Process data reconciliation in the presence of non-uniform measurement errors

The goal of flow rates reconciliation was to adjust measured values and estimate unmeasured streams so as to balance both measured and unmeasured values, identify gross errors and detect leaks and losses. Thus, data reconciliation plays a key role in the monitoring of industrial plants for the early detection of critical events which might cause environmental and economic damages, and it is,therefore, an essential component of any clean technology process. Consequently, any method that improves the accuracy of the reconstructed data by considering more realistic assumptions on the statistical nature of the data can add considerably to the overall reliability of the process. The reconciliation procedure is statistical in nature and requires adequate information on the structure of the random errors of the flow rates measured. A frequent assumption is the homoscedasticity and the independence of the errors affecting different streams. This assumption leads to efficient algorithms based on advanced linear algebra decompositions, such as QR or Singular Value Decomposition, but it frequently leads to biased estimates, especially when the values of flow rates vary over two or more orders of magnitude. The goal of this article was to show the importance of considering general heteroscedasticity when reconciling flow rates. Errors are supposed to be normally distributed according to \(\varepsilon_{\text{i}} \cong N(0,\,\sigma_{0} L_{\text{i}}^{ 2\eta } )\), where \(L_{\text{i}}\) is the measurement of the ith flow rate and \(\theta\) = \(\left\{ {\theta |\sigma_{0} ,\eta } \right\}\) is a set of two parameters to be estimated along with the adjustments to the measured flow rates. Therefore, the overall variance–covariance is characterised by 3 parameters \(\sigma_{0} ,\eta\) and the correlation factor among measurement errors ρ. The algorithm here proposed is based on conditional optimality, and it carries out the whole optimisation in terms of the parameters \(\theta\) only, the unknown adjustments being expressed at each iteration as functions of \(\theta\).