Enhancing Insights into Intermittent Water Supply Systems: Uncertainty and Sensitivity Analyses of Hydraulic Model ^†

Abstract

So far, many researchers have attempted to tackle issues associated with intermittent water supply (IWS) systems, such as the inequitable distribution of water, by employing deterministic models that rely on multiple assumptions about input parameters. However, owing to the diverse practices and operations associated with IWS systems, significant uncertainty prevails in various aspects, including user water consumption, supply characteristics, and household tank sizes. In this work, a novel uncertainty quantification framework for assessing uncertain model input parameters is proposed.

1. Introduction

Hydraulic modelling of IWS systems is a crucial step in analysing and addressing the problems that arise. It has been used for the design, improvement of equitable supply, conversion to continuous supply, and rehabilitation of IWS systems. In all these applications where the IWS network model is used to support decision making, understanding the relative magnitude of risk and uncertainty is critical to accurately assessing system behaviour. To simulate realistic scenarios, develop and validate model-based methodologies, and make informed decisions, it is important to better understand the similarities and differences between various IWSs and their operating modes. To develop more representative models, the first step is to analyse all factors affecting the quantity of water a consumer receives over time, and the second step is to integrate uncertainties that arise for each case into the modelling processes. Failure to account for uncertainty can lead to incorrect conclusions, which underscores the importance of a thorough analysis that takes into account the various uncertainties. In short, the accurate determination of the network model’s parameters and the consideration of their uncertainty are essential components of developing reliable and effective solutions.

To the best of the authors’ knowledge, there have been few attempts to address the diverse practices and analyse the unique characteristics of IWS systems through hydraulic modelling [1,2]. An issue arises with these modelling techniques, as they are unable to fully simulate the crucial processes of network filling and draining that dictate consumer supply access, and only a select set of uncertain parameters are examined under specific circumstances. A further limitation lies in the numerous simplifying assumptions made about consumer behaviour and the hydraulic model itself, which still awaits calibration and verification against real-world data.

IWS behaviour investigation with a suitable modelling approach in the context of uncertainty analysis is a relatively new research topic that this paper attempts to cover.

2. Uncertainty Quantification Framework

The proposed uncertainty quantification framework was adapted from [3]’s general probabilistic framework (GPF) for the uncertainty and global sensitivity analyses of deterministic models. In the initial phase, uncertainty analysis (UA), each uncertain parameter is characterised with probabilistic-based approaches (Section 2.1), and Monte Carlo simulations are used to investigate the impact of uncertain model input parameters on the network performance indicators (Section 2.2). In the second phase, a global sensitivity analysis (SA) (Section 2.3) is conducted to identify the primary sources of uncertainty impacting the simulation results and to identify the influential and non-influential parameters.

2.1. IWS Modelling with Uncertain Input Parameters

• The uncertainty in supply characteristics was characterised using probability distribution functions (PDFs). These PDFs were derived from the water supply schedules of Delhi, dated 12 May 2020 [4].
• The uncertainty in the household tank size was characterised by a truncated normal distribution considerin the IWS practices, with a mean value (µ) of 1.0 m³ and standard deviation (σ) of 0.8. Each realisation had the minimum tank size restricted to 0.2 m³, and the sampled tank sizes were rounded to 1 decimal place.
• For the time series of the water consumption from the household tank, several demand patterns were generated by employing Gaussian process (GP) emulators using a given hourly-based demand pattern to account for the correlation between successive demand multipliers.

2.2. Performance Indicators

Of the nine indicators described by [5], four key metrics were selected to assess the uncertainties in model outputs: indicator I₁—the proportion of the number of effective hours a user is served; indicator I₂—the proportion of the volume of water supplied to users; indicator I₃—the level of pressures at consumption nodes; and indicator I₄—the level of equity in the supply.

2.3. Global Sensitivity Analysis

Sobol’s method [6] was employed in this study to examine the influence of both individual uncertain parameters and their interactions on the model’s performance. The objective was to understand the behaviour of the network in response to the interactions between household water consumption, household tank size, and supply characteristics.

Utilising the Saltelli sequence, which is specifically designed for Sobol’s SA [7], 213 = 8192 samples were used. This led to a total of N × (2D + 2) = 65,536 simulations being executed to calculate the Sobol sensitivity indices (where N is the number of samples, and D is the number of uncertain input parameters, here, 3).

3. Results and Discussion

The proposed framework implementation was demonstrated by its application to an IWS network detailed in [8]. The EPA-SWMM-based IWS modelling approach was used for hydraulic simulation [9]. The simulation spanned 24 h for both SA and UA, covering a total of 12 scenarios split into two groups based on the household tank sizes at each node.

1.

Same-sized household tank (SSHT) for each node: In this situation, each node was characterised by household tanks of the same size.

2.

Different-sized household tanks (DSHT) for each node: In this situation, household tanks of different sizes were assigned to each node.

These scenarios were designed to examine two primary aspects: the reduction in the water supplied to the network and the variability in water demands met at the nodes. In this context, “demand” was defined as the average water consumption from the household tanks. Each scenario, amounting to 12 in total, was executed 65,536 times.

Table 1. Scenarios for uncertainty and sensitivity analyses.

Scenario Name	Scenario ID	Water Supplied	Demand Met
Same-sized household tank for each node (SSHT)	SSHT-1	-	-
	SSHT-2	70%	-
	SSHT-3	-	50%
	SSHT-4	70%	50%
	SSHT-5	-	0
	SSHT-6	-	25%
Different-sized household tanks for each node (DSHT)	DSHT-1	-	-
	DSHT-2	70%	-
	DSHT-3	-	50%
	DSHT-4	70%	50%
	DSHT-5	-	0
	DSHT-6	-	25%

outlines the scenario names and IDs, along with the alterations made to the water supplied in the network and to the demand values. Percentages represent the modification relative to the default value, where a “0” value indicates an assumption of no water consumption from the household tanks. A scenario with no water consumption does not reflect real-world conditions; however, it was included to investigate the interplay between uncertain parameters and to understand the system dynamics under hypothetical extreme conditions.

3.1. Uncertainty Analysis (UA)

Figure 1 shows the variations in the performance indicators in the 12 scenarios to ascertain the overall uncertainty in the network’s performance. It can be seen from the figure that the scenarios themselves do not impose significant impacts on the network’s overall performance. I₄ has the highest levels of uncertainty, with values above 0.4, which indicates a significant relationship with the uncertain input parameters.

3.2. Sensitivity Analysis (SA)

Figure 2 shows the Sobol indices for the base case scenario, SSHT-1, specifically for indicator I₂. The figure clearly indicates that the most sensitive parameter affecting the model is the supply characteristics. This is primarily because the quantity of water delivered to the system is determined by the duration of the supply.

4. Conclusions

It is challenging to anticipate system performance without comprehensive field data, but uncertainty analysis helps reveal the system behaviour and leads towards modelling improvements. The performance indicators display variability under uncertainties, emphasising the necessity of incorporating uncertain input parameters into relevant decision-making processes. Further data collection is needed for more accurate modelling, particularly regarding supply characteristics.