## 3 Research Method
### Data Reconciliation Principle  
In order to monitor the operating parameters in different position of the system, it is necessary to set sensors to obtain the measured values, and the data collected by these sensors is merged into the measurement data base as the input to data reconciliation. However, deviations in these measured values are unavoidable. Data reconciliation is a method to modify the measurement data under the premise of minimum total correction, so that all the measurement data can meet the constraint equations of the system. It can also solve the unknown data that is difficult to measure. When the whole thermal system is in a steady state, the vector composed of all known variables including characteristic data and measurement data with only random errors is defined as:  
\[\mathbf{x}=\mathbf{x_{0}}+\delta \tag{3}\]  
where \(\mathbf{x_{0}}\) is the true value vector of the known variable, and \(\delta\) is the random error vector. For the known variables in the thermal system, the problem that data reconciliation can solve is to minimize the sum of squares of the differences between the reconciliated values and the initial value under the constraint equations of energy and mass balance. Mathematically, this optimization problem can be converted to solving the least square solution that satisfies multiple constrained equations:  
\[ \begin{cases}\underset{\hat{x}}{\min}\xi(\hat{\mathbf{x}})=\sum\limits_{i=1}^{n} \dfrac{\left(\hat{x}_{i}-x_{i}\right)^{2}}{\sigma_{i}^{2}}\\ \text{s.t. }\mathbf{F}(\hat{\mathbf{x}},\mathbf{u})=\mathbf{0}\end{cases}  \tag{4}\]  
where \(x_{i}\) is the component of known variable \(\mathbf{x}\) in Equation (3), \(\hat{x}_{i}\) is the reconciliated value of the known variable in group \(i\) corresponding to \(x_{i}\) (which compose the reconciled vector \(\hat{\mathbf{x}}\)), \(\sigma_{i}\) is the mean square deviation (standard deviation) corresponding to \(x_{i}\), \(\xi(\hat{\mathbf{x}})\) is the object function of the optimization problem, \(\mathbf{F}\) is a equation vector representing a group of algebraic equations, and \(\mathbf{u}\) is the vector of the unknown variable. If \(\mathbf{F}\) has a linear function to \(\hat{\mathbf{x}}\), then there is an analytical solution to this linear constrained problem after using the method of the unknown variable mentioned in Section 3.5. However, for general thermal systems, due to the existence of complex functions such as the property function of the working medium and the characteristic function of the equipment components, \(\hat{\mathbf{x}}\) to \(\mathbf{u}\) should be nonlinear, so this problem is a quadratic solution problem with nonlinear constraints. The value of the unknown variable cannot be determined by the optimization problem and must be eliminated in the process of solving. In this paper, the improved Lagrange multiplier method proposed by [PERSON] [36] is used for the calculation. This method converts constrained optimization into unconstrained optimization and nonlinear constraints into linear constraints in this problem in essence. After eliminating the effects of the unknown variable, the correction vector of the initial value vector is given as follows [36]:  
\[\mathbf{v}=\mathbf{S_{x}}\mathbf{J_{F,x}}\mathbf{S_{F}}^{-1}\mathbf{F}^{(0)} \tag{5}\]  
where \(\mathbf{S_{x}}\) is the covariance matrix of the known variable vector, \(\mathbf{J_{F,x}}\) is the Jacobian matrix of equation vector \(\mathbf{F}\) to \(\mathbf{x}\), \(\mathbf{S_{F}}^{-1}\) is the inverse matrix of the covariance matrix of equation vector \(\mathbf{F}\), which can be calculated as follows:  
\[\mathbf{S_{F}}=\mathbf{J_{F,x}}\mathbf{S_{x}}\mathbf{J_{F,x}} \tag{6}\]  
and \(\mathbf{F}^{(0)}\) presents the initial residual vector of \(\mathbf{F}\) whose notation in the upper right corner indicates that it has not been iterated.