In the previous blog post, I had attempted to create a linear model to predict the reference for the filter capacitor voltage such that the current injected into the grid will track it's reference. And when I saw the linear model, I had the feeling, it would probably not work. And it didn't.
The linear model that the LinearRegression module with scikit.linear_model gave me was:
vcf_ref = -0.01157 + 0.04176*igrid_ref + 1.0587*vgrid
Seems elegant and when I tried to fit it to the test set, I got a cluster that fell along a linear line. All this seemed wonderful as an application of machine learning to power electronics. But looking at it from a power electronics perspective, what did I get?
vcf_ref = vdc + Remu*igrid_ref + K*vgrid
Where vdc is a dc bias that apparently the linear model feels is needed. Though there should not be a dc bias needed in an ac system unless there is a dc component due to some severe unbalanced harmonic load, this dc bias is fairly small and can be neglected.
The Remu is actually what the linear model approximated the inductor as. The actual equation is:
vcf = vgrid + igrid*R + L* digrid/dt
The linear model has approximated (igrid*R + L* digrid/dt) by Remu (emulated resistance). And this Remu = 0.04176 which is very small given the fact that an inductor is being modeled by this resistor.
The next major problem is the 1.0587*vgrid. We are scaling the grid voltage by 5.8%. And compared to the other two factors, this is the major factor. So really, all that this linear model does to try to create a reference filter capacitor voltage is to scale up the grid voltage by 5.8%. And this is totally inadequate.
Suppose, the current injected into the grid will have to be in-phase with the grid voltage and this would be a requirement for many distributed energy resources. The filter capacitor voltage would need a phase advance with respect to the grid voltage. Or else, it would not be able to produce a grid current in phase with the grid voltage. Because the current through the inductor would be in quadrature and lagging behind the voltage drop across it.
With the above linear model, the filter capacitor voltage would be scaled up by 5% and so in phase with the grid voltage. The voltage drop across the grid inductor would be in phase with the grid voltage and therefore, the current injected into the grid would be lagging behind the grid voltage by 90 degrees. Which is not what we want.
My hypothesis is that what I need is a more detailed and accurate model for the differential operator. And that may need more complicated techniques that a LinearRegression. But before jumping to that, I would like to exhaust any possible solution that linear models have just to gain knowledge.
In all fairness, the linear model that I got from LinearRegression above was a flawed one. Because what I did to train the model was simplistic. To begin with, I had chosen the equilibrium point to be a modulation signal that was the grid voltage scaled by the dc bus:
m = vgrid/vdc
So, the filter capacitor voltage is very close to the grid voltage in magnitude and most importantly in phase as well. To this modulation signal, I had added a random disturbance.
m + mrand*sin(wt)
Here was the mistake - I was giving the random modulation signal perturbation the same phase as the modulation signal. And so, the filter capacitor voltage voltage only changes in magnitude.
And this is exactly why the linear model we ended up with has the major component that scales up the grid voltage by 5%. Because the linear model saw that in a vast majority of cases, that's all there was to it - the capacitor voltage scaled up by small amount and the grid current in quadrature with the grid voltage. So, the linear model assumed this was the ultimate truth and gave that very convenient equation.
So, my second attempt would be to train the linear model making sure that the filter capacitor voltage differed from the grid voltage by not just magnitude by also in phase.
So, check out these two current snapshots in the previous training case:
In both cases, the current is always 90 degrees lagging behind the grid voltage because filter capacitor voltage is in-phase with the grid voltage. So, the linear model thinks this is the norm, this is always how it is.
In contrast, now I am going to add two components to the random shuffle to the modulation index - an in-phase component and a quadrature component. So, the grid current will have a varying phase angle difference with the grid voltage. Below is an example. With this, I am going to try and retrain the linear model. Probably still won't work, but let's see how it changes.
The linear model that the LinearRegression module with scikit.linear_model gave me was:
vcf_ref = -0.01157 + 0.04176*igrid_ref + 1.0587*vgrid
Seems elegant and when I tried to fit it to the test set, I got a cluster that fell along a linear line. All this seemed wonderful as an application of machine learning to power electronics. But looking at it from a power electronics perspective, what did I get?
vcf_ref = vdc + Remu*igrid_ref + K*vgrid
Where vdc is a dc bias that apparently the linear model feels is needed. Though there should not be a dc bias needed in an ac system unless there is a dc component due to some severe unbalanced harmonic load, this dc bias is fairly small and can be neglected.
The Remu is actually what the linear model approximated the inductor as. The actual equation is:
vcf = vgrid + igrid*R + L* digrid/dt
The linear model has approximated (igrid*R + L* digrid/dt) by Remu (emulated resistance). And this Remu = 0.04176 which is very small given the fact that an inductor is being modeled by this resistor.
The next major problem is the 1.0587*vgrid. We are scaling the grid voltage by 5.8%. And compared to the other two factors, this is the major factor. So really, all that this linear model does to try to create a reference filter capacitor voltage is to scale up the grid voltage by 5.8%. And this is totally inadequate.
Suppose, the current injected into the grid will have to be in-phase with the grid voltage and this would be a requirement for many distributed energy resources. The filter capacitor voltage would need a phase advance with respect to the grid voltage. Or else, it would not be able to produce a grid current in phase with the grid voltage. Because the current through the inductor would be in quadrature and lagging behind the voltage drop across it.
With the above linear model, the filter capacitor voltage would be scaled up by 5% and so in phase with the grid voltage. The voltage drop across the grid inductor would be in phase with the grid voltage and therefore, the current injected into the grid would be lagging behind the grid voltage by 90 degrees. Which is not what we want.
My hypothesis is that what I need is a more detailed and accurate model for the differential operator. And that may need more complicated techniques that a LinearRegression. But before jumping to that, I would like to exhaust any possible solution that linear models have just to gain knowledge.
In all fairness, the linear model that I got from LinearRegression above was a flawed one. Because what I did to train the model was simplistic. To begin with, I had chosen the equilibrium point to be a modulation signal that was the grid voltage scaled by the dc bus:
m = vgrid/vdc
So, the filter capacitor voltage is very close to the grid voltage in magnitude and most importantly in phase as well. To this modulation signal, I had added a random disturbance.
m + mrand*sin(wt)
Here was the mistake - I was giving the random modulation signal perturbation the same phase as the modulation signal. And so, the filter capacitor voltage voltage only changes in magnitude.
And this is exactly why the linear model we ended up with has the major component that scales up the grid voltage by 5%. Because the linear model saw that in a vast majority of cases, that's all there was to it - the capacitor voltage scaled up by small amount and the grid current in quadrature with the grid voltage. So, the linear model assumed this was the ultimate truth and gave that very convenient equation.
So, my second attempt would be to train the linear model making sure that the filter capacitor voltage differed from the grid voltage by not just magnitude by also in phase.
So, check out these two current snapshots in the previous training case:
In both cases, the current is always 90 degrees lagging behind the grid voltage because filter capacitor voltage is in-phase with the grid voltage. So, the linear model thinks this is the norm, this is always how it is.
In contrast, now I am going to add two components to the random shuffle to the modulation index - an in-phase component and a quadrature component. So, the grid current will have a varying phase angle difference with the grid voltage. Below is an example. With this, I am going to try and retrain the linear model. Probably still won't work, but let's see how it changes.
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