Sustainable Water Management: Optimization & Rainwater Harvesting

Computers and Chemical Engineering 121 (2019) 158–173
Contents lists available at ScienceDirect
Computers and Chemical Engineering
journal homepage: www.elsevier.com/locate/compchemeng
A multi-objective optimization approach for sustainable water
management for places with over-exploited water resources
Salvador I. Pérez-Uresti
a
, José María Ponce-Ortega
b
, Arturo Jiménez-Gutiérrez
a , ∗
a
Departamento de Ingeniería Química, Instituto Tecnológico de Celaya, Celaya, Gto. 38010, Mexico
b
Facultad de Ingeniería Química, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán 58060, Mexico
a r t i c l e i n f o
Article history:
Received 14 May 2018
Revised 23 September 2018
Accepted 6 October 2018
Available online 12 October 2018
Keywords:
Rainwater harvesting
Over-exploited aquifers
Optimization
Water supply
Water networks
a b s t r a c t
Rainwater harvesting (RWH) is analyzed in this work as an option for water supply for places with over-
exploited water resources. The model is formulated under a multi-period, multi-objective optimization
model. The objective is two fold, first to assess the potential of RWH as an alternative water source, and
second to design an optimal water distribution network in which both natural and alternative sources
work as an integrated system. The problem is formulated to account for three different objectives, namely
maximum profit, minimum groundwater usage and minimum investment cost. A case study for the city
of Queretaro in Mexico was considered to show the applicability of the proposed approach. The results
show that as much as 27% of the domestic demand in Queretaro City could be supplied by RWH, which
would lead to a significant recovery of deep wells currently under depletion.
©2018 Elsevier Ltd. All rights reserved.
1. Introduction
Water supply has become an important issue in several regions
worldwide, and some countries have been implementing several
strategies to mitigate this problem. For instance, in Hong Kong a
policy for water usage has combined the consumption of freshwa-
ter for potable purposes and of seawater for toilet flushing. Under
such a scheme, it has been estimated that 20% of the total water
supply can be saved ( Leung et al., 2012 ). Ross (2017) proposed the
application of a water conjunctive approach that consists in the
combined use of groundwater and surface water as a solution to
meet the policy goals in Australia.
Reclaimed water has an enormous potential as an alternative
water source. For instance, the water discharged for industry in
countries like China has been reported to potentially cover 50 to
80% of the water needed in urban zones ( Yi et al., 2011 ), although
the production of reclaimed water in that country is only 5% of the
domestic demand ( Lyu et al., 2016 ). Singapore has implemented a
NEWater program, under which the use of reclaimed water into a
global integration system has reached a level higher than 30% of
the total water needs ( Lee and Tan, 2016 ). Zhang et al. (2013) de-
veloped a mathematical formulation to design a water distribution
system using reclaimed water for irrigation purposes, taking into
account uncertainties in water demand.
∗Corresponding author.
E-mail address: [e-mail] (A. Jiménez-Gutiérrez).
Aquifer recharge using surplus of either reclaimed water or
desalination plants has been identified as a potential option to
mitigate the water scarcity ( Zekri et al., 2014; Ghaffour et al.,
2013 ). This approach is known as Manage Aquifer Recharge (MAR).
Missimer et al. (2014) carried out an economic evaluation for two
types of water supply in rural zones of Saudi Arabia, desalination
via reverse osmosis and aquifer recharge using reclaimed water,
where MAR reduces significantly the cost of both irrigation water
and drinking water supply.
Greywater, wastewater generated by domestic users in showers,
baths or dish washes, has also shown a great potential for its use
in activities where high water quality is not required. For instance,
it has been reported that 41–91% of domestic water is converted to
greywater, and it could supply 9–46% of the total water needed for
activities such as toilet flushing, garden irrigation and floor clean-
ing ( Boyjoo et al., 2013 ). The advantage of its application for irriga-
tion purposes has also been addressed in some works ( Rodda et al.,
2011; Coelho et al. , 2011; Al-Mashaqbeh et al., 2012 ).
