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3
Electrotechnical Complexes and Systems
UDC 621.763: 621.74.047 https://doi.org/10.20998/2074-272X.202L5.02
P.D. Andrienko, O.V. Nemykina, A.A. Andrienko, R.E. Mokhnach
RESEARCH OF OPERATING MODES OF CONDUCTORS IN POWER SUPPLY SYSTEMS OF CRANES WITH INDUCTION FEED, TAKING INTO ACCOUNT THE INFLUENCE OF HIGHER HARMONICS OF THE CURRENT
Purpose. Investigation of the influence of higher harmonics of current on current distribution, voltage and power losses in the supply systems of crane trolleys and development of a calculation method for practical use. Methodology. The analytical method and the results of the modeling method were used for research. Results. Analytical relationships have been obtained that make it possible to determine the current distribution, voltage and power losses in the systems of induction feeding of crane trolleys, taking into account the composition and amplitude of the higher harmonics of the current. Originality. For the first time, analytical dependences are obtained that take into account the effect of changing the trolley parameters on the frequency in the feed systems. Numerical values have been determined for the most commonly used induction feed systems for cranes. It is shown that with an increase in the cross-section of the feed bar there is a decrease in the main, and especially additional, losses. Practical value. Theoretical relationships have been obtained that can be used to calculate the optimization of induction feed systems in the presence of higher harmonic currents arising in power systems during operation of crane semiconductor controlled electric drives. References 13, tables 4, figures 6.
Key words: induction feed system, trolleys, feed bus, current distribution, power and voltage losses.
У cmammi викладена методика розрахунку розподту струму по струмопроводам, втрат напруги i потужностi з урахуванням вищих гармотк струму в системах живлення кратв з тдукцтним тдживленням. Отриман необхiднi аналтичш залежностi, що пов 'язують параметри струмопроводiв з вiдносними значеннями частоти вищих гармоншних i визначають Их вплив на струморозподт, втрати напруги та потужностi. Показано, що зi збтьшенням перетину шин тдживлення вiдбуваeться зниження втрат напруги i додаткових втрат, в тому чиmi i вiд струмiв вищих гармотк, за рахунок перерозподыу цих струмiв i втрат вiд них в шину подачi, що мае практично незалежний вiд частоти активний отр. Показано, що основна частина додаткових втрат визначаеться амплтудами гармотк з порядком n<7. Методика може бути застосована для систем живлення залiзничного транспорту i розподтьних систем, виконаних з застосуванням сталемiдних i сталеалюмтевих струмопроводiв. Бiбл. 13, табл. 4, рис. 6.
Ключовi слова: система шдукцшного живлення, тролея, шина живлення, розподш струму, втрати потужност та напруги.
В статье изложена методика расчёта токораспределения по токопроводам, потерь напряжения и мощности с учётом высших гармоник тока в системах питания кранов с индукционной подпиткой. Получены необходимые аналитические зависимости, связующие параметры токопроводов с относительными значениями частоты высших гармонических и определяющие их влияние на токораспределение, потери напряжения и мощности. Показано, что с увеличением сечения шин подпитки происходит снижение потерь напряжения, потерь мощности, в том числе и от токов высших гармоник, за счёт перераспределения этих токов и потерь от них в шину подпитки, обладающей практически независимым от частоты активным сопротивлением. Показано, что основная часть добавочных потерь определяется амплитудами гармоник с порядком n<7. Методика применима для систем питания железнодорожного транспорта и распределительных систем, выполненных с применением сталемедных и сталеалюминиевых токопроводов. Библ. 13, табл. 4, рис. 6.
Ключевые слова: система индукционной подпитки, троллея, шина подпитки, токораспределение, потери мощности и напряжения.
Introduction. Energy saving in electrical networks is a priority area, both worldwide and in Ukraine. The widespread introduction of semiconductor converters leads to an increase in higher harmonics of current and voltage distortion, which increases voltage and power losses in electrical networks and leads to a deterioration in power quality indicators [1-3], and also has a significant impact on the operation of converters connected to these networks [4]. Determination of the composition and amplitude of the higher harmonics of the current is carried out by calculation, experimental and modeling methods [5-9]. To determine the influence of higher current harmonics on the supply network, it is necessary to know the parameters of the equivalent circuit. For low-voltage workshop networks, the values of active and inductive resistances are determined mainly by an analytical method. For complex wired conductor systems containing ferromagnetic elements and protective shields, analytical calculations are difficult. For these cases, modeling methods are used [5, 10].
