Swept Blade Dynamic Investigations for a 100 kW Small Wind Turbine

Abstract

Most small–medium-sized turbine studies have focused on presenting new design methods and corresponding performance improvements rather than detailed dynamic investigations. This paper presents comprehensive dynamic investigations of a straight and a swept-back blade for a 100 $\mathrm{k}$$\mathrm{W}$ turbine by performing modal analysis, dynamic load analysis, and flutter analysis. The considered load cases include steady wind and operational conditions under normal and extreme turbulence. Modal results show that although both blades have similar natural frequencies, their mode shapes are quite different due to the couplings in flapwise-torsion directions introduced by the back-swept geometry. This coupling alters the aeroelastic response of the blade, which results in different loads in the operational conditions. The load analysis results show that the blade damage equivalent fatigue loads for the swept blade are much lower (up to 29% for the flapwise bending moment and 31% for the edgewise bending moment) than the straight blade. For the ultimate loads, blade root edgewise load for the swept blade is almost 50% lower than the straight blade while the flapwise ultimate load is similar for both blades. Moreover, both blades have no aeroelastic instability near the operational conditions, and the flutter limit for the swept-back blade is lower than the straight blade.

1. Introduction

Wind energy is one of the most widely used renewable energy sources. The Cost of Energy (CoE) generated by wind power is dramatically reduced and it is cheaper than the energy generated from existing coal and gas plants [1]. Research projects worldwide continue to improve wind turbine performance, further pushing towards larger and lighter rotors. Multi-disciplinary design optimization approaches are used to design the wind turbine rotor, considering many different aspects, such as aerodynamics, structural dynamics, materials, controller, dynamics, etc. [2,3,4,5,6]. Most recently, the IEA 15MW Offshore Reference Wind Turbine model, using multi-disciplinary design optimization frameworks, such as WISDEM [7] and HawtOpt2 [3], was introduced [8]. The main design objectives for these studies were to increase the rotor size to capture more energy from wind without increasing the applied loads and/or the total weight. Generally, one can conclude that the increased costs of larger contemporary turbines grow slower than the increased revenue they generate.

In addition to multi-megawatt-sized wind turbine systems, small–medium-sized wind turbine systems are receiving attention, i.e., they are being used in urban and rural areas with isolated electric networks [9]. The cumulative installed capacity of small wind turbines was about 1727 $\mathrm{M}$$\mathrm{W}$ until 2018 worldwide [10]. It is worth noting that the rate of growth was more than 50% from 2013 to 2018. The design considerations for small wind turbines can differ somewhat from much larger machines in that they are not necessarily required to run at optimal tip speed ratios until a ‘rated power’. For example, small/medium wind turbines could be designed to produce power over a broad range of operating conditions, regardless of optimum wind conditions [11]. Therefore, small/medium wind turbine should have good start-up performances at low wind speed ranges to increase power generation. There were various studies performed to design wind turbine blades, to generate more power at low wind speed ranges.

Pourrajabian et al. performed design and optimization study for a small wind turbine to improve the power output and reduce the starting time [12]. Suresh and Rajakumar introduced a 2 kW wind turbine blade for low wind speed applications. They tested various airfoils at different Reynolds numbers for low wind speed turbine design. They concluded that SD7080 airfoil was the most suitable airfoil to produce the maximum power in low wind speed application [13]. Tahir et al. also introduced an optimum blade using improved blade element momentum theory including the Viterna–Corrigan stall model with the objective to yield low cut-in speed and high power level [14]. Verelst investigates the performance of a downwind free-yawing stall controlled 140 kW wind turbine. The focus is on the analysis of the free-yawing stability as function of blade coning angle [15]. Sharifi and Nobari announced a 300 kW wind turbine blade with optimized sectional pitch angle to maximize the power output [16]. Sessarego and Wood applied a multi-dimensional design optimization approach in which the optimization can be of any combination of maximum power and minimum noise, starting time, and structure. However, their study contained relatively simple structural design, aerodynamic design and simplified load analysis [17].

In order to improve small/medium wind turbine performances, various advanced active and passive blade control techniques were investigated. For the active control methods, references [18,19] introduced an innovative trailing edge flap for Vestas V27 225 $\mathrm{k}$$\mathrm{W}$ wind turbine and performed full scale field tests. From these studies, it was revealed that an average 14% of the flapwise blade root load reduction and a 20% reduction of the amplitude of the 1P loads were observed; with the developed active trailing edge flap, a clear load reduction could be observed. There were two full-scale experiment studies with plasma actuators. The first one was performed with Dielectric barrier discharge (DBD) actuators located at the leading edge [20]. The production increment (at least three times higher than the actuator power supply) during plasma activity was presented. The second one had actuators located at mid-chord or close to the trailing edge on the suction side, applied to a 20 $\mathrm{k}$$\mathrm{W}$ ( $9.5$ $\mathrm{m}$ in diameter) [21]. A 2 to 6% power gain was observed.

For the passive control methods, Ashwill designed a Sweep Twist Adaptive Rotor (STAR) blade to reduce the applied loads resulting in a larger rotor to produce more power output. A prototype of the $26.2$ $\mathrm{m}$ blade was manufactured and assembled with the Zond 750 Z48 turbine. Three months of field tests were performed in the Tehachapi Mountain area of California in 2008. Moreover, it showed that the STAR blade produced more power and lower loads compared to the Z48 baseline blade ( $23.2$ $\mathrm{m}$ length) [22]. Recently, McWilliam et al. applied a Multi-disciplinary Design Optimization (MDO) method to design a bend-twist coupled blade for a 100 $\mathrm{k}$$\mathrm{W}$ wind turbine where both the material coupled method and geometric coupled method were adapted. It was concluded that both the material and geometric coupled blade performed better than the optimum baseline blade. Moreover, the geometric coupled blade (swept blade) showed better potential to increase Annual Energy Production (AEP) than the material coupled blade [23].

