A new measure of the degree of controllability for linear system with external disturbance and its application to wind turbines

Abstract

This paper is concerned with the measure of degree of controllability (DOC) for linear system with external disturbance. A new measure of DOC, in which the initial condition is regarded as a random vector, is proposed in this paper by solving the fixed-time expected minimum-energy transfer control problem. Since this new measure is dependent on the statistical information of initial condition rather than its estimated value, it is more suitable to apply the proposed measure in the design and optimization of the structural parameters of controlled plants. Furthermore, the simulations on the NREL (National Renewable Energy Laboratory) CART3 wind turbine demonstrate that the relation of the proposed measure to turbine parameters (including rotor inertia and optimum tip speed ratio) coincides with that of the MPPT efficiency to turbine parameters. This indicates that the proposed measure is applicable to guide the design and optimization of the structural parameters of wind turbines. Meanwhile, a mass-spring-damper system is also simulated to validate the proposed measure.

Keywords

Degree of controllability external disturbance minimum-energy transfer control problem wind turbines mass-spring-damper

1. Introduction

As a quantitative measure of controllability, the degree of controllability (DOC) describes how controllable the system is. Currently, several different definitions of DOC have been proposed, which are based on the control energy (Kalman et al., 1962; Muller and Weber, 1972; Lee and Park, 2014), the distance from controllable to uncontrollable set (Paige, 1981), the minimum of 2-norms of initial conditions (Viswanathan et al., 1984) and the modal degree of controllability (Roh and Park, 1997)), respectively. This paper only addresses the issue of the control energy-based DOC. For linear systems without external disturbance, Muller and Weber (1972) described the controllability of plants quantitatively with a physically meaningful value: control energy and proposed three measures of DOC, in which the capabilities for stabilizing the system is regarded as the minimum energy required to regulate the system from any initial state to the origin within a limited time.

External disturbance is usually neglected in the above definitions of DOC. In fact, external disturbance appears in many practical systems, which is often attributed as the major cause of unfavorable performance and instability. As an important property of a plant, disturbance rejection is often the main objective of process control. Therefore, a measure representing DOC of disturbance rejection (Kang et al., 2009) was presented based on the expected total transfer energy in linear systems with external disturbance (Lee et al., 2011, 2012; Marghub and Johannes, 1999).

However, since the measure of Kang et al. (2009) focuses on the control energy contributing to disturbance rejection, it can only be employed to indicate the controllability of the plant, in which rejecting disturbance must be the major control objective. Assume that if the external disturbance disappears (i.e., the linear system with external disturbance degenerates into the one without disturbance), this measure is constantly equal to zero and cannot be applied to guide the design and optimization of controlled plants (Muller and Weber, 1972). Besides, the zero value of this measure which means that the system is to be controlled without consuming energy is inconsistent with the measure of Muller and Weber (1972) in which the control energy for stabilizing performance is only expressed. Therefore, the control energy has not been comprehensively considered in the definition of the measure presented in Kang et al. (2009). In addition, when the DOC measure in Kang et al. (2009) is applied to wind turbines, it is found that the variation of this measure with turbine parameters is opposite to that of the control performance of maximum power point tracking (MPPT) (Peng et al., 2014) which is commonly represented as the MPPT efficiency (Femia et al., 2005; Boubekeur and Houria, 2011) with turbine parameters.

In Lee et al. (2011, 2012), by considering both disturbance-rejection and system stabilization simultaneously, a DOC measure was presented, in which the initial condition is assumed to be estimated by the sensor. However, since the DOC measure is closely related to the initial condition, its value is variable and consequently unsuitable for the design and optimization of structural parameters of controlled plants, when the initial condition is a random vector. For example, the initial condition of wind turbines changes randomly because of the intermittent and random fluctuation of wind speed.

Hence, to remove the unexpected dependency of the specific initial condition reasonably, a new DOC measure, in which the initial condition is regarded as a random vector, is proposed in this paper for linear system with external disturbance. Since the new DOC measure is dependent on the statistical information of initial condition instead of the estimated value, it is more suitable to be applied to the design and optimization of the structural parameters of controlled plants. Furthermore, the relationship of the proposed DOC measure versus the MPPT efficiency is investigated by the simulations on the NREL (National Renewable Energy Laboratory) CART3 wind turbine. It is observed that the larger DOC measure is, the higher MPPT efficiency of wind turbines can be derived. Moreover, the larger DOC measure can be obtained by reducing rotor inertia and optimum tip-speed ratio, which is coincided with the engineering observations (Johan et al., 2006; Chun et al., 2013) that a higher MPPT efficiency can be achieved by the wind turbine with smaller rotor inertia and optimum tip-speed ratio. Meanwhile, a mass-spring-damper system (Kang et al., 2009) is also simulated to validate the proposed measure. These indicate that the applicability of the proposed measure is improved.

The remainder of this paper is organized as follows. Before presenting the main results, basic definitions and motivation are given in section 2.1. In section 2.2, the new DOC measure is derived for linear system with external disturbance. The application and verification of the new measure to wind turbines are given in section 3. In section 4, a mass-spring-damper system is employed to show the effectiveness of the new measure. Conclusions are drawn in the last section of the paper.

Notation: Standard notations will be used throughout this paper. $λ min (X)$ and $λ max (X)$ are minimum and maximum eigenvalues of matrix X, respectively, and $| | \cdot | | 2$ stands for the spectral 2-norm. The superscripts T and $- 1$ represent the transpose and inverse of matrix, respectively. $tr (X)$ denotes the summation of the diagonal of matrix X. $E (w (t))$ is the expected value of w(t).

