Model-Based Optimization of EconomicalGrade Changes for the Borealis

Borstar R© Polyethylene Plant

Per-Ola Larsson∗, Johan Åkesson∗,Niclas Carlsson†, Niklas Andersson∗

*Lund University, Sweden†Borealis AB, Sweden

Jan. 26, 2012

Background

◮ Polyethylene◮ Most widely used plastic in the world◮ Several different grades, i.e., types◮ Specified by quality variables:

◮ density◮ melt index◮ molecular weight distribution◮ . . .

◮ Why Grade Changes?◮ Market demands◮ Raw material and product pricing◮ Technology supports several grades

◮ How to Perform a Grade Change◮ Manipulate inflows of raw material in a continuous manner

◮ No “stop-and-go”◮ Desired: Economically optimal◮ Obey safety and constraints on e.g., flows, conc., grades.

PE3 at Borealis AB, Stenungsund

Pre-poly.reactor

Loopreactor

Gas-phasereactor

uc1

ufp3

ue1uh1up1

ue2uh2up2 uol

uop

To propanebuffer

Lightscolumn

Propanecolumn

Heaviescolumn

Product

uflare

Waste

ue3uh3ub3up3un3

◮ Inflows: catalyst, ethylene, hydrogen, propane and nitrogen◮ Outflows: propane, off-gases, flare, product, (and waste)

Modelica Model and Model Size

◮ Modelica library constructed

◮ Number of◮ states x: 46◮ algebraic variables w: 167◮ inputs u: 15◮ equations in F(⋅): 213◮ quality variables y: 7◮ operational variables z: 2

0 = F (x,x,w,u)

y = gy (x,w,u)

z = gz (x,w,u)

x(t0) = x0

Grade Definitions and Prices/Costs

Grade X he1 MI2 MImix ρmix S1 S2 S3 Ej

A 1.00 1.00 1.00 1.000000 1.000 1.000 1.000 1.24B 0.37 6.50 3.51 1.001065 1.000 1.132 0.917 1.46

±% ±5 ±5 ±5 ±0.1 ±0.5 ±0.5 ±0.5 -

◮ Grade definitions have target values and intervals.◮ Inside intervals = on-grade [ sell price Ej .◮ Outside intervals = off-grade [ sell price Eoff.◮ Costs Ci of inflows, offgas sell price Eo� and off-grade

polymer sell price Eoff.

Cc Ce Ch Cb Cp Cn Eo� Eoff

214.6 1.000 8.003 1.419 0.501 0.044 0.266 0.880

Stationary Optimization Problem◮ Instantaneous profit R j in stationarity production of grade j

R j =

production revenue︷ ︸︸ ︷

Ejws3 +

offgas revenue︷ ︸︸ ︷

Eo�uol + Eo�uop+

propane revenue︷ ︸︸ ︷

Cpwbp6

−∑

i∈{c,e,h,p}

Ciui1 −∑

i∈{e,h,p}

Ciui2 −∑

i∈{e,h,b,p,n}

Ciui3 − Cpufp3

︸ ︷︷ ︸

inflow costs

◮ Optimization problem solved using JModelica.org

maxx,x,w,u

R j

s.t. 0 = F (x,x,w,u) (dynamics)

y j = gy (x,w,u) (on specification)

z j = gz (x,w,u) (pressures in GPR)

x = 0 (stationarity)

xmin ≤ x ≤ xmax,wmin ≤ w ≤ wmax, umin ≤ u ≤ umax

Stationary Optimization – Some results

◮ Specifications on grade variables sets reactant ratios.◮ Profitable to produce[ production level at maximum.◮ Off-gases at minimum and flare closed.◮ Minimizes expensive components in off-gases.◮ Gives economically optimal operating points for grade j.

