1. 1 Active constraint regions for optimal operation of chemical processes Magnus Glosli Jacobsen PhD defense presentation November 18th, 2011

2. 2 Outline 1.Optimal operation of chemical processes 2.Identifying active constraint regions 3.Case studies 1.Reactor-separator-recycle process 2.Distillation 4.Optimization of natural gas liquefaction plants Challenges in optimization of liquefaction processes Active constraint regions for a simple liquefaction process 5.Conclusions and suggestions for future work

3. 3 Outline 1.Optimal operation of chemical processes 2.Identifying active constraint regions 3.Case studies 1.Reactor-separator-recycle process 2.Distillation 4.Optimization of natural gas liquefaction plants Challenges in optimization of liquefaction processes Active constraint regions for a simple liquefaction process 5.Conclusions and suggestions for future work

4. 4 Optimal operation of chemical processes Optimal operation: Use process inputs to 1.Satisfy constraints 2.Maximize profit Processes are subjected to disturbances Disturbances will change the optimal conditions! –Consider an example with two inputs, linear constraints and a quadratic objective function:

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8. 8 Mathematical description J is the cost function which we seek to minimize f contains the model equations, which give x for given u and d c contains the constraints imposed on the problem x, u and d are internal states, inputs and disturbances, respectively

9. 9 Active constraint regions At the optimal solution, some c will be equal to 0, and some will be smaller than 0. Those which are equal to 0 are called the active constraints. As d changes, the optimal solution changes, and so does the active set. The disturbance space can be divided into regions with different active sets. Next: Why are these regions important?

10. 10 Self-optimizing control In order to have self-optimizing control, we need to control: 1.Active constraints! 2.Other variables whose optimal values are insensitive to disturbances. When the set of active constraints (1) changes, the variables included in (2) may change as well Thus we need to know the regions

11. 11 Regions example Two disturbances d 1 and d 2, two constraints c 1 ≤c 1,max and c 2 ≤c 2,max

12. 12 Outline 1.Optimal operation of chemical processes 2.Identifying active constraint regions 3.Case studies 1.Reactor-separator-recycle process 2.Distillation 4.Optimization of natural gas liquefaction plants Challenges in optimization of liquefaction processes Active constraint regions for a simple liquefaction process 5.Conclusions and suggestions for future work

13. 13 Finding the regions Simple problems: Just optimize at many points across the entire disturbance range Chemical engineering problems are usually not simple For N independent constraints that may either be active or inactive, we may have N 2 regions We need a strategy to simplify the problem!

14. 14 Finding the regions II At any solution u opt of the optimization problem, the Karusch- Kuhn-Tucker conditions are satisfied:

15. 15 Finding the regions III The optimal u, c and λ are dependent on d At a particular d, for the i’th constraint, either c i or λ i is zero When the i’th constraint switches from active to inactive, we have that c i + λ i = 0 In other words, to find when a constraint changes from active to inactive, we solve the following equation by interpolation:

16. 16 Finding the regions IV Still, we need to optimize a number of times How can we simplify the task? Use process and problem knowledge! –Identify the most important disturbances –Are any constraints independent of one disturbance? –Are any combinations of active constraints physically impossible or unrealistic? –Do we know the bounds of the feasible part of the disturbance space?

17. 17 Constraint lines/curves A constraint curve is a curve which divides the disturbance space into a part where a given constraint is active, and a part where it is inactive If the constraint curve is straight, we will call it a constraint line Each region is defined by the curves which bound it Thus, we need to draw the constraint curves

18. 18 Summary of the method 1.Find out which disturbances d are important, and their range 2.Find, from process and problem knowledge, whether any constraints will be active for all d 3.Find out if some constraint curves/lines will be independent of one disturbance In the two-dimensional case this will lead to a vertical or horizontal curve (line) segment 4.Then find sufficiently many points to draw each curve

