Implementation of Single-phase pq Theory Mehrdad Tarafdar Haque*, and Toshifumi he** Electrical Engineering Department, Tabriz University, Tabriz, IRAN* Phone & Fax.: 98-41 1-3356025 e-mail:
[email protected] Graduate School of Engineering, Osaka University, Osaka, JAPAN** Phone: 8 1-6-6879-7695 Fax.: 81-6-6879-7263 e-mail:
[email protected]
Abstract Instantaneous reactive power compensation in singlephase scheme is implemented. The analytical analysis is presented. The proposed single-phase based theory can be usedfor reactive power compensation of single-phase as well as three-phase systems, successfully. This method generates sinusoidal reference current waveform in the utility side in imbalanced utility voltage andor imbalanced load current cases. The validity of presented method and its comparison with original p q theory is studied through simulation results.
only in three-phase systems. This subject results in dependency of reference compensation current in one phase to the current and voltage waveform of other two phases. The main objective of this paper is presentation of single-phase pq theory and using it for instantaneous reactive power compensation of each of the phases of a three-phase system, independently. Single-phase pq theory has two main advantages over existing original pq theory as follows: (a) Single-phase pq theory compensates for instantaneous reactive power of single-phase as well as three-phase systems. It can be used in singlephase loaded three-phase systems, too. But, original pq theory generates incorrect compensating currents in remaining phases of a three-phase system, which is loaded by a single-phase load on one of its phases. (b) Single-phase pq theory generates sinusoidal reference currents in the utility side when utility voltages and/or load currents are imbalance but original pq theory cannot generate sinusoidal current waveform in above-mentioned conditions.
Key words: single-phase, pq theory 1 Introduction In modem power systems the requirement for reactive power compensation is becoming more and more rigorous. During the last decades, voltage source inverters (VSIs) due to advancements in power electronics and control methods have attracted a great deal of attention for fast dynamic reactive power compensation. It is possible canceling out the reactive power of load using injection of specified current waveform to the utility by VSI. Obviously, the most important subject in the operation of these inverters is the strategy of generation of reference current waveforms, which the inverter should inject in each of the phases. Between the various methods in this field, the instantaneous pq theory has gained considerable attention [ 11. After the presentation of original pq theory [ 1,2] some instantaneous power based theories presented [3-61. The main objectives of these articles were extension of pq theory to three-phase three-wire and/or three-phase fourwire systems considering different utility voltage and/or load current conditions. These conditions include subjects such as imbalance and/or harmonic polluted utility voltage, imbalance and/or harmonic polluted load current and neutral current compensation [7-91. Considering this fact that the pq theory is a threephase system based theory, it is obvious that all of the researches which are based on this theory, are usable 0-7803-7156-9/02/$10.0002002 IEEE
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Theory of operation
The basic idea of original pq theory begins with transforming the instantaneous space vectors of three phase utility voltages and load currents from a-b-c coordinates into a-p coordinates using (1) and (2) [ 11.
The instantaneous real power, p(t) and imaginary power, q(t) of load is defined as follows:
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From (3), the reference compensating currents on the a-j3 coordinates, icompa and icompp are given as follows:
Considering this fact that the voltages and currents on the p axis are x/2phase lag in respect with the quantities on a axis, it is possible writing the following equations:
-1
Transforming these currents into a-b-c coordinates results in the following equation:
[
*lm1//(1)] fcomp/,(t)
[
1
= - -0.5
icompc(f)
-0.5
0
&/2 -&/2
][
icompa(t)
1
(5)
icomLJp(‘)
Where, ;compa(t) Icompb(1) and icompc(O are the instantaneous compensating currents which should be absorbed by voltage source PWM inverter from the utility in each of the phases a, b and c, respectively. Considering a there-phase utility which has a only one single-phase load on phase “a” it is possible writing the following expression: i b ( t ) = ic(t) = 0
(6)
Using (2) to (4) results in the following equation: (7)
Obviously, situating compensating currents of (7) into (5) result in compensating currents not only in the phase “a” but also in remaining phases “b” and “c”. But, the phases b and c were unloaded and it is not necessary absorbing such a compensating currents from these phases. In other words, original pq theory generates incorrect compensating currents in remaining phases of a three-phase system, which has a single-phase load on one of its phases. Using the presented single-phase pq theory solves this problem. Considering the transformation formulas of a-p coordinates into a-b-c coordinates gives the following expressions for current and voltage on c1 axis:
On the other hand, from (3), (4) and (5) we have:
These equations are correct in absence of negative sequence of load currents and utility voltages. In (1 1) and (12), v’, (t) and i’, (t) lead v, (t) and i, (t) by x/2, respectively. Substituting of (S), (9), (1 1) and (12) in (10) gives the following equation:
In eq. (13), v‘, (t) and i‘, (t) lead v,(f) and i , ( t ) by d 2 , respectively. This equation determines the reference compensating current for instantaneous reactive power compensation in phase “a”. The compensating current of (1 3) depends on the utility voltage and load current in the phase “a”. Considering this fact that compensating cument of phase “a” is independent on the voltages and currents of other two phases makes it possible using this equation for instantaneous reactive power compensation in singlephase systems as well as three-phase systems with single-phase load. In a similar mariner for obtaining (1 3), it is possible concluding compensating currents in the phases b and c using (14) and (15), respectively. These currents are dependent only on the utility voltages and load currents of the phases “b” and “c”, respectively.
