# A multi-objective optimization model to plan city-scale water systems with economic and environmental objectives: a case study in Santiago, Chile

TL;DR: In this paper, a multi-objective mixed-integer programming (MOP) problem is formulated to minimize the environmental and economic impact of the network, minimizing water extracted from natural sources and total cost.

About: This article is published in Journal of Cleaner Production.The article was published on 2021-01-10 and is currently open access. It has received 6 citations till now. The article focuses on the topics: Water scarcity & Reuse.

## Summary (4 min read)

Jump to: [1. Introduction] – [2. Literature review and novelty of this paper] – [3. Problem structure] – [4. Mathematical model] – [4.1. Mass balances] – [4.1.2. Mass balances for each node] – [4.2. Covering the demand] – [4.3.1. New drinking water treatment plant capacity] – [4.3.2. New wastewater treatment plant capacity] – [4.4. Logical relationships of existence] – [4.5. Cost] – [4.6. Objective functions] – [5. Multi-objective optimization strategy] – [6. Case study] – [7. Results] – [8. Discussions] and [9. Conclusions]

### 1. Introduction

- Climate change has become an important issue in several re gions worldwide since human behavior is conditioned by the ef fects of Global Warming on Earth (Santibanez, 2018).
- Under uncertainty has been a priority area of research aiming at improving large scale regional water systems (Vucetic and Simonovic, 2011).
- Optimization techniques can be a valuable tool for water resource management in order to redesign regional water systems.
- This topic constitutes the main focus of this project.
- Previous research shows that there are efforts focused on different scales.

### 2. Literature review and novelty of this paper

- Lovelady and El Halwagi (2009) developed a model to plan water management among multiple processes in a common EIP facility.
- McGivney and Kawamura (2008) researched installation and operating costs of various water treat ment technologies, including drinking water and wastewater treatment technologies.
- The present investigation attempts to minimize both impacts in the same model.
- The present investi gation uses Goal programming technique to solve the multi objective problem.
- The following sections present the formulation and solution of a Multi objective optimization model so as to redesign a city scale water network with environmental and economic objectives.

### 3. Problem structure

- The methodology used is shown in Fig. 1. First, an investigation of the current structure is carried out, and then possible changes in the network, its characteristics and requirements are studied.
- Each sub region has its own population center to assign the location of the participants in the network.
- Each consumption node has its own demand and in the base case all the demands are satisfied with drinking water.
- Modified plants are positioned in the original location of the existing ones.
- With these new treatment plants it is possible to reduce water consumption from the natural source and create new possible connections and recycling within the system.

### 4. Mathematical model

- The proposed model is based on the superstructure shown in Fig.
- The model consists of a set of mass balances in the treatment plant nodes, distribution and collection nodes, and consumption nodes.
- The sets, variables and subscripts used in the model are defined in Nomenclature section.

### 4.1. Mass balances

- It is considered that only total mass balances are required.
- This assumption implies that water quality at the exit of each treatment plant and network consumption satisfy the quality constraints for their respective user.
- Stationary state is also assumed for each node.
- Thus, the incoming flow rates will be the same as the outgoing flow rates.
- Finally, each flow density is assumed constant, then volu metric balances can be made.

### 4.1.2. Mass balances for each node

- At any node, the incoming flow rates will be the same as the outgoing flow rates, as a result of the steady state assumption.
- Fig. 3 represents the flows and sets involved in the mass balance of commercial and residential consumption.
- Inflows of large existing WWTPs come from collection nodes and the outflows go into natural discharge courses and to the sink.

### 4.2. Covering the demand

- As mentioned above, each consumption node has an associated water demand.
- Depending its consumption type and location, the demand is different.
- To satisfy the demand of each node, Equation (19) must be respected.
- X i2ONj Fi/j DM fj;pg;c j2ADj;c p2CT (19) ONj, j being the current node, i.e. where the demand must be satisfied (ADj), which can be a residential, commercial, industrial, agricul tural, and urban park irrigation consumption demand; and p represent each district.