19. 19 Outline 1.Optimal operation of chemical processes 2.Identifying active constraint regions 3.Case studies 1.Reactor-separator-recycle process 2.Distillation 4.Optimization of natural gas liquefaction plants Challenges in optimization of liquefaction processes Active constraint regions for a simple liquefaction process 5.Conclusions and suggestions for future work

20. 20 Case study I: Stirred tank reactor with distillation and recycle

21. 21 Process description The reactant A reacts to B (desired product) and C (undesired by-product) in a CSTR: A  B (1) A  2C (2) B is the heavier product, and is separated from A and C in a distillation column (B leaves in the bottom) A fraction of the distillate stream is purged, the rest is mixed with fresh A and fed back to the reactor.

22. 22 Process description Reaction (1) is favored by low temperature C is the most volatile component, followed by A Nominal feed rate of pure A is F 0 = 1 mol/s

23. 23 Degrees of freedom Feed flow rate is considered a disturbance, so there are five degrees of freedom: –Recycle/purge ratio P/D –Reactor holdup M r –Reactor temperature Tr –Column reboiler duty Q R –Column condenser duty Q C

24. 24 Optimization objective The objective is written as follows: J = p F F + p V V – p B B –p P P with the following prices:

25. 25 Constraints 1.x B ≥ X B,B,min 2.T r ≤ T r,max 3.M r ≤ Mr,max 4.V ≤ V max 5.P ≤ P max In addition, all flows must be larger than zero

26. 26 Disturbances Feed flow rate F: If the process is part of a large plant, e.g. a refinery, the feed flow rate may be set elsewhere. If it is set locally, it may be considered a degree of freedom. Energy price p V : This parameter may potentially vary a lot, for example if the plant operator must change between alternative energy sources.

27. 27 Nominal operating point F, x B,B, T r, M r, L and P/D were set The remaining variables were calculated

28. 28 Results: Constraint regions

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30. 30 Optimal values of variables

31. 31 Summary The interpolation method worked well, but: Care had to be taken when choosing end points for interpolation Only one curve was found to have a straight segment (between regions IV and V)

32. 32 Outline 1.Optimal operation of chemical processes 2.Identifying active constraint regions 3.Case studies 1.Reactor-separator-recycle process 2.Distillation 4.Optimization of natural gas liquefaction plants Challenges in optimization of liquefaction processes Active constraint regions for a simple liquefaction process 5.Conclusions and suggestions for future work

33. 33 Distillation case studies 1.One distillation column with two products and fixed product prices 2.One distillation column with two products and variable price for the more valuable product 3.Two distillation columns with three products and fixed product prices

34. 34 One-column case studies Degrees of freedom: Two steady-state degrees of freedom (e.g. L and V) Disturbances: Feed flow F and energy price p V Constraints: 1.V ≤ V max 2.x D ≥ x D,min 3.x B ≥ x B,min (mole fractions of main components)

35. 35 One column: Objectives The two case studies have different objective functions: Case study Ia uses fixed prices: J = p F F + p V V – p B B –p D D Case study Ib: No payment for impurity in distillate, J = p F F + p V V – p B B –p D Dx D

36. 36 One column: Data

37. 37 Finding regions for distillation Case 1a: x D is always active because of the ”avoid product giveaway rule”. Border between ”only V 1 active” and ”V 1 plus one purity constraint” regions must be vertical, because: When V 1 is at its maximum, the optimal solution is independent of p V Borders between regions where only purity constraints are active, will be horizontal (independent of F), because: Optimal flow ratios depend on p V only

38. 38 Regions for case study Ia

39. 39 Regions for case study Ib

40. 40 Summary, one column There is a physical bottleneck at F = 1.44 mol/s, above this we cannot satisfy all constraints The use of a priori knowledge reduces the number of optimizations needed The change in price policy turns the order of purity constraints ”upside down”: –In case 1a, x B is active only at high values of p V –In case 1b, x B becomes active before x D as p V increases

