optimal controlled variable selection for individual process units
DESCRIPTION
Optimal controlled variable selection for individual process units. Ramprasad Yelchuru Sigurd Skogestad. Outline. Problem formulation, c = Hy Convex formulation (full H) CVs for Individual unit control (Structured H) MIQP formulations Distillation Case study Conclusions. - PowerPoint PPT PresentationTRANSCRIPT
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 1
Optimal controlled variable selection for individual process units
Ramprasad YelchuruSigurd Skogestad
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 2
Outline
1. Problem formulation, c = Hy
2. Convex formulation (full H)
3. CVs for Individual unit control (Structured H)
4. MIQP formulations
5. Distillation Case study
6. Conclusions
CV – Controlled VariablesMIQP - Mixed Integer Quadratic Programming
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 3
Optimal steady-state operation
( , ) ( , )opt optL J u d J u d
Ref: Halvorsen et al. I&ECR, 2003 Kariwala et al. I&ECR, 2008
1. Problem Formulation
21/2 1( )yavg uu F
L J HG HY
Loss is due to(i) Varying disturbances(ii) Implementation error in controlling c at set point cs
31( , ) ( , ) ( ) ( ) ( )
2T
opt u opt opt uu optJ u d J u d J u u u u J u u
1[( ) ]y yuu ud d d nY G J J G W W
u
J
( )opt ou d
Loss
min ( , )u
J u d'd
Controlled variables,c yH
ydG
cs = constant +
+
+
+
+
- K
H
yG y
'yn
c
u
dW nW
d
optu
Assumptions: (1) Active constraints are controlled(2) Quadratic nature of J around uopt(d)(3) Active constraints remain same throughout the analysis
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 4
2. Convex formulation (full H)1/2 1min ( )yuu FH
J HG HY Seemingly Non-convex
optimization problem
-1 -1 -1 1 -11 y 1 y y y (H G ) H = (DHG ) DH = (HG ) D DH = (HG ) H
1H DH
D : any non-singular matrix
Objective function unaffected by D.So can choose freely.
We made H unique by adding a constraint as
yHG
1/2yuuHG J
Hmin HY F
subject to 1/ 2yuuHG J
Full HConvex
optimization problem
Global solution Problem is convex in decision matrix H
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 5
Vectorization
X
1 2
1 2 2*
( 1)* 1 ( 1)* 2 *
ny
ny ny ny
nc ny nc ny nc ny nu ny
x x x
x x xH
x x x
TX H
Hmin HY F
subject to 1/ 2yuuHG J
min
.
T T
X
T
X Y Y X
st G X J
Problem is convex QP in decision vector
1
2
* ( * ) 1nu ny nu ny
x
xX
x
1 1 ( 1)* 1
2 2 ( 1)* 2
2* *
ny nc ny
ny nc ny
ny ny nc ny ny nu
x x x
x x xX
x x x
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 6
Full H
11
1 1
1 1 1
1
1 1 1
1
1
1, 1 1,
2, 1
1,1
2,
1,1 1,
,1 ,2 ,
1, 1 1,
,
1,
1 ,
1
,
,
y y
y y
u y u y
u y u
u u y
u u u y y
y
u u y
n
n n n n
n n n n
Bottom T
n n
op
n n
n n
n n n
n n
n
n n
h
h h
h h
h
h h
h h
hh
h h hh
H
h
u yn n
T1, T2, T3,…, T41
Tray temperaturesqF
Top sectionT21, T22, T23,…, T41
Bottom sectionT1, T2, T3,…, T20
c = Hy
1
2
cc
c
1
2
41
T
Ty
T
Binary distillation column
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 7
Need for structural constraints (Structured H)
11
1 1
1 1 1
1
1 1 1
1
1
1, 1 1,
2, 1
1,1
2,
1,1 1,
,1 ,2 ,
1, 1 1,
,
1,
1 ,
1
,
,
y y
y y
u y u y
u y u
