# Optimization Based on Computational Fluid Dynamics and Machine Learning for the Performance of Diffuser-Augmented Wind Turbines with Inlet Shrouds

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

- BPNN algorithm: ML is a branch of artificial intelligence. In the modern era of software, ML depends on the prediction of datasets based on various algorithms for different software modules. A neural network is an ML model that implements a learning/training rule, i.e., when the input nodes are activated, synaptic weights are updated and forwarded to output nodes. Different training algorithms are available, like backpropagation, genetic, and krill herd algorithms [53]. The BPNN is such an ML model, one inspired by the biological neural network, and it is one of the oldest supervised-learning multilayer feed-forward neural network algorithms [54], having been proposed by Rumelhart, Hinton, and Williams in 1986 [55]. As introduced in the 1980s, it quickly became a focal point in neural network research due to its outstanding learning capability and adaptability [56]. Due to its backpropagating ability, it is highly suitable for problems with no relationships between the outputs and inputs [54]. Its flexibility, learning, and powerful fitting capabilities make it a robust tool for addressing complex problems [54,56]. Lillicrap et al. [57] pointed out that neural networks trained with backpropagation of error are at the heart of the recent successes of ML, including state-of-the-art speech and image recognition and language translation. In addition, backpropagation of error even underpins recent progress in unsupervised learning problems such as image and speech generation, language modeling, and other next-step prediction tasks. Over the years, BPNN has been proven to be the best algorithm among the multilayer perceptron algorithms [58]. Thus, the multilayer perceptron neural network trained with BPNN is the most popular and widely used network paradigm employed by engineering applications to solve practical problems, and it has demonstrated exceptional performance [53,54,56,59,60]. Inevitably, the traditional BPNN algorithm has some shortcomings, such as low convergence speed and an easy fall to the local minimum, but some remedies have been proposed to solve these problems [61]. In this study, the BPNN model is selected based on the abovementioned survey and then employed on the MATLAB platform, which will promptly update the latest modifications to the model.
- NSGA-II algorithm: Inspired by Darwin’s theory of species evolution, John Holland proposed the genetic algorithm (GA) in 1975, which is widely used in various fields, including artificial intelligence, logistics distribution, and engineering science applications. It can be employed as an optimization algorithm that simulates the biological evolution process for multi-objective optimization problems (MOPs) [62]. In the real world, it is challenging to determine optimal solutions over MOPs with multiple conflicting objectives in complex systems. In such a situation, it is impossible to compute a single optimal solution. Therefore, the most common solution concept is to compute a set of Pareto optima, solutions that cannot be improved in one objective without accepting a worsening in others, and then let a decision-maker select the final solution based on their preference [63]. As a mainstream method for solving MOPs, the development and application of evolutionary algorithms (EAs) has attracted thousands of researchers since the 1950s [64]. EAs profit from their general ability to work with sets of solutions, are the standard approach to MOPs, and have many successful applications [63]. The NSGA-II algorithm, a model initially proposed by Deb et al. [65] in 2002, is considered the most prominent multi-objective EA [63,66] with the most popular GA framework [64], and has served as a powerful decision-space exploration engine, based on GA, used to solve MOPs [67]. So far, it has been cited more than 50,000 times on Google Scholar [68] and is becoming one of the most widely used algorithms for solving MOPs in various applications in different fields [12,68,69,70]. It has been verified that the Pareto frontier obtained by the NSGA-II algorithm is evenly distributed and has good convergence and robustness. [71]. We take advantage of its high competence, efficiency, and strength in dealing with most MOPs and adopt it in this research.

#### 2.1. Computational Fluid Dynamics (CFD)

_{n}is the velocity component normal to the surface, v

_{i}is the surface velocity component, v

_{n}is the surface velocity component normal to the surface, a

_{0}is the far-field sound speed, δ(f) is the Dirac delta function, and p′ is the sound pressure at the far field (p′ = p − p

_{0}).

