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Suggested power definition and measurement due to harmonic load
Electric Power Systems Research 53 (2000) 73 – 81 www.elsevier.com/locate/epsr
Suggested power definition and measurement due to harmonic load Shun-Li Lu a, Chin E. Lin b,*, Ching-Lien Huang a b
a Department of Electrical Engineering, National Cheng Kung Uni6ersity, Tainan, Taiwan, ROC Department of Aeronautics and Astronautics, National Cheng Kung Uni6ersity Tainan70101, Taiwan, ROC
Received 12 August 1997; received in revised form 13 October 1998; accepted 27 October 1998
Abstract This paper resolves conventional power content into incident, reflected, and transmitted power components that can be measured in relation to power factor compensation and harmonic elimination problems in power distribution systems. Simultaneously, the proposed concept can successfully overcome the drawbacks of Budeanu reactive and distortion power. From the power engineer’s viewpoint, the proposed reactive and distortion powers possess the attributes of non-sinusoidal waveforms in power systems, and provide the information necessary for compensation of both reactive power and harmonics. An analytical method is proposed in this paper, and examples and case studies illustrate the physical and engineering significance of active, reactive and distortion power. Finally, this paper proposes a concept of ‘reflected power loss factor’, also termed as ‘G-factor,’ for harmonic loads. It represents a figure of merit in defining the power transfer capability of a harmonic load. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Power definition; Incident power; Reflected power; Transmitted power; Harmonic
1. Introduction An accurate definition of power due to harmonics resulting from non-linear loads in power systems has become more important as the significance of harmonics produced from non-linear loads have increased. However, voltage and current waveform distortion resulting from harmonics pollution has made the theory of power delivery a matter of serious complexity Since the 1920s non-linear loads have been developed into a major part of electric utility system architecture, many controversial discussions over the various definitions of power involving non-sinusoidal waveforms. The disputation over this concept has continued [1]. The Bibliography of Power System Harmonics [2] includes many valuable papers which discussed this dispute. Many proposed power definitions for non-sinusoidal systems were discussed and used [3]. Because there were many disagreements regarding the definition of the various power components, many papers have attempted to define and measure the particular power * Corresponding author.
quantities referring to the conventional power theory [4–11]. The present accepted theory of harmonics assumes that, in a power supply system with non-sinusoidal voltage and/or current, the apparent power consists of active, reactive and distortion power. These power definitions can be found in the current IEEE Dictionary [12], and has remained virtually unchanged until now. Via the above references and discussion, we consider that (1) there are not any fully accepted, clear definitions or methods regarding apparent, active, reactive and distortion power due to harmonic loads; (2) the accepted definitions do not possess clear engineering and physical significance from the viewpoint of power flow; (3) there are no power definitions that can successfully overcome the shortcomings of Budeanu’s reactive and distortion power. In this paper, we will introduce a new power definition and measurement method applying the fast fourier transform (FFT) algorithm. Our interests focus on active, reactive and distortion power and their effects on power distribution systems. This paper presents an attempt to clarify these issues. Simultaneously, the presented idea is implemented by the use of a digital
0378-7796/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 7 8 - 7 7 9 6 ( 9 8 ) 0 0 1 7 1 - 0
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scheme for either sinusoidal or non-sinusoidal waveforms in distribution power systems. Finally, these concomitant power line losses and the possibilities of their compensation are discussed.
2. Proposed analytic method Non-linear loads in power distribution systems include three components: (1) active power (measured in W); (2) reactive power (measured in var); (3) distortion power (measured in var; some engineer suggested the var in the 1930s). Fundamental active or reactive power components are generated from power plants, whereas harmonic power can be generated by non-linear loads. Distortion power is always associated with harmonic loads. Reactive and distortion power are also considered as non-active power components in power systems.
2.1. Power theory analysis 2.1.1. Sinusoidal 6oltage and harmonic load The proposed concept of power in the presence of harmonic load is based on the following assumptions: (l) pure sinusoidal voltage power supply; (2) harmonic current injected back into system causing voltage distortion [13,14]. Therefore, the following summary holds. In Fig. 1, the load terminal voltage (also termed transmitted voltage) 6t(t) can be divided into two components, incident voltage 6i(t) and reflected voltage 6r(t), such that the 6t(t) may be expressed as: 6t(t) = 6i(t)+6r(t),
(1)
n is harmonic order, Vn represents the RMS values of the nth harmonic voltage, an represents the phase angles of the nth harmonic voltage and current. Similarly, the load current (also termed transmitted current) if it(t) can be split into two components, incident current ii(t) end reflected current ir(t). Therefore, the it(t) can be expressed as: it(t)= ii(t)+ ir(t), where ii(t)= 2V1 sin(vt +b1),
ir(t)= % 2In sin(nvt + bn ), n=2
and n is harmonic order, In represents RMS values of the nth harmonic current, bn represents phase angles of the nth harmonic current. The apparent power of load, or termed transmitted apparent power, Sb t can be calculated as: Sb t = Vb tIb *= (Vb i + Vb r)(Ib i + Ib r)*=Sb i + Sb r, t
6i(t) = 2V1 sin(vt + a1),
(3)
where Sb i and Sb r are incident power and reflected power, and the asterisk denotes a conjugate-complex number. If the DC component is absent, the square of St can be expressed as follows [15]:
n=1
n=1
n=2
n=2
n=2
n=2
S 2t = V 2t I 2t = % V 2n % I 2n = V 21I 21 + % V 2n % I 2n + V 21 % I 2n + I 21 % V 2n, (4a)
% V 2nI 2n + %
%
n = 1 m = 1,m " n
n=1
where
(2)
V 2nI 2m
n=2
n=2
= V 21I 21 + % V 2nI 2n + V 21 % I 2n + %
%
n = 2 m = 2,m " n
V 2nI 2m
+ I 21 % V 2n.
6r(t) = % 2Vn sin(nvt +an ),
(4b)
n=2
n=2
In accordance with the above equation, we will derive the suggested power components as follows.
2.1.1.1. Acti6e power. The transmitted active power Pt due to harmonics load is given as: Pt =
1 T
&
Vt(t)it(t) dt= % VnIn cos un = Pi + Pr, n=1
where un = an − bn, Pi = V1I1 cos u1, and
Fig. 1. Sinusoidal voltage and harmonic load.
Pr = % VnIn cos un. n=2
(5)
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where Pi is the incident active power into harmonic load terminals and has positive sign, Pr is the reflected active power back to system and has negative sign, and Pt is the transmitted active power into harmonic load. Note that Eqs. (1)– (3) and Eq. (5) were suggested in [13,14].
2.1.1.2. Reacti6e power. Similarly, the transmitted reactive power Qt can be given as:
Qt = % VnIn sin un =Qi +Qr,
(6)
n=1
Fig. 2. The proposed vector diagram of power concept.
where The transmitted distortion power Dt, which is the vector sum of Di and Dr corresponding to the Eq. (3), and can be defined as:
Qi = V1I1 sin u1, and
Dt = D 2i + D 2r =
Qr = % VnIn sin un. n=2
Similarly, the terms Qi, Qr and Qt are the reactive power components that are indispensable to reactive power flow. Qi is incident reactive power into harmonic load terminal and has positive sign, and can be completely compensated by shunt capacitor Qr is the reflected harmonic reactive power back to distribution system and with negative sign, and Qt is transmitted reactive power into harmonic loads.
2.1.1.3. Distortion power. The distortion power occurs as a consequence of currents and voltages of dissimilar frequencies. However, it can be split into incident, reflected and transmitted distortion power, too. Therefore, the incident distortion power Di is consisted of incident voltage and reflected current distortion power Di,i i − 6 r and incident current and reflected voltage distortion power Di,i i − 6 r. Note that the Di,6 i − 6 r and Di,i i − 6 r are also suggested current distortion power and voltage distortion power [13]. Their scalar quantity are given as:
' '
Di,6 i − i r = ViIr = V1 Di,i i − 6 r = VrIi = I1
% I 2n,
(7a)
n=2
% V 2n.
(7b)
n=2
'
%
%
n = 1 m = 1,m " n
V 2nI 2m.