### 4.3.1. New drinking water treatment plant capacity

- As mentioned above, there are large and small treatment plants.
- The problem becomes mixed integer, with continuous and discrete variables.
- In fact, the plants do not have a fixed ca pacity only for the consumption of its district inhabitants, its ca pacity varies depending on the requirements of the participants of the systems.
- This flexibility is given by the parameter ‘m’.
- I corresponds to each new small drinking water treatment plant in each district, j corresponds to all consumers requiring drinking water in the district p’, and p’ corresponds to each district, in particular, where the plant i is located.

### 4.3.2. New wastewater treatment plant capacity

- For new big wastewater treatment plants, Equation (22) must be respected.
- M corresponds to the parameter mentioned above, i corresponds to each new small wastewater treatment plant in each district p’, j corresponds to each sewer user in the district p’, and p’ corresponds to each district, in particular where the plant i is located.
- This variation does not imply an extension of existing plants.
- On the other hand, the plant can treat 23% more than the current flow, which is the summer variation.

### 4.4. Logical relationships of existence

- If the plant does not exist, then the incoming flowsmust be zero.
- This can be written mathematically trough Equation (28) applying the BigM method (Song, 2015).
- This value has to be greater than all the flow rates being treated.

### 4.5. Cost

- The costs in the problem are divided in operational costs (OpC) and capital costs (CapC).
- The OpC are estimated by water transport costs, while CapC are estimated by the cost of installing new plants or of modifying the existing ones.

### 4.6. Objective functions

- The problem has two opposing objective functions to be mini mized: water flow used from the water source and the total cost.
- Thus, these two objective functions FO1 and FO2 can be repre sented by Equations (32) and (33), respectively.
- Where G0 is the total fresh water consumed, OpC is the operational cost, and CapC is the investment cost.

### 5. Multi-objective optimization strategy

- All other constraints are also respected, from equation (1) to equation (31).
- The indices id and nid correspond to ideal and non ideal solutions respectively, and represent the different ideal and non ideal points outside the Pareto curve in the goal programming methodology.
- Pa rametersw1 andw2 represent the relativeweights of each objective function.

### 6. Case study

- The model presented was applied in the city of Santiago, capital of Chile.
- Santiago is Chilean political, economic, and institutional center.
- The system is represented by one large and one small drinking water treatment plant and one large and one small wastewater treatment plant.
- These 4 plants can treat all the real city flow, since all the plants were added in the 4 that are represented in the problem.
- Population was distributed geographically according to the districts demographic information (INE, 2018b).

### 7. Results

- The MIP model of the case study has 484 constrains, 1075 var iables (including 50 binary variables), andwas executed in an INTEL CORE i7 7700 HQ computer with 16 GB of memory.
- With the results of Table 2, the multi objective problem is formulated obtaining the Pareto curve shown in Fig. 22, where the values at the extremes of the curve were removed to make other intermediate valuesmore clearly visible.
- The mass balance for the complete system is shown in Fig. 26.
- The annualized total cost of the water network grows a 3.2% when compared with the solution at the economic extreme of the Pareto curve, when the importance of the economic objective function is complete.

### 8. Discussions

- With respect to the obtained results, the current network is not optimal for water treatment, under the assumption that both ob jectives have the same importance.
- The optimal result includes the installation of a new small DWTP instead of a new large DWTP, in order to supply drinking water consumers.
- An interesting observation is that the model tries to reduce the G0 and TC, so the water extracted is mainly used for irrigation and large industrial consumption, since there are no costs associated with transport, nor losses fromwater treatment.
- It is possible to make the model more complex by adding other costs, such as treatment plant operating costs and pipeline installation costs.

### 9. Conclusions

- This paper deals with the management of water resources by integrating new water treatment plants to find the optimal configuration of the water network, applied to the case study of Santiago, Chile.
- With parameters of demands, consumption, losses, locations, and costs, it is possible to characterize water use of the sets present in the model, which allowed to establish the optimal configuration for the problem.
- Their respective economic and environ mental indicators are defined.
- These re sults show that (iv) it is more environmentally and economically convenient to reuse water for irrigation and drinking consumption rather than recycling water to the natural source.
- Model implementation, analysis, data curation, writing, review, and edition, also known as Daniela Gormaz-Cuevas.

