ISSN 10683712, Russian Electrical Engineering, 2011, Vol. 82, No. 6, pp. 332–336. © Allerton Press, Inc., 2011. Original Russian Text © G.S. Zinoviev, 2011, published in Elektrotekhnika, 2011, No. 6, pp. 54–58.
Expansion of Set of Power Efficiency Factors of Power Electronics Installations G. S. Zinoviev Received May 23, 2011
Abstract—Based on the author’s direct calculation methods of powerproducing processes, new quality indi cators of nonsinusoidal voltages and currents in the form of integral and differential coefficients of harmonics of voltage and current have been obtained. The advisability of introducing them into the standards for the quality of the power network voltage and the consumption current, as well as also into the method for calcu lating of the partial participation of the consumer and the power supply system in the change in the overall quality of the electric energy has been shown. Keywords: harmonic factors, partial participation factors, electricpower quality standards. DOI: 10.3103/S1068371211060137
The energysaving problem in electrical devices in general and power electronics installations in particu lar is determined in many respects by the quality of the electric energy. A more profound understanding of the causes of losses conditioned by low quality of electric energy determines the need to introduce more advanced characteristics of the quality of electric energy compared to those set according to the various standards (GOST State Standards, IEC, ES). In Rus sia, this is above all GOST State Standard 13109–97 for the quality of the supplyline voltage and GOST State Standard 51317.3.2–99 for the quality of the consumption current. This is indicative that first stan dard with every new revision increases the number of critical quality factors of the electric energy (voltage) from seven in the first editing (1965) to 11 in the last editing (1997). The second standard, which is new for Russia, is analogous to the International Standard IEC 6100032–95, which has no integral character istics for the current quality and contains only spectral characteristics. In particular, the overall expansion of power elec tronics devices, which, in developed countries, con vert more than 50% of the total energy output, accen tuates problems related to generating higherorder harmonics of the voltage and current at the input and output of semiconductor converter installations and the estimation of losses related with that. There are many publications on these problems, the summaries of which are presented in [1–3]. In given work, the effectiveness of new integral factors of quality of non sinusoidal processes that are missing in the Russian State Standards, is shown and the rationality of intro ducing them into new standards for the quality of elec tric energy is discussed.
NEW FACTORS IN VOLTAGE QUALITY It is known from [4, 5] that the current harmonics factors in the inductive elements, asynchronous machines, and capacitive elements are identically determined as follows: in the first case, they are deter mined by the integral coefficient of the firstorder har monics of voltage K hν ; in the second case, they are determined by the multiplicity of the starting current of asynchronous motor Kst multiplied by the integral coefficient of harmonics of voltage K hν ; and, in the third case, they are determined by the differential ˆ of the first coefficient of the harmonics of voltage K hν order, i.e., K hcL = K hν ,
ˆ , K hcC = K hν
K hcIM = K st K hν ,
(1)
where (q)
K hν =
∞
⎞ ∑ ⎛⎝ nU⎠
2
q
n=2
ˆ (q) = K hν
Un
∞
(q)
1
⎞ ∑ ⎛⎝ n U⎠
n=2
q
= ω U hh /U 1 ;
q Un
(2) 2
(q) q = Uˆ hh /ω U 1
1
are the integral and differential coefficients of the qorder harmonics of voltage, respectively, and n is harmonic’s number. The first expression may be con sidered to be common to both coefficients of harmon ics, which are obtained from this expression at q > 0 and q < 0, respectively. The integral and differential coefficients of qorder harmonics may be calculated through the harmonics of the voltage or through the effective meaning of a qrepeated integrated or differ
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EXPANSION OF SET OF POWER EFFICIENCY FACTORS (q)
entiated voltage of higherorder harmonics U hh and ˆ ( q ) , respectively. Then, U hh
T (q) U hh
=
T
ˆ (q) = U hh
q 2 1 ( u hh ) dt ; T
∫
q 2 1 ( uˆ hh ) dt . T
0
∫ 0
The instantaneous values of qrepeated integrated and differentiated voltage of higherorder harmonics are determined as follows: q
u hh =
∫
It is also rational to use integral (differential) coef ficients of the harmonics of the voltage (and current) of fraction orders, the determination of which follows from the general formula for integral (differential) coefficients of harmonics in accordance with (2) at fraction q. However, the IEC 892 Standard contains new fac tor of voltage quality, i.e., the harmonic variation fac tor (HVF), as follows: 1 HVF = u1
∫
… ( q ) ( – times ) ( u – u 1 ) dt; q
d ( u – u1 ) q . uˆ hh = q dt The quality of the current is determined without calculating its instantaneous value as early as at the design stage by new factors of the voltage quality, which, for typical forms of converter’s voltages, are known beforehand. The new quality factors of voltage for a particular active mains supply can be measured and, using the results, one can predict the current quality of a potential consumer. In electric circuits that are modeled by an norder electric circuit and described in the form of the norder differential equation n
l
di a l l = dt l=0
∑
m
l
du
, ∑ b dt l
(3)
l
l=0
the current quality in any j element of the circuit at the transformation of the differential equation into alge braic form by the ADU2 method [5] is determined by the circuit parameters and set m integral coefficients of the harmonics of the voltage, starting with the n–m
333
n = 13
∑
n=2
2
Un . n
(5)
This factor allows one to measure the harmonics of the lowfrequency part of the voltage spectrum, which is important for, e.g., asynchronous motors. The new GOST State Standard 5131732–2008 [6] contains a more common factor, known as the partly weighted coefficient of harmonic components (PWHD): n max
PWHD =
2
⎞ , ∑ n ⎛⎝ U⎠ Un
(6)
1
n min
where nmin, nmax may be selected in the range of values of 2–40. It is appropriate to note here that Western calcula tion methods often use the determination of the weighted coefficient harmonics of voltage WTHD, which is determined similarly to the presented integral coefficient of harmonics of the first order.
