Your SlideShare is downloading. ×
0
Process Optimization Mathematical Programming and Optimization of Multi-Plant Operations and Process Design Ralph W. Pike ...
Process Optimization <ul><li>Typical Industrial Problems </li></ul><ul><li>Mathematical Programming Software </li></ul><ul...
New Results <ul><li>Using one computer language to write and run a program in another language </li></ul><ul><li>Cumulativ...
Design vs. Operations <ul><li>Optimal Design </li></ul><ul><li>  − Uses flowsheet simulators and SQP </li></ul><ul><ul><li...
Plant Problem Size Contact  Alkylation  Ethylene 3,200 TPD 15,000 BPD 200 million lb/yr Units     14   76   ~200 Streams  ...
Optimization Programming Languages <ul><li>GAMS -  G eneral  A lgebraic  M odeling  S ystem </li></ul><ul><li>LINDO - Wide...
Software with Optimization Capabilities <ul><li>Excel – Solver </li></ul><ul><li>MATLAB </li></ul><ul><li>MathCAD </li></u...
Mathematical Programming <ul><li>Using Excel – Solver </li></ul><ul><li>Using GAMS </li></ul><ul><li>Mathematical Basis fo...
Simple Chemical Process P – reactor pressure R – recycle ratio
Excel Solver Example Showing the equations in the Excel cells with initial values for P and R Solver optimal solution
Excel Solver Example
Excel Solver Example Not  Not the minimum for C
Excel Solver Example Use Solver with these values of P and R
Excel Solver Example optimum  Click to highlight to generate reports
Excel Solver Example Information from Solver Help is of limited value
Excel Solver Answer Report management report format constraint status slack variable   values at the optimum
Excel Sensitivity Report Solver uses the generalized reduced gradient optimization algorithm Lagrange multipliers used for...
Excel Solver Limits Report Sensitivity Analysis provides limits on variables for the optimal solution to remain optimal
GAMS
GAMS S O L V E  S U M M A R Y MODEL  Recycle  OBJECTIVE  Z TYPE  NLP  DIRECTION  MINIMIZE SOLVER  CONOPT  FROM LINE  18 **...
GAMS <ul><li>LOWER  LEVEL  UPPER  MARGINAL </li></ul><ul><li>---- EQU CON1  9000.000  9000.000  9000.000  117.284  </li></...
GAMS Solvers 13 types of  optimization problems LP - Linear Programming linear economic model and linear constraints NLP –...
GAMS Solvers 32 Solvers new global optimizer DICOPT One of several MINLP optimizers MINOS a sophisticated NLP optimizer  d...
Mathematical Basis for Optimization is the Kuhn Tucker Necessary Conditions General Statement of a Mathematical Programmin...
Kuhn Tucker Necessary Conditions x n+i   are the slack variables used to convert  the inequality constraints to equalities...
Kuhn Tucker Necessary Conditions Necessary conditions for a relative minimum at  x *
Lagrange Multipliers <ul><li>Treated  as an: </li></ul><ul><li>Undetermined multiplier  – multiply constraints by  λ i  an...
Lagrange Multipliers optimize:  y(x 1 , x 2 ) subject to:  f(x 1 , x 2 )  =  0
Lagrange Multipliers Rearrange the partial derivatives in the second term
Lagrange Multipliers Call the ratio of partial derivatives in the (  ) a Lagrange multiplier,  λ Lagrange multipliers are ...
Lagrange Multipliers Define L = y + λ f , an unconstrained function and by the same procedure Interpret  L as an unconstra...
Lagrange Multipliers Optimize:  y(x 1 ,x 2 ) Subject to:  f(x 1 ,x 2 ) = b Manipulations give: ∂ y   =  -  λ   ∂ b Extends...
Geometric Representation of an LP Problem max: 3A + 4B = P s.t.  4A + 2B  <   80 2A + 5B  <  120 Maximum at vertex P = 110...
LP Example <ul><li>Maximize:  </li></ul><ul><li>    x 1 + 2x 2                 =   P </li></ul><ul><li>Subject to:      </...
Final Step in Simplex Algorithm <ul><li>Maximize:             - 3/2 x 4  - 1/2 x 5          =  P - 10   P = 10  </li></ul>...
Lagrange Multiplier Formulation <ul><li>Returning to the original problem </li></ul><ul><li>Max:  (1+2 λ 1 +  λ 2   -  λ 3...
Lagrange Multiplier Interpretation <ul><li>Maximize: 0x 1  +0x 2   +0 x 3  - 3/2  x 4  - 1/2  x 5   +0x 6   =  P - 10    P...
  General Statement of the Linear Programming Problem   Objective Function : Maximize:      c 1 x 1  + c 2 x 2  + ... + c ...
LP Problem with Lagrange Multiplier Formulation Multiply each constraint equation, (4-1b), by the Lagrange multiplier  λ i...
Sensitivity Analysis <ul><li>Use the results from the final step in the simplex method to determine the range on the varia...
Nonlinear Programming Three standard methods –  all use the same information Successive Linear Programming Successive Quad...
Generalized Reduced Gradient Direction Reduced Gradient Line Specifies how to change x nb  to have the largest change in y...
Generalized Reduced Gradient Algorithm Minimize:   y(x) =  y( x)    Y [x k,nb  +  α   Y(x k )] = Y( α ) Subject to:  f i (...
Generalized Reduced Gradient Trajectory Minimize : -2x 1  - 4x 2  + x 1 2  + x 2 2  + 5  Subject to: - x 1  + 2x 2     <  ...
On-Line Optimization <ul><li>Automatically adjust operating conditions with the plant’s distributed control system </li></...
 
 
 
 
 
 
Data Reconciliation Heat  Exchanger Chemical  Reactor y 1  730 kg/hr x 1 y 2 718 kg/hr x 2 y 3 736 kg/hr x 3 Material Bala...
Data Reconciliation y 1  730 kg/hr x 1 y 3 736 kg/hr x 3 Heat  Exchanger Chemical  Reactor y 1  730 kg/hr x 1 y 2 718 kg/h...
Data Reconciliation using Least Squares Analytical solution using LaGrange Multipliers Q  = diag[  i ]
Data Reconciliation Measurements having only random errors - least squares
 
 
 
 
 
 
 
Key Elements of On-Line Optimization
 
 
 
Mosaic-Monsanto Sulfuric Acid Plant 3,200 tons per day of 93% Sulfuric Acid, Convent, Louisiana
Motiva Refinery Alkylation Plant 15,000 barrels per day, Convent, Louisiana, reactor section, 4 Stratco reactors
Steady State Detection Execution frequency must be greater than the plant settling time (time to return to steady state).
On-Line Optimization - Distributed Control System Interface Plant must at steady state when data extracted from DCS and wh...
 
