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If in the expression (2) in order to replace the shifted copy of voltage U(t + x) with the shifted copy of

current I(l+x), we will get the mutual correlation function of voltage with the current:

*„,<«■>=«[<>(0 ■/(<+«■)] =

/((+t)-»,6*= (io) .l\v«yi(t+z)dt-- "% •/(>«)* +

Similarly, if in the expression (2) to replace the shifted copy of current I(t +x) with the shifted copy of

voltage U(l +x), we will get the mutual correlation function of current with the voltage:

ir,„(r)=Ju|</(<+r)-/( rtj. (11)

At a temporary shift change eąual to the zero, formulas (10) and (11) determine energy of voltage and current interaction, expressed by mutual correlation functions:

Kli (* = 0) = yjV(0 • I(t)dt-m_v -m,, (12)

K_lu(x = 0)=-jl(t) U(t)dt-m, m_u . (23)

At the same time it is also known that according to the theoretical electrical engineering, active-power D is determined as arithmetical mean of instantaneous power p(l) = U(t) ■ I(t) for the interval [0, / ]:

P4)

Comparing expressions (12) and (13) with the formula (14) we come to the conclusion that active-power £) is determined by the mutual correlation function of voltage with a current K_ai(x) or current with voltage Km (T) at a temporary shift X = 0 as

P = K_ul(x = 0) + m_um_l.    (15)

Then inactive power after Fryzę will be written as

J{k_l, (t = 0) + ml, )x x(*r,(l = 0) + m,^!)--[ir„(T = 0) + m„ m,Y

(A,₊KUD,+»i)-    ₍₁₆₎

- \K_U (x = 0) + m_v ■ m, ]²

3. Results of numeral calculations and their ana-lysis

With the use of the above-mentioncd correlations productive and non-productive losses of electric power were estimated in the wires of traction network of feeder zones in the areas of one of Ukrainian railways electrified on a direct current (+3.0 kB). With this purpose we have got time dependences of the rectified feeder voltage and currents, performed their cross-correlation-spectral treatment and determined electric indexes and losses of electric power A W_b and A , daily dependences of which (in one of areas) are presented in fig.2 but absolute and relative values in three areas are given in Table.

Fig. 1 - Daily average (for one month) values of basie A W_b aiłd additional AfK_ld losses of electric power in a traction network of one of feeder areas

Fig. 2 and Table 1 shows that daily average values of non-productive losses of electric power, at first, exceed productive losses A W_b by many days. Secondly, non-productive losses are 5.62...9.93 % (and maximal values - up to 15 %) of consumed (active) energy W by electro-mobile composition and up to 140 % in comparison with productive losses. Conseąuently, non-productive (additional) losses exceed the productive losses of electric power almost by 40%.

POJAZDY SZYNOWE NR 3/2011

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