Ultra-Short-Term Prediction of Monopile Offshore Wind Turbine Vibration Based on a Hybrid Model Combining Secondary Decomposition and Frequency-Enhanced Channel Self-Attention Transformer

Abstract

Ice loads continue to pose challenges to the structural safety of offshore wind turbines (OWTs), while the rapid development of offshore wind power in cold regions is enabling the deployment of OWTs in deeper waters. To accurately simulate the dynamic response of an OWT under combined ice–wind loading, this paper proposes a Discrete Element Method–Wind Turbine Integrated Analysis (DEM-WTIA) framework. The framework can synchronously simulate discontinuous ice-crushing processes and aeroelastic–structural dynamic responses through a holistic turbine model that incorporates rotor dynamics and control systems. To address the issue of insufficient prediction accuracy for dynamic responses, we introduced a multivariate time series forecasting method that integrates a secondary decomposition strategy with a hybrid prediction model. First, we developed a parallel signal processing mechanism, termed Adaptive Complete Ensemble Empirical Mode Decomposition with Improved Singular Spectrum Analysis (CEEMDAN-ISSA), which achieves adaptive denoising via permutation entropy-driven dynamic window optimization and multi-feature fusion-based anomaly detection, yielding a noise suppression rate of 76.4%. Furthermore, we propose the F-Transformer prediction model, which incorporates a Frequency-Enhanced Channel Attention Mechanism (FECAM). By integrating the Discrete Cosine Transform (DCT) into the Transformer architecture, the F-Transformer mines hidden features in the frequency domain, capturing potential periodicities in discontinuous data. Experimental results demonstrate that signals processed by ISSA exhibit increased signal-to-noise ratios and enhanced fidelity. The F-Transformer achieves a maximum reduction of 31.86% in mean squared error compared to the standard Transformer and maintains a coefficient of determination (R²) above 0.91 under multi-condition coupled testing. By combining adaptive decomposition and frequency-domain enhancement techniques, this framework provides a precise and highly adaptable ultra-short-term response forecasting tool for the safe operation and maintenance of offshore wind power in cold regions.

1. Introduction

Amidst the clean energy revolution, offshore wind turbines (OWTs) have emerged as a pivotal pillar in the energy transition due to their zero-carbon emission advantage [1]. As global energy systems undergo a profound transformation, OWTs demonstrate robust growth potential as a key driver. According to DNV [2], annual OWT generation is projected to surge from 70 TWh in 2018 to 7400 TWh by 2050, with its contribution to global electricity demand leaping from 1270 TWh to 17,840 TWh—a 14-fold increase. Compared to onshore wind, OWTs offer superior resource reserves and effectively overcome land constraints, providing sustainable energy solutions for coastal nations. Notably, this green energy shift is reshaping traditional ocean economies [3]. Data indicates that global offshore oil production will contract by 51% by 2050 relative to 2019, while OWT generation is expected to intersect historically with oil’s energy contribution. This transition marks humanity’s accelerated shift from fossil fuels to renewable-dominated systems [2]. With the intensive development of near-shore wind farm resources, the global wind power industry is accelerating its strategic deployment into deep and remote seas [4].

However, this expansion into colder regions introduces unique technical challenges. In these frigid waters abundant with wind energy potential, the extreme low-temperature environment and complex hydrodynamic properties collectively pose unprecedented engineering difficulties—particularly the persistent interaction between ice floes and OWT foundations that triggers ice-induced vibration (IIV) phenomena [5]. Such dynamic responses have caused structural failures and major accidents. Field data from bottom-founded jackets in Cook Inlet (Alaska) and Bohai Bay (China) demonstrate that when fragmented ice continuously impacts multi-legged structures under tidal forces, dynamic ice loads not only trigger resonance but also subject critical joints to concurrent ultimate strength exceedance and fatigue accumulation [6,7].

Recognizing IIV hazards, substantial research efforts have been made through various approaches. Model-scale tests replicated vibration mechanisms in structures like the JZ-20-2 jacket platform [8] and Molikpaq caisson [9]. Other studies focused on general ice–structure interaction development, numerical validation, and dimensionless formulations [10,11,12,13,14]. Recent advances in ice–structure interaction modeling further illuminate these mechanisms, including DEM-based simulations of moored ship dynamics under ice ridge impacts [15]. These investigations collectively highlight that as OWTs proliferate in cold seas like the Baltic and Bohai, their monopile foundations exhibit distinct IIV sensitivity compared to traditional offshore structures due to their tall, flexible configurations with low natural frequencies and slender ice contact profiles.

The critical need for real-time vibration prediction emerges from these findings, as it enables active control strategies to mitigate load fluctuations and enhance resilience in harsh environments. Accurate forecasting further supports digital twin development for structural health monitoring and maintenance planning. These imperatives necessitate urgent advancement in vibration prediction methods for OWTs under combined wind–ice loading.

To address these challenges, our study integrates advancements from three key domains: high-fidelity ice modeling, adaptive signal processing, and frequency-aware deep learning. Precise prediction first requires reliable simulation data. Building on prior work, this study employs Discrete Element Method (DEM) for ice load computation, which effectively captures the discontinuous ice-crushing processes and provides realistic dynamic ice loads. For turbine modeling, we establish integrated structures incorporating rotor dynamics to capture aerodynamic effects on OWT responses, while embedding control systems to enable real-time pitch adjustment in turbulent winds. The coupling of ice load calculation with the integrated turbine model through the DEM-WTIA multifield co-simulation approach generates high-fidelity data for subsequent analysis.

After obtaining simulation data, this study addresses the challenge of accurate and efficient OWT vibration forecasting. However, research on vibration response prediction for monopile OWTs in cold regions remains scarce, primarily due to the complex nature of wind–ice coupled loads that generate tower-top displacement signals exhibiting strong non-stationarity (transient ice impacts) and multi-scale characteristics (low-frequency turbulence modulation superimposed by high-frequency IIV resonance).

We therefore bridge this gap by drawing inspiration from ultra-short-term ship motion prediction studies in similarly complex marine environments [16,17,18,19,20,21,22,23]. While these approaches show promise, they face limitations when applied to OWT dynamics: Single neural networks (e.g., LSTM, TCN) perform well in ship motion prediction but fail to capture frequency-domain energy distribution features of ice-impacted OWTs due to pure time-domain modeling mechanisms. Hybrid models combining secondary decomposition with Transformer-MLR [24] improve long-sequence processing but introduce high-frequency components through traditional Fourier Transform, causing Gibbs phenomena and limiting adaptability to dynamic uncertainties in cold regions.

However, a critical gap remains between high-fidelity physical simulation and efficient predictive modeling. While DEM-FEM coupling [25] offers advancements in ice–structure interaction analysis, it often prioritizes structural response over the integrated aeroelastic control system dynamics of a complete wind turbine. This limitation hinders its ability to generate fully representative data for OWT vibration prediction under operational conditions. Conversely, in the realm of forecasting, although models like the Transformer [26] and its variants excel in capturing long-term dependencies in time series, their reliance on time-domain analysis alone renders them insufficient for processing the complex, frequency-rich signals generated by wind–ice coupled loads. The inability of these methods to seamlessly integrate high-fidelity physical simulation with frequency-aware deep learning forms the key motivation for this study.