2. The new DOC measure

In this section, a new measure of the degree of controllability (DOC) will be introduced for linear system with external disturbance.

2.1. Basic definitions and motivation

Let Γ denote the linear time-invariant control system with external disturbance

Γ : x \cdot = Ax + Bu + Dw y = Cx

(1)

where

x \in R n

is the state vector,

u \in R m

is the control input, and

w \in R l

is the external disturbance. A, B, D and C are constant matrices with appropriate dimensions. Assume that the disturbance is Gaussian white noise with the correlation function

R w (τ) = E [w (t) w T (t + τ)] = S w δ (τ)

(2)

expected value is

E [w (t)] = 0

(3)

and the variance satisfies that

E [w (t) 2] < \infty

where

δ (τ)

is a dirac-delta-function.

Denoting $e A (t - t 0)$ as the state transition matrix of A, the controllability Grammian is defined as

W (t) = \int_{t 0}^{t} e A (t 0 - τ) BB T e A T (t 0 - τ) d τ

(4)

Similarly, the disturbance-sensitivity Grammian is defined as

Σ (t) = \int_{t 0}^{t} e A (t 0 - τ) DS w D T e A T (t 0 - τ) d τ

(5)

Definition 1

(Muller and Weber, 1972) For linear time-invariant control system without external disturbance, the DOC measure is given as

μ e = λ min (W (t f))

(6)

Remark 1

This measure represents the minimum energy required to regulate the system from any initial state to the origin within a limited time (Muller and Weber, 1972).

Definition 2

(Kang et al., 2009) The DOC measure of linear time-invariant control system with external disturbance can be expressed as

κ = tr (e - A T (t f - t 0) W (t f) - 1 Σ (t f) e A T (t f - t 0))

(7)

Remark 2

This measure describes the capabilities for disturbance-rejection quantitatively with a physically meaningful value: control energy (Kang et al., 2009).

Remark 3

According to the measures of definition 1 and 2, it is found that:

On the one hand, external disturbances, which do exist in many practical systems and even mainly cause the unfavorable performance and instability, are not considered in the measure of definition 1. On the other hand, the measure of definition 2 focuses only on the control energy for rejecting external disturbance, but the one for stabilizing the system is neglected, which results in the inapplicability of this measure to the controlled plants.

Assume that the external disturbance to the plant decays to zero (i.e., the linear system with external disturbance degenerates into the one without disturbance), the DOC of system (1) still exists according to the definition 1, but the measure of definition 2 is constantly equal to zero, which has no physical significance. Besides, the zero value of this measure is unable to the design and optimization of structural parameters of controlled plants (Muller and Weber, 1972).

In addition, since the DOC measure proposed in Lee et al. (2011, 2012) is dependent on the estimated value of the initial condition, it cannot be well suitable for the design and optimization of the structural parameters of controlled plants, especially when the initial condition is a random vector. Therefore, in order to improve the applicability of the existing measures, a new DOC measure is proposed in the following section.

2.2. The proposing of a new DOC measure

Consider the fixed-time expected minimum-energy transfer control problem as

σ = min u (t) E [\int_{t 0}^{t f} u T (t) u (t) d t], s . t . x (t 0) = x 0, x (t f) = 0

(8)

where x₀ is assumed to be a random vector with a given distribution and independent of

w (t)

. Then, if the optimal solution of (8) exists, definition 3 and Theorem 1 can be given in the following.

Definition 3

For the linear time-invariant control system with external disturbance in (1), the DOC measure μ can be defined as

μ - 1 = min u (t) E [\int_{t 0}^{t f} u T (t) u (t) d t]

Theorem 1

Suppose that the optimization problem in (8) is solvable. Then, a DOC measure can be obtained as

μ = \frac{1}{tr (W (t f) - 1 Λ) + tr (W (t f) - 1 Σ (t f))}

(9)

where

Λ = E (x 0 x_{0}^{T})

, which is the covariance matrix of x₀, can be calculated according to the statistical information of x₀.

Proof

Since the optimization problem in (8) is solvable for each initial condition x₀, then the optimal solution of (8) can be given by minimum principle as

u * (t) = - B T e A T (t 0 - t) W (t f) - 1 (x 0 + \int_{t 0}^{t f} e A (t 0 - τ) Dw (τ) d τ)

(10)

for all

t 0 \leq t \leq t f

. The process for obtaining the optimal solution is shown in Kailath (1980). Hence, the input covariance matrix is shown by

R u * (t) = E [u * (t) (u * (t)) T] = E [B T e A T (t 0 - t) W (t f) - 1 (x 0 + \int_{t 0}^{t f} e A (t 0 - τ) Dw (τ) d τ) \cdot (x_{0}^{T} + \int_{t 0}^{t f} w (s) D T e A T (t 0 - s) d s) W (t f) - 1 e A (t 0 - t) B]

(11)

According to (2) and (3), rearranging (11) yields

R u * (t) = B T e A T (t 0 - t) W (t f) - 1 E (x 0 x_{0}^{T}) W (t f) - 1 e A (t 0 - t) B + B T e A T (t 0 - t) W (t f) - 1 \int_{t 0}^{t f} e A (t 0 - τ) DS w D T e A T (t 0 - τ) d τ W (t f) - 1 e A (t 0 - t) B

(12)

Using equation (5), the input covariance matrix can be expressed as

R u * (t) = B T e A T (t 0 - t) W (t f) - 1 E (x 0 x_{0}^{T}) W (t f) - 1 e A (t 0 - t) B + B T e A T (t 0 - t) W (t f) - 1 Σ (t f) W (t f) - 1 e A (t 0 - t) B

(13)

