Simplified equations for the rotational speed response to inflow velocity variation in fixed-pitch small wind turbines

The governing equation of rotational speed ω is as follows:

$$\begin{eqnarray}&&I\frac{{\rm d}\omega }{{\rm d}{{t}^{r}}}=T_{Q}^{r}-T_{L}^{r},\end{eqnarray} \tag{ 1 }$$

where I and t^r are the moment of inertia and dimensional time, and T_Q ^r and T ^r_L are the aerodynamic torque and load torque, respectively.

The aerodynamic torque T_Q ^r in the steady-state analysis is given as follows:

$$\begin{eqnarray}&&T_{Q}^{r}=\frac{1}{2}\rho \pi {{R}^{3}}\;{{C}_{Q}}(\lambda ,{\rm Re}){{U}^{2}},\end{eqnarray} \tag{ 2 }$$

where ρ, R, ${{C}_{Q}}(\lambda ,{\rm Re})$, and U are the air density, disk radius, coefficient of aerodynamic torque, and inflow velocity, respectively. ${{C}_{Q}}(\lambda ,{\rm Re})$ is a function whose variables are the tip-speed ratio λ and Reynolds number $\operatorname{Re}$, based on a chord length. The tip-speed ratio λ is defined as follows:

$$\begin{eqnarray}&&\lambda \equiv \frac{R\omega }{U}.\end{eqnarray} \tag{ 3 }$$

We only focus on the rotational speed response to a slow variation of inflow velocity. Pitch angle control, in which the pitch rate is a parameter characterizing the dynamic inflow effect (Snel and Schepers 1991), is not considered in this study. The present study focuses on how the rotational speed varies with the inflow velocity, and we assume that the flow field is quasisteady, as also assumed in previous studies (Karasudani et al 2007, 2008, 2010). Note that the steady-state analysis does not describe the phenomena by which the time scale is smaller than the time scale D/U, where D is disk diameter and is equal to 2R. In common small wind turbines, D/U is of the order of 0.01 ∼0.1 [s]. For example, the magnitude of the time scale in previous wind turbines discussed in a later section is approximately 0.02 ∼ 0.06 [s] (Snel and Schepers 1991, Manwell et al 2010). Note also that, since $c/U$ is smaller than $D/U$, the smaller time scale includes a time scale characterized by $c/U$ implicitly, where c is a chord length.

We focus on a case in which the tip-speed ratio deviates from a constant tip-speed ratio slightly. Because of this deviation, the magnitude of tip-speed ratio λ deviates slightly from the constant tip-speed ratio ${{\lambda }_{o}}$. We expand the torque coefficient in (2) as follows:

$$\begin{eqnarray}&&{{C}_{Q}}(\lambda )={{C}_{Q}}({{\lambda }_{o}})+{{\frac{{\rm d}{{C}_{Q}}}{{\rm d}\lambda }}_{{{\lambda }_{o}}}}(\lambda -{{\lambda }_{o}})+\frac{1}{2}{{\frac{{{{\rm d}}^{2}}{{C}_{Q}}}{{\rm d}{{\lambda }^{2}}}}_{{{\lambda }_{o}}}}{{(\lambda -{{\lambda }_{o}})}^{2}}+\cdot \cdot \cdot ,\end{eqnarray} \tag{ 4 }$$

where we focus on a condition satisfying $\partial {{C}_{Q}}/\partial \lambda \gg \partial {{C}_{Q}}/\partial ({\rm Re})$ using the condition that Re is larger than 10⁵. A previous study focused only on the first derivative term of the torque coefficient (Karasudani et al 2007). We introduce the second-order leading term additionally to examine the role of the term.

There are some merits to using the expansion for the tip-speed ratio. We introduce the expansion of the torque coefficient for the tip-speed ratio by focusing on these merits. The nonlinearity of the derived governing equation is reduced and the torque coefficient, which is included in the governing equation, consists of a high-degree polynomial or trigonometric function (e.g., Karasudani et al 2007, Munteanu et al 2008). Therefore, the governing equation can be nonlinear owing to the formulation of the torque coefficient. It will be difficult to yield a general solution of such a nonlinear governing equation. We consider that there is a possibility that the governing equation modified by focusing mainly on the differential coefficient can be more easily solved. Therefore, it can be anticipated that the general solution and/or spatial solutions are yielded from the modified governing equation.

There are several formulations of the torque coefficient with variable tip-speed ratios. The profile of the torque coefficient varies owing to wind turbine aerodynamics. This fact implies that the effects of the formulation of the torque coefficient on the rotational speed response will vary for each wind turbine. In other words, universal results cannot be given as long as the profile of the torque coefficient is individually used. Therefore, a general conclusion will not be reached as long as the formulation of the torque coefficient is the sole focus. We focus on the differential coefficient of the formula of the torque coefficient. Therefore, we consider that reaching a general conclusion is easier by focusing on the differential coefficient even if the functional forms of the torque coefficient differ from each other.

We consider an additional merit of using the expansion. Upon expanding the torque coefficient for the tip-speed ratio using the differential coefficient, the modified governing equation includes parameters based on the differential coefficient. By focusing on the term that consists of the parameter, we can discuss the roles of the parameter for essential natures such as the linearity/nonlinearity in the governing equation. In other words, we can discuss the essential role of the parameters based on a differential coefficient by introducing the expansion.

The proposed method using this expansion needs the C_Q profile in advance. Blade element momentum (BEM) is a common theory which can form C_Q profiles in a steady state. Using the BEM methodology, the C_Q profiles for the various conditions can be calculated. Accuracy of the BEM methodology is sufficient as used for fundamental research as well as for practical application.

