Economic optimization in Model Predictive Control
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1 Economic optimization in Model Predictive Control Rishi Amrit Department of Chemical and Biological Engineering University of Wisconsin-Madison 29 th February, 2008 Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
2 Outline 1 Incentives for process control 2 Preliminaries 3 Motivating the idea 4 Current work 5 Future work 6 Conclusions Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
3 Incentives for process control Incentives for process control Production specifications Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
4 Incentives for process control Incentives for process control Production specifications Operational constraints / Environmental regulations Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
5 Incentives for process control Incentives for process control Production specifications Operational constraints / Environmental regulations Safety Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
6 Incentives for process control Incentives for process control Production specifications Operational constraints / Environmental regulations Safety Economics Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
7 Incentives for process control Economic incentive Economic Incentive Production of a plant depends heavily on plant s limitations and operating constraints Operating conditions keep changing plant production Under all variations and restrictions, plant must do the best it can: Process optimization Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
8 Incentives for process control Global production maximum Economic incentive Profit ($) $ Loss Higher profit expected when band of variation is reduced Profit ($) $ Loss Allows operation at/near the optimum for more time Smoother operation = Higher profit Process parameter Time Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
9 Incentives for process control Global production maximum Economic incentive Profit ($) $ Loss Higher profit expected when band of variation is reduced Profit ($) $ Loss Allows operation at/near the optimum for more time Smoother operation = Higher profit Process parameter Time Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
10 Incentives for process control Maximum production at bound Economic incentive Profit ($) $ Loss Higher profit when band of variation is reduced Profit ($) $ Loss Allows operation at/near the optimum for more time Smoother operation = Higher profit Process parameter Time Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
11 Incentives for process control Economic incentive Profit $$ Savings! Poor Control Better Control Higher fluctuations: Poor disturbance rejection Forces the mean operating state to be away from optimum to meet the constraints Solution: Reduce fluctuations and go nearer to the optimal Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
12 Process model Preliminaries Process model u Process x y Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
13 Process model Preliminaries Process model u Process x y Process model governing process dynamics dx dt = f (x(t), u(t)) y(t) = g(x(t)) Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
14 Process model Preliminaries Process model u Process x y Process model governing process dynamics dx dt = f (x(t), u(t)) y(t) = g(x(t)) Steady state: f (x s, u s ) = 0 y s = g(x s ) Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
15 Objective translation Preliminaries Process model Economic objectives are translated into process control objectives Notion of setpoints / targets x u Economic optimum Economic profit function Φ(x, u) Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
16 Objective translation Preliminaries Process model Economic objectives are translated into process control objectives Notion of setpoints / targets x u Constraints Economic optimum Economic profit function Φ(x, u) Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
17 Objective translation Preliminaries Process model Economic objectives are translated into process control objectives Notion of setpoints / targets x u Constraints Economic optimum Economic profit function Φ(x, u) Target Steady state curve f (x s,u s) = 0 Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
18 Objective translation Preliminaries Process model Economic objectives are translated into process control objectives Notion of setpoints / targets x u Constraints Economic optimum Economic profit function Φ(x, u) Target Steady state curve f (x s,u s) = 0 Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
19 Model Predictive Control Preliminaries Model Predictive Control Control/Prediction Horizon Output target Measured Output Measured Input k Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
20 Model Predictive Control Preliminaries Model Predictive Control Control/Prediction Horizon Output target Measured Output Optimized Future Input Trajectory u k Measured Input k k + 1 REGULATION Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
21 Model Predictive Control Preliminaries Model Predictive Control Control/Prediction Horizon Output target Measured Output Predicted Output u k u k+1 Optimized Future Input Trajectory Measured Input k k + 1 REGULATION Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
22 Problem definition Preliminaries MPC Optimization problem Get to the steady economic optimum (target): Minimize the distance from the target (stage cost) L(x, u) = (x x t ) Q(x x t ) + (u u t ) R(u u t ) Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
