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.