Abstract
Active damped LCL-filter-based inverters have been widely used for grid-connected distributed generation (DG) systems. In weak grids, however, the phase-locked loop (PLL) dynamics may detrimentally affect the stability of grid-connected inverters due to interaction between the PLL and the controller. In order to solve the problem, the impact of PLL dynamics on small-signal stability is investigated for the active damped LCL-filtered grid-connected inverters with capacitor voltage feedback. The system closed-loop transfer function is established based on the Norton equivalent model by taking the PLL dynamics into account. Using an established model, the system stability boundary is identified from the viewpoint of PLL bandwidth and current regulator gain. The accuracy of the ranges of stability for the PLL bandwidth and current regulator gain is verified by both simulation and experimental results.
Similar content being viewed by others
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’.
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:
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).
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:
From Fig. 2, the system variable relationships can be described by the following open-loop transfer functions [29]:
where the transfer functions H 1(s) ~ H 6(s) are
Based on the above equations, the grid current can be obtained as:
which in vector matrix form is:
where g t(s) and y i(s) can be expressed as:
The small-signal transfer function can be expressed as:
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:
where k pPLL and k iPLL are the proportional and integral gains of the PI regulator, respectively.
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):
Based on (1), the capacitor voltage can be written as
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:
where G PLL(s) is given in the following equation:
The vector matrix form of \( \Delta \varvec{V}_{\text{C}}^{\text{c}} \) is
Similarly, the grid current disturbance ∆I cg can be expressed as follows:
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:
With (12), (13) and (14), the inverter input impedance Z eq can be obtained:
Then with (14) and (15), the grid current can be expressed as
The equivalent Norton model of the inverter system is shown in Fig. 4.
In Fig. 4, Z L1(s) denotes the grid-side filter inductance and Z g(s) denotes the grid line impedance:
The final transfer function of the closed-loop system can be obtained from Fig. 4 as follows:
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
References
Pan D, Ruan X, Bao C et al (2015) Optimized controller design for LCL-type grid-connected inverter to achieve high robustness against grid-impedance variation. IEEE Trans Ind Electron 62(3):1537–1547
Guo X, Jia X (2016) Hardware-based cascaded topology and modulation strategy with leakage current reduction for transformerless PV systems. IEEE Trans Ind Electron 62(12):7823–7832
Shuai Z, Huang W, Shen C et al (2017) Characteristics and restraining method of fast transient inrush fault currents in synchronverters. IEEE Trans Ind Electron. doi:10.1109/TIE.2017.2652362
Guo X, Liu W, Zhang X et al (2015) Flexible control strategy for grid-connected inverter under unbalanced grid faults without PLL. IEEE Trans Power Electron 30(4):1773–1778
Wu W, Sun Y, Huang M et al (2014) A robust passive damping method for LLCL-filter-based grid-tied inverters to minimize the effect of grid harmonic voltages. IEEE Trans Power Electron 29(7):3279–3289
Li F, Zhang X, Zhu H et al (2015) An LCL-LC filter for grid connected converter: topology, parameter, and analysis. IEEE Trans Power Electron 30(9):5067–5077
Beres RN, Wang X, Liserre M et al (2016) A review of passive power filters for three-phase grid-connected voltage-source converters. IEEE J Emerg Sel Top Power Electron 4(1):54–69
Pan D, Ruan X, Bao C et al (2014) Capacitor-current feedback active damping with reduced computation delay for improving robustness of LCL-type grid-connected inverter. IEEE Trans Power Electron 29(7):3414–3427
Wang X, Blaabjerg F, Loh PC (2016) Grid-current-feedback active damping for LCL resonance in grid-connected voltage source converters. IEEE Trans Power Electron 31(1):213–223
