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
Process Optimization
•
•
•
•
•
•
•
Typical Industrial Problems
Mathematical Programming Software
Mathematical Basis for Optimization
Lagrange Multipliers and the Simplex Algorithm
Generalized Reduced Gradient Algorithm
On-Line Optimization
Mixed Integer Programming and the Branch
and Bound Algorithm
• Chemical Production Complex Optimization
New Results
• Using one computer language to write and
run a program in another language
• Cumulative probability distribution instead
of an optimal point using Monte Carlo
simulation for a multi-criteria, mixed integer
nonlinear programming problem
• Global optimization
Design vs. Operations
• Optimal Design
−Uses flowsheet simulators and SQP
– Heuristics for a design, a superstructure, an
optimal design
• Optimal Operations
– On-line optimization
– Plant optimal scheduling
– Corporate supply chain optimization
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
761
1,579
~400,000
28
50
~10,000
43
125
~300
732
1,509
~10,000
11
64
~100
Constraints
Equality
Inequality
Variables
Measured
Unmeasured
Parameters
Optimization Programming Languages
• GAMS - General Algebraic Modeling System
• LINDO - Widely used in business applications
• AMPL - A Mathematical Programming
Language
• Others: MPL, ILOG
optimization program is written in the form of an
optimization problem
optimize: y(x)
economic model
subject to: fi(x) = 0 constraints
Software with Optimization Capabilities
•
•
•
•
•
•
Excel – Solver
MATLAB
MathCAD
Mathematica
Maple
Others
Mathematical Programming
•
•
•
•
Using Excel – Solver
Using GAMS
Mathematical Basis for Optimization
Important Algorithms
– Simplex Method and Lagrange Multipliers
– Generalized Reduced Gradient Algorithm
– Branch and Bound Algorithm
Simple Chemical Process
minimize: C = 1,000P +4*10^9/P*R + 2.5*10^5R
subject to: P*R = 9000
P – reactor pressure
R – recycle ratio
Excel Solver Example
Solver optimal solution
C
P*R
P
R
3.44E+06
9000.0
6.0
1500.0
Example 2-6 p. 30 OES A Nonlinear Problem
minimize: C = 1,000P +4*10^9/P*R + 2.5*10^5R
subject to: P*R = 9000
Solution
C = 3.44X10^6
P = 1500 psi
R=6
Showing the equations in the Excel cells with initial values for P and R
C
P*R
P
R
=1000*D5+4*10^9/(D5*D4)+2.5*10^5*D4
=D5*D4
1
1
Excel Solver Example
Not the minimum
for C
Excel Solver Example
N
o
t
Use Solver with these
values of P and R
Excel Solver Example
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
values at the
optimum
constraint
status
slack
variable
Excel Sensitivity Report
Solver uses the
generalized reduced
gradient optimization
algorithm
Lagrange multipliers used
for sensitivity analysis
Shadow prices ($ per unit)
Excel Solver Limits Report
Sensitivity Analysis provides
limits on variables for the optimal
solution to remain optimal
GAMS
GAMS
SOLVE
SUMMARY
MODEL Recycle
TYPE NLP
SOLVER CONOPT
OBJECTIVE Z
DIRECTION MINIMIZE
FROM LINE 18
**** SOLVER STATUS 1 NORMAL COMPLETION
**** MODEL STATUS
2 LOCALLY OPTIMAL
**** OBJECTIVE VALUE
3444444.4444
RESOURCE USAGE, LIMIT
ITERATION COUNT, LIMIT
EVALUATION ERRORS
0.016
1000.000
14
10000
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.
Lagrange
multiplier
GAMS
•
LOWER
• ---- EQU CON1
• ---- EQU OBJ
9000.000 9000.000 9000.000 117.284
.
.
.
1.000
LEVEL
UPPER
•
LOWER
LEVEL
• ---- VAR P
• ---- VAR R
• ---- VAR Z
1.000
1.000
-INF
1500.000
6.000
3.4444E+6
UPPER
+INF
+INF
+INF
MARGINAL
MARGINAL
.
EPS
.
values at the
optimum
• **** REPORT SUMMARY :
0 NONOPT
•
0 INFEASIBLE
•
0 UNBOUNDED
900 page Users Manual
•
0 ERRORS
GAMS Solvers
13 types of
optimization
problems
NLP – Nonlinear Programming
nonlinear economic model and
nonlinear constraints
LP - Linear Programming
linear economic model
and linear constraints
MIP - Mixed Integer Programming
nonlinear economic model and
nonlinear constraints with
continuous and integer variables
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
Mathematical Basis for Optimization
is the Kuhn Tucker Necessary Conditions
General Statement of a Mathematical Programming Problem
Minimize: y(x)
Subject to: fi(x) <
0
for i = 1, 2, ..., h
fi(x) = 0 for i = h+1, ..., m
y(x) and fi(x) are twice continuously
differentiable real valued functions.
Kuhn Tucker Necessary Conditions
Lagrange Function
– converts constrained problem to an unconstrained one
   f ( x)  x
h
L ( x,  )  y ( x) 
i
i
i 1
  
m
2
ni
i
fi ( x)
λi are the Lagrange multipliers
xn+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*
1.
h
m

y(x *)

f i (x *)

fi ( x* )
—
+  i —
+  i —
=0

xj

xj

xj
i=1
i= h+1
2.
f i(x *) 0
for i = 1, 2, ..., h
3.
f i(x *) = 0
for i = h+ 1, ..., m
4.
i fi( x* ) = 0
for i = 1, 2, ..., h
5.
i > 0
for i = 1, 2, ..., h
6.
i is unrestricted in sign
for i = h+ 1, ..., m
for j = 1,2,..,n
Lagrange Multipliers
Treated as an:
• Undetermined multiplier – multiply
constraints by λi and add to y(x)
• Variable - L(x,λ)
• Constant – numerical value computed
at the optimum
Lagrange Multipliers
optimize:
subject to:
dy 
0
y
 x1
f
 x1
dx 1 
dx 1 
y(x1, x2)
f(x1, x2) = 0
y
x2
f
x2
dx 2
dx 2
f
dx 2  
 x1
f
x2
dx 1
Lagrange Multipliers
f
dy 
y
 x1
dx 1 
 y  x1
x2 f
x2
dx 1
Rearrange the partial derivatives in the second term
Lagrange Multipliers