Most of the small/medium-sized turbine design studies so far have focused on presenting new design methods and corresponding performance improvements rather than detailed dynamic investigations for the designed blade. Therefore, in this paper, a comprehensive dynamic investigation with an optimum swept blade design from [23] will be considered. Details about the straight and swept blade models and analysis methods are addressed in Section 2 and Section 3, respectively. In Section 4, numerical results, including comprehensive discussions, are presented for different load cases and stability analyses. The main conclusion is provided in Section 5.

2. Models

Two blade designs for a 100 $\mathrm{k}$$\mathrm{W}$ turbine were studied in this paper. The first blade, called Blade-1, had a straight geometry. The second blade, called Blade-2, had a back-swept design and it had more of a complex three-dimensional shape than Blade-1. Both blade designs were obtained via an optimization process with some constraints; details can be found in [23], where the controller parameters were kept constant during the optimization process. The flexibilities of both blades were very similar; their first flapwise and edgewise natural frequencies were quite close. Although both blade designs had ‘prebend’, only Blade-2 had an in-plane backward sweep, which contributed more flapwise torsion coupling than a straight blade design (Blade-1). The main properties of the turbine model and its detailed features are provided in this section. The turbine models had rigid tower and shaft models whereas their blade models were flexible. A representative shaft-generator inertia (or drive train) was added as well. In this paper, HAWCStab2 [24] was used to calculate the aeroelastic mode shapes [25,26,27] and HAWC2 [28,29,30] was used to calculate the dynamic response of the turbine in the time domain. Both simulation tools were able to capture large blade deflections, which is important to compute the blade and whole turbine response accurately. The tower bottom was clamped and the shaft had a bearing constraint that enabled it to rotate around its own axis. The general dimensions of the turbine structures without the blades are given in

Table 1. General dimensions of the turbine model without the blade information.

Tower height ($\mathrm{m}$)	24
Tower-shaft dist. ($\mathrm{m}$)	1.13
Shaft length ($\mathrm{m}$)	1.67
Hub radius ($\mathrm{m}$)	0.43
Hub height ($\mathrm{m}$)	25.25
Tilt (${}^{\circ }$)	4.0
Shaft-Generator inertia ($\mathrm{k}$$\mathrm{g}$$\mathrm{m}$${}^2$)	$2.04\times {10}^4$
Hub mass ($\mathrm{k}$$\mathrm{g}$)	$8.00\times {10}^2$

.

The general dimensions of both blade designs are given in

Table 2. Blade dimensions and mass properties for both blades.

	Blade-1	Blade-2
Blade length [$\mathrm{m}$]	11.06	10.55
Prebend at tip [$\mathrm{m}$]	0.89	1.00
Sweep at tip [$\mathrm{m}$]	0.0	0.5
Blade mass [$\mathrm{k}$$\mathrm{g}$]	255.9	254.8
CoG from blade root [$\mathrm{m}$]	2.5	2.7

. The blade without the back sweep is named Blade-1 and the blade with the back sweep is labeled Blade-2; these new labels are used hereafter. Blade-2 was $0.5$ $\mathrm{m}$ shorter and had more initial prebend and sweep than Blade-1. Blade-2 prebend reached almost 10% of its length and its backward sweep reached almost 5% of the its length at the tip, whereas Blade-1 only had prebend that reached 8% of its length at the blade tip. The blades had very similar mass and center of gravity Center of Gravity (CoG) positions. Although Blade-1 was longer, its CoG was much closer to its root than Blade-2. This was due to the mass distribution along the span (shown in Figure 4d).

Figure 1 visualizes both blade designs from an upper view with a scale ratio 4 between x- and y-axis. The blades look very similar at root sections but they have different tip parts. Blade-1 goes straight towards, to its tip, whereas Blade-2 goes in the backward direction after $8.2$ $\mathrm{m}$ of the blade span. Blade-1 has a curved length of $11.186$ $\mathrm{m}$, which is 1% more than its tip radius of $11.06$ $\mathrm{m}$. The total curve length of Blade-2 is $10.785$ $\mathrm{m}$, which means it is 2.2% more than its tip radius of $10.55$ $\mathrm{m}$.

Figure 2 shows the airfoil names and their thickness ratios used and the thickness ratio distribution over the normalized span length for both blades. The same airfoils are used in both blade designs but their distributions are different over the span. The airfoil properties, blade aerodynamic layout, structural beam properties, and all load case definitions are available as supplementary files (formatted according to the HAWC2 input file conventions, see supplementary files).

Figure 3 shows the half chord positions in the flapwise and edgewise directions, the aerodynamic twist, and chord lengths along the normalized span for both designs. Half chord flapwise and edgewise locations indicate the distribution of the backward sweep and prebend along the span in the first row of Figure 3. The blades have very similar flapwise half chord positions, aerodynamic twist, and chord lengths along the span. The edgewise half chord positions are quite different due to back sweep and the aerodynamic twist after 80% of the span having different trends and values.

Figure 4 shows flapwise, edgewise, and torsional stiffness and mass per unit length distributions along to the normalized span. The main differences occur at the root and transition regions, especially for edgewise stiffness values. The circular cross section at the root turns into airfoil shapes, gradually, at the transition section. Blade-1 is much stiffer in the edgewise direction at the transition region and its unit mass is also larger than Blade-2 for the same region. On the other hand, Blade-2 seems heavier than Blade-1 between 20% and 40% of the span. Although the blades have similar total mass (see Table 2), Blade-2 Damage Equivalent Load (DEL) is further from the root than Blade-1 because of the difference in mass distribution. Although there are some mismatched regions for edgewise and unit mass plots, the stiffness and mass values generally look similar for both blades.

