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

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

PROBLEM SOLVING FOR OPTIMIZATION

Decisions:

1. Purchase feed type A

2. Process feed to max.

utilization of machinery

Max Profit

s.t. Max Rate Fi

The results from the mathematical analysis must satisfy the needs in the real world!

Reality Model Results interpretation

Implementation


Before we begin this process, we will be confident that

• The problem involves optimization

- Some degrees of freedom remain after safety, etc.

• A model-based approach is appropriate

- Not an empirical approach

• The likely benefit is worth the effort

- Answer is not obvious

- Changes will likely yield substantial improvement

PROBLEM SOLVING FOR OPTIMIZATION


Step 1 - EngageStep 2 - DefineStep 3 - ExploreStep 4 - DiagnosisStep 5 - ImplementStep 6 - Lookback

PROB. SOLV. FOR OPTIMIZATION

USING THE SIX-STEP PROBLEM SOLVING METHOD

It’s circular, not linear

We look back after eachstep.

1

2

34

5

6

1

2

34

5

6

If step is complex, can apply all six steps inside one major step



1. Engage

2. Define

a. Current, desired,deviation

b. Define obj, constr., variables

c. Define the scenarios

3. Explorea. Mental model

b. Model variables

c. Prior experience

d. Qualitative aspects/bounds

4. Plan Solution a. Model-based,

b. Model complexity

5. Do it a. Solution method

b. Debug

c. Evaluate scenarios

d. Sensitivity analysis

6. Evaluate: Look backa. Meets goals?

b. Extra benefits/problems?

c. Develop heuristics

d. Communicate & document


1. ENGAGE - Be confident and calm

a. Read and listen to information from all stakeholders

b. Do not be concerned about proposing “silly” solutions

I want to and I can!


Manage your time;

Apply your problem-solving skills,

and

Be confident!


2. DEFINE - Concentrate on the objective of the optimization study

Do not “force” the problem definition to be suitable for a specific solution method at this step


I only know linear programming, so this must be an LP problem!


2. DEFINE - Concentrate on the objective of the optimization study (cont’d)

a. Sketch the problem and label variablesObserve the real system, if possible

b. Confirm/establish goals with priorities. We must understand all goals more important than the objective function so that we will satisfy these: for example,- Safety- Product quality




c. Specify the following aspects of the problem:- objective function- variables to be predicted- number of “degrees of freedom” for optimization- inequality constraints

You should be able to state these is words and explain them, as well as specify mathematical relationships

d. Look for/eliminate inconsistencies




e. Define range of conditions to be investigated:

• production rates, • product qualities, • feed materials, • economics, • likely model errors (parametric and structural), • solution variables (e.g., types of equipment, temperatures)• steady-state or dynamic




f. Be sure that hard constraints are true. We can often violate “normal policies” or investment limits for a good reason.

g. Define the desired solution in words

h. Establish facts from opinions. Collect initial evidence.



3. EXPLORE - Form a rich mental image of the problem

a. See if you can determine qualitative aspects of the problem- define the “system” and likely balances- shape of the feasible region (operating window)- contours of the objective function- likely location of the optimum (interior or boundary)


In this step, use simple models to establish bounds on the possible solutions.


3. EXPLORE - Form a rich mental image of the problem (cont’d)

b. Determine all of the variables needed to solve the optimization problem with sufficient accuracy, e.g.,

- Intermediate variables (concentration along a PFR)- Feed properties (which concentrations, enthalpy, etc.)- Environmental variables (cooling water)- physical properties (what accuracy?)

Why? This will help when we formulate a model.



3. EXPLORE - Form a rich mental image of the problem (cont’d)

c. Challenge the assumption that optimization is needed. Try your best to find the solution using simple principles and models.- Why can’t you solve the problem without the model? - What must the model tell you and with what accuracy? - Is there an obvious (sub-) optimal solution?- Can you determine the best values of a subset of the variables?

d. Find relevant prior experience- Literature - Colleagues - Prior solutions




4. PLAN - Formulate the optimization model

Now, we make a major decision - the model formulation that determines the model accuracy!

How accurate is good enough? We might have to

• formulate and solve the model• perform a sensitivity analysis to determine whether

the accuracy is good enough for the decisions • if accuracy not acceptable, iterate with other model




1. First, we need to decide the basic approach

Does information for a model exist?

Model-based optimization

Empirical optimization

N

F

T

• process must exist!• experiments are costly• more delay for modelling• possible w/o model• slow but presistent and

accurate

Lots of applications of both!

• New process can be investigated

• experiments are not needed• model required• fast if model exists• depends on model accuracy

Y




2. Model complexity

Model-based variables needed?

Only input / output for each process

Detailed, fundamental models

Typically, models are greatly simplified models of complex processes

Typically, models are based on fundamental balances, phys. prop., rate expressions, etc. These can be quite complex

This is more accurate, but requires more time and $. Is it (1) possible and (2) required?

Many intermediate variables




Input/output models

Linear Non-linear

Inputs = manipulated and disturbance variables

Outputs = key dependent variables (flows, quality, energy consumption, etc.)

