Instituto Tecnológico y de Estudios Superiores de Monterrey Campus Monterrey

School of Engineering and Sciences

Optimal design of water allocation networks in highly altered basins:

The Guandu River case, Brazil.

A thesis presented by

Oswaldo Adolfo Saucedo Ramírez

Submitted to the

School of Engineering and Sciences

in partial fulfillment of the requirements for the degree of

Master of Science In

Engineering Science

Monterrey Nuevo León, December 3^{rd}, 2020

**iv **

**Dedication **

**To my parents: **

Eva Ramírez Rosas Federico Saucedo Osorio

**To my brother and his family: **

Edgar Iván Hernández Ramírez Daniela Isabel Luján Ramírez Emmanuel Hernández Luján Miguel Ángel Hernández Luján Ivanna Hernández Luján

**For your unconditional support, love, and for always believing in me. **

**To my friends: **

Ali Ghahraei Figueroa

Ana Karen Sánchez González Brian de Jesús Herberth Guerrero Daniela Maritza Carrasco Juárez Erick Daniel Pavón Ceja

Herón Zurita Vázquez Lorena Que García

Luis Alberto Valenzuela Reynoso Mauro Guillermo Bonilla Morales

**For your words of encouragement, for your support and loyalty. For all the **
**excellent and bad moments that we have been through, **

**I love you all. **

**v **

**Acknowledgments **

To CONACyT for economic support during the realization of my master’s degree.

To ITESM for the full scholarship during the realization of my master’s degree.

To my advisor, Ph. D. Ramón González Bravo, for all your support, trust, knowledge, and words of encouragement to realize this project.I will always be grateful to you, Doc!

To my advisor, Ph. D. Jürgen Mahlknecht, for the opportunity to join the group and for all your support during the realization of this work. Thank you so much.

To my committee members, Ph. D. Aldo Iván Ramírez Orozco and Ph. D. Frank J.

Loge, for all your concise recommendations for improving this research.

To my friend Ángel Manuel Villalba Rodríguez for your support, friendship, and help.

To my friend Iris Cassandra Cámara Gutierrez for all the support and help during the classes together and for the laughs. Thank you so much.

To my team Erika Karolina Zazueta Padilla, for all the good times during the master’s degree. For your help, for the projects and homework achieved from the challenging classes. Thank you, sis.

To my friend Norberto Emmanuel Naal Ruíz for all the support and good moments during the master’s degree.

**viii **

**Optimal design of water allocation networks in highly altered ** **basins: The Guandu River case, Brazil.**

**by **

**Oswaldo Adolfo Saucedo Ramírez **

**Abstract **

Water scarcity is present in many regions around the world. Factors affecting water availability include, but are not limited to, population growth, resource depletion, alteration of natural ecosystems, and climate change. Said this, the study of water allocation networks is taking an important role worldwide due to the importance of the liquid for the development of human activities. The application of optimization models represents an opportunity to create new approaches to water resources management and to guarantee the sustainability of natural resources. Many previous optimization models focused on studying the hydraulic elements of the water allocation networks (e.g., pipes, pumps, and storage) to maximize economic profit or minimize distribution costs. These past approaches often neglected the hydrological aspects which describe the behavior of natural ecosystems. The aim of this research is to develop a multiobjective optimization model that incorporates parameters and equations for hydrological processes for the design of water allocation networks in highly altered basins. The model is applied to the Guandu basin in Brazil, one of the most altered watersheds worldwide (receiving 96% of its volume from surrounding basins). This basin supplies 9 million inhabitants of the Rio de Janeiro metropolitan area. Simultaneously, this basin is characterized by a strong relationship with the energy sector, i.e., around 25 % of the city's energy is produced in the basin through a hydropower complex. The results show that water transfer can be optimized by integrating water storage and reuse/recycling elements to satisfy water demands throughout the year. Also, through optimal allocation networks, it is possible to avoid saline intrusion downstream of the Guandu river, even if there is a reduction in the volume transferred from nearby basins. The developed tool is a highly feasible option for decision-making in water resources planning and management.

**ix **

**Content **

List of Figures ... x

List of Tables ... xi

1. Introduction ... 1

2. Objectives ... 4

3. Theoretical Framework ... 5

3.1. Water situation ... 5

3.2. Hydrologic Processes ... 6

3.3. Basin alterations ... 7

3.4. Hydrologic Models ... 7

3.4.1. Hydrologic Conceptual Model ... 9

3.4.2. Software for Hydrologic Modelling... 10

3.5. Optimization ... 11

3.5.1. Optimization Models ... 12

3.5.2. Software for Optimization Models ... 13

3.6. Piping and Pumping Calculations ... 15

4. Methodology ... 15

4.1. The Hydrological Model ... 16

4.2. Superstructure ... 18

4.3. Mathematical Model for Water Allocation ... 20

5. Case study ... 37

6. Results and Discussion ... 45

6.1. Initial Considerations ... 45

6.2. First scenario ... 46

6.3. Second scenario. ... 49

6.4. Third scenario ... 52

6.5. Comparison of Scenarios ... 55

7. Conclusions and recommendations ... 57

Appendix A ... 59

References ... 66

**x **

**List of Figures **

Figure 1. Water Cycle (NASA,2020) ... 6

Figure 2. Hydrological models classification ... 8

Figure 3. Concept of Temez model ... 9

Figure 4. AQUATOOL interface ... 10

Figure 5. Classification of optimization problems ... 12

Figure 6. Optimization Procedure in GAMS... 14

Figure 7. Methodology steps ... 16

Figure 8. Proposed superstructure for water allocation in the highly altered basin 19 Figure 9. Guandu Basin Location ... 37

Figure 10. Geographical distribution of sub-basins of the Guandu river for this study. ... 38

Figure 11. Schematic representation of the case study and identification labels for the optimization model. ... 42