3. Analysis

The response of turbine models were investigated for steady and turbulent wind cases. Steady state responses, aeroelastic responses, and controller tuning processes were performed by HAWCStab2 [24]. The optimum operational points were determined by evaluating power production, loads, and blade deflections for different operating points. When the operating points were selected, the controller parameters were computed at those points. After a steady state analysis, load analyses for operational conditions under normal and extreme turbulent cases were performed with HAWC2 [28,29,30]. The load cases were selected according to International Electrotechnical Commission (IEC) standard 61400-1 document [31]. Finally, flutter analyses for both blade models were performed, as per a runaway case proposed by [32], to find a critical flutter speed at which an aeroelastic turbine mode becomes negatively damped.

3.1. Steady Wind Analysis

HAWCStab2 uses the Blade Element Momentum (BEM) theory for aerodynamic calculations and co-rotational formulations with Timoshenko beam elements to compute structural deflections [24,25]. First, operational points were computed with the high order aeroelastic model in HAWCStab2 and then controller parameters were computed for the selected operational points by using a reduced order model [26]. The high order aeroelastic model of HAWCStab2 can be represented as

$$\begin{array}{c}\mathbf{M}{\ddot{\mathbf{x}}}_{\mathbf{s}}+(\mathbf{C}+\mathbf{G}+{\mathbf{C}}_{\mathbf{a}}){\dot{\mathbf{x}}}_{\mathbf{s}}+(\mathbf{K}+{\mathbf{K}}_{\mathbf{sf}}+{\mathbf{K}}_{\mathbf{a}}){\mathbf{x}}_{\mathbf{s}}+{\mathbf{A}}_{\mathbf{f}}{\mathbf{x}}_{\mathbf{a}}={\mathbf{f}}_{\mathbf{s}},\end{array}$$

(1a)

$$\begin{array}{c}{\dot{\mathbf{x}}}_{\mathbf{a}}+{\mathbf{A}}_{\mathbf{d}}{\mathbf{x}}_{\mathbf{a}}+{\mathbf{C}}_{\mathbf{sa}}{\dot{\mathbf{x}}}_{\mathbf{s}}+{\mathbf{K}}_{\mathbf{sa}}{\mathbf{x}}_{\mathbf{s}}={\mathbf{f}}_{\mathbf{a}},\end{array}$$

(1b)

where Equation (1a) governs structural response and Equation (1b) governs the aerodynamic states, which includes the unsteady aerodynamics of the blades for stability analysis. Vector ${\mathbf{x}}_{\mathbf{s}}$ includes elastic degrees of freedom variables and vector ${\mathbf{x}}_{\mathbf{a}}$ contains aerodynamic state variables. The matrices $\mathbf{M}$, $\mathbf{K}$, and $\mathbf{C}$ are the mass, stiffness, and structural damping matrix, respectively. The stiffness matrix $\mathbf{K}$ includes both elastic stiffness and centrifugal stiffness effects. The matrix $\mathbf{G}$ includes the gyroscopic force terms and ${\mathbf{C}}_{\mathbf{a}}$ is the aerodynamic damping matrix. The aerodynamic stiffness contributions are split into the geometric stiffness matrix ${\mathbf{K}}_{\mathbf{sf}}$ due to steady aerodynamic forces and the aerodynamic stiffness matrix ${\mathbf{K}}_{\mathbf{a}}$ due to changes in aerodynamic forces when the blade is deflected, such as torsion deflection. The aerodynamic states are coupled with the structural states through the terms ${\mathbf{A}}_{\mathbf{f}}$${\mathbf{x}}_{\mathbf{a}}$. The structural dynamics excites the aerodynamic states through both a velocity, ${\mathbf{C}}_{\mathbf{sa}}{\dot{\mathbf{x}}}_{\mathbf{s}}$, and a translation or rotation, ${\mathbf{K}}_{\mathbf{sa}}{\mathbf{x}}_{\mathbf{s}}$ terms. Matrix ${\mathbf{A}}_{\mathbf{d}}$ [33] describes the lag on the aerodynamic loads for unsteady aerodynamic loads. Structural and aerodynamic force vectors are shown by ${\mathbf{f}}_{\mathbf{s}}$ and ${\mathbf{f}}_{\mathbf{a}}$, respectively. They include pitch actuator loads and changes in the wind speed.

Figure 5 shows the flow chart for the HAWCStab2 steady state computation. Equation (1a) is used with only stiffness terms and force terms to compute internal loads and blade deflections in a steady case. There are two convergence loops in the solution scheme; first, the structural loads are converged by updating the stiffness matrix and external forces in the inner loop. Then the aerodynamic loads are updated in the outer loop of HAWCStab2.

3.2. Controller Tuning

In order to simulate a dynamic analysis, a controller should be tuned. An open source Proportional Integral (PI) controller, called the basic DTU wind energy controller, was used in this study. The details about the controller can be found from [34]. HAWCStab2 can compute PI control parameters for the given operational states. After the optimum operational conditions are computed as explained in Section 3.1, PI control parameters are computed for those states. In this study, three control regions are considered: optimum power region, constant speed variable power region, and constant speed constant power region. The details about each region are illustrated below.

3.2.1. Region-1 (Optimum Power)

This region covers wind speeds from cut-in to slightly lower wind speeds than the rated wind speed, so there is not enough energy in the flow to produce the rated power of the turbine. Turbine operational points are determined to maximize the power production by controlling the generator torque. Here, the term K determines the optimum torque constant for each operational point defined with $\theta$ (pitch) and $\lambda$ (tip speed ratio) for region-1. In other words, K is used to set the generator torque, which results in the highest power production by adjusting the rotational speed at a given operating point. Equation (2) shows how K is calculated.

$$\begin{array}{c}\hfill K={\displaystyle \frac{\eta \rho A{R}^3{C}_P({\theta }_{opt},{\lambda }_{opt})}{2{\lambda }_{opt}^3}},\end{array}$$

(2)

where $\eta$ is the generator efficiency, $\rho$ is the air density, A is the rotor area, R is the rotor radius, and $C_P$ is the power coefficient, which is a function of $\theta$ (pitch) and $\lambda$ (tip speed ratio). So, the optimum torque constant for generator (K) can be found when the optimum lambda and pitch angles are found and they are known from HAWCStab2 steady state simulations.

