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Abstract

Wind energy is known as one of the most efficient clean renewable energy sources and has attracted extensive research interests in both academic and industry fields. In this study, the effects of turbulent wind and voltage disturbance on a wind turbine drivetrain are analyzed, and a wind turbine drivetrain dynamic model combined with the electric model of a doubly fed induction generator is established. The proposed model is able to account for the dynamic interaction between turbulent wind, voltage disturbance, and mechanical system. Also, the effects of time-varying meshing stiffness, transmission error, and bearing stiffness are included in the mechanical part of the coupled dynamic model. From the resultant model, system modes are computed. In addition, by considering the actual control strategies in the simulation process, the effects of turbulent wind and voltage disturbance on the geared rotor system are analyzed. The computational results show that the turbulent wind and voltage disturbance can cause adverse effects on the wind turbine drivetrain, especially the gearbox. A series of parametric studies are also performed to understand the influences of generator and gearbox parameters on the drivetrain system dynamics. Finally, the appropriate generator parameters having a positive effect on the gearbox in alleviating the extreme loads and the modeling approach for investigating the transient performance of generator are discussed.

Keywords

Wind turbine gearbox dynamics doubly fed induction generator

Introduction

Today, with the rapid development of wind power generation technology, in order to guarantee a reliable and robust wind turbine generation system design, research on the interaction of different parts of a wind turbine in the presence of turbulent wind and voltage disturbance is warranted. In a wind turbine system, mechanical transmission is normally used to connect the electrical machine and the wind load in a low-speed, high-torque application. Hence, the transmission is the most important component that can directly affect the safety and stability of the wind turbine. This geared rotor system is usually working under the coupled excitations of wind and electric loads as well as the internal excitation due to the gear transmission error and shaft dynamics. Therefore, to ensure the reliability of the wind turbine design, a dynamic model of the wind turbine drivetrain is proposed.

In the past few years, the dynamic problems of the wind turbine drivetrain have been studied extensively. For example, by building the planetary gear train models and considering the internal excitation caused by time-varying meshing stiffness, Kahraman^1,2 and Parker et al.³ analyzed the vibratory responses of the planetary gear train. In 2007, a research group from National Renewable Energy Laboratory used different methods to explore approaches to improve the reliability of a wind turbine gearbox.⁴ Helsen et al.^5,6 conducted numerical simulations of the dynamic responses of a wind turbine gearbox by employing different modeling methods, and the effects of the different supporting types on the gearbox were studied. Todorov et al.⁷ acquired the natural frequencies, mode shapes, and amplitude–frequency characteristics of a wind turbine drivetrain by building a torsional model that consists of 10 rigid bodies and 8 degrees of freedom (DOFs). Li et al.⁸ analyzed the transient stability of a grid-connected wind turbine with induction generator by building a detailed squirrel cage induction generator (SCIG) model and considering flexible blades. Qin et al.^9,10 applied mechanical system dynamics methods to investigate vibrations and load situations of the gears and bearings in the gearbox and made a great contribution to the optimization problems about gear transmission systems.

The above studies provided valuable guidance for the optimum design of wind turbine drivetrain systems and the assessment of their reliability. However, a coupled model that considers the effects of the dynamic interactions between electrical and mechanical components and wind load is rarely mentioned in the previous publications. In this work, a dynamic model that combines all electrical, mechanical, and aerodynamic effects of a 1-MW wind turbine drivetrain is developed. Applying this model, first, the modes of the drivetrain are calculated, and second, the interaction between turbulent wind, voltage disturbance, and drivetrain is analyzed. In addition, the effect of variation in generator parameters on the gearbox and the effect of different degrees of freedom of the gearbox on the transient performance of the generator are investigated.

Mathematic modeling

The 1-MW wind turbine drivetrain mainly consists of a wind rotor, a low shaft, a gearbox, a high shaft, and a generator system as shown in Figure 1. The generator stator is directly connected to the power grid, and the generator rotor excitation current is controlled by two bidirectional pulse-width modulation (PWM) converters. The electrical power can be transmitted into the power grid by the stator and the rotor.

Figure 1.

Brief model of a grid-connected wind turbine with DFIG.

Wind rotor model

The wind rotor model is formulated in this section. The aerodynamic power $P_{v}$ , which is captured by the wind rotor, is expressed by¹¹

P_{v} = \frac{1}{2} ρ π R_{w}^{2} v^{3}

(1)

where $ρ$ is the air density, $R_{w}$ is the blade radius, and v is the wind speed.

The tip speed ratio $λ$ is defined as

λ = \frac{ω_{w} R_{w}}{v}

(2)

where $ω_{w}$ is the angular velocity of wind rotor.

The power coefficient $C_{p}$ is the function of the blade pitch angle $β$ and the tip speed ratio $λ$ as shown in Figure 2.

Figure 2.

$C_{p} (λ, β)$ curve.

The output power $P_{o}$ of the wind rotor is

P_{o} = C_{p} (λ, β) P_{v} = \frac{1}{2} ρ π R_{w}^{2} v^{3} C_{p} (λ, β)

(3)

Furthermore, the aerodynamic torque $T_{wtr}$ can be defined as

T_{wtr} = \frac{1}{2} ρ π R_{w}^{3} v^{2} C_{q} (λ, β)

(4)

where $C_{q} (λ, β) = C_{p} (λ, β) / λ$ .

