16th International Middle- East Power Systems Conference -MEPCON'2014 Ain Shams University, Cairo, Egypt, December 23 - 25, 2014
A new Approach for Defining Three -Phase Power Components based on the Instantaneous Power Theory: Part 1- Analytical Study Hossam K. M. Youssef
Ayman A. Eisa, A. H. Elbahrawy ,and Omar Fathy F.
Member, IEEE Faculty of Engineering Cairo university , Egypt
Egyptian Atomic Energy Authority ,Egypt
[email protected]
Abstract - This paper proposes a new approach for definition of power components in three-phase four-wire balanced system under sinusoidal and non-sinusoidal condition. This new instantaneous power approach for the calculation of power components extends and adapts, for three-phase conditions, the procedure applied by Ref. [1] for the quantification of power components in single-phase two-wire systems. In this study, a new approach which based on the analysis of three-phase instantaneous power flows of both fundamental and all harmonics of signals is proposed. Index Term - power theory, power definitions, power components, sinusoidal and non-sinusoidal conditions.
I.
INTRODUCTION
Industrial applications based on three-phase static converters and adjustable speed drivers are examples of threephase balanced non-linear loads that produce distortion in the current waves. The presence of voltage and current components at frequencies other than the fundamental one is harmful to the equipment connected in the power system network. Harmonics can be produced by the loads only, by the source only or by both the source and the load. Power components such as active, reactive and apparent power are well defined under sinusoidal operating conditions in linear balanced three-phase systems, at present the definitions and physical meanings of power components under nonsinusoidal conditions are still debatable and none of the approaches can lead to power component definitions that satisfy all required power properties under non-sinusoidal.
II.
LITERATURE REVIEW
In recent years, many papers on this topic [1–17] have been issued, but to date, only two approaches have dominated, the American which is recommended by IEEE, and the European which is accepted by IEC. The first one, also called the practical approach [6], is presented in the Standard IEEE Std 1459-2000 [7]. The second approach is sometimes referred to as the theoretical approach [6] or FBD (Fryze–Buchholz– Depenbrock) method [8]. There are many articles have been
published trying to compare the two methods by focusing on their divergences and convergences. In Ref. [9], surveys previously published literature of power theories and definitions for non- sinusoidal situations. Recently, authors have motivated to use more coordinate transformations and different types of projections in order to determine some of instantaneous power components. Generally, most of them decompose the load currents and power signal into different divisions, such as: active and reactive [10]; orthogonal active and residual [11]; active, reactive, scattered and generated components [12]; instantaneous active and instantaneous reactive [13]; active and inactive [14]; distortion-less and distortion [15]; real and imaginary [16]; compensable and noncompensable [17]; fundamental and nonfundamental [18]; active, reactive, distortion, non-active and apparent power [7]. In Ref. [19], the authors propose a new definition of the unbalance power using an instantaneous approach based on the well-established concepts used in IEEE Std. 1459 for the definition of active and reactive power in single-phase systems. In Ref. [20], authors have developed an exhaustive analysis of the p–q instantaneous reactive power theory based on a new formulation without using mapping matrices. In Ref. [21], the authors presented a comparison on power decompositions in a simple single-phase circuit with nonsinusoidal waveforms of voltage and/or currents. In Ref. [22], a generalized instantaneous reactive power theory formulation has been presented, this form calculates the instantaneous reactive power, which can be applied not only to three-phase power systems. In Ref. [23], defined a general theory of instantaneous power for multi-phase, m-wire systems with distortion, unbalance and direct current components that can be calculated at any instant from instantaneous values of voltage and currents, without prior information of the voltages and currents. In Ref. [24], extends, still using vector algebra, the decomposition of current components in the instantaneous power domain for any m-wire system to decomposition in the RMS domain. In Ref. [25], the authors proposed a new definition of the non-fundamental effective apparent power based on an analysis of instantaneous power flows and which avoids the problems in the harmonic active power calculation.