(4)
NEW QUALITY FACTORS OF CONSUMPTION CURRENT It has been shown in [5] that, in the presentation of the reactivity of the power network by inductance L, the coefficient of the harmonics of the power network voltage will be determined through the consumption current by the formula ˆ , (7) K = K K
The formulas for calculating the coefficients of current harmonics for the circuit elements of any order can be taken from [5]. Thus, determining the current quality in a circuit with a nonsinusoidal volt age of the known form is reduced to calculations using a ready formula in the known set of integral and differ ential coefficients of the harmonics of applied voltage. Since the majority of the engineering models of elec trical devices are interpreted by a circuit of no more than the second (third) order, it seems sufficient to introduce norms with regard to the integral (differen tial) coefficients of the harmonics of the powernet work voltage up to the second (third) order into the standards for the quality factors of the voltage.
where Ksc is the shortcircuit current multiplicity of the power network with respect to the effective value of ˆ is the first harmonic of the consumption current; K hν the differential coefficient of the firstorder harmonics of the consumption current, which is determined sim ilarly (and accurately within the notations) to the dif ferential coefficient of the firstorder harmonics of the voltage in accordance with expression (2). Using the direct calculation method [5], it is easy to show that, in the interpretation of the powernet work inductance by a secondorder circuit (LC filter), the coefficient of harmonics of the powernetwork voltage is determined through the integral coefficient of the firstorder harmonics of the consumer current.
(n – m)
(n – m + 1)
order, i.e., K hν etc., and in total
and, correspondingly, K hν
K hcj = f ( a n , a n – 1, …, a 0, b m, b m – 1, …, b 0, U 1 ; (n – m)
K hν
(n – m + 1)
, K hν
(m)
, K hν ).
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hc
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Standard 13109–97) presents the following relation to calculate this participation:
Lc e
i1
i2
in
~
QF off b = 1 – , QF on
u
Equivalent circuit of power network with several consumers.
The second factor of the electromagnetic compatibil ity of the consumer and power network is the coeffi cient of harmonics of the current of the power network, which is determined by the integral coefficient of second order harmonics of the consumption current. Thus, for ease and speed of estimating the reverse influence of the consumer with a nonsinusoidal cur rent on the voltage and current of the power network, it has been proposed in [2, 4] to introduce differential and integral coefficients of the firstorder harmonics of the consumption current into the standard for the quality of the consumption current. However, the intrinsic Russian standard on the quality of the con sumption current quality is absent. GOST State Stan dard 5131732–99 is based on the corresponding IEC Standard (IEC 6100032–95). In these standards, the GCT factor is determined as the common coeffi cient of the influence of harmonic components, which is calculated similar to the differential coefficient of the firstorder harmonics of the consumption current, which were previously determined as follows: 40
GCT =
∑
n=2
2 2 I n ⎛ n⎞ . ⎝ I 1⎠
(8)
In addition, the influence of partly weighted coef ficient of harmonic components (PWCHC) is also introduced: 40
PWCHC =
∑
n = 14
2 2 I n ⎛ n⎞ . ⎝ I 1⎠
(9)
This coefficient allows one to measure the harmon ics of the highfrequency part of the current spectrum, rather than the HVF factor, in accordance with (5), which measures the harmonics of the lowfrequency part of the spectrum. FACTORS OF REVERSE INFLUENCE OF SEVERAL CONSUMERS ON QUALITY OF POWERNETWORK VOLTAGE When there are several consumers with nonsinuso idal currents fed from a common power network, the question is raised of the partial participation of indi vidual consumers in common voltage distortion. Our standard for electricenergy quality (GOST State
(10)
where QFon and QFoff are the quality factors of the elec tric energy at the switchon and switchoff states of a given consumer, respectively. However, modeling [9–13] has shown that this approach leads to an error of up to 50%; therefore, it may not always be used, since the given consumer may be switched off. Instead, a method for adequately determining the participations in the load and power system in the change in the voltage quality at the point of common connection (PCC) has been proposed that does not require switching off by the consumer and may be applied at any number of consumers. The method is also applicable at any number of branches containing EMF sources adjoined to the PCC. Figure 1 shows the calculated equivalent circuit of a power network with several consumers of nonsinuso idal current at the PCC and with one branch with the EMF. The differential equation for the voltage of higher order harmonics in the PCC initially subject to only two consumers has the form di 1a di 2a u a = e a – L – L , dt dt
(11)
where the variables with index a denote the abnormal components of the processes in given case the high order harmonics of currents of two consumers i1a, i2a and highorder harmonics of EMF ea of the power network. After transforming a differential equation into algebraic form in accordance with the ADU2 method (by squaring an average over a period) and separating the left and righthand sides of the equation on the squared effective value of the highorder harmonic of voltage (Ua)2, the following algebraic equation is obtained: T
