Integration of Scheduling and Control Operationsegon.cheme.cmu.edu/ewo/docs/EWO_Seminar_09_08_2011.pdfIntegration of Scheduling and Control Operations Antonio Flores T. with colaborations

Integration of Scheduling and ControlOperations

Antonio Flores T.with colaborations from:

Sebastian Terrazas-Moreno, Miguel Angel Gutierrez-Limon, Ignacio E. Grossmann

Universidad Iberoamericana, Mexico

September 7, 2011

Antonio Flores T. with colaborations from: Sebastian Terrazas-Moreno, Miguel Angel Gutierrez-Limon, Ignacio E. Grossmann Universidad Iberoamericana, MexicoIntegration of Scheduling and Control Operations

Outline

I Introduction

I Aims of this talk

I Problem Definition

I Problem Assumptions

I Mixed-Integer Dynamic Optimization Approach

I Solving Mixed-Integer Dynamic Optimization Problems

I Single Stage Cycle Scheduling Formulation

I Full Discretization Optimal Control Formulation

I Simultaneous Scheduling and Control problems

I Single Stage Scheduling and Control Examples

I Parallel Lines Continuous Scheduling Problems

I MultiObjective Scheduling and Control Problems

I Conclusions

I Future Work


Introduction

I Enterprise Wide Optimization (EWO) deals with theoptimization of supply, manufacturing and distributionactivities so to reduce costs and inventories

I At the manufacturing level EWO considers planning,scheduling and control operations

I EWO activities are at the interface between ChemicalEngineering and Operations Research

I Nonlinear mathematical models are commonly used inmanufacturing operations


Introduction

Quoting Shobrys and White 1:

I Planning: Defines desired changes to current businessaffecting access to raw material, production/distributioncapacity, long term objectives

I Scheduling: Deals with the timing and volume of certainoperations (i.e. start time, unit to be used,processing time,production level), medium term objectives

I Control: Rejection of upsets, tracking of signals to meetcertain quality and production goals, short term objectives

1”Planning, scheduling and control systems: why cannot they worktogether”, Donald E. Shobrys, Douglas C. White, Comput.Chem.Eng. 26(2002) 149160


Introduction

PLANNING

SCHEDULING

CONTROL

PLANNING

SCHEDULING

REAL-TIME OPTIMIZATION

MPC

BASIC REGULATORY CONTROL


Introduction

I Two main approaches have been used for dealing withScheduling and Control problems

I The first one formulates the problem as a Mixed-IntegerDynamic Optimization (MIDO) problem

I MIDO problems can also incorporate logic decisions to switchamong different conditions

I The second one uses agents (i.e. heuristics) to model optimalsystem behavior

I Published SC problems include polymerization reactors, watertreatment plants, distillation columns and tubular reactors


Aims of this Talk

In this work, we propose a simultaneous approach to address schedulingand control problems for a set of continuous plants. We take advantageof the rich knowledge of scheduling and optimal control formulations,and we merge them so the final result is a formulation able to solvesimultaneous scheduling and control problems.We cast the problem as an optimization problem. In the proposedformulation:

I Integer variables are used to determine the best production sequence

I Continuous variables take into account production times, cycle time,and inventories

Because dynamic profiles of both manipulated and controlled variablesare also decision variables, the resulting problem is cast as amixed-integer dynamic optimization (MIDO) problem.


Problem Definition

Given are:

I A number of products to be manufactured in a single CSTR

I Lower bounds for the product demands

I Steady-state operating conditions for each desired product

I Cost of each product

I Inventory and raw materials costs

The problem consists in:

Simultaneous determination of a cyclic schedule (i.e. production wheel)and the control profile such that a given cost function is minimized

Major decisions involve:

I Selecting the cyclic time and Sequence in which the products will bemanufactured

I The transition times, Production rates, Length of processing times

I Amounts manufactured of each product

I Manipulated variables profiles for the transition


Problem Assumptions

I All products are manufactured ina single continuous reactor

I Products sequence follows aproduction wheel

I Cyclic time is divided into a seriesof slots

Two operations occur inside each slot:

I Transition period: dynamictransitions between two productstake place

I Production period: a givenproduct is manufactured aroundsteady-state conditions

