Transcript Presentation

Fresh-water minimization through constrained topoly
design with Genetic Algorithm
Dr. Vasile LAVRIC, Drd. Petrica IANCU, Prof. Valentin PLESU
UNIVERSITY POLITEHNICA OF BUCHAREST, CENTRE FOR TECHNOLOGY TRANSFER IN THE PROCESS INDUSTRIES
1, Polizu Street, Building A, Room A056, Sector 1, RO-011061, Bucharest, Romania, email: [email protected]
THE ORIENTED GRAPH
SCHEMATIC MODEL OF UNIT OPERATION I
ABSTRACT
The present paper deals with the design of a water usage network using Genetic
Algorithm as optimization approach subject to the above mentioned restrictions.
This approach generates the best water network topology with a minimum fresh
water usage, complying, in the same time, with all restrictions. An optimal water
network could be viewed as a graph, starting from unit operations with the lowest
contaminant concentrations at entrance, each unit operation "i" receiving streams
from possibly (but not inevitably) all the other operations "j" (j=1, 2, … ji, … N),
and sending streams to probably (but not necessarily) all the other operations "j"
(j=1, 2, … ji, … N), except those which have the imposed level of contaminants
at entrance close or equal to the fresh water level. The mathematical model
describing the unit "i" is based upon total and contaminant species mass
balances, together with the input and output constraints. Solving this optimization
problem is not trivial, since the unknowns' number outcomes the equations'
number. The GA optimization uses each internal flow as a gene, defining a
chromosome from all these flows. The restrictions are cope with naturally, during
the population generation, simply eliminating those individuals outside the
feasible domain. The individuals interbreed according to their frequency of
selection, using one-point crossover method, and then mutation is applied to
randomly selected individuals. The objective function is the total fresh water
consumption, which should be minimized. Comparison with the results of water
pinch and mathematical programming methods is made. .
•
Xji - water flow from j to i operation
(likewise Xij);
Ckj - k pollutant concentration
coming from j operation;
Cki- k pollutant concentration going
to i operation;
Wi - exit flow from unit operation i;
Li - possible flow loss;
mki - k pollutant released in water in
unit operation i;
fi- fresh water flow to i operation;
•
•
•
•
•
•
i 1
THE MATHEMATICAL MODEL
•
•
GENETIC ALGORITHM PROCEDURE
1. Choose the elements of the upper triangular matrix X as genes;
2. Form a chromosome, defining an individual, from all de genes of the matrix
X;
3. Choose the alleles a gene can have (in our case, a continuous domain,
bounded by some convenient chosen values);
4. Choose the magnitude of the population formed of individuals;
5. Choose the appropriate number of generations a population should have;
6. Impose the main parameters of the genetic algorithm:
6.1 favoring the best factor
9. Make a new generation, interbreeding individuals according to
6.2 fitness factor
6.3 crossover probability
10. their frequency of selection, using one-point crossover method;
6.4 mutation probability
11. Apply mutation to randomly selected individuals;
7. Generate the initial population pool
12. If the minimum objective function is attained, stop the algorithm;
8. With each individual:
otherwise, restart from 8.
8.1 solve the mathematical model
8.2 compute the objective function
8.3 convert the objective function to a scaled fitness
8.4 convert the scaled fitness to an expected frequency of
selection as parent
Total mass balance
Partial mass balance, for k
pollutant species
The first set of constraints
The second set of constraints
The objective function
•
•
•
•
From first set of constraints - for
any k species, the equal sign is
valid for some minimum fresh water
flow, dependent of mki
In order for the inequality to hold for
the worst case of k pollutant, the
maximum value for fi should be
picked-up
From second set of constraints for any k species, the equal sign is
valid for some minimum fresh water
flow
In order for the inequality to hold,
the maximum value for fi should be
picked-up
The objective function, in order to
minimize the total amount of fresh
water
•
•
•
•
j 1
N


X jiCkj  mki   Wi  Li   X ij  Cki , k  1, 2

j 1
j i 1


i 1
Cki 
X
ji
j 1

 Wi  Li 


X ij 

j i 1

N
C 
in
ki
10 Units with 6 pollutants,
contaminated supply water (same
order as in table, ppm : 160, 30,
150, 10, 10, 240)
a) For a single contaminated source,
the level of contamination, for all
pollutants, is subtracted from the
maximum allowable input/output;
then, the oriented graph
methodology is applied
Process
Number
1
2
3
4
b) For multiple contaminated
sources, eliminate, first, the over
polluted sources; then, apply the
free graph algorithm
5
6
7
8
9
10
max
C out
(ppm)
500
350
280
15
12
400
400
300
150
50
50
1000
400
300
150
10
12
630
400
300
150
50
50
1000
400
300
150
10
12
630
j 1
ji
Ckj  mki
i 1


 f i   X ji 
j 1


X
j 1
ji
C
out,max
ki
Ckj
C
i 1
fi   X ji
in,max
ki
j 1
fob   max  f
N
i 1 in,out
8
min
i ,in
8
3
,f
min
i ,out

