I have taken hourly-based wind power data, by taking its periodogram after making it stationary in R. It gives me four seasonal patterns at periods of 24, 12, 08, and 06, as shown in the figure below.
Is it possible to have sub-hourly seasonality in my wind power data?
The data is given as follows:
Many things are possible. (Here is one possible data-generating process that might lead to such sub-daily seasonalities, but no, I'm not really serious about that.)
I am not an expert in wind power generation as such, however, I find such a periodicity extremely unlikely. I understand that wind could have intra-daily patterns (differences between night and day), i.e., a seasonal period of 24 hours, and intra-yearly patterns (differences between summer and winter), i.e., a seasonal period of about 8766 hours. (These are examples of multiple-seasonalities - you may find the tag wiki helpful.)
However, other patterns simply don't make a lot of sense from a data generation point of view. Yes, the moon might have some very weak effect on the wind, perhaps via tides, so we might have intra-monthly patterns. I see no reason why the wind itself should have intra-weekly effects. Conversely, wind power generation might actually have such patterns: since electricity demand has weekly patterns, it might be that wind farms get cycled up or down with a certain weekly influence. But subdaily patterns at all these periodicities sound like an artifact or an error to me.
I have to admit that I am a forecaster and think like one. To a forecaster, the question is not so much whether a structure is present in a time series, but whether it helps us forecast the series better. As such, I naturally turn to papers about forecasting wind power, e.g., something connected with a good journal like the International Journal of Forecasting, and then I start skimming them on whether they report this kind of seasonality. Here is a paper that sampled in 10-minute buckets and found daily seasonality ($24\times 6 = 144$) periods per cycle - nothing about the kinds of seasonality you found. The GEFCom2014 did include a competition on wind power forecasting, but the summary paper does not go into details on what seasonalities were modeled by the winning methods. You might be able to get a better idea of what other people have found with a deeper literature search.
TLDR; The signal period is $24$ hours, eventhough you have your power spectrum indicating components with a smaller period. The components with period $6, 8, 12$ hrs also repeat themselves every $4 \times 6 = 24, 3 \times 8 =24, 2 \times 12 =24 $ hrs. So they all have, in a way, a common period of 24 hrs.
It gives me four seasonal patterns at periods of 24, 12, 08, and 06
This sounds like you get overtones. If your periodogram is a sort of Fourier spectrum, then this is not weird. It means that the daily pattern consists of more structure than just a single sine wave.
This doesn't mean that the period of the signal is smaller than 1 day. Below is an example signal constructed with several overtones (wave lengths smaller than 1 day), and you can see that the period of the signal is 1 day. The higher frequency signals influence the shape but not the period of the signal.
The period of a function that is a sum of periodic functions is the least common multiple of the periods of the functions in the sum.
### t is time for one week of data sampled every ten minutes
t = seq(0,7*24*60,10)
### some example measurement of data that depends on sin waves with multiple sub-daily periods
Td = (24*60)/(2*pi) ### daily period
y = 2 + sin((t+1000)/(Td)) + 0.4* sin((t+1200)/(Td/2)) + 0.1* sin((t+800)/(Td/3)) + 0.1* sin((t+1000)/(Td/4))
### plot
plot(t/24/60,y+rnorm(7*24*6,0,0.2),
type = "l", xlab = "time in days", ylab = "signal",
main = "example of signal with a daily period, but several overtones")
When looking at time-series data in the frequency domain, it is common for periodic signals in the data to consist of a main frequency and then a set of harmonics. The harmonic frequencies are integer multiples of the main frequency of the signal. This occurs because each of the spikes in the frequency domain correspond to a sinusoidal wave in the time domain, and the actual periodic signal in the time domain is often not a perfect sinusoidal wave. Thus, what you have here sugggests that your wind power data has a periodic daily signal that is not a perfect sinusoidal wave (but which can be created as a weighted sum of a main sinusoidal wave and then four harmonics that are smaller sinusoidal waves at the harmonic frequencies).