Mechanical model

The wind turbine mechanical model consists of a wind rotor, a drivetrain, and a generator rotor. The drivetrain has a gearbox with three stages, which includes a low-speed planetary gear stage (three identical planets with spur teeth, sun and fixed ring wheel) and two high-speed parallel gear stages (spur gear pairs). Applying lumped parameter modeling, the dynamic model can be realized as shown in Figure 3. The model consists of blades, a hub, a planet carrier, gears, a generator rotor, and elastic shafts. It considers the effects of the bending flexibility of the blades, meshing stiffness, meshing error, supporting stiffness, and shaft torsional stiffness. The blade break point where the spring is assumed to be placed is the end of blade. All the rotational coordinates associated with the drivetrain components are included, but only the radial translations of planet carrier and gears are considered. Thus, the model has 12 rigid bodies and 30 DOFs.

Figure 3.

Dynamic model of transmission system with 30 degrees of freedom.

As shown in Figure 3, $T_{wtr}$ and $T_{em}$ represent the aerodynamic torque and electromagnetic torque, respectively; $J_{a} (a = b, h, c, s, p_{i}, 1, 2, 3, 4, g, i = 1, 2, 3)$ represent the inertias of blades, the hub, the planet carrier, gears, and the generator rotor; $m_{b} (b = c, s, p_{i}, 1, 2, 3, 4, i = 1, 2, 3)$ represent the masses of the planet carrier and gears; $K_{bh}$ and $C_{bh}$ represent the bending stiffness and damping of blades; and $K_{m}$ and $C_{m}$ represent torsional stiffness and damping of the low-speed shaft between the hub and the planet carrier. Furthermore, $K_{1 s}$ and $C_{1 s}$ represent the torsional stiffness and damping of the intermediate shaft I between the sun gear and the planet carrier; $K_{23}$ and $C_{23}$ represent the torsional stiffness and damping of the intermediate shaft II between gear 2 and gear 3; $K_{c}$ and $C_{c}$ represent torsional stiffness and damping of the high-speed shaft between gear 4 and the generator rotor; $K_{uv} (t)$ and $C_{uv} (t) [(u, v) = (r, p_{i}; p_{i}, s; 1, 2; 3, 4)]$ represent the meshing stiffness and damping of the gear pairs; and $e_{uv} (t)$ represents the static transmission error of the gear pairs.

The rotational directions of the components are shown in Figure 3. The planet gears adopt the $o η ζ$ moving coordinate system, and the rest of components adopt the oyz fixed coordinate system. Let the compressive deflection between interacting elastic components be $Δ$ , the meshing deflection of the gear pairs $p_{i} - s$ , $r - p_{i}$ , $g_{1} - g_{2}$ , and $g_{3} - g_{4}$ , expressed as $Δ_{p_{i} s}$ , $Δ_{r p_{i}}$ , $Δ_{12}$ , and $Δ_{34}$ , respectively, can be written as

{\begin{array}{l} \begin{matrix} Δ_{p_{i} s} = r_{b p} θ_{p_{i}} - r_{b s} θ_{s} + r_{b c} θ_{c} \cos a_{p s} + η_{p_{i}} \cos a_{p s} - ζ_{p_{i}} \sin a_{p s} - Y_{s} \cos (a_{p s} + ψ_{i}) \\ + Z_{s} \sin (a_{p s} + ψ_{i}) + Y_{c} \cos (a_{p s} + ψ_{i}) - Z_{c} \sin (a_{p s} + ψ_{i}) - e_{p_{i} s} \end{matrix} \\ \begin{matrix} Δ_{r p_{i}} = - r_{b p} θ_{p_{i}} + r_{b c} θ_{c} \cos a_{r p} + η_{p_{i}} \cos a_{r p} - ζ_{p_{i}} \sin a_{r p} + Y_{c} \cos (ψ_{i} - a_{r p}) \\ - Z_{c} \sin (ψ_{i} - a_{r p}) - e_{r p_{i}} \end{matrix} \\ Δ_{12} = r_{b 1} θ_{1} - r_{b 2} θ_{2} + Y_{1} \sin a_{12} - Z_{1} \cos a_{12} - Y_{2} \sin a_{12} + Z_{2} \cos a_{12} - e_{12} \\ Δ_{34} = r_{b 3} θ_{3} - r_{b 4} θ_{4} + Y_{3} \sin a_{34} + Z_{3} \cos a_{34} - Y_{4} \sin a_{34} - Z_{4} \cos a_{34} - e_{34} \end{array}

(5)

where $ψ_{i} = 2 π (i - 1) / 3$ is the circumferential angle of the ith planet gear; $a_{ps}$ , $a_{rp}$ , $a_{12}$ , and $a_{34}$ are meshing angles of the gear pairs; $r_{bj} (j = c, p_{i}, s, 1, 2, 3, 4)$ are the base radius; $θ_{a}$ is the angular displacement of the drivetrain components; $Y_{n}$ , $Z_{n} (n = c, s, 1, 2, 3, 4)$ , and $η_{p_{i}}$ , $ζ_{p_{i}}$ are radial translations of the planet carrier and the gears.

The dynamic meshing forces $F_{kuv} (t)$ and dynamic meshing damping force $F_{cuv} (t)$ are expressed by¹²

{\begin{matrix} F_{kuv} (t) = K_{uv} (t) x_{uv} \\ F_{cuv} (t) = C_{uv} (t) {x'}_{uv} \end{matrix}

(6)

In this formulation, the supporting stiffness and damping for the planet carrier and every gear are also considered. As shown in Figure 4, $K_{ny}$ , $K_{p η}$ , $K_{nz}$ , and $K_{p ζ}$ are the supporting stiffness for the planet carrier and the gears, respectively. Also, $C_{ny}$ , $C_{p η}$ , $C_{nz}$ , and $C_{p ζ}$ are the damping parameters for the planet carrier and gears, respectively.

Figure 4.

Model of bearing supporting stiffness and damping.

