1 Introduction

LCL-filter-based grid-connected inverters have been widely discussed in [1,2,3,4]. One of the important issues is their stability. Ref. [5] proposed a robust passive damping method for LLCL-filter-based grid-connected inverters, so as to minimize the effect of harmonic voltages on the grid. Another interesting filter was proposed in [6] for optimizing the system performance and stability. A recent review of passive filters for grid-connected converters was presented in [7], and different passive damping methods were discussed. In order to avoid the losses resulting from passive damping, capacitor-current feedback active damping was proposed in [8] to enhance the stability of LCL-type grid-connected inverters. Ref. [9] presented an interesting active damping method that can handle stability with only grid-current feedback control. In [10], a virtual RC damping method was presented with extended selective harmonic compensation. Note that these solutions are for voltage-source inverters. In practice, there are also current-source inverters [11,12,13]. An interesting active damping solution was presented in [14]. However, while the above methods are insightful for stability analysis, the impact of phase-locked loop (PLL) dynamics on the stability of grid-connected inverters has been neglected. The PLL is used to track the phase angle of the voltage at the point of common coupling (PCC) and thus to achieve grid synchronization for the inverter [15]. In case of high grid impedance, it has been found that the stability of a voltage source converter (VSC) can be affected by PLL parameters [16, 17]. In VSC-based high voltage direct current (HVDC) transmission systems, the PLL parameters have been found to limit the maximum power transfer capability [18]. In [19], the impact of short-circuit ratio and phase-locked-loop parameters on small-signal behavior is investigated, However, it only takes L-filter-based inverters into account.

In summary, these early works have discussed the influence of PLL parameters on the system stability. However, they only focused on grid-connected inverter systems with L or LC filters. In fact, an LCL filter is preferred in grid-connected inverters, due to its better higher frequency attenuation and reduced physical size of the inductor [20]. In practice, the LCL filter usually suffers from resonance problems, which result in system instability. In order to deal with the problem, active damping controllers have been used [21].

Under weak grid conditions, the current regulator and the PLL dynamics of the active damped inverter may interact as the grid current passes the grid impedance, especially in case of a high bandwidth PLL. In fact, fast grid synchronization and PLL is important for operating the grid-connected inverters. Increasing PLL bandwidth is helpful for the fast and accurate grid synchronization. The PLL bandwidth should be large enough for the fast grid synchronization, and this is why diverse PLL designs have been proposed, as shown in [22,23,24,25,26,27,28]. However, when the PLL bandwidth is increased beyond a certain limit, the system may become unstable due to resonance between the inverter and the weak grid. Therefore, the interaction between fast PLL dynamics and current regulation deserves further investigation.

The objective of this paper is to contribute both analytical and experimental verification on the interaction of PLL dynamics and current regulation in three-phase grid-connected active damped inverters with LCL filters under weak grid conditions. Using the Norton Theorem, a closed-loop transfer function is established to model the whole grid-connected inverter system including the PLL dynamics. By analyzing the poles of the closed-loop transfer function, the ranges of stability of the PLL bandwidth and the current regulator gain are obtained. Both a simulation and experimental results from a three-phase grid-connected inverter are provided to verify the effectiveness of the proposed model. The remainder of this paper is organized as follows. In Sect. 2, a small-signal model is established for the three-phase grid-connected LCL-filter-based inverter with capacitor voltage feedback and active damping. In Sect. 3, the small-signal stability of the inverter is discussed in terms of the PLL dynamics and the current regulator. In Sect. 4, the analytical results are verified through simulations and experimental tests. Finally, conclusions are presented in Sect. 5.

2 Inverter system model

Figure 1 shows the structure of a typical three-phase grid-connected inverter system. S 1 ~ S 6 are the switches; L 1 is the inverter-side filter inductance; C is the filter capacitance; L 2 is the grid-side filter inductance; L g is the grid line inductance; V Ca, V Cb, V Cc are the capacitor voltages in the grid frame ‘abc’; I ga, I gb, I gc are the grid currents in the grid frame ‘abc’.

