Hybrid SDE-Neural Networks for Interpretable Wind Power Prediction Using SCADA Data

Abstract

Wind turbine power forecasting is crucial for optimising energy production, planning maintenance, and enhancing grid stability. This research focuses on predicting the output of a Senvion MM92 wind turbine at the Kelmarsh wind farm in the UK using SCADA data from 2020. Two approaches are explored: a hybrid model combining Stochastic Differential Equations (SDEs) with Neural Networks (NNs) and Deep Learning models, in particular, Recurrent Neural Networks (RNNs), Long Short-Term Memory (LSTM), and the Combination of Convolutional Neural Networks (CNNs) and LSTM. Notably, while SDE-NN models are well suited for predictions in cases where data patterns are chaotic and lack consistent trends, incorporating stochastic processes increases the complexity of learning within SDE models. Moreover, it is worth mentioning that while SDE-NNs cannot be classified as purely “white box” models, they are also not entirely “black box” like traditional Neural Networks. Instead, they occupy a middle ground, offering improved interpretability over pure NNs while still leveraging the power of Deep Learning. This balance is precious in fields such as wind power prediction, where accuracy and understanding of the underlying physical processes are essential. The evaluation of the results demonstrates the effectiveness of the SDE-NNs compared to traditional Deep Learning models for wind power prediction. The SDE-NNs achieve slightly better accuracy than other Deep Learning models, highlighting their potential as a powerful alternative.

1. Introduction

Wind energy has experienced rapid growth in recent years and has significantly contributed to global renewable electricity generation. According to the International Energy Agency (IEA), wind power was the world’s largest non-hydro renewable electricity source globally in 2020, accounting for $6\%$ of global electricity generation. The IEA predicts that wind energy’s share in global electricity production will continue to grow, potentially reaching $46\%$ by 2030 and becoming the leading source of electricity generation by 2050 in scenarios aligned with international climate goals [1].

Wind energy production is characterised by random fluctuations, typically showing high volatility, which creates challenges for the operation and management of wind power generation systems. The stochastic nature of wind energy arises from its dependence on complex atmospheric processes, leading to high variability across multiple time scales [2]. This variability appears as rapid fluctuations in wind speed and direction, influenced by, e.g., terrain nature, temperature gradients, and large-scale weather patterns [3]. Consequently, wind power output can change dramatically within short periods, presenting significant challenges for grid integration and energy market operations. The uncertainty in wind forecasts increases with the prediction horizon, complicating long-term planning and resource allocation. Moreover, the nonlinear relationship between wind speed and power output further exacerbates the difficulty in accurately predicting wind energy production, necessitating advanced forecasting techniques that can capture these complex dynamics.

Accordingly, it is then mandatory to derive accurate and precise wind power predictions for grid operators to efficiently plan power generation schedules and maximise the operational benefits of wind farms [4,5]. Wind turbine power prediction models can be broadly categorised into three main approaches: physical models, such as blade element momentum theory; statistical models; and AI-driven solutions, including advanced techniques like Deep Learning. In particular, we have the following:

The physical approach, such as blade element momentum (BEM) theory, is fundamental to wind turbine aerodynamics. It combines momentum theory with blade element theory to analyze the performance of wind turbine rotors. BEM is widely used due to its simplicity and computational efficiency compared to more complex methods like computational fluid dynamics (CFD). However, this method is limited in accurately capturing complex aerodynamic phenomena, as it assumes uniform inflow and neglects 3D flow effects—factors typically accounted for in more advanced computational techniques [3].

Statistical models such as Autoregressive (AR) models, Vector Autoregression (VAR), Autoregressive Moving Average (ARMA) models, and Autoregressive Integrated Moving Average (ARIMA) models have been widely used in wind speed, wind turbine power prediction, and energy consumption [6,7,8]. These models focus on identifying patterns between historical input and output data from wind farms to forecast wind power. The accuracy of these models can be limited when faced with the highly variable and nonlinear characteristics of wind power generation. The limitations of statistical models in wind power forecasting, particularly in dealing with nonlinearity and capturing complex structures and unpredictability, have been widely recognised in the literature [9,10].

Machine Learning (ML) and Deep Learning (DL) are subsets of artificial intelligence that allow systems to learn from data. Deep Learning models have demonstrated exceptional capabilities in capturing complex, nonlinear patterns in wind power data, making them highly effective for wind power production forecasting. These models, particularly Recurrent Neural Networks (RNNs) and their variants like Long Short-Term Memory (LSTM) networks [11], as well as integrated models like CNN-LSTM [12], excel at identifying intricate temporal and spatial patterns in wind data. These Deep Learning techniques can significantly reduce prediction errors, improving decision-making processes in power system management [13]. However, despite offering superior accuracy, Deep Learning models often have the drawback of slower response times compared to traditional Machine Learning methods. For the sake of completeness, it is worth noting that Machine Learning models, such as Gradient Boosting Regression, have also been applied in wind power forecasting and energy consumption prediction, demonstrating potential in addressing complex energy tasks [14,15]. However, Deep Learning models are gaining preference for their ability to model the complexity inherent in wind power data.

Within the depicted scenario, it is worth mentioning that wind power prediction methods can generally be classified into two main approaches: direct and indirect. Both have advantages and limitations, depending on the availability of data and the specific forecasting goals. In particular, we have the following:

The indirect approach first predicts wind speed using meteorological and historical data, then converts the predicted speed into wind power output using power curve models. While this method can utilise abundant wind speed data, it introduces errors at the wind speed prediction and the wind speed-to-power conversion stages [16].