3.2.2. Region-2 (Constant Speed Variable Power)

The rotational speed is constant but power is variable in this region and it covers wind speeds in the vicinity of the rated wind speed. The pitch angle is set to the optimum angle for maximum power and the generator torque is used to regulate the rotor speed at the rated rotor speed. The controller is tuned by using the below equations in HAWCStab2. The integral gain parameters in Region-2 can be written as,

$$\begin{array}{c}\hfill \underset{inertia}{\underbrace{(I_r+n_g^2{I}_g)}}\ddot{\varphi }+\underset{damping}{\underbrace{{\displaystyle \frac{1}{\eta }}{k}_{Pg}}}\dot{\varphi }+\underset{stiffness}{\underbrace{{\displaystyle \frac{1}{\eta }}{k}_{Ig}}}\varphi =0,\end{array}$$

(3)

where the proportional and integral gains, which are shown as $k_{Pg}$ and $k_{Ig}$, respectively, are part of damping and stiffness terms. The rotor and generator inertia are shown by $I_r$ and $I_g$. The gear box ratio is shown by $n_g$ and $\eta$ is the generator efficiency. The rotor rotation angle, speed and acceleration are shown as $\varphi$, $\dot{\varphi }$, and $\ddot{\varphi }$, respectively. The proportional and integral gain terms can be computed for given natural frequency ($\omega$) and damping ratio ($\zeta$) of the system, as shown in Equation (4). High frequency values result in quick response and high damping means less overshoot while reaching the target rotational speed. The limitations for these numbers arise from the dynamics of the whole system, so that the frequency should be far from the system frequencies to avoid any resonance in the turbine structure.

$$\begin{array}{c}\hfill k_{Ig}={\omega }^2\eta (I_r+n_g^2{I}_g),k_{Pg}=2\zeta \omega \eta (I_r+n_g^2{I}_g)\end{array}$$

(4)

3.2.3. Region-3 (Constant Speed Constant Power)

In this region, the blade pitch is controlled to regulate the rotational speed and reduce the aerodynamic loads. The generator torque $Q_g$ is also applied to regulate the power so that the power and rotational speed are kept constant. The controller gains and second order dynamic equation for pitch angle can be written as,

$$\begin{array}{c}\hfill \underset{inertia}{\underbrace{(I_r+n_g^2{I}_g)}}\ddot{\varphi }+\underset{damping}{\underbrace{\left({\displaystyle \frac{1}{\eta }}{\left.{\displaystyle \frac{\partial Q_g}{\partial \mathsf{\Omega }}}\right|}_{{\mathsf{\Omega }}_o}-{\left.{\displaystyle \frac{\partial Q}{\partial \theta }}\right|}_{{\theta }_o}{k}_P\right)}}\dot{\varphi }-\underset{stiffness}{\underbrace{{\left.{\displaystyle \frac{\partial Q}{\partial \theta }}\right|}_{{\theta }_o}{k}_I}}\varphi =0,\end{array}$$

(5)

where the rotor and generator torque are represented by Q and $Q_g$, respectively. The partial derivative ${\displaystyle \frac{\partial Q_g}{\partial \mathsf{\Omega }}}$ represents the derivative of generator torque with respect to rotor speed $\mathsf{\Omega }$ and ${\displaystyle \frac{\partial Q}{\partial \theta }}$ shows the rotor torque derivative with respect to the pitch angle ($\theta$). The derivatives are taken at the operational rotor speed (${\mathsf{\Omega }}_r$) and pitch angle ${\theta }_o$. The proportional and integral gain terms can be computed by Equation (6)

$$\begin{array}{c}\hfill k_P={\displaystyle \frac{2\zeta \omega (I_r+n_g^2{I}_g)-{\displaystyle \frac{1}{\eta }}{\left.{\displaystyle \frac{\partial Q_g}{\partial \mathsf{\Omega }}}\right|}_{{\mathsf{\Omega }}_o}}{-{\left.{\displaystyle \frac{\partial Q}{\partial \theta }}\right|}_{{\theta }_o}}},k_I={\displaystyle \frac{{\omega }^2(I_r+n_g^2{I}_g)}{-{\left.{\displaystyle \frac{\partial Q}{\partial \theta }}\right|}_{{\theta }_o}}}.\end{array}$$

(6)

Following a constant power strategy, the generator torque is defined as $Q_g={\displaystyle \frac{P_r}{\mathsf{\Omega }}}$ and the term ${\left.{\displaystyle \frac{\partial Q_g}{\partial \mathsf{\Omega }}}\right|}_{{\mathsf{\Omega }}_o}$ becomes $-{\displaystyle \frac{P_r}{{\mathsf{\Omega }}_r^2}}$. The rated power is represented by $P_r$. The aerodynamic gain term represented by ${\left.{\displaystyle \frac{\partial Q}{\partial \theta }}\right|}_0$ needs to be computed to determine the controller parameters and they are computed for each wind speed in HAWCStab2. A curve fit with a second order polynomial is done in HAWCStab2 for simplification of the function. The polynomial fit constants for both blades are given in

Table 3. Second order polynomial fit constants for aerodynamic gains computed in HAWCStab2.

	Const. for ${\mathit{x}}^2$	Const. for ${\mathit{x}}^1$	Const. for ${\mathit{x}}^0$
Blade-1	−0.000910226	−0.14235716	−1.26338789
Blade-2	−0.002363050	−0.11380048	−1.31816826

. The actual aerodynamic gain values and the curve fit results are shown in Figure 6. Both blades have similar aerodynamic gain values. In other words, the pitch angle change of blades results in similar rotor torque variation for both blades.

The selected natural frequency for PI parameters should be far from the natural frequency of the system. Some of the important frequencies come from tower modes. Although there is a rigid tower in the analysis, the terms are selected for a realistic turbine case by setting $\omega =$ $0.1$ Hz and $\zeta =$ 0.7 for controller tuning in HAWCStab2.

