## 3 Research Method
### Gross Error Identification  
Under normal circumstances, data reconciliation has a good coordination and optimization effect for random errors obeying statistical distribution law, but it may have a poor effect for gross errors disobeying statistical distribution law. The reason is that gross errors will be dispersed to all related data in the system during the process of data reconciliation, which may lead to a greater degree of overall deviation in the reconciliated data. Gross error identification usually adopts the statistical hypothesis test method, which is only based on the operation data, has the characteristics of strong applicability, economy and simple operation, and has been widely studied and applied. Different test statistics correspond to the hypothesis statistical test methods, which commonly include the global test method, the constraint equation test method and the measurement test method. The data reconciliation model established in this paper adopts the global test method and the measurement test method. The global test method is a method to judge whether a gross error in the whole system calculation model exists. In the global test method, the global test statistic can be defined as follows [38]:  
\[K=\mathbf{v}^{T}\mathbf{S}_{\mathbf{x}}^{-1}\mathbf{v} \tag{13}\]If the gross error does not exist in the whole system, the test statistic \(K\) following the chi-square distribution should meet the following requirements:  
\[K<\chi^{2}_{1-\alpha,n} \tag{14}\]  
where \(\chi^{2}_{1-\alpha,n}\) is the chi-square distribution critical value. Normalization of this formula which is easy to use results in the following [38]:  
\[Q=\frac{K}{\chi^{2}_{1-\alpha,n}}<1 \tag{15}\]  
where \(Q\) means quality factor is used to characterize the gross error of the whole system. Even if the whole system satisfies Equation (15), there may still be gross errors in some parameters, and the measurement test method is used to identify the gross errors of parameters. In the measurement test method, in order to realize the identification of suspicious data with gross errors, the test statistics of variable \(x_{i}\) based on the correction vector \(\mathbf{\
u}\) can be constructed as follows [36]:  
\[r_{i}=\frac{|v_{i}|}{\sqrt{max\bigg{(}\sigma^{2}_{v,i^{2}}\frac{\sigma^{2}_{v, i^{2}}}{10}\bigg{)}}} \tag{16}\]  
where \(v_{i}\) is the correction value of variable \(x_{i}\). For the right side of Equation (16), when \(|v_{i}|\) reaches the allowable error maximum \(\xi_{max}\), there is the following equation:  
\[r_{i}<\mathcal{Z}_{1-\alpha/2} \tag{17}\]  
where \(\mathcal{Z}_{1-\alpha/2}\) is the lower \(\alpha\) fraction of the normal distribution, which can be obtained by looking up the statistics table. In the process of systematic data reconciliation calculation, given \(\alpha=0.05\) which means the confidence level is 95%, it can be seen from the table that \(\mathcal{Z}_{1-\alpha/2}=1.96\). Under normal circumstances, this should be:  
\[r_{i}<1.96 \tag{18}\]  
This means that there are no anomalies in this set of data. If Equation (18) is not satisfied, it means that the confidence probability of the reconciliated value is less than 5%, and it can be considered that the parameter is suspicious with gross error. At this time, it is reasonable to think that an instrument error or equipment breakdown may have occurred, and further inspection must be carried out. Therefore, how do we use practical identification methodology in data reconciliation process? First, we can build the overall model of heat balance equations and redundant constraint equations after determining known and unknown variables. At this time, we can give the global test statistic \(K\) of the model and the test statistics \(r_{i}\) of each variable respectively according to above formulas, which can be used as criteria for testing gross errors.