THE MOST ADVANTAGEOUS RATIO BETWEEN THE INSTALLED CAPACITY OF A WIND POWER PLANT AND THE LOAD

V. M. Osipov

(Presented by Professor, Doctor of Technical Sciences I. D. Kutyavin)

With the broad use of wind energy for purposes of agricultural electrification, the principal mode of operation of wind power plants (WPPs) will be their operation in a system together with power plants of other types. In this connection, the question of the effective use of WPP energy in a system assumes great importance. The present work attempts to solve this question in a sufficiently general case. In the presence of local consumers, the ideal regime would be one in which at any moment in time the capacity of the WPP would be equal to the load capacity. In practice, such a regime cannot be created without imposing substantial restrictions on the load curve. In the general case, owing to the random character of WPP output, power flows will be observed in the networks of the system, the magnitude and direction of which depend on the ratio between the WPP output and the load at each given moment in time, and also on the location of the consumers in the system. In this case the annual load-duration curve for the power flowing in the transmission line through which the WPP, together with the local consumers, is connected to the system will be characterized by the presence of a negative region and will have the form shown in Fig. 1. The energy generated by the WPP will be fully used by the local consumers if its installed capacity is equal to or less than the minimum load. However, such a ratio cannot be regarded as advantageous, since in this case an incomparably larger part of the load curve will be supplied by the system, and the role of the WPP will be reduced to a minimum. Conversely, with an installed capacity considerably exceeding the maximum load, the greater part of the WPP energy will be delivered to the system for supplying distant consumers, which likewise cannot be regarded as a successful solution.

Thus, the criterion for the effective use of WPP energy may be the degree of its utilization by consumers located near the WPP installation. The greater the share of WPP energy in supplying the nearest consumers, the smaller the transmission of WPP energy into the system and, consequently, the higher the annual technical-and-economic performance indicators of its operation. If the system is divided into two parts in such a way that one of them contains the WPP with the nearby consumers, and the other contains everything else, then the magnitude of the power flows in the section of the line connecting these parts will characterize the share of WPP energy participation in covering the local load curve, i.e., the effectiveness of its utilization. In the general case, the curve of the power flowing in the indicated section of the line will have the form shown in Fig. 1. From what has been said it is clear that the condition

a criterion for the most advantageous ratio between the load and the installed capacity of the wind power plant may be the equality

\[ M+N=\min., \]

where \(M\) is the annual share of the system in covering the load curve;
\(N\) is the energy of the wind power plant not used by local consumers during the year.

Finding the minimum of the sum of these two energies in general form is practically impossible, since the load curves (distribution functions) are unknown. It must be noted, however, that the installed capacity of the wind power plant found in this way will not take into account by what means the reduction of the areas \(M\) and \(N\) is achieved.

The same reduction of these areas may be achieved both by reducing the duration of small powers and by cutting off the maxima. The second method is preferable, since it improves the technical-economic performance of the transmission line (losses are reduced, the quality of energy is improved). A quantity characterizing, on average, the magnitude of energy flows in the line and also taking into account to some extent the circumstance noted above may be the root-mean-square value of the power at the beginning of the transmission line \((P_{\text{l.n.r.m.s.}})\). The root-mean-square value of a random variable is nothing other than the root-mean-square deviation of the latter from its zero value. The smaller it is, the smaller the dispersion, the lower the probability of large deviations, and the smaller the magnitude of energy flows. The installed capacity of the wind power plant found from the condition of the minimum of \(P_{\text{l.n.r.m.s.}}\) (or \(P_{\text{l.n.r.m.s.}}^{2}\), which is the same) will, of course, differ from the installed capacity found from the condition of the minimum of the sum \(M+N\); however, this difference cannot be large, and, moreover, the circumstance noted above gives us the right to consider as the most advantageous the installed capacity found from the condition of the minimum of \(P_{\text{l.n.r.m.s.}}\). This problem can be solved without special difficulties, using the simplest methods of probability theory [1], as well as relations from the theory of wind-power calculations.