Hence, the expected total energy of (8) can be calculated by

σ = \int_{t 0}^{t f} E [(u * (t)) T u * (t)] d t = tr {\int_{t 0}^{t f} E [(u * (t) u * (t)) T] d t} = tr {\int_{t 0}^{t f} B T e A T (t 0 - t) W (t f) - 1 E (x 0 x_{0}^{T}) W (t f) - 1 e A (t 0 - t) B + B T e A T (t 0 - t) W (t f) - 1 Σ (t f) W (t f) - 1 e A (t 0 - t) B d t}

(14)

Applying the property of the trace and rearranging terms, equation (14) yields

σ = tr {\int_{t 0}^{t f} E (x 0 x_{0}^{T}) W (t f) - 1 e A (t 0 - t) BB T e A T (t 0 - t) W (t f) - 1 + W (t f) - 1 Σ (t f) W (t f) - 1 e A (t 0 - t) BB T e A T (t 0 - t) d t} = tr {E (x 0 x_{0}^{T}) W (t f) - 1 + W (t f) - 1 Σ (t f)}

(15)

Therefore, σ provides a quantitative measure of controllability of (1) depending on $[t 0, t f]$ and x₀. It can be seen that a system $Γ 1$ with initial condition x₀ is more controllable at $[t 0, t f]$ than a system $Γ 2$ with initial condition x₀ if the corresponding minimum energy in (15) is smaller. For (15), $σ 1 = tr (E (x 0 x_{0}^{T}) W (t f) - 1)$ is denoted as the energy for the stabilizing performance and $σ 2 = tr (W (t f) - 1 Σ (t f))$ is expressed as the energy for the rejection of external disturbance.

Let $Λ = E (x 0 x_{0}^{T})$ , then the measure is defined as

μ = \frac{1}{tr (W (t f) - 1 Λ) + tr (W (t f) - 1 Σ (t f))}

(16)

Remark 4

It is obvious that the DOC measure of uncontrollable systems is 0. If the system (1) is uncontrollable in $[t 0, t f]$ , the value of (15) is ∞ and the μ defined in (16) is 0. Thus, it is reasonable that the measure (15) is replaced by (16).

Besides, Λ which is the covariance matrix of x₀, can be calculated according to the statistical information of x₀. When the distribution of x₀ is given, Λ is a constant matrix. For example, the initial condition x₀ usually obeys a Gaussian distribution in some stochastic systems (Fang, 2005).

As shown in (16), the new measure μ is dependent on t₀ and t_f. Without loss of generality, the initial time t₀ is assumed to be zero. Furthermore, to eliminate the dependence of this measure on t_f, the measure μ at steady state is defined below.

Definition 4

For the asymptotically stable system, assume that $μ (t f)$ is the DOC measure in $[0, t f]$ , then the DOC measure at steady state $μ (\infty)$ is defined as

μ (\infty) = lim t f \to \infty μ (t f)

In Kang et al. (2009), the measure for disturbance rejection at steady state is expressed as:

κ = tr (Σ ¯ \cdot W ¯ - 1)

(17)

where

W ¯

and

Σ ¯

satisfy the equations (18) and (19) for asymptotically stable systems, respectively.

A W ¯ + W ¯ A T + BB T = 0

(18)

A Σ ¯ + Σ ¯ A T + DS w D T = 0

(19)

Theorem 2

Assume that the system is asymptotically stable, then the DOC measure at steady state $μ (\infty)$ can be calculated as

μ (\infty) = \frac{1}{tr (Σ ¯ W ¯ - 1)}

(20)

Proof

Note that, according to the definition 4, the DOC measure at steady state $μ (\infty)$ can be expressed as

lim t f \to \infty μ (t f) = lim t f \to \infty \frac{1}{{tr ((\int_{0}^{t f} e - A τ BB T e - A T τ d τ) - 1 E (x 0 x_{0}^{T})) + tr ((\int_{0}^{t f} e - A τ BB T e - A T τ d τ) - 1 \int_{0}^{t f} e - A τ DS w D T e - A T τ d τ)}} = lim t f \to \infty 1 / {(tr (e A T t f (\int_{0}^{t f} e A (t f - τ) BB T e A T (t f - τ) d τ) - 1 e At f E (x 0 x_{0}^{T})) + tr (e A T t f (\int_{0}^{t f} e A (t f - τ) BB T e A T (t f - τ) d τ) - 1} e At f e - At f (\int_{0}^{t f} e A (t f - τ) DS w D T e A T (t f - τ) d τ) e - A T t f)) = lim t f \to \infty 1 / {(tr (e A T t f (\int_{0}^{t f} e A (t f - τ) BB T e A T (t f - τ) d τ) - 1 e At f E (x 0 x_{0}^{T})) + tr ((\int_{0}^{t f} e A (t f - τ) BB T e A T (t f - τ) d τ) - 1} (\int_{0}^{t f} e A (t f - τ) DS w D T e A T (t f - τ) d τ))), = 1 / {(lim t f \to \infty tr (e A T t f (\int_{0}^{t f} e A (t f - τ) BB T e A T (t f - τ) d τ) - 1 e At f E (x 0 x_{0}^{T})) + lim t f \to \infty tr ((\int_{0}^{t f} e A (t f - τ) BB T e A T (t f - τ) d τ) - 1} (\int_{0}^{t f} e A (t f - τ) DS w D T e A T (t f - τ) d τ)))

