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CALCULATION AND IDENTIFICATION OF REACTIVE POWER IN NETWORKS
WITH NON-SINUSOIDAL (NONLINEAR) LOADS
Article · February 2013
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Ivano-Frankivsk National Technical University of Oil and Gas
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CALCULATION AND IDENTIFICATION OF REACTIVE POWER IN NETWORKS
WITH NON-SINUSOIDAL (NONLINEAR) LOADS
О. V. Solomchak, Ph.D.
Ivano-Frankivsk National Technical University
Semiconductor devices and transformers, rectifiers, power electronics , dynamic loading,
valve electro-technological installations cause distortion of the curved currents and voltage
in the systems of power supply.
The considerable increase of electro-transceivers with the nonlinear loading, that
generate higher order harmonics, complicates the process of calculation of reactive power
and choice of compensating devices.
Number of researches are devoted to the problem of systematization of
approaches to identification and calculation of reactive power at non-sinusoidal
currents [1-3]. However, the common methodology hasn’t been developed, especially
from the point of view of practical realization.
The purpose of work is experimental researches of current and voltage curves of
nonlinear electro-transceivers and complex unit with part of the nonlinear loads, analysis
and developing the practical approaches to the calculation of reactive power and
determination of compensating devices from the point of view of peculiarities of account of
reactive electric power and practical application.
The electric networks of enterprises and organizations with the nonlinear loading
have become the research object.
Means and methods of researches
A computer informatively-measuring complex which allowed to get the graphs of
instantaneous currents and voltage in the three-phase network of 0,38 kW was used to
receive reliable information and analysis of nature of reactive power.
Fundamental theoretical backgrounds
In the circles of alternating sinusoidal current it is accepted to define complete power
as:
(1)
S  P  jQ , VA
2
2
2
(2)
S  P  Q , VA
where P  UI cos  - is an active power, W; Q  UI sin  - is a reactive power, VAr.
Actually power in the circle of periodic current is characterized by the mean value of
power for the period, which is named an active power:
T
T
1
1
P   pdt   uidt ;
(3)
T 0
T 0
However in the circles of non-sinusoidal current even at the purely active loading
correlation (1) and (2) are not fulfilled. In such circles S  P even when cos   1 . There
appear certain power T which is connected with correlation
(4)
S 2  P2  T 2 .
Taking into account, that S  UI it is possible to define T :
(5)
T  (UI ) 2  P 2 .
Last years a discussion devoted to identification of the generally accepted concept of
reactive power is carried on in scientific editions. The authors assume that the most clear
and acceptable for the practical calculations determination of reactive power and
mathematical correlations is given in [3] . In circles with active, inductive and nonlinear
electro-transceivers the following determinations and correlations are offered:
(6)
S 2  P2  D2  P2  Q2  T 2 ,
where
P  UI1 cos 1 - is active power;
Q  UI1 sin 1 - is reactive power of displacement;
T U
n
I
i 2
2
i
- is reactive power of distortion;
D  Q 2  T 2 - is total reactive power;
I i - it is current of i harmonic.
It is thus assumed that voltage is sinusoidal.
The total reactive power can be defined from (6) :
(7)
(8)
(9)
(10)
(11)
D  (UI ) 2  P 2 .
But exactly this correlation (11) is considered to be the basis of calculation algorithm
of reactive power by electronic meters. Thus at the purely active loading, but nonsinusoidal current which is caused by the nonlinear loading, electronic electric power
meters will fix the consumption of reactive electric power. Taking into account the extra
charge of pay by the power supply company for the consumption of reactive electric
power, it becomes natural that the firms have come to a logical decision to compensate
this reactive power by capacitor installations or other compensating devices. However
appearance of displacement reactive power of capacitor batteries, according to (6), will
result in the yet greater consumption of reactive electric power.
The presence of non-sinusoidal electro-transceivers can result also in wrong work of
automatic capacitor installation with the regulators of power coefficient, the algorithm of
work of which is given in (11).