To bridge this gap, we propose a novel hybrid framework that synergistically integrates three innovative components: (1) the DEM-WTIA co-simulation approach for generating high-fidelity physical data that encapsulates full turbine dynamics, (2) the CEEMDAN-ISSA secondary decomposition mechanism for adaptive signal denoising and feature extraction, and (3) the F-Transformer prediction model, which incorporates a Frequency-Enhanced Channel Attention Mechanism (FECAM) to mine latent periodic features. This integrated approach represents a significant departure from and advancement over existing DEM-FEM or standalone Transformer models, providing a comprehensive solution for ultra-short-term vibration forecasting in OWTs subjected to combined wind–ice loading. This integrated framework—combining advanced ice modeling through DEM, adaptive signal decomposition via CEEMDAN-ISSA, and frequency-aware deep learning with F-Transformer—represents a comprehensive solution to the challenging problem of OWT vibration prediction in ice-prone environments. The following sections detail how these components converge to provide a robust prediction framework that addresses the unique challenges posed by combined wind–ice loading on offshore wind turbines.

2. Multiphysics Modeling Development for OWT

2.1. Ice Load Calculation Methods

Ji established a DEM-FEM coupled model [27], and the computational accuracy of the discrete element method (DEM) under this model was validated through experimental comparisons. This study adopts the self-developed discrete element calculation software ICE-SDEM (Version 2.2) created by the aforementioned research team for ice load calculations.

Level-ice is constructed using spherical discrete particles of identical size and mass, which are arranged in a regular hexagonal close packing configuration. To realistically simulate the boundary conditions of sea ice in actual marine environments, spherical particles on both sides of the model are subjected to fixed displacement constraints in the y and z directions, while a constant velocity is applied to the rear side of the model, as shown in Figure 1. To better replicate the macroscopic continuous characteristics of sea ice, the parallel bonding model [28,29] is adopted. Particles are bonded via elastic bonding disks, and forces and torques between adjacent particles are transmitted through these disks. The maximum normal and shear stresses between parallel bonding models are calculated using beam theory.

$${\partial }_{max}={\displaystyle \frac{-F_n}{A}}+{\displaystyle \frac{\left|M_S\right|}{I}}R$$

(1)

$${\tau }_{max}={\displaystyle \frac{|F_{s|}}{A}}+{\displaystyle \frac{\left|M_n\right|}{J}}R$$

(2)

In the equations, $F_n$ and $F_s$ epresent the normal force and shear force between bonded particles, respectively; $M_n$ and $M_S$ denote the normal torque and shear torque of bonded particles, respectively; $A$, $R$, $I$ and $J$ are the cross-sectional area, radius, moment of inertia, and polar moment of inertia of the elastic bonding disk, respectively, with their specific calculation formulas as follows:

$$A=\pi R^2,I={\displaystyle \frac{1}{4}}\pi R^4,A=\pi R^2$$

(3)

The DEM model employs a tensile-shear partitioned fracture criterion to determine particle bond failure, as shown in Figure 2. When the maximum normal bond force ${\sigma }_{max}$ or maximum shear stress ${\tau }_{max}$ between two bonded particles reaches the material tensile failure strength ${\sigma }^t$ or shear failure strength ${\tau }^t$, the bond between the particles fails. The tensile failure strength and shear failure strength can be expressed as

$${\sigma }^t={\sigma }_b^n$$

(4)

$${\tau }^t={\sigma }_b^s+{\mu }_b{\sigma }_{max}$$

(5)

In the formula, ${\sigma }_b^n$ and ${\sigma }_b^s$ represent the normal bond strength and tangential bond strength, respectively. According to the mechanical properties of sea ice, the ratio ${\sigma }_b^n/{\sigma }_b^s=1$ is adopted. ${\mu }_b$ denotes the friction coefficient between bonded particles. This study employs the self-excitation vibration theory (SVT), and the compressive strength of sea ice is determined by the multivariate expression for sea ice compressive strength summarized by Määttänen [29]:

$${\sigma }_c=\left\{\begin{array}{c}0 \dot{\sigma }\le 0\\ \sqrt{{\displaystyle \frac{A_0}{A}}}\left({\displaystyle \genfrac{}{}{0pt}{}{2.0+7.8\dot{\sigma }-18.57{\dot{\sigma }}^2}{+13.0{\dot{\sigma }}^3+2.91{\dot{\sigma }}^4}}\right) 0<\dot{\sigma }\le 1.3 \\ 1 \dot{\sigma }>1.3\end{array}\right.$$

(6)

In the formula, ${\sigma }_c$ denotes the compressive strength of sea ice; $\dot{\sigma }$ represents the stress rate; $A_0$ is the reference loading area (taken as $A_0=1{ \mathrm{m}}^2$); and $A$ indicates the ice load action area. The stress rate is determined by the relative velocity between ice and the OWT structure, as well as the structural dimensions, and can be expressed by the following equation [30]:

$$\dot{\sigma }=\left(v_c-\dot{x}\right){\displaystyle \frac{8{\sigma }_0}{\pi D}}$$

(7)

In the formula, $v_c$ represents the sea ice motion velocity; $\dot{x}$ denotes the structural velocity at the action position of the sea ice; ${\sigma }_0$ is the reference compressive strength; $D$ stands for the structural diameter. For large-diameter structures (e.g., OWT), one or two times the ice thickness can be used to replace the structural diameter.

2.2. Integrated Analysis Methodology for Wind Turbine

The dynamic equilibrium of the integrated OWT model can be expressed by the following formula:

$$R^I\left(r,\ddot{r},t\right)+R^D\left(r,\dot{r},t\right)+R^S\left(r,t\right)=R^E\left(r,\dot{r},t\right)$$

(8)

where $R^I$ represents the inertia force vector; $R^D$ denotes the damping force vector; $R^S$ is the internal structural restoring force vector, defined as the product of the global stiffness matrix K and the displacement vector r, i.e., $R^S$ = K · r; $R^E$ corresponds to the external force vector; r, ṙ, r¨ are the structural displacement, velocity, and acceleration vectors, respectively.

The inertia force vector $R^I$ can be expressed by the following equation:

$$R^I\left(r,\ddot{r},t\right)=\left[M^S+M^F\left(r\right)+M^H\left(r\right)\right]\ddot{r}$$

(9)

where $M^S$ denotes the structural mass matrix; $M^F\left(r\right)$ represents the mass matrix considering internal fluid flow; $M^H\left(r\right)$ is the displacement-dependent hydrodynamic mass matrix, which treats the structural acceleration term in the Morison equation as added mass in local directions.

The damping force vector $R^D$ is expressed as

$$R^D\left(r,\dot{r}\right)=\left[C^S\left(r\right)+C^H\left(r\right)+C^D\left(r,\dot{r}\right)\right]\dot{r}$$

(10)

where $C^S\left(r\right)$ denotes the internal structural damping matrix; $C^H\left(r\right)$ represents the hydrodynamic damping matrix; $C^D\left(r,\dot{r}\right)$ corresponds to the specified discrete damper matrix dependent on displacement and velocity.