Fig. 1.

It should be noted that determining the installed capacity generally does not require great accuracy, since the final choice of it is made according to the scale of rated capacities, taking into account the prospective growth of loads.

Let us consider a sufficiently general case of a one-way connection of a wind power plant with a system in the presence of intermediate power take-offs. Let the number of take-offs be equal to two (Fig. 2). The power on the main section, without taking losses into account, will be equal to:

\[ P_{34}=P_1+P_2+P_3-P_{\text{WP}}=P_{\Sigma n}-P_{\text{WP}}, \tag{1} \]

where \(P_{\Sigma n}=P_1+P_2+P_3\) is the total load.

The power at the beginning of the main section (i.e., at the system buses) will obviously be equal to:

\[ P_c=P_{\Sigma n}-P_{\text{WP}}+\Delta P_{\Sigma} =P_{\Sigma n}-P_{\text{WP}}+\Delta P_{\Sigma p}+\Delta P_{\Sigma Q}, \tag{2} \]

where \(\Delta P_{\Sigma p}\) is the total losses in the transmission line due to active-power flows;

\(\Delta P_{\Sigma Q}\) is the same due to reactive-power flows.

Express the losses from active-power flows in terms of the loads and the electric power of the wind power plant \((P_{\mathrm{WPP}})\).

The power flows in the individual sections, without taking losses into account, will be equal to (Fig. 2):

\[ \left. \begin{aligned} P_{12} &= P_1 - P_{\mathrm{WPP}};\\ P_{23} &= P_1 + P_2 - P_{\mathrm{WPP}};\\ P_{34} &= P_1 + P_2 + P_3 - P_{\mathrm{WPP}}. \end{aligned} \right\} \tag{3} \]

The total losses from these flows will be proportional to the sum:

Fig. 2.

\[ \Delta P_{\Sigma p} \equiv P_{12}^{2} r_{12} + P_{23}^{2} r_{23} + \]

\[ + P_{34}^{2} r_{34} = P_{\mathrm{WPP}}^{2}(r_{12}+r_{23}+r_{34}) - \]

\[ - 2P_{\mathrm{WPP}}\left[P_1(r_{12}+r_{23}+r_{34}) + P_2(r_{23}+\right. \]

\[ \left. + r_{34}) + P_3 r_{34}\right] + K_p, \]

where

\[ K_p = P_1^2 r_{12} + (P_1+P_2)^2 r_{23} + (P_1+ \]

\[ + P_2 + P_3)^2 r_{34}. \]

Taking into account that

\[ r_{12}+r_{23}+r_{34}=r_{\mathrm{tot}}; \]

\[ r_{23}+r_{34}=r_{24}, \]

we can finally write:

\[ \left. \begin{aligned} \Delta P_{\Sigma p} &= \frac{ P_{\mathrm{WPP}}^{2} r_{\mathrm{tot}} -2\left[P_1 r_{\mathrm{tot}}+P_2 r_{24}+P_3 r_{34}\right]P_{\mathrm{WPP}} +K_p }{U_{\mathrm{av}}^{2}} \\[6pt] \text{Similarly,}\qquad \Delta P_{\Sigma Q} &= \frac{ Q_B^{2} r_{\mathrm{tot}} -2\left[Q_1 r_{\mathrm{tot}}+Q_2 r_{24}+Q_3 r_{34}\right]Q_B +K_Q }{U_{\mathrm{av}}^{2}}, \end{aligned} \right\} \tag{4} \]

where

\[ K_Q = Q_1^2 r_{12} + (Q_1+Q_2)^2 r_{23} + (Q_1+Q_2+Q_3)^2 r_{34}. \]

From (2) we obtain:

\[ P_c^2 = (P_{\Sigma n}-P_{\mathrm{WPP}})^2 +2(P_{\Sigma n}-P_{\mathrm{WPP}})(\Delta P_{\Sigma p}+\Delta P_{\Sigma Q}) +(\Delta P_{\Sigma p}+\Delta P_{\Sigma Q})^2. \]