(10)
It implies that the Dt can be split into two orthogonal contents, as the incident component Di and the reflected component Dr.Thus, they are mutually orthogonal components in jb − kb plane, as shown in Fig. 2. Therefore, the apparent power may be expressed in matrix form as:
Æ Sb i Ç Æ P i Q i 0 ÇÆ ib Ç ÆD iÇ ÃSb rà = ÃP r Q r 0 Ãà jb à + à 0 Ãm ÈSb tÉ ÈP t Q t D tÉÈkb É È 0 É Æ0Ç + ÃD rÃfb , È0É
(11)
where each component of this equation is shown as Fig. 2; ib , jb and kb are mutually orthogonal unit vectors in three dimensional space; m and fb represent Di and Dr unit vectors in the jb − kb plane, respectively. The result of Eq. (11) is very important. It shows that apparent power may be represented by five unit vectors in three-dimensional vector space. An equation of this form does not create difficulties expressing the power units accepted in practice. It allows us to use the standard units, VA, W, var, and the previous vad unit.
Therefore, the scalar quantity Di can be defined as: Di = V I + V I = 2 2 i r
2 2 r i
'
2 1 n=2
2 1 n=2
V % I +I % V . 2 n
2 n
(8)
The reflected distortion power Dr, which is formed by vector sum of dissimilar frequencies reflected voltage 6r(t), and reflected current ir(t) RMS product, and is defined as: Dr =
'
%
%
n = 2 m = 2,m " n
V 2nI 2m.
(9)
2.1.1.4. Three-phase system. The proposed concept can easily be extended into a three-phase power distribution system including three phase three-wire or four-wire system. (a)Three-phase three-wire system Fig. 3(a) shows a–b–c three-phase three-wire power distribution system whether the load is wye- or deltaconnected balanced or unbalanced harmonic loads. The combination of current symbols consist of one of ii, ir, it, with footnote a, b, c, for incident, reflected, and
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transmitted line current corresponding to a – b – c threephase loads respectively, such as ii,a. Similarly, the combination of voltage symbols consist of one of 6i, 6r, 6t, with footnote ab, bc and ca, for incident, reflected, and transmitted line-to-line voltage corresponding to a – b, b–c, and c–a terminals, respectively, such as 6i,ab. However, if we assume voltage reference point at load terminal of phase-c, we can easily calculate power delivery. Similar to the well-known two-wattmeter measurement of three-phase power method, the three-phase total transmitted apparent power is: S 2t =S 2t,ac +S 2t,bc = V 2t,acI 2t,a +V 2t,bcI 2t,b. Similar to the Eq. (11), the threephase three-wire apparent power can be expressed as:
Æ Sb i Ç ÃSb rà = ÈSb tÉ Æ P i,ac + Pi,bc Q i,ac +Qi,bc Ç 0 ÃP r,ac + Pr,bc Q r,ac +Qr,bc 0 Ã È P t,ac + Pi,bc Q t,ac +Qt,bc D t,ac +Dt,bcÉ Æ i Ç ÆD i,ac +Di,bcÇ Æ Ç 0 à jb à + à 0 Ãm + ÃD r,ac +Dr,bcÃfb (12) Èkb É È É È É 0 0 where the composition of active power symbols consist of one of Pi, P0, Pt, with footnote ac and bc for
incident, reflected and transmitted active power, reactive power and distortion power corresponding to the a–c, and b–c terminals of three-phase loads respectively, such as Pi,ac. And, these reactive power and distortion power symbols also have similarly expressions, too. (2) Three-phase four-wire system Fig. 3(b) shows a–b–c three-phase four-wire power distribution system, the load is wye-connected balanced or unbalanced harmonic loads, and with a neutral point N connected to source side via a neutral line. The combination of current symbols consists of one of ii, ir, it, with footnote a, b and c for incident, reflected, and transmitted line current corresponding to a–b–c threephase loads, respectively, such as ii,a. Similarly, the combination voltage symbols consist of one of 6i, 6r, 6t, with footnote a, b and c for incident, reflected, and transmitted line-to-neutral-point voltage corresponding to a–N, b–N, and c–N terminals respectively, such as 6i,a. An additional neutral line also includes incident current ii,N, reflected current ir,N and transmitted current it,N. However, if the neutral line is without any power source, it implies ii,N = 0 and ir,N = it,N Thus it,N +it,a + it,b + it,c = 0, and the reflected current ir,N = it,N = − (it,a + it,b + it,c). It is a zero-sequence current. However, it can be expressed as:
ir,N(t)= 2 % Ir,Nn sin(nvt + fn ),
(13)
n=1
where Ir,Nn and fn represent the nth harmonic RMS value and phase angle of ir,N(t), respectively. Therefore, the corresponding reflected active power Pr,N and reflected reactive power Qr,N quantities are expressed as:
Pr,N = − % I 2r,NnRN,n,
(14a)
n=1
Qr,N = − % I 2r,NnXN,n,
(14b)
n=1
where RN,n and XN,n represent the nth neutral line harmonic resistance and harmonic reactance, respectively. Similar to the first and second case, the three-phase four-wire apparent power can be expressed as: Æ Sb i Ç ÃSb rà = ÈSb É t P i,a +Pi,b +Pi,c Æ ÃP r,a +Pr,b +Pr,c +Pr,N È P t,a +Pt,b +Pt,c
Fig. 3. (a) Three-phase three-wire voltage source and harmonic loads. (b) Three-phase four-wire voltage source and harmonic loads.
Q i,a +Qi,b +Qi,c Q r,a +Qr,b +Qr,c +Qr,N Q t,a +Qt,b +Qt,c
0 Ç 0 Ã D t,a +Dt,b +Dt,cÉ
0 Æ ib Ç ÆD i,a +Di,b +Di,cÇ Æ Ç Ã jb à +à 0 Ãm + ÃD r,a +Dr,b +Dr,cÃfb Èkb É È É È É 0 0
(15)
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where these composition of active power symbols consist of one of Pi, Pr, Pt, with footnote a, b and c for incident, reflected and transmitted active power, reactive power and distortion power corresponding to the a – N, b–N and c– N terminals of three-phase loads respectively, such as Pi,a. And, these reactive power and distortion power symbols have similarly expression, too.
2.1.2. Non-sinusoidal 6oltage and harmonic load If the couple point a – a% has an injected harmonic current ij(t) from other non-linear loads, as shown in Fig. 1, which causes harmonic voltage Vj(t), the load current and terminal voltage can be expressed as: it(t) =ii(t)+ir(t)+ij(t) = ii(t) + i%r(t),
(16)
6t(t) = 6i(t)+6r(t)+6j(t) = 6i(t) + 6%r(t).
(17)
where i %(t) and 6 %(t) represent the reflected current and r r voltage, respectively. In this situation, we must emphasis that each harmonic active and/or reactive power may have either positive sign or negative sign because harmonic contents interact between harmonic loads. The direction of the active and/or reactive harmonic power components may be the same as or opposite to the direction of the incident active and/or reactive power. And, the suggested Pi and Pr must be modified as following: Pt = % Pf + % Pb =Pi +Pr.
(18)
where Pf and Pb represent positive harmonic active power content and negative harmonic active power content respectively. Therefore, Pi and Pr represent the sum of positive active power components and the sum of negative active power components respectively. Similarly, the suggested Qi and Qr are also modified as follows: Qt = % Qf + % Qb =Qi +Qr.
(19)
where Qf and Qb represent positive harmonic active power content and negative harmonic active power content respectively. Therefore, Qi and Qr represent the sum of positive active power components and the sum of negative active power components respectively. However, the noted definition of distortion power contents in Eqs. (7a) and (7b) – (10) are adequate without any modification. Therefore, the previous suggested definitions are easily modified for non-sinusoidal voltage source.