Figure 12. Location of water demands and aquifers along the Guandu basin. ... 43

Figure 13. Water consumption by domestic, industrial, and agricultural users. ... 44

Figure 14. Water allocation network in Scenario 1 ... 48

Figure 15. Comparative water volume discharged at Guandu river mouth (Scenario 1 versus normal conditions)... 49

Figure 16. Water Allocation Network Configuration Scenario 2 ... 51

Figure 17. Comparative water volume discharged at Guandu river mouth in (Scenario 2 versus normal conditions) ... 52

Figure 18. Water Allocation Network Configuration Scenario 3 ... 54

Figure 19. Comparative water volume discharged at Guandu river mouth (Scenario 3 versus normal conditions)... 55

**xi **

**List of Tables **

Table 1. Optimization models for water allocation ... 12

Table 2. Components and sub-basins of the Guandu river, Brazil ... 39

Table 3. Water volume granted for the Guandu river basin. ... 40

Table 4. Operators of the monitoring network in Brazil. ... 40

Table 5. Correlation factors ... 41

Table 6. Optimal results of the proposed three scenarios ... 56

**1 **

**1. Introduction **

Water is one of the most essential resources for humanity and supports the ecosystem’s health. Its availability can trigger or limit social, economic, and technological development in a region (UN-Water, 2020). In recent years, many megacities worldwide have been facing a water resource crisis due to the increase of water demand caused by fast urbanization (including population growth and migration), industrialization, and food production (agriculture and livestock) (Cruz et al., 2018). The rapid modernization versus an irregular natural water distribution in time and space has aggravated the water competition and generated conflicts between different stakeholders at local and regional levels (Cruz et al., 2018; I-Shing et al., 2020; Joseph et al., 2019; Zhang and Guo, 2016). Under this panorama and having currently more than four billion people living in cities with persistent or seasonal water scarcity, representing over 70% of the 'world's river basins, will suffer water-related problems by 2050.

Climate change, pollution, and basin alterations have exacerbated the water stress in these important areas. Undoubtedly, climate change causes shifts in precipitation patterns, in which these variations directly affect the most critical variables of the hydrological cycle, such as humidity, evapotranspiration, and precipitation affecting the availability of the natural water resource in a basin or region (Martínez-Austria and Patiño-Gómez, 2012; Törnqvist et al., 2014). On the other hand, the deterioration of water quality has become a cause of concern worldwide due to the high levels of uncontrolled waste and wastewater discharge (Santos, 2014). At the same time, basin alteration is a phenomenon that has increased in the last decades.

These alterations due to the unplanned and chaotic growth of urban areas modify the natural state of the basin, affecting the hydrological and catchment processes (infiltration, evapotranspiration, groundwater recharge, and discharge) (Daneshi et al., 2020).

**2 **

In recent years, it has become clear that one of the most critical challenges in water -deficit areas is allocating available water since competition for its service between agricultural, urban, industrial, and environmental demand has intensified (Liu et al., 2011). The availability of water has a strong relationship with adequate planning and management, which determines the allocation of water for a region (Cosgrove and Loucks, D., 2015). Thus, in areas with water stress, optimization techniques represent a viable option for evaluating water planning and management alternatives, contributing to sustainable water resources. Optimal water management becomes a major priority (Müller et al., 2020; WWAP, 2019).

Several optimization models have been developed to improve the planning and management of water resources. There exist models that have focused on optimal water resource allocation in agricultural systems for irrigation, because agriculture is traditionally the user with the highest water demand (Gong et al., 2020; Huang et al., 2020). Other models considered multiple users in a region for the optimal allocation to supply demands (González-Bravo et al., 2016). Others have focused on the reduction of cost and meeting the quality required for each user (Senante et al., 2013). Given the complexity of extensive water allocation networks, multiobjective models have been developed considering the creation of sub-networks to manage and distribute the liquid (Hajebi, 2014). The Hazen-Williams equation was used to optimize the water allocation network and find the optimal elements for the correct distribution of water (Caballero and Ravagnani., 2019). In another study, a macroscopic optimization model has been developed, which may consider the variable availability of water sources such as deep wells depending on groundwater table levels (Olivarez-Areyan et al., 2020).

These previous optimization models for water resources management allow us to find technical and economically viable solutions to distribute the vital liquid. However, there is a need to include the hydrological aspects that better represent the natural water cycle in optimization schemes and thus improve them. These aspects consider precipitation, evaporation, evapotranspiration, infiltration, among others.

**3 **

Additionally, existing approaches related to water allocation networks do not consider the implications of basin modifications. Thus, this research aims to develop an optimization model that considers hydrologic processes and offers an improved optimization tool for the appropriate use and distribution of water in water-stressed areas.

**4 **

**2. Objectives **

The objective of this project is to incorporate parameters and equations that represent the hydrological processes in an optimization model for water resources allocation in a highly altered basin.

Particular objectives are:

• To develop a mathematical model that includes considerations and equations from the hydrological point of view.

• To optimize the water resource management in highly altered basins based on incorporating new elements to save water for periods with reduced water availability.

• To generate a tool for decision-making in the planning, allocation, and management of water resources.

• To apply the model in a water-scarce and highly altered basin and analyze scenarios to develop feasible solutions focusing on economic and sustainability aspects.

• To propose recommendations based on the obtained results to improve water resources allocation in the selected basin.

**5 **

**3. Theoretical Framework **

**3.1. Water situation **

Water is widely distributed on Earth, though, in different quantities and qualities. It plays a vital role in environmental health and human well-being (Shiklomanovy, 1991). The water resources are present in all the water bodies (oceans, rivers, groundwater, lakes, etc.) to supply human demands in finite volume (Pimentel et al., 2004), and its availability carries enormous weight for the development in a region (UN-Water, 2020).

