Introducción a la Optimización de procesos químicos. Curso 2005/2006 UNIT 2: FORMULATING THE OPTIMIZATION PROBLEM Variables The objective function Equality

Introducción a la Optimización de procesos químicos. Curso 2005/2006

UNIT 2:FORMULATING THE OPTIMIZATION

PROBLEM

• Variables• The objective function• Equality constraints• Inequality constraints• Degrees of freedom• Formulating the problem


FORMULATING THE OPTIMIZATION PROBLEM

Objective function

This is the general formulation that we will be using throughout the course

min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £

Equality constraints

Variable Bounds

Inequality constraints


Variables can be grouped into two categories

“decision” or “optimization” variables

These are the variables in the system that are changed independently to modify the behavior of the system.

dependent variables

whose behavior is determined by the values selected for the independent variables.

DESIGN:

OPERATIONS:

MANAGEMENT:

Although they can be grouped this way to help understanding, thesolution method need not distinguish them. We need to solve a setof equations involving many variables.

min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £


VARIABLESmin maxx x x£ £

reactor volume, number of trays, heat exch. area, …

temperature, flow, pressure, valve opening, …

feed type, purchase price, sales price, ..


Some comments on variables

• Many variables are continuous, but some are discrete or integer.

Exercise: Should we model the following as continuous or discrete?

- Ball bearings in a plant that manufactures 10,000/day

- Crew on an airplane

- Automobiles in the Missassauga Ford plant

Exercise: Give some additional examples of each.

min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £

VARIABLES




• Typically, we do not define the “decision” and “dependent” variables.

•Since we solve a set of simultaneous equations, all variables are evaluated together.

Exercise: Identify variables in each category.

1

2

3

15

16

17LC-1

LC3

dP-1

dP-2

To flare

T5

T6

TC7

AC1

LAHLAL

PAH

PC-1

P3

FC4

FC7

FC 8

F9

PV-3

T10

L4

T20

LAHLAL

F

30

P20

P23

T45

T44

P21

T

22

P11

P12

T30

T29

min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £


VARIABLES



We should always place bounds on variables.

Exercise: Why place bounds?

Exercise: Propose bounds for variables in the process.

1

2

3

15

16

17LC-1

LC3

dP-1

dP-2

To flare

T5

T6

TC7

AC1

LAHLAL

PAH

PC-1

P3

FC4

FC7

FC 8

F9

PV-3

T10

L4

T20

LAHLAL

F

30

P20

P23

T45

T44

P21

T

22

P11

P12

T30

T29

min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £


VARIABLES


OBJECTIVE FUNCTION

min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £This is the goal or objective, e.g., - maximize profit (minimize cost) - minimize energy use - minimize polluting effluents - minimize mass to construct a vessel

We will formulate most problems with a scalar objective function

This should represent the full effect of x on the objective. Forexample, $/kg is not a good objective unless kg is fixed. Whenneeded, include time-value of money.

Also, we need a quantitative measure, not “good” or “bad”.

The symbol “x” represents the variables. It is a vector.

max ( )xf x


How should I formulate these?


Some comments on the objective function

• A scalar is preferred for solving. However, multiple objectives are typical in real life.

• Note that Max (f) is the same as Min (-f)

- Therefore, no fundamental or practical difference between max and min problems. The same algorithm and software can solve both.

• Sometimes we use a simple, physical variable, such as yield of a key product. This assumes that max (profit) is the same as Max(yield), which might not always be true.


OBJECTIVE FUNCTION

min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £


Some comments on the objective function (continued)

• We have difficulty when the models are inaccurate, for example, the tradeoff between current reactor operation and long-term catalyst activity.

• Modelling the market response to improved product quality, etc is difficult.

• We want a “smooth” objective function.


min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £

OBJECTIVE FUNCTION

• The objective function can be a function of indexed variables

Exercise: Write the expression for an objective function that depends on all variables x(i) and the cost associated with each variable is c(i).

- Express the answer as a summation of indexed variables

- Express the answer as a product of vectors


EQUALITY CONSTRAINTS. .

( ) 0

s t

h x =

This means “subject to”. The expressions below limit (orconstrain) the allowable values of the variables x. They define thefeasible region

These are equality constraints, e.g., - material, energy, force, current, … - equilibrium - decisions by the engineer ( F1 - .5 F2 = 0 ) - behavior enforced by controls TC set point = 231

BALANCES

By convention, we will write the equations with a zero rhs (right hand side).