Production period

Slot 1 Slot 2 Slot Ns

Transition period

u

Time

Dyn

am

ic s

yst

em

beh

avio

ur

Production periodTransition period

x


Mixed-Integer Dynamic Optimization Approach

I Binary variables: Production sequence (i.e. slot productassignment)

I Continuous variables: Processing times, Transition times,Amounts manufactured

I Dynamic behaviour: During product transitions a dynamicmodel is used

I Mixed-Integer Dynamic Optimization is the naturalalgorithmic tool for SC problems

I Advantages: nonlinear behaviour, reliable MINLP solvers

I Disadvantages: MIDO problems are hard to solve, onlymedium size/small problems, off-line control policies, nouncertainties


Solving Mixed-Integer Dynamic Optimization Problems

MIDO Problem

Discretize DAE system

using Orthogonal Collocation

on Finite Elements

MINLP


Solving Mixed-Integer Dynamic Optimization Problems

I For solving Scheduling and Control problems we will show firsthow to handle the solution of Scheduling problems and afterthe numerical solution of dynamic optimization problems willbe addressed

I Scheduling: single stage and multiple stages

I Control: lumped and distributed parameters system

I Other MIDO solution approaches proposed


Single Stage Cycle Scheduling Formulation

In this section we will describe a scheduling formulation for continuous plants asproposed by Pinto and Grossmann. The formulation assumes a cyclic productionsequence. Let us assume that a given plant manufactures the products A,B,C in thesequence A→ B→ C:

ABC ABC ABC

1 2 N

A B C

..............

I Objective function

φ =

Np∑i=1

C pi Wi

Tc−

Np∑i

Np∑i′

Ns∑k

C ti′

iZ

ii′

k

Tc−

Np∑i=1

C si

(Gi −Wi/Tc )

Tc

ti

2(1)

where C pi is the product cost, C t

i′

iis the transition cost from product i to

product i′, C s

i is the inventory cost, Tc is the total cycle time, Np is the numberof products, Ns is the number of slots.


I Product assignment

Ns∑k=1

yik = 1, ∀i (2a)

Np∑i=1

yik = 1, ∀k (2b)

y′ik = yi,k−1, ∀i , k 6= 1 (2c)

y′i,1 = yi,Ns , ∀i (2d)

Equation 2a states that, within each production wheel, any product can only bemanufactured once, while constraint 2b implies that only one product ismanufactured at each time slot. Due to this constraint, the number of productsand slots turns out to be the same. Equation 2c defines backward binary

variable (y′ik ) meaning that such variable for product i in slot k takes the value

assigned to the same binary variable but one slot backwards k − 1. At the firstslot, Equation 2d defines the backward binary variable as the value of the samevariable at the last slot. This type of assignment reflects our assumption of

cyclic production wheel. The variable y′ik will be used later to determine the

sequence of product transitions.


I Amounts manufactured

Wi ≥ Di Tc , ∀i (3a)

Wi = Gi Θi , ∀i (3b)

Gi = F o (1− Xi ), ∀i (3c)

Equation 3a states that the total amount manufactured of each product i (Wi )must be equal or greater than the desired demand rate (Di ) times the durationof the production wheel, while Equation 3b indicates that the amountmanufactured of product i is computed as the product of the production rate(Gi ) times the time used (Θi ) for manufacturing such product. The productionrate is computed from Equation 3c as a simple relationship between the feedstream flowrate (F o ) and the conversion (Xi ).

I Processing times

θik ≤ θmax yik , ∀i , k (4a)

Θi =

Ns∑k=1

θik , ∀i (4b)

pk =

Np∑i=1

θik , ∀k (4c)

The constraint given by Equation 4a sets an upper bound on the time used formanufacturing product i at slot k (θik ). Equation 4b is the time used formanufacturing product i , while Equation 4c defines the duration time at slot k(pk ).


I Transitions between products

zipk ≥ y′pk + yik − 1, ∀i, p, k (5)

The constraint given in Equation 5 is used for defining the binary production transition variable zipk . Ifsuch variable is equal to 1 then a dynamic transition will occur from product i to product p within slot k,zipk will be zero otherwise.