Rainwater Harvesting (RWH) is an attractive opportunity for
places with drought conditions ( Handia et al., 2003; Aladenola and
Adeboye, 2010; Mahmoud et al., 2014; Campisano et al., 2017 ).
However, RWH is generally subject to high levels of contamination,
so that it has to be treated before consumption ( Ward et al., 2017 ).
In addition, to consider it as a sustainable option, it is necessary
to take into account catchment characteristics, quality of rainwa-
ter and alternative resources in the site. Such water resources have
to work as an integrated system in order to provide a successful
supply system ( Kahinda et al., 2007 ). RWH has been used in UK
https://doi.org/10.1016/j.compchemeng.2018.10.003
0098-1354/© 2018 Elsevier Ltd. All rights reserved.

S.I. Pérez-Uresti et al. / Computers and Chemical Engineering 121 (2019) 158–173 159
Nomenclature
Parameters
A
a
n
Collection area in location n for artificial ponds a
at time period t (m
2
)
A
ai
w
Collection area in location w for industrial artificial
ponds ai (m
2
)
A
max
n
Maximum capacity of artificial ponds A in location
n (m
3
)
A
s
l
Collection area in location l for storage tank s (m
2
)
A
si
b
Collection area in location b for industrial storage
tanks si at time period t (m
2
)
A
ROW
k
Collection area for natural sources k (m
2
)
AI
max
w
Maximum capacity of industrial artificial ponds AI
in location w (m
3
)
AC
t Aqueduct water flowrate (m
3
/month)
ASC Unit water cost for agricultural use ($/m
3
)
Ce
y ( t )
Runoff coefficient for natural sources during year t
Ce
s Runoff coefficient for storage tanks s
Ce
a Runoff coefficient for artificial ponds a
Ce
si Runoff coefficient for industrial storage si
Ce
ai Runoff coefficient for industrial artificial ponds ai
CTND Unit treatment cost for natural sources with do-
mestic use ($/m
3
)
CTNA Unit treatment cost for natural sources with agri-
cultural use ($/m
3
)
CTNI Unit treatment cost for natural sources with indus-
trial use ($/m
3
)
CTAD Unit treatment cost of rainwater for domestic use
($/m
3
)
CTAA Unit treatment cost of rainwater for agricultural
use ($/m
3
)
CTAI Unit treatment cost of rainwater for industrial use
($/m
3
)
CTPE Unit treatment cost of wastewater regeneration for
final disposal ($/m
3
)
d
ds
j,t
Domestic user j demand at time period t
(m
3
/month)
d
as
h,t
Agricultural user h demand at time period t
(m
3
/month)
d
di
u,t
Industrial user u demand at time period t
(m
3
/month)
DPWV
k, t
Water collected from direct precipitation in natu-
ral sources k at time period t (m
3
/month)
DSC Unit water sale price for domestic use ($/m
3
)
ISC Unit water sale price for industrial use ($/m
3
)
Ka Parameter for type of land and use land
K
F
Factor to take into account annualized investment
index
P
t Precipitation over the time period k (m
3
/month)
P
Total Annual precipitation in year y (m 3
/year)
PCSTD Unit cost of transport from storage tank l to do-
mestic sink j ($/m
3
)
PCASD Unit cost of transport from artificial pond n to do-
mestic sink j ($/m
3
)
PCSTA Unit cost of transport from storage tank l to agri-
cultural sink h ($/m
3
)
PCASA Unit cost of transport from artificial pond n to
agricultural sink h ($/m
3
)
PCSTI Unit cost of transport from industrial storage tank
b to industrial sink u ($/m
3
)
PCASI Unit cost of transport from industrial artificial
pond w to industrial sink u ($/m
3
)
PCND Unit cost of transport from natural source k to do-
mestic main ($/m
3
)
PCNA Unit cost of transport from natural source k to
agricultural main ($/m
3
)
PCNI Unit cost of transport from natural source k to in-
dustrial main ($/m
3
)
PCACD Unit cost of transport from aqueduct to domestic
user ($/m
3
)
PCACI Unit cost of transport from aqueduct to industrial
user ($/m
3
)
r
m, k, t
Segregated flowrate from tributaries m to natural
sources k in time period t (m
3
/month)
p
g
k,t
Water collected from direct precipitation and
runoff water in sources kin time period t
(m
3
/month)
ROWV
k, t
Runoff water collection in natural sources kin
time period t (m
3
/month)
S
max
l
Maximum capacity of storage tanks S in location
(m
3
)
SI
max
b
Maximum capacity of industrial tanks SI in loca-
tion (m
3
)
VP Factor to consider the value of investment
L
max Maximum allowable of domestic and agricultural
storage tank to be installed in a single period .
