Optimization Design of Lazy-Wave Dynamic Cable Configuration Based on Machine Learning

Abstract

The safe and efficient design of dynamic submarine cables is critical for the reliability of floating offshore wind turbines, yet traditional time-domain simulation-based optimization approaches are computationally intensive and time consuming. To address this challenge, this study proposes a closed-loop optimization framework that couples machine learning with intelligent optimization algorithms for a dynamic cable configuration design. A high-fidelity surrogate model based on a backpropagation (BP) neural network was trained to accurately predict cable dynamic responses. Three optimization algorithms—Particle Swarm Optimization (PSO), Ivy Optimization (IVY), and Tornado Optimization (TOC)—were evaluated for their effectiveness in optimizing the arrangement of buoyancy and weight blocks. The TOC algorithm exhibited superior accuracy and convergence stability. Optimization results show an 18.3% reduction in maximum curvature while maintaining allowable effective tension limits. This approach significantly enhances optimization efficiency and provides a viable strategy for the intelligent design of dynamic cable systems. Future work will incorporate platform motions induced by wind turbine operation and explore multi-objective optimization schemes to further improve cable performance.

1. Introduction

As an essential direction in the development of renewable energy, the offshore wind power industry is undergoing a critical phase of strategic transformation [1]. Nearshore wind power development faces multiple constraints: limited resources, stringent ecological protection requirements, spatial competition with other marine economic activities, and relatively scattered site distributions. In particular, when the water depth exceeds 60 m, the economic viability of fixed foundations declines significantly [2,3]. In contrast, deep-sea regions offer distinct advantages, including abundant wind energy resources, vast development space, and fewer restrictive factors. Floating foundations provide a more cost-effective solution for harnessing wind energy in deep-sea areas, making floating offshore wind power technology a key approach for deep-sea offshore wind energy development and one of the main trends in the evolution of floating foundations for deep-sea wind power.

Submarine cables, serving as the conduit for transmitting electrical power from floating wind turbines to the shore, can be categorized into dynamic and static cables based on their operating conditions [4]. Dynamic cables span from floating wind turbines to subsea anchoring devices. Unlike fixed-bottom wind turbines, floating wind turbines allow the platform to move within a certain range. A well-designed dynamic cable configuration can mitigate the loads experienced by the cable in harsh environments and extend its fatigue life. Therefore, optimizing cable configurations is of significant importance.

In the field of flexible riser and umbilical cable configuration optimization, which shares similar performance characteristics with dynamic submarine cables, scholars worldwide have established a relatively comprehensive research framework. Tanaka et al. [5] formulated an objective function for minimizing dynamic stress and systematically evaluated the multi-condition performance of lazy-wave risers, innovatively introducing an analytical framework that couples marine environmental parameters with wave–current directions. Li et al. [6] developed a multi-objective optimization model for buoyancy block placement in steep-wave risers based on the NSGA-II algorithm for floating production storage and offloading (FPSO) systems, effectively addressing design challenges under extreme operational constraints. Cunliffe et al. [7] overcame the path dependence of traditional optimization methods on initial conditions by developing an adaptive optimization framework based on genetic algorithms. Vieira et al. [8] further incorporated cost control into the objective function, achieving the multi-parameter collaborative optimization of segment lengths in lazy-wave configurations. Chen et al. [9] proposed a radial basis function surrogate model strategy, significantly enhancing the design efficiency of ultra-shallow-water risers by optimizing Latin hypercube sampling and hybrid optimization algorithms. Liu et al. [10] established a three-dimensional objective space encompassing the drift region, configuration curvature, and economic cost. They applied a multi-objective optimization approach to quantify the optimal buoyancy block configuration for steep-wave risers.

Comparatively, systematic research on the design and optimization of typical dynamic cable configurations is still in its developmental stage. Zhao et al. [11] demonstrated the superiority of the double-wave configuration in floating offshore wind power systems through hydrodynamic performance experiments and fatigue damage analysis. Shi et al. [12] conducted a parameter sensitivity analysis to reveal the critical failure mechanisms of W-shaped dynamic cables under extreme environmental conditions. Yu et al. [13] established a comparative analysis framework for shallow-water lazy-wave and lazy-S dynamic cables using the OrcaFlex 9.7a software, while Zou et al. [14] developed a multi-objective optimization model for steep-wave configurations based on the simulated annealing algorithm. Poirette et al. [15] proposed a multi-scale optimization method that integrates a cross-sectional stress–global load coupling model to optimize the life-cycle cost of lazy-wave dynamic cables for offshore wind power applications.