Fig. 1
figure 1

Structure of a three-phase grid-connected inverter system

In the grid dq frame, θ 1 is the PLL output phase angle; ω 1 is the angular frequency; V C, I g, I L, V PCC, V g are the voltage vector at the filter capacitors, grid current vector, inverter-side filter current vector, voltage vector at the point of common coupling, and voltage vector of the grid-side voltage sources, respectively. In the inverter dq frame, the PLL output phase angle is denoted as θ, which is almost the same as θ 1 in steady-state conditions. Other variables in the inverter dq frame are labeled with a superscript c, when referred in the grid dq frame. As current references are constant, in both dq frames, they are denoted as I ref.

Let V sC denote the capacitance voltage in the αβ frame. \( \varvec{V}_{\text{C}}^{{}} = {\text{e}}^{{ - {\text{j}}\theta_{1} }} \varvec{V}_{\text{C}}^{\text{s}} \) when it is in the grid dq frame, and V cC  = ejθ V sC in the inverter dq frame. Then, the following relationship is obtained:

$$ \left\{ \begin{array}{l} V_{\text{C}}^{\text{c}} = {\text{e}}^{ - j\Delta \theta } V_{\text{C}} \hfill \\ \Delta \theta = \theta - \theta_{1} \hfill \\ \end{array} \right. $$
(1)

where ∆θ is the phase angle difference in these two dq frames and is approximately equal to zero when the grid-connected inverter operates in steady state. The PLL dynamics, however, make ∆θ depart from zero. Consequently, it is important to investigate the impact of the PLL on the dynamic stability of the grid-connected inverter under weak grid conditions.

2.1 Current regulation with active damping

As shown in Fig. 2, the current regulator for the inverter system includes a PI regulator and an active damping controller, whose transfer function is F(s).

Fig. 2
figure 2

Structure of inverter current regulator

Note that an active damped LCL filter increases the system complexity while increasing the stability of the inverter. Passive damping can also be used, but it incurs additional power losses. Therefore, in this paper, active damping is used due to avoid these power losses, and it is based on capacitor voltage feedback, where the capacitor voltage is differentiated by s and multiplied by a gain K before being fed back to the current regulator output. The transfer function \( F\left( s \right) \) is \( - KCs \).

In practice, the differential term may result in a high-frequency disturbance. To avoid this problem, a partially differentiation is used in the form \( s \approx \frac{s}{\tau s + 1} \), where τ is a time constant parameter. It is close to a differential element when τ is very small. This gives the active damping controller a transfer function of the form \( F\left( s \right) = - K\frac{Cs}{\tau s + 1} \).

The inverter output voltage is described as follows:

$$ \varvec{V}_{\text{PWM}} \left( s \right) = \left[ {\left( {\varvec{I}_{\text{ref}} \left( s \right) - \varvec{I}_{\text{g}}^{\text{c}} \left( s \right)} \right)\left( {k_{\text{p}} + \frac{{k_{\text{i}} }}{s}} \right) + F\left( s \right)\varvec{V}_{\text{C}}^{\text{c}} \left( s \right)} \right]\frac{{V_{\text{dc}} }}{2} $$

From Fig. 2, the system variable relationships can be described by the following open-loop transfer functions [29]:

$$ \begin{aligned} \varvec{I}_{\text{L}}^{\text{c}} \left( s \right) = H_{1} \left( s \right)\varvec{V}_{\text{PWM}} \left( s \right) + H_{2} \left( s \right)\varvec{V}_{\text{PCC}}^{\text{c}} \left( s \right) \hfill \\ \varvec{I}_{\text{g}}^{\text{c}} \left( s \right) = H_{3} \left( s \right)\varvec{V}_{\text{PWM}} \left( s \right) + H_{4} \left( s \right)\varvec{V}_{\text{PCC}}^{\text{c}} \left( s \right) \hfill \\ \varvec{V}_{\text{C}}^{\text{c}} \left( s \right) = H_{5} \left( s \right)\varvec{V}_{\text{PWM}} \left( s \right) + H_{6} \left( s \right)\varvec{V}_{\text{PCC}}^{\text{c}} \left( s \right) \hfill \\ \end{aligned} $$
(2)