For practical wind turbines, tip-speed ratio is often controlled to be optimal. When users only focus on wind turbines operated at the optimal tip-speed ratio without the use of the second-order derivative, there are solutions which become universal. In this study, an exact solution of rotational speed response to the step-change of inflow velocity is derived, as shown later. The gain of the linear solution is independent of the C_Q profile (31). This universal nature is an additional merit of the method we propose.

The constant tip-speed ratio ${{\lambda }_{o}}$ consists of the constant rotational speed ${{\omega }_{o}}$ and constant inflow velocity U_o. The constant inflow velocity U_o, which may often be taken to be rated, is not necessary to be rated. We introduce the deviation of ${{C}_{Q}}(\lambda )$ from that at constant tip-speed ratio (4). To formulate the deviation of the tip-speed ratio λ from ${{\lambda }_{o}}$, we additionally define each deviation of the rotational speed and inflow velocity. The deviations of the rotational speed and inflow velocity, ω and U, from their constant magnitude, ${{\omega }_{o}}$ and U_o, respectively, are defined as follows:

$$\begin{eqnarray}&&{{f}_{\omega }}\equiv \frac{\omega }{{{\omega }_{o}}}\ {\rm and}\ {{f}_{U}}\equiv \frac{U}{{{U}_{o}}},\end{eqnarray} \tag{ 5 }$$

where the magnitude of ${{f}_{\omega }}$ and f_U is of the order of unity, and these are nondimensional quantities. Using the above definitions, the tip-speed ratio λ is related to the constant tip-speed ratio ${{\lambda }_{o}}$ as follows:

$$\begin{eqnarray}&&\lambda =\frac{{{f}_{\omega }}}{{{f}_{U}}}{{\lambda }_{o}}.\end{eqnarray} \tag{ 6 }$$

As shown above, we formulate the definitions of the deviation as a relative quantity. Figure 1 shows the present flow model and quantities defined. We define the deviations as a relative quantity to improve the conciseness of the equation for the frequency response, as shown below.

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The relation between the present definition and an additive definition of deviation, which is found in previous studies, is clarified by introducing the following additional quantities:

$$\begin{eqnarray}&&f_{\omega }^{{\prime}}\equiv {{f}_{\omega }}-1\ {\rm and}\ f_{U}^{{\prime} }\equiv {{f}_{U}}-1.\end{eqnarray} \tag{ 7 }$$

Using these definitions, the relations are clarified as follows:

$$\begin{eqnarray}&&{{f}_{\omega }}=\frac{{{\omega }_{o}}}{{{\omega }_{o}}}+f_{\omega }^{{\prime} }\ {\rm and}\ {{f}_{U}}=\frac{{{U}_{o}}}{{{U}_{o}}}+f_{U}^{{\prime} },\end{eqnarray} \tag{ 8 }$$

where the above relation is similar to the Reynolds decomposition and the second terms are considered the additive terms of each derivation. Using the definitions for the deviations of the rotational speed and inflow velocity, the expansion of the torque coefficient is rewritten as follows:

$$\begin{eqnarray}&&{{C}_{Q}}(\lambda )={{C}_{Q}}({{\lambda }_{o}})+{{a}_{1}}{{\lambda }_{o}}\left( \frac{{{f}_{\omega }}}{{{f}_{U}}}-1 \right)+\frac{1}{2}{{a}_{2}}\lambda _{o}^{2}\left( \frac{f_{\omega }^{2}}{f_{U}^{2}}-2\frac{{{f}_{\omega }}}{{{f}_{U}}}+1 \right),\end{eqnarray} \tag{ 9 }$$

where

$$\begin{eqnarray}&&{{a}_{1}}\equiv {{\frac{{\rm d}{{C}_{Q}}}{{\rm d}\lambda }}_{{{\lambda }_{o}}}}\ {\rm and}\ {{a}_{2}}\equiv {{\frac{{{{\rm d}}^{2}}{{C}_{Q}}}{{\rm d}{{\lambda }^{2}}}}_{{{\lambda }_{o}}}}.\end{eqnarray} \tag{ 10 }$$

Here, a₁ and a₂ are constants defined as the first-order and the second-order differential coefficients at a constant tip-speed ratio.

It should be noted that the sign of a₁ will be negative at the optimal tip-speed ratio ${{\lambda }_{{\rm opt}}}$. At the optimal tip-speed ratio, the first-order derivative of the power coefficient is zero. Using the useful relation ${{C}_{p}}=\lambda {{C}_{Q}}$, the following is obtained:

$$\begin{eqnarray}&&{{\frac{{\rm d}{{C}_{P}}}{{\rm d}\lambda }}_{{{\lambda }_{{\rm opt}}}}}={{\lambda }_{{\rm opt}}}\;{{a}_{1}}+{{C}_{Q}}({{\lambda }_{{\rm opt}}})=0.\end{eqnarray} \tag{ 11 }$$

From the above relation, the sign of a₁ is considered negative because ${{C}_{Q}}\gt 0$ and ${{\lambda }_{{\rm opt}}}\gt 0$.

