Cameron M. Crowe

Data reconciliation — Progress and challenges

Abstract Measurements in a chemical process are subject to errors, both random and systematic, so that the laws of conservation of mass and energy are not obeyed. In order to record the performance of the process, these measurements are adjusted in order that they conform to the conservation laws and any other constraints imposed upon them. This procedure is known as data reconciliation. Advances in the theory and application of data reconciliation are reviewed and current problems are highlighted. In addition to defining the basic problem, we discuss the detection of gross errors in data and of pre-adjustment of data, finding departures from steady state, estimation of the variance structure of the data, observability of unmeasured quantities and redundancy of measurements.

Observability and redundancy of process data for steady state reconciliation

Abstract Two important questions arise in the evaluation of a set of measurements of flow rates and concentrations in a steady state process. First, can an unmeasured quantity be uniquely calculated from the data? If so, it is called observable. If not, it is unobservable and no such calculated value should be trusted. Second, could a measurement be uniquely calculated from the other measurements if it were measured? If so, it is called redundant. Without some redundant measurements, no separate check on measurements is possible and data reconciliation is not feasible. Direct methods of classifying unmeasured quantities into observable and unobservable sets and of identifying redundant measurements are presented, based on matrices describing plant structure and on any set of values consistent with the constraints. The measurements of flows and concentration can be arbitrarily located in the process and chemical reactions, flow splitters and pure energy flows can be treated. Simple test are derived to determine whether each measurement is redundant.

Formulation of linear data reconciliation using information theory

The reconciliation of process measurements, subject to linear constraints, has usually involved finding the minimum weighted sum of squares of adjustments to the measurements. In order to do statistical tests, the data are most commonly assumed to follow a multivariate normal distribution. In this paper, linear data reconciliation is reformulated by maximizing the information entropy to obtain probability distributions of the data with the minimum incorporation of prior knowledge. Then the reconciled measurements are obtained by maximum likelihood, subject to the process constraints. Two cases are presented, first with only the bounds on the data being specified and second with the variance-covariance matrix of the data additionally being specified. The first case provides a means of performing data reconciliation in the absence of information on the variance-covariance matrix of the data. In the second case, reconciliation using maximum likelihood is formally identical to the conventional least-squares solution. The least-prejudiced probability distribution is a truncated normal distribution, which for reasonably precise data essentially coincides with the multivariate normal distribution. A major difference from conventional reconciliation is that by assuming prior bounds on the measurements, one also should apply those bounds to the reconciled values. An example is used to illustrate the practical implications of the two cases.

Optimization of reactors with catalyst decay and the constant conversion policy

Abstract For continuous stirred-tank and plug-flow catalytic reactors in which the catalyst activity decreases with time, Szepe [1] showed that the optimal temperature-time policy can lead to a policy of constant exit conversion under specific conditions. These conditions included a single irreversible reaction with a separable rate equation and a rate of decay of catalyst activity also separable and independent of composition. It is the purpose of this paper to extend the results of Szepe and to establish: (a) A necessary and sufficient condition for a constant-conversion optimal policy in a continuous stirred-tank reactor, and (b) A proof of the constant-conversion policy for a plug-flow reactor with distributed control of temperature for a general irreversible reaction with separable kinetics. In both cases, the rate of decay is assumed to depend on composition.

The numerical solution of bilinear data reconciliation problems using unconstrained optimization methods

A new approach to solving steady-state data reconciliation problems with bilinear constraints is proposed. Previously it was shown by Crowe that these problems can be solved using the method of matrix projection. In this work the objective function and its constraints are put into unconstrained form using Lagrange multipliers. Unconstrained optimization methods based on analytical derivatives are then used to solve the unconstrained formulation. The unconstrained approach is tested on two different reconciliation problems from the literature using number of Newton-type unconstrained optimization methods. The unconstrained methods are compared in terms of robustness and efficiency in solving the example problems. The performance of the unconstrained approach is compared to that of the method of matrix projection and the projected Lagrangian algorithm implemented in GAMS/MINOS.

On a relationship between Quasi-Newton and dominant eigenvalue methods for the numerical solution of non-linear equations

Abstract In order to compare Quasi-Newton methods with the dominant eigenvalue (DE) method of convergence acceleration, the latter is modified so that an acceleration step is taken at every iteration and the method of calculation of the required coefficients can be suitably chosen. A relationship is derived between the class of rank-one Quasi-Newton methods (QN1) and modified DE methods such that each QN1 method corresponds to a particular modified DE method. However, it was not possible to find a QN1 formula which would correspond to the DE method applied at every iteration. Such a formula would appear to require information in advance, from iteration yet to be done. The advantages of using the equivalent modified DE method instead of Broydens QN1 method are examined. A new algorithm of QN1 methods is proposed, based on the equivalent modified DE method, which was at least one order of magnitude faster, for Broydens mehtod[1] on an example problem, than the conventional implementation.