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24th may 200624th may 2006 11
Use of genetic algorithm Use of genetic algorithm for designing redundant for designing redundant
sensor networksensor network
Carine GerkensCarine Gerkens
Systèmes chimiques et conception de procédésSystèmes chimiques et conception de procédésDépartement de chimie appliquéeDépartement de chimie appliquée
Université de LiègeUniversité de Liège
2224th may 200624th may 2006
SynopsisSynopsis
ObjectivesObjectives
Data validationData validation
Algorithm descriptionAlgorithm description
OptimizationOptimization
Case studyCase study
ParallelizationParallelization
Global parallelizationGlobal parallelization
Distributed genetic algorithmsDistributed genetic algorithms
ConclusionsConclusions
3324th may 200624th may 2006
ObjectivesObjectives
Using concepts for data reconciliation in chemical Using concepts for data reconciliation in chemical processes, create an algorithm able to processes, create an algorithm able to Design a sensor network that allows toDesign a sensor network that allows to
Limit the annualised cost of the measurement system for a Limit the annualised cost of the measurement system for a chemical plantchemical plant
Evaluate process key variables with a prescribed accuracyEvaluate process key variables with a prescribed accuracy
Secure redundancy even in case of one sensor failureSecure redundancy even in case of one sensor failure
Give the solution for quite large plants within a Give the solution for quite large plants within a reasonable timereasonable timeProblem solved by Bagajewicz Problem solved by Bagajewicz (linear mass balances)(linear mass balances) and and Madron Madron (graph oriented method)(graph oriented method)
4424th may 200624th may 2006
Data validationData validationAll measurements are erroneousAll measurements are erroneousSome important variablesSome important variables (efficiency, conversion…) can not be (efficiency, conversion…) can not be measuredmeasured
Data validationData validation
Thanks to redundancy: correct each measurement as slightly Thanks to redundancy: correct each measurement as slightly as possible to verify all conservation equations (linear or not, as possible to verify all conservation equations (linear or not, mass and energy balances, link equations)mass and energy balances, link equations)
Estimate non measured variables and their accuracy from Estimate non measured variables and their accuracy from reconciled measured variables and accuraciesreconciled measured variables and accuracies
Hypothesis: Hypothesis: Gaussian distribution of measurement errors
Accuracies influenced by the number, the location and the Accuracies influenced by the number, the location and the precision of sensorsprecision of sensors
Sensor network designSensor network design
5524th may 200624th may 2006
RedundancyRedundancyRedundancy is more than repeating the same measurement on the same variable several times (temporal redundancy)
It is also
installing several identical sensors (spacial redundancy)
estimating the same variable thanks to different sensors (stuctural redondancy)
F
F1P
F2 = f(P) T1 T2
Q
F3 = Q / Cp (T2-T1)
6624th may 200624th may 2006
Data validation(2)Data validation(2)Constrained optimization problem (linearised eq)Constrained optimization problem (linearised eq)
' '
,
0
T
X Y
X X W X X
st AX BY C
Min
' '
, ,
2T T
X Y
X X W X X AX BY CMinL
Unconstrained optimization problem (LagrangeUnconstrained optimization problem (Lagrange))
0
0
0
dL
dXdL
dYdL
d
Optimality
7724th may 200624th may 2006
Data validation(3)Data validation(3)
Solve the system:Solve the system:'0
0 0 0
0
T
T
W A X X WX
B Y M Y
A B C
1 ' 1,, ,
1 1
1 ' 1,, ,
1 1
pm
i j j j ki j i n m kj k
pm
i j j j kn i j n i n m kj k
X M W X M C
Y M W X M C
Solution:Solution:
8824th may 200624th may 2006
Data validation(4)Data validation(4)
Validated accuraciesValidated accuracies
21
,
'1
21
,
'1
mi j
ij j
mn i j
ij j
MVar X
Var X
MVar Y
Var X
9924th may 200624th may 2006
Algorithm descriptionAlgorithm description
1var,
1i
j
Ni j
Yj y
MVar
Var
Sensor databaseBelsim-Vali validation model
Optimization criteria
Key variables
Sensor network optimization
Problem feasability?