23 Problem definition Preliminaries MPC Optimization problem Get to the steady economic optimum (target): Minimize the distance from the target (stage cost) L(x, u) = (x x t ) Q(x x t ) + (u u t ) R(u u t ) Minimize the stage cost summed over a chosen control horizon (number of moves into the future: N) L(x, u) N 1 min u i=0 subject to the process model x k+1 = Ax k + Bu k Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
24 Current practice: RTO Preliminaries Implementation strategies Planning and Scheduling Real time optimization Steady State Optimization Validation Model Update Reconciliation Controllers Plant Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
25 Current practice: RTO Preliminaries Implementation strategies Planning and Scheduling Steady State Optimization Validation Model Update Reconciliation Real time optimization Two layer structure used to address economically optimal solution RTO generated setpoints passed to lower level controller Controllers try to track the targets provided to it Controllers Plant Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
26 Current practice: RTO Preliminaries Implementation strategies Planning and Scheduling Steady State Optimization Validation Controllers Plant Model Update Reconciliation Real time optimization Two layer structure used to address economically optimal solution RTO generated setpoints passed to lower level controller Controllers try to track the targets provided to it Drawbacks Lower sampling rate Adaptation of operating conditions is slow Consequence: Loss in economics Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
27 Motivating the idea Motivating the idea Profit Input (u) State (x) Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
28 Motivating the idea Motivating the idea Profit Input (u) State (x) Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
29 Motivating the idea Profit ($) Maximum profit $ Profit Global economic optimum not being a steady state introduces high potential areas of transient operation Translation of economic objective to control objective loses the information about maximum profit possible Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
30 Motivating the idea Motivating the idea What is not the primary objective of feedback control Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
31 Motivating the idea Motivating the idea What is not the primary objective of feedback control Tracking setpoints or targets Tracking dynamic setpoint changes Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
32 Motivating the idea Motivating the idea What is not the primary objective of feedback control Tracking setpoints or targets Tracking dynamic setpoint changes Setpoints/Targets: Translating economic objectives to process control objectives Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
33 Motivating the idea Motivating the idea What is not the primary objective of feedback control Tracking setpoints or targets Tracking dynamic setpoint changes Setpoints/Targets: Translating economic objectives to process control objectives Process Economics Steady state economics Loss of economic information due to two layer approach Control objective: L(x, u) = (x x t ) Q(x x t ) +(u u t ) R(u u t ) Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
34 Motivating the idea Motivating the idea What is not the primary objective of feedback control Tracking setpoints or targets Tracking dynamic setpoint changes Setpoints/Targets: Translating economic objectives to process control objectives Process Economics Loss of economic information due to two layer approach Control objective: L(x, u) = P(x, u) Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
35 Motivating the idea Make money or chase target? Due to disturbances and constraints, the economic optimum is not a steady state in general System stabilizes at the steady target estimated from the steady state optimization During system transients, system may or may not pass through the economic optimum Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
36 Motivating the idea The contest The closer the system gets to the economic optimum, the more profitable it is Who gets closest to the global economic optimum? Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
37 Motivating the idea The contest The closer the system gets to the economic optimum, the more profitable it is Who gets closest to the global economic optimum? Tracking controllers: Rush to the target (away from non steady economic optimum) Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
38 Motivating the idea The contest The closer the system gets to the economic optimum, the more profitable it is Who gets closest to the global economic optimum? Tracking controllers: Rush to the target (away from non steady economic optimum) Tracking speed chosen through penalties, but still the objective remains to drive away from non steady economic optimum! Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
39 Motivating the idea The contest The closer the system gets to the economic optimum, the more profitable it is Who gets closest to the global economic optimum? Tracking controllers: Rush to the target (away from non steady economic optimum) Tracking speed chosen through penalties, but still the objective remains to drive away from non steady economic optimum! Economics optimizing controller: Expected to get closer to the optimum with eventual setting at the steady target Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
