Cost Minimization of a WWTP Using an
Augmented Lagrangian Pattern Search
Based Solver
1Isabel
Espírito Santo,
1Edite
Fernandes,
1Madalena
Araújo,
2Eugénio
Ferreira
1Systems
and Production Department, 2Center of Biological Engineering
Minho University, Braga, Portugal
1{iapinho;emgpf;mmaraujo}@dps.uminho.pt, [email protected]
With the growing need to decrease the installation and operation costs of a wastewater treatment
plant (WWTP), the search for a minimum cost design is becoming more and more challenging. To be
able to achieve an optimal WWTP design an optimization procedure that considers the mathematical
modelling of the secondary treatment, an activated sludge system that consists of an aeration tank
and a secondary settler, and the definition of a cost function is carried out. The impact of the primary
treatment on the cost of the secondary treatment is also reported.
In the presence of non-smooth functions as encountered in some of the mathematical equations involved in the model, a derivative-free
optimization technique is the most appropriate. So, a pattern search algorithm is proposed. This algorithm relies on an augmented Lagrangian
function in order to obtain a solution that satisfies the equality and inequality constraints of the problem.
influent
effluent
secondary settler
aeration tank
recycle
waste
Methods
constraints:
objective function:
-mass balances around the Aeration Tank (ASM1)
The total cost (TC) function results from the sum of the
investment and operation costs
-constraints of the Secondary Settler (combination of ATV
and double exponential model)
TC 
-composite variables, quality constraints, flow and mass
balances around the system, simple bounds
1.07
174 .2Va
 12487
0.62
GS
 114 .8G S 
0.97
955 .5 As
 41 .3( As h)
1.07
Augmented Lagrangian pattern search
Augmented Lagrangian penalty function:
2


p
m
g
(
x
)
1 m



2
2

i
 ( x;  ,  ,  )  f ( x )   i bi ( x ) 
 bi ( x )   max  0,  i 
  i

i 1
2  i 1
2 i 1





mathematical formulation:
minimize
  x : b( x)  0, g ( x)  0, l  x  u
f ( x)
x    
n
Lagrangian multiplier procedure
Initialize variables and algorithm parameters
while convergence criteria are not satisfied do
pattern search procedure: approximately solve subproblem
pattern search procedure
k
Initialize with x  
while termination criterion is not satisfied do
k k
Compute  s using Hooke and Jeeves exploratory moves
k
k k
j j j
such
that
x   s 
min n ( x;  ,  ,  ) ,   {x : l  x  u}
k
k
k k
k 1
k
k k
k 1
k
if ( x ;.)  ( x   s ;.)  0 then x  x   s else x  x
x
Update multipliers j and j, penalty parameter j, j  j  1 end if
k
k
end while
Update  and s , k  k  1
end while
Results and conclusions
Primary
treatment
efficiency
TSS
Va
GS
As
h
COD
N
total cost
(106 €)
Alijó
0
40
22.8
11.2
2632
1738
108
100
654
811
23.8
16.4
41.1
44.6
6.4
6.5
2.79
2.41
Murça
0
40
10.4
24.2
1599
913
102
101
746
371
16.5
16.9
12.8
74.7
3.1
9.5
2.24
1.25
Sabrosa
0
40
20.1
14.6
1344
914
100
133
707
557
17.5
9.6
88.7
28.7
9.1
9.7
2.14
1.36
Sanfins
do Douro
0
40
24.0
16.3
1263
1179
103
158
703
404
14.0
10.7
51.7
51.2
10.5
9.7
1.9
1.28
Location of the WWTPs
Alijó
Murça
Sabrosa
Sanfins do
Douro
6850
3850
2750
3100
1050
885
467.5
530
peak flow (m3/h)
108
86.4
48.6
54
COD (kg/m3)
2000
Pop. equivalent
influent flow
(m3/d)
750
TSS (kg/m3)
1750
660
1250
610
1250
610
This work presents a new derivative-free optimization algorithm for solving a problem that considers the WWTP design, in which the objective
is to minimize the costs associated with the installation and operation of an activated sludge system. Four real WWTPs were analyzed and
the mathematical modelling of the activated sludge system was carried out using the ASM1 model for the aeration tank and a combination of
the ATV and double exponential models for the settling tank. The cost function was obtained using real data provided by a portuguese WWTP
company builder. This mathematical programming problem was solved using an augmented Lagrangian pattern search method.
The obtained optimal WWTP designs are robust and economically attractive. The presence of the primary treatment causes a reduction in the
total cost: from 13.6% (Alijó) to 44.2% (Murça).
References:
. Ekama G.A., Barnard J.L., Günthert F.W., Krebs P., McCorquodale J.A., Parker D.S. and Wahlberg E.J. (1978). Secondary Settling Tanks: Theory, Modeling, Design and Operation. Technical Report 6, IAWQ – International Association on Water Quality
. Hooke R. and Jeeves T. A. (1961). Direct Search Solution of Numerical and Statistical Problems, Journal on Associated Computation, 8, pp. 212-229
. Lewis R. M. and Torkzon V. (1999) Pattern Search Algorithms for Bound Constrained Minimization, SIAM Journal on Optimization, 9(4), pp. 1082-1099
. Takács I., Patry G. G. and Nolasco D. (1991). A Dynamic Model of the Clarification-Thickening Process, Water Research, Vol. 25, no. 10, pp. 1263-1271
. Tyteca D., Smeers Y. and Nyns E. J. (1977). Mathematical Modeling and Economic Optimization of Wastewater Treatment Plants, CRC Critical Reviews in Environmental Control, 8 (1), pp. 1-89
10th IWA Specialised Conference Design, Operation and Economics of Large Wastewater Treatment Plants, Vienna, 9-13 September 2007