There can be many of these equations, so that h(x) is a vector.


min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £

For example?


1. Define Goals

2. Prepare information

3. Formulate the model

4. Determine the solution

5. Analyze Results

6. Validate the model

• What decision?• What variable?• Location

• Sketch process• Collect data• State assumptions• Define system

Component Material

Accumulation of component component

component mass mass in mass out

generation of

component mass

ì ü ì ü ì ü= -í ý í ý í ý

î þ î þ î þ

ì ü+í ýî þ

Energy

{ } { }

s

AccumulationH PE KE in H PE KE out

U PE KE

Q-W

ì ü= + + - + +í ý

+ +î þ+

• What type of equations do we use first?

Conservation balances for key variable

• How many equations do we need?

Degrees of freedom = NV - NE = 0

• What after conservation balances?

Constitutive equations, e.g.,

Q = h A (T)

rA = k 0 e -E/RT

Typically, the solution and optimization are achieved simultaneously.


min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £

EQUALITY CONSTRAINTS


Some comments on equality constraints

• The key balances must be strictly observed. If we do not ensure that they are “closed”, the optimizer will find a way to create mass and energy!

• The constraints can also have indices. For example, the index could be a location (tray).

( ) 0

( ) 0

ji

ji

h x j

h x for all j

= "

=

å

å

These are equivalent statements




Exercise: F(m,n) is the total mass flow rate leaving unit m and going to unit n.

Formulate the constraints for material balance for every unit.


min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £


Some comments on equality constraints

• Balances can be on a wide range of entities, e.g.- material- time- boxes in a warehouse- people working in sections of a plant

• The models can change. For example, a heat exchanger could have either one or two phases, with the number of phases depending on the optimization decisions.

This makes a solution very difficult!


INEQUALITY CONSTRAINTS. .

( ) 0

s t

g x £min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £


These are “one-way” limits to the system, e.g.,

- maximum investment available - maximum flow rate due to pump limit - minimum liquid flow rate on tray # 24 - minimum steam generation in a boiler for stable flame - maximum pressure of a closed vessel

We must be careful to prevent defining a problem incorrectly withno feasible region.

By multiplying by (-1), we can change the inequality to g(x)<=0So, these two forms are equivalent.

for example?


DEGREES OF FREEDOM (DOF)

Can we determine the DOF for an optimization problem using the relationship below?

DOF = (# variables) - (# equations)

# variables =

# equations =


min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £

For optimization, what value(s) do we expect for the DOF?

The answer explains why optimization is so widely applied!


Often, we will think of the problem as having

#Opt Var = # var - #equality constr.

We can plot this if only two dimensions.

Opt Var1

Opt

Var

2

feasibleregion

What about points inside?

Which is the best?



min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £


Opt

Var

2Opt Var1Opt Var1

Opt

Var

2

Case A Case B

We can plot values of the objective function as contours.

Exercise: Where is the optimum for the two cases shown below?




This is a typical feasible region for a CSTR with reaction

A B

with reactant and coolant adjusted.

Explain the shape of the feasible region.

From Marlin, Process Control, McGraw-Hill, New York, 1995

T

A

Reactant

Solvent

Coolant




The feasible region depends on the degrees of freedom, i.e., the number of variables that are adjusted independently.

We revisit the CSTR, but only the coolant flow can be adjusted.

What is different?

T

A

Reactant

Solvent

Coolant

From Marlin, Process Control, McGraw-Hill, New York, 1995





min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £

How do we select the appropriate “system” for a specific problem?




min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £

How do we define a scalar that represents performance, including

• Economics

• Safety

• Product quality

• Product rates (contracts!)

• Flexibility

• …...




How accurately must we model the physical process?

• Macroscopic

• 1,2 3, spatial dimensions

• Steady-state or dynamic

• Physical properties

• Rate models (U(f), k0e-E/RT, ..

min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £




What limits the possible solutions to the problem?

• Safety

• Product quality

• Equipment damage (long term)

• Equipment operation

• Legal/ethical considerations

min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £




min max

max ( )

. .

( ) 0

( ) 0

xf x

s t

h x

g x

x x x

=

£

£ £

Factoid: Many process simulations and optimizations have a large number of variables and constraints. Why?