40 A motivating formulation Current work Quadratic economics Consider a CSTR A V dc A dt V dc B dt = F (C Af C A ) kc A V = F (C Bf FC B ) + kc A V F, C Af V States: C A, C B Input: F A B A, B Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
41 A motivating formulation Current work Quadratic economics Consider a CSTR A F, C Af V V dc A dt V dc B dt = F (C Af C A ) kc A V = F (C Bf FC B ) + kc A V States: C A, C B Input: F The simplest form of profit: A B A, B P = α A F (C A C Af ) + α B F (C B ) = [ ] [ ] α C A C A B F α α A C Af F B α A : Cost of A α B : Cost of B State-Input Cross term! Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
42 Current work SISO Example Example: Single input single output Consider a linear system x k+1 = 0.3x k + u k Profit function: 3xk 2 5u2 k 2x ku k + 98x k + 80u k Objective: Maximize Profit! Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
43 Current work SISO Example Example: Single input single output Consider a linear system x k+1 = 0.3x k + u k Profit function: 3x 2 k 5u2 k 2x ku k + 98x k + 80u k Objective: Maximize Profit! Scheme one: Evaluate the best economic target at every sample time (RTO) Controller tracks the target given to it Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
44 Current work SISO Example Cost contours Steady state line State Input Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
45 Current work SISO Example Cost contours Steady state line State Tracking contours Input Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
46 Current work SISO Example Cost contours Steady state line targ-mpc State Tracking contours Input Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
47 Current work SISO Example Profit function: 3x 2 k 5u2 k 2x ku k + 98x k + 80u k Objective: Maximize Profit! Scheme two: Controller minimizes the negative of profit Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
48 Current work SISO Example Cost contours Steady state line targ-mpc State Tracking contours Input Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
49 Current work SISO Example Cost contours Steady state line targ-mpc eco-mpc State Tracking contours Input Performance Measures targ-mpc eco-mpc (index)% Loss a $642.6 $ a Reference: Maximum profit = 0 Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
50 Current work Effect of disturbance Profit 0-5 p = 0-10 p = 5 p = Input Disturbance model: x k+1 = Ax k + Bu k + B d p k Disturbance shifts the steady state cost curve The steady state target changes System transients from previous target to the new target Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
51 Current work Effect of disturbance Profit 0-5 p = 0-10 p = 5 p = Input Disturbance model: x k+1 = Ax k + Bu k + B d p k Disturbance shifts the steady state cost curve The steady state target changes System transients from previous target to the new target Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
52 Current work Effect of disturbance 8 6 State 4 2 p = Input Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
53 Current work Effect of disturbance 8 6 p = 5 State 4 2 p = Input Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
54 Current work Effect of disturbance 8 6 p = 5 targ-mpc State 4 2 p = Input Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
55 Current work Effect of disturbance 8 6 p = 5 targ-mpc eco-mpc State 4 2 p = Performance Measures Input targ-mpc eco-mpc (index)% Loss a $ $ a Reference: Maximum profit = 0 Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
56 Current work Effect of disturbance Random disturbance corrupts state evolution All states assumed measured Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
57 Current work Effect of disturbance State Input Random disturbance corrupts state evolution All states assumed measured targ-mpc eco-mpc eco-opt Time targ-mpc eco-mpc Time Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
58 Current work Effect of disturbance targ-mpc eco-mpc (index)% Loss a $ $ a Reference: Maximum profit = 0 Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
59 Maximum throughput Current work Linear economics Consider a typical profit function for the plant: ( L) = j p Pj P j i p Fi F i k p Qk Q k P j : Product flows F i : Feed flows Q k : Utility duties Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
60 Maximum throughput Current work Linear economics Consider a typical profit function for the plant: ( L) = j p Pj P j i p Fi F i k p Qk Q k P j : Product flows F i : Feed flows Q k : Utility duties Assume all feed flows set in proportion to throughput (F ), constant efficiency in the units and constant intensive variables F i = k F,i F P j = k P,j F Q k = k Q,k F Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
61 Maximum throughput Current work Linear economics Consider a typical profit function for the plant: ( L) = j p Pj P j i p Fi F i k p Qk Q k P j : Product flows F i : Feed flows Q k : Utility duties Assume all feed flows set in proportion to throughput (F ), constant efficiency in the units and constant intensive variables F i = k F,i F P j = k P,j F Q k = k Q,k F ( L) = j p Pj k P,j i p Fi k F,i k p Qk k Q,k F = pf p: operational profit per unit feed F processed Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