The last term of this equality is very small in comparison with the first two terms, and it may be neglected; after this we obtain:

\[ P_c^2 = (P_{\Sigma n}-P_{\mathrm{WPP}})^2 +2(P_{\Sigma n}-P_{\mathrm{WPP}})(\Delta P_{\Sigma p}+\Delta P_{\Sigma Q}). \tag{5} \]

Taking into account the preceding equalities and carrying out a number of transformations, we shall have:

\[ P_c^2 = (P_{\Sigma n}-P_{\mathrm{WPP}})^2 +2\,\frac{ P_{\Sigma n} P_{\mathrm{WPP}}^{2} r_{\mathrm{tot}} - P_{\mathrm{WPP}}^{3} r_{\mathrm{tot}} - P_{\mathrm{WPP}}\left[3P_1^2 r_{\mathrm{tot}}+ }{U_{\mathrm{av}}^{2}} \]

\[ \frac{ +3P_2^2 r_{24}+3P_3^2 r_{24} +2P_1P_2(2r_{24}+r_{\mathrm{tot}}) +2P_1P_3(2r_{34}+r_{\mathrm{tot}}) +2P_2P_3(2r_{34}+r_{24})\right] }{U_{\mathrm{av}}^{2}} + \]

\[ +2\,\frac{ P_{\mathrm{WPP}}^{2}(P_1 r_{\mathrm{tot}}+P_2 r_{24}+P_3 r_{34}) }{U_{\mathrm{av}}^{2}} +2(P_{\Sigma n}-P_{\mathrm{WPP}})\Delta P_{\Sigma Q}. \]

Keeping in mind that the total load \((P_{\Sigma n})\) and the output of the wind power plant \((P_{WE})\) are independent random variables, we find the mathematical expectation of the quantity \(P_c^2\).

\[ \begin{aligned} M[P_c^2] = P_{c.av.ms}^2 &= P_{\Sigma n.av.ms}^2 - 2P_{\Sigma n.av}P_{WE.av} + P_{WE.av.ms}^2 \\ &\quad + 2(P_{\Sigma n.av} - P_{WE.av})\Delta P_{\Sigma Q,av} \\ &\quad + 2\,\frac{ P_{\Sigma n.av}P_{WE.av.ms}^{2}r_{tot} - P_{WE.av.cub}^{3}r_{tot} - P_{WE.av}M[S] }{U_{av}^{2}} \\ &\quad + \frac{ 2P_{WE.av.ms}^{2}\left(P_{1av}r_{tot}+P_{2av}r_{24}+P_{3av}r_{34}\right) }{U_{av}^{2}} . \end{aligned} \tag{6} \]

The letter \(S\) denotes the expression

\[ \begin{aligned} S &= 3P_1^2 r_{tot} + 3P_2^2 r_{24} + 3P_3^2 r_{34} + 2P_1P_2(2r_{24}+r_{tot}) \\ &\quad + 2P_1P_3(2r_{34}+r_{tot}) + 2P_2P_3(2r_{34}+r_{24}) . \end{aligned} \]

The mathematical expectation of the quantity \(S\) will obviously be equal to the sum of the mathematical expectations of the individual terms, i.e.,

\[ \begin{aligned} M[S] &= 3P_{1av.ms}^{2}r_{tot} + 3P_{2av.ms}^{2}r_{24} + 3P_{3av.ms}^{2}r_{34} \\ &\quad + 2M[P_1P_2](2r_{24}+r_{tot}) + 2M[P_1P_3](2r_{34}+r_{tot}) \\ &\quad + 2M[P_2P_3](2r_{34}+r_{24}) . \end{aligned} \]