16th International Middle- East Power Systems Conference -MEPCON'2014 Ain Shams University, Cairo, Egypt, December 23 - 25, 2014 All these efforts are appreciated, but their produced concepts for power definition have been analyzed and criticized among each other. It is obvious that instantaneous power is the most accurate value represents the energy content of a system since the law of ―conservation of energy‖, a fundamental principle of science, is satisfied by this power in ac system. From the literature review presented in this paper, there are several works in which instantaneous power flows are used in the analysis of electrical systems. Ref. [1] introduced new definitions of power components which presents together the form and amplitude of the instantaneous power, and define it in Phasor form without the need of sophisticated mathematical models or theories, and without changing the vector plane by new coordinate transformation. It deals with single-line systems for residential applications. This work is applying that approach for threephase balanced systems. The new definitions was developed to give guidance with respect to the quantities that should be measured or monitored for revenue purposes, engineering economic decisions, and determination of major harmonic polluters [7].
III.
ANALYSIS OF POWER COMPONENTS IN BALANCED THREE-PHASE SYSTEMS UNDER SINUSOIDAL CONDITIONS
In this case, the line-to-neutral instantaneous voltages at time instant t are as follows: ( () ) (1) √ ( ) () (2) √ ( ) () √ (3)
()
()
()
()
(10)
Where: s(t) is defined as the instantaneous phasor power for each phase p(t) is defined as the instantaneous active power for each phase ( ) is the average or active power (true power) for each phase, which is also the amplitude of p(t) q(t) is defined as the instantaneous reactive power for each phase, ( ) is the amplitude of q (t) The instantaneous power per phase ua(t) follows a sinusoidal oscillation with a double frequency (2f) shifted up by the average active power per phase P. The sinusoidal oscillation is characterized by a function defined as the instantaneous phasor power per phase s(t), which can be decomposed into two sinusoidal orthogonal functions p(t) and q(t). These two functions are defined as instantaneous active and reactive powers respectively. The sinusoidal power components, mentioned above, can be represented vectorially as phasor quantities on a phasor diagram as shown in fig. 1.
V Q
I
S
The line instantaneous currents at time instant t are as follows: () ( ) (4) √ ( ) () (5) √ ( ) () √ (6) Using equations (1) and (4), the instantaneous power per phase is defined as the product of instantaneous voltage and instantaneous current per phase. () () () (7) For three –phase sinusoidal balanced system; () () () The total instantaneous power ( ) is defined as: () () () () () () ( ) ( ) ( ) ( ) (8) () ( ) ⁄ ) ( () ( ) ( )] (9) () ( ⁄ ) ( )]
P
Note: Dotted line … for V&I vectors with rotating speed Solid line _____ for power vectors with rotating speed fig. 1. Phasor diagram for the vectors of power components per phase
This diagram is familiar and acceptable among the power engineering society. The phase angels shown are taken when we use the same plane axis of the V & I phasor diagram as a reference plane, keeping in mind that the speed of rotation for power components vectors is twice the basic supply frequency. If we take the phase ―a‖ current phasor as the reference for angle; that is, to choose ia= 0 , so that phasor Ia lies along the real axis, the phasor diagram plane of the power components can be appeared as in fig. 2.
16th International Middle- East Power Systems Conference -MEPCON'2014 Ain Shams University, Cairo, Egypt, December 23 - 25, 2014
Active power
S VI P VI cos0
Reactive power
Q VI sin / 2
Phasor power
V Q I
P
S
Thus, the instantaneous phasor power per phase s(t) have been represented as a vector S in a power phasor diagram. This diagram has two perpendicular axises represent the reference directions of active and reactive powers flow. IV.