We formulate the load torque $T_{{\rm L}}^{r}$ by focusing on equation (12) of Karasudani et al (2007), and we introduce the following load torque:

$$\begin{eqnarray}&&T_{{\rm L}}^{r}=K\omega _{o}^{2},\ {\rm where}\ K=\frac{1}{2}\rho \pi {{R}^{2}}\frac{{{C}_{Q}}({{\lambda }_{o}}){{R}^{3}}}{\lambda _{o}^{2}},\end{eqnarray} \tag{ 12 }$$

where K is the constant gain of the load torque. Dividing

$$\begin{eqnarray}&&\frac{1}{2}\rho \pi {{R}^{3}}U_{o}^{2}\left( =\frac{1}{2}\rho \pi {{R}^{2}}U_{o}^{3}\times R/{{U}_{o}} \right),\end{eqnarray} \tag{ 13 }$$

the nondimensional load torque ${{T}_{{\rm L}}}$ is obtained as follows:

$$\begin{eqnarray}&&{{T}_{{\rm L}}}={{C}_{Q}}({{\lambda }_{o}}).\end{eqnarray} \tag{ 14 }$$

As shown in equation (14), the nondimensional form of ${{T}_{{\rm L}}}$ is equal to the coefficient of the aerodynamic torque at a constant tip-speed ratio.

In wind turbines in practical applications the load torque may vary with rotational speed. The fixed load torque which is proportional to the square of the rotational speed can maintain an optimal tip-speed ratio, and is conventional in small wind turbines. Load torque can change due to other factors, such as the temporal characteristics of the electricity grid. By modifying the form of the load torque the present method can include these effects. To include effects of the former, we rewrite the form of the load torque (equation (12)) to $T_{{\rm L}}^{r}=K{{\omega }^{2}}$, where definition of K is the same with the equation. Using nondimensionalization, the form, $T_{{\rm L}}^{r}=K{{\omega }^{2}}$, is rewritten as ${{T}_{{\rm L}}}={{C}_{Q}}({{\lambda }_{o}})f_{\omega }^{2}$. Note that the simplified governing equation with this load torque is more difficult to solve analytically because this load torque is nonlinear. When the effects of a sudden change of the load torque are included, a form of the load torque is unknown or will be too complicated to solve analytically. In this case we cannot seek the exact solution in the simplified equation. Thus, numerical simulation should be selected to analyze the effect of this kind of load torque.

There will be two forms of governing equations. The dimensional form of the governing equation generally has parameters such as inflow velocity U_o and rotor diameter R. The number of dimensional parameters to be set is larger than the number of nondimensional parameters to be introduced. By introducing nondimensionalization into the governing equation, the number of parameters to be set can be decreased, which helps to simplify the description of the phenomena focused on. Therefore, we derive the nondimensional form of the governing equation and focus on its merits.

The nondimensional parameters introduced into the governing equation are the constant tip-speed ratio ${{\lambda }_{o}}$ and energy ratio T_o, defined as follows:

$$\begin{eqnarray}&&{{T}_{o}}\equiv \frac{\frac{1}{2}\rho \pi {{R}^{2}}U_{o}^{3}\times R/{{U}_{o}}}{\frac{1}{2}I\omega _{o}^{2}}.\end{eqnarray} \tag{ 15 }$$

This parameter characterizes the magnitude of the effect of the inflow velocity on the rotational speed ${{\omega }_{o}}$ of the actuator disk. The numerator of T_o includes the energy flux incoming to the actuator disk, $\frac{1}{2}\rho \pi {{R}^{2}}U_{o}^{3}$, and characteristic time, $R/{{U}_{o}}$, defined by the characteristic velocity and length, U_o and R, respectively. Note that the former relates to the output power P of the wind turbine with power coefficient C_P as $P=\frac{1}{2}{{C}_{P}}\;\rho \pi {{R}^{2}}U_{o}^{3}$. The denominator of T_o, $\frac{1}{2}I\omega _{o}^{2}$, is the kinetic energy of the actuator disk with rotational speed ${{\omega }_{o}}$.

By introducing the nondimensional parameters, the number of parameters to be set is decreased. Table 1 compares the number of parameters to be introduced between dimensional and nondimensional forms of the equation. As listed in the table, the number of parameters can be decreased from five to two.

Table 1.  Comparison of the parameters without and with nondimensionalization, where $I,{{\omega }_{o}},R,{{U}_{o}}$, and ρ denote moment of inertia, constant rotational speed, disk radius, inflow velocity, and air density, respectively. T_o and ${{\lambda }_{o}}$ are the nondimensional parameters.

	number of parameters	parameters
without nondimensionalization	5	$I,{{\omega }_{o}},R,{{U}_{o}}$, and ρ
with nondimensionalization	2	To and ${{\lambda }_{o}}$

Table 2.  Value of parameters of Toshimitsu et al (2012) and Karasudani et al (2007). Notation of I, ω, R, U_o, and ρ is the same as in table 1, where $\rho =1.2\ {\rm kg}\;{{{\rm m}}^{-3}}$. ${{\lambda }_{{\rm opt}}}$ denotes the optimal tip-speed ratio

Case	$I\;[{\rm kg}\ \cdot \ {{{\rm m}}^{2}}]$	${{\omega }_{o}}\;[{\rm rad}\;{{{\rm s}}^{-1}}]$	$R\;[{\rm m}]$	${{U}_{o}}\;[{\rm m}\;{{{\rm s}}^{-1}}]$	${{T}_{o}}\;[-]$	${{\lambda }_{opt}}\;[-]$
Toshimitsu et al (2012)	1.52 × 10−4	257.7	0.097	5	0.022	2.5
Karasudani et al (2007)	0.1	78.6	0.7	11	0.0080	5.0

The value of T_o depends on both the specific parameters of the wind turbines and the parameters characterizing the wind turbine operation, U_o and ${{\omega }_{o}}$, respectively. Therefore, we define the ratio of the inertia moment,

$$\begin{eqnarray}&&{{r}_{{\rm im}}}\equiv \frac{\rho \pi {{R}^{3}}\times {{R}^{2}}}{I},\end{eqnarray} \tag{ 16 }$$

and we divide the form of T_o into the specific parameters of wind turbines and parameters characterizing wind turbine operation;

$$\begin{eqnarray}&&{{T}_{o}}=\frac{{{r}_{{\rm im}}}}{\lambda _{o}^{2}}.\end{eqnarray} \tag{ 17 }$$