There is a correlation relationship between the quantities \(P_1\), \(P_2\), and \(P_3\); therefore \(M[P_1P_2]\ne M[P_1]M[P_2]\), etc. \([1,2]\). However, taking into account the insignificant influence of the quantity \(M[S]\) on \(P_{c.av.ms}^{2}\), one may assume that

\[ \begin{aligned} M[S] &= 3P_{1av.ms}^{2}r_{tot} + 3P_{2av.ms}^{2}r_{24} + 3P_{3av.ms}^{2}r_{34} + 2P_{1av}P_{2av}(2r_{24}+r_{tot}) \\ &\quad + 2P_{1av}P_{3av}(2r_{34}+r_{tot}) + 2P_{2av}P_{3av}(2r_{34}+r_{24}) . \end{aligned} \tag{7} \]

The quantity \(P_{c.av.ms}^{2}\) is a function of the installed capacity of the wind power plant, since

\[ \left\{ \begin{aligned} P_{WE.av} &= \frac{1}{T}\int_{0}^{P_{inst}} t\,dP_{WE}, \\[4pt] P_{WE.av.ms}^{2} &= \frac{2}{T}\int_{0}^{P_{inst}} tP_{WE}\,dP_{WE}, \\[4pt] P_{WE.av.cub}^{3} &= \frac{3}{T}\int_{0}^{P_{inst}} tP_{WE}^{2}\,dP_{WE}. \end{aligned} \right. \tag{8} \]

Taking the last equalities into account, we take the first derivative of \(P_{c.av.ms}^{2}\) with respect to \(P_{inst}\) and set it equal to zero:

\[ \begin{aligned} \frac{dP_{c.av.ms}^{2}}{dP_{inst}} &= -2\frac{t_{inst}}{T}P_{\Sigma n.av} + 2\frac{t_{inst}}{T}P_{inst} + 2\frac{t_{inst}}{T} \frac{2P_{\Sigma n.av}P_{inst}r_{tot}}{U_{av}^{2}} \\ &\quad + 2\frac{t_{inst}}{T} \frac{ 4(P_{1av}r_{tot}+P_{2av}r_{24}+P_{3av}r_{34})P_{inst} - 3P_{inst}^{2}r_{tot} - M[S] }{U_{av}^{2}} \\ &\quad - 2\frac{t_{inst}}{T}M[\Delta P_{\Sigma Q}] = 0 . \end{aligned} \]

Reducing by \(2\dfrac{t_{ust}}{T}\), we obtain a quadratic equation with respect to \(P_{ust}\):

\[ A P_{ust}^{2} - B P_{ust} + C = 0, \]

where

\[ \left. \begin{aligned} A &= 3 \frac{r_{obshch}}{U_{sr}^{2}};\\ B &= 1 + \frac{2P_{\Sigma n.sr} r_{obshch}}{U_{sr}^{2}} + \frac{4\left(P_{1sr} r_{obshch} + P_{2sr} r_{24} + P_{3sr} r_{34}\right)}{U_{sr}^{2}};\\ C &= \frac{M[S]}{U_{sr}^{2}} + \Delta P_{\Sigma Qsr} + P_{vn.sr}. \end{aligned} \right\} \tag{9} \]

The roots of the equation are

\[ P_{ust1,2} = \frac{B \mp \sqrt{B^{2} - 4AC}}{2A}. \tag{10} \]

Consider the expression \(B^{2} - 4AC\):

\[ B^{2} - 4AC = \left\{ 1 + \left[ \frac{2P_{\Sigma n.sr} r_{obshch}}{U_{sr}^{2}} + \frac{4\left(P_{1sr} r_{obshch} + P_{2sr} r_{24} + P_{3sr} r_{34}\right)}{U_{sr}^{2}} \right] \right\}^{2} - \]

\[ - 12 \frac{P_{\Sigma n.sr} r_{obshch}}{U_{sr}^{2}} - 12 \frac{\Delta P_{\Sigma Qsr} r_{obshch}}{U_{sr}^{2}} - 12 \frac{M[S] r_{obshch}}{U_{sr}^{4}} . \]