where the transfer functions H 1(s) ~ H 6(s) are

$$ \begin{aligned} H_{1} \left( s \right) = \frac{{L_{2} Cs^{2} + 1}}{{L_{1} L_{2} Cs^{3} + \left( {L_{1} + L_{2} } \right)s}},\quad H_{2} \left( s \right) = - \frac{1}{{L_{1} L_{2} Cs^{3} + \left( {L_{1} + L_{2} } \right)s}}, \hfill \\ H_{3} \left( s \right) = \frac{1}{{L_{1} L_{2} Cs^{3} + \left( {L_{1} + L_{2} } \right)s}},\quad H_{4} \left( s \right) = - \frac{{L_{1} Cs^{2} + 1}}{{L_{1} L_{2} Cs^{3} + \left( {L_{1} + L_{2} } \right)s}}, \hfill \\ H_{5} \left( s \right) = \frac{{L_{2} s}}{{L_{1} L_{2} Cs^{3} + \left( {L_{1} + L_{2} } \right)s}},\quad H_{6} \left( s \right) = \frac{{L_{1} s}}{{L_{1} L_{2} Cs^{3} + \left( {L_{1} + L_{2} } \right)s}}. \hfill \\ \end{aligned} $$

Based on the above equations, the grid current can be obtained as:

$$ \varvec{I}_{\text{g}}^{\text{c}} \left( s \right) = \varvec{G}_{\text{T}} \left( s \right)\varvec{I}_{\text{ref}} \left( s \right) - \varvec{Y}_{\text{i}} \left( s \right)\varvec{V}_{\text{C}}^{\text{c}} \left( s \right) $$
(3)

which in vector matrix form is:

$$ \varvec{I}_{\text{g}}^{\text{c}} \left( s \right) = \underbrace {{\left[ {\begin{array}{*{20}c} {g_{\text{t}} \left( s \right)} & 0 \\ 0 & {g_{\text{t}} \left( s \right)} \\ \end{array} } \right]}}_{{\varvec{G}_{\text{T}} \left( s \right)}}\varvec{I}_{\text{ref}} \left( s \right) - \underbrace {{\left[ {\begin{array}{*{20}c} {y_{\text{i}} \left( s \right)} & 0 \\ 0 & {y_{\text{i}} \left( s \right)} \\ \end{array} } \right]}}_{{\varvec{Y}_{\text{i}} \left( s \right)}}\varvec{V}_{\text{C}}^{\text{c}} \left( s \right) $$
(4)

where g t(s) and y i(s) can be expressed as:

$$ \left\{ \begin{aligned} g_{\text{t}} = \frac{{V_{\text{dc}} \left( {k_{\text{p}} + \frac{{k_{\text{i}} }}{s}} \right)H_{3} H_{6} - V_{\text{dc}} \left( {k_{\text{p}} + \frac{{k_{\text{i}} }}{s}} \right)H_{4} H_{5} }}{{2H_{6} + V_{\text{dc}} \left( {k_{\text{p}} + \frac{{k_{\text{i}} }}{s}} \right)H_{3} H_{6} - V_{\text{dc}} \left( {k_{\text{p}} + \frac{{k_{\text{i}} }}{s}} \right)H_{4} H_{5} }} \hfill \\ y_{\text{i}} = \frac{{V_{\text{dc}} F(s)H_{4} H_{5} - 2H_{4} - V_{\text{dc}} F(s)H_{3} H_{6} }}{{2H_{6} + V_{\text{dc}} \left( {k_{\text{p}} + \frac{{k_{\text{i}} }}{s}} \right)H_{3} H_{6} - V_{\text{dc}} \left( {k_{\text{p}} + \frac{{k_{\text{i}} }}{s}} \right)H_{4} H_{5} }} \hfill \\ \end{aligned} \right. $$
(5)

The small-signal transfer function can be expressed as:

$$ \Delta \varvec{I}_{\text{g}}^{\text{c}} = - \underbrace {{\left[ {\begin{array}{*{20}c} {y_{\text{i}} \left( s \right)} & 0 \\ 0 & {y_{\text{i}} \left( s \right)} \\ \end{array} } \right]}}_{{\varvec{Y}_{\text{i}} \left( s \right)}}\Delta \varvec{V}_{\text{C}}^{\text{c}} $$
(6)

where \( \Delta \varvec{I}_{\text{g}}^{\text{c}} \) is a grid current disturbance and \( \Delta \varvec{V}_{\text{C}}^{\text{c}} \) is a capacitor voltage disturbance in the inverter dq frame.