We derive a nondimensional form of the modified governing equation because of the advantage expected. Using equations (1), (2) and (9) with the appropriate nondimensionalization, a modified equation for the rotational speed response is derived as follows:

$$\begin{eqnarray}&&\frac{{\rm d}{{f}_{\omega }}(t)}{{\rm d}t}=P_{eq}^{2}{{f}_{\omega }}{{(t)}^{2}}+P_{eq}^{1}(t){{f}_{\omega }}(t)+{{Q}_{eq}}(t),\end{eqnarray} \tag{ 18 }$$

where P²_eq, $P_{eq}^{1}(t)$, and Q_eq(t) are formulated as follows:

$$\begin{eqnarray}\begin{array}{rcl} P_{eq}^{2} & = & \frac{1}{4}{{a}_{2}}\lambda _{o}^{3}{{T}_{o}},\ P_{eq}^{1}(t)=\frac{1}{2}{{a}_{1}}\lambda _{o}^{2}{{T}_{o}}{{f}_{U}}(t),\ {\rm and} \\ {{Q}_{eq}}(t) & = & \frac{1}{2}{{\lambda }_{o}}{{T}_{o}}\left[ \left( {{C}_{Q}}({{\lambda }_{o}})-{{a}_{1}}{{\lambda }_{o}} \right)f_{U}^{2}(t)-{{C}_{Q}}({{\lambda }_{o}}) \right], \\ \end{array}\end{eqnarray} \tag{ 19 }$$

where t is the nondimensional time, defined as $t\equiv {{t}^{r}}{{U}_{o}}/R$.

Note that the modified equation for the rotational speed response is a linear equation when ${{a}_{2}}=0$. The linearity of this equation shows that the phenomenon of rotational speed response focusing only on a first-order differential coefficient is linear, and a solution would generally be derived for this phenomenon. Therefore, discussions of the phenomena of rotational speed response will be made significantly simpler by using the general solution expected. In contrast, when ${{a}_{2}}\ne 0$, the model equation displays the nonlinear nature of the rotational speed response. In this regard, the second-order differential coefficient will also play a significant role in rotational speed response.

Exact solutions will be helpful to analyze the characteristics of rotational speed of wind turbines. Solving the equation based on the first-order differential coefficient is much easier than solving the other equations, because the equation is linear. We can derive a solution of rotational speed response to the step-change of inflow velocity as shown in this work because the equation is linear, which facilitates an analytical solution.

The equation which is based on the first- and second-order differential coefficients is nonlinear. This equation is Riccatiʼs differential equation and may be solved analytically. However, solving Riccatiʼs differential equation is more difficult, and the solutions of Riccatiʼs differential equation are more complicated. Deriving the exact solutions of the time evolution of Riccatiʼs differential equation will not be suitable for wind turbines in practical application. Neglecting the rate of change of ${{f}_{\omega }}$, ${\rm d}{{f}_{\omega }}/{\rm d}t$, in the equation, Riccatiʼs differential equation results in a quadratic equation of ${{f}_{\omega }}$ and becomes easier to solve analytically. The application of this equation based on both differential coefficients will be limited to issues in which the rotational speed is steady.

The original equation is also nonlinear and does not have known forms of torque coefficient. In practical wind turbines there will be a variety of torque coefficients and so we should not attempt to seek exact solutions in the original equation because of this difficulty.

First we show the results of the temporal variation of rotational speed for the parameters of Toshimitsu et al (2012). The power coefficient profile of Toshimitsu et al (2012) is given for a real wind turbine for the use of wind tunnel testing. In this numerical simulation, there are two formulations of f_U which we set. The first formulation consists of a sinusoidal function (Karasudani et al 2010). In this case, f_U is formulated as follows,

$$\begin{eqnarray}&&{{f}_{U}}(t)=1+\Delta {{f}_{U}}(t)=1+\sqrt{2}{{A}_{U}}\;{\rm sin} \left( 2\pi t/T \right),\end{eqnarray} \tag{ 23 }$$

where $\Delta {{f}_{U}}(t)$, A_U, and T are the velocity variation and constant parameters. Note that the value $\sqrt{2}$ is introduced to equal A_U to the rms of f_U. The magnitude of A_U is set to be ${{A}_{U}}=0\sim 0.1$, and is intended to be the rms of realistic turbulent inflow. We set T = 500, 1000, and 2000, and their magnitudes are similar with those of Toshimitsu et al (2012). The second formulation of $\Delta {{f}_{U}}(t)$ consists of a random-number sequence whose probability density function is Gaussian. The rms of the normal-random-number sequence is set to be in the same range.

Figure 3 shows temporal variations of f_w(t) normalized by $\sqrt{2}{{A}_{U}}$. Note that we normalize f_w(t) by $\sqrt{2}{{A}_{U}}$ not by A_U because we focus mainly on the amplitude of the sinusoidal-like variation. As shown in figure 3(a), a temporal variation given by the original governing equation has a transient nature which is due to the initial magnitude; with initial magnitude of f_w set to be zero. This transient nature is found in temporal variations given by the other governing equations. This nature occurs in the other conditions, as shown in figures 3(b), (c), and (d). In a sufficiently large time, f_w(t) varies as a sinusoidal-like variation responding to the sinusoidal variation of the inflow velocity.

