Optimizing sensor placement for enhanced observability in water distribution networks Open Access

Water plays a crucial role in sustaining life on Earth, by supporting ecosystems, regulating the climate, and promoting human health. It is essential in agriculture, industry, and daily consumption, playing a critical role in food production and hygiene maintenance. The accessibility of drinkable water directly impacts the quality of life and economic progress of communities. In the absence of adequate management, water scarcity has the potential to result in conflicts, displacement, and health crises. Hence, preserving and effectively utilizing water is imperative for guaranteeing a sustainable future. According to the Organization for Economic Cooperation and Development, the demand for this limited resource is projected to increase by up to 55% by 2050 (OECD 2021; Ilango & Sridharan 2022).

In areas facing water scarcity, conservation and efficient management are crucial, as access to clean and sufficient water is a daily challenge. The water distribution networks (WDNs) in urban areas are crucial for ensuring continuous and safe access to potable water in cities. These infrastructures enable the efficient delivery of treated water to homes, schools, hospitals, industries, and other essential services, thereby ensuring hygiene, public health, and the well-being of the population. Moreover, a well-managed water supply system is fundamental for economic development and urban sustainability, as it supports both domestic and commercial activities, fostering a healthy and productive urban environment (Bello et al. 2019; Kanakoudis & Tsitsifli 2019; Tzanakakis et al. 2020).

Leakage volumes in WDNs vary by country and depend on network conditions and maintenance. Poorly maintained systems can lose up to 70–80% of the input volume, while well-monitored networks reduce losses to around 7%. In 2018, global non-revenue water, including leakages and unbilled consumption, reached 126 billion cubic meters, valued at USD 39 billion (Serafeim et al. 2024). Detecting and managing leaks effectively in WDNs poses a critical challenge. Undetected leaks lead to significant water loss, reduced efficiency, higher energy consumption, and potential infrastructure damage. Persistent leaks further increase maintenance costs and threaten the long-term reliability of the system (Heryanto et al. 2021).

Mathematical modeling is essential for maintaining the integrity and efficiency of urban WDNs because it provides a systematic framework for identifying and localizing leaks that may otherwise go undetected (Nimri et al. 2023). By simulating the hydraulic behavior of the network under normal and abnormal conditions, these models enable the identification of inconsistencies in pressure and flow, which are often indicative of leaks (Islam et al. 2022; Vrachimis et al. 2022).

For any decision-making, monitoring, and control techniques to be effective, it is crucial to ensure system observability. This capability allows the internal states of the system to be inferred from its external outputs, providing the necessary information for informed decisions and precise adjustments. Without good observability, it becomes difficult to detect internal issues and act promptly, which can compromise the effectiveness and reliability of the system (Díaz et al. 2016; Berardi & Giustolisi 2021). To guarantee proper monitoring, it is common to incorporate sensors into the real system. Nevertheless, usually, not all the state variables can be measured, because the instrumentation budget is limited, physical constraints are present, or the sensor does not exist (Yoo et al. 2018; Hu et al. 2020; Geelen et al. 2021).

Many WDNs are modeled as nonlinear dynamical systems (Hu et al. 2021; Price et al. 2022). These systems exhibit complex behavior because of their nonlinear characteristics, challenging their analysis and control. The observability analysis of these systems is particularly intricate, as it necessitates a detailed mathematical framework to accurately deduce the internal states from external measurements. Typically, this analysis is based on Lie Algebra, a sophisticated branch of mathematics that studies algebraic structures and their applications to differential equations and dynamical systems (Yano 2020). For real systems, comprising dozens of state variables, however, this analysis is not tractable and linearization techniques need to be used.

For linear, time-invariant systems, the observability property is determined by the rank of the observability matrix (OM) (Montanari & Aguirre 2020; Rodriguez et al. 2021). This matrix is a fundamental tool in monitoring theory used to evaluate whether all states of a system can be deduced from its outputs. However, challenges arise when dealing with systems that have limited observed outputs. Here, this matrix only indicates whether the system is observable or not, without providing information about the system observability degree (SOD) (Geelen et al. 2021; Bernard et al. 2022).