u u y
u u u y y
y
u u y
n
n n n n
n n n n
Bottom T
n n
op
n n
n n
n n n
n n
n
n n
h
h h
h h
h
h h
h h
hh
h h hh
H
h
u yn n
Binary distillation column
T1, T2, T3,…, T41
Tray temperaturesqF
Transient response for 5% step change in boil up (V)
Top sectionT21, T22, T23,…, T41
Bottom sectionT1, T2, T3,…, T20
*Compositions are indirectly controlledby controlling the tray temperatures
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 8
Need for structural constraints (Structured H)
T1, T2, T3,…, T41
Tray temperaturesqF
Top sectionT21, T22, T23,…, T41
Bottom sectionT1, T2, T3,…, T20
Individual Unit
control1
1 1 1
1
1 1 1
1
1,1 1,
,1 ,
, 1 1,
, 1 ,
0 0
0 0
0 0
0 0
0 0
u y u
u y
y
u u y
y
u y u y
Bot
n
tom T
n
n
op
IUn n n
n n n
n
n
n
n
n
h h
h hh
h
H h
h
Transient response for 5% step change in boil up (V)
Binary distillation column
Structured H is required for better dynamics and controllability
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 9
3. CVs for Individual Unit control (Structured H)
1/2 1min ( )yuu FH
J HG HY
1H DH
-1 -1 -1 1 -11 y 1 y y y (H G ) H = (DHG ) DH = (HG ) D DH = (HG ) H
1H DHD : any non-singular matrix
So we can use D to match certain elements of toyHG 1/2
uuJ
For individual unit control HIU
only block diagonal D preserve the structure in H and
1 1 1
1
1 1
1
1
1, 1 1,
,
Re
1, 1
1
1 ,
,1 ,
,
0 0
0 0
0 0
0 0
0 0
y
u u
u y u y
u y u y
y
u y
n n
actor Separ
n n
n n n n
n
n n n
ator
IU
n n
h h
H h h
h
h
h
h
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 10
2,3 2,4 22
1,1 1,2 110 0 0;
0 0 0
h h
dH
h
dD
h
332,
11 121,1 1,2
2,1 2,2 21 22
3 2,4
00 0
0 0 ; 0
0 00 0 dh h
d dh h
h h d dH D
22
11 11
1 2
3
1,1 1,1 1,2 2,1 1,1 1,2 1,2 2,2
2,3 3,1 2,4 4,1 2,3 3,2 2,4 4,222
4
( ) ( )
( ) ( )
y y y y
y
y y y y
h G h G h G h GHG
h G h G h G h G
z
z
d
d
d
z
d
z
1
2
1 2 3 4
1 2
3 4
{0,1} 1,2, , 4
2
1
1
j
nz
nz
nz
z j
z z z z n
z z n
z z n
CVs for Individual Unit control (Structured H)
Example 1 : Example 2 :
11 1,
22 2,
1 11 1
3 22 2,1
,2
4
0 0
0 0 d h d hH D
d h d hH
This results in convex upper bound1/2yuuHG J
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 11
Controlled variable selectionOptimization problem :
Minimize the average loss by selecting H and CVs as
(i) best individual measurements
(ii) best combinations of all measurements
(iii) best combinations with few measurements
Minimize the average loss by selecting H and CVs as
(i) best individual measurements of disjoint measurement sets
(ii) best combinations of disjoint measurement sets of all measurements
(iii) best combinations of disjoint measurement sets with few measurements
st.
min
.
T T
X
T
X Y Y X
st G X J
H
min HY F
1/ 2yuuHG J1/2 1min ( )y
uu FHJ HG HY
1
1,1 1,4 1,1 1,4
2,1 2,4 2,1 2,4
0 0 1 0 00; ;
0 0 0 0 01IU
h h h hH D H DH
h h h h
1/2 1min ( )yuu FH
J HG HY
1
1 1
1
1
1
1 1
1, 1
R
1,
, 1 ,
1,1 1,
,
e
1 ,
0 0
0 0
.
0 0
0 0
0 0.
y
u u y
u
u y u y
u y u y
y
n n n n
actor Separato
n n n
n
n
n
n
n n
r
ns
h h
h hh
t Hh
h h
Individual unit control
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 12
4. MIQP Formulation (full H)
{0,1}
1,2, ,i
i ny
( 1)*
min
.