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##### Citations

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TL;DR: In this article, a non-linear programming (NLP) model was developed to optimize water regeneration and reuse network, as well as biogas generation from selected wastewater streams.

17 citations

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TL;DR: In this article, a non-linear autoregressive exogenous neural network (NARX) was used for forecasting sediment concentrations at the exit of Francia Creek watershed (Valparaiso, Chile).

Abstract: Estimating and forecasting suspended sediments concentrations in streams constitutes a valuable asset for sustainable land management. This research presents the development of a non-linear autoregressive exogenous neural network (NARX) for forecasting sediment concentrations at the exit of Francia Creek watershed (Valparaiso, Chile). Details are presented on input data selection, data splitting, selection of model architecture, determination of model structure, NARX training (optimization of model parameters), and model validation (hindcasting and forecasting). The study explored if the developed artificial neural network model is valid for forecasting daily suspended sediment concentrations for a complete year, capturing seasonal trends, and maximum and baseflow concentrations. Francia Creek watershed covers approximately 3.24 km2. Land cover within the catchment consists mainly of native and exotic vegetation, eroded soil, and urban areas. Input data consisting of precipitation and stream flow time-series were fed to a NARX network for forecasting daily suspended sediments (SST) concentrations for years 2013–2014, and hindcasting for years 2008–2010. Training of the network was performed with daily SST, precipitation, and flow data from years 2012 and 2013. The resulting NARX net consisted of an open-loop, 12-node hidden layer, 100 iterations, using Bayesian regularization backpropagation. Hindcasting of daily and monthly SST concentrations for years 2008 through 2010 was successful. Daily SST concentrations for years 2013 and 2014 were forecasted successfully for baseflow conditions (R2 = 0.73, NS = 0.71, and Kling-Gupta efficiency index (K-G) = 0.84). Forecasting daily SST concentrations for year 2014 was within acceptable statistical fit and error margins (R2 = 0.53, NS = 0.47, K-G = 0.60, d = 0.82). Forecasting of monthly maximum SST concentrations for the two-year period (2013 and 2014) was also successful (R2 = 0.69, NS = 0.60, K-G = 0.54, d = 0.84).

5 citations

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TL;DR: Multi-objective optimization has become a powerful tool to aid the decision-making mechanism in the synthesis, design, operation and control of such processes as mentioned in this paper , and the solution to the mathematical models provides the necessary tools to asses the system performance in terms of different metrics and evaluate the trade-offs between the objectives in conflict.

Abstract: Industrial processes provide several of the products and services required for society. However, each industry faces different challenges from different perspectives, all of which must be reconciled to obtain profitable, productive, controllable, safe and sustainable processes. In this context, multi-objective optimization has become a powerful tool to aid the decision-making mechanism in the synthesis, design, operation and control of such processes. The solution to the mathematical models provides the necessary tools to asses the system performance in terms of different metrics and evaluate the trade-offs between the objectives in conflict. The number of applications of multi- objective optimization in industrial processes is ample and each application has its own challenges. In the present literature review, a broad panorama of the applications in multi-objective optimization is presented, including future perspectives and open questions that still need to be addressed.

4 citations

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TL;DR: In this paper , the authors compared four strategies to simplify the nonlinear constraints generated by the extended Bernoulli equation with the Darcy-Weisbach friction term in a city-scale water network, and the best results were obtained with a two-stage strategy to address large MINLP multiobjective problems.

