# Multi-Objective Optimization of a Small Horizontal-Axis Wind Turbine Blade for Generating the Maximum Startup Torque at Low Wind Speeds

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## Abstract

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## 1. Introduction

_{p}) [13]. For this purpose, the analytical correlations presented in the literature [14] or optimization algorithms can be employed. In this regard, in recent years, the use of nature-inspired metaheuristic approaches to optimize wind turbine blades has substantially grown. As the name suggests, these algorithms are inspired by the behavior of animals or other natural processes. Some examples of popular nature-inspired metaheuristic algorithms are Cuckoo Search [15], Jumping Frogs Optimization [16], Bat Algorithm [17], Genetic Algorithm (GA) [18], Ant Colony Optimization [19], and Particle Swarm Optimization [20]. Soni et al. [21] compared the performance of twelve types of nature-inspired algorithms from the perspectives of accuracy, speed, convergence, and efficiency. Their results showed the GA provides rapid speed and also a great accuracy. This point has caused the widespread use of this algorithm for optimizing wind turbine blades [22,23].

_{p}and startup time (T

_{s}) has been carried out by Wood [10] and Pourrajabian et al. [22]. The results have shown that increasing the θ and c in the root region can reduce the T

_{s}of the turbine. In another study, Pourrajabian et al. [24] suggested the use of hollow blades to reduce the blade inertia (J) for reducing the T

_{s}of SWTs. Akbari et al. [25] investigated the effect of ten different airfoils on the performance of an SWT in terms of T

_{s}and C

_{p}. The results showed that SG6043 and BW-3 airfoils have the best performance in windy regions and regions with low wind speeds, respectively. The optimization of the control systems [26], minimizing mass [27], cost [23], and aerodynamic noise [28] are some of the other design and optimization objectives of wind turbine blades that have been examined by researchers.

_{p}of linear and nonlinear models is 0.426 and 0.446 respectively. The blade with a linear distribution started to rotate at a wind speed of 5 m/s, while this value for the blade with a nonlinear distribution was found to be 6 m/s. Rector et al. [30] investigated the effect of solidity, the number of blades (N = 3, 6), and pitch angle on the performance of a SHAWT. The results showed that by increasing the solidity and the number of blades, the turbine performance improves at low wind speeds, but the rotor with three blades performs better at high wind speeds. The results of Eltayesh et al. [5] revealed that raising the number of blades reduces the maximum C

_{p}and blade tip speed ratio, but it raises the startup torque and reduces the required wind speed for the start of the rotation. Similar results were observed in the research conducted by Wang and Chen [31]. In the recent study by Bourhis et al. [6], it was highlighted that with a fixed number of blades (N = 8), the increase in solidity due to the increase in the c leads to a better startup torque. They showed that by raising the solidity from 0.5 to 1.25, the required wind speed for the start of rotation is decreased from 5.8 m/s to 3.8 m/s. In their study, the effect of θ on the startup torque was not studied, while according to the work of Astolfi et al. [32], the θ is one of the most effective parameters in the startup process of wind turbines.

## 2. The Base Wind Turbine

## 3. Numerical Methodology

#### 3.1. Calculating Design Goals

_{p}as well as the startup torque. This method was proposed by Glauert [35] in 1935 [14,36]. In the BEM, the blade is separated into various individual elements, which have a particular θ and c values. Using conservation laws, including the continuity correlation and the momentum correlations for these elements, the amount of local forces is computed for the entire blade, and by summing the loadings along the blade, the total torque (Q) is obtained which is employed to calculate the C

_{p}as follows:

_{s}) can be calculated as follows [24]:

_{s}on a stationary blade can be calculated.

#### 3.2. Multi-Objective Optimization

_{p}and Q

_{s}, the following equation was considered as the objective function:

_{p}and the maximum Q

_{s}of the blades, respectively.