I Timing relations

θtk =

Np∑i=1

Np∑p=1

ttpi zipk , ∀k (6a)

ts1 = 0 (6b)

tek = ts

k + pk +

Np∑i=1

Np∑p=1

ttpi zipk , ∀k (6c)

tsk = te

k−1, ∀k 6= 1 (6d)

tek ≤ Tc , ∀k (6e)

tfck = (f − 1)θt

k

Nfe

+θt

k

Nfe

γc , ∀f , c, k (6f)

Equation 6a defines the transition time from product i to product p at slot k. It should be remarked that the termttpi stands only for an estimate of the expected transition times. Equation 6b sets to zero the time at the beginning

of the production wheel cycle corresponding to the first slot. Equation 6c is used for computing the time at the endof each slot as the sum of the slot start time plus the processing time and the transition time. Equation 6d statesthat the start time at all the slots, different than the first one, is just the end time of the previous slot. Equation 6eis used to force that the end time at each slot be less than the production wheel cyclic time. Finally, Equation 6f isused to obtain the time value inside each finite element and for each internal collocation point.


Simple Single Line Scheduling Example

Let us assume that a given plant facility manufactures three products: A,B,C. We would like to compute theoptimal cyclic production sequence that maximizes the process profit while meeting the demand rate of eachproduct.

C

B

A

C

B

A

C

B

A

1 2 3

Transition times(Transition cost)

A B C

A10

(4000)20

(8000)

B15

(3500)30

(6000)

C10

(7000)25

(5500)

Process dataProduct Demand Price Process Inventory

rate Products rates costA 2.1 320 8 1.5B 3 430 9 1C 4 450 12 2

Optimal Cyclic Scheduling Results: Profit=329Slot Product Process Amount Start End

time Manufactured time time1 B 235.3 2117.7 0 265.32 A 185.3 1482.4 265.3 460.63 C 235.3 2823.5 460.6 705.9


Full Discretization Optimal Control Formulation

DAE Optimization Variational Approach

Sequential Approach

Multiple Shooting Simultaneous Approach

Pontryagin

Inefficient for constrained problems

state variables

Discretize some

Handles instabilities

Large NLP

Discretize Controls

Small NLP

Can not handle instabilities properly

state variables

Discretize all

NLP problem

Efficient for constrained problems

Large NLP

Handles instabilities


Discretizing ODEs using Orthogonal Collocation

Given an ODE system:

dx

dt= f(x, u, p), x(0) = xinit

where x(t) are the system states, u(t) is the manipulated variable and p are the systemparameters.The aim is to approximate the behaviour of x and u by Lagrange interpolationpolynomials (of orders K+ 1 and K, respectively) at collocation or discretizationpoints tk:

xk+1(t) =K∑

k=0

xk`xk(t), `x

k(t) =K∏

j=0j 6=k

t− tj

tk − tj

uk(t) =K∑

k=1

uk`uk(t), `u

k(t) =K∏

j=1j 6=k

t− tj

tk − tj

xN+1(tk) = xk, uN(tk) = uk

F(x)

x

Collocationpoints

true behavior

approximated behavior

Therefore replacing into the original ODE system, we get the system residual R(tk):

R(tk) =K∑

j=0

xjd`j(tk)

dt− f(xk, uk, p) = 0, k = 1, ..,K


Transformation of a Dynamic Optimization problem into a NLP

Original dynamic optimization problem

minx,u

φ(x, u)

s.t.dx(t)

dt= F (x(t), u(t), t, p)

x(0) = x0

g(x(t), u(t), p) ≤ 0

h(x(t), u(t), p) = 0

xl ≤ x ≤ xu

ul ≤ u ≤ uu

Discretized NLP

minxk,uk

φ(xk, uk)

s.t.K∑

j=0

xj˙j(tk)− F (xk, uk) = 0, k = 1, ...,K

x0 = x(0)

g(xk, uk, p) ≤ 0, k = 1, ...,Kh(xk, uk, p) = 0, k = 1, ...,Kxl ≤ x ≤ xu

ul ≤ u ≤ uu


Approximation of a Dynamic Optimization Problem using Orthogonal Collocation ofFinite Elements

Sometimes it is convenient to use Orthogonal Collocation on Finite Elements toapproximate the behavior of systems exhibiting fast dynamics.

