1. Optimization of the Pulp Mill Economical Efficiency; study on the behavior effect of the economically significant variables Alexey Zakharov, Sirkka-Liisa Jämsä-Jounela

2. Content The Pulp Mill benchmark problem (developed by F. Doyle) Idea of the optimization Approximation methods Comparison of the results Conclusion

3. The scheme of the process

4. List of the Setpoints

5. List of the economically significant variables

6. Scheme of the control strategy MPC controllers regulate 14 important quality and environmental variables. A number of SISO controllers are used to stabilize the open-loop unstable modes of the process. The setpoints of the basic control loops are partly generated by the MPC and partly defined as inputs of the plant. Optimization of Economical Efficiency Model Predictive Control SISO Control Loops Pulp Mill Process MPC inputsSISO inputs Free inputs Measurements

7. Content The Pulp Mill benchmark problem (developed by F. Doyle) Idea of the optimization Approximation methods Comparison of the results Conclusion

8. Introduction A row of papers exists concentrated on the optimization of a single unit operations or a single factor optimization (such as ClO 2 minimization). New approach, proposed by F. Doyle: the whole plant economical efficiency optimization (with respect to production and quality, minimization of energy, chemical consumption)

9. Direction of the optimization Setpoints (Decision variables) Manipulated variables (construction of the approximations) Economical Efficiency (Profit rate) Optimization of the Economical Efficiency Testing quality of the solution

10. The linear approximations The model: where V i, U j are the i-th economically significant variable and j-th decision variable, and V i 0, U j 0 are their nominal values. The elements of the matrix K i,j are defined as partial derivatives of the economically significant variables with respect to the decision variable: The values of the matrix elements are identified using a number of the plant tests and setpoints changes.

11. The problem formulation The following profit of the plant is used as the objective function for the optimization: The problem includes the lower and upper non-equality constraints both to the Economically Significant and Decision variables. The problem also includes the equality constraints related to the dependences of the steady states of the Economically Significant variables on the Decision variables. These constraints could be:  linear  quadratic

12. Content The Pulp Mill benchmark problem (developed by F. Doyle) Idea of the optimization Approximation methods Comparison of the results Conclusion

13. Results of optimization for the linear approximations The real dependences between economically significant variables and the decision variables may be non linear. As a result, the insufficient reliability of the approximations is the essential drawback of the approach: the LP forecast of the profit increase is about 26.7 USD/min the simulation show only 12.2 USD/min profit increase

14. Examples of the Economically Significant variables behavior Dependence of the steady states of the economically significant variables 12 (D1 steam flow), 13 (E Caustic flow), 15 (D2 ClO2 flow) on the decision variable 9 (E Washer [OH]) value

15. Examples of the Economically Significant variables behavior Dependence of the steady states of the economically significant variable 9 (O Steam flow 3) on the decision variable 9 (E Washer [OH]) value

16. The 1-dimensional quadratic approach The model: The elements of the matrix L are defined as the second partial derivatives. The values of the matrix elements are identified using a number of the plant tests and setpoints changes. The set of nominal setpoints variations, that has been used in the linear case, gives significant errors of the second derivatives estimations. As a result, the 25% variations are used for the quadratic approximations case instead of 5% variations, used in the linear case.

17. Multidimensional quadratic approximations The method: The approach includes all terms from the previous one, since the one dimensional quadratic terms are covered by the case j equals k. The approach includes the interactions between the decision variables. The approach requires a lot of simulations to perform.

18. Content The Pulp Mill benchmark problem (developed by F. Doyle) Idea of the optimization Approximation methods Comparison of the results Conclusion

19. Comparison of results Forecast of the profit Simulated profit Nominal stedy state105.9 Linear approximations132.6118,1 $/min 1Dim quadratic approximations 124.3 $/min120 $/min Multi Dim quadratic approximations 123.2 $/min121.9 $/min

20. The explanation of the bias of the profit forecast for the linear approximations case The matrix L contains the following elements:  83 elements of the L matrix are bigger than 0.1 (taking into account the sign of the chemicals costs)  45 elements are smaller than -0.1 (taking into account the sign of the chemicals costs) Since positive second derivatives increase the values of the economically significant outputs in comparison with the linear approximation, the linear approximations are too optimistic.

21. Comparison of the accuracies Comparison of the errors of different approximation approaches in the L1 sense: The one dimensional quadratic approach always performs better, than the linear one. The multidimensional quadratic approach has a good quality of approximation for the first and the second solutions, but its quality falls seriously for the third setpoint. LA solution QA 1D Solution QA MD Solution Linear approx3.593.647.38 1D Quad approx2.942.656.72 MD Quad approx2.952.508.25

22. Comparison of the approaches The real profit and its Linear approximations and MultiDim QA forecasts

23. Conclusion The quadratic approximations were constructed (the most important thing that requires the most efforts and computational time) The optimization of the economical efficiency was performed. The profit has been improved at about 4% (compared Linear approximations). The quality of the approximations is decided to be sufficiently high (the error of the profit forecast based on approximations is not significant).