Key Calibration Strategies for Mitigation of Water Scarcity in the Water Supply Macrosystem of a Brazilian City

Abstract

This study focuses on Fortaleza, the largest metropolis in Brazil’s semi-arid region. Due to recurrent droughts, massive infrastructure like high-density reservoir networks, inter-municipal and interstate water transfer systems, and a seawater desalination plant have been implemented to ensure the city’s water security. To evaluate the quantitative and qualitative impact of introducing these diverse water sources into Fortaleza’s water supply macrosystem, adequate calibration of the operating and demand parameters is required. In this study, the macrosystem was calibrated using the Particle Swarm Optimization (PSO) method based on hourly data from 50 pressure head monitoring points and 40 flow rate monitoring points over two typical operational days. The calibration process involved adjusting the operational rules of typical valves in large-scale Water Distribution Systems (WDS). After parameterization, the calibration presented the following results: R² of 88% for pressure head and 96% for flow rate, with average relative errors of 13% for the pressure head and flow rate. In addition, with NSE values above 0.80 after calibration for the flow rate and pressure head, the PSO method suggests a significant improvement in the simulation model’s performance. These results offer a methodology for calibrating real WDS to simulate various water injection scenarios in the Fortaleza macrosystem.

1. Introduction

Water Distribution Systems (WDS) play a strategic role in addressing water deficits. Through effective design, operational control, and the understanding of hydraulic parameters, it is possible to reduce losses throughout the system. These losses can reach millions of cubic meters yearly worldwide, with more severe impacts in developing countries [1,2]. Moreover, the rapid population growth driven by urbanization intensification and the adverse effects of extreme climate events pose significant challenges to ensuring the water supply for large urban centers [3,4]. These challenges are even more important in semi-arid regions, characterized by high temperatures and minimal annual precipitation. In such regions, investments in water infrastructure are essential to meet multiple uses and ensure an adequate water supply [5,6]. Computational hydraulic simulation models are useful as management tools for water systems, enabling visualizing and predicting hydraulic and quality indicators and enabling scenario anticipation.

Before using the results from hydraulic simulation, the calibration step is a key part of the process. It involves minimizing errors between monitored and calculated data by adjusting the input parameters of the hydraulic network model, often including pipe roughness, node base demand, water quality properties, and the operational rules of pumps and valves [7,8]. Calibration is frequently supported by optimization tools that adjust WDS model parameters to achieve optimal results and typically verified using pressure head, flow rate, and chemical concentration data. Reliable hydraulic models developed by various calibration and optimization methods can enhance WDS operations [9,10]. Such models may help reduce energy costs [11], expand the water supply with adequate pressure heads [12], detect leaks and pipeline obstructions [13,14,15], and rehabilitate WDS with fewer interventions and greater precision [16].

Calibration studies for WDS models are generally complex when applied to large-scale real networks (models with over 500 pipes and covering areas larger than 20 km², as classified by 17) due to limited data availability, operational rules complexity, network topology, and size uncertainties. These challenges lead many studies to focus on hypothetical or benchmark WDS models instead [7,17,18,19]. However, hypothetical WDS models introduce simplifications that may limit the application of proposed methodologies for real WDS calibrations. Real hydraulic model calibration presents various difficulties, including the large number of nodes and pipes to optimize, the limited distribution of monitored data, dynamic operational rules, and data errors or omissions that force researchers to disregard existing information [9,20]. Furthermore, despite optimization methods and modelers’ knowledge, sensitivity and experience remain crucial in tasks such as parameter selection, anomaly identification, operational rule adjustments, network behavior analysis, and other aspects [21].

Several WDS calibration methods are highlighted in the literature, particularly automatic approaches known for their precision and efficiency in real networks [8,22]. These automatic calibration methods are broadly categorized into numerical and optimization-based approaches. Numerical approaches directly solve continuity and energy equations using numerical methods such as weighted least squares and singular value decomposition, making calibration complex and time-consuming for large-scale networks [20,23]. On the other hand, optimization-based approaches reduce differences between simulated and observed results by adjusting decision variables through nonlinear hydraulic equations, a highly complex process [22,24]. To address these challenges, Evolutionary Algorithms (EAs) have been employed due to their heuristic flexibility and ability to solve nonlinear, non-convex, and multimodal problems, making them particularly suitable for WDS calibration [22,25].