The direct method aims to forecast wind power output directly from historical wind power data and other influencing factors. It bypasses the need to predict wind speed first, potentially reducing computational complexity. However, it requires high-quality historical wind power data for training [17].

In this article, we adopt the direct approach based on AI, focusing on predicting wind power output directly from historical time series data. Such a choice is justified by the fact that we can exploit high-quality historical wind power data to train the models and follow an efficient prediction process. The models employed are RNN, LSTM, CNN-LSTM, and an SDE-based $\mathsf{\alpha }$-stable model. The first three models are standard Deep Learning models that have demonstrated success in sequential data modelling. In contrast, the $\mathsf{\alpha }$-stable SDE model belongs to a class of hybrid stochastic–neural approaches that integrate Stochastic Differential Equations with Neural Networks to better model uncertainty, irregular dynamics, and non-Gaussian noise in complex time series.

Recent neural–SDE frameworks such as Neural Jump SDEs [18], SDE-Net [19], and CGNSDE [20] demonstrate that combining stochastic modelling with Neural Network parameterisation enhances learning from irregular and partially physical and dynamic systems. Additionally, deep BSDE methods have been extended toward heavy-tailed SDE modelling, and recent theoretical work addresses convergence and ergodicity for $\mathsf{\alpha }$-stable driven models [21]. By employing an $\mathsf{\alpha }$-stable SDE framework, our model is designed to capture the heavy-tailed, non-Gaussian statistical features of wind fluctuations, delivering more robust multi-step forecasts under stochastic uncertainty. In particular, $\mathsf{\alpha }$-stable SDEs extend classical Brownian-motion-driven SDEs by incorporating Lévy processes, introducing heavy tails and jump discontinuities—properties especially suitable for modelling abrupt transitions and asymmetric fluctuations in wind power generation. Such jump-driven dynamics have been rigorously studied by Peter Imkeller et al. [22] and convergence behavior under Hölder drifts driven by $\mathsf{\alpha }$-stable Lévy noise [23]. These hybrid models leverage the approximation power of Neural Networks to learn drift and diffusion terms, while retaining the SDE formulation to capture stochastic temporal evolution.

For the sake of completeness, let us summarise the paper’s structure. In particular, the article is organised as follows: In Section 2, we discuss the above-mentioned Deep Learning models, specifically, the $\mathsf{\alpha }$-stable SDE model. In Section 3, we describe the considered dataset, which provides insights into wind turbine operations at Kelmarsh Wind Farm, outlining the data exploration and preparation steps for accurate wind power prediction: Key trends are identified, inconsistencies are removed through data cleaning, and preprocessing is performed in view of Deep Learning models application, ensuring that the data can be used for effective model training and robust forecasting. In Section 4, we analyse the performance of different models, including RNN, LSTM, CNN-LSTM, and SDE-Alpha Stable models, also highlighting key findings through error metrics and correlation analysis. We conclude with Section 5 summarising the study’s findings and emphasising the balance between accuracy and interpretability in model performance.

2. Methods and Formulation

In this section, we explore the structure and formulation of Recurrent Neural Networks (RNNs) and their variant Long Short-Term Memory (LSTM) networks, designed to handle sequential data, making them particularly effective for time series prediction tasks. Additionally, we analyse the CNN-LSTM hybrid model, which combines the spatial pattern recognition of Convolutional Neural Networks (CNNs) with the temporal dynamics captured by LSTMs. Finally, we introduce our proposal, namely, the Stochastic Differential Equations Neural Networks (SDEs-NN) method. This powerful approach incorporates stochasticity to model the inherent uncertainty in wind power data, enhancing the robustness and accuracy of the predictions, as we demonstrate in Section 3. Accordingly, for completeness, let us provide detailed formulations of the aforementioned models.

2.1. Recurrent Neural Networks (RNN)

Recurrent Neural Networks (RNNs) are widely used for modelling sequential data where dependencies over time are essential. RNNs are designed to process sequential data by maintaining a hidden state, the hidden state $h_t$ evolves recursively based on the current input $x_t$ and the previous hidden state $h_{t-1}$, that captures information about previous inputs. The basic architecture consists of an input layer, a hidden layer, and an output layer. Unlike feedforward neural networks, RNNs have recurrent connections, as shown in Figure 1, allowing information to cycle within the networks. At each time step t, the RNN takes an input vector $x_t$ and updates its hidden state $h_t$ using the following equation:

$$\begin{array}{cc}\hfill h_t& ={\mathsf{\varphi }}_h(W^{\top }{h}_{t-1}+U^{\top }{x}_t),\hfill \\ \hfill y_t& ={\mathsf{\varphi }}_o\left(V^{\top }{h}_t\right),\hfill \end{array}$$

(1)

where W, U, and V are weight matrices, and ${\mathsf{\varphi }}_h$, ${\mathsf{\varphi }}_o$ are activation functions. The network is trained to minimise a loss between predicted and actual outputs over time steps using backpropagation through time [24].

In the context of wind power prediction, $x_t$ typically represents the historical wind power at time t, while $y_t$ predicts the future power output $P_t$. The hidden state $h_t$ captures temporal dependencies and fluctuations inherent to wind dynamics, enabling the model to learn delayed effects and sequential trends in the power output time series.