The expression in square brackets is small in comparison with unity; therefore its square may be neglected. After expanding the braces, we obtain:

\[ B^{2} - 4AC \simeq 1 + 8 \frac{P_{1sr} r_{obshch} + P_{2sr} r_{24} + P_{3sr} r_{34}}{U_{sr}^{2}} - 8 \frac{P_{\Sigma n.sr} r_{obshch}}{U_{sr}^{2}} - \]

\[ - 12 \frac{\Delta P_{\Sigma Qsr} r_{obshch}}{U_{sr}^{2}} - 12 \frac{M[S] r_{obshch}}{U_{sr}^{4}} . \]

This expression is greater than zero; otherwise the dependence \(P_{s.sr.kv} = f(P_{ust})\) would have no minimum, which cannot be the case.

By the rule for approximate calculation of roots, one may obtain:

\[ \sqrt{B^{2} - 4AC} \simeq 1 - 4 \frac{P_{\Sigma n.sr} r_{obshch}}{U_{sr}^{2}} - 6 \frac{\Delta P_{\Sigma Qsr} r_{obshch}}{U_{sr}^{2}} - \]

\[ - 6 \frac{M[S] r_{obshch}}{U_{sr}^{4}} + 4 \frac{P_{1sr} r_{obshch} + P_{2sr} r_{24} + P_{3sr} r_{34}}{U_{sr}^{2}} . \]

Next we find:

\[ P_{ust1} = \frac{B - \sqrt{B^{2} - 4AC}}{2A} \simeq P_{\Sigma n.sr} + \Delta P_{\Sigma Qsr} + \frac{M[S]}{U_{sr}^{2}}, \tag{11} \]

\[ P_{ust2} = \frac{B + \sqrt{B^{2} - 4AC}}{2A} \simeq \frac{U_{sr}^{2}}{3r_{obshch}} + \frac{4}{3} \left[ P_{1sr} + P_{2sr}\frac{r_{24}}{r_{obshch}} + P_{3sr}\frac{r_{34}}{r_{obshch}} \right] - \]

\[ - \frac{2}{3} P_{\Sigma n.sr} - \Delta P_{\Sigma Qsr} - \frac{M[S]}{U_{sr}^{2}} . \tag{12} \]

We shall find the value of the second derivative under condition (10)

\[ \left. \frac{d^{2}P^{2}_{\text{a.a.wpp}}}{dP^{2}_{\text{inst}}} \right|_{P_{\text{inst}}=P_{\text{inst}1,2}} = 2\frac{t_{\text{inst}}}{T}\left(B-2AP_{\text{inst}}\right) \bigg|_{P_{\text{inst}}=P_{\text{inst}1,2}} . \]

For \(P_{\text{inst}}=P_{\text{inst}1}\) we have:

\[ \left. \frac{d^{2}P^{2}_{\text{a.a.wpp}}}{dP^{2}_{\text{inst}}} \right|_{P_{\text{inst}}=P_{\text{inst}1}} = 2\frac{t_{\text{inst}}}{T}\sqrt{B^{2}-4AC}>0. \]

For \(P_{\text{inst}}=P_{\text{inst}2}\)

\[ \left. \frac{d^{2}P^{2}_{\text{a.a.wpp}}}{dP^{2}_{\text{inst}}} \right|_{P_{\text{inst}}=P_{\text{inst}2}} = -2\frac{t_{\text{inst}}}{T}\sqrt{B^{2}-4AC}<0. \]

Thus, under condition (11) we have a minimum, and under (12) a maximum. The latter may be represented as the moment when the increase of \(P^{2}_{\text{a.a.wpp}}\) from increasing the installed capacity is entirely compensated by the growth of losses in the transmission line. Thus, the optimal installed capacity is determined by equality (11):

\[ P_{\text{inst.opt}} = P_{\text{l.a.}} + \Delta P_{\Sigma Q\text{av}} + \frac{M[S]}{U^{2}_{\text{av}}}. \]