2.3. The Coupling Method of DEM-WTIA

This study establishes a full process analytical model for sea ice-OWT foundation dynamic response based on a coupled framework integrating the DEM and computational modules of integrated wind turbine analysis methodologies. During theoretical model construction, the parallel bond model is employed to accurately characterize the constitutive properties of sea ice materials. High-precision DEM simulations dynamically track the collision process between sea ice and OWT foundation, thereby obtaining ice load spectra with spatiotemporal distribution characteristics. It should be noted that the ice force discussed in this study primarily manifests as positive pressure on the surface of the monopile structure. Subsequently, through data interface transmission, DEM calculation results are integrated into the WTIA module. Based on multi-body dynamics principles, the global dynamic response of wind turbines under ice load excitation is systematically analyzed. After multi-physics coupling iterative calculations, a three-dimensional visualization model for ice-induced vibration processes is constructed. This enables precise acquisition of key parameter variation curves for OWT system components (including blades, tower, and foundation displacements) under ice load effects. The coupled workflow design is illustrated in the attached Figure 3.

2.4. Establishment and Validation of OWT Model

The 5 MW monopile-supported OWT from NREL [31] has been selected as the research subject for ice-induced vibrations in this study. As shown in Figure 4, this OWT configuration comprises five principal structural components: rotor blades, nacelle assembly, hub subsystem, tapered tower structure, and monopile foundation.

The main parameters of the model are listed in

Table 1. Main Parameters of 5 MW Monopile-type OWT.

Parameter	Value
Power	5 MW
Rotor direction and number	Upwind, 3 blades
Rotor and hub diameter	126 m, 3 m
Hub height	90 m
Cut-in, rated, cut-out wind speed	3 mps, 11.4 mps, 25 mps
Cut-in, rated rotor speed	6.9 rpm, 12.1 rpm
Rated tip speed	80 mps
Rotor mass	110,000 kg
Nacelle mass	240,000 kg
Tower mass	347,500 kg
Overall center of mass coordinates (CM)	(−0.2 m, 0.0 m, 64.0 m)

. The operational water depth of this project is selected as 20 m.

Modal analysis of the OWT was performed in SIMA. After obtaining the natural frequencies and steady-state behavior of the entire OWT system, the mode shapes of the first ten modes are listed in the following Figure 5.

Among them: The first two modes are the first eigenfrequencies of the tower, corresponding to the first-order fore-aft and first-order side-to-side vibrations of the tower, respectively. The third mode is the first eigenfrequency of the drive system, representing the first-order torsional vibration.

The fourth to eighth modes are the first eigenfrequencies of the blades, corresponding to the first-order flapwise vibration, the first-order asymmetric flap-pitch vibration, the first-order asymmetric flap-yaw vibration, the first-order asymmetric edge-pitch vibration, and the first-order asymmetric edge-yaw vibration of the blades, respectively. The ninth and tenth modes are the second eigenfrequencies of the blades, corresponding to the second-order flapwise vibration and the second-order asymmetric flap-pitch vibration of the blades on the tower, respectively.

The first ten natural frequencies of the OWT model in FAST (version7) and ADAMS software (version 2005–2010) were obtained from NREL literature [31] and compared with those derived from SIMA (version4.4-00), as shown in Figure 6.

From Figure 6, it can be observed that the integrated wind turbine model (OWT) demonstrates excellent agreement with the other two models in terms of the first natural frequency. Significant discrepancies only emerge at the second natural frequency. As observed, deviations in the first natural frequency (Mode 1) remained below 15%, while those in the second natural frequency (Mode 2) were controlled within 13%. However, the deviation in the ninth eigenfrequency (Mode 9), classified as a second eigenfrequency, increased significantly to 17.1%. Specifically, the modal prediction-based second natural frequency generally exceeds that predicted by the multibody and finite element method (FEM) models across most scenarios. This indicates that divergences among different modeling approaches become pronounced within higher frequency ranges.

2.5. Environmental Condition Analysis

The data source for this study comprises the dynamic responses of a 5 MW monopile OWT under multifield coupling. The operational environment is set in cold-region seas, where coupled ice and wind loads are simulated on the OWT model. During model construction, ice loads are precisely applied at the design waterline position to simulate ice collision effects, with structural dynamic responses captured through a 650-s dynamic simulation. To address potential transient response issues in the initial phase of time-domain simulation, a staged loading strategy is adopted: the first 400 s are allocated for system stabilization, followed by sustained ice load application from 400 to 540 s (140 s duration) to ensure acquisition of reliable steady-state thrust data. Ice loads are directionally synthesized and applied at 0°. In practical ocean environments, the presence of sea ice significantly suppresses wave activity. The ice cover absorbs and dissipates wave energy, resulting in a substantially smaller significant wave height in areas with severe ice conditions compared to ice-free periods. Thus, under the extreme ice loading conditions considered in this study, wave loads are no longer the dominant environmental loads. Consequently, this research focuses specifically on investigating the ice resistance performance of an integrated monopile offshore wind turbine, and the influence of wave loads was not included in the analysis. Wave loads are intentionally excluded based on research objectives, while ocean currents are simplified to a linear distribution model ranging from ice drift velocity (sea surface) to zero velocity (seabed).

2.5.1. Sea Ice Parameter Selection

An ice thickness of 40 cm is selected to represent severe conditions. The ice drift velocity is set at 0.02 m/s, informed by the Strain-rate effect Theory (SVT): when ice strain rate increases, its material properties undergo a ductile-to-brittle transition. At approximately 0.02 m/s drift velocity, sea ice reaches its ductile-to-brittle transition threshold, exhibiting maximum compressive strength. Consequently, this yields stronger ice loads with heightened structural impact. The time-history curve of dynamic ice loads over the 140-s loading phase is explicitly illustrated in Figure 7.

2.5.2. Wind Velocity Selection

For wind load modeling, turbulent wind fields are generated using the Kaimal wind spectrum, constructing a multi-gradient wind velocity system. Three boundary thresholds are selected as representatives: 3 m/s (cut-in), 11.4 m/s (rated), and 25 m/s (cut-out). Through the superposition of ice and wind loads, an analysis matrix encompassing three typical operational conditions is ultimately formed, as shown in

Table 2. Combined Environmental Loads.

Environmental Condition	Wind Spectrum	Wind Velocity (m/s)	Ice Thickness (m)	Ice Velocity (m/s)
EC1	Kaimal	3	0.4	0.02
EC2	Kaimal	11.4	0.4	0.02
EC3	Kaimal	25	0.4	0.02

.

Following simulation computations yielding multiple OWT response parameters, this study selects the tower-top displacement—critical for safety performance evaluation—as the research focus. The time-domain responses under three operational conditions are illustrated in Figure 8 below.

3. Construction of Ensemble Learning-Based Prediction Framework

3.1. CEEMDAN Method

Empirical Mode Decomposition (EMD), proposed by Norden E et al. [32], is an adaptive signal processing technique. Its core principle decomposes complex signals into a finite set of Intrinsic Mode Functions ($IMFs$). Unlike traditional methods (e.g., Fourier Transform), EMD requires no predefined basis functions but adaptively decomposes signals based on local characteristics, making it particularly suitable for non-stationary and nonlinear signals. Despite its advantages, EMD suffers from mode mixing, where oscillations of similar scales appear across different $IMFs$ or disparate amplitudes coexist within one $IMF$. The Ensemble EMD (EEMD) algorithm mitigates mode mixing by adding Gaussian white noise to the signal [33]. However, EEMD fails to fully eliminate residual noise during signal reconstruction, causing reconstruction errors. To address this, the complete integrated empirical modal decomposition with adaptive noise (CEEMDAN) has been proposed as an enhanced version [34]. This method decomposes the original time series into multiple $IMFs$ and a residual, where each $IMF$ captures features at distinct time scales. Owing to its exceptional capability in handling nonlinear time-series data, CEEMDAN is widely adopted for complex sequence analysis and demanding decomposition tasks. It leverages uniformly distributed white noise to enhance separation of features across scales. The CEEMDAN procedure is as follows:

1.