It is assumed that the ring is fixed. According to Newton’s second law of motion, the wind turbine dynamic system equations are described as:

1. Wind rotor and low-speed shaft equations

\begin{matrix} J_{b} {\overset{\cdot\cdot}{θ}}_{b} + C_{bh} ({\overset{\cdot}{θ}}_{b} - {\overset{\cdot}{θ}}_{h}) + K_{bh} (θ_{b} - θ_{h}) = T_{wtr} \\ J_{h} {\overset{\cdot\cdot}{θ}}_{h} + C_{m} ({\overset{\cdot}{θ}}_{h} - {\overset{\cdot}{θ}}_{c}) + K_{m} (θ_{h} - θ_{c}) + C_{bh} ({\overset{\cdot}{θ}}_{h} - {\overset{\cdot}{θ}}_{b}) \\ + K_{bh} (θ_{h} - θ_{b}) = 0 \end{matrix}

(7)

2. Gearbox and high-speed shaft equations

\begin{array}{l} J_{c} {\overset{\cdot\cdot}{θ}}_{c} + C_{m} ({\dot{θ}}_{c} - {\dot{θ}}_{h}) + K_{m} (θ_{c} - θ_{h}) = - \sum_{i = 1}^{3} (F_{k p_{i} s} + F_{c p_{i} s}) r_{b c} - \sum_{i = 1}^{3} (F_{k r p_{i}} + F_{c r p_{i}}) r_{b c} \\ m_{c} {\overset{\cdot\cdot}{Y}}_{c} + C_{c y} {\dot{Y}}_{c} + K_{c y} Y_{c} = - \sum_{i = 1}^{3} (F_{k p_{i} s} + F_{c p_{i} s}) \cos (α_{p s} + ψ_{i}) - \sum_{i = 1}^{3} (F_{k r p_{i}} + F_{c r p_{i}}) \cos (ψ_{i} - α_{r p}) \\ m_{c} {\overset{\cdot\cdot}{Z}}_{c} + C_{c z} {\dot{Z}}_{c} + K_{c z} Z_{c} = \sum_{i = 1}^{3} (F_{k p_{i} s} + F_{c p_{i} s}) \sin (α_{p s} + ψ_{i}) + \sum_{i = 1}^{3} (F_{k r p_{i}} + F_{c r p_{i}}) \sin (ψ_{i} - α_{r p}) \\ J_{s} {\overset{\cdot\cdot}{θ}}_{s} + C_{1 s} ({\dot{θ}}_{s} - {\dot{θ}}_{1}) + K_{1 s} (θ_{s} - θ_{1}) = \sum_{i = 1}^{3} (F_{k p_{i} s} + F_{c p_{i} s}) r_{b s} \\ m_{s} {\overset{\cdot\cdot}{Y}}_{s} + C_{s y} {\dot{Y}}_{s} + K_{s y} Y_{s} = \sum_{i = 1}^{3} (F_{k p_{i} s} + F_{c p_{i} s}) \cos (α_{p s} + ψ_{i}) \\ m_{s} {\overset{\cdot\cdot}{Z}}_{s} + C_{s z} {\dot{Z}}_{s} + K_{s z} Z_{s} = - \sum_{i = 1}^{3} (F_{k p_{i} s} + F_{c p_{i} s}) \sin (α_{p s} + ψ_{i}) \\ J_{p} {\overset{\cdot\cdot}{θ}}_{p_{i}} = - (F_{k p_{i} s} + F_{c p_{i} s}) r_{b p} + (F_{k r p_{i}} + F_{c r p_{i}}) r_{b p} \\ m_{p} {\overset{\cdot\cdot}{η}}_{p_{i}} + C_{p ς} {\dot{η}}_{p_{i}} + K_{p ς} η_{p_{i}} = - (F_{k p_{i} s} + F_{c p_{i} s}) \cos α_{p s} - (F_{k r p_{i}} + F_{c r p_{i}}) \cos α_{r p} \\ m_{p} {\overset{\cdot\cdot}{ζ}}_{p_{i}} + C_{p η} {\dot{ζ}}_{p_{i}} + K_{p η} ζ_{p_{i}} = (F_{k p_{i} s} + F_{c p_{i} s}) \sin α_{p s} - (F_{k r p_{i}} + F_{c r p_{i}}) \sin α_{r p} \\ J_{1} {\overset{\cdot\cdot}{θ}}_{1} + C_{1 s} ({\dot{θ}}_{1} - {\dot{θ}}_{s}) + K_{1 s} (θ_{1} - θ_{s}) = - (F_{k 12} + F_{c 12}) r_{b 1} \\ m_{1} {\overset{\cdot\cdot}{Y}}_{1} + C_{1 y} {\dot{Y}}_{1} + K_{1 y} Y_{1} = - (F_{k 12} + F_{c 12}) \sin α_{12} \\ m_{1} {\overset{\cdot\cdot}{Z}}_{1} + C_{1 z} {\dot{Z}}_{1} + K_{1 z} Z_{1} = (F_{k 12} + F_{c 12}) \cos α_{12} \\ J_{2} {\overset{\cdot\cdot}{θ}}_{2} + C_{23} ({\dot{θ}}_{2} - {\dot{θ}}_{3}) + K_{23} (θ_{2} - θ_{3}) = (F_{k 12} + F_{c 12}) r_{b 2} \\ m_{2} {\overset{\cdot\cdot}{Y}}_{2} + C_{2 y} {\dot{Y}}_{2} + K_{2 y} Y_{2} = (F_{k 12} + F_{c 12}) \sin α_{12} \\ m_{2} {\overset{\cdot\cdot}{Z}}_{2} + C_{2 z} {\dot{Z}}_{2} + K_{2 z} Z_{2} = - (F_{k 12} + F_{c 12}) \cos α_{12} \\ J_{3} {\overset{\cdot\cdot}{θ}}_{3} + C_{23} ({\dot{θ}}_{3} - {\dot{θ}}_{2}) + K_{23} (θ_{3} - θ_{2}) = - (F_{k 34} + F_{c 34}) r_{b 3} \\ m_{3} {\overset{\cdot\cdot}{Y}}_{3} + C_{3 y} {\dot{Y}}_{3} + K_{3 y} Y_{3} = - (F_{k 34} + F_{c 34}) \sin α_{34} \\ m_{3} {\overset{\cdot\cdot}{Z}}_{3} + C_{3 z} {\dot{Z}}_{3} + K_{3 z} Z_{3} = - (F_{k 34} + F_{c 34}) \cos α_{34} \\ J_{4} {\overset{\cdot\cdot}{θ}}_{4} + C_{c} ({\dot{θ}}_{4} - {\dot{θ}}_{g}) + K_{c} (θ_{4} - θ_{g}) = (F_{k 34} + F_{c 34}) r_{b 4} \\ m_{4} {\overset{\cdot\cdot}{Y}}_{4} + C_{4 y} {\dot{Y}}_{4} + K_{4 y} Y_{4} = (F_{k 34} + F_{c 34}) \sin α_{34} \\ m_{4} {\overset{\cdot\cdot}{Z}}_{4} + C_{4 z} {\dot{Z}}_{4} + K_{4 z} Z_{4} = (F_{k 34} + F_{c 34}) \cos α_{34} \end{array}