N
max Maximum allowable of domestic and agricultural
ponds to be installed in a single period .
B
max Maximum allowable of industrial storage tanks be
installed in a single period .
W
max Maximum allowable of industrial pond be installed
in a single period .
Binary variables
z
s
l,t
Binary variable for installing the storage tank s in
location l over time period t
z
a
n,t
Binary variable to select the installation of artificial
pond a in location n over the time period t
z
si
b,t
Binary variable to select the installation of industrial
storage tank si in location b over the time period t
z
ai
w,t
Binary variable to select the installation of industrial
artificial pond ai in location w over the time period
t
Variables
A
n, t Existing water in artificial pond A
in location n in time period t
(m
3
/month)
A
n,t−1 Existing water in artificial pond A
in location n in time period t-1
(m
3
/month)
a
in
n,t
Water obtained from rainfall sent to
artificial pond a in location n in time
period t (m
3
/month)
a
out ,d
n,j,t
Segregated flowrate from artificial
pond a in location n sent to do-
mestic user j in time period t
(m
3
/month)
a
out ,a
n,h,t
Segregated flowrate from artificial
pond a in location n sent to agri-
cultural user h in time period t
(m
3
/month)
AI
w, t Existing water in artificial pond AI
in location w in time period t
(m
3
/month)

160 S.I. Pérez-Uresti et al. / Computers and Chemical Engineering 121 (2019) 158–173
A I
w,t−1 Existing water in artificial pond AI
in location w in time period t-1
(m
3
/month)
ai
in
w,t
Water obtained from rainfall sent to
industrial artificial pond ai in loca-
tion w in time period t (m
3
/month)
ai
out ,i
w,u,t
Segregated flowrate from industrial
artificial pond ai in location w sent
to agricultural user u in time period
t (m
3
/month)
Cost
s
l
Cost of storage tank s in location l
($)
Cost
a
n
Cost of artificial pond a in location n
($)
Cost
si
b
Cost of storage tank si in location b
($)
Cost
ai
w
Cost of industrial artificial pond ai in
location w ($)
cw
d
j,t
Water consumed and lost in do-
mestic sinks j in time period t
(m
3
/month)
cw
di
u,t
Water consumed and lost in in-
dustrial sinks u in time period t
(m
3
/month)
G
k, t
Existing water in natural source k at
tome period t (m
3
/month)
G
k,t−1
Existing water in natural source k at
tome period t-1 (m
3
/month)
g
d
k,t
Segregated flowrate from natural
source k to main domestic d in time
period t (m
3
/month)
g
a
k,t
Segregated flowrate from natural
source k to main agricultural a in
time period t (m
3
/month)
g
i
k,t
Segregated flowrate from natural
source k to main industrial I in time
period t (m
3
/month)
Drop
g
k,t
Water that exceeds the maximum
capacity of natural source k in time
period t (m
3
/month)
f
j, t
Segregated flowrate sent from the
domestic main to the domestic user
j in time period t (m
3
/month)
int
in
j,t
Wastewater produced in domestic
sink j in time period t (m
3
/month)
inti
in
u,t
Wastewater produced industrial sink
u in time period t (m
3
/month)
P
s
l,t
Available precipitation in location l
for storage tank s in time period t
(m
3
/month)
P
a
n,t
Available precipitation in location n
for artificial pond a in time period t
(m
3
/month)
P
si
b,t
Available precipitation in location b
for industrial storage tank si in time
period t (m
3
/month)
P
ai
w,t
Available precipitation in location w