In contrast, the use of the observability Gramian matrix (OGM) provides significant advantages over the OM in dynamic systems. First, it offers a more comprehensive assessment by quantifying how accurately states can be estimated from output measurements. Additionally, this tool allows for flexibility in evaluating precise system monitoring with various sets of observed outputs. Moreover, the OGM plays a crucial role in devising effective strategies for the precise estimation of unmeasured states in monitoring applications (Taha et al. 2021; Yamada et al. 2023).

A significant number of indices based on geometric properties of the OGM to determine observability exist. One of these indices uses the smallest eigenvalue of this matrix. The sensor configuration that maximizes this value maximizes the observability of the least observable directions of the system. Other indices maximize the maximum eigenvalue, unlike the previous index a large value refers to the dominant direction of the system. Also, the trace of the OGM can be interpreted as the sum of the eigenvalues of this matrix. Increasing this index increases the overall monitoring effectiveness of a system. In the first index, the directions of minimum observability have greater preponderance; in contrast, in the last two, the maximum directions hold more importance (Georges 1995).

Other types of observability analysis of nonlinear dynamical systems use the empirical OGM concept. Unlike the theoretical OGM, the calculation is based on measurements when performed for a given sensor configuration, or through numerical simulation for a new one. Although this methodology was successfully used for sensor localization in chemical processes, it was not used in WDN (Rodriguez et al. 2021).

Solving a sensor location problem (SLP) involves selecting the system variables to be measured, and the sensor characteristics to be used (cost, precision). This process requires a thorough understanding of the system dynamics and the critical points where measurements are most beneficial. The selection of system variables is crucial because it determines the accuracy and reliability of the data collected. Additionally, choosing appropriate sensor characteristics such as cost and precision ensures that the sensors meet the specific needs of the system without exceeding budget constraints (Boatwright et al. 2018; Rodriguez et al. 2021). Usually, this is not a simple task.

A widespread strategy for WDN uses a numerical simulation of the system, but the sensor networks obtained are susceptible to the precision of the mathematical model structure and parameters, process noise, etc. (Quiñones-Grueiro et al. 2018). Some methods use leak representation based on the leak signature space and formulate the SLP as an integer optimization problem, which is solved by using the genetic algorithms and particle swarm optimization (Casillas et al. 2013).

In the study of Geelen et al. (2021) an optimal weight analysis algorithm was proposed to solve the problem of sensor placement. In contrast to previous methodologies, an optimal SLP for WDN monitoring was presented using a state space-based approach. The system is modeled using a linearized version of the hydraulic network model, considering the pressure dynamics and hydraulics of the networks. The design criterion is to maximize the output energy of the states, calculated as the minimum eigenvalue of the OGM. However, no constraints were used on the instrumentation budget and the algorithm to solve the problem was not mentioned. Other SLPs in WDN addressed in the literature include abnormal and contaminant detection (Shahsavandi et al. 2024).

To summarize, urban WDNs are crucial for providing continuous and safe access to potable water. One primary challenge in these networks is to detect leaks, which can be predicted and detected using mathematical models that analyze hydraulic dynamics, reducing water loss and operational costs. Ensuring observability is essential for effective monitoring; however, achieving this in nonlinear systems involves complexities dependent on specific assumptions, such as system linearization, model accuracy, and stability. Additionally, real-world conditions, like variability in operational parameters, can significantly impact observability analysis. For linear, time-invariant systems, the OM is used, but it has limitations. In contrast, the use of the OGM offers a more comprehensive assessment and flexibility, aiding in precise state estimation.

Despite the scarcity of water resources and its increasing demand, the literature review in this paper shows that SLPs formulation for WDN received little attention, in particular, the problem of maximizing the SOD, calculated through the OGM, subject to cost constraints, or the problem of minimizing the cost subject to SOD constraints. This requires finding the most cost-effective sensor configuration that still ensures efficient and effective system monitoring.