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The amplitude of the sinusoidal variation given by the original governing equation is about unity in figure 3(a). The amplitude given by the other governing equation is about unity and agrees with that obtained by the original governing equation. This result implies that amplitude of the sinusoidal variation is not affected by the nonlinear effect, which is first introduced by the second-order differential coefficient. This agreement is found in the other figures 3(b), (c) and (d). Thus, these results suggest that the insensitivity of the amplitude to the nonlinear effect holds for other inflow conditions.

In figure 3(a), there is no phase lag between temporal variations obtained by the original nonlinear equation and the other equations which are modeled by focusing on the differential coefficients. This result is also found in figures 3(b), (c), and (d). This result will be insensitive to flow conditions.

The mean value of the sinusoidal variation given by the linear equation is slightly larger than that given by the original governing equation but this difference can be reduced by adding the leading term based on the second-order differential coefficient. The governing equation based on the second-order differential coefficient is nonlinear and the difference shows that the mean magnitude is affected by the nonlinear effect of the governing equation. This difference is also found in figure 3(b). In contrast to figures 3(a) and (b), in the figures 3(c) and (d) the difference in the mean magnitude is small and can be negligible. From this result, the difference in the mean magnitude depends mainly on the magnitude of A_U.

We use a more realistic variation of inflow velocity to compare the rotational speed variation between the governing equations. Figure 4 compares the temporal variation of the rotational speed for the normal-random inflow velocity, where the rms of the normal-random inflow velocity is 0.01. As shown in the figure, temporal variations given by the modeled governing equations are similar to those given by the original governing equation, except for the magnitude of the mean value. The mean magnitude obtained by the linear equation may be found to be slightly larger than that given by the original governing equation. These results agree qualitatively with results for the sinusoidal variation of the inflow velocity.

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Nonlinearity is found only in the mean magnitude of the temporal variation of rotational speed as shown in the previous results. To quantify these results, we introduce three parameters: ${{A}^{+}}$, ${{A}^{-}}$, and $\Delta T$, where the magnitude of these parameters is calculated in a sufficiently large time to avoid transient effects. $\sqrt{2}{{A}^{+}}$ and $\sqrt{2}{{A}^{-}}$ are the local maximum and local minimum magnitudes, and $\Delta T$ is the phase lag between ${{f}_{\omega }}$ and f_U, respectively. Figure 5 illustrates the meaning of these parameters using a sinusoidal variation. The amplitude and mean value of the sinusoidal variation are simply formulated as follows;

$$\begin{eqnarray}&&{\rm amplitude}=\frac{{{A}^{+}}+{{A}^{-}}}{2}\ {\rm and}\ {\rm mean}\ {\rm value}=1+\frac{{{A}^{+}}-{{A}^{-}}}{\sqrt{2}}.\end{eqnarray} \tag{ 24 }$$

Note that the magnitude of the amplitude is equal to the rms of the sinusoidal variation because of the parameter $\sqrt{2}$ included in the relation.

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In the previous results, the amplitude is found to be insensitive to the nonlinear effect. Figure 6(a) shows rms amplitude profiles depending on the rms of variation A_U for three magnitudes of T, T = 500, 1000, and 2000. As shown in the figure, the rms amplitude increases with A_U for all magnitudes of T. The profiles given by the linear governing equation agree well with those given by the original governing equation in the whole magnitude of A_U, though there may be a slight deviation of the former from the latter. The slight deviation is decreased by adding the leading term based on the second-order differential coefficient for small magnitude of A_U. Thus, the slight deviation is due to the nonlinear effect. In the range of large magnitudes of A_U, the amplitude for the second-order differential coefficient could not be calculated because the rotational speed diverges. In this regard, the solution to the linear governing equation is more robust than that of the nonlinear governing equation. An additional term which is based on the third-order differential coefficient may be useful to restrain the divergence of the variation.

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The phase lag between ${{f}_{\omega }}$ and f_U is less sensitive to the nonlinear effect. As shown in figure 6(b), there is no difference in the phase lag between the original governing equation and the modeled governing equations. This result is found for all magnitudes of T.

In contrast to the results of the amplitude and phase lag, the mean value is affected by the nonlinear effect as noted in the previous results. Figure 7(a) shows the mean value which varies depending on A_U. As shown in the figure, the profiles obtained by the linear equation deviate from that obtained by the original governing equation. This deviation is decreased by adding a term of the second-order differential coefficient. This deviation is found for all magnitudes of T.

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The nonlinear effect on the mean value is also found in a more real condition. Figure 7(b) compares the mean value of ${{f}_{\omega }}$ between the three governing equations in a condition of normal-random inflow velocity. As shown in the figure, the results of the mean value are similar to that of the mean value for the sinusoidal variation.

The above results of the mean value suggest that the nonlinear nature of the governing equation affects the operating point of the tip-speed ratio significantly for large magnitudes of rms of inflow velocity variation. When rotational speed variation is discussed for the range of velocity rms, introducing the additional term will contribute to discussion of rotational speed response in larger magnitudes of the rms. For smaller magnitudes of inflow velocity rms, the governing equation focusing on the second-order differential coefficient as well as the first-order differential coefficient gives accurate results of the mean value. When the mean value as well as amplitude and phase lag are focused on, the linear equation may not be suitable to predict the rotational speed response even if the magnitude of inflow velocity rms is small, because the nonlinear effect is not included in the equation.

All analytical solutions can be given from the linear equation. This is one of the most notable advantages of the use of the linear equation. As shown in these results, the amplitude and phase lag can be predicted with sufficient accuracy even if the nonlinear effect is included. Therefore, the linear equation will be suitable to discuss the nature of rotational speed response when only the two quantities are the focus.

