# Time series analysis of electricity load questions

I have hourly data of electricity load (MW) that span 8 months (that is, 5760 data points). I also have predictions from a regression model for the same period. My goal is:

1. to examine some properties of the time series (stationarity, seasonality, trend)
2. to assess the performance of the regression model.

To check whether the time series is stationary I used the Dickey-Fuller test. The results of the test are:

Test Statistic                                                          -3.85795
p-value                                                               0.00236705
#Lags Used                                                                    34
Number of Observations Used                                                 5725
Critical Values                {'5%': -2.86204498529, '1%': -3.43149274838,...


It seems that it is safe to reject the null hypothesis of the test (i.e. that the time series is non-stationary) as the test statistic lies beyond the critical values. I have also plotted the ACF values. I used lags up to T/4 (that is 1400) and I am not sure that this makes much sense. The plot is shown below:

Finally, a residual plot is shown:

The plot does not seem to indicate problem with the equality of variance assumption.

My questions are:

1. Is it possible to draw any conclusions from the ACF plot and is this the proper way to do it (i.e. use the complete time series and a lag up to 1400 or would it be possible to perform the test using a subset of the series)?

2. Do the findings in the ACF plot in any way contradict or reinforce the results of the Dickey-Fuller test?

3. What would the "proper" approach to analyzing such time series data be?

Electric load typically exhibits intra-daily seasonality, as well as intra-weekly seasonality (weekends have different power demand patterns than weekdays), plus yearly seasonality (high power demands for heating in winter, higher power demands for air conditioning in summer). Plus time-shifting holidays.

I'd say your ACF and Dickey-Fuller are fully consonant with these seasonalities. (It's hard to see it in your ACF plot, but I assume the peaks are at multiples of 24?)

Anyway, these seasonalities are so typical and prevalent for electricity demands that, to be honest, I would not be too interested in diagnostics checking these. I'd be more interested in (P)ACFs and tests for residuals after accounting for such seasonalities.

That is, starting from observations $y_t$, you would create a model that accounts for multiple seasonalities and yields in-sample fits $\tilde{y}_t$. If this model truly captures the full seasonal pattern, then the residuals $y_t-\tilde{y}_t$ should not exhibit any remaining seasonality - which you can then test by applying (P)ACF and statistical tests to these residuals. (More precisely, the model that yields $\tilde{y}_t$ should also capture trends and other sources of nonstationarity - but the multiple seasonalities will usually be the strongest source of explainable variation, which is why I'm concentrating on them.)

As to how to deal with and forecast electric load, this is an active research topic. Googling "electric load forecasting" and similar will yield quite a number of relevant hits, such as Cho et al. (2013, JASA). The most important point is of course to capture the overlapping seasonalities, as paradigmatically done by Taylor (2003, JORS). You could also browse our previous questions on multiple seasonal patterns.

Finally, Weron (2014, IJF) is a recent review for electricity price forecasting, which of course is different from load forecasting, but it may be inspirational.

EDIT: I just read Hong & Fan (2016, International Journal of Forecasting), which is probably the very best review on electric load forecasting far and wide. Very much recommended, indeed. Of course, anything by Tao Hong is, who I'd say is the top expert in the field.