 
Mixed Integer Programming Numerous Applications Batch Processing Pinch Analysis Optimal Flowsheet Structure Branch and Bou...
Mixed Integer Process Example Produce C from either Process 2 or Process 3 Make B from A in Process 1 or purchase B
Mixed Integer Process Example   operating cost   fixed cost feed cost sales max: -250 F 1 A  - 400 F 6 B  - 550 F 7 B  - 1...
Branch and Bound Algorithm LP Relaxation Solution Max:  5x 1  + 2x 2  =P P = 22.5 Subject to:   x 1  + x 2   <  4.5 x 1   ...
Branch and Bound Algorithm Max:  5x 1  + 2x 2  =P   Max:  5x 1  + 2x 2  =P Subject to:    x 1  + x 2   <  4.5 Subject to: ...
Branch and Bound Algorithm Max:  5x 1  + 2x 2  =P   Max:  5x 1  + 2x 2  =P Subject to:   x 1  + x 2   <  4.5   Subject to:...
Branch and Bound Algorithm 22.5 Infeasible 21.0 19.5 20 LP relaxation solution X 1  = 4.5 X 2  = 0 X 1   >  5 X 1   <  4 X...
Branch and Bound Algorithm inf 13.7 17 Inf 12 16.8 inf 15.6 17.4 15 inf Integer solution Integer solution – optimal solution
Mixed Integer Nonlinear Programming Flow Chart of GBD Algorithm to Solve MINPL Problems ,  Duran and Grossmann, 1986, Math...
Triple Bottom Line Triple Bottom Line =    Product Sales    -  Manufacturing Costs (raw materials,  energy costs, others) ...
Optimization of Chemical Production Complexes <ul><li>Opportunity </li></ul><ul><ul><li>New processes for conversion of su...
Plants in the lower Mississippi River Corridor Source: Peterson, R.W., 2000
Some Chemical Complexes in the World <ul><li>North America </li></ul><ul><ul><li>Gulf coast petrochemical complex in Houst...
Plants in the lower Mississippi River Corridor, Base Case. Flow Rates in Million Tons Per Year
Commercial Uses of CO 2 <ul><li>Chemical synthesis in the U. S. consumes 110 million m tons per year of CO 2   </li></ul><...
Surplus Carbon Dioxide <ul><li>Ammonia plants produce 0.75 million tons per year in lower Mississippi River corridor. </li...
Greenhouse Gases as Raw Material From Creutz and Fujita, 2000
Some Catalytic Reactions of CO 2 Hydrogenation Hydrolysis and Photocatalytic Reduction CO 2  + 3H 2     CH 3 OH + H 2 O  ...
Methodology for Chemical Complex Optimization with New Carbon Dioxide Processes <ul><li>Identify potentially new processes...
Twenty Processes Selected for HYSYS Design <ul><li>Chemical Synthesis Route Reference </li></ul><ul><li>Methanol CO2 hydro...
Integration into Superstructure <ul><li>Twenty processes simulated </li></ul><ul><li>Fourteen processes selected based on ...
New Processes Included in Chemical Production Complex <ul><li>Product Synthesis Route Value Added Profit (cents/kg) </li><...
Application of the Chemical Complex Analysis System to Chemical Complex in the Lower Mississippi River Corridor <ul><li>Ba...
Chemical Complex Analysis System
Plants in the lower Mississippi River Corridor, Base Case. Flow Rates in Million Tons Per Year
Superstructure
Plants in the Superstructure <ul><li>Plants in the Base Case </li></ul><ul><li>Ammonia </li></ul><ul><li>Nitric acid </li>...
Superstructure Characteristics <ul><li>Options </li></ul><ul><li>-  Three options for producing phosphoric acid  </li></ul...
Some of the Raw Material Costs, Product Prices and Sustainability Cost and Credits Raw Materials Cost Sustainable Cost and...
Optimal Structure
Plants in the Optimal Structure from the Superstructure
Comparison of the Triple Bottom Line for the Base Case and Optimal Structure
Carbon Dioxide Consumption in Bases Case  and Optimal Structure   All of the carbon dioxide was not consumed in the optima...
Multi-Criteria or Multi-Objective Optimization Subject to: f i (x) = 0 min: cost max: reliability min: waste generation ma...
Multi-Criteria Optimization - Weighting Objectives Method Subject to: f i (x) = 0 with  ∑ w i   =  1 Optimization with a s...
Multicriteria Optimization   P= Σ Product Sales - Σ Manufacturing Costs  - Σ Environmental Costs max:   S = Σ Sustainable ...
Multicriteria Optimization Convert to a single criterion optimization problem max:  w 1 P + w 2  S  subject to: Multi-plan...
Multicriteria Optimization W 1  =1 only profit W 1  = 0 no profit Debate about which weights are best
Monte Carlo Simulation <ul><ul><li>Used to determine the sensitivity of the optimal solution to the costs and prices used ...
Monte Carlo Simulation Triple Bottom Line Mean $513million per year Standard deviation - $109 million per year  50% probab...
Conclusions ●  The optimum configuration of plants in a chemical production complex was determined based on the triple bot...
Transition from Fossil Raw Materials to Renewables   Introduction of ethanol into the ethylene product chain. Ethanol can ...
Global Optimization Locate the global optimum of a mixed integer nonlinear programming problem  directly. Branch and bound...
Acknowledgements Collaborators Dean Jack R. Hopper Professor Helen H. Lou Professor Carl L. Yaws Graduate Students Industr...
www.mpri.lsu.edu
Upcoming SlideShare
Loading in...5
×