Sta

te

First element Second Element Third Element

Time

Points

CollocationInternal System behavior

minxk,uk

φ(x, u)

s.t.K∑

j=0

xij˙j(τk)− hiF (xik, uik) = 0,

i=1,...,NE

k = 1, ...,NC

x10 = x(0)

g(xik, uik, p) = 0, i = 1, ...,NE; k = 1, ..,NC

xlij ≤ xij ≤ xu

ij, i = 1, ...,NE; k = 1, ..,NC

ulij ≤ uij ≤ uu

ij, i = 1, ...,NE; k = 1, ..,NCwhere NE is the number of finite elements, NC is the number of internal collocationpoints, hi is the length of each element.


Dynamic optimal transition between two steady-states: Hicks CSTR

Let us consider the dimensionless mathematical model of a non-isothermal CSTR asproposed by Hicks and Ray modified for displaying nonlinearities:

dC

dt=

1− C

θ− k10e−N/TC

dT

dt=

yf − T

θ+ k10e−N/TC− αU(T− yc)

Parameter valuesθ 20 Residence time

Tf 300 Feed temperatureJ 100 (−∆H)/(ρCp)

k10 300 Preexponential factorcf 7.6 Feed concentrationTc 290 Coolant temperatureα 1.95x10−4 Heat transfer areaN 5 E1/(RJcf )

Desired Transition B→ A

50 100 150 200 250 300 350 400 450 5000.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Cooling flowrate

Dim

en

sio

nle

ss t

em

pe

ratu

re

y1=0.0944, y2=0.7766, u=340 (s)y1=0.1367, y2=0.7293, u=390 (s)y1=0.1926, y2=0.6881, u=430 (u)y1=0.2632, y2=0.6519, u=455 (u)

A

B C

D

Desired dynamic transitionC T U

Initial (B) 0.1367 0.7293 390Final (A) 0.0944 0.7766 340

C = Concentration (c/cf ), T = temperature (Tr/Jcf ), yc = Coolant temperature (Tc/Jcf ), yf = feed temperature (Tf/Jcf ), U = Cooling flowrate. c and Tr arenondimensionless concentration and reactor temperature.


Simultaneous approach for optimal control problems

I Objective functionAs objective function the requirement of minimum transition time between theinitial and final steady-states will be imposed:

Min

∫ tf

0

α1(C(t)− Cdes)2 + α2(T(t)− Tdes)2 + α3(U(t)− Udes)2

dt

the subscript ”des” stands for the final desired values. αi, i = 1, 2, 3 areweighting factors. The above integral is approximated using a Radau quadratureprocedure:

Min Φ =

Ne∑i=1

hi

Nc∑j=1

Wj

[α1(Cij − Cdes)2 + α2(Tij − Tdes)2 + α3(Uij − Udes)2

]Ne is the number of finite elements (Ne=3), Nc is the number of collocationpoints including the right boundary in each element (so in this case Nc = 3), Cij

and Tij are the dimensionless concentration and temperature values at eachdiscretized ij point, hi is the finite element length of the i−th element, Wj arethe Radau quadrature weights.


I Mass balance constraintsThe value of the dimensionless concentration at each one of the discretizedpoints (Cij) is approximated using the following monomial basis representation:

Cij = Coi + hiθ

Nc∑k=1

AkjdCik

dt, i = 1, ...,Ne; j = 1, ...,Nc

Coi is the concentration at the beginning of each element, Akj is the collocation

matrix. Note that Co1 stands for the initial concentration. The length of each

finite element (hi)can be computed as:

hi =1

Ne

I Energy balance

Tij = Toi + hiθ

Nc∑k=1

AkjdTik

dt, i = 1, ...,Ne; j = 1, ...,Nc

similarly, Toi is the temperature at the beginning of each element. Again, note

that To1 stands for the initial reactor temperature.


I Mass balance continuity constrains between finite elementsOnly the system states must be continuous when crossing from one finiteelement to the next one. Algebraic and manipulated variables are allowed toexhibit discontinuous behaviour between finite elements. To force continuousconcentration profiles all the elements at the beginning of each element(Ci, i = 2, ...,No

e) are computed in terms of the same monomial basis usedbefore:

Coi = Co

i−1 + hi−1θ

Nc∑k=1

Ak,Nc

dCi−1,k

dt, i = 2, ...,Ne

I Energy balance continuity constrains between finite elements

Toi = To

i−1 + hi−1θ

Nc∑k=1

Ak,Nc

dTi−1,k

dt, i = 2, ...,Ne


I Approximation of the dynamic behaviour of the mass balance at eachcollocation pointThe first order derivatives of the concentration at each collocation point (ij) areobtained from the corresponding continuous mathematical model:

dCi,j

dt=

1− Cij

θ− k10e−N/Tij Cij, i = 1, ...,Ne; j = 1, ...,Nc

I Approximation of the dynamic behaviour of the energy balance at eachcollocation point

dTi,j

dt=

yf − Tij

θ+ k10e−N/Tij Cij − αUij(Tij − yc), i = 1, ...,Ne; j = 1, ...,Nc

I Initial values constraints

Co1 = Cinit

To1 = Tinit

the subscript ”init” stands for the initial steady-state values from which theoptimal dynamic transition will be computed.