Table 4. Controller constants for three regions computed in HAWCStab2.

	Blade-1	Blade-2
K	$1.95747\times {10}^2$	$1.68602\times {10}^2$
$k_{Pg}$	$2.84795\times {10}^4$	$2.87368\times {10}^4$
$k_{Ig}$	$1.27816\times {10}^4$	$1.28970\times {10}^4$
$k_P$	$4.20796\times {10}^{-1}$	$4.06719\times {10}^{-1}$
$k_I$	$1.76573\times {10}^{-1}$	$1.70765\times {10}^{-1}$

shows the computed controller parameters for Blade-1 and Blade-2.

There is no need to respond to every change in wind speed in real life; hence, there is a low pass filter for controller wind speed reading. On the other hand, the back-swept effects occur naturally with blade deflections, so the effects are generally much faster than the controller response. These effects are presented in the response of the whole system, such as blade loads, deflections, and generator torque.

3.3. Load Analysis

The load analyses of the turbines with two different blade models were performed with HAWC2, which is a state-of-the-art aero-servo-elastic wind turbine simulation tool. HAWC2 uses the BEM formulation [30,35,36], including effects of dynamic stall, dynamic inflow, wind shear on induction, tip loss, tower shadow, and large blade deflections for aerodynamic load calculations. A PI controller [34] algorithm, which controls the generator torque and pitch angle, as explained in Section 3.2, was used. The pitch actuators were modeled as a second order dynamical system with an appropriate given frequency and damping. HAWC2 used multibody formulation based on an augmented Floating Reference Frame (FRF) method [37] for structural models. The continuous structures were modeled as bodies and connected by constraints, which allowed motions in certain directions or kept the relative distance and motion the same along the analysis. The bodies consisted of linear classical isotropic or anisotropic Timoshenko beam elements [29] to capture the elastic deflections of the structures. The HAWC2 structural model is able to capture large deflections of a continuous structure [38].

HAWC2 solves the equations of motion of the system with constraints, given in Equation (7) for ith time step, $t_i$. There are N generalized coordinates in vector $u$. Their first and second time derivatives (velocities and accelerations) are given in vectors $\dot{u}$ and $\ddot{u}$. The inertia, damping, and stiffness matrices are shown by $\mathit{M},\mathit{C}\in {\mathbb{R}}^{N\times N},\mathrm{and}\mathit{K}\in {\mathbb{R}}^{N\times N}$. The mass matrix is a function of $u$ in the FRF formulation. Bodies are connected to a point, or to each other, by constraints that are represented by vector $\mathit{g}\in {\mathbb{R}}^{N_c}$, where $N_c$ is the number of constraints in the model. The Jacobian of $\mathit{g}$ shown by ${\mathit{G}}_{\mathit{u}}\in {\mathbb{R}}^{N_c\times N}$ is used in the equation of motion together with Lagrange multipliers given in vector $\mathbf{\Lambda }\in {\mathbb{R}}^{N_c}$. Generalized external forces and quadratic velocities, including Coriolis and gyroscopic forces, are collected in vectors $\mathit{f}$ and ${\mathit{f}}_{\mathit{v}}$. HAWC2 computes $\mathit{u}$, $\dot{\mathit{u}}$, $\ddot{\mathit{u}}$ and $\mathbf{\Lambda }$ at each time step and send $\mathit{u}$, $\dot{\mathit{u}}$, $\ddot{\mathit{u}}$ to the aerodynamic solver to calculate the corresponding aerodynamic loads used in the external force vector, $\mathit{f}$. This update process happens at every iteration at each time step until it converges. The pitch actuators defines the displacement constraint for the blade pitch position and the generator model determines the torque at the end of the shaft

$$\begin{array}{c}\hfill \begin{array}{c}\mathit{M}\left(\mathit{u}\right)\ddot{\mathit{u}}\left(t_i\right)+\mathit{C}\dot{\mathit{u}}\left(t_i\right)+\mathit{K}\mathit{u}\left(t_i\right)+{{\mathit{G}}_{\mathit{u}}}^T(\mathit{u},t_i)\mathbf{\Lambda }\left(t_i\right)=\mathit{f}(\mathit{u},\dot{\mathit{u}},t_i)+{\mathit{f}}_{\mathit{v}}(\mathit{u},\dot{\mathit{u}},t_i),\\ \mathit{g}(\mathit{u},t_i)=0,{\mathit{G}}_{\mathit{u}}(\mathit{u},t_i)=\frac{\partial \mathit{g}(\mathit{u},t_i)}{\partial \mathit{u}\left(t_i\right)}.\end{array}\end{array}$$

(7)

The load cases are selected for normal operating conditions according to IEC standard [39]. Design Load Case (DLC) 1.2 includes normal operating conditions under normal turbulence and it is the most dominant fatigue load case for wind turbines. DLC 1.3 includes extreme turbulence operating conditions and it is used to determine the ultimate loads. These load cases are very similar, except the wind turbulence intensity and each load case includes 216 scenarios in total. The wind speeds vary from 4 $\mathrm{m}$/$\mathrm{s}$ to 26 $\mathrm{m}$/$\mathrm{s}$ with 2 $\mathrm{m}$/$\mathrm{s}$ steps, which gives 12 different wind speeds. For each speed, there are 3 different yaw angles for wind direction; $-10$${}^{\circ }$, 0${}^{\circ }$, and 10${}^{\circ }$, and there are 6 different turbulence seeds for each yaw angle at each wind speed. The simulation time for each load case is set to 700 $\mathrm{s}$ where the first 100 $\mathrm{s}$ is used for the initialization period and the last 600 $\mathrm{s}$ is used for the load and response calculation of the turbine. DLC1.2 covers more than 90% of the turbine fatigue life [38] and gives the annual energy production for the turbine. DLC1.3 gives the ultimate loads for extreme turbulent cases during operation, resulting in the ultimate blade root moments for DTU10MW reference turbine [40,41].