For the most common and typical circuits, as a rule, analytical methods for calculating voltage and power losses are used [2, 11]. Such circuits include crane installations where variable frequency drives (VFDs) are used when modernizing old ones or designing new ones. The use of VFDs with semiconductor converters in crane power systems leads to a significant content of higher harmonic currents in the supply network, which are taken into account by the total harmonic distortion (THDI) in accordance with the requirements of the International Standards IEEE 519-1992, IEC 61000-3-12:2012 and IEC 61000-3-12:2004. Higher harmonic currents lead to additional voltage and power losses in shop networks [6]. This circumstance attracts more and more attention to the study of operating modes of nonlinear loads, taking into account the higher harmonics of the current [4-9, 12, 13].
The implementation of the requirements for limiting the generation of higher harmonics in the network required research and development of circuit solutions for
© P.D. Andrienko, O.V. Nemykina, A.A. Andrienko, R.E. Mokhnach
converters, passive active filters [1, 3]. From an economic point of view, the power distortion compensation is carried out at the load nodes: switchgear 6, 10 kV or switchgear 0.4 kV. However, in shop networks supplying electrical receivers with converters, the influence of higher harmonics turns out to be significant [5] and requires its solution.
In [6, 7], the authors proposed a method for studying the influence of higher harmonics of current in power supply systems of crane installations using steel trolleys and aluminum buses for the current conductor. It is shown that the presence of higher harmonics of the current leads to an increase in voltage losses by 3.2-4 times and power losses by 1.26-1.43 times in comparison with sinusoidal current for steel trolleys.
In the power supply system of heavy duty cranes and relatively long working spans, to ensure the operating voltage within the permissible limits, the main trolley is fed. The most widely used induction feed system is the least expensive. With induction feeding, an aluminum bus is usually laid in parallel to the trolleys [11].
The presence of higher harmonics of current in the power supply systems of the cranes leads to a change in the impedances of individual conductors and, accordingly, in the current distribution in them.
The goal of the paper is to investigate the influence of higher harmonics of currents on current distribution, voltage and power losses in the conductors of the induction feed system of cranes, and offer recommendations for reducing losses from higher harmonics.
Main research material.
Initial data. According to the generally accepted technique, the current of the fundamental harmonic of the trolleys It is determined from the condition of the permissible voltage losses in the working section of the crane operation, according to the relationship [6, 11]:
I _ max
AU „
lt 'AUa V3• lt • Rt-(cos^ + tgpfl • sin^)' (1)
I = I — I
-1 s 1 max -11'
where AUmax, AUt1 are the permissible voltage losses and voltage losses per 1 m of trolley section length, respectively, at a given trolley current; lt is the working length of the trolley; Imax, Is are the maximum current of the system and feed bus, respectively; tg^t1 = Xt1/Rt1; where Xt1, Rt1 are the inductive and active resistance of the trolley for the fundamental harmonic with frequency of 50 Hz; ^ is the shift angle of the fundamental harmonic.
To ensure the permissible voltage losses AUmax < 5 %, an aluminum bus is laid parallel to the trolley in the working area of the crane operation. The current distribution along the conductors in the feed system is determined by the ratio of the impedances at the fundamental harmonic [11].
Ratio of currents in conductors using the superposition method for components with harmonic n
Yn _
Zh
Rs
Rt,
i
1+tg
1 + tg Vtn
(2)
where Z.
(t) = -[Rs(t)n^^s(t)n corresponding conductor (s - bus, t - trolley) for n haimonic; = Xtn + X"n + X'tn ; Xsn = XSn + XS"n is the inductive resistance: X' - internal; X" - external; X '"' is the resistance to mutual inductance of the trolley and the feed bus.
The parameters of the conductors of the most common induction feed systems are given in Table 1 for distance between trolleys of 250 mm, made with a corner of 50 x 50 x 5 mm.
The inductive resistance of the conductors is indicated taking into account the mutual inductance of the trolley and the feed bus [11].