_{ε}= 0.1, $I\left(\overrightarrow{y}\right)=\frac{{A}_{c}\left(\overrightarrow{y}\right)}{12{\rho}_{0}\pi {a}_{0}^{3}}\overline{{\left[\frac{\partial p}{\partial t}\right]}^{2}}$. Equation (6) is Proudman’s formula, which indicates the quadrupole source strength related to turbulent shear stress and can be used to calculate the acoustic power. Equation (7) is the boundary layer noise source model, which can simulate the dipole source strength related to pressure fluctuations and can be used to calculate the acoustic surface power.

#### 2.2. Backpropagation Neural Network (BPNN)

_{ij}is the weight between neurons i and j. BPNN uses the least-squares method for weight adjustment, and its process is as follows: 1. randomly initialize the weights; 2. use the current weights to calculate the output value; 3. calculate the difference, i.e., error, between the output value and the target value; 4. re-adjust the weights; 5. repeat steps 2.~4. until convergence.

_{k}is the target value of the kth neuron in the output layer. The weight is adjusted according to the two formulas ${w}_{ij}={w}_{ij}+\eta \times {\delta}_{j}\times ne{t}_{i}$ and ${w}_{jk}={w}_{jk}+\eta \times {\delta}_{k}\times ne{t}_{j}$, where η is the learning rate between 0 and 1, ${\delta}_{k}=\left({D}_{k}-ne{t}_{k}\right)\times \left[ne{t}_{k}\times \left(1-ne{t}_{k}\right)\right]$, and ${\delta}_{j}={\displaystyle \sum _{k=1}^{K}\left({\delta}_{k}\times {w}_{jk}\right)\times \left[ne{t}_{j}\times \left(1-ne{t}_{j}\right)\right]}$.

#### 2.3. Multi-Objective Genetic Algorithm-NSGA-II

_{i}(x) is the i-th objective function to be minimized, x is the solution vector, and Ω is the solution space. Usually, the objective functions are contradictory, i.e., the improvement of one objective function requires the improvement of another objective function to be lowered as a price. If a and b are two sets of feasible solutions to the above m objective minimization problem, then if $\forall i{f}_{i}\left(a\right)\le {f}_{i}\left(b\right)\mathrm{and}\exists j:{f}_{j}\left(a\right){f}_{j}\left(b\right)$ is satisfied; a can be said to dominate b ($a\succ b$). Any feasible solution in the solution space, if other feasible solutions do not dominate it, is called a non-dominated solution, also called a Pareto-optimal solution. This solution is not unique but belongs to the Pareto-optimal set. The solutions in this set are indistinguishable, and the line connected in the solution space is called the Pareto front. Therefore, solving multi-objective optimization problems aims to find the complete Pareto front.

#### 2.4. Experimental Setup and Measurement

_{i}is the relative uncertainty of item i, and u

_{i,j}is the relative uncertainty of i in the result due to uncertainty in j. According to the above calculation, the uncertainty of the measured power output u

_{W}is 2.7%.

## 3. Results and Discussion

#### 3.1. Numerical Validation

^{6}was employed for the following simulations. Figure 8 displays the mesh distribution around the rotor blade and the corresponding y+ values, whose maximum value is lower than 10. The predicted turbine power of the BWT with design rotational speed (ω = 500 rpm) under various wind speeds (U = 5–7 m/s) is compared with the experimental data presented by Sign and Ahmed [72], as shown in Figure 9. It can be found that the order of magnitude and trend of the predictions are generally consistent with the experimental data but are slightly overpredicted. The reason may be that the numerical simulation does not consider the energy conversion losses of the generator, so the predicted values are somewhat higher than the experimental values.

#### 3.2. Optimized Results Obtained by Taguchi Method

_{25}(5

^{4}) is yielded, creating 25 different geometric designs for the diffuser, so that the CFD simulations can be used to produce datasets for ML. At this step, a set of optimized parameters can also be gained using the Taguchi method for single-objective optimization. The Taguchi method uses a signal-to-noise (S/N) ratio, which renders a design quality and response graph to obtain the optimal parameters. Based on the 25 sets of generated power data from the CFD simulations, the corresponding S/N ratios can be calculated via Equation (14) for the cases of larger-the-better (LTB) characteristics.