With the breakthrough advancements in artificial intelligence, data-driven surrogate modeling techniques have opened new avenues for optimizing the design of flexible pipes and cables. Rasoul et al. [16] employed a Bayesian prediction model based on Gaussian process regression, improving the efficiency of fatigue failure probability analysis in the touchdown zone by five orders of magnitude. Yetkin et al. [17] utilized the NARX (nonlinear autoregressive with exogenous inputs) model to accurately capture the nonlinear characteristics of mooring systems. Yan et al. [18] proposed an interpolation–exceedance joint prediction method, effectively addressing challenges in predicting stochastic load responses. Guarize et al. [19] developed a hybrid ANN-FEM (Artificial Neural Network–Finite Element Method) computational framework, which reduced the time cost of dynamic response analysis by approximately 80% using a neural network surrogate model. Depina et al. [20,21] systematically compared the predictive performance of the NARX and WNN (Wavelet Neural Network) models, demonstrating the superior performance of the latter. Kim [22] introduced a quadratic Volterra model that successfully decoupled the nonlinear dynamic response characteristics of the Morison equation. In recent years, Cortina et al. [23] developed an FEM-ANN integrated prediction system, achieving efficient time-series predictions of riser touchdown point tension and bending curvature. Li [24] proposed an LSTM-based fatigue analysis framework that significantly improved fatigue analysis efficiency.

In summary, most existing studies on dynamic cable configuration optimization rely on traditional hydrodynamic software for time-domain analysis. However, due to the complexity of solving nonlinear governing equations, a single complete analysis consumes substantial computational resources. This study proposes a closed-loop solution that integrates rapid dynamic response prediction with coordinated configuration parameter optimization, enabled by the deep coupling of a neural network prediction model and optimization algorithms. The objective is to achieve a significant improvement in the efficiency of dynamic cable configuration design. Specifically, a high-accuracy surrogate model based on a BP (backpropagation) neural network is developed to replace the time-consuming traditional time-domain analysis. By thoroughly learning the input–output mapping relationships generated from conventional numerical simulations, this surrogate model is capable of replicating the dynamic response prediction function of traditional time-domain analysis with exceptionally high computational efficiency. Meanwhile, the introduction of intelligent optimization algorithms helps overcome the dimensional limitations of conventional parameter search methods. The organic integration of these two approaches provides a feasible strategy for the rapid prediction of dynamic cable responses and the improvement of overall configuration optimization efficiency. The detailed flowchart is shown in Figure 1.

2. Dynamic Cable Environmental Conditions and Structural Parameters

2.1. Environmental Conditions

2.1.1. Wave

Waves are generated by wind exerting force on the sea surface and can be characterized by the length between wave crests and troughs, influenced by factors such as wind speed, fetch, duration, and wave height. Based on in situ hydrological measurements provided by the project partner for a specific sea area in Hainan Province, the wave conditions are established in

Table 1. Wave conditions.

Operating Conditions	Wave Height (m)	Wave Period (s)	Wave Direction	Wind Speed (m/s)	Wind Direction
100-Year Return Period Wave under High Tide Level	11.8	9.4	NW	40	NW
100-Year Return Period Wave under Low Tide Level	10.62	9.1	NW	40	NW
Multi-Year Average Wave	1.0	3	NE	6	NE

.

2.1.2. Current

Ocean currents refer to the continuous movement of seawater in a specific direction, driven by various factors such as wind, density differences, temperature, gravity, salinity, and coastal topography. Based on in situ hydrological measurements provided by the project partner for a specific sea area in Hainan Province, the current velocity conditions are established in

Table 2. Current velocity conditions.

Type	One-Year Return Period Current Velocity (m/s)	Fifty-Year Return Period Current Velocity (m/s)
Surface Current	1.24	2.18
Mid-Layer Current	1.00	1.75
Near-Bottom Current	0.55	0.97

.

2.1.3. Establishment of Analysis Conditions

In this study, the dynamic cable is simulated under extreme conditions to investigate its dynamic response under wave and current effects. The total length of the dynamic cable is set to 200 m, with an average operating water depth of 119.5 m. The wave and current directions are aligned with the dynamic cable at an angle of 0°. The environmental conditions are composed of a 100-year return period wave and a 50-year return period current. The specific parameters for the simulation conditions are listed in

Table 3. Operating condition parameters.

Parameters		Extreme Condition
Wave	Wave Height (m)	11.8
	Wave Period (s)	9.4
Current	Surface Current (m/s)	2.18
	Mid-Layer Current (m/s)	1.75
	Near-Bottom Current (m/s)	0.97

.