 y


y  x2
dy  

  x1   f


 x2





  f  dx
  x1  1




 y
f 
dy  

 dx 1
 x1 
  x1
( )=λ
Call the ratio of partial derivatives in the ( ) a Lagrange multiplier, λ
Lagrange multipliers are a ratio of partial derivatives at the optimum.
Lagrange Multipliers
dy 
( y  f )
 x1
dx 1  0
Define L = y +λf , an unconstrained function
L
 x1
0
and by the same
procedure
L
x2
0
Interpret L as an unconstrained function, and the partial derivatives set
equal to zero are the necessary conditions for this unconstrained function
Lagrange Multipliers
Optimize: y(x1,x2)
Subject to: f(x1,x2) = b
Manipulations give:
∂y = - λ
∂b
Extends to:
∂y = - λi shadow price ($ per unit of bi)
∂bi
Geometric Representation of an LP Problem
Maximum at vertex
P = 110
A = 10, B = 20
max: 3A + 4B = P
s.t. 4A + 2B < 80
2A + 5B < 120
objective function is a plane
no interior optimum
LP Example
Maximize:
x1+ 2x2
Subject to:
2x1 + x2 + x3
x1 + x2
+ x4
-x1 + x2
+ x5
-2x1 + x2
+ x6
= P
= 10
= 6
= 2
= 1
4 equations and 6 unknowns, set 2 of the xi =0 and solve for 4 of the xi.
Basic feasible solution: x1 = 0, x2 = 0, x3 = 10, x4 = 6, x5 = 2, x6 =1
Basic solution:
x1 = 0, x2 = 6, x3 = 4, x4 = 0, x5 = -4, x6 = -5
Final Step in Simplex Algorithm
Maximize:
Subject to:
x1
- 3/2 x4 - 1/2 x5
x3 - 3/2 x4 + 1/2 x5
1/2 x4 - 3/2 x5 + x6
+ 1/2 x4 - 1/2 x5
x2 + 1/2 x4 + 1/2 x5
= P - 10 P = 10
=
=
=
=
2
1
2
4
x3 = 2
x6 = 1
x1 = 2
x2 = 4
x4 = 0
x5 = 0
Simplex algorithm exchanges variables that are zero with ones
that are nonzero, one at a time to arrive at the maximum
Lagrange Multiplier Formulation
Returning to the original problem
Max: (1+2λ1+ λ2 - λ3- 2λ4) x1
(2+λ1+ λ2 + λ3 +λ4)x2 +
λ 1 x 3 + λ 2 x 4 + λ 3 x 5 + λ 4x 6
- (10λ1 + 6λ2 + 2λ3 + λ4) = L = P
Set partial derivatives with respect to x1, x2, x3, and x6 equal
to zero (x4 and x5 are zero) and and solve resulting
equations for the Lagrange multipliers
Lagrange Multiplier Interpretation
(1+2λ1+ λ2 - λ3- 2λ4)=0
(2+λ1+ λ2 + λ3 +λ4)=0
λ3=-1/2
λ2=-3/2
λ4=0
λ1=0
Maximize: 0x1 +0x2 +0 x3 - 3/2 x4 - 1/2 x5
Subject to:
x3 - 3/2 x4 + 1/2 x5
1/2 x4 - 3/2 x5 + x6
x1
+ 1/2 x4 - 1/2 x5
x2 + 1/2 x4 + 1/2 x5
+0x6 = P - 10
=
=
=
=
2
1
2
4
P = 10
x3 = 2
x6 = 1
x1 = 2
x2 = 4
x4 = 0
x5 = 0
-(10λ1 + 6λ2 + 2λ3 + λ4) = L = P = 10
The final step in the simplex algorithm is used to evaluate the Lagrange
multipliers. It is the same as the result from analytical methods.
General Statement of the Linear Programming Problem
Objective Function:
Maximize:
c1x1 + c2x2 + ... + cnxn
=p
(4-1a)
Constraint Equations:
Subject to:
a11x1 + a12x2 + ... + a1nxn < b1
a21x1 + a22x2 + ... + a2nxn <
...
...
...
...
am1x1 + am2x2 + ... + amnxn <
xj > 0 for j = 1,2,...n
(4-1b)
b2
...
...
bm
(4-1c)
LP Problem with Lagrange Multiplier Formulation
Multiply each constraint equation, (4-1b), by the Lagrange multiplier λi and add to the
objective function
Have x1 to xm be values of the variables in the basis, positive numbers
Have xm+1 to xn be values of the variables that are not in the basis and are zero.
equal to zero from
∂p/∂xm=0
not equal to zero, negative
positive
in the
basis
equal to zero
not in basis
Left hand side = 0 and p = - ∑biλi
Sensitivity Analysis
• 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:
• bi availability of raw materials demand for
product, capacities of the process units
• cj sales price and costs
• See Optimization for Engineering Systems book
for equations at www.mpri.lsu.edu
Nonlinear Programming
Three standard methods – all use the same information
Successive Linear Programming
Successive Quadratic Programming
Generalized Reduced Gradient Method
Optimize: y(x)
Subject to: fi(x) =0
∂y(xk)
∂xj
x = (x1, x2,…, xn)
for i = 1,2,…,m n>m
∂fi(xk) evaluate partial derivatives at xk
∂xj
Generalized Reduced Gradient Direction
Reduced Gradient Line
Specifies how to change xnb
to have the largest change in
y(x) at xk
x nb  x k , nb    Y ( x k )
Generalized Reduced Gradient Algorithm
Minimize: y(x) = y(x)
Subject to: fi(x) = 0
(x) = (xb,xnb)
Y[xk,nb + α Y(xk)] = Y(α)
m basic variables, (n-m) nonbasic variables
Reduced Gradient
1
b
 Y ( x k )   y nb ( x k )   y b ( x k ) B B nb
T
T
Reduced Gradient Line
x nb  x k , nb    Y ( x k )
Newton Raphson Algorithm
x i 1,b  x i ,b  B
1
b
f ( x i , b , x nb )
B 
fi ( xk )
x j
Generalized Reduced Gradient Trajectory
Minimize : -2x1 - 4x2 + x12 + x22 + 5
Subject to: - x1 + 2x2 < 2
x1 + x2 < 4
On-Line Optimization
• Automatically adjust operating conditions with the plant’s distributed
control system
• Maintains operations at optimal set points
• Requires the solution of three NLP’s in sequence
gross error detection and data reconciliation
parameter estimation
economic optimization
BENEFITS
• Improves plant profit by 10%
• Waste generation and energy use are reduced
• Increased understanding of plant operations
s e tp o in ts
fo r
c o n tro lle rs
p la n t
m e a s u re m e n ts
D is trib u te d C o n tro l S y s te m
s a m p le d
p la n t d a ta
o p tim a l
o p e ra tin g
c o n d itio n s
s e tp o in t
ta rg e ts
G ro s s E rro r
D e te c tio n
and
D a ta R e c o n c ila tio n
re c o n c ile d
p la n t d a ta
O p tim iz a tio n A lg o rith m
E c o n o m ic M o d e l
P la n t M o d e l
e c o n o m ic m o d e l
p a ra m e te rs
u p d a te d p la n t
p a ra m e te rs
P a ra m e te r
E s tim a tio n
S o m e C om p an ie s U s in g O n -L in e O p tim iza tio n
U n ited S tates
E u rope
T e xac o
O M V D eutsc hlan d
Am oc o
D ow B en elu x
C on oc o
S h ell
L yon del
O EM V
S u noc o
P en ex
P h illips
B orealis A B
M arathon
D S M -H ydrocarbons
D ow
C h evron
P yr otec/K T I
N O V A C hem ic als (C anada)
B ritis h P etroleum
A p p licatio n s
m ain ly c ru de u nits in refineries an d eth ylen e
plan ts
C om pan ies P rovidin g O n-Lin e O ptim ization
A s pen Tech n ology - A s pen P lu s O n -Lin e
- D M C C orporation
- S etpoin t
- H yprotech L td.
S im ulation S cien ce - R O M
- S h ell - R om eo
P rofim atic s - O n-O pt
- H oneyw ell
L itwin P rocess A u tom ation - F A C S
D O T P rodu cts, In c. - N O V A
D is tribu ted C ontrol S ys tem
R uns c on tro l a lgo rith m three tim e s a s econ d
T a gs - c ontain a bo ut 2 0 va lu es fo r e ac h
m e asurem en t, e .g. s et p o int, lim its, a larm
R efinery a n d la rg e c hem ica l p lan ts h ave 5 ,000
- 1 0,000 ta gs
D ata H is torian
S tores insta ntan eous va lues o f m e asure m en ts
fo r e a ch ta g e very five s econds o r as sp e cified .
In clud es a re lationa l d a ta b a se fo r la bo rato ry
a n d o the r m easu rem ents n o t fro m th e D C S
V a lu es a re s tored fo r o n e y ea r, a n d re quire
h u ndreds o f m eg abites
In form ation m ad e a vaila ble o ve r a L A N in
va rious fo rm s, e .g. a ve rag es, E xce l file s.
K ey E lem en ts
G ro ss E rro r D etec tio n
D ata R econc iliatio n
P aram eter E stim atio n
E conom ic M od el
(P ro fit F unc tio n )
P lan t M od el
(P roc ess S im u latio n )
O p tim izatio n A lgo rithm
D ATA
R E C O N C IL IA T IO N
A dju s t proces s data to s atisfy m aterial an d
en ergy balan ces .
M eas urem ent error - e
e = y -x
y = m eas ured proc es s variables
x = tru e values of th e m eas ured variables
~
x
= y+ a
a - m easurem en t adju stm ent
Data Reconciliation
y1
730 kg/hr
Heat
Exchanger
x1
y2
718 kg/hr
Chemical
Reactor
x2
Material Balance
x 1 = x2
Steady State
x2 = x3
y3
736 kg/hr
x3
x1 - x2 = 0
x2 - x3 = 0
Data Reconciliation
y1
y3
730y1kg/hr
Heat
Exchanger
730xkg/hr
1
x1
1

