Wavelet based reference current calculation method for active compensation systems/ Vilneliu saltinio sroves apskaiciavimo metodo taikymas aktyviosioms kompensavimo sistemoms.
Active compensation and filtering systems have been becoming more important day by day for ensuring high quality in power generation and reducing its cost. Passive compensation and filter circuits have been replaced by SCR-based circuits as a result of development in semi- conductor technology. Determining reference current for these circuits is quite important for filter and compensation circuits [1-12].
Reference current can be defined as total of all instantaneous active and reactive currents drawn due to unbalanced loads or harmonic distortions. An active compensation circuit is run in a way that it generates this reference current. Most frequently preferred methods for obtain reference current are p-q theory, Peng theory, Park Power theory etc. PWM technique is generally preferred in compensator current control [1-12]. p-q theory, which is most prevalent calculation method, was took reference in this study.
p-q theory is based on finding power values by converting voltage and current from a, b, c phase plane into [alpha], [beta], 0 plane [7,13,14].
Voltage and current vectors are expressed as the following to show three-phase instantaneous voltage values [v.sub.a], [v.sub.b], [v.sub.c] and three-phase instantaneous current values [i.sub.a], [i.sub.b], [i.sub.c]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (1)
This voltage's and current's a, p and 0 values are expressed as (2-3).
Instantaneous active power is found by scalar multiplication (*) of voltage and current vectors while reactive power is found by vector multiplication of them (x) (4).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (4)
Instantaneous active and reactive power can be expressed as the following for the systems without 0 component
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (5)
In this equation, [bar.p] stands for mean instantaneous active power while [??] represents alternating active power, [bar.q] represents mean instantaneous reactive power and [??] represents alternating reactive power. All powers must be compensated except [bar.p] [1-14]. Fundamental current drawn from the source ([[??].sub.f]) and shunt-compensator's reference current are calculated by using [bar.p] mean instantaneous active power
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)
Discrete Wavelet Packet Transformation
Wavelet transformation is one of the most recently developed methods for signal analysis. It is prevalently used in many areas like mathematics, physics and engineering .
Wavelet packet transformation is an expression of digital signal with time scale. The signal resolution is changed by filtering process while its scale is replaced by down sampling process. Output of cascaded filter banks depends on the main wavelet and high- and low-frequency components of the system. The procedure is started by passing a N-length discrete signal s(n) through an impulse response high-pass filter h(n) and an impulse response low-pass filter g (n) . Responses of high-pass and low-pass filters constitute first-stage decomposition of the discrete signal and expressed as the following [17, 18]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)
Extension functions ([[phi].sub.j,j]k](x) and [[psi].sub.jk](x)) are not seen in this equation. This allows simulations to be performed through digital filter banks on computer environment [17, 18]. [s.sup.1.sub.0](n) and [s.sup.1.sub.1](n) sequences are more decomposed at second stage of wavelet packet transformation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)
The entire wavelet packet decomposition can be produced by following similar procedure;
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)
The decompositions from the sequence [s.sup.i.sub.n] (n) at stage i into the sequences [s.sup.i+1.sub.m0](n) and [s.sup.i+1.sub.m1] at stage i +1 using the two decomposition quadrature filters are given by [17, 18]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (10)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (11)
The proposed method is, first of all, based on calculation of instantaneous active and reactive powers according to Peng's generalized reactive power theory with no need for p-q transformation. Instantaneous active power is calculated as the following [6, 7]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)
Instantaneous active power is a scalar magnitude according to the equation above. Instantaneous reactive power is calculated as the following [6,7]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)
Reactive power is a vector magnitude and [absolute value of [??]] represents magnitude of instantaneous reactive power for three phases [6,7]. DWPT decompositions of instantaneous active power at third stage can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)
DWPT decompositions of instantaneous reactive power for each phase at third stage can be written as;
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)
where instantaneous active currents drawn for each decomposition of instantaneous active power:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
Sum of these currents yields value of the drawn instantaneous active current
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)
where instantaneous reactive currents drawn for each decomposition of instantaneous reactive power:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
Sum of these currents yields value of the drawn instantaneous reactive current
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)
Frequency ranges of current decompositions for sampling rate of 0.000625 seconds are given in Table 1 .