77
2.2. Engineering significance of suggested power definition 2.2.1. Apparent power Apparent power in the IEEE dictionary is defined simply as the product of RMS terminal voltage and load current, and possesses the property of magnitude only. Apparent power is an available term for pure sinusoidal waveform cases. For example, a transformer rating is generally expressed in terms of apparent power rather than active power. In non-sinusoidal waveforms situations, the product of RMS voltage and current is not sufficiently accurate for rating purposes in many cases. For example, the AC–DC converter load produces harmonic current, injecting harmonic current back to the source side, and causing voltage distortion. A voltage harmonic source can result in increased iron loss, and current harmonic source can result in increased copper loss and stray flux in equipment. To study these effects, the conventional concept of apparent power may be inadequate for power definition analysis. A new analytic method to overcome this is thus proposed. The proposed apparent power is split into three components, namely incident, reflected, and transmitted apparent power. They have the properties of magnitude and definite direction of power now, as shown in Fig. 2. Sb r represents the useless but necessary apparent power during energy transmission process. 2.2.2. Acti6e power The engineering significance of the current component ii(t) is rather clear. It is the current of a harmonic load with whose voltage 6i(t) supplies the active power Pi forward to non-linear load. The reflected current ir(t) is produced from transmission process of active power to the load which then reflects active power Pr back to source side, thereby causing voltage distortion and excessive heat loss across source impedance. Since active power is transmitted into load by transmitted current it(t) only, therefore reflected current ir(t) contributes no active power transmission. Such ir(t) current is similarly useless to reactive current in a sinusoidal system. The useless reflected current ir(t) increases the RMS values of incident current ii(t) and the incident apparent power Sb i. The proposed active power consists of Pi, Pr and Pt. The total power entering the harmonic load is Pi, and yields the reflected harmonic power Pr back to the source side. Pr, may be dissipated in the source side. The energy transferred by Pr is entirely converted into heat loss. For example, the significant losses due to harmonic currents in transformer, cables and shielding may be included in Pr [8]. Pt is the active power transmitted forward to load.
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2.2.3. Reacti6e power The concept of reactive power is clear for linear loads with a sinusoidal voltage power supply. In this situation, the reactive power does not contribute to the energy transfer. To delve more deeply into the physical meaning of reactive power in non-sinusoidal circuit, we will describe the suggested reactive power as following. The term reactive power Qt is also Budeanu’s reactive power. In the IEEE Dictionary has well described. However, the term Qi has strong impact on RMS value of the incident current ii(t). Therefore, we recommend that the incident reactive power can be represented by Qi. The reciprocating energy transmission, between source and load, increases the incident current when the RMS reflected current increases. This is related to the presence of the sum of the negative harmonic reactive power components, and can be called the ‘reflected reactive power’. The effect can be explained as some power harmonic frequency may cause capacitive in load. Budeanu’s definition of reactive power is simply the sum of the generalized instantaneous reactive power amplitude and its respective harmonic frequency component [6]. Though each of Budeanu’s component terms has a distinctive physical meaning their sum loses meaning completely. In particular, it can be equal to zero at non-zero values. However, the proposed definitions wholly overcome this limitation of Budeanu’s concept. 2.2.4. Distortion power Another type of power is called ‘distortion power’. Budeanu’s distortion power DB implies a measure of waveforms distortion in power systems. If load terminal voltage and current waveforms are sinusoidal, it is equal to zero. But it is also equal to zero if the distorted voltage waveform is applied to a resistive load [6]. Now, we will demonstrate why our definition can wholly overcome the flaw by example 2 and 3 of the [6]. In example 2, load terminal voltage 6t(t) and current it(t) are given as: 6t(t) = Vi(t)+ 6r(t)= 2V1 sin 6t + 2V3 sin 36t. And,
it(t) =ii(t)+ir(t)
= 2I1 sin 6t +
p p + 2I3 sin 36t + 2 2
= 2V1 sin 6t +
p p + 2V3 sin 36t + . 2 2
Therefore, the Budeanu distortion power is DB =0. But our definitions are:Di = 2V1V3, Dr =0 and Dt =
2V1V3.In example 3, the same voltage Vt(t) as in example 2, and
Fig. 4. The voltage and current waveforms of case one: (- - -) voltage; ( — ) current.
it(t)= ii(t)+ ir(t)
= 2V1 sin 6t +
p p + 2V3 sin 36t − . 2 2
In such a case, the Budeanu distortion power is DB = 2V1V3. However, our suggested distortion powers are always the same as Di = 2V1V3, Dr = 0 and Dt =
2V1V3. Via these results, it implies that DB does not provide any information about waveforms distortion[6], but this flaw has wholly overcome with the suggested distortion power definition. In fact, both circuit models in these examples always exist a 36 notch LC-filter, as shown in Fig. 2 and Fig. 4 in example 2 and 3, respectively [6]. This causes Dr = 0, and implies Di =Dt in a linear load. In other words, it implies 36 notch LC-filter provides − Di distortion power. Simultaneously, this analysis result successfully proves that distortion power flow (another type non-active power) exists in power systems. Distortion power, or also Budeanu’s distortion power, is defined as a scalar quantity in the IEEE Dictionary. The engineering significance of distortion power is a matter of contention. The sign of distortion power, according to some, is to be taken as the same as that of reactive power, while others maintain it can be given either sign. The IEEE Dictionary does not state what type of information is given by the sign, and no physical meaning can be attributed to it. However, distortion power is formed from harmonic loads, and LC-filter or active power filter has capability of eliminating distortion power. Additionally, harmonic load must be considered as bi-directional power flow element. Therefore, we must emphasize that the proposed distortion power has one specific meaningful sign, not either sign as in the conventional reactive power concept. We suggest that Di has the same sign as the positive direction of the unit vector m . This causes Dr and Dt have to the same sign as the positive direction of
S.-L. Lu et al. / Electric Power Systems Research 53 (2000) 73–81
the unit vector fb and kb , respectively, as shown in Fig. 2. The equivalent of Di but with inverted sign can be provided by ideal passive L – C filter bank or active power filter. Thus Di and its sign can have real and practical meaning. Furthermore, the use of a 3-D vector system for our proposed distortion power components overcomes the conventional ambiguity or uselessness regarding sign and, at the same time, successfully models observed phenomena. However, in harmonic loads, it is clear from Eq. (8) that as Ir decreased by LC-filter, it cause Vr decreased. Then Di is also decreased, it corresponding to Dt and Dr are also decreased. In such condition, the incident RMS current, Ii, are also decreased. This leads reduced harmonic loss in a power circuit. A further significance of distortion power (i.e. Dt, Di, and Dr) is that it is another type non-active component of apparent power.
3. Summary
Pt . St
(21)
The object of power factor correction is to increase the active power percentage in relation to the apparent power. If the maximum value of both PFt and PFi are equal, it implies that the power system has reached optimum power factor operation. (4) The proposed reflected power loss factor, the G-factor, is defined as: G=
Pr . Pi
(22)
The G-factor represents a figure of merit of excessive power consumption due to harmonic loading. A high value primarily indicates poor utilization of the sourcepower capacity by the load. Therefore, it is one of superior index rather than %THDv and %THDi in power quality estimation. 4. Experimental results
(1) The incident voltage generates the incident active power Pi from which Pt, enters the load, and Pr is converted into harmonic power (e.g. the harmonic load acting as a power frequency harmonics generator) flowing back to the source. Therefore, the proposed active power (i.e. Pi, Pr and Pt) concept has three different physical quantities, each with real physical significance, that satisfy the principles of energy and power conversion associated with different circuit components as a sum of the terminal power. Similarly, reactive power and distortion power do, too. (2) However, Qi represents one part of Sb i, which can be compensated by a shunt capacitor bank. Similarly, Di represents one part of Sb i, which can be completely eliminated by passive L – C power filters and/or active power filters. In practical power distribution system, the power harmonic filter provides both Qi compensation and Di elimination properties. Therefore, power factor correction becomes a complicated problem, and our suggested definition can provide the basis of an effective analytic method. (3) From the power factor viewpoint, any power distribution system with linear or non-linear loads and either sinusoidal or non-sinusoidal voltage sources shall contain two basic parameters, active power and apparent power. The presented decomposition is particularly useful from the viewpoint of power factor definition in power distribution systems with harmonic loads. The incident power factor PFi can be defined as: P PFi = i, Si
PFt =
79
(20)
and the transmitted power factor PFt is defined as:
Data acquisition and analysis were implemented via a portable PC instrument. Three case studies were described as follows:
4.1. Case 1 A prototype circuit as shown in Fig. 1 was built in our laboratory. A TRIAC circuit was designed to generate, for controlled experiments, different THD values for load currents with 60 Hz 110 V power supplies. A 2.6-mHz inductor was used to model transmission line and transformer leakage inductance. In addition, one 5-A load was used. Fig. 4 show the prototype’s experimental voltage and current waveforms when operated with lagging power factor and THDi = 19.05% load current. Analysis of data in this experiment under four different THD conditions is as shown in Table 1. Conditions 1–4 represent the load current with (1) THDi= 19.05%; (2) THDi= 38.32%; (3) THDi= 64.45%; (4) THDi= 81.8%, respectively. Our proposed powers were measured and are also shown in Table 1.
