, (X_{{i2}}^{{New}}) represents the position after boundary processing of (X_{i}^{{{text{new}}}}).
In the WMA algorithm, the position updating of the juvenile whale depends on the average location and the position of the neighboring whale. If either type of whale performs poorly, the juvenile whales will deviate from the population. This deviation reduces population diversity, which in turn weakens the algorithm’s overall optimization capability. To overcome this limitation, this paper introduces a Lévy flight strategy with dynamic factor adjustment to improve the position update mechanism of young whales, enhancing both search efficiency and stability.
Lévy flight, as a random walk pattern, has step lengths that follow a heavy-tailed distribution⁵¹. Compared with Gaussian walk, it can generate larger step lengths with a certain probability, which enables individuals break away from local optima solutions and expand the search scope. Its random path satisfies:
In practical applications, the Mantegna algorithm is often used to generate Lévy steps, and the specific calculation formula is:
where, (mu) and (nu) follow a normal distribution:
where, ({delta _u}) and ({delta _v}) are respectively:
where, (beta) is usually taken as 1.5.
In traditional Lévy flight, the step length parameter (alpha) is usually a fixed value, which limits the flexibility of the search. Accordingly, this study introduces a dynamic factor adjustment strategy that enables adaptive variation of (alpha) throughout the iterative process. When the population diversity decreases, increasing (alpha) can enhance the exploration intensity and prevent premature convergence. This paper adopts the Sigmoid function to control the attenuation process of (alpha), with its specific mathematical expression formulated as follows:
where, (z=t/T) represents the normalized iterative process. (alpha) is the initial step size. Set the constant(k=0.05). When the condition (f(X_{i}^{{new}})) < ( f({X_i})) is satisfied, then the position update formula for the i-th less-experienced whale is adjusted as follows:
where, ({X_{i – 1}}) represents the position of the previous whale, and ({X_i}) denotes the current position of the whale. The symbol (otimes) stands for the dot product, while (alpha) is defined by Eq. (24).
Subsequently, boundary constraint processing is applied to the updated positions.
where, (X_{{i2}}^{{New}}) represents the position after boundary processing of (X_{i}^{{{text{new}}}}).
Compared to the original WMA, IWMA retains its core mechanism while introducing a chaotic mapping, dynamic inertial weighting, and a Lévy flight strategy with dynamic factor adjustment. This addresses the shortcomings of WMA’s fixed parameters, such as slow convergence and difficulty escaping local optima. It enhances optimization efficiency and precision. Moreover, all the newly added hyperparameters of the IWMA are parameters with clear physical meanings and fixed value ranges, requiring no additional tuning. Compared with the no-hyperparameter design of the standard WMA, it only introduces a small number of new parameters without the need for complex debugging, balancing algorithm performance and ease of use. This makes the IWMA algorithm better suited to meet the hyperparameter optimization requirements of models in photovoltaic power forecasting.
Pseudo-code of the proposed IWMA.
Suppose the population quantity denoted as ({N_{pop}}), the uppermost limit of iterations is (MaxIt), and the dimensionality of the problem is D. The time complexity of the IWMA algorithm primarily stems from two components: population initialization and iterative update. The complexity of the original WMA algorithm across all iterations is (O({N_{pop}} times D times MaxIt)). At the initial phase, the IWMA algorithm exhibits the same time complexity as the WMA algorithm, both being (O({N_{pop}} times D)). Throughout the iteration steps, IWMA introduces a dynamic inertia weight and a dynamic factor-adjusted Lévy flight strategy. Both the exploration phase and the exploitation phase share an identical time complexity of (O({N_{pop}}/2 times D)). Since each stage performs (MaxIt) iterations, the IWMA algorithm’s total time complexity stays(O({N_{pop}} times D times MaxIt)). Therefore, by implementing enhanced strategies and structural enhancements, the IWMA algorithm boosts its performance, all the while preserving a similar time complexity level to that of the original algorithm.