(8)

3. Generator rotor dynamic equations

J_{g} {\overset{\cdot\cdot}{θ}}_{g} + C_{c} ({\overset{\cdot}{θ}}_{g} - {\overset{\cdot}{θ}}_{4}) + K_{c} (θ_{g} - θ_{4}) = - T_{em}

(9)

The wind turbine system dynamic equations can also be written as

M \overset{\cdot\cdot}{X} + C \overset{\cdot}{X} + KX = T + F

(10)

where $X = [θ_{b}, θ_{h}, θ_{c}, Y_{c}, Z_{c}, θ_{s}, Y_{s}, Z_{s}, θ_{p_{i}}, Y_{p_{i}}, Z_{p_{i}}, θ_{1}, Y_{1}, Z_{1}, θ_{2}, Y_{2}, Z_{2}, θ_{3}, Y_{3}, Z_{3}, θ_{4}, Y_{4}, Z_{4}, θ_{g}]^{T}$ , which is the vector of the displacements of the system; M is the matrix of the masses of the system; C is the matrix of the damping of the system; and K is the matrix of the stiffness of the system. In addition, T is the vector of the external excitations, including the aerodynamic torque $T_{wtr}$ and electromagnetic torque $T_{em}$ , and F is the vector of the external excitations, including bearing supporting stiffness, time-varying meshing stiffness and static transmission error, and elastic deformation of shaft.

Bearing supporting stiffness and damping

In this analysis, an empirical formula is adopted to calculate the bearing stiffness. The supporting stiffness $K_{b}$ is expressed by¹³

K_{b} = 4.77 \times 10^{- 6} Z D^{1 / 2} \cos^{5 / 2} β δ_{r}^{1 / 2}

(11)

where $δ_{r} = (4.62 \times 10^{- 5} F_{r}^{2 / 3}) / (Z^{2 / 3} D^{1 / 3} \cos^{5 / 3} β)$ , $F_{r}$ is the radius force, Z is the number of bearing rollers, D is the diameter of the roller, and $β$ is the contact angle.

The supporting damping $C_{b}$ is expressed by⁹

C_{b} = a_{R} m + b_{R} K_{b}

(12)

where $a_{R}$ and $b_{R}$ are the coefficients of Rayleigh and m is the mass of a component.

Mesh stiffness and damping

In the meshing process of spur gear, there are a couple of meshing conditions that are single and double tooth meshing situations. Therefore, it causes meshing stiffness $K_{uv} (t)$ to process an obvious characteristic of periodic variation. Using Fourier series expansion, the time-varying meshing stiffness $K_{uv} (t)$ can be expressed by¹²

K_{uv} (t) = K_{0} + \sum_{n = 1}^{N} [a_{n} \cos (n ω_{m} t) + b_{n} \sin (n ω_{m} t)]

(13)

where $K_{0}$ is the average meshing stiffness, $a_{n}$ and $b_{n} (n = 1, 2, \dots, N)$ are the coefficients of Fourier series expansion, and $ω_{m}$ is the meshing frequency.

The values of the meshing damping of the gear pairs $C_{uv} (t)$ are calculated by¹²

C_{uv} (t) = 2 ζ_{uv} \sqrt{K_{uv} (t) \frac{r_{bu}^{2} r_{bv}^{2} J_{u} J_{v}}{r_{bu}^{2} J_{u} + r_{bv}^{2} J_{v}}}

(14)

where $ζ_{uv}$ is the ratio of the meshing damping;¹⁴ $r_{bu}$ and $r_{bv}$ are the base radius of the driving gears and driven gears, respectively; and $J_{u}$ and $J_{v}$ are the inertias of the driving gears and driven gears, respectively.

Static transmission error

The transmission error of the gear pairs $e_{uv} (t)$ is produced by manufacturing errors and assembly errors. In this work, the meshing error is simulated using a sine function. The meshing error of the gear pairs $e_{uv} (t)$ is expressed by¹²

e_{uv} (t) = e_{0} + e_{a} \sin (ω_{m} t + ϕ)

(15)

where $e_{0}$ is the average value of the meshing errors, $e_{a}$ is the maximum value of the meshing errors, and $ϕ$ is the phrase angle.

