Modeling of truncated normal distribution for estimating hydraulic parameters in water distribution systems: taking nodal water demand as an example.

The normal probability density function (PDF) is widely used in parameter estimation in the modeling of dynamic systems, assuming that the random variables are distributed at infinite intervals. However, in practice, these random variables are usually distributed in a finite region confined by the physical process and engineering practice. In this study, we address this issue through the application of truncated normal PDF. This method avoids a non-differentiable problem inherited in the truncated normal PDF at the truncation points, a limitation that can limit the use of analytical methods (e.g., Gaussian approximation). A data assimilation method with the derived formula is proposed to describe the probability of parameter and measurement noise in the truncated space. In application to a water distribution system (WDS), the proposed method leads to estimating nodal water demand and hydraulic pressure key to hydraulic and water quality model simulations. Application results to a hypothetical and a large field WDS clearly show the superiority of the proposed method in parameter estimation for WDS simulations. This improvement is essential for developing real-time hydraulic and water quality simulation and process control in field applications when the parameter and measurement noise are distributed in the finite region.

A noise adaptive approach for nodal water demand estimation in water distribution systems.

Hydraulic models have emerged as a powerful tool for simulating the real behavior of water distribution systems (WDSs). In using the models for estimating nodal water demands, measurement uncertainty must be considered. A common approach is to use the covariance of measurement noises to quantify the measurement uncertainty. The noise covariance is typically assumed constant and estimated a priori. However, such an assumption is frequently misleading as actual measurement accuracies are affected by measuring instruments and environmental noises. In this study, we develop a variational Bayesian approach for real-time estimation of noise covariance and nodal water demands. The approach can adaptively adjust the noise covariance with the variation of the noise intensity, thereby efficiently avoiding model overfitting. The measurement residual decomposition reveals that this new approach is effective in determining model structural errors caused by topological structure parameterization.

Numerical approach for water distribution system model calibration through incorporation of multiple stochastic prior distributions.

The use of water distribution system (WDS) hydraulic models facilitates the design and operation of such systems. For offline or online model applications, nodal water demands-variables with the highest levels of uncertainty-should be carefully calibrated because these can considerably affect the accuracy of model outputs in terms of hydraulics and water quality. With the increasing utilization of automatic water metering technology, nodal water demands can be modeled with high time resolution in certain forms of probability distributions. However, the fusion of various demand probability distributions with conventional measurements to improve the accuracy of WDS hydraulic models is a difficult problem. To resolve this, a numerical approach that incorporates various probability distributions and field measurements to calibrate nodal water demands based on Bayesian theory is proposed. In particular, the linearization of the exponential family prior distribution is well elaborated in this paper. The application of this proposed approach in two cases demonstrates that the technique is more accurate than methods that merely utilize measurements or prior information. Because this technique can avoid the overfitting of measurement noise and allow the retention of calibrated nodal water demands with stochastic nature, it is robust when errors or uncertainties exist in prior demand distribution or measurements. This method is expected to improve the WDS model accuracy relative to the increasing use of automatic water metering technology.

Time-Series-Based Leakage Detection Using Multiple Pressure Sensors in Water Distribution Systems.

Leak detection is nowadays an important task for water utilities as leakages in water distribution systems (WDS) increase economic costs significantly and create water resource shortages. Monitoring data such as pressure and flow rate of WDS fluctuate with time. Diagnosis based on time series monitoring data is thought to be more convincing than one-time point data. In this paper, a threshold selection method for the correlation coefficient based on time series data is proposed based on leak scenario falsification, to explore the advantages of data interpretation based on time series for leak detection. The approach utilizes temporal varying correlation between data from multiple pressure sensors, updates the threshold values over time, and scans multiple times for a scanning time window. The effect of scanning time window length on threshold selection is also tested. The performance of the proposed method is tested on a real, full-scale water distribution network using synthetic data, considering the uncertainty of demand and leak flow rates, sensor noise, and so forth. The case study shows that the scanning time window length of 3-6 achieves better performance; the potential of the method for leak detection performance improvement is confirmed, though affected by many factors such as modeling and measurement uncertainties.

Real-Time Water Distribution System Hydraulic Modeling Using Prior Demand Information by Formal Bayesian Approach.

Real-time water distribution system (WDS) hydraulic models are used in water utilities to facilitate the planning and operation of the water distribution system. As a critical model input, spatiotemporally varying nodal water demands significantly affect the performance and applicability of such WDS models. Thus, real-time nodal demands must be calibrated for reliability before their use. The main difficulty for real-time calibration is the lack of observed data sufficient to determine thousands of nodal demands accurately in a network. To address the difficulty, this study proposes a formal Bayesian approach to determine nodal demands in WDS hydraulic modeling by explicitly taking prior water demand information into account and coupling more information to constrain the nodal water demand modeling. Application of the approach on a simple hypothetical network and a field network in a city of eastern Zhejiang Province, China demonstrates that by adding prior information, the nodal demand can be uniquely determined in real time. The approach limits uncertainty propagation and improves the robustness of the real-time model calibration and analysis.

A Greedy Sampling Design Algorithm for the Modal Calibration of Nodal Demand in Water Distribution Systems.

This paper presents a greedy optimization algorithm for sampling design to calibrate WDS hydraulic model. The proposed approach starts from the existing sensors and sequentially adds one new sensor at each optimization simulation step. In each step, the algorithm tries to minimize the calibration prediction uncertainty. The new sensor is installed in the location where the uncertainty is greatest but also sensitive to other nodes. The robustness of the proposed approach is tested under different spatial and temporal demand distribution. We found that both the number of sensors and the perturbation ratio affect the calibration accuracy as defined by the average nodal pressure deviation itself and its variability. The plot of the calibration accuracy versus the number of sensors can reasonably guide the trade-off between model calibration accuracy and number of sensors placed or the cost. This proposed approach is superior in calibration accuracy and modeling efficiency when compared to the standard genetic algorithm (SGA) and Monte Carlo Sampling algorithm (MCS).