The above presented power definitions are based on an average value concept. Some loads, e.g. an arc furnace or power electronic loads, may be dynamically changing so that the infor-mation of how the load is behaving during the cycle in average is not enough, but the informa-tion on the instantaneous quantities is needed. Typically, the instantaneous values are needed in the compensation of the unwanted components of the load current. Such actively controlled compensator is known as the active filter, see e.g. (Akagi, 1996; Mattavelli, 2001; Sonnen-schein and Weinhold, 1999). Further, the control systems of modern line converters, even without any harmonic compensation capabilities, process instantaneous active power and re-active power quantities, see e.g. (Malinowski et al., 2001; Noguchi et al., 1998; Pöllänen et al., 2003). Let us summarize two theories of the instantaneous power components to help us understand what these quantities are. The first theory is the Fryze-Buchholz-Depenbrock method, which is a generally applicable method to analyze the relations connecting currents, voltages, instantaneous-, active-, nonactive-, and apparent power quantities in nonsinusoidal, unbalanced polyphase systems (Depenbrock, 1993). The second method is the theory of in-stantaneous reactive power in three-phase circuits introduced by Akagi et al. (1983, 1984).
3.2.1 The Fryze-Buchholz-Depenbrock method
The Fryze-Buchholz-Depenbrock method has its roots in the single-phase system study of Fryze (1932). According to Depenbrock and Staudt (1998), the work of Fryze was extended to polyphase systems by Buchholz in 1950. In 1962 Depenbrock published his dissertation, which presented extensions to the theory. The FBD-method was summed up in (Depenbrock, 1971). In English, the FBD-method was published in (Depenbrock, 1993). The FBD-method uses the sets of collective quantities, called collective magnitudes, denoted with a subscript
‘Σ’. The instantaneous collective values of voltage and current are calculated from the in-stantaneous phase quantities as
uΣ = s
X
k
u2k (3.18)
iΣ = s
X
k
i2k , (3.19)
where the voltages are between the phase and a virtual star point so that alwaysP
kuk = 0.
Also, the currents must add up to zero, satisfyingP
kik = 0. The theory assumes n-terminal circuit, which means that the neutral conductor, if it exists, is treated similarly to the phase conductors. The collective RMS valuesUΣandIΣmay be calculated analogously
UΣ = s
X
k
Uk,2rms (3.20)
IΣ = s
X
k
Ik,2rms . (3.21)
3.2 Instantaneous power components 33
The total collective apparent powerSΣis defined as in (3.16). The instantaneous powerpΣ(t) is calculated as
pΣ(t) =X
k
ukik . (3.22)
The phase currents are split up into the so called power currents ik, p and zero-power, or
“powerless”, currentsik, z so thatik = ik, p+ik, z. The power currents produce, with the corresponding voltages, the same instantaneous collective powerpΣ(t)as the complete cur-rents and have the smallest possible collective magnitude. Hence,pΣ(t) = P
kukik, p and P
kukik, z= 0. The power currents are determined in phases as
ik, p=Gp(t)uk (3.23)
and as a collective quantity as
iΣ, p=Gp(t)uΣ , (3.24)
whereGp(t)is the equivalent time dependent conductance calculated as Gp(t) = pΣ(t)
u2Σ . (3.25)
The zero-power current in phases is given as
ik, z =ik−ik, p , (3.26)
and in a collective quantity as
iΣ, z =iΣ−iΣ, p. (3.27)
The power currents can further be divided into the active currents ik, a and the variation currentsik, v. The currents delivering a given mean value of collective instantaneous power pΣ(t)with the smallest possible collective RMS valueIΣ, aare called the active currentsik, a. The mean value of the collective instantaneous power is the active power, or power,PΣ. The following equations are given
ik, a = Guk (3.28)
Ik,rms, a = GUk,rms (3.29)
pk, a(t) = Gu2k (3.30)
iΣ, a = GuΣ, a (3.31)
IΣ, a = GUΣ (3.32)
pΣ, a(t) = Gu2Σ , (3.33)
where pk, a(t) and pΣ, a(t) are the phase and collective instantaneous active powers, re-spectively. Instantaneous collective powerpΣ(t)may be split up into the active component pΣ, a(t)and the variation componentpΣ, v(t)as
pΣ(t) =pΣ, a(t) +pΣ, v(t). (3.34) The phase instantaneous power may be split up analogously. The equivalent active conduc-tanceGin (3.28)–(3.33) is given as
G=pΣ(t) u2Σ = PΣ
UΣ2 , (3.35)
whereu2Σ is the mean value of the squared collective instantaneous voltage. It should be noted thatGcan not be determined instantaneously because it involves the averaged values PΣandUΣ2, which generally can not be calculated for a shorter period than one fundamental cycle. Therefore, the instantaneous active currentsik, acan be correctly estimated only in the steady-state conditions. The FBD-method decomposes the phase currents to the following components
ik = ik, a
|{z}
active
+ ik, v
|{z}
variation
| {z }
power current, ik, p
+ ik, z
|{z}
zero power
. (3.36)
Also, the phase currents may be decomposed to the active and the nonactive components as ik= ik, a
|{z}
active
+ ik, v
|{z}
variation
+ ik, z
|{z}
zero power
| {z }
nonactive current, ik, na
. (3.37)
The collective quantities may be decomposed similarly.
