Structural parameters optimization based on the neural network predicting the torque coefficient for H-type vertical-axis wind turbines

Abstract

The wind energy utilization of vertical-axis wind turbines (VAWTs) is closely related to the wind wheel geometry. Neural networks can quickly and accurately describe the nonlinear relationships between the structural parameters and aerodynamic performance of VAWTs. Hence, a novel optimization method using the neural network to predict the torque coefficient is developed for the structural design of VAWTs. The statistical parametric mapping initializes the particles and the sine algorithm, variable spiral search, and Levy flight modify the population position update for the Dung Beetle Optimization (DBO) algorithm. The improved DBO algorithm and Binary Particle Swarm Optimization algorithm optimize weight switches, weights, and thresholds of the single hidden layer back-propagation (BP) neural network, and the BP algorithm further optimizes weights and thresholds. Based on the built SMLDBO-BPNN predicting the torque coefficient, the structural parameters of the VAWT are optimized by the Particle Swarm Optimization algorithm. The pressure, vorticity, and torque coefficient of VAWTs before and after optimization are studied by the computational fluid dynamics method. The results show that after the optimization, the high- and low-pressure regions of blades in the windward region increase, the impact of shed vortexes is reduced, and the average torque coefficient increases by 15%.

Keywords

vertical-axis wind turbine structural parameters optimization neural network average torque coefficient

Introduction

Wind energy with the properties of wide distribution, inexhaustibility, and sustainability has the potential to become a major contributor meeting the growing green energy demand in the global community (Lebsir et al., 2025; Su et al., 2025). The vertical-axis wind turbine (VAWT) has many advantages such as simple structure and low operating cost, but the low wind energy utilization rate limits its wide application (Le and Mulleners, 2024; Rahman et al., 2024). As well known, the wind wheel realizes the conversion of wind energy, that is, the output power of the VAWT is closely related to the wind wheel geometry (Xu et al., 2021). Hence, it is necessary to optimize the structural parameters of the VAWT, but the model remeshing in the computational fluid dynamics (CFD) method will result in large calculation consumption in the optimization process.

Much effort has been spent on the structural optimization without using the CFD method for VAWTs. Bedon et al. (2013) used the blade element-momentum theory to predict the power of wind turbines, and then used the genetic algorithm (GA) to solve the optimization model with the thickness and chord of the airfoil as design variables and the maximum power coefficient of wind turbines as the design objective. Tahani et al. (2016) proposed a novel continuous optimization algorithm, and combined it with the Double Multiple Stream Tube model (DMST) to solve an optimization model with the wind wheel diameter, airfoil type, chord, and blade number as design variables and the maximum power coefficient as the design objective. Kavade and Ghanegaonkar (2019) used the DMST to predict the aerodynamic performance of VAWTs, and then combined the linear interpolation to solve an optimization model for maximizing the power coefficient at different tip speed ratios, in which the rotor radius, blade height, blade number, and chord length are design variables. Chen et al. (2020) introduced the Covariance Matrix Adaptive Evolutionary Strategies algorithm (CMAES) to the DMST model, and solved the optimization model to maximize the power coefficient, in which the turbine radius, the ratio of radius over half-height, and the blade number are design variables. Trentin et al. (2022) established a response surface model between the power coefficient and the blade number, chord length, pitch angle, and rotor diameter based on the CFD method for a VAWT, and optimized four parameters mentioned above using the Non-dominated Sorting Genetic Algorithm (NSGA-II) to achieve the maximum power coefficient. Taken together, the optimal design of structural parameters improves the aerodynamic performance of VAWTs, and the relationship between the structural and aerodynamic parameters is described by the DMST model or response surface method. The neural network has strong nonlinear mapping ability and self-learning ability, and can accurately describe the multivariate relationship between the aerodynamic performance and multiple structural parameters of wind turbines (Leimeister et al., 2021; Zhang et al., 2024). However, to date, it has not been adopted during the structural optimization process of wind turbines.

The back-propagation (BP) neural network has strong applicability, but the redundancy of neural connections in this network causes overfitting, resulting in a decrease in generalization ability. It is necessary to improve the network structure and optimize the structural parameters. Leung et al. (2003) constructed a BP neural network with weight switches and used an improved GA to optimize the network parameters. Tsai et al. (2006) used the Hybrid Taguchi Genetic Algorithm (HTGA) combined with the PSO algorithm, and Zhao and Qian (2011) used the Binary Particle Swarm Optimization algorithm (BPSO) algorithm combined with the PSO algorithm, to optimize the parameters of a three-layer BP neural network containing weight switches. In addition, the Dung Beetle Optimization (DBO) algorithm has the advantages of fast global search speed and wide applicability, which is more recognized in recent years. The DBO algorithm and its improved variants have been widely used to address the slow convergence and low solution accuracy of BP neural networks. Qiao et al. (2023) used the DBO algorithm combined with the PSO algorithm, and Yi et al. (2024) used the DBO algorithm alone, to optimize the structural parameters within a BP neural network, and, respectively, established the DBO-PSO-BP and DBO-BP neural networks. Zhang and Zhu (2023) used the Improved Dung Beetle Optimization algorithm (IDBO) to optimize the parameters of the BP neural network, and found that the prediction accuracy of the IDBO-BPNN was higher than that of the DBO-BPNN. Taken together, the BP neural network with weight switches has the function of self-learning and high-speed optimization search, and quickly approximates the nonlinear relationship. But the gradient descent is prone to occur during training, making the model to fall into the local optimum. Such cases can be effectively avoided by utilizing the intelligent algorithms to optimize the weight switches, weights, and thresholds of networks. However, the imbalance between the global search and local development capabilities exists in the DBO algorithm. Hence, it is necessary to improve the DBO algorithm first and then apply it to the optimization of the BP neural network.