In recent years, major cities worldwide have faced a water resource crisis due to
increased water demand caused by fast urbanization, industrialization, and food
production (Cruz and Martinez, 2015). This situation has become worse due to
climate change, which may cause a modification of all water cycle elements. Cities
are particularly vulnerable to insufficient water provision in quantity and quality,
sanitation, and drainage (Lossouarn et al., 2016).^{ }

Mega-cities represent major social and economic development hubs; however, overpopulation represents the biggest challenge to the water supply at all scales, putting stress on natural resources and its consequent pollution. Most of these megacities are found in Latin America, Asia, and Africa (Mendoza, 2020). For example, in Mexico City, apart from overpopulation and pollution, the water scarcity problems are aggravated because 40% of the water is wasted due to problems in the pipe system (Barkin, 2004). The city of Bangalore, India, has had rapid growth in terms of new urbanization, which has caused problems of water shortage, added to the fact that most of its lakes are so polluted that 85% of the water can only be used for industrial cooling and irrigation in agriculture, and not for drinking or bathing (Ramachandra, 2015). Beijing is another city that has problems with water shortages, mainly due to the high contamination of its rivers and lakes. In 2015, 40%

of water was so contaminated that it could not be used for industry or agriculture (Wang, 2015). Cities like Sao Paulo and Rio de Janeiro in Brazil, with 23.5 and 11.9

**6 **

million inhabitants, share their water resources and have presented multiple water supply crises starting in 2015 because of the overpopulation, basin modification, and the increasing industrialization demands (Britto, 2016).As can be seen, this problem is typically caused by insufficient water and wastewater planning and management, in which alterations of the basins have also modified the hydrological cycle. In consequence, the pollution reduces the water quantity and quality, and this increases the water stress in those areas.

**3.2. Hydrologic Processes **

The hydrologic cycle describes water changes from different states and storage between the biosphere, atmosphere, lithosphere, and hydrosphere. These changes occur through evaporation, condensation, precipitation, deposition, runoff, infiltration, sublimation, transpiration, melting, and groundwater flow (Mohammed, 2018). (Figure 1).

**Figure 1. Water Cycle (NASA,2020) **

**7 **

**3.3 Basin alterations **

A basin is an area of land that drains all the streams and rainfall to a common outlet:

a river (USGS, 2020). This area is the scenario for most of the natural processes associated with the hydrological cycle. In some cases, the alteration inland, such as intensive deforestation, mining, and farming (Carlson and Arthur., 2000), including the construction of dams, channels, and irrigation systems, modify the natural water cycle (Ferguson and Maxwell, 2012). These alterations modify the natural state of the basins, which are irreversible in most cases. Trying to revert them could jeopardize water availability. The alteration of basins is a phenomenon that has increased in the last decades. Also, altered basins deal with increasing water stress, water quality impairments, and lack of well-functioning infrastructure (Krueger et al., 2019). Nonetheless, these modifications are, in part result of the construction of infrastructure for the benefit of human society: reservoirs to face the variability in space and time of water availability (Dong, N et al., 2020), hydropower dams to cope with increasing demand for energy due to economic development and population growth (Jiang et al., 2018), among other benefits.

**3.4. Hydrologic Models **

Hydrological processes can be simulated using mathematical equations; good models give a close approximation of reality using the least parameters and low complexity (Devia, G et al., 2015). A hydrologic model is a simplification of a natural system and must meet two essential conditions: the calibration data must be accessible, and the calibration must be simple (Mendoza, M., 2020).

Hydrologic models are classified according to the structure shown in Figure 2. There are deterministic and stochastic models: in the case of deterministic models, randomness is not considered, and an input calculates the same output; while in the case of stochastic models, they produce partial random production, based on statistic predictions. The deterministic models are based on space discretization and are divided into three types: lumped, semi-distributed, and distributed.

**8 **

Lumped (parameter) models are generally applied to regions without dimension to simulate various hydrological processes. The parameters used in the lumped model represent the average characteristics of a system (Dwarakish et al., 2015).

In the case of semi-distributed models, as the basin's size increases, the soil types and their characteristics will vary throughout the basin. It is possible to discriminate zones with similar hydrological behavior in hydrologic response units, analyze each of them independently, and then combine their results (Cabrera, 2012).

In distributed models, the method is similar to the semi-distributed, however, to achieve the greatest representativeness. The basin is divided into much smaller elements called grids. The model calculates the balance in each grid, then transmitting its effect to adjacent grids.

**Figure 2. Hydrological models classification **

### HYDROLOGIC MODELS

### Stochastic Models Deterministic Models

### Spatial Discretization

### Lumped

### Semi-distributed

### Distributed

**9 **

**3.4.1. Hydrologic Conceptual Model **

Several authors have stated the equations that represent the hydrological
processes. For this study, the selected model was proposed by Temez et al. (Temez,
1977), which is a lumped hydrological model that assumes that the soil profile is
divided into an upper unsaturated zone and a lower saturated zone (Oñate-
Valdivieso et al., 2016). The whole process is governed by the principle of continuity
and mass balance and transfer between the different terms of the balance (Figure
**3) (Arquiola et al., 2014). Due to its simplicity and the few parameters involved for **
its realization, the Temez model is an excellent option to be applied in an
optimization model.

**Figure 3. Concept of Temez model **

PRECIPITATION

SOIL MOISTURE

REAL EVAPOTRANSPIRATION

TEMPERATURE

POTENTIAL EVAPOTRANSPIRATION

### INFILTRATION

GROUNDWATER DISCHARGE

### DIRECT SURFACE

### TOTAL RUNOFF

AQUIFER VOLUME STORAGE

**10 **

**3.4.2. Software for Hydrologic Modelling **

A variety of software or codes for hydrologic modeling are available: HEC-HMS, TETIS, SWAT, AQUATOOL, among others. These programs consist of preprocessing, processing, interactive graphical user interface (GUI), and postprocessing units that help the user to conceptualize, enter data, calibrate, and run scenarios and show its numeric and graphical output.

