Tension–Torsion Coupling Analysis and Structural Parameter Optimization of Conductor Based on RBFNN Surrogate Model

Abstract

To mitigate the impact of the conductor’s inherent tension–torsion coupling effect on conductor quality during tension stringing, a method for tension–torsion analysis and structural parameter optimization of conductors is proposed based on the radial basis function neural network (RBFNN) surrogate model. The layer-wise lay ratios of conductors are selected as the structural parameters. Using the tension–torsion coupling computational method for conductors, the layer-wise lay ratios are sampled by Latin hypercube sampling (LHS) to construct the sample data by computing conductor torque under different combinations. The RBFNN surrogate model is trained with the data, and its shape parameter is optimized through Leave-One-Out Cross-Validation (LOOCV), achieving a coefficient of determination R² close to 1 with minimal errors. Targeting torque minimization, the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) is employed to identify the optimal combination of conductor lay ratio parameters, reducing conductor torque by approximately 18% under the same axial tension. For practical applications, prioritize the optimal combination for JL/G1A-630/45-45/7 and analogous conductors, and adopt the RBFNN model for rapid torque prediction. The proposed method also serves as a reference for design optimization of conductor structural parameters.

1. Introduction

As China advances the development of a new-type power system and energy system, the coverage of transmission lines has expanded steadily, driven in particular by the large-scale construction of high-voltage (HV), extra-high-voltage (EHV), and ultra-high-voltage (UHV) transmission lines [1]. As the carrier for electric power transmission, overhead transmission conductors are chiefly fabricated by “concentric-lay stranding” of single wires in successive helical layers, with opposite lay directions in adjacent layers. The strand material, together with the structural parameters, determines the conductor’s overall performance [2]. Due to the helically stranded construction of the strands, axial tension induces a torque in the same-layer strands opposite to the lay direction, namely, a tension–torsion coupling effect, which directly affects the conductor’s stress state and deformation during tension stringing in construction of transmission lines. Therefore, it is necessary to investigate how conductor structural parameters influence tension–torsion coupling.

Extensive studies have been conducted domestically and internationally on conductors and analogous helically stranded wires through theoretical analysis, experimental measurement, and numerical simulation. Early studies relied predominantly on theoretical analysis and started with stranded wires of simple lay configurations. Costello and Phillips [3,4,5] employed the theory of an elastic rod to derive and analyze the mechanical response of stranded wires subjected to tension, bending, and torsion. Li [6] analyzed the conductor torque caused by axial tension based on a simplified model for the distribution of axial tension in aluminum conductor steel reinforced (ACSR). Hong [7] proposed an analytical model for steel cables subjected to axial tension and small bending, and obtained an approximate upper limit of the cable curvature. Foti [8] proposed an analytical calculation method for wire ropes under tensile, bending and torsional loads. With the advancement of computer technology and experimental measurement techniques, new methods have emerged for the structural analysis and computation of conductors. Dang [9] provides a detailed account of research advances on overhead conductors in mechanical, thermo-mechanical coupling, and electro-thermal coupling, and future directions for overhead conductor simulation. Xiang [10] proposed a computational model for multi-strand steel wire ropes that incorporates the elastoplastic behavior of strands, with predictions showing close agreement with DIC measurements. Yang [11] laser-grooved the surfaces of aluminum strands and embedded fiber Bragg gratings (FBGs). Then, layer-wise strand stresses of ACSR under axial tension were obtained experimentally. Menezes [12] employed a finite element numerical method to model steel wire ropes using the finite element method. By comparing the results with those of the analytical model, the calculation accuracy of the beam element model was verified. Xue [13] proposed two simplified computational models for steel wire rope slings based on the strand-slip theory. Zheng [14] proposed an analytical method for predicting the hysteretic bending behavior of helical strands. Its validity was demonstrated through comparison with finite element results and experimental data. Chen [15] used the finite element method to analyze the stress relaxation behavior of a three-layer steel wire rope under combined tension-bending loading. Wan [16] investigated the influence of tension stringing process parameters on conductor strand loosening and, using finite element simulation, concluded that conductor torque is the primary factor leading to conductor strand loosening. Qin [17,18] conducted theoretical analyses, finite element simulations, and experiments on defects during tension stringing, such as lantern-shaped strands and breakage, and demonstrated that conductor torque has a significant impact on the formation of these defects. Further research is required to analyze and optimize the influence of conductor structural parameters on tension–torsion coupling.

