CC BY
52
Electrotechnical Complexes and Systems. Power Electronics
UDC 621.316.12 doi: 10.20998/2074-272X.2019.3.04
P.D. Andrienko, O.V. Nemykina, A.A. Andrienko
HIGH CURRENT HARMONICS INFLUENCE ON THE CHOICE OF CONDUCTORS OF CRANE POWER SUPPLY SYSTEMS
Purpose. To study the effect of high current harmonics on the power and voltage losses in the conductive lines of the crane power supply systems and the development of an account method for this influence in practical calculations. Methodology. For research analytical methods and methods of simulation are used. Results. Analytical calculations have been performed for power losses and voltage losses for the conductors of crane power supply systems in the conditions of high harmonic generation for frequency-controlled drives. Originality. For the first time, the authors have obtained the analytical expressions and graphical dependencies in relative units for practical calculations that allow determining the effect of high harmonics to the values of power losses and voltage losses for crane supply systems, while the parameters of steel conductors are nonlinear for load current and frequency. We have established that the values of power losses and voltage losses increase for crane power supply systems. It is shown that the power losses lead to a decrease the efficiency of crane supply systems up to 7 %, which must be taken into account when choosing electric drive systems and its payback period. Practical value. The obtained theoretical expressions can be used for calculations, design, optimization of crane power supply systems in terms of high harmonic generation. References 12, tables 2, figures 4.
Key words: high harmonics, voltage losses, power losses, conductors, steel materials, aluminum tires, crane power systems.
У cmammi проведено дослдження впливу вищих гармонш струму на втрати напруги i потужностi в струмопроводах систем живлення кричи;. Отримаш необхiднi розрахунковi спiввiдношення для визначення парамеmрiв сmрумопроводiв при наявносmi вищих гармотк. На прикладi найбтьш розповсюджених частотно-регульованих приводiв показано, що в тролейних лнях 3i сталевих маmерiалiв втрати напруги i поmужносmi зростають до 4 разiв i до 1,43 рази, вiдповiдно. Показано, що наявнсть нелiнiйноi залежносmi активного опору сталевих сmрумопроводiв вид струму навантаження i частоти призводить до збшьшення розрахунковоТ величини втрат поmужносmi в порiвняннi з розрахунком через коефщкнт спотворення струму. Встановлено, що величина tgfa1 може бути використана як конструктивний показник струмопроводу. Наявнiсmь втрат поmужносmi призводить до зниження ККД систем живлення кранв до 7 %, що необхiдно враховувати при виборi систем електроприводу i його термну окупносmi Бiбл. 12, табл. 2, рис. 4.
Ключовi слова: вищi гармошки, втрати напруги, втрати потужносп, струмопроводи, сталевi матерiали, алюмшieвi шини, системи живлення кратв.
В статье проведено исследование влияние высших гармоник тока на потери напряжения и мощности в токопроводах систем питания кранов. Получены необходимые расчетные соотношения для определения параметров токопроводов при наличии высших гармоник. На примере наиболее часто встречающихся частотно-регулируемых приводов показано, что в троллейных линиях из стальных уголков потери напряжения и мощности возрастают до 4 раз и до 1,43 раза, соответственно. Показано, что наличие нелинейной зависимости активного сопротивления стальных токопроводов от тока нагрузки и частоты приводит к увеличению расчетной величины потерь мощности по сравнению с расчетом через коэффициент искажения тока. Установлено, что величина tgfm1 может быть использована как конструктивный показатель токопровода Наличие потерь мощности приводит к снижению КПД систем питания кранов до 7 %, что необходимо учитывать при выборе систем электропривода и его срока окупаемости. Библ. 12, табл. 2, рис. 4.
Ключевые слова: высшие гармоники, потери напряжения, потери мощности, токопроводы, стальные уголки, алюминиевые шины, системы питания кранов.