The response of ${{f}_{\omega }}(t)$ to a step input is a simple issue in addressing the rotational speed response. In this subsection, we address the rotational speed response to a step input of inflow velocity using the equation we derived.

The step input of f_U(t) is given as follows:

$$\begin{eqnarray}{{f}_{U}}(t)=\left\{ \begin{array}{ccccccccccccccc} 1 & (t\lt 0) \\ 1+\Delta {{f}_{U}} & (t\geqslant 0) \\ \end{array} \right.,\end{eqnarray} \tag{ 25 }$$

where $\Delta {{f}_{U}}$ is a constant parameter characterizing the magnitude of the step. The relative rotational speed ${{f}_{\omega }}(t)$ responding to the step input of f_U(t) is derived from the equation. A solution of ${{f}_{\omega }}$ is obtained as follows:

$$\begin{eqnarray}&&{{f}_{\omega }}(t)=1+\Delta {{f}_{U}}K\left[ 1-{\rm exp} \left( -\frac{t}{\tau } \right) \right],\end{eqnarray} \tag{ 26 }$$

where the gain K and time constant τ are derived as follows:

$$\begin{eqnarray}&&K=1-\frac{{{C}_{Q}}(2+\Delta {{f}_{U}})}{{{a}_{1}}{{\lambda }_{o}}(1+\Delta {{f}_{U}})}\ {\rm and}\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&\tau =-\frac{1}{(1+\Delta {{f}_{U}})}\frac{2}{{{a}_{1}}\lambda _{o}^{2}{{T}_{o}}}=-\frac{1}{(1+\Delta {{f}_{U}})}\frac{2}{{{a}_{1}}{{r}_{im}}}.\end{eqnarray} \tag{ 28 }$$

As derived in equations (26) and (28), the rotational speed response is derived as an exact solution of the modified governing equation.

The form of equation (26) is considered to be a form of first-order time lag. Thus, the governing equation derived by focusing on the first-order derivative term of the tip-speed ratio gives the rotational response of the first-order time lag. The solution of the rotational speed response has two parameters: gain K and time constant τ. The two parameters characterize the response of the rotational speed to the step inflow velocity. As shown in the above equation, the form of the time constant τ is rewritten to a form including an energy ratio T_o. As shown in equation (28), the form of the time constant is simplified using a parameter r_im for the formulation.

In this study we focus on the magnitude of the gain. As shown in equation (26), $\Delta {{f}_{\omega }}=\Delta {{f}_{U}}K$ when the presently derived equation is solved, where ${{f}_{\omega }}=1+\Delta {{f}_{\omega }}$. Figure 8 shows the results of $\Delta {{f}_{\omega }}$ as a function of $\Delta {{f}_{U}}$ for the cases of Toshimitsu et al (2012) and Karasudani et al (2007). Each exact solution agrees with that of the numerical result for the same governing equation. From this agreement, each exact solution can be verified. These exact solutions agree well with the numerical solution of the original nonlinear equation in the region of smaller $\Delta {{f}_{U}}$. In the region of large $\Delta {{f}_{U}}$, the difference between the exact solution and the numerical solution of the nonlinear equation is larger because the nonlinearity is not negligible, as is found in both cases. The results of the two cases are quite similar to each other.

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To simplify the discussion for the gain K, we additionally derive the first-order approximation of the exact solution for $\Delta {{f}_{U}}$. The first-order approximation of K for $\Delta {{f}_{U}}$ is derived as follows:

$$\begin{eqnarray}&&K=1-\frac{2{{C}_{Q}}}{{{a}_{1}}{{\lambda }_{o}}}+\frac{{{C}_{Q}}}{{{a}_{1}}{{\lambda }_{o}}}\Delta {{f}_{U}}+\cdot \cdot \cdot .\end{eqnarray} \tag{ 29 }$$

At the optimal tip-speed ratio, the following relation holds:

$$\begin{eqnarray}&&{\rm d}{{C}_{P}}/{\rm d}{{\lambda }_{{{\lambda }_{{\rm opt}}}}}={\rm d}(\lambda {{C}_{Q}})/{\rm d}{{\lambda }_{{{\lambda }_{{\rm opt}}}}}=0\Leftrightarrow {{C}_{Q}}=-{{a}_{1}}{{\lambda }_{{\rm opt}}}.\end{eqnarray} \tag{ 30 }$$

Thus, $-{{C}_{Q}}/{{a}_{1}}{{\lambda }_{{\rm opt}}}=1$ at the optimal tip-speed ratio. As listed in table 3, the magnitudes of $-{{C}_{Q}}/{{a}_{1}}{{\lambda }_{{\rm opt}}}$ of the two previous studies are unity. Using equation (30), equation (29) is rewritten as follows:

$$\begin{eqnarray}&&K={{K}_{o}}-\Delta {{f}_{U}}+\cdot \cdot \cdot ,\ {\rm where}\ {{K}_{o}}=3.\end{eqnarray} \tag{ 31 }$$

As shown in equation (31), the magnitude of K is approximately three.

It should be noted that the magnitude of the first-order approximation of K includes no parameters characterizing the rotational speed response; this term does not include T_o and ${{\lambda }_{{\rm opt}}}.$ This result suggests that the magnitude of the gain K will not depend on T_o and ${{\lambda }_{{\rm opt}}}$ and is approximately three for this type of common wind turbine.

From equation (26), to obtain the result of the tip-speed ratio from the initial value of the constant tip-speed ratio for $t\to \infty$, the following is needed using equation (6):

$$\begin{eqnarray}&&\lambda =\frac{1+\Delta {{f}_{U}}K}{1+\Delta {{f}_{U}}}{{\lambda }_{o}}={{\lambda }_{o}}\Leftrightarrow K=1.\end{eqnarray} \tag{ 32 }$$

As derived above, the magnitude of the gain K should be unity to return the tip-speed ratio to the constant tip-speed ratio. As shown in equation (31), the magnitude of the gain K is not equal to unity. This deviation suggests that the tip-speed ratio cannot result in the initial magnitude of the constant tip-speed ratio.