2.2. Long Short-Term Memory (LSTM)

Long Short-Term Memory (LSTM) networks are a type of RNN designed to capture both short- and long-term dependencies in sequential data by addressing the vanishing gradient problem. Each LSTM cell includes three gates—forget, input, and output—that regulate the flow of information. In Figure 2, we can see the structure of the LSTM network.

The core LSTM equations are

$$\begin{array}{cc}\hfill f_t& =\mathsf{\sigma }(W_f{x}_t+U_f{h}_{t-1}+b_f)\hfill \\ \hfill i_t& =\mathsf{\sigma }(W_i{x}_t+U_i{h}_{t-1}+b_i)\hfill \\ \hfill o_t& =\mathsf{\sigma }(W_o{x}_t+U_o{h}_{t-1}+b_o)\hfill \\ \hfill {\tilde{C}}_t& =tanh(W_c{x}_t+U_c{h}_{t-1}+b_c)\hfill \\ \hfill C_t& =f_t\odot C_{t-1}+i_t\odot {\tilde{C}}_t\hfill \\ \hfill h_t& =o_t\odot tanh\left(C_t\right)\hfill \end{array}$$

(2)

To explain Equation (2), at each time step t, the LSTM decides what information to keep, update, or share. It uses three gates: The forget gate $f_t$ decides what old information to discard, the input gate $i_t$ determines what new information ${\tilde{C}}_t$ to add, and the output gate $o_t$ controls what information to pass on. The cell state $C_t$ is updated by combining the previous cell state $C_{t-1}$ with the new candidate information ${\tilde{C}}_t$. Finally, the hidden state $h_t$ is produced by filtering the updated cell state through the output gate, allowing the network to remember important information over time.

Here, $x_t$ is the input, $h_t$ is the hidden state, and $C_t$ is the cell state at time t. $\mathsf{\sigma }$ is the sigmoid activation, and ⊙ denotes element-wise multiplication [26]. In wind power forecasting, LSTM can model temporal dynamics by learning patterns from past power outputs ($x_t$) to predict future power values ($P_t$). This makes it well suited for capturing fluctuations and memory effects in power generation.

2.3. Convolutional Neural Networks (CNN)—LSTM

In the context of wind power forecasting, CNN-LSTM architectures combine the feature extraction strength of CNN with the temporal modelling capabilities of LSTM networks [27].

CNN Component extracts short-term temporal features from historical wind power data using convolution and pooling operations:

$$y_j^l=\mathsf{\sigma }\left(b_j^l+\sum _{m=1}^M{w}_{mj}^l·y_m^{l-1}\right),p_j^l=\underset{0\le r<s}{max}{y}_{j,sr+r}^l$$

(3)

the parameter $y_j^l$ represents the activation output of the jth filter in the lth convolutional layer, while $b_j^l$ denotes the bias term associated with this filter at layer l. The weight of the convolutional kernel connecting the mth input feature map from the previous layer $(l-1)$ to the jth filter in the current layer l is denoted by $w_{mj}^l$. The input feature map m from the previous layer is represented by $y_m^{l-1}$. The function $\mathsf{\sigma }(·)$ corresponds to a nonlinear activation function, such as the ReLU function. The pooling output $p_j^l$ is obtained by applying a max-pooling operation to the activations $y_j^l$, where s indicates the size of the pooling window used in this operation.

Figure 3 shows the 1D CNN structure, where the input time series data are passed through convolutional and pooling layers to extract relevant local features.

The LSTM component models long-term dependencies within the extracted feature sequence, allowing the network to capture seasonal trends and time-lagged effects in wind turbine output. The LSTM cell operations follow Equation (2) as previously defined.

The question here is how we can take advantage of the combination of CNN and LSTM in the context of wind power prediction. This hybrid model is particularly effective for predicting wind turbine power output from historical power measurements, as it captures both local fluctuations and broader temporal dynamics. In the CNN-LSTM architecture, the convolutional layers serve as feature extractors that process raw input sequences—such as time series of wind speed, temperature, and historical power output—to identify local temporal patterns. These extracted features are organised as sequences (feature vectors over time), which are then passed as input to the LSTM layer. Essentially, the CNN transforms the original input $X=[x_1,x_2,\dots ,x_T]$ into a higher-level representation $Z=[z_1,z_2,\dots ,z_T]$, where each $z_t$ captures local characteristics over a small time window around time step t. The LSTM then takes this feature sequence Z and models the temporal dependencies across time steps. This connection enables the model to capture short-term patterns jointly (via CNN) and long-term dependencies (via LSTM), making it especially effective for time series forecasting tasks such as wind power prediction [29].

2.4. SDE-Neural Models and Network Models

Finally, we consider our proposal based on Stochastic Differential Equation (SDE) models, which are widely used to investigate complex phenomena in various noisy systems [30]. Let us recall that SDE-based models have gained significant importance in real-world scientific and engineering applications because of their effectiveness in capturing inherent variability and uncertainty. For wind speed prediction, Iverson applied the SDE model for short-term prediction [30], Di Persio et al. [31] developed an SDE predictive model for wind energy production in Italy, while Yang et al., used the Neural Network Stochastic Differential Equation models for the applications in financial data forecasting [32].