#### 3.3. Configuring the Input Variables

_{p}at a rated wind speed of 10 m/s. According to the experimental test results of this turbine, which are mentioned in the results section (validation of the BEM code), this value is equal to 10.16. The wind speed for the startup process was set at 5 m/s. It is noteworthy that although changing the startup wind speed (${U}_{s}$) can completely alter the Q

_{s}, it does not affect the geometric shape of the optimal blades because Equation (7) is independent of the Reynolds number. The considered range for the design variables based on the fabrication restrictions is listed in Table 1.

## 4. Results and Discussion

#### 4.1. Validity of the BEM Technique

_{d}and C

_{l}) coefficients of the NACA4412 profile, at different α and various Re numbers, are shown in Figure 5. These coefficients are tabulated and considering the Re and α of each element, the related lift, and drag values are calculated and called in the BEM and also optimization stages.

#### 4.2. Validity of the Optimization Technique

_{p}.

_{l}and α should be determined at the maximum L/D ratio. For the NACA4412 airfoil, at $\mathrm{Re}=3\times {10}^{5}$, these values are 6° and 0.9, respectively [10]. By placing n = 1 in the objective function (Equation (8)), the distributions of $\theta $ and $c$ which are obtained from the optimization algorithm are compared with those from the ideal correlations (Equations (9) and (10)) in Figure 7.

#### 4.3. Performing the Optimization

_{p}, the blades with n = 1 and n = 0.95 are recommended, while the blade with n = 0.75 is the best choice for generating a high Q

_{s}for areas with low wind speeds. Figure 9 shows the distribution of $\theta $ and $c$ for the optimal blades.

_{s}, is in direct relationship with c. Thus, the optimization algorithm tries to increase Q

_{s}by raising c, of course in the design limit that was specified in Table 1. But the remarkable point regarding Figure 9 is that the shape of the blades is considerably similar at r/R ≥ 0.52, the reason for which is explained in detail in Section 4.4.

_{s}to work at low wind speeds, the blade with n = 0.75 is selected as the optimal blade for further analysis. It should be noted that the blade with n = 0.7 was also considered in the objective function, but the optimization algorithm could not produce a smooth distribution for θ and c in the outer part of the blade. Because by reducing n in Equation (8), the optimization algorithm mostly focuses on the evolution of design variables on the inner part of the blade to raise the Q

_{s}, which results in discrepancies in the outer part of the blade and a sharp reduction in the C

_{p}. Thereby, it was ensured that the blade with n = 0.75 is the most suitable blade based on the design objectives.

_{p}and maximizing the Q

_{s}) and the optimal blade designed for a 0.75 kW turbine by Pourrajbian et al. [24] (with the goals of maximizing the C

_{p}and minimizing T

_{s}).

_{s}and requires an increase in θ and c values, and the second half is responsible for generating the output power; for that part, the θ and c profiles can be obtained from ideal correlations.

#### 4.4. The Power Generation Analysis

_{p}, as well as the aerodynamic torque generated along the blades, is examined. Figure 11 shows the geometry of the base and designed blades.

_{p}values obtained by the BEM and CFD methods for the base/designed blades. This table also shows the experimental C

_{p}of the base turbine. Both methods predict a slight reduction in the C

_{p}of the designed blade compared with the base blade, but this reduction is not noticeable, especially in the CFD method. The BEM technique, which is the basis of the present study calculations, predicts only a 1.5% reduction.

#### 4.5. The Startup Behavior Analysis

_{s}value of the base/designed blades is shown in Figure 19. The results show a 140% increase in the value of this parameter for the designed blade compared with the base blade. For further examination, the distribution of Q

_{s}generated by each element is shown in Figure 20. By observing this figure, it is clear that most of the Q

_{s}is generated in the inner half of the blade ($r/R<0.52$).