The aim of this work is to investigate and demonstrate a strategy for sensor location with observability purposes in WDNs. This work presents new SLPs for monitoring purposes in WDNs and the solution methods for these problems. Sensor quality criteria were not considered, and it was assumed that all state variables could be measured without any spatial restrictions on sensor placement. First, the SOD is calculated through the eigenvalues of the OGM, and then by taking into account the eigenvalues of all directions. This observability index is optimized and subject to budget constraints. Second, it addresses the tradeoff between the cost of the sensor network and the SOD, introducing the concept of SOD degradation and formulating the SLPs accordingly. These formulations are important when it is desired to optimize the SOD and the instrumentation cost.

The remaining part of this paper is structured as follows. In the next section, the WDN is modeled using continuity and momentum equations. These equations are presented in state-space form considering the dynamic behavior around a given hydraulic scenario. The SODs are calculated through the eigenvalues of the OGM. Then, the SLP formulations and the resolution methods are addressed. The Hanoi WDN is considered a case study, and the optimization problems are solved using genetic algorithms. Finally, the obtained results are discussed, and the conclusion and future works are presented.

Mathematical models are essential for monitoring WDNs as they enable performance evaluation, support decision-making, and simulate various scenarios. They help analyze networks and identify issues like water losses, pressure fluctuations, and infrastructure failures.

The variable p represents the pressure (Pa), V is the directional flow velocity (m/s), and |V| is its magnitude, ρ is the fluid density (kg/m³), c is the elastic wave velocity (m/s), g is the gravitational acceleration (m/s²), θ is the conduit slope, f is the Darcy–Weisbach friction factor, and D is the internal diameter (m).

Using the minimum eigenvalue helps identify critical directions within a system. A small value of points out that there are directions in the state-space that are difficult to monitor from q. Conversely, a large value suggests that all states of the system are easily observable. Moreover, in numerical computations, a large improves the numerical condition of algorithms that depend on the inverse of the OGM, making the results more accurate and reliable.

For simplicity, in the rest of this article ⁠, ⁠, ⁠, and are used to represent ⁠, and and ⁠, respectively.

Let be the n-dimensional set containing all possible measured state variables of the system, being n = a + b. The SLP consists of selecting the elements of that optimize a certain criterion while satisfying specific constraints. Proper choice of the location of these sensors is necessary to ensure that all necessary data for system observability is captured.

Water flows and piezometric heads are measured with flowmeters and pressure sensors, respectively. Generally, flowmeters are more expensive due to their advanced technology requirements, such as measuring flow velocity, pressure, and turbulence. They also endure harsher conditions, increasing production and maintenance costs. In contrast, piezometric sensors have simpler, more robust designs as they mainly measure static pressure.

If all the state variables are measured ny = 65 (see Figure 3), the measurement matrix is the identity matrix ⁠, and ⁠.

To solve the SLPs, the algorithm uses a GA with binary variables. The initial population of 50 random bit strings ensures diverse solutions. In each iteration, 20% of the fittest solutions undergo crossover, and 40% experience mutations to maintain diversity. Additionally, 20% of the population is replaced with new random solutions. This process repeats 50 times, updating the best solution in each iteration using a fitness function to evaluate sensor placement effectiveness.

Considering several budgets, SLPs (16) for are executed for the proposed case study. In addition, a ratio ⁠.

Table 1 presents the solutions obtained for ranging from 0.1 to 0.9, and ⁠. Values of β₁ near 1 correspond to high instrumentation budgets.