2.4.1. Definition of SDE

In our analysis, we assume that $(\Omega ,\mathcal{F},P)$ is a probability space, and ${\left\{W_t^H\right\}}_{t\ge 0}$ represents a fractional Brownian motion (fBm) on $\mathbb{R}$, adapted to its natural filtration ${\mathcal{F}}_t:=\mathsf{\sigma }(W_s^H:s\le t)$. Accordingly, we exploit a one-dimensional SDE of Itô type [31], as in Equation (4):

$$d{X}_t=b(t,X_t)dt+\mathsf{\sigma }(t,X_t)d{W}_t^H,X_0=x_0$$

(4)

where ${\left\{X_t\right\}}_{t\ge 0}$ is a continuous-time stochastic process representing wind power output on $\mathbb{R}$, adapted to ${\mathcal{F}}_t$. $b(t,X_t)$ is the drift term, representing the deterministic component of wind power changes. It describes the average or expected rate of change of the process over time, incorporating long-term trends or seasonality observed in the data. In practical terms, the drift captures systematic influences such as daily cycles or persistent weather patterns that drive the general direction of the wind power output.

On the other hand, $\mathsf{\sigma }(t,X_t)$ is the diffusion term, which quantifies the magnitude of random fluctuations around the deterministic path defined by the drift. This term is responsible for modelling short-term volatility, measurement noise, and any other unpredictable variations in wind behaviour. A higher diffusion value indicates more variability or uncertainty in the wind power output at that point in time.

The process $W_t^H$ is a fractional Brownian motion, the stochastic part of the equation, which generalises standard Brownian motion by incorporating memory effects through the Hurst parameter $H\in (0,1)$. This makes it especially suitable for modelling wind-related processes, as it captures the persistence or anti-persistence observed in real wind speed and power generation time series.

We can rewrite the above SDE in its integral form:

$$X_t=x_0+{\int }_0^{t}b(s,X_s)ds+{\int }_0^t\mathsf{\sigma }(s,X_s)d{W}_s^H$$

(5)

This representation separates the evolution of the process into two parts: The first integral reflects the accumulated deterministic dynamics over time, while the second integral accounts for the cumulative effect of stochastic disturbances driven by the fractional Brownian motion.

2.4.2. Residual Networks (ResNets) and ResNet as a Discretisation of ODEs

Residual Networks (ResNets) were introduced by He et al. In Figure 4, we can see a simple architecture of the ResNet block [33]. ResNets are a class of Deep Neural Networks designed to enable the training of very deep architectures by mitigating the vanishing gradient problem. The central innovation of ResNets is the introduction of residual connections, or skip connections, that allow the network to learn a residual mapping rather than a direct transformation.

Specific Neural Network structures can be understood as approximations or discretisations of differential equations, which can provide insight into their behaviour and properties. For example, Lu et al. [35] demonstrated that various Neural Network architectures are different forms of numerical discretisations of ordinary differential equations (ODEs).

Considering Figure 4, the mapping of $f(x_t;\mathsf{\theta })$ can be realised by feedforward Neural Networks, and the output of the identity mapping x is added to the output of the stacked layers [32,33]. Therefore, the hidden state of the network can be described by the following equation:

$$x_{t+1}=x_t+f(x_t;\mathsf{\theta })$$

(6)

where $x_t$ is the hidden state of the t-th layer and $t\in \{0,\dots ,T\}$. By adding more layers and taking smaller steps, the continuous dynamics of the hidden units can be expressed as follows [36], where ResNet can be understood as an Euler discretisation of ordinary differential equations [32]:

$${\displaystyle \frac{dx}{dt}}=f(x;\mathsf{\theta })$$

(7)

Applying the forward Euler discretisation with step size $\Delta t$, we recover the ResNet update rule:

$$x_{t+1}=x_t+\Delta t·\mathcal{F}(x_t,{\mathsf{\theta }}_t),$$

(8)

where, typically, $\Delta t=1$. This connection led to the formulation of neural ordinary differential equations (Neural ODEs) by Chen et al. [36], where hidden states evolve according to a continuous ODE model, solved using numerical integration techniques.

In the context of wind turbine power prediction, the hidden state $x_t$ can be interpreted as a latent representation of the wind power system at time t, incorporating relevant features such as recent wind speed, direction, air pressure, and turbine operating conditions. The transformation function $\mathcal{F}(h_t,{\mathsf{\theta }}_t)$ models the underlying dynamics that govern changes in the power output, capturing, e.g., how variations in wind speed or turbulence influence the turbine’s mechanical and electrical response. The parameters ${\mathsf{\theta }}_t$ denote the learnable weights of the network, trained on historical wind and power data to approximate the system’s behaviour over time.

Viewing this structure as a discretised ordinary differential equation (ODE) implies that the model treats wind power generation as a dynamical process, where each layer approximates the evolution of the system’s state over a small time increment. This continuous-time interpretation provides not only a powerful predictive model but also a framework for incorporating domain knowledge and physical constraints directly into the network structure. For instance, turbine efficiency curves, cut-in and cut-out wind speeds, or inertia effects can be encoded into the model to enhance interpretability and generalisation. Consequently, ResNet-inspired ODE models offer a flexible and physically informed approach to forecasting wind power under dynamically changing atmospheric conditions.

2.4.3. Euler–Maruyama Approximation for SDE

The Euler–Maruyama approximation is a numerical method for simulating the paths of solutions to Stochastic Differential Equations (SDEs). It is advantageous when an analytical solution to the SDE is unavailable, as is often the case in real-world applications. The method discretises the continuous SDE into a discrete-time stochastic process.