The conductor has a complex structure. The analysis involves nonlinear handling, rendering direct analysis and optimization time-consuming and making it difficult to achieve satisfactory results. Since their introduction, surrogate modeling methods have become an important means for tackling the optimization of complex models [19,20]. A surrogate model is a mathematical model constructed from sample points, with representative forms such as the response surface model (RSM), the Kriging model, and the artificial neural network (ANN) model [21,22,23]. The sample points are commonly generated through design of experiments methods such as orthogonal experimental design and Latin hypercube sampling [24,25]. RSM exhibits poor computational efficiency and accuracy for high-dimensional, complex non-linear problems. The Kriging model is highly susceptible to noise in the sample data. ANN constructs a mathematical model by emulating the synaptic connectivity of the brain. Zhang [26] developed an RBFNN surrogate model for the structural parameters of the groove geometry in a cylindrical gas film seal. Zhang [27] employed a radial basis surrogate model in conjunction with a multi-objective particle swarm optimization (MOPSO) algorithm to perform design optimization of high-speed train parameters. Peng [28] applied an adaptive surrogate model technique to design optimization of semi-rigid connection joints and the structure of transmission towers, improving the efficiency of design optimization.

As indicated above, current research on the tension–torsion coupling effect inherent to the conductor’s intrinsic structure and its implications for conductor stringing quality remains relatively scarce. Therefore, this study aims to develop an efficient structural parameter optimization method to mitigate the impact of conductor torque induced by tension–torsion coupling on conductor stringing quality. First, the RBFNN modeling approach based on the variable-power spline basis function and the computational method for multi-layer stranded conductors are systematically elaborated. Second, a high-precision surrogate model for structural parameter-torque mapping is established. Finally, the NSGA-II algorithm is employed to optimize the model, minimizing conductor torque under different tension levels. The validity of the proposed method and results is further verified through rigorous analysis.

2. RBFNN Surrogate Model and Calculation Method for Tension–Torsion Coupling of Conductor

2.1. RBFNN Surrogate Modeling Method

RBFNN is a three-layer feedforward network model comprising an input layer, a hidden layer, and an output layer. Its core mechanism is to achieve function fitting through the linear combination of radial basis functions of spatial distance.

Let the input layer be a d-dimensional input vector X = [x₁, x₂, …, x_d]. Following the mapping action of the radial basis function ${\phi }_i\left(\mathit{X}\right)$ in the hidden layer, the output layer yields the target value y as the sum of a linear combination of these functions and a bias term.

The hidden layer consists of a set of radial basis functions that are symmetric about their centers. This study adopts a variable-power spline basis function, defined as:

$${\phi }_i\left(\mathit{X}\right)={‖\mathit{X}-{\mathit{X}}_i‖}^c,$$

(1)

where X_i denotes the center vector, i.e., the i-th sample vector, i = 1, 2, …, n (where n is the number of samples); $‖\mathit{X}-{\mathit{X}}_i‖$ is the Euclidean distance to the center X_i, given by ${\left(\mathit{X}-{\mathit{X}}_i\right)}^T\left(\mathit{X}-{\mathit{X}}_i\right)$; and c is the shape parameter, 0.2 ≤ c ≤ 3.

At the output layer, the interpolation function for the target output y is:

$$y\left(\mathit{X}\right)={\displaystyle \sum _{i=1}^n{a}_i{\phi }_i\left(\mathit{X}\right)}+b,$$

(2)

where $a_i$ is the weight coefficient, ${\displaystyle \sum _{i=1}^n{a}_i=0}$; n is the number of hidden nodes (i.e., the number of sample points); and b is the bias term.