4. Process Optimization

2,803

Published on

0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
2,803
On Slideshare
0
From Embeds
0
Number of Embeds
1
Actions
Shares
0
Downloads
108
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide
  • To determine the optimum configuration of plants in the chemical complex, a value-added economic model was expanded to account for environmental and sustainable costs. This extended value-added economic model is referred to as the “triple bottom line” and is the difference between sales and economic costs (raw materials and utilities), environmental costs and sustainable credits/costs as given in the equation.
  • There are opportunities to use carbon dioxide as raw materials in new, energy-efficient processes. The Chemical Complex Analysis System is a methodology and computer program that has been used evaluate designs of plants that converts greenhouse gases into new products and integrate them into existing chemical production complexes. This technology is being applied to the chemical production complex in the lower Mississippi River corridor.
  • This map shows the location of about 150 plants in the lower Mississippi River corridor. These plants that consume about 1.0 quad (1x1015 Btu/yr) of energy and generate about 215 million pounds of pollutants annually.
  • The system not only can be applied to the chemical complex in the lower Mississippi River corridor, but can be applied to other places in the world. Gulf coast petrochemical complex in Houston area (U.S.A.) is the largest petrochemical complex in the world, supplying nearly two-thirds of the nation’s petrochemical needs
  • A base case of existing plants was developed with the assistance of industrial collaborators. It includes sources and consumers of carbon dioxide. This is the process flow diagram for these existing plants in the lower Mississippi River Corridor that make up a chemical production complex. Each block represents several plants. For example, the sulfuric acid production unit contains five plants owned by two companies. There are thirteen production units plus associated utilities for power, steam and cooling water and facilities for waste treatment.
  • This slide shows the commercial use of CO2 and illustrates that it is an important industrial chemical.
  • In this complex, ammonia plants produce 0.75 million tons per year, and methanol and urea plants consume 0.14 million tons per year of this carbon dioxide. The 0.61 million tons per year of surplus high purity carbon dioxide is exhausted to atmosphere. This excess carbon dioxide is available in pipelines that can be sent to new plants that use carbon dioxide as a raw material for new products.
  • This information categorizes the carbon dioxide reactions that produce industrially important products. Hydrogenation reactions produce alcohols, hydrocarbon synthesis reactions produce paraffins and olefins, and amine synthesis produce methyl and higher order amines. Hydrolysis reactions can produce alcohols and organic acids. Carbon dioxide serves as an oxygen source in the ethylbenzene to styrene reaction. Propane dehydrogenation can provide hydrogen required for some of these reactions. Recently, BASF has started up a 350,000 metric ton per year plant for propylene in Sonatrac PropanChem S.A., located at Tarragona, Spain.
  • This slide describes the overall methodology. After new processes were evaluated, they were designed using HYSYS. Energy requirements were estimated from the HYSYS process flow diagram A value added economic model was evaluated Processes were selected to be incorporated in the superstructure for optimization.
  • This slide describes the overall methodology. After new processes were evaluated, they were designed using HYSYS. Energy requirements were estimated from the HYSYS process flow diagram A value added economic model was evaluated Processes were selected to be incorporated in the superstructure for optimization
  • This slide gives the references for these processes. All of the details are given in Sudheer Indala’s thesis
  • From the twenty HYSYS simulated processes, fourteen new processes were selected based on the value added profit. These processes were incorporated into the superstructure using Chemical Complex Analysis System.
  • This slide lists the processes included in the complex with their value added profit These processes include methanol (4), ethanol, dimethyl ether, formic acid, acetic acid, styrene, methylamines, graphite, synthesis gas and propylene (2).
  • The base case of existing plants was expanded into a superstructure with the new plants. Then the optimal configuration of plants was determined by solving amixed integer nonlinear programming problem.
  • A base case of existing plants was developed with the assistance of industrial collaborators. It includes sources and consumers of carbon dioxide. This is the process flow diagram for these existing plants in the lower Mississippi River Corridor that make up a chemical production complex. Each block represents several plants. For example, the sulfuric acid production unit contains five plants owned by two companies. There are thirteen production units plus associated utilities for power, steam and cooling water and facilities for waste treatment.
  • This diagram shows Superstructure that was developed by adding more processes. I am sorry that this slide is too detailed to be clearly seen. The detailed information is in Sudheer Indala’s thesis. These plants are summarized on the next slide.
  • This shows the plants in the base case and the plants added in the superstructure. There are 14 plants that use carbon dioxide as a raw material and 4 that are potentially new plants for other chemicals in the complex.
  • This slide summarizes the options incorporated in the superstructure. Also, it gives the size of the mixed integer nonlinear programming problem.
  • This table gives some of the sale prices for products and costs of raw material that were used in the economic model of the complex. All of them are given in Sudheer Indala’s thesis. Also shown are sustainable costs and credits. Sustainable costs were from the AIChE/CWRT report where carbon dioxide emissions had a sustainable cost of $3.25 per ton of carbon dioxide. A cost of $3.25 per ton was charged as a cost to plants that emit carbon dioxide, and plants that consume carbon dioxide were given a credit of twice this cost or $6.50 per ton. These costs are arbitrary but conservative. Emissions trading cost of carbon dioxide is about $50.00 per ton.
  • This slide gives the diagram of the optimal configuration of plants obtained from the superstructure. I am sorry that this slide is too detailed to be clearly seen. The detailed information is in Amin Xu’s dissertation. The next slide gives a summary of the results.
  • Seven new processes in the optimal structure were selected from eighteen new processes in the superstructure. These included acetic acid, graphite, formic acid, methylamines, propylene (2) and synthesis gas production. The new acetic acid process replaced the commercial acetic acid plant in the chemical complex. The processes for dimethyl ether, styrene, and methanol were not selected in the optimal structure. It was more profitable to have the corresponding commercial processes present. The commercial process for methanol does not use expensive hydrogen as a raw material, but the new methanol processes does.
  • Comparison of the sales and costs associated with the triple bottom line are shown here for the base case and the optimal structure. The triple bottom line profit increased from $412 to $574 million dollars per year or about 40% from the base case to the optimal solution. Sales increased from additional products from carbon dioxide, and there were corresponding increases in the other costs associated with producing these products by the companies. Cost to society improved since sustainable credits increased from $21 to $24 million dollars per year from the credits given for using carbon dioxide and increased energy efficiency.
  • The increased use of carbon dioxide is shown here for the optimal structure. However, it was not optimal to consume all of the carbon dioxide available, and 0.24 million metric tons per year is vented to the atmosphere, down from 0.61 million metric tons per year or 61%.
  • The objective is to find optimal solutions that maximize companies’ profits and minimize costs to society. Company’s profits are sales minus economic and environmental costs. The costs to society are measured by sustainable costs. Sustainable credits are awarded for reductions in emissions. Multi-criteria optimization obtains solutions that are called efficient or Pareto optimal solutions. These are optimal points where attempting to improving the value of one objective would cause another objective to decrease.
  • To locate Pareto optimal solutions, multi-criteria optimization problems are converted to one with a single criterion by either of two methods. One method is by applying weights the each objective and optimizing the sum of the weighted objectives as shown here.
  • The Chemical Complex Analysis System was used to determine the Pareto optimal solutions for the weights using w1 + w2 = 1. The results are shown here for increments of 0.001 for w1. Company profits are an order of magnitude larger than sustainability credits, and sustainability credits decline and company profits increase as the weight on company profits increase. It is another decision to determine the weight that is acceptable to all concerned
  • Monte Carlo simulation is used to determine the sensitivity of the optimal solution to the costs and prices. One of the results is the cumulative probability distribution, a curve of the cumulative probability as a function of the triple bottom line. This curve is used to determine the upside and downside risks. For a Monte Carlo simulation, mean prices and costs along with an estimate of their standard deviations are required. The standard deviations were estimated from cost and price fluctuations from the sources used for costs and prices.
  • Monte Carlo simulation was used with 1,000 iterations to obtain the cumulative probability distribution shown here for the profit. The lines on the diagram show that there is 60% probability that the profit is equal to or less than $310 million/year. Sustainable costs and credits were constant, and sensitivity to these values is to be determined in a subsequent evaluation
  • This slide summarizes the results of the work to date. The capabilities of the Chemical Complex Analysis System have been demonstrated by determining the optimal configuration of units based on economic, environmental and sustainable costs. This methodology could be applied to other chemical complexes in the world for reduced emissions and energy savings.
  • are proceeding to apply the methodology developed for carbon dioxide processes to processes for carbon nanotubes. These potentially new processes are operate at high temperature, are energy intensive and generate hazardous and toxic wastes
  • Transcript of "4. Process Optimization"