The software AQUATOOL (Figure 4) is a Decision Support System (DSS) created in Spain; it consists of several modules that allow the analysis of different approaches in water resources systems (Pedro-Monzonís et al., 2016). The process in this platform is based mostly on the knowledge of the contributions of precipitation and evapotranspiration, gauging stations, the geographical location of rivers and reservoirs, and their technical operation policies (González-Bravo, R., 2018). Given its simplicity and the capacity of integrations of the aspects mentioned before, it is an excellent software for developing a model. The EVALHID module can be added to the environment for the development of rainfall-runoff models in complex basins.

It evaluates the produced water resources in the basin based on the Temez model.

**Figure 4. AQUATOOL interface **

**11 **

**3.5. Optimization **

Water availability is related to the geographical distribution of water resources and its modification in a basin. Policy and economic decisions determine the water availability for a particular region or area (Peña-García, 2007). In recent years, it has become clear that one of the most critical challenges in water deficit areas is allocating available water since competition for its service between agricultural, urban, industrial, and environmental demand has intensified (Liu et al., 2011). In areas with water scarcity, optimization techniques represent a viable option for improving water planning and management, thereby contributing to water resources' sustainable use. Optimization techniques focus on determining a set of values that the factors take to regulate the system's performance, with this continue to maximize or minimize the system's response. In general, this response is an indicator of the type "profit, cost, value", etc., which is a function of the selected policy. This answer is called the objective, and the associated function is called the objective function (Baquela, 2013).

Optimization models can be classified into three categories (Figure 5): Linear Programming (LP), Nonlinear Programming (NLP), and Mixed-Integer Nonlinear Programming:

• LP: refers to optimization problems that, given the characteristics, are limited to only taking into consideration linear equations (Lewis, 2008).

• NLP: the nature of these optimization problems is the inclusion of nonlinear equations in the objective function and/or the constraints in search of the best result (Bradley, 1977).

• MINLP: optimization problems are characterized by the inclusion of continuous and discrete variables and nonlinear functions in the objective function or the constraints (Lee and Leyffer, 2011).

MINLP problems arise in applications in various fields, including engineering, finance, and manufacturing (Lee and Leyffer, 2011). Given the complexity of current problems, the behavior of the constraints or the objective function is not always

**12 **

linear, and the inclusion of new elements is needed for improving the model. MINLP is the best option for its development.

**Figure 5. Classification of optimization problems **

**3.5.1. Optimization Models **

Optimization models have been developed to improve the planning and management of water resources. Different approaches and types of optimization models are presented in Table 1.

**Table 1. Optimization models for water allocation **

Approach Type of

optimization model

Source

Agriculture Demand LP (Gong et al., 2020)

Agriculture Demand LP (Huang et al., 2020)

Agriculture, Industrial

and Domestic Demands MINLP (González-Bravo et al., 2016)

Water Allocation LP (Senante et al., 2013)

Water Allocation LP (Hajebi, 2014)

Water Allocation

Network Elements MINLP (Caballero and Ravagnani, 2019) Industrial and Domestic

Demands MINLP (Olivarez-Areyan et al., 2020)

Optimization

MINLP LP

NLP

Limited to linear equations

Use nonlinear equations

Nonlinear equations + Discrete variables

**13 **

Previous optimization models allow to find technical and economically viable solutions for the allocation of water resources. However, there is a need to include hydrological parameters that better represent the natural water cycle in optimization schemes. These aspects can be precipitation, evaporation, evapotranspiration, infiltration, among others.

**3.5.2. Software for Optimization Models **

Several applications have been designed to assist in the insertion of an optimization model, compile it, solve it, and show the results (GAMS, LINDO, GEKKO, AMPL, Julia, etc.). These programs are based on local and/or global optimization, the first one finds the best solution among several sets of solutions, and the latter is focused on finding the absolute solution for a selected objective function (Müller, 2004).

The General Algebraic Modeling System (GAMS) (Figure 6) is a high-level modeling system for mathematical programming and optimization. It consists of a language compiler and a range of associated solvers (SBB, ANTIGONE, BARON, LINDO, CPLEX, etc.). The GAMS modeling language allows modelers to translate real-world optimization problems into computer code quickly. The GAMS language compiler then translates this code into a format the solvers can understand and solve. This architecture provides excellent flexibility by allowing changing the solvers used without changing the model formulation (GAMS, 2020). Its adaptation to robust models, based on global optimization and its ease of using different solvers, is an excellent option for a model of this nature.

**14 **

**Figure 6. Optimization Procedure in GAMS**

## Data Gathering

### Mathemati cal Model to GAMS language

### Set the constraints

### Establish the objective

### function

### Selection of solvers

### Optimal

### solution

**15 **

**3.6. Piping and Pumping Calculations **

The modeling of pumping and piping elements is one of the most important considerations in analyzing the optimal allocation of water resources. The methodology used to determine the economic pipe diameter is based on an equation proposed by Dupuit (Gama et al., 2019). Given the simplicity of the equation, it is a useful tool for the calculation of the economic diameter for new elements in a water distribution network:

*D k Q*= _{(1) }

*Where D is the diameter (m), k is a dimensionless factor based on the prices of *
*pipes, labor, and installation services, and Q is the flow through the pipe (m³/s) *
For the calculation of head loss at a pipe, the Hazem-Williams equation (Yıldırım
and Özger, 2009) is considered, given its adaptation to complex models and showing
outstanding result in the performance:

1.852 1.852 4.871)

*HW*

*Hf KL* *Q*

*C* *D*

= (2)

Where D is the diameter (m), K is a conversion factor to standardize the units, Q is the flow through the pipe (m³/s), L is the length of the pipe, and C is the Hazem- William Lost Factor.