Let $E_j(·)$ negotiate the operator extracting the $j$ $IMF$ via EMD. Define white noise ${\omega }_i,i=\mathrm{1,2},\dots ,I$. ${\omega }_i~N\left(0, 1\right)$

2.

Add white noise to the original signal $S\left(t\right)$ with noise coefficient ${\epsilon }_0$ (typically small):

3.

Perform EMD on $S_i\left(t\right)$

$$S_i\left(t\right)=S\left(t\right)+{\epsilon }_0{\omega }_i$$

(11)

and extract the first $IMF$. Define the first mode as

$$\overline{IM{F}_1}={\displaystyle \frac{1}{I}}{\sum }_{i=1}^{I}IM{F}_{i1}$$

(12)

4.

Compute the first residual:

$$r_1\left(t\right)=S\left(t\right)-\overline{IM{F}_1}$$

(13)

5.

Decompose the residual $r_1\left(t\right)+{\epsilon }_1{E}_1\left[{\omega }_i\left(t\right)\right]$ to extract the second $IMF$.

6.

Repeat until the final residual $R_M\left(t\right)$ is obtained:

$$R_M\left(t\right)=S\left(t\right)-{\sum }_{j=1}^M\overline{IM{F}_j}$$

(14)

where M is the total number of IMFs.

3.2. Multi-Scale Adaptive SSA Method

3.2.1. SSA

Singular Spectrum Analysis (SSA) maps a 1D time series to a high-dimensional space via an embedding matrix, extracting principal components through Singular Value Decomposition (SVD). This effectively identifies periodicities, trends, and local features. The SSA procedure is as follows.

For a length-N time series $x\left(t\right)$, construct embedding matrix $X$ using window size $L$ and lag $D$. The specific steps are as follows:

1.

Convert x(t) to matrix $X\in R(L-D)\times D$, where each column represents a local segment of the time series:

$$X_{ij}=x\left[t+\left(i-1\right)D\right]$$

(15)

2.

Apply SVD to $X$:

$$X=U\Sigma V^T$$

(16)

where $U$, $V^T$ are left/right singular vectors, and $\Sigma$ contains singular values ${\sigma }_k$.

3.

Select the top K singular values to reconstruct principal components:

$$X_K=U_K{\Sigma }_K{V}_K^T$$

(17)

4.

By retaining modes corresponding to larger singular values, noise is reduced in the signal to obtain the denoised output $\widehat{x}\left(t\right)$:

$$\widehat{x}\left(t\right)=\sum _{k=1}^K{m}_k\left(t\right)$$

(18)

where $m_k\left(t\right)$ denotes the k-th mode of the signal, and $K$ represents the number of selected principal modes. Through this process, noise components within the signal are eliminated while preserving its core features.

3.2.2. Improved SSA (ISSA)

However, SSA method exhibits inherent defects. Firstly, the sensitivity of window length L to periodic components constitutes a significant flaw. When the window length L is not an integer multiple of the signal’s dominant period T, it induces a spectral aliasing effect, causing periodic components to scatter across multiple adjacent components in the singular spectrum. This leads to inaccurate extraction of periodic information. Secondly, the energy thresholding method frequently fails in practical applications. This is because transient events and noise exhibit overlapping energy characteristics in non-stationary signals, making it difficult for a single energy threshold to effectively distinguish valid signal components from noise components. Consequently, signal screening based on energy thresholds struggles to reliably separate valid signals from noise. Finally, SSA suffers from static parameterization: a fixed L value cannot adapt to the time-varying spectral characteristics of signals. This severely limits the effectiveness and applicability of SSA when processing complex, dynamically changing signals. To circumvent these defects, this study proposes the following Improved SSA (ISSA) method.

1.

Dynamic Window Optimization Mechanism

Introduce permutation entropy-driven adaptive window selection. Specifically, determine the optimal window length $L_{opt}$ by minimizing the permutation entropy function $H_p$:

$$L_{opt}=arg\underset{L\in \left[L\mathit{min},L\mathit{max}\right]}{\mathit{min}}{H}_p\left[X\left(L\right)\right]$$

(19)

where the permutation entropy function $H_p$ is defined as

$$H_p\left(X\right)=-\sum _{j=1}^{m!}{p}_j\mathit{ln}{p}_j$$

(20)

Here, $p_j$ represents the probability of occurrence of the j-th permutation pattern in the symbolic sequence of the trajectory matrix X.

2.

Multi-Feature Fusion Anomaly Detection

Feature Extraction Model

For each singular component $s_i$, extract a three-dimensional feature vector $f_i$

$$f_i=\left[\begin{array}{c}{E}_i\\ H_i\\ R_i\end{array}\right]=\left[\begin{array}{c}\sum s_i^2\\ H_p\left(s_i\right)\\ corr\left[s_i\left(t\right),s_i\left(t+1\right)\right]\end{array}\right]$$

(21)

where $E_i$ is the energy feature, $H_i$ is the permutation entropy feature, and $R_i$ is the autocorrelation coefficient feature.

Anomaly Detection Model

An unsupervised detection method based on isolation forest is used to identify anomalous components. Specifically, compute the anomaly score $\varphi \left(f_i\right)$ for each feature vector $f_i$:

$$\varphi \left(f_i\right)={\displaystyle \frac{1}{T}}\sum _{t=1}^T{h}_t{f}_i$$

(22)

where $h_t$ is the path length of the t-th isolation tree. If $\varphi \left(f_i\right)<\tau$, then $s_i$ is classified as a noise component.

3.

Secondary Processing of Noise Components

Sample Entropy Classifier

Utilize Sample Entropy to further classify noise components. Sample Entropy is defined as

$$S_{amp}{E}_n(m,r)=-ln{\displaystyle \frac{A^m\left(r\right)}{B^m\left(r\right)}}$$

(23)

where $B^m\left(r\right)$ is the number of template matches, and $A^m\left(r\right)$ is the number of offset matches.

Classification Decision

Classify components according to the value of Sample Entropy:

$$IM{F}_i\in \left\{\begin{array}{c}Low frequency,if S_{amp}{E}_n\left(IM{F}_i\right)\le \theta \\ High frequency,otherwise\end{array}\right.$$

(24)

The flowchart of the ISSA algorithm is shown in Figure 9.

3.3. Enhanced Transformer Prediction Model

3.3.1. Frequency-Enhanced Channel Attention Mechanism (FECAM)

Fourier Transform (FT) introduces high-frequency components due to periodicity assumptions, leading to Gibbs phenomena (boundary information artifacts), while complex computations and inverse transforms increase computational overhead. To address these limitations, FECAM leverages the Discrete Cosine Transform (DCT), fundamentally circumventing Gibbs phenomena and eliminating unnecessary inverse transformation costs. Furthermore, to better exploit inter-variable relationships in multivariate time series, we propose the Frequency-Enhanced Channel Attention Mechanism—a universal framework that significantly enhances mainstream models’ forecasting capabilities on real-world temporal data, as shown in Figure 10.