Table 1

Results for SLP (16)

.	.	Solution .	.
0.9	0.8872	1; 5; 7–23; 25–65	10.4310
0.8	0.7970	1; 2; 5; 7; 9–19; 21–24; 26 28–31 33 35–65	10.4310
0.7	0.6842	1; 4; 5; 9–11; 13–19; 21–23; 29–31; 33; 35–65	10.4310
0.6	0.5940	2; 4; 5; 8–12; 14; 16–18; 24; 26; 29; 31; 35–65	10.4156
0.5	0.4812	3; 8; 10–16; 27; 30; 35–65	10.4102
0.4	0.3985	1; 3; 8; 13; 19; 25; 31; 33; 35–42; 44–52; 54–65	9.7716
0.3	0.2932	1; 16; 27; 32; 33; 35; 36; 38–43; 45–48; 50; 51; 53; 54; 56–59; 61–64	7.9672
0.2	0.1955	8; 35; 37–42; 44–49; 51–57; 61; 63; 65	7.4544
0.1	0.0977	35; 41; 44–47; 53; 54; 57–60; 62	3.3540

.	.	Solution .	.
0.9	0.8872	1; 5; 7–23; 25–65	10.4310
0.8	0.7970	1; 2; 5; 7; 9–19; 21–24; 26 28–31 33 35–65	10.4310
0.7	0.6842	1; 4; 5; 9–11; 13–19; 21–23; 29–31; 33; 35–65	10.4310
0.6	0.5940	2; 4; 5; 8–12; 14; 16–18; 24; 26; 29; 31; 35–65	10.4156
0.5	0.4812	3; 8; 10–16; 27; 30; 35–65	10.4102
0.4	0.3985	1; 3; 8; 13; 19; 25; 31; 33; 35–42; 44–52; 54–65	9.7716
0.3	0.2932	1; 16; 27; 32; 33; 35; 36; 38–43; 45–48; 50; 51; 53; 54; 56–59; 61–64	7.9672
0.2	0.1955	8; 35; 37–42; 44–49; 51–57; 61; 63; 65	7.4544
0.1	0.0977	35; 41; 44–47; 53; 54; 57–60; 62	3.3540

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Similar behavior is observed when analyzing the solutions of the SLP (16) for presented in Table A2 in the Supplementary Material section. This is because the piezometric head sensors contribute more significantly to the SOD, effectively balancing cost and performance. Including costly sensors, though rare, enhances the design. The algorithm balances cost and performance, leveraging expensive sensors when their benefits justify the investment.

The SLP (16) can be compared with other approaches in the literature, such as the one by Casillas et al. (2013), which addresses sensor placement for leak detection in WDNs. Their method maximizes leak isolability but may compromise system observability and state estimation. Its effectiveness depends on the chosen isolability criterion, potentially limiting its applicability. Additionally, its adaptation to other isolation schemes may not be straightforward. Since budget constraints are not explicitly considered, it may require more sensors than economically viable. In contrast, our SLPs maximize observability while accounting for cost, offering a more balanced approach.

Also, the SLP (16) proposed in this work can be related to problem 12 presented in Geelen et al. (2021) as both maximize β₁. However, in the latter, budget constraints are not formally imposed. Another key aspect worth highlighting is that the only solution reported by Geelen et al. (2021) for this case study includes variable 51 (a pressure sensor). This solution emerges as a specific case of solving SLP (16) when is set equal to the of a single pressure sensor. This highlights not only the greater flexibility of the SLP formulation in this work but also its ability to accommodate various sensor configurations, making it more adaptable to different network conditions and optimization criteria.

Subsequently, SLP (17) was solved for ⁠; and different ⁠. More instruments are needed as the constraints for become smaller, consequently increasing ⁠. Tables 2, A3, and A4 show the results obtained from solving the aforementioned SLP under different RSC conditions.

The results show that pressure sensors significantly impact the SOD, making them the top choice even at RSC = 1. Sensors 45 and 62 are consistently selected, emphasizing their importance in observability optimization.

After implementing cost constraint ⁠, using a ⁠, and solving SLP 18 for ⁠, the solutions obtained exhibit a comparable pattern to those obtained from solving SLP 12, as depicted in Table 3. Sensors 47, 50, and 63 appear in all solutions, while sensors 39, 45, 54, 56, and 57 appear in 8 of 9, highlighting their importance for SOD maximization. Under budget constraints, pressure sensors contribute most to observability.