2.2 Small-signal model of PLL dynamics

Figure 3 illustrates the PLL closed-loop control system used in the three-phase grid-connected inverter. A PI regulator is used to enable the q-axis element of the capacitor voltage to follow the zero input reference V q*, while V cCa ,V cCb ,V cCc are the PLL inputs; ω 0 is the base phase angular velocity; f and θ are the PLL output frequency and phase angle, respectively. The output of the PI regulator is the instantaneous angular velocity disturbance ∆ω in the following form:

$$ \Delta \omega = \underbrace {{\left( {k_{\text{pPLL}} + \frac{{k_{\text{iPLL}} }}{s}} \right)}}_{{F_{\text{PLL}} \left( s \right)}}\text{Im} \left\{ {\varvec{V}_{\text{C}}^{\text{c}} } \right\} $$
(7)

where k pPLL and k iPLL are the proportional and integral gains of the PI regulator, respectively.

Fig. 3
figure 3

PLL system

It can be interpreted from Fig. 3 that, by integrating ∆ω + ω 0, the following relationship is obtained, where \( F_{\text{PLL}} \left( s \right) \) is defined in (7):

$$ \frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \omega_{0} + \Delta \omega = \omega_{0} + F_{\text{PLL}} \left( s \right)\text{Im} \left\{ {\varvec{V}_{\text{C}}^{\text{c}} } \right\} $$
(8)

Based on (1), the capacitor voltage can be written as

$$ \begin{aligned} \varvec{V}_{\text{C}}^{\text{c}} & = {\text{e}}^{{ - {\text{j}}\Delta \theta }} \varvec{V}_{\text{C}} \approx \left( {1 - {\text{j}}\Delta \theta } \right)\left( {V_{\text{C}}^{ 0} + \Delta \varvec{V}_{\text{C}} } \right) \\ & = V_{\text{C}}^{ 0} + \Delta \varvec{V}_{\text{C}} - {\text{j}}\Delta \theta V_{\text{C}}^{ 0} \\ & \Rightarrow \text{Im} \left\{ {\varvec{V}_{\text{C}}^{\text{c}} } \right\} = \text{Im} \left\{ {\Delta \varvec{V}_{\text{C}} } \right\} - \Delta \theta V_{\text{C}}^{ 0} \\ \end{aligned} $$
(9)

where V 0C is the steady-state capacitor voltage and ∆V C is the capacitor voltage disturbance. The capacitor voltage disturbance in the inverter dq frame ∆V cC is:

$$ \Delta \varvec{V}_{\text{C}}^{\text{c}} = \Delta \varvec{V}_{\text{C}} - {\text{j}}\Delta \theta V_{\text{C}}^{ 0} = \Delta \varvec{V}_{\text{C}} - {\text{j}}V_{\text{C}}^{ 0} G_{\text{PLL}} \left( s \right)\text{Im} \left\{ {\Delta \varvec{V}_{\text{C}} } \right\} $$
(10)

where G PLL(s) is given in the following equation:

$$ \Delta \theta = \underbrace {{\frac{{F_{\text{PLL}} \left( s \right)}}{{s + V_{\text{C}}^{ 0} F_{\text{PLL}} \left( s \right)}}}}_{{G_{\text{PLL}} \left( s \right)}}\text{Im} \left\{ {\Delta \varvec{V}_{\text{C}} } \right\} $$
(11)

The vector matrix form of \( \Delta \varvec{V}_{\text{C}}^{\text{c}} \) is

$$ \Delta \varvec{V}_{\text{C}}^{\text{c}} = \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & {1 - V_{\text{C}}^{ 0} G_{\text{PLL}} \left( s \right)} \\ \end{array} } \right]\Delta \varvec{V}_{\text{C}} $$
(12)

