## 5 Conclusions  
In this paper, a comprehensive analysis framework was developed for data reconciliation and gross error detection in the S-CO\({}_{2}\) systems. The proposed approach integrated thermal calculation models and robust optimization algorithms to enhance the reliability of real-world measurements. Key findings included the following:  
1. This study applied robust data reconciliation into an S-CO\({}_{2}\) recompression cycle for the first time and explored hybrid SQP and IPM solutions for the nonlinear optimization problem. Results proved the effectiveness of the proposed framework in the convergence and correction of the measurements.
2. Five estimators, including the classical WLS estimator and four robust estimators (fair, logistic, Cauchy, and Welsch), were evaluated under 2000 simulated conditions with various numbers and ranges of gross errors in the S-CO\({}_{2}\) system. Results demonstrated the superiority of robust estimators over the WLS estimator in the correction of both measurements and unknown parameters, where the MRE of all measurements can be reduced from 1.02% to 0.39% (Welsch estimator), and the MRE of the gross errors can be reduced from 4.79% down to only 1.11% (Welsch estimator).
3. Judging by the MRE and RMSE results, the Welsch estimator achieved optimal performance for measurements and unknown parameter estimations but may fail to correct excessive errors of measurements under very few conditions. Compared with the Welsch estimator, the Cauchy estimator was able to acquire a stable performance for most conditions.  
In general, as the most preferable robust estimator studied in this work, the Welsch estimator had the best MRE and RMSE results for almost all parameters, which is consistent with the conclusion drawn from the previous study for a simpler chemical system [31]. This proves the applicability of such a robust estimator in the relatively complex thermal system.  
The robust data reconciliation framework constructed in this paper can lay a solid foundation for the subsequent research on data processing and operation optimization  
Figure 17: RMSEs of each unknown parameter in the S-CO\({}_{2}\) system under all conditions.  
for supercritical carbon dioxide systems. Experimental validation was not carried out for the proposed study due to the lack of a comprehensive test rig of the S-CO\({}_{2}\) thermal cycle, which should be taken seriously at the next stage of the study. In addition, dynamic conditions were not considered in the current research, which should be necessary to apply the proposed framework to reality. Further research is expected to consider the optimal configuration of the number and position of the sensors, as well as the dynamic model of the important facility in the system under non-steady conditions, and experimental studies, as well as sensitivity or stability analysis, are necessary to verify the practical performance of the robust data reconciliation method and the optimization solvers.  
Conceptualization, Y.L.; formal analysis, J.Y.; funding acquisition, Y.X.; methodology, J.Y.; project administration, Y.X.; validation, Y.L.; writing--original draft, J.Y.; writing--review and editing, Y.L. and Y.X. All authors have read and agreed to the published version of the manuscript.  
The work was fully supported by the Major Project of Shaanxi Province Key Research Project (2024 PT-ZCK-47).  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare