# Fault-Tolerant Model Predictive Control Applied to a Sewer Network
## 5 Results and Discussion
### Fault Detection, Diagnosis and MPC Reconfiguration Tests
#### 5.3.2 Scenario 3 Results  
* Fault in the tank 1 level sensor: alarm percentage before a fault detection: 2.44%. Detection instant: 2.26 days. MPC reconfiguration is performed in the same way as in Section 5.3.1 for the tank 1 level sensor. Figure 15 shows the \(Q\) threshold and the value of the \(Q\) statistic calculated online. Detection occurs when \(Q\) exceeds the threshold 20 consecutive times. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Data** & **No Control** & **Normal MPC** & **MPC with u3 Fault** & **Reconfigured MPC** \\ \hline \(N_{\text{ev},1}\) & 0 & 3 & 3 & 3 \\ \(N_{\text{ev},2}\) & 0 & 1 & 3 & 3 \\ \(N_{\text{ev},3}\) & 1 & 3 & 4 & 4 \\ \(N_{\text{ev},4}\) & 3 & 3 & 3 & 3 \\ \(N_{\text{ev},5}\) & 1 & 4 & 4 & 3 \\ \(N_{\text{ev},WWTP}\) & 6 & 5 & 6 & 6 \\ \(T_{\text{ev},1}\) & 0 & 0.0396 & 0.0396 & 0.0375 \\ \(T_{\text{ev},2}\) & 0 & 0.0146 & 0.0354 & 0.0396 \\ \(T_{\text{ev},3}\) & 0.0146 & 0.0437 & 0.6896 & 0.6896 \\ \(T_{\text{ev},4}\) & 0.0771 & 0.0917 & 0.0792 & 0.0813 \\ \(T_{\text{ev},5}\) & 0.0083 & 0.0064 & 0.0563 & 0.0500 \\ \(T_{\text{ev},WWTP}\) & 0.3563 & 0.5854 & 0.5271 & 0.5250 \\ \(V_{\text{ev},1}\) & 0 & 5,3998 \(\times\) 10\({}^{3}\) & 4.6476 \(\times\) 10\({}^{3}\) & 4.1425 \(\times\) 10\({}^{3}\) \\ \(V_{\text{ev},2}\) & 0 & 83.5089 & 716.7745 & 907.9697 \\ \(V_{\text{ev},3}\) & 556.261 & 1.8417 \(\times\) 10\({}^{3}\) & 1.6621 \(\times\) 10\({}^{3}\) & 1.6522 \(\times\) 10\({}^{3}\) \\ \(V_{\text{ev},4}\) & 7.9044 \(\times\) 10\({}^{3}\) & 1.3097 \(\times\) 10\({}^{4}\) & 9.1273 \(\times\) 10\({}^{3}\) & 9.2190 \(\times\) 10\({}^{3}\) \\ \(V_{\text{ev},5}\) & 314.0450 & 1.2763 \(\times\) 10\({}^{4}\) & 1.1609 \(\times\) 10\({}^{4}\) & 1.1453 \(\times\) 10\({}^{4}\) \\ \(V_{\text{ev},WWTP}\) & 5.9698 \(\times\) 10\({}^{4}\) & 148.6536 & 134.7366 & 134.5941 \\ \(V_{\text{ev}}\) & 6.8473 \(\times\) 10\({}^{4}\) & 2.7935 \(\times\) 10\({}^{4}\) & 3.5235 \(\times\) 10\({}^{4}\) & 3.5162 \(\times\) 10\({}^{4}\) \\ \(Q_{\text{uv}}\) & 3.2379 \(\times\) 10\({}^{4}\) & 3.5974 \(\times\) 10\({}^{4}\) & 3.5092 \(\times\) 10\({}^{4}\) & 3.5114 \(\times\) 10\({}^{4}\) \\ \(G_{\text{u}}\) & 53.9648 & 59.9560 & 58.4874 & 58.5230 \\ \(S\) & - & 6.8276 \(\times\) 10\({}^{10}\) & 6.7277 \(\times\) 10\({}^{10}\) & 6.7843 \(\times\) 10\({}^{10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: System performance in every case: fault in tank 3 gate (scenario 2). Figure 15: \(Q\) statistic and threshold calculated in level fault situation (scenario 3). Table 7 provides the comparative data of system performance in every case. These results allow to achieve the same conclusions as the scenario 2 results:  
Regarding the main indices (\(V_{\text{ov}}\), \(G_{\text{u}}\) and \(S\)), the results discussion is like previous cases, with the normal MPC controller having the best performance. Furthermore, the MPC with the fault considered reduces the system performance, but by comparing with no control case, its performance is better since \(G_{\text{u}}\) is 67.28% vs. 61.67%, \(V_{\text{ov}}\) is 4.77 \(\times\) 10\({}^{4}\) vs. 8.74 \(\times\) 10\({}^{4}\) (m\({}^{3}\)). Furthermore, by comparing the MPC with a fault with FTCMPC, this one improves the system performance slightly since the total overflow volume is reduced from 4.77 \(\times\) 10\({}^{4}\) to 4.41 \(\times\) 10\({}^{4}\) and the degree of WWTP utilization increases from 67.28% to 68.81%, but \(S\) is worse because the system needs greater control efforts. * Fault in the tank 3 gate: alarm percentage before a fault detection: 3.05%. Detection instant: 2.271 days. MPC reconfiguration is performed in the same way as in Section 5.3.1 for the tank 3 gate. Figure 16 shows the \(Q\) threshold calculated online and the value of the \(Q\) statistic for fault detection and allows knowing the detection instant. Detection occurs when \(Q\) exceeds the threshold 20 consecutive times. Table 8 presents the comparative data of system performance in each case.