# Simulation of correlated stochastic processes based on some time series

I'd like to simulate power output time series of other wind farms based on measured outputs of one (or more) wind farms. Specifically, I have data on power outputs of wind farm A:

$$\mathbf{x}^A = [x^A_1, x^A_2, ..., x^A_T]$$

How can I simulate $$\mathbf{x}^B$$, $$\mathbf{x}^C$$, ... during time $$[1, T]$$?

• There must be some correlations between $$\mathbf{x}^A$$, $$\mathbf{x}^B$$, $$\mathbf{x}^C$$, which doesn't matter as long as it makes sense.
• It is hard to obtain satisfying statistical models for $$\mathbf{x}^A$$. Even if you do, those must be based on lots of other factors like wind speeds, angles, temperatures, etc ... which will make the simulation too complicated.
• I have thought about bootstrapping for dependent data, but the method is to resample from the same object, so I cannot simulate $$\mathbf{x}^B$$ by bootstrapping $$\mathbf{x}^A$$.
• I am trying to reproduce woods2013simulation, but it is based on methods in frequency domain, with which I am not familiar. Does anyone have simpler methods?
• Why I want to simulate outputs of other farms is that I don't have enough data.

## Example

My intention can be illustrated by data from Aneroid Energy. Power outputs from three wind farms in NSW Australia are considered.

Their latest power outputs on April 24, 2020, can be visualized by the following three coloured lines. The values are normalized by their own capacities so the y axis is the capacity factor. Note that the black line indicates the average wind power output in Australia. Because SAPHWF1 and WRWF1 are close to each other, so they are expected to have similar fluctuations.

Now, suppose I only have the series of SAPHWF1, how can I simulate those of WRWF1 and BODWF1? Moreover, what if I don't have historical data at all? For example, I imagine there is a new wind farm next to BODWF1, how can I simulate a reasonable realization?

## Reference

• [woods2013simulation] Woods, M.J., Russell, C.J., Davy, R.J. and Coppin, P.A., 2012. Simulation of wind power at several locations using a measured time-series of wind speed. IEEE Transactions on Power Systems, 28(1), pp.219-226.
• [Aneroid Energy] Andrew Miskelly, 2020
• To me model-free simulation sounds like an oxymoron... Apr 23 '20 at 17:31
• It is from the answer for How do you do bootstrapping with time series data?. "Model-free" means you use block bootstrapping directly on the original data without any model. Apr 24 '20 at 6:51
• Bootstrapping is validated by assumptions guaranteeing the empirical distribution is converging. Otherwise, "rubbish in, rubbish out" to paraphrase Babbage. Apr 24 '20 at 8:11
• I'll keep that in mind when I use bootstrapping. But I don't think it can be used here. Apr 24 '20 at 8:47

## 1 Answer

Time series simulation based on a given time series might help explain how data can be generated for a user-specified model.

In this regard it is possible to specify a model without having generated one from a given time series. Is that what you meant by your question ?

Generating a simulation from histogram-like input using the percentage for each interval is a feature of a commercially available software that I have helped to develop. It is straightforward to program a similar feature in R or Python .

• If it's easy to establish a general model for power output, the new series can be simulated by MCMC based on a new model with different parameters. The power outputs are non-stationary. Apr 24 '20 at 10:15