2.2. Floating Platform

The floating platform parameters are provided by the project partner. The platform used in this simulation is a semi-submersible wind power platform with a radius of 37.5278 m, a structural depth of 32 m, a design draft of 13.5 m, and a displacement of 13,029 tons. The corresponding parameters of the floating platform are input into professional software, and a model is constructed based on the coordinates of various key points. The final modeling result is shown in Figure 2.

2.3. Dynamic Cable Structural Parameters

The three-core dynamic cable studied in this paper adopts a four-layer armored design. The cross-sectional structure of the cable is shown in Figure 3. The relevant cable parameters are referenced from the published literature [25], with specific details listed in

Table 4. Dynamic cable structural parameters.

Parameters	Values	Parameters	Values
Outer Diameter (mm)	Approximately 147.3	Torsional Stiffness (kN·m2)	209.8
Dry Weight (kg/m)	Approximately 45.90 (in air)	Minimum Bending Radius (m)	1.80
Wet Weight (kg/m)	Approximately 28.43 (underwater)	Minimum Breaking Force (kN)	1351.5
Axial Stiffness (MN)	631.7	Maximum Operating Force (kN)	330.5
Bending Stiffness (kN·m2)	10.3	Design Lifetime (years)	30

.

3. BP Neural Network Prediction Model

3.1. Dataset Construction and Preprocessing

3.1.1. Data Collection Based on Numerical Simulation

The data used in this study were collected through numerical simulations. Due to the high cost of experimental verification, this study did not directly validate the numerical model with experimental data. The numerical model used in this study is based on commercially available hydrodynamic software widely used for simulating dynamic cable responses. The modeling methods (including cable structure parameters, fluid dynamic coefficients, and boundary condition settings) follow established practices in the existing literature (e.g., references [13,26,27]). The construction process of the numerical simulation model involves the following key steps: First, the vessel module in the hydrodynamic software is used to model the floating platform, inputting the platform’s structural parameters and defining its geometric shape. Second, in the environment interface, environmental parameters such as the water depth, wave characteristics, ocean current velocity, and direction are specified to simulate real ocean conditions. Finally, the lines module is used to establish the numerical model of the dynamic cable. The line type section is configured with the key parameters of the dynamic cable, including outer diameter, unit length mass, axial stiffness, and bending stiffness. To simulate the connection characteristics between the dynamic cable and the floating platform, a bend stiffener is added at the top end of the dynamic cable where it connects to the platform. Additionally, the attachments module is used to represent the buoyancy blocks in the lazy-wave configuration. The hydrodynamic analysis of the dynamic cable in this study was conducted using the random wave method, with the JONSWAP spectrum employed to simulate irregular waves. The interaction between the seabed and the cable was modeled using the Coulomb friction model, with the sliding friction coefficient set to 0.5. Structural and hydrodynamic damping were represented by Rayleigh damping coefficients. The influence of the floating platform motion on the dynamic cable was considered, while the thrust effect of the wind turbine was neglected. Therefore, the numerical model of the floating platform was established without incorporating the wind turbine. The finalized numerical model of the lazy-wave dynamic cable is shown in Figure 4.

By performing simulations on the established numerical model, the dynamic response data of the dynamic cable under different configuration parameters are collected, including curvature distribution and effective tension distribution. These datasets serve as the sample data for the subsequent prediction model, providing critical insights for the study of dynamic cable behavior and optimization design.

3.1.2. Configuration Parameter Selection

In the preliminary design phase of dynamic cables, it is essential not only to prevent various functional failures of the cables but also to ensure that the fatigue life of the dynamic cable meets the relevant regulatory requirements. Among these factors, the curvature and effective tension of the dynamic cable have the greatest impact on its lifespan. Meanwhile, when the configuration parameters such as the cable length are determined, the arrangement of the buoyancy blocks and weight blocks is a key factor influencing the dynamic response of the lazy-wave dynamic cable system. Therefore, this study investigates the influence of the buoyancy block and weight block parameters on the curvature and effective tension of the dynamic cable. Relevant parameters include the length of the suspension section L₁ from the suspension point to the starting position of the buoyancy block, the number of buoyancy blocks S₁, the spacing between buoyancy blocks L₂, the starting position of the weight block L₃, the number of weight blocks S₂, and the spacing between weight blocks L₄. Specific parameters are illustrated in Figure 5. To ensure that the dynamic cable forms a lazy-wave configuration, the parameter ranges are determined as shown in

Table 5. Value range of parameters.

