During Transients
Since the input voltage is shifted by an angle
then
where
and
is a voltage shift and
is the initial value of the current.
Transient components of
are
Using the
function, we can calculate the RMS value of the currents both during the transient and steady state,
Figure 2.
In generally, in the case of non-harmonic current,
But
can be determined by [
8], where there are more ways to do it:
- -
Using continuous-time filter;
- -
Using digital filtering;
- -
Using integral calculus;
- -
Using a discrete Fourier transform;
- -
Using artificial neural networks [
20].
The use of the continuous-time filter seems to be the fastest way to process a signal, but it requires the use of auxiliary hardware for conversion and processing (DAC-ADC converter, multiplier, integrator, etc.). So, the time of calculation also depends on the settling time, time of conversion of DAC converter, and soon. The last two items are not directly bound to the ip-iq method.
By this, we can use Fourier coefficients for the maximal value calculation of ip-iq currents and their time waveforms.
The first term in Equation (11)
or calculating the sliding window using the square or (more exactly) meander rule
And the first term under sqrt in (11)
So, for single power components, we can write:
For other power components analogically:
For blind reactive power
although
It can be also calculated as
which is, in steady state, the same.
Since the distortion reactive power is, in steady state, equal to zero, distortion power in the transient can be determined as
And it can also be calculated as
In practical simulation or implementation, the distortion power component
, however, is not zero due to the different calculated time of
and
or
, respectively. The principal time waveforms of the power components S, P, Q, D (k) by using the movRMS method are shown in both transient states’ start-up (a) and recovery (b),
Figure 3.
As can be seen from the figure, the apparent power (black) is not equal to the square of the sum of the active and reactive ones (blue dashes). Let us name the difference between these power components the fictitious distortion component (purple).
Let us note, in this regard, using the Fourier transform on Equation (9a) for the total transient current waveform
which consists of steady and transient components (two-times
). Since the equation consists of steady- and transient-state components, we can write
Taking into account Tabs in [
8], the Fourier transform of the
function can be derived (a simplified approach [
8]), namely, using the time-shift rule
where
.
Then, for the steady-state component
And, for the transient component
This equation shows the distortion component is not quite fictitious.
- (a)
In the case of harmonic supply and non-harmonic current:
The total non-harmonic current can be decomposed
We need to know
,
to determine the rms value of the sum of higher harmonics
By this, we can use Fourier coefficients for the maximal value calculation of
currents and their time waveforms
The time waveforms of decomposed current components are shown in
Figure 4.
Using Equations (13)–(16), we obtain the active and blind reactive power components , , and .
A classical non-harmonic rectangular time waveform causes under R-L load a non-harmonic current,
Figure 5.
The voltage and the current can be decomposed into p- and q- components.
Since the supply voltage is non-harmonic
We need to know
,
to determine the rms value of the sum of higher harmonics
The first term under sqrt
and
By this, we can use Fourier coefficients for the maximal value calculation of the
voltages and their time waveforms
The time waveforms of the decomposed current components are shown in
Figure 6.
Figure 6 shows non-harmonic input voltage (inverter voltage) u(t), its fundamental harmonic, and its decomposition into u1p, u1q components (a), and also the decomposition of the total current fundamental harmonic into i1p, i1q components, respectively. Decomposition (a) follows Equations (29)–(30) and decomposition (b) follows Equations (23) and (24), in which the ip-iq method is included. So, again using Equations (13)–(16), we obtain the active and blind reactive power components
,
, and
.
The calculated power components , , and are applied in the next application section.
While for a non-symmetric system, the inequality holds
because of including a zero-phase sequence component
into the Clarke transform [
13].
where
−1/2 +
and
is the transformation constant.
We can also use the method of symmetrical components
, [
7,
11].
The inverse transformation into the
system can be obtained using the inverse transformation matrix. Consequently, the power in a non-symmetrical system also features an instantaneous zero-sequence component
.
The phase power components can be expressed by the inverse transform of Equation (34).