Particle Swarm Optimization (PSO) is one of the AEs with strong applicability in WDS, as it is particularly effective due to its ability to converge quickly to solutions, making it highly practical for real WDS hydraulic models [9,26]. PSO is a random global quasi-optimal solution search algorithm used to minimize the differences between monitored and simulated data. In addition, the method is inspired by the theory of swarm intelligence, which considers the exchange of information and the coevolution of species in a colony, as observed in bird migrations or fish schooling behaviors [7,26,27]. Over time, PSO and its more than 22 variants have been applied in WDS calibrations and other diverse hydraulic applications [28].

This study evaluates the PSO method for calibrating real, large-scale WDS by focusing on the water supply macrosystem of a major Brazilian city with water availability challenges. The calibration process incorporated 48 h monitoring data from 50 pressure head points and 40 flow rate points, as well as operational rules of 48 pressure head-reducing valves. This approach mirrors real-world complexities, contrasting with many studies that overlook such operational data. However, it also emphasizes that classical calibration alone is insufficient for accurate model performance. Inferring a priori knowledge about system behavior, such as operational constraints and infrastructure specifics, is crucial for improving calibration outcomes. Calibration was assessed using five hydraulic modeling indicators and comparative pre- and post-calibration graphs. The results demonstrate significant hydraulic model improvement, enhancing its predictive capabilities and adaptability to various scenarios, such as integrating new water supply sources or responding to severe water crises.

2. Materials and Methods

2.1. Study Area

Fortaleza, the capital of Ceará state in Brazil, has approximately 2.5 million inhabitants and is located in the country’s semi-arid Northeast region. This region is characterized by low annual precipitation, high evaporation rates, elevated temperatures, limited groundwater resources, intermittent rivers, and irregular rainfall patterns, resulting in water constraints throughout most of the year [29,30]. The city’s water infrastructure includes two Water Treatment Plants (WTP Oeste and WTP Gavião) and eight reservoirs: Gavião, Pacoti, Riachão, Pacajus, Aracoiaba, Banabuiú, Orós, and Castanhão, with a combined storage capacity exceeding 11,000 hm³. These reservoirs are integrated through two major water supply systems: the “Canal do Trabalhador” and the “Eixão das Águas” [31]. Together, the treatment plants have a maximum productivity of 15 m³/s; however, during the multi-year 2012–2017 drought, this capacity was reduced to 8.2 m³/s. In response to the city’s ongoing water scarcity challenges, Fortaleza began the development of a seawater desalination plant project (DESAL) with a projected production capacity of 1 m³/s, which is expected to supply water to approximately 300,000 inhabitants [32,33]. Figure 1 illustrates Fortaleza’s water system.

2.2. Macrosystem of the Metropolitan Region of Fortaleza (MMF)

The water supply macrosystem model of Fortaleza comprises 479 junctions, one reservoir, and four tanks, resulting in 484 nodes. In addition, the model includes 393 pipes, 19 pumps, and 103 valves, resulting in 515 links. The total length of the pipes is 202.59 km and occupies an area of around 350 km², with most of this extension composed of pipes more than 1000 mm in diameter, as shown in

Table 1. Pipes distribution in the analyzed network by diameter.

Diameter	Number	Extension (km)
≤200 mm	61	1.73
200–400 mm	43	9.94
400–600 mm	105	46.19
600–1000 mm	107	60.06
≥1000 mm	77	84.67
Total	393	202.59

. The average topographic elevation of the hydraulic elements is 33.6 m, with the highest point at an elevation of 103.4 m and the lowest point at an elevation of just 7.5 m. The base demand of the 84 consumption nodes ranges from 0.37 to 868.3 L/s, according to information provided by the state water and sewage agency. In turn, the system is equipped with 48 pressure head-reducing valves governed by hourly operating rules.

The water supply macrosystem model was provided by the state water and wastewater agency (CAGECE) in a format compatible with EPANET 2.2 software. The schematic representation of this water distribution macrosystem model and the location of the monitored points are shown in Figure 2.