To this end, this study proposes an optimization method for the structural parameters of VAWTs, where the torque coefficient is predicted using an optimized three-layer BP neural network. Section 2 improves the DBO algorithm, combines it with the BPSO algorithm to optimize the weight switches, weights, and thresholds of a three-layer BP neural network, and employed the BP algorithm to optimize the weights and thresholds again, thereby constructing a new BPNN to predict the average torque coefficient. Section 3 establishes the optimization model for the structural parameters of VAWTs and solves it by the PSO algorithm, and employs the CFD method to analyze the pressure, vorticity, and torque coefficient. Section 4 gives the final conclusions summarizing the most important achievements of the study.

New BPNN establishment and torque coefficient prediction for VAWTs

Improvement of the DBO algorithm

The DBO algorithm is a meta-heuristic algorithm proposed in 2022, and its populations include the ball-rolling dung beetle, breeding dung beetle, small dung beetle, and stealing dung beetle (Xue and Shen, 2023). The ball-rolling dung beetle adopts the ball-rolling and dancing behaviors to update the position, and other beetle populations update their positions through a unique behavior. This diversified position update strategy makes the DBO algorithm have global and local search capabilities. However, an imbalance between the global search and local exploitation ability exists in the DBO algorithm. This algorithm is prone to fall into local optimum, and its global search ability is weak. Hence, the DBO algorithm is modified by integrating the statistical parametric mapping (SPM), improved sine algorithm (MSA), variable spiral search, and Levy flight, making the algorithm have faster convergence speed and higher solving accuracy.

Population initialization based on the SPM

In the DBO algorithm, the initial positions of four dung beetle populations are random, which may cause the uneven distribution of individual positions, thus reducing the population diversity. The SPM with simple parameters and uniform chaotic distribution can produce more uniformly distributed random values (Ban et al., 2020). To expand the distribution range of initial dung beetle populations in the solution space and make the dung beetle populations traversable and random, four dung beetle populations are initialized by the SPM:

X_{n + 1} = {\begin{array}{c} \mod (\frac{X_{n}}{η} + τ \sin (π X_{n}) + r_{0}, 1) 0 \leq X_{n} < η \\ \mod (\frac{X_{n} (0.5 - η)}{η} + τ \sin (π X_{n}) + r_{0}, 1) η \leq X_{n} < 0.5 \\ \mod (\frac{(1 - X_{n}) (0.5 - η)}{η} + τ \sin (π (1 - X_{n})) + r_{0}, 1) 0.5 \leq X_{n} < 1 - η \\ \mod (\frac{(1 - X_{n})}{η} + τ \sin (π (1 - X_{n})) + r_{0}, 1) 1 - η \leq X_{n} < 1 \end{array}

(1)

where

X_{n + 1}

and

X_{n}

are the chaos numbers at the (n+1)-th and n-th iterations, respectively,

η \in (0, 1)

τ \in (0, 1)

, and

r_{0} = r a n d (0, 1)

Position update of the ball-rolling dung beetle

The MSA utilizes the sine function to perform the iterative optimization, and has strong global exploration capability. And, the adaptive variable inertia weight coefficient is introduced in the position update process, making the algorithm fully search the local region. These achieve a good balance between the global exploration and local exploitation capabilities (Pan et al., 2023). Hence, the MSA is adopted to replace the dancing behavior of the ball-rolling dung beetle, and the position update of the ball-rolling dung beetle is as follows:

X_{i_{1}}^{Roll} (t + 1) = {\begin{array}{c} X_{i_{1}}^{Roll} (t) + 0.1 \times ς \times X_{i_{1}}^{Roll} (t - 1) + β \times Δ X \\ w_{ada} X_{i_{1}}^{Roll} (t) + f_{1} \times \sin r_{1} \times [r_{2} \times X_{i_{1}}^{Roll} (t - 1) - X_{i_{1}}^{Roll} (t)] \end{array} \begin{array}{l} δ < ε \\ δ \geq ε \end{array}

(2)

where t is the current iteration number;

{X_{i_{1}}}^{Roll} (t)

is the position of the

i_{1}

-th ball-rolling dung beetle at the t-th iteration;

ς

is a natural coefficient which is assigned −1 or 1;

β

a constant value belonging to

(0, 1)

;

Δ X

is used to simulate the changes of light intensity and

Δ X = | X_{i_{1}}^{Roll} (t) - X_{worst} |

, and

X_{worst}

is the global worst position;

w_{ada}

is the adaptive variable strategy inertia weight and

w_{ada} = w_{\max} - (w_{\max} - w_{\min}) \times t / t_{total}

w_{\max}

and

w_{\min}

are the maximum and minimum inertia weights, and

t_{total}

is the total number of iterations;

r_{1} \in [0, 2 π]

;

r_{2} \in [- 2, 2]

;

f_{1}

is a nonlinear decreasing function and

f_{1} = (w_{\max} - w_{\min}) / 2 \times \cos (π t / t_{total}) + (w_{\max} + w_{\min}) / 2

;

ε

is a constant value and

ε \in (0.5, 1]

;

δ

is a random number belonging to (0, 1).

The ball-rolling dung beetle performs the ball-rolling behavior to realize the global search when $δ < ε$ . It has no rolling direction and needs to use the sine function to adjust the movement direction when $δ \geq ε$ . The position update mode improvement of the DBO algorithm avoids much randomness, making the ball-rolling dung beetle be compared with the current optimal individual, and expands the scope of searching to some extent, improving the global search ability.