5
34
1
20.96
50
46.93
6
1.02
8
9
142.94
4
41.4
2
54.85
8
5.46
10
4.85
65.44
3.43
9.31
32.08
50
2.22
23.35
7.27
7
12.73
27.09
8
30.51
b) Optimal solution with Genetic Algorithm
Original optimal solution from Savelski et al.
TEST CASE STUDY B
max
C in
(ppm)
160
30
150
10
10
240
400
150
150
10
12
630
400
150
150
10
12
630
400
150
150
10
12
630
400
150
150
10
12
630

X
K
i 1
167.49
Efficient Use And Reuse of Water in
Refineries and Process Plants
M. J. Savelski, M. Rivas and M. J. Bagajewicz
ENPROMER'99
Florianópolis - Santa Catarina - Brasil
Inspect/change input data buttons:
maximum input/output unit concentrations,
contaminants load per unit, regeneration
unit input/output concentrations,
contaminants load in fresh water
Contaminants
Load
(kg/hr)
132.6
124.8
50.7
1.95
0.78
62.4
0
42
0
11.2
10.64
103.6
0
14.55
0
0
0
0
0
12
0
3.2
3.04
29.6
0
1.95
0
0
0
0
i 1
Ckj  mki
When active, reorder units in the network
descending, according to the maximum
fresh water needed per unit
Process
Number
 Li  0
i 1
10 units with 3 contaminants (A, B and C),
as published in:
When active, reorder units in the network
descending, according to the maximum
contaminant load per unit
max
C out
(ppm)
800
300
500
100
100
2035
800
300
500
100
100
2035
400
300
250
100
100
2035
400
300
200
100
100
2035
600
500
200
50
50
2035
j i 1
ij
2.81
When active, neglect internal flows
under a specified value (1 t/h,
customary)
max
C in
(ppm)
400
200
400
30
30
630
400
200
400
30
30
630
160
35
200
30
30
240
160
35
200
30
30
240
160
35
200
30
30
240
X
TEST CASE STUDY A
When active, propagate the best
individual thru generations; after crossover, randomly generate individuals from
a shrinking vicinity of the best individual
Contaminants
Load
(kg/hr)
31.2
7.8
7.8
5.46
5.46
109.59
16.4
4.1
4.1
2.87
2.87
57.605
1.2
1.325
0.25
0.35
0.35
8.975
4.8
5.3
0
1.4
1.4
35.9
2.64
2.79
0
0.12
0.12
10.77
N
fi   X ji  Wi 
Observations
Input data file for a network of 10 units
and 6 contaminants. Fresh water from
Source-1, contaminated
APPLICATION’S INTERFACE
The water network is an
oriented graph, with multiple
starting knots (unit operations
1,2 & 3) with no contaminant
at entrance, each knot i of
the graph receiving streams
from possibly all previous m
(1 & 2) knots only (m=1, 2 …
i-1), and sending streams to
probably all next n (j & N-1)
knots (n=i+1,i+2 … N).
CONCLUSIONS
2.037
3
(Fwtotal = 389.87 ton/hr)
1.443
5
1.52
53.862
284.251
11.978
8
13.471
67.143
3.067
72.433
9
2.182
6
20
4
4.148
7
5
13.887
42.203
2
14.316
1
137.04
3
389.064
10.738
1.734
2.016
6
3
5
35.453
6
12.875
377.224
27.076
10
278.811
36.202
4.093
15.852
Optimal solution (Fwtotal = 910.778 ton/hr)
35.368
•GA technique was implemented to find the optimal water network topology
together with the minimum fresh water consumption, observing for each unit
operation:
•The maximum allowable pollutant input concentration;
•The maximum allowable pollutant output concentration;
•GA performs better than other current techniques (mathematical programming,
tree search methodology with branch cutting, water pinch);
•GA can handle an indefinite number of pollutants;
•GA can handle contaminated sources of supply water;
•GA could be generalized to non-oriented graphs, if all knots have non-zero
input pollutant concentrations;