Similarly, the grid current disturbance ∆I cg can be expressed as follows:

$$ \Delta \varvec{I}_{\text{g}}^{\text{c}} = \Delta \varvec{I}_{\text{g}} - {\text{j}}\Delta \theta I_{\text{g}}^{ 0} = \Delta \varvec{I}_{\text{g}} - {\text{j}}I_{\text{g}}^{ 0} G_{\text{PLL}} \left( s \right)\text{Im} \left\{ {\Delta \varvec{V}_{\text{C}} } \right\} $$
(13)

where I 0g is the steady-state value of the grid current and ∆I g is the current disturbance in the grid dq frame. The vector matrix form of (13) can be written as:

$$ \Delta \varvec{I}_{\text{g}}^{\text{c}} = \Delta \varvec{I}_{\text{g}} - \left[ {\begin{array}{*{20}c} 0 & 0 \\ 0 & {I_{\text{ref}} G_{\text{PLL}} \left( s \right)} \\ \end{array} } \right]\Delta \varvec{V}_{\text{C}} $$
(14)

With (12), (13) and (14), the inverter input impedance Z eq can be obtained:

$$ \varvec{Z}_{\text{eq}} = - \Delta \varvec{V}_{\text{C}} \left( {\Delta \varvec{I}_{\text{g}} } \right)^{ - 1} = - \left[ {\begin{array}{*{20}c} 0& 0\\ 0& {I_{\text{ref}} G_{\text{PLL}} \left( s \right)} \\ \end{array} } \right]{ + }\left[ {\begin{array}{*{20}c} {y_{\text{i}} \left( s \right)} & 0\\ 0& {y_{\text{i}} \left( s \right)} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & {1 - V_{\text{C}}^{ 0} G_{\text{PLL}} \left( s \right)} \\ \end{array} } \right] $$
(15)

Then with (14) and (15), the grid current can be expressed as

$$ \varvec{I}_{\text{g}} \left( s \right) = \varvec{G}_{\text{T}} \left( s \right)\varvec{I}_{\text{ref}} \left( s \right) - \varvec{Z}_{\text{eq}} \left( s \right)^{ - 1} \varvec{V}_{\text{C}} \left( s \right) $$
(16)

The equivalent Norton model of the inverter system is shown in Fig. 4.

Fig. 4
figure 4

Equivalent Norton model of inverter system

In Fig. 4, Z L1(s) denotes the grid-side filter inductance and Z g(s) denotes the grid line impedance:

$$ \left\{ {\begin{array}{*{20}l} {\varvec{Z}_{\text{L1}} \left( s \right) = \left[ {\begin{array}{*{20}c} {{1 \mathord{\left/ {\vphantom {1 {L_{2} s}}} \right. \kern-0pt} {L_{2} s}}} & 0 \\ 0 & {{1 \mathord{\left/ {\vphantom {1 {L_{2} s}}} \right. \kern-0pt} {L_{2} s}}} \\ \end{array} } \right]} \hfill \\ {\varvec{Z}_{\text{g}} \left( s \right) = \left[ {\begin{array}{*{20}c} {L_{\text{g}} s} & 0 \\ 0 & {L_{\text{g}} s} \\ \end{array} } \right]} \hfill \\ \end{array} } \right. $$
(17)

The final transfer function of the closed-loop system can be obtained from Fig. 4 as follows:

$$ \varvec{G}\left( s \right) = \varvec{V}_{\text{PCC}} \left( s \right)\left[ {\varvec{I}_{\text{ref}} \left( s \right)} \right]^{ - 1} = \varvec{G}_{\text{T}} \frac{{\left( {\varvec{Z}_{\text{L1}} + \varvec{Z}_{\text{g}} } \right)\varvec{Z}_{\text{eq}} }}{{\varvec{Z}_{\text{eq}} + \varvec{Z}_{\text{L1}} + \varvec{Z}_{\text{g}} }} + \frac{{\varvec{Z}_{\text{eq}} }}{{\varvec{Z}_{\text{eq}} + \varvec{Z}_{\text{L1}} + \varvec{Z}_{\text{g}} }}\varvec{V}_{\text{g}} \left( {\varvec{I}_{\text{ref}} } \right)^{ - 1} $$
(18)

3 Small signal stability analysis

In general, a weak grid condition means the grid has a high impedance. In this section, a small signal stability analysis is carried out for the grid-connected inverter system with high grid impedance. More specifically, the ranges of the PLL bandwidth and the current regulator gain k p required to keep the inverter system stable are investigated. It should be noted that optimized design of the current regulator and the PLL gains is beyond the scope of this paper.