_{i}is the i-th experiment quality characteristics. Figure 10 displays the response graph of the SN ratio for the LTB analysis of DAWT-generated power. Consequently, an optimized parameter combination can be obtained, as shown in Table 5, and labeled as DAWT-Taguchil, alongside an original design labeled as DAWT-Origina. The original one is set as level 1 of the studied parameters for reference. Figure 10 also indicates that the impact of the rotor’s axial position on the power output is the most significant, followed by the diffuser length, flange angle, and flange height.

#### 3.3. Optimized Results Obtained by Multi-Objective Optimization

- Training dataset: The training set comprises a subset of the generated datasets obtained from the CFD simulations. This dataset trains the neural network model using the backpropagation algorithm. During training, the neural network learns the underlying patterns and the relationships between the input parameters (flange height/angle, diffuser length, rotor axial position) and the corresponding outputs (power output and generated noise of the DAWT). The Levenberg–Marquardt algorithm was employed in this phase to obtain a lower mean squared error [61].
- Validation dataset: This dataset, separate from the training dataset, tunes hyperparameters and assesses the model’s performance during training. After each epoch of training iterations, the model’s performance is evaluated on the validation dataset to monitor the model’s generalization ability and prevent overfitting.
- Test dataset: This dataset, separate from the training and validation datasets, is used to evaluate the final performance of the trained model and assess how well the trained model generalizes on unseen data.

_{a}is the atmospheric pressure, p

_{b}is the back pressure behind the diffuser, p

_{2}is the pressure behind the rotor, u

_{1}is the wind speed in front of the rotor, and U is the upstream wind speed. Watanabe and Ohya [31] indicated that the back-pressure coefficient is significant for the performance of DAWTs and pointed out that a lower C

_{pb}and a higher C

_{pd}are desired for a high-performance DAWT. In addition, Table 7 displays the tip clearance for various designs, and it can be found that its original value of 10 mm will change with the selected parameter combinations during the optimization process, as mentioned before. Table 7 also displays these tip clearance values as divided by the rotor radius, which fall within a reasonable range, as presented in the literature [32,39].

#### 3.4. Flow and Acoustic Fields: Analysis and Comparison

_{pb}(Table 7). However, such a geometric arrangement causes it to have the poorest C

_{pd}and exhibit a weaker pressure gradient near the rotor (Figure 13e). Such flow characteristics are helpful to noise reduction and enable DAWT-NSGAII-B to balance both design objectives. Similar phenomena, e.g., higher K does not produce higher power output, can be observed with the DAWT-Original and DAWT-NSGAII-C, but another reason causes this. The rotor axial positions of both designs are not too different (1 and 1.4, as indicated in Table 5). Because K is an area-averaged value, it cannot reflect the different space distributions of the flow field. By observing Figure 14b,f, it can be determined that the high-velocity regions of both DAWT-Original and DAWT-NSGA-II-C are closer to the diffuser peripheral and farther from the rotor. Based on the observation of pressure and velocity fields, it can be concluded that adding a diffuser enhances the mass flow rate of the airflow that passes through the rotor, and increases its power output.

^{2}/s

^{2}), which has almost the same form as the acoustic power distribution. Figure 20 displays the acoustic surface power, calculated by Equation (7), which is related to the pressure fluctuations in the flow field. Observing this figure shows that the intensity of acoustic surface power on the blade surface is almost the same for all cases, including the BWT. It can be inferred that the primary noise source is the pressure fluctuation produced by the rotating blade, and the imposition of a diffuser will mitigate its contribution to the overall noise in the frequency band of 200 Hz to 800 Hz.

_{p}) of the DAWTs analyzed in this study and compares them with the ones of typical BWTs presented in Ref. [8]. The definition of C

_{p}is shown in Equation (16).