Position updates of breeding dung beetle and small dung beetle

The breeding dung beetle population and small dung beetle population both update their positions in a specific local area, reducing the population diversity thus making the DBO algorithm fall into local optimum. The Whale Optimization Algorithm (WOA) has strong global search capability because the whales use a spiral search strategy to update the prey position (Zhang et al., 2021). The spiral search strategy is used to update the position of the breeding dung beetle:

X_{i_{2}}^{Breed} (t + 1) = X^{lbest} + e^{l C_{spi}} \cos (2 π l) \times b_{1} \times (X_{i_{2}}^{Breed} (t) - L b^{bre}) + e^{l C_{spi}} \cos (2 π l) \times b_{2} \times (X_{i_{2}}^{Breed} (t) - U b^{bre})

(3)

where

X_{i_{2}}^{Breed} (t)

is the position of the

i_{2}

-th breeding dung beetle at the t-th iteration;

X^{lbest}

is the current local optimum;

C_{spi}

is the dynamic spiral search parameter and

C_{spi} = e^{5 \times \cos (π \times t / t_{total})}

; l is a random number belonging to (−1,1);

L b^{bre}

and

U b^{bre}

are the lower and upper boundaries of the breeding dung beetle spawning area,

L b^{bre} = \max {X^{lbest} \times (1 - C_{dyn}), L b}

U b^{bre} = \min {X^{lbest} \times (1 + C_{dyn}), U b}

, and Lb and Ub are the lower and upper boundaries of the search space;

C_{dyn}

is the dynamic coefficient and

C_{dyn} = 1 - t / t_{total}

;

b_{1}

and

b_{2}

are random vectors of size

1 \times D_{1}

, and

D_{1}

is the dimension of the search space.

The same spiral search strategy is also used to update the position of the small dung beetle:

X_{i_{3}}^{Small} (t + 1) = e^{l C_{spi}} \cos (2 π l) \times X_{i_{3}}^{Small} (t) + C_{1} \times (X_{i_{3}}^{Small} (t) - L b^{sma}) + b_{3} \times (X_{i_{3}}^{Small} (t) - U b^{sma})

(4)

where

X_{i_{3}}^{Small} (t)

is the position of the

i_{3}

-th small dung beetle at the t-th iteration,

C_{1}

is a random number that conforms to a normal distribution;

L b^{sma}

and

U b^{sma}

are the lower and upper boundaries of the dung beetle foraging area,

L b^{sma} = \max {X^{gbest} \times (1 - C_{dyn}), L b}

U b^{sma} = \min {X^{gbest} \times (1 + C_{dyn}), U b}

, and

X^{gbest}

is the global optimum;

b_{3}

is a random vector whose range falls between 0 and 1.

Position update of the stealing dung beetle

The stealing dung beetle population updates its position information by referring to the global optimum before the position update of the ball-rolling dung beetle, resulting in weak global search capability thus easily falling into local optimal. After introducing the step length generated by Levy flight (i.e., a probability distribution with long-tailed characteristics), the stealing dung beetles perform random walks in the search space to increase the exploration area (Cheng et al., 2023; Viswanathan et al., 2001). This mechanism helps the stealing dung beetle escape local optimum during the search process, thus increasing the global search capability. The Levy flight strategy makes the distance moved by the population individual obey the normal distribution, and ensures that the movement direction is randomized, effectively improving the global search ability and convergence speed of the DBO algorithm. Hence, the position of the stealing dung beetle, updated by the Levy flight strategy, is as follows:

X_{i_{4}}^{Steal} (t + 1) = C_{Levy} \times X^{gbest} + g \times 0.5 \times (| X_{i_{4}}^{Steal} (t) - X^{lbest} | + | X_{i_{4}}^{Steal} (t) - w_{dy} \times X^{gbest} |)

(5)

where

X_{i_{4}}^{Steal} (t)

is the position of the

i_{4}

-th stealing dung beetle at the t-th iteration;

g

is a random vector with the size of

1 \times D_{2}

, and obey the normal distribution;

w_{dy}

is the dynamic inertia weight and

w_{dy} = (e^{2 \times (1 - t / t_{total})} - e^{- 2 \times (1 - t / t_{total})}) / (e^{2 \times (1 - t / t_{total})} + e^{- 2 \times (1 - t / t_{total})})

;

C_{Levy}

is the step length obeying the Levy flight,

C_{Levy} = 0.01 \times u \times r_{3} / {| λ |}^{- ζ}

r_{3}

is a random number belonging to (0,1),

λ \sim (0, 1)

ζ \in (0, 2)

u \sim (0, σ_{u})

σ_{u} = {(Γ (1 + ζ) \times \sin (0.5 π ζ) \times ζ \times 2^{0.5 \times (ζ - 1)} / Γ ((1 + ζ) / 2))}^{1 / ζ}

, and

Γ ()

is the gamma function.

Improved DBO algorithm and its test

Initialize the dung beetle populations by the SPM, modify the position update mode of different dung beetle populations by the improved sine algorithm, variable spiral search, and Levy flight, and thus establish a new DBO algorithm, namely, the SMLDBO algorithm. The flowchart of the SMLDBO algorithm is shown in Figure 1.

Figure 1.

Flowchart of the SMLDBO algorithm.