Torsional stiffness and damping of shafts

In this analysis, elastic deformation of shaft only considers torsional deformation. The torsional stiffness $K_{t}$ of a shaft is expressed by¹⁰

K_{t} = \frac{G I_{x}}{l}

(16)

where G is the shear modulus of the shaft, $I_{x}$ is the cross-section moment of inertia, and l is the length of the shaft.

The torsional damping $C_{t}$ of a shaft is expressed by¹²

C_{T} = 2 ζ_{T} \sqrt{K_{T} \frac{J_{1} J_{2}}{J_{1} + J_{2}}}

(17)

where $ζ_{T}$ is the ratio of the torsional damping¹⁴ and $J_{1}$ and $J_{2}$ are the inertias of the components.

Doubly fed induction generator model

In this derivation, the effects of magnetic saturation and iron losses are neglected in the synchronous rotating reference frame ( $d - q$ frame). Based on these assumptions, a detailed transient model of doubly fed induction generator (DFIG) (voltage equations) can be expressed by^15,16

{\begin{matrix} u_{sd} = R_{s} i_{sd} + \begin{matrix} ρ \end{matrix} ψ_{sd} - ω_{1} ψ_{sq} \\ u_{sq} = R_{s} i_{sq} + \begin{matrix} \begin{matrix} ρ \end{matrix} \end{matrix} ψ_{sq} + ω_{1} ψ_{sd} \\ u_{rd} = R_{r} i_{rd} + \begin{matrix} \begin{matrix} ρ \end{matrix} \end{matrix} ψ_{rd} - (ω_{1} - ω_{r}) ψ_{rq} \\ u_{rq} = R_{r} i_{rq} + \begin{matrix} \begin{matrix} ρ \end{matrix} \end{matrix} ψ_{rq} + (ω_{1} - ω_{r}) ψ_{rd} \end{matrix}

(18)

where $u_{sd}$ , $u_{sq}$ , $u_{rd}$ , and $u_{rq}$ are $d -$ and $q -$ axis of stator and rotor voltages; $i_{sd}$ , $i_{sq}$ , $i_{rd}$ , and $i_{rq}$ are $d -$ and $q -$ axis of stator and rotor currents; $ψ_{sd}$ , $ψ_{sq}$ , $ψ_{rd}$ , and $ψ_{rq}$ are $d -$ and $q -$ axis of stator and rotor fluxes; $ω_{1}$ is stator angular electrical frequency, $ω_{1} = 2 π f_{1}$ ; $f_{1}$ is the power grid frequency; $ω_{r}$ is rotor electrical angular velocity; and $ρ$ is the $d / dt$ operator. The corresponding flux equations are

{\begin{matrix} ψ_{sd} = L_{s} i_{sd} + L_{m} i_{rd} \\ ψ_{sq} = L_{s} i_{sq} + L_{m} i_{rq} \\ ψ_{rd} = L_{m} i_{sd} + L_{r} i_{rd} \\ ψ_{rq} = L_{m} i_{sq} + L_{r} i_{rq} \end{matrix}

(19)

where $L_{m}$ is magnetizing inductance and $L_{s}$ and $L_{r}$ are rotor and stator inductances. The electromagnetic torque $T_{em}$ equation is

T_{e} = \frac{3}{2} p L_{m} (i_{sq} i_{rd} - i_{sd} i_{rq})

(20)

where p is the number of pole pairs. Finally, the stator active power $P_{s}$ and reactive power $Q_{s}$ are described as

{\begin{matrix} P_{s} = 1.5 (u_{sd} i_{sd} + u_{sq} i_{sq}) \\ Q_{s} = 1.5 (u_{sq} i_{sd} - u_{sd} i_{sq}) \end{matrix}

(21)

Control strategy of the wind turbine

Pitch control

Above the rated wind speed, due to the limitation of the mechanical loads and the capacity of electrical devices, the output power of the wind rotor should be limited. As mentioned in section “Wind rotor model,” the aerodynamic power is related to wind speed and power coefficient $C_{p}$ . When generator mechanical rotor speed exceeds the rated rotor speed, the power coefficient $C_{p}$ can be reduced to guarantee the safe operation for the wind turbine generation system. As shown in Figure 2, because changing the blade pitch angle can rapidly change $C_{p}$ , it is the most effective method to control the aerodynamic load and limit the rotation speed.

In this analysis, in order to simplify the pitch control strategy, a proportional–integral–derivative (PID) controller is implemented for controlling the pitch angle. The angle command is limited to the range of $β_{\min}$ to $β_{\max}$ , where $β_{\min}$ is the optimum pitch angle at which tracking the largest wind energy of the wind turbine is achieved. $Δ β_{\max}$ is the maximum pitch angle speed. The actual generator mechanical rotor speed $ω_{m}$ will be used as the control signal for the pitch control system. The control strategy frame is shown in the following figure.

As shown in Figure 5, $ω_{m}^{*}$ is the desired generator mechanical angular velocity. It is a function of wind speed v as shown in Figure 6.

Figure 5.

Control structure of pitch control.

Figure 6.

Desired generator mechanical rotor speed curve.

At point A, wind speed is 3 m/s, which is the cut-in wind speed at which the doubly fed induction generator begins to output electrical power to grid. At point B, wind speed is 12 m/s, which is the rated wind speed at which the DFIG arrives at the rated power.

Stator flux–oriented control

The DFIG is controlled in the synchronous rotating $d - q$ frame with attaching the $d -$ axis to the stator flux. It achieves the decoupled control of active power and reactive power.