To validate the comprehensive optimization ability of the IWMA algorithm after introducing the optimization strategies, this paper selects the WMA algorithm and six newly proposed intelligent optimization algorithms as comparison algorithms, including the Butterfly Optimization Algorithm (BOA)⁵², the Whale Optimization Algorithm (WOA)⁵³, the Greylag Goose Optimization (GGO)⁵⁴, the Dung Beetle Optimizer (DBO)⁵⁵ and the Parrot Optimizer (PO)⁵⁶. The experiment employs eleven benchmark test functions for performance evaluation. Among them, functions F1 to F5 are unimodal functions with a single global optimum, which are utilized to evaluate the rate of convergence and convergence accuracy of the algorithm during the iterative process. Functions F6 to F9 are multimodal and have numerous local optimal solutions. They can serve as a basis for evaluating the algorithm’s ability to avoid getting trapped in local optima. Functions F10 to F11 are the fixed-dimension multimodal functions, and they can be employed to further assess the algorithm’s robustness within intricate high-dimensional spaces. Table 2 details the mathematical definitions, search spaces, dimensions, and ideal values of 11 test functions.
In order to ensure fairness in the experimental procedure and comparability of the results, we uniformly set the population quantity of all algorithms to 30 and define the maximum number of iterations as 1000. We independently conduct 30 runs of the experiments on each of the eleven test functions, and then record the average value (Avg) and standard deviation (Std) obtained from these 30 runs. The average value indicates the optimization accuracy of the algorithm. The closer this value is to the ideal optimum; the superior the algorithm’s search capability will be. Standard deviation is a metric to measure an algorithm’s robustness. A smaller value means the algorithm is more stable. The detailed parameter settings of other algorithms are presented in Table 3. All experiments are conducted on a 64-bit Windows 10 system with an Intel(R) Core (TM) i5-10500 CPU @ 3.10 GHz processor, and all algorithms are implemented using MATLAB 2024a. The best results obtained in this paper are shown in bold. The specific experimental outcomes are exhibited in Table 4.
As presented in Table 4, we can observe that the IWMA algorithm demonstrates superior compared to the other algorithms in overall performance. Especially in multiple test functions, the average value and standard deviation of this algorithm are close to the theoretical optimal values, demonstrating outstanding optimization ability. On unimodal test functions, the IWMA algorithm is significantly superior to comparison algorithms for convergence accuracy and standard deviation. For instance, for F4, the mean value of the IWMA algorithm is 7.62 × 10^–149, which is 149 orders of magnitude higher than that of the original WMA algorithm. Although it does not attain the theoretical optimal value in certain test functions, its solution accuracy and error are still markedly superior to those of other algorithms.
Among the multimodal test functions, the IWMA algorithm achieves the superior performance in many metrics for some functions. This means the changes we made help the algorithm get away from local optima easily and make it more robust. On fixed-dimension multimodal functions, the mean value of IWMA is closest to the theoretical best optimum, and its standard deviation also performs well. Although GGO has a slightly smaller standard deviation on the F6 function, the IWMA maintains a comparable magnitude with negligible discrepancy. Furthermore, taking the F11 test function as an example, the mean of IWMA is −1.01 × 10¹. This value is very close to the theoretical optimum of −10. Also, its standard deviation is 1.51 × 10⁻¹, which is the smallest among all algorithms. So, we can see that IWMA has excellent search ability and stability when dealing with high-dimensional complex problems.
Overall, the IWMA algorithm demonstrates outstanding performance across all benchmark functions. In this paper, we simulate the convergence of seven algorithms to different test functions. The convergence curves are presented in Fig. 4.
Among the unimodal functions shown in Fig. 4(a) to (e), the IWMA algorithm consistently maintains the strengths of fast convergence speed and high precision throughout the entire iterative process, outperforming other algorithms. Among the multimodal and fixed-dimensional multimodal functions illustrated in Fig. 4(f) to (k), the IWMA algorithm can rapidly approach the optimal solution within a relatively small number of iterations, achieving the best convergence efficiency. In particular, in Fig. 4(h) and (k), the IWMA algorithm can achieve the optimal fitness value at the very beginning and exhibits a high-precision convergence state. The experimental results demonstrate that IWMA has good adaptability and convergence for different types of test functions, and can avoid local optimal solutions, enabling it to approach the global optimal value quickly and stably. In this way, the effectiveness of the improved strategies and structural optimization introduced in this study for enhancing the algorithm’s search-optimization ability is verified.
Current photovoltaic power forecasting methodologies require further improvement in their adaptability to variable meteorological conditions and complex geographical environments. In short-term forecasting, the VMD model enhances prediction accuracy and model robustness by decomposing photovoltaic power sequences into multiple sub-sequences with distinct characteristics and relative stationarity. The CNN-KELM hybrid model can perform deep feature extraction from data, effectively capturing the nonlinear variations in photovoltaic power output. Furthermore, the IWMA algorithm leverages its superior global search capability and high-precision convergence properties to facilitate further enhancement of the model’s predictive performance. On the basis of above analysis, this paper constructs a short-term photovoltaic power prediction model that integrates the VMD model, IWMA, and the CNN-KELM hybrid model, aiming to comprehensively enhance the model’s forecasting performance.