Introduction. The main number of cranes is powered by alternating current of power frequency, which is decisive when choosing the type of used electric drives of cranes. The modern state of crane production industry is characterized by the introduction of semiconductor converters, significantly changing the quality of the crane electric drive, providing uniformly accelerated start and stop of mechanisms, which contributes to the reliability and durability of their mechanical structures and moving parts with significant energy saving [1-3]. In most cases, the advantage is given to the use of a variable frequency drive (VFD). The presence of semiconductor converters leads to the emergence of higher harmonics currents in the power supply system of cranes (PSSC), which in turn negatively affects the quality of electric power, electromagnetic compatibility, leads to a drop in voltage and power losses. When choosing a type of adjustable drive, an economic assessment is made by comparing
their efficiency, cost, without taking into account losses in AC PSSC, which cause deterioration of the efficiency of the PSSC-VFD system [4-6].
The goal of the paper is to study the effect of higher current harmonics on the power losses and voltages in the conductors of the power supply systems of cranes and the development of methods for taking this influence into account in practical calculations.
Main materials of investigations.
1. Initial data. In existing practice, the AC power supply system is mainly used, and for the implementation of adjustable electric drives in crane installations, controlled rectifiers with DC motors or VFDs based on a two-stage frequency converter with an independent voltage inverter, having an uncontrolled, controlled or active bridge rectifier are used. The presence of bridge rectifiers leads to the appearance of higher harmonics of
© P.D. Andrienko, O.V. Nemykina, A.A. Andrienko
n=6k±1 order (k is the natural number k = 1, 2, 3, 4, ...) in the AC network, which leads to an increase in the calculated current by an amount taken into account by the distortion coefficient v or the harmonic factor THD/ in accordance with the requirements of International Standards IEEE 519-1992 or IEC 61000-3-12:2012 [5-7].
To assess the effect of higher harmonics in the AC PSSC using a two-stage frequency converter (FC), an equivalent circuit has been used (Fig. 1). The power source (G) in the equivalent circuit is a symmetric system of limited power voltages. The power supply network is simulated by successive RLi chains.
I»-1 Jtn In, «a La Rn. L„ , R H..Jm
J ^ M1...M-
Fig. 1. Equivalent circuit of the AC PSSC
Each chain includes a corresponding inductance and a resistance: of a transformer - Rm Lm, of a cable line -Rkl Lkl, of a section of a conductor RT1LT1 - respectively to the first crane in the span.
When modernizing cranes, the VFD circuit with the common rectifier (R) which is located on the crane is most often used. Depending on the type of crane, conductors of profiled steel, aluminum tires, flexible cables are used.
The following assumptions were made during the analysis:
• the constancy of the amplitude, the sinusoidal shape and symmetry of the three phase voltages on the secondary winding of the power transformer;
• the inductive and active resistances of all network elements, except the main conductor, are considered constant and independent on current;
• taking into account the influence of higher harmonics of the current is carried out by the simulation results for the PSSC with a VFD or by analytical relations, while the parameters of the conductor depend on the frequency and load current.
The choice of conductor sections is made according to the heating condition and is checked for voltage losses at the most distant point [8, 9].
The rated current of the conductor at non-sinusoidal load is determined based on the value of the calculated power (P) by the relation [9]:
P P I —-¡=-—-¡=-, (1)
V3 •!•Unom V3 • cos^i-v-Unom where Unom is the nominal voltage, Unom = 380 V; X, cos^i, v are, respectively, the power factor for the non-sinusoidal circuit, the power factor of the main harmonic and the distortion factor.
For crane installations with controlled rectifiers and DC motors or two-stage frequency converters and VFD operating in intermittent mode, the input power factor is X = 0.5-0.6 [2]. Ensuring electromagnetic compatibility (EMC) of the frequency converter (FC) with the supply network is achieved by installing an input smoothing reactor and/or an active rectifier. In this case, the input power factor of the main harmonic increases and can
reach unity. Further studies were performed with cos^ = 0.5-1 [4].
In the case of generation of higher harmonics, the calculated current of the conductor is determined by the relation [10]:
I =
L
i=1
In
(2)
where In is the value of the calculated current of the harmonic of the n-th order in conductors.
