1 Combination of Measurements as Controlled Variables for Self-optimizing Control Vidar Alstad † and Sigurd Skogestad Department of Chemical Engineering, Norwegian University of Science and Technology, Trondheim, Norway † Presented at ESCAPE 13, Lappeenranta, Finland, June

2 Escape 13 - Lapperanta - June Outline Introduction and motivation –Formulation of operational objectives Implementation of optimal operation –Strategies Self-optimizing control –Introduction –Illustrating example Optimal selection of controlled variables –Optimal linear combination of measurements Examples –Toy example –Gas allocation in oil production

3 Escape 13 - Lapperanta - June Introduction and Motivation Optimal operation for a given disturbance d Generally two classes of problems –Constrained: All DOF (u’s) optimally constrained → Implementation easy by active constraint control –Unconstrained: Some DOF (u’s) unconstrained (Focus here)

4 Escape 13 - Lapperanta - June Implementation Real-time optimization –Requires detailed on-line model Self-optimizing control (feedback control) – easy implementation

5 Escape 13 - Lapperanta - June Self-optimizing Control –Self-optimizing control is when acceptable loss can be achieved using constant set points (c s ) for the controlled variables c (without re- optimizing when disturbances occur). Define loss:

6 Escape 13 - Lapperanta - June Self-optimizing Control – Illustrating Example Optimal operation of Marathon runner, J=T –Any self-optimizing variable c (to control at constant setpoint)? c 1 = distance to leader of race c 2 = speed c 3 = heart rate c 4 = level of lactate in muscles

7 Escape 13 - Lapperanta - June Controlled variables Controlled variables c to be selected among all available measurements y, Goal: Find the optimal linear combination (matrix H):

8 Escape 13 - Lapperanta - June Candidate Controlled Variables: Guidelines Requirements for good candidate controlled variables (Skogestad & Postlethwaite, 1996) 1.Its optimal value c opt (d) is insensitive to disturbances 2.It should be easy to measure and control accurately 3.The variables c should be sensitive to change in inputs 4.The selected variables should be independent

9 Escape 13 - Lapperanta - June Optimal Linear Combination - Linearized Want optimal value of c insensitive to disturbances: To achieve Always possible if: where “sensitivity”

10 Escape 13 - Lapperanta - June Example – Toy example Consider the scalar unconstrained problem The following measurements are available Controlling y 1 gives perfect self-optimizing control. Is there a combination of y 2 and y 3 with the same properties? (Yes, should be because we have

11 Escape 13 - Lapperanta - June Example – Toy example (cont.) Select y 2 and y 3 : Gives the optimal controlled variable: Loss

12 Escape 13 - Lapperanta - June Example – Gas Lift Allocation - Introduction Wells produce gas and oil from sub-sea reservoirs Gas injection: –used to increase production –Additional cost of compressing gas Limited gas processing capacity top-side –Limits the rate of gas from the reservoirs and injection Case studied –2 production wells –Gas injection into each well –1 transportation line

13 Escape 13 - Lapperanta - June Example – Gas Lift Allocation (cont.)

14 Escape 13 - Lapperanta - June Example – Gas Lift Allocation (cont.) Objective –Maximize profit Constraints –Maximum gas processing capacity –Valve opening

15 Escape 13 - Lapperanta - June Example – Gas Lift Allocation (cont.)

16 Escape 13 - Lapperanta - June Example – Gas Lift Allocation (cont.) The loss for c LC with the combined uncertainty is due to non- linearities Evaluation of loss for different control structures

17 Escape 13 - Lapperanta - June Choice of measurements y

18 Escape 13 - Lapperanta - June Conlusion Controlled variables: Derived simple method for optimal measurement combination Find sensitivity of optimal value of measurements to disturbances Select the controlled variables as: Illustrated on two examples –Toy example –Gas injection in oil production