Convergence curve of the test function for the algorithm.
In the CNN-KELM hybrid model, the settings of the kernel parameter (delta) and the regularization coefficient C of KELM directly affects its prediction performance⁵⁷. So, hyperparameter optimization is particularly crucial. This paper introduces the IWMA algorithm to optimize the hyperparameters of the CNN-KELM hybrid model. The goal is to make the model’s prediction accuracy better. Here is the detailed optimization procedure:
Step 1. Divide the original PV power signal into training and testing sets, and do normalization processing.
Step 2. Set the population size and maximum iteration of the IWMA algorithm, and set the search range for kernel parameters and regularization coefficients.
Step 3. Initialize the CNN-KELM model, input the parameters of each individual in the population into the model for training and prediction, and use the root mean square error (RMSE) as the objective function to evaluate the performance of each parameter combination of the model. The RMSE is calculated using the following formula:
where, n denotes the total sample size, ({y_i}) denotes the true value of the i-th sample, and ({hat {y}_i}) denotes the predicted value of the model. The algorithm performs iterative optimization with the objective of minimizing the RMSE value of the output combination parameters, with a theoretical optimal value of 0.
Step 4. Update the fitness values in each iteration. If the algorithm finds a superior solution, update the population positions and fitness values, and record the current optimal kernel parameters and regularization coefficients.
Step 5. Check whether the iteration count has hit the predefined maximum. If it has, bring the iteration process to an end and present the result; if not, continue carrying out the search.
Step 6. Input the training dataset and the test dataset into the optimized CNN-KELM model, complete the training and output the prediction results.
Based on the above process, the process of IWMA optimizing the hyperparameters of the CNN-KELM hybrid model is shown in Fig. 5.
Flowchart of IWMA optimized CNN-KELM model.
External environmental factors have a substantial impact on photovoltaic power generation, leading to large data fluctuations and poor stability, which restricts the performance of prediction models. Therefore, this paper introduces a multi-step prediction model of VMD-CNN-IWMA-KELM to strengthen the forecasting precision and generalization ability. This model first filters out noise through VMD decomposition, extracts the periodic characteristics of signals in different frequency bands, and improves the quality of input data. The detailed procedure of VMD decomposition is presented in Fig. 6 Subsequently, deep feature mining and selection are performed using CNN-KELM to enhance feature representation capabilities. Furthermore, the IWMA is introduced to optimize the hyperparameters of the CNN-KELM hybrid model, thereby overcoming the constraints imposed by manual parameter tuning on model performance. The specific steps for the prediction of the VMD-CNN-IWMA-KELM model are as follows:
Step 1. Collect the original photovoltaic power data and meteorological data.
Step 2. Break down the original photovoltaic power into multiple relatively stable IMFs through VMD, and normalize each component to eliminate the influence of dimensions.
Step 3. Establish an independent CNN-KELM hybrid model for each component.
Step 4. Utilize the IWMA to optimize the (delta) and C of the CNN-KELM models corresponding to each subsequence, respectively. Upon meeting the termination condition, the optimal prediction models for each component are obtained.
Step 5. By superimposing the prediction outcomes of each component, the final photovoltaic power prediction value is generated.
In this model, each sub model takes historical meteorological data and historical power data as inputs, producing multi-step forecast values for corresponding components as outputs. This approach does not rely on previous forecast results, enabling direct prediction. The core advantage of this strategy lies in avoiding the propagation of errors from recursive predictions. Each step’s prediction independently relies on historical input features, eliminating the cumulative risk of prior step errors influencing subsequent steps. The flow diagram of the VMD-CNN-IWMA-KELM is illustrated in Fig. 7.
VMD decomposition structure diagram.
