Designing a PI controller for my power system

Question

I have a power system which is given as:

enter image description here

Using MATLAB, I found the system's response to be: enter image description here

As you can see, the system is not stable. So, I need to design a PI controller, so that, the transient performance of the \$\ p(s) \$ can be improved.

Therefore, my new system will be: enter image description here

Please, know that:

• \$\ P(s) = C [sI-A]^{-1} B \$
• \$\ \Delta x ' = A \Delta x + B u \$
• \$\ Y= C \Delta x \$

Since we are dealing with a real power plant system, we should consider the input equations to be non-linear. Just for calculation purposes, let us assume that:

\$\ \Delta x = [\Delta \delta, \Delta \omega. \Delta e_q' , \Delta E_d ] ^T \$

Let the angular speed of the rotor ω to be our input. (Just to make everything easier)

Therefor, C= [0 1 0 0]

Where ω can be defined as:

\$ ω' = \frac{1}{M} (P_M - P_E - D(ω-ω_0))\$

Please, know that all of these values are arbitrary, except for ω. (These values will be given to me later by the manufacturer of P(s))

Anyhow, I tried to find \$\ K_I \$ and \$\ K_P \$ using the above equations. So, after using Laplace transforms and many other calculations, I reached to the point where

\$\ H(\lambda)= \frac{\lambda T_W}{1+ \lambda T_W} (K_P + \frac{K_I}{\lambda}) \$

The question is:

What are the best values for \$\ K_I \$ and \$\ K_P\$ so that my power system is stable? And how can I know that?

• aiao: What is your P(s) transfer funtion?You need also to provide a value for \$T_w\$ which is first order time delay constant, which depends on the latency of your controller? sisotool command in matlab should solve your problem. The solution for \$ K_I\$ and \$K_P \$ will be a range and not a mere single point. sisotool in matlab will let you play with \$ K_I\$ and \$K_P \$ until you are happy with the output.
• James Mitch: I don't know what the value of \$T_w \$ is, it is just an arbitrary value. I don't care what number it has. The same thing goes with the other matrices (A, B and C), and I already give you the value of \$P(s)\$ in terms of these matrices. All I need is just to know: how can I chose the best values for \$K_I \$ and \$K_P \$? I think I should learn more about the pole-placement method, maybe this is the solution. I will think about it later. Regarding the second part, thank you very much for the help. The sisotool command was really helpful. Thank you very much

Answer 1

I suppose you don't have the P(s) transfer function. It can be an academic problem or an exercise over a real plant which an unknown model, but at least you should know WHAT INPUT makes P(s) to have that output. Is it an impulse, a step, a sine?

Lacking mathematical knowledge, you can follow some empirical rules, well spread in the industrial field, google for "ziegler nichols tuning method" or begin here.

• James Mitch: Yeah, this is the problem. I don't know what is the exact expression for P(s). But as for the input value, I have updated the question. I assumed that the angular speed of the rotor is our input. Finally, thank you very much for the help, I will read more about the Ziegler–Nichols method. I hope it could help me. Thank you
• Gossamer: Well, Ziegler-Nichols is good method when you are allowed to bring your system to unstable phase, as it is required. By definition of ZN, you need to set the Kp to the value when you are experiencing the oscillations and then set the Ki from the table. But if you are dealing with the real thing, that is, most of the time, unacceptable.