It is not difficult to write the condition for the minimum for any number of intermediate power takeoffs. Let us note that in all expressions \(U_{\text{av}}\) is the average voltage of the transmission line. It is easy to see that the optimal installed capacity of the WPP is entirely independent of the form of the WPP power-distribution function (i.e., of the form of the wind-speed distribution function and of the annual average speed), and is determined mainly by the total annual average load. It can also be seen that the value of the second derivative

\[ \frac{d^{2}P^{2}_{\text{a.a.wpp}}}{dP^{2}_{\text{inst}}}, \]

calculated under condition (11), is determined by the magnitude of \(t_{\text{inst}}\) for a given load and line parameters, i.e., depends on the form of the wind-speed distribution function and on the annual average speed. As the latter increases, \(t_{\text{inst}}\) will grow, and, consequently, the minimum of \(P^{2}_{\text{a.a.wpp}}\) will be manifested more strongly. In other words, the degree of utilization of WPP energy by local consumers increases with the growth of the annual average wind speed.

The installed capacity of a WPP can be found not only from the condition of minimum energy transfers on the section of the line connecting the WPP and the local load with the system. In the general case, connection of the WPP substantially changes the power flows on all sections of the transmission line, and at a certain value of the installed capacity a minimum of the total losses from active-power flows \((\Delta P_{\Sigma})_{p}\) is attained. Although in high-voltage agricultural networks the relative magnitude of losses is large, the question of reducing them is not of great importance because of the insignificant absolute magnitude of these losses [3]. Nevertheless, it is of some interest to dwell on this question in somewhat greater detail.

Previously we found [see (4)]:

\[ \Delta P_{\Sigma p} = P^{2}_{\text{WPP}} r_{\text{tot}} - 2P_{\text{WPP}}\left[ P_{1}r_{\text{tot}}+P_{2}r_{24}+P_{3}r_{34} \right] + K_{p}. \]

The total annual average losses will obviously be proportional to

\[ \Delta P_{\Sigma p.\text{av}} = P^{2}_{\text{WPP.a.a.}} r_{\text{tot}} - 2P_{\text{WPP.av}}\left[ P_{1\text{av}}r_{\text{tot}}+P_{2\text{av}}r_{24}+P_{3\text{av}}r_{34} \right] + M[K_{p}]. \]

or, taking equalities (8) into account, we may write:

\[ \Delta P_{\Sigma p.c.p} = 2r_{\mathrm{tot}}\frac{1}{T} \int_{0}^{P_{\mathrm{inst}}} tP_{\mathrm{WE}}\,dP_{\mathrm{WE}} - \]

\[ -2\left[ P_{1cp}r_{\mathrm{tot}}+ P_{2cp}r_{24}+ P_{3cp}r_{34} \right] \frac{1}{T} \int_{0}^{P_{\mathrm{inst}}} t\,dP_{\mathrm{WE}} + M[K_p]. \]

Equating the first derivative of \(\Delta P_{\Sigma p.c.p}\) with respect to \(P_{\mathrm{inst}}\) to zero, we find:

\[ 2\frac{t_{\mathrm{inst}}}{T}P_{\mathrm{inst}}r_{\mathrm{tot}} - 2\frac{t_{\mathrm{inst}}}{T} \left[ P_{1cp}r_{\mathrm{tot}}+ P_{2cp}r_{24}+ P_{3cp}r_{34} \right] =0, \]

whence

\[ P_{\mathrm{inst}} = P_{1cp} + \frac{r_{24}}{r_{\mathrm{tot}}}P_{2cp} + \frac{r_{34}}{r_{\mathrm{tot}}}P_{3cp} \tag{13} \]

and in general, with \(n\) loads,

\[ P_{\mathrm{inst}} = \sum_{i=1}^{i=n+1} \frac{r_{i(n+1)}}{r_{1(n+1)}}P_{icp}. \tag{14} \]

The second derivative

\[ \frac{d^2\Delta P_{\Sigma p.c.p}}{dP_{\mathrm{inst}}^2} \]

will be proportional to the quantity \(2\frac{t_{\mathrm{inst}}}{T}>0\) under condition (13); consequently, the total annual losses have a minimum. The condition for the minimum, as before, does not depend on the form of the distribution function of wind speed, but is determined by the average annual loads and the active resistances of the sections of the tie line.