The stator flux equations are rewritten as vector equations

Ψ_{s} = L_{s} I_{s} + L_{m} I_{r} = L_{m} I_{ms}

(22)

where $I_{s}$ and $I_{r}$ are the current vectors of the stator and the rotor, $I_{s} = i_{sd} + j i_{sq}$ , $I_{r} = i_{rd} + j i_{rq}$ , and $I_{ms}$ is the equivalent exciting current vector, $I_{ms} = ((L_{s} / L_{m}) I_{s} + I_{r})$ .

Attaching the $d -$ axis to the stator flux, the stator flux equations and excitation current equations are expressed as

{\begin{matrix} ψ_{sd} = | Ψ_{s} | = ψ_{s} = L_{s} i_{sd} + L_{m} i_{rd} \\ ψ_{sq} = 0 = L_{s} i_{sq} + L_{m} i_{rq} \end{matrix}

(23)

{\begin{matrix} i_{msd} = | I_{ms} | = i_{ms} = \frac{ψ_{s}}{L_{m}} \\ i_{msq} = 0 \end{matrix}

(24)

where $ψ_{s}$ is the amplitude of the stator flux vector and $i_{ms}$ is the amplitude of the stator exciting current vector.

Thus, the stator current equations are expressed as

{\begin{matrix} i_{sd} = \frac{L_{m}}{L_{s}} (i_{ms} - i_{rd}) \\ i_{sq} = - \frac{L_{m}}{L_{s}} i_{rq} \end{matrix}

(25)

Neglecting the stator resistance, the $d -$ axis component of the stator voltage is zero. Thus, the stator voltage equations are expressed as

{\begin{matrix} u_{sd} = 0 \\ u_{sq} = | U_{s} | = u_{s} \end{matrix}

(26)

where $u_{s}$ is the amplitude of the stator voltage. Based on equations (21), (25), and (26), the stator active power $P_{s}$ and reactive power $Q_{s}$ equations are rewritten as

{\begin{matrix} P_{s} = - \frac{3}{2} \frac{L_{m}}{L_{s}} u_{s} i_{rq} \\ Q_{s} = \frac{3}{2} u_{s} \frac{L_{m}}{L_{s}} (i_{ms} - i_{rd}) \end{matrix}

(27)

Based on equation (27), using the stator flux–oriented control strategy, the stator active power is controlled by the $q -$ axis component of the rotor current, and the stator reactive power is controlled by the $d -$ axis component of the rotor current. The rotor current control achieves the decoupled control of active power and reactive power.

The rotor flux equations are rewritten as

{\begin{matrix} ψ_{rd} = \frac{L_{m}}{L_{s}} ψ_{s} + σ L_{r} i_{rd} = \frac{L_{m}^{2}}{L_{s}} i_{ms} + σ L_{r} i_{rd} \\ ψ_{rq} = σ L_{r} i_{rq} \end{matrix}

(28)

where $σ$ is the leakage coefficient of the generator, $σ = 1 - (L_{m}^{2} / L_{r} L_{s})$ .

So, the rotor voltage equations are rewritten as

{\begin{matrix} u_{rd} = R_{r} i_{rd} + σ L_{r} \frac{d i_{rd}}{dt} - ω_{s} σ L_{r} i_{rq} \\ u_{rq} = R_{r} i_{rq} + σ L_{r} \frac{d i_{rq}}{dt} + ω_{s} (\frac{L_{m}}{L_{s}} ψ_{s} + σ L_{r} i_{rq}) \end{matrix}

(29)

In this derivation, the control of active and reactive power is based on equation (27), while the rotor current controller is designed based on equation (29). Hence, the overall wind turbine model is built, as shown in Figure 7.

Figure 7.

Simulation of the wind turbine by MATLAB/Simulink.

Numerical simulations

In this section, numerical simulations are conducted. The main parameters of the wind turbine are shown in Table 1, the main parameters of the drivetrain are shown in Table 2, and the main parameters of the generator are shown in Table 3.

Table 1.

Parameters for wind rotor.

Main parameter	Value
Air density, ρ (kg/m³)	1.225
Rotor radius, $R_{w}$ (m)	32.025
Cut-in wind speed, $v_{in}$ (m/s)	3
Rated wind speed, $v_{rated}$ (m/s)	12
Cut-out wind speed, $v_{out}$ (m/s)	25
Minimum pitch angle, $β_{\min}$ (rad)	0.04538 (2.6°)
Maximum pitch angle, $β_{\max}$ (rad)	1.5708 (90°)
Maximum pitch speed, $Δ β_{\max}$ (rad/s)	0.1658 (9.5°/s)

Table 2.

Parameters for driven train.