_J R
+ X2
is the impedance of the
Table 1
Parameters of conductors of feed systems
Dimensions, mm Parameters
Steel corner 50x50x5 mm Rti, Q/km Xt1, Q/km Zt1, Q/km Xt1 + Xt1, Q/km Xt'^ Q/km tg9t1 Yl, P-u-
1,65 1,263 2,08 0,339 0,924 0,765
Aluminum bus RsU Q/km Xsb Q/km ZsU Q/km — —
20x3 0,513 0,277 0,583 — — 0,54 0,28
30x3 0,342 0,253 0,425 — — 0,74 0,204
40x3 0,256 0,237 0,348 — — 0,926 0,161
50x3 0,205 0,225 0,32 — — 0,11 0,147
60x4 0,128 0,213 0,248 — — 1,664 0,119
80x5 0,077 0,195 0,21 — — 2,53 0,101
The most common sources of higher harmonics are uncontrolled (for variable frequency drives) and controlled (for DC drives) rectifiers. The relative values of the n-order harmonics of the input current of the bridge rectifier are determined from the relationship:
In = Kn • ~T = Kn — = Kn--(3)
I1 n f*
where Kn is the coefficient that takes into account the ratio of the ripple amplitude in a real rectifier to an ideal one [6] (with inductance Ld in the rectifier link Ld = ® Kn = 1); In, I1 are the current values of the n-order harmonic and fundamental harmonic in current conductors, respectively; fn = fn / 50 is the relative frequency of the n-order harmonic.
In [6] it was shown that the resistance of aluminum buses is related by the following ratios for the n harmonic component relative to the main one:
*
Xsn ~ Xs1fn ;
Xn n (4)
Rsn - Rsi;
tg Vs,
-Rsn - tg^i/;
Rsn
The resistance of the steel corners is related by the ratios for the n harmonic component relative to the main one:
Rtn - Rti\/t
x,
tg V
:Xn
Rtn
(( + Xt'i + Xtl )f Xti + 0,56^/ + Xti ^ /,
(5)
R-i
The maximum current taking into account higher harmonics is determined by the relationship [2, 6]
I m
n-6k±i
X Kn-k -0
n=6k±i
X
k-0
K,
i
r*2
(6)
fn
where k is the series of integers i, 2, 3, etc. In this case, we assume that the fundamental harmonic is equal to the fundamental harmonic of the sinusoidal current of the trolley without feed. Research results.
1. Distribution of currents in the feed conductors.
Transforming expression (2), taking into account the considered relations (3), we have:
Yn =-
R
si
M fn
i + (g Vsifn f
i+
X'ti + 0,56^/ + Xfi) /*
(7)
Rt2i
Analysis of the relationship (7), taking into account the values of the parameters for calculating the conductors, summarized in Table 1 showed that for
fnn >7, relationship (7) with sufficient accuracy can be reduced to the form:
Xs
Yn =-
Lsi
(8)
I
sn
i
i
I
Itn -
max Itn
max
i + Yn /*
Yn . J_
i *
i + Yn /*
(9)
Yn, P-u.
20x3 50x3
1 80x5
1
/l P-u-
I
Fig. 1. Dependences yn = f( f*) for the steel corner 50x50x5 mm with feed bus 20x3, 50x3, 80x5 mm
Figure 2 shows the relative values of the currents in the induction feed system: a corner 50 x 50 x 5 mm with a feed bus 80 x 5 mm for the n harmonic component.
i
OS 66 0.4 02 0
1n , P-U.
lSn
\
\
\ *
\ Itn
10 a
1 10 :o f;, p.u.
Fig. 2. Dependences In* = f (f„) for the steel corner 50x50x5 mm with feed bus 80x5 mm
Table 2 shows the relative values of the currents in the trolley made of a corner 50 x 50 x 5 mm with a feed bus.
Table 2
Relative values of currents
0,56RrtV fn
Thus, the distribution of currents along the conductors is practically directly proportional to the inductive resistance of the feed buses at the fundamental harmonic and inversely proportional to the square root of
the frequency f*, i.e. with increasing frequency, yn decreases monotonically, which indicates an increase in high-frequency components in the feed bus (Fig. 1).
It is not difficult to show, using the second equation in expression (1) and relation (2), that the relative value of the bus current Isn* and trolley current Itn* for the n harmonic component has the form:
Bus sizes, mm Parameter
I * P-u- I * P-u- I * 1 snZ ' n>5, p-u- I * hi, P-u- I * P-u- I * ltnL , n>5, P-u-
20x3 0,781 0,8i9 0,024 0,2i9 0,221 0,026
50x3 0,872 0,908 0,252 0,i28 0,i3 0,021
80x5 0,908 0,943 0,255 0,092 0,094 0,018
Analysis of Table 2 shows that with an increase in the cross-section of the feed bus, the current of the
n
trolleys significantly decreases including the decrease
in the high-frequency component I*nIi. 2. Voltage losses.
Since the trolleys are selected according to the permissible voltage losses at a given current (1), then we check the influence of higher harmonics for the trolleys.
