5. Robustness of the controller

Until now, we have developped a current controller that gives appropriate performance under ideal conditions. In particular, we have assumed:

1. perfect knowledge of the system behavior. In particular, perfect knowlegdge of the R and L parameters.
2. sinusoidal grid voltages: no harmonics
3. balanced three-phase grid voltages: \(v_a, v_b, v_c\) are exactly the same signals, shifted by 120°.

In reality, each of these assumptions will not be entirely satisfied in practice.

Question: why ?

For each of these assumptions, find possible reasons that could explain why they may not be entirely satisfied in practice.

5.1. Imperfect knowledge of the grid filter

The series resistance of inductances may not always be well specified by its manufacturer.

Question

For a given setting of PI controller, study the effect of varying the resistance between 0 and 10 times its value (for which the controller was tuned):

• in the frequency domain, with an open loop behavior diagram (Bode, Black or Nyquist).
  • effect on the phase margin.
  • effect on the gain at low frequencies gain
• in the time domain, with a step response (Simulink or a step command)
  • effect on current overshoot
  • effect on the current settling time

In the end, it may be necessary to change the PI tuning and trade off a bit of performance with the nominal system for more performance with the imperfect system.

TODO: in future version of the assigment, it may be interesting to compare the control with an inductor-capacitor-inductor (“LCL”) filter instead of the single inductor used so far (cf. [Lis2005]).

5.2. Harmonics in the grid voltages

In this section, we consider that the grid voltages have some harmonics (voltages are closed to, but not exactly sinusoidal).

If we consider that the harmonics are the same (but shifted) on each phase, the grid voltages can be redefined as:

• \(v_a = V_{grid} \cos(\omega t) \qquad\quad + v_n \cos(n\omega t + \phi_n)\)
• \(v_b = V_{grid} \cos(\omega t - 2\pi/3) + v_n \cos(n\omega t + \phi_n - 2n\pi/3)\)
• \(v_c = V_{grid} \cos(\omega t - 4\pi/3) + v_n \cos(n\omega t + \phi_n - 4n\pi/3)\)

if we only consider one harmonic component of rank n (integer greater than two). $v_n$ is the amplitude of the n-th harmonic compare and \(\phi_n\) its lag. Notice that the harmonic on phase b (respectively c) is shifted by 1/3 (resp. 2/3) of the fundamental electric period, not the harmonic period. So the phase lag is \(2n\pi/3\) instead of \(2\pi/3\) (resp. \(4n\pi/3\) instead of \(4\pi/3\)).

We propose to study harmonics of rank 3 and 5 [1], with amplitudes of about 5%.

5.3. Imbalanced three-phase voltages

In this section, we consider a small imbalance in the grid voltage. For that purpose, we redefine the grid voltages:

• \(v_a = \qquad V_{grid} \cos(\omega t)\)
• \(v_b = (1+a) V_{grid} \cos(\omega t - 2\pi/3)\)
• \(v_c = (1+b) V_{grid} \cos(\omega t - 4\pi/3)\)

where the unbalance is parametrized by the pair of real numbers \(a, b\) that will slightly deviate from zero. For example, we can use a test case with (+1%, -1%).

Question: observation

Implement the unbalanced voltages in a physical simulation of the closed-loop system.

What can you see on grid voltages, on the current ? What measurement can be used to quantify the defect?

5.3.1. Investigating the problem

The unbalanced grid voltages can affect the current controller in two ways: in a direct and indirect manner.

• the indirect effect is related to the grid synchronization process: the grid phase angle needs to be estimated to feed the dq transorms. The imbalance can perturb this estimation and yield a disturbed phase angle. This in turn would perturb all the dq transformed values.
• the direct effect: the unbalanced voltages are perturbation input on the closed loop system.

Question

Compute (numerically) the dq tranform of the unbalanced grid voltages, assuming that the phase reference is ideally estimated (\(\theta = \omega t\)). You should observe that dq components are not constant anymore.

For a given value of the imbalance (e.g. +1%, -1%), describe the perturbation quantitatively (amplitude, …).

Coming back the system equations in the dq reference frame, the imbalance can be seen to act as an additive perturbation \(w\) on the voltage commands.

Question

Add the perturbation in the DC block diagram simulation of the inverter. Does the simulated behavior reproduce the one observed with the physical simulation?

The effect of the perturbation can be studied using transfer functions, which will help understanding how to reject it.

Question

Derive the expression of the “Perturbtion → Output” transfer function (\(I(s)/W(s)\)).

Evaluate the transfer at the frequency of the perturbation.

5.3.2. Rejecting the perturbation

Using the expression of the transfer function, we can get a clue on how to tune the frequency behavior of the controller to reject the perturbation caused by the grid voltage imbalance.

Question

How should the controller be tuned to better reject the perturbation ?

Question

Implement the new tuning of the current controller and check:

• the effect on the perturbation rejection (should be better!)
• the effect on the stability (e.g. current overshoot)

Note

About robust control design

The design of a robust controller is the subject of many research articles. See for example the book [Zho2013] which contains several examples (H∞ design, repetitive or resonant controller).

Footnotes

[1]	The most commonly found grid voltage harmonics are the ones with low odd rank (n = 3, 5, 7…) They are a consequence of the non sinusoidal current of typical non linear loads like diode rectifiers.