Given that our dataset contains SCADA (Supervisory Control And Data Acquisition) hourly data, the Euler–Maruyama method is appropriate for approximating the process $X_t$ at discrete time points. The time interval $[0,T]$ is divided into N equal subintervals of length $\Delta t={\displaystyle \frac{T}{N}}$, and the approximation is given by the recursive formula

$$X_{t+\Delta t}=X_t+a(X_t,t)\Delta t+b(X_t,t)\Delta W_t$$

(9)

In the above equation, $X_{t+\Delta t}$ is the state of the process at the future time $t+\Delta t$. $X_t$ is the state at the current time t. $a(X_t,t)$ is the drift coefficient, determining the deterministic trend. $b(X_t,t)$ is the diffusion coefficient, determining the stochastic volatility. $\Delta t$ is a small time increment. $\Delta W_t$ is the increment of the Wiener process (Brownian motion). This approximation method allows us to simulate the dynamics of wind speed and power output using SDEs.

2.4.4. Lévy-Based Stochastic Differential Equations

In wind power forecasting, wind speed and wind power often exhibit sudden spikes or jumps that cannot be captured by standard Brownian motion. To model such phenomena, we employ a jump-diffusion model using Lévy-based SDEs. Lévy motion introduces jumps, allowing us to consider the model’s discontinuities and, above all, capture extreme variations critical in wind power systems from a mechanical point of view.

Let us recall that a stochastic process $X=\left(X\right(t),t\ge 0)$ is said to be a Lévy motion if it satisfies the following properties:

$X\left(0\right)=0$ a.s. (almost surely), X has independent and stationary increments, and X is stochastically continuous, i.e., for all $a>0$ and all $s\ge 0$,

$$\underset{t\to s}{lim}\mathbb{P}\left(\right|X\left(t\right)-X\left(s\right)|>a)=0.$$

(10)

In this context, $\mathsf{\alpha }$-stable Lévy motion ($0<\mathsf{\alpha }<2$) provides a more accurate representation of wind speed variations than Gaussian models, especially in cases where rapid fluctuations or rare, high-impact events occur.

A random variable X is called a symmetric $\mathsf{\alpha }$-stable random variable if its distribution is denoted as $X\sim S_{\mathsf{\alpha }}(\mathsf{\sigma },0,0)$. When $\mathsf{\alpha }=1$, the variable becomes a standard symmetric $\mathsf{\alpha }$-stable random variable, denoted as $X\sim S_{\mathsf{\alpha }}(1,0,0)$. These stable distributions are characterised by their stability parameter $\mathsf{\alpha }$, where $0<\mathsf{\alpha }<2$.

A symmetric $\mathsf{\alpha }$-stable scalar Lévy motion $L_t^{\mathsf{\alpha }}$, with $0<\mathsf{\alpha }<2$, is a stochastic process with the following properties: $L_0^{\mathsf{\alpha }}=0$ a.s., $L_t^{\mathsf{\alpha }}$ has independent increments, the increments $L_t^{\mathsf{\alpha }}-L_s^{\mathsf{\alpha }}\sim S_{\mathsf{\alpha }}({(t-s)}^{{\displaystyle \frac{1}{\mathsf{\alpha }}}},0,0)$, and $L_t^{\mathsf{\alpha }}$ has stochastically continuous sample paths, meaning for every $s>0$, $L_t^{\mathsf{\alpha }}\to L_s^{\mathsf{\alpha }}$ in probability as $t\to s$.

2.4.5. SDE with Lévy Motion and Neural Networks

Subsequently, to introduce the Lévy setting, we modify the continuous-time SDE dynamics by incorporating Neural Networks into our model. In particular, we consider the following (see [32]):

$$dx\left(t\right)=f(x\left(t\right);{\mathsf{\theta }}_f)dt+g(x\left(t\right);{\mathsf{\theta }}_g)d{L}_t^{\mathsf{\alpha }},x\left(0\right)=x_0$$

(11)

where $dx\left(t\right)$ represents the infinitesimal change in $x\left(t\right)$ at time t, $f(x\left(t\right);{\mathsf{\theta }}_f)dt$ is the drift term, which represents the deterministic part of the process and is modelled by a Neural Network with parameters ${\mathsf{\theta }}_f$, $g(x\left(t\right);{\mathsf{\theta }}_g)d{L}_t^{\mathsf{\alpha }}$ is the diffusion term, representing the stochastic part of the process, modelled by another Neural Network with parameters ${\mathsf{\theta }}_g$, and $L_t^{\mathsf{\alpha }}$ is the increment of the symmetric $\mathsf{\alpha }$-stable Lévy process.

Because we aim at obtaining an efficient numerical scheme, we derive the following Euler–Maruyama discretisation scheme for the considered SDE incorporating Lévy motion:

$$x_{k+1}=x_k+f(x_k;{\mathsf{\theta }}_f)\Delta t+g(x_k;{\mathsf{\theta }}_g){(\Delta t)}^{1/\mathsf{\alpha }}{L}_k,k=0,1,\dots ,N-1$$

(12)

$x_k$ is the state at time step k, $f(·)$ and $g(·)$ are Neural Networks modelling the drift and diffusion, respectively, $\Delta t=T/N$ is the fixed time step and $L_k\sim S_{\mathsf{\alpha }}(1,0,0)$ are i.i.d. symmetric $\mathsf{\alpha }$-stable random variables.

