Transfer function

[NjegosJankovic]

Can the transfer function be obtained between two points in the system? I looked through documentation and examples, but I have not found anything on this topic. What I have in mind is the way to select two points in the system, and get the numerical representation of the system between these two points in the transfer function. I assume that the symbolical expression would be more challenging to obtain and less useful.

[cuihantao]

I'm not sure what you mean by "numerical representation of the system in the transfer functions". Are you looking to observe the numerical output for a given set of inputs?

The software is not like Simulink, which is causal. It's a non-causal numerical solution of DAE models with discontinuities, so I'm sure if theoretically the transfer functions betwen two points be obtained.

[NjegosJankovic]

Thank you for your quick reply. Sorry for not describing the question clearly enough. What I had in mind is a system frequency response. Let's say, for a certain operating point the system frequency response is measured from one signal in the system to the other. I know that something similar exists in PowerFactory. There, the result is frequency response in form of a Bode plot.
In this discussion I saw your comment about obtaining the B matrix manually. What about C and D matrices? If these can be obtained also, I can calculate the frequency response.

[cuihantao]

Thank you for the clarification. I understand you are looking to obtain a bode plot from the model. The current program is not structured in a way that represents a dynamic power network as an input-output system. When we look at the whole system, it is autonomous, or say self-contained, with no input or output. That's why the state matrix A is for the whole system, with no mention of B, C or D.

The y specific remains to be defined for the subsystem of interest. Note that the whole differential-algebraic equations defines algebraic variables with a coincidental name y, but they are not necessarily a linear combination of x and u. Voltage at a bus is an algebraic variable, but there is no explicit solution for it. The DAE's y also includes the controller outputs, which can be linear. Their equations can be used to find out C by expressig the outputs as a function of x.

To obtain the Bode plot between selected points, some nontrivial work is needed to define the I/O for the selected subsystem, extract the partial derivatives Fx, Fy, Gx, and Gy from the Jacobian matrix, and reconstruct the ABCD matrices.

[NjegosJankovic]

Thank you very much for your quick and detailed response. I understand your point and I agree that such a requirement is not common and not needed in the majority of power system studies. At the moment I might go to another tool with such an option, as I need it relatively soon. Nonetheless, I might come back to the Andes in future. I really like the idea and the way it is presented and structured.

[cuihantao]

I just called a friend in control studies. He corrected me that B, C, and D matrices can be defined based on the need to control and observe the system. I need to walk back from my previous argument; there is no need to manipulate the Jacobian matrices. Since defining the input and outputs will not change the system characteristics (the linearization), the A matrix should stays unchanged.

So using the TGOV1 model as an example (https://docs.andes.app/en/latest/modeling/examples.html#tgov1), we want to observe the output Pout following an input \Delta Pref to the leftmost summation block. We can manually substitute the second and third algebraic equation into the first differential equation to find out the coefficient for the input that goes to B. Of course, the indices for the x_{LG} differential variable needs to be retrieved to construct B.

In terms of the output, we will need to solve for P_{out} as a function of the states. This can be trivial for LTI systems. Oftentimes, a symbolic toolbox can help.

So in all, this can be done with relatively small efforts. A can be obtained from calc_As() function in the eigenvalue routine.

[NjegosJankovic]

This procedure does seem feasible. I will definitely give it a go.
Thank you very much for all the answers.