2.3. Methodology

This study calibrated the proposed WDS model using observed pressure head and flow rate data provided by the state water and wastewater agency (CAGECE). The calibration relied on data from 50 pressure head monitoring points and 40 flow rate monitoring points over two typical operational days, totaling 2400 pressure head data points and 1920 flow rate data points employed in the analysis. Hourly flow rate measurements were recorded by stations installed at the network nodes by the state water and wastewater agency (CAGECE). Since EPANET calculates flow rates only for pipes, it was necessary to transfer the measured flows from the outlet nodes to the upstream pipes of each node. Modifications to valve operation rules were also implemented. The model’s 48 pressure head-reducing valves, each with hourly operation rules, introduced a level of complexity to the calibration process. This complexity was addressed through manual adjustments in EPANET software during the calibration in collaboration with the state water and wastewater agency. These adjustments involved analyzing the operational rules through generated graphs and assessing their influence on network nodes. The valve operation rules were adjusted (e.g., increasing or decreasing pressure head control values) or, in some cases, removed to better match the observed monitoring point values while avoiding potential negative pressure heads at any network node during the modeled period. It is important to highlight that each pressure head-reducing valve has operational rules for each time step, meaning that 2304 operation rules were analyzed. Any modification to a single rule impacted the network’s behavior. Therefore, for each calibration iteration, these operation rules were considered, as it would not have been possible to achieve a minimally calibrated model without these adjustments. This highlights the complexity of calibrating large-scale hydraulic WDS models [34].

Initially, the process relied on pre-calibration data provided by the state water and wastewater agency (CAGECE). That is, the initial values for absolute roughness and base demands were considered based on the agency’s data and operational experience with the network. Moreover, this study prioritized the calibration of the base demand factors, ranging from 0.1 to 4, before addressing the calibration of pipe absolute roughness (ε), ranging from 0.0001 mm to 15 mm. After optimizing the objective function for base demand calibration, these values were updated in the hydraulic model, and the network conditions were checked for the presence of negative pressure heads. This sequence was chosen because base demand significantly influences flow rate behavior in hydraulic models, thereby affecting flow rates and hydraulic head losses. In addition, this parameter is more variable than roughness [35], introducing high uncertainty into the network, with any variation altering its behavior, making base demand a dominant factor in hydraulic networks [36]. Once the hydraulic model is adjusted, absolute roughness values can be calibrated, updated in the hydraulic model, and verified for negative pressure heads within the network. The precise determination of the pipe materials in the macrosystem was not possible due to the lack of detailed information about the materials used in each network segment. However, based on prior knowledge and the practices of the agency responsible for maintenance, PVC and Cast Iron are known to be the most commonly used materials. Additionally, it is important to highlight that absolute roughness values of 0.0001 mm are not realistic for water distribution networks. This value was used in this study solely to broaden the search for optimal results during the calibration process. Furthermore, it is noteworthy that the objective function scores were computed using the absolute difference between the modeled and simulated values and divided by the product of the number of monitored points and the total simulation time of 48 h. Additionally, no constraint was imposed on the equality between the Base Demand and the tank flow rates. In summary, the calibration process utilized in this study required pre-calibration operational adjustments to achieve a hydraulic model that closely represents real-world conditions.

The PSO method aims to minimize the objective function by iteratively changing and updating the position of particles within the search space, thereby seeking the best individual and collective positions. Thus, a particle $i$ can be represented by two vectors: one representing the spatial position $x_i$ and the other representing the search velocity $v_i$. Initially, the particles are randomly distributed, generating the first solution. Subsequently, the particles are moved so that those farther from the best solution converge toward it, storing the best position based on both individual and collective best-known position of the swarm [7,37,38]. This iterative process is described by Equations (1) and (2).

$$v_i^{t+1}=w{v}_i^t+c_1{r}_1{\displaystyle \frac{\left(x_i^t-l_i^t\right)}{\Delta t}}+c_2{r}_2{\displaystyle \frac{\left(x_i^t-g_i^t\right)}{\Delta t}},$$

(1)

$$x_i^{t+1}=x_i^t+v_i^{t+1}\Delta t,$$

(2)

where $i$ represents the particles of the swarm; $r_1$ e $r_2$ are the values of the random distribution, with values varying from 0 to 1; $c_1$ is the cognitive factor related to the information of the particle’s best positions; $c_2$ is the social factor related to the information of the best positions experienced by the group; w is the inertia coefficient responsible for weighting the velocities of the particles; $l_i^t$ represents the vector of the best local solution; and $g_i^t$ represents the vector of the best global solution or the swarm’s best solution.