Dynamic Optimization – Ideal Profit◮ Ideal instantaneous profit R j for grade j

R j =

Effective sell price︷ ︸︸ ︷((Ej − Eoff)θ j(y) + Eoff

)ws3+Eo�uol+Eo�uop+Cpw

bp6

−∑

i∈{c,e,h,p}

Ciui1 −∑

i∈{e,h,p}

Ciui2 −∑

i∈{e,h,b,p,n}

Ciui3 − Cpufp3,

where θ j(y) is the ideal on-grade function for grade j

θ j(y) =

{

1 if yminji ≤ yi ≤ y

maxji , i ∈ {1, . . . , 7}

0 otherwise,

◮ Define a time interval [t0, t1] and a transition time tT

R =

{

RA, t0 ≤ t ≤ tT

RB, tT < t ≤ t1.

◮ Goal: Maximize cumulative profit Veco =t1∫

t0

R dt.

Ideal Effective Sell Price – Visualization

0.9

0.95

1

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Time S3 (split GPR)

Effe

ctiv

ese

llpr

ice

tT

t1

t0

Dynamic Optimization – Difficulties and a Solution

◮ Difficulties◮ Cost function is not differentiable.◮ Might be optimal to be on quality variable interval border.◮ Desired to be on-target with grade variables.◮ Desired to be on-target with operational variables.

◮ A solution◮ Smooth approximation of ideal on-grade function θ j(y).◮ Economical incentives, i.e., “price peaks” at target values.

◮ Rational function of◮ grade variables◮ target values◮ grade intervals

◮ Parameters in rational function control◮ Smoothness◮ Price peak

Smooth approximation with peaks – Visualization

0.9

0.95

1

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Time S3 (split GPR)

Effe

ctiv

ese

llpr

ice

tT

t1

t0

Dynamic Optimization Problem

maxu

∫ t1

t0

(

R − uTUdu)

dt

s.t. 0 = F(x,x,w,u), u =∫ t

t0

u dτ

y = gy (x,w,u)

z = gz (x,w,u)

xmin ≤ x ≤ xmax, wmin ≤ w ≤ wmax

umin ≤ u ≤ umax, ymin ≤ y ≤ ymax

zmin ≤ z ≤ zmax, umin ≤ u ≤ umax

x(t0) = xs, u(t0) = us

u = ue, t1 − Tc ≤ t ≤ t1◮ Control flow derivatives u as decision variables.◮ uTUdu penalizing highly varying control flows.◮ Control flows fixed at an end interval, avoiding all fresh

inflows closing.◮ Solved using JModelica.org.

Results – Quality variables

0 10 20 30 40 50 60

1

2

3

4

0 10 20 30 40 50 60

0.999

1

1.001

1.002

ρm

ix,

ρm

ixMI m

ix,MI

mix

Time

0 10 20 30 40 50 600.99

1

1.01

0 10 20 30 40 50 60

1

1.05

1.1

1.15

0 10 20 30 40 50 60

0.920.940.960.98

1

S1,S1

S2,S2

S3,S3

Time

◮ Economical incentive for on-target production largeenough.

◮ Over-/undershoots in instantaneous quality variables.◮ Preparations performed prior grade change.

Results – Control flows

0 10 20 30 40 50 600.95

1

1.05

1.1

1.15

0 10 20 30 40 50 60

1

2

3

0 10 20 30 40 50 60

0.8

0.9

1

ue2

uh2

up2

Time

0 10 20 30 40 50 600.99

1

1.01

0 10 20 30 40 50 601

2

3

4

0 10 20 30 40 50 601

2

3

ufla

reuop

uol

Time

◮ Change for a longer time period than off-grade period.◮ Significant over-/undershoots for rapid grade change.◮ Flare never used due to pure economical loss.◮ Off-gases remove hydrogen at economically beneficial

times.

Results – Production and Profit

0 10 20 30 40 50 60

0

0.5

1

0 10 20 30 40 50 600

20

40

0 10 20 30 40 50 60

0.97

0.98

0.99

1

1.01

On/

offg

rade

Cum

.pr

ofit

Pro

duct

ion

Time

◮ Most profitable product produced the longest time.◮ High production rate [ small reactor hold-up times.◮ Economically optimal transition has three phases:

1. Preparation2. Transition3. Completion

Summary

◮ A Modelica library has been constructed forthe Borealis Borstar R© Polyethylene Plant.

◮ Cost functions for optimization of economical gradechanges have been designed.

◮ Economically optimal grade changes havebeen characterized.