Table 3

Results for SLP (18), ⁠,

.	.	Solution .	.
0.9	0.8958	45; 47; 50; 52; 56; 57; 63	0.053
0.8	0.7954	36; 45; 47; 50; 52; 54; 56; 57; 63	0.068
0.7	0.6674	36–39; 45; 47; 50; 53; 54; 56; 57; 63	0.075
0.6	0.5615	36; 39; 42; 45; 47; 50; 53; 54; 56; 57; 63	0.083
0.5	0.4973	37; 39; 40; 45; 47; 50; 52; 54; 55–57; 59; 60; 63; 65	0.113
0.4	0.3921	35; 37; 39; 41; 43; 47; 50; 52; 54–57; 59; 63	0.113
0.3	0.2816	36; 38; 39; 43, 45; 47; 50–52; 54–57; 59; 60; 63; 65	0.128
0.2	0.1905	35; 36; 39–41; 43; 46; 47; 50–52; 54; 56; 57; 60; 63; 65	0.135
0.1	0.0851	35–37; 39–40; 43; 45; 47; 48; 50–54; 56; 57; 59; 60; 63; 65	0.150

.	.	Solution .	.
0.9	0.8958	45; 47; 50; 52; 56; 57; 63	0.053
0.8	0.7954	36; 45; 47; 50; 52; 54; 56; 57; 63	0.068
0.7	0.6674	36–39; 45; 47; 50; 53; 54; 56; 57; 63	0.075
0.6	0.5615	36; 39; 42; 45; 47; 50; 53; 54; 56; 57; 63	0.083
0.5	0.4973	37; 39; 40; 45; 47; 50; 52; 54; 55–57; 59; 60; 63; 65	0.113
0.4	0.3921	35; 37; 39; 41; 43; 47; 50; 52; 54–57; 59; 63	0.113
0.3	0.2816	36; 38; 39; 43, 45; 47; 50–52; 54–57; 59; 60; 63; 65	0.128
0.2	0.1905	35; 36; 39–41; 43; 46; 47; 50–52; 54; 56; 57; 60; 63; 65	0.135
0.1	0.0851	35–37; 39–40; 43; 45; 47; 48; 50–54; 56; 57; 59; 60; 63; 65	0.150

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The SLP (17) and (18) strategies offer significant advantages over SLP (16) and Problem (12) of Geelen et al. (2021), where observability is maximized under cost constraints. Unlike these approaches, SLP (17) and (18) enable us to efficiently solve the SLP by achieving significant reductions in the instrumentation budget without severely compromising the SOD. Moreover, these formulations offer greater flexibility in sensor placement, making them more practical under budget constraints.

In this work, new SLP formulations for sensor placement in WDNs are presented, aiming to balance instrumentation costs with system observability. The SOD was quantified using the eigenvalues of the OGM. One SLP aimed to maximize SOD within a budget, while another minimized the budget under ODDF constraints. Genetic algorithms were employed to solve the optimization problems, and sensitivity analysis was conducted to assess performance under different RSC scenarios.

The Hanoi WDN was used as a case study in order to validate the proposed SLP formulations. The results show that pressure sensors have the greatest influence on the SOD, consistently being prioritized, even when RSC is 1. Sensors like 45 and 62 are regularly chosen when solving SLP 16–18, emphasizing their importance in observability optimization. The greater selection of pressure sensors than flow sensors is due to their crucial role in determining the state of the system they provide. The termination criteria for the GA were initially set at 50 iterations; however, it has been observed that the solutions to the SLPs typically converge around iteration 40.

The SLP techniques can be adapted by replacing the proposed observability indices with other metrics, such as the condition number, allowing for customization to fit different contexts and specific requirements. This flexibility optimizes monitoring system design for WDN observability needs. Additionally, a GA-VNS hybrid approach is being explored to improve the search for optimal solutions.

The authors wish to thank the financial support of CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas), UNSJ (Universidad Nacional de San Juan).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.