Limits	L1/m	S1/unit	L2/m	L3/m	S2/unit	L4/m
Minimum	80	8	3	20	4	3
Maximum	130	14	5	40	8	5

. To study the significance of the impact of each parameter on the curvature and effective tension of the dynamic cable, and to select the input parameters for the subsequent dataset, different buoyancy block and weight block parameters are chosen. Simulations of the dynamic cable under extreme conditions are performed to study the effects of these parameters on the maximum curvature and maximum effective tension.

3.1.3. Dataset Construction and Preprocessing

In this study, a dataset is constructed using the orthogonal design method, ensuring that the parameters remain within the specified ranges to form a lazy-wave configuration of the dynamic cable. The orthogonal design method is an efficient experimental design approach that allows comprehensive experiments with fewer experimental combinations, covering multiple factors and levels while significantly reducing computational efforts. A total of six configuration-related parameters were considered in this research: the length of the suspended section L₁ (ranging from 80 to 130 with an interval of 10), the number of buoyancy blocks S₁ (ranging from 8 to 14 with an interval of 1), the spacing between buoyancy blocks L₂ (ranging from 3 to 5 with an interval of 0.5), the starting position of the weight blocks L₃ (ranging from 20 to 40 with an interval of 5), the number of weight blocks S₂ (ranging from 4 to 8 with an interval of 1), and the spacing between weight blocks L₄ (ranging from 3 to 5 with an interval of 0.5). Among these six factors, one has 7 levels, one has 6 levels, and the remaining four have 5 levels each. To ensure the scientific rigor of the experimental design and the broad applicability of the results, a standard orthogonal table was used to determine the parameter combinations. In total, 87 data samples were generated for the dataset.

The different parameter combinations generated based on the orthogonal design method are used as input parameters for simulations, utilizing the numerical model discussed earlier. The output parameters, such as the curvature distribution and effective tension distribution of the dynamic cable, are then obtained. The complete dataset is divided into training, validation, and test sets in a ratio of 7:1.5:1.5. The distribution of key features was confirmed to be balanced across the subsets. During model training, to accelerate convergence, improve accuracy, prevent gradient vanishing or explosion, and enhance the model’s generalization ability, the dataset was normalized to the [0, 1] range. Normalization not only improves training efficiency but also enhances the model’s performance, providing reliable data support for subsequent optimization designs.

3.2. BP Neural Network Model Design

The BP (backpropagation) neural network is a classic multilayer feedforward neural network, trained through the error backpropagation algorithm. Its network structure is shown in Figure 6. The network consists of an input layer, hidden layers, and an output layer, with each layer comprising several neurons. The layers are interconnected through weights and thresholds to form the network structure.

There is no simple one-to-one mapping relationship between the dynamic cable configuration parameters and the dynamic response; rather, it is a complex relationship constrained by high nonlinearity. In this study, a BP neural network is used, which can achieve global nonlinear mapping between the configuration parameters and dynamic response without the need for explicitly defining the inherent relationships between the variables. This method, through training, can simulate a more accurate dynamic response distribution, characterizing the nonlinear functional relationship between the two. The BP neural network is used to build an approximate predictive model for dynamic cable configuration parameters and dynamic responses. This model can be applied to predict the dynamic response of the dynamic cable, providing references for the configuration design and optimization of the cable, helping to improve design efficiency and reduce experimental costs.

When training the BP neural network, it is necessary to adjust the parameters to determine the final structure of the network. Different problems may require different structures. The BP neural network selected in this study consists of 9 hidden layers, with the number of neurons in each hidden layer, from left to right, being 10, 10, 20, 40, 60, 40, 20, 10, and 10. The other parameters for the BP neural network are set as follows: the number of training iterations is 2000, the learning rate is 0.01, and the maximum number of validation failures is set to 20. During the training process, the error backpropagation algorithm is used to automatically adjust the weights (WH) between the input layer neurons and hidden layer neurons, as well as the thresholds (WO) between the hidden layer neurons and output layer neurons, to optimize network performance. The sample data generated using the orthogonal design method and the corresponding results obtained from simulations are input into the neural network for training. After multiple iterations and optimizations, a trained neural network model capable of accurately predicting the dynamic response of dynamic cables is obtained. This predictive model effectively characterizes the nonlinear relationship between dynamic cable configuration parameters and dynamic responses, providing theoretical support for the design and optimization of dynamic cables.