0
y2
718 kg/hr
Chemical
Reactor
736y3kg/hr
736xkg/hr
3
x2
1
1
 x1 
0    0 
  x2    
 1
0 
 x 3 
x3
Ax  0
Data Reconciliation using Least Squares
 yi  xi 

min :  
i 
x
i 1 
n
Subject to: Ax  0
2
Q=
diag[i]
Analytical solution using LaGrange Multipliers
1
x  y  QA ( AQA )
T
T
x  [728 728 728]
T
Ay
Data Reconciliation
Measurements having only random errors - least squares
 yi  xi 

Minimize: 
x
i 
i 1 
n
2
Subject to: f(x)  0
f(x) - process model
- linear or nonlinear
 i  standard deviation of yi
T ypes of G ross E rrors
S ource : S . Naras im han and C . J ordache , D ata Reconciliation a nd G ross
E rro r Detection, G ulf P ublishing C om pany, Hous to n, TX (2 000 )
C o m b ine d G ro ss E rro r D e tec tio n a n d D ata R e con c ilia tio n
M e a sure me nt Te st M e tho d - le a st s qua re s
M inimize:
(y - x ) T Q -1 (y - x ) = e T Q -1e
x, z
S ubjec t to:
f(x , z , ) = 0
x L x x U
z L z z U
Te s t s tatistic :
if 
e i =y i-x i
/i > C m e asure m e nt c ontains a gro ss e rro r
L ea st s qua re s is ba se d o n o nly ra ndo m e rrors be ing present G ro ss e rro rs
c a use nume ric a l diffic ultie s
Ne e d me thods t hat a re no t s ensitive to gross e rro rs
M e th o d s In sen sitive to G ro ss E rro rs
Tjao-B ieg ler’s C on tam inated G aus s ian
D is tribution
P (yi x i ) = (1 -η)P (y i xi , R ) + η P (yi x i, G )
P (y i x i , R ) = pro ba bility dis tributio n func tio n fo r the ra ndo m e rro r
P (y i x i , G ) = pro ba bility dis tributio n func tio n fo r the gros s e rro r.
G ro s s e rro r o c c ur w it h pro ba bility η
G ro s s E rro r D istributio n F unctio n
(y x)
P (y 
x, G )

1
2πb σ
e
2 2
2b σ
2
T ja o -B ie g le r M e th o d
Ma xim iz in g th is d is tribu tio n fu n c tio n o f m e a su rem en t
e rro rs o r m in im iz in g th e n e g a tive lo g a rith m su b je c t to th e
c on s tra in ts in p la n t m o d el, i.e .,
M in im iz e :
x
( y x
i

ln (1 
)e
i
)2
( y x
i
2
2 i

i
Su b je c t to :
f(x ) = 0
x L x x U

e
b
i
2
)2
2
2 b i
ln
2 i
p la n t m od e l
b o un d s o n th e p ro c e s s
va riab le s
A N L P, an d va lu e s a re n e ed e d fo r a n d b
T es t for G ros s E rrors
If P( y i
x i, G ) (1 -)P (y i
x i, R ), gros s error
p robability o f a
g ross e rro r



i
p robab ility o f a
rand om e rro r
y ix i
2b 2



 

 b
i
>
2
1
ln
b(1 )

R o b u st F u nc tio n M e tho d s
- [ (y i, x i) ]
i
f(x ) = 0
M inim ize:
x
S u bject to:
x L x x U
L orentzian distribu tion
1
(
) 
i
1 
1
2
F air fu n ction
(
, c)  c
i
2
2

i






i
i
log 1 
c
c
c is a tuning pa ra me te r
T est statistic
i = ( y i - xi )/i
Pa ram ete r E s tim ati o n
E r ro r-in -Variables M eth o d
L eas t s qu ares
M in im ize: (y - x) T -1 (y - x) = e T -1 e

