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

Abstract: Climate Change and its effects in water scarcity has become an important challenge for cities with water management problems. These problems require an integral planning of the city, which can be supported by optimization. The main goal of the research is to provide a regional optimization model for water networks, including new treatment options. The model is formulated as a multi-objective mixed-integer programming problem, focused on environmental and economic impact of the network, minimizing water extracted from natural sources and total cost. The formulation is developed with the goal-programming methodology. The model covers a complete existing city-scale water network, including 4 different options of water reuse within the city: drinking water, fresh water, irrigation, and discharge in natural courses. The case study is Santiago, capital of Chile, which is the political, economic, and institutional center of Chile. If both objective functions have equal importance to configure the solution, the following ideas characterize the optimal water network: (i) it is more environmentally and economically convenient to reuse water within the network rather than recycling water to the natural source; (ii) the reuse of water is preferred in the form of irrigation and drinking qualities rather than industrial qualities to reduce transport costs, and (iii) the modification of the current treatment plants is preferred, because of the high cost of installation of new plants. An environmental and cost-effective solution for Santiago, Chile, can reduce the source water extraction in 35.7%. The model can be implemented in other contexts, providing orientations to decision-makers so as to plan city-scale water networks with simultaneous environmental and economic considerations.

## Summary (4 min read)

### 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.

Did you find this useful? Give us your feedback

...read more

##### Citations

2 citations

1 citations

1 citations

##### References

952 citations

381 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])....

[...]

176 citations

135 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])....

[...]

118 citations