Additional constraints allow DR to remain a linear optimization problem

In general, data reconciliation solves the following constrained, weighted, least squares optimization problem:

where

In the case of the simple mass balance constraint, the problem is a linear optimization. Adding component and enthalpy constraints the balance creates a nonlinear constrained optimization problem. Nonlinear algorithms are a higher order of complexity and take longer to compute, sometimes becoming unsolvable in a practical amount of time.

Our experiences building the data reconciliation models of many plants have taught us that the main purpose of data reconciliation is to determine the true mass quantity. Other balance constraints, such as component and enthalpy balances, help to calculate accurate mass quantities but are not in themselves a major driver for implementation of data reconciliation software.

Our approach is to add additional balance models for component and enthalpy and add the new constraints between the values in the new models and the mass balance to the linear optimization expression. The problem, though larger, remains linear:

We shall conceptually explain the new approach with the following example of a simple mass and enthalpy balance model.

Example

An example of a simple mass and enthalpy balance model.

Figure: An example of a simple mass and enthalpy balance model

Where:

Xi is the mass quantity

Hi is the specific enthalpy

The data reconciliation problem is defined as the following objective function using the constraints of the mass and enthalpy balance. Below we show the traditional approach:

The enthalpy balance constraint shown above is the nonlinear form used to determine the mass quantity and specific enthalpy simultaneously. The data reconciliation as posed above is a nonlinear optimization problem.

Our new approach models the mass balance and enthalpy balance separately and imposes additional constraints between the mass balance and enthalpy balance.

Figure: The mass balance model Figure: The enthalpy balance model

Thus, we have changed a nonlinear constraint data reconciliation problem into a linear constraint problem by adding additional models and constraints. The reduced sets of linear constraints are determined by the linear projection matrix method, which does not involve any unmeasured variables. The exact solution can then be obtained by the Lagrange multiplier method. The mass balance constraint is solved as well and the solution time remains of the same order as that required for solving only the mass balance.

Another advantage of this new approach is that if the data reconciliation algorithm employs gross error detection and elimination methods in the mass balance, those same methods should be applicable to the augmented expression. Faulty component composition and specific enthalpy values can more easily be found in this way than when nonlinear constrained optimization is employed, for which gross error detection methods are much more complex.