Obviously, that it is wrong to calculate power-factor as
P
(12)
cos   .
S
Taking into account (7-11), the correct calculation is
P
.
(13)
UI1
For description of part of displacement reactive power it is more correctly to use the
coefficient of reactive power
Q
(14)
tg  .
P
cos  
Experimental researches
Experimental researches for the number of nonlinear electro-transceivers and
complex un it of loading with inductive and non-sinusoidal electro-transceivers have been
carried on to determine the actual character of loading and to confirm the abovementioned theoretical calculations.
The instantaneous values of voltage were measured by means of transformer of
voltage of compensative type CV3-1000 production of LEM with accuracy class 0,2, and
instantaneous values of current were measured by means of current measuring pliers
АТА-2502 production of АКТАКОМ with accuracy class 2. Input of secondary signals in
COMPUTER was carried out by 14-bit digit ADC NI USB - 6009 production of National
Instruments with accuracy class 0,5. The discreteness of reading the showings was 100
values for 1 period (5 кHz).
Analysis of form of voltage
Fig.1 demonstrates the oscillogram of voltage and first harmonics, received at
decomposition in the Fourier series.
Fig. 1. Oscillogram of voltage and main harmonic Fig. 2. Spectral composition (%) of network voltage
As it is shown in Fig.1 2 , real curve of voltage is some distorted and differs from
ideal. Fig. 2 shows spectral composition of the curved voltage after decomposition in
the Fourier series.
Effective voltage, calculated from the experimental curve, is 215,01 V , and value of
the maim harmonic of sinusoidal voltage is 214,93 V. Difference does not exceed 0,041%,
that is a far fewer error of measuring transformers and devices. The spectral composition
shows that higher order harmonics is insignificant. Thus, calculating the power, deviation
of voltage from a sinusoidal form can be ignored.
Analysis of currents and power
A number of typical electro-transceivers was chosen for the analysis of influence
of the form of curved current on the calculation of power.
Rheostat, connected in series with a diode
From the point of view of the electrical engineering it is quite active element.
Oscillograms of current and voltage in relative units (in relation to effective values) at
the connecting through a diode is shown in Fig.3. Fig. 4 shows the experimental and
main harmonic of current, and Fig. 5 shows the spectral composition of current.
Fig. 3. Oscillograms of experimental current
and voltage of rheostat with a diode
Fig. 4. Experimental curve and main harmonic
of current
Fig.. 5. Spectral composition (%) of current of
rheostat at connecting through a diode
Table 1. Effective values of harmonic of current of rheostat with a diode
Harmonic
0
1
2
3
Current, A
1,215
1,307
0,558 0,016 0,107 0,035 0,030 0,009 0,023 0,005 0,007 0,002
%
64,85
69,77
29,76
0,82
4
5,74
5
6
1,87
1,57
7
0,51
8
1,22
9
0,27
10
0,38
11
0,12
Effective (root mean square) values of voltage are 215.9 V, of current - 1.874 А.
While calculating the complete power according to S  UI , we get 404.6 VA, and active
power, calculated according to (3), is 280.6 W. As one may see, in this circuit S  P ,
although, as it can be seen from the oscillograms, the displacement of phases between
current and voltage is absent and sin   0 . At a classic record (1), (2) Q  UI sin   0 .
The given example confirms, that for non-sinusoidal circuits expressions (1), (2), (12)
are improper.
According to (7) Q  UI1 sin 1  0 .
If using expression (9) then T  U
n
I
i 2
2
i
 123.19 VAr . According to expression (6)
S  306 .48 VA. But it does not correspond to the above calculated value according to
S  UI  404 .6 VA.
As it is shown in Fig.4, this chart has a considerable permanent constituent of
current. If it is taken into account at calculation
n
T  U I 02   I i2 ,
(15)
i 2
Then we will get T  289 .82 VAr, and according to (6) S  403 .43 VA, that practically
fully coincides with value 404.6 VA.