Sensor requirements
Sensor database
- Cost
- Accepted range
- UncertaintyKey variables
- Variables
- Required standart deviation
Sensor requirements
- existing sensor
- impossible placement
Optimization criteria
- Cost
- Target accuracies
- Singularity of the validation problem
- Safeguard against one sensor failure
- Several operating points
Sensor network optimization
Problem feasability?
For all possible sensors :
-M singular?
-Target accuracies reached?
Belsim-Vali validation model
-For all operating points
-Validated values of all variables
-Jacobian matrices A and B
101024th may 200624th may 2006
OptimizationOptimizationThe problem to optimise isThe problem to optimise is
Large scale
Generally multimodal
Not derivable
Genetic algorithmGenetic algorithm Developed by John Holland
Solution described by a set of binary decisions (genes) corresponding to the decision to install a sensor at a given location
111124th may 200624th may 2006
Optimization(2)Optimization(2)
Random search algorithm based on reproduction and natural selection mechanisms
Biological systems are robust, efficient and flexible
Artificial systems try to translate nature but less perfomant
Robustness of GA proved empirically
Combine survival of best individuals with information exchange
Exploit parents information to create better children
121224th may 200624th may 2006
Optimization(3)Optimization(3)
Mecanisms used: Selection
Reproduction (50%)
One-point-crossover (50%)
Jump mutation (1%)
01011 10110
11010 01100 11010 10110
01011 01100
Before single point crossover
After single point crossover
Father
Mother
First child
Second child
01011 1011001011 10110
11010 0110011010 01100 11010 10110
01011 01100
11010 1011011010 10110
01011 0110001011 01100
Before single point crossover
After single point crossover
Father
Mother
First child
Second child0101110110 0101100110
Before jump mutation
After jump mutation
01011101100101110110 01011001100101100110
Before jump mutation
After jump mutation
131324th may 200624th may 2006
Optimization(2)Optimization(2)
First population chosen randomly with a high First population chosen randomly with a high probability for each sensor to be chosenprobability for each sensor to be chosenPopulation of 20 chromosomesPopulation of 20 chromosomesEvaluation of each chromosome’s fitnessEvaluation of each chromosome’s fitness
Sensitivity matrix inversion Sensitivity matrix inversion (Chen et (Chen et Stadherr) Stadherr) sparse matrixsparse matrix
The best individual is kept at each generationThe best individual is kept at each generationStop criterion: best remains unchanged during x Stop criterion: best remains unchanged during x generationsgenerationsFinal solution better than initial one Final solution better than initial one (but not (but not necessary the global minimum)necessary the global minimum)
141424th may 200624th may 2006
Ammonia synthesis loop 224 variables 178 constraint equations 117 potential sensors 58 key parameters
Case study: ammonia loopCase study: ammonia loop
Optimes.exe
151524th may 200624th may 2006
Search history: ammonia loop (2)Search history: ammonia loop (2)Case of redundant sensor networkCase of redundant sensor network
Processor M (dothan) 1.6 GHz Processor M (dothan) 1.6 GHz
Stop criterion : 200 generationsStop criterion : 200 generations 100 objective function evaluations per second100 objective function evaluations per second Solution obtained after 76 secondsSolution obtained after 76 seconds
Objective function : 1822.9
Cost : 1850 units
361 generations
7241 objective function evaluations
Evolution of the objective function
-10000-8000-6000-4000-2000
0
0 100 200 300 400
Generations
Fit
nes
s
161624th may 200624th may 2006
Solution : Solution : case of redundant sensor case of redundant sensor networknetwork
39 sensors :
1 chromatograph,
7 mass flowmeters,
20 temperature sensors,
11 pressure gauges
171724th may 200624th may 2006
Solution : Solution : case of one sensor failurecase of one sensor failure
67 sensors :
2 chromatograph,
13 mass flowmeters,
31 temperature sensors,
21 pressure gauges
Computing time : 1h45
181824th may 200624th may 2006
Case study: reformerCase study: reformer
1263 variables1116 constraint equations473 potential sensors9 key parametersCase of redundant sensor networkCase of redundant sensor network
Processor Pentium IVProcessor Pentium IV Stop criterion : 200 generationsStop criterion : 200 generations 9 objective function evaluations per minut9 objective function evaluations per minut Solution obtained after 6 daysSolution obtained after 6 days
191924th may 200624th may 2006
Case study: reformer (2)Case study: reformer (2)
Objective function : 1955.1 unitsCost : 1960 units1618 generations77665 objective function evaluationsSolution: 72 sensors :
3 chromatographs, 10 mass flowmeters, 45 temperature sensors, 13 pressure gauges,1 density sensor
202024th may 200624th may 2006
ParallelisationParallelisationWhy?Why?