In the presence of higher harmonics, the relative increase in voltage losses in trolleys relative to the voltage losses at the fundamental harmonic AUt1 is determined taking into account expressions (1), (9):
AU =
Vau2 + AU/5 +... + AU213 AU»
"Il
n=6k±i
Zau=,
k=0
n=6k±1
H+ Z Kf
k=1 Jn V
where AUt1 = ^Wt( + tg^sin^). 100% ;
U „
Ii = I m
' ri ^
AUt =
n=6k±1 , f
i + Z ft
k=1 J n V
rn
A
i+r
n y
2/ \2
' 1+V
ri
AU*n , Pu
1
OS
oc
OJ ÜJ 0
20x3
\ 50x3 80x5
! V
; V c os^i = 1
1 ïj "J, 1 f—— J
1
10
AU*n , Pu
1.1 1
0.9 Û.S 0.7 0.6
cos^i = 0,5
20x3 50x3 80x5
1
10
/*, pu
Fig. 4. Dependences AUt* = //*) in the trolley with feed bus 20x3, 50x3, 80x5 mm; cos^ = 0.5
1 + rn
\2/ \2 / t . '1 ' " lcosffi + \gq>tifn sinyj
i+ri
ri
(cOs^+ tg^ti sin^>! )2
(10)
, tgm = Xh + 0,56^ + x;i ;
J+n.
Unom is the trolley rated voltage.
For the case of an ideal uncontrolled rectifier Kn = 1, cos^1 ~ 1 which corresponds to a rectifier with an LC filter (distortion factor v = 0.955 which corresponds to THDi = 31.05 %)), the voltage losses are:
(11)
The dependence of the relative values of voltage losses in the trolley with a feed bus as a function of frequency /* are shown in Fig. 3.
The relative values of AUt5* and AUtl* are about 22 % and 16 % of the voltage losses at the fundamental harmonic, and the relative values of AUt11* and AUi25* are 10 % and 5 %, respectively.
/*, p.u.
Fig. 3. Dependences AUt* = / /*) in the trolley with feed bus 20x3, 50x3, 80x5 mm; cos^ = 1
The dependence of the relative values of the voltage losses in the trolley with feed bus on the frequency is shown in Fig. 4 at cos^1 = 0.5.
As follows from Fig. 4, the relative values of the voltage losses AUtn* for n>5 harmonic components at cos^1 = 0.5 increase significantly which is explained by the influence of the component (cos^+tg^- /* •sin^1) in
expression (10). An increase in AUtn* is noted for n>5 with an increase in the cross-section of the feed bus which is caused by the redistribution of the ratio of the relative values of the currents of the fundamental harmonic of the trolley It1* and high-frequency components Itnj**. This ratio increases as the cross-section of the feed bus decreases.
Table 3 shows the relative values of the voltage losses in the trolley at cos^1 = var, made of the corner 50 x 50 x 5 mm for some combinations of feed at /* <25.
Table 3
Relative value of voltage losses
"""""---...bus, mm cos^1 20x3 50x3 80x5
1 1,051 1,033 1,025
0,9 1,36 1,63 2,081
0,8 1,54 1,91 2,31
0,7 1,69 2,16 2,53
0,6 1,853 2,39 2,83
0,5 2,022 2,65 3,14
Analysis of Table 3 shows that the relative value of the voltage losses of the corner at cos^1 = 1 with 20 x 3 mm bus increases by 5.1 %, and with 80 x 5 mm bus - by 2.5 %. The relative value of the voltage losses reaches its maximum value at cos^1 = 0.5: with 20 x 3 mm bus it increases by 2.022 times, and with 80 x 5 mm bus - by 3.1 times. Therefore, AUmax in expression (1) should be reduced by an appropriate value.
Dependences AU*= fcos^) in the trolley with feed bus are shown in Fig. 5.
AU*, p.u.
20x3 50x3
80x5 \-
_ - . \
0.6
oj
o.s
. ^ cos^i 1
Fig. 5. Dependences AU, = /cos^) in the trolley with feed bus 20x3, 50x3, 80x5 mm
An analysis of the dependence AUt*= /cosyij) for the trolley with feed bus shows that with a decrease in cos^1, the values of the relative voltage losses in the trolley increase with an increase in the cross-section of the feed bus.
r
11
Note that the relative value of the voltage losses in the trolley with feed bus, depending on cos^i, is lower in the same trolley without feed bus in the presence of higher harmonics [6].