3.2.2 The instantaneous reactive power theory
The instantaneous reactive power theory, or the p-q-theory, was introduced in the early 80’s by Akagi et al. (1983, 1984). The theory introduced an interesting concept of instantaneous reactive power, or instantaneous imaginary power as this quantity was named by the authors.
However, some conceptual limitations of this theory were pointed out by Willems (1992), particularly if zero-sequence currents or voltages are present. A modified theory was pro-posed by Nabae et al. in 1993. The modified theory is also presented in (Nabae et al., 1995) and summarized in (Akagi et al., 1999). However, as analyzed by Depenbrock et al. (2003), both theories yield incorrect results in a four-wire case compared to the generally valid FBD-method. However, if the zero-sequence components are absent the results of both instanta-neous power theories correspond to the results of the FBD-method. This is a typically valid assumption in three-phase three-wire systems.
Let us summarize the original theory of instantaneous power in three-phase three-wire sys-tems without zero-sequence components. The instantaneous power may be calculated with instantaneous phase currents and voltages as
p=uaia+ubib+ucic . (3.38) If the zero-sequence components are absent, the instantaneous power may be expressed by using the peak-value scaled space-vector representation as
p= 3
2(uαiα+uβiβ). (3.39)
Akagi et al. (1983, 1984) name this quantity the instantaneous real power. In order to de-fine the instantaneous reactive power Akagi et al. (1983, 1984) introduced the instantaneous imaginary power space-vector, the magnitude of which is the instantaneous imaginary power1
q=3
2(uβiα−uαiβ). (3.40)
1Akagi et al. (1983, 1984) originally defined instantaneous imaginary power as negation of (3.40).
3.2 Instantaneous power components 35
Expressing with phase quantities by applyingαβζ-to-abc transform yields q=− 1
As clearly pointed out by Akagi et al. (1983, 1984) the instantaneous imaginary power is not a conventional electrical quantity, because the termsuβiαanduαiβ are products of the instantaneous current in one axis and the instantaneous voltage in another axis. Therefore, Akagi et al. (1983, 1984) suggested that a new dimension should be introduced forq. The authors proposed “imaginary watt”. The instantaneous powers are conveniently written as
p
The inverse is calculated as iα
The instantaneous currents may be written as iα = 2
iαpis theα-axis instantaneous active current iβpis theβ-axis instantaneous active current iαqis theα-axis instantaneous reactive current iβqis theβ-axis instantaneous reactive current.
It should be noted that the instantaneous active current here is different from the instantaneous active current in the FBD-method. Further, the instantaneous powers in theα- and theβ-axis, pαandpβ, respectively, may be expressed as
pα = 3
Using (3.46)–(3.51) the power in the three-phase three-wire circuit may be expressed as
pαpis theα-axis instantaneous active power pβpis theβ-axis instantaneous active power pαqis theα-axis instantaneous reactive power pβqis theβ-axis instantaneous reactive power.
Also, in here it is noted that the concept of the instantaneous active power is different from the instantaneous active power of the FBD-method. From (3.53) it is noticed that always pαq+pβq= 0. Hence
p = uαiαp+uβiβp =pαp+pβp (3.54) 0 = uαiαq+uβiβq=pαq+pβq. (3.55) It is interesting to remark that in (3.55) thepαq andpβq are calculated as a product of the voltage and the current in the same axis, and that their dimension is Watt. Akagi et al. (1999) explained thatpαq contributes to the energy transfer within theα-circuit only and similarly pβqcontributes to the energy transfer within theβ-circuit only. If theαβ-circuit is considered as a whole these energy transfers add up to zero meaning thatpαq andpβqdo not contribute to the energy transfer to or from theαβ-circuit but only to the circulating energy flow inside theαβ-circuit. At the same time, they increase the current magnitudes in both axes leading to increased line losses. This is quite an intriguing view on the essence of reactive power.
Akagi et al. (1984) stated that in a balanced sinusoidal case the instantaneous reactive power is numerically equal to three times the conventional reactive power per one phase. Later, Ferrero and Superti-Furga (1991) developed the Akagi’s instantaneous power concept further and derived interesting properties.2By utilizing complex Fourier series representation of the space-vector Ferrero and Superti-Furga (1991) stated that in a steady state the average value of instantaneous power is
whereuhandihare the coefficients of the complex Fourier series as shown in (2.12). Further, the average value of the instantaneous reactive power is
q= 3
2Ferrero and Superti-Furga (1991) used the name Park-powers.