4.2. Case 2 Here, a power distribution system serves a mainly chemical industrial load. An AC 11.4-kV feeder has two 1042-kW DC drives, which have large numbers of DC motors located at downstream 750 V busses, each fed by 6-pulse AC–DC converters. These converters are connected through two similar transformers, one connected delta–delta, and the other connected delta–wye. Therefore, harmonics (e.g. 5th, 7th, 17th, 19th etc.) circulate between the two transformer banks but do not appear on the 11.4 kV AC side. Fig. 5 shows the tested
S.-L. Lu et al. / Electric Power Systems Research 53 (2000) 73–81
80 Table 1 The results of case studies Case power
Si (VAi) Sr (VAr) St (VAt) Pi (Wi) Pr (Wr) Pt (Wt) Qi (vari) Qr (varr) Qt (Vart) Di (vadi) Dr (vadr) Dt (vadt) Pfi PFt G-factor THDv(%) THDi(%) a
One Condition 1
Condition 2
Condition 3
Condition 4
508.9 8.0 506.31 493.41 −1.14 492.27 61.73 −5.30 56.43 103.85 7.61 104.12 0.97 0.972 −0.0023 8.4% 19.2%
521.64 30.02 497.94 439.16 −9.26 429.95 175.35 −21.81 153.54 196.74 28.19 198.75 0.841 0.863 −0.021 15.87% 38.46%
528.33 49.05 470.83 304.03 −14.52 289.53 282.75 −47.5 235.20 283.31 47.5 287.26 0.575 0.614 −0.048 22.15% 64.7%
545.88 67.19 461.63 193.13 −22.08 171.05 333.49 −64.2 269.3 327.32 64.69 333.65 0.353 0.371 −0.114 26.5% 81.2%
Two
Three
806.5 Ka 5.2 K 802.8 K 638.1 K −265.6 637.8 K 467.6 K −1.67 K 465.9 K 143.2 K 4.47 K 143.3 K 0.974 0.975 0.0004 3.9% 17.7%
22.7 K 7.6 K 23.8 K 413 −7.33 K −6.92 K 27.6 −2.54 K −2.52 K −22.66 K −454 −22.67 K 0.018 −0.29 −17.7 2.68% 294%
K, kilo.
It must be noted in Table 1 that the power unit footnote i, r and t represent incident, reflected, and transmitted power, respectively. Examining these results, it can be seen that the non-linear loading effects on power phenomena are more pronounced than predicted by conventional power concepts. Another observation in these cases is the fact that our suggested power concept is superior to conventional power definition.
5. Conclusion This paper first presents a discursive bibliography of prior attempts at defining power components. The idea Fig. 5. The voltage and current waveforms of case two (- - -) voltage; (—) current.
voltage and current waveforms of single-phase content on the 11.4 kV side and Table 1 shows the measured power components.
4.3. Case 3 The third case illustrates a 5th harmonic LC filter that is added to a 220 V harmonic load bus. In this case, a 40 kVAR 5th filter bank is switched on the presence of a downstream load. The flow of power components occurring in this case is interesting because the filter can generate reactive power and eliminate distortion power at the same time. Fig. 6 shows the measured voltage and current waveforms of single-phase content, and the measured power components are shown in Table 1.
Fig. 6. The voltage and current waveforms of case three (- - -) voltage; ( — ) current.
S.-L. Lu et al. / Electric Power Systems Research 53 (2000) 73–81
of our suggested power components, the active, reactive, and distortion powers in non-sinusoidal power systems, was derived from this bibliographic study. It further seemed to us that power supply physical phenomena should be clearly and usefully recognized in a practical definition. Thus, this paper proposes a clear and useful definition of power and its components and presents a series of interesting experimental applications of the proposed definitions to harmonic loads in power systems. The authors emphasize that each of these proposed power components has a clear physical interpretation in power system circuits. The power engineering community is slowly being convinced that Budeanu’s reactive and distortion power lost physical meaning and cannot be used when optimizing power factor operation. However, we have made a step forward with our presented concept of incident and transmitted power factor operation in power system circuits. The following comments are aimed at a more explicit discussion of the statement from the above discussion that our formulas are capable of straight forward calculation of the suggested power components. It is shown that: (1) the apparent power in a non-sinusoidal system can be represented as three dimensional vector space; (2) active, reactive and distortion power, as defined here, consist of incident, reflected and transmitted power components. Examples and case studies lead us to understand the application of the proposed power theory. The proposed reactive and distortion power successfully overcome the failings of Budeanu reactive and distortion power, and possess attributes that are directly related to the physical phenomena in harmonic polluted power systems. The distortion power was successfully split into the incident, reflected, and transmitted distortion power via mathematics model. They provide information about waveform distortion and clear physical meaning. Therefore, the proposed concept is useful in power theory regarding distribution
.
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systems with non-sinusoidal waveforms, and provides new insight and technique to the power engineer. References [1] P.S. Filipski, Y. Baghzouz, M.D. Cox, Discussion of power definitions contained in the IEEE dictionary, IEEE Trans. Power Delivery 9 (3) (1994) 1237 – 1244. [2] IEEE Power System Harmonics Working Group, Bibliography of Power System Harmonics, report, part I, IEEE Trans. Power Apparatus System, PAS-109, 9 (1984) 2460 – 2462, part II, ibid. [3] C.P. Steinnetz, Theory and Calculation of Alternating Current Phenomena, McGraw, New York, 1908. [4] L. Curtis, F.B. Silsbee, Definition of power and related quantities, AIEE Trans. 54 (1935) 394 – 404. [5] H. Watanabe, R.M. Stephan, M. Aredes, New concepts of instantaneous active and reactive powers in electrical systems with generic loads, IEEE Trans. Power Delivery 8 (2) (1993) 697–703. [6] L.S. Czanecki, What is wrong with the Budeanu concept of reactive and distortion power why it should be abandoned, IEEE Trans. Instrument. Measure. IM-36 (3) (1987) 834 – 837. [7] L.S. Czamecki, Considerations on the reactive power in non-sinusoidal situations, IEEE Trans. Instrument. Measure. IM-34 (3) (1985) 399 – 404. [8] A.E. Emanuel, Power in non-sinusoidal situations a review of definitions and physical meaning, IEEE Trans. Power Delivery 5 (3) (1990) 1377 – 1389. [9] M.A. Slonim, J.D. VanWyk, Power components in a system with sinusoidal and non-sinusoidal voltages and/or currents, IEEE Proc. 135 (2) (1995) 76 – 84. [10] C.H. Page, Reactive power in non-sinusoidal situations, IEEE Trans. Instrument. Measure. 29 (4) (1980) 420 – 426. [11] L.S. Czanecki, Power related phenomena in three-phase unbalanced system, IEEE Trans. Power Delivery 10 (3) (1995) 1168– 1176. [12] IEEE Standard Dictionary of Electrical and Electronics Terms ANSI/EKE std 100 – 1988, IEEE, New York, 1988. [13] A.E. Emanuel, On the assessment of harmonic pollution, IEEE Trans. Power Delivery 10 (3) (1995) 1693 – 1698. [14] IEEE Working Group on Non-sinusoidal Situations Effects on Meter Performance and Definitions of Power, Practical definitions for powers in systems with non-sinusoidal waveforms and unbalanced loads: a discussion, IEEE Trans. Power Delivery 11 (1) (1996) 79 – 101. [15] K. Olejniczak, G.T. Heydt, Basic mechanisms of generation and flow of harmonic signals in balanced and unbalanced three-phase power systems, IEEE Trans. Power Delivery 4 (4) (1989) 2162– 2168.
