Representation of Photovoltaic cell and BESS in Microgrid by first order transfer function

Question

I am working in the are of frequency control of Microgrid and have come across many publications (For example: https://www.sciencedirect.com/science/article/abs/pii/S0142061515005220 (Pg 3,5)), wherein the transfer function that relates power output and the solar insolation in a photovoltaic (PV) cell is represented as a first order lag transfer function. Also, the transfer function that relates the deviations in power output of the battery energy storage system (BESS) and the deviations in frequency by the first order transfer function with lag. Like: $$G_{bess} = \frac{\Delta P_{bess}}{\Delta f} =\frac{K_{bess}}{1 + sT_{bess}}$$

My question is: How to mathematically prove starting from the physical equations that such relations can be represented by a first order transfer function? Is there a mathematical proof or mathematical justification using which we can obtain these transfer functions?

Answer 1

For very few real energy systems will you be able to prove that the system can be represented perfectly by a first order transfer function. The reality is that energy systems, especially those with additional control systems managing them, are much more complex.

However, in some cases, a first order transfer function may represent the actual, much more complicated transfer function, within an acceptable tolerance level for your purposes. In this case you wouldn't be looking to mathematically prove a match, but rather you'd be better off to evaluate the goodness of fit between the candidate first order transfer function and the detailed transfer function (which you would derive starting from physical equations of the system) under steady-state and transient conditions relevant to your application (e.g., a step change in frequency for the BESS response to frequency example). This then comes down to determining what metric for goodness of fit you want to use (e.g., least squares fit, etc.), simulating the scenario you care about, and evaluating the metric. See the answer Fitting of continuous transfer function for more context on that.

Also, keep in mind that the real system may not even be fully representable by a linear time invariant (LTI) transfer function (i.e., non-linearities are not uncommon).