Thus at the calculation of reactive power of distortion it is necessary to take into
account the higher order harmonics of current and permanent constituent.
Computer
This device is equipped with the switched-mode power supply with the typical
curve of current. Most power modules are not equipped with the filters of harmonic
distortion.
The oscillograms of current and voltage in relative units (in relation to effective
values) are shown in Fig 6. Fig 7 shows the experimental curve and main harmonic of
current, and Fig 8 shows the spectral composition of current.
Fig. 6. Oscillograms of experimental current
and voltage of computer
Fig. 7. Oscillograms of computer current
Fig. 8. Spectral composition (%) of computer
current
Table 2. Effective values of harmonic of computer current
Harmonic
0
1
2
3
4
5
6
7
8
9
10
11
Current, A
0,000 0,429 0,005 0,339 0,008 0,211 0,004 0,071 0,004 0,008 0,003 0,019
%
0,00 72,51 0,87 57,34 1,45 35,80 0,56 12,11 0,75
1,36
0,53
3,26
Effective (root mean square) values of voltage are 217,8 V, of current - 0,591 А.
Calculating complete power according to S  UI , we get 128,7 VA, and active power,
calculated according to (3), is 92,1 W. As one may see, in this circuit S  P too,
although, as it is seen from oscillograms, the displacement of phases between the main
harmonic of current and voltage is absent, and thus the reactive power of displacement
is practically equal to zero.
According to (7) Q  0 . According to (15) T  88.42 .
If by mistake the reactive power of distortion is taken for the reactive power of
displacement and if to use a capacitor with power of 88,42 VAr to compensate it, then
according to (6) we will get:
S  P 2  Q 2  T 2  92.12  (88.42)2  88.422  155.3 VA.
As we may see, it will result in the yet greater increase of complete power and
current.
Active resistance with thyristor regulators of voltage
This device is equipped with a thyristor (TRIAC) regulator which opens thyristors at
certain angle.
The oscillograms of current and voltage in relative units are shown in Fig 9. Fig 10
shows an experimental curve and main harmonic of current, and Fig. 11 shows the
spectral composition of current.
Fig. 9. Oscillograms of experimental current and
voltage of thyristor regulator
Fig. 10. Oscillograms of current of thyristor
regulator
Fig. 11. Spectral composition (%) of current of
thyristor regulator
Table 3. Effective values of harmohic of current of voltage thyristor regulator
Harmonic
Current, A
%
0
1
2
3
4
5
6
7
8
9
10
11
-0,029 8,579 0,029 5,905 0,006 2,758 0,013 1,947 0,014 1,733 0,006 1,143
-0,25 74,87 0,25 51,53 0,05 24,07 0,12 16,99 0,12 15,12 0,06
9,98
Effective (root mean square) values of voltage are 215,0 V , of current - 11,46 А.
Calculating complete power according to S  UI , we get 2463,62 VA, and active
power, calculated according to (3), is 1312,86 W. As one may see, in this circuit S  P
too, although, at the active resistance the reactive power of displacement must be
equal to zero.
However approximation by Fourier series of current curve shows that the main
harmonic is displaced in relation to voltage at angle 44,8 degree which causes the
appearance of reactive power displacement Q  1300 VAr. According to (15)
T  1510 VAr.
Thus, cutting the current sinusoid by thyristors (TRIAC) leads to appearance of
reactive power of displacement even at the pure active loading.
Administrative and educational building of higher educational establishment
This consumer is characterized by the complex loading, which contains lighting
installations, computers, office equipment, heater devices.
The oscillograms of current and voltage in relative units are shown in Fig.12.
Fig.13 shows the experimental curve and main harmonic of current, and Fig.14 shows
the spectral composition of current.