Suggested power definition and measurement due to harmonic load Shun-Li Lu a, Chin E. Lin b,*, Ching-Lien Huang a b
a Department of Electrical Engineering, National Cheng Kung Uni6ersity, Tainan, Taiwan, ROC Department of Aeronautics and Astronautics, National Cheng Kung Uni6ersity Tainan70101, Taiwan, ROC
Received 12 August 1997; received in revised form 13 October 1998; accepted 27 October 1998
Abstract This paper resolves conventional power content into incident, reflected, and transmitted power components that can be measured in relation to power factor compensation and harmonic elimination problems in power distribution systems. Simultaneously, the proposed concept can successfully overcome the drawbacks of Budeanu reactive and distortion power. From the power engineer’s viewpoint, the proposed reactive and distortion powers possess the attributes of non-sinusoidal waveforms in power systems, and provide the information necessary for compensation of both reactive power and harmonics. An analytical method is proposed in this paper, and examples and case studies illustrate the physical and engineering significance of active, reactive and distortion power. Finally, this paper proposes a concept of ‘reflected power loss factor’, also termed as ‘G-factor,’ for harmonic loads. It represents a figure of merit in defining the power transfer capability of a harmonic load. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Power definition; Incident power; Reflected power; Transmitted power; Harmonic
1. Introduction An accurate definition of power due to harmonics resulting from non-linear loads in power systems has become more important as the significance of harmonics produced from non-linear loads have increased. However, voltage and current waveform distortion resulting from harmonics pollution has made the theory of power delivery a matter of serious complexity Since the 1920s non-linear loads have been developed into a major part of electric utility system architecture, many controversial discussions over the various definitions of power involving non-sinusoidal waveforms. The disputation over this concept has continued [1]. The Bibliography of Power System Harmonics [2] includes many valuable papers which discussed this dispute. Many proposed power definitions for non-sinusoidal systems were discussed and used [3]. Because there were many disagreements regarding the definition of the various power components, many papers have attempted to define and measure the particular power * Corresponding author.
quantities referring to the conventional power theory [4–11]. The present accepted theory of harmonics assumes that, in a power supply system with non-sinusoidal voltage and/or current, the apparent power consists of active, reactive and distortion power. These power definitions can be found in the current IEEE Dictionary [12], and has remained virtually unchanged until now. Via the above references and discussion, we consider that (1) there are not any fully accepted, clear definitions or methods regarding apparent, active, reactive and distortion power due to harmonic loads; (2) the accepted definitions do not possess clear engineering and physical significance from the viewpoint of power flow; (3) there are no power definitions that can successfully overcome the shortcomings of Budeanu’s reactive and distortion power. In this paper, we will introduce a new power definition and measurement method applying the fast fourier transform (FFT) algorithm. Our interests focus on active, reactive and distortion power and their effects on power distribution systems. This paper presents an attempt to clarify these issues. Simultaneously, the presented idea is implemented by the use of a digital
0378-7796/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 7 8 - 7 7 9 6 ( 9 8 ) 0 0 1 7 1 - 0
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scheme for either sinusoidal or non-sinusoidal waveforms in distribution power systems. Finally, these concomitant power line losses and the possibilities of their compensation are discussed.
2. Proposed analytic method Non-linear loads in power distribution systems include three components: (1) active power (measured in W); (2) reactive power (measured in var); (3) distortion power (measured in var; some engineer suggested the var in the 1930s). Fundamental active or reactive power components are generated from power plants, whereas harmonic power can be generated by non-linear loads. Distortion power is always associated with harmonic loads. Reactive and distortion power are also considered as non-active power components in power systems.
2.1. Power theory analysis 2.1.1. Sinusoidal 6oltage and harmonic load The proposed concept of power in the presence of harmonic load is based on the following assumptions: (l) pure sinusoidal voltage power supply; (2) harmonic current injected back into system causing voltage distortion [13,14]. Therefore, the following summary holds. In Fig. 1, the load terminal voltage (also termed transmitted voltage) 6t(t) can be divided into two components, incident voltage 6i(t) and reflected voltage 6r(t), such that the 6t(t) may be expressed as: 6t(t) = 6i(t)+6r(t),
(1)
n is harmonic order, Vn represents the RMS values of the nth harmonic voltage, an represents the phase angles of the nth harmonic voltage and current. Similarly, the load current (also termed transmitted current) if it(t) can be split into two components, incident current ii(t) end reflected current ir(t). Therefore, the it(t) can be expressed as: it(t)= ii(t)+ ir(t), where ii(t)= 2V1 sin(vt +b1),
ir(t)= % 2In sin(nvt + bn ), n=2
and n is harmonic order, In represents RMS values of the nth harmonic current, bn represents phase angles of the nth harmonic current. The apparent power of load, or termed transmitted apparent power, Sb t can be calculated as: Sb t = Vb tIb *= (Vb i + Vb r)(Ib i + Ib r)*=Sb i + Sb r, t
6i(t) = 2V1 sin(vt + a1),
(3)
where Sb i and Sb r are incident power and reflected power, and the asterisk denotes a conjugate-complex number. If the DC component is absent, the square of St can be expressed as follows [15]:
n=1
n=1
n=2
n=2
n=2
n=2
S 2t = V 2t I 2t = % V 2n % I 2n = V 21I 21 + % V 2n % I 2n + V 21 % I 2n + I 21 % V 2n, (4a)
% V 2nI 2n + %
%
n = 1 m = 1,m " n
n=1
where
(2)
V 2nI 2m
n=2
n=2
= V 21I 21 + % V 2nI 2n + V 21 % I 2n + %
%
n = 2 m = 2,m " n
V 2nI 2m
+ I 21 % V 2n.
6r(t) = % 2Vn sin(nvt +an ),
(4b)
n=2
n=2
In accordance with the above equation, we will derive the suggested power components as follows.
2.1.1.1. Acti6e power. The transmitted active power Pt due to harmonics load is given as: Pt =
1 T
&
Vt(t)it(t) dt= % VnIn cos un = Pi + Pr, n=1
where un = an − bn, Pi = V1I1 cos u1, and
Fig. 1. Sinusoidal voltage and harmonic load.
Pr = % VnIn cos un. n=2
(5)
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where Pi is the incident active power into harmonic load terminals and has positive sign, Pr is the reflected active power back to system and has negative sign, and Pt is the transmitted active power into harmonic load. Note that Eqs. (1)– (3) and Eq. (5) were suggested in [13,14].
2.1.1.2. Reacti6e power. Similarly, the transmitted reactive power Qt can be given as:
Qt = % VnIn sin un =Qi +Qr,
(6)
n=1
Fig. 2. The proposed vector diagram of power concept.
where The transmitted distortion power Dt, which is the vector sum of Di and Dr corresponding to the Eq. (3), and can be defined as:
Qi = V1I1 sin u1, and
Dt = D 2i + D 2r =
Qr = % VnIn sin un. n=2
Similarly, the terms Qi, Qr and Qt are the reactive power components that are indispensable to reactive power flow. Qi is incident reactive power into harmonic load terminal and has positive sign, and can be completely compensated by shunt capacitor Qr is the reflected harmonic reactive power back to distribution system and with negative sign, and Qt is transmitted reactive power into harmonic loads.
2.1.1.3. Distortion power. The distortion power occurs as a consequence of currents and voltages of dissimilar frequencies. However, it can be split into incident, reflected and transmitted distortion power, too. Therefore, the incident distortion power Di is consisted of incident voltage and reflected current distortion power Di,i i − 6 r and incident current and reflected voltage distortion power Di,i i − 6 r. Note that the Di,6 i − 6 r and Di,i i − 6 r are also suggested current distortion power and voltage distortion power [13]. Their scalar quantity are given as:
' '
Di,6 i − i r = ViIr = V1 Di,i i − 6 r = VrIi = I1
% I 2n,
(7a)
n=2
% V 2n.
(7b)
n=2
'
%
%
n = 1 m = 1,m " n
V 2nI 2m.