S u b je c t to :
f(x , ) = 0
 - p la n t p a ram e ters
S im u lta ne o u s d a ta re co n cilia tio n a n d p a ram e te r
e s tim a tio n
M in im ize: (y - x) T -1 (y - x) = e T -1 e
x, 
S u b je c t to :
f(x , ) = 0
a n o th e r n o n line a r p ro gram m in g p ro blem
T h re e S im ila r O p tim iz ation P ro ble m s
O p tim ize :
S u b je c t to :
m od e l
O b je c tiv e fu nc tion
C o ns tra ints a re th e p la n t
O b je cti ve fu n ctio n
d a ta re c on cilia tio n - d is trib utio n fu n ctio n
p a ra m ete r e s tim atio n - le a s t s q u a re s
e co n om ic o p tim iza tio n - p ro fit fu n ctio n
C o n strain t e q u a tio n s
m a te ria l a n d e n e rg y b a lan ce s
c he m ica l re a ctio n ra te e q u a tion s
th e rm o d yn am ic e q u ilibrium re la tio n s
c ap a citie s o f p roce ss u n its
d e m a n d fo r p ro d uc t
a va ilab ility o f ra w m a te ria ls
Key Elements of On-Line Optimization
P lan t m o de l
P lan t da ta
fro m D C S
S im ultan eo u s d ata
rec on ciliation a nd
p aram eter estim ation
C o m b in ed g ro ss
err or de te ction a nd
d ata r eco n cilia tio n
P lan t
ec on o m ic
o ptim iza tio n
O p tim ization
alg orith m
C ite d B e ne fits:
˜
˜
Id en tifyin g in stru m en t
m a lfu n ction s
P ro cess m o n ito ring
˜
˜
Im p ro ved e qu ipm ent
p erfo rm a nc e
P ro cess m o n ito ring
˜
˜
Im p ro ved p lan t p rofit
R ed u ced e m issio n a nd
en erg y u se
O p tim al
se tp o in ts
to D C S
In teractive O n -Lin e O p tim izatio n P ro gram
1.
C on duct com bined g ross error d etection an d data
recon ciliat ion to d etect and rectify g ross errors in
p lant d ata sam pled from d istributed con trol system
u sing th e Tjoa-B iegler's m ethod (th e con tam inated
G aussian d istribution) or robu st m ethod (Lorentzian
d istribution).
T h is step g enerates a set o f m easu rem en ts co n tainin g
o n ly ran do m erro rs fo r p aram eter estim atio n.
2.
U se th is set of m easurem ents for sim ultaneous
p aram eter estim ation an d d ata reconciliation u sing
th e least squares m ethod.
T h is step p rovid es th e u pdated p aram eters in th e
p lan t m o del fo r eco nom ic o ptim ization .
3.
G enerate optim al set p oints for th e distributed con trol
system from th e econ om ic op tim ization u sing th e
u pd ated p lant an d econ om ic m odels.
Interac tiv e O n-Line O ptim ization P rogram
Process a n d e con omic m od e ls a re e nte re d a s
e q uation s in a fo rm sim ila r to F ortra n
Th e p ro gra m w rites a n d ru ns three G AM S
p rog ram s.
R esults a re p resen ted in a s umma ry fo rm , o n a
p rocess flow shee t a nd in the fu ll G AM S o utput
Th e p ro gra m a n d u sers m an ua l (1 20 p a ges ) c an b e
d o wn lo aded fro m th e L S U M in era ls Processing
R esearch Institu te w e b s ite
U R Lhttp://w w w.mpri.lsu.e du
Mosaic-Monsanto Sulfuric Acid Plant
3,200 tons per day of 93% Sulfuric Acid, Convent, Louisiana
Ai r
Ai r
In let
Dr yer
Ma in
Co mp-
Su lfu r
W ast e
Bu rne r
He at
re sso r
S upe r-
SO 2 t o
He ate r
SO3
Co nve rt er
Bo ile r
Ho t &
C old
H eat
Fi nal
Ga s t o
Gas
Ec ono -
He at
EX .
mi zer s
&
In ter pa ss
To wer s
SO 3
SH '
E’
4
3
D R Y A IR
2
H
C
1
SO 2
S u lfu r
E
BLR
SH
C o o le r
W
D ry A c id C o o le r
W
93% H 2SO4
p ro d u ct
A cid T o we rs
Pump T ank
98% H 2SO4
A cid D ilut ion Ta n k
93% H 2SO4
Motiva Refinery Alkylation Plant
15,000 barrels per day, Convent, Louisiana, reactor section, 4 Stratco reactors
1
M -3
5
O lefin s Feed
M -2
M -4
H C 28
H C 31
H C 24
H C 26
C 301
H C 27
H C 25
C 401
5C -614
H C 01
5E -628
ST FD
H C 30
M -24
5E -629,
630
Fresh A cid
M -1
H C 03
M -11
M -7
A C 02
5E -633
S-7
S-5
A C 09
A C 18
H C 11
A C 20
5C -625
5C -623
H C 29
2
A C 07
H C 07
S-2
3’
H C 14
H C 08
H C 06
C 403
A C 23
A C 12
H C 32
H C 02
H C 04
A C 15
A C 05
C 402
3
A cid Settler
5C -632
A cid Settler
5C -631
R1
Isobutan e
R 29
4
ST R A T C O
R eactor
R 20
R3
R7
R2
R6
S-19
H C 38
H C 34
H C 33
M -15
S-23
H C 23
R 10
H C 22
A cid Settler
5C -634
A cid Settler
5C -633
H C 40
A C 26
Spen t A cid
A C 37
A C 34
A C 45
M -17
M -13
A C 23
S-11
5C -627
A C 40
H C 19
A C 29
A C 31
H C 16
5C -629
A C 42
H C 14
R 19
R 12
S-27 H C 41
R 11
R 16
H C 45
R 15
op tim iza tio n
Steady State Detection
op tim iza tio n
s ettlin g
op tim iza tio n
tim e
s ettlin g
tim e
ou tp ut
v aria ble
Execution frequency must
be greater than the plant
settling time (time to
return to steady state).
ex ec ution
ex ec ution
fre que nc y
fre que nc y
tim e
a . T im e b e tw e e n o p tim iza tio n s is lo n g e r th a n se ttlin g tim e
op tim iza tio n
op tim iza tio n
op tim iza tio n
s ettlin g
tim e
ou tp ut
v aria ble
ex ec ution
ex ec ution
fre que nc y
fre que nc y
tim e
b . T im e b e tw e e n o p tim iza tio n s is le ss th a n se ttlin g tim e
On-Line Optimization - Distributed Control
System Interface
No
W a it
1 m in u te
P la n t S te a d y?
S e le cte d p la n t
ke y m e a s u re m e n ts
Plant must at steady state
when data extracted from
DCS and when set points
sent to DCS.
Plant models are steady state
models.
P la n t M o d e l:
M e a su re m e n ts
E q u a lity c o n st ra in t s
D a ta V a lid a t io n
V a lid a te d m e a su re m e n ts
P la n t M o d e l:
E q u a lity c o n st ra in t s
Coordinator program
P a ra m e te r E stim a tio n
U p d a te d p a ra m e te rs
P la n t m o d e l
E c o n o m ic m o d e l
C o n tro lle r li m its
E c o n o m ic O p tim iza tio n
No
P la n t S te a d y?
S e le cte d p la n t
m e a su re m e n ts &
co n tro lle r lim its
Im p le m e n t O p tim a l
S e tp o in ts
L in e -O u t P e rio d
9 0 m in u te s
S om e O the r C onside rations
R edundan cy
O bserveability
Va rian ce e stim ation
C losing the loo p
D ynam ic d a ta re concilia tion
a n d p aram ete r e stim ation
Ad ditional O bservation s
Most d ifficult p art of on-lin e op tim ization is d eveloping an d
validating th e p rocess an d econom ic m odels.
Most valuable in form ation obtain ed from on -lin e op tim ization is a
m ore th orough u nderstanding of the p rocess
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
Mixed Integer Process Example
F 4B
F lo w ra t e o f B
purch as ed (t o ns/h r)
F 1A
F lo w ra t e o f A
( t ons /hr)
1
6
F 5B
2
A B
8
P roces s 2
B  C
4
P r oc e ss 1
F 8C
F lo w ra t e o f C ( t o ns/h r)
F 6B
9
F 9B