Because A is stable and $E (x 0 x_{0}^{T})$ is a constant matrix when the distribution of x₀ is given, $lim t f \to \infty tr (e A T t f (\int_{0}^{t f} e A (t f - τ) BB T e A T (t f - τ) d τ) - 1 e At f E (x 0 x_{0}^{T}))$ is equal to zero. Then,

lim t f \to \infty μ = 1 / lim t f \to \infty tr ((\int_{0}^{t f} e A (t f - τ) BB T e A T (t f - τ) d τ) - 1 \times (\int_{0}^{t f} e A (t f - τ) DS w D T e A T (t f - τ) d τ)) = 1 / tr ((\int_{0}^{\infty} e A (t f - τ) BB T e A T (t f - τ) d τ) - 1 \times (\int_{0}^{\infty} e A (t f - τ) DS w D T e A T (t f - τ) d τ)) = 1 / tr (W ¯ - 1 Σ ¯)

Remark 5

If A is stable, $tr (W (t f) - 1 E (x 0 x_{0}^{T}))$ tends to zero when t_f approaches infinity, which is consistent with the conclusion that the control input energy for regulating a stable system (e.g. three-degree-of-freedom mass-spring-damper system) in an infinite time period is zero (Chow and Yorke, 1974). In addition, by comparing equation (17) and (20), it is observed that the measure at steady state in Kang et al. (2009) (as expressed by (17)) is the reciprocal of the proposed one (20). Therefore, these two measures are equivalent for indicating the DOC of asymptotically stable systems at steady state.

3. The application and verification of the new measure to wind turbines

In this section, the new measure is applied to wind turbines. The MPPT efficiency (Femia et al., 2005; Boubekeur and Houria, 2011), as an engineering performance index of wind turbines, is adopted to test the DOC measure. In other words, if the variation of the DOC measure is in accordance with the one of MPPT efficiency with change of turbine parameters (i.e., the MPPT efficiency increases when the DOC measure becomes larger), it can be verified that the new DOC measure is effective for wind turbines.

The procedure for testing the proposed DOC measure is shown in Figure 1. Note that the MPPT efficiency is calculated by simulations on NREL’s FAST (Fatigue, Aerodynamics, Structures, and Turbulence) turbine model. And, the expression of DOC measure is derived from the single/two-mass model of wind turbines which is obtained by simplifying the FAST model. The details of each step are provided in section 3.1–3.4.

Figure 1.

The procedure of testing the DOC measure by NREL’s FAST.

3.1. NREL CART3 wind turbine and turbulent wind

The FAST code (Jonkman and Buhl, 2005), which is developed by NREL based on the bladed-element-momentum (BEM) theory and multi-body dynamics and has passed the German Lloyd's certification, is employed in this paper to perform the numerical simulation of wind turbines and calculate the MPPT efficiency. The FAST model of NREL CART3 wind turbine (Darrow, 2009), whose parameters are listed in Table 1, is selected to obtain the relationship between MPPT efficiency and structural parameters through simulations.

Table 1.

The parameters of CART3 wind turbine.

Parameters	J_r	J_g	R	$C_{p}^{max}$	$λ opt$	K_r	K_g	n_g	$B ls$	$K ls$
Value	549206.4	34.4	20	0.467	5.8	0	0	43.165	1.6043e7	0

In addition, according to the regulation of IEC 61400-1 (International EC, 2005), the turbulent wind whose parameters are listed in Table 2, is generated by the TurbSim (Jonkman, 2009). As illustrated in Figure 2, the turbulent wind is used to conduct simulations on the CART3 wind turbine and derive the statistical distribution of initial condition required in the proposed DOC measure.

Figure 2.

The turbulent wind generated by the TurbSim.

Table 2.

Parameters for generating turbulence wind.

Parameters	Value
Mean wind speed(m/s)	5.64
Time series length(sec.)	600
Turbulence model	IEC Kaimal
Time step(sec.)	0.05
Turbulence integral length scale(m)	150
Turbulence class	A

3.2. The derivation of new DOC measure based on the single/two-mass model of wind turbines

In this section, the new DOC measure with statistical distribution of initial condition is derived based on the single/two-mass model of wind turbines. Considering that wind turbines always operate in turbulence, the initial condition of wind turbines is closely relevant to wind speed. Hence, by deducing the functional relation of $Λ = E (x 0 x_{0}^{T})$ to $E (v 2)$ , $E (v 3)$ and $E (v 4)$ , the statistical distribution of initial condition in the proposed DOC measure is replaced by the one of turbulent wind speed.

3.2.1. The single/two-mass model and its linearization

Several types of wind turbine models, including the aero-servo-elasticity model (Bottasso et al., 2009; Bottasso et al., 2006), the multi-modal model (Fitzgerald and Basu, 2013; Staino and Basu, 2013) and the simplified model (i.e. the single/two-mass model (Chun et al., 2013; Boubekeur and Houria, 2011) have been established. Since the simplified model developed for wind turbine controller design are expressed as state space equations, this paper focuses on the single/two-mass model and derives the proposed DOC measure of wind turbines based on them. Note that because of the intermittent and random fluctuation of wind speed, two linearization models with wind speed regarded as external disturbance (Farzad et al., 2012) are deduced based on the single/two-mass model.

The single mass model and its linearization

If the stiffness, damping and the dynamic of electrical part of wind turbines are neglected, a single-mass model with following dynamic equations in MPPT stage is shown in Figure 3 (Jovan et al., 2014).

