To describe this concept through simulations, I'll make the "b" phase component of voltage to be zero.
The green line is the "b" phase voltage. This is a fairly extreme case of unbalance and so this is chosen as a benchmark to design a notch filter. The PLL output without a notch filter is:
The above figure shows the double frequency creeping into all control signals and in the above case the angular frequency. Additionally, the double frequency can also be seen in then "d" and "q" components of the transformed voltage as below.
Now that the problem has been described, the notch filter has been inserted before the PI controller. Instead of feeding the "q" component of the transformed voltage to the PI controller, it is passed through a notch filter that removes the 120 Hz component and the filtered voltage is fed to the controller.
The voltage "vq" is the same as above, but the output of the notch filter is:
The 120 Hz component has been attenuated significantly. The output of the PLL is now:
With this done, the next step is to simulate a grid connected inverter feeding a controlled current into the grid.
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