Nine test functions are adopted to test the performance of the SMLDBO algorithm, and the results are compared with those of standard DBO algorithm, Particle Swarm Optimization (PSO) algorithm, and WOA. The specific expressions of nine test functions are given in Table 1. The optimal values, average values, and standard deviations of test functions and the iterative training curves from different algorithms are shown in Table 2 and Figure 2, respectively, wherein the optimal results are displayed in bold. The accuracy of the SMLDBO algorithm is obviously superior to those of DBO algorithm, PSO algorithm, and WOA except for F₂, F₆, and F₇. The SMLDBO algorithm has faster convergence speed in comparison with other algorithms except for F₇ and F₉.

Table 1.

Description of test functions.

Function characteristics	Function expressions
Unimodal benchmark	$F_{1} (x) = \sum_{i = 1}^{d} x_{i}^{2}$
Unimodal benchmark	$F_{2} (x) = \sum_{i = 1}^{d - 1} [100 {(x_{i + 1} - x_{i}^{2})}^{2} + {(1 - x_{i})}^{2}]$
Unimodal benchmark	$F_{3} (x) = \sum_{i = 1}^{d} i x_{i}^{2}$
Unimodal benchmark	$F_{4} (x) = \sum_{i = 1}^{d} \sum_{j = 1}^{i} x_{j}^{2}$
Multimodal benchmark	$F_{5} (x) = - 20 \exp (- 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {x_{i}}^{2}}) - \exp (\frac{1}{n} \sum_{i = 1}^{n} \cos (2 π x_{i})) + 20 + e$
Multimodal benchmark	$F_{6} (x) = 418.9829 d - \sum_{i = 1}^{d} x_{i} \sin (\sqrt{\| x_{i} \|})$
Multimodal benchmark	$F_{7} (x) = \sin^{2} (π x_{1}) + \sum_{i = 1}^{d - 1} {(x_{i} - 1)}^{2} [1 + 10 \sin^{2} (π x_{i} + 1)] + {(x_{d} - 1)}^{2} [1 + \sin^{2} (2 π x_{d})]$
Multimodal benchmark	$F_{8} (x) = 10 d + \sum_{i = 1}^{d} [x_{i}^{2} - 10 \cos (2 π x_{i})]$
Multimodal benchmark	$F_{9} (x) = \sum_{i = 1}^{d} \frac{x_{i}^{2}}{4000} - \prod_{i = 1}^{d} \cos (\frac{x_{i}}{\sqrt{i}}) + 1$

Table 2.

Optimization results obtained by different algorithms.

Functions	Statistical values	DBO	PSO	WOA	SMLDBO
F₁	Optimal value	3.7276E-160	3.50048E-16	1.92429E-16	0
	Average value	4.497E-100	1.14573E-13	5.81087E-15	0
	Standard deviation	1.0056E-99	1.3747E-13	6.91678E-15	0
F₂	Optimal value	24.63397929	4.606223031	28.60178786	28.21426622
	Average value	25.047902	1216.003784	28.74907967	28.5523497
	Standard deviation	0.264030772	1652.748239	0.089394117	0.318949679
F₃	Optimal value	2.1038E-159	7.56776E-15	3.50555E-18	0
	Average value	1.1938E-126	40	1.47024E-15	0
	Standard deviation	2.6695E-126	89.4427191	1.6737E-15	0
F₄	Optimal value	1.0528E-144	1.24956E-13	3.021E-15	0
	Average value	1.4525E-107	3.38879E-13	3.16813E-14	0
	Standard deviation	3.248E-107	2.85758E-13	4.1179E-14	0
F₅	Optimal value	4.44089E-16	7.99179E-08	6.00771E-09	4.44089E-16
	Average value	4.44089E-16	1.100142795	1.41574E-08	4.44089E-16
	Standard deviation	0	1.036538941	5.94102E-09	0
F₆	Optimal value	2029.759454	3358.839005	61.08440354	1976.092489
	Average value	3207.099944	3673.169348	1665.010865	3319.198079
	Standard deviation	1012.766449	331.5991859	1701.418858	1632.104169
F₇	Optimal value	12.54863283	0.454324019	9.154365762	78.05565939
	Average value	27.78345336	3.152523331	256.0024668	89.04446453
	Standard deviation	15.67456876	3.420836024	498.9968831	8.832600847
F₈	Optimal value	0	46.76301015	0	0
	Average value	4.97478722	65.66717165	2.96154E-10	0
	Standard deviation	11.1239624	15.95034985	6.52194E-10	0
F₉	Optimal value	0	4.32987E-14	0	0
	Average value	0	0.003943406	8.52429E-14	0
	Standard deviation	0	0.005673534	1.79734E-13	0

Figure 2.

Iterative training curves of test functions optimized by different algorithms.

Parameters optimization of the BPNN

Construction of the BPNN

The weight switches are added between the input layer and hidden layer as well as between the hidden layer and output layer, and a new BPNN model with weight switches is established, as shown in Figure 3.

Figure 3.

Three-layer BPNN model with weight switches.

The BPNN model adopts the sigmoid activation function, and the input and output of hidden layer neurons are given by:

{\begin{cases} I N_{j}^{J} = \sum_{α = 1}^{D_{3}} ω_{j h}^{α} p_{j h}^{α} s_{h}^{α} \\ O U T_{j}^{J} = \tan s i g (I N_{j}^{J} - V_{j}^{1}) = \frac{2}{1 + e^{- 2 (I N_{j}^{J} - V_{j}^{1})}} - 1 \end{cases}

(6)

where

I N_{j}^{J}

is the input value of the j-th neuron in the hidden layer,

j \in (1, J)

, and J is the number of hidden layer layers;

{S_{h}}^{α} = [(s_{1}^{1}, \dots, s_{1}^{α}), \dots, (s_{h}^{1}, \dots, s_{h}^{α})]

is the input layer of the BPNN (i.e., the structural parameter dataset of the VAWT),

h \in (1, N)

N

is the number of datasets,

α \in (1, D_{3})

, and

D_{3}

is the dimension of the dataset;

{ω_{j h}}^{α}

is the weight from the h-th neuron in the input layer to the j-th neuron in the hidden layer;

{p_{j h}}^{α}

is the weight switch from the h-th neuron in the input layer to the j-th neuron in the hidden layer; and

{p_{j h}}^{α}

takes the value of 0 or 1;

O U T_{j}^{J}

is the output value of the j-th neuron in the hidden layer;

V_{j}^{1}

is the j-th threshold of the neuron in the hidden layer.