J r ω \cdot r = Δ T = T r (v, ω r) - T g (ω r)

(21)

T r (v, ω r) = \frac{1}{2} ρ π R 3 C Q (λ) v 2

(22)

T g (ω r) = {0 ω r < ω bgn k m ω_{r}^{2} ω r \geq ω bgn

(23)

where

C Q = C p (λ) / λ

ω r

is rotor speed, J_r is rotor inertia, ρ is air density, R is rotor radius, v is wind speed,

λ = ω r R / v

is tip-speed ratio, and

C P (λ)

is rotor power coefficient. In MPPT stage, the blade pitch angle β is generally assumed to be zero. T_g is generator torque, when rotor speed is larger than initial rotor speed

ω bgn

. The optimal torque control method (Kim et al., 2013) is commonly used to R realize MPPT control, and T_g is adjusted as the optimum torque curve. In addition,

k m = 0.5 ρ π R 5 C_{P}^{max} / λ_{opt}^{3}

is relevant to ρ,

C_{P}^{max}

and

λ opt

, respectively. Since ρ is a constant, k_m varies with R,

C_{P}^{max}

and

λ opt

, where

λ opt

is optimum tip-speed ratio, and

C_{P}^{max}

is optimum wind energy capture coefficient corresponding to

λ opt

Figure 3.

The single-mass model of wind turbine system.

In Xia et al. (2014) a linearization model of wind turbines has been given in which the wind speed v is assumed to be constant near the equilibrium point. Nevertheless, because of the intermittent and random fluctuation of wind speed, it is more reasonable to regard wind speed as external disturbance. Therefore, a linearization model of wind turbines with external disturbance is derived from equation (21). Denoting the equilibrium point of wind turbines as $(v ¯, ω ¯ r)$ , the variations are $Δ ω r = ω r - ω ¯ r$ , $Δ v = v - v ¯$ , respectively. The linearization of $T r (v, ω r)$ at the point of $(v ¯, ω ¯ r)$ is as follows:

Δ T r = \frac{1}{2} ρ π R 4 v ¯ \frac{\partial C Q}{\partial λ} | (v ¯, ω ¯ r) Δ ω r + \frac{1}{2} ρ π R 3 v ¯ (2 C Q - λ \frac{\partial C Q}{\partial λ}) | (v ¯, ω ¯ r) Δ v

Thus, the linearization model of wind turbines with external disturbance can be obtained as

x \cdot = A x + B u + D w

(24)

where

x = Δ ω r

u = Δ T g

w = Δ v

A = ρ π R 4 v ¯ (\partial C Q / \partial λ) | (v ¯, ω ¯ r) / 2 J r

B = - 1 / J r

, and

D = ρ π R 3 v ¯ (2 C Q - λ (\partial C Q / \partial λ)) | (v ¯, ω ¯ r) / 2 J r

The two-mass model and its linearization

The two-mass model with following dynamic equations in MPPT stage is shown in Figure 4 (Boubekeur and Houria, 2011).

T r = \frac{1}{2} ρ π R 3 C Q (λ) v 2, C Q = \frac{C P (λ)}{λ}

(25)

K r ω r + J r ω \cdot r = T r - T ls

(26)

B ls (θ r - \frac{θ g}{n g}) + K ls (ω r - \frac{ω g}{n g}) = T ls

(27)

T hs = \frac{T ls}{n g}

(28)

K g ω g + J g ω \cdot g = T hs - T g

(29)

where K_r is rotor external damping,

T ls

is low-speed shaft torque. The

B ls

K ls

θ r

θ g

and n_g in (27) are low-speed shaft stiffness, low-speed shaft damping, rotor-side angular deviation, generator-side angular deviation, and gearbox ratio, respectively. J_g is rotor inertia of drive-train, K_g is generator external damping, T_g is generator (electromagnetic) torque,

T hs

is high-speed shaft torque,

ω r

and

ω g

is rotor speed and generator speed, respectively.

Figure 4.

The two-mass model of wind turbine system.

Denoting the equilibrium point of wind turbines as $(v ¯, ω ¯ r)$ , the variations are $Δ ω r = ω r - ω ¯ r$ , $Δ v = v - v ¯$ , respectively. The linearization of T_r at the point of $(v ¯, ω ¯ r)$ is as follows:

Δ T r = \frac{1}{2} ρ π R 4 v ¯ \frac{\partial C Q}{\partial λ} | (v ¯, ω ¯ r) Δ ω r + \frac{1}{2} ρ π R 3 v ¯ (2 C Q - λ \frac{\partial C Q}{\partial λ}) | (v ¯, ω ¯ r) Δ v

(30)

Assume a state vector $x^= [ω^r, ω^g, T^ls] T$ , where $x^= x - x eq$ and $x eq$ is the equilibrium point vector of states. Therefore, the linearization model of wind turbines with external disturbance according to (25)-(30) can be obtained as

^\cdot x = A x^+ B u^+ D w^

(31)

where the plant matrices are defined as

A = [\frac{1}{2 J r} ρ π R 4 v ¯ \frac{\partial C Q}{\partial λ} | (v ¯, ω ¯ r) - \frac{K r}{J r} 0 - \frac{1}{J r} 0 - \frac{K g}{J g} \frac{1}{n g J g} {\frac{K ls}{2 J r} ρ π R 4 v ¯ \frac{\partial C Q}{\partial λ} | (v ¯, ω ¯ r) + \frac{J r B ls - K r K ls}{J r}} \frac{K ls K g - J g B ls}{n g J g} - \frac{J r K ls + n_{g}^{2} J g \cdot K ls}{n_{g}^{2} J g J r}]

(32)

B = [0 - \frac{1}{J g} \frac{K ls}{n g J g}], D = [\frac{1}{2 J r} ρ π R 3 v (2 C Q - λ \frac{\partial C Q}{\partial λ} | (v ¯, ω ¯ r)) 0 \frac{K ls}{2 J r} ρ π R 3 v (2 C Q - λ \frac{\partial C Q}{\partial λ} | (v ¯, ω ¯ r))]

(33)

The control input is defined as $u^= T g - T ¯ g$ , where $T ¯ g$ is the generator torque at equilibrium points. And the external disturbance is $w^= v - v ¯$ , $v ¯$ is the wind speed at equilibrium points.

