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