    1. 1. Process Optimization Mathematical Programming and Optimization of Multi-Plant Operations and Process Design Ralph W. Pike Director, Minerals Processing Research Institute Horton Professor of Chemical Engineering Louisiana State University Department of Chemical Engineering, Lamar University, April, 10, 2007
    2. 2. Process Optimization <ul><li>Typical Industrial Problems </li></ul><ul><li>Mathematical Programming Software </li></ul><ul><li>Mathematical Basis for Optimization </li></ul><ul><li>Lagrange Multipliers and the Simplex Algorithm </li></ul><ul><li>Generalized Reduced Gradient Algorithm </li></ul><ul><li>On-Line Optimization </li></ul><ul><li>Mixed Integer Programming and the Branch and Bound Algorithm </li></ul><ul><li>Chemical Production Complex Optimization </li></ul>
    3. 3. New Results <ul><li>Using one computer language to write and run a program in another language </li></ul><ul><li>Cumulative probability distribution instead of an optimal point using Monte Carlo simulation for a multi-criteria, mixed integer nonlinear programming problem </li></ul><ul><li>Global optimization </li></ul>
    4. 4. Design vs. Operations <ul><li>Optimal Design </li></ul><ul><li> − Uses flowsheet simulators and SQP </li></ul><ul><ul><li>Heuristics for a design, a superstructure, an optimal design </li></ul></ul><ul><li>Optimal Operations </li></ul><ul><ul><li>On-line optimization </li></ul></ul><ul><ul><li>Plant optimal scheduling </li></ul></ul><ul><ul><li>Corporate supply chain optimization </li></ul></ul>
    5. 5. Plant Problem Size Contact Alkylation Ethylene 3,200 TPD 15,000 BPD 200 million lb/yr Units 14 76 ~200 Streams 35 110 ~4,000 Constraints Equality 761 1,579 ~400,000 Inequality 28 50 ~10,000 Variables Measured 43 125 ~300 Unmeasured 732 1,509 ~10,000 Parameters 11 64 ~100
    6. 6. Optimization Programming Languages <ul><li>GAMS - G eneral A lgebraic M odeling S ystem </li></ul><ul><li>LINDO - Widely used in business applications </li></ul><ul><li>AMPL - A M athematical P rogramming L anguage </li></ul><ul><li>Others: MPL, ILOG </li></ul><ul><li>optimization program is written in the form of an </li></ul><ul><li>optimization problem </li></ul><ul><li>optimize: y( x ) economic model </li></ul><ul><li>subject to: f i ( x ) = 0 constraints </li></ul>complete
    7. 7. Software with Optimization Capabilities <ul><li>Excel – Solver </li></ul><ul><li>MATLAB </li></ul><ul><li>MathCAD </li></ul><ul><li>Mathematica </li></ul><ul><li>Maple </li></ul><ul><li>Others </li></ul>
    8. 8. Mathematical Programming <ul><li>Using Excel – Solver </li></ul><ul><li>Using GAMS </li></ul><ul><li>Mathematical Basis for Optimization </li></ul><ul><li>Important Algorithms </li></ul><ul><ul><li>Simplex Method and Lagrange Multipliers </li></ul></ul><ul><ul><li>Generalized Reduced Gradient Algorithm </li></ul></ul><ul><ul><li>Branch and Bound Algorithm </li></ul></ul>
    9. 9. Simple Chemical Process P – reactor pressure R – recycle ratio
    10. 10. Excel Solver Example Showing the equations in the Excel cells with initial values for P and R Solver optimal solution
    11. 11. Excel Solver Example
    12. 12. Excel Solver Example Not Not the minimum for C
    13. 13. Excel Solver Example Use Solver with these values of P and R
    14. 14. Excel Solver Example optimum Click to highlight to generate reports
    15. 15. Excel Solver Example Information from Solver Help is of limited value
    16. 16. Excel Solver Answer Report management report format constraint status slack variable values at the optimum
    17. 17. Excel Sensitivity Report Solver uses the generalized reduced gradient optimization algorithm Lagrange multipliers used for sensitivity analysis Shadow prices ($ per unit)
    18. 18. Excel Solver Limits Report Sensitivity Analysis provides limits on variables for the optimal solution to remain optimal
    19. 19. GAMS
    20. 20. GAMS S O L V E S U M M A R Y MODEL Recycle OBJECTIVE Z TYPE NLP DIRECTION MINIMIZE SOLVER CONOPT FROM LINE 18 **** SOLVER STATUS 1 NORMAL COMPLETION **** MODEL STATUS 2 LOCALLY OPTIMAL **** OBJECTIVE VALUE 3444444.4444 RESOURCE USAGE, LIMIT 0.016 1000.000 ITERATION COUNT, LIMIT 14 10000 EVALUATION ERRORS 0 0 C O N O P T 3 x86/MS Windows version 3.14P-016-057 Copyright (C) ARKI Consulting and Development A/S Bagsvaerdvej 246 A DK-2880 Bagsvaerd, Denmark Using default options. The model has 3 variables and 2 constraints with 5 Jacobian elements, 4 of which are nonlinear. The Hessian of the Lagrangian has 2 elements on the diagonal, 1 elements below the diagonal, and 2 nonlinear variables. ** Optimal solution. Reduced gradient less than tolerance.
    21. 21. GAMS <ul><li>LOWER LEVEL UPPER MARGINAL </li></ul><ul><li>---- EQU CON1 9000.000 9000.000 9000.000 117.284 </li></ul><ul><li>---- EQU OBJ . . . 1.000 </li></ul><ul><li>LOWER LEVEL UPPER MARGINAL </li></ul><ul><li>---- VAR P 1.000 1500.000 +INF . </li></ul><ul><li>---- VAR R 1.000 6.000 +INF EPS </li></ul><ul><li>---- VAR Z -INF 3.4444E+6 +INF . </li></ul><ul><li>**** REPORT SUMMARY : 0 NONOPT </li></ul><ul><li>0 INFEASIBLE </li></ul><ul><li>0 UNBOUNDED </li></ul><ul><li>0 ERRORS </li></ul>Lagrange multiplier values at the optimum 900 page Users Manual
    22. 22. GAMS Solvers 13 types of optimization problems LP - Linear Programming linear economic model and linear constraints NLP – Nonlinear Programming nonlinear economic model and nonlinear constraints MIP - Mixed Integer Programming nonlinear economic model and nonlinear constraints with continuous and integer variables
    23. 23. GAMS Solvers 32 Solvers new global optimizer DICOPT One of several MINLP optimizers MINOS a sophisticated NLP optimizer developed at Stanford OR Dept uses GRG and SLP
    24. 24. Mathematical Basis for Optimization is the Kuhn Tucker Necessary Conditions General Statement of a Mathematical Programming Problem Minimize: y(x) Subject to: f i (x) <  0 for i = 1, 2, ..., h f i (x) = 0 for i = h+1, ..., m y(x) and f i (x) are twice continuously differentiable real valued functions.
    25. 25. Kuhn Tucker Necessary Conditions x n+i are the slack variables used to convert the inequality constraints to equalities. Lagrange Function – converts constrained problem to an unconstrained one λ i are the Lagrange multipliers
    26. 26. Kuhn Tucker Necessary Conditions Necessary conditions for a relative minimum at x *
    27. 27. Lagrange Multipliers <ul><li>Treated as an: </li></ul><ul><li>Undetermined multiplier – multiply constraints by λ i and add to y( x ) </li></ul><ul><li>Variable - L(x, λ ) </li></ul><ul><li>Constant – numerical value computed at the optimum </li></ul>