41. 41 II: Two distillation columns

42. 42 Description A is the most volatile component and C the least volatile B is most valuable product Maximum boilup in column 2 is half of maximum boilup in column 1

43. 43 Objective and constraints Objective function: J = p F F + p V (V 1 +V 2 ) – p A D 1 – p B D 2 –p C B 2 Constraints: 1.V 1 ≤ V 1,max 2.V 2 ≤ V 2,max 3.x A ≥ x A,min 4.x B ≥ x B,min 5.x C ≥ x C,min 5 constraints potentially gives 2 5 = 32 regions

44. 44 Regions map for case II

45. 45 Control of two columns in series

46. 46 Outline 1.Optimal operation of chemical processes 2.Identifying active constraint regions 3.Case studies 1.Reactor-separator-recycle process 2.Distillation 4.Optimization of natural gas liquefaction plants Challenges in optimization of liquefaction processes Active constraint regions for a simple liquefaction process 5.Conclusions and suggestions for future work

47. 47 Optimization of LNG plants Recall the general optimization problem f: Model equations who describe the relationship between states x, inputs u and disturbances d c: Constraints, conditions that we impose on the problem.

48. 48 Optimization of LNG plants The objective is often simple – either: –Minimize energy consumption for a given production rate, or –Maximize the production rate The optimization problem: –Is nonlinear –May be non-convex –May have discontinuous constraints (e.g. when phase changes occur)

49. 49 Different approaches to solving In chemical engineering, the model equations (f) are often solved for x by a flowsheet simulator (i.e. Hysys, Aspen, Pro II) The optimization solver then only changes u, not x, and the solver handles a problem with only inequality constraints Some model equations may be left for the optimization solver

50. 50 Different approaches to solving Which equations to include in the optimization problem, and which ones to leave to the simulator? –Will the simulator be able to solve the equations for all allowed values of u? –Will the overall solution of the problem be more efficient when leaving equations to the optimization solver? Example: Solving a simple counter-current heat exchanger model

51. 51 Example: Heat exchanger model

52. 52 Example: Heat exchanger model

53. 53 Different approaches to solving Many process models include recycles/tear streams Recycle convergence: –Include in optimization problem? –Keep within simulator? This depends on which solver is more likely to reach convergence Test: Liquefaction part of a C3-MR process

54. 54 C3-MR process Most widely used process for liquefaction of natural gas Uses propane (C3) for precooling down to -40°C Precooling loop has three pressure levels Uses a mixed refrigerant (MR) for liquefaction in a spiral-wound heat exchanger (MCHE) MR consists of methane, ethane, propane and N 2 Natural gas leaves this exchanger at -157°C

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56. 56 Model, liquefaction part

57. 57 Different model formulations 1.Specify temperatures in the MCHE model Must add heat exchanger area specifications as equality constraints in optimization, but the recycles are not needed 2.Specify heat exchanger UA values, and include recycle convergence in optimization problem 3.Specify heat exchanger UA values, and let the simulator solve recycles Gives the lowest dimension for the optimization problem as seen by the optimization solver

58. 58 Testing of model formulations How frequently would the simulator fail to solve ”its part”? Each of six disturbances was varied over a range around their nominal values: –High and low MR pressure –MR flow rate –MR mole fractions of ethane, propane and nitrogen

59. 59 Result, testing of model formulations in Unisim

60. 60 Testing of model formulations: Solving remaining equations In Formulation I, we need to solve UA calculated (T)=UA specified In formulation II, we need to converge the recycles x recycle,guessed = x recycle,calculated This is basically a comparison between external equation solvers and the internal solvers of the simulator

61. 61 Results Tolerances set to match internal simulator tolerances Formulation I (table 5.4 from thesis): MATLAB solves for UA calculated (T)=UA specified Formulation II (table 5.5 from thesis): MATLAB solves recycles

62. 62 Summary, testing of different model formulations Best solution overall: Using a dedicated equation- solver to converge recycles Best solution with optimization solver: Use model formulation I together with an algorithm which honours bounds on decision variables