p
Speed upEfficiency
N 1
p
TSpeed up
T
Large computing time for middle size problemsLarge computing time for middle size problems
share the computing work between several share the computing work between several processorsprocessors
reduce the computing timereduce the computing time
techniques are compared by efficiencytechniques are compared by efficiency
ParallelisationParallelisation allows toallows to
Impossible to deal with larger size problemsImpossible to deal with larger size problems
212124th may 200624th may 2006
Global parallelisationGlobal parallelisation
Evaluation of chromosomes’fitnessEvaluation of chromosomes’fitness
Population evolution, comparison of Population evolution, comparison of fitnessfitness
Best efficiency if Best efficiency if integer1processors
chromosomes
processors
k nn k
nk
Fastest operations
Use of MPI (Message Passing Interface)Use of MPI (Message Passing Interface)
Carried out by master processor
Weak loss of efficiency
Shared between processors
Slowest operations
222224th may 200624th may 2006
0
50
100
150
200
250
300
350
400
450
0 0.2 0.4 0.61/processors number
Tim
e (s
ec)
40
50
60
70
80
90
100
Eff
icie
ncy
(%
)
Master time
Elapsed time
Efficiency
Global parallelisation Global parallelisation (2)(2)
chromosomes processorsn k nCase of a redundant sensor network:
232324th may 200624th may 2006
Distributed Genetic AlgorithmDistributed Genetic AlgorithmGlobal parallelisation : fall of efficiency with the number of Global parallelisation : fall of efficiency with the number of
processorsprocessors
DGA : better efficiency?DGA : better efficiency?Chromosomes distributed in sub-populationsChromosomes distributed in sub-populations
Migration operator : chromosomes transfertMigration operator : chromosomes transfert
Migrating chromosomes chosen randomlyMigrating chromosomes chosen randomly
Parameters : Parameters : Sub-populations’size: 10 chromosomesSub-populations’size: 10 chromosomes
Number of migrating chromosomes : 2Number of migrating chromosomes : 2
Number of sub-populations : 5Number of sub-populations : 5
Number of generations before migration : 5Number of generations before migration : 5
Influence of the number of iterations between two migrations
0
500
1000
1500
2000
0 5 10 15 20 25
Number of iterations between two migrations
Total number of iterations
Master time(sec)
Objectif function
Influence of the number of 10 chromosoms sub-populations
0300600900
120015001800
0 5 10
Number of sub-populations
Total number of iterations
Master time (sec)
Objectif function
242424th may 200624th may 2006
Results comparisonResults comparisonTime comparison between global
parallelisation and distributed algorithm
0100
200300
400500
600700
800
0 2 4 6 8 10
Number of processors or sub-populations
Tim
e (s
ec)
Global parallelisation : master time
Global parallelisation : elapsed timeDistributed algorithm : master time
Distributed algorithm : elapsed time
Efficiency comparison
0.7
0.8
0.9
1
1.1
1.2
1.3
0 2 4 6 8 10
Number of sub-populations or processors
Eff
icie
ncy
Global parallelisation: efficiency
Distributed algorithm: efficiency
252524th may 200624th may 2006
ConclusionsConclusions
The solution found is better than the initial The solution found is better than the initial network but there is no guarantee of network but there is no guarantee of overall optimumoverall optimumAccuracies on key parameters are Accuracies on key parameters are acceptableacceptableBoth parallelization techniques allow to Both parallelization techniques allow to reduce the computing timereduce the computing timeDistributed genetic algorithm gives better Distributed genetic algorithm gives better results than global parallelizationresults than global parallelization
262624th may 200624th may 2006
Future workFuture work
Adaptation to dynamic problemsAdaptation to dynamic problems
Create an algorithm of dynamic data Create an algorithm of dynamic data validationvalidation
Design of networks able to identify Design of networks able to identify process faultsprocess faults
272724th may 200624th may 2006
AcknowledgementsAcknowledgements
Walloon RegionWalloon Region
European Social FundsEuropean Social Funds