As seen in the table, currents are decomposed into 100 Hz equal frequency ranges. Therefore, currents of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] are active and reactive currents drawn at fundamental frequency of 50 Hz. These currents cause [bar.p] ve [bar.q] powers from which the load requires from the source and consequently, all currents must be compensated except [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] [19, 20, 21]. If is is the source current, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the load current and iC is the compensation current then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)
Because the load current is equal to sum of active and reactive currents then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)
Because we aim that only the basic frequency [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] active current is drawn from the source, all other current components account for shunt compensator's reference current ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]). Thus,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)
Working principle of DWPT-based reactive power and harmonic compensator is given in Fig. 1.
[FIGURE 1 OMITTED]
Calculations and simulations were carried out according to p-q theory, as first, and then, according to the proposed method. Then, the results were compared. db20 was used in all DWPT simulations.
Phase-neutral effective value of source voltage is 2400 V and frequency is 50 Hz. Fig. 2 shows configuration of the compensation system. It was assumed that the compensator can track the reference current at instantaneously at infinite speed. Fig. 3 shows the source voltage and non-linear load current.
[FIGURE 2 OMITTED]
[FIGURE 3 OMITTED]
p and q powers produced through (1-5) based on p-q theory are given in Fig. 4.
[FIGURE 4 OMITTED]
To find [bar.p], p signal was made pass through a low-pass filter, whose cut-off frequency was 30 Hz. Fig. 5 shows fundamental frequency instantaneous active current, ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) and compensator's reference current ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) obtained by using (6).
[FIGURE 5 OMITTED]
[FIGURE 6 OMITTED]
[FIGURE 7 OMITTED]
Fig. 8 and 9 show DWPT decompositions of instantaneous active and reactive currents found by using (17-18) according to generalized reactive power theory.
[FIGURE 8 OMITTED]
[FIGURE 9 OMITTED]
Fig. 10 shows compensator's reference current [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] values calculated according to the proposed method considering [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].
[FIGURE 10 OMITTED]
Fig. 11 shows the compensation results obtained through both of the methods
[FIGURE 11 OMITTED]
The biggest difference value in of both compensation currents is around [+ or -]7 A (Fig. 12).
[FIGURE 12 OMITTED]
In this study, a DWPT-based method was proposed for calculating reference current for active compensation systems. Unlike classical methods, mean instantaneous active power [(bar.p]) was obtained directly through DWPT and moreover, instantaneous active power signal (p) does not require to be made pass through low-pass filter. Fundamental frequency instantaneous active current ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]), reference current ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]) and other currents can be calculated easily by using proposed method. In comparison of the obtained compensation current with that, which is obtained according to p-q theory, maximum variation in current is [+ or -] 7 A. This accounts for very small and negligible error as 0. 43. according to maximum compensation current, [+ or] 1600 A. Thus, the proposed method can be used in active harmonic and compensation systems.
Received 2010 10 22
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G. Gokmen Department of Electrical Education, Technical Education Faculty, Marmara University, Goztepe Campus, 34722 Kadikoy, Istanbul, Turkey, phone: +90 216 336 5 7 70 (265), e-mail:firstname.lastname@example.org
Table 1. Wavelet packet decompositions of currents and their frequency ranges Decomposition Frequency Harmonic Range (Hz) Order [i.sup.3.sub.p000], [i.sup.3.sub.p000] 700~800 15 [i.sup.3.sub.p001], [i.sup.3.sub.p001] 600~700 13 [i.sup.3.sub.p010], [i.sup.3.sub.p010] 500~600 11 [i.sup.3.sub.p011], [i.sup.3.sub.p011] 400~500 9 [i.sup.3.sub.p100], [i.sup.3.sub.p100] 300~400 7 [i.sup.3.sub.p101], [i.sup.3.sub.p101] 200~300 5 [i.sup.3.sub.p110], [i.sup.3.sub.p110] 100~200 3 [i.sup.3.sub.p111], [i.sup.3.sub.p111] 0~100 1
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|Title Annotation:||ELECTRICAL ENGINEERING/ELEKTROS INZINERIJA|
|Publication:||Elektronika ir Elektrotechnika|
|Date:||Feb 1, 2011|
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