3. Power losses.
Power losses in the induction feed system have two components: losses in the trolley APt and in the feed bus AP, which are equal, respectively:
n=6k±1 n=6k±1
APt = 3 • X RtnItn and APS = 3 • X Rs1ll. k=0 k=0
In relative units, power losses are determined taking into account expressions (1), (5), (9): APt + AP,
* * -.AP = AP +APs = -
AP
n=6k ±1
= Z *2
k=0 fn
n -*2
Yn 1 + Y
n y
1
1+Yn
22
Kn *2
Jn
R n=6k±" x2 2 (12)
+— X
Rt1 k=0
where AP1 = 3Rt1 It2 are the losses in trolleys without feed.
The relative values of the power losses in the induction feed system are shown in Fig. 6.
0.04 0.03 0.02 0.01
APn , p u.
0.04 -
•
-, 003
•
002 aP,*„
• \
, 0.01
* AP*n APtn
0 1
1 * o /
APtn
1
10
20 f*, p u.
Fig. 6. Dependences APn = f f*) in the trolley with feed bus 80x5 mm
The relative values of power losses in the trolley with feed bus for an ideal uncontrolled rectifier Kn = 1 are summarized in Table 4.
Table 4
Relative values of power losses
Bus sizes, mm Parameter, P-u-
APt* AP„z , n>5 AP* ap; AP* Z, sn Z n>5 AP* s AP*
20x3 0,048 0,0018 0,0498 0,189 0,012 0,2 0,248
50x3 0,016 0,0007 0,0167 0,118 0,0061 0,118 0,135
80x5 0,0084 0,00044 0,00884 0,038 0,0018 0,039 0,047
from harmonics n<7. Accounting of the coefficient K2 , according to [6], leads to an increase in additional losses by about 1.5 times. Therefore, the calculation of the losses should be made taking into account the real values of the higher harmonics obtained experimentally [8] or by modeling [6].
Note that in order to reduce voltage and power losses in systems with crane installations that operate in heavy duty with a large number of starts, relatively expensive non-inductive feed systems are used, in which the feed bus is made of aluminum wires laid in pipes [11]. Analysis of these feed systems shows that with cross-section of wires of 50-150 mm2 and with number of cores equal to 3, the inductive resistances decrease by 2-3 times. This leads, according to expression (12), to a decrease in additional voltage and power losses in the trolleys. This circumstance partially or completely compensates the primary capital costs for building a non-inductive feed system, which are determined by a technical and economic calculation.
The proposed technique for calculating voltage and power losses can be used to calculate voltage and power losses in steel-copper and steel-aluminum wires used in railway transport and distribution networks.
A feature of AC power supply systems in railway transport is the significant value of the currents of the 3rd and the 5th harmonics, which reach 60 and 30 % of the fundamental one, respectively [13] which significantly affects the distribution of currents and the value of additional power losses and voltage losses.
Conclusions.
Research results show that in induction feed systems, due to the redistribution of higher harmonic currents between the feed bus and the trolley, there is a decrease in voltage losses, main and additional power losses.
When determining the permissible voltage losses, the reduction factor of the value of the permissible voltage losses 1.051-1.025, and 2.022-3.14 should be used, depending on the change in the power factor in the range of cos^1 = 1.0-0.5 and depending on the cross-section of the feed buses, respectively.
The use of an induction feed system allows to reduce the total power losses by 4-21.3 times depending on the cross-section while the relative additional power losses are no more than 5.5 % of the total power losses.
The proposed technique for calculating current distribution, voltage losses and power losses can be used to calculate the modes of steel-aluminum and steel-copper conductors.
Conflict of interests. The authors declare no conflicts of interest.
Analysis of Table 4 shows that relative to the first harmonic of the system current, the power losses in the induction feed systems AP* decrease depending on the cross-section of the feed buses by 4; 7.4 and 21.3 times, respectively. In this case, the relative additional power losses ( AP*x + AP*n X) are 5.5-4.6 % of the total losses.
Analysis of losses from high-frequency components shows that the main share of additional losses is losses
REFERENCES
1. Lumbreras D-, Galvez E-, Collado A-, Zaragoza J- Trends in Power quality, harmonic mitigation and standards for light and heavy industries: a review- Energies, 2020, vol- i3, no- 2i, P-5792- doi: httPs://doi-org/i0-3390/eni32i5792-
2. Zhezhelenko I-V-, Saenko Yu-L- Pokazateli kachestva i ih kontrol na promyishlennyih predpriyatiyah [Quality indicators and their control at industrial enterPrises]- Moscow, Energoatomizdat Publ-, 2000- 252 P- (Rus)-
3. Kalair A., Abas N., Kalair A.R., Saleem Z., Khan N. Review of harmonic analysis, modeling and mitigation techniques. Renewable and Sustainable Energy Reviews, 2017, vol. 78, pp. 1152-1187. doi: https://doi.org/10.1016/j.rser.2017.04.121.