3. Case Study and Experiment

Exploiting what has been developed so far, we present the considered dataset and the data exploration and preparation steps used to develop accurate wind power prediction models in this section. The dataset utilised for this study provides detailed information about the operational conditions of wind turbines at Kelmarsh Wind Farm, allowing for an in-depth analysis of wind power generation patterns. We begin by exploring the data to identify key trends, followed by data cleaning procedures to remove any inconsistencies or errors. Finally, we preprocess the data to prepare them for input into the Deep Learning models, ensuring that the models can effectively learn from the historical wind power data for robust forecasting.

3.1. Dataset

The dataset used in this study was obtained from the Zenodo website [37], which provides data related to the wind turbine’s working conditions in Kelmarsh Wind Farm from 2014 to 2020. Kelmarsh Wind Farm, shown in Figure 5, is a notable renewable energy installation in the United Kingdom near Kelmarsh, Northamptonshire.

This wind farm consists of six 2.05 MW Senvion MM92 horizontal-axis wind turbines. An example of the MM92 wind turbine is shown in Figure 6. Detailed technical specifications of these turbines are available on the following website [39]. The Senvion MM92 wind turbines have a SCADA (Supervisory Control and Data Acquisition) system. It plays a crucial role in monitoring and controlling turbine operations in real time by collecting data from various sensors and equipment. The data were originally recorded at 10-min intervals, but for our analysis, we aggregated them into hourly data by taking the average of each hour. Also, the dataset selected for analysis pertains to Turbine Number 1 and 2020.

3.2. Data Exploration

Wind speed is a crucial factor in wind power generation. However, the variability of wind speed poses challenges for consistent and reliable energy production. Figure 7 and Figure 8 represent the performance of the Senvion MM92 2.05 MW wind turbine based on daily average power output and monthly wind speed statistics. Figure 7 represents monthly wind speed statistics, maximum, minimum, and average wind speeds, along with standard deviations. The highest wind speeds are recorded in winter months, especially in February. Figure 8 shows the daily average power throughout the year, categorised by seasons, where winter, represented in blue, consistently produces the highest and most variable power output. In contrast, in red, summer shows the lowest and most stable generation, while spring (green) and autumn (orange) provide intermediate power levels. Visualising such data together allows us to capture the peculiar characteristics of each energy production period immediately. Moreover, in terms of wind energy production, stronger winds in winter and autumn contribute to higher power generation.

3.3. Data Cleaning and Preprocessing

In wind power data, cleaning the data recorded from a Wind Turbine in operating condition is particularly crucial because wind turbine performance analysis, maintenance planning, and power curve modelling rely on accurate and reliable data. But the question is, how can we solve the problem of cleaning wind power data?

There are three different types of abnormal data in the context of wind power data [41]. Type I abnormal data are the negative abnormal data whose values are negative or close to zero when the wind speed is larger than the cut-in speed, which is the minimum wind speed at which the turbine should begin producing electricity. The main reasons for generating Type I abnormal data are small wind speeds, the maintenance of wind turbines, the failure of wind turbines, and curtailment. Type II abnormal data are scattered and randomly distributed around the standard data. This type of abnormal data usually has a low frequency of occurrence and is generally caused by signal propagation noise, sensor fault, or extreme weather conditions. Since these random factors may appear or recover quickly, type fluctuation is also variable and random. Type III abnormal data are the stacked abnormal data that consistently appear continuously and stack in a line on one side of the power curve, caused by wind curtailment command, communication failures, etc. Wind curtailment command is a series of control procedures to curtail the output power of the wind turbine, which is caused by the limited capacity of wind power accommodation of the current power system.

Figure 9 illustrates the power output of the Kelmarsh wind turbine during operational conditions. The figure highlights the detection of abnormal data, specifically Type I and Type II anomalies. The green closed line graphically represents anomalies. In these denoting instances, the power output exhibits negative or near-zero values when the wind speed exceeds the cut-in speed. Additionally, Type II anomalies are depicted by a series of black closed lines dispersed throughout the graph. These types of anomalies (Type II) tend to occur infrequently and are often a result of signal problems, issues related to the sensors, and unusual weather conditions. In Figure 10, the Manufacturing Power Curve is represented by a solid black line, while the cleaned Actual Power Curve is depicted using a continuous colour format.

After data cleaning, preprocessing the wind power output data is essential to preparing the data for input to the Deep Learning models. One common technique is to normalise or scale the data to a similar range, which can help improve the models’ training speed and performance. For example, the data can be scaled to a range of 0 to 1 using the min–max normalisation technique:

$$X_{norm}={\displaystyle \frac{X-X_{min}}{X_{max}-X_{min}}}$$

(13)

where X is the original data, $X_{min}$ and $X_{max}$ are the minimum and maximum values of the data, and $X_{norm}$ is the scaled data.