_{s}in the tip elements of the blades, which is due to the negative θ in this region. Another parameter that has a cardinal role in the operation of SWTs at low wind speeds is T

_{s}[24]. Indeed, generating a high Q

_{s}becomes sensible as a practical improvement when the turbine has a low T

_{s}so that power generation is not delayed. Raising the c in the inner part of the designed blade increases its moment of inertia, which can affect the rotational acceleration and hence the T

_{s}of the turbine. Thereby, it is necessary to perform a T

_{s}analysis. In this regard, the changes in $\lambda $ over time during the startup process can be considered as follows [24]:

_{s}is calculated [10]. In this study, the Adams–Moulton method was employed to calculate this equation, and the trapezoidal integration technique was used to calculate the blade inertia (Equation (12)). By considering ${\rho}_{b}=550\mathrm{kg}/{\mathrm{m}}^{3}$ and also ${Q}_{r}=0.5\mathrm{Nm}$ for the turbine of the present study [10,33], the T

_{s}of the base/designed blades was investigated. The results are summarized in Table 4 for various wind speeds. In this table, the J values of the blades are also given.

_{s}) of 4 and 5 m/s. On the other hand, the designed blade, despite having a higher J than the base blade, starts to rotate at U

_{s}= 4 m/s and can be used to generate power. Figure 21 shows the changes of $\lambda $ during the startup process and Figure 22 shows the changes of Q

_{s}up to $\lambda =2$ for the base and designed blades.

_{s}slightly diminishes and then begins to rise. It should be noted that although the Q

_{s}of the designed blade at U

_{s}= 4 m/s is greater than the Q

_{s}of the base blade at U

_{s}= 6 m/s (see Figure 22), it has a higher T

_{s}than the base blade. This is due to the lower J value of the base blade.

## 5. Conclusions

_{p}) but also for the maximum startup torque (Q

_{s}) because these turbines usually lack a pitch control mechanism to adjust the optimal angle of attack. For this purpose, a genetic multi-objective optimization algorithm was employed. The twist angle (θ) and chord length (c) were considered as the design variables and the BEM method was utilized to calculate the design objectives. Validation studies were carried out for both the BEM technique and also the optimization algorithm. A weighting factor was used to make a compromise between the C

_{p}and the Q

_{s}, and different blades were designed by changing it. The outline of the research can be highlighted as follows:

- By increasing the c and θ values in r/R < 0.52 and also following the ideal c and θ profiles in r/R ≥ 0.52, with only a 1.5% reduction in the C
_{p}, the Q_{s}of the designed blade was augmented by 140% compared with the base blade. This increase in the Q_{s}reduced the startup wind speed from 6 m/s in the base blade to 4 m/s in the designed blade. This makes the turbine appropriate to use in more regions; - In blade design, to obtain the maximum C
_{p}, the genetic algorithm and the Schmitz equations reached a similar c and θ profiles, which indicates the high accuracy and capability of the genetic algorithm in optimizing wind turbine blades; - The blade geometry designed with ideal equations produces the lowest Q
_{s}, so these types of blades are not suitable for application in regions with low wind speed; - The aerodynamic torque for power generation increases along the length of the blade from the hub to r/R = 0.89, and a good agreement can be seen between the CFD and BEM results up to this point, but after that, the aerodynamic torque decreases due to the formation of tip vortices. In this regard, the CFD method predicts a greater drop than Prandtl’s tip correction factor;
- Considering the Q
_{s}in the objective function is very useful because, firstly, it leads to c and θ distributions without discrepancies, and, secondly, it results in a significant increase in the Q_{s}against a slight decrease in the output power. This compromise is very attractive for low-wind-speed areas; - When the blade is stationary, the tip elements of the blade not only do not contribute to the Q
_{s}but also can produce a negative torque which is due to the negative θ in this region.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

