The file TN123_ActiveDamping_Simulation.zip contains the simulation/code generation files:
parameters.m: Initialization file (system parameters and calculation of the closed-loop controllers).TN123_ActiveDamping: Base Simulink file.TN123_ActiveDamping_PLECS_viewer: Simulink file compatible with PLECS viewer.TN123_ActiveDamping_PLECS_viewer_Matlab2015a: Simulink file compatible with PLECS viewer and Matlab Simulink above 2015a.
Hardware configuration
The following configuration is considered. The DC bus voltage is assumed to be constant and the control objective is to control the grid current Ig (energy transfer from DC bus to grid) while damping the filter.
Resonance and damping
Since the grid is assumed to be an ideal voltage source, the transfer function of the LCL filter is
| \begin{array}{l}\displaystyle \displaystyle\frac{i_{g}(s)}{U_{i}(s)}\bigg|_{U_g(s)=0}=\displaystyle\frac{1}{(L_1L_2C_f)s^3+(L_1+L_2)s}\end{array} |
In this example (L1=L2=2.5 mH, Cf=3µF), the cutoff frequency is 2.6kHz. It reflects a natural resonance, which means that, at this frequency, the impedance is zero and the current is tends to be infinite. In practice, as the grid voltage source isn’t ideal and the cables and inductors have parasitic resistances, the current can't become infinite, but may be high enough to damage the system. Consequently, the system must be damped in order to avoid the impedance from being zero at this frequency.
Active damping
As detailed in DN122, the damping of the LCL filter can be achieved with several approaches (passive, active, hybrid)[1-3]. Active damping consists in modifying the closed-loop transfer function to remove the resonance phenomenon.
Feedback control techniques are generally used for this alteration of the transfer function, relying on either the inverter current, the capacitor voltage, or the grid current. Notably, the feedback of the grid current is interesting as it can modify the closed-loop transfer function to be equivalent to that of an LCL filter with a resistor on the grid side. This equivalence authorizes a simple understanding of the damping dynamics.
Following this approach, the control of the LCL filter can be subdivided into two parts:
A proportional and resonant controller in the stationary reference frame,
The active damping using feedback on the grid current.
Grid current control
First, a proportional and resonant controller is developed for the grid current. This topic is further detailed in TN110. The base circuit equations yield:
| \begin{array}{l}\displaystyle V_{reg,\alpha\beta}(s)=G_{I\alpha\beta}(s)\left(I_{g,\alpha\beta}^*-I_{g,\alpha\beta}\right)\end{array} |
| \begin{array}{l}\displaystyle G_{I\alpha\beta}(s)=k_p+k_r\displaystyle\frac{s}{s^2+\omega_{grid}^2}\end{array} |
The discretization with sample time Ts gives:
| \begin{array}{l}\displaystyle G_{I\alpha\beta}(z)= k_p+k_r \displaystyle\frac{\sin(\omega_{grid}T_s)}{2\omega_{grid}} \displaystyle\frac{1-z^{-2}}{1-2z^{-1}\sin(\omega_{grid}T_s)+z^{-2}}\end{array} |
Current feedback loop
Second, the damping is achieved using a feedback loop on the grid current, as proposed in [4].
A feedback with the transfer function s2 allows inserting a virtual resistor in parallel with the inductor L2. However, the implementation of a double derivative is uneasy. Alternatively, the following transfer function can be used, which replaces the virtual resistor with a virtual inductor, itself being in parallel with an inductor and a resistor.
| \begin{array}{l}\displaystyle G_{AD\alpha\beta}(s)=-\displaystyle\frac{L_1L_2R_vs}{L_v(L_vs+R_v)}\end{array} |
If the added virtual inductor is chosen equal to the grid-side inductor L2, the added grid-side part (L2 Lv Rv) is virtually replaced by a virtual resistor Rv in series with the grid-side inductor L2.
| \begin{array}{l}\displaystyle G_{AD\alpha\beta}(s)=-\displaystyle\frac{L_1R_vs}{L_2s+R_v}=-\displaystyle\frac{k_{ad}s}{s+\omega_{ad}}\end{array} |
The following illustration shows the possible circuits to be added in parallel to L2.
The discretization with sample time \begin{array}{l}T_s\end{array} gives:
| \begin{array}{l}\displaystyle G_{AD\alpha\beta}(z)=\displaystyle\frac{-2k_{ad}\left(z^{-1}-1\right)}{\left(\omega_{ad}T_s+2\right)+\left(\omega_{ad}T_s-2\right)z^{-1}}\end{array} |
Simulation results
The system and the control presented in this note were simulated using Matlab Simulink and PLECS. The system parameters are summarized in the next table. They correspond to the inductors and capacitors of the passives rack. The latter indeed contains inductors and a three-phase LC filter, which can be easily connected together to form a full LCL filter.
The existing LC filter is preconfigured with 1Ω damping resistors, but these are generally insufficient to provide satisfying damping in a broad range of conditions.
Parameters | Values | Parameters | Values |
|---|---|---|---|
Grid voltage (line-to-line) | 400 VRMS | Grid frequency | 50Hz |
Grid inductor Lg | 0.5mH | Grid side inductor L2 | 2mH |
Capacitor Cf | 3µF | Serie resistor Rf | 1Ω |
Inverter side inductance L1 | 2.5mH | DC bus voltage | 750V |
Inverter PWM frequency | 20kHz | Sample time Ts | 50µs |
Proportional gain (PR controller) kp | 5V/A | Resonant gain (PR controller) kr | 523Vs/A |
Active damping gain kAD | 40Ω | Active damping pulsation ωAD | 16493 rad/s |
The following figures show the result of the deactivation of the active damping at 0.2s. As it can be seen, the system becomes rapidly unstable.
In practice, if such a phenomenon would occur, the over-current protection thresholds on the controller (e.g. BBox RCP) would block the generation of PWM signals, hence almost instantly blocking the grid currents and damaging consequences.
The following figures show the system response to a step of reference current, with the active damping properly configured. During the transition, an oscillation of 0.2A peak-peak appears which is damped in 2ms with a damping frequency at 2.5kHz (8 times lower than the switching frequency).
Experimental results
Experimental results related to this active damping method are available in the application note AN005.
References
[1] A. K. Balasubramanian, V. John “Analysis and design of split capacitor resistive inductive passive damping for LCL filters in grid-connected inverters,” IET Power Electronics, vol. 6, November. 2013
[2] J. Dannehl, F. W. Fuchs, S. Hansen, P. B. Thogersen, “Investigation of Active Damping Approaches for PI-Based Current Control of Grid-Connected Pulse Width Modulation Converters With LCL Filters”, IEEE Trans. on Industry Applications, vol. 46, August 2010
[3] J. Wang, J. Yan, L. Jiang, J. Zou, “Delay-dependent stability of single-loop controlled grid-connected inverters with LCL filters,” IEEE Trans. on Power Electronics, vol. 31, pp. 743–757, January 2016
[4] X. Wang, F. Blaabjerg, P. C. Loh, “Grid-Current-Feedback Active Damping for LCL Resonance in Grid-Connected Voltage-Source Converters”, IEEE Trans. on Power Electr., vol. 31, January 2010