_{d}is the cross-section area of the diffuser. It can be found that the optimized designs proposed in this study provide reasonable power performance. By inspecting the definition presented in Equation (16), it can be recognized that DAWTs implemented with high flange height will decrease their C

_{p}value if their output power increases cannot compensate for the increased area. Figure 22 shows the experimental measurement results of the scaled-down DAWT models manufactured by 3D printing. It can be found that the sequence of the magnitudes of measured output power almost displays the same trend as the full-scale CFD simulations of the optimized designs. Thus, this can verify the effectiveness of the optimization methodology proposed in this study even though all the results presented in this paper are achieved without considering the manufacturing costs, structural strength, energy conversion efficiency of the electric generator, friction loss of the gearbox, and bearing power loss. In addition, both the BPNN and NSGA-II algorithms are potent tools for predictive and optimization modeling. However, they also have certain limitations; e.g., BPNN may suffer from issues about local minima, overfitting, gradient vanishing/exploding, hyperparameter sensitivity, etc.; NSGA-II may encounter problems with convergence, premature convergence, scalability, parameter sensitivity, etc. In summary, while both BPNN and NSGA-II offer valuable capabilities for optimization and modeling tasks, researchers and practitioners should be mindful of their limitations and employ them with careful and deliberate testing and evaluation, using the latest-version models and suitable platforms, to address specific challenges effectively.

## 4. Conclusions

- 1.
- This study successfully integrates CFD, BPNN, and NSGA-II to conduct multi-objective optimization of DAWT, using output power and noise as objective functions. Performance evaluation and verification are carried out for the optimized designs. Finally, the diffuser configurations that meet different requirements for power and noise are proposed.
- 2.
- The influence of design parameters can be evaluated through the Taguchi method. It was found that the impact of the rotor’s axial position on the power output of DAWT is the most significant, followed by those of the diffuser length, flange angle, and height.
- 3.
- When employing a cycloid diffuser profile, varying the design parameters, i.e., flange height/angle, diffuser length, and rotor axial position, allows for the indirect incorporation of additional parameters, i.e., tip clearance, diffuser opening angle, and the adding of inlet shroud, allowing these values to be optimized together.
- 4.
- It was evident that a well-designed diffuser requires the acceleration of airflow while maintaining high-pressure recovery.
- 5.
- Under the conditions of this study, introducing a diffuser can reduce the noise in the frequency band of 200 Hz to 800 Hz, but if the induced tip vortex is too strong, it will have the opposite effect on the noise reduction. This finding can be used to interpret the positive and negative impacts on noise of installing a diffuser.
- 6.
- The flange height should not be too high or too low. If it is too low, it will not be able to generate a vortex with sufficient strength behind it to accelerate the airflow inside the diffuser, which will provide limited help to the power output. However, if the flange is too high, it will cause the recirculation zone to be far away from the rotor inside the diffuser, resulting in limited benefit to power output. Moreover, if the flange height is too high, it would lead to structural damage and deformation, resulting in higher maintenance costs and making it unsuitable for practical applications.
- 7.
- An appropriate flange angle can induce vortices to drive and accelerate airflow within the diffuser. Negative flange angles balance power and noise demands, resulting in smoother flow fields with weaker pressure fluctuations and lower noise levels.
- 8.
- If the diffuser length is too long, it is unsuitable for practical applications because it can cause the DAWT to be too heavy for placement in elevated locations. Additionally, excessive length can hinder the vortices generated by the flange from influencing the flow field near the rotor inside the diffuser.
- 9.
- The rotor should ideally be positioned behind the throat (approximately halfway between the throat and the outlet) to receive the accelerated flow field induced by the vortices generated by the flange, thereby enhancing its power output. Placing the rotor near this position between the throat and the outlet yields better results for maximizing power output. However, positioning the rotor in front of the throat is more effective for noise reduction.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Illustration of the principal components and geometric parameters of the studied diffuser-augmented wind turbine with inlet shroud.