Wang X, Blaabjerg F, Loh PC (2015) Virtual RC damping of LCL filtered voltage source converters with extended selective harmonic compensation. IEEE Trans Power Electron 30(9):4726–4737
Guo X (2017) Three phase CH7 inverter with a new space vector modulation to reduce leakage current for transformerless photovoltaic systems. IEEE J Emerg Sel Top Power Electron 5(2):708–712
Guo X, Xu D, Wu B (2016) Common-mode voltage mitigation for back-to-back current-source converter with optimal space-vector modulation. IEEE Trans Power Electron 31(1):688–697
Guo X (2017) A novel CH5 inverter for single-phase transformerless photovoltaic system applications. IEEE Trans Circuits Syst II Express Br. doi:10.1109/TCSII.2017.2672779
Liu F, Wu B, Zargari NR et al (2011) An active damping method using inductor-current feedback control for high-power PWM current source rectifier. IEEE Trans Power Electron 26(9):2580–2587
Cespedes M, Sun J (2014) Impedance modeling and analysis of grid-connected voltage-source converters. IEEE Trans Power Electron 29(3):1254–1261
Shuai Z, Hu Y, Peng Y et al (2017) Dynamic stability analysis of synchronverter-dominated microgrid based on bifurcation theory. IEEE Trans Ind Electron. doi:10.1109/TIE.2017.2652387
Zhang D, Wang Y, Hu J et al (2016) Impacts of PLL on the DFIG-based WTG’s electromechanical response under transient conditions: analysis and modeling. CSEE J Power Energy Syst 2(2):30–39
Zhang L, Harnefors L, Nee HP (2010) Power-synchronization control of grid-connected voltage-source converters. IEEE Trans Power Syst 25(2):809–820
Zhou JZ, Ding H, Fan S et al (2014) Impact of short-circuit ratio and phase-locked-loop parameters on the small-signal behavior of a VSC-HVDC converter. IEEE Trans Power Del 29(5):2287–2296
Guo X, Wu W, Gu H (2010) Modeling and simulation of direct output current control for LCL-interfaced grid-connected inverters with parallel passive damping. Simul Model Pract Theory 18(7):946–956
Pan D, Ruan X, Wang X et al (2017) Analysis and design of current control schemes for LCL-type grid-connected inverter based on a general mathematical model. IEEE Trans Power Electron 32(6):4395–4410
Zheng L, Geng H, Yang G (2016) Fast and robust phase estimation algorithm for heavily distorted grid conditions. IEEE Trans Ind Electron 63(11):6845–6855
Liu H, Xing Y, Hu H (2016) Enhanced frequency-locked loop with a comb filter under adverse grid conditions. IEEE Trans Power Electron 31(12):8046–8051
Guo X, Guerrero JM (2016) Abc-frame complex-coefficient filter and controller based current harmonic elimination strategy for three-phase grid connected inverter. J Mod Power Syst Clean Energy 4(1):87–93. doi:10.1007/s40565-016-0189-4
Wu F, Zhang L, Duan J (2015) A new two-phase stationary-frame-based enhanced PLL for three-phase grid synchronization. IEEE Trans Circuits Syst II Express Br 62(3):251–255
Guo X, Liu W, Lu Z (2017) Flexible power regulation and current-limited control of grid-connected inverter under unbalanced grid voltage faults. IEEE Trans Ind Electron. doi:10.1109/TIE.2017.2669018
Guan Q, Zhang Y, Kang Y et al (2017) Single-phase phase-locked loop based on derivative elements. IEEE Trans Power Electron 32(6):4411–4420
Guo X, Wu W, Chen Z (2011) Multiple-complex-coefficient-filter- based phase-locked loop and synchronization technique for three-phase grid interfaced converters in distributed utility networks. IEEE Trans Ind Electron 58(4):1194–1204
He J, Li YW, Bosnjak D et al (2013) Investigation and active damping of multiple resonances in a parallel-inverter-based microgrid. IEEE Trans Power Electron 28(1):234–246
Acknowledgements
This work was supported by Science Foundation for Distinguished Young Scholars of Hebei Province (No. E2016203133) and Hundred Excellent Innovation Talents Support Program of Hebei Province (No. SLRC2017059).
Author information
Authors and Affiliations
Corresponding author
Additional information
CrossCheck date: 23 May 2017
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
GUO, X., LIU, S. & WANG, X. Impact of phase-locked loop on stability of active damped LCL-filter-based grid-connected inverters with capacitor voltage feedback. J. Mod. Power Syst. Clean Energy 5, 574–583 (2017). https://doi.org/10.1007/s40565-017-0302-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40565-017-0302-3