3.2.2. The derivation of the new measure with statistical information of initial condition

Based on the single-mass model of wind turbines, the new measure is expressed as: i.e.

μ = (1 - e \frac{1}{J r} ρ π R 4 v ¯ \frac{\partial C Q}{\partial λ} | (v ¯, ω ¯ r) (t 0 - t f)) / (J r ρ π R 4 v ¯ \frac{\partial C Q}{\partial λ} | (v ¯, ω ¯ r) E (v 2) λ_{opt}^{2} / R 2 + 0.25 ρ 2 π 2 R 6 v ¯ 2 (2 C Q - λ \frac{\partial C Q}{\partial λ} | 2 (v ¯, ω ¯ r) S w (1 - e \frac{1}{J r} ρ π R 4 v ¯ \frac{\partial C Q}{\partial λ} | (v ¯, ω ¯ r) (t 0 - t f)))

where

Λ = E (x 0 x_{0}^{T}) = λ_{opt}^{2} E (v 2) / R 2

. For wind turbines, it is assumed that the initial point x₀ is equal to the optimal reference

ω ropt

(Boubekeur and Houria, 2011) and correspondingly

Λ = E (ω_{ropt}^{2})

. Then, according to the definition of

λ opt = ω ropt R / v

E (ω_{ropt}^{2})

can be rewritten as

λ_{opt}^{2} E (v 2) / R 2

and

Λ = λ_{opt}^{2} E (v 2) / R 2

For purpose of the comparison, the measure κ in (7) is replaced by its reciprocal $μ 1$ , and the relationship between $μ 1$ and structure parameters of wind turbines can be derived as

μ 1 = \frac{1}{κ} = \frac{4 λ_{opt}^{2}}{9 S w ρ 2 π 2 R 6 v 2 (C_{P}^{max}) 2}

(35)

Based on the two-mass model of wind turbines, the new measure is expressed as:

μ = \frac{1}{tr (W (t f) - 1 Λ) + tr (W (t f) - 1 Σ (t f))}

(36)

where

Λ = E (x 0 x_{0}^{T})

W (t f) = \int_{t 0}^{t f} e A (t 0 - τ) BB T e A T (t 0 - τ) d τ

, and

Σ (t f) = \int_{t 0}^{t f} e A (t 0 - τ) DS w D T e A T (t 0 - τ) d τ

. The matrix A, B and D are defined in (32) and (33).

Based on the similar assumption that the initial point x₀ is equal to the optimal reference $[ω ropt ω gopt T lsopt] T$ where

ω ropt = λ opt v / R = k 1 v, ω gopt = n g λ opt v / R = k 2 v T lsopt = \frac{1}{2} ρ π R 3 v 2 C_{p}^{max} / λ opt - K r λ opt v / R = k 3 v 2 - k 4 v

Λ in (36) can be calculated as

Λ = E ([ω ropt ω gopt T lsopt] [ω ropt ω gopt T lsopt]) = E ([k_{1}^{2} v 2 k 1 k 2 v 2 {k 1 v (k 3 v 2 - k 4 v)} k 1 k 2 v 2 k_{2}^{2} v 2 {k 2 v (k 3 v 2 - k 4 v)} {k 1 (k 3 v 2 - k 4 v)} {k 2 v (k 3 v 2 - k 4 v)} {(k 3 v 2 - k 4 v) 2}]) = [k_{1}^{2} E (v 2) k 1 k 2 E (v 2) {k 1 k 3 E (v 3) - k 1 k 4 E (v 2)} k 1 k 2 E (v 2) k_{2}^{2} E (v 2) {k 2 k 3 E (v 3) - k 2 k 4 E (v 2)} {k 1 k 3 E (v 3) - k 1 k 4 E (v 2)} {k 2 k 3 E (v 3) - k 2 k 4 E (v 2)} {k_{3}^{3} E (v 4) - 2 k 3 k 4 E (v 3) + k_{4}^{2} E (v 2)}]

The measure $μ 1$ in Kang et al., (2009) is expressed as

μ 1 = \frac{1}{tr (e - A T (t f - t 0) W (t f) - 1 Σ (t f) e A T (t f - t 0))}

(37)

where

W (t f) = \int_{t 0}^{t f} e A (t 0 - τ) BB T e A T (t 0 - τ) d τ

and

Σ (t f) = \int_{t 0}^{t f} e A (t 0 - τ) DS w D T e A T (t 0 - τ) d τ

. The matrix A, B and D are defined in (32) and (33).

From (34)-(37), it is observed that the DOC measures are relevant to structural parameters, including J_r, $λ opt$ , $B ls$ and $K ls$ . Since the control performance of wind turbines mainly depends on J_r and $λ opt$ (Johan et al., 2006; Chun et al., 2013), sensitivity analysis of the DOC measures with respect to J_r and $λ opt$ is discussed by simulations in section 3.4.

3.3. The variation of the MPPT efficiency versus structural parameters obtained through simulations on NREL’s FAST turbine model

The MPPT efficiency, which is commonly used in the design and optimization of wind turbines, is chosen as the performance index of the MPPT control. And the formula for calculating the MPPT efficiency is (Femia et al., 2005; Boubekeur and Houria, 2011)

η = \frac{\int_{0}^{t} P real d t}{\int_{0}^{t} P max d t} = \frac{\int_{0}^{t} v 3 (t) C p (λ (t)) d t}{\int_{0}^{t} v 3 (t) C_{p}^{max} d t}

(38)

where

P real

is the actual captured wind power,

P max

is the maximal captured wind power based on the blade element momentum theory, and t is the simulation time.