(10)
It implies that the Dt can be split into two orthogonal contents, as the incident component Di and the reflected component Dr.Thus, they are mutually orthogonal components in jb − kb plane, as shown in Fig. 2. Therefore, the apparent power may be expressed in matrix form as:
Æ Sb i Ç Æ P i Q i 0 ÇÆ ib Ç ÆD iÇ ÃSb rà = ÃP r Q r 0 Ãà jb à + à 0 Ãm ÈSb tÉ ÈP t Q t D tÉÈkb É È 0 É Æ0Ç + ÃD rÃfb , È0É
(11)
where each component of this equation is shown as Fig. 2; ib , jb and kb are mutually orthogonal unit vectors in three dimensional space; m and fb represent Di and Dr unit vectors in the jb − kb plane, respectively. The result of Eq. (11) is very important. It shows that apparent power may be represented by five unit vectors in three-dimensional vector space. An equation of this form does not create difficulties expressing the power units accepted in practice. It allows us to use the standard units, VA, W, var, and the previous vad unit.
Therefore, the scalar quantity Di can be defined as: Di = V I + V I = 2 2 i r
2 2 r i
'
2 1 n=2
2 1 n=2
V % I +I % V . 2 n
2 n
(8)
The reflected distortion power Dr, which is formed by vector sum of dissimilar frequencies reflected voltage 6r(t), and reflected current ir(t) RMS product, and is defined as: Dr =
'
%
%
n = 2 m = 2,m " n
V 2nI 2m.
(9)
2.1.1.4. Three-phase system. The proposed concept can easily be extended into a three-phase power distribution system including three phase three-wire or four-wire system. (a)Three-phase three-wire system Fig. 3(a) shows a–b–c three-phase three-wire power distribution system whether the load is wye- or deltaconnected balanced or unbalanced harmonic loads. The combination of current symbols consist of one of ii, ir, it, with footnote a, b, c, for incident, reflected, and
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transmitted line current corresponding to a – b – c threephase loads respectively, such as ii,a. Similarly, the combination of voltage symbols consist of one of 6i, 6r, 6t, with footnote ab, bc and ca, for incident, reflected, and transmitted line-to-line voltage corresponding to a – b, b–c, and c–a terminals, respectively, such as 6i,ab. However, if we assume voltage reference point at load terminal of phase-c, we can easily calculate power delivery. Similar to the well-known two-wattmeter measurement of three-phase power method, the three-phase total transmitted apparent power is: S 2t =S 2t,ac +S 2t,bc = V 2t,acI 2t,a +V 2t,bcI 2t,b. Similar to the Eq. (11), the threephase three-wire apparent power can be expressed as:
Æ Sb i Ç ÃSb rà = ÈSb tÉ Æ P i,ac + Pi,bc Q i,ac +Qi,bc Ç 0 ÃP r,ac + Pr,bc Q r,ac +Qr,bc 0 Ã È P t,ac + Pi,bc Q t,ac +Qt,bc D t,ac +Dt,bcÉ Æ i Ç ÆD i,ac +Di,bcÇ Æ Ç 0 à jb à + à 0 Ãm + ÃD r,ac +Dr,bcÃfb (12) Èkb É È É È É 0 0 where the composition of active power symbols consist of one of Pi, P0, Pt, with footnote ac and bc for
incident, reflected and transmitted active power, reactive power and distortion power corresponding to the a–c, and b–c terminals of three-phase loads respectively, such as Pi,ac. And, these reactive power and distortion power symbols also have similarly expressions, too. (2) Three-phase four-wire system Fig. 3(b) shows a–b–c three-phase four-wire power distribution system, the load is wye-connected balanced or unbalanced harmonic loads, and with a neutral point N connected to source side via a neutral line. The combination of current symbols consists of one of ii, ir, it, with footnote a, b and c for incident, reflected, and transmitted line current corresponding to a–b–c threephase loads, respectively, such as ii,a. Similarly, the combination voltage symbols consist of one of 6i, 6r, 6t, with footnote a, b and c for incident, reflected, and transmitted line-to-neutral-point voltage corresponding to a–N, b–N, and c–N terminals respectively, such as 6i,a. An additional neutral line also includes incident current ii,N, reflected current ir,N and transmitted current it,N. However, if the neutral line is without any power source, it implies ii,N = 0 and ir,N = it,N Thus it,N +it,a + it,b + it,c = 0, and the reflected current ir,N = it,N = − (it,a + it,b + it,c). It is a zero-sequence current. However, it can be expressed as:
ir,N(t)= 2 % Ir,Nn sin(nvt + fn ),
(13)
n=1
where Ir,Nn and fn represent the nth harmonic RMS value and phase angle of ir,N(t), respectively. Therefore, the corresponding reflected active power Pr,N and reflected reactive power Qr,N quantities are expressed as:
Pr,N = − % I 2r,NnRN,n,
(14a)
n=1
Qr,N = − % I 2r,NnXN,n,
(14b)
n=1
where RN,n and XN,n represent the nth neutral line harmonic resistance and harmonic reactance, respectively. Similar to the first and second case, the three-phase four-wire apparent power can be expressed as: Æ Sb i Ç ÃSb rà = ÈSb É t P i,a +Pi,b +Pi,c Æ ÃP r,a +Pr,b +Pr,c +Pr,N È P t,a +Pt,b +Pt,c
Fig. 3. (a) Three-phase three-wire voltage source and harmonic loads. (b) Three-phase four-wire voltage source and harmonic loads.
Q i,a +Qi,b +Qi,c Q r,a +Qr,b +Qr,c +Qr,N Q t,a +Qt,b +Qt,c
0 Ç 0 Ã D t,a +Dt,b +Dt,cÉ
0 Æ ib Ç ÆD i,a +Di,b +Di,cÇ Æ Ç Ã jb à +à 0 Ãm + ÃD r,a +Dr,b +Dr,cÃfb Èkb É È É È É 0 0
(15)
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where these composition of active power symbols consist of one of Pi, Pr, Pt, with footnote a, b and c for incident, reflected and transmitted active power, reactive power and distortion power corresponding to the a – N, b–N and c– N terminals of three-phase loads respectively, such as Pi,a. And, these reactive power and distortion power symbols have similarly expression, too.
2.1.2. Non-sinusoidal 6oltage and harmonic load If the couple point a – a% has an injected harmonic current ij(t) from other non-linear loads, as shown in Fig. 1, which causes harmonic voltage Vj(t), the load current and terminal voltage can be expressed as: it(t) =ii(t)+ir(t)+ij(t) = ii(t) + i%r(t),
(16)
6t(t) = 6i(t)+6r(t)+6j(t) = 6i(t) + 6%r(t).
(17)
where i %(t) and 6 %(t) represent the reflected current and r r voltage, respectively. In this situation, we must emphasis that each harmonic active and/or reactive power may have either positive sign or negative sign because harmonic contents interact between harmonic loads. The direction of the active and/or reactive harmonic power components may be the same as or opposite to the direction of the incident active and/or reactive power. And, the suggested Pi and Pr must be modified as following: Pt = % Pf + % Pb =Pi +Pr.
(18)
where Pf and Pb represent positive harmonic active power content and negative harmonic active power content respectively. Therefore, Pi and Pr represent the sum of positive active power components and the sum of negative active power components respectively. Similarly, the suggested Qi and Qr are also modified as follows: Qt = % Qf + % Qb =Qi +Qr.
(19)
where Qf and Qb represent positive harmonic active power content and negative harmonic active power content respectively. Therefore, Qi and Qr represent the sum of positive active power components and the sum of negative active power components respectively. However, the noted definition of distortion power contents in Eqs. (7a) and (7b) – (10) are adequate without any modification. Therefore, the previous suggested definitions are easily modified for non-sinusoidal voltage source.