First, FECAM partitions the input feature map along the channel dimension into n subgroups: [v₀, v₁,……, v_n−1], where each subgroup Vⁱ = R^1×L, $i\in \{0, 1,\dots \dots , n - 1\}$, n = N_v. Each subgroup is then processed by its corresponding DCT frequency components in ascending order (lowest to highest frequency). Within subgroups, identical frequency components are applied per channel according to

$$Fre{q}^i=DC{T}_J\left(V^i\right)=\sum _{j=0}^{j=L_{s-1}}\left(V_l^i\right)B_l^j$$

(25)

where $i\in \{0, 1,\dots \dots , Nv-1\}$,$j \in \{0, 1,\dots \dots , Ls-1\}$, with j denoting the one-dimensional frequency component index corresponding to subgroup Vⁱ. Each Freqⁱ $\in$R^L represents the L-dimensional vector obtained via Discrete Cosine Transform. The full channel attention vector is then constructed through the stack operation:

$$Freq=DCT\left(V\right)=stack\left(\left[Fre{q}^0,Fre{q}^1,\dots ,Fre{q}^{n-1}\right]\right)$$

(26)

where $Freq\in R^{C*L}$ denotes the attention vector for $V\in R^{C*L}$. The frequency-enhanced channel attention framework is formally expressed as

$$F_c-att=\sigma \left(W_2\delta \left(W_{1}DCT\left(V\right)\right)\right)$$

(27)

This mechanism establishes interactions between channel features and spectral components, comprehensively capturing critical temporal information in the frequency domain. Consequently, it enhances feature diversity during network extraction while maintaining computational efficiency.

3.3.2. F-Transformer: A Parallel Framework Integrating FECAM and Transformer

The Transformer architecture, originally proposed for sequence analysis, eliminates recurrence and convolution by relying solely on attention mechanisms [35]. Compared to conventional approaches, the self-attention mechanism demonstrates superior capability in capturing intrinsic dependencies within both data and feature representations. This empowers the Transformer to model complex input–output positional relationships while eliminating sequential processing constraints. Although initially designed for machine translation, Transformer variants now permeate diverse domains. Similarly to natural language, oscillatory signals constitute one-dimensional sequential data composed of superimposed positive and negative waveforms. Consequently, owing to its exceptional long-sequence modeling capabilities, the Transformer architecture has gained widespread adoption in time-series-related tasks in recent years [36]. The Transformer model fundamentally comprises three core components: an embedding layer, encoder, and decoder [37].

Conventional Transformer models face two principal limitations in time-series forecasting applications: computationally prohibitive complexity and inadequate positional encoding. To address these constraints, we introduce a streamlined architecture specifically optimized for univariate time-series prediction. The modified workflow proceeds as follows.

1.

Input Embedding

Given an input sequence $X\in R^{L\times 1}$ (where $L$ denotes sequence length), we first project it into a $d_{model}$-dimensional space via linear transformation:

$$E=X{W}_e+b_e$$

(28)

where $W_e\in R^{1\times d_{model}} \mathrm{a}\mathrm{n}\mathrm{d} b_e\in R^{d_{model}}$ are learnable parameters.

2.

Positional Encoding

To preserve temporal order information of the sequence, we inject positional encodings. Positional encodings have the same ${\mathrm{d}}_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}$ as the embedding vectors and are generated using the sine and cosine functions:

$$P{E}_{\left(pos, 2i\right)}=\mathit{sin}\left({\displaystyle \frac{pos}{{1000}^{2i/d_{model}}}}\right)$$

(29)

$$P{E}_{\left(pos, 2i+1\right)}=\mathit{cos}\left({\displaystyle \frac{pos}{{1000}^{2i/d_{model}}}}\right)$$

(30)

where $pos$ denotes the time step position and $i$ represents the dimension index. The positional encodings are then additively combined with the embedding vectors:

$$Z=E+P$$

(31)

where $\mathrm{P}\in R^{\mathrm{L}\times d_{model}}$ constitutes the positional encoding matrix.

3.

Multi-Head Self-Attention

The self-attention mechanism computes contextual representations by evaluating pairwise dependencies between sequence elements. First, the input $Z$ is projected into query (Q), key (K), and value (V) subspaces:

$$Q=Z{W}_Q,K=Z{W}_K,V=Z{W}_V$$

(32)

where projection matrices satisfy $Q,K,V\in R^{d_{model}\times d_k}$ and $d_k=d_{model}/h$ ($h$ is number of attention heads).

Scaled dot-product attention is then computed:

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

(33)

Multi-head attention concatenates outputs from parallel attention heads and projects:

$$MultiHead\left(Q,K,V\right)=Contact\left(hea{d}_1,\dots ,hea{d}_n\right)W_O$$

(34)

where $W_O\in R^{\mathrm{h}·d_k\times d_{model}}$. Each attention head operates independently:

$$hea{d}_i=Attention\left(Q_i,K_i,V_i\right)$$

(35)

4.

Feed-Forward Neural Network

Each Transformer layer incorporates a position-wise feed-forward network (FFN) composed of two linear transformations with intermediate activation:

$$FFN\left(x\right)=max\left(0,x{W}_1+b_1\right)W_2+b_2$$

(36)

where ${\mathrm{W}}_1\in {\mathrm{R}}^{{\mathrm{d}}_{\mathrm{f}\mathrm{f}}\times {\mathrm{d}}_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}}, {\mathrm{W}}_2\in {\mathrm{R}}^{{\mathrm{d}}_{\mathrm{f}\mathrm{f}}\times {\mathrm{d}}_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}},{\mathrm{d}}_{\mathrm{f}\mathrm{f}}$ denoting the hidden dimension (typically ${\mathrm{d}}_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}$).

5.

Layer Normalization and Residual Connections

Residual connections and layer normalization are applied around each sublayer (self-attention and FFN):

$$z=LayerNorm\left[x+Sublayer\left(x\right)\right]$$

(37)

6.

Output Layer

The position-indexed output representation at the final time step is projected to generate predictions:

$$\widehat{y}=z_L{W}_d+b_d$$

(38)

where $z_L$ denotes the output at the last time step from the final Transformer layer, ${\mathrm{W}}_{\mathrm{d}}\in {\mathrm{R}}^{1\times {\mathrm{d}}_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}}}$.

7.

Temporal Optimization Strategies

We employ a compact architecture with reduced model scale (e.g., $d_{model}$, encoder layers = 2) to minimize computational overhead.

The Gaussian Error Linear Unit (GELU) replaces standard activations in embedding layers, demonstrating superior performance over ReLU in Transformer architectures. Its formulation is

$$GELU\left(x\right)=x\Phi \left(x\right)$$

(39)

where $\Phi \left(x\right)$ denotes the cumulative distribution function of the standard Gaussian distribution.

The integrated framework combining FECAM and Transformer (parenthetically termed F-Transformer) effectively synergizes their respective strengths in feature extraction and temporal forecasting. The incorporation of Transformer enables accurate time-series prediction with reduced computational complexity.

As depicted in Figure 11, the architecture applies the Frequency-Enhanced Channel Attention Mechanism (FECAM) between the encoder and decoder. Through Discrete Cosine Transform, FECAM applies adaptive weighting to encoder outputs, directing attention to the most critical sequence components. By adaptively modeling frequency-wise interdependencies across channels via DCT, this mechanism fundamentally circumvents high-frequency noise induced by problematic periodicity inherent in Fourier Transform processes, thereby enhancing the Transformer’s predictive performance. This integration strategy simultaneously leverages the Transformer’s global information extraction capability and FECAM’s local temporal relationship modeling capacity.