4. Catenary Optimization Based on Multiple Optimization Algorithms

In this study, the maximum curvature of the dynamic cable is used as the optimization objective, while the effective tension is constrained to ensure that it does not exceed the allowable tension limits. BP neural networks are used to establish the mapping relationship between dynamic cable configuration parameters and dynamic responses, and intelligent optimization methods are employed to optimize the cable configuration parameters. Intelligent optimization methods are favored in engineering optimization due to their unique advantages, with two key features: domain independence and fast random search capabilities. Commonly used intelligent optimization algorithms include genetic algorithms, Particle Swarm Optimization (PSO), the Ivy Algorithm, and the Tornado Optimization Algorithm. In this paper, three optimization algorithms—Particle Swarm Optimization, Ivy Algorithm, and Tornado Optimization—are used to optimize the dynamic cable configuration parameters. Among them, the Particle Swarm Optimization (PSO) algorithm features the fastest convergence speed and simple implementation, making it suitable for rapid prototyping in the early design stages and low-dimensional parameter optimization. However, it is prone to premature convergence and is more applicable in scenarios that prioritize real-time performance over high precision. The Ivy Optimization Algorithm (IVY), which mimics the growth behavior of plants, demonstrates strong global search capabilities in multi-modal optimization problems and is well suited for engineering applications involving multiple local optima. The Tornado Optimization Algorithm (TOC), leveraging the Coriolis force mechanism and a hierarchical optimization strategy, exhibits outstanding performance in high-dimensional, complex problems with strong constraints and in multi-objective optimization tasks. The set of parameters with the best optimization results is selected as the final optimization solution.

4.1. Particle Swarm Optimization Algorithm

The Particle Swarm Optimization (PSO) algorithm, known for its simplicity, fast convergence speed, good global optimization performance, and few control parameters, has been widely applied in engineering optimization [28]. PSO simulates the foraging behavior of a bird flock, where each potential solution in the problem-solving space is represented as a particle. The state of a particle is represented by its position vector and velocity vector, and its fitness is determined by the objective function. During the algorithm’s iterations, each particle dynamically adjusts its velocity and position based on the individual best solution and the global best solution. The velocity and position of the particle are updated using Equations (1) and (2). This information-sharing and cooperative evolution mechanism enables the particle swarm to gradually approach the global optimal solution and perform an efficient search within the solution space. In this study, the parameters are set to 300 iterations and 50 particles.

$$x_i(t+1)=x_i(t)+v_i(t+1)$$

(1)

$$v_i(t+1)=w{v}_i(t)+c_1\mathrm{r}.()[\mathrm{pb}(t)-x_i(t)]+c_2\mathrm{r}.()[\mathrm{gb}(t)-x_i(t)]$$

(2)

where x_i represents the current position of the i-th particle, where i = 1, 2, …, N, and N is the population size. t represents the current iteration number, v_i is the preset velocity of the particle, which is a key variable in the algorithm’s search process, w is the inertia weight, c₁ is the individual learning factor, c₂ is the group learning factor, r.() is a random function that generates random numbers in the range of [0, 1], increasing the randomness of the search, pb(t) records the individual best solution found by the i-th particle in the first t iterations, and gb(t) represents the global best solution discovered by the entire population in the first t iterations.

4.2. Ivy Optimization Algorithm

The Ivy Optimization Algorithm (IVY) is inspired by the orderly and coordinated growth and diffusion evolution process of ivy plants. Known for its powerful global search capability and adaptive dynamic adjustment mechanism, IVY performs excellently in complex optimization problems [29]. IVY simulates the coordinated population growth, propagation, and evolutionary behavior of ivy plants, where each potential solution in the problem-solving space is represented as an ivy individual. Each individual determines its growth direction based on the information from nearby ivy plants and improves itself by selecting the nearest and most significant neighbors, thus mimicking the behavior of ivy in nature. The fitness of each individual is determined by the objective function. During the algorithm’s iterations, each individual dynamically adjusts its position and growth direction based on the current best solution and neighborhood cooperation information. The position and growth vector of each individual are updated using Equations (3) and (4). This mechanism enables the algorithm to efficiently search the solution space and progressively approach the global optimal solution. In this study, the population size is set to 50, and the number of iterations is set to 300.

$$x_i^{\prime }=\left\{\begin{array}{ll}{x}_i+\left|r_1\right|\cdot (x_{i+1}-x_i)+r_2\cdot g_i\hfill & f(x_i)<\beta \cdot f(x_{\mathrm{best}})\hfill \\ x_{\mathrm{best}}\cdot (r_3+r_4\cdot g_i)\hfill & f(x_i)\ge \beta \cdot f(x_{\mathrm{best}})\hfill \end{array}\right.$$

(3)

where $x_i^{\prime }$ represents the updated position of the i-th individual, x_i represents the current position of the i-th individual, x_best represents the current global optimal solution, g_i is the growth vector of the i-th individual, r₁, r₂, and r₄ are random numbers generated from the standard normal distribution, r₃ is a random number generated from a uniform distribution, f (x_i) represents the fitness value of the i-th individual, and β is the control parameter.

$$g_i^{\prime }=g_i\cdot (r_5^2\cdot r_6)$$

(4)

where $g_i^{\prime }$ represents the updated growth vector of the i-th individual, g_i represents the current growth vector of the i-th individual, r₅ is a random number generated from a uniform distribution, and r₆ is a random number generated from a standard normal distribution.