    28. 28. Lagrange Multipliers optimize: y(x 1 , x 2 ) subject to: f(x 1 , x 2 ) = 0
    29. 29. Lagrange Multipliers Rearrange the partial derivatives in the second term
    30. 30. Lagrange Multipliers Call the ratio of partial derivatives in the ( ) a Lagrange multiplier, λ Lagrange multipliers are a ratio of partial derivatives at the optimum. ( ) = λ
    31. 31. Lagrange Multipliers Define L = y + λ f , an unconstrained function and by the same procedure Interpret L as an unconstrained function, and the partial derivatives set equal to zero are the necessary conditions for this unconstrained function
    32. 32. Lagrange Multipliers Optimize: y(x 1 ,x 2 ) Subject to: f(x 1 ,x 2 ) = b Manipulations give: ∂ y = - λ ∂ b Extends to: ∂ y = - λ i shadow price ($ per unit of b i ) ∂ b i
    33. 33. Geometric Representation of an LP Problem max: 3A + 4B = P s.t. 4A + 2B < 80 2A + 5B < 120 Maximum at vertex P = 110 A = 10, B = 20 objective function is a plane no interior optimum
    34. 34. LP Example <ul><li>Maximize: </li></ul><ul><li>   x 1 + 2x 2             =   P </li></ul><ul><li>Subject to:     </li></ul><ul><li> 2x 1 +  x 2 + x 3                        = 10 </li></ul><ul><li> x 1 +  x 2          + x 4               = 6 </li></ul><ul><li> -x 1 + x 2                 + x 5         =  2 </li></ul><ul><li>-2x 1 + x 2                        + x 6   =  1 </li></ul><ul><li>4 equations and 6 unknowns, set 2 of the x i =0 and solve for 4 of the x i. </li></ul><ul><li>Basic feasible solution: x 1 = 0, x 2 = 0, x 3 = 10, x 4 = 6, x 5 = 2, x 6 =1 </li></ul><ul><li>Basic solution: x 1 = 0, x 2 = 6, x 3 = 4, x 4 = 0, x 5 = -4, x 6 = -5 </li></ul>
    35. 35. Final Step in Simplex Algorithm <ul><li>Maximize:             - 3/2 x 4 - 1/2 x 5        =  P - 10   P = 10 </li></ul><ul><li>Subject to:   </li></ul><ul><li>             x 3 - 3/2 x 4 + 1/2 x 5       =  2      x 3 = 2 </li></ul><ul><li>                  1/2 x 4 - 3/2 x 5 + x 6   =  1     x 6 = 1 </li></ul><ul><li>x 1            + 1/2 x 4 - 1/2 x 5        =  2      x 1 = 2 </li></ul><ul><li>     x 2     + 1/2 x 4 + 1/2 x 5        =  4      x 2 = 4 </li></ul><ul><li>             x 4 = 0 </li></ul><ul><li>           x 5 = 0 </li></ul><ul><li>Simplex algorithm exchanges variables that are zero with ones that are nonzero, one at a time to arrive at the maximum </li></ul>
    36. 36. Lagrange Multiplier Formulation <ul><li>Returning to the original problem </li></ul><ul><li>Max:  (1+2 λ 1 + λ 2 - λ 3 - 2 λ 4 ) x 1   </li></ul><ul><li>  </li></ul><ul><li> (2+ λ 1 + λ 2 + λ 3 + λ 4 )x 2 + </li></ul><ul><li>   λ 1 x 3 + λ 2 x 4 + λ 3 x 5 + λ 4 x 6   </li></ul><ul><li> - (10 λ 1 + 6 λ 2 + 2 λ 3 + λ 4 ) = L = P </li></ul><ul><li>Set partial derivatives with respect to x 1, x 2 , x 3 , and x 6 equal to zero (x 4 and x 5 are zero) and and solve resulting equations for the Lagrange multipliers </li></ul>
    37. 37. Lagrange Multiplier Interpretation <ul><li>Maximize: 0x 1 +0x 2  +0 x 3 - 3/2 x 4 - 1/2 x 5  +0x 6  =  P - 10    P = 10 </li></ul><ul><li>Subject to:   </li></ul><ul><li>             x 3 - 3/2 x 4 + 1/2 x 5       =  2     x 3 = 2 </li></ul><ul><li>                  1/2 x 4 - 3/2 x 5 + x 6   =  1     x 6 = 1 </li></ul><ul><li>x 1            + 1/2 x 4 - 1/2 x 5        =  2     x 1 = 2 </li></ul><ul><li>     x 2     + 1/2 x 4 + 1/2 x 5          =  4     x 2 = 4 </li></ul><ul><li>             x 4 = 0 </li></ul><ul><li>             x 5 = 0 </li></ul><ul><li>-(10 λ 1 + 6 λ 2 + 2 λ 3 + λ 4 ) = L = P = 10 </li></ul><ul><li>The final step in the simplex algorithm is used to evaluate the Lagrange multipliers. It is the same as the result from analytical methods . </li></ul>(1+2 λ 1 + λ 2 - λ 3 - 2 λ 4 )=0 (2+ λ 1 + λ 2 + λ 3 + λ 4 )=0 λ 1 =0 λ 2 =-3/2 λ 3 =-1/2 λ 4 =0
    38. 38. General Statement of the Linear Programming Problem Objective Function : Maximize:     c 1 x 1 + c 2 x 2 + ... + c n x n   = p                     (4-1a) Constraint Equations : Subject to:  a 11 x 1 + a 12 x 2 + ... + a 1n x n   <   b 1        (4-1b) a 21 x 1 + a 22 x 2 + ... + a 2n x n   <    b 2 . . .                   . . .                   . . . . . .                   . . .                   . . . a m1 x 1 + a m2 x 2 + ... + a mn x n   <   b m x j > 0   for j = 1,2,...n                    (4-1c)
    39. 39. LP Problem with Lagrange Multiplier Formulation Multiply each constraint equation, (4-1b), by the Lagrange multiplier λ i and add to the objective function Have x 1 to x m be values of the variables in the basis, positive numbers Have x m+1 to x n be values of the variables that are not in the basis and are zero. positive in the basis not equal to zero, negative equal to zero not in basis equal to zero from ∂p/∂x m =0 Left hand side = 0 and p = - ∑b i λ i
    40. 40. Sensitivity Analysis <ul><li>Use the results from the final step in the simplex method to determine the range on the variables in the basis where the optimal solution remains optimal for changes in: </li></ul><ul><li>b i availability of raw materials demand for product, capacities of the process units </li></ul><ul><li>c j sales price and costs </li></ul><ul><li>See Optimization for Engineering Systems book for equations at www.mpri.lsu.edu </li></ul>
    41. 41. Nonlinear Programming Three standard methods – all use the same information Successive Linear Programming Successive Quadratic Programming Generalized Reduced Gradient Method Optimize: y( x ) x = (x 1 , x 2 ,…, x n ) Subject to: f i ( x ) =0 for i = 1,2,…,m n>m ∂ y ( x k ) ∂f i ( x k ) evaluate partial derivatives at x k ∂ x j ∂x j
    42. 42. Generalized Reduced Gradient Direction Reduced Gradient Line Specifies how to change x nb to have the largest change in y(x) at x k
    43. 43. Generalized Reduced Gradient Algorithm Minimize: y(x) = y( x) Y [x k,nb + α Y(x k )] = Y( α ) Subject to: f i (x) = 0 (x) = (x b ,x nb ) m basic variables, (n-m) nonbasic variables Reduced Gradient Line Reduced Gradient Newton Raphson Algorithm
    44. 44. Generalized Reduced Gradient Trajectory Minimize : -2x 1 - 4x 2 + x 1 2 + x 2 2 + 5 Subject to: - x 1 + 2x 2   < 2 x 1 +  x 2   < 4
    45. 45. On-Line Optimization <ul><li>Automatically adjust operating conditions with the plant’s distributed control system </li></ul><ul><li>Maintains operations at optimal set points </li></ul><ul><li>Requires the solution of three NLP’s in sequence </li></ul><ul><li> gross error detection and data reconciliation </li></ul><ul><li> parameter estimation </li></ul><ul><li>economic optimization </li></ul><ul><li>BENEFITS </li></ul><ul><li>Improves plant profit by 10% </li></ul><ul><li>Waste generation and energy use are reduced </li></ul><ul><li>Increased understanding of plant operations </li></ul>