F lo w ra t e o f B
unrea ct ed (t o ns/ hr)
5
F 2B
F lo w ra t e o f B
( t ons /hr)
3
F 7B
7
P roces s 3
12
F 12 C
F lo w ra t e o f C
pro duct ( to ns/ hr)
10
B  C
F 10 C
F lo w ra t e o f C ( t o ns/h r)
F 3A
F lo w ra t e o f A
unrea ct ed (t o ns/ hr)
11
F 11 B
F lo w ra t e o f B
unrea ct ed (t o ns/ hr)
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 F1A - 400 F6B - 550 F7B - 1,000y1 - 1,500y2 - 2,000y3 -500 F1A - 950 F4B + 1,800 F12C
subject to:
mass yields
-0.90 F1A + F2B = 0
-0.10 F1A + F3A = 0
-0.82 F6B + F8C = 0
-0.18 F6B + F9B = 0
-0.95 F7B + F10C = 0
-0.05 F7B + F11B = 0
F2B + F4B - F5B = 0
node MB
F5B = F6B - F7B = 0
F8C + F10C - F12C= 0
availability of A
F1A < 16 y1
availability of B
F4B < 20 y4
demand for C
F8C < 10 y2
Demand for C from either Process 2,
F10C < 10 y3
stream F8C or Process 3, stream F10C
y2 + y3 = 1
Select either Process 1 or Purchase B
y1 + y4 = 1
Select either Process 2 or 3
integer constraint
Availability of raw material A to make B
Availability of purchased material B
Branch and bound algorithm used for optimization
Branch and Bound Algorithm
LP Relaxation Solution
Max: 5x1 + 2x2 =P
Subject to:
P = 22.5
x1 + x2 < 4.5
x1 = 4.5
-x1 +2x2 < 6.0
x2 = 0
x1 and x2 are integers > 0
Branch on x1, it is not an integer in the LP Relaxation Solution
Form two new problems by adding constraints x1>5 and x1<4
Max: 5x1 + 2x2 =P
Subject to:
Max: 5x1 + 2x2 =P
x1 + x2 < 4.5
Subject to: x1 + x2 < 4.5
-x1 +2x2 < 6.0
-x1 + 2x2 < 6.0
x1
x1
>5
<4
Branch and Bound Algorithm
Max:
Subject to:
5x1 + 2x2 =P
Max: 5x1 + 2x2 =P
x1 + x2 < 4.5
Subject to:
x1 + x2 < 4.5
-x1 +2x2 < 6.0
-x1 +2x2 < 6.0
x1
>5
x1
<4
infeasible
LP solution P = 21.0
no further evaluations required
x1 = 4
x2 = 0.5
branch on x2
Form two new problems by adding constraints x2 > 1 and x2< 0
Max:
Subject to:
5x1 + 2x2 =P
x1 + x2 < 4.5
-x1 +2x2 < 6.0
x1
<4
x2 > 1
Max: 5x1 + 2x2 =P
Subject to: x1 + x2 < 4.5
-x1 +2x2 < 6.0
x1
<4
x2 < 0 =0
Branch and Bound Algorithm
Max: 5x1 + 2x2 =P
Subject to:
x1 + x2 < 4.5
-x1 +2x2 < 6.0
x1
<4
x2 > 1
P = 19.5
x1 = 3.5
x2 = 1
Max: 5x1 + 2x2 =P
Subject to: x1 + x2 < 4.5
-x1 +2x2 < 6.0
x1
<4
x2 < 0
P = 20
x1 = 4
x2 = 0
optimal solution
Branch and Bound Algorithm
22.5
LP relaxation solution
X1 = 4.5
X2 = 0
21.0
Infeasible
X1 > 5
X1 < 4
19.5
20
X1 < 4
X2 > 1
X1 <
4
X2 <
Branch and Bound Algorithm
17.4
13.7
Inf
17
12
16.8
inf
Integer solution
15.6
15
Integer solution –
optimal solution
inf
inf
Mixed Integer Nonlinear Programming
M IN L P P roblem
F ix B ina ry V ariab les 'Y '
N e w V alue s of Y
S olve R ela xed N L P P ro blem
T o G et U p per B ou nd Z U
S olve M IL P M as ter P roblem
T o G e t L ow er B ound z L
Yes
Is z L  z
U
?
No
O p tim al S olution
Flow Chart of GBD Algorithm to Solve MINPL Problems,
igure
1.1(b). F 1986,
low CMathematical
hart of G BDProgramming,
and O A /ER Vol.
A lgorithm
to
DuranFand
Grossmann,
36, p. 307-339
S olve M IN LP Problem s.
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
Optimization of Chemical Production Complexes
• Opportunity
– New processes for conversion of surplus carbon
dioxide to valuable products
• Methodology
– Chemical Complex Analysis System
– Application to chemical production complex in
the lower Mississippi River corridor
Plants in the lower Mississippi River Corridor
Source: Peterson, R.W., 2000
Some Chemical Complexes in the World
• North America
– Gulf coast petrochemical complex in Houston area
– Chemical complex in the Lower Mississippi River
Corridor
• South America
– Petrochemical district of Camacari-Bahia (Brazil)
– Petrochemical complex in Bahia Blanca (Argentina)
• Europe
– Antwerp port area (Belgium)
– BASF in Ludwigshafen (Germany)
• Oceania
– Petrochemical complex at Altona (Australia)
– Petrochemical complex at Botany (Australia)
Plants in the lower Mississippi River Corridor, Base Case. Flow Rates in Million Tons Per Year
claysettling
ponds
reclaim
old mines
phosphate
rock
rock slurry
[Ca3(PO4)2...]
slurry water
rain
decant
water
decant water
fines
(clay, P2O5)
100's of
evaporated
acres of
Gypsum
gypsum
Stack
slurried gypsum
tailings
(sand)
bene-fici-ation
plant
>75 BPL
<68 BPL
5.3060
2.8818
mine
rock
Frasch
mines/
wells
Claus
sulfur
air
BFW
H2O
1.1891
7.6792
5.7683
0.7208
1.1891
sulfuric
acid
plant
0.5754
recovery
3.6781 H2SO4
5.9098
vent
1.9110 LP steam
0.4154 blowdown
2.8665
0.0012 others
H2SiF6
H2O
others
4.5173
3.6781
phosphoric
acid
plant
2.3625
0.2212
rock
1.0142
0.3013
2.6460
cooled
LP
H2O
vapor
P2O5
0.5027
inert
0.1238
H3PO4 selling
2.3625
1.8900
HP steam
from HC's
IP
fuel
BFW
0.0501
1.2016
power
gene-ration
3.8135
LP
0.8301
H2O
0.1373
CO2
1,779 elctricity
TJ
P2O5
NH3
0.7518
vent
0.9337 air
air
air
natural gas
0.7200
0.2744
steam
0.5225
ammonia
plant
NH3
CO2
0.6581
0.7529
H2O
purge
0.0938
0.0121
0.0995 H2O
0.0536
nitric
acid plant
0.0493
other use
3.2735
0.0567
0.0732
LP steam
0.0374
urea
urea
H2O
cw
NH3
CO2
0.2184
Ammonium
NH4NO3
NH3 Nitrate plant H2O
0.0483
0.0331
0.0256
0.0742
0.0299
0.0374
0.0001
0.0001
CO2
steam
0.0629
0.0511
0.0682
methanol
plant
vent 0.0008
CH3OH
0.1814
0.5833
0.2278
0.0507
ethylbenzene
0.0000
0.8618
ethylbenzene
ethylbenzene
0.8618
styrene
0.7533
0.0355
0.0067
0.0156
0.0507
0.0045
0.0044
0.1771
CH4
benzene
ethylene
benzene
MAP [11-52-0]
0.2931
DAP [18-46-0]
1.8775
0.0279
urea
0.0326
UAN
plant
UAN
0.0605
urea [CO(NH2)2]
0.0416
CO2
0.6124
vent
0.0097
HF
0.3306
0.3306
urea
plant
0.0265
H2O
0.7137
Mono& DiAmmonium
Phosphates
granulation
0.7487
GTSP [0-46-0]
AN [NH4NO3]
HNO3
NH3
CO2
for DAP %N
control
NH3
urea
2.1168
0.4502
0.0256
inert
0.2917
0.1695
Granular
Triple
Super
Phosphate
styrene
fuel gas
toluene
C
benzene
0.0005
acetic
acid
0.0082
acetic acid
H2O
0.0012
Commercial Uses of CO2
Chemical synthesis in the U. S. consumes
110 million m tons per year of CO2
− Urea (90 million tons per year)
–
–
–
–
–
Methanol (1.7 million tons per year)
Polycarbonates
Cyclic carbonates
Salicylic acid
Metal carbonates
Surplus Carbon Dioxide
• Ammonia plants produce 0.75 million tons per
year in lower Mississippi River corridor.
• Methanol and urea plants consume 0.14
million tons per year.
• Surplus high-purity carbon dioxide 0.61
million tons per year vented to atmosphere.
• Plants are connected by CO2 pipelines.
Greenhouse Gases as Raw Material
From Creutz and Fujita, 2000
Some Catalytic Reactions of CO2
Hydrogenation
