3.1. Demand for the Inverter Output Impedance
To suppress the oscillation, the inverter output impedance should be designed relatively high at the harmonic oscillation frequency while relatively low at other frequencies. For this purpose, the virtual impedances are added to be connected with the original output impedance, as shown in
Figure 3.
Figure 3a–c presents parallel, series, and parallel-series virtual impedances, respectively. The above three forms can reach the same performance for oscillation suppression. For the first and second forms, it is relatively complicated to introduce one virtual impedance to realize that the inverter output impedance shows high at the harmonic oscillation frequency while relatively low at other frequencies. However, the third form with two virtual impedances is relatively easy to achieve this purpose. Thus, the third form is mainly discussed in this paper.
In
Figure 3c,
Zoj is the self-impedance when the parallel-series virtual impedances are not added. The equivalent impedance
Zj (
Zj = 1/
Yj) is formed by
Zoj in parallel with
Zpj, and then in series with
Zsj. The grid equivalent impedance
Zsgj consists of the virtual impedance
Zsj and the grid impedance
Zg connected in series.
if/hj,
if/h1j,
if/h2j, and
if/h3j are total fundamental/high-frequency harmonic current, fundamental/high-frequency harmonic current of
Zoj branch, fundamental/high-frequency harmonic current of
Zpj branch, and fundamental/high-frequency harmonic current of
Zsgj branch, respectively.
if/hj = if/h1j + if/h2j + if/h3j. The total fundamental frequency current
ifj and the total high-frequency harmonic current
ihj are determined by the shunt circuit, consisting of the self-impedance
Zoj, parallel virtual impedance
Zpj, and grid equivalent impedance
Zsgj. From the PCC, the Norton equivalent circuit of
Figure 3c is refined into the forms of
Figure 3d.
On the basis of the proposed basic technique, the parallel virtual impedance Zpj should be designed to show low impedance at the harmonic oscillation frequency. By doing so, most high-frequency harmonic current will flow into the parallel virtual impedance Zpj branch; it effectively suppresses the oscillation of paralleled multi-inverter system. In the meantime, to improve the power quality of grid-connected current, the series virtual impedance Zsj should be designed to display low impedance at the fundamental frequency. This way, most fundamental frequency current flows into the grid branch with relatively low impedance.
3.2. Oscillation Suppression Method
The parallel-series virtual impedances in
Figure 4 can be realized by introducing two notch filters. In
Figure 4, two virtual impedances are added in parallel and series with inverter output impedance, respectively.
Gi is the grid-connected current loop proportional resonant (PR) controller,
GPWM is the equivalent gain of the inverter,
ZL1j =
sL1j +
RL1j,
ZC1j = 1/
sC1j,
ZL2j =
sL2j +
RL2j.
The parallel virtual impedance
Zpj and series virtual impedance
Zsj can be expressed as
where
r1 and
r2 are the proportional coefficient and
GN is the notch filter.
The grid-connected current loop PR controller
Gi can be expressed as
where
kp is the proportional coefficient of quasi-proportional resonant controller,
ki1 is the resonance gain of quasi proportional resonance controller,
ωc is the cut-off angular frequency, and
ωo is the fundamental angular frequency.
The effects of dead-time of switching devices in paralleled multi-inverter system are regarded as a disturbance, which have a constant amplitude and an alternative direction depending on the inverter-side inductor current
iL1j [
25]. It is notable that the disturbance can be seen as the controlled current source
idj in Norton equivalent circuit, which can be presented as
where
and
.
According to Reference [
25],
Uej in Equation (7) can be expressed as
where
UTj and
UDj are the on-state voltage drop of switching devices and diodes,
Tsj,
tdj,
tonj and
toffj are switching period, dead-time, turn-on time, and turn-off time of switching devices.
Meanwhile, sign(
iL1j) in Equation (7) can be expressed as
From
Figure 4, the closed-loop transfer function of the system can be expressed as
where
irefj is the reference current of single inverter,
Gj is the current source equivalent coefficient of single inverter, and
Yj is the equivalent admittance of single inverter.
According to Equation (10), the refined equivalent output impedance model is shown in
Figure 5. The current source
ij is equivalent to the current source
i1j in parallel with the current source
idj. Two virtual impedances are added to be in parallel and in series with the original inverter output impedance, respectively. Thus,
Figure 5 is equivalent to the refinement of
Figure 3d, which can achieve the proposed approach.
From
Figure 5, the equivalent model of the m-th inverter can be expressed as
where
irefm,
Gm,
Ym,
idm,
ZL1m,
ZC1m,
ZL2m,
Zpm, and
Zsm are the variables of the m-th inverter.