This will yield a LP problem that can be solved reliably for large problems

This will yield a non-linear optimization problem. Typically, the problem that can be solved reliably for large problems. (No promises for NL models!)



4. PLAN - Formulate the optimization modelLinear I/O Models

FD = 0.50FFB = F - FDQrb = (*2.2)FQc = F(2.2)

• FD FDmax

Represents “standard” operating conditions. Constraint models are approximate. These models can be determined from plant data or simplifications of fundamental models.



4. PLAN - Formulate the optimization modelNon-linear I/O Models

F = FD + FBF(XF) = FD(XD) + FB(XB)V = (RR+1) FDQc = VRm = [XD/XF-(1-XD)/(1-XF)]/(1- )

• FD FDmax

Model obeys fundamental balances and uses correlations for complex aspects of the process. Key non-linearities are represented, but model is not necessarily highly accurate .

See EHL pg 453





Lumped Parameter

Distributed Parameter

Part. Diff. Equations

ODE or PDE

Ord. Diff. equations

Algebraic equations

s-s dynamicdynamic s-s

PF Reactor

CSTR




• Mat. Balance on trays

• Component Mat. balance on trays

• Energy balance on trays

• Physical properties

• Heat exchangers

• Tray hydraulics (flooding)

•

•

Can determine the best operation with high accuracy. Flooding constraints on individual trays can be modelled. The interaction between distillation and heat exchange can be optimized.




4. PLAN - Formulate the optimization modelBefore proceeding, we must check our

formulation.

1. Does the model address the issues in Step 2, DEFINE?

2. Does the model contain the qualitative behaviors that you have predicted in Step 3, EXPLORE?

3. Is this solvable?Here is where we mighthave to compromise!

Iterate between Steps 2 + 4

Iterate between Steps 3 + 4

Carefully evaluate the formulation to find reasonable simplifications.



5. DO IT - Solve the optimization problem

a. Select a solution method. The chart below shows some criteria and selections for problems with continuous variables.

Linear equations

Non-linear with small problem, black-box model

Non-linear with large problem, open equation model

Linear Program Non-linear program using analytical

derivatives

Non-linear program using numerical

derivatives




b. Select a software package. The chart below shows some criteria and selections for problems with continuous variables.

Linear equations

Non-linear with small problem, black-box model

Non-linear with large problem, open equation model

Linear Program• Small - Excel• Large - GAMS

Non-linear program

• GAMS

Non-linear program

Using existing program: optimizer from library using the same language. e.g., MATLAB




c. Check your formulation and solution method: DEBUG.

1. Cross check solutions with • previously published, • other methods, • qualitative understanding (from EXPLORE), • changing convergence tolerances, etc.2. Change the sign of the objective function

- does the solution change?3. Try other initial conditions (for NLP)4. Check balances independently5. Solve smaller parts of problem; then, combine.




d. Solve all scenarios specified in Step 2, DEFINE

1. Check solver diagnostics for lack of convergence, alternative solutions - anything not indicating unique optimum.

2. Collect results in side-by-side tables, so that you can see how variables and constraint activities change in scenarios.

3. Provide an explanation for every scenario - NEVER report the numbers alone!

4. Summarize the conclusions and likely benefits, if any, for implementing results




e. Build confidence in your results - concentrate on the decision variables

1. Assumptions - Does the solution obey all key assumptions, or have variables moved “too far”?

2. Sensitivity - Evaluate the sensitivity of the decisions and objective to changes in key parameters that are uncertain.

3. Relative importance - Evaluate the importance of each decision variable. Could you gain the benefits with a subset of opt. variables?




f. If you have the opportunity, implement the results

1. Potential Problem Analysis - Evaluate how these decisions influence other issues, including safety. How can problems be eliminated or mitigated?

2. Schedule - Develop a sequence for implementation.3. Maintenance - Establish how the decisions will be

maintained; in a plant, is control required?4. Monitoring - Closely monitor the behavior to ensure that it

follows the predictions of the analysis.



6. EVALUATE - Look back and learn

a. Have you solved the problem?

1. Done? - Was the model and optimization approach appropriate? If not, must iterate with a different approach.- If the limit was accuracy, try more accurate model- If the limit was computing, try simpler model

2. Simple solution? - Now that you have optimization results, can you find a heuristic that would give similar results in the future without (or with limited) mathematical analysis?




b. Check results of this problem

1. Consistency - Is the implementation consistent with all prior parts of the PS method?

2. Objective - Has the goal been achieved? If not, what factors have limited us? How can we improve further?

3. Engineering - What uncertainty in model structure or values has limited the achievements?

4. Unexpected factors - Have you encountered unexpected safety, legal, ethical issues?




c. Guidelines (experience factors) for the future

1. Problem Insights - What have you learned about this problem that could be used in the future?- Model accuracy - Parameter uncertainty- Variables changed - Useful objective function

2. General Insights - What can be used in many problems?- Performance of optimization method- Performance of software- Sensitivity analysis




d. Spreading the word

1. Engineering - How can you teach others about the use of optimization through this example?

2. Maintenance - How can we monitor, evaluate, decide when to optimize again?- Personnel training - Additional sensors- real-time calculations

3. Communication - How will you communicate these complex calculations to the people running the plant? What about management?

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