    46. 52. Data Reconciliation Heat Exchanger Chemical Reactor y 1 730 kg/hr x 1 y 2 718 kg/hr x 2 y 3 736 kg/hr x 3 Material Balance x 1 = x 2 x 1 - x 2 = 0 Steady State x 2 = x 3 x 2 - x 3 = 0
    47. 53. Data Reconciliation y 1 730 kg/hr x 1 y 3 736 kg/hr x 3 Heat Exchanger Chemical Reactor y 1 730 kg/hr x 1 y 2 718 kg/hr x 2 y 3 736 kg/hr x 3
    48. 54. Data Reconciliation using Least Squares Analytical solution using LaGrange Multipliers Q = diag[  i ]
    49. 55. Data Reconciliation Measurements having only random errors - least squares
    50. 63. Key Elements of On-Line Optimization
    51. 67. Mosaic-Monsanto Sulfuric Acid Plant 3,200 tons per day of 93% Sulfuric Acid, Convent, Louisiana
    52. 68. Motiva Refinery Alkylation Plant 15,000 barrels per day, Convent, Louisiana, reactor section, 4 Stratco reactors
    53. 69. Steady State Detection Execution frequency must be greater than the plant settling time (time to return to steady state).
    54. 70. On-Line Optimization - Distributed Control System Interface Plant must at steady state when data extracted from DCS and when set points sent to DCS. Plant models are steady state models. Coordinator program
    55. 73. Mixed Integer Programming Numerous Applications Batch Processing Pinch Analysis Optimal Flowsheet Structure Branch and Bound Algorithm Solves MILP Used with NLP Algorithm to solve MINLP
    56. 74. Mixed Integer Process Example Produce C from either Process 2 or Process 3 Make B from A in Process 1 or purchase B
    57. 75. Mixed Integer Process Example operating cost fixed cost feed cost sales max: -250 F 1 A - 400 F 6 B - 550 F 7 B - 1,000 y 1 - 1,500 y 2 - 2,000 y 3 -500 F 1 A - 950 F 4 B + 1,800 F 12 C subject to: mass yields -0.90 F 1 A + F 2 B = 0 -0.10 F 1 A + F 3 A = 0 -0.82 F 6 B + F 8 C = 0 -0.18 F 6 B + F 9 B = 0 -0.95 F 7 B + F 10 C = 0 -0.05 F 7 B + F 11 B = 0 node MB F 2 B + F 4 B - F 5 B = 0 F 5 B = F 6 B - F 7 B = 0 F 8 C + F 10 C - F 12 C = 0 availability of A F 1 A < 16 y 1 Availability of raw material A to make B availability of B F 4 B < 20 y 4 Availability of purchased material B demand for C F 8 C < 10 y 2 Demand for C from either Process 2, F 10 C < 10 y 3 stream F 8 C or Process 3, stream F 10 C integer constraint y 2 + y 3 = 1 Select either Process 1 or Purchase B y 1 + y 4 = 1 Select either Process 2 or 3 Branch and bound algorithm used for optimization
    58. 76. Branch and Bound Algorithm LP Relaxation Solution Max: 5x 1 + 2x 2 =P P = 22.5 Subject to: x 1 + x 2 < 4.5 x 1 = 4.5 -x 1 +2x 2 < 6.0 x 2 = 0 x 1 and x 2 are integers > 0 Branch on x 1 , it is not an integer in the LP Relaxation Solution Form two new problems by adding constraints x 1 > 5 and x 1 < 4 Max: 5x 1 + 2x 2 =P Max: 5x 1 + 2x 2 =P Subject to: x 1 + x 2 < 4.5 Subject to: x 1 + x 2 < 4.5 -x 1 +2x 2 < 6.0 -x 1 + 2x 2 < 6.0 x 1 > 5 x 1 < 4
    59. 77. Branch and Bound Algorithm Max: 5x 1 + 2x 2 =P Max: 5x 1 + 2x 2 =P Subject to: x 1 + x 2 < 4.5 Subject to: x 1 + x 2 < 4.5 -x 1 +2x 2 < 6.0 -x 1 +2x 2 < 6.0 x 1 > 5 x 1 < 4 infeasible LP solution P = 21.0 no further evaluations required x 1 = 4 x 2 = 0.5 branch on x 2 Form two new problems by adding constraints x 2 > 1 and x 2 < 0 Max: 5x 1 + 2x 2 =P Max: 5x 1 + 2x 2 =P Subject to: x 1 + x 2 < 4.5 Subject to: x 1 + x 2 < 4.5 -x 1 +2x 2 < 6.0 -x 1 +2x 2 < 6.0 x 1 < 4 x 1 < 4 x 2 > 1 x 2 < 0 =0
    60. 78. Branch and Bound Algorithm Max: 5x 1 + 2x 2 =P Max: 5x 1 + 2x 2 =P Subject to: x 1 + x 2 < 4.5 Subject to: x 1 + x 2 < 4.5 -x 1 +2x 2 < 6.0 -x 1 +2x 2 < 6.0 x 1 < 4 x 1 < 4 x 2 > 1 x 2 < 0 P = 19.5 P = 20 x 1 = 3.5 x 1 = 4 x 2 = 1 x 2 = 0 optimal solution
    61. 79. Branch and Bound Algorithm 22.5 Infeasible 21.0 19.5 20 LP relaxation solution X 1 = 4.5 X 2 = 0 X 1 > 5 X 1 < 4 X 1 < 4 X 2 > 1 X 1 < 4 X 2 < 0
    62. 80. Branch and Bound Algorithm inf 13.7 17 Inf 12 16.8 inf 15.6 17.4 15 inf Integer solution Integer solution – optimal solution
    63. 81. Mixed Integer Nonlinear Programming Flow Chart of GBD Algorithm to Solve MINPL Problems , Duran and Grossmann, 1986, Mathematical Programming, Vol. 36, p. 307-339
    64. 82. Triple Bottom Line Triple Bottom Line = Product Sales - Manufacturing Costs (raw materials, energy costs, others) - Environmental Costs (compliance with environmental regulations) - Sustainable Costs (repair damage from emissions within regulations) Triple Bottom Line = Profit (sales – manufacturing costs) - Environmental Costs + Sustainable (Credits – Costs) (credits from reducing emissions) Sustainable costs are costs to society from damage to the environment caused by emissions within regulations, e.g., sulfur dioxide 4.0 lb per ton of sulfuric acid produced. Sustainable development: Concept that development should meet the needs of the present without sacrificing the ability of the future to meet its needs
    65. 83. Optimization of Chemical Production Complexes <ul><li>Opportunity </li></ul><ul><ul><li>New processes for conversion of surplus carbon dioxide to valuable products </li></ul></ul><ul><li>Methodology </li></ul><ul><ul><li>Chemical Complex Analysis System </li></ul></ul><ul><ul><li>Application to chemical production complex in the lower Mississippi River corridor </li></ul></ul>
    66. 84. Plants in the lower Mississippi River Corridor Source: Peterson, R.W., 2000
    67. 85. Some Chemical Complexes in the World <ul><li>North America </li></ul><ul><ul><li>Gulf coast petrochemical complex in Houston area </li></ul></ul><ul><ul><li>Chemical complex in the Lower Mississippi River Corridor </li></ul></ul><ul><li>South America </li></ul><ul><ul><li>Petrochemical district of Camacari-Bahia (Brazil) </li></ul></ul><ul><ul><li>Petrochemical complex in Bahia Blanca (Argentina) </li></ul></ul><ul><li>Europe </li></ul><ul><ul><li>Antwerp port area (Belgium) </li></ul></ul><ul><ul><li>BASF in Ludwigshafen (Germany) </li></ul></ul><ul><li>Oceania </li></ul><ul><ul><li>Petrochemical complex at Altona (Australia) </li></ul></ul><ul><ul><li>Petrochemical complex at Botany (Australia) </li></ul></ul>
    68. 86. Plants in the lower Mississippi River Corridor, Base Case. Flow Rates in Million Tons Per Year
    69. 87. Commercial Uses of CO 2 <ul><li>Chemical synthesis in the U. S. consumes 110 million m tons per year of CO 2 </li></ul><ul><li> − Urea (90 million tons per year) </li></ul><ul><ul><li>Methanol (1.7 million tons per year) </li></ul></ul><ul><ul><li>Polycarbonates </li></ul></ul><ul><ul><li>Cyclic carbonates </li></ul></ul><ul><ul><li>Salicylic acid </li></ul></ul><ul><ul><li>Metal carbonates </li></ul></ul>