Hydrolysis and Photocatalytic Reduction
CO2 + 3H2  CH3OH + H2O
methanol
CO2 + 2H2O CH3OH + O2
2CO2 + 6H2  C2H5OH + 3H2O
ethanol
CO2 + H2O  HC=O-OH + 1/2O2
CO2 + H2  CH3-O-CH3
dimethyl ether
CO2 + 2H2O  CH4 + 2O2
Hydrocarbon Synthesis
CO2 + 4H2  CH4 + 2H2O
methane and higher HC
2CO2 + 6H2  C2H4 + 4H2O
ethylene and higher olefins
Carboxylic Acid Synthesis
Other Reactions
CO2 + H2  HC=O-OH
formic acid
CO2 + ethylbenzene styrene
CO2 + CH4  CH3-C=O-OH
acetic acid
CO2 + C3H8  C3H6 + H2 + CO
dehydrogenation of propane
CO2 + CH4  2CO + H2 reforming
Graphite Synthesis
CO2 + H2  C + H2O
CH4  C + H2
CO2 + 4H2  CH4 + 2H2O
Amine Synthesis
CO2 + 3H2 + NH3  CH3-NH2 + 2H2O
methyl amine and
higher amines
Methodology for Chemical Complex Optimization
with New Carbon Dioxide Processes
•
•
•
•
•
Identify potentially new processes
Simulate with HYSYS
Estimate utilities required
Evaluate value added economic analysis
Select best processes based on value added
economics
• Integrate new processes with existing ones to
form a superstructure for optimization
Twenty Processes Selected for HYSYS Design
Chemical
Synthesis Route
Reference
Methanol
CO2 hydrogenation
CO2 hydrogenation
CO2 hydrogenation
CO2 hydrogenation
CO2 hydrogenation
Nerlov and Chorkendorff, 1999
Toyir, et al., 1998
Ushikoshi, et al., 1998
Jun, et al., 1998
Bonivardi, et al., 1998
Ethanol
CO2 hydrogenation
CO2 hydrogenation
Inui, 2002
Higuchi, et al., 1998
Dimethyl Ether
CO2 hydrogenation
Jun, et al., 2002
Formic Acid
CO2 hydrogenation
Dinjus, 1998
Acetic Acid
From methane and CO2
Taniguchi, et al., 1998
Styrene
Ethylbenzene dehydrogenation
Ethylbenzene dehydrogenation
Sakurai, et al., 2000
Mimura, et al., 1998
Methylamines
From CO2, H2, and NH3
Arakawa, 1998
Graphite
Reduction of CO2
Nishiguchi, et al., 1998
Hydrogen/
Synthesis Gas
Methane reforming
Methane reforming
Methane reforming
Methane reforming
Song, et al., 2002
Shamsi, 2002
Wei, et al., 2002
Tomishige, et al., 1998
Propylene
Propane dehydrogenation
Propane dehydrogenation
Takahara, et al., 1998
C & EN, 2003
Integration into Superstructure
• Twenty processes simulated
• Fourteen processes selected based
on value added economic model
• Integrated into the superstructure for
optimization with the System
New Processes Included in Chemical Production Complex
Product
Synthesis Route
Value Added Profit (cents/kg)
Methanol
Methanol
Methanol
Methanol
Ethanol
Dimethyl Ether
Formic Acid
Acetic Acid
Styrene
Methylamines
Graphite
Synthesis Gas
Propylene
Propylene
CO2 hydrogenation
CO2 hydrogenation
CO2 hydrogenation
CO2 hydrogenation
CO2 hydrogenation
CO2 hydrogenation
CO2 hydrogenation
From CH4 and CO2
Ethylbenzene dehydrogenation
From CO2, H2, and NH3
Reduction of CO2
Methane reforming
Propane dehydrogenation
Propane dehydrogenation with CO2
2.8
3.3
7.6
5.9
33.1
69.6
64.9
97.9
10.9
124
65.6
17.2
4.3
2.5
Application of the Chemical Complex Analysis
System to Chemical Complex in the Lower
Mississippi River Corridor
• Base case – existing plants
• Superstructure – existing and
proposed new plants
• Optimal structure – optimal
configuration from existing and
new plants
Chemical Complex Analysis System
Plants in the lower Mississippi River Corridor, Base Case. Flow Rates in Million Tons Per Year
claysettling
ponds
reclaim
old mines
phosphate
rock
rock slurry
[Ca3(PO4)2...]
slurry water
rain
decant
water
decant water
fines
(clay, P2O5)
100's of
evaporated
acres of
Gypsum
gypsum
Stack
slurried gypsum
tailings
(sand)
bene-fici-ation
plant
>75 BPL
<68 BPL
5.3060
2.8818
mine
rock
Frasch
mines/
wells
Claus
sulfur
air
BFW
H2O
1.1891
7.6792
5.7683
0.7208
1.1891
sulfuric
acid
plant
0.5754
recovery
3.6781 H2SO4
5.9098
vent
1.9110 LP steam
0.4154 blowdown
2.8665
0.0012 others
H2SiF6
H2O
others
4.5173
3.6781
phosphoric
acid
plant
2.3625
0.2212
rock
1.0142
0.3013
2.6460
cooled
LP
H2O
vapor
P2O5
0.5027
inert
0.1238
H3PO4 selling
2.3625
1.8900
HP steam
from HC's
IP
fuel
BFW
0.0501
1.2016
power
gene-ration
3.8135
LP
0.8301
H2O
0.1373
CO2
1,779 elctricity
TJ
P2O5
NH3
0.7518
vent
0.9337 air
air
air
natural gas
0.7200
0.2744
steam
0.5225
ammonia
plant
NH3
CO2
0.6581
0.7529
H2O
purge
0.0938
0.0121
0.0995 H2O
0.0536
nitric
acid plant
0.0493
other use
3.2735
0.0567
0.0732
LP steam
0.0374
urea
urea
H2O
cw
NH3
CO2
0.2184
Ammonium
NH4NO3
NH3 Nitrate plant H2O
0.0483
0.0331
0.0256
0.0742
0.0299
0.0374
0.0001
0.0001
CO2
steam
0.0629
0.0511
0.0682
methanol
plant
vent 0.0008
CH3OH
0.1814
0.5833
0.2278
0.0507
ethylbenzene
0.0000
0.8618
ethylbenzene
ethylbenzene
0.8618
styrene
0.7533
0.0355
0.0067
0.0156
0.0507
0.0045
0.0044
0.1771
CH4
benzene
ethylene
benzene
MAP [11-52-0]
0.2931
DAP [18-46-0]
1.8775
0.0279
urea
0.0326
UAN
plant
UAN
0.0605
urea [CO(NH2)2]
0.0416
CO2
0.6124
vent
0.0097
HF
0.3306
0.3306
urea
plant
0.0265
H2O
0.7137
Mono& DiAmmonium
Phosphates
granulation
0.7487
GTSP [0-46-0]
AN [NH4NO3]
HNO3
NH3
CO2
for DAP %N
control
NH3
urea
2.1168
0.4502
0.0256
inert
0.2917
0.1695
Granular
Triple
Super
Phosphate
styrene
fuel gas
toluene
C
benzene
0.0005
acetic
acid
0.0082
acetic acid
H2O
0.0012
H2O
reducing gas
air
gyp
vent
CaCO3
H2O
S
SO2
S & SO2
recovery
plant
water
air
rock
SiO2
C
air
sulfuric
dioxide
recovery
plant
wood gas
gyp
vent
CaO
H2O
HCl
SO2
rock
rain
decant
water
claysettling
ponds
reclaim
old mines
phosphate
rock
rock slurry
[Ca3(PO4)2...]
slurry water
fines
>75BPL
bene-fici-ation
plant
<68 BPL
H2O
Claus
H2O
evaporated
gypsum
H2SiF6
H2O
others
rock
mine
SO2
S
air
BFW
HF
CaCl2
P2O5
others
H2O
slurried
gypsum
tailings
Frasch
mines/
wells
CaSiO3
CaF2
P2O5
CO2
rock
(clay, P2O5)
(sand)
HCL
to phosacid
100's of
acres of
Gypsum
Stack
decant water
Superstructure
vent
electric
furnace
phosphoric
acid
plant
H2SO4
sulfuric
acid
plants
vent
LP steam
blowdown
others
vapor
cooled LP
H2O
Granular
Triple
Super
Phosphate
P2O5
P2O5
P2O5
LP
HF
GTSP [0-46-0]
others
P2O5
recovery
HP steam
from HC's
P2O5
P2O5
IP
fuel
BFW
power
gene-ration
LP
H2O
H2O
CO2
electricity
vent
air
P2O5
NH3
urea
for DAP %N P2O5
control
air
air
NH3
CO2
natural gas
steam
ammonia
plant
NH3
nitric
acid
HNO3
NH3
H2O
purge
Ammonium
Nitrate
NH4NO3
H2O
urea
NH3
CO2
LP steam
urea
plant
other use
H2O
MonoMAP [11-52-0]
& Diothers
Ammonium DAP [18-46-0]
Phosphates
granulation
NH3
AN [NH4NO3]
UAN
plant
UAN
urea
urea
H2O
cooled LP
NH3 purge
CO2 purge
CH3OH
vent
CO2
steam
CH4
methanol
plant
CH3OH
CO2
CH4
acetic