Table 5. General inputs for load cases. There are, in total, 216 simulation cases for each DLC.

	DLC1.2	DLC1.3
Wind speed (wsp)	4–26 $\mathrm{m}$/$\mathrm{s}$ with steps of 2 $\mathrm{m}$/$\mathrm{s}$	4–26 $\mathrm{m}$/$\mathrm{s}$ with steps of 2 $\mathrm{m}$/$\mathrm{s}$
Yaw	−10/0/10 deg	−10/0/10 deg
Turbulence	Normal turbulence with 6 seeds per wsp and yaw	Extreme turbulence with 6 seeds per wsp and yaw
Wind shear	Vertical and exponent of 0.2	Vertical and exponent of 0.2
Gust	None	None
Fault	None	None

shows the general input load cases. Statistical results are computed for each DLC independently. There is no fault in the turbine, such as stuck pitch angle for a blade or grid loss, and no wind gust for the considered DLCs.

4. Results and Discussion

The results are given for the modal analysis of the blades, steady wind analysis results from HAWCStab2, and statistical results from turbulent wind load cases computed in HAWC2. The steady wind analysis is useful to evaluate wind turbine performance, including controller behaviors. The turbulent wind cases represent the response and performance of the turbine under realistic conditions.

4.1. Modal Analysis Results

The design of the blades, in terms of geometry and stiffness/mass distribution, determines the structural response of the blades. Before aeroelastic and load analysis, a structural eigenvalue analysis for both blade designs were performed using HAWCStab2 to see the structural features of the blades (a more detailed description of this method can be found in [24,25,26,27]). The blade mode shapes and natural frequencies were determined and compared. The eigenvalue analysis was conducted for clamped boundary conditions so the blade root had zero translations and rotations. There was no external load on the blade. Damping effects were not included for the eigenvalue analysis. In other words, mode shapes were computed for zero damping values at undeformed blade states.

Table 6. First four natural frequencies of both blade models without damping. The frequencies are computed for the clamped blade root configuration.

	Blade-1	Blade-2
First mode freq. (Hz)	1.95	1.89
Second mode freq. (Hz)	4.49	4.92
Third mode freq. (Hz)	5.88	6.24
Fourth mode freq. (Hz)	11.62	11.25

shows the first four natural frequencies for both blades. The blade natural frequencies are very close to each other, but it does not mean they have similar mode shapes. It is hard to name the mode shapes for complex shape structures, such as blades, due to strong couplings in motions. Therefore, modes are referred by their rank in the natural frequency sorting list, given below.

The flapwise, edgewise and torsion modal amplitudes of the first mode shape are shown in Figure 7 for both designs. The maximum modal amplitude is adjusted to be 1, which is in a flapwise direction in this case. The flapwise and edgewise motions are very similar for both blades. The flapwise motion is the dominant direction for the first mode shape. The torsion plot shows that Blade-2 has very large torsion coupling with its flapwise motion due to its backward sweep. This effect can be very useful to reduce the loads when the blade sees sudden changes in the wind speed. The controller cannot respond to these sudden changes since they are too quick for the controller. Although Blade-1 does not have any back sweep, it has a torsion value less than 1${}^{\circ }$ at the tip, since there are some offsets between the elastic center locations and shear center locations. A sudden increase of wind speed increases the Angle of Attack (AoA) and causes deflections, especially in the flapwise direction. The flapwise–torsion coupling similar to Blade-2 can lead to a decrease AoA as blades deflect more in the flapwise direction. This can be thought of as a passive load control mechanism.

The prebend causes coupling between edgewise and torsion motions which can be seen in the second mode shape of the blades where the edgewise motion is the dominant. Figure 8 shows the flapwise, edgewise and torsion amplitudes for the second mode shape. The edgewise motions of both blades are similar whereas its coupling with flapwise and torsion looks different for the blades. Blade-1 has very small flapwise magnitudes up to 80% of its span. On the other hand, Blade-2 has much larger flapwise motion than Blade-1 starting from 30% of blade span. The coupling between edgewise and torsion is much stronger for Blade-2 than Blade-1 since Blade-2 has larger prebend than Blade-1 (see Table 2).

Figure 9 shows the third mode shape amplitudes. Flapwise motions are dominant for both blades but the amplitudes of the motion over the blade span is different unlike the first two mode shapes shown in Figure 7 and Figure 8. Other important dissimilarities between two blades are edgewise and torsion amplitudes. The Blade-2 edgewise amplitude almost reaches to 1 at the blade tip and its distribution is very similar to the second mode shape edgewise motion shown in Figure 8. So, there is a strong coupling between the second flapwise mode shape motion and the first edgewise mode shape motion. The torsion motion seems to be the result of both flapwise and edgewise motions in the mode shape and it is larger for Blade-2 than Blade-1.

Figure 10 shows the fourth mode shape amplitudes. Both blades have different main direction for the fourth mode shape; Blade-1 dominant direction is in the flapwise rotation which is not shown in mode shape plots. As a result of this fact, the flapwise motion has the largest amplitude among three direction for Blade-1. On the other hand, Blade-2 dominant direction is torsion for this mode shape and its strongly coupled with flapwise and edgewise motions. The torsion value at Blade-2 tip reaches 1 $\mathrm{rad}$ (57.3 deg).

The coupling between edgewise and torsion motion varies as blades deform in the flapwise direction. In other words, the edgewise–torsion coupling is a function of the flapwise deflection. It is particularly important for wind turbine blades since they deflect mainly in a flapwise direction in their operational conditions. The significance of this coupling and its variation for aeroelastic and load analysis of flexible blades is investigated in [25,27]. Figure 11 shows the second mode shape at the 10 $\mathrm{m}$/$\mathrm{s}$ wind speed by including effects coming from aerodynamic forces, gyroscopic forces, and deflections from HAWCStab2. Although flapwise and edgewise motions are similar to Figure 8, the torsion amplitudes are significantly lower compared to the undeflected mode shape in Figure 8. We should note that geometrically nonlinear blade models are critical to capture this variation. HAWC2 and HAWCStab2 are able to capture this effect, which is important for accurate aeroelastic response and load results.