    70. 88. Surplus Carbon Dioxide <ul><li>Ammonia plants produce 0.75 million tons per year in lower Mississippi River corridor. </li></ul><ul><li>Methanol and urea plants consume 0.14 million tons per year. </li></ul><ul><li>Surplus high-purity carbon dioxide 0.61 million tons per year vented to atmosphere. </li></ul><ul><li>Plants are connected by CO 2 pipelines. </li></ul>
    71. 89. Greenhouse Gases as Raw Material From Creutz and Fujita, 2000
    72. 90. Some Catalytic Reactions of CO 2 Hydrogenation Hydrolysis and Photocatalytic Reduction CO 2 + 3H 2  CH 3 OH + H 2 O methanol CO 2 + 2H 2 O  CH 3 OH + O 2 2CO 2 + 6H 2  C 2 H 5 OH + 3H 2 O ethanol CO 2 + H 2 O  HC=O-OH + 1/2O 2 CO 2 + H 2  CH 3 -O-CH 3 dimethyl ether CO 2 + 2H 2 O  CH 4 + 2O 2   Hydrocarbon Synthesis CO 2 + 4H 2  CH 4 + 2H 2 O methane and higher HC 2CO 2 + 6H 2  C 2 H 4 + 4H 2 O ethylene and higher olefins   Carboxylic Acid Synthesis Other Reactions CO 2 + H 2  HC=O-OH formic acid CO 2 + ethylbenzene  styrene CO 2 + CH 4  CH 3 -C=O-OH acetic acid CO 2 + C 3 H 8  C 3 H 6 + H 2 + CO dehydrogenation of propane   CO 2 + CH 4  2CO + H 2 reforming Graphite Synthesis CO 2 + H 2  C + H 2 O CH 4  C + H 2   CO 2 + 4H 2  CH 4 + 2H 2 O Amine Synthesis CO 2 + 3H 2 + NH 3  CH 3 -NH 2 + 2H 2 O methyl amine and higher amines
    73. 91. Methodology for Chemical Complex Optimization with New Carbon Dioxide Processes <ul><li>Identify potentially new processes </li></ul><ul><li>Simulate with HYSYS </li></ul><ul><li>Estimate utilities required </li></ul><ul><li>Evaluate value added economic analysis </li></ul><ul><li>Select best processes based on value added economics </li></ul><ul><li>Integrate new processes with existing ones to form a superstructure for optimization </li></ul>
    74. 92. Twenty Processes Selected for HYSYS Design <ul><li>Chemical Synthesis Route Reference </li></ul><ul><li>Methanol CO2 hydrogenation Nerlov and Chorkendorff, 1999 </li></ul><ul><li>CO2 hydrogenation Toyir, et al., 1998 </li></ul><ul><li>CO2 hydrogenation Ushikoshi, et al., 1998 </li></ul><ul><li>CO2 hydrogenation Jun, et al., 1998 </li></ul><ul><li>CO2 hydrogenation Bonivardi, et al., 1998 </li></ul><ul><li>Ethanol CO2 hydrogenation Inui, 2002 </li></ul><ul><li>CO2 hydrogenation Higuchi, et al., 1998 </li></ul><ul><li>Dimethyl Ether CO2 hydrogenation Jun, et al., 2002 </li></ul><ul><li>Formic Acid CO2 hydrogenation Dinjus, 1998 </li></ul><ul><li>Acetic Acid From methane and CO2 Taniguchi, et al., 1998 </li></ul><ul><li>Styrene Ethylbenzene dehydrogenation Sakurai, et al., 2000 </li></ul><ul><li>Ethylbenzene dehydrogenation Mimura, et al., 1998 </li></ul><ul><li>Methylamines From CO2, H2, and NH3 Arakawa, 1998 </li></ul><ul><li>Graphite Reduction of CO2 Nishiguchi, et al., 1998 </li></ul><ul><li>Hydrogen/ Methane reforming Song, et al., 2002 </li></ul><ul><li>Synthesis Gas Methane reforming Shamsi, 2002 </li></ul><ul><li>Methane reforming Wei, et al., 2002 </li></ul><ul><li>Methane reforming Tomishige, et al., 1998 </li></ul><ul><li>Propylene Propane dehydrogenation Takahara, et al., 1998 </li></ul><ul><li>Propane dehydrogenation C & EN, 2003 </li></ul>
    75. 93. Integration into Superstructure <ul><li>Twenty processes simulated </li></ul><ul><li>Fourteen processes selected based on value added economic model </li></ul><ul><li>Integrated into the superstructure for optimization with the System </li></ul>
    76. 94. New Processes Included in Chemical Production Complex <ul><li>Product Synthesis Route Value Added Profit (cents/kg) </li></ul><ul><li>Methanol CO 2 hydrogenation 2.8 </li></ul><ul><li>Methanol CO 2 hydrogenation 3.3 </li></ul><ul><li>Methanol CO 2 hydrogenation 7.6 </li></ul><ul><li>Methanol CO 2 hydrogenation 5.9 </li></ul><ul><li>Ethanol CO 2 hydrogenation 33.1 </li></ul><ul><li>Dimethyl Ether CO 2 hydrogenation 69.6 </li></ul><ul><li>Formic Acid CO 2 hydrogenation 64.9 </li></ul><ul><li>Acetic Acid From CH 4 and CO 2 97.9 </li></ul><ul><li>Styrene Ethylbenzene dehydrogenation 10.9 </li></ul><ul><li>Methylamines From CO 2 , H 2 , and NH 3 124 </li></ul><ul><li>Graphite Reduction of CO 2 65.6 </li></ul><ul><li>Synthesis Gas Methane reforming 17.2 </li></ul><ul><li>Propylene Propane dehydrogenation 4.3 </li></ul><ul><li>Propylene Propane dehydrogenation with CO 2 2.5 </li></ul>
    77. 95. Application of the Chemical Complex Analysis System to Chemical Complex in the Lower Mississippi River Corridor <ul><li>Base case – existing plants </li></ul><ul><li>Superstructure – existing and proposed new plants </li></ul><ul><li>Optimal structure – optimal configuration from existing and new plants </li></ul>
    78. 96. Chemical Complex Analysis System
    79. 97. Plants in the lower Mississippi River Corridor, Base Case. Flow Rates in Million Tons Per Year
    80. 98. Superstructure
    81. 99. Plants in the Superstructure <ul><li>Plants in the Base Case </li></ul><ul><li>Ammonia </li></ul><ul><li>Nitric acid </li></ul><ul><li>Ammonium nitrate </li></ul><ul><li>Urea </li></ul><ul><li>UAN </li></ul><ul><li>Methanol </li></ul><ul><li>Granular triple super phosphate </li></ul><ul><li>MAP and DAP </li></ul><ul><li>Sulfuric acid </li></ul><ul><li>Phosphoric acid </li></ul><ul><li>Acetic acid </li></ul><ul><li>Ethylbenzene </li></ul><ul><li>Styrene </li></ul><ul><li>Plants Added to form the Superstructure </li></ul><ul><li>Acetic acid from CO 2 and CH 4 </li></ul><ul><li>Graphite and H 2 </li></ul><ul><li>Syngas from CO 2 and CH 4 </li></ul><ul><li>Propane dehydrogenation </li></ul><ul><li>Propylene from propane and CO 2 </li></ul><ul><li>Styrene from ethylbenzene and CO 2 </li></ul><ul><li>Methanol from CO 2 and H 2 (4) </li></ul><ul><li>Formic acid </li></ul><ul><li>Methylamines </li></ul><ul><li>Ethanol </li></ul><ul><li>Dimethyl ether </li></ul><ul><li>Electric furnace phosphoric acid </li></ul><ul><li>HCl process for phosphoric acid </li></ul><ul><li>SO 2 recovery from gypsum </li></ul><ul><li>S and SO 2 recovery from gypsum </li></ul>
    82. 100. Superstructure Characteristics <ul><li>Options </li></ul><ul><li>- Three options for producing phosphoric acid </li></ul><ul><li>- Two options for producing acetic acid </li></ul><ul><li>Two options for recovering sulfur and sulfur dioxide </li></ul><ul><li>Two options for producing styrene </li></ul><ul><li>Two options for producing propylene </li></ul><ul><li>Two options for producing methanol </li></ul><ul><li>Mixed Integer Nonlinear Program </li></ul><ul><li>843 continuous variables </li></ul><ul><li>23 integer variables </li></ul><ul><li>777 equality constraint equations for material and energy balances </li></ul><ul><li>64 inequality constraints for availability of raw materials </li></ul><ul><li>demand for product, capacities of the plants in the complex </li></ul>