For a bridge rectifier, the relative values of the higher harmonics of the input current are determined from the relation:
I = k
In
1
(3)
— kn — kn - js , h n f*
where kn is the coefficient taking into account the relative
value of the amplitude of the n-th harmonic at different
pulsations of the rectified current.
For the ideal rectifier Ld=<x>, kn=1 the input current distortion factor v = 0.955.
For a three-phase bridge rectifier in the input link of the FC and a capacitive filter, the values of higher harmonic currents were obtained by the simulation method [11], which was performed in the Matlab software package (Table 1).
The relation (2) in relative units with regard to (3) takes the form:
I * =
L
i=1
kn -
n y
fn
(4)
Table 1
Higher harmonic input current values
In/h, % n=5 n=7 n=11 n=13 n=17 n=19 V
38.3 % 2,2 % 7 % 3 % 3 % 2 % 0.926
Ld=™, kn=1 20 % 14 % 9 % 7 % 6 % 5 % 0.955
kn 1.91 0.87 0.77 0.42 0.5 0.4 -
2. Determination of the parameters of conductors taking into account higher harmonics. A feature of conductors in terms of the generation of higher harmonics is the dependence of their active resistance on the magnetic permeability of the steel conductor and the frequency of the current.
The value of the active resistance of conductors made of corner steel is determined by the relations [9]:
Ro> — K®R0 — Ko> 'P50 - -ST St
Ka= 1 + 0,84 •ßw, at ßw < 1 Km = 0,758 + J 1,34 ßw - 0,83, at 1 < ßw
at ßw > 3
<3
Ka = 0,758 + 1,159ßw
(5)
Pw — 2 -102 P • P M
P v P50 ,
where S, P, l are, respectively, the section, cm2, perimeter, cm, length, m; R0 is the ohmic resistance of steel conductor to direct current, Q/km; ^ is the relative magnetic permeability of steel conductor, which is determined by the curves [10], depending on the magnetic field strength H = 0.4-n-I/P, A/cm; p50 is the resistivity to
2
2
direct current, S-mm2/m; I is the current in the conductor, A, Si is the section, mm2.
For trolley lines made of corner steel 50x50x5 and 75x75x10, according to the condition of permissible heating, the magnetic field strength values H are within 6-23.5 A/cm, which corresponds to the value i = 1500-750. According to relations (5), for the frequency 50 Hz (the first harmonic) takes the values = 3.6-3.2 and fiw1 = 6.5-5.8, respectively. The specified values of fiw1 correspond to the coefficient Km1 = 5-4.5 and Km1 = 8.25-7.6. Active resistance to alternating current Rm1 = 1.5-1.4 Q/km and Ra1 = 0.85-0.78 Q/km for corner steel 50x50x5 and
Parameters of the i
75x75x10, while the ohmic resistance of the corners to direct current is 5 and 8.25 times less, respectively (Table 2).
The value of the inductive resistance of trolley lines of corner steel for the first harmonic is determined by the relation [9]:
Xo1 = X1 + X11 « X1 + 0,56Ro1, (6)
where X1 and X11 are the internal and external inductive resistances of trolley lines, respectively.
The values of active and internal inductive resistances of trolley lines made of corner steel Rm1 and X11, according to relations (5), (6), depend on the load current and frequency. The value of the internal inductive resistance X1 does not depend on the load current.
Table 2
:stigated conductors
Dimensions, mm Load current, A R0, fi/km Rmi, fi/km Xm1, Q/km 1 A tgpmi
X1, fi/km X11, fi/km
Trolleys 50x50x5 100/170 0.3 1.5/1.36 0.216 0.85/0.77 328 0.71/0.725
75x75x10 200/360 0.103 0.85/0.78 0.18 0.49/0.44 542 0.788/0.795
Tires 40x4 - 0.192 0.222 0.214 475 0.96
120x10 - 0.0255 0.0331 0.153 2070 4.6
The parameters of the most used conductors in the form of steel trolley lines and aluminum tires for the first harmonic are given in Table 2. Analysis of data of Table 2 shows that for trolley lines made of corner steel, tgyoA varies in the range of 0.71-0.795 and practically does not depend on their section; with a slight error, tg^m1 ~ 0.75 = = const can be taken, while for tires, tgqoA increases with increasing section of tires.