3 Control Circuit of Single-phase pq Theory Fig. 1 shows the control circuit of presented method in one of the phases. In this figure v(t), i(t) and icomp(t) stand for the utility phase voltage, load current and reference compensating current, respectively. The notations of v’(t) and i’(t) show d2 phase leaded waveform of v(t) and i(t), respectively. From theoretical point of view, the control block of ‘ W 2 phase lead” should be a quadrature filter, which its transfer hnction is as follows [lo]:
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”“‘+
phase lead i comp 0)
Fig. 1. Control circuit single-phase pq theory The operation of this filter is equivalent with implementation of Hilbert transform in the frequency domain. The output function of this filter is orthogonal to its input signal in steady state as well as transient case. Unfortunately, from practical point of view, it is not possible making this filter, exactly. In this paper the following all pass filter is used for approximation of operation of quadrature filter: 1-6s
H ( s ) = --
1+6S
This circuit has four stages of operation as follows: 1-
From zero to 50 (ms) the load of utility is only the three-phase balanced R-L load. This load would result in reactive power burden on utility. 2The second three-phase balance R-L load switches ON to the utility at 50 (ms). Switching this load ON can be used for studying the dynamic operation of control circuit. 3At 100 (ms), the single-phase load on the phase “c” switches ON. This load generates imbalance load current case. Indeed, the single-phase reactive power compensation ability of control circuit is studied from 100 (ms) to 150 (ms). 4At 150 (ms) to the end of simulation time, the magnitude of voltage of phase “a” is reduced to 50% of its rated value. This generates a voltage imbalance case. On the other hand, the current of phase “a” decreases and the current imbalance case of previous interval changes to a new state.
(17)
Where, T I = l / o l and o1 stands for angular fundamental frequency. In three-phase systems, it is possible using the proposed control circuit of Fig. 1 in each of the phases, independently. In this way, the control circuit compensates for instantaneous reactive power of each of the phases, independently. For example, considering a single-phase loaded three-phase system and using (6), ( 1 4) and ( 1 5) results in zero compensating currents in the phases b and c. The other feature of single-phase pq theory is generation of sinusoidal reference current waveforms in the utility side in imbalance utility voltages and/or imbalance load side currents cases. Obviously, this is because of single-phase based operation of presented theory. The simulation results of following section has been used to explain the operation of single-phase pq theory.
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Fig. 2. Simulated power system
Simulation Results
The operation of presented single-phase pq theory is studied through simulation results. Three similar singlephase pq theory based control circuits (see Fig. 1) are used in each of the phases a, b and c, independently. Fig. 2 shows the simulated power system. The source consists of a 3-phase 4-wire 120 V (rms, L-L), 60 Hz utility. The load consists of a three-phase balance R-L load, another three phase balance R-L load which, can be switched ON and OFF using a three phase thyristor switch and a resistive load which can be switched ON and OFF using a single-phase thyristor switch.
Fig. 3 shows the simulation results. Fig. 3(a) shows the utility voltages. This figure shows the voltage imbalance case from 150 (ms) to 200 (ms). Fig. 3(b) shows the load side currents. These currents are balanced from zero to 50 (ms) and show reactive power burden of load. From 50 (ms) to 100 (ms) these currents are balanced again but reactive power burden of load is increased. From 100 (ms) to 150 (ms) there is a current imbalance case and from 150 (ms) to 200 (ms) there is current and voltage imbalance case, simultaneously. The control circuit uses the voltage and current waveforms of F i g s 3(a) and 3(b) as input signals for generation of reference reactive power compensating current waveforms. Fig.s 3(c) and 3(d) show the reference source side currents after the operation of pq theory and single-phase scheme of pq theory, respectively. Considering Fig. 3(c) shows that original pq theory cannot generate sinusoidal waveform at the source side in load current and/or utility voltage cases (i.e. from 100 (ms) to 200 (ms)). On the other hand, Fig. 3(d) shows that the operation of
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.