T
∫
∫
2 2 1 1 e 2 dt + 1 L ⎛ di 1a⎞ dt 1 = a 2 2 Ua T 0 U a T 0 ⎝ dt ⎠ 2
T
2
2
T
1 L ⎛ di 2a⎞ dt + 2 L di 1a di 2a + dt 2 T ⎝ dt ⎠ 2 Ua 0 U a T 0 dt dt
∫
∫
T
d( i 1a + i 2a ) 2 L – e a dt = OPC * ( e a )OPC * ( i 1a ) 2T dt Ua 0
∫
– OPC * ( i 2a ) + MPC * ( i 1a, i 2a ) + MPC * ( e a, ( i 1a + i 2a ) ). Here, OPC(ea) is the partial participation of a corre sponding branch (e.g., ea) that adjoins to the node in
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the total change in the voltage quality in the node and MPC(i1a, i2a) is the mutual partial participation of two branches adjoining the node in the total change in the voltage quality in the node. The physical meaning of the factor of the partial participation OPC() of a specific branch adjoining node of the power network consists of determining the loss of voltage quality in the node of the power network that would arise in the absence of a loss of voltage qual ity in another branches. The physical meaning of the factor of the mutual partial participation MPC() of two branches adjoining the node consists of determining the change in voltage quality in the node of the power network (deterioration or improvement) that is condi tioned by the summation of voltage abnormalities in the node of every branch. The coefficient of the partial participation of the electric system in the voltage distortion in the PCC is determined as follows: K ( e a ) = OPC * ( e a ) + MPC * ( e a, ( i 1a + i 2a ) ) OPC * ( e a ) × , OPC * ( e a ) + OPC * ( i 1a ) + OPC * ( i 2a )
(12)
which takes into account both the partial participation of the system and its mutual participation in the inter action with consumers. The coefficient of the partial participation of the first nonlinear consumer is determined analogously taking into account both the system’s participation and its mutual participation with other consumers and the electric system as follows: K ( i 1a ) = OPC * ( i 1a ) + MPC * ( i 1a, i 2a ) OPC * ( i 1a ) × + MPC * ( e a, ( i 1a + i 2a ) ) OPC * ( i 1a ) + OPC * ( i 2a ) (13) OPC * ( i 1a ) – . OPC * ( e a ) + OPC * ( i 1a ) + OPC * ( i 2a ) The coefficient of the partial participation of the second nonlinear consumer is found as follows: K ( i 2a ) = OPC * ( i 2a ) + MPC * ( i 1a, i 2a ) OPC * ( i 2a ) × + MPC * ( e a, ( i 1a + i 2a ) ) OPC * ( i 1a ) + OPC * ( i 2a ) (14) OPC * ( i 2a ) – . OPC * ( e a ) + OPC * ( i 1a ) + OPC * ( i 2a ) It is evident that these coefficients are related by the common equation K ( e a ) + K ( i 1a ) + K ( i 2a ) = 1, RUSSIAN ELECTRICAL ENGINEERING
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which allows one to calculate (measure) only these two coefficeints, and find the third using this equation. This equation may also be used to test the accuracy of calculating (measuring) the partial participation coef ficients. When the number of consumers is greater than two, one can similarly construct more complicated formu las for calculating their coefficients of partial partici pation. Another possibility of conducting calculations using the above relationships consists of the initial sep aration of one consumer and the consideration of all other consumers as a resulting second consumer. Then, the calculations are repeated for another sepa rate partial consumer. The coefficients of partial par ticipation will not be determined simultaneously for separate consumers. One can also construct formulas for a more complicated form of complex resistance in the power network. It should also be noted that the above approach to determining the partial participation of consumers in a common voltage distortion based on highorder har monics at a PCC can also be expanded to other types of voltage distortion (asymmetry, oscillations, flicker, etc.). The integral approach proposed here is easy to use, rather simple to calculate, and provides more single value results than the Western approach to determin ing the partial participations of separate consumers in a common voltage distortion in the network based on a singleharmonic analysis [14]; however, it is inferior to the spectral approach in the possibility of differen tiating the results and the determination of more deli cate characteristics of the quality of electric energy on their basis. It seems that both approaches mutually supplement each other. It is advisable to use integral factors at the first stage of the express analysis of the quality of electric energy, then to carry out more extensive studies of the voltage quality at the second stage using spectral representation. The proposed technique for determining the partial participation of consumers has been tested using a mathematical model of a threephase network with two threephase controllable bridge rectifiers. The error in determining the partial participation of sepa rate rectifiers in the total coefficient of voltage har monics of the power network using the existing GOST State Standard technique in accordance with balance equation (14) equaled 32%, whereas in calculations according to the proposed technique, this error is only 1.4%. The error in the partial participation of separate rectifiers in the total integral coefficient of the voltage harmonics of the power network according to the bal ance equation using the existing State Standard tech nique equaled 50% and less than 0.1% using the pro posed technique [9, 12, 13].
