As the number of interconnected wind farms increases, sixteen response variables from the training data are modeled through the linear regression models: five slopes (not including the third slope), the first and last PSD values, the variability, the STD, the maximum wind power, minimum wind power, the mean of future output, the monthly ramp size, the monthly ramp rate, the amplitude of the daily cycle, and the mean of the daily cycle. These variables are fitted using a quadratic function, and regression coefficients are estimated by the least squares method [112]. The explanatory variable is selected as the capacity. It is assumed that wind farms in ERCOT will keep a similar configuration of turbine types and a similar configuration of wind farms in the future, although this may not be true and will affect result.
Figure 3.7(a) shows the slope of the first affine function whose frequency range has period between two days and one month. It is observed that the first slope does not change much as capacity increases, which means that the power outputs over longer periods are highly correlated with many wind farms. In Figure 3.7(b), the slope of the second affine function whose frequency range has period between eight hours and two days is shown. It is fixed around −2.4 relatively independent of capacity. Since the second slope is around −2.4, power dissipation to the higher frequency starts from a period of two days.
Figure 3.8(a) shows the slope of the third affine function whose fre- quency range has period between six hours and eight hours. This segment works as a joint between the PSD pattern in the low and middle frequency.
0 2000 4000 6000 8000 10000 −4 −3 −2 −1 0 Slope 1 in 2010/4 Capacity [MW] Slopes y = − 6.6e−10*x 2 + 1.1e−05*x − 0.16 2010/4 Quadratic (a) 0 2000 4000 6000 8000 −4 −3 −2 −1 0 Slope 2 in 2010/4 Capacity [MW] Slopes y = 3.5e−09*x2 − 5.4e−05*x − 2.2 2010/4 Quadratic (b)
Figure 3.7: Slopes of first and second segments. (a) The slope of the first segment is plotted against the total installed capacity of wind power. (b) The slope of the second segment is plotted against the total installed capacity of wind power.
0 2000 4000 6000 8000 10000 −4 −3 −2 −1 0 Slope 3 in 4/2010 Capacity [MW] Slopes (a) 0 2000 4000 6000 8000 10000 −3.5 −3 −2.5 −2 −1.5 Slope 4 in 2010/4 Capacity [MW] Slopes 2010/4 Power Function (b)
Figure 3.8: Slopes of third and fourth segments. (a) The slope of the third segment is plotted against the total installed capacity of wind power. Slopes do not follow a specific trend. (b) The slope of the fourth segment is plotted against the total installed capacity of wind power.
The first two segments start from the first PSD and have a distinct pattern. On the contrary, the last three segments start from the last PSD value and also have a distinct pattern. To satisfy two patterns at the same time, a piecewise function in a short frequency range is required to connect these two patterns. Since the third affine function will be used as a degree of freedom, patterns and fitted functions are not searched for the third slope. Figure 3.8(b) shows the slope of the fourth affine function whose frequency range has pe- riod between 15 minutes and six hours. It is observed that the fourth slope drastically decreases as capacity increases. The frequency range of the fourth segment might correspond to periods of short-term weather events in differ- ent local areas. Since those weather events are not strongly correlated, wind power fluctuation in this frequency range is canceled out when wind power in those areas is aggregated. Therefore, the power level in this frequency range is reduced. Fourth slopes appear to converge to a constant negative value, but further investigation with more wind power data is necessary. If it converges to a certain negative value, it means that wind power variability could not be reduced significantly with greatly increased total capacity. It also means that the geographical smoothing effect is saturated, so there is always a certain amount of wind power variability.
Figure 3.9(a) shows the slope of the fifth affine function whose frequency range has period between five minutes and 15 minutes. To detect trends of the fifth slope clearly, 100 wind farm integration orders are simulated, the mean of linear slopes of fifth slopes is measured. Then, it is observed that
the fifth slopes slightly increase as wind power capacity increases with a slope of 0.017 per GW. Figure 3.9(b) shows the slope of the sixth affine function whose frequency range has period between two minutes and five minutes. It should be noted that signals corresponding to the sixth slope are affected by independent local weather events. Sixth slopes are also fitted by a quadratic function.
In summary, the five slopes (not including the third slope) are fore- casted through the regression analysis. The first, second, fifth, and sixth slopes are fitted to a linear function, but the fourth slope is fitted to a quadratic func- tion. Furthermore, the initial and final PSD are fitted to a quadratic func- tion. Other statistical characteristics are also analyzed through the regression model. Variability of actual wind power is linearly proportional to the square root of capacity. The following statistical characteristics are all linearly pro- portional to the capacity: the maximum value, minimum value, mean, STD of wind power, amplitude of the daily pattern, STD of the monthly ramp size, and STD of the monthly ramp rate.