3.4. Hybrid Forecasting Framework Development

Following data preparation, the processed time series is fed into our proposed hybrid model. This framework addresses core challenges in non-stationary time series forecasting—noise interference, multi-scale features, and long-term dependencies—through a three-tier architecture:

• Decomposition Layer: Resolving Non-stationarity via CEEMDAN

Non-stationary signals are decomposed into finite intrinsic mode functions (IMFs). Based on sample entropy thresholds, components are automatically classified: low-entropy IMFs are retained as trend terms, while high-entropy IMFs are designated as noise components requiring purification.

2.

Denoising Layer: Mitigating Noise Contamination via ISSA

An adaptive window selection mechanism minimizes permutation entropy to construct a three-dimensional feature space (energy-entropy-correlation). Unsupervised anomaly detection is then performed using Isolation Forest, followed by secondary decomposition and denoising to generate refined signal components.

3.

Prediction Layer: Modeling Long-term Dependencies via Transformer

Denoised high-frequency and low-frequency components are fused for multi-step forecasting. Under shared-weight learning of universal temporal patterns, independent predictions are generated for each component. Final reconstruction integrates all component forecasts to produce the output prediction.

The integrated data processing and hybrid forecasting architecture is illustrated in Figure 12.

4. Model Performance Evaluation

4.1. CEEMDAN Decomposition Results

This study employs the data source described in Section 2.5 as input, specifically analyzing response data from 400 s to 540 s during stable OWT thrust under combined wind–ice loading. Prior to denoising, signals undergo CEEMDAN decomposition. Taking the tower-top displacement of EC2 as a representative case—where maximum wind and ice loads induce the strongest stochasticity—this scenario maximizes performance differentiation between models. Initial data preprocessing includes standardization to eliminate dimensional effects and enhance algorithmic stability:

$${Signal}_{norm}={\displaystyle \frac{Signal-{Signal}_{mean}}{{Signal}_{std}}}$$

(40)

where ${Signal}_{mean}$ denotes the mean and ${Signal}_{std}$ the standard deviation of the raw signal.

To characterize unmodeled dynamics and environmental stochasticity in numerical simulations, reflect transient energy dissipation of ice loads, and enhance correlation between simulated data and field-measured responses under actual sea conditions, we introduce low-frequency noise components conforming to ocean environmental spectra into displacement signals. The noise amplitude was calibrated via parametric sensitivity analysis to 18% of the signal’s standard deviation. A hybrid pink-Gaussian noise model was employed across 300 ensemble iterations. Sample entropy parameters were configured with: embedding dimension $m=2$, tolerance $r=0.2$, and entropy threshold = 1.2 (components with sample entropy > 1.2 classified as high-entropy). Preprocessing, CEEMDAN decomposition, and computational results are presented in Figure 13 and

Table 3. Computed Signal Characteristics Post-CEEMDAN Decomposition.

Type	Mean	Kurtosis	Sample Entropy	High Entropy
IMF1	−2.910699 × 10−3	3.057349	1.376894	Yes
IMF2	2.944852 × 10−3	1.944058	0.885758	No
IMF3	−7.187959 × 10−3	−0.443757	0.580185	No
IMF4	2.890170 × 10−4	−0.144773	0.507726	No
IMF5	−2.045220 × 10−2	−0.045354	0.158690	No
IMF6	−4.088522 × 10−2	−0.602115	0.075034	No
IMF7	6.820146 × 10−2	−1.286228	0.022810	No
Residue	3.984172 × 10−19	3.623493	0.000000	No

.

The residual component exhibits substantial amplitude fluctuations post-decomposition, with IMF1 demonstrating a sample entropy of 1.376894 that exceeds the 1.2 threshold. This indicates persistent strong nonlinearity and low predictability in the CEEMDAN-decomposed signal, necessitating additional decomposition and denoising procedures.

4.2. Effectiveness Analysis of Improved Singular Spectrum Analysis (ISSA)

To enhance the nonlinear characteristics of high-entropy component IMF1 obtained from CEEMDAN decomposition, ISSA-based denoising is applied. Addressing spectral leakage caused by fixed window lengths in conventional SSA, we propose a permutation entropy-minimized dynamic window optimization mechanism. As illustrated in Figure 14, the algorithm automatically determines the optimal window length that minimizes permutation entropy by systematically evaluating its variation trend $L_{opt}$ within the prescribed interval $\left[\mathrm{10,100}\right]$. When the optimal window length $L_{opt}=10$ is attained, the permutation entropy value reaches its global minimum $1.0782$—evidencing maximal signal orderliness and optimal capture of deterministic structures. ISSA dynamically optimizes window length through permutation entropy minimization, thereby achieving optimal signal regularity. This adaptive window selection mechanism significantly enhances SSA’s adaptability to complex signals, particularly accommodating the non-stationary characteristics of offshore wind turbine (OWT) response signals. When confronting complex environmental influences, ISSA leverages three key capabilities: (1) real-time decomposition granularity adjustment via adaptive windows to mitigate wind speed fluctuations; (2) enhanced transient feature preservation through optimal windows to withstand surge impacts; and (3) effective suppression of spurious component generation using low-entropy windows to resolve turbulent noise interference.

Figure 15 illustrates the core principle of the multi-feature fusion anomaly detection mechanism. Within the three-dimensional feature space constructed by energy, permutation entropy, and autocorrelation, the Isolation Forest algorithm effectively discriminates between valid components and noise components. As evidenced, noise components predominantly occupy low-energy regions with minimal physical information content, whereas valid components cluster in high-energy zones carrying essential signal information—all exhibiting permutation entropy values below 1. Crucially, significant overlap in permutation entropy values between both component types highlights the limitation of single-threshold energy methods. The distinct separability achieved in this 3D feature space validates the efficacy of the multi-feature fusion strategy, providing a robust foundation for anomaly detection.

Following processing via the improved Singular Spectrum Analysis (ISSA), we obtain eight decomposed and denoised components, as illustrated in Figure 16.

After treatment with the improved SSA method, the noise level of IMF1 is reduced from 0.0973 to 0.0187. The resulting eight SIMF components will enter the F-Transformer prediction module alongside the eight low-frequency IMF components derived from CEEMDAN decomposition.

To validate the superiority of Improved SSA over conventional SSA, this study processes IMF1 using traditional SSA for comparative analysis. Figure 17 demonstrates the denoising performance comparison between conventional SSA and Improved SSA on the IMF1 component. The outcome demonstrates clearly that Improved SSA more effectively suppresses high-frequency noise while preserving critical signal features. Compared to the residual high-frequency oscillations in conventional SSA results, the residual signal generated by the improved method exhibits significantly reduced amplitude and more uniform distribution. This phenomenon confirms that the enhanced approach—through its adaptive window selection mechanism and multi-feature fusion anomaly detection—achieves superior balance between signal fidelity and noise suppression.

Figure 18 and Figure 19 reveal fundamental differences in component independence between the two methods. Conventional SSA (Figure 18) exhibits significant component coupling. A particualr evidence, a correlation coefficient of 1.00 between SIMF1 and SIMF2, indicates component redundancy. In contrast, the correlation coefficient matrix of Improved SSA (Figure 19) demonstrates a more desirable diagonal-dominant pattern, with substantial improvements in off-diagonal elements. This optimization stems from the precise identification and elimination of noise components by the isolation forest algorithm, ensuring high independence among valid components and establishing a more reliable foundation for subsequent fault feature extraction.

