Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Volume 98　　　　　　　　　　　　　　　　　　　　　　　　　　　1960

On the Question of the Most Advantageous Aerodynamic Operating Mode of a Wind Power Plant (WPP)

V. M. Osipov

(Presented by Prof. Dr. of Technical Sciences I. D. Kutyavin)

By the most advantageous aerodynamic operating mode of a WPP is meant the establishment of such rotation speeds of the wind wheel at which the annual generation of electric energy will be maximal. At present this question is solved by comparing various calculation variants, which substantially complicates the analysis and makes such calculations very labor-intensive. Below a method is set forth for directly determining the parameters of the most advantageous mode.

Let us consider an ideal WPP having no losses in the conversion of mechanical energy into electrical energy. We shall assume that the characteristic of the wind motor is specified. Let us also suppose that the power of the WPP is limited by its installed (rated) value. The power of the WPP will be equal to [2]:

$$ P = C_0 D^2 v^3 \xi = \varphi (v_{\mathrm{okr}}, v), \ \mathrm{kW}. \tag{1} $$

The equation of the curve by duration (the decreasing integral distribution curve) is obtained [2] if, to the expression for power, we add the relation

$$ t = T \int_v^\infty f(v)\,dv. \tag{2} $$

In the mode with variable rotation speed, when \(v_{\mathrm{okr}} = z_n v\), and \(\xi = \xi_m\), the area under the curve has the maximum possible value. In the operating mode with constant rotation speed, by assigning various values of \(v_{\mathrm{okr}}\), one can obtain a family of duration curves. On each of these curves there is a single point at which \(\xi = \xi_m\), and consequently,

$$ P\big|_{v_{\mathrm{okr}}=\mathrm{const}} = P\big|_{v_{\mathrm{okr}}=v_{av}} . $$

This equality holds at the wind speed \(v = v_k = \dfrac{v_{\mathrm{okr}}}{z_n}\); at any other points (i.e., at any other wind speed) we shall have the inequality

$$ P\big|_{v_{\mathrm{okr}}=\mathrm{const}} < P\big|_{v_{\mathrm{okr}}=v_{av}} . $$

Obviously, the common point can only be a point of tangency and, consequently, the curve for continuity at \(v_{okr}=v_{ar}\) is the envelope of the family of curves at \(v_{okr}=\mathrm{const}\). We must find the most advantageous value of the peripheral velocity, i.e., from the entire family choose the curve that would bound the maximum area. This curve will correspond to a completely definite point of tangency and, consequently, the problem can be reduced to finding this point. This is the idea of the method that we shall apply. Let us represent the equation of the curves in the form

\[ t=\Phi(p). \]

Expanding this function in a power series at \(p=p_k\), we obtain

\[ t=t_k+\Phi'(p_k)\frac{p-p_k}{1}+\Phi''(p_k)\frac{(p-p_k)^2}{1\cdot2}+\ldots, \tag{3} \]

where \(t_k\) and \(p_k\) are the coordinates of the point of tangency at which \(\xi=\xi_m\). Since the function \(\Phi(p)\) is continuous, the resulting series will converge uniformly. The absolute value of the terms of the series will decrease, tending to zero; consequently, the series will be absolutely convergent. The annual energy output will be equal to

\[ A=\int_{0}^{p_p} t\,dp=\int_{0}^{p_p} t\,d(p-p_k)\ \text{kWh}, \tag{4} \]

Keeping expansion (3) in mind and carrying out the integration, we obtain

\[ A=t_k(p-p_k)+\Phi'(p_k)\frac{(p-p_k)^2}{1\cdot2} +\Phi''(p_k)\frac{(p-p_k)^3}{1\cdot2\cdot3}+\ldots\Bigg|_{0}^{p_p}. \tag{5} \]