This study uses solar energy data from a renewable energy generation forecasting competition provided by a regional power supply company of the State Grid Corporation of China. Two sets of 30-day data are selected from the datasets of two power stations as the experimental subjects. The two power stations are designated as Dataset A and Dataset B, with rated capacities of 50 MW and 35 MW, respectively. The time ranges are from February 10, 2019, to March 11, 2019, and from August 1, 2020, to August 30, 2020, respectively. These periods cover both winter and summer seasons, fully reflecting the impact of different seasonal meteorological conditions on photovoltaic power output and exhibiting strong representativeness. Considering that effective solar radiation mainly occurs between 8:00 and 20:00 and that photovoltaic modules generate negligible power outside this period, only data from this period are selected for analysis. With a sampling interval of 15 min, each day contains 49 samples, yielding a total of 1,470 samples over 30 days. The simulation experiment divides the data using a chronological split, with the first 24 days used as the training set and the last 6 days as the prediction set. The data file is in CSV format with GBK encoding. The input data include meteorological variables such as total solar irradiance (W/m²), direct normal irradiance (W/m²), global horizontal irradiance (W/m²), air temperature (°C), atmospheric pressure (hpa), and relative humidity (%), while the output is the photovoltaic power (MW). The power variation trends of Dataset A and Dataset B are presented in Fig. 8. It shows that the photovoltaic power data, on the basis of its periodic characteristics, also exhibits certain non-stationary and fluctuating features. Among them, Dataset B exhibits a greater degree of fluctuation and is suitable for validating the robustness of the model. To characterize the original power sequence in detail, its specific characteristics are displayed in Table 5. Although the training and test datasets differ in sample size, their maximum values, means, and standard deviations are generally comparable, indicating that the training and test datasets exhibit good statistical consistency in their characteristics. This allows the patterns learned by the model during training to be more easily applied to the test set, thereby reducing overfitting and making the prediction results more reliable.
Flow chart of photovoltaic power generation forecasting model.
Original photovoltaic power sequence (a) Dataset A; (b) Dataset B.
To provide a unified comparison scale for data of different magnitudes and units, prevent the analysis results from being dominated by the dimensional differences of individual variables, and ensure the objectivity of the experimental conclusions, all data need to be normalized. The specific normalization operation is as follows:
where, (x^{prime}) is the normalized data, ({x_{hbox{max} }}) and ({x_{hbox{min} }}) are respectively the maximum and minimum values of the original input variables, x is the original input data.
In this study, we carry out the simulation experiments using an Intel (R) Core (TM) i5-10500 CPU processor with a clock speed of 3.10 GHz. The programming software used is MATLAB 2024a, and the drawing tools are MATLAB and VISIO. The parameter settings for each model are detailed in Table 6.
To mitigate the randomness and volatility present in the data, this paper employs the Variational Mode Decomposition (VMD) method to process the initial photovoltaic power signals in the dataset. By converting the photovoltaic power data into multiple stationary single-frequency IMFs, the fluctuation patterns of the original data are simplified, providing more distinguishable input features for subsequent prediction models. Taking Dataset B as an example, for the two important parameters in VMD, namely the number of mode components K and the penalty factor (alpha), the proposed IWMA algorithm is employed to automatically determine their optimal values. The maximum number of iterations and population size of the algorithm are set to 30 and 15, respectively. The optimization range for the mode number K is [2, 15], and for the penalty factor (alpha) is [200, 6000]. The variation of the algorithm’s best fitness value (RMSE) during the iteration process is shown in the Fig. 9. The algorithm converges at the 20th generation, with the output parameter combination [(alpha), K] = [10, 1500].
Convergence Curve of VMD Parameter Optimization.
To verify the strong robustness of the parameter combination optimized by IWMA, parameter perturbation experiments are designed in this study for validation. Taking the optimized optimal parameters as the baseline, different offset values are set around them to compare the variations in forecasting performance. The specific experimental design and results are presented in Table 7. It can be observed that when the parameter combination is [10, 1500], the corresponding RMSE value is the smallest. Therefore, the parameter selection is reasonable. This sensitivity analysis experiment verifies that the parameter combination optimized by the IWMA algorithm exhibits strong robustness.
The two groups of photovoltaic power sequences are decomposed into 10 IMFs using Variational Mode Decomposition (VMD), and the decomposition results are shown in Fig. 10. We can see that each component has an inconsistent frequency but exhibits certain regularity and periodicity. By decomposition, the local features of power data can be presented more clearly. This way, it becomes easier for the model to make predictions.
To verify the feasibility of the VMD-CNN-IWNA-KELM hybrid model proposed in this study, we compare the prediction results of each model with the actual photovoltaic power values and calculate multiple evaluation metrics for verification.
VMD model decomposition results (a) Dataset A; (b) Dataset B.