The quantitative superiority of the proposed ISSA method is unequivocally demonstrated in Figure 20. Its drastic performance improvement, particularly in noise suppression (76.4% reduction) and SNR enhancement (705.3% improvement), stems directly from its core innovations: the permutation entropy-driven adaptive windowing that eliminates spectral leakage, and the multi-feature fusion anomaly detection that intelligently discriminates noise from signal in a 3D feature space. This makes ISSA a robust and indispensable preprocessing step for noisy OWT vibration data.

From an engineering perspective, the 76.4% noise suppression rate and 31.86% reduction in MSE directly contribute to more accurate fatigue load estimation. By effectively isolating high-frequency noise components induced by ice impacts and turbulent wind, this method provides cleaner vibration signals for fatigue analysis. This leads to more reliable stress cycle counting and damage accumulation calculations, ultimately extending the predicted fatigue life of critical components such as the monopile foundation and tower welds. In practical engineering terms, this means that conservative safety margins in design can be reduced, and the intervals between structural inspections can be extended.

In summary, the ISSA method resolves spectral leakage issues through permutation entropy-driven adaptive windowing while significantly enhancing physical information extraction in high-noise environments via intelligent discrimination in three-dimensional feature space, providing a more robust preprocessing solution for OWT tower-top displacement prediction. This approach is particularly suitable for processing non-stationary signals affected by complex marine disturbances, pioneering new pathways for wind turbine structural health monitoring.

4.3. Prediction Performance Testing

This section conducts prediction experiments on power output and tower-top displacement for a 5 MW monopile offshore wind turbine. First, multi-step predictions for power and tower-top displacement are performed within a single operational condition to validate model accuracy. Subsequently, multi-condition coupled predictions simulate the turbine’s response in realistic complex offshore environments, thereby verifying the model’s applicability under intricate conditions. To investigate the effectiveness of the improved F-Transformer prediction model, the Transformer model is selected for comparative analysis.

Three metrics evaluate model performance: Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and Coefficient of Determination (R²). MSE represents the average squared difference between predicted and actual values. R² ranges [0, 1], where 1 indicates perfect prediction and 0 denotes complete failure to explain data variance. RMSE equals the square root of MSE. Their formulations are

$$MSE={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\left(y_i-\widehat{y_i}\right)}^2$$

(41)

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

(42)

$$R^2=1-{\displaystyle \frac{{\left(y_i-\widehat{y_i}\right)}^2}{{\left(y_i-\overline{y_i}\right)}^2}}$$

(43)

where $y_i$ is actual value, $\widehat{y_i}$ is predicted value, $\widehat{y_i}$ is mean of actual values, and $n$ is sample size.

4.3.1. Hyperparameter Settings

To eliminate non-model structural influences and validate the effectiveness of the decomposition-denoising method, input data undergoes preprocessing via CEEMDAN-ISSA. Identical activation functions, optimizers, and hyperparameters are applied to all models as specified in

Table 4. Hyperparameter Settings.

Hyperparameter Settings
Model Dimension	32
Attention Heads	8
Encoder Blocks	2
Encoder Blocks	1
Decoder Blocks	128
FNN Dimension	$2\sqrt{2}$
Attention Factor	Gelu
Activation Function	0.001
Learning Rate	Adam
Optimizer	1
Output Dimension	64

. Deep learning models were developed in Python 3.11 and PyTorch 2.0. Since optimal neuron counts remain indeterminate, model configurations reference established research. Network hyperparameters—including model dimension, activation function, training set size, optimizer, and attention factor—impact prediction performance variably. To ensure test comparability, both models use identical hyperparameters from Table 4.

4.3.2. Single Operational Condition Prediction Performance Testing

In practical applications, marine environments typically do not change significantly over short periods. Therefore, data collected under similar or identical sea conditions is often used to predict OWT vibration in current scenarios. This approach relies on historical observations, assuming minimal environmental variations to effectively infer OWT behavior under present conditions. Compared to the Transformer model, the F-Transformer model learns governing patterns of tower-top vibration from data, enabling a more comprehensive understanding of nonlinear dynamics and maintaining superior persistence as prediction horizons extend. To validate the proposed F-Transformer’s superiority in OWT vibration prediction accuracy, this section uses EC2 tower-top displacement for testing, with the first 80% of data as the training set and the remaining 20% as the prediction set.

As shown in

Table 5. Accuracy Comparison of Transformer and F-Transformer Models across Prediction Steps in EC2.

Prediction Step	F-Transformer			Transformer			Compared to Transformer Prediction
	MSE	RMSE	${\mathbf{R}}^2$	MSE	RMSE	${\mathbf{R}}^2$	MSE	RMSE	${\mathbf{R}}^2$
1	0.000077	0.008786	0.941057	0.000082	0.009044	0.937544	+5.63%	+2.85%	+0.37%
10	0.000088	0.009385	0.930868	0.000153	0.012387	0.879558	+42.60%	+24.24%	+5.83%
20	0.000110	0.010504	0.913950	0.000149	0.012217	0.883600	+26.07%	+14.02%	+3.43%
30	0.000101	0.010071	0.921787	0.000131	0.011457	0.898773	+22.74%	+12.10%	+2.56%
40	0.000121	0.010993	0.907319	0.000129	0.011356	0.901086	+6.30%	+3.20%	+0.69%
50	0.000120	0.010940	0.908494	0.000147	0.012113	0.887822	+18.43%	+9.68%	+2.33%
60	0.000116	0.00824	0.912199	0.000156	0.012508	0.881450	+25.94%	+13.94%	+3.49%
70	0.000111	0.010530	0.916703	0.000163	0.012756	0.877764	+31.86%	+17.45%	+4.44%

, the F-Transformer achieves an average 21.1% reduction in MSE, indicating significantly minimized squared deviations between predicted and actual values; a 10.2% average decrease in RMSE demonstrates substantially reduced prediction error dispersion; and a 2.4% average increase in R² confirms enhanced capability to explain data variance. Under EC2, the F-Transformer consistently outperforms Transformer in prediction accuracy, particularly in medium-to-long-term forecasts (e.g., steps 60 and 70), where it maintains accuracy improvements. Within the 60–80 step range, the F-Transformer exhibits stable performance advantages. Unlike Transformer’s accelerated performance degradation with increasing step lengths, the F-Transformer shows more gradual degradation. Its superiority persists across extended prediction horizons, demonstrating robust and stable enhancements in error control and goodness-of-fit. This occurs because the conventional Transformer relies solely on time-domain attention, which is insensitive to periodic features. In contrast, the F-Transformer integrates FECAM to introduce frequency information via Discrete Cosine Transform (DCT) into channel attention, mitigating information loss during time-series feature extraction. The FECAM comprehensively captures periodic and fluctuating characteristics, particularly effective for complex oscillatory patterns. The fusion of FECAM and Transformer enables simultaneous modeling of short-term fluctuations and long-term trends. Consequently, the F-Transformer maintains high accuracy as prediction horizons expand.

The visual evidence presented in Figure 21. powerfully complements the quantitative metrics in Table 5. The markedly tighter confidence intervals of the F-Transformer, especially at long prediction horizons, are a direct visual manifestation of its superior performance. This enhanced stability is a key contribution of our work and is attributable to the FECAM module, which enriches the Transformer’s attention mechanism with critical frequency-domain information, allowing the model to better learn the underlying periodicities and transients in ice-induced vibration signals, thus reducing error propagation over time.