ANALYSIS OF POWER COMPONENTS IN BALANCED THREE-PHASE SYSTEMS UNDER NONSINUSOIDAL CONDITIONS
In this case, the line-to-neutral instantaneous voltages at time instant t are as follows: Fig. 2 Phasor diagram for power components, while choosing the phase a current vector as a reference axis
In this case, -
v - i = v is the phase difference between V&I reactive power Q will always be with zero phase shift active power P will always be with –/2 phase shift phasor power S will always be with /2 phase shift
This appearance stems from the power triangle, S, P, Q, and is useful in power flow studies. Thus, rotating all the power vectors in figure 2 by /2 anticlockwise direction will lead to the conventional power flow directions as interpreted in literature [7], and summarized in fig. 3.
()
∑√
(
()
∑√
(
()
∑√
(
)
(
)
)
(
)
)
(
)
The line instantaneous currents at time instant t are as follows: ()
∑√
(
()
∑√
(
)
()
∑√
(
)
)
(
)
(
)
(
)
Where: j&k are integer numbers the rms value of the jth harmonic voltage for each phase the rms value of the kth harmonic current for each phase The phase angle of the jth harmonic voltage with respect to fundamental current of phase a. the phase angle of kth harmonic current with respect to fundamental current of phase a.
Fig. 3. Four-quadrant power flow directions [7]
While adding /2 to the power components phase angles, the following phasors representing instantaneous power components are applicable:
We will take the fundamental current of phase ―a‖ ( ) as reference axis i.e. . This selection is useful because all the harmonic powers can be [1]: Plotted in the same two – dimensional plane (with different speed rotation); Compared (in magnitude and direction) with power generated by fundamental voltage and current (useful power). Using equation (11), (14), the instantaneous power per phase is defined as the product of instantaneous voltage and instantaneous current per phase.
16th International Middle- East Power Systems Conference -MEPCON'2014 Ain Shams University, Cairo, Egypt, December 23 - 25, 2014 ()
()
()
()
∑∑
*
)
((
∑∑
*
)
)
(( ()
Equation (21) is very similar to (10), the case of sinusoidal conditions, except that the sinusoidal power components p, q, and s are with an angular speed equal to 2nω (this is why they are subscripted by 2n).
(17)
)+
)
((
B. The Sum of Power Components Performed by The Voltage and Current of Different Harmonic
)
The portion of u(t) that is obtained by the voltage and current of different harmonic , i.e. when j≠k, will be called distortion power ( ),will equal
)
(( )+
(
)
()
For three –phase non -sinusoidal balanced system The total instantaneous power ( ) is defined as
[
)
((
()
(
() ∑ *
(
)
)
)
((
)
For j < k
)+
) ∑
((
For j >k
)
( ((
)
( ) is the component of distortion power that has frequency equal to the nth harmonic frequency.
() () () () (19) Using equation (18), (19), the total instantaneous power ( ) is defined as: () ∑∑
∑
[
(
)
) (
)]
(
)
(
)
)] A. The Sum of Power Components Performed by the Voltage and Current of The Same Harmonic order "n" The portion of u(t) that is obtained by the voltage and current of the same harmonic order n, i.e. when j=k=n, will be called distortionless power ( ) ()
∑
(
)
∑
[
(
)
()
)
(
)]
∑
()
()
∑
()
∑
( )
(
)
(
)
∑
()
( ) ∑
Where, - While n is an even number
()
()
Because the voltage and current harmonic order n is always an integer number, we can divide u (t) into odd and even values
()
()
()
∑
)
( ) is defined as
The total instantaneous power () () ()
( ) (
(
()
()
(
)
16th International Middle- East Power Systems Conference -MEPCON'2014 Ain Shams University, Cairo, Egypt, December 23 - 25, 2014
( )
(
[
)
(
)
(
)]
Vn/2 Qn
In/2 I1
Dn Sn
( )
( *
( )
Pn
(
)
(
)
(
)+
( [
Fn
)
⁄
)
⁄
(
Note: Dotted line ……… for V&I vectors with rotating speed Solid line _____ for power vectors with rotating speed
)
(
)
(
)]
(
⁄
)
(
⁄
)
- While n is an odd number () () () Adding distortion power ( ) to phasor power result fictitious power ( )
Fig. 4 Phasor diagram for vectors representing the nth harmonic power components per phase
( ) will
() () () (26) This will be in case of even harmonic. In case of odd harmonics, ( ) will only equal to ( ). Finally, we can introduce the total instantaneous power ( ) as: () Where, ()
∑
()
(
)
()
phasor diagram for phasor quantities of the nth harmonic sinusoidal power components can be represented as in fig. 4.
