3.1.1 Single-phase system
Let us, at first, consider a general single phase ac-system with instantaneous voltage and cur-rent asu(t)andi(t). Let us assume that the waveforms are periodic with a fundamental period T. Such a waveform can be decomposed into the harmonic components and is presented in the frequency domain by its Fourier series. The apparent powerSis defined as
S=UrmsIrms , (3.1)
whereUrmsandIrmsare the RMS values of the voltage and the current. The RMS values (or the so called true RMS values) are calculated as shown in the case of the voltage
Urms= s
1 T
Z T 0
u(t)2dt . (3.2)
The RMS value can be expressed with the RMS values of individual harmonic components as
Urms= s
X
h
Uh2 , (3.3)
whereUhis the RMS value of the harmonic component of the orderh, (h=0, 1, 2,. . .). The active power is calculated as
P= 1 T
Z T 0
u(t)i(t)dt . (3.4)
In a sinusoidal case, where only the fundamental components of the voltage and the current, U1 andI1, are present the apparent power may be split up into the active and the reactive parts as
P1 = S1cos(φ1) =U1I1cos(φ1) (3.5) Q1 = S1sin(φ1) =U1I1sin(φ1), (3.6) whereφ1 is the phase shift between the fundamental voltage and the fundamental current components. Further, it holds that
S12=P12+Q21 . (3.7)
The power factor is generally accepted as the measure of the power transfer efficiency and is defined as
λ=P
S . (3.8)
For the fundamental wave quantities we have λdpf= P1
S1 = U1I1cos(φ1)
U1I1 = cos(φ1), (3.9) which is known as the displacement power factor. In a nonsinusoidal case, the definition for Pis not altered, obviously, because it has a clear physical interpretation as the rate of change of the energy. Further, the definition of the apparent power in nonsinusoidal case may be written as
S2=P2+Q2F , (3.10)
whereQFis Fryze’s reactive power. This quantity, introduced by Fryze (1932), is defined as QF=p
S2−P2 . (3.11)
There is, however, different definitions for the reactive power in nonsinusoidal case. Another formulation, called Budeanu’s reactive power, is defined by summing the reactive power con-tributions of each individual harmonic components
QB=X
h
UhIhsin(φh), (3.12) whereφhis a phase shift between thehth harmonic voltage and current. Further, to decom-pose the apparent power a new quantity D is introduced, which may be called distortion power. The apparent power is now divided as
S2=P2+Q2B+D2 . (3.13)
The definition ofQBin (3.12) seems well justified by recognizing that the active power may be expressed as
P=X
h
UhIhcos(φh). (3.14) Budeanu’s reactive power has also been adopted by the IEEE in IEEE Std 100–1984 (1984).
However, Czarnecki (1987) has shown that the Budeanu’s reactive powerQB and the dis-tortion power D do not possess attributes, which can be related to the power phenomena
3.1 Averaged power components 31
in circuits with nonsinusoidal waveforms. Moreover, Czarnecki (1987) notes that the dis-tortion power does not provide information related to waveform disdis-tortion. The Budeanu’s concept has also been criticized by Filipski et al. (1994) and by Emanuel (1990) for being misleading and having no physical meaning in nonsinusoidal situations. In (Pretorius et al., 2000) Budeanu’s definition was found to be impractical and concluded to yield erroneous results in some cases. As pointed out by Czarnecki (1987), Pretorius et al. (2000) and also noted by Filipski et al. (1994), the reactive powers in different harmonic frequencies may have opposite signs because of the sine term in (3.12) and hence be canceled out. As a re-sult, Budeanu’s reactive power may yield a zero value when there is reactive power in some harmonic frequencies (calculated asQh=UhIhsin(φh)). Even though each termQhhas a physical meaning, their sumQBhas not. Further, Budeanu’s reactive power is defined in the frequency domain, which is not very practical. The conclusion is that the Budeanu’s reactive power is not sensible in nonsinusoidal cases and it should not be used.
Also, (3.10) and (3.13) are not the only decompositions of the apparent powerS presented.
Arrillaga and Watson (2003) give also a decomposition proposed by Shepherd and Zakikhani and a decomposition by Kusters and Moore.
3.1.2 Three-phase system
Several definitions of apparent power in three-phase systems using Budeanu’s definition and Fryze’s definition of reactive power have been proposed (Arrillaga and Watson, 2003). Prob-ably the most practical ones are the definitions not involving Budeanu’s reactive power. The apparent power calculated per phase basis is
Sppb=X
k
Uk,rmsIk,rms=X
k
q
Pk2+Q2F, k , (3.15) wherekdenotes a phase. The system apparent power that considers a three-phase network as unit is calculated as
SΣ= s
X
k
Uk,2rms s
X
k
Ik,2rms=UΣIΣ= q
PΣ2+Q2F, (3.16) wherePΣis the average three-phase power in the fundamental cycle. Eq. (3.16) is equiva-lent to the total apparent power used in the Fryze-Buchholz-Depenbrock (FBD) method de-scribed in (Depenbrock, 1993). Further, it corresponds to the IEEE Working Group approach (IEEE Working Group on Nonsinusoidal situations, 1996), to which, however, Depenbrock and Staudt (1996) suggested some enhancements for the three-phase four-wire case. In (Pre-torius et al., 2000) some methods of power resolution were compared considering their prac-ticability and the use of the IEEE Working Group approach was recommended. Generally
SΣ≥Sppb , (3.17)
and further, the corresponding power factors relate asλΣ ≤ λppb (Arrillaga and Watson, 2003). The active power in a three-phase system is, obviously, the sum of the active powers of the individual phases. The three-phase reactive power may be defined as a sum of Fryze’s reactive powers of individual phases, as in (3.15) or as the total Fryze’s reactive power of the three-phase system as in (3.16). Also the sum of Budeanu’s reactive powers of individual phases may be used, although not recommended.