4.3. Tornado Optimization Algorithm

The Tornado Optimization Algorithm (TOC) is a heuristic optimization algorithm based on the natural phenomenon of tornadoes [30]. This algorithm simulates the interactions between tornadoes, thunderstorms, and windstorms, incorporating the physical phenomenon of the Coriolis force to guide the search process and ultimately find the global optimal solution. The TOC algorithm simulates the following natural phenomena:

Tornadoes: represent the optimal solution in the current population, with strong attraction capabilities.

Thunderstorms: represent suboptimal solutions, with certain local search abilities.

Windstorms: represent ordinary solutions, responsible for exploring the search space.

By simulating the interactions between these natural phenomena (such as windstorms evolving into tornadoes and thunderstorms) and combining the physical effects of the Coriolis force, the TOC algorithm can efficiently balance exploration and exploitation in the search space. At the initialization of the algorithm, a certain number of windstorms, thunderstorms, and tornadoes are generated. Each individual’s position is randomly distributed in the search space, and its fitness value is calculated. Based on the fitness values, the best individual is selected as the tornado, the second best as the thunderstorm, and the remaining individuals as windstorms. The updates for the windstorm positions and speeds are performed according to Equations (5)–(7). The parameters for this study are set as a population size of 50 and an iteration count of 300.

The storm position update consists of two parts: evolution toward the tornado and evolution toward the thunderstorm. The evolution toward the tornado is as follows:

$$x_{i,j}=x_{i,j}+2\cdot \alpha \cdot (x_{\mathrm{tornado},j}-x_{i,j})+v_{i,j}$$

(5)

where x_i,j is the position of the i-th storm in the j-th dimension, x_tornado is the position of the tornado in the j-th dimension, α is a random weight, and v_i,j is the velocity of the i-th storm in the j-th dimension.

The evolution toward the thunderstorm is as follows:

$$x_{i,j}=x_{i,j}+2\cdot \mathrm{rand}\cdot (x_{\mathrm{thunderstorm},j}-x_{i,j})+2\cdot \mathrm{rand}\cdot (x_{\mathrm{tornado},j}-x_{i,j})$$

(6)

where x_{thunderstorm,j} is the position of the thunderstorm in the j-th dimension, x_tornado is the position of the tornado in the j-th dimension, rand is a random number generated from a uniform distribution.

The speed update of the storm is influenced by the Coriolis force, and the formula is as follows:

$$v_{i,j}=\eta \cdot \left(\kappa \cdot v_{i,j}-c\cdot {\displaystyle \frac{f\cdot R}{2}}+\sqrt{CF}\right)$$

(7)

where v_i,j is the velocity of the i-th storm in the j-th dimension, η is the scaling factor, κ is the inertia weight, c is the random coefficient, f is the Coriolis force coefficient, R is the radius parameter, and CF is the Coriolis force term.

5. Results

5.1. Selection Results of Cable Configuration Parameters

By selecting different parameters related to the buoyancy block and weight block, simulations of dynamic cables were performed under extreme conditions to investigate the significance of each parameter’s influence on the dynamic cable’s curvature and effective tension. The values of the relevant parameters and the analysis results are shown in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12.

When S₁ = 11, L₂ = 4, L₃ = 30, S₂ = 6, and L₄ = 4, the length of the suspension section L₁ varied between 80 and 130 with a step size of 10, and numerical simulations were performed. As shown in Figure 7, the variation in the length of the suspension section significantly affects the maximum curvature and maximum effective tension of the cable. As the suspension section length increases, the maximum curvature of the cable gradually decreases, and the maximum effective tension generally shows an increasing trend.

When L₁ = 110, L₂ = 4, L₃ = 30, S₂ = 6, and L₄ = 4, the number of buoyancy blocks S₁ varied between 8 and 14 with a step size of 1, and numerical simulations were performed. As shown in Figure 8, the increase in the number of buoyancy blocks significantly affects the cable curvature but has a less notable effect on the maximum effective tension of the cable. As the number of buoyancy blocks increases, the maximum curvature gradually increases, whereas the change in maximum effective tension is relatively small.