Substitute Equation (1) into Equation (11), and Equation (11) can be rewritten as
where
Gselfm is the transfer relationship between the grid-side inductance current
iom of the m-th inverter and reference current
irefm of the m-th inverter,
Gparalm,j is the transfer relationship between the grid-side inductance current
iom of the m-th inverter and reference current
irefj of the
jth inverter, and
Gserim is the transfer relationship between grid-side inductance current
iom of the m-th inverter and grid voltage
ug.
When a similar analysis method is adopted to other inverters, the system can be expressed using a closed-loop transfer function matrix with reference currents, grid voltage and controlled current sources as inputs and grid-side inductor currents as outputs
It is obvious that the strong coupling between the inverters and the grid may exist and introduce harmonic oscillation currents under weak grid condition.
According to
Figure 4,
Figure 6 gives the control block diagram of the oscillation suppression method by two notch filters for parallel inverters. The first notch filter is introduced into the feedforward path of PCC voltage and the second notch filter is introduced into the feedback path of the grid-side inductor current. The feedback path of the grid-side inductor current with the notch filter is equivalent to a virtual impedance in series with inverter output impedance, which can effectively improve the power quality of the grid-connected current. The feedforward path of PCC voltage with the notch filter equals to a virtual impedance in parallel with inverter output impedance. This can effectively restrain the parallel inverters’ harmonic current from flowing into the grid and avoid the oscillation phenomenon.
H1j is the feedback coefficient of the grid-side inductor current, and
H2j is the feedforward coefficient of PCC voltage.
From
Figure 6, the equivalent closed-loop transfer function of the system can be expressed as
where
Gjeq is the equivalent coefficient of the current source after the single inverter transformation,
Yjeq is the inverter equivalent admittance after the single inverter transformation,
,
, and
.
In order to achieve the same purpose of
Figure 4 and
Figure 6, the equivalent coefficient of the current source and the inverter equivalent admittance in Equation (10) are equal to those in Equation (14). Thus, it can be expressed as
From Equation (15), the feedback coefficient of the grid-side inductor current
H1j and the feedforward coefficient of PCC voltage
H2j can be expressed as
At the specific frequency, the amplitude of the notch filter is greatly attenuated while the amplitude at other frequencies is almost non-destructive. The notch filter
GN can be expressed as
where
fo is the fundamental frequency and
Q is the quality factor of the notch filter.
The analysis diagrams of the notch filter with
Q = 0.25, 0.5, and 1 are shown in
Figure 7. From
Figure 7a, the larger
Q is, the better the notch characteristic of the notch filter, but the worse the frequency adaptability. From
Figure 7b, when
Q = 0.25, the characteristic equation has two unequal real poles on the negative real axis of the s plane, which is over-damping. When
Q = 0.5, the characteristic equation has two equal real poles on the negative real axis of the s plane, which is critical-damping. When
Q = 1, the characteristic equation has the conjugate complex poles on the left half plane, which is under-damping. From
Figure 7c, the tuning time of notch filter with
Q = 0.5 is better than
Q = 0.25, 1. Therefore, considering the notch characteristics and dynamics comprehensively, the
Q value was selected to be 0.5.
The Bode diagrams of parallel virtual impedance
Zpj and series virtual impedance
Zsj are shown in
Figure 8. At the fundamental frequency, the parallel virtual impedance
Zpj shows a high impedance, and the series virtual impedance
Zsj displays a low impedance, so that the fundamental current flows into the grid. At the high frequency, the parallel virtual impedance
Zpj shows a low impedance, and the series virtual impedance
Zsj displays a high impedance, so that the high-frequency harmonic current flows into the parallel virtual impedance
Zpj branch.
The Bode diagrams of the inverter self-impedance
Zoj, parallel virtual impedance
Zpj, grid equivalent impedance
Zsgj, and grid impedance
Zg are shown in
Figure 9. Combined with
Figure 5, the grid equivalent impedance
Zsgj is much lower than the self-impedance
Zoj and the parallel virtual impedance
Zpj at the fundamental frequency, thus most fundamental frequency current flows into the grid branch with relatively low impedance, which improves the power quality of grid-connected current. At the high frequency, the parallel virtual impedance
Zpj is much lower than the self-impedance
Zoj and grid equivalent impedance
Zsgj, thus most high-frequency harmonic current flows into the parallel virtual impedance
Zpj branch with relatively low impedance, and it effectively suppresses the oscillation of the paralleled multi-inverter system.