4.2. Steady Wind Results

The operating points at different wind speeds are determined by HAWCStab2 where the entire operating wind speed range (4–25 m/s with 1 $\mathrm{m}$/$\mathrm{s}$ wind step) is considered. Figure 12 shows the power, rotor speed, pitch angle, and thrust force for different wind speeds. The rotor speeds are limited at 70 rpm for both turbine and the controller parameters are determined for the same optimum tip speed ration, which is 9.77. Both blades have similar power and thrust force curves. Since Blade-1 is slightly longer than Blade-2, its rotational speed is a bit lower than Blade-2’s rotational speed at below-rated wind speeds for the same tip speed ratio (9.77). Both turbines reached the rated power at 10 $\mathrm{m}$/$\mathrm{s}$ where the controller was activated. The pitch angles for both cases are similar. However, Blade-2, swept blade, required a bit more pitch, resulting in less thrust after the rated wind speed, 10 $\mathrm{m}$/$\mathrm{s}$.

Figure 13 shows edgewise and flapwise displacements, torsion deflections, and radial (spanwise) displacements at the blade tip according to the wind speed. The edgewise and flapwise plots are not deflections, but displacements, as, in case of a non-zero pitch, the blade tip can have rigid body motions, which cannot be counted as deflection. In other words, for zero pitch values (below 10 $\mathrm{m}$/$\mathrm{s}$ wind speeds), edgewise and flapwise displacements actually equal to deflections, but it is not correct for non-zero pitch values. Blade-2 had larger displacements in all directions than Blade-1 in the plots. The largest displacement occurred in a flapwise direction and it reached more than 10% of the blade length at the rated wind speeds for both blades. The edgewise displacement curves changed their trend after the pitch activation at 10 $\mathrm{m}$/$\mathrm{s}$. The reason for these trend changes is that the pitch axis and elastic center have an off-set at the blade tip, resulting in rigid body motions due to pitch. The torsional deflection of Blade-2 show a correlation with its flapwise deflection due to the swept-back blade geometry creating flap–torsion coupling effects. The radial displacement is a secondary effect of flapwise and edgewise deflections. Since the actual total curve length of the blade does not change with flapwise and edgewise deflections, the radial tip position changes to keep the curve length constant. The tip prebend decreases with flapwise deflections and increases the blade tip radius, so we see positive displacement values in the tip radial displacement plot. Blade-2 radial displacements are larger compared to Blade-1 due to its larger deflections and prebend. The radial displacements are mainly due to flapwise deflections since it is the main deflection direction for the blades. The positive displacement values indicate that the blade’s effective rotor radius is larger than the original effective radius. This difference is up to $0.125$ $\mathrm{m}$ for Blade-2 and $0.08$ $\mathrm{m}$ for Blade-1.

Figure 14 shows edgewise and flapwise unit load distributions over the normalized span at the rated wind speed (10 $\mathrm{m}$/$\mathrm{s}$). Flapwise loads are 10 times larger than edgewise loads and they reach their maximum around 85% of the blade span. Blade-2 has higher unit loads after 60% of the span compared to Blade-1 and both blades produce the same amount of power at 10 $\mathrm{m}$/$\mathrm{s}$ wind speed. Higher steady loads for Blade-2 do not mean it will see higher loads in turbulent wind. Fatigue and ultimate loads come from the load fluctuations around the steady load values due to wind change. More detailed fatigue and ultimate load comparisons are described in the next section.

4.3. Turbulent Wind Results

The statistics from 216 turbulent wind load cases from DLC 1.2 and DLC 1.3 are presented in this section. Figure 15 shows the Damage Equivalent Load (DEL) and maximum and minimum ultimate moment loads in Flapwise (FW) and Edge-Wise (EW) directions over the normalized span. 1 $\mathrm{Hz}$ DEL results are computed from DLC 1.2 and the ultimate loads are determined among all load cases, including DLC 1.2 and DLC 1.3, but they come from DLC 1.3 as expected. The first and second rows in Figure 15 show the 1 $\mathrm{Hz}$ DELs and the ultimate loads for FW and EW bending moments, respectively. As it was explained in [23], both Blade-1 and Blade-2 had been obtained to maximize the AEP compared to the baseline blade with the same design constraints. Although Blade-2 was not optimized based on Blade-1, it is clearly seen that the optimum swept blade, Blade-2, has 29% lower flapwise DEL and 31% lower edgewise DEL at the blade root compared to the optimum straight blade, Blade-1. Moreover, the Blade-2 edgewise ultimate load range is also 2 times lower than Blade-1 at the blade root. On the other hand, the Blade-2 flapwise ultimate load range is slightly larger than Blade-1 at the blade root.

It is well known that the swept blade may produce higher torsional loads compared to the straight blade. Therefore, the Damage Equivalent Load (DEL) and maximum and minimum ultimate torsion moment loads over the normalized span are compared in Figure 16. It is very interesting to see that the swept blade, Blade-2, produces much less torsion DEL compared to Blade-1 (more than 50%) and its ultimate load range at 16% span is 2.5 times lower than Blade-1 in this study. We should note that one reason for this difference might be that Blade-2 is not optimized/redesigned from Blade-1. Both Blade-1 and Blade-2 were optimized from the baseline 10 $\mathrm{m}$ blade [23]. Therefore, both blades show different structural property distributions and different dynamic responses even under the same load conditions.

