# A Latent-Factor System Model for Real-Time Electricity Prices in Texas
## 4 Results
The \(Y\)-axis represents the mean price values ($/MWH). Each value on this axis is the mean price from each of the eight zones during the years 2015-2018. Focus on the 4:00 a.m. panel at the top. The interquartile range (IQR) of the mean prices of the eight zones in ERCOT at this hour is $16.18 to $16.61; see the left-most box in blue. Next, assume wind generation increases \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Coefficient** & \(p\)**-Value** & **95\% Confidence Intervals** \\ \hline \multicolumn{5}{c}{Latent Factor Equation} \\ \(\mathbf{f}_{t-1}\) & 0.1326 & 0.0010 & 0.0571 & 0.2080 \\ SystemLoad & 1.8513 & 0.00001 & 1.6365 & 2.0661 \\ Lag(WtPr) & 0.2243 & 0.0010 & 0.0899 & 0.3587 \\ \multicolumn{5}{c}{Houston Equation} \\ \(\mathbf{f}_{t}\) & 0.4302 & 0.00001 & 0.4117 & 0.4487 \\ Wind & \(-\)0.1761 & 0.00001 & \(-\)0.2085 & \(-\)0.1438 \\ Nuclear & \(-\)0.7292 & 0.00001 & \(-\)0.8540 & \(-\)0.6043 \\ Solar & 0.0703 & 0.00001 & 0.0413 & 0.0993 \\ Henry Hub & 0.3608 & 0.00001 & 0.2006 & 0.5211 \\ Dummy & 2.0212 & 0.00001 & 1.8419 & 2.2004 \\ \multicolumn{5}{c}{Austin Equation} \\ \(\mathbf{f}_{t}\) & 0.4334 & 0.00001 & 0.4169 & 0.4499 \\ Wind & \(-\)0.2222 & 0.00001 & \(-\)0.2520 & \(-\)0.1924 \\ Nuclear & \(-\)0.6829 & 0.00001 & \(-\)0.8060 & \(-\)0.5599 \\ Solar & 0.0482 & 0.00001 & 0.0211 & 0.0752 \\ Henry Hub & 0.3430 & 0.00001 & 0.1933 & 0.4927 \\ Dummy & 1.8556 & 0.00001 & 1.6916 & 2.0195 \\ \multicolumn{5}{c}{West Equation} \\ \(\mathbf{f}_{t}\) & 0.4615 & 0.00001 & 0.4307 & 0.4922 \\ Wind & \(-\)0.4326 & 0.00001 & \(-\)0.4827 & \(-\)0.3825 \\ Nuclear & \(-\)0.5835 & 0.00001 & \(-\)0.7351 & \(-\)0.4319 \\ Solar & 0.0473 & 0.0320 & 0.0040 & 0.0906 \\ Henry Hub & 0.5272 & 0.00001 & 0.2893 & 0.7651 \\ Dummy & 1.2448 & 0.00001 & 0.9629 & 1.5268 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: ML coefficients for the 4:00 p.m. sample. by 10%. Using the MLE estimates of the price elasticities for wind generation for each of the eight zones from our latent-factor system model, we adjust the mean prices in the blue box and construct the resulting change in prices due to increased wind generation. The corresponding distribution of the adjusted mean prices in ERCOT is shown as the second box in red. From the caption, the IQR for the prices, after accounting for increased wind generation, is between $15.59 and $16.07. Finally, we do a similar adjustment to energy prices using the parameter estimates for nuclear generation; this is shown as the green box in the top row of Figure 7. The IQR is between $14.98 and $15.42. Observing the three panels, it is also interesting to note that there is less volatility in the mean prices in the entire ERCOT system during the off-peak hour. Consider the middle panel which corresponds to the 12:00 p.m. hour. While the reduction in energy prices is less now, wind and nuclear generation still have a measurable  
Figure 7: ERCOT merit-order effects for wind and nuclear generation. impact on real-time prices in ERCOT as a whole. Also, there is more volatility in real-time prices during this peak hour. Finally, the bottom panel shows the impact on prices due to the merit-order effects at 4:00 p.m. Nuclear generation is much more influential than wind at this hour of the day; its boxplot barely intersects with the boxplot from wind generation. Also, the volatility in ERCOT's prices is lesser at 4:00 p.m. when compared to 12:00 p.m.