0.2 0.1
0 -0.1 m
9
V.L.
1
0
1
1
1
0.05
0.1
0.15
0.2
0.15
0.2
(a) Utility Voltages 0
la
O b
IC
A
0.04
0.02 0
-0.02
-0.04 0
0.05
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(b) Load Side Currents
0.03 0.02
lqsoua
0
(W
0
lqsoub
A
Iqsouc
0.01
0 -0.01 -0.02 -n n? -V.VY
1
- 1
0
0.05
8
8
1
0.1
0.15
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(c) Source Currents by pq Theory 0
'
lasou
0
lbsoo
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lcsou
0.03 0.02 0.01 0 -0.01
-0.02 -n n7 ".W
8
1
1
1
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(d) Source Currents by Presented Method
Fig. 3. Simulation results
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presented single-phase scheme of pq theory not only compensates for instantaneous reactive power but also results in sinusoidal current waveform in all of the load currents and source voltages cases. Obviously, such ability is due to single-phase based operation of this control circuit. It should be noticed that this control strategy would not generate balanced current waveforms in the utility side.
5
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[41
Conclusions
The demerits of original pq theory in compensation for a single-phase loaded three-phase utility are discussed through analytical expressions. Single-phase pq theory is presented and its application in abovementioned condition is studied through analytical expressions. The formula of single-phase reference instantaneous reactive power compensation current is derived. The validity of presented control strategy is proved through simulation results. These results show valuable dynamic response of single-phase pq theory. The independent operation of single-phase pq theory in each of the phases is other feature of presented control strategy, which results in sinusoidal current waveform in imbalance load current and/or utility voltage cases.
[71
6 References [l]
[2]
H. Akagi, Y. Kanazawa, A. Nabae, “Generalized Theory of the Instantaneous Reactive Power in Three-phase Circuits,” Proc. of IEEJ International Power Electronics Conference. (IPEC-Tokyo), 1983, pp. 13751386. H. Akagi, Y. Kanazawa, A. Nabae, “Instantaneous Reactive Power Compensators Comprising Switching Devices without Energy Storage components,” IEEE Trans. Ind. Appl., Vol. 20, No. 3, MayIJune 1984, pp. 625-630.
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J. L. Willems, “A New Interpretation of the Akagi-Nabae Power Components for Nonsinusoidal Three-phase Situations”, IEEE Trans. Instrum. Meas., Vol. 41, No. 4, Augest. 1992, pp. 523-527. A. Nabae, H. Nakano and S. Togasawa, “An Instantaneous Distortion Current Compensator without any Coordinate Transformation,” Proc. of IEEJ International Power Electronics Conf. (IPEC-Yokohama), 1995, pp. 1651-1655. F. Z. Peng and J. S. Lai, “Generalized Instantaneous Reactive Power for Three-phase Power Systems,” IEEE Trans. Instrum. Meas., Vol. 45, No. 1, Feb. 1996, pp. 293-297. M. T. Haque, S. H. Hosseini and T. Ise, “A Control Strategy for Parallel Active Filters Using Extended pq Theory and Quasi Instantaneous Positive sequence Extraction Method,” International Symposium on Industrial Electronics (ISIE 2001-Pusan-Korea), Vol. 1, pp. 348-353. S. J. Huang and J. C. Wu, “A Control Algorithm for Three-phase Three-Wired Active Power Filters under Non Ideal Main Voltages,” IEEE Trans. Power Electronics, Vol. 14, No. 4, July 1999, pp. 753-760. Y. Komatsu and T. Kawabata, “A Control Methods of Active Power Filter in Unsymmetrical and Distorted Voltage System,” Proc. of PCC - Nagaoka’97, pp. 16 1- 168. M. T. Haque, T. Ise and S. H. Hosseini, “A Novel Control strategy for Active Filters Usable in harmonic polluted and/or Imbalanced utility voltage Case of 3-Phase 4-wire Distribution systems,” 9’th Intl. Conf. on Harmonics and Quality of Power, Vol. I , Oct. 2000, Orlando, USA, pp. 239-244. J. G. Proakis and M. Salehi, “Communication Systems Engineering,” Prentice-Hall Intl., Inc. 1994.