Carrying out the idea stated above, we take the derivative with respect to \(p_k\) and set it equal to zero; then, after transformations, we obtain

\[ \frac{p-p_k}{p_k} = \sqrt{ \frac{ 1-\alpha(p_k)\dfrac{p_k}{3}+\ldots }{ 1+\alpha(p_k)\dfrac{p_p-p_k}{3}+\ldots }}, \tag{6} \]

where

\[ \alpha(p_k)= \frac{ \dfrac{d\Phi''(p_k)}{dp_k}-\Phi'''(p_k) }{ \dfrac{d\Phi'(p_k)}{dp_k}-\Phi''(p_k) }. \tag{7} \]

The greatest deviation of the peripheral velocity from the most advantageous (optimal) one within \(\pm 10\%\) does not lead to a noticeable decrease in the annual output [1]. This determines the zone of the most advantageous aerodynamic regime. The magnitude of the peripheral velocity will lie in the indicated zone if the radical expression varies over a very wide interval from 0.22 to 2.7. In view of this

Under these circumstances, the condition for maximum annual output can approximately be represented in the form

\[ \frac{p_p}{p_k}=\frac{2}{\left(1+\frac{\alpha}{6}-\frac{\alpha^2}{9}\right)^3}. \tag{8} \]

Hence, taking into account that \(p_k=c_0D^2v_k^3\xi_m\), we find the wind speed at which the coefficient of utilization of wind energy reaches its maximum value:

\[ v_k=\left(1+\frac{\alpha}{6}-\frac{\alpha^2}{9}\right)\sqrt[3]{\frac{p_p}{2c_0D^2\xi_m}} . \tag{9} \]

and, consequently, the optimal circumferential speed will be equal to

\[ v_{\mathrm{circ.\,opt.}}=\left(1+\frac{\alpha}{6}-\frac{\alpha^2}{9}\right)z_n\sqrt[3]{\frac{p_p}{2c_0D^2\xi_m}} . \tag{10} \]

When \(\alpha\) varies within the limits from \(-0.4\) to \(+0.7\), the value of \(v_{\mathrm{circ.\,opt.}}\) changes by only \(6\text{--}8\%\); therefore, in approximate calculations one may take

\[ v_{\mathrm{circ.\,opt.}}=z_n\sqrt[3]{\frac{p_p}{2c_0D^2\xi_m}} \tag{11} \]

and

\[ v_k=v_1=\sqrt[3]{\frac{p_p}{2c_0D^2\xi_m}} . \tag{12} \]

Thus, the most advantageous circumferential speed of rotation of the wind wheel is determined by the speed ratio of the wind motor, the rated power, the diameter of the wind wheel, and the maximum value of the coefficient of utilization of wind energy, and depends very little on the wind-speed distribution function (wind frequency). The latter circumstance is especially important, since it makes it possible to determine the most advantageous speed of rotation rather accurately without having data on wind frequency. A more exact value of \(v_{\mathrm{circ.\,opt.}}\) can be found if the dependence of \(\alpha\) on \(v_k\) is taken into account. Calculating successively the derivatives of the integral curve in parametric form according to the equation (i.e., from equations (1) and (2)) and substituting their value into equality (7), and also taking into account that \(\alpha=\alpha'(p_k)\dfrac{p_k}{3}\), we obtain, for a normal wind-speed distribution,

\[ \alpha=-\frac{2(v_k-v_{\mathrm{av}})v_k}{9s^2}+\frac{2z_n^2}{9(z_0-z_n)^2}. \tag{13} \]

For the Gullen distribution we find [2]:

\[ \alpha=\frac{2\left[av_k-3av_k\operatorname{tg}(av_k)\right]}{9\operatorname{tg}(av_k)} +\frac{2z_n^2}{9(z_0-z_n)^2}. \tag{14} \]

If these values of \(a\) are substituted into (9), we obtain a rather complicated equation, which will be difficult to solve. One can proceed otherwise, with practically no loss of accuracy. Since \(v_k\) differs very little from \(v_1\), one may compute \(a\) for \(v_k = v_1\) and substitute this value into (9) or (10). Knowing the circumferential speed, it is easy to obtain the values of the initial \((v_0)\) and rated \((v_p)\) wind speeds. In our case, when there are no losses,

\[ v_0 = \frac{v_{\mathrm{circ}}}{z_0}, \]

where \(z_0\) is the synchronous speed ratio.