for artificial pond ai in time period t
(m
3
/month)
S
l, t
Existing water in storage tank S
in location l in time period t
(m
3
/month)
S
l,t−1
Existing water in storage tank S
in location l in time period t-1
(m
3
/month)
s
in
l,t
Water obtained from rainfall sent to
storage tank s in location l in time
period t (m
3
/month)
s
out ,d
l,j,t
Segregated flowrate from storage
tank s in location l sent to domestic
user j in time period t (m
3
/month)
s
out ,a
l,h,t
Segregated flowrate from storage
tank s in location l sent to agri-
cultural user h in time period t
(m
3
/month)
si
in
b,t
Water obtained from rainfall sent to
industrial storage tank si in location
b in time period t (m
3
/month)
r
h, t
Segregated flowrate sent from the
agricultural main to the agricultural
user h in time period t (m
3
/month)
q
u, t Segregated flowrate sent from the
industrial main to the industrial user
u in time period t (m
3
/month)
v
s
l,t
Water lost in storage tank s in time
period t (m
3
/month)
v
a
n,t
Water losses in artificial pond a in
time period t (m
3
/month)
v
si
b,t
Water lost in industrial storage tank
si in time period t (m
3
/month)
v
ai
w,t
Water lost in industrial artificial
pond ai in time period t (m
3
/month)
SI
b, t
Existing water in industrial storage
tank SI in location b in time period
t (m
3
/month)
S I
b,t−1
Existing water in industrial storage
tank SI in location b in time period
t-1 (m
3
/month)
si
out ,i
b,u,t
Segregated flowrate from industrial
storage tanks si in location b sent to
industrial users u in time period t
(m
3
/month)
ACDD
j, t
Segregated flowrate from aqueduct
to domestic users j in time period t
(m
3
/month)
ACII
u, t Segregated flowrate from aqueduct
to industrial users u in time period
t (m
3
/month)
Greek symbols
a Exponent to take into account economies scale
δs,max
l
Maximum capacity of storage tank s in location l
(m
3
)
δa,max
n
Maximum capacity of artificial pond a in location n
(m
3
)
δsi,max
b
Maximum capacity of industrial storage tank si in
location b (m
3
)
δai,max
w
Maximum capacity of industrial artificial pond ai in
location w (m
3
)
Sets
B set for location of industrial storage tanks
( { b| b = 1 , ... , B } )
H set for agricultural sinks ( { h | h = 1 , . . . , H } )
J set for domestic sinks ( { j| j = 1 , . . . , J } )
K set for natural sources ( { k | k = 1 , . . . , K } )

S.I. Pérez-Uresti et al. / Computers and Chemical Engineering 121 (2019) 158–173 161
L set for location of storage tanks ( { l| l = 1 , . . . , L } )
M set for tributaries ( { m | m = 1 , . . . , M } )
N set for location of artificial ponds ( { n | n = 1 , . . . , N } )
T set for time period in months ( { t| t = 1 , . . . , T } )
U set for industrial sinks ( { u | u = 1 , . . . , U } )
W set for location of industrial artificial ponds
( { w | w = 1 , . . . , W } )
Y set for time periods in years ( { y | y = 1 , . . . , Y } )
to provide non-potable water, and in places with restricted use
to fresh water sources such as small islands in the Pacific Ocean
( Quigley et al., 2016 ). According to Donahue et al. (2017), up to
60,0 0 0 inhabitants in Hawaii Island depend on RWH to cover their
basic domestic needs. The use of RWH systems could benefit from
recent developments, which include the generation of potable wa-
ter through a treatment method that combines filtration, UV and
ozonation ( Melville-Shreeve et al., 2016 ).