The weight coefficient ${\alpha }_i$ and the bias term b associated with each output target y are determined by solving the following extended matrix form:

$$\left[\begin{array}{cccc}{\phi }_1\left({\mathit{X}}_1\right)& \cdots & {\phi }_n\left({\mathit{X}}_1\right)& 1\\ ⋮& \ddots & ⋮& ⋮\\ {\phi }_1\left({\mathit{X}}_n\right)& \cdots & {\phi }_n\left({\mathit{X}}_n\right)& 1\\ 1& \cdots & 1& 0\end{array}\right]\cdot \left[\begin{array}{c}{a}_1\\ ⋮\\ a_n\\ b\end{array}\right]=\left[\begin{array}{c}y\left({\mathit{X}}_1\right)\\ ⋮\\ y\left({\mathit{X}}_n\right)\\ 0\end{array}\right],$$

(3)

As indicated in Equation (3), the key to the aforementioned model lies in the choice of the shape parameter c. Specifically, the choice of c should be guided by the expected smoothness of the underlying function and the density of the sample points:

• For small c (i.e., c is close to 0.2), ${\phi }_i\left(\mathit{X}\right)$ becomes highly localized and sharp, which enables the model to capture abrupt variations or discontinuities in the response, but may also lead to over-fitting and ill-conditioning of the coefficient matrix.
• For large c (i.e., c is close to 3), ${\phi }_i\left(\mathit{X}\right)$ behaves more globally and smoothly, which improves stability and is suitable for approximating harmonic or periodic functions, but may fail to resolve steep gradients.

For the above reasons, an adaptive selection strategy replaced the fixed value, with the parameter c (0.2 ≤ c ≤ 3) being automatically optimized through LOOCV. The value of c is optimized by minimizing the prediction error ${\epsilon }_{LOOCV}\left(c\right)$ when each point is sequentially omitted from the fitting set:

$${\epsilon }_{LOOCV}={\displaystyle \frac{1}{K}}{\displaystyle \sum _{i=1}^K\left|y\left({\mathit{X}}_i\right)-y^{\prime }\left({\mathit{X}}_i\right)\right|}={\displaystyle \frac{1}{K}}{\displaystyle \sum _{i=1}^K\left|y_i-y_i^{\prime }\right|},$$

(4)

where K denotes the number of sample points validated, 1 ≤ K ≤ n. When K = n, all sample points are validated. When the sample size is large, a random subset of K samples may be selected to accelerate validation. $y\left({\mathit{X}}_i\right)$ or $y_i$ denotes the output of the RBFNN surrogate model trained on the full sample set at sample point ${\mathit{X}}_i$, namely the true value at this point; $y^{\prime }\left({\mathit{X}}_i\right)$ or $y_i^{\prime }$ denotes the output of the RBFNN surrogate model obtained after removing the sample point ${\mathit{X}}_i$.

The calculation process for optimizing the shape parameter c is as follows:

Step 1: Extract the cross-validation dataset. K samples are randomly selected from the full sample set containing n samples to constitute the cross-validation dataset.

Step 2: Generate candidate c values. A sequence of candidate c values is generated within the bounds 0.2 ≤ c ≤ 3. In our implementation, this was achieved using uniformly spaced sampling to explore the parameter space.

Step 3: Calculate the LOOCV error. For each candidate value c, the LOOCV procedure is executed: for every sample point X_i in the cross-validation dataset, an RBFNN model is fitted to the remaining n-1 sample points of the full sample set using the current c, and its prediction error at point X_i is recorded. The ${\epsilon }_{LOOCV}$ is then calculated as the mean of these individual absolute errors, according to Formula (4).

Step 4: Select the optimal c. The candidate value c that yields the minimum ${\epsilon }_{LOOCV}$ is selected as the optimal shape parameter. This data-driven approach ensures the model adapts to the specific characteristics of the training data.