3.1 Impact of PLL bandwidth

When the current regulator gain k p = 0.1 and the PLL regulator gain k pPLL is varied within the range from 2 to 22, the root loci of the closed-loop transfer function (18) are shown in Fig. 5. The arrows indicate the direction of increasing PLL bandwidth, and the intersection with the vertical axis is at k pPLL = 12. As can be seen from the root loci, the poles of the closed-loop system are all in the left half side of the complex plane, and the system is stable, when k pPLL is within the range from 2 to 12 (equally, the PLL bandwidth is less than 90 Hz). It should be noted that when the current regulator gain is fixed, the system tends to become less stable as the PLL bandwidth increases. Therefore, under weak grid conditions, a relatively small PLL bandwidth should be chosen to ensure system stability for the inverter.

Fig. 5
figure 5

Root locus of the system as PLL bandwidth changes

3.2 Impact of current regulator gain

When the PLL bandwidth is fixed at 50 Hz and the current regulator gain k p is varied within the range from 0 to 0.1, the root loci of the closed-loop transfer function (18) are shown in Fig. 6. The arrows indicate the direction of decreasing gain, and the intersection with the vertical axis is at k p = 0.04. As can be seen from this figure, the poles of the closed-loop system are all in the left half side of the complex plane, and the system is stable when k p is greater than 0.04. It can be observed that, when the PLL bandwidth is constant, the system tends to be less stable as the current regulator gain decreases.

Fig. 6
figure 6

Root locus of the system as current regulator gain changes

4 Simulation and experimental verifications

Based on the results of the small-signal stability analysis, it can be seen that the PLL dynamics may significantly affect the inverter system stability under weak grid conditions. In this section, time-domain simulations and experimental tests are conducted to verify the theoretical results.

Parameters of a three-phase grid-connected inverter system with the structure shown in Fig. 1 are listed in Tables 1 and 2.

Table 1 System parameters
Table 2 Control parameters

4.1 Simulation results

In order to verify the effectiveness of the active damping controller, the following two test cases were simulated with sampling period \( T_{\text{s}} \) = 50 μs and with integral gain of the PI-type current regulator k i = 10:

Case 1

The inverter is initially only equipped with a PI-type current regulator, then the active damping function is added at t = 0.05 s.

Case 2

The inverter is initially installed with a PI-type current regulator and the active damping function, then the damping function is removed at t = 0.1 s.

As shown in Fig. 7, the system becomes stable as soon as the active damping function is added, and it becomes unstable after the active damping function is removed.

Fig. 7
figure 7

Simulated input current waveform in the grid-connected inverter

To evaluate the impact of the PLL bandwidth, with a fixed current regulator gain k p = 0.1, the simulations were conducted with PLL bandwidth of 30 Hz and 125 Hz. The resulting dynamics of grid current I g and voltage at PCC V PCC are shown in Figs. 8 and 9 respectively. In Fig. 8, when the PLL bandwidth is 30 Hz, the inverter is stable and the oscillations of I g and V PCC are small. However, in Fig. 9, when the PLL bandwidth is increased to 125 Hz, the dynamics of both I g and V PCC diverge and the system becomes unstable. Based on these simulation results, for a given current regulator gain, it can be seen that the stability margin of the inverter system is bigger when the PLL bandwidth is smaller. This conclusion verifies the results of the small-signal analysis.

Fig. 8
figure 8

Simulated dynamics of I g and V PCC when PLL bandwidth is 30 Hz

Fig. 9
figure 9

Simulated dynamics of I g and V PCC when PLL bandwidth is 125 Hz

To evaluate the effect of the current regulator gain, keeping the PLL bandwidth fixed at 50 Hz, the simulations were conducted with k p = 0.1 and k p = 0.02. The resulting dynamics of grid current I g and voltage at PCC V PCC are shown in Figs. 10 and 11 respectively. The grid-connected inverter system is stable when k p = 0.1, as shown in Fig. 10. However, the dynamics of both I g and V PCC diverge and the grid-connected inverter system becomes unstable when k p = 0.02, as shown in Fig. 11. The conclusion again verifies the results of the small-signal analysis. These observations also indicate that the inverter system may become unstable when the PLL bandwidth is beyond its stable range, even when the current regulation gain is within its stable range. Therefore, it is important to take the PLL dynamics into account.