The input and output of neurons in the output layer are, respectively:

{\begin{cases} I N_{k}^{K} = \sum_{j = 1}^{J} μ_{k j} q_{k j} O U T_{j}^{J} \\ O U T_{k}^{K} = I N_{k}^{K} - V_{k}^{2} = \sum_{j = 1}^{J} μ_{k j} q_{k j} O U T_{j}^{J} - V_{k}^{2} \end{cases}

(7)

where

I N_{k}^{K}

is the input value of the k-th neuron in the output layer,

k \in (1, K)

, and

K

is the number of output layers;

μ_{k j}

is the weight from the j-th neuron in the hidden layer to the k-th neuron in the output layer;

q_{k j}

is the weight switch from the j-th neuron in the hidden layer to the k-th neuron in the output layer, and

q_{k j}

takes the value of 0 or 1;

O U T_{k}^{K}

is the output value of the k-th neuron in the output layer;

V_{k}^{2}

is the threshold of the k-th neuron in the output layer.

The error of the k-th output layer neuron, $m_{k}$ , and the total error of all output layer neurons, $M_{k}$ are given by:

{\begin{cases} m_{k} = Y_{k} - O U T_{k}^{K} \\ M_{k} = \sum_{k = 1}^{K} m_{k} \end{cases}

(8)

where

Y_{k}

is the desired output value of the k-th output layer neuron.

Parameters optimization of the BPNN

To improve the fitting capability of the BPNN model, an optimization with the weight switches, weights, and thresholds as design variables is first performed by the BPSO algorithm integrated with the SMLDBO algorithm to minimize the training error. Based on initial optimized weight switches, weights, and thresholds, a subsequent optimization with the weights and thresholds as design variables is performed by the BP algorithm to minimize the testing error. The input dataset is divided into training dataset and testing dataset, denoted as $S_{XL} = (s_{1}, s_{2}, \dots, s_{XL})$ and $S_{CS} = (s_{XL + 1}, s_{XL + 2}, \dots, s_{N})$ , where N is the sample size of the input dataset and XL is the number of training data.

Initial optimization of BPNN parameters

Based on the BPSO algorithm in collaboration with the SMLDBO algorithm, the parameters of the BPNN are firstly optimized using the training data $S_{XL}$ as the learning sample. The design variables of the initial optimization model are the weight switches, weights, and thresholds, and the objective function is to minimize the average error of training data, $E_{train}$ :

f (A_{1}) = Min E_{train} = Min (\frac{1}{XL} \sum_{k = 1}^{XL} | Y_{k} - O U T_{k}^{K} |)

(9)

where

A_{1}

is a design variable vector and

A_{1} = (p, q, ω, μ, V^{1}, V^{2})

p, q, ω, μ, V^{1}

and

V^{2}

are the vectors composed of

p_{j h}^{α}, q_{k j}, ω_{j h}, μ_{k j}, V_{j}^{1}

and

V_{k}^{2}

, and

O U T_{k}^{K}

is the prediction value corresponding to the training data.

The improved BPSO algorithm from Liu et al. (2011) has better local search capability in dealing with discrete optimizations, and is used to optimize the weight switches of the BPNN. The position of the ib-th particle is updated by:

x_{i b} (t b + 1) = {\begin{cases} x_{i b} (t b) \\ 0 \\ 1 \end{cases} \begin{array}{l} rand () > S (v_{i b} (t b + 1)) \\ rand () \leq S (v_{i b} (t b + 1)) & v_{i} (t b + 1) < 0 \\ rand () \leq S (v_{i b} (t b + 1)) & v_{i} (t b + 1) > 0 \end{array}

(10)

where

x_{i b} (t b + 1)

and

x_{i b} (t b)

are the positions of the ib-th particle at the (tb+1)-th and tb-th iterations, respectively; rand() is a random number on [0,1]; when

v_{i b} (t b + 1) \leq 0

S (v_{i b} (t b + 1)) = 1 - 2 \times {[1 + \exp (- v_{i b} (t b + 1))]}^{- 1}

; when

v_{i b} (t b + 1) > 0

S (v_{i b} (t b + 1)) = 2 \times {[1 + \exp (- v_{i b} (t b + 1))]}^{- 1}

;

v_{i b} (t b + 1)

is the velocity of the ib-th particle at the (tb+1)-th iteration.

The update formula of $v_{i b} (t b + 1)$ is:

v_{i b} (t b + 1) = w v_{i b} (t b) + r_{1} a_{1} (P b e s t - x_{i b} (t b)) + r_{2} a_{2} (G b e s t - x_{i b} (t b))

(11)

where w is the inertia weight,

a_{1}

and

a_{2}

are learning weight factors,

r_{1}

and

r_{2}

are random numbers on [0,1],

P b e s t

is the individual best particle, and

G b e s t

is the global best particle.

