## 4 Simulation Analysis
### Beam Demand Variation Analysis  
When using time series models, it is generally necessary for the time series to be stationary. Therefore, when studying a time series, the first step is usually to conduct a stationarity test. In addition to visually inspecting the demand sequence graph as described in the previous section, quantitatively judging whether a given sequence is stationary can be achieved through a \"unit root test\". The commonly used rigorous statistical test method is the Augmented Dickey-Fuller (ADF) test, also known as the unit root test.  
The ADF test determines whether a sequence has a unit root: if the sequence is stationary, there is no unit root; otherwise, there is a unit root. The null hypothesis (H_0) of the ADF test is the presence of a unit root (corresponding to non-stationary sequences). If the obtained test statistic is less than the thresholds of the three confidence levels (10%, 5%, 1%), there is a corresponding certainty (90%, 95%, 99%) to reject the null hypothesis.  
By using the demand data of a single beam over 14 days as the test data, we obtain the following test results:  
ADF statistic: \(-5.585057543276498\)  
Confidence level: 1%, Threshold: \(-3.4306750563862054\)  
Confidence level: 5%, Threshold: \(-2.8616836635455414\)  
Confidence level: 10%, Threshold: \(-2.566846468171587\)  
Comparing the ADF test statistic value with the thresholds of the three confidence levels, since the statistic value is less than the threshold at the 1% confidence level, we have 99% confidence in rejecting the null hypothesis and conclude that the demand sequence is stationary.  
Meanwhile, it is essential to consider additional steps to ensure the robustness and reliability of this analysis. It is also important to analyze the autocorrelation in the demand sequences. Autocorrelation measures how the data points in a time series are related to each other over different time lags. Plotting the Autocorrelation Function (ACF) is an effective method to observe the correlation of the time series with its past values. The sequences were analyzed using the ACF, and the following plot was obtained.  
As shown in Figure 4, one-tenth of the data length was selected to plot the ACF graph, ensuring it effectively displays the autocorrelation without excessive smoothing. The horizontal axis represents the lags, which are the number of periods between observations (essentially the data points). The vertical axis represents the autocorrelation, ranging from \(-1\) to \(1\). A value of 1 indicates perfect positive correlation, meaning that as one data point increases, the next one increases proportionally. Conversely, a value of \(-1\) indicates perfect negative correlation, meaning that as one data point increases, the next one decreases proportionally. The blue shaded area represents the confidence interval. If the autocorrelation coefficients fall outside this interval, the autocorrelation at those lags is considered significant. The results indicate that most of the lag values' autocorrelation coefficients fall well outside this interval, underscoring their statistical significance. The high autocorrelation and periodic fluctuations observed in the data suggest the presence of long-term trends and seasonal components.  
Additionally, we have conducted an alias detection analysis on the frequency spectrum. The spectrum showed a high amplitude near 0, indicating a significant presence of low-frequency components, while the energy of the high-frequency components was very low. There were no noticeable high-frequency components appearing in the low-frequency position, and no significant asymmetries were observed, indicating that there was no folding of high-frequency components into the low-frequency region. This confirms that our signal has adhered to the Nyquist criterion, with a sufficiently high sampling frequency to avoid aliasing.