\[ v_p = \frac{v_{\mathrm{circ}}}{z_p}. \]

However, \(z_p\) is unknown. From (6), taking into account the relation \(\dfrac{v_k}{v_p} = \dfrac{z_p}{z_n}\), we find

\[ \xi_p = \xi(z_p) = \frac{2 \xi_m}{\left(1+\dfrac{a}{6}-\dfrac{a^2}{9}\right)^3} \left(\frac{z_p}{z_n}\right)^3 . \]

The point of intersection of this curve with the wind-engine characteristic \(\xi(z)\) will determine \(\xi_p\) and \(z_p\). Let us now turn to the question of determining the annual output of a WPP in the most advantageous aerodynamic mode. From consideration of the duration curves it is seen that, with an increase in the mean annual wind speed, the curvature of the curves decreases, approaching zero; then the sign of the curvature for the initial part of the curve changes to the opposite, although the curve itself differs little from a straight line. This is observed for both theoretical distributions. At sufficiently high mean annual wind speeds, replacing the curves by straight lines will introduce a comparatively small error into the determination of annual output. Such a replacement is equivalent to taking the first two terms of series (3). If, however, four terms are taken, then high accuracy will be obtained in practically all cases. Thus, it may be considered that \(t = \Phi(p)\) is a complete cubic polynomial and, consequently, the theorem is applicable to integral (4), according to which

\[ A = \int_{0}^{p_p} \Phi(p)\,dp = \frac{p_p}{6} \left[ \Phi(0) + 4\Phi\left(\frac{p_p}{2}\right) + \Phi(p_p) \right] \]

or

\[ A = \frac{p_p}{6}\,[t_0 + 4t_1 + t_p]\ \mathrm{kWh}, \tag{15} \]

where \(t_0\), \(t_1\), and \(t_p\) are the durations corresponding to the wind speeds \(v_0\), \(v_1\), and \(v_p\), which may be determined by formula (2) or from the curves [2].

One can obtain a much simpler, but less accurate, formula for determining the annual output. In expansion (5), when

\[ p_k = \frac{p_p}{2}, \]

the first term will play the main role, i.e.,

\[ A \cong p_p t_1\ \mathrm{kWh}, \tag{16} \]

which is equivalent to replacing the curve, over its length, by a straight line that is tangent at the point $\left(\dfrac{P_p}{2}, t_1\right)$. The formula is suitable for preliminary calculations.

The formulas presented remain valid also in the case when the overall efficiency of the wind-power installation does not depend on wind speed; only the determination of the initial speed $v_0$ must be carried out with allowance for losses. In the case of variable losses the problem becomes substantially more complicated, but the proposed method is in principle applicable in this case as well. Analysis of this question shows that the most advantageous peripheral speed increases somewhat, and the increase is greater the larger the ratio of the nominal total efficiency of the wind-power installation to the value of the efficiency at $v = v_0$.

Conclusions

1. The most advantageous peripheral rotational speed of the wind wheel depends in practice only slightly on the form of the wind-speed distribution function and its parameters, and is determined by the tip-speed ratio of the wind engine, the rated power, and the value of the maximum coefficient of utilization of wind energy.

2. The proposed graphical-analytical method for calculating the parameters of the most advantageous aerodynamic regime of a wind-power installation makes it possible, with practically the same accuracy as the previous method, but much more rapidly, to carry out the calculation and to determine the influence of various factors on this regime. The relations obtained can be successfully used for calculating a regime different from the most advantageous one.

Literature

1. Pokataev A. I. Questions of parallel operation of wind-power installations with high-speed stabilizer-type wind engines through a synchronous generator into the grid. Dissertation. Moscow, 1952.

2. Osipov V. M. Some questions of wind-power calculations. Proceedings of the Tomsk Polytechnic Institute, vol. 94, TЭI, 1958.