After constructing the surrogate model, its accuracy must be validated. Error assessment may employ the average absolute error (AE), the maximum absolute error (ME), the root-mean-square error (RMSE), and the coefficient of determination (R-Squared or R²). By normalizing the first three errors (appending the suffix N to their abbreviations), the corresponding error levels across different outputs can be compared. Since the RBFNN model adopted in this study is an interpolation model, the model accuracy is evaluated using the validation set selected by the aforementioned LOOCV method. In Formulas (5)–(8), y_max and y_min denote the maximum and minimum values of the target output values for the full sample set, respectively.

$$AEN=\left({\displaystyle \frac{1}{K}}{\displaystyle \sum _{i=1}^k\left|y_i-y_i^{\prime }\right|}\right)/\left(y_{\max }-y_{\min }\right),$$

(5)

$$MEN=\max \left(\left|y_i-y_i^{\prime }\right|\right)/\left(y_{\max }-y_{\min }\right)\hspace{1em}\hspace{1em}\left(i=1,2,\cdots n\right),$$

(6)

$$RMSEN=\sqrt{{\displaystyle \frac{1}{K}}{\displaystyle \sum _{i=1}^K{\left(y_i-y_i^{\prime }\right)}^2}}/\left(y_{\max }-y_{\min }\right),$$

(7)

$$R^2=1-\left({\displaystyle \sum _{i=1}^K{\left(y_i-y_i^{\prime }\right)}^2}\right)/\left({\displaystyle \sum _{i=1}^K{\left(y_i-{\overline{y}}_i\right)}^2}\right),$$

(8)

It is generally considered that the model’s predictive accuracy is acceptable when AEN ≤ 0.2, MEN ≤ 0.3, and RMSEN ≤ 0.2. The coefficient of determination should satisfy R² ≥ 0.9, which is the minimum acceptable threshold for model reliability in this research field; a value closer to 1 indicates a higher fitting accuracy.

2.2. Calculation Method for Tension–Torsion Coupling of Conductor

Overhead transmission conductors are predominantly fabricated by helically stranding multiple layers of strands in a “concentric-lay stranding” configuration. Adjacent layers are laid in opposite directions, resulting in a relatively complex structure, as shown in Figure 1.

For a conductor of length h, the rotation angle of the strands in layer i from the start to the end is φ; the radius of the central strand is r₀; the diameter of the strand in layer i is r_i; the helical radius is R_i; the lay ratio is m_i; and the lay angle is β. Then

$${\beta }_i=\arctan ({\displaystyle \frac{R_i{\phi }_i}{h}})=\arctan \left[{\displaystyle \frac{\pi }{m_i}}\left({\displaystyle \frac{R_i}{R_i+r_i}}\right)\right],$$

(9)

The curvatures of the strand in layer i along the m and n directions and the torsion along the l direction, before and after deformation, are [17]:

$${\kappa }_i=0, {\kappa }_i^{\prime }={\displaystyle \frac{{\sin }^2{\beta }_i}{R_i}}, {\tau }_i={\displaystyle \frac{\cos {\beta }_i\sin {\beta }_i}{R_i}},$$

(10)

$${\overline{\kappa }}_i=0, {\overline{{\kappa }_i}}^{\prime }={\displaystyle \frac{{\sin }^2{\overline{\beta }}_i}{{\overline{R}}_i}}, {\overline{\tau }}_i={\displaystyle \frac{\cos {\overline{\beta }}_i\sin {\overline{\beta }}_i}{{\overline{R}}_i}},$$

(11)

where ${\kappa }_i$, ${\kappa }_i^{\prime }$ and ${\tau }_i$ denote the curvature in the m and n directions and the torsion rate in the l direction of the strand in the i-th layer before deformation, respectively. ${\overline{\kappa }}_i$, ${\overline{{\kappa }_i}}^{\prime }$ and ${\overline{\tau }}_i$ denote the curvature in the m and n directions and the torsion rate in the l direction of the strand in the i-th layer after deformation, respectively.