When the magnitude of the constant tip-speed ratio is not optimal, $-{{C}_{Q}}/{{a}_{1}}{{\lambda }_{o}}$ is not equal to unity. In this case, $-{{C}_{Q}}/{{a}_{1}}{{\lambda }_{o}}$ is formulated as $-{{C}_{Q}}/{{a}_{1}}{{\lambda }_{o}}$ = $1-\left( {\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}} \right)\times {{\left( {{\lambda }_{o}}{\rm d}{{C}_{Q}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}} \right)}^{1}}$. Therefore, the magnitude of the gain K_o at which the tip-speed ratio is not optimal is given as follows:

$$\begin{eqnarray}&&{{K}_{o}}=3-2\frac{{\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}}{{{\lambda }_{o}}\;{\rm d}{{C}_{Q}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}}.\end{eqnarray} \tag{ 33 }$$

As shown above, the profile of C_P and C_Q affects the magnitude of K_o, which is equal to that of K for zero $\Delta {{f}_{U}}$. This formulation implies that the profile of C_P and C_Q affects the magnitude of change of the rotational speed owing to the change in the inflow velocity.

The magnitude of ${{\lambda }_{o}}$ giving ${\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}$ is not equal to that giving ${\rm d}{{C}_{Q}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}$. Therefore, the profiles with ${{\lambda }_{o}}$ determine whether K_o is smaller than three. In the regions where ${\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}\gt 0$ and ${\rm d}{{C}_{Q}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}\gt 0$ and ${\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}\lt 0$ and ${\rm d}{{C}_{Q}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}\lt 0$, the magnitude of K_o is smaller than three because the ratio of the differential coefficients in the second terms is positive. In contrast, the magnitude of K_o is smaller than three in the region of ${\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}\gt 0$ and ${\rm d}{{C}_{Q}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}\lt 0$ because the ratio is negative. This discussion is summarized in table 4.

Table 4.  Magnitude of K_o and sign of ${\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}/({{\lambda }_{o}}{\rm d}{{C}_{Q}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}})$, which are characterized by sign of ${\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}$ and ${\rm d}{{C}_{Q}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}$, where ${{K}_{o}}$ = $3-2\left[ {\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}/({{\lambda }_{o}}{\rm d}{{C}_{Q}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}) \right]$.

Region	Value of Ko	sign of $\frac{{\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}}{{{\lambda }_{o}}{\rm d}{{C}_{Q}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}}$	sign of $\frac{{\rm d}{{C}_{p}}}{{\rm d}\lambda }{{\mid }_{{{\lambda }_{o}}}}$	sign of $\frac{{\rm d}{{C}_{Q}}}{{\rm d}\lambda }{{\mid }_{{{\lambda }_{o}}}}$
Region (I)	${{K}_{o}}\lt 3$	positive	positive	positive
Region (II)	${{K}_{o}}\gt 3$	negative	positive	negative
Region (III)	${{K}_{o}}\lt 3$	positive	negative	negative

Karasudani et al (2007) discussed the rotational speed response to inflow velocity variation with positive magnitude of the first-order differential coefficient of the torque coefficient. They noted that unrealistic flow field phenomena needed to be assumed for converging rotational speed with positive magnitude of a₁. The sign of a₁ in region (I) is positive. Thus, this previous discussion suggests that region (I) may not be suitable for wind turbine operation. The first-order differential coefficient of torque coefficient, a₁, is positive in this region. For positive magnitude of a₁, the magnitude of the time constant is negative, as shown in equation (28). The second term of equation (26) diverges when the time constant is negative. Thus, the present wind turbine will not operate properly in region (I).

In contrast to the rotational speed response in region (I), the sign of a₁ in region (III) is negative. Thus, the second term of equation (26) will converge as t increases. Karasudani et al (2007) showed the similar nature of the rotational speed response for a negative value of a₁. Therefore, region (III) is appropriate for the operation of small wind turbines.

In realistic situations, ${\rm d}{{C}_{Q}}/{\rm d}\lambda$ can be zero. Thus, the second terms of equation (33) cannot be calculated for all magnitudes of ${{\lambda }_{o}}$. Thus, we rewrite equation (33) as follows:

$$\begin{eqnarray}&&{{K}_{o}}=3-2\frac{{\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}}{{\rm d}{{C}_{P}}/{\rm d}\lambda {{\mid }_{{{\lambda }_{o}}}}-{{C}_{Q}}({{\lambda }_{o}})}.\end{eqnarray} \tag{ 34 }$$

The above relation makes it easier to calculate the magnitude of K_o for all magnitudes of ${{\lambda }_{o}}$. Figure 9(a) shows Toshimitsu et alʼs (2012) and Karasudani et alʼs (2007) profiles of K_o. As shown in the figure, the magnitude of both profiles is smaller than three in the region where the tip-speed ratio is larger than the optimal tip-speed ratio. In addition, both profiles of K_o for the two previous works decrease as ${{\lambda }_{o}}/{{\lambda }_{{\rm opt}}}$ increases.