63. 63 Summary, challenges in optimization of liquefaction processes How we formulate the problem, has an impact on the likelihood of solving it If the calculations are carried out on several levels, the bottom level should be defined in such a way that it is certain to converge In the thesis, another example is given (consequence of discontinuous constraint functions)

64. 64 Outline 1.Optimal operation of chemical processes 2.Identifying active constraint regions 3.Case studies 1.Reactor-separator-recycle process 2.Distillation 4.Optimization of natural gas liquefaction plants Challenges in optimization of liquefaction processes Active constraint regions for a simple liquefaction process 5.Conclusions and suggestions for future work

65. 65 Active constraint regions for a simple liquefaction process The PRICO process is a simple liquefaction process with one multi-stream heat exchanger.

66. 66 Nominal conditions Feed flow rate: 15 170 kmol/h Temperature of gas and refrigerant after water cooling: 30°C Turbine outlet pressure: 10.29 barg Compressor inlet pressure: 4.445 barg Compressor speed: 1000 rpm Refrigerant flow rate: 69 300 kmol/h

67. 67 Key variables, compared to Jensen and Skogestad (2008) Jensen’s work is done using gPROMS, this work using Honeywell Unisim

68. 68 Objective and degrees of freedom Optimization objective: Minimize compressor work (so J = W s ) Five degrees of freedom for operation: –Amount of cooling in condenser –Compressor speed –Turbine speed –Main choke valve opening –Active charge (level in liquid receiver)

69. 69 Constraints 1.Exit temperature for natural gas, T NG,out ≤ -157°C (always active) 2.Refrigerant to compressor must be superheated, ΔT sup > 5°C 3.Compressor must not operate in surge, ΔM surge > 0 4.Maximum compressor work, W s ≤ 132MW 5. Turbine exit stream must be liquid only, ΔP sat > 0 6.Maximum compressor speed: ω comp ≤ ω max = 1200rpm

70. 70 Disturbances Feed flow rate (relative to nominal), F/F nominal Ambient temperature T amb in °C Since cooling water is cheap compared to compression power, natural gas and refrigerant entering the main heat exchanger are cooled as much as possible Thus, the disturbance sets the temperature of these streams directly

71. 71 Optimum at nominal F and T amb

72. 72 Regions for PRICO process

73. 73 Regions for PRICO process II There are five regions, with the following active constraints: I. T NG,out and ΔM surge are active II. T NG,out, ΔT sup and ΔM surge are active III. T NG,out, ΔT sup, ΔP sat and ω max are active IV. T NG,out, ΔT sup and ω max are active V. T NG,out and ω max are active

74. 74 Control of PRICO process 5 regions  5 control structures? ΔP sat is optimally close to zero in all regions –Keep it at zero to simplify Surge margin and max speed are closely connected –Use compressor speed to control ΔM surge when ω opt < ω max We can use two control structures! –One for Regions I and V (where ΔT sup is inactive) –One for Regions II, III and IV (where ΔT sup is active)

75. 75 Control structure

76. 76 Conclusions, PRICO case study 5 different active constraint regions were found Near-optimal control can be achieved with only two control structures

77. 77 Outline 1.Optimal operation of chemical processes 2.Identifying active constraint regions 3.Case studies 1.Reactor-separator-recycle process 2.Distillation 4.Optimization of natural gas liquefaction plants Challenges in optimization of liquefaction processes Active constraint regions for a simple liquefaction process 5.Conclusions and suggestions for future work

78. 78 Conclusions A method for identifying active constraint regions has been described The method has been illustrated with case studies in distillation and natural gas liquefaction Challenges in optimization of liquefaction processes have been addressed

79. 79 Directions for future work A more analytical approach to finding active constraint regions may be taken –Use ideas from multiparametric programming? For LNG: Look into self-optimizing control –Can the results for the PRICO process be generalized to more complex processes?