After deformation, the moments $G_i$, $G_i^{\prime }$, $H_i$ on the strand cross-section along m, n and l directions and the axial force $T_i$ along the l direction, are:

$$G_i={\displaystyle \frac{E_i\pi r_i^4}{4}}({\overline{\kappa }}_i-{\kappa }_i)=0,$$

(12)

$$G_i^{\prime }={\displaystyle \frac{E_i\pi r_i^4}{4}}({\overline{\kappa }}_i^{\prime }-{\kappa }_i^{\prime }),$$

(13)

$$H_i={\displaystyle \frac{E_i\pi r_i^4}{4(1+{\nu }_i)}}({\overline{\tau }}_i-{\tau }_i),$$

(14)

$$T_i=E_i\pi {\overline{r}}_i^2{\epsilon }_i=E_i\pi r_i^2{(1-{\nu }_i{\epsilon }_i)}^2{\epsilon }_i,$$

(15)

where E_i, ε_i and ν_i denote the elastic modulus, strain, and Poisson’s ratio of the strand in layer i, respectively.

Considering mutual contact both between adjacent layers and among strands within the same layer, the following hold:

$${\overline{d}}_i=d_i-2{\nu }_i{\epsilon }_i{r}_i-{\delta }_i^s,$$

(16)

$${\overline{R}}_i-{\overline{R}}_{i-1}=r_i(1-{\nu }_i{\epsilon }_i)+r_{i-1}(1-{\nu }_{i-1}{\epsilon }_{i-1})-{\delta }_i^d,$$

(17)

where $d_i$, ${\overline{d}}_i$ and ${\delta }_i^s$ denote, respectively, the inter-strand distance within the same layer before deformation, the inter-strand distance within the same layer after deformation, and the contact depth. When ${\delta }_i^s>0$, contact occurs between strands. ${\overline{R}}_{i-1}$, ${\overline{R}}_i$ and ${\delta }_i^d$ denote, respectively, the helical radii of deformed strands in layers i − 1 and i and the contact depth. When ${\delta }_i^d>0$, contact occurs between strands.

The distributed contact forces exerted by strands in layers i − 1 and i + 1 on the strand in layer i are denoted X_iⁱ⁻¹ and X_iⁱ⁺¹, respectively (excluding the core wire and the outermost layer strand, i.e., i = 1,…, g − 1). The resultant contact force between adjacent strands within layer i is denoted Y_i. Accordingly, the resultant external force F_i on the strand in layer i along the m direction can be expressed as:

F_i = X_iⁱ⁻¹ + X_iⁱ⁺¹ + Y_i (i = 1,…,g − 1),

(18)

The resultant external force on the central core wire F₀ is 0. The strand in the outermost layer (layer g) is subjected only to the contact forces exerted by the strands of layers g − 1 and g, i.e.,

F_g = X_g^g−1 + Y_g,

(19)

where F_g donates the resultant external force acting on the outermost layer (layer g) of strands.

Based on the force analysis of the strand, it follows that

$$\left\{\begin{array}{l}{F}_i=N_i^{\prime }{\overline{\tau }}_i-T_i{\overline{\kappa }}_i^{\prime }\\ N_i^{\prime }=-G_i^{\prime }{\overline{\tau }}_i+H_i{\overline{\kappa }}_i^{\prime }\end{array}\right.,$$

(20)

Let $U({\epsilon }_i,{\overline{\tau }}_i,{\overline{\kappa }}_i^{\prime })$ denote, respectively, the axial force, bending moment, and torque components of the strand in layer i along the m direction, then

$$U({\epsilon }_i,{\overline{\tau }}_i,{\overline{\kappa }}_i^{\prime })=(-G_i^{\prime }{\overline{\tau }}_i+H_i{\overline{\kappa }}_i^{\prime }){\overline{\tau }}_i-T_i{\overline{\kappa }}_i^{\prime },$$

(21)

Accordingly, the force equilibrium equation for the strand along the m direction is:

$$F_i=U({\epsilon }_i,{\overline{\tau }}_i,{\overline{\kappa }}_i^{\prime }),$$

(22)

Solving the system of nonlinear equations yields the stress state and deformation of the strands in each conductor layer. Accordingly, the terminal tension and torque of the conductor can be expressed as:

$$F_{end}=\pi r_0^2{E}_0{\epsilon }_0+{\displaystyle \sum _{i=2}^g{s}_i(T_i\cos {\overline{\beta }}_i+N_i^{\prime }}\sin {\overline{\beta }}_i),$$