The adopted evaluation metrics include Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), maximum prediction error, and coefficient of determination (R²). Among them, the first three indicators are used to describe the deviation between the predicted value and the true value. The smaller their values are, the better the model’s predictive ability for extreme situations and the more stable the prediction. We use R² to measure the model’s forecasting accuracy. The closer its value is to 1, the stronger the model’s fitting ability. The specific calculation formulas for each evaluation indicator are provided below:
where, n denotes the total number of samples, ({y_i}) represents the actual photovoltaic power of the i-th sample, ({hat {y}_i}) is its predicted value, and (bar {y}) represents the average power of the entire sample set.
This paper comprehensively validates the predictive performance and generalization capability of the VMD-CNN-IWMA-KELM model across two power plant datasets through five analytical approaches: comparison of actual and predicted values, model evaluation metrics analysis, linear regression analysis, ablation studies, and model complexity analysis.
Figure 11 presents the simulation diagrams of the predicted values and the true values of the four single-step models, BP, ELM, SVR, and KELM, as well as the CNN-KELM hybrid model on the two datasets. Through observation, we can see that the predicted curve of the single-step model deviates significantly from the actual value, especially in the rapid fluctuation range of power, where the gap is relatively large. Its adaptability to the complex fluctuation characteristics of photovoltaic power is insufficient. Compared with the KELM model, the fit between the prediction curve of the CNN-KELM hybrid model and the actual value is improved. Especially on dataset B, which exhibits greater volatility, its predicted values consistently align more closely with the actual values, demonstrating superior robustness. This indicates that the sequential local pattern capture and spatio-temporal correlation feature extraction capabilities of CNN are effectively complementary to the efficient nonlinear fitting capabilities of KELM. As a result, this makes up for the deficiencies of the KELM model in feature extraction and nonlinear adaptability, and improves the accuracy of PV power forecasting. Therefore, the necessity of using CNN-KELM as the baseline model is validated.
Predictive simulation diagrams for BP, ELM, SVR, KELM, CNN-KELM. (a) Dataset A; (b) Dataset B.
Figure 12 presents the simulation diagrams of the KELM, CNN-KELM, CNN-WMA-KELM and CNN-IWMA-KELM models on two datasets. We can observe that after the CNN-KELM model initially improves the performance of the KELM model, the introduction of an intelligent optimization algorithm further enhances the model’s fitting ability. With the introduction of IWMA algorithm, its predicted value curve is closer to the true value curve, and the prediction effect of the model is better. For the strong fluctuating trends in the detailed diagrams, CNN-IWMA-KELM can accurately capture them. This result demonstrates that by using the IWMA algorithm to optimize the hyperparameters in the CNN-KELM hybrid model, during the parameter optimization process, the model can accurately find the parameters that are most suitable for the model, which can effectively alleviate the limitations of parameter selection on the model performance. Accordingly, the model’s prediction accuracy is improved, while its robustness is also enhanced.
Predictive simulation diagrams for KELM, CNN-KELM, CNN-WMA-KELM, CNN-IWMA-KELM. (a) Dataset A; (b) Dataset B.
Figure 13 presents the prediction simulation diagrams of the KELM, CNN-KELM, and VMD-CNN-KELM models. By comparison, it can be observed that that the performance of the model on the two data sets is consistent after the inclusion of the VMD method. Its prediction performance is further improved compared with the KELM and CNN-KELM models. Especially in the multiple rapid power fluctuation intervals present in Dataset B, the VMD-CNN-KELM model can accurately capture the variations, achieving the closest overlap with the actual value curve. This indicates that VMD greatly reduces the nonlinearity and complexity of the data by disintegrating the original complex power sequence into multiple modal components of simple frequencies. At this stage, the CNN model in VMD-CNN-KELM doesn’t need to learn the complex multi-scale features of the original sequence anymore. It only needs to focus on the individual fluctuation patterns of each modal component. This not only reduces the learning difficulty but also improves the accuracy of feature extraction. Therefore, the decomposition operation enhances the learnability of features. This enables the model to effectively enhance its ability to capture fluctuation segments and its prediction accuracy. Therefore, the effectiveness of introducing the VMD module is verified.
Predictive simulation diagrams for KELM, CNN-KELM, VMD-CNN-KELM. (a) Dataset A; (b) Dataset B.