By the same way, as in sinusoidal conditions, rotating all the power vectors by /2 anticlockwise direction will lead to the conventional power flow reference directions for power components generated by system frequency (fundamental harmonic) voltage and current V1&I1. These power components will be second order harmonic power (they will be subscripted by number 2). By this way, power components for each harmonic order can be compared, in magnitude and direction, with power generated by V1&I1 (useful power).
V.
CONCLUSION
This paper has introduced a new notion in power theory for periodic current and voltage waveforms for any threephase four wire balanced system under sinusoidal and nonsinusoidal conditions. It is an expansion of previously published paper concerning the single-phase system. According to this suggested notion, for each n order harmonic power, different components of power are proposed. It is obvious that some terms in the equations, such as s(t), p(t), q(t) are lead to zero value because of the phase balancing. However, the authors intended to write the complete equations to be useful in case of analyzing the unbalanced situations in future work. This paper presents a method to determine and quantifying power components, which will be used in the completion paper (part 2) which introduces a method for determining
16th International Middle- East Power Systems Conference -MEPCON'2014 Ain Shams University, Cairo, Egypt, December 23 - 25, 2014 customer and utility harmonic contribution at the PCC through a case study mentioned in ref. [7] as an example. REFERENCE [1] Ayman A. Eisa, M.M. Abdel Aziz, and Hosam K. M. Youssef, ―New Notions Suggested to Power Theory Development - Part 1: Analytical Derivation‖, IEEE/PES General Meeting 2008. [2] A. Ferrero, ―Definitions of electrical quantities commonly used in nonsinusoidal conditions‖, Eur. Trans. Electr. Power (ETEP) 8 (1998) 249–257. [3] P. Tenti, P. Mattavelli,‖ A time-domain approach to power term definitions under non-sinusoidal conditions‖, 6th International Workshop on Power Def. and Meas. under Non-sinusoidal Conditions, IEEE Instrum. and Meas. Society, North Italy Chapter, Milano, Italy, 2003, pp. 41–50. [4] M.-T. Chen, C.-L. Huang, L.-W. Fei, H.-Y. ChU, ―Power-component definitions and measurements for a harmonic-polluted power circuit‖, IEEE PROCEEDINGS-C, Vol. 138, No. 4, JULY 1991. [5] F. Ghassemi,‖ What the wrong with the electric power theory and how it should be modified,‖ presented at the Inst. Elect. Eng.,9th Int. Conf. Metering and Tarrif for Energy Supply, Birmingham,U.K.,May 1999. [6] S. Pajie, A.E. Emanuel, ―Modern apparent power definitions: theoretical versus practical approach—the general case‖, IEEE Trans. Power Deliv. 21 (4) (2006) 1787–1792. July 1996. [7] IEEE Trial-Use Standard Definitions for the Measurement of Electric Power Quantities Under Sinusoidal, Non-sinusoidal, Balanced or Unbalanced Conditions, June 2000. IEEE PES, Power System Instrum. and Meas. Committee, IEEE Std. 1459-2000. [8] M. Depenbrock, "The FBD-method, a generally applicable tool for analyzing power relations", IEEE Trans. Power Syst. 8 (2) (1993) 381– 387. [9] Walid G. Morsi, and M. E. El-Hawary,‖ Defining Power Components in