(23)

$$M_{end}={\displaystyle \frac{\pi r_0^4{E}_0}{4(1+{\nu }_0)}}{\displaystyle \frac{d\phi }{h}}+{\displaystyle \sum _{i=2}^g{s}_i\left(H_i\cos {\overline{\beta }}_i+G_i^{\prime }\sin {\overline{\beta }}_i+T_i{\overline{R}}_i\sin {\overline{\beta }}_i-N_i^{\prime }{\overline{R}}_i\cos {\overline{\beta }}_i\right)}$$

(24)

On the basis of the above analysis, once the conductor specification is fixed, structural parameters such as the conductor diameter, the number of stranding layers, the number of strands in each layer, and the strand diameter are determined. Only the layer-wise lay ratio m_i (corresponding to a change in lay angle) may be selected within a specified range. Therefore, in analyzing conductor structural parameters, the primary approach is to investigate how variations in the layer-wise lay ratio affect the conductor’s tension–torsion coupling performance.

3. RBFNN Surrogate Model for Tension–Torsion Coupling of Conductor

3.1. Construction of RBFNN Surrogate Model

Based on the aforementioned analytical and computational method for conductor tension–torsion coupling, RBFNN is employed to construct a surrogate model that relates the layer-wise lay ratio to the conductor’s overall torque under a specified tensile load. The construction workflow is shown in Figure 2.

3.1.1. Sample Data Acquisition

Constructing an approximation model requires an appropriate set of sample points, and the number and distribution of these points have a significant impact on the model’s accuracy. An optimized Latin hypercube sampling (OLHS) method based on the maximin distance criterion is employed in this paper to disperse the samples as uniformly as possible across the entire design space, thereby improving the surrogate model’s fitting accuracy.

To analyze the effect of the layer-wise lay ratio on the tension–torsion coupling of the JL/G1A-630/45-45/7 conductor, 80 sample sets were sampled using OLHS method over the layer-wise lay ratio distribution ranges specified in

Table 1. Technical Parameters of Conductor.

Layer Order	No. of Strands	Diameter (mm)	Lay Ratio	Elastic Modulus (GPa)	Poisson’s Ratio
Layer 0	1	2.81	/	196	0.28
Layer 1	6	2.81	16–26	196	0.28
Layer 2	9	4.22	10–16	59	0.3
Layer 3	15	4.22	10–16	59	0.3
Layer 4	21	4.22	10–12	59	0.3

.

The distribution of a subset of the sample data (normalized) is shown in Figure 3, where m₁–m₄ correspond, in order, to the lay ratios of strands in Layer 1–Layer 4. The results indicate that the sampled layer-wise lay ratio data are uniformly distributed across the entire design space along every dimension.

Using the aforementioned analytical and computational method for tension–torsion coupling, torques M₁₅, M₂₀ and M₂₅ of the JL/G1A-630/45-45/7 conductor were obtained under tensile levels of 15% RTS, 20% RTS, and 25% RTS (where RTS denotes the rated tensile strength) for 80 distinct sets of lay ratio distribution, thus completing the sample data acquisition.

By calculating the Pareto effect rate of the layer-wise lay ratio m_i of the conductor to the torque, it is obtained that the layer-wise lay ratio m₁ of the steel core layer has a relatively small influence on the conductor torque. Moreover, the layer-wise lay ratio m_i of the steel core layer and m₃ of the outer-adjacent aluminum layer have a positive effect on the conductor torque (the torque increases as the layer-wise lay ratio increases). Figure 4 shows the results of the Pareto effect rate of layer-wise lay ratio of the conductor to the torque under an axial force of 15% RTS. Blue indicates negative effects, while orange indicates positive effects.

3.1.2. Model Construction and Accuracy Verification

The sample data were trained using an RBFNN to obtain an RBFNN surrogate model that maps the variation in the conductor’s overall torque M₁₅, M₂₀ and M₂₅ with the layer-wise lay ratio. The model’s fitting accuracy was computed according to the evaluation metrics defined in Equations (5)–(8), and the results are presented in

Table 2. Fitting Accuracy of RBFNN Model.