77
2.2. Engineering significance of suggested power definition 2.2.1. Apparent power Apparent power in the IEEE dictionary is defined simply as the product of RMS terminal voltage and load current, and possesses the property of magnitude only. Apparent power is an available term for pure sinusoidal waveform cases. For example, a transformer rating is generally expressed in terms of apparent power rather than active power. In non-sinusoidal waveforms situations, the product of RMS voltage and current is not sufficiently accurate for rating purposes in many cases. For example, the AC–DC converter load produces harmonic current, injecting harmonic current back to the source side, and causing voltage distortion. A voltage harmonic source can result in increased iron loss, and current harmonic source can result in increased copper loss and stray flux in equipment. To study these effects, the conventional concept of apparent power may be inadequate for power definition analysis. A new analytic method to overcome this is thus proposed. The proposed apparent power is split into three components, namely incident, reflected, and transmitted apparent power. They have the properties of magnitude and definite direction of power now, as shown in Fig. 2. Sb r represents the useless but necessary apparent power during energy transmission process. 2.2.2. Acti6e power The engineering significance of the current component ii(t) is rather clear. It is the current of a harmonic load with whose voltage 6i(t) supplies the active power Pi forward to non-linear load. The reflected current ir(t) is produced from transmission process of active power to the load which then reflects active power Pr back to source side, thereby causing voltage distortion and excessive heat loss across source impedance. Since active power is transmitted into load by transmitted current it(t) only, therefore reflected current ir(t) contributes no active power transmission. Such ir(t) current is similarly useless to reactive current in a sinusoidal system. The useless reflected current ir(t) increases the RMS values of incident current ii(t) and the incident apparent power Sb i. The proposed active power consists of Pi, Pr and Pt. The total power entering the harmonic load is Pi, and yields the reflected harmonic power Pr back to the source side. Pr, may be dissipated in the source side. The energy transferred by Pr is entirely converted into heat loss. For example, the significant losses due to harmonic currents in transformer, cables and shielding may be included in Pr [8]. Pt is the active power transmitted forward to load.
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2.2.3. Reacti6e power The concept of reactive power is clear for linear loads with a sinusoidal voltage power supply. In this situation, the reactive power does not contribute to the energy transfer. To delve more deeply into the physical meaning of reactive power in non-sinusoidal circuit, we will describe the suggested reactive power as following. The term reactive power Qt is also Budeanu’s reactive power. In the IEEE Dictionary has well described. However, the term Qi has strong impact on RMS value of the incident current ii(t). Therefore, we recommend that the incident reactive power can be represented by Qi. The reciprocating energy transmission, between source and load, increases the incident current when the RMS reflected current increases. This is related to the presence of the sum of the negative harmonic reactive power components, and can be called the ‘reflected reactive power’. The effect can be explained as some power harmonic frequency may cause capacitive in load. Budeanu’s definition of reactive power is simply the sum of the generalized instantaneous reactive power amplitude and its respective harmonic frequency component [6]. Though each of Budeanu’s component terms has a distinctive physical meaning their sum loses meaning completely. In particular, it can be equal to zero at non-zero values. However, the proposed definitions wholly overcome this limitation of Budeanu’s concept. 2.2.4. Distortion power Another type of power is called ‘distortion power’. Budeanu’s distortion power DB implies a measure of waveforms distortion in power systems. If load terminal voltage and current waveforms are sinusoidal, it is equal to zero. But it is also equal to zero if the distorted voltage waveform is applied to a resistive load [6]. Now, we will demonstrate why our definition can wholly overcome the flaw by example 2 and 3 of the [6]. In example 2, load terminal voltage 6t(t) and current it(t) are given as: 6t(t) = Vi(t)+ 6r(t)= 2V1 sin 6t + 2V3 sin 36t. And,
it(t) =ii(t)+ir(t)
= 2I1 sin 6t +
p p + 2I3 sin 36t + 2 2
= 2V1 sin 6t +
p p + 2V3 sin 36t + . 2 2
Therefore, the Budeanu distortion power is DB =0. But our definitions are:Di = 2V1V3, Dr =0 and Dt =
2V1V3.In example 3, the same voltage Vt(t) as in example 2, and
Fig. 4. The voltage and current waveforms of case one: (- - -) voltage; ( — ) current.
it(t)= ii(t)+ ir(t)
= 2V1 sin 6t +
p p + 2V3 sin 36t − . 2 2
In such a case, the Budeanu distortion power is DB = 2V1V3. However, our suggested distortion powers are always the same as Di = 2V1V3, Dr = 0 and Dt =
2V1V3. Via these results, it implies that DB does not provide any information about waveforms distortion[6], but this flaw has wholly overcome with the suggested distortion power definition. In fact, both circuit models in these examples always exist a 36 notch LC-filter, as shown in Fig. 2 and Fig. 4 in example 2 and 3, respectively [6]. This causes Dr = 0, and implies Di =Dt in a linear load. In other words, it implies 36 notch LC-filter provides − Di distortion power. Simultaneously, this analysis result successfully proves that distortion power flow (another type non-active power) exists in power systems. Distortion power, or also Budeanu’s distortion power, is defined as a scalar quantity in the IEEE Dictionary. The engineering significance of distortion power is a matter of contention. The sign of distortion power, according to some, is to be taken as the same as that of reactive power, while others maintain it can be given either sign. The IEEE Dictionary does not state what type of information is given by the sign, and no physical meaning can be attributed to it. However, distortion power is formed from harmonic loads, and LC-filter or active power filter has capability of eliminating distortion power. Additionally, harmonic load must be considered as bi-directional power flow element. Therefore, we must emphasize that the proposed distortion power has one specific meaningful sign, not either sign as in the conventional reactive power concept. We suggest that Di has the same sign as the positive direction of the unit vector m . This causes Dr and Dt have to the same sign as the positive direction of
S.-L. Lu et al. / Electric Power Systems Research 53 (2000) 73–81
the unit vector fb and kb , respectively, as shown in Fig. 2. The equivalent of Di but with inverted sign can be provided by ideal passive L – C filter bank or active power filter. Thus Di and its sign can have real and practical meaning. Furthermore, the use of a 3-D vector system for our proposed distortion power components overcomes the conventional ambiguity or uselessness regarding sign and, at the same time, successfully models observed phenomena. However, in harmonic loads, it is clear from Eq. (8) that as Ir decreased by LC-filter, it cause Vr decreased. Then Di is also decreased, it corresponding to Dt and Dr are also decreased. In such condition, the incident RMS current, Ii, are also decreased. This leads reduced harmonic loss in a power circuit. A further significance of distortion power (i.e. Dt, Di, and Dr) is that it is another type non-active component of apparent power.
3. Summary
Pt . St
(21)
The object of power factor correction is to increase the active power percentage in relation to the apparent power. If the maximum value of both PFt and PFi are equal, it implies that the power system has reached optimum power factor operation. (4) The proposed reflected power loss factor, the G-factor, is defined as: G=
Pr . Pi
(22)
The G-factor represents a figure of merit of excessive power consumption due to harmonic loading. A high value primarily indicates poor utilization of the sourcepower capacity by the load. Therefore, it is one of superior index rather than %THDv and %THDi in power quality estimation. 4. Experimental results
(1) The incident voltage generates the incident active power Pi from which Pt, enters the load, and Pr is converted into harmonic power (e.g. the harmonic load acting as a power frequency harmonics generator) flowing back to the source. Therefore, the proposed active power (i.e. Pi, Pr and Pt) concept has three different physical quantities, each with real physical significance, that satisfy the principles of energy and power conversion associated with different circuit components as a sum of the terminal power. Similarly, reactive power and distortion power do, too. (2) However, Qi represents one part of Sb i, which can be compensated by a shunt capacitor bank. Similarly, Di represents one part of Sb i, which can be completely eliminated by passive L – C power filters and/or active power filters. In practical power distribution system, the power harmonic filter provides both Qi compensation and Di elimination properties. Therefore, power factor correction becomes a complicated problem, and our suggested definition can provide the basis of an effective analytic method. (3) From the power factor viewpoint, any power distribution system with linear or non-linear loads and either sinusoidal or non-sinusoidal voltage sources shall contain two basic parameters, active power and apparent power. The presented decomposition is particularly useful from the viewpoint of power factor definition in power distribution systems with harmonic loads. The incident power factor PFi can be defined as: P PFi = i, Si
PFt =
79
(20)
and the transmitted power factor PFt is defined as:
Data acquisition and analysis were implemented via a portable PC instrument. Three case studies were described as follows:
4.1. Case 1 A prototype circuit as shown in Fig. 1 was built in our laboratory. A TRIAC circuit was designed to generate, for controlled experiments, different THD values for load currents with 60 Hz 110 V power supplies. A 2.6-mHz inductor was used to model transmission line and transformer leakage inductance. In addition, one 5-A load was used. Fig. 4 show the prototype’s experimental voltage and current waveforms when operated with lagging power factor and THDi = 19.05% load current. Analysis of data in this experiment under four different THD conditions is as shown in Table 1. Conditions 1–4 represent the load current with (1) THDi= 19.05%; (2) THDi= 38.32%; (3) THDi= 64.45%; (4) THDi= 81.8%, respectively. Our proposed powers were measured and are also shown in Table 1.
