## 4 Results
### Overall Correction on Measurements  
To better investigate the effects of different estimators, it was necessary to evaluate the correction results on the measurements of all sensors that existed in the S-CO\({}_{2}\) system. The MREs of all measurements under 2000 conditions are shown in Figure 8. By average, the original error brought to the sensors was rather small, only about 1%, which meant the gross error only happened for a small portion of the measurements. After data reconciliation, the MREs of all measurements were reduced across all estimators. However, the WLS estimator yielded the highest MRE, while the Welsch estimator achieved the lowest MRE. All robust estimators performed better than the WLS estimator, in which the Welsch estimator achieved only a 0.39% MRE, almost half of that of the WLS estimator, which proved the advantage of robust data reconciliation over standard data reconciliation. To be more specific, the MREs of each sensor before and after correction under 2000 conditions are demonstrated in Figure 9. It was obvious that for most sensors, the error of measurement was significantly reduced under any types of estimators. For example, for sensors T1, M7, and M8, the original MRE can be up to about 2%, and after data reconciliation under any robust estimator, it can be suppressed to less than 0.5%. Among all  
Figure 6: Computation cost distributions of different estimators. Figure 7: RMSEs of constraints after data reconciliation under different estimators. estimators, the Welsch estimator performed the best in most sensors, and it was followed by Cauchy, logistic, and fair estimators, successively. The WLS estimator was the worst for all sensors, except the sensors related to water (state points 13 and 14). For temperature sensors T6, T7, T8, and T10 and the power G, the WLS estimator even generated a larger error than the original data, which is unacceptable, as these sensors are all very important for S-CO\({}_{2}\) system performance monitoring. Generally, the performance order was consistent in almost every sensor, indicating the distinctive differences in the correction performances of these robust estimators. The MRE of the measurements can reveal the general performance of robust data reconciliation, yet it may ignore the defective results happening in the change. To capture such problems, it was necessary to evaluate the RMSEs of the measurements before and after robust data reconciliation. The RMSEs of all measurements are shown in Figure 10. Compared with the original value, the RMSEs under all estimators were significantly reduced after data reconciliation, still following the successive order: WLS was the worst, followed by fair and logistic estimators, while the Cauchy estimator and Welsch estimator performed the best. It should be noted that the RMSEs of fair and logistic estimators were very close to the WLS estimator, suggesting that they may not control excessive gross errors under a number of conditions. The RMSEs of each sensor are shown in Figure 11. Compared with Figure 9, the most obvious distinction was that the Welsch estimator had worse results for some sensors (e.g., P4), where the RMSE under the Welsch estimator was close to the WLS estimator, worse than all other robust estimators. For sensors M13 and M14, the Welsch estimator performed terribly, where the RMSE was almost the same as the original RMSE without data reconciliation. The possible reason was that these measurements were not strongly related to the S-CO\({}_{2}\) system, and few redundant measurements and constraints were imposed upon them. Nevertheless, the Welsch estimator still performed much worse than other robust estimators, which implies its strong dependencies on the redundancy of the measurements. Compared with the Welsch estimator, the Cauchy estimator performed more stably than the RMSEs of most sensors, including M13 and M14.