0 0 0 0
0 0 0 0
1,2, ,
0 0 0 0
0,1
aug
T
Taug aug
x
ynew aug
aug
i
ny i
i i
nu ny i
i
x Fx
st G x
x n
xM MxM M
for i ny
M Mx
δ
P
J
1
2
( * ) 1
aug
ny nu ny ny
X
x
[ ( , )]
[ ( * , )]
[ (1, * ) (1, )]
max( ) / min( )
T
T
y Tnew
y
F Y Y zeros ny ny
G G zeros nu ny ny
zeros nu ny ones ny
upper bound for M J G
P
1 2
1 2
1 2 2*
( 1)* 1 ( 1)* 2 *
ny
ny
ny ny ny
nc ny nc ny nc ny nu ny
x x x
x x xH
x x x
We solve this MIQP for n = nu to ny
Big M approach
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 13
MIQP Formulation (Structured H)
1
2
( * ) 1u uu y y u u
aug
N
n n n n n n n
X
z
x z
z
[ ( , )]
[ ( , )]
[ (1, ) (1, )]
max( ) / min( )
T
TN y u u y u u
y TN u y y u u
yN u y
y
F Y Y zeros n n n n n n
G G zeros n n n n n
zeros n n ones n
upper bound for M J G
P
We solve this MIQP for n = nu to ny
Big M approach
1
2
1 2 3 4
1 2
3 4
{0,1} 1,2, , 4
2
1
1
j
nz
nz
nz
z j
z z z z n
z z n
z z n
Matching elements
Selecting measurements
Structured H
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 14
5. Case Study : Distillation Column
T1, T2, T3,…, T41
Tray temperaturesqF
Binary Distillation ColumnLV configuration(methanol & n-propanol)
41 Trays
Level loops closed with D,B
2 MVs – L,V41 Measurements – T1,T2,T3,…,T41
3 DVs – F, ZF, qF
*Compositions are indirectly controlledby controlling the tray temperatures
2 2
, ,
, ,
D D s B B s
D s B s
y y x xJ
y x
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 15
Case Study : Individual section control
T1, T2, T3,…, T41
Tray temperaturesqF
Top sectionT21, T22, T23,…, T41
Bottom sectionT1, T2, T3,…, T20
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 16
Case Study : Distillation Column21/2 11
( ( ) )2
yavg uu F
L J HG HY
10.83 -10.96 5.85 11.17 10.90
15.36 -15.55 8.30 15.86 15.473.88 3.88
; ;3.89 3
13.01 -12.81 5.85 13.10 12.90
8.76 -8.62 3.94 8.82 8.68
y yd uuG G J
0.2 0 0
1.96 3.96 3.88; ; 0 0.1 0 ; (0.5* (41,1))
.90 1.97 3.97 3.890 0 0.1
ud dJ W Wn diag ones
1[( ) ]y yuu ud d d nY G J J G W W
41 2 41 3 2 2 3 3 3 41 41; ; ; ; ;y yd uu ud d nG G J S J W W
Results
Data
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 17
Case Study : Distillation Column
The proposed methods are not exact (Loss should be same for H full, H disjoint with individual measurements)
Proposed method provide tight upper bounds
1 1 1
1
1
1 1 1
1
1 1
1
1
1,1 1,
,1 ,
1, 1
1,1 1,
,1 ,2 ,
1,
2, 1 2
1, 1 1,
, 1 ,
,
Re
y y
y y
u y u
y
u
yu u y
u u y
u
u y
y
u u y
n
n n n
n
n
n n
n n
n
n
act
n n n
n n
n n n n
or Separator
n n
h h
h
h h
h h
h h
h h
h
H
hh
h h
h
u yn n
1 1 1
1
1
1 1 1
1, 1
1,1 1,
,1 ,
,
1 ,
e
1
,
R
0
0 0
0
0
0
0
0 0
0
y
u u
u y u y
u y u y
y
u y
n n
actor Separ
n n
n n n n
n
n n n
ator
IU
n n
h h
H h h
h
h
h
h
Ramprasad Yelchuru, Optimal controlled variable selection for Individual process units, 18
6. Conclusions
Using steady state economics of the total plant, the optimal controlled
variables selection as
optimal individual measurements from disjoint/(individual unit)
measurement sets
combinations of optimal fewer measurements from disjoint/(individual
unit) measurement sets
is solved using MIQP based formulations.
The proposed methods are not exact, but provide upper bounds to Loss
to find CVs as combinations of measurements from individual units.