A Comparative Study Of Deterministic And Stochastic Optimization Methods For Integrated Design Of Processes Mario Francisco a, Silvana Revollar b, Pastora.

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

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

Introduction

Classical process design: Sequential procedure

Synthesis and Design

Control system design

Selection of the optimal process structureDimensioning, and determination of working point

T

P V

Design might result in plants difficult to control

$$ $

Introduction

Integrated designThe integrated-process-and-control-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.

Process Design

Controllability Analysis

Process Synthesis

Better controllable plants:

Trade off between design and control

Open loop and closed loop indices are considered for design

Introduction

The mathematical formulation for the integrated design results into a non-linear dynamical optimisation problem which considers controllability constraints and dynamical performance indices.

Min f (x,y) Constraints:

h(x) = 0

g(x) 0

g(t,x) 0

x

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

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.

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

Formulation of the Optimization Problem

ASU1 ASU2 ASU3 ASU4 ASU5

RASS

Nitrate internal recycle

waste

EFFLUENT

Physical characteristics

5 biological tanks in series with a secondary settler

Operational characteristics

ASU1 and ASU2 unaereated but fully mixed

Nitrate internal recycle

RAS recycle from the underflow of the secondary settler

Formulation of the Optimization Problem

The control of this process aims to keep the substrate at the output (s1) below a legal value despite the large variations of the flow rate and the substrate concentration of the incoming water (qi and si). A PI controller was chosen

Si disturbances Qi disturbances

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

Formulation of the Optimization Problem

Objective function: Investment and operation cost

12 2 2 2f (x) w V w A w fk w q

1 2 3 4 2

Activated sludge process superstructure

Mass balances constraints

2

1 1 1 1max d c 1 1

s 1 1

dx s x x qy K K x xir x

dt K s s V

2

1 1 1 1max kd d kd c 1 1

s 1 1

ds s x x qf K f K x sir s

dt K s s V

2

1 1la s 1 01 max 1

s 1

dc x qK fk c c K c

dt (K s ) V

b1 sal b 2 b d b

b

dx 1qx q x q x Avs x Avs x

dt Al

r2 b 2 r b

r

dx 1q x q x Avs x

dt Al

dsal b sal d d

d

dx 1q x q x Avs x

dt Al

Formulation of the Optimization Problem

Objective function: Investment and operation cost

12 2 2 2f (x) w V w A w fk w q

1 2 3 4 2

Activated sludge process superstructure

Residence times and mass loads in the aeration tanks:

Limits in the relationship between the input, recycled and purge flow rates:

Limits in hydraulic capacity and sludge age in the settler

5.1A

q 2 1 r r

p r

vx Al x3 10

q x 24

07.0q

q03.0

2

p 9.0

q

q5.0

i

2

06.0vx

sqsq001.0

1

11ii

1

12

v2.5 8

q

Formulation of the Optimization Problem

dw

G max w

12 2 2 2f (x) w V w A w fk w q

1 2 3 4 2

Objective function: Investment and operation cost

Activated sludge process superstructure

Controllability Constraints:The H∞ norm

The disturbance transfer function:

d 2

2

G j dDs

d

For the closed loop design:The ISE norm as a dynamical performance index

T max

2

1r 1

t 0

ISE s s dt

Good disturbance rejection

dG

Ds

ISE

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

Genetic Algorithms

Parameters used for solving the problem: Population size of 60 individuals and a maximum generation number of 300.

p

1k

m

1l

2k

2k xh)x(g,0maxR)x(f)x(F

i2i1i yxz

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

Techniques to deal with constraints:

Chromosome codification: Real coded -The variables are normalised

Open loop

Closed loop

Stronger penalty function

Crossover technique:

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

1 ( ) ( )

( ) ( )exp ( ) ( )

if f j f i

P accept j f i f jif f j f i

c

Starting point

New state

Acceptance probability

Parameters used for solving the problem: Linear cooling schedule for c, decreasing rate 0.88

Codification: Real coded -The variables are normalised

Sequential Quadratic Programming

Optimal plant parameters with the best controller

Controller parameters

Plant dimensions Steady state point

PLANT DESIGN (optimization of f1)

SQP algorithm

CONTROLLER TUNING (optimization of f2)

SQP algorithm

2f (x) w ISE5

Optimal plant parametersController parameters: Kp, Ti constant

Optimal PI controller parameters: Kp Ti Plant parameters constant

12 2 2 2f (x) w V w A w fk w q

1 2 3 4 2

For closed loop design: A methodology consisting of an iterative two steps approach is proposed to solve closed loop Integrated Design. (Suboptimal solution) Step 1: Performs the plant design optimizing f1

Step 2: Performs he controller tuning optimizing f2

For open loop design: Optimization of function f1 considering ISE< is sufficient

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

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

Results

Parameters SQP GA SA GA refined V (m3) 5046 5939 4829 5066 A (m2) 1885 1980 1775 1887 S1 (mg/l) 87.5 86.21 85.30 87.47 Cost 0.040 0.059 0.035 0.040 Constraints satisfaction

Hight Acceptable Low Hight

Parameters SQP GA SA GA refined V (m3) 7772 7784 6777 5968 A (m2) 2172 2447 2165 2990 S1 (mg/l) 51.26 51.41 51.42 51.32 Cost 0.083 0.104 0.0706 0.0863 Norm H 0.1600 0.16 0.16 0.1600 Constraints satisfaction

Hight Acceptable Low Hight

Integrated Design without controllability

Open loop integrated design Norm H <

Parameters SQP GA GA refined V (m3) 8570 9664 8611 A (m2) 3084 2405 3026.1 S1 (mg/l) 47.38 59.00 38.63 ISE 79791 75114 79771 Cost 0.1335 0.1355 0.1292 Hight Hight Hight

Closed loop integrated design ISE <

Results

Integrated Design without controllability Open loop Integrated Design Norm H <

V (m3): 5046A (m2): 1885S1 (mg/l): 87.5ISE: 588790Cost : 0.040Ds (1): 2.342Ds (2): 2.700Norm H: 0.2900

V (m3): 7772A (m2): 2172S1 (mg/l): 51.26ISE: 185350Cost : 0.083Ds (1): 1.349Ds (2): 1.510Norm H: 0.1600

V (m3): 8611A (m2): 3026.1S1 (mg/l): 38.63ISE: 79771Cost : 0.1292Kp: -7.33Ti: 415.1Norm H: 0.1080

Closed Loop Integrated Design

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

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.

Disturbances Gains

Parameters SQP GA SA GA refined V (m3) 7270 6317 6215 6762 A (m2) 2372 3075 3013 2615 S1 (mg/l) 50.72 51.52 50.06 50.53 ISE 186230 179110 175520 184330 Cost 0.0807 0.0985 0.0895 0.0812 Ds (1) 1.354 1.378 1.379 1.366 Ds (2) 1.499 1.494 1.497 1.500 Norm H 0.151 0.156 0.160 0.1573