1. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 1/16 MIQP formulation for optimal controlled variable selection in Self Optimizing Control Ramprasad Yelchuru Prof. Sigurd Skogestad MIQP - Mixed Integer Quadratic Programming

2. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 2/16 Outline 1.Motivation 2.Problem formulation 3.MIQP formulation 4.Evaporator Case study 5.Comparison of MIQP & customized BAB 6.Conclusions

3. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 3/16 1.Motivation  Want to minimize cost J  Which two - individual measurements or - measurement combinations should be selected as controlled variables (CVs) to minimize the cost J? y = candidate measurements; H = selection/combination matrix c = H y, H=? Combinatorial problem 1. Exhaustive search (10C 2,10C 3,…) 2. customized BAB 3. MIQP 2 MVs – F 200, F 1 Steady-state degrees of freedom 10 candidate measurements – P 2, T 2, T 3, F 2, F 100, T 201, F 3, F 5, F 200, F 1 3 DVs – X 1, T 1, T 200

4. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 4/16 Optimal steady-state operation Ref: Halvorsen et al. I&ECR, 2003 Kariwala et al. I&ECR, 2008 2. Problem Formulation Loss is due to (i) Varying disturbances (ii) Implementation error in controlling c at set point c s u J Loss Controlled variables, c s = constant + + + + + - K H y c u d

5. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 5/16 Non-convex optimization problem (Halvorsen et al., 2003) D : any non-singular matrix st Convex optimization problem Global solution - Do not need Juu - And Q is used as degrees of freedom for faster solution st Improvement 1 (Alstad et al. 2009) Improvement 2 (this work)

6. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 6/16 Vectorization subject to Problem is convex QP in decision vector

7. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 7/16 Controlled variable selection Optimization problem : Minimize the average loss by selecting H to obtain CVs as (i) best individual measurements (ii) best combinations of all measurements (iii) best combinations with few measurements st.

8. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 8/16 3. MIQP Formulation We solve this MIQP for n = nu to ny Big M approach high value M => high cpu time

9. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 9/16 4. Case Study : Evaporator System 2 MVs – F 200, F 1 10 candidate measurements – P 2,T 2,T 3,F 2,F 100,T 201,F 3,F 5,F 200,F 1 3 DVs – X 1, T 1, T 200

10. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 10/16 Case Study : Results Results Controlled variables (c) Optimal individual measurements Loss 2 = 3.7351 Loss 4 = 0.4515 Data Optimal 4 measurement combinations

11. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 11/16 Case Study : Results

12. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 12/16 Case Study : Computational time ** Branch and bound (BAB): Kariwala and Cao, IEEE Trans. (2010) No. Meas Optimal Measurements MIQP cpu time (sec) Downwar ds BAB cpu time (sec) Partial BAB cpu time (sec) Exhau stive cpu time (sec)*Loss 2 [F 3 F 200 ]0.03100.07810.06000.453.7351 3 [F 2 F 100 F 200 ]0.01600.00000.14061.20.6501 4 [F 2 T 201 F 3 F 200 ]0.04700.0313 2.10.4515 5 [F 2 F 100 T 201 F 3 F 200 ]0.03200.00000.03132.520.3373 6 [F 2 F 100 T 201 F 3 F 5 F 200 ]0.01600.00000.03132.10.2857 7 [P 2 F 2 F 100 T 201 F 3 F 5 F 200 ]0.01600.03130.00001.20.2532 8 [P 2 T 2 F 2 F 100 T 201 F 3 F 5 F 200 ]0.0000 0.07810.450.2296 9 [P 2 T 2 F 2 F 100 T 201 F 3 F 5 F 200 F 1 ]0.0000 0.10.2100 10 [P 2 T 2 T 3 F 2 F 100 T 201 F 3 F 5 F 200 F 1 ]0.00000.03130.00000.010.1936

13. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 13/16 5. Comparison of MIQP, Customized Branch And Bound (BAB) methods  MIQP formulations can accommodate wider class than monotonic functions (J)  MIQPs are solved using standard cplex routines  MIQPs are simple and are easy to incorporate few structural constraints  MIQPs are computationally intensive than BAB methods  Single MIQP formulation is sufficient for the described problems  Customized BAB methods can handle only monotonic cost functions (J)  Customized routines are required  BABs require a deeper understanding of the customized routines to solve problems with structural constraints  Computationally faster than MIQPs as they exploit the monotonic properties efficiently  Monotonicity of the measurement combinations needs to be checked before using PB 3 for optimal measuement subset selections

14. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 14/16 MIQP formulation with structural constraints If the plant management decides to procure only 5 sensors (1 pressure, 2 temperature, 2 flow sensors) 2 MVs – F 200, F 1 3 DVs – X 1, T 1, T 200 10 candidate measurements – P 2,T 2,T 3,F 2,F 100,T 201,F 3,F 5,F 200,F 1 Loss 5-sc = 0.5379

15. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 15/16 6. Conclusions  The self optimizing control non-convex problem is reformulated as convex problem  MIQP based formulation is presented for  Selection of CVs as optimal individual measurements  Selection of CVs as combinations of all measurements  Selection of CVs as combinations of optimal measurement subsets  MIQPs are more simple, intuitive and are easy compared to customized Branch and Bound methods  MIQPs are computationally intensive than customized Branch and Bound methods

16. Ramprasad Yelchuru, MIQP formulations for optimal controlled variables selection in Self Optimizing Control, 16/16