Abstract: Water resource management is a crucial issue today when global warming is advancing, generating water shortages in several world countries. Mathematical and optimization tools have addressed these problems, including new alternative water sources. Water networks are designed to cover consumption. In this context, pumping is critical in modeling water networks through optimization techniques because of the nonlinear terms from the extended Bernoulli equation with the Darcy–Weisbach friction term. This paper compares four strategies to simplify these terms, focusing on the nonlinear constraints generated by the extended Bernoulli equation with the Darcy–Weisbach friction term. This equation has a significant impact on operation costs because of pumping power. The four simplification strategies were compared with a focus on (i) solution, (ii) error, and (iii) execution time. The best results were obtained with a two-stage strategy to address large MINLP multiobjective problems. This strategy is applied to a model with three objective functions (freshwater inlet, global warming potential, and total cost) to illustrate the simplification performance in a city-scale water network. The problem is focused on a case study in Santiago, Chile, and is based on a previous formulation. By solving the multiobjective problem, some results and changes in the network are obtained. When the three objective functions have all the same importance, the results show the following: (i) The current location of water treatment plants is suboptimal. (ii) Water recycling in the city is the best option, with drinking and irrigation qualities. (iii) With the optimal configuration, Santiago can reduce their water consumption by 30%, increasing the economic cost by 108% and the global warming potential by 49%. Finally, this model can be implemented in other contexts to approach nonlinearities by the extended Bernoulli equation with the Darcy–Weisbach friction term in large-scale water networks.

1 citations

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TL;DR: This paper determines a constant M0 and proves that the big M method is convergent to an optimal solution of the primal problem when MM0.

Abstract: In the simplex method of linear programming,there is a big M method(the penalty factor method) for finding an initial feasible basis.The current textbooks of operations research only explain that the big M method is efficient when M is large enough,and never give precise evaluation to the parameter M.This paper determines a constant M0 and proves that the big M method is convergent to an optimal solution of the primal problem when MM0.

1 citations

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01 Jan 2014TL;DR: This chapter discusses the fundamental principles of multi-objective optimization, the differences between multi-Objective optimization and single-objectives optimization, and describes a few well-known classical and evolutionary algorithms for multi- objective optimization.

Abstract: Multi-objective optimization is an integral part of optimization activities and has a tremendous practical importance, since almost all real-world optimization problems are ideally suited to be modeled using multiple conflicting objectives. The classical means of solving such problems were primarily focused on scalarizing multiple objectives into a single objective, whereas the evolutionary means have been to solve a multi-objective optimization problem as it is. In this chapter, we discuss the fundamental principles of multi-objective optimization, the differences between multi-objective optimization and single-objective optimization, and describe a few well-known classical and evolutionary algorithms for multi-objective optimization. Two application case studies reveal the importance of multi-objective optimization in practice. A number of research challenges are then highlighted. The chapter concludes by suggesting a few tricks of the trade and mentioning some key resources to the field of multi-objective optimization.

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TL;DR: In this article, the authors present a global scale assessment of the impact of climate change on water scarcity, using the Water Crowding Index (WCI) and the Water Stress Index to calculate exposure to increases and decreases in global water scarcity.

Abstract: This paper presents a global scale assessment of the impact of climate change on water scarcity. Patterns of climate change from 21 Global Climate Models (GCMs) under four SRES scenarios are applied to a global hydrological model to estimate water resources across 1339 watersheds. The Water Crowding Index (WCI) and the Water Stress Index (WSI) are used to calculate exposure to increases and decreases in global water scarcity due to climate change. 1.6 (WCI) and 2.4 (WSI) billion people are estimated to be currently living within watersheds exposed to water scarcity. Using the WCI, by 2050 under the A1B scenario, 0.5 to 3.1 billion people are exposed to an increase in water scarcity due to climate change (range across 21 GCMs). This represents a higher upper-estimate than previous assessments because scenarios are constructed from a wider range of GCMs. A substantial proportion of the uncertainty in the global-scale effect of climate change on water scarcity is due to uncertainty in the estimates for South Asia and East Asia. Sensitivity to the WCI and WSI thresholds that define water scarcity can be comparable to the sensitivity to climate change pattern. More of the world will see an increase in exposure to water scarcity than a decrease due to climate change but this is not consistent across all climate change patterns. Additionally, investigation of the effects of a set of prescribed global mean temperature change scenarios show rapid increases in water scarcity due to climate change across many regions of the globe, up to 2 °C, followed by stabilisation to 4 °C.

538 citations

### "A multi-objective optimization mode..." refers background in this paper

...A direct effect is water scarcity ([2]), considered a global risk by the World Economic Forum ([3])....