62 Current work Linear economics Economic optimum Maximizing throughput Linear economics: Unconstrained problem unbounded Constrained problem: Optimal solution lies on the process bounds 10 9 Steady state line 8 Linear economic contours 7 State Input Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
63 Current work Linear economics Economic optimum Maximizing throughput Linear economics: Unconstrained problem unbounded Constrained problem: Optimal solution lies on the process bounds 10 9 Steady state line 8 Linear economic contours 7 State Input Tracking contours Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
64 Current work Linear economics Economic optimum Maximizing throughput Linear economics: Unconstrained problem unbounded Constrained problem: Optimal solution lies on the process bounds 10 9 Steady state line targ-mpc eco-mpc 8 Linear economic contours 7 State Input Tracking contours Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
65 Example Current work Example: Transient to steady state x k+1 = [ ] [ ] x k + u k Input constraint: 1 u 1 L eco = α x + β u α = [ 3 2 ] β = 2 L track = x x 2 Q + u u 2 R Q = 2I 2 R = 2 x = [ 60 0 ] u = 1 Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
66 Current work Example: Transient to steady state x targ-mpc x 1 Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
67 Current work Example: Transient to steady state x targ-mpc eco-mpc x 1 Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
68 Current work Example: Transient to steady state State targ-mpc State targ-mpc Input targ-mpc Time Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
69 Current work Example: Transient to steady state State State targ-mpc eco-mpc targ-mpc eco-mpc Input targ-mpc eco-mpc Time Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
70 Current work Example: Effect of disturbance Example: Transient to steady state Random disturbance affecting the state evolution All states assumed measured System started at the steady optimum with zero disturbance Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
71 Current work Effect of disturbance State State targ-mpc targ-mpc Input targ-mpc Time Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
72 Current work Effect of disturbance State State targ-mpc eco-mpc targ-mpc eco-mpc Input targ-mpc eco-mpc Time Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
73 Future work Future work Theoretical Issues Investigate economic models Presented idea banks on a good economic measure Translation of objectives needs deep investigation Need to define a good representative of the process economics Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
74 Future work Future work Theoretical Issues Investigate economic models Presented idea banks on a good economic measure Translation of objectives needs deep investigation Need to define a good representative of the process economics Establish asymptotic stability and convergence properties for broader class of cost functions Steady state cost maybe nonzero = Infinite horizon cost is unbounded Costs corresponding to the optimal input sequence may not be monotonically decreasing Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
75 Future work Software development Update the software tools to handle the new class of problems Efficient software tools critical to the evaluation of the new class of problems The existing tools handle quadratic objective functions Economics may not be quadratic and hence the tools have to be capable of handling more general cost functions Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
76 Future work Set up the problem for a realistic scenario and test using industrial data Simulations, like the ones shown, just predict the possible advantages of the new scheme The idea must be tested for a physical system with well defined economics Collaborate for the distributed version Distributed control schemes allow more robust and flexible control The new scheme can be implemented in distributed scenario Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
77 Conclusions Conclusions Profit depends heavily on steady state economic optimization layer Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
78 Conclusions Conclusions Profit depends heavily on steady state economic optimization layer A separate layer causes a loss in economic performance during transient Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
79 Conclusions Conclusions Profit depends heavily on steady state economic optimization layer A separate layer causes a loss in economic performance during transient Opportunity to rethink distribution of functionality between layers Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
80 Conclusions Conclusions Profit depends heavily on steady state economic optimization layer A separate layer causes a loss in economic performance during transient Opportunity to rethink distribution of functionality between layers Merging the economics with the controller objective reduces the loss of economic information Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
81 Conclusions Conclusions Profit depends heavily on steady state economic optimization layer A separate layer causes a loss in economic performance during transient Opportunity to rethink distribution of functionality between layers Merging the economics with the controller objective reduces the loss of economic information Economic optimizing control expected to capture the potential profitable areas of operation Rishi Amrit (UW-Madison) Economic Optimization in MPC 29 th February, / 37
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