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CONCLUSIONS (1) It is advisable to supplement the standards for the quality of electric energy (GOST State Standard 13109–97 and analogous International Standards) by the norms for the integral and differential coefficients of voltage harmonics, since they enable one to esti mate the quality of nonsinusoidal current in consum ers without calculating the instantaneous value or the current spectrum. (2) It is rational to supplement the standards on the electriccurrent quality (GOST State Standard 51317.3.2–99 and analogous IEC 6100032–95 International Standard) with the norms on the integral and differential coefficients of highorder harmonics of consumed current, which enable one to estimate the voltage quality of the power network with regard to highorder harmonics without calculating the instan taneous value or voltage spectrum. (3) In the standard on the quality of electric energy (GOST State Standard 13109–97), it is advisable to change the method for determining the partial partici pation of the load in the change in the voltage quality, which yields large errors for nonorthogonal currents of the consumers, for the technique proposed in this paper, which is based on the author’s direct calculation methods and is adequate for any value of the current.
4. Zinoviev, G.S., Results of Solving Some EMC Prob lems of Valve Converters, Elektrotekhn., 2000, no. 11, pp. 12–16. 5. Zinoviev, G.S., Osnovy silovoi elektroniki (Foundations of Power Electronics), Novosibirsk: NGTU, 2009. 6. GOST (State Standard) no. 51317.4.7–2008 (IEC 6100047–2002): Obshchee rukovodstvo po sredstvam izmerenii i izmereniya garmonik i intergarmonik dlya sistem elektrosnabzheniya i podklyuchaemykh k nim tekhnicheskikh sredstv (General Manual on Measuring Instruments and for Harmonics and Interharmonics Measuring for Power Supply Systems and Connected Devices), Moscow: Standartinform, 2009. 7. Zinoviev, G.S., Direct Method for Calculating Powers in Circuits with Valve Converters, Elektrichestvo, 1989, no. 6, pp. 70–75. 8. Zinoviev, G.S and Mukhacheva, G.L., The Way to Cal culate the Electric Energy Q–Factors in the Mains with Valve Converters, Elektrotekhn., 1989, no. 7, pp. 62–64. 9. Zinoviev, G.S and Popov, V.I., New Approach for Esti mating the Electromagnetic Compatibility of Valve Converters with the Main and Load, Elektrichestvo, 2007, no. 8, pp. 29–34. 10. Zinoviev, G.S., Popov, V.I., and Zotov, L.G., Develop ment of Direct Methods of Definition of Contributions of Power Converters in Common Distortion of Voltage of Mains, Proc. CPE2005, Gdansk, 2005. 11. Zinoviev, G.S. and Gnatenko, M.A., RF Patent no. 2191392 RF, Byull. Izobret., 2002, no. 29.
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12. Alekseev, V.Yu., Zinoviev, G.S., and Popov, V.I., RF Patent no. 2307364 RF, Byull. Izobret., 2007, no. 27. 13. Zinoviev, G.S., Baulin, V.G., Volkov, A.V., et al., Elek tromagnitnaya sovmestimost’ ustroistv silovoi elektroniki (Electromagnetic Compatibility of Power Electronic Devices), Novosibirsk: NGTU, 2006, part 2. 14. Chen, C., Lin, X., Koval, D., Xu, W., and Tajasanant, T., Critical Impedance Method – A New Detecting Har monics Source Method in Distribution Systems, IEEE Trans. Power Deliv., 2004, vol. 19, no. 1, pp. 288–297.
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