Main parameter	Value
Inertia of three blades, $J_{b}$ (kg m²)	2.034E6
Inertia of the hub, $J_{h}$ (kg m²)	5226.27
Inertia of the carrier, $J_{c}$ (kg m²)	214.07
Inertia of the sun gear, $J_{s}$ (kg m²)	1.55
Inertia of the planet gear, $J_{p_{i}}$ (kg m²)	5.51
Inertia of gear 1, $J_{1}$ (kg m²)	113.23
Inertia of gear 2, $J_{2}$ (kg m²)	0.55
Inertia of gear 3, $J_{3}$ (kg m²)	18.96
Inertia of gear 4, $J_{4}$ (kg m²)	0.06
Inertia of the generator rotor, $J_{g}$ (kg m²)	58
Mass of the carrier, $m_{c}$ (kg)	1726.18
Mass of the sun gear, $m_{s}$ (kg)	100.12
Mass of the planet gear, $m_{p_{i}}$ (kg)	150.88
Mass of gear 1, $m_{1}$ (kg)	857.47
Mass of gear 2, $m_{2}$ (kg)	75.92
Mass of gear 3, $m_{3}$ (kg)	318.64
Mass of gear 4, $m_{4}$ (kg)	20.70
Carrier radius, $r_{bc}$ (mm)	375
Sun gear base radius, $r_{bs}$ (mm)	140.954
Planet gear base radius, $r_{bp}$ (mm)	202.974
Gear 1 base radius, $r_{b 1}$ (mm)	432.671
Gear 2 base radius, $r_{b 2}$ (mm)	109.471
Gear 3 base radius, $r_{b 3}$ (mm)	309.101
Gear 4 base radius, $r_{b 4}$ (mm)	71.811
Meshing angle of the planet gear and the sun gear, $α_{ps}$ (°)	23.49
Meshing angle of the ring gear and the planet gear, $α_{rp}$ (°)	21.22
Meshing angle of the first high-speed parallel gear stage, $α_{12}$ (°)	22.21
Meshing angle of the second high-speed parallel gear stage, $α_{34}$ (°)	22.09

Table 3.

Parameters for DFIG.

Main parameter	Value
Rated power, P (MW)	1.05
Rated line-to-line voltage, U (kV)	0.69
Rated rotor speed, $ω_{r}$ (rad/s)	188.5
Frequency, $f_{1}$ (Hz)	50
Stator resistance, $R_{s}$ (Ω)	0.012
Rotor resistance, $R_{r}$ (Ω)	0.013
Stator leakage reactance, $L_{s}$ (Ω)	0.092
Rotor leakage reactance, $L_{r}$ (Ω)	0.115
Magnetizing reactance, $L_{m}$ (Ω)	3.94
Poles, p	2

Modal analysis

Neglecting the system damping and meshing error, the meshing stiffness and the bearing supporting stiffness are the average value. Thus, the system un-damped free vibration differential equation is defined as

M \overset{\cdot\cdot}{X} + KX = 0

(30)

It is assumed that the mode shape matrix is A , thus the mode shape equation is defined as

M ω^{2} A + KA = 0

(31)

where $ω$ is the un-damped natural frequency. The frequency equation is defined as

| K + M ω^{2} | = 0

(32)

The un-damped natural frequencies are shown in Table 4. The mode shapes are shown in Figure 8.

Table 4.

Categorized natural frequencies for the wind turbine.

30 DOF Multibody dynamical model
Rotational modes (Hz)	Translational modes (Hz)	Rotational–translational modes (Hz)
0.22	786.56	66.63
29.85	931.35	264.34
577.27	2425.37	465.36
610.56		663.89
856.89		706.36
1202.37		821.36
1832.33		867.42
10,398.68		873.56
69,908.78		884.03
176,230.94		941.28
		1138.68
		1156.06
		1157.65
		2012.23
		2269.89
		3230.69
		3569.65

DOF: degree of freedom.

Figure 8.

Mode shapes of the wind turbine drivetrain.

As shown in Figure 8 and Table 4, the mode shapes of the wind turbine drivetrain consist of rotational modes, translational modes, and rotational–translational modes. The amplitude of rotational vibration is greater than translational vibration, so the main pattern of the system vibration is represented as rotational vibration. Hence, on the condition of the external exciting torques such as the aerodynamic torque and the electrical torque, the rotational vibration amplitude of the system may increase, especially when resonance occurs. Besides, the severe rotational vibration can influence the fatigue life of shafts and gears.

Dynamic response

In the condition of the three-phase ideal voltage, two kinds of wind speed models are considered as the external excitation of the wind turbine system, especially gearbox. The wind models include a steady state (12 m/s) and turbulent one (average wind speed is 12 m/s and turbulent intensity is 20%), as shown in Figure 9. The turbulent one is based on the linear auto-regressive wind speed model considering the variety of field conditions, the wind spectrum characters, and the traits of buildings, which is regarded as more representative than others.

Figure 9.

Wind time series.

According to equations from sections “Mathematic modeling” and “Control strategy of the wind turbine,” under the conditions of the two kinds of wind models, the response of the model, as shown in Figure 7, is computed. Figure 10 illustrates that by comparing with the steady wind condition, the turbulent wind will cause more disturbances on pitch angle, generator mechanical angular velocity, generator power, and electrical torque. Besides, the calculation results show that under the condition of turbulent wind, the components in the gearbox will vibrate more intensively, such as the planet carrier shown in Figure 11. The dynamic meshing force and amplitudes counting are shown in Figure 12. Because of the limitation of space, only the meshing force between gear 3 and gear 4 is given. The meshing force has significant time variation. The wind speed fluctuation will aggravate the trait. In the turbulent wind case, the range of meshing force of a gear pair rise by 60% over the force in the steady wind case so that it has a negative influence on the fatigue life of gears. The statistic results will make a contribution to provide a theoretical basis for the gear parameter optimization and reliability assessment.

Figure 10.

Pitch angle and generator parameters curve: (a) pitch angle, (b) generator mechanical angular velocity, (c) generator stator active power, (d) generator rotor active power, (e) generator stator reactive power, and (f) generator electromagnetic torque.

Figure 11.

Vibration angular acceleration of planet carrier.

Figure 12.

Meshing force between gear 3 and gear 4 and amplitude counting results: (a) dynamic meshing force of gear 3 and gear 4 and (b) meshing force amplitude counting.

Finally, under the turbulent wind condition, the effects of voltage sag on the dynamic responses of the drivetrain are also studied. The power grid voltage sags to 20% at $t = 70 s$ and recovers to normal voltage level at $t = 71 s$ , as shown in Figure 13. Propose the situation after this voltage sag do not trigger the rotor protection device. The control schemes of the DFIG operate as normal and try to restore the normal operation after the disturbance is cleared.