4. Egorov A.N., Kharitonov Y.S., Shevchuk V.A., Semenov A.S. Influence of high harmonics on a frequency converter operation in underground mining. Bulletin o/ the Tomsk Polytechnic University. Geo Assets Engineering, 2020, vol. 331, no. 6, pp. 141-151. (Rus). doi: https://doi.org/10.18799/24131830/2020/6/2683.
5. Zmieva K.A. Modeling of an industrial enterprise power supply system using direct current. Russian Electrical Engineering, 2015, vol. 86, no. 5, pp. 239-245. doi: https://doi.org/10.3103/s1068371215050120.
6. Andrienko P.D., Nemykina O.V., Andrienko A.A. High current harmonics influence on the choice of conductors of crane power supply systems. Electrical Engineering & Electromechanics, 2019, no. 3, pp. 24-29. doi: https://doi.org/10.20998/2074-272x.2019.3.04.
7. Andrienko P.D., Nemykina O.V., Andrienko A.A. Comparative analysis of crane electrotechnical complexes. Bulletin of NTU "KhPI". Series: Problems of Electrical Machines and Apparatus Perfection. The Theory and Practice, 2019, no. 2, pp. 3-7. (Rus). doi: https://doi.org/10.20998/2079-3944.2019.2.01.
8. Proykov M., Simeonova N. Investigation of the influence of current harmonics generated by crane systems on some parameters of the power supply systems. 2021 17th Conference on Electrical Machines, Drives and Power Systems (ELMA), 2021, pp. 1-4. doi: https://doi.org/10.1109/elma52514.2021.9503048.
9. Semenov A.S., Semenova M.N., Fedorov O.V. The results of the implementation of the system for monitoring the quality of electricity in mining enterprises. 2019 1st International Conference on Control Systems, Mathematical Modelling, Automation and Energy Efficiency (SUMMA), 2019, pp. 644649. doi: https://doi.org/10.1109/summa48161.2019.8947601.
10. Milykh V.I., Polyakova N.V. Determination of electromagnetic parameters and phase relations in turbogenerators by the automated calculation of the magnetic field in the software environment FEMM. Electrical Engineering & Electromechanics, 2016, no. 1, pp. 26-32. doi: https://doi.org/10.20998/2074-272X.2016.L05.
11. Benerman V.I., Lovtskii N.N. Proektirovanie silovogo elektrooborudovaniia promyshlennykh predpriiatii [Design of power electrical equipment for industrial enterprises]. Moscow-Leningrad, GEI Publ., 1960. 385 p. (Rus).
12. Shevyrev Y.V., Pichuev A.V., Shevyreva N.Y. Improving energy performance in networks with semiconductor converters. 2019 International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM), 2019, pp. 1-6. doi: https://doi.org/10.1109/icieam.2019.8743020.
13. Sychenko V.H., Kuznetsov V.H., Bosyi D.O., Sablin O.I. Enerhetyka tiahovykh merezh [Power engineering of traction networks]. Dnipro, Standart-Servis Publ., 2017. 210 p. (Ukr). doi: https://doi.org/10.15802/978-617-73-82-04-0.
Received 11.07.2021 Accepted 23.09.2020 Published 26.10.2021
P.D. Andrienko1, Doctor of Technical Science, Professor, O. V. Nemykina1, PhD, Associate Professor, A.A. Andrienko1, Postgraduate Student, R.E. Mokhnach1,
1 Zaporizhzhia Polytechnic National University,
64, Zhukovsky Str., Zaporizhzhia, Ukraine, 69063,
e-mail: andrpd@ukr.net (Corresponding author),
olganemikina@ukr.net,
vamoseandrey @mail.ru,
etkmpk@gmail.com
How to cite this article:
Andrienko P.D., Nemykina O.V., Andrienko A.A., Mokhnach R.E. Research of operating modes of conductors in power supply systems of cranes with induction feed, taking into account the influence of higher harmonics of the current. Electrical Engineering & Electromechanics, 2021, no. 5, pp. 11-16. doi: https://doi.org/10.20998/2074-272X.2021.5.02.