When L₁ = 110, S₁ = 11, L₃ = 30, S₂ = 6, and L₄ = 4, the buoyancy block spacing L₂ varied between 3 and 5 with a step size of 0.5, and numerical simulations were performed. As shown in Figure 9, the increase in buoyancy block spacing has a significant impact on the effective tension of the cable, while its effect on the maximum curvature is less significant. Although the maximum curvature tends to gradually increase as the buoyancy block spacing increases, the overall change is very small. In contrast, the change in maximum effective tension is more substantial, with the maximum effective tension gradually increasing as the buoyancy block spacing increases.

When L₁ = 110, S₁ = 11, L₂ = 4, S₂ = 6, and L₄ = 4, the starting position of the weight block L₃ varied between 20 and 40 with a step size of 5, and numerical simulations were performed. As shown in Figure 10, the starting position of the weight block significantly affects both the maximum curvature and the maximum effective tension of the cable. As the starting position increases, the maximum curvature gradually increases, while the maximum effective tension gradually decreases.

When L₁ = 110, S₁ = 11, L₂ = 4, L₃ = 30, and L₄ = 4, the number of weight blocks S₂ varied between 4 and 8 with a step size of 1, and numerical simulations were performed. As shown in Figure 11, the increase in the number of weight blocks demonstrates a dual impact on the cable performance. As the number of weight blocks increases, the maximum curvature gradually decreases, indicating that weight blocks can effectively reduce local bending. However, at the same time, the maximum effective tension shows a gradual increase, suggesting that the increase in the number of weight blocks could impose a greater vertical load on the dynamic cable. Therefore, controlling the number of weight blocks is crucial for the safe design of dynamic cables.

When L₁ = 110, S₁ = 11, L₂ = 4, L₃ = 30, and S₂ = 6, the weight block spacing L₄ varied between 3 and 5 with a step size of 0.5, and numerical simulations were performed. As shown in Figure 12, the weight block spacing has a significant regulatory effect on the maximum effective tension of the cable, while its impact on the maximum curvature is minimal. As the spacing increases, the maximum effective tension gradually decreases, while the maximum curvature shows a decreasing trend but with a small variation.

From the above simulation analysis, it can be observed that various parameters have varying degrees of influence on the maximum curvature and maximum effective tension of the dynamic cable. Among them, the length of the suspension section, the number of buoyancy blocks, the starting position of the weight block, and the number of weight blocks have a significant impact on the maximum curvature. The length of the suspension section, the buoyancy block spacing, the starting position of the weight block, the number of weight blocks, and the spacing have a significant impact on the maximum tension. Therefore, in the dynamic cable prediction model, these six parameters are selected as input parameters, and during the dynamic cable configuration optimization, these parameters will be optimized for design.

5.2. Model Prediction Results

The relevant parameters selected in previous studies were chosen as input parameters, and a dataset was constructed based on the method described in this paper. A BP neural network prediction model was then established. During the training iteration process, the BP neural network exhibited good convergence characteristics. In the 27th training round, the validation set reached the maximum validation failure count (20 times), and the training was concluded. By observing the training process, it can be noted that the mean squared error of the training set continuously decreased throughout the training process, while the mean squared error of the validation set reached its minimum value of 0.051396 in the seventh round. At this point, the neural network model exhibited the best performance, and the training stopped. For the training set, the root mean square error (RMSE) and mean absolute error (MAE) of the maximum curvature of the dynamic cable are 0.048063 and 0.027641, respectively, and the RMSE and MAE of the maximum effective tension are 6.4331 and 4.274, respectively. For the testing set, the RMSE and MAE of the maximum curvature are 0.045886 and 0.025424, and the RMSE and MAE of the maximum effective tension are 9.427 and 5.1645, respectively. These results indicate that the model demonstrates high fitting performance and prediction accuracy and is capable of accurately capturing the nonlinear relationship between the cable configuration parameters and the dynamic response. The prediction results for test set samples 1 to 5, compared with the target values, are shown in Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17. The figures clearly demonstrate a high degree of agreement between the predicted values and the target values, further validating the reliability and accuracy of the neural network model in predicting dynamic cable responses. This provides a solid foundation for subsequent optimization designs.