Given that the SMLDBO algorithm has great improvement in both global and local search abilities, it is used to optimize the weights and thresholds in the BPNN. Among, the scale factors of ball-rolling dung beetle and breeding dung beetle are both set to 0.3, and those of small dung beetle and stealing dung beetle are both set to 0.2. That is, the ball-rolling dung beetle, breeding dung beetle, small dung beetle, and stealing dung beetle account for 30%, 30%, 20%, and 20% of the total population, respectively, as shown in Figure 4.

Figure 4.

Distribution map of various dung beetle populations.

Subsequent optimization of BPNN parameters

Based on the BP algorithm, the parameters of the BPNN are subsequently optimized using the test data $S_{CS}$ as the learning sample. The design variables of the subsequent optimization model are the weights and thresholds, and the objective function is to minimize the average error of the testing data, $E_{test}$ :

f (A_{2}) = Min E_{test} = Min (\frac{1}{CS} \sum_{k = XL + 1}^{N} | Y_{k} - O U T_{k}^{K} |)

(12)

where

A_{2}

is a design variable vector and

A_{2} = (ω, μ, V^{1}, V^{2})

, and CS is the number of test data.

Torque coefficient prediction for the VAWT

Structural parameters-torque coefficient dataset construction

The Latin Hypercube Sampling method is used to uniformly sample from the value ranges of blade installation angle γ, chordwise connective position G, height-to-diameter ratio H/D, blade number B, and airfoil chord length c, and 20 sets of sample points are obtained. The random distribution relationships between γ and c, between G and c, between H/D and c, and between B and c are presented in Figure 5. The sample points selected are well-distributed and do not overlap, indicating that the clustering of sample points is avoided. To increase the dataset size, the other 20 sets of sample points are randomly selected within the ranges of five structural parameters. That is, the dataset includes 40 sets of sample points. The average torque coefficients corresponding to 40 sets of sample points are calculated using the CFD method, and so the structural parameters-torque coefficient dataset is constructed, wherein the percentage split between training and testing is 8:2.

Figure 5.

Scatter plots of structural parameters obtained by the Latin Hypercube Sampling method.

Aerodynamic calculation of the VAWT

Computational domain and grid division

The components, like connecting rod and tower, etc., have less influence on the flow near the wind wheel. The wind turbine model containing only blades is established, reducing the number of grids thus saving computational cost. The geometric model, computational domain, and grids of the VAWT designed in Liu (2018) are generated by the ICEM software, as shown in Figures 6 and 7. The computational domain consisting of a rotational subdomain and a static subdomain is a cuboid with the size of 14R × 8R × H₁, where R is the radius of the wind wheel, and H₁ is the blade height. The rotational subdomain is a cylinder with a radius of R + c and a height of H₁, that is, the red part shown in Figure 6, and the static subdomain is a graph between the cuboid and the rotational subdomain. The rotational center of the wind turbine is at a distance of 4R from the left, front, and rear boundaries, 10R from the right boundary, and 0.5H₁ from the upper boundary. The 3D grid model is obtained using the structural grid to discretize the computational domain. A boundary layer with the first-row height of 0.000031 m, the growth factor of 1.3, and the layer row of 40 is attached to the airfoil to encrypt the near-wall surface due to the complexity of the airfoil surface, the grid division is outward, and $y^{+} = 1.016523$ .

Figure 6.

Computational domain of the VAWT.

Figure 7.

Grid division of the VAWT.

Boundary conditions, turbulence model, and solver setting

The Realizable k-ε turbulent model, which is likely to provide superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, and separation, is adopted. The left and right sides of the computational domain are the velocity-inlet and pressure-outlet, respectively. The upper and lower surfaces of static subdomain and rotational subdomain are symmetry boundaries. The boundary condition of blade surfaces is moving wall, and the rotational and static subdomains are coupled together using the mesh interface. The pressure-velocity coupling equation is solved by the SIMPLEC algorithm, and the momentum, turbulence kinetic energy, and specific dissipation rate are all discretized using the second-order upwind scheme. The rotational speed of the wind turbine is 200 r/min, the inlet velocity is 9 m/s, and the turbulence intensity and viscosity ratio are 10% and 1. In each rotation cycle of the wind wheel, the number and size of time steps are 400 and 0.005 s, and 40 iterations are performed per time step.

Grid independence validation

The quality and quantity of grids have significant influence on the accuracy of the CFD calculation, and the grid quantity is ensured by y⁺>1. Thus, the grid independence analysis is conducted on the wind wheel. The torque coefficient $C_{m}$ is calculated when the number of grids is 5105511, 5879733, and 6707811, as shown in Figure 8. The torque coefficients corresponding to the grids of 5879733 and 6707811 exhibit a higher level of agreement, and vary periodically with the azimuth angle. The average torque coefficient is 0.201, 0.205, and 0.207 for three sets of grid numbers, and increases with the increase of the grid number. It increases by 2% and 3% when the number of grids increases from 2105511 to 5879733 and from 2105511 to 6707811, respectively. That is, the increase in the average torque coefficient decelerates after a certain number of grids. Considering the accuracy and efficiency of numerical simulation comprehensively, the meshing method used in the grid number of 5879733 is selected for subsequent grid division.

Figure 8.

Grid independence verification.

Numerical reliability verification

The same CFD framework is applied to calculate the power coefficient of the VAWT from Castelli et al. (2010) under different tip speed ratios, and the obtained simulation results are compared with the experimental and numerical data presented in Castelli et al. (2010), as shown in Figure 9. Compared to the numerical results in Castelli et al. (2010), the power coefficients calculated by the CFD framework in this paper agree better with the experimental data in Castelli et al. (2010), especially in the tip speed ratio range of [2.3-3.3].