The development of models to assess the potential of RWH in-
tegration into a global water supply system is a rather complicated
task because factors such as climate, land use and rainfall-runoff
affect the performance of the system ( Kahinda et al., 2008 ). To con-
tribute to such an understanding, Silva and Ghisi (2016) carried out
a sensitivity analysis on the design variables involved in RWH. De-
mand and roof area were identified as the most relevant variables
for selecting the optimal tank capacity and providing potable water
savings.
For evaluating and optimizing the RWH performance,
Adham et al. (2016) proposed a model that takes into account
both the harvesting system and the hydrological process. They
point out that understanding the relationship between rainfall
and runoff water in catchments is mandatory to determine the
potential water savings. The macroscopic design of RWH systems
has been addressed in several works. Hashim et al. (2013) de-
veloped a mathematical model to determine the optimal size of
storage tanks, rooftop area and reliability system for its applica-
tion in Malaysia. To design a sustainable RWH system, Li et al.
(2017) developed a multi-objective model for optimizing the
construction areas of green rooftops, porous pavements and green
lands, taking into account economic and environmental issues; this
approach led to an overall secure and sustainable urbanization.
Lopes et al. (2017) presented a stochastic approach to evaluate the
performance of RWH systems under several climate patterns of a
city in Brazil. They concluded that even under unfavorable RWHS
conditions, a significant share of non-potable water demand could
be met. Nápoles-Rivera et al. (2013) developed a mathematical
program to determine the optimal scheduling distribution for
different users in a macroscopic system considering rainwater
harvesting and water reclamation as alternative water sources. In
later works, Nápoles-Rivera et al. (2015) incorporated uncertainty
in rainwater patterns, whereas Rojas-Torres et al. (2015) imple-
mented that model to carry out a multi-objective optimization for
a case study in Mexico.
It should be noticed that none of the above mentioned ap-
proaches have presented a general optimization formulation to sat-
isfy the future water demand in places with over-exploited water
resources (particularly where the aquifers are significantly over-
exploited). Therefore, this paper presents a general optimization
model based on a multi-objective and multi-period formulation to
satisfy the future water demand in places with over-exploited re-
sources. The main objective is to meet the future water demand
and at the same time to improve the levels of the aquifers within
the next few years. The model formulated in this work would al-
low for a better water use and distribution by including alternative
water sources such as rain water harvesting and reuse of reclaimed
water, considering also the proper storage and distribution of these
important resources. A case study for the city of Queretaro in Mex-
ico is considered in order to reduce the groundwater usage. The
evaluation is addressed from three perspectives, namely maximum
revenue, minimum groundwater usage, and minimum investment
cost.
2. Problem statement
The problem addressed in this work is stated as follows. Given
a set of sources and sinks, as shown in Fig. 1 , it is required to
find the optimal planning and scheduling of resources manage-
ment that satisfies the demand of users (sinks) while maximizing
profit and minimizing groundwater usage. The following items are
taken into account.
Three different types of sinks are considered, domestic, indus-
trial, and agricultural, with water demand subjected to seasonal
variations, such that for hot months water demand is considered
to be higher than that for cold months. As far as industrial users,
their water demand remains almost constant, normally supplied by
groundwater. The main natural source is groundwater, and it can
be recharged by runoff water and natural tributaries. Water is ex-
tracted and sent to central facilities, where a proper treatment is
carried out in order to meet quality specifications as required by
each user. Domestic and industrial wastewater is treated in cen-
tralized facilities. Another source of water is provided by Aqueduct
II; currently, such a source is restricted by federal law to a level of
47 Mm
3
/y (DOF, 2009; CONAGUA, 2011 ) and it is used to supply
water to industrial and domestic users.
