A Comparative Study Of Deterministic And Stochastic Optimization Methods For Integrated Design Of Processes

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A Comparative Study Of Deterministic And
Stochastic Optimization Methods For Integrated
Design Of Processes
Mario Franciscoa, Silvana Revollarb, Pastora
Vegaa, Rosalba Lamannab a Departamento de
Informática y Automática. Universidad de
Salamanca. Spain b Universidad Simón Bolívar.
Dpto. de Procesos y Sistemas. Venezuela
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Schedule

• Introduction
• Description of the process and plant controller
• Formulation of the optimization problem
• Process constraints
• Controllability constraints
• Solving the problem by deterministic and
  stochastic methods
• Sequential Quadratic Programming
• Genetic algorithms
• Simulated annealing
• Hybrid method
• Integrated design results
• Open loop design
• Closed loop design
• Conclusions

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Introduction
Classical process design Sequential procedure
Selection of the optimal process structure
Dimensioning, and determination of working point
Synthesis and Design
Design might result in plants difficult to
control
Control system design
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Introduction
Integrated design The integrated-process-and-contr
ol-system-design lies in the systematic study of
the influence of the process design on the
stability and controllability of the system, even
before the process flowsheet is defined.
Open loop and closed loop indices are considered
for design
Better controllable plants Trade off between
design and control
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Introduction
Open loop controlability contraints

• Open loop eigenvalues analysis
• Analysis of controllability indices derived from
  system linearized model to determine disturbance
  rejection capability

Closed loop criteria

• Proper tuning of the controller parameters to
  ensure
• closed loop stability
• good disturbance rejection
• optimization of dynamical performance indexes

Min f (x,y) Constraints h(x) 0 g(x) ?
0 g(t,x) ? 0 x ? ?
The mathematical formulation for the integrated
design results into a non-linear dynamical
optimisation problem which considers
controllability constraints and dynamical
performance indices.
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Introduction
Objective

• Perform the Integrated Design of an activated
  sludge process considering controllability
  indices such as disturbance sensitivity gains,
  the H? norm, and dynamical performance indices as
  the ISE norm.
• Apply and compare stochastic and deterministic
  optimization methods to solve the dynamical
  optimisation non-linear problem that emerges from
  the Integrated Design.
• Propose an hybrid methodology that combines both
  deterministic and stochastic optimisation methods
  for the solution of the optimisation problem.

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Schedule

• Introduction
• Description of the process and plant controller
• Formulation of the optimization problem
• Process constraints
• Controllability constraints
• Solving the problem by deterministic and
  stochastic methods
• Sequential Quadratic Programming
• Genetic algorithms
• Simulated annealing
• Hybrid method
• Integrated design results
• Open loop design
• Closed loop design
• Conclusions

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Formulation of the Optimization Problem
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Formulation of the Optimization Problem
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Schedule

• Introduction
• Description of the process and plant controller
• Formulation of the optimization problem
• Process constraints
• Controllability constraints
• Solving the problem by deterministic and
  stochastic methods
• Sequential Quadratic Programming
• Genetic algorithms
• Simulated annealing
• Hybrid method
• Integrated design results
• Open loop design
• Closed loop design
• Conclusions

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Formulation of the Optimization Problem
Objective function Investment and operation
cost
Activated sludge process superstructure
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Formulation of the Optimization Problem
Objective function Investment and operation
cost
Activated sludge process superstructure
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Formulation of the Optimization Problem
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Schedule

• Introduction
• Description of the process and plant controller
• Formulation of the optimization problem
• Process constraints
• Controllability constraints
• Solving the problem by deterministic and
  stochastic methods
• Sequential Quadratic Programming
• Genetic algorithms
• Simulated annealing
• Hybrid method
• Integrated design results
• Open loop design
• Closed loop design
• Conclusions

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Genetic Algorithms
Genetic algorithms are general optimization
methods which mimics principles of natural
evolution

• Chromosome codification Real coded -The
  variables are normalised

Open loop
Closed loop
Techniques to deal with constraints
Stronger penalty function
Crossover technique
Parameters used for solving the problem
Population size of 60 individuals and a maximum
generation number of 300.
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Simulated Annealing
The simulated annealing is inspired in the
annealing process to get minimum energy states in
a solid. The states represent candidate solutions
and the energy is the cost associated to each
state

Codification Real coded -The variables are
normalised
Starting point
Acceptance probability
New state
Parameters used for solving the problem Linear
cooling schedule for c, decreasing rate 0.88
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Sequential Quadratic Programming

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Hybrid method

• Genetic Algorithms have the advantage of
  avoiding local minima and the ability of
  providing solutions when dealing with complex
  problems, but sometimes, do not arrived to
  feasible solutions.
• SQP have been broadly applied obtaining good
  solutions in a reasonable amount of computing
  time, mainly if the search starts near the
  optimum, but might not converge to any solution
  when dealing with complex problems.

Hybrid method
Step 1 Genetic Algorithm
Step 2 SQP
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Schedule

• Introduction
• Description of the process and plant controller
• Formulation of the optimization problem
• Process constraints
• Controllability constraints
• Solving the problem by deterministic and
  stochastic methods
• Sequential Quadratic Programming
• Genetic algorithms
• Simulated annealing
• Hybrid method
• Integrated design results
• Open loop design
• Closed loop design
• Conclusions

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Results
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Results
Open loop Integrated Design Norm H? lt?
Integrated Design without controllability

Closed Loop Integrated Design
V (m3) 8611 A (m2) 3026.1 S1
(mg/l) 38.63 ISE 79771 Cost
0.1292 Kp -7.33 Ti 415.1 Norm H? 0.1080
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Schedule

• Introduction
• Description of the process and plant controller
• Formulation of the optimization problem
• Process constraints
• Controllability constraints
• Solving the problem by deterministic and
  stochastic methods
• Sequential Quadratic Programming
• Genetic algorithms
• Simulated annealing
• Hybrid method
• Integrated design results
• Open loop design
• Closed loop design
• Conclusions

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Conclusions

• The Integrated Design of an activated sludge
  process considering controllability indices and
  dynamical performance indices as the ISE norm was
  successfully performed.
• The stochastic methods (SA and GA) and
  deterministic (SQP) showed good results in open
  loop design and closed loop Integrated Design
  with PI controllers.
• Hybrid optimization starting with GA and
  refining solutions with SQP has also been
  developed, combining advantages of both methods,
  and giving also good results for Integrated
  Design.
• GA seems very suitable for solving MINLP
  problems, these results are encouraging for the
  application of the hybrid method to solve the
  problems derived from process synthesis, or
  Integrated Design with model predictive
  controllers, that also involves integer
  variables.

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Disturbances Gains