Dynamic Transitions profiles for the Hicks CSTR example

0 1 2 3 4 5 6 7 8 9 100.09

0.1

0.11

0.12

0.13

0.14

0.15

Time

Co

nce

ntr

atio

n

0 1 2 3 4 5 6 7 8 9 100.72

0.73

0.74

0.75

0.76

0.77

0.78

0.79

0.8

Time

Te

mp

era

ture

0 1 2 3 4 5 6 7 8 9 100

100

200

300

400

500

600

Time

Co

olin

g f

low

rate


Simultaneous Scheduling and Control problems

I We have shown how to handle the solution of Scheduling andOptimal Control problems

I Because of strong interactions between Scheduling andControl Problems better optimal solutions can be found bythe simultaneous solution of these problems

I No need to neglect process dynamics (Scheduling)

I No need to fix production sequence (Control)

I Merge both formulations and solve MIDO problem


Simultaneous Scheduling and Control problems

I Full discretization approaches lead to large size MINLPproblems

I Reliable MINLP solvers are needed

I Problem solution is highly-dependent on MINLP initializationscheme

I Use Scheduling and Control MINLP solutions for providingacceptable initial guesses of the decision variables

I Develop your Scheduling and Control formulation step by step


Single Stage Scheduling and Control Examples

Case Study 1: Single line Multiproduct CSTR




CPU time: 254 s, SbbAntonio Flores T. with colaborations from: Sebastian Terrazas-Moreno, Miguel Angel Gutierrez-Limon, Ignacio E. Grossmann Universidad Iberoamericana, MexicoIntegration of Scheduling and Control Operations





Case Study 2: Single line HIPS Polymerization CSTR



Statistics: 6666 Constraints, 6267 continuous variables, 25binaries, 7640 s CPU time (IBM 1.6 Ghz), Dicopt





Case Study 3: Single line HIPS Polymerization Plant








Case Study 4: Single line Multiproduct Tubular Catalytic Reactor


Statistics: 40537 equations, 40952 cont. vars, 72 discrete vars, 12h CPU time, Sbb


Case Study 4: Single line Tubular Catalytic Reactor


Parallel Lines Continuous Scheduling Problems

We used Sahinidis and Grossmann scheduling formulation forparallel lines continuous systems


Case Study 5: Two Series-Connected Nonisothermal CSTRs


Statistics: 5011 equations, 5995 cont. vars, 64 discrete vars, 1 hCPU time, Sbb


Case Study 5: Two Series-Connected Nonisothermal CSTRs


A Multiobjective Optimization Approach for SC problems

I Many Engineering problems feature conflicting objectives

I Until now dynamic transitions have been converted intoeconomic profits by using weighting functions so a singleobjective function was used

I However improved optimal solutions can be obtained bysolving SC problems as Multiobjective optimization problemsand the subjective choice of weighting functions is avoided

I Instead of computing a single optimal solution the Paretofront is computed

I The designer chooses his/her preferred optimal solution

I Other measurements such as minimum distance from Utopiaregion to a point on the Pareto front can be used



For dealing with single objective scheduling and control problemsthe following objective function (Ω) was employed:

Ω = ϕ1 − ϕ2

where the individual objective functions ϕ1 and ϕ2 read as follows,

ϕ1 =

Np∑i=1

Cpi Wi

Tc−

Np∑i=1

C si (Gi −Wi/Tc )

2Θi

ϕ2 =

∫ tf

0

∑i

∆xi (t)2dt

where the first part of the ϕ1 term corresponds to the earningsconcerning the sales of the products, whereas the second partrepresents the inventory costs and ϕ2 is a function related with theoff-set or deviation from the target steady-states and it is ameasure of the dynamic performance of the processing system.