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This decreasing value of K_o with ${{\lambda }_{o}}/{{\lambda }_{{\rm opt}}}$ is qualitatively related to the profile of T_o. As shown in equation (17), the value of T_o decreases as ${{\lambda }_{o}}/{{\lambda }_{{\rm opt}}}$ increases (figure 9(b)), where we rewrite T_o as ${{T}_{o}}={{r}_{im}}/({{\lambda }_{{\rm opt}}}^{2}{{({{\lambda }_{o}}/{{\lambda }_{{\rm opt}}})}^{2}})$. As shown in the figure, the profile of K_o at large tip-speed ratio is qualitatively related to the profile of T_o.

Figure 10 shows the temporal variation of the rotational speed respondse to a step input of the inflow velocity, where $\Delta {{f}_{U}}=0.01$. As shown in this figure, the results of the exact solution, which are verified by numerical results in the figure, agree well with those of the nonlinear original equation. The magnitude of the difference is relatively around 5%. The results of the exact solution for a small magnitude of $\Delta {{f}_{U}}$ are also shown in the figure. The difference in the magnitude of the exact solutions is sufficiently small and can be negligible.

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In a linear-time-invariant system, the time constant plays a significant role because the form of the output response to the input is formulated based on an exponential function including the time constant. The time constant remains a useful parameter even if the system in question is not completely linear-time-invariant. The time constant has been considered a parameter characterizing the response to an input.

The difference of the rotational speed response between the two cases is characterized by the difference in the time constant. The magnitudes of the time constant (equation (28)) for Toshimitsu et al (2012) and Karasudani et al (2007) are approximately $3.9\times {{10}^{2}}$ and $6.6\times {{10}^{2}}$, respectively. The magnitude of the time constant in Toshimitsu et al (2012) is smaller than that in Karasudani et al (2007).

To simplify the form of the time constant (equation (28)), the form is also approximated as follows:

$$\begin{eqnarray}&&\tau =-\frac{2}{{{a}_{1}}\lambda _{o}^{2}{{T}_{o}}}\left( 1+\Delta {{f}_{U}}+\cdot \cdot \cdot \right).\end{eqnarray} \tag{ 35 }$$

In contrast to the form of the gain K, the time constant depends on three parameters: ${{a}_{1}},{{T}_{o}}$, and ${{\lambda }_{o}}$. As shown in equation (35), the magnitude of the first derivative of the torque coefficient directly characterizes the magnitude of the time constant.

Karasudani et al (2007) derived a form of the time constant as follows:

$$\begin{eqnarray}&&\tau _{u}^{r}=\frac{I{{\lambda }_{o}}}{{{C}_{Q}}\rho \pi {{R}^{3}}{{U}_{o}}}.\end{eqnarray} \tag{ 36 }$$

We attempt to relate the present form of the time constant (equation (28)) to the previous form. We rewrite this form with a slight modification, which focuses on $\frac{1}{2}\rho \pi {{R}^{3}}U_{o}^{2}$ rather than on $\rho \pi {{R}^{3}}U_{o}^{2}$, as follows:

$$\begin{eqnarray}&&{{\tau }_{u}}\equiv \frac{4{{U}_{o}}}{R}\tau _{u}^{r}=\frac{2\frac{1}{2}I\omega _{o}^{2}}{{{C}_{Q}}{{\lambda }_{o}}\frac{1}{2}\rho \pi {{R}^{3}}U_{o}^{2}}=\frac{1}{{{C}_{Q}}}\frac{2}{{{\lambda }_{o}}{{T}_{o}}},\end{eqnarray} \tag{ 37 }$$

where ${{\tau }_{u}}\;[-]$ is a nondimensional form of the time constant with the characteristic quantities in Karasudani et al (2007). Using equation (30), equation (37) at the optimal tip-speed ratio can be rewritten as follows:

$$\begin{eqnarray}&&\tau =-\frac{1}{{{a}_{1}}{{\lambda }_{{\rm opt}}}}\frac{2}{{{\lambda }_{{\rm opt}}}{{T}_{o}}}.\end{eqnarray} \tag{ 38 }$$

This form of the time constant is the same as that of equation (28) with the condition that $\Delta {{f}_{U}}=0$. The present time constant is the same as the time constant for small magnitudes of $\Delta {{f}_{U}}$ in Karasudani et al (2007). In addition, the Karasudani et al (2007) time constant is noted to be a specific case of the present time constant with sufficiently small magnitude of velocity variation $\Delta {{f}_{U}}$. As mentioned in this subsection, the previous form of the time constant is a specific case of the present form of the time constant.

The time constant is related to the power of wind turbines. At the optimal tip-speed ratio, the present form of the time constant is rewritten for zero magnitude of $\Delta {{f}_{U}}$ as

$$\begin{eqnarray}&&\tau =-\frac{2}{{{a}_{1}}{{r}_{im}}}=-\frac{1}{{{a}_{1}}{{\lambda }_{{\rm opt}}}}\frac{2}{{{\lambda }_{{\rm opt}}}{{T}_{o}}}=\frac{2}{{{C}_{P}}({{\lambda }_{{\rm opt}}}){{T}_{o}}}.\end{eqnarray} \tag{ 39 }$$

As shown above, the time constant is directly related to the power coefficient C_P. Using a dimensional form of the time constant, ${{\tau }^{r}}$, from the above form of τ, the following is obtained:

$$\begin{eqnarray}&&P\times {{\tau }^{r}}=2\times \frac{1}{2}I\omega _{o}^{2}\Leftrightarrow T_{Q}^{r}\times {{\tau }^{r}}=I{{\omega }_{o}},\end{eqnarray} \tag{ 40 }$$

where P is the power of a wind turbine, and there exists the relation $P=T_{Q}^{r}{{\omega }_{o}}$. Thus, from the above relation, the form of the time constant is interpreted to imply an integral time relating aerodynamic characteristics to characteristics of the rotating rotor of a wind turbine.