A | The surface area of the airfoil (m^{2}) |

C_{l} | Lift coefficient |

C_{p} | Power coefficient |

c | Blade chord (m) |

F | Prandtl tip loss factor |

f | Term in Prandtl tip loss factor |

J | Rotational inertia (kg·m^{2}) |

N | Number of blades |

n | Weighting factor |

Q | Torque (kg·m^{2}·s^{−2}) |

Q_{r} | Resistive torque (kg·m^{2}·s^{−2}) |

R | Blade tip radius (m) |

R_{e} | Reynolds number |

r | Radial coordinate along blade (m) |

r_{h} | Hub radius (m) |

S | The swept area of blades (m^{2}) |

T_{s} | Startup time (s) |

t | Time (s) |

U | Wind velocity (m·s^{−1}) |

y^{+} | Non-dimensional distance |

Greek Symbols | |

α | Angle of attack |

θ | Blade twist angle |

λ | Tip speed ratio |

λ_{r} | Local tip speed ratio |

ρ | Density (kg·m^{−3}) |

ϕ | Blade inflow angle |

ω | Angular velocity (s^{−1}) |

Subscripts | |

b | Blade |

s | Startup |

Abbreviations | |

BEM | Blade element momentum |

CFD | Computational fluid dynamic |

HAWT | Horizontal-axis wind turbine |

NACA | U.S. National Advisory Committee on Aeronautics |

SST | Shear stress transport |

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**Figure 2.**The geometry of the base turbine [33].

**Figure 5.**The variations in the drag and lift coefficients of the NACA4412 profile [10].

**Figure 6.**The changes of ${C}_{P}$ at various $\lambda $ for the current research and the study of Anderson et al. [33].

**Figure 7.**The distributions of $\theta $ (

**top**) and $c$ (

**bottom**) obtained from the optimization algorithm and ideal correlations.

**Figure 10.**The θ and c profiles of the optimal blade in the research of Pourrajabian et al. [24] and the optimal blade of the present study.

**Table 1.**The considered range for the design variables [24].

Variable | Lower Limit | Upper Limit |
---|---|---|

$\theta $ (°) | −5 | 25 |

c/R | 0.01 | 0.2 |

**Table 2.**The input values in the GA [22].

Variable | Value |
---|---|

Population | 3000 |

Number of generations | 500 |

Selection rate | 0.1 |

Mutation rate | 0–0.1 |

Method | ${\mathit{C}}_{\mathit{p}}$ | |
---|---|---|

Base Blade | Designed Blade | |

Anderson et al. [33] | 0.454 | - |

BEM | 0.460 | 0.453 |

CFD | 0.453 | 0.451 |

Case | J (kg·m^{2}) | T_{s} (s) | ||
---|---|---|---|---|

U_{s} = 4 m/s | U_{s} = 5 m/s | U_{s} = 6 m/s | ||

Base blade | 0.798 | - | - | 10.63 |

Designed blade | 1.691 | 12.59 | 6.69 | 4.71 |

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**MDPI and ACS Style**

Akbari, V.; Naghashzadegan, M.; Kouhikamali, R.; Afsharpanah, F.; Yaïci, W.
Multi-Objective Optimization of a Small Horizontal-Axis Wind Turbine Blade for Generating the Maximum Startup Torque at Low Wind Speeds. *Machines* **2022**, *10*, 785.
https://doi.org/10.3390/machines10090785

**AMA Style**

Akbari V, Naghashzadegan M, Kouhikamali R, Afsharpanah F, Yaïci W.
Multi-Objective Optimization of a Small Horizontal-Axis Wind Turbine Blade for Generating the Maximum Startup Torque at Low Wind Speeds. *Machines*. 2022; 10(9):785.
https://doi.org/10.3390/machines10090785

**Chicago/Turabian Style**

Akbari, Vahid, Mohammad Naghashzadegan, Ramin Kouhikamali, Farhad Afsharpanah, and Wahiba Yaïci.
2022. "Multi-Objective Optimization of a Small Horizontal-Axis Wind Turbine Blade for Generating the Maximum Startup Torque at Low Wind Speeds" *Machines* 10, no. 9: 785.
https://doi.org/10.3390/machines10090785