Nonsinusoidal Unbalanced Polyphase System: The Issues ―,IEEE Trans. Power Del., vol. 22, no. 4, pp. 2428-2438, Oct. 2007. [10] Akira Nabae, Toshihiko Tanaka, ―A new definition of instantaneous active-reactive current and power based on instantaneous space vectors on polar coordinates in three-phase‖, IEEE Trans. On Power Delivery, Vol. 11, No. 3, July 1996. [11] C. H. page,‖ Reactive power in non-sinusoidal situations‖, IEEE Trans. Instrum. Meas., vol. IM-29, no. 4, pp. 420–423, Dec. 1980. [12] L.S. Czarnecki,‖ Orthogonal decomposition of the currents in a 3-phase nonlinear asymmetrical circuit with a nonsinusoidal voltage source‖, IEEE Trans. Instrum. Meas. 37 (1) (1988) ,pp. 30–34. [13] L. Rosseto and P. Tenti, ―Using AC-Fed PWM converters as instantaneous reactive‖, IEEE Trans. Power Electron., vol. 7, no. 1, pp. 224-230, jan.1992. [14] A. M. Stankovic, H. Lev-Ari, ―Frequency domain observations on definitions of reactive power‖, M.E. El-Hawary Editor, Power Engineering Letters, IEEE power engineering review, June 2000. [15] S. Q. Sun, Q. R. Xiang, ―Waveform distortion and distortion power‖, IEE Proceedings, Vol. 139, No. 4, July 1992. [16] P. Salmeron, J. C. Montano, ―Instantaneous power components in polyphase systems under non-sinusoidal conditions‖, IEE Proceedings, Vol. 143, No. 2, March 1996 [17] Renato Sasdelli, Gian Carlo Montanari, ― Compensable power for electrical systems in non-sinusoidal conditions‖, IEEE Trans. On Instrumentation and measurement, Vol. 43, No. 4, August 1994. [18] Alexander Eigeles Emanuel, ―Powers in non-sinusoidal situations: a review of definitions and physical meaning‖, IEEE Trans. On Power Delivery, Vol. 5, No. 3, July 1990.
[19] Segui-Chilet S, Gimeno-Sales FJ, Orts S, Garcerá G, Figueres E, Alcañiz M, ―Approach to unbalance power active compensation under linear load unbalances and fundamental voltage asymmetries. nt J Electrical Power Energy Syst 2007;29(7):526–39. [20 ] Patricio Salmeron, Reyes S. Herrera, Jesus R. Vazquez,‖A new approach for three-phase loads compensation based on the instantaneous reactive power theory‖, Elsevier, Electrical Power Systems Research , Jun. 2007, vol. 87, pp 605-617. [21] M. Erhan Balci, M. Hakan Hocaoglu, ―Quantitative comparison of power decompositions‖, Elsevier, Electrical Power Systems Research , Mar. 2007, vol. 78, pp 318-329. [22] P. Salmerón, R.S. Herrera,‖ Instantaneous reactive power theory—A general approach to poly-phase systems‖, Elsevier, Electrical Power Systems Research, Apr. 2009, vol. 79, pp 1263-1270. [23] M. Malengret, C.T. Gaunt," General theory of instantaneous power formulti-phase systems with distortion, unbalance and direct current components", Elsevier, Electrical Power Systems Research , Jun. 2011, vol. 81, pp 1897-1904. [24] M. Malengret, C.T. Gaunt," General theory of average power for multiphase systems with distortion, unbalance and direct current components", Elsevier, Electrical Power Systems Research , Dec. 2011, vol. 84, pp 224–230. [25] N. Munoz-Galeano, J.C. Alfonso-Gil, S. Orts Grau, S. Segui Chilet, F.J. Gimeno," Non-fundamental effective apparent power defined through an instantaneous power approach", Elsevier, Electrical Power and Energy Systems, Oct. 2011, vol. 33, pp 1711–1720.
