Determining chaotic characteristics and forecasting tall tower wind speeds in Missouri using empirical dynamical modeling (EDM)

Abstract The chaotic characteristics of the tall tower wind speed data within Missouri was investigated using both quantitative and qualitative methodologies. The phase space diagrams were constructed using the method of time delay. The two parameters needed in the construction of the attractor are the embedding dimension and the time delay. The former was determined using the Cao Algorithm and the latter by Average Mutual Information (AMI). Qualitatively, the phase portraits display chaos for all the wind speed time series for the various stations and height levels. They did not illustrate periodicity nor were they random motions, rather, they depicted a single attractor representative of chaos. Quantitatively the Largest Lyapunov Exponent (LLE) was evaluated. It was determined that for the Columbia station the wind speeds display chaotic characteristics representative of the positive LLEs. However, the increasing level of chaos characteristics did not coincide with the increasing height levels of the tall tower. Thereafter, a simple non-linear prediction algorithm was used to forecast wind speeds using a moving window. The attractor was constructed using the first 56 days and the subsequent 6 h or 36 (10 min) time steps were predicted. The preceding forecast was done when the attractor was reconstructed using the training data of 56 days starting from a 6-h delay from the previous run. The RMSE, MAE and Correlation were investigated for the model with the errors evaluated cumulatively for all of the 1st through 36st predictions. It was determined that the errors increase as the forecasting steps increased for all stations and height levels. The RMSE plateaus at higher wind speeds for increasing height levels with the exception of the station, Neosho, where it plateaued at all height levels at approximately 3.0 ms − 1 . For Columbia at all height levels, after the 20th time step or 3.33 h, the model's normalized errors exceeds 1 or 100%. However, using a 50% normalized error cap, it was noted that these values occurred for Columbia's height levels after the 1st, 2nd and 3rd time steps respectively. For Blanchard, this value was given by the 2nd time step for both heights whilst for Neosho, at all heights this percentage occurred after at most, 2 time steps. From the Lyapunov exponent, the prediction horizons or the time limits to obtain accurate predictions from the chaotic system were determined to be 6 time steps for all the height levels in the Columbia station using a 95% confidence band. When a range of confidence bands was used, it was shown that for the 90% confidence, this value was decreased to 4 time steps. This model was compared to the benchmark model of persistence where it was determined that the EDM is comparable to persistence and it beats it in the very short-term range of one time step for Columbia and Blanchard. Seasonality and diurnal cycle analyses were also accomplished. Seasonality was investigated by slicing the results every 6 h or extracting every 36th forecast error. It was shown that four of the eight stations' height levels had the season of summer incurring the lowest magnitude of average errors and standard deviations. The diurnal cycle was examined by extracting every four of the 6 time slices done previously. The time of day was analysed by lagging these slices by 6, 12 and 18 h. It was determined that there was no evident trend where a particular time of day the model incurred more errors and had greater standard deviations for all stations and heights.