UQWorld

Bayesian calibration for dynamic wind turbine model

Hello UQWorld,

I am starting to look into a Bayesian inference problem for a dynamic wind turbine model using UQLab. The goal is to calibrate the model parameters for time-dependent model output. I am attaching herewith a figure for easier interpretation of my model response. In my model, I need to incorporate a dynamic response at time-step (t_t) for a simulation run for t_n. This means: the model inputs need to change at t_t than that used to initialize the simulation at t_0. Could someone guide me to any literature where a similar problem was dealt with using the UQLab?

dynamic

Best,
Vinit

Hi @Vinit

I am not sure I fully understand your problem. Do you, in fact, just have two models, where

Y_t = \begin{cases} \mathcal{M}_1(\mathbf{X}_1, t), \quad t \le t_t,\\ \mathcal{M}_2(\mathbf{X}_2, t), \quad t > t_t,\\ \end{cases}

where \mathbf{X}_1 and \mathbf{X}_2 are the input parameters of the first and second models, respectively. Or does t_t depend on your model output?

Hi Paul-Remo,

There is only one model. Mathematically: Y =\mathcal{M}(\mathbf{X}_1, t), \quad t \le t_t\ \quad and \quad \mathcal{M}(\mathbf{X}_2, t), \quad t > t_t\ .
So, in this case, the model inputs \mathbf{X}_1 changes to \mathbf{X}_2 at t_t. Having said that, the model parameters \theta (to be calibrated) are meant to have constant values independent of the model inputs.

Hi @Vinit

Where does \mathbf{\theta} enter your model?

I would rephrase the mathematical statement then: Y = \mathcal{M}(\mathbf{X}_1(t), \theta), \quad t \le t_t\ \quad and \quad \mathcal{M}(\mathbf{X}_2(t), \theta), \quad t > t_t\ ,
where \mathbf{X}_1 and \mathbf{X}_2 are operating conditions and are time (t) dependent. \theta comprises the vector of model parameters to be calibrated.

So for a given t, \mathbf{X}_1 and \mathbf{X}_2 are known exactly?

In this case I don’t think there is an issue. You can just rewrite your model to depend only on \mathbf{\theta} and let it return a vector of discrete responses at each time step, such that

Y_t = \mathcal{M}_t(\mathbf{\theta}), \quad\text{where}\quad t_t\in[t_0,t_n].

Would that work?

Yes, for a given t, \mathbf{X}_1 and \mathbf{X}_2 are known.
Yes, that sounds very much correct. Any relevant literature on the same lines that I could refer?

Alright, then this is a standard case. Just have a look at the inversion manual, specifically Section 2.3.