The results of the calibrated network are compared with the results of the pre-calibration network using quantitative evaluation parameters in addition to a graphical analysis of the flow rate and pressure head behavior. The quantitative evaluation was conducted through the Mean Absolute Error (MAE), Average Relative Hourly Error (ARHE), Nash-Sutcliffe Efficiency Coefficient (NSE), Coefficient of Determination (R²), and Percentual Bias (PBIAS). The MAE is calculated as the sum of the absolute difference between the monitored and modeled values divided by the total number of data points. The ARHE evaluates the absolute difference between the monitored and modeled values normalized by the monitored value and the number of simulated hours (48 h). In this way, the ARHE considers the deviation of the monitored and modeled data for each hour of the simulation, thus allowing a more detailed analysis of the deviation behavior. The NSE evaluates the model’s efficiency by comparing the squared difference between the monitored and modeled values with the squared deviation of the monitored values from their mean. The range spans from 1 (perfect model representation) and −∞ (the mean of observed data is a better predictor than the model itself). This metric is one of the main indicators of the accuracy of hydrological models, but it can also be applied in hydraulic modeling [39]. The R² reflects the proportion of variance explained by the model, representing the degree of alignment between observed and modeled values. It ranges from 0 (poor regression fit) to 1 (perfect regression fit), indicating how well the model predicts observed data [39]. The last indicator used was the PBIAS, which considers the trend of the mean modeled values in relation to the monitored values. For this indicator, the closer the value is to zero, the better the model. Negative PBIAS values indicate that the model is overestimating the results, while positive PBIAS values suggest an underestimated model [40].

Table 2. Relationship of model indicators and their equations.

Indicators	Equations
Mean Absolute Error	$MAE={\displaystyle \frac{{\sum }_{i=1}^{n}abs\left(O_i-M_i\right)}{n·h}}$
Average Relative Hourly Error (%)	$ARHE={\displaystyle \frac{{\sum }_{i=1}^{n}abs\left(\frac{O_i-M_i}{O_i}\right)}{n·h}}·100$
Nash–Sutcliffe Efficiency Coefficient	$NSE=1-{\displaystyle \frac{{\sum }_{i=1}^n{\left(O_i-M_i\right)}^2}{{\sum }_{i=1}^n{\left(O_i-O_i^{mean}\right)}^2}}$
Coefficient of Determination	$R2={\left({\displaystyle \frac{{\sum }_{i=1}^n\left(O_i-O_i^{mean}\right)\left(M_i-M_i^{mean}\right)}{\sqrt{{\sum }_{i=1}^n{\left(O_i-O_i^{mean}\right)}^2·{\sum }_{i=1}^n{\left(M_i-M_i^{mean}\right)}^2}}}\right)}^2$
Percentual Bias (%)	$PBIAS={\displaystyle \frac{{\sum }_{i=1}^n\left(O_i-M_i\right)}{{\sum }_{i=1}^n{O}_i}}·100$

Notes: where $O_i$ are the observed or monitored data and $O_i^{mean}$ is their respective mean, $M_i$ are the values obtained from the modeling and $M_i^{mean}$ is their respective mean, $n$ is the number of observed or monitored nodes, and $h$ is the number of hours used in the simulation.

presents the list of equations used for the model indicators:

The calibration process lasted a year and a half, accounting for code adjustments, data collection from the state agency, outlier analysis, and modifications to the valve operation rules. Additionally, each execution of the calibration code took an average of seven days to produce results, even when using a computer with a Dell 12th Gen Intel(R) Core(TM) i7-12700 processor and 16 GB of RAM, Sobral, Brazil. Several modifications were made to both the network and the code to achieve the results presented in this study, with the most exhaustive adjustment involving the operation rules of the various valves in the network.

3. Results

The results showed absolute roughness values ranging from approximately 15 mm to 0.0001 mm, with a mean of 6.2 mm and a median of 1.45 mm. For the base demands, the values ranged from 0.46 L/s to 868.3 L/s, with a mean of 90.2 L/s and a median of 30.38 L/s. Although the age of the pipes is unknown, the results, particularly the absolute roughness, suggest the presence of old or severely deteriorated pipes. In the monitored pipes, the flow velocities had a mean of 2.03 m/s and a median of 1.12 m/s. These values indicate relatively high flow velocities in the network, suggesting that the pressure in the pipes is not strongly dependent on the topographical elevations provided by the agency. In other words, the high flow velocity reduces the influence of terrain height variations on pressure distribution, implying that the network operates with less sensitivity to changes in topography. The values of the five chosen indicators used to evaluate the calibration of roughness and base demands through pressure heads and flow rates are summarized in

Table 3. General results of the calibration indicators for pressure heads.

	MAE	ARHE	NSE	R2	PBIAS
Before Calibration	7.10 m	31.13%	0.597	0.623	2.64%
After Calibration	3.53 m	12.96%	0.873	0.879	3.38%

and

Table 4. General results of the calibration indicators for flow rates.