    83. 101. Some of the Raw Material Costs, Product Prices and Sustainability Cost and Credits Raw Materials Cost Sustainable Cost and Credits Cost/Credit Products Price ($/mt) ($/mt) ($/mt) Natural gas 235 Credit for CO2 consumption 6.50 Ammonia 224 Phosphate rock Debit for CO2 production 3.25 Methanol 271 Wet process 27 Credit for HP Steam 11 Acetic acid 1,032 Electro-furnace 34 Credit for IP Steam 7 GTSP 132 Haifa process 34 Credit for gypsum consumption 5.0 MAP 166 GTSP process 32 Debit for gypsum production 2.5 DAP 179 HCl 95 Debit for NOx production 1,025 NH4NO3 146 Sulfur Debit for SO2 production 192 Urea 179 Frasch 53 UAN 120 Claus 21 Phosphoric 496 Sources: Chemical Market Reporter and others for prices and costs, and AIChE/CWRT report for sustainable costs.
    84. 102. Optimal Structure
    85. 103. Plants in the Optimal Structure from the Superstructure
    86. 104. Comparison of the Triple Bottom Line for the Base Case and Optimal Structure
    87. 105. Carbon Dioxide Consumption in Bases Case and Optimal Structure All of the carbon dioxide was not consumed in the optimal structure to maximize the triple bottom line Other cases were evaluated that forced use of all of the carbon dioxide, but with a reduced triple bottom line
    88. 106. Multi-Criteria or Multi-Objective Optimization Subject to: f i (x) = 0 min: cost max: reliability min: waste generation max: yield max: selectivity
    89. 107. Multi-Criteria Optimization - Weighting Objectives Method Subject to: f i (x) = 0 with ∑ w i = 1 Optimization with a set of weights generates efficient or Pareto optimal solutions for the y i (x). There are other methods for multi-criteria optimization, e.g., goal programming, but this method is the most widely used one Efficient or Pareto Optimal Solutions Optimal points where attempting to improving the value of one objective would cause another objective to decrease.
    90. 108. Multicriteria Optimization P= Σ Product Sales - Σ Manufacturing Costs - Σ Environmental Costs max: S = Σ Sustainable (Credits – Costs) subject to: Multi-plant material and energy balances Product demand, raw material availability, plant capacities
    91. 109. Multicriteria Optimization Convert to a single criterion optimization problem max: w 1 P + w 2 S subject to: Multi-plant material and energy balances Product demand, raw material availability, plant capacities
    92. 110. Multicriteria Optimization W 1 =1 only profit W 1 = 0 no profit Debate about which weights are best
    93. 111. Monte Carlo Simulation <ul><ul><li>Used to determine the sensitivity of the optimal solution to the costs and prices used in the chemical production complex economic model. </li></ul></ul><ul><ul><li>Mean value and standard deviation of prices and cost are used. </li></ul></ul><ul><li>The result is the cumulative probability distribution, a curve of the probability as a function of the triple bottom line. </li></ul><ul><li>A value of the cumulative probability for a given value of the triple bottom line is the probability that the triple bottom line will be equal to or less that value. </li></ul><ul><li>This curve is used to determine upside and downside risks </li></ul>
    94. 112. Monte Carlo Simulation Triple Bottom Line Mean $513million per year Standard deviation - $109 million per year 50% probability that the triple bottom line will be $513 million or less Optimal structure changes with changes in prices and costs
    95. 113. Conclusions ● The optimum configuration of plants in a chemical production complex was determined based on the triple bottom line including economic, environmental and sustainable costs using the Chemical Complex Analysis System. ● Multcriteria optimization determines optimum configuration of plants in a chemical production complex to maximize corporate profits and maximize sustainable credits/costs. ● Monte Carlo simulation provides a statistical basis for sensitivity analysis of prices and costs in MINLP problems. ● Additional information is available at www.mpri.lsu.edu
    96. 114. Transition from Fossil Raw Materials to Renewables Introduction of ethanol into the ethylene product chain. Ethanol can be a valuable commodity for the manufacture of plastics, detergents, fibers, films and pharmaceuticals. Introduction of glycerin into the propylene product chain. Cost effective routes for converting glycerin to value-added products need to be developed. Generation of synthesis gas for chemicals by hydrothermal gasification of biomaterials. The continuous, sustainable production of carbon nanotubes to displace carbon fibers in the market. Such plants can be integrated into the local chemical production complex. Energy Management Solutions: Cogeneration for combined electricity and steam production (CHP) can substantially increase energy efficiency and reduce greenhouse gas emissions.
    97. 115. Global Optimization Locate the global optimum of a mixed integer nonlinear programming problem directly. Branch and bound separates the original problem into sub-problems that can be eliminated showing the sub-problems that can not lead to better points Bound constraint approximation rewrites the constraints in a linear approximate form so a MILP solver can be used to give an approximate solution to the original problem. Penalty and barrier functions are used for constraints that can not be linearized. Branch on local optima to proceed to the global optimum using a sequence of feasible sets (boxes). Box reduction uses constraint propagation, interval analysis convex relations and duality arguments involving Lagrange multipliers. Interval analysis attempts to reduce the interval on the independent variables that contains the global optimum Leading Global Optimization Solver is BARON, Branch and Reduce Optimization Navigator, developed by Professor Nikolaos V. Sahinidis and colleagues at the University of Illinois is a GAMS solver. Global optimization solvers are currently in the code-testing phase of development which occurred 20 years ago for NLP solvers.
    98. 116. Acknowledgements Collaborators Dean Jack R. Hopper Professor Helen H. Lou Professor Carl L. Yaws Graduate Students Industry Colleagues Xueyu Chen Thomas A. Hertwig, Mosaic Zajun Zang Michael J. Rich, Motiva Aimin Xu Sudheer Indala Janardhana Punru Post Doctoral Associate Derya Ozyurt Support Gulf Coast Hazardous Substance Research Center Department of Energy
    99. 117. www.mpri.lsu.edu
    1. A particular slide catching your eye?

      Clipping is a handy way to collect important slides you want to go back to later.

    ×