Fig. 10
figure 10

Simulated dynamics of I g and V PCC when k p = 0.1

Fig. 11
figure 11

Simulated dynamics of I g and V PCC when k p = 0.02

4.2 Experimental results

A 3 kVA experimental platform system was built to further verify the impacts of the PLL dynamics and the current regulator. The schematic diagram of the experimental setup is shown in Fig. 12.

Fig. 12
figure 12

Schematic diagram of the experimental setup

The inverter is controlled by a 32-bit fixed-point 150 MHz TMS320F2812 DSP. The system parameters are the same as those simulated above, listed in Tables 1 and 2. The effectiveness of the active damping controller was verified using two test cases similar to those simulated, as follows:

Case 1

The inverter system is initially only equipped with a PI type current regulator, then the active damping function is added at t = 0.023 s.

Case 2

The inverter system is initially installed with a PI type current regulator and the active damping function, then the damping function is removed at t = 0.25 s.

As shown in Fig. 13, the system becomes stable as soon as the active damping function is added, and it becomes unstable after the damping function is removed. These results verify the effectiveness of active damping regarding the resonance which results from the LCL filters.

Fig. 13
figure 13

Experimental waveforms for the grid-connected inverter

The interaction between the PLL dynamics and the current regulator was also experimentally tested. First of all, the current regulator gain was fixed as k p = 0.1, and experimental tests were carried out with the PLL bandwidth set to 30 Hz and 125 Hz. The results are shown in Figs. 14 and 15 respectively. It can be observed that the waveforms of V PCC, I g, and i L are stable when the PLL bandwidth is 30 Hz. However, when the PLL bandwidth is 125 Hz, V PCC, I g, and i L start diverging and operate unstably. This is consistent with the theoretical analysis and simulation results that, for a fixed current regulator gain, the system is more stable with a smaller PLL bandwidth. The same good agreement with theory was observed when the PLL bandwidth was changed online, as shown in Fig. 16.

Fig. 14
figure 14

Experimental waveforms when the PLL bandwidth is 30 Hz and the current regulator gain is k p = 0.1

Fig. 15
figure 15

Experimental waveforms when the PLL bandwidth is 125 Hz and the current regulator gain is k p = 0.1

Fig. 16
figure 16

Experimental waveforms when changing the PLL bandwidth

Secondly, the PLL bandwidth was fixed at 50 Hz, and experimental tests were carried out with the current regulator gain set to k p = 0.1 and k p = 0.02. The results are shown in Figs. 17 and 18 respectively.

Fig. 17
figure 17

Experimental waveforms when the current regulator gain k p = 0.1 and the PLL bandwidth is 50 Hz

Fig. 18
figure 18

Experimental waveforms when the current regulator gain k p = 0.02 and the PLL bandwidth is 50 Hz

It can be observed from Fig. 17 that the waveforms of V PCC, I g, and i L are stable when k p = 0.1. However, the system becomes unstable when k p = 0.02, with V PCC, I g, and i L diverging and showing irregular behavior. This is consistent with the theoretical analysis and simulation results that, when the PLL bandwidth is fixed, the system is more stable for a larger current regulator gain. When the current regulator gain was changed online, as shown in Fig. 19, the same good agreement with theory was observed.

Fig. 19
figure 19

Experimental waveforms when changing the current regulator gain k p

5 Conclusion

In this paper, the interaction of phase-locked loop (PLL) dynamics and the current regulator on system stability has been investigated for active damped LCL-filter-based grid-connected inverters with capacitor voltage feedback under weak grid conditions. To efficiently model the system including the PLL dynamics, an equivalent Norton model was applied to obtain the closed-loop transfer function of the whole system. By investigating the poles of the transfer function, it has been found that the system tends to be instable as the PLL bandwidth increases with a fixed current regulator gain. Also, when the PLL bandwidth is fixed, the system tends to be instable as the current regulator gain decreases. The ranges of stability of the PLL bandwidth and the current regulator gain have been identified. The effectiveness of the designed active damping control method has been verified by both simulations and experimental results using a three-phase grid-connected inverter system. Future research can include stability analysis with a low sampling frequency under different control and operating modes.