To calculate the MPPT efficiency, the simulation on CART3 wind turbine under turbulence is conducted by the FAST code (Jonkman and Buhl, 2005) and the MPPT control (i.e. the optimal torque control Kim et al., 2013) is applied.

Here, the flow chart for obtaining the relationship between the MPPT efficiency and structural parameters (i.e. J_r and $λ opt$ ) through FAST simulations is shown in Figure 5, respectively. Since the procedures for two parameters are similar, the one for J_r is described from Step 1 to Step 7 as an example.

Figure 5.

Flow chart for obtaining the relationship between the MPPT efficiency and structural parameters through FAST simulations.

Step 1: Initialization.

Step 1.1: Choose the CART3 turbine, the parameters has been shown in Table 1.

Step 1.2: The variation interval of J_r is set to $[4 \times 105, 7 \times 105]$ with an increasing step of $3 \times 103$ . Thus, an array of J_r is obtained.

Step 2: Choose a $J r i$ from the array.

Step 3: Obtain the mass of each blade-element corresponding to $J r i$ . In the CART3 turbine, the blade is divided into 22 blade-elements. $J r i$ can be modified by uniformly scaling the mass of each blade-element based on the initial blade-element.

Step 4: Substitute the mass of each blade-element into FAST model.

Step 5: Calculate the MPPT efficiency with respect to $J r i$ according to simulations on FAST model.

Step 6: If all J_r are calculated, go to Step 7. Otherwise, return to step 2.

Step 7: Export the MPPT efficiencies corresponding to different J_r. Then, the variation of the MPPT efficiency versus J_r is obtained and shown in section 3.4.

3.4. Simulation and verification

According to section 3.2 and 3.3, the relations of DOC measures and MPPT efficiency to structural parameters (i.e. J_r and $λ opt$ ) are respectively obtained. Then, by comparing the variation of DOC measures with that of MPPT efficiency, the DOC measure proposed in this paper is verified.

3.4.1. The relationship between the DOC measures and structural parameters

According to the expressions of DOC measures derived in section 3.2.2, the relations of the DOC measures to J_r and $λ opt$ are respectively calculated. Note that only one of the aforementioned parameters is changed while all the others are fixed to their initial design values listed in Table 1 and the equilibrium point is assumed to be the optimal rotor speed at a certain time. According to the constraints of the structural design of wind turbines, the variation interval of J_r is chosen as $[4 \times 105, 7 \times 105]$ , and that of $λ opt$ as [5.0, 9.0]. Simulation results are plotted in Figure 6–12.

The variation of the DOC measures versus J_r and $λ opt$ based on the single-mass model

Figure 6.

The relationship between the DOC $μ 1$ and optimum tip-speed ratio $λ opt$ .

Based on the single-mass model, the relations of the new measure μ proposed in the paper to J_r and $λ opt$ are plotted in Figure 7–8. It is observed from Figure 7–8 that the proposed measure μ increases with the decrease of J_r and $λ opt$ . However, as shown in Figure 6, the measure $μ 1$ proposed in Kang et al. (2009) is positively correlated with the $λ opt$ .

The variation of the DOC measures versus J_r and $λ opt$ based on the two-mass model

Figure 7.

The relationship between the DOC μ and rotor inertia J_r.

Figure 8.

The relationship between the DOC μ and optimum tip-speed ratio $λ opt$ .

Based on the two-mass turbine model, the relationships of the measure $μ 1$ proposed in Kang et al. (2009) to J_r and $λ opt$ are shown in Figure 9–10. Meanwhile, the variations of the new measure μ proposed in the paper versus J_r and $λ opt$ are depicted in Figure 11–12. It is seen from Figure 11–12 that the new measure μ increases with progressive decreasing of J_r and $λ opt$ . However, the measure $μ 1$ is positively correlated with the J_r and $λ opt$ .

Figure 9.

The relationship between the measure $μ 1$ and J_r.

Figure 10.

The relationship between the measure $μ 1$ and $λ opt$ .

Figure 11.

The relationship between the measure μ and J_r.

Figure 12.

The relationship between the measure μ and $λ opt$ .

3.4.2. The relationship between the MPPT efficiency and structural parameters

According to the flow chart presented in section 3.3, the variations of MPPT efficiency versus J_r and $λ opt$ are obtained, as plotted in Figure 13–14. It is observed from Figure 13–14 that the MPPT efficiency increases with progressive decreasing of J_r and $λ opt$ , which is coincided with the engineering observations (Johan et al., 2006; Chun et al., 2013) that a higher MPPT efficiency can be achieved by decreasing J_r and $λ opt$ .

Figure 13.

The relationship between MPPT efficiency and J_r.

Figure 14.

The relationship between MPPT efficiency and $λ opt$ .

3.4.3. The relationship between the DOC measures and MPPT efficiency

Based on the results obtained from section 3.4.1 and 3.4.2, the variations of DOC measures and MPPT efficiency versus structural parameters are further compared, as shown in Figure 15–22.

The relation of the DOC measures based on the single-mass model to MPPT efficiency

Figure 15.

The relationship between the DOC $μ 1$ and MPPT efficiency when $λ opt$ is changed.

First, the variation of the DOC measure $μ 1$ in Kang et al. (2009) to MPPT efficiency and the variation of the energy to $λ opt$ are depicted in Figure 15–16, respectively. It can be observed that the MPPT efficiency monotonously decreases with increasing of $μ 1$ (as shown in Figure 15). However, according to the definition 2 for the measure κ, the higher control performance should be obtained by increasing $μ 1$ . Obviously, the simulation results of the measure $μ 1$ are not agreed with the practical ones of wind turbines. These imply that the measure $μ 1$ is not suitable for the design and optimization of the structural parameters of wind turbines.