Figure 14 presents the prediction simulation diagrams of the nine models selected in this study on two datasets. From the R² results, the single-step prediction model has deficiencies in prediction accuracy on the two test sets, capture of fluctuations, and trend fitting. In contrast, the multi-step prediction models demonstrate superior predictive capability. Among them, the VMD-CNN-IWMA-KELM model has a prominent performance in the prediction of photovoltaic power generation. Its prediction curve shows a superior fit to the actual value, and it captures the shape of the power peak, the amplitude of the valley and the trend of the rapid fluctuation section with the highest precision. Through the deep collaboration of multi-modules of feature extraction, hyperparameter exploration and sequence decomposition, this model markedly surpasses the others for forecasting accuracy and robustness. Therefore, it fully validates that the VMD-CNN-IWMA-KELM model possesses outstanding predictive capability in photovoltaic power forecasting applications.
Predictive simulation diagrams for nine models. (a) Dataset A; (b) Dataset B.
The simulation experiments present the detailed results of the nine forecasting models for root mean square error (RMSE), mean absolute error (MAE), maximum prediction error (({delta _{hbox{max} }})), and coefficient of determination (R²) in Table 8. To clearly visualize the variations in these metrics, we present histograms of the evaluation indicators for the nine models in Figs. 15 and 16.
After comparing the results of the single-step models and the CNN-KELM model, we can see that all single-step models have R² values below 0.9 and 0.8, respectively, and exhibit relatively large errors. Although the KELM model outperforms the ELM model, it still exhibits poor robustness and low accuracy, failing to meet the prediction expectations. In contrast, the CNN-KELM model, which combines the advantages of CNN and KELM, can effectively enhance the model’s prediction performance. Relative to the KELM, the CNN-KELM model increases the R² by 4.05% and 4.66% on the two power station datasets, respectively. Meanwhile, it significantly reduces the model’s error metrics. Thus, this verifies the synergistic advantages of the dual-module architecture combining CNN’s local feature extraction and KELM’s nonlinear processing.
By comparing the indicators of the CNN-KELM, CNN-WMA-KELM and CNN-IWMA-KELM models, we can know that intelligent optimization algorithms can improve the prediction accuracy. Among them, the optimization effect of IWMA algorithm proposed in this paper is more obvious. Its RMSE and MAE are reduced, and compared with the CNN-KELM, the R² of the CNN-IWMA-KELM model is increased by 3.91% and 5.52%, respectively. This improvement stems from the iterative optimization mechanism of the IWMA algorithm. It can search the hyperparameters corresponding to the optimal fitness through continuous iterative update, which effectively alleviates the problem of model inefficiency caused by the randomness of hyperparameters and improves the prediction ability.
By comparing the prediction results of the CNN-KELM and VMD-CNN-KELM, it can be observed that the evaluation indexes of the model are further optimized by incorporating the VMD signal decomposition method. In the two power station datasets, compared with the CNN-KELM, their RMSE is reduced by 25.85% and 25.44%, and MAE is reduced by 15.93% and 27.73%, respectively. R² values for Dataset A and Dataset B are 93.15% and 91.14%, indicating a satisfactory prediction performance. Therefore, the model can effectively reduce prediction errors caused by the nonlinear and non-stationary characteristics of the original photovoltaic power series through adaptive decomposition using VMD, thus improving its forecasting accuracy.
A comprehensive comparison of nine forecasting model evaluation metrics indicates that CNN-KELM, CNN-IWMA-KELM, VMD-CNN-KELM, and VMD-CNN-IWMA-KELM all demonstrate outstanding performance in PV power forecasting. Specifically, CNN-KELM leverages the powerful feature extraction capability of CNN to fully exploit the advantages of the hybrid model. With parameter optimization by IWMA, the CNN-IWMA-KELM model achieves R² values of 91.45% and 89.57% on the two datasets, approaching the optimal performance under the given architectural constraints. By smoothing noise through modal decomposition, the VMD-CNN-KELM model further improves the R² values to 93.15% and 91.14%. By integrating the advantages of all three components, the VMD-CNN-IWMA-KELM model achieves R² values of 96.71% and 92.33%, effectively capturing power variation trends and fully demonstrating the superiority of the proposed model.
Comparison of evaluation indicators for dataset A.
Comparison of evaluation indicators for Dataset B.