	MAE	ARHE	NSE	R2	PBIAS
Before Calibration	43.7 L/s	320.79%	0.754	0.780	11.80%
After Calibration	16.3 L/s	13.09%	0.965	0.965	1.06%

. The results indicate that the calibration process successfully met the proposed objectives. For the pressure heads, four out of the five indicators demonstrated a better fit between the proposed model and the monitored data. Similarly, all five indicators for flow rates showed an improved alignment between the model and the observations.

Figure 3 shows an example of the typical average behavior of a monitored point for the pressure heads and flow rates of the analyzed WDS hydraulic model. Figure 3 shows the pressure head and flow rate scale presented in the Fortaleza microsystem and how well the calibrated model tracks the behavior of the monitored data over the two-day period. It is important to note that, in Figure 3, the y-axis does not start at zero to enhance visualization.

Figure 4, Figure 5, Figure 6 and Figure 7 show the behavior of the pipes and nodes represented by the Box and Whiskers diagram, which aims to represent the variability of the data and summarize various value distributions using the median, quartiles, minimum, maximum, and outliers [41,42]. Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13 consist of scatter plots addressing the relationship between a set of binary data, representing the behavior of the evaluation parameters for each monitored point and its respective modeled value, allowing the visualization of the network’s overall behavior.

Figure 4 and Figure 5 show the improvement in the hydraulic model after calibration, considering the error between the monitored and calculated data. The Box and Whiskers diagram for the MAE demonstrates a reduction in the number of outliers and a clustering of values closer to zero on the y-axis after calibration. In Figure 4, monitored points 24 and 25 highlight the reduction in pressure head errors after calibration. Similarly, Figure 5 shows a reduction of outliers above 300 L/s after calibration, a value that corresponds to nearly 30% of the highest monitored value for flow rates. For both pressure head and flow rate, the MAE was reduced by over 50%. Furthermore, considering the maximum allowable error of 2 m for pressure heads, as defined by the [43], the calibrated model achieves approximately 60% of pressure heads within this limit before calibration, compared to only 33% prior to calibration.

As the ARHE represents the average relative hourly error, this indicator reflects the behavior of the errors in hourly pressure heads and flow rates, allowing for a more detailed analysis of the parameters considered. Figure 6 shows the reduction in the number of relative errors above 1 for pressure heads after calibration compared to before. Similarly, Figure 7 highlights how much calibration reduced flow rate relative errors, which decreased from a maximum of approximately 8 m before calibration to below 2 m after calibration. Therefore, the improvement is more noticeable for the flow rates, as the reduction is higher than 95%.

The NSE and R² showed similar results, indicating that the calibration adequately represents the monitored values. The NSE and R² for the pressure head are 0.87, and for the flow rate, it is 0.96 after calibration. That is, values above 0.5 are considered acceptable for a model, and values above 0.8 indicate a very well-adjusted model [39]. Through Figure 8, Figure 9, Figure 10 and Figure 11, it is possible to see the increase in the number of pipes and nodes above 0.5, with the most significant increase observed in Figure 8, where there are only three nodes above NSE = 0.5 before calibration, and after calibration, there are 22 nodes above this mark. For R², this increase in the number of values close to 1 is also visible in the graph, where a significant number of points near 0 can be seen before calibration. In comparison with the work by [44], which calibrated a hydraulic model of a WDS of a small town in the UK using 672 observed pressure heads and resulting in R² = 0.999, this work utilized more than three times the amount of observed pressure heads (2400) and obtained an R² of 0.87, indicating a hydraulic model close to ideal, despite the large amount of observed data. It is important to note that the scale of Figure 8 had to be expanded because some values before calibration were quite divergent (with small NSE values), and it would not have been possible to view the graph in detail.

For the PBIAS, an adverse behavior was observed, as the value for the pressure head increased after calibration, while for flow rate, the reduction was predominant, as shown in Table 3 and Table 4. This increase in PBIAS for pressure heads can be justified by the PBIAS equation itself, where the summation of the difference between observed and monitored data considers the sign of these subtractions. Thus, for the PBIAS value after calibration in the pressure heads to be higher than before calibration, it may indicate a slight tendency toward greater underestimation. Nevertheless, the PBIAS values after calibration can still be considered very good, as they are below 10% [45]. In Figure 12, it is emphasized that the graph had to be enlarged to view more details, as before calibration, the PBIAS values were significantly lower than the others. This highlights the improvement of the PBIAS indicator after calibration.