**4. Methodology **

The methodology used in this work consists of seven main stages (Figure 7):

• The first stage consists of obtaining the parameters related to the hydrological processes (infiltration, runoff, evapotranspiration, etc.); these parameters are obtained from the calibration of precipitation, temperature, and streamflow data from a series of hydro-meteorological stations.

• The second stage involves developing a superstructure, which is a graphic representation that includes all the existing elements and new elements that will enable us to obtain the optimal water allocation network.

**16 **

• In stage three, a mathematical model is proposed considering all the possibilities described in the superstructure in the previous stage.

• The fourth stage consists of the insertion, codification, compilation, and solution of the mathematical model in GAMS software.

• The fifth stage comprises the development of 3 scenarios representing the characteristics in the search for the region's optimal solution; these scenarios include both economic and sustainability aspects.

• The sixth step involves the analysis of the results and the comparative analysis of the results obtained for each scenario.

• The last section describes the general conclusions of the work and provides some recommendations for decision-makers.

**Figure 7. Methodology steps **

**4.1 The Hydrological Model **

The hydrological model was stated based on the equations proposed in equations (3) to (13). The proposed model calculation requires a series of parameters for each

**17 **

subbasin to be assessed. These parameters were obtained from a previous study (Gonzalez-Bravo et al., 2020):

• Precipitation (*P** _{t}*)

• Potential evapotranspiration (*ETP**t*)

• Maximum Soil Moisture (Hmax)

• Maximum Infiltration Capacity (Imax_{) }

• Recession coefficient of the aquifer (α)

• Area of the subbasin (S)

• Initial soil moisture (H0_{) }

• Initial volume in the aquifer (V0_{) }

The mathematical model requires the addition of the subindex b representing the sub-basin where the hydrological processes occur. The equations are stated as follows:

0* _{b t}*, C (H

*max*

_{b}

_{b}*, 1)*

_{b t}*P* = −*H* _{−} _{(3) }

, Hmax* _{b}* , 1 ,

*b t* *H**b t* *ETP**b t*

###

= −_{−}+

_{(4) }

, ,

,

, 0

, 0

,

, , 0

0;

( )

( 2 )

*b t*
*b t*

*b t*

*b t*
*b t* *b t*

*b t* *b t*

*P* *P*
*P* *P*
*T*

*P* *P*

−

= + −

(5)

, 1 , , ,

,

, 1 , , ,

0; _{b t}_{b t}_{b t}* _{b t}* 0

*b t* *b t*

*b t* *b t* *b t*

*H* *P* *T* *ETP*

*H* *H* *P* *T* *ETP*

−

−

+ − −

= + − − (6)

, 1 , , ,

, , 1 , , ,

0; _{b t}_{b t}_{b t}* _{b t}* 0

*b t* *b t* *b t* *b t* *b t*

*ET* *P* *T* *ETP*

*ET* *ET* *P* *T* *ETP*

−

−

+ − −

= + − − (7)

, max ,

, max

I ^{b}* ^{b t}*I

_{b}*b t*

*b t*

*I* *T*

= *T*

+ (8)

**18 **
sup*b t*, *b t*, *b t*,

*A* =*T* −*I* _{(9) }

, , 1 ^{b}^{b}* ^{b t}*, (1

*)*

^{b}*b t* *b t*

*b*

*V* *V e*^{−} ^{−}^{} *S I* *e*^{−}^{}

= + − _{(10) }

, , 1 , ,

*sub**b t* *b t* *b t* *b t*

*A* =*V* _{−} −*V* +*I* _{(11) }

, sup, ,

*b t* *b t* *b t*

*total* *sub*

*A* =*A* +*A* _{(12) }

Once the total contribution of a sub-basin is obtained, it is multiplied by its surface to obtain the volume of contribution, as follows:

, ,

*b t* *b t*

*aport* *b* *total*

*V* =*AREA A* _{(13) }

**4.2 Superstructure **

The proposed superstructure is presented in Figure 8. This graphic represents all
the water allocation systems' possibilities to satisfy the users’ demands, including
existing and new infrastructure. The incorporation of hydrological processes and
*water transferences from other basins are also considered. The agricultural (g), *
*domestic (r), and industrial (o) demands of water can be supplied by existing *
*treatment plants (j), new treatment plants (k), existing storage tanks (p), and new *
*storage tanks (q). The treatment plants can be satisfied with water from aquifers (i) *
*and water runoff (rivers) (b). The dams (d) and hydroelectric plants (n) are *
considered for hydropower production. The model includes equations to consider
recycling/reuse water by installing new graywater (GWTP) and new blackwater
(BWTP) treatment plants to reuse water in agriculture and industry. Also, the return
of treated blackwater to the environment is considered.

**19 **
**Figure 8. Proposed superstructure for water allocation in the highly altered basin **

**20 **

**4.3 Mathematical Model for Water Allocation **

The proposed mathematical model is based on the superstructure shown in Figure
**8. The model includes: **

• Accumulation balances in subbasins, aquifers, dams, existing and new storage tanks, and new reservoirs.

• Volume balances for the satisfaction of the demands of domestic, agricultural, and industrial users.

• Binary variables to determine the optimal network considering or not the existence of new elements such as new storage tanks, new treatment plants, reuse/recycling treatment plants, new reservoirs, and new pipelines.