Metric	AEN (≤0.2)	MEN (≤0.3)	RMSEN (≤0.2)	R2 (≥0.9)
M15 fitting accuracy	0.005	0.012	0.007	1.000
M20 fitting accuracy	0.005	0.011	0.006	1.000
M25 fitting accuracy	0.004	0.010	0.005	1.000

.

As shown in Table 2, the fitted RBFNN model exhibits very small errors, with a coefficient of determination R² close to 1, indicating excellent fitting accuracy. The model is suitable for subsequent analysis and computation.

3.2. Results

Computations based on the fitted RBFNN model show that, under tensile levels of 15% RTS, 20% RTS, and 25% RTS, the variation trends of the conductor’s overall torque with the layer-wise lay ratio are essentially consistent. Accordingly, selected response surfaces of the overall torque M₁₅ with respect to the layer-wise lay ratio m_i under a tensile level of 15% RTS are plotted, as shown in Figure 5.

As shown in Figure 5, the conductor torque M decreases with increasing strand lay ratios of the inner aluminum layer m₂ and the outer aluminum layer m₄, whereas it increases with increasing strand lay ratios of the steel core layer m₁ and the adjacent outer aluminum layer m₃.

4. Design Optimization of Conductor Structural Parameters Based on RBFNN Surrogate Model

In the tension stringing of overhead transmission lines, the buildup of conductor torque is a major contributor to defects such as strand loosening and lantern-shaped strands. A portion of the conductor torque is attributable to the intrinsic tension–torsion coupling of its structure. Accordingly, the layer-wise lay ratios of the conductor are optimized to determine the lay-ratio parameters that minimize torque under tension–torsion coupling, thereby suppressing the occurrence of construction defects. The core optimization idea is to utilize the RBFNN surrogate model (with R² close to 1) to establish a high-precision mapping between lay ratios and torque, avoiding the inefficiency of direct nonlinear analysis of complex conductor structures, and adopting NSGA-II to solve the multi-tension-level (15% RTS, 20% RTS, 25% RTS) torque minimization problem under engineering constraints.

4.1. Optimization Objective

Based on the RBFNN surrogate model for the torque M of the JL/G1A-630/45-45/7 conductor, the NSGA-II [29] is employed to determine the layer-wise lay ratios that minimize M. NSGA-II is a widely adopted evolutionary algorithm for multi-objective optimization, known for its ability to maintain a diverse set of Pareto-optimal solutions through non-dominated sorting and crowding-distance computation. In this study, the default implementation was adopted.

The optimization model is as follows:

Find: m₁, m₂, m₃, m₄

Min: M₁₅ (m₁, m₂, m₃, m₄), M₂₀ (m₁, m₂, m₃, m₄),

M₂₅ (m₁, m₂, m₃, m₄)

s.t.: m₁ ∈ [16,26], m₂ ∈ [10,16], m₃ ∈ [10,16], m₄ ∈ [10,12],

m₁-m₂ ≥ 0, m₂-m₃ ≥ 0, m₃-m₄ ≥ 0

The NSGA-II parameters are set as follows: population size = 20, maximum iterations = 200, and crossover probability = 0.9. These parameters are determined through preliminary tests to balance optimization efficiency and solution accuracy. The lay ratio sequence constraint is integrated into the initial population generation to exclude non-compliant solutions in advance, improving optimization efficiency. In addition, a lay-ratio constraint of adjacent layers was imposed in accordance with GB/T 1179–2017 [30], which stipulates that, “for multi-layer stranded wires, the lay ratio of any layer must not exceed that of the immediately inner layer”.

4.2. Optimization Results and Comparative Analysis

As shown in Figure 6, iterative optimization with NSGA-II yields the conductor’s layer-wise lay ratio distribution that minimizes torque M under 15% RTS, 20% RTS, and 25% RTS. The results are presented in

Table 3. Optimization Results.