• Entire model is repeated for many locations, e.g., trays in a tower.

• Model repeated for many components in a stream.

• Model repeated for many times in a dynamic system




We use the term “tractable” to describe whether we can

1. Solve the mathematical optimization problem

2. Achieve desired accuracy in the “Real World”

- This prevents us from using a useless, simple model

3. Calculate the solution in an acceptable time. The allowable time depends on the problem.




Very accurate over wide range of conditions

Longer computing

More complex

Less accurate over a narrow range of conditions

Shorter computing

Less complex

The engineer must select the appropriate balance for each problem. The problem must be tractable. Intractable problems have to be reformulated.



Decisions to be made

modelSolver method and software

Solution

Does the formulation and solution method support the solution?


Not known with certainty

• Structure of equations

• Parameter values

“The truth”

• Measurement error

• Disturbances

Uncertainty: We must recognize uncertainty in our methods and estimate bounds of the effects of out solutions.



Optimization Formulation: Workshop #1


We want to schedule the production in two plants, A and B, each of which can manufacture two products: 1 and 2. How should the scheduling take place to maximize profits while meeting the market requirements based on the following data:

How many days per year should each plant operate processing each kind of material?

Material processed (kg/day)

Profit(€/kg)

Plant 1 2 1 2

A MA1 MA2 SA1 SA2

B MB1 MB2 SB1 SB2




Suppose the flow rates entering and leaving a process are measures periodically. Determine the best value for stream A in kg/h for the process shown from the three hourly measurements indicated of B and C in the figure, assuming steady-state operation at a fixed operating point.

Material reconciliation

(a) 11.1kg/h

(b) 10.8kg/h

(c) 11.4kg/h

(a) 92.4kg/h

(b) 94.3kg/h

(c) 93.8kg/h

A

C

Bplant




Consider the process diagram of the figure where each product (E,F,G) requires different amounts of reactants according to the table shown in the next slide.

The table below show the maxium amount of reactant available per day as well as the cost per kg.

Material flows allocation

A

B

C

E

F

G

Raw material

Maximum available (kg/day)

Cost (€/kg)

A 40000 1.5

B 30000 2.0

C 25000 2.5

1

2

3


Process

Product

Reactants requirements (kg/kg product)

Processing cost (product) (€/kg)

Selling price (product) (€/kg)

1 E 2/3A,1/3B 1.5 4.0

2 F 2/3A,1/3B 0.5 3.3

3 G 1/2A,1/6B,1/3C 1.0 3.8

Formulate the optimization problem. The objective function is to maximize the total operating profit per day in units of €/day

Optimization Formulation: Workshop #3 (cont’d)



Dump to safe location

FC

LC

CW

TC

fc

fo

fo

L LAHLAL

TAH

T T

TY>

PC

fo

PAH

Problem formulation: We love chemical reactors. Formulate an economic optimization for the reactor in the figure.

The reaction is

A B C

with first order, irreversible rate expressions and arrhenius temperature dependence.

Include the objective function, equality constraints, inequality constraints, and variable bounds.




1

2

3

15

16

17LC-1

LC3

dP-1

dP-2

To flare

T5

T6

TC7

AC1

LAHLAL

PAH

PC-1

P3

FC4

FC7

FC 8

F9

PV-3

T10

L4

T20

LAHLAL

F

30

P20

P23

T45

T44

P21

T

22

P11

P12

T30

T29

Formulation: Describe the major components of a steady-state optimization model for the distillation tower in the figure.

• Define the objective function

• Identify continuous and discrete variables

• Identify dependent and independent variables

• Give examples of each category of equality constraints

• Give examples of each category of inequality constraints

• Discuss advantages for indexed variables and constraintsFORMULATING THE OPTIMIZATION

PROBLEM



T

A

Reactant

Solvent

Coolant

Formulation: Let’s consider a semi-batch chemical reactor.

• Discuss the major difference in this model from others in this section.

• Formulate the model.

• Describe how you would optimize the temperature, feed rates, etc. after you have a computer program to solve the model.


Documents

Introducción a la Optimización de procesos químicos. Curso 2005/2006 UNIT 2: FORMULATING THE OPTIMIZATION PROBLEM Variables The objective function Equality