# Fault-Tolerant Model Predictive Control Applied to a Sewer Network
## 5 Results and Discussion
### Fault Detection, Diagnosis and MPC Reconfiguration Tests  
In this section, to assess the MPC reconfiguration performance, comparative results of the control system in four cases will be shown:  
* Case 1: sewer network without control, that is, always with all the gates open.
* Case 2: sewer network controlled with MPC in the absence of faults.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Fault Type** & \multicolumn{2}{c}{**Fault on the 2 nd Day**} & \multicolumn{2}{c}{**Fault on the 5 th Day**} & \multicolumn{2}{c}{**Fault on the 8 th Day**} \\ \hline  & **Detection (Day)** & **Diagnosis (Variable)** & **Detection (Day)** & **Diagnosis (Variable)** & **Detection (Day)** & **Diagnosis (Variable)** \\ \hline Level sensor 1 & 2.26 & h1 & 5.26 & u1 & 8.26 & h1 \\ Level sensor 2 & 2.194 & u1 & 5.26 & h3 & 8.26 & h2 \\ Level sensor 3 & 2.26 & h3 & 5.31 & h3 & 8.26 & h4 \\ Level sensor 4 & 2.865 & h4 & 5.26 & h4 & 8.76 & u4 \\ Level sensor 5 & 2.302 & h5 & 5.594 & h5 & 8.896 & h2 \\ Gate 1 & 2.26 & u1 & 5.365 & u1 & 8.579 & h1 \\ Gate 2 & 3.146 & u2 & 5.537 & u3 & 8.419 & h3 \\ Gate 3 & 2.271 & u3 & 5.485 & u3 & 8.387 & h3 \\ Gate 4 & 2.219 & u1 & 5.735 & u1 & 8.677 & h4 \\ Gate 5 & 2.26 & u5 & 5.604 & u3 & 8.438 & h4 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Fault detection and diagnosis results for scenario 3.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Fault Type** & \multicolumn{2}{c}{**Fault on the 2 nd Day**} & \multicolumn{2}{c}{**Fault on the 5 th Day**} & \multicolumn{2}{c}{**Fault on the 8 th Day**} \\ \hline  & **Detection (Day)** & **Diagnosis (Variable)** & **Detection (Day)** & **Diagnosis (Variable)** & **Detection (Day)** & **Diagnosis (Variable)** \\ \hline Level sensor 1 & 2.285 & h1 & 5.26 & h1 & 8.26 & h1 \\ Level sensor 2 & 2.521 & h2 & 5.26 & u2 & 8.26 & h2 \\ Level sensor 3 & 2.625 & u3 & 5.26 & h3 & 8.271 & h3 \\ Level sensor 4 & 2.75 & h4 & 5.812 & u1 & 8.26 & h4 \\ Level sensor 5 & 3.083 & h2 & 5.521 & h5 & 8.521 & h5 \\ Gate 1 & 2.438 & u3 & 5.312 & h5 & 8.464 & h5 \\ Gate 2 & 2.346 & u3 & 5.277 & u2 & 8.327 & u2 \\ GateCase 3: sewer network controlled with MPC in the presence of a certain fault.  
Case 4: sewer controlled with reconfigured MPC (FTMPC). Once the fault is correctly detected and identified, the controller is reconfigured to improve system performance compared to the previous case.  
In each case, scenarios 2 and 3 will be considered. Two of the most representative faults have been selected:  
Fault in the tank 1 level sensor, in which its gain is reduced to 10% of its normal value.  
Fault in the tank 3 gate, which is supposed to be blocked at 20% of its total opening.  
Furthermore, to better evaluate the effect of the fault and the reconfiguration of the system, it will be assumed that, in all cases, the fault is generated on the second day of the 10-day simulation period considered for each scenario.  
The performance evaluation criteria will be the same as those detailed in [40]. In summary, these criteria are number of overflows (\(N_{\text{ov}}\)), duration of overflow (\(T_{\text{ov}}\)) in _days_, volume overflowed (\(V_{\text{ov}}\)) in m\({}^{3}\), degree of utilization of WWTP (\(G_{\text{u}}\)) in %, and smoothness in the application of control signals (\(S\)) in m\({}^{3}\)/d.