Internal inductive resistance for the corners of 50x50x 5 and 75x75x 10 is in the range of 0.216-0.18 Q/km and with a slight error X1 ~ 0.195 = const can be taken.
When generating harmonic currents n>5 in the steel conductor, the coefficient ¡.3w, expression (5), therefore, with an accuracy of up to 10%, Km„ ~ 1.159-ftw„ can be taken.
The relative value of the active resistance of the conductor for the harmonics of the n-th order, taking into account expressions (5):
R* = R°™
Pw
- = V fn
(7)
Ro1 Ko1 Pw1
where f„ = fn/f is the relative frequency of the harmonic of the n-th order; fn, f are the frequency of the harmonic of the n-th order and the fundamental frequency, respectively.
Active and inductive resistance of trolley lines for the harmonics of the n-th order:
Rom = Ro>1 ^^fn , (8)
Xan = Xo1 • fl = ( X1 + 0,56Ro1 • fj • fl. (9) Using relations (8) and (9) we express the value:
tgvan=Rr-
0,195 + 0,56 • Rml -J f* I. fr
R®1 'V Jn
. (10)
At fn>5, the value of the internal inductive resistance of the trolley lines is small compared with the external inductive resistance, therefore, with enough accuracy for practice, it is possible to use the relation:
tgVcon = tgVco1 • fn ~ 0,56 • fn
(11)
Note that for aluminum tires and copper conductors in the frequency range under study, the manifestation of the skin effect is insignificant, therefore the tires resistance is constant Rmn = Rm1.
The tg^mn value for harmonics of the n-th order of aluminum tires and copper conductors is determined by the relation tg^ = tg^f
3. Determination of voltage losses in conductors taking into account higher harmonics. In the general case, the voltage losses are determined by the relation [10]:
AU =
lAU 2
(12)
i=1
where AUn are the voltage losses for harmonics of the nth order in conductors:
AUn =
l • sinfft)
Un
•100 =
' Rwn 'l + tgVcon ' sin»1)
(13)
Un
•100,
where I„, are the current value of the n-th harmonic and the shift angle of the main harmonic, respectively; l is the conductor length; kmax is the coefficient taking into account the increase in peak current relative to the calculated value of current.
The value of cos^ is determined by the switching angle y for rectifiers installed in the input link of the FC.
Using the previously accepted assumptions, we transform the relation (12). For steel conductors, the relation takes the form:
n
AU
" AU 2n
AU
(15)
n=6k±1
J1 - S (n )f
| k=1 Jn
( ncc r* ■ ^2
cos^i + 0,56 • Jn • sm^i
cos^i + tgq>oA ■ sin^ For aluminum tires the relation (15) takes the form:
AU * =
n=6k±1
11 + S (k„ )2
k=1
*
cos^ + • fn • sin^1
cos^+ tgq)oA • sin^1
(16)
In the study of voltage losses in terms of the generation of higher harmonics, it was found that their value is determined mainly by the product tgpm1f „-sinp1, which is part of the function /(p) = = (cos^i+tg^/n-simpi).
The dependencies of the function /(p)=(cosp1+tgpm1-sinp1) for the main harmonic are shown in Fig. 2.
1.3
1.2
1.1
f('P)
= 0,75 \
\
0.(5 0.7 0.S 0.9 a
1
COS Pi
m
__ № ei = 4,6
X
tgflu 1=1,7 ■-V N
tg<?a i = 0,9 6 \
0.6 0.7
0.S b
0.9
1
COS <p 1
relations (15) and (16), are shown in Fig. 3. Analysis of dependencies shows that at the same harmonic composition of the current, the relative value of the voltage losses in conductors of the steel corner is much higher than in tires. This is explained by the fact that for trolley lines the component AU n n is inversely proportional to f n, and for tires to f n)2. When reducing the shift coefficient to cosp1 = 0.5, which is typical for controlled rectifiers, voltage losses increase 4 times for steel corners, and 2.5 times for tires.