Figure 9.

Comparison of power coefficients from CFD simulation and experiment.

Torque coefficient prediction and SMLDBO-BPNN performance evaluation

The established SMLDBO-BPNN model is used to predict the average torque coefficient $C_{m}^{ave}$ of VAWTs corresponding to 40 sample points mentioned above, as shown in Figure 10. The parameters of the SMLDBO-BPNN model are as follows: the number of hidden layers is 1, the number of neurons in the hidden layer is 10, the training iterations are 4000, and the learning rate is 0.014. The parameters of the SMLDBO algorithm are as follows: the population size is 30, and the maximum number of iterations is 600. The parameters of the BPSO algorithm are as follows: the population size is 30, the learning weight factors $a_{1} = 2$ and $a_{2} = 2$ , the inertia weight coefficient $w$ is improved using the linear decreasing weight method and $w = w_{\max} - (w_{\max} - w_{\min}) t / t_{\max}$ , $w_{\max}$ and $w_{\min}$ are the maximum and minimum weight coefficients, $t$ and $t_{\max}$ are the current and maximum number of iterations, and $w_{\max} = 1.2$ , $w_{\min} = 0.4$ , $t_{\max} = 600$ . According to Figure 10, the predicted results of the SMLDBO-BPNN model are basically consistent with the calculated values of the CFD method.

Figure 10.

Comparison of SMLDBO-BPNN prediction and CFD calculation.

Moreover, the prediction performance of SMLDBO-BPNN is evaluated using the mean absolute error (MAE), the mean square error (MSE), and the coefficient of determination (R²), and the corresponding curves are shown in Figure 11. For the training dataset, the MAE, MSE, and R² of SMLDBO-BPNN are 0.0281, 0.0014, and 0.9968, respectively, while for the testing dataset, they are 0.0315, 0.0014, and 0.9933, all satisfying convergence requirements. R² is close to 1 for both datasets, indicating excellent prediction accuracy. The model achieves low MAE and MSE at the beginning due to the initial optimization of weight switches, weights, and thresholds by the combined BPSO and SMLDBO algorithms. Subsequently, the MAE and MSE decrease rapidly and remain low, whereas R² steadily increases and stays high. Furthermore, the MAE, MSE, and R² curves of the testing dataset show good consistency with those of the training dataset. Overall, the SMLDBO-BPNN model achieves low errors and high R² on both training and testing datasets, and the metrics of the testing dataset closely match those of the training dataset. These demonstrate that the model not only captures the nonlinear mapping relationships effectively but also exhibits strong generalization capability.

Figure 11.

Training curves of evaluation indicators.

Structural optimization of the VAWT

Optimization method

The built SMLDBO-BPNN model coupled with the PSO algorithm is used to optimize the structural parameters of the VAWT, and the flowchart is shown in Figure 12.

Figure 12.

Flowchart for optimizing the structural parameters of the VAWT.

The average torque coefficient is an important index to evaluate the aerodynamic performance of wind turbines, and so the design objective is to maximize the average torque coefficient:

F (A_{3}) = \max (C_{m}^{ave})

(13)

where

A_{3}

is the vector of design variables.

The structural parameters of the VAWT, that is, blade installation angle, chordwise connective position, H/D, blade number, and chord length, are adopted as the design variables:

A_{3} = (γ, G, H / D, B, c)

(14)

According to Yang and Xi (2017), Liu et al. (2024), Li (2009), the optimal blade installation angle, blade number, and H/D of the H-type VAWT generally fall within [−3°, 8°], [2, 5], and [0.5, 3], respectively. The chord length formulation from Zhang et al. (2020) is combined with H/D to determine the chord length range. According to Miao et al. (2023) and Bianchini et al. (2016), the optimal blade-rod connective position is around 0.25c for the H-type VAWT. Taken together, the specific ranges of design variables are given in Table 3.

Table 3.

Ranges of design variables.

Design variables	Ranges
Installation angle γ/(°)	−3 to 8
Chordwise connective position G	0.15c–0.35c
Height-to-diameter ratio H/D	0.5–3
Number of blades B	[2,3,4,5]
Chord length c/(m)	0.095–0.234

Optimization results

The VAWT designed in Liu (2018) is chosen as the testbed to optimize the structural parameters, where the radius of the wind wheel, R, is 0.85 m and the height of the wind wheel, H, is 1 m. The specific parameters of the PSO algorithm are as follows: the number of particles is 5,

t_{\max} = 200

a_{1} = a_{2} = 1.5

w_{\max}

= 0.9, and

w_{\min}

= 0.4. The optimization results and iterative history are presented in Table 4 and Figure 13. After the optimization, the height-to-diameter ratio, installation angle, blade number, and chord length all decrease, and the distance between the blade-rod connective position and the airfoil leading-edge point becomes larger. The iteration curve increases rapidly in the early stage, indicating efficient global exploration, and gradually flattens after 27 iterations, indicating a shift from global to local search, and remains unchanged after 175 iterations, indicating a stable convergence state.

Table 4.

Un-optimized and optimized values of design variables.

Design variables	Original values	Optimized values
Installation angle γ/(°)	0	−2.5
Chordwise connective position G	0.25c	0.29c
Height-to-diameter ratio H/D	0.588	0.5
Number of blades B	5	4
Chord length c/(m)	0.22	0.156

Figure 13.

Iterative curve for the structural optimization of the VAWT.