In order to quantify the linear correlation and error distribution between the predictive model outputs and the actual photovoltaic power, linear regression analysis is introduced as an evaluation method to intuitively reflect the matching degree between the forecasted and the actual outputs, with the results shown in Fig. 17. In both datasets, the VMD-CNN-IWMA-KELM model exhibits better goodness of fit, with values all above 90%, and a more concentrated error distribution. Compared with multi-step models, single models exhibit a more discrete error distribution in both data types. Furthermore, through signal decomposition and algorithm optimization, by integrating the advantages of VMD and IWMA into the CNN-KELM model, the forecasting accuracy of the model is improved to varying degrees. Therefore, the VMD-CNN-IWMA-KELM model proposed in this study demonstrates superior adaptability and robustness across different test sets, and exhibits strong predictive capability in photovoltaic power prediction scenarios.
Linear regression graph of predicted and true values (a) Dataset A; (b) Dataset B.
To quantify the impact of each part in the VMD-CNN-IWMA-KELM model, we design ablation experiments. We adopt CNN-KELM as the baseline model and then separately introduce the IWMA optimization algorithm and VMD decomposition model to construct four comparative models: CNN-KELM, CNN-IWMA-KELM, VMD-CNN-KELM, and VMD-CNN-IWMA-KELM. We keep the parameter settings of all models consistent with those in Table 6 and present the experimental results in Table 9.
Based on the experimental results obtained from two datasets, introducing the optimization algorithm and the signal decomposition module, respectively, improves the prediction performance to varying degrees compared with the CNN-KELM baseline model. Above all, integrating the IWMA optimization algorithm into the CNN-KELM model reduces the RMSE, MAE, and ({delta _{hbox{max} }}) on both datasets. Specifically, the RMSE and MAE of dataset A are reduced by 17.19% and 6.99%, respectively. The RMSE and MAE of dataset B are reduced by 19.13% and 26.71%, respectively. In addition, the R² increases by 3.91% and 5.52%, respectively. This indicates the necessity of using IWMA algorithm for hyperparameter optimization. Then, after incorporating the VMD signal decomposition module, the error metrics for both datasets are further reduced. It also improves the fitting ability. Finally, we integrate these two components to form the VMD-CNN-IWMA-KELM model. The proposed model has a smaller maximum error than the CNN-KELM model. On Dataset A, the error drops from 22.1424 to 8.1332. On Dataset B, it drops from 11.5586 to 6.7756. The R² values increase to 96.71% and 92.33%, respectively. The results indicate that incorporating VMD and IWMA into the CNN-KELM model can effectively enhances prediction accuracy of the model by combining the advantages of signal decomposition and parameter optimization. Therefore, the ablation experiments verify that each module of the model exhibits its own unique role. Using data from two power stations, the generalization ability and robustness of the model are verified, further demonstrating the effectiveness of the proposed VMD-CNN-IWMA-KELM model.
The time consumption of the model serves as a crucial indicator for performance evaluation. This paper analyzes the time complexity of the 9 selected models, and the results are presented in Table 10. We can observe that compared with the single-step model, the VMD-CNN-IWMA-KELM prediction model proposed in this paper requires a longer running time for implementation. This is due to the introduction of the intelligent optimization algorithm, which increases the overall operating cost of the model due to factors such as population size, number of iterations, and the internal computational structure of the algorithm. Although the model takes a long time to run, in the deployment system of practical engineering applications, the model training module will be completed in advance before building the prediction system, and the prediction stage can be finished in just a few seconds. Therefore, VMD-CNN-IWMA-KELM model can achieve more accurate prediction outcomes within a reasonable time.
Photovoltaic power generation power prediction is a key prerequisite for ensuring the stable operation of the power system and promoting the large-scale development of photovoltaic power. However, photovoltaic power has intermittency and non-stationarity, which seriously affects the prediction accuracy. Therefore, this paper takes short-term photovoltaic power as the research object and constructs a multi-step forecasting model based on VMD and IWMA algorithm. This model takes CNN-KELM as the basic framework. Firstly, VMD is utilized to decompose the original power signal to suppress noise interference and improve the quality of input data. Furthermore, the Improved Whale Migration Algorithm (IWMA) optimized by the Lévy flight strategy based on chaotic mapping, dynamic inertia weight, and dynamic factor adjustment is adopted to optimize the hyperparameters of the CNN-KELM hybrid model, thereby enhancing the model’s prediction accuracy. Finally, the multi-step hybrid prediction model proposed in this study is evaluated and compared through a series of experiments, which validates its effectiveness. We can draw the following conclusions:
We combine the CNN model with the KELM model. This fixes the problem that the single KELM model is weak at extracting features. Leveraging their complementarity, it achieves a complete functionality from automatic feature extraction to efficient nonlinear fitting. The experimental results confirm that, compared with the KELM single-step model, the CNN-KELM hybrid model demonstrates higher prediction accuracy in the prediction. The R² improvement on the two datasets is 4.05% and 4.66%, respectively, highlighting the advantages of the CNN-KELM hybrid structure over the single-step model.
