# Peak Energy Reduction in Flow Shop including Switch-Off Policies and Battery Storage
## 5 Numerical Analysis  
The first analysis of the results is conducted using the Analysis of Variance (ANOVA) to know the source of variation of the simulation experiments. The main effects of the battery capacity, charging speed, power fluctuation, and switch-off policy are presented in Table 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Source of Variance** & **Sum of Square** & **Degree of Freedom** & **Mean of Square** & **F-Ratio** & **\(p\)-Value** \\ \hline  & \multicolumn{4}{c}{**Energy Peak**} & & \\
**Battery Capacity** & 62,729,371 & 2 & 31,364,685 & 1.28 & 0.285 \\
**charge rate** & 4,203,607,727 & 2 & 2,101,803,863 & 85.48 & 0.000 \\
**Power fluctuations** & 7,152,612,496 & 2 & 3,576,306,248 & 145.45 & 0.000 \\
**Models** & 207,676,538 & 2 & 103,838,269 & 4.22 & 0.018 \\
**Error** & 1,770,323,905 & 72 & 24,587,832 & & \\ \hline  & \multicolumn{4}{c}{**Peak Power**} & & \\
**Battery Capacity** & 8.37 & 2 & 4.18 & 6.63 & 0.002 \\
**charge rate** & 130.52 & 2 & 65.26 & 103.48 & 0.000 \\
**Power fluctuations** & 2430.02 & 2 & 1215.01 & 1926.46 & 0.000 \\
**Models** & 3.85 & 2 & 1.93 & 3.06 & 0.053 \\
**Error** & 45.41 & 72 & 0.63 & & \\ \hline  & \multicolumn{4}{c}{**Average electricity grid utilization**} & & \\
**Battery Capacity** & 0.1549 & 2 & 0.07743 & 1.60 & 0.209 \\
**charge rate** & 8.6491 & 2 & 4.32456 & 89.24 & 0.000 \\
**Power fluctuations** & 6.4681 & 2 & 3.23404 & 66.74 & 0.000 \\
**Models** & 0.3074 & 2 & 0.15371 & 3.17 & 0.048 \\
**Error** & 3.4890 & 72 & 0.04846 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: ANOVA analysis. The battery capacity affects the peak power and the average energy stored in the battery. The charging speed and power fluctuations have a relevant effect on all measures analyzed. Finally, the switch-off (or not switch-off) policies affect mainly the energy consumed over the peak power. Figure 3 shows the main effect plot of the parameters analyzed that highlights the difference of importance among the parameters. In this figure, NO indicates that no policy has been considered, whereas UP and UPD indicate that the Upstream policy and the Upstream and Downstream policy have been adopted, respectively. Figure 4 shows the benefits obtained by the introduction of the battery without a switch-off policy. These figures report the energy used over the power peak constraint compared to the model without battery as a benchmark. For each power variation, the surface reports the reduction of energy over the peak power constraint, changing the battery capacity and charge rate. The benefits are relevant for each experimental class studied. As the reader can notice, the reduction of the energy over the peak power constraint is about 85% for the higher charge rate for the three power variations. The use of a lower charge rate reduces this energy from 55% to 25% from the lower to higher power variation. The slope of the surface highlights how the charge rate has a relevant impact on the reduction of the energy over the peak power. The slope of the surface over the charge rate is more evident when the variation of the power is higher. The three capacities of the battery  
Figure 4: Energy over peak power. Figure 3: Graphs of the response mean for each factor level. tested have a minor impact on the energy over the peak power. The above results suggest how it is more important to invest on the charge rate than the capacity of the battery. Table 3 shows the improvements of the switch-off policies comparing the reduction of the models with switch-off policies with no switch-off using the energy storage system. The negative percentage reported in Table 3 is the reduction of energy used over the peak power constraints. The behavior is very different when the power fluctuation changes. For the case with 10% of power fluctuation, the UP policy is more robust to the battery capacity and charging speed. The UP&D policy performs better with a lower charging speed. The UP&D leads to better performance than the UP policy for medium and high-power fluctuations.