In order to reduce the depletion of natural sources, the installa-
tion of different types of storage devices in the city was assumed,
which would be used to collect, treat and distribute rainwater har-
vested to the final user. Four types of storage devices are included,
which consist of storage tanks for general purpose (domestic and
agricultural), storage tanks for industrial use, artificial ponds for
general purpose, and artificial ponds for industrial users. The main
difference among these options lies on the size and capacity to col-
lect rainwater. It is worth pointing out that part of the problem
consists in determining the optimal location of the storage devices.
3. Mathematical model
The model was developed under a multi-objective, multi-period
optimization formulation, and based on mass balances over mixing
and splitting points of the superstructure shown in Fig. 1 . A set
of equations used to determine whether or not a storage device
should be installed is also included. The following assumptions are
considered:
•The water quality of each outflow of splitting points satisfies
environmental and quality requirements imposed by users.
•The increase in water demand is correlated to population
growth.
•Money changes value over time
•The historical data of precipitation are used to determine the
amount of water from rain that can potentially be harvested.
•Other specifications related to sustainability and economic cri-
teria are included.
3.1. Mass balance for natural water sources
In order to determine the accumulation G
k, t
in natural source k
over a time period t , it is necessary to consider some aspects such
as tributary sources r
m, k, t
(for instance, groundwater recharge), as
well as the direct precipitation and runoff water ( p
g
k,t
) for inlets.
For outlets, g
d
k,t
, g
a
k,t
and g
i
k,t
represent the water sent to domestic,

162 S.I. Pérez-Uresti et al. / Computers and Chemical Engineering 121 (2019) 158–173
Fig. 1. Superstructure for water distribution.
agricultural and industrial mains, respectively. Also, v
d
k,t
and Drop
g
k,t
are the water losses due to evaporation, filtration and losses in the
distribution process. In this work, these losses are assumed as 20%
of the inlets for each period.
G
k,t
−G
k,t−1
= 
mεM
r
m,k,t
+ p
g
k,t
−g
d
k,t
−g
a
k,t
−g
i
k,t
−v
d
k,t
−Drop
g
k,t
, ∀ k ∈ K , ∀ t ∈ T (1)
Direct precipitation can be calculated as follows,
p
g
k,t
= ROW V
k,t
+ DP W V
k,t
, ∀ k ∈ K , ∀ t ∈ T (2)
where ROWV
k, t
is the runoff water, which depends on a runoff co-
efficient, annual precipitation and collection area,
ROW V
k,t
= P
t
·A
ROW
k
·C e
y
(
t
) ∀ k ∈ K , ∀ t ∈ T (3)
Ce
y ( t )
is a function of both total annual precipitation and a pa-
rameter K, which takes into account the type and use of terrain.
The runoff coefficient is calculated using the method proposed by
the Mexican Norm NOM-001CNA-2000 (CONAGUA, 20 0 0).
Ce =
Ka
P
Total
−250
20 0 0
, Ka ≤0 . 15 (4)
Ce =
Ka
P
Total
−250
20 0 0
+
Ka −0 . 15
1 . 5
, Ka > 0 . 15 (5)
Direct precipitation ( DPWV
k, t
) is calculated as a function of
both the collecting area and monthly precipitation. It is worth not-
ing that this variable only applies to superficial water sources.
DP W V
k,t
= P
t
·A
DPW
k
, ∀ k ∈ K , ∀ t ∈ T (6)
Finally, the water losses are calculated as follows,
v
d
k,t
= 0 . 2
mεM
r
m,k,t
+ p
g
k,t
, ∀ k ∈ K , ∀ t ∈ T (7)
3.2. Balance for harvested rainwater
The amount of water that can potentially be harvested is a
function of precipitation, the runoff coefficient and the collection