The IWMA algorithm proposed in this paper effectively boosts the optimization ability of WMA the algorithm. By applying the IWMA algorithm to the hyperparameter optimization of the CNN-KELM model, it significantly improves model prediction performance.
Variational Mode Decomposition (VMD) can adaptively decompose the original PV generation power sequence into multiple single-frequency stationary mode components (IMFs). This method effectively alleviates the mode aliasing problem, reduces the prediction error caused by data noise, enhances the fitting ability of strong nonlinear and non-stationary signals, and then improves the quality of model input.
On two power station datasets, comparative experiments are carried out on nine models. The results show that the proposed VMD-CNN-IWMA-KELM multi-step prediction model has a closest fit between the predicted curve and the actual values, achieving the best fitting performance. The goodness-of-fit values are 96.71% and 92.33%, respectively, and the model’s error metrics are the smallest. This fully demonstrates its notable advantages in PV power sequence decomposition and feature extraction, and can achieve more accurate PV power prediction.
Overall, the VMD-CNN-IWMA-KELM multi-step forecasting model can overcome the limitations of single models, comprehensively enhance the model’s prediction performance, and demonstrate high adaptability in photovoltaic power prediction. This model can also be extended to complex prediction and classification tasks that require simultaneously handling multi-scale patterns, local features, and nonlinear relationships, such as bearing fault diagnosis, power load forecasting, arrhythmia detection, and hazard evaluation.
Although this study has constructed a VMD-CNN-IWMA-KELM multi-step short-term photovoltaic power prediction model and achieved favorable results, it still has certain limitations. First, the data utilized in this experiment covers two seasons, namely winter and summer. While it encompasses most common weather conditions, the predictive performance of the model has not been validated under extreme weather scenarios such as severe sandstorms, strong winds, and heavy snowstorms. Second, during the model validation process, the number of comparative models currently selected is relatively limited, and they do not yet cover predictive architectures such as CNN-LSTM, CNN-GRU, or those based on Transformer. Given these limitations, future research will incorporate additional parameter variables and further enhance the model’s generalization capability and stability under extreme weather conditions. Meanwhile, CNN-LSTM, CNN-GRU, and Transformer-based prediction models will be incorporated for comparative analysis to fully demonstrate the superiority of the VMD-CNN-IWMA-KELM model. In addition, the IWMA algorithm will continue to be optimized to meet the demand for high-precision parameter optimization under extreme meteorological conditions, thereby further enhancing the model’s predictive performance and robustness.
The datasets generated and analyzed in the present study are available from the corresponding author upon reasonable request.
Alternating direction method of multipliers
Autoregressive integrated moving average model
Autoregressive moving average model
Bidirectional long short-term memory
Back propagation
Butterfly optimization algorithm
Convolutional neural network
Dung beetle optimizer
Extreme learning machine
Empirical mode decomposition
Fractional whale optimization algorithm
Greylag goose optimization
Intrinsic mode function
Improved whale migration algorithm
Improved dung beetle optimizer
Improved moth-flame optimization
Kernel extreme learning machine
Least squares support vector machine
Long short-term memory
Multilayer perceptron
Maximum power point tracking
Parrot optimizer
Photovoltaic
Power law model
Single-diode model
Support vector machine
Support vector regression
Time – variable filtering
Variational mode decomposition
Whale migration algorithm
Wavelet transform
Whale optimization algorithm
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College of Science, Xi’an University of Science and Technology, Xi’an, 710054, China
Mengling Zhao, Shan Wu & Yibo Hu
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Funding is provided by National Natural Science Foundation of China (Grant no: 12401265).
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Zhao, M., Wu, S. & Hu, Y. A multi-step short-term photovoltaic power prediction model based on an improved whale migration algorithm. Sci Rep 16, 10537 (2026). https://doi.org/10.1038/s41598-026-41673-2
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Version of record: 30 March 2026
DOI: https://doi.org/10.1038/s41598-026-41673-2
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