I ϕ1 and ϕ2 are conflicting objectives

I Large ϕ1 values mean systems with high profit that commonlylead to poor dynamic performance (i.e. large ϕ2 values): moreattention is paid to selecting a good scheduling strategy withless emphasis on process dynamics

I Ideally, we would like to achieve large ϕ1 values and small ϕ2

values

I Since this is not possible a trade-off between the twoobjectives ought to be established

I Although there are some methods for computing the Paretofront, the ε-constraint method was used due to is simplicity



I In the ε-constraint method one of the objectives is selected tobe optimized and the others are converted into constraintsbounded by a parameter ε

I An advantage of the ε-constraint method over the weightingmethod to solve Multiobjective problems is that theε-constraint method can find a Pareto optimal solution evenfor non convex problems

I Following the ε-constraint approach, we separated the originalobjective function and formed the next Multiobjectiveoptimization problem

max Ω = ϕ1

subject to

ϕ2 ≤ ε

g(x , y , u) ≤ 0


Case Study 6: CSTR with Simultaneous Reactions and Input Multiplicities

(1) 2R1k1−−→ A; (2) R1 + R2

k2−−→ B; (3) R1 + R3k3−−→ C

dCR1

dt=

(QR1C i

R1− QCR1

)

V+ <r1

dCR2

dt=

(QR2C i

R2− QCR2

)

V+ <r2

dCR3

dt=

(QR3C i

R3− QCR3

)

V+ <r3

dCA

dt=

Q(C iA − CA)

V+ <A

dCB

dt=

Q(C iB − CB )

V+ <B

dCC

dt=

Q(C iC − CC )

V+ <C

<A = k1C 2R1,<B = k2CR1

CR2

<C = k3CR1CR3

,<r1= −<A − <B − <C

<r2= −<B ,<r3

= −<C

Q = QR1+ QR2

+ QR3

where QR1, QR2

, and QR3are the feed stream volumetric flow rates of reactants R1, R2, and R3, respectively.


Table: Operating Conditions Leading to the Manufacture of the A, B,and C Products of the Second Case Study

Product Demand rate (Kg/h) Product cost ($/Kg) Inventory cost ($/Kg)A 5 500 1.0B 10 400 1.5C 15 600 1.8

0 5 10 15 20 25 30

1.6

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

x 104

φ2 x 10

−5

φ1

2

1

Figure: Pareto curve for the second case of study. The coordinates forthe first and second points are: [φ1

2, φ11] = [5x10−5, 25590] and [φ2

2, φ21]

= [2.5x10−4, 35250], respectively.Antonio Flores T. with colaborations from: Sebastian Terrazas-Moreno, Miguel Angel Gutierrez-Limon, Ignacio E. Grossmann Universidad Iberoamericana, MexicoIntegration of Scheduling and Control Operations

Table: Results for the the first optimal operating point. The objectivefunction values are: φ1

2 = 5x10−5 and φ11 = 25590.Total cycle time is

659.3 h.

Process Prod rate w Transition T start T endSlot Prod time (min) (Kg/min) (Kg) time (min) (min) (min)

1 A 49.423 66.700 3296.519 10 0.000 59.4232 B 92.456 71.310 6593.038 10 59.423 161.8793 C 447.425 89.520 40053.458 50 161.879 659.304

Table: Results for the the second optimal operating point. The objectivefunction values are: φ2

2 = 2.5x10−4 and φ21 = 35250.Total cycle time is

327.8 h.

Process Prod rate w Transition T start T endSlot Prod time (min) (Kg/min) (Kg) time (min) (min) (min)

1 B 45.969 71.310 3278.079 10 0.000 55.9692 A 24.573 66.700 1639.039 10 55.969 90.5433 C 227.265 89.520 20344.778 10 90.543 327.808


0 2 4 6 8 100

0.25

0.5

0.75

C, m

ol/lt

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

u

0 2 4 6 8 100

0.1

0.2

0.3

0.4

C,

mol/lt

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

u

0 10 20 30 40 500

0.2

0.4

0.6

0.8

time, m

C,

mo

l/lt

0 10 20 30 40 50−0.1

0

0.1

0.2

0.3

0.4

time, m

u

CA

CB

CC

u2

u3

AB

BC

CA

Figure: Optimal dynamic transition profiles for reactor concentration andvolumetric flow rate for the first point of the Pareto front.