• All the equations are described as follows:

The change in the total water volume in a subbasin (*BS**b t*, −*BS**b t*, 1_{−}) over a particular
time is equal to the sum of the natural runoff in the sub-basin (*APORT**b t*, ), the water
accumulation of upstream subbasin(s) (*ACUM**b t*−1, ), the transbasin water
transferred (WT* _{b t}*, ), the water received from hydropower plants (

*out*, , ,

*b d n t*

*HP* ) minus the
water sent to dams (*WDAM**b d t*, , ), the water sent to exiting water treatment plants (

, ,
*b j t*

*BDTP* ), the water sent to new water treatment plants (*BNDTP**b k t*, , ) and the water
sent to new reservoirs (*BNTS**b e t*, , _{): }

, , 1 , 1, , , , ,

, , ', , ', , ', ,

, , 1

WT ^{out}

*b t* *b t* *b t* *b* *t* *b t* *b d n t*

*d D n N*

*b d t* *b j t* *b k t* *b e t*

*d D* *j J* *j J* *k K* *e E*

*BS* *BS* *APORT* *ACUM* *HP*

*WDAM* *B BDTP* *BNDTP* *B*

*b B t T*

*N*
*t*

*TS*

− −

+

+

− = +

− − + +

###

###

^{(14) }

The change in the total water volume in a subbasin (*BS**b t*', −*BS**b t*', 1− ) over a certain
time is equal to the sum of the natural runoff in the sub-basin (*APORT**b t*', ), the water

**21 **

sent to existing drinking treatment plants (*BDTP** _{b j t}*', , ), the water sent to new drinking
treatment plants (

*BNDTP*

*b k t*', , ) and the water sent to new reservoirs (

*BNTS*

*b e t*', ,

_{) : }

, , 1 A T , , , , , , ,

' ', , 1

*b t* *b t* POR *b t* *b j t* *b k t* *b e t*

*j J* *k K* *e E*

*BS* *BS* *BDTP* *BNDTP* *BNTS*

*b B t T t*

−

− = − − −

###

(15)

Where b' is a subset of the subbasins with no interconnection and draining into the sea.

The accumulation of water of subbasins interconnected is:

, , 1

, *b*

*b t* *t* *b t*

*A**CU**M* =*B**S* −*BS* − (16)

The water volume in an aquifer over a certain time (*W W**i t*, − *i t*, 1− ) is equal to the sum of
water infiltration (*WA**,it*) minus the water sent to existing water treatment plants (

, ,
*i j t*

*AWTP* ), the water sent to new water treatment plants (*ANWTP**i k t*, , ), and the water
sent to new reservoirs (*AWTPS**i e t*, , _{): }

, , 1 , , , , , , ,

, , 1

*i t* *i t* *i t* *i j t* *i k t* *i e t*

*j J* *k K* *e E*

*W* *W* *WA* *AWTP* *ANWTP* *AWTPS*

*i I t T t*

−

− = − −

###

−###

(17)

The water volume in existing dams over a specific time (*DAM**d t*, −*DAM**d t*, 1− ) is equal
to the sum of water received from the basin (*WDAM**b d t*, , ) minus the water sent to the
hydropower plant (HP_{b d n t}^{in}_{, , ,} ), the water lost by infiltration in the dam (ID* _{d t}*, ), and the
water lost by evaporation of the dam (EV

*,*

_{d t}_{): }

**22 **

, , 1 , , , , , , ,

, , 1

HP* ^{in}* ID EV

*d t* *d t* *b d t* *b d n t* *d t* *d t*

*b B* *b B n N*

*DAM* *DAM* *W* *M*

*d D t* *t*

*A*
*T*

− *D*

− = −

− −

###

(18)

The volume of water that enters in hydropower plant (*HP*_{b d n t}^{in}_{, , ,} ) is equal to the volume
of water that leaves the hydropower plant (*HP*_{b d n t}^{out}_{, , ,} ) plus the water lost in the process
(LF* _{n t}*,

_{): }

, , , , , , ,

,

*in* *out*

*b d n t* *b d n t*

*n N* *n N* *n t*

*n N t T*

*HP* *HP* *LF*

= +

###

(19)

The power generated by existing hydropower plants (*PW**n t*, ) is given by the water
volume that enters into the hydropower plant (*HP*_{b d n t}^{in}_{, , ,} ) multiplied by an energy factor
(EFn_{): }

, EFn

,

*n t* *b,d,n,t**in*

*b B d D*

*n N t T*

*PW* *HP*

=

###

(20)Existing hydropower plants are limited by their maximum power generating capacity:

, PWnmax

*PW **n t* (21)

The cost of operation and maintenance of an existing hydropower plant (OpCost^{hp}_{n}_{) }
is a function of a unit operation cost (F1), the hydropower generating plant capacity
(PWn^{max}), and a factor used to account for the operational time annually (AF ):

**23 **

1 nmax

OpCost^{hp}* _{n}* =AF F PW (22)

The total cost of operation and maintenance of hydropower plant (HPOpCost) is
given by the sum of operation cost of existing hydropower plants (OpCost^{hp}_{n}_{): }

HPOpCost OpCost^{hp}_{n}

*n N*

=

###

^{(23) }

Energy sales can be calculated as follows:

2 ,

EnergySales AF F _{n t}

*t T n N*

*PW*

=

###

^{(24) }

The water volume treated in the drinking water treatment plant (*DWTP**,jt*) is the sum
of the water received from subbasins (*BDTP**b j t*, , ), the water obtained from aquifers (

, ,
*i j t*

*AWTP* ), and the water received from new reservoirs (*TDWTPR**j e t*, , _{): }

, , , , , , ,

,

*j t* *b j t* *i j t* *j e t*

*b B* *i I* *e E*

*j J t T*

*DWTP* *BDTP* *AWTP* *TDWTPR*

= + +

###

(25)

The water volume in the existing drinking water treatment plant (*DWTP**,jt*) is the sum
of the water sent to domestic users (*DOMWTP**r j t*, , ), the water sent to industrial users
(*INDWTP**o j t*, , ), the water sent to agricultural users (*AGRWTP**g j t*, , ), the water sent to
existing tanks (*SWTP**p j t*, , ), and the water sent to new tanks (*NSWTP**q j t*, , _{): }