4. Discussion

It is also worth highlighting that several adjustments were made to the operating rules of the pressure head-reducing valves, including the removal of some rules and the hourly modification of certain valve operation rules. Moreover, for each modification made to the operating rules, it was necessary to check the consistency of the hydraulic model. These verifications involved modifying the valve operation rules, base demand factors, and absolute roughness values of the hydraulic model, followed by running the EPANET simulation for a 48 h period and checking for possible negative pressure heads at any point in the 484 nodes every hour. These analyses confirm that reaching the hydraulic model used in the calibration requires meticulous interventions by the modeler [21]. As a result, several modifications and preliminary calibrations were made to arrive at the final calibrated model, highlighting the inherent complexity of calibrating real, large-scale WDSs. The progress of this calibration process for both pressure heads and flow rates demonstrates the evolution of the indicator results alongside their respective calibration versions. For each version, the valve operation rules, the calibrated parameters, or the number of monitoring points were modified. For instance, from version 1 to version 2, the pressure head monitoring points were reduced from 57 to 56 due to inconsistent data at one point, in addition to changes made to the pressure head values of some valves. This procedure was repeated until version 8, which resulted in the final configuration of this study.

The exclusion of certain monitoring points and the increase in the values of the indicators were also evaluated. After excluding only six pressure head monitoring points based on the worst MAE results, leaving 44 monitoring points, it was observed that the evaluation parameters for the calibrated model presented the following results: MAE = 2.59 m, ARHE = 10.25%, NSE = 0.94, R² = 0.94, and PBIAS = 1.84%. When the same procedure was performed excluding only six flow rate monitoring points based on the worst MAE results and leaving 34 monitoring points, the following results were presented: MAE = 6.34 L/s, ARHE = 11.06%, NSE = 0.99, R² = 0.99, and PBIAS = −0.11%. Therefore, all indicators showed improvements in the results after the removal of just six monitoring points. Even with this removal, this work presents the number of monitored points used in the calibration as higher than that found in most of the analyzed literature, thus highlighting that the entire calibration process and the amount of data used in this work resulted in a consistent hydraulic model [7,46,47,48,49].

It is important to highlight that several innovative techniques for calibrating water supply networks have been developed, such as graph-based metamodels [12] and artificial neural networks [7], among others. However, the decision to calibrate the Fortaleza macrosystem using more traditional strategies from the literature was driven by the complexity of working with a real, dynamic system in collaboration with the state water and wastewater agency responsible for its management. Furthermore, the calibration’s representativeness could be enhanced by incorporating additional parameters. One example is substitute roughness, which accounts not only for actual roughness but also for local resistances and eccentricities caused by reductions or cross-section changes [7,8,15]. Given the Brazilian context of increasing water demand and the impacts of climate change on water resources [50]—particularly in the largest metropolis of the Brazilian semi-arid region, which frequently experiences drought events [6]—this calibrated network represents a critical tool. It will support the state water and wastewater agency in evaluating the impacts of introducing new water sources, operational changes, and climate change scenarios, as well as increasing demand on the network’s performance.

5. Conclusions

This study conducted a calibration of the absolute roughness (ε) and base demand multiplication factors of a large-scale real Water Distribution System (WDS) located in a metropolitan area of Brazil facing water risks. The calibration process involved 50 pressure head monitoring points and 40 flow rate monitoring points over a 48 h period. Additionally, the hydraulic model considered the operating rules of several pressure head-reducing valves within the network, highlighting the inherent challenges of real WDS and underscoring the importance of the modeler in calibrating large-scale hydraulic models with operational rules. The PSO method was used as the algorithm to minimize the objective function.

Moreover, the calibration development included adjustments to the valve operation rules, analyzing generated graphs and the influence of these rules on the network nodes. Modifications such as increasing, reducing, or eliminating pressure head control values were made to meet the monitored points and avoid negative pressure heads. In total, 2304 operation rules were evaluated, with each adjustment directly impacting the network behavior. These adjustments were essential for obtaining a minimally calibrated model, highlighting the complexity of calibrating large-scale WDS hydraulic models.

The results indicate that the calibrated model showed good performance for the five indicator parameters used (MAE, ARHE, NSE, R², and PBIAS), which were evaluated for both the entire network and each monitored point. Notably, 60% of the pressure head errors are below 2 m; the study includes a large number of monitoring points, highlighting the robustness of the calibration for a complex hydraulic model.

Finally, it is expected that the work in this study will contribute to improvements in the water distribution system of Fortaleza City by using the calibrated model for operation scenario analysis and network modifications. Moreover, it will support any analysis by the managing entities of the WDS on the impact of integrating new water supply sources into the network.