**Figure 6.**Schematic diagram of the rotor blade: (

**a**) AF300 airfoil and constructed solid model; (

**b**) chord distribution along the rotor radius; (

**c**) twist angle distribution along the rotor radius.

**Figure 7.**Schematic diagram of the computational domain (not to scale) and employed boundary conditions.

**Figure 9.**Comparisons of the simulated turbine power with the experimental data for BWT [72].

**Figure 13.**Pressure field distributions (Y/D = 0): (

**a**) BWT; (

**b**) DAWT-Original; (

**c**) DAWT-Taguchi; (

**d**) DAWT-NSGA-II-A; (

**e**) DAWT-NSGA-II-B; and (

**f**) DAWT-NSGA-II-C.

**Figure 14.**Z-component velocity field distributions (Y/D = 0): (

**a**) BWT; (

**b**) DAWT-Original; (

**c**) DAWT-Taguchi; (

**d**) DAWT-NSGA-II-A; (

**e**) DAWT-NSGA-II-B; and (

**f**) DAWT-NSGA-II-C.

**Figure 15.**Y-component vorticity field and velocity vector distributions (Y/D = 0): (

**a**) BWT; (

**b**) DAWT-Original; (

**c**) DAWT-Taguchi; (

**d**) DAWT-NSGA-II-A; (

**e**) DAWT-NSGA-II-B; and (

**f**) DAWT-NSGA-II-C.

**Figure 16.**Original noise spectrum: (

**a**) BWT; (

**b**) DAWT-Original; (

**c**) DAWT-Taguchi; (

**d**) DAWT-NSGA-II-A; (

**e**) DAWT-NSGA-II-B; and (

**f**) DAWT-NSGA-II-C.

**Figure 17.**A-weighting noise spectrum: (

**a**) BWT; (

**b**) DAWT-Original; (

**c**) DAWT-Taguchi; (

**d**) DAWT-NSGA-II-A; (

**e**) DAWT-NSGA-II-B; and (

**f**) DAWT-NSGA-II-C.

**Figure 18.**Acoustic power distributions (X/D = 0): (

**a**) BWT; (

**b**) DAWT-Original; (

**c**) DAWT-Taguchi; (

**d**) DAWT-NSGA-II-A; (

**e**) DAWT-NSGA-II-B; and (

**f**) DAWT-NSGA-II-C.

**Figure 19.**Turbulent kinetic energy distributions (X/D = 0): (

**a**) BWT; (

**b**) DAWT-Original; (

**c**) DAWT-Taguchi; (

**d**) DAWT-NSGA-II-A; (

**e**) DAWT-NSGA-II-B; and (

**f**) DAWT-NSGA-II-C.

**Figure 20.**Acoustic surface power distributions: (

**a**) BWT; (

**b**) DAWT-Original; (

**c**) DAWT-Taguchi; (

**d**) DAWT-NSGA-II-A; (

**e**) DAWT-NSGA-II-B; and (

**f**) DAWT-NSGA-II-C.

**Figure 21.**Comparison of power coefficients of the designed DAWTs and typical BWTs [8].

**Figure 22.**Comparison of experimentally measured generated power of designed DAWTs and a BWT, both made by 3D printing.

Diameter | Hub Diameter | Twist Angle | Rotor Solidity | Design Rotational Speed |
---|---|---|---|---|

D = 1.26 m | H = 0.13 m | β = 20° − 3° = 17° | σ = 8.27% | ω = 500 rpm |

Flange Height | Flange Angle | Diffuser Length | Rotor Axial Position |
---|---|---|---|

h/D= 0.05–0.25 | θ= −15°–15° | L_{t}/D = 0.132–0.223 | Five specific positions ^{1} |

^{1}The positions are indicated in Figure 1 and fall inside the range of adopted diffuser lengths.