From an engineering perspective, the improved ultra-short-term prediction capability (R² > 0.91 under multi-condition testing) enables proactive maintenance scheduling and early detection of abnormal vibration patterns. By accurately forecasting tower-top displacements up to 70 steps ahead, operators can anticipate extreme load events and initiate preventive measures (e.g., yaw adjustment or temporary shutdown) before structural limits are exceeded. This predictive capacity minimizes the risk of catastrophic failures and reduces unplanned downtime, thereby enhancing overall energy availability and economic viability.

4.3.3. Multi-Condition Coupled Prediction Performance Testing

In practical marine environments, the dynamic variability of sea conditions is particularly critical for OWT vibration forecasting. This stems from the fundamental distinction between ocean environments and controlled laboratory settings: laboratories operate under relatively stable conditions, while marine environments are highly dynamic due to multiple interacting factors. These include tidal cycles, storm-induced disturbances, ice drift velocity variations, and shifting ocean current patterns. Such complex, uncertain, and interdependent factors render marine systems inherently unstable.

Consequently, these time-varying elements demand that prediction models not only capture environmental changes in real-time but also maintain accuracy amid fluctuations. A model’s ability to adapt to such dynamics directly determines forecast reliability. Failure to address these complexities may increase prediction errors and compromise operational safety. Thus, prediction models must both accommodate diverse sea-state transitions and sustain high precision under uncertainty.

This adaptability is vital for forecasting technology advancement. As marine conditions evolve and technology progresses, models require continuous refinement to enhance their responsiveness to complex scenarios. High-accuracy predictions also facilitate optimal resource utilization—enabling timely yaw adjustments and emergency shutdowns during extremes, thereby minimizing resource waste.

To validate the F-Transformer’s robustness in real sea conditions, this experiment uses a dataset combining three distinct operational conditions (EC1–EC3). Crucially, data junctions retain abrupt transitions without smoothing preprocessing, intentionally introducing significant instantaneous mutations. If F-Transformer maintains accurate predictions despite these shocks, it confirms practical applicability. As shown in Figure 22, the first 500 data points from each condition are merged into a 2000-point dataset. Model hyperparameters follow Section 4.3.1, with an 80:20 train–test split.

Table 6. Prediction Accuracy of F-Transformer Model across Steps under Coupled Conditions.

Prediction Step	MSE	RMSE	${\mathbf{R}}^2$
1	0.000169	0.012995	0.964791
10	0.000340	0.018452	0.927990
20	0.000375	0.019366	0.919627

presents MSE, RMSE, and R² for multi-step predictions under coupled conditions. Despite non-smooth data with abrupt junctions, all prediction metrics remain excellent. This demonstrates F-Transformer’s capacity to maintain high accuracy even during sudden environmental shifts in actual sea states.

Figure 23 compares prediction results across varying step lengths under coupled conditions. As prediction horizons extend, the divergence between forecasted and observed values gradually increases. Specifically, MSE and RMSE progressively rise while R² shows a decreasing trend, indicating declining model precision over longer forecasting steps.

This trend is more pronounced in the prediction confidence interval plots. As the prediction step length increases, the distribution of forecasted values gradually deviates from the median line of true values and exhibits greater dispersion, indicating that prediction uncertainty also amplifies with longer horizons. In other words, prediction results become more scattered and accuracy progressively diminishes as step lengths extend.

However, despite the increased prediction errors at longer steps, the F-Transformer model maintains high forecasting precision overall when handling coupled conditions. Even under extended prediction horizons, its forecasts remain closely aligned with true values, demonstrating robust performance within certain prediction horizons. Collectively, the F-Transformer model delivers reliable forecasts despite dynamically changing marine operating conditions.

From an engineering perspective, the robustness of the F-Transformer under coupled and transient environmental conditions (e.g., abrupt transitions between EC1–EC3) ensures reliable performance in real-world scenarios. The model’s ability to maintain high accuracy amid sudden ice load shifts and wind variations supports safer operation in ice-prone waters. By providing timely and accurate vibration forecasts, the system aids in avoiding resonance conditions and excessive load cycles, which are critical for preventing structural failures and ensuring crew safety.

5. Conclusions

This study addresses the challenge of predicting dynamic responses for OWT in cold regions under combined wind–ice loading by proposing an ultra-short-term forecasting framework integrating CEEMDAN-ISSA secondary decomposition and F-Transformer. Through systematic validation, key conclusions are drawn: The developed DEM-WTIA model successfully achieves end-to-end simulation of ice-breaking processes and turbine dynamic responses, providing high-fidelity data sources for prediction. The CEEMDAN-ISSA secondary decomposition mechanism resolves mode-mixing issues in non-stationary signals, with the ISSA method—featuring dynamic window optimization and multi-feature anomaly detection—achieving a 76.4% noise suppression rate while significantly outperforming conventional methods in signal fidelity. This mechanism demonstrates strong robustness against transient impacts and turbulent noise in complex marine environments. The proposed F-Transformer model integrates the Frequency-Enhanced Channel Attention Mechanism (FECAM), introducing Discrete Cosine Transform (DCT) into the Transformer architecture to excavate hidden frequency-domain features and effectively capture latent periodicity in discontinuous data. By fusing these features with the Transformer’s self-attention-based outputs, the system comprehensively learns deep temporal patterns. Experimental evaluations confirm F-Transformer’s excellence: in single-condition tower-top displacement prediction, it substantially enhances long-term stability versus standard Transformer with a 31.86% reduction in MSE and 4.44% increase in R² at 70-step forecasts, alongside tighter confidence interval distributions. Multi-condition coupled tests further validate its generalization capability in dynamic sea states, maintaining high accuracy (R² > 0.91 at 20-step predictions) despite abrupt environmental shifts.

Crucially, the performance enhancements of this framework can be directly translated into tangible engineering value. The improved prediction accuracy and stability can form the core of an early warning system for ice-induced vibrations, providing crucial lead time for operational adjustments—such as yaw control and power curtailment—to mitigate extreme loads. This capability is essential for reducing the accumulation of fatigue damage, extending the service life of critical components, and enhancing the operational safety and reliability of OWTs in ice-prone regions. Furthermore, highly accurate predictions can be seamlessly integrated into digital twin systems, enabling proactive maintenance planning and optimizing the lifecycle cost of offshore wind farms.

Although promising results have been achieved, it is imperative to acknowledge the limitations of this study to objectively contextualize its contributions and scope. A primary limitation stems from the exclusion of wave loads in the environmental loading conditions. While such a simplification is commonly employed in initial research phases to isolate the fundamental mechanisms of wind–ice–structure interaction—and is partly justified by the wave-damping effect of sea ice—it inevitably reduces the model’s completeness in representing real-world combined loading scenarios. Moreover, the validation in this study relies exclusively on high-fidelity numerical simulations. Although the numerical model was rigorously calibrated, the absence of field-measured data remains a constraint that limits the immediate generalizability of the findings. Furthermore, the proposed hybrid forecasting framework (CEEMDAN-ISSA and F-Transformer) is inherently data-driven, targeting the essential challenge of modeling non-stationary and nonlinear time series. This characteristic enhances its generalizability and suggests potential applicability to dynamic response prediction in other offshore structures, such as floating wind turbines, jacket platforms, and ship motions. Future work should focus on validating its performance across these diverse structural types.