Figure 13.

Voltage sag.

As shown in Figure 14, the voltage will cause severe fluctuation of generator electromagnetic torque which directly acts on the gearbox through a coupling. So, the fluctuation may damage gear tooth and influence the gearbox life.

Figure 14.

Electromagnetic torque and meshing force: (a) generator electromagnetic torque and (b) dynamic meshing force of gear 3 and gear 4.

Parametric studies

First, the effects of rotor and stator resistance variation on the gearbox during voltage sag are discussed. Because of the limitation of space, only effects on meshing force between gear 3 and gear 4 are given.

As shown in Figure 15, under the condition of voltage sag shown in Figure 13 and steady wind shown in Figure 9, different values of resistance have significant effects on the maximum meshing force between gear 3 and gear 4. With the increase in resistance, the maximum meshing force will reduce. Therefore, in the process of generator design, not only can electric capacity be considered but also the interaction between gearbox and generator should be contained.

Figure 15.

Effects of generator parameters on gearbox during voltage sag: (a) the relationship between maximum meshing force and stator resistance and (b) the relationship between maximum meshing force and rotor resistance.

And then, the effects of different freedom models of the gearbox on the transit performance of the generator system are also investigated. In the section, a simplified gearbox model is proposed. Based on the detail gearbox model, the simplified model only considers the rotational freedom of components of gearbox, so its dynamic equation can be expressed as

M \overset{\cdot\cdot}{X} + C \overset{\cdot}{X} + KX = T + F

(33)

where $X = [θ_{b}, θ_{h}, θ_{c}, θ_{s}, θ_{p_{i}}, θ_{1}, θ_{2}, θ_{3}, θ_{4}, θ_{g}]^{T}$ .

Under the conditions of wind speed 12 m/s, the power grid voltage sags to 80% at $t = 0.5 s$ and the fault is cleared at $t = 0.6 s$ . The results are shown in Figure 16.

Under the conditions of wind speed 12 m/s, the power grid voltage sags to 10% at $t = 0.5 s$ and the fault is cleared at $t = 0.6 s$ . The results are shown in Figure 17.

Under the conditions of wind speed 7 m/s, the power grid voltage sags to 10% at $t = 0.5 s$ and recover to normal voltage level at $t = 0.6 s$ . The results are shown in Figure 18.

Under the conditions of wind speed 12 m/s, the power grid voltage respectively sags to 10% at $t = 0.5 s$ and recover to normal voltage level at $t = 1 s$ . The results are shown in Figure 19.

Figure 16.

Transient response of the generator system: (a) the transient response of angular velocity of generator rotor and (b) the transient response of active power of generator.

Figure 17.

Transient response of the generator system: (a) the transient response of angular velocity of generator rotor and (b) the transient response of active power of generator.

Figure 18.

Transient response of the generator system: (a) the transient response of angular velocity of generator rotor and (b) the transient response of active power of generator.

Figure 19.

Transient response of the generator system: (a) the transient response of angular velocity of generator rotor and (b) the transient response of active power of generator.

It can be seen in these figures that the fault of power grid both for brief and detailed model can lead to large oscillation of active power and rotational speed of generator rotor. Furthermore, the results clearly show a few differences in the transient responses with different freedom models of gearbox. In the high wind speed and extreme fault of power grid, the effects of freedom of gearbox on transient responses of the generator system are significant and cannot be neglected. The freedom of gearbox plays a dominant role in transient responses of generator system; therefore, in order to deeply research the transient stability of a wind power generation system, a detailed model of gearbox should be carefully considered.

Conclusion

In this study, in order to accurately analyze the effects of turbulent wind and voltage sag on the 1-MW wind turbine gearbox, a detailed wind turbine drivetrain model which has 12 rigid bodies and 30 DOFs has been proposed by considering the bending flexibility of the blades, meshing stiffness and damping, meshing error, bearing stiffness and damping, and torsional flexibility of the drivetrain shaft.

First, modal analysis is computed, and the results show that the main pattern of the system vibration is represented as rotational vibration. Under the influence of external exciting torques, the system has the potential possibility for resonance. Second, by incorporating the aerodynamic torque calculation model, the electromagnetic transient models of DFIG, and control strategies, the dynamic behaviors of the gearbox system can be analyzed. The results show that the disturbance produced by turbulent wind will increase the vibration level of components inside and also expand the variation range of meshing force. This potentially can cause premature fatigue of the gearbox system. Third, under the condition of turbulent wind, the effect of voltage sag is analyzed. The results show that the electromagnetic torque fluctuation produced by voltage sag will cause severe impact on the gears. Finally, the effects of generator parameter variation on the gearbox and effects of different freedom models of the gearbox on transient responses of the generator system are researched. The results show that parameters of the generator play an important role in decreasing gearbox loads in extreme situation and the importance of the dynamics of gearbox system. This analysis also demonstrated the salient features of the proposed wind turbine drivetrain model coupled incorporating aerodynamic and electrical effects and its usefulness in predicting design reliability.

Footnotes

Academic Editor: José Tenreiro Machado

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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Dynamic responses of a wind turbine drivetrain under turbulent wind and voltage disturbance conditions

Abstract

Keywords

Introduction

Mathematic modeling

Wind rotor model

Mechanical model

Bearing supporting stiffness and damping

Mesh stiffness and damping

Static transmission error

Torsional stiffness and damping of shafts

Doubly fed induction generator model

Control strategy of the wind turbine

Pitch control

Stator flux–oriented control

Numerical simulations

Modal analysis

Dynamic response

Parametric studies

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References