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01 Feb 2017

TL;DR: This paper presents a meta-anatomical architecture for multi-Objective Optimization of multi-Product Microbial Cell Factory for Multiple Objectives and some of the principles used in this architecture were previously described in the book “Optimal Design of Chemical Processes for Multiple Economic and Environmental Objectives.”

Abstract: Introduction (G P Rangaiah) Multi-Objective Optimization Applications in Chemical Engineering (Masuduzzaman & G P Rangaiah) Techniques: Multi-Objective Evolutionary Algorithms: A Review of the State of the Art and Some of Their Applications in Chemical Engineering (A L Jaimes & C A Coello Coello) The Jumping Gene Adaptations of Multi-Objective Genetic Algorithm and Simulated Annealing (M Ramteke & S K Gupta) Multi-Objective Optimization Using Surrogate-Assisted Evolutionary Algorithm (T Ray) Why Use Interactive Multi-Objective Optimization in Chemical Process Design? (K Miettinen & J Hakanen) Net Flow and Rough Set: Two Methods for Ranking the Pareto Domain (J Thibault) Applications: Multi-Objective Optimization of Gas-Phase Refrigeration Systems for LNG (N Shah et al.) A Multi-Objective Evolutionary Algorithm for Practical Residue Catalytic Cracking Feed Optimization (K C Tan et al.) Optimal Design of Chemical Processes for Multiple Economic and Environmental Objectives (E S Q Lee et al.) Multi-Objective Emergency Response Optimization around Chemical Plants (P S Georgiadou et al.) Array Informatics Using Multi-Objective Genetic Algorithms: From Gene Expressions to Gene Networks (S Garg) Multi-Objective Optimization of a Multi-Product Microbial Cell Factory for Multiple Objectives - A Paradigm for Metabolic Pathway Recipe (F C Lee et al.).

184 citations

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TL;DR: In this article, an optimization-based approach to the design and integration of eco-industrial parks (EIPs) is presented, where a source-interception-sink structural representation is used to embed potential configurations of interest, such as direct recycle, material exchange, mixing and segregation of different streams, separation and treatment in interception units, and allocation to process users.

Abstract: This work is aimed at developing an optimization-based approach to the design and integration of eco-industrial parks (EIPs). Focus is given to the management of water among multiple processes in a common EIP facility. Recycle, reuse, and separation using interception devices are considered as possible strategies for managing wastewater. A source-interception-sink structural representation is used to embed potential configurations of interest. The representation accounts for the possibilities of direct recycle, material (waste) exchange, mixing and segregation of different streams, separation and treatment in interception units, and allocation to process users (sinks). Then, the EIP design problem is formulated as an optimization program whose objective is to minimize cost of the EIP while determining optimal recycle and separation strategies. A case study is solved to illustrate the applicability of the devised approach.

150 citations

### "A multi-objective optimization mode..." refers background in this paper

...Lovelady and El-Halwagi (2009) developed a model to plan water management among multiple processes in a EIP facility ([9])....

[...]

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01 Dec 2005TL;DR: This approach is described by analyzing the conflicts that may arise between profitability, variable costs of production and pumping of groundwater for a hypothetical irrigation area by linking bio-economic objectives with the optimum use of all water resources under conflicting demands.

Abstract: The management of river basins is complex especially when decisions about environmental flows are considered in addition to those concerning urban and agricultural water demand. The solution to these complex decision problems requires the use of mathematical techniques that are formulated to take into account conflicting objectives. Many optimization models exist for water management systems but there is a knowledge gap in linking bio-economic objectives with the optimum use of all water resources under conflicting demands. The efficient operation and management of a network of nodes comprising storages, canals, river reaches and irrigation districts under environmental flow constraints is challenging. Minimization of risks associated with agricultural production requires accounting for uncertainty involved with climate, environmental policy and markets. Markets and economic criteria determine what crops farmers would like to grow with subsequent effect on water resources and the environment. Due to conflicts between multiple goal requirements and the competing water demands of different sectors, a multi-criteria decision-making (MCDM) framework was developed to analyze production targets under physical, biological, economic and environmental constraints. This approach is described by analyzing the conflicts that may arise between profitability, variable costs of production and pumping of groundwater for a hypothetical irrigation area.

133 citations