Figure 16.

The comparison between two kinds of energy for wind turbine system.

Furthermore, according to Figure 16, the energy for stabilizing performance, which is not considered in the definition of $μ 1$ , plays a primary role in the MPPT control of wind turbines. Therefore, the neglect of the energy for stabilizing the system results in the inapplicability of the measure of Kang et al. (2009) to wind turbines.

As shown in Figure 17–18, the new measure μ is positively correlated with the MPPT efficiency. Therefore, it can be concluded from Figure 13–14 and Figure 17–18 that the relation of the proposed measure to turbine parameters is coincided with the engineering observations (Johan et al., 2006; Chun et al., 2013) that a higher MPPT efficiency can be achieved by decreasing J_r and $λ opt$ . It is demonstrated that the new DOC measure is effective for wind turbines.

The relation of the DOC measures based on the two-mass model to MPPT efficiency

Figure 17.

The relationship between the DOC μ and MPPT efficiency when J_r is changed.

Figure 18.

The relationship between the DOC μ and MPPT efficiency when $λ opt$ is changed.

Similarly, based on the two-mass model, the variations of the measure $μ 1$ versus MPPT efficiency and the ones of the new measure μ are given in Figures 19–20 and Figures 21–22, respectively. As shown in Figures 21–22, the new measure μ is positively correlated with the MPPT efficiency. However, it can be observed from Figures 19–20 that the measure $μ 1$ is negatively correlated with the MPPT efficiency.

Figure 19.

The relationship between the DOC $μ 1$ and MPPT efficiency when J_r is changed.

Figure 20.

The relationship between the DOC $μ 1$ and MPPT efficiency when $λ opt$ is changed.

Figure 21.

The relationship between the DOC μ and MPPT efficiency when J_r is changed.

Figure 22.

The relationship between the DOC μ and MPPT efficiency when $λ opt$ is changed.

In summary, according to the above simulation results, the variation of the new DOC measure proposed in this paper is in accordance with the one of MPPT efficiency with change of turbine parameters. This indicates that the new DOC measure rather than the measure in Kang et al. (2009) is suitable for the design and optimization of structural parameters of wind turbines.

4. The application and verification of the new measure to Mass-spring-damper system

In this section, a mass-spring-damper system with a matched or unmatched disturbance (Kang et al., 2009) is analyzed, as shown in Figure 23.

Figure 23.

Three-degree-of-freedom mass-spring-damper system.

The state-space representation of the mass-spring-damper system is expressed as

x \cdot = Ax + Bu + Dw

(39)

where

x \in R 6

and

u \in R 1

are the state and control input, respectively.

w \in R 1

is a disturbance with covariance

E [w T] = 1

. By calculation, the system is controllable and asymptotically stable.

If the disturbance is matched, the matrix D is given by

D = [000010] T

and the DOC measure (17) is obtained as

κ = tr (Σ ¯ \cdot W ¯ - 1) = 6

(40)

If the disturbance is unmatched, the matrix D is given by

D = [000001] T

and the DOC measure (17) is equal to

κ = tr (Σ ¯ \cdot W ¯ - 1) = 234

(41)

Obviously, it is demonstrated by equations (40) and (41) that the measure in Kang et al. (2009) is suitable for the aforementioned mass-spring-damper system. As mentioned in Remark 5, because the DOC measure proposed in this paper is equivalent to the measure in Kang et al. (2009) for indicating the DOC of asymptotically stable systems at steady state, the former is also applicable to the mass-spring-damper system.

In addition, to validate Theorem 2, further analysis is performed on the same example. Without loss of generality, assume that the initial time t₀ is 0 and

E (x 0 x_{0}^{T}) = I

is satisfied. Then, the variation of

μ - 1

to the final time t_f is listed in Table 3.

Table 3.

The variation of $μ - 1$ to the final time t_f.

Final time t_f	0.01	0.1	1	10	100	∞
$μ - 1$ for the matched disturbance	2.6255 × 10²²	8.1026 × 10¹⁹	2.2765 × 10¹⁰	6.8643	6.0625	6
$μ - 1$ for the unmatched disturbance	2.6368 × 10²²	8.3153 × 10¹⁹	3.2468 × 10¹⁰	2.9920 × 10²	2.3406 × 10²	234

It is observed from Table 3 that when the final time t_f approaches infinity, $μ - 1$ tends to $tr (W ¯ - 1 Σ ¯)$ , This verifies the deduction of the new measure at steady state presented in Theorem 2.

5. Conclusions

In this paper, a new measure of DOC, in which the initial condition is denoted as a random vector, is proposed for linear system with external disturbance. Then, a wind turbine system and a mass-spring-damper system are given to verify the effectiveness of the proposed measure. For wind turbines, it can be observed from the simulation results that the variation of the proposed DOC measure to turbine parameters is consistent with that of MPPT efficiency with turbine parameters.

It is also observed that when using the same MPPT control, the higher MPPT efficiency can be achieved by the wind turbine with the larger value of DOC measure which can be obtained by reducing rotor inertia and optimum tip-speed ratio. This implies that the proposed DOC measure can be further employed to guide the design and optimization of the structural parameters of wind turbines.

Footnotes

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by National Natural Science Foundation of China (Grant No. 61174038, 61573186, 61203129, 61473151) and the Fundamental Research Funds for the Central Universities (30915011104, 30920140112005)

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