AU\%
5
= 0,75
COS Çi
2.5
1.5 0.6 0.7 0.8 0.9 1
a
A U*,%
1.5
__/£<
= 0,96
0.6
0.7
0.8
0.9
1
COS (¡>\
b
Fig. 2. Dependencies f(p) = (cosp^tgp^-sinp^
for conductors made by trolley lines (a) and tires (b)
For trolley lines (tgpm1=0.75) at changes 0.6 < cosp1 < 0.95, the function /(p) can be approximated with enough accuracy for practice by the value /p)~1.2, which greatly simplifies the calculations. The maximum value of the function /(p)~1.25 takes at cosp1=0.8.
When using tires tgpm1 varies in the range of 0.96-4.6.
For tgpm1=0.96, at 0.5 < cosp1 <0.95, the values /(p) ~ 1.37. For tgpm1 > 1.7, which is typical for tires of 50^6 mm and more, the function monotonously increases with decreasing cosp1. The minimum value the function f(p) takes at cosp1 = 1.
The dependencies of the relative values of A^*=/(cosp1) for trolley lines and tires, calculated by
Fig. 3. Dependencies AU ^fcos^O for conductors at v = 0.955 (solid line) and v = 0.926 (dotted line), made by trolley lines (a) and tires (b)
This circumstance confirms the need to take into account the effect of higher harmonics when calculating the voltage losses.
The decrease in the distortion coefficient v from 0.955 to 0.926 leads to a decrease in the voltage losses in the conductors, which indicates a non-linear dependence of AU* on the amplitudes of the harmonic components of the current curve.
Therefore, when designing PSSCs that have semiconductor converters (rectifiers, FCs, voltage regulators), it is necessary to determine the harmonic composition by simulation modeling. Dependencies of AU* have a number of characteristic points.
For conductors of corner steel, the relation (15) takes the form:
- at v < 0.95 and cospi = 1
AU =
1 +
n=6k±1
S (kn )
k=1
2 J_
f*
n
(17)
2
1
■ at sini = 1 (cos^i = 0)
AU =
1 +
n=6k±1
Z (n )
k=1
(18)
- at kn = 1,
The absolute value of the voltage losses is proportional to the calculated value of In and the active resistance Rmn according to (13). Since the tire resistance is less than steel corners one, at an equal value of the calculated current, the absolute value of the voltage drop in the tires is significantly less.
Note that the use of cable conductors for powering portal [12] and gantry cranes provides a significant reduction in voltage losses due to their relatively low
tgVcA.
4. Determination of power losses in conductors taking into account higher harmonics. The power losses in the AC conductors for the first harmonic is
determined by the relation [9]:
( \2
Ap = 3(1 )2 Rai = 3
P
V3 • c
R
e>1 ■
(19)
• cos^1 •Unom J
where I1 is the calculated value of the main harmonic current.
The relative value of the additional losses in the AC conductor in the conditions of generation of higher harmonics is determined from the relationship:
AP* =
Z
n=6k±1 k=1
APn
AP1
n=6k±1
■= Z R* k=1
n
(20)
n=6k±1
APZ = Z kn k=1
(21)
ASP = 1 +
(1+ApZ) •
(22)
In conductors made of aluminum tires (for example, for powering portal cranes), the active resistance value is not significantly dependent on the presence of higher harmonics, therefore, with enough accuracy for practice, power losses can be determined using the standard technique: ASP*= APi/v2 = 1/v2.
This circumstance confirms the need to take into account the effect of higher harmonics when calculating power losses in conductors.
5. Influence of power losses in conductors on the efficiency of power supply systems of cranes. Figure 4 shows the dependence of the relative value of the main harmonic power losses (AP1* =AP1/P1) in conductors 100 m long at load currents and the parameters of the conductors given in Table 2 when changing the values of cos^1 = 0.5-1.