3.4. Evaluation Index

Since it is a crucial point to assess how well a model predicts outcomes compared to actual observed values, we consider several (standard) metrics, i.e.:

The Mean Absolute Error (MAE) measures the average magnitude of prediction errors. It is calculated as the mean of the absolute differences between predicted and actual values, representing the degree of average deviation between predicted and actual values. A lower MAE indicates better model accuracy.

$$MAE={\displaystyle \frac{1}{n}}\sum _{i=1}^n|y_i-{\widehat{y}}_i|$$

(14)

The Root Mean Square Error (RMSE) is a metric derived from the square root of the mean squared error. By squaring the errors, the RMSE emphasises more significant errors, making it particularly sensitive to outliers. This emphasis creates a more pronounced significance when errors are smaller, which is beneficial for optimising algorithms, as it guides them towards the optimal parameter values. The RMSE also reflects how much the predictions deviate from the actual values on average, providing insight into the model’s overall accuracy. It helps assess whether the features used in the model contribute to better predictions. A lower RMSE indicates a more accurate model, suggesting that the predictions closely match the actual values.

$$RMSE=\sqrt{MSE}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}$$

(15)

The Coefficient of Determination (R²) indicates the percentage of variation in the dependent variable that is explained by the regression model, ranging from 0 to 1:

R² = 1 means the model’s predictions perfectly match the observed data, indicating an excellent fit.

R² = 0 means the model fails to explain any variation in the data.

$$R^2=1-{\displaystyle \frac{{\sum }_{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{{\sum }_{i=1}^n{(y_i-\overline{y})}^2}}$$

(16)

4. Results and Discussion

Figure 11 shows the time series data representing the wind power generation of Senvion MM92 in Kelmarsh Wind Farm, divided into two distinct sections: 80% for training (the blue colour) and the remaining 20% (the red colour) for testing. The models under consideration—RNN, LSTM, CNN-LSTM, and the Alpha-Stable SDE model, which is based on Lévy processes—will be trained using the training set. These models will then be evaluated on the test set to assess their performance. Finally, a comparative analysis of the models will be conducted to determine their relative effectiveness.

4.1. RNN Performance

Figure 12 compares the actual and predicted power curves in the test section, and Figure 13 displays the power curve and Coefficients of Determination for the RNN model in the corresponding section. In Figure 13 dots on the red dashed line indicate perfect predictions, where predicted values equal actual values, while dots above the line show the model overpredicted and dots below the line show the model underpredicted. Additionally, the color intensity of the dots represents the density of predictions—darker areas indicate multiple overlapping predictions at the same or nearby values.

4.2. LSTM Performance

Figure 14 compares the actual and predicted power curves in the test section, and Figure 15 displays the Coefficients of Determination for the LSTM model in the corresponding section. In this Figure, dots on the red dashed line represent perfect predictions. Dots above indicate overprediction, dots below show underprediction, and darker areas show higher density of overlapping predictions.

4.3. CNN-LSTM Performance

Figure 16 and Figure 17 compare the actual and predicted power curves and correlation analysis in the test section using CNN-LSTM and LSTM models, respectively.

4.4. SDE-Alfa Stable

The parameters $\mathsf{\alpha }$ and $\beta$ are critical in the framework of Stochastic Differential Equations (SDEs) driven by $\mathsf{\alpha }$-stable distributions. These parameters define the shape and characteristics of the underlying $\mathsf{\alpha }$-stable distribution, known for its ability to model systems with jumps or extreme events, such as wind power.

Calibrating parameters is crucial since it ensures the model’s performance is optimised, improving its ability to generalise and accurately predict outcomes and minimise errors. The parameter $\mathsf{\alpha }$ governs the tail behaviour of the distribution, controlling the frequency and magnitude of extreme events. Smaller values of $\mathsf{\alpha }$ correspond to heavier tails, which indicate a higher likelihood of extreme jumps. This makes $\mathsf{\alpha }$ crucial for capturing the variability of stochastic processes where extreme fluctuations are significant.

The parameter $\mathsf{\alpha }$, defined within the range $(0,2]$, determines the degree of extreme jumps in the data. Values of $\mathsf{\alpha }<1$ correspond to extreme jumps, rare events in wind speed and wind power distributions. As a result, in this case study, we do not consider the values of $\mathsf{\alpha }$ lower than 1. So, we restricted $\mathsf{\alpha }$ to the range $[1,2]$ to focus on moderate to strong variations.

Considering the $\beta$ parameter, we first plot the wind speed and power distributions to observe their skewness. Based on the distribution, we determine the value of $\beta$, identifying whether the data are right-skewed ($\beta >0$) or left-skewed ($\beta <0$). The parameter of $\beta$, with a range of $[-1,1]$, controls the asymmetry or skewness of the distribution. Negative $\beta$ values indicate left-skewed distribution, while positive $\beta$ values suggest right-skewed distribution. For our analysis, $\beta$ was limited to the range $[0,1]$ as negative values were not relevant for this case since both wind speed and wind power, according to Figure 18, show right-skewed distributions. So, what we have is the parameter $\mathsf{\alpha }$ considered within the range $[1,2]$, while the parameter $\beta$ is constrained between 0 and 1, reflecting distributions with moderate to heavy tails and right-skewness.

To identify the best values, we divided their respective ranges into increments of 0.1. Specifically, $\mathsf{\alpha }$ was varied from 1.0 to 2.0 ($\mathsf{\alpha }=\{1.0,1.1,1.2,\dots ,2.0\}$), and $\beta$ was varied from 0.0 to 1.0 ($\beta =\{0.0,0.1,0.2,\dots ,1.0\}$). The model’s performance was evaluated for each combination of $\mathsf{\alpha }$ and $\beta$ using two metrics: the Mean Absolute Error and the Root Mean Square Error. Following this calibration process, the optimal parameter values were identified as $\mathsf{\alpha }=1.8$ and $\beta =0.8$. This combination minimised the MAE and RMSE, demonstrating superior model performance in capturing the underlying wind power distribution dynamics.