Figure 17 shows the mean generator torque, shaft axial force, and flapwise and torsion deflections and their standard deviations by error bars for DLC 1.2. The generator torque is an important parameter for power production and control purposes. The axial shaft load is an important load for shaft bearings and it is generally related to the total thrust force. Since the maximum flapwise loads occurred around 85% of the span as shown in Figure 14, a $9.5$ $\mathrm{m}$ blade span that is in the vicinity of 85% of the span for both blades was selected to monitor the torsion and flapwise deflections. Torsion deflection directly contributes to the AoA and leads to changes in the aerodynamic loads. The generator torque and shaft axial force results look similar for both blades. Blade-2 has slightly higher mean shaft force values than Blade-1 for above-rated wind speeds. On the other hand, Blade-2 has larger mean flapwise and torsion deflections than Blade-1. Blade-2 deflections have wider standard deviation values than Blade-1. Since Blade-2 has a $0.11$ $\mathrm{m}$ larger prebend at the blade tip than Blade-1 and the flapwise deflection differences between the two blades are not that big, the tower clearance values are very similar for both blades. Blade-2 deflections lead to lower DELs as shown in Figure 15, and this shows the back-swept design works as intended, which is a mitigation of aerodynamic loads under turbulent wind conditions by flapwise–torsion coupling.

Figure 18 shows similar results to Figure 17, but the results are for DLC 1.3 here. The standard deviations are wider for DLC 1.3 than DLC 1.2. This is an expected result due to higher turbulence intensity for DLC 1.3 than DLC 1.2. The generator torque and shaft axial load results are again very similar for both blades. Although Blade-2 deflects more than Blade-1 in flapwise and torsion directions, its ultimate torsion and edgewise loads are much lower than Blade-1 and have similar flapwise ultimate loads (see Figure 15 and Figure 16). In other words, larger deflections do not always mean higher ultimate loads, especially if they are intended to control loads, such as in Blade-2.

Blade-1 has $0.9$% more AEP than Blade-2. AEP is computed from DLC 1.2 results for a Weibull wind distribution with $8.5$ $\mathrm{m}$/$\mathrm{s}$ mean speed and $k=2$. Figure 19 shows mean power and pitch values with their standard deviations as error bars for DLC 1.2. Results show that the differences of power production mean values are very small, which is consistent with very small differences in AEP. The mean pitch values are consistent with the steady wind results shown in Figure 12. There is no clear difference between Blade-1 and Blade-2 standard deviations in power and pitch. This is particularly important since these results show that Blade-2 pitch actuators are not more active than Blade-1 pitch actuators. This also shows that passive load mechanism due to flapwise–torsion coupling works in different time scale than the controller and does not lead aggressive controller signals.

Finally, instability analyses by over-speed load cases are performed for each blade design. The instability analysis model includes all turbines without a generator and controller. The pitch angle is fixed at zero degrees (0${}^{\circ }$). Since there is no generator torque, the turbine is free to rotate and accelerate with increasing wind speed. A constant 5 $\mathrm{m}$/$\mathrm{s}$ wind speed is defined for the first 100 $\mathrm{s}$ then it increases at a rate of $0.0136$ $\mathrm{m}$/$\mathrm{s}$ every second. The wind speed changes are very slow since the aim is to obtain a quasi-steady analysis at different wind speeds. Figure 20 shows flapwise and torsion moments at the blade root with respect to rotor rotation speed. At some point, a strong vibration starts on both blades. These points are marked as the instability rotation speeds and they are identified at 133 rpm for Blade-2 and 144 rpm for Blade-1. These rotational speeds correspond to 153 $\mathrm{m}$/$\mathrm{s}$ wind speed at Blade-2 tip and 173 $\mathrm{m}$/$\mathrm{s}$ at Blade-1 tip. Both designs have 70 rpm at the rated wind speed; hence, the instability speeds are too far from their design rotational speeds. The backward sweep effect is similar to the bend-twist coupling effect introduced by material; hence, the results are consistent with [27] where flap-torsion feather coupling results in lower damping values. Back-swept aircraft wings have lower flutter speeds than straight aircraft wings [42], which is also consistent with the results shown in Figure 20.

5. Conclusions

Comprehensive dynamic investigations of two blade models for a 100 $\mathrm{k}$$\mathrm{W}$ turbine are performed in this study. Both blades are optimized for the turbine by using the optimization algorithm. One of them has a passive control mechanism introduced by the back-swept tip section (Blade-2). The back-swept geometry causes a coupling between flapwise deflections and torsion deflections so that the flapwise deflection reduces the angle of the attack. The investigation shows that the passive load control mechanism can lead to significantly lower blade damage equivalent loads although the blade deflections are larger than the straight blade design (Blade-1). The selected load cases include steady wind speeds and operational conditions under normal and extreme turbulence. HAWCStab2 is used to compute operational states for steady wind speeds and computing the controller parameters whereas the turbulent wind case results are obtained from HAWC2.

The structural dynamic analysis results show that although both blades have similar natural frequencies, their mode shapes are quite different, especially in torsion direction. The steady wind performance and aerodynamic gains of the blades are also very similar. The differences occur in deflections especially in torsion and flapwise directions. Turbulent wind cases show the effect of flap-torsion coupling (passive load control mechanism) in load results. Damage equivalent loads are much lower with the passive load control mechanism (Blade-2) than the straight blade design (Blade-1). Blade-2 has a 29% lower flapwise moment DEL at the blade root than Blade-1. Both blades have similar performances so that the Blade-2 annual energy production is just $0.9$% lower than Blade-1. Hence, the results show that the loads can be reduced by passive load mechanisms (aeroelastic tailoring) without compromising performance or having a more aggressive controller.

Aeroelastic instability analyses show that blades show some instabilities including edgewise, flapwise, and torsion motions, but they do not show a classic flutter. The back-swept design (Blade-2) leads to a lower stability limit than Blade-1; however, the instability speeds for both blades are much larger than their design rotational speeds.

Future studies can investigate other load case scenarios; the flow nonlinearities by high fidelity flow solvers and similar effects for different turbine sizes and applications. Similar aeroelastic effects are expected to be observed with large blades, but this should also be proven by an comprehensive study. Since floating turbines have different instability problems than onshore turbines, such as pitch instability at above rated wind speeds, the effect of back-swept on floating wind turbine stability should also be investigated.