5.3. Optimized Configuration Results

In the optimization of the dynamic cable configuration, this study sets the minimization of the maximum curvature of the dynamic cable as the optimization objective. Additionally, considering the practical operational requirements of the dynamic cable, a validation step for the maximum effective tension is incorporated. During each iteration, the maximum effective tension corresponding to the current configuration parameters is calculated and compared with the allowable effective tension. If the value exceeds the permissible range, the solution is marked as invalid and removed from the optimization process, ensuring that only valid solutions meeting the requirements are retained. By running the three optimization algorithms described earlier, the optimal configuration parameters for the dynamic cable are obtained. The variations in the maximum curvature of the dynamic cable during the iteration process for the PSO, IVY, and TOC algorithms are shown in Figure 18, Figure 19, and Figure 20, respectively. A comparison of the initial configuration and the optimized configurations obtained by the three algorithms is presented in

Table 6. Comparison of initial and optimized cable configurations.

Parameters	Initial Configuration	Optimized Configuration
		PSO	IVY	TOC
L1/m	110	130	130	105.5
S1/unit	11	8	8.2038	8
L2/m	4	4	3.9431	3
L3/m	30	20	20	28.3
S2/unit	6	6	5.0584	8
L4/m	4	3	3	3.4
Maximum Curvature/m−1	0.5234	0.4319	0.4313	0.4276
Maximum Effective Tension/kN	98.766	110.297	115.4649	106.5653

.

From the analysis of the figures and tables, it can be observed that the initial configuration exhibits a maximum curvature of 0.5234 m⁻¹, which is close to the allowable curvature limit of the dynamic cable (0.55 m⁻¹). After optimization with the three algorithms (PSO, IVY, and TOC), the maximum curvature of the dynamic cable is significantly reduced, staying well below the allowable limit. Among them, the PSO algorithm achieves the fastest optimization speed but lacks convergence precision. The IVY Algorithm provides slightly better convergence accuracy than PSO. The TOC algorithm demonstrates superior overall performance, achieving rapid convergence while maintaining a stable optimization process.

The Tornado Optimization Algorithm (TOC) performs best in dynamic cable design due to its effective balance between exploration and exploitation, achieved through the Coriolis force mechanism and a hierarchical search strategy (tornado, thunderstorm, and storm phases). The Coriolis force mechanism enables the algorithm to avoid local optima while maintaining fast convergence. In contrast, the traditional PSO algorithm tends to converge prematurely, missing the global optimum, while the IVY Algorithm overly relies on random perturbations, leading to instability. The hierarchical search strategy effectively prevents the “blind following” issue in PSO and outperforms the IVY Algorithm’s single plant growth simulation.

Comparing the optimization results of the three algorithms, the optimized dynamic cable configuration obtained using the TOC algorithm achieves the best performance in both key performance indicators: maximum curvature and maximum effective tension. The optimized maximum curvature is reduced to 0.4276 m⁻¹, approximately an 18.3% reduction from the initial configuration. Meanwhile, the optimized maximum effective tension is 106.5653 kN, representing an approximate 7.9% increase compared to the initial value but still within the allowable tension limits. Therefore, the optimized configuration successfully reduces maximum curvature while ensuring that the tension requirements of the dynamic cable are met, providing a better solution for dynamic cable deployment.

6. Conclusions

This study focuses on the optimization design of the lazy-wave dynamic cable configuration and innovatively integrates machine learning methods with optimization algorithms, proposing a systematic optimization approach. This method provides a novel solution for improving the performance and reliability of dynamic cables. The main conclusions of this study are as follows:

A dynamic response prediction model for dynamic cables was constructed based on the BP neural network. This study shows that this model effectively captures the nonlinear characteristics of the dynamic cable response, demonstrating high prediction accuracy and generalization capability. Through training and validation with numerical simulation data, the model can rapidly evaluate the curvature and effective tension distribution of dynamic cables under different configuration parameters, providing a reliable tool for dynamic cable performance analysis.

In the optimization design of dynamic cable configurations, this study compares the performance of three intelligent optimization algorithms: PSO, IVY, and TOC. The results indicate that the TOC algorithm achieves the best optimization accuracy, efficiently finding the global optimum and significantly improving optimization efficiency. By optimizing the buoyancy block and weight block arrangement parameters using the TOC algorithm, the maximum curvature of the dynamic cable is reduced by 18.3%, effectively improving its bending performance.

The research findings hold significant application value in offshore wind power and other marine engineering fields. The proposed optimization design approach is not only applicable to lazy-wave dynamic cables but can also be extended to other types of marine dynamic cable systems. However, this study has certain limitations. First, the influence of wind turbines on floating platform motion has not been considered. Future research can incorporate this effect to enhance the model’s realism. Second, multi-objective optimization methods can be introduced to balance cable performance and economic factors. Additionally, while this study focuses on predicting the dynamic response of the cable, future research could extend to directly predicting the fatigue life of dynamic cables, constructing more precise models to identify potential failure points in advance.