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