0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

C, m

ol/lt

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

u

0 2 4 6 8 100

0.2

0.4

0.6

0.8

C,

mo

l/lt

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

u

0 2 4 6 8 100

0.1

0.2

0.3

0.4

time, m

C,

mo

l/lt

0 2 4 6 8 100

0.1

0.2

0.3

0.4

time, m

u

CA

CB

CC

u2

u3

BA

AC

CB

Figure: Optimal dynamic transition profiles for reactor concentration andvolumetric flow rate for the second point of the Pareto front.


Conclusions

I An algorithmic framework for addressing Scheduling andControl problems using a MIDO formulation was presented

I The SC MIDO formulation was successfully applied to severalcase studies of varying complexity such as chemical reactorsfeaturing strong nonlinear behavior, bifurcation points,unstable operating regions, multiple steady-states

I Presently the SC MIDO formulation only computes off-lineoptimal control policies, no consideration of the effect ofuncertainties or disturbances


Future Work

I Real-Time Optimization for Scheduling and Control

I Stochastic Scheduling and Control

I Large-Scale Scheduling and Control

I Integration of Planning, Scheduling and Control


1. Antonio Flores-Tlacuahuac, I.E. Grossmann, “An Effective MIDO Approach for the Simultaneous CyclicScheduling and Control of Polymer Grade Transition Operations”, In W. Marquardt and C. Pantelides,

editors. In 16th European Symposium on Computer Aided Process Engineering and 9th InternationalSymposium on Process System Engineering, pages 1221-1226, Elsevier, 2006. ISBN: 978-0-444-52970-1.

2. Antonio Flores-Tlacuahuac, Ignacio E. Grossmann, “Simultaneous Cyclic Scheduling and Control of aMultiproduct CSTR Reactor”, Industrial and Engineering Chemistry Research, 45,20,6698-6712(2006)

3. Sebastian Terrazas-Moreno, Antonio Flores-Tlacuahuac, Ignacio E. Grossmann, “Simultaneous CyclicScheduling and Optimal Control of Polymerization Reactors”, AIChE Journal, 53,9,2301-2315(2007)

4. Sebastian Terrazas-Moreno, Antonio Flores-Tlacuahuac, Ignacio E. Grossmann, “A Lagrangean Heuristicfor the Scheduling and Control of Polymerization Reactors”, AIChE Journal, 54,1,163-182(2008)

5. Sebastian Terrazas-Moreno, Antonio Flores-Tlacuahuac, Ignacio E. Grossmann, “Simultaneous Design,Scheduling and Optimal Control of a Methyl-Methacrylate Continuous Polymerization Reactor”, AmericanInstitute of Chemical Engineers Journal, 54,12,3160-3170(2008)

6. Antonio Flores-Tlacuahuac, Ignacio E. Grossmann, “Simultaneous Scheduling and Control of MultiproductContinuous Parallel Lines”, Industrial and Engineering Chemistry Research, 49,17,7909-7921(2010)

7. Antonio Flores-Tlacuahuac, Ignacio E. Grossmann, “Simultaneous Cyclic Scheduling and Control ofTubular Reactors: Single Production Lines”, Industrial and Engineering Chemistry Research,49,22,11453-11463(2010)

8. Antonio Flores-Tlacuahuac, Ignacio E. Grossmann, “Simultaneous Cyclic Scheduling and Control ofMultiproduct Tubular Reactors in Parallel Lines”, Industrial and Engineering Chemistry Research, 50 (13),8086-8096, 2011

9. Miguel Angel Gutierrez-Limon, Antonio Flores-Tlacuahuac, Ignacio E. Grossmann, “A MultiObjectiveOptimization Approach for the Simultaneous Single Line Scheduling and Control of CSTRs”, Submitted to:Industrial and Engineering Chemsirty Research, 2011.

10. Miguel Angel Gutierrez-Limon, Antonio Flores-Tlacuahuac, Ignacio E. Grossmann, “A Real TimeOptimization Approach for Simultaneous Single Line Scheduling and Control Problems”, Underpreparation, 2011.


Documents

Integration of Scheduling and Control Operationsegon.cheme.cmu.edu/ewo/docs/EWO_Seminar_09_08_2011.pdfIntegration of Scheduling and Control Operations Antonio Flores T. with colaborations