Item		Before Optimization	Optimized Result
m1 (16–26)		21	16
m2 (10–16)		13	16
m3 (10–16)		13	12
m4 (10–12)		11	12
Conductor torque M15 (N·m)	RBFNN computed value	19.66	15.87
	Theoretical value	19.67	16.10
	Relative error	−0.05%	−1.43%
Conductor torque M20 (N·m)	RBFNN computed value	26.24	21.21
	Theoretical value	26.25	21.55
	Relative error	−0.04%	−1.58%
Conductor torque M25 (N·m)	RBFNN computed value	32.90	26.54
	Theoretical value	32.91	26.99
	Relative error	−0.03%	−1.67%

.

The results indicate that, for the minimum conductor torque M obtained by optimizing the RBFNN surrogate model with NSGA-II, the maximum absolute relative error with respect to the theoretical value (calculated via Section 2.2 tension–torsion coupling method) is 1.67%, which is lower than the 5% threshold required in engineering practice and verifies the reliability of the optimized parameters. This further demonstrates the computational accuracy of the RBFNN surrogate model. After optimization, the torque M of the JL/G1A-630/45-45/7 conductor under 15% RTS, 20% RTS, and 25% RTS is reduced by approximately 18%.

A comparison of the axial stress of the strands in each conductor layer before and after optimization is shown in Figure 7. In the figure, Core and Layer 1 denote the steel core, while Layer 2 to Layer 4 correspond to the aluminum strands from the inner to the outer layer.

The results show that, after optimization, the stresses in the steel core (Core and Layer 1) decrease by 4.2–6.3% and 1.1–1.4%, respectively. By contrast, the stresses of the aluminum strands in the inner layer (Layer 2) and the outer layer (Layer 4) increase by 2.2–2.4% and 1.5–1.6%, respectively, whereas the aluminum strands in the adjacent outer layer (Layer 3) decrease by 2.4–2.5%. The overall stress ratio between the steel core and the aluminum strands remains essentially unchanged. It is noted that the marginal stress increases in certain aluminum layers are within the material’s design allowance. The primary optimization goal of torque reduction is achieved while maintaining the structural integrity and load-sharing ratio of the conductor.

5. Conclusions

Based on the RBFNN surrogate modeling method and the analytical and computational method for conductor tension–torsion coupling, sample points were sampled in this paper using the OLHS method to construct a surrogate model that relates the overall torque to the conductor structural parameters under tension–torsion coupling. On this basis, design optimization of conductor structural parameters was completed using NSGA-II.

• An OLHS method based on the maximin distance criterion was adopted to generate 80 sets of layer-wise lay ratios for the conductor. The torque response corresponding to each set of lay ratios was then computed using the analytical and computational method for tension–torsion coupling. The results indicate that the sample points are uniformly dispersed across the design space, and that the torque is affected primarily by the lay ratios of the inner aluminum layer (Layer 2), the aluminum adjacent outer layer (Layer 3), and the outer aluminum layer (Layer 4).
• Using the RBFNN surrogate modeling method, the sample data were used to train an RBFNN surrogate model that maps the variation in the conductor’s overall torque with the layer-wise lay ratio. Validation shows the model exhibits very small errors, with a coefficient of determination R² close to 1, indicating excellent accuracy.
• The RBFNN surrogate model was optimized using NSGA-II. After optimization, the conductor’s overall torque decreased by approximately 18% relative to the pre-optimization baseline. Meanwhile, the RBFNN optimization results agree closely with the theoretical computation results, with a maximum relative error of only 1.67%, indicating that the optimization is feasible. It effectively reduces the torque induced by the conductor’s intrinsic tension–torsion coupling and serves as a reference for the structural design of stranded conductors.
• For the JL/G1A-630/45-45/7 conductor (and analogous multi-layer stranded conductors), the optimized layer-wise lay ratio combination (m₁ = 16, m₂ = 16, m₃ = 12, m₄ = 12) can be directly adopted in the conductor design and production stage. This combination reduces torque while maintaining the stress balance between the steel core and aluminum strands, effectively suppressing construction defects during tension stringing.

In the future, we will simplify the RBFNN surrogate model construction process by optimizing the sampling method and model structure, and develop a user-friendly engineering calculation tool to facilitate the popularization and application of the proposed method in engineering practice.