The minimum of energy losses, just like the minimum of \(P_{c.p.kv}^2\), becomes more pronounced as the average annual wind speed increases. Comparing the conditions for these minima, it is easy to see that the minimum of losses is attained at a considerably smaller value of the installed capacity of the wind power plant.

This, however, does not exhaust the compensating effect of the wind power plant on the energy losses of the tie. By acting on the excitation of the synchronous generator of the wind power plant, we can substantially influence the magnitude of the reactive-power flows in the line and, consequently, the magnitude of the losses from these flows. This question, however, is connected with voltage regulation at the terminals of the wind power-plant generator. In rural networks the ratio of active resistance to reactance (i.e., \(r/x\)) is sufficiently large; therefore it may turn out that the counterflow of reactive power of the wind power plant, necessary to maintain a specified voltage level, exceeds the available reactive power of the generator and regulation becomes impossible. In addition, when the period of minimum loads coincides with a period of strong winds (and this is always possible), increasing the power of the wind power-plant generator \((P_{\mathrm{WE}})\) in the presence of an excitation regulator will lead to a more rapid decrease in the reactive power \(Q_{\mathrm{B}}\). The generator will enter an underexcited mode, as a result of which the internal limit of the transmitted power will sharply decrease. The circumstances indicated give grounds to consider that voltage regulation at the terminals of the wind power-plant generator, even if it is possible, is undesirable from the standpoint of the stability of parallel operation. In the absence of a voltage regulator, the reactive power of the wind power-plant generator will remain constant throughout the year; its value may be chosen from the condition of minimum losses, although this question requires additional-

...in this study. Entirely analogously to the preceding case, it can be found that the minimum losses due to reactive-power flows with \(n\) withdrawals is attained under the condition:

\[ Q_{B}=\sum_{i=1}^{i=n+1}\frac{r_{i(n+1)}}{r_{1(n+1)}}Q_{i\mathrm{cp}}. \]

Thus, generally speaking, a wind power plant can substantially reduce energy losses in the transmission line. The relations obtained can be successfully used for the case when the parameters of the transmission line depend on the current flowing through it.

An experimental verification of the basic propositions and relations, carried out in the laboratory of the Tomsk Polytechnic Institute, has fully confirmed their correctness.

Conclusions

1. The optimum installed capacity of a wind power plant, found from the condition of minimum energy transfers over the section of line connecting the wind power plant and the local load with the system, is determined mainly by the total average annual load and is completely independent of the form of the wind-speed distribution function and of the average annual speed. The parameters of the load curves and the active resistances of the line sections have only an insignificant influence.

2. As the average annual wind speed increases, the degree of utilization of the energy of the wind power plant by nearby consumers increases.

3. A wind power plant has a significant compensating effect on energy losses in the transmission line. The installed capacity of the wind power plant, found from the condition of minimum energy losses, is determined by the average annual loads and the active resistances of the line sections, and is completely independent of the form of the wind-speed distribution function and of the average annual speed.

4. The degree of compensation depends on the average annual wind speed. As the latter increases, the compensation increases.

5. In view of the considerable magnitude of the ratio \(\dfrac{r}{x}\), the voltage regulator at the generator of a wind power plant impairs the stability of parallel operation and, from this point of view, is undesirable.

6. The reactive power of the wind power plant, found from the condition of minimum energy losses due to reactive-power flows, is determined by the average annual reactive load and the active resistances of the line sections.

Literature

1. Gnedenko B. V., A Course in Probability Theory, Gostekhizdat, Moscow, 1951.
2. Boev G. P., Probability Theory, Gostekhizdat, Moscow—Leningrad, 1950.
3. Zalesskii A. M., Transmission of Electric Energy, Gosenergoizdat, 1948.