1. Cycle Detection and Removal in Electricity Markets “Lunch at Lab” Presentation Matt Lyle Department of mathematics&statistics University of Calgary, Alberta

2. Outline ● Characteristics of electricity spot prices ● Electricity price function ● The Seasonal Function ● The FFT ● The sweeping method ● Noise characteristics ● Some ending remarks

3. Electricity spot price characteristics ● Very high volatility ● Numerous price spikes ● High rate of mean reversion ● Seasonal (cyclical) behaviour ● Cannot be economically be stored

4. Hrly prices for AB and PJM markets

5. Electricity Price Function ● Suppose that the spot price is as follows Where is the deterministic component is the Stochastic component is the spot price

6. Electricity Price Function ● We would like to be able to model directly, but it is complicated… ● Instead we try to model its components separately and then combine them afterwards

7. Seasonal Function Since in this presentation we are concerned with the deterministic component of the price, we begin by assuming Where can be estimated using a linear fit on the data set And is the seasonal or cyclical term that needs to be determined

8. Seasonal Function Going back to we have

9. The Fast Fourier Transform ● The continuous time Fourier Transform is defined as ● Discretizing ● We get the discrete Fourier Transform

10. The Fast Fourier Transform With now de-trended we can now perform the FFT on the data

11. The sweeping method ● With the cyclical components identified we now need to separate them from the noise ● We can do this by first noting the location of the spikes in the frequency domain. We use the first eight most dominant spikes. ● And then remove those spikes from the set

12. The Sweeping Method After removing the eight spikes from the set we can see that spikes still remain

13. The Sweeping Method We again identify the eight most dominant spikes and remove them

14. The Sweeping Method

16. After the third sweep we stop. But more or less sweeps can be applied for different data sets

17. The Sweeping Method

19. ● We know have our cyclical and stochastic components separated ● By taking the inverse FFT or IFFT we can restore the seasonal and stochastic components to the time domain ● And we now have the stochastic component by itself

20. Noise Characteristics The time domain of the noise

21. Noise Characteristics Density of the Alberta spot prices

22. Noise Characteristics Density of PJM spot prices

23. Ending Remarks ● Some advantages with this method: 1) Robust 2) Allows for a visual interpretation of the cycles 3) FFT algorithm is common in many computational packages 4) Calibration time is reduced

24. Ending Remarks ● Some disadvantages with this method: 1) Hard to identify low frequency components 2) More work at the front end of model construction 3) Deciding how many sweeps to do is hard to identify

25. Thank you ● mrlyle@math.ucalgary.ca