4.2. Case 2 Here, a power distribution system serves a mainly chemical industrial load. An AC 11.4-kV feeder has two 1042-kW DC drives, which have large numbers of DC motors located at downstream 750 V busses, each fed by 6-pulse AC–DC converters. These converters are connected through two similar transformers, one connected delta–delta, and the other connected delta–wye. Therefore, harmonics (e.g. 5th, 7th, 17th, 19th etc.) circulate between the two transformer banks but do not appear on the 11.4 kV AC side. Fig. 5 shows the tested
S.-L. Lu et al. / Electric Power Systems Research 53 (2000) 73–81
80 Table 1 The results of case studies Case power
Si (VAi) Sr (VAr) St (VAt) Pi (Wi) Pr (Wr) Pt (Wt) Qi (vari) Qr (varr) Qt (Vart) Di (vadi) Dr (vadr) Dt (vadt) Pfi PFt G-factor THDv(%) THDi(%) a
One Condition 1
Condition 2
Condition 3
Condition 4
508.9 8.0 506.31 493.41 −1.14 492.27 61.73 −5.30 56.43 103.85 7.61 104.12 0.97 0.972 −0.0023 8.4% 19.2%
521.64 30.02 497.94 439.16 −9.26 429.95 175.35 −21.81 153.54 196.74 28.19 198.75 0.841 0.863 −0.021 15.87% 38.46%
528.33 49.05 470.83 304.03 −14.52 289.53 282.75 −47.5 235.20 283.31 47.5 287.26 0.575 0.614 −0.048 22.15% 64.7%
545.88 67.19 461.63 193.13 −22.08 171.05 333.49 −64.2 269.3 327.32 64.69 333.65 0.353 0.371 −0.114 26.5% 81.2%
Two
Three
806.5 Ka 5.2 K 802.8 K 638.1 K −265.6 637.8 K 467.6 K −1.67 K 465.9 K 143.2 K 4.47 K 143.3 K 0.974 0.975 0.0004 3.9% 17.7%
22.7 K 7.6 K 23.8 K 413 −7.33 K −6.92 K 27.6 −2.54 K −2.52 K −22.66 K −454 −22.67 K 0.018 −0.29 −17.7 2.68% 294%
K, kilo.
It must be noted in Table 1 that the power unit footnote i, r and t represent incident, reflected, and transmitted power, respectively. Examining these results, it can be seen that the non-linear loading effects on power phenomena are more pronounced than predicted by conventional power concepts. Another observation in these cases is the fact that our suggested power concept is superior to conventional power definition.
5. Conclusion This paper first presents a discursive bibliography of prior attempts at defining power components. The idea Fig. 5. The voltage and current waveforms of case two (- - -) voltage; (—) current.
voltage and current waveforms of single-phase content on the 11.4 kV side and Table 1 shows the measured power components.
4.3. Case 3 The third case illustrates a 5th harmonic LC filter that is added to a 220 V harmonic load bus. In this case, a 40 kVAR 5th filter bank is switched on the presence of a downstream load. The flow of power components occurring in this case is interesting because the filter can generate reactive power and eliminate distortion power at the same time. Fig. 6 shows the measured voltage and current waveforms of single-phase content, and the measured power components are shown in Table 1.
Fig. 6. The voltage and current waveforms of case three (- - -) voltage; ( — ) current.
S.-L. Lu et al. / Electric Power Systems Research 53 (2000) 73–81
of our suggested power components, the active, reactive, and distortion powers in non-sinusoidal power systems, was derived from this bibliographic study. It further seemed to us that power supply physical phenomena should be clearly and usefully recognized in a practical definition. Thus, this paper proposes a clear and useful definition of power and its components and presents a series of interesting experimental applications of the proposed definitions to harmonic loads in power systems. The authors emphasize that each of these proposed power components has a clear physical interpretation in power system circuits. The power engineering community is slowly being convinced that Budeanu’s reactive and distortion power lost physical meaning and cannot be used when optimizing power factor operation. However, we have made a step forward with our presented concept of incident and transmitted power factor operation in power system circuits. The following comments are aimed at a more explicit discussion of the statement from the above discussion that our formulas are capable of straight forward calculation of the suggested power components. It is shown that: (1) the apparent power in a non-sinusoidal system can be represented as three dimensional vector space; (2) active, reactive and distortion power, as defined here, consist of incident, reflected and transmitted power components. Examples and case studies lead us to understand the application of the proposed power theory. The proposed reactive and distortion power successfully overcome the failings of Budeanu reactive and distortion power, and possess attributes that are directly related to the physical phenomena in harmonic polluted power systems. The distortion power was successfully split into the incident, reflected, and transmitted distortion power via mathematics model. They provide information about waveform distortion and clear physical meaning. Therefore, the proposed concept is useful in power theory regarding distribution
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systems with non-sinusoidal waveforms, and provides new insight and technique to the power engineer. References [1] P.S. Filipski, Y. Baghzouz, M.D. Cox, Discussion of power definitions contained in the IEEE dictionary, IEEE Trans. Power Delivery 9 (3) (1994) 1237 – 1244. [2] IEEE Power System Harmonics Working Group, Bibliography of Power System Harmonics, report, part I, IEEE Trans. Power Apparatus System, PAS-109, 9 (1984) 2460 – 2462, part II, ibid. [3] C.P. Steinnetz, Theory and Calculation of Alternating Current Phenomena, McGraw, New York, 1908. [4] L. Curtis, F.B. Silsbee, Definition of power and related quantities, AIEE Trans. 54 (1935) 394 – 404. [5] H. Watanabe, R.M. Stephan, M. Aredes, New concepts of instantaneous active and reactive powers in electrical systems with generic loads, IEEE Trans. Power Delivery 8 (2) (1993) 697–703. [6] L.S. Czanecki, What is wrong with the Budeanu concept of reactive and distortion power why it should be abandoned, IEEE Trans. Instrument. Measure. IM-36 (3) (1987) 834 – 837. [7] L.S. Czamecki, Considerations on the reactive power in non-sinusoidal situations, IEEE Trans. Instrument. Measure. IM-34 (3) (1985) 399 – 404. [8] A.E. Emanuel, Power in non-sinusoidal situations a review of definitions and physical meaning, IEEE Trans. Power Delivery 5 (3) (1990) 1377 – 1389. [9] M.A. Slonim, J.D. VanWyk, Power components in a system with sinusoidal and non-sinusoidal voltages and/or currents, IEEE Proc. 135 (2) (1995) 76 – 84. [10] C.H. Page, Reactive power in non-sinusoidal situations, IEEE Trans. Instrument. Measure. 29 (4) (1980) 420 – 426. [11] L.S. Czanecki, Power related phenomena in three-phase unbalanced system, IEEE Trans. Power Delivery 10 (3) (1995) 1168– 1176. [12] IEEE Standard Dictionary of Electrical and Electronics Terms ANSI/EKE std 100 – 1988, IEEE, New York, 1988. [13] A.E. Emanuel, On the assessment of harmonic pollution, IEEE Trans. Power Delivery 10 (3) (1995) 1693 – 1698. [14] IEEE Working Group on Non-sinusoidal Situations Effects on Meter Performance and Definitions of Power, Practical definitions for powers in systems with non-sinusoidal waveforms and unbalanced loads: a discussion, IEEE Trans. Power Delivery 11 (1) (1996) 79 – 101. [15] K. Olejniczak, G.T. Heydt, Basic mechanisms of generation and flow of harmonic signals in balanced and unbalanced three-phase power systems, IEEE Trans. Power Delivery 4 (4) (1989) 2162– 2168.