acid
CH3COOH
H2O
CO2
CO2
CH4
CO2
CH4
graphite
&
H2
H2O
C
H2
syngas
CO
H2
H2
CO2
H2
H2
propane
propene
&
H2
propene
CO2
H2
methanol
Bonivardi
formic acid
methylamines
CO
propylene
H2O
H2
CO2
H2
EtOH
new
styrene
plant
CO
styrene
H2O
CO2
H2
DME
styrene
styrene
fuel gas
toluene
C
benzene
propylene
plant
CO2
CO2
ethylbenzene
benzene
ethylene
benzene
ethylbenzene
ethylbenzene
CO
MeOH
H2O
formic
acid
NH3
propane
CH3COOH
H2
CO2
CO2
CH4
new
acetic
acid
CO
MMA
DMA
H2O
EtOH
H2O
CO
DME
MeOH
H2O
Plants in the Superstructure
Plants in the Base Case
• Ammonia
• Nitric acid
• Ammonium nitrate
• Urea
• UAN
• Methanol
• Granular triple super
phosphate
• MAP and DAP
• Sulfuric acid
• Phosphoric acid
• Acetic acid
• Ethylbenzene
• Styrene
Plants Added to form the Superstructure
• Acetic acid from CO2 and CH4
• Graphite and H2
• Syngas from CO2 and CH4
• Propane dehydrogenation
• Propylene from propane and CO2
• Styrene from ethylbenzene and CO2
• Methanol from CO2 and H2 (4)
• Formic acid
• Methylamines
• Ethanol
• Dimethyl ether
• Electric furnace phosphoric acid
• HCl process for phosphoric acid
• SO2 recovery from gypsum
• S and SO2 recovery from gypsum
Superstructure Characteristics
Options
-
Three options for producing phosphoric acid
Two options for producing acetic acid
Two options for recovering sulfur and sulfur dioxide
Two options for producing styrene
Two options for producing propylene
Two options for producing methanol
Mixed Integer Nonlinear Program
843
23
777
64
continuous variables
integer variables
equality constraint equations for material and energy balances
inequality constraints for availability of raw materials
demand for product, capacities of the plants in the complex
Some of the Raw Material Costs, Product Prices and
Sustainability Cost and Credits
Raw Materials
Cost
($/mt)
Sustainable Cost and Credits
Natural gas
235
Credit for CO2 consumption
6.50
Ammonia
224
Debit for CO2 production
3.25
Methanol
271
11
Acetic acid 1,032
Phosphate rock
Wet process
27
Credit for HP Steam
Electro-furnace
34
Credit for IP Steam
Haifa process
34
GTSP process
HCl
Cost/Credit
($/mt)
Products Price
($/mt)
7
GTSP
132
Credit for gypsum consumption
5.0
MAP
166
32
Debit for gypsum production
2.5
DAP
179
95
Debit for NOx production
1,025
NH4NO3
146
Debit for SO2 production
192
Urea
179
120
Sulfur
Frasch
53
UAN
Claus
21
Phosphoric 496
Sources: Chemical Market Reporter and others for prices and costs,
and AIChE/CWRT report for sustainable costs.
claysettling
ponds
reclaim
old mines
phosphate
rock
rock slurry
[Ca3(PO4)2...]
slurry water
rain
decant
water
decant water
fines
(clay, P2O5)
100's of
evaporated
acres of
Gypsum
gypsum
Stack
slurried gypsum
tailings
(sand)
bene-fici-ation
>75 BPL
<68 BPL
rock
Frasch
mines/
wells
Claus
sulfur
air
BFW
H2O
1.1891
7.6792
5.7683
0.7208
sulfuric
acid
1.1891
0.5754
recovery
Optimal Structure
5.3060
2.8818
mine
3.6781 H2SO4
5.9098
vent
1.9110 LP steam
0.4154 blowdown
2.8665
0.0012 others
H2SiF6
H2O
others
4.5173
3.6781
phosphoric
acid
(wet process)
2.3625
0.2212
rock
vapor
1.0142
0.3013
2.6460
cooled
LP
H2O
P2O5
2.3625
1.8900
0.5027
0.1238
inert
H3PO4 selling
from HC's
0.1695
Granular
Triple
Super
Phosphate
0.7487
GTSP [0-46-0]
0.0097 HF
0.0265
H2O
HP steam
IP
fuel
BFW
0.1068
2.5639
power
gene-ration
5.0147
LP
0.9910
H2O
0.2929
CO2
2,270 elctricity
TJ
P2O5
NH3
0.7518
vent
air
0.0995 H2O
0.9337 air
air
natural gas
0.7200
0.2744
NH3
CO2
0.6581
0.7529
H2O
purge
0.0938
0.0121
0.0283
nitric
acid
0.0493
CO2
other use
4.4748
0.2250
vent
CO2
steam
0.0567
0.0732
LP steam
0.0374
0.0629
0.0511
0.0682
methanol
MAP [11-52-0]
0.2931
DAP [18-46-0]
1.8775
NH3
0.3306
0.2184
Ammonium
NH4NO3
NH3
Nitrate
H2O
0.0483
0.0331
0.3306
urea
plant
0.7137
Mono& DiAmmonium
Phosphates
granulation
AN [NH4NO3]
HNO3
NH3
ammonia
steam
0.5225
2.1168
0.4502
0.0256
for DAP %N
0.2917
control urea
inert
urea
urea
H2O
cw
NH3
CO2
0.0279
0.0256
0.0742
0.0299
0.0374
0.0001
0.0001
urea
0.0326
UAN
plant
UAN
0.0605
urea [CO(NH2)2]
0.0416
vent 0.0008
CH3OH
0.1814
CO2
0.0060
CH4
0.0022
new
acetic
acid
0.0082 CH3COOH
0.3859
CO2
0.0679
CH4
0.0367
CO2
CH4
0.1174
0.0428
0.0438
propane
0.0439
propane
CO2
benzene
ethylene
benzene
0.5833
0.2278
0.0507
graphite
&
H2
syngas
propene
&
H2
H2O 0.0556
C
0.0460
H2
0.0030
CO
H2
H2 sale
0.1494
0.0108
CO2
H2
0.0745
0.0034
0.0020 H2
CO2
H2
0.1042
0.0134
NH3 0.0254
0.0418 propene
propylene
plant
0.0140
0.0419
0.0090
0.0010
CO
propene
H2O
H2
styrene
0.7533
0.0355
0.0067
0.0156
0.0507
styrene
fuel gas
toluene
C
benzene
0.0219
0.0000
ethylbenzene
0.8618
ethylbenzene
0.8618
0.0000
0.0779 formic acid
formic
acid
methylamines
0.0068
0.0264
0.0288
0.0809
CO
MMA
DMA
H2O
Plants in the Optimal Structure from the Superstructure
Existing Plants in the Optimal Structure
Ammonia
Nitric acid
Ammonium nitrate
Urea
UAN
Methanol
Granular triple super phosphate (GTSP)
MAP & DAP
Power generation
Contact process for Sulfuric acid
Wet process for phosphoric acid
Ethylbenzene
Styrene
Existing Plants Not in the Optimal
Structure
Acetic acid
New Plants in the Optimal Structure
Formic acid
Acetic acid – new process
Methylamines
Graphite
Hydrogen/Synthesis gas
Propylene from CO2
Propylene from propane dehydrogenation
New Plants Not in the Optimal Structure
Electric furnace process for phosphoric acid
HCl process for phosphoric acid
SO2 recovery from gypsum process
S & SO2 recovery from gypsum process
Methanol - Bonivardi, et al., 1998
Methanol – Jun, et al., 1998
Methanol – Ushikoshi, et al., 1998
Methanol – Nerlov and Chorkendorff, 1999
Ethanol
Dimethyl ether
Styrene - new process
Comparison of the Triple Bottom Line for the Base Case and Optimal
Structure
Income from Sales
Economic Costs
(Raw Materials and Utilities)
Raw Material Costs
Utility Costs
Environmental Cost
(67% of Raw Material Cost)
Sustainable Credits (+)/Costs (-)
Triple Bottom Line
Base Case
million dollars/year
1,316
560
Optimal Structure
million dollars/year
1,544
606
548
12
365
582
24
388
21
412
24
574
Carbon Dioxide Consumption in Bases Case
and Optimal Structure
CO2 produced by NH3 plant
CO2 consumed by methanol,
urea and other plants
CO2 vented to atmosphere
Base Case
million metric tons/year
0.75
0.14
Optimal Structure
million metric tons/year
0.75
0.51
0.61
0.24
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
Multi-Criteria or Multi-Objective Optimization
 y1 ( x ) 