The dependency analysis (see Fig. 4) shows that at cos^1 = 1 and l = 100 m, the relative power losses in the trolley lines are 5% and 6.1%, respectively for the corners 50x50x5 and 75x75 x10, for aluminum tires 1.4 % and 1.03%, respectively, for sections 40x4 mm and 120x10 mm. At cos^1 = 0.5, the relative losses in trolley lines are 20 % and 25 %, respectively for 50x50x5 and 75x75x10, the losses in tires are 4.1 % and 5.5 %, respectively for sections 120x10 mm and 40x4 mm.
where AP1 are the power losses at the main harmonic in the AC conductor.
Taking into account expressions (3) and (7), the relative value of the additional losses in the conductor:
APi ,%
\
0.5 0.6 0.7 O.S 0.9 a
APi ,%
1
COS If J
Relative total losses taking into account the first harmonic:
After summing up the series (21) for the conductor under consideration at v = 0.955, we obtain the value AP*S = 0.26. The relative total losses ASP* = 1.26 according to (22).
When calculating using the distortion coefficient v, the relative total losses:
ASP* = AP1/v2 = 1/v2= 1/0.95 = 1.11.
The resulting value by the ratio (22) 1.26/1.11 = 1.135 times more compared with the well-known conventional approach.
When the distortion coefficient v = 0.926 according to the data of Table 1 relative value of additional losses according to the expression (21) AP*S = 0.436. Relative total losses taking into account the main harmonic ASP* = 1.436.
When calculating using the distortion factor: ASP* = 1/0.932 = 1.15.
The value of the relative total losses increase by 1.436/1.15 = 1.25 times.
b
0.6
0.7
O.S b
0.9
1
cos <3j
Fig. 4. Dependencies AP! =/(cospi) for conductors made by trolley lines (a) 50x50x5 mm at P = 59 kW (solid line) and 75x75x10 mm at P = 125 kW (dotted line); as well as made by tires (b), 40x4 mm at P =100 kW (solid line) and 120x10 mm at P = 500 kW (dotted line)
Thus, the relative losses in trolley lines increase by 3-3.5 times as compared with tires. At real lengths of 60-70 m and the location of the nodes for feeding trolley lines, the losses in them are 3-5 %, depending on the angle of the corner. Taking into account the additional losses from higher harmonics and real power factors, the power losses increase to 4.5-7 %.
At tire lengths up to 300 m, which is typical of gantry cranes, the losses amount to 4.2 %, taking into account the additional losses the power losses increase to 5 %.
This circumstance leads to a decrease in the efficiency of the PSSC with VFD, which must be taken into account when justifying the payback period of the electric drive system.
Conclusions.
1. The presence of higher harmonics in the conductors of power supply systems of cranes at changing values of cos^1 = 0.5 - 0.95 causes:
- an increase in voltage losses of 1.8-2.5 times as compared with sinusoidal current for tires tg^ra1 and of 3.2-4 times for steel trolley lines;
- an increase in power losses of 1.26-1.43 times compared with sinusoidal current at using steel trolley lines.
2. It is shown that the decisive parameter affecting the voltage and power losses is the tangent of the conductor tg^ra1, which can be used as a design parameter of the conductor. Conductors with minimal tgq>m1 have minimal losses.
3. The presence of power losses in conductors made by trolley lines, taking into account the additional losses leads to a decrease in the efficiency of power supply systems of cranes to 4.5-7 % at using trolley lines and up to 5 % when using tires, which must be taken into account during the feasibility study of electric drive systems. The presence of additional losses from higher harmonics in the power systems of AC cranes leads to an increase in the cost of the implementation of conductors.
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Received 07.01.2019
P.D. Andrienko1, Doctor of Technical Science, Professor, O.V. Nemykina1, Candidate of Technical Science, Associate Professor,
A.A. Andrienko1, Postgraduate Student, 1 Zaporozhye National Technical University, 64, Zhukovsky Str., Zaporozhye, 69063, Ukraine, e-mail: andrpd@ukr.net, olganemikina@ukr.net
How to cite this article:
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: 10.20998/2074-272X.2019.3.04.