Flow characteristics and aerodynamic performance of un-optimized and optimized VAWTs

The pressure, vortex, and torque coefficient of un-optimized and optimized VAWTs are studied by the CFD method so as to further understand the improvement effect of the structural optimization.

Pressure distribution

The pressure distributions before and after the wind wheel optimization are presented in Figure 14. As for the un-optimized wind wheel in the windward region, the high-pressure region on the blade pressure-surface expands from the leading-edge to the center firstly and then gradually moves to the trailing-edge, and its influence range significantly decreases. The low-pressure region on the suction-surface gradually expands to the leading-edge, and the other new low-pressure region forms at the trailing-edge. Then, two low-pressure regions move behind the trailing-edge, and the influence range significantly decreases. In the leeward region, the high-pressure region is always concentrated at the leading-edge, while the rest of pressure- and suction-surfaces consistently maintain the low-pressure.

Figure 14.

Pressure distributions of un-optimized and optimized wind turbines.

As for the optimized wind wheel in the windward region, the high-pressure region on the pressure-surface and the low-pressure region on the suction-surface gradually move to the leading-edge, and their influence range continuously decreases. In the leeward region, the high-pressure region is always concentrated at the leading-edge of the pressure-surface, while the low-pressure region gradually expands from the entire suction surface to the center and trailing-edge of the pressure-surface.

The vertical color bar scale in the right part of Figure 14 indicates different pressure values. The color of the pressure contour after the optimization is darker than that before the optimization, indicating that the pressure generated by the optimized wind wheel is greater. After the optimization, only one high-pressure region and one low-pressure region occur near the leading-edge of each blade, and the ranges of high- and low-pressure regions in the windward region are significantly larger, which have a positive impact on the start-up performance of the wind turbine. Moreover, more concentrated high- and low-pressure regions are closer to the incoming flow and blade leading-edge, which indicates that the main aerodynamic loads are generated earlier and more stably during the blade azimuthal cycle. To sum up, these changes contribute to a stronger and more sustained torque production.

Vortex distribution

The vortex distributions before and after the wind wheel optimization are presented in Figure 15. As for the un-optimized wind wheel in the windward region, the influence region of the wake vortex gradually increases firstly and then decreases, that is, the wake shape gradually becomes narrow and long firstly and then becomes wide and short. During the process, the flow separation occurs and gradually intensifies. In the leeward region, the influence region of the wake vortex gradually increases, and the wake shape gradually becomes narrow and long.

Figure 15.

Vortex distributions of un-optimized and optimized wind turbines.

As for the optimized wind wheel in the windward region, the influence region of the wake vortex gradually decreases, the wake shape gradually becomes narrow and long, and more concentrated wake vortexes occur, which implies that the vortex dissipation reduce and the more incoming wind energy is utilized. In the leeward region, the influence region of the wake vortex gradually increases, and the wake shape gradually becomes longer. Furthermore, the wake rotates outward and has a weaker impact on downstream blades, thereby reducing periodic inter-blade interference. These demonstrate that the optimized configuration is conducive to more effective torque generation.

Torque coefficient

The torque coefficients of un-optimized and optimized wind wheels at one rotating period are presented in Figure 16. The torque coefficients before and after the optimization show the same periodic variation pattern, but their peak positions are slightly different. The torque peaks of the un-optimized wind wheel occur at 36°, 108°, 180°, 252°, and 324°, while those of the optimized wind wheel occur at 66°, 156°, 246°, and 336°. The reasons are as follows: the “phase” of the torque peak depends on the number of blades because blade spacing determines how long each blade interacts with the incoming wind. The number of blades before and after optimization are 5 and 4, respectively. The reduction of blade number increases this interaction time, delaying the azimuthal angle of the maximum torque thus shifting the overall peak backward.

Figure 16.

Comparison of torque coefficients before and after the optimization.

Based on Figure 16, the average torque coefficient is calculated and is 0.2 and 0.23 before and after the optimization, respectively. That is, it increases by 15%. The higher torque helps the wind turbine start faster and reach a stable state under low wind speeds, which improves the operational efficiency and response speed.

Conclusion

In this study, a structural optimization method, based on the optimized three-layer BPNN with weight switches predicting the torque coefficient, is proposed for the VAWT. And, the flow characteristics and output performance of un-optimized and optimized wind turbines are also investigated. The main conclusions are as follows:

(1) The obtained SMLDBO algorithm calculates the Sphere function, Rastrigin function, and Shekel function, and shows significant improvements in the convergence accuracy, velocity, and stability compared to the DBO algorithm.

(2) The average torque coefficients predicted by the established SMLDBO-BPNN agree well with the CFD results, and the linear correlation coefficient between the predicted and calculated results is 0.9908, indicating that this network has excellent predictive capability.

(3) After the wind wheel optimization, the pressure around blades obviously increases, and the pressure distribution around each blade becomes regular, that is, only one high-pressure region and one low-pressure region occur near each blade. The width of the concentrated vortex generated by blades significantly reduces, exerting little influence on rear blades. Thus, the flow separation is reduced, and the average torque coefficient increases by 15%.

Footnotes

ORCID iD

Xu Zhang

Author contributions

Zhang X: conceptualization, methodology, investigation, formal analysis, data curation, writing-original draft preparation, writing-review and editing, project administration, funding acquisition. Qi Y L: methodology, software, data curation, formal analysis, writing-review and editing. Wang Z T: software, formal analysis, writing-original draft preparation. Ruan J T: validation, supervision. Wang Z: software. All authors have read and agreed to the published version of the manuscript.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of Tianjin (Grant No. 23JCZDJC00460) and the National Natural Science Foundation of China (Grant No. 51805369).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.*

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