Fig. 12. Oscillograms of experimental current
and voltage of administrative andeducational
building
Fig. 13. Oscillograms of current
Fig. 14. Spectral composition (%) of current
Table 4. Effective values of current harmohic of administrative and educational building
Accordion
Current, A
%
0
1
2
3
4
5
6
7
8
9
10
11
-0,033 20,635 0,066 2,511 0,002 1,570 0,025 0,519 0,054 0,048 0,035 0,115
-0,16
98,95
0,32 12,04 0,01
7,53
0,12
2,49
0,26
0,23
0,17
0,55
Effective (root mean square) values of voltage are 209,8 V , of current - 20,85 А.
Calculating complete power according to S  UI , we get 4374,75 VA, and active
power, calculated according to (3), is 4273,08 W. As one may see, in this circuit S  P
too. Reactive power of displacement Q  624 . According to ( 15 ) T  631.22 VAr,
D=937.67 VAr.
As we may see, reactive power of displacement and distortion are practically
commensurate.
Oscillograms and spectral composition of current in a zero wire are shown in
Fig.15 and 16.
Fig. 15. Oscillograms of current in zero wire
Fig 16. Spectral composition (%) of current of
neutral
In spite of the fact that effective values of currents in phases were practically
equal, current in a zero wire is 14.03 A (67.3 % of phase). It is caused by imposition of
the third harmonic of phase currents, the period of which is equal to the displacement
of phases.
Conclusions:
1. The systems of power supply of enterprises, organizations, city owned utilities
electro-transceivers contain nonlinear (non-sinusoidal) elements, part of which constantly
grows. That’s why the concepts of reactive power of displacement and distortion must be
approved at the official level (6), (8), (15).
2. The digital electricity meters measure the total reactive power D : displacement
and distortion, which is correct, as both of them cause the additional losses of active
power in electric networks. Electromechanical induction meters measure the reactive
power of displacement only.
3. Calculation of value of reactive power of displacement according to the digital
electricity meters without harmonic filter is improper.
4. Deviation of the form of the curved voltage from a sinusoidal insignificantly
influences the calculations of effective value and powers.
5. A power-factor must be determined according to (13) for the basic harmonics of
voltage and current, including in the algorithm of work of automatic regulators of powerfactor. Use of coefficient of reactive power tg  is considered to be more evident and
more correct for characterizing the reactive power of displacement.
6. It is necessary to take into account the higher order harmonicss of current and
permanent constituent (15) while calculating reactive power of distortion.
7. Compensation of reactive power of distortion by capacitors results in the increase
of complete power and current.
8. For compensation of reactive power of distortion it is necessary to use filters of
higher order harmonics and Power Factor Correction.
9.Cutting the current sinusoid by thyristors (TRIAC) leads to appearance of
reactive power of displacement even at the pure active loading, which can be
compensated by capacitors.
10. It is necessary to introduce standards [4,5] in relation to limitation of level of
higher order harmonics of current of electrical equipment and providing the
electromagnetic compatibility (ЕМС) of nonlinear consumers.
References:
1. S. Fryze, “Active, Reactive, and Apparent Power in Non-Sinusoidal Systems,” Przeglad
Elektrot, no. 7, pp. 193-203, 1931. (in Polish)
2. Sayenko Y.L. Reactive power in the systems of power supply with the nonlinear loading:
Dr.Sc. Diss / Pryazovskyi state technical University.- Mariupol, 2003.- 42 p.
3. Segeda M.S. The Electric networks and systems: Textbook. - Lviv: lviv polytechnic
national university, 2007.- 488 с.
4. IEC 61000-3-2: 2004 Electromagnetic compatibility. Part 3-2. Norms are on emission of
accordions of current(for strength of entrance current of equipment not more than 16 And
on a phase).
5. IEC 61000-6-2:2004 Electromagnetic compatibility. Part is a 6-2. immunity of
industrial equipment to the hindrances. General technical requirements.
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