 y2 ( x) 

opt  




 y ( x)
 p

Subject to: fi(x) = 0
min: cost
max: reliability
min: waste generation
max: yield
max: selectivity
Multi-Criteria Optimization - Weighting Objectives Method

opt w1 y1 ( x )  w 2 y 2 ( x )     w p y p ( x )

Subject to: fi(x) = 0
with ∑ wi = 1
Optimization with a set of weights generates efficient
or Pareto optimal solutions for the yi(x).
Efficient or Pareto Optimal Solutions
Optimal points where attempting to improving the value of one objective
would cause another objective to decrease.
There are other methods for multi-criteria optimization,
e.g., goal programming, but this method is the most widely used one
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
Multicriteria Optimization
Convert to a single criterion optimization problem
max:
subject to:
w 1P + w 2 S
Multi-plant material and energy balances
Product demand, raw material availability,
plant capacities
S u s ta in a b le C r e d it/C o s t ( m illio n d o lla r s /y e a r )
Multicriteria Optimization
W1 = 0 no profit
Debate about which
weights are best
25
20
15
10
5
0
W1 =1 only profit
-5
-1 0
-1 5
0
100
200
300
400
500
P r o f it ( m illio n d o lla r s /y e a r )
600
Monte Carlo Simulation
• Used to determine the sensitivity of the optimal solution to the
costs and prices used in the chemical production complex
economic model.
•Mean value and standard deviation of prices and cost are used.
• The result is the cumulative probability distribution, a curve of
the probability as a function of the triple bottom line.
• 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.
• This curve is used to determine upside and downside risks
Monte Carlo Simulation
Triple Bottom Line
Mean $513million per year
Standard deviation - $109 million per year
C u m u la tiv e P r o b a b ility ( % )
100
80
60
50% probability that the triple bottom
line will be $513 million or less
40
20
Optimal structure changes with
changes in prices and costs
0
200
400
600
800
1000
1200
T r ip le B o tto m L in e ( m illio n d o lla r s /y e a r )
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
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.
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.
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
www.mpri.lsu.edu
Descargar

Optimization - Louisiana State University