So before I discussed the control of a UPS which a dc-ac converter (single phase for now) with an LC filter at the output. The objective of the controller was to regulate the voltage across the filter capacitor. The state feedback controller was designed using Linear Quadratic Regulator (LQR) and in addition to that a correction was added to the reference to get rid of the steady state error.
The next stage is to control a dc-ac converter with an LCL filter at the output with the objective of the controller being to regulate the current through the final inductor that connects the system to the grid. This is a problem that I have never been able to solve comprehensively and am now going to take another crack at it.
The difference between control of a UPS and the current control of an inverter with LCL filter is the sensitivity of the output with respect to the input. In the case of the UPS, the input is the output of the dc-ac converter which is a voltage in the case of a Voltage Source Converter (VSC). The output of the system is the voltage across the filter capacitor. These ratio of the output to the input is close to unity. And therefore, the controller was much easier to design, the behavior of the controller system much more predictable and the correction of the steady state error was fairly easy.
But, in the case of a dc-ac converter with LCL filter, the input is the converter voltage while the output of the system is the current through the final inductor. The ratio of the output to the input is very low. Therefore, the controller is quite sensitive. Small changes in the modulation index can result in large changes in the output/grid current.
Suppose the filters were named as follows - Lf the inductor immediately after the converter, Cf the capacitor after Lf, and Lc the inductor that connects the system to the grid. The equation that defines the grid current is:
vf - ic*Rc - Lc*dic/dt - vgrid = 0
The difference in the voltage (vf - vgrid) is what defines the grid current. So, if the state feedback control law is:
m = k1*(if - ifref) + k2*(vf - vfref) + k3*(ic - icref)
The problem is with the second term. The second term will dominate the controller action because of the nature of inputs and quite interestingly the value of k2. I have been using Scilab again to design the state feedback gains and the results are quite interesting.
When the weighting giving to the state variables was Q = diag(0.1, 0.1, 1) which means low weightage to if and vf and high weightage to ic, the state feedback gains were:
k1=3.1852838, k2=4.3873536, k3=7.2589122
Given the magnitudes of vf with respect to if and ic, a value of k2 as above will result in dominating the control output. So, it turns out that the reference for vf becomes critical.
We know the reference for ic as that is the objective of control. It should be possible to do:
vfref = vgrid + Lc*dicref/dt
But you never want to use d/dt in control because it magnifies disturbance. For a sinusoidal fundamental harmonic reference icref, it could be possible to do a decomposition and peform:
vfref_real + j vfref_imag = vgrid + jwLc*(icref_real + icref_imag)
But then, the control cannot be applied to harmonic compensation. So, the question - can we map a function:
vfref = f(icref)
I have only considered using electrical network laws and their approximation so far to do something like the above. But now, with my limited forays into data science and linear regression, this seems like a ripe problem for machine learning. Can we fit a model that is reasonably accurate based on simulations from a vast number of cases? This is what I am going to try to do next - combine machine learning with power electronics!
This a simulation result of what I have achieved so far in terms of current control. As with the previous case - considerable error but most important stable response within a cycle. A PI controller could take 6-8 cycles to stabilize in contrast. If we solve the problem of steady state error, we could have an incredibly robust controller!
The next stage is to control a dc-ac converter with an LCL filter at the output with the objective of the controller being to regulate the current through the final inductor that connects the system to the grid. This is a problem that I have never been able to solve comprehensively and am now going to take another crack at it.
The difference between control of a UPS and the current control of an inverter with LCL filter is the sensitivity of the output with respect to the input. In the case of the UPS, the input is the output of the dc-ac converter which is a voltage in the case of a Voltage Source Converter (VSC). The output of the system is the voltage across the filter capacitor. These ratio of the output to the input is close to unity. And therefore, the controller was much easier to design, the behavior of the controller system much more predictable and the correction of the steady state error was fairly easy.
But, in the case of a dc-ac converter with LCL filter, the input is the converter voltage while the output of the system is the current through the final inductor. The ratio of the output to the input is very low. Therefore, the controller is quite sensitive. Small changes in the modulation index can result in large changes in the output/grid current.
Suppose the filters were named as follows - Lf the inductor immediately after the converter, Cf the capacitor after Lf, and Lc the inductor that connects the system to the grid. The equation that defines the grid current is:
vf - ic*Rc - Lc*dic/dt - vgrid = 0
The difference in the voltage (vf - vgrid) is what defines the grid current. So, if the state feedback control law is:
m = k1*(if - ifref) + k2*(vf - vfref) + k3*(ic - icref)
The problem is with the second term. The second term will dominate the controller action because of the nature of inputs and quite interestingly the value of k2. I have been using Scilab again to design the state feedback gains and the results are quite interesting.
When the weighting giving to the state variables was Q = diag(0.1, 0.1, 1) which means low weightage to if and vf and high weightage to ic, the state feedback gains were:
k1=3.1852838, k2=4.3873536, k3=7.2589122
Given the magnitudes of vf with respect to if and ic, a value of k2 as above will result in dominating the control output. So, it turns out that the reference for vf becomes critical.
We know the reference for ic as that is the objective of control. It should be possible to do:
vfref = vgrid + Lc*dicref/dt
But you never want to use d/dt in control because it magnifies disturbance. For a sinusoidal fundamental harmonic reference icref, it could be possible to do a decomposition and peform:
vfref_real + j vfref_imag = vgrid + jwLc*(icref_real + icref_imag)
But then, the control cannot be applied to harmonic compensation. So, the question - can we map a function:
vfref = f(icref)
I have only considered using electrical network laws and their approximation so far to do something like the above. But now, with my limited forays into data science and linear regression, this seems like a ripe problem for machine learning. Can we fit a model that is reasonably accurate based on simulations from a vast number of cases? This is what I am going to try to do next - combine machine learning with power electronics!
This a simulation result of what I have achieved so far in terms of current control. As with the previous case - considerable error but most important stable response within a cycle. A PI controller could take 6-8 cycles to stabilize in contrast. If we solve the problem of steady state error, we could have an incredibly robust controller!
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