Data reconciliation theory

If we focus our attention on the solution of linear data reconciliation problems, we can state the reconciliation problem mathematically as:

The object function showed above is expressed as matrices and is the same as:

In the generalized constraint:

• A1 is the Measured Flow Incidence Matrix,

• Xr is the vector of reconciled values for measured streams,

• A2 is the Unmeasured Flow Incidence Matrix,

• Y is the vector of reconciled values for unmeasured streams.

To explain data reconciliation theory we are going to use the following model. Streams are named as Yn (n is the number) for a better understanding. The red circles means that is a measured stream.

In this example we have a tank. The tank is expressed as another node, and it has two streams. The Start Inventory is the tank’s input and is named as Ty (Tank yesterday). The End Inventory is the tank’s output and is named as Tt (Tank today).

Streams and nodes will be expressed in a matrix. Nodes are the rows and streams are the columns. When a stream is entering to a node the value is 1. When a stream is going out of the node the value is -1. By doing this we can obtain A1 and A2.

• The next step is obtaining which unmeasured streams are solvable and which ones are unsolvable. To do this we need to calculate the Reduced Row Echelon Form (rref) of A2.

The dependent columns in an rref matrix are non solvable flows. The independent columns are solvable flows. As we noticed, the rref above has only independent flows. All unmeasured streams (y2, y6, y7 and y9) are solvable.

• The generalized constraint is A1*Xr + A2*Y=0. To obtain Xr (vector of reconciled values for measured streams) we are going to use Matrix Projection.

  Let P be a matrix such that P*A2=0. This matrix is called Projection matrix, and if we multiply P with the generalized constraint we have:

  

  P is defined as:

  

• Once we have P, is easy to calculate B (Reduced Balance matrix).

  

  The rows in B are the three resulting constraints of redundant flows:

  

• To have Xr (reconciled values of measured streams) we are going to use another equation.

  

  X is the vector with measured values.

  

• Q is the variance matrix of X and its diagonal have the tolerance of each measured stream. In this case the tolerance is the 5% of the measurement value. The tolerance for a constant flow is 0. Consequently Ty and Tt have a tolerance =0.

  

• The imbalance of the three nodes using in the constraints can be calculate as:

  

  

• H is the covariance matrix of imbalance in node, and its calculated as:

  

• As we have Q, B, H and x, we can calculate Xr.

  

  

• The last step is to find the value of the unmeasured streams. As we found all unmeasured streams in this model are solvable. The next equations are use to find the values of solvable flows.

  

• The reconciled values of unmeasured flows are:

  

• If you reconcile the model you can check the values that have been found.