Figure 19 compares the actual and predicted power curves obtained using the SDE-Alpha Stable model in the test section. In Figure 20, the red dashed line marks where predictions would be perfect, meaning the predicted and actual values are exactly the same. Points above the line indicate that the model overestimated the actual values, while points below the line indicate underestimation. It is important to note that the alpha ($\mathsf{\alpha }$) and beta ($\beta$) parameters for the current figures are $\mathsf{\alpha }=1.8$ and $\beta =0.8$.

From Figure 21, the results for all models show that RMSE values range between 206 and 230 kW, corresponding to approximately 10–11% of the maximum power output (2050 kW). MAE values range from 141 to 162 kW, representing about 7–8% of the maximum power. The difference between the RMSE and MAE for each model indicates the presence of some larger errors or outliers, as the RMSE is more sensitive to significant errors. The model performance ranked from best to worst based on the RMSE shows that the RNN performed the best with an RMSE of 206.66 and an MAE of 141.56. This is followed by the SDE-Alfa Stable model with an RMSE of 210.73 and an MAE of 148.23, and the LSTM (Long Short-Term Memory) with an RMSE of 211.78 and an MAE of 148.17. The CNN-LSTM had the highest errors, with an RMSE of 230.05 and an MAE of 161.75.

Meanwhile, R² values above 0.86 demonstrate predictive solid power for wind power output. The RNN slightly leads with an R² of 0.8973, closely followed by LSTM and SDE models, which differ by less than 0.5%. The SDE model stands out for its high accuracy and better interpretability potential. The CNN-LSTM, with an R² of 0.8646, underperforms slightly, suggesting its complexity may not be advantageous for this task.

The SDE-Alfa Stable model performs remarkably well, with an RMSE of 210.73 and an MAE of 148.23. It outperforms the LSTM model and is close to the best-performing RNN model. This demonstrates that SDE models can indeed predict wind turbine power output with an accuracy comparable to, and in some cases better than, that of traditional Neural Networks. SDE models often offer better interpretability than black-box Neural Networks, which can be crucial in energy systems modelling.

5. Conclusions

We considered the prediction task for wind power produced by a 2.05 MW horizontal-axis wind turbine, solving the problem by implementing four Deep Learning models. The first three rely on carefully implementing RNN, LSTM, and CNN-LSTM models, whose performance has been compared to that of our Lévy-driven SDE-based proposal.

The results demonstrate that all models exhibit robust predictive capabilities, with R² values exceeding 0.86, indicating solid explanatory power for wind power output variance. The error metrics, notably, the Root Mean Square Error (RMSE), ranging from 206 to 231 kW (approximately 10–11% of maximum turbine output), and the Mean Absolute Error (MAE), between 141 and 162 kW (7–8% of maximum output), underscore the models’ high accuracy in this complex forecasting domain.

Meanwhile, the SDE-Alfa Stable Neural Network model achieved a performance comparable to, and in some metrics surpassing, that of traditional Neural Network architectures. This hybrid model, which integrates Stochastic Differential Equations with the pattern recognition capabilities of Neural Networks, offers a promising balance between accuracy and interpretability. Its performance (RMSE of 210.73 kW, MAE of 148.23 kW, and R² of 0.8941) nearly matches that of the best-performing RNN model while potentially offering more profound insights into the underlying physical processes that drive wind power generation.

We would like to underline that the present paper introduces a novel approach to wind power forecasting by integrating Stochastic Differential Equations (SDEs) with Neural Networks (NNs), offering a hybrid model that bridges the gap between traditional physical models and purely data-driven methodologies. In particular, we emphasise a balance between interpretability and predictive performance, deriving a solution within the middle ground between conventional black-box Deep Learning models and physically informed systems. A key contribution is represented by incorporating Lévy-based stochastic processes, which effectively capture the inherent randomness and occasional extreme fluctuations in wind power data, addressing the limitations of traditional Gaussian-based models. We also rigorously evaluate the performance of various Deep Learning architectures, including Recurrent Neural Networks (RNNs), Long Short-Term Memory networks (LSTMs), Convolutional Neural Networks (CNNs), and hybrid CNN-LSTM models, alongside the proposed SDE-based approach, showing that it has comparable or superior accuracy in terms of RMSE and MAE metrics while maintaining a higher degree of interpretability. Specifically, the model’s ability to incorporate stochastic elements allows it to better capture complex, nonlinear dependencies within wind power data, yielding robust predictions even in the presence of data irregularities. However, SDE-based Neural Network models face key challenges, such as higher computational cost, lack of standardised tools, and difficulty in calibrating parameters—especially for complex processes like $\mathsf{\alpha }$ -stable distributions, when compared to standard Deep Learning models.

The SDE-NN model is further refined through detailed parameter calibration to optimally capture the statistical properties of wind power distributions, particularly by leveraging $\mathsf{\alpha }$ - stable distributions to model heavy tails and skewness in the data. The findings highlight that including these stochastic elements enhances prediction accuracy and provides deeper insights into the underlying dynamics of wind energy generation. Summing up, we establish the SDE-NN framework as a powerful and interpretable alternative for wind power forecasting, demonstrating its potential for applications in energy systems management and advancing the state of the art in renewable energy modelling.