Flange Height h/D | Flange Angle θ | Diffuser Length L _{t}/D | Rotor Axial Position z/D |
---|---|---|---|

0.05–0.2 [14] | −25°–25° [24] | 0.1–0.4 [14] | 0.04–0.16 [32] |

0–0.05 [19] | −15°–15° [27] | 0.1–0.4 [22] | 0.4–0.8 [34] |

0.025–0.35 [22] | −15°–0° [35] | 0.5–1.25 [30] | |

0.05–0.2 [33] | 0.013–0.556 [32] | ||

0–0.3 [34] | 0.1–0.371 [33] |

Levels | Flange Height h/D | Flange Angle θ | Diffuser Length L _{t}/D | Rotor Axial Position Labeled Number |
---|---|---|---|---|

1 | 0.05 | 15° | 0.132 | 1 |

2 | 0.1 | 10° | 0.152 | 2 |

3 | 0.15 | 0° | 0.183 | 3 |

4 | 0.2 | −10° | 0.211 | 4 |

5 | 0.25 | −15° | 0.223 | 5 |

**Table 5.**Parameter combinations for different diffuser designs according to different optimization algorithms or goals.

Case | Flange Height, h/D | Flange Angle, θ | Diffuser Length, L_{t}/D | Rotor Axial Position |
---|---|---|---|---|

DAWT-Original | 0.05 | 15 | 0.132 | 1 |

DAWT-Taguchi | 0.1 | 15 | 0.152 | 4 |

DAWT-NSGA-II-A | 0.13 | 2.4 | 0.181 | 3.5 |

DAWT-NSGA-II-B | 0.06 | −8 | 0.149 | 4.3 |

DAWT-NSGA-II-C | 0.07 | 13.7 | 0.185 | 1.4 |

**Table 6.**Comparison of values of power and noise predicted by BPNN and CFD for the cases after completion of multi-objective optimization.

Case | Prediction | Power (W) | Noise (dBA) |
---|---|---|---|

DAWT-NSGA-II-A | BPNN | 154.51 | 46.91 |

CFD | 154.44 | 47.2 | |

Error | 0.04% | 0.6% | |

DAWT-NSGA-II-B | BPNN | 151.08 | 44.92 |

CFD | 151.85 | 43.7 | |

Error | 0.5% | 2.7% | |

DAWT-NSGA-II-C | BPNN | 135.41 | 42.39 |

CFD | 134.26 | 38.4 | |

Error | 0.8% | 9.4% |

**Table 7.**Generated power and noise associated with various designs, and their corresponding performance coefficients.

Case | Tip Clearance (mm) | Power (W) | Noise (dBA) | C_{pb} ^{2} | C_{pd} ^{3} | K ^{4} |
---|---|---|---|---|---|---|

BWT | — | 98.12 | 44.24 | — | — | 0.875 |

DAWT-Original | 10 (1.6%) ^{1} | 135.31 | 43.3 | −0.778 | 0.366 | 0.975 |

DAWT-Taguchi | 25.7 (4.1%) | 152.08 | 45.8 | −0.98 | 0.211 | 1.14 |

DAWT-NSGA-II-A | 14.1 (2.2%) | 154.44 | 47.2 | −0.924 | 0.304 | 1.142 |

DAWT-NSGA-II-B | 37.8 (6%) | 151.85 | 43.7 | −0.974 | 0.224 | 1.149 |

DAWT-NSGA-II-C | 21.7 (3.4%) | 134.26 | 38.4 | −0.834 | 0.311 | 1.006 |

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**MDPI and ACS Style**

Hwang, P.-W.; Wu, J.-H.; Chang, Y.-J.
Optimization Based on Computational Fluid Dynamics and Machine Learning for the Performance of Diffuser-Augmented Wind Turbines with Inlet Shrouds. *Sustainability* **2024**, *16*, 3648.
https://doi.org/10.3390/su16093648

**AMA Style**

Hwang P-W, Wu J-H, Chang Y-J.
Optimization Based on Computational Fluid Dynamics and Machine Learning for the Performance of Diffuser-Augmented Wind Turbines with Inlet Shrouds. *Sustainability*. 2024; 16(9):3648.
https://doi.org/10.3390/su16093648

**Chicago/Turabian Style**

Hwang, Po-Wen, Jia-Heng Wu, and Yuan-Jen Chang.
2024. "Optimization Based on Computational Fluid Dynamics and Machine Learning for the Performance of Diffuser-Augmented Wind Turbines with Inlet Shrouds" *Sustainability* 16, no. 9: 3648.
https://doi.org/10.3390/su16093648