**24 **

, , , , ,

, , , , , ,

,

*j t* *r j t* *o j t*

*r R* *o O*

*g j t* *p j t* *q j t*

*g G* *p P* *q Q*

*D*

*T*

*WTP* *DOMWTP* *INDWTP*

*AGRWTP* *SWTP* *NS T*

*t*

*W*
*j J*

*P*

+

+ +

=

###

+###

###

^{(26) }

The capacity of the existing drinking water treatment plant is limited by the maximum
plant capacity (DWTP^{max}* _{j}* ):

, DWTPmax

*j t* *j*

*DWTP * _{(27) }

The cost of operation and maintenance of an existing drinking water treatment plant
(OpCost^{dp}* _{j}* ) is a function of a unit operation cost (F3), the volume of water treated (

*DWTP**,jt*), and a factor used to account for the operational hours per year (AF ):

3 ,

OpCost^{dp}* _{j}* AF F

_{j t}*t T*

*DWTP*

=

###

^{(28) }

The total cost of water treatment operation and maintenance (WTOpCost) for
existing drinking water treatment plants is the sum of the operation cost of each
existing drinking water treatment plant (OpCost^{dp}* _{j}* ):

WTOpCost OpCost^{dp}_{j}

*j J*

=

###

^{(29) }

The existence, location, and capacity of the new drinking water treatment plants are
modeled through binary variables (*y**k** ^{ndp}*). If the binary variable is equal to one, then
the drinking water treatment plant is needed. If the binary variable is equal to zero,
the drinking water treatment plant is not required. Also, a new drinking water

**25 **

treatment plant has a maximum capacity (WCAPk^{max}) and a minimum capacity (

kmin

WCAP _{): }

min max max

k k k

WCAP NDP WCAP

*ndp* *ndp*

*k* *k*

*y* *y* _{ } _{(30) }

The water volume treated in new drinking water treatment plants (*NDWTP**k t*, ) is the
sum of the water received from subbasins (*BNDTP**b k t*, , ), the water obtained from
aquifers (*ANWTP**i k t*, , ), and the water received from new reservoirs (*NTDWTP**e k t*, , _{): }

, , , , , , ,

,

*k t* *b k t* *i k t* *e k t*

*b B* *i I* *e E*

*N*

*T*

*DWTP* *BNDTP* *ANWTP* *NTD*

*k K*

*WTP*
*t*

= +

+

###

(31)

The total water volume in existing drinking water treatment plant (*NDWTP**k t*, _{) is the }

sum of the water sent to domestic users (*DOMNWTP**k r t*, , ), the water sent to industrial
users (*INDNWTP**k o t*, , ), the water sent to agricultural users (*AGRNWTP**g k t*, , ), the water
sent to existing tanks (*SNWTP**p k t*, , ) and the water sent to new tanks (*NSNWTP**q k t*, , _{): }

, , , , ,

, , , , , ,

,

*k t* *k r t* *k o t*

*r R* *o O*

*g k t* *k p t* *k q t*

*g G* *p P* *q Q*

*N*

*T*

*DWTP* *DOMNWTP* *INDNWTP*

*K*

*AGRNWTP* *SNWTP* *NSNW*
*k*

*P*
*t*

*T*

= +

+ +

+

###

###

^{(32) }

The capacity of a new drinking water treatment plant is limited by the maximum plant
capacity (*NDP**k*^{max}_{): }

, max

*k t* *k*

*NDWTP* *NDP* _{(33) }

**26 **

The installation cost of a new drinking water treatment plant (InstCost^{ndp}_{k}_{) is a }
function of unit costs (F _{4} ), the factors (β1, ,β γ ,2 1 γ2), the maximum capacity of the
new plant (*NDP**k*^{max}), and the element to annualize the inversion (kF_{): }

1 2

γ γ

max max

F 4 1 2

InstCost^{ndp}* _{k}* =k F

*y*

_{e}*+β (*

^{ndp}*NDP*

*) +β (*

_{k}*NDP*

*) (34)*

_{k}The cost of operation and maintenance of a new drinking water treatment plant (
OpCost^{ndp}* _{k}* ) is a function of a unit operation cost (F4), the volume of water treated (

,

*NDWT**k t*), and a factor used to account for the operational time per year (AF ):

4 ,

OpCost^{ndp}* _{k}* AF F

_{k t}*t T* *NDWT*

=

###

^{(35) }

The installation cost of new drinking water treatment plants (NWTInstCost) can be calculated using the following equation:

NWTInstCost InstCost^{ndp}_{k}

*k K*

=

###

^{(36) }

The operation and maintenance cost of new drinking water treatment plants ( NWTOpCost) can be calculated using the next equation:

NWTOpCost OpCost^{ndp}_{k}

*k K*

=

###

^{(37) }

The existence, location, and capacity of the new reservoirs are modeled through
binary variables *y**e** ^{res}*. If the binary variable is equal to one, then the reservoir is
needed. If the binary variable is equal to zero, the reservoir is not required. Also, the

**27 **

new reservoir has a minimum volume of water MINRLe and a maximum volume MAXRLefor its existence:

e max e

MINRL MAXRL

*res* *res*

*e* *e* *e*

*y* *NRES* *y* _{ } _{(38) }

The change in water volume in new reservoirs (*NRES**e t*, −*NRES**e t*, 1_{−} ) is equal to the
sum of water received from subbasins (*BNTPS**b e t*, , ), the water received from aquifers
(*AWTPS**i e t*, , ) minus the water sent to existing water treatment plants (*TDWTPR**e j t*, , _{) }

and new drinking water treatment plants (*NTDWTP ): **e k t*, ,

, , 1 , , , ,

, , , ,

, , 1

*e t* *e t* *b e t* *i e t*

*b b* *i I*

*e j t* *e k t*

*j J* *k K*

*NRES* *NRES* *BNTPS* *AWTPS*
*TDWTPR* *NTDWTP*

*e E t T t*

−

− = +

−

−

###

###

^{(39) }

New reservoirs maximum capacity (*NRES**e*^{max}) is calculated as follows:

*,et* *e*max

*NRES* *NRES* _{(40) }

The installation cost of a new reservoir (InstCost^{res}* _{e}* ) is a function of unit costs (F5

_{), }

the factors (β ,γ3 3)y, the maximum reservoir capacity (*NRES**e*^{max}), and a factor to
obtain the annualized inversion (kF_{): }

γ3

F 5 3 max

InstCost^{res}* _{e}* =k [F

*y*

_{e}*+ β (*

^{res}*NRES*

*) ]*

_{e}_{ }

_{(41) }

**28 **

The reservoirs installation cost (ReservoirCost) can be calculated as the sum of all
the cost of installation of new reservoirs InstCost_{e}^{res}_{: }

ReservoirCost InstCost^{res}_{e}

*e E*

=

###

^{(42) }

The water available in the water treatment plant to satisfy the demands of domestic
users (*wad**,rt*) is the sum of the water received from existing water treatment plants (

, ,
*r j t*

*DOMWTP* ) and new water treatment plants (*DOMNWTP**r k t*, , _{): }

, , , , ,

,

*r t* *r j t* *r k t*

*j J* *k K*

*r R t T*

*wad* *DOMWTP* *DOMNWTP*

+

=

###

_{(43) }

The water available to satisfy the demands of industrial users (*wai**o t*, ) is the sum of
the water received from existing water treatment plants (*INDWTP**j o t*, , ) and new water
treatment plants (*INDNWTP**k o t*, , _{): }

, , , , ,

,

*o t* *j o t* *k o t*

*j J* *k K*

*o O t T*

*wai* *INDWTP* *INDNWTP*

+

=

###

_{ }

_{(44) }

The water available to satisfy the demands of agricultural users (*wag*_{g t}_{,} )is the sum
of the water received from existing drinking water treatment plants (*AGRWTP**j k t*, , _{) and }

new drinking water treatment plants (*AGRNWTP**k g t*, , _{): }

, , , , ,

,

*g t* *j k t* *k g t*

*j J* *k K*

*g G t T*

*wag* *AGRWTP* *AGRNWTP*

= +

###

_{ }

_{(45) }

**29 **

The change in water volume in storage tanks in a certain time (*S**p t*, −*S**p t*, 1_{−} ) is equal
to water received from existing drinking water treatment plants (*SWTP**p j t*, , ), the water
obtained from new drinking water treatment plants (*SNWTP**p k t*, , ) minus the water sent
to domestic users (*DOMSTO**p r t*, , ), the water sent to industrial users (*INDSTO**p o t*, , _{) and }

the water sent to agricultural users (*AGRSTO**p g t*, , ):

, , 1 , , , ,

, , , , , ,

, , 1

*p t* *p t* *p j t* *p k t*

*j J* *k K*

*p r t* *p o t* *p g t*

*r R* *o O* *g G*

*S* *S* *SWTP* *SNWTP*

*DOMSTO* *INDSTO* *AGRSTO*
*p P t T t*

−

− = +

− − −

###

###

^{(46) }

The capacity of the existing tank is limited by the maximum tank capacity (S^{max}p ):

, Smaxp

*S **p t* (47)

The existence, location, and capacity of new storage tanks in each subbasin are
modeled through binary variables (*y*_{q}* ^{sto}*). If the binary variable is equal to one, the
tank is needed; if the binary variable is equal to zero, the tank is not required; also;

a new storage tank is subject to a maximum capacity of the tank (TCAPq^{max}) and a
minimum capacity (TCAPq^{min}):

min max max

q q q

TCAP NS TCAP

*sto* *sto*

*q* *c*

*y* *y* _{(48) }

The change in water volume in new storage tanks in a certain period (*NS**p t*, −*NS**p t*, 1_{−}

) is equal to water received from existing water treatment plants (*NSWTP**q j t*, , _{), the }

water obtained from new water treatment plants (*NSNWTP**q k t*, , ), minus the water sent

**30 **

to domestic users (*DOMNSTO**q r t*, , ), the water sent to industrial users (*INDNSTO**q o t*, , _{) }

and the water sent to agricultural users (*AGRNSTO**q g t*, , _{): }

, , 1 , , , ,

, , , , , ,

, , 1

*p t* *p t* *q j t* *q k t*

*j J* *k K*

*q r t* *q o t* *q g t*

*r R* *o O* *g G*

*NS* *NS* *NSWTP* *NSNWTP*

*DOMNSTO* *INDNSTO* *AG*
*q*

*RN*
*Q t T*

*STO*
*t*

−

− = +

−

− −

###

###

^{(49) }

The maximum capacity of the new tanks can be calculated as follows:

, max

*q t* *q*

*NS* *NS* _{(50) }

The installation cost of a new storage tank (InstCost_{q}* ^{sto}*) is a function of the unit costs
(F6

_{) and (}F

_{7}), the maximum capacity of the new storage tank (

*NS*

_{q}^{max}) and a factor to obtain the annual inversion kF

_{: }

F 7 max

k F6 F ( )

*sto* *sto*

*q* *q* *q*

*InstCost* = *y* + *NS* (51)

The installation cost of new storage tanks (StorageCost) can be calculated as follows:

StorageCost InstCost^{sto}_{q}

*q Q*

=

###

^{(52) }

The domestic demands domdemr,t can be satisfied by the water available (*wad* *,rt*_{), the }

water in existing storage for domestic users (*DOMSTO**p r t*, , ), and the water in new
storage for domestic users (*DOMNSTO**q r t*, , _{): }