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

Journal Pre-proofs Numerical Approach for Water Distribution System Model Calibration through Incorporation of Multiple Stochastic Prior Distributions Shipeng Chu, Tuqiao Zhang, Yu Shao, Tingchao Yu, Huaqi Yao PII: DOI: Reference:

S0048-9697(19)34556-5 https://doi.org/10.1016/j.scitotenv.2019.134565 STOTEN 134565

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Please cite this article as: S. Chu, T. Zhang, Y. Shao, T. Yu, H. Yao, Numerical Approach for Water Distribution System Model Calibration through Incorporation of Multiple Stochastic Prior Distributions, Science of the Total Environment (2019), doi: https://doi.org/10.1016/j.scitotenv.2019.134565

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Numerical Approach for Water Distribution System Model Calibration through Incorporation of Multiple Stochastic Prior Distributions Shipeng Chua, Tuqiao Zhangb, Yu Shaoc*, Tingchao Yud, Huaqi Yaoe

aCollege

of Civil Engineering and Architecture, Zhejiang University, Hangzhou, CHINA,

310058. E-mail: [email protected] bCollege

of Civil Engineering and Architecture, Zhejiang University, Hangzhou, CHINA,

310058. E-mail: [email protected] c*Corresponding

author, College of Civil Engineering and Architecture, Zhejiang University,

Hangzhou, CHINA, 310058. E-mail: [email protected] dCollege

of Civil Engineering and Architecture, Zhejiang University, Hangzhou, CHINA. E-mail:

[email protected] eCollege

of Civil Engineering and Architecture, Zhejiang University, Hangzhou, CHINA,

310058. E-mail: [email protected]

Numerical Approach for Water Distribution System Model Calibration through Incorporation of Multiple Stochastic Prior Distributions Abstract: 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. Keywords: Prior information; Automatic water meter; Real-time; Bayesian framework;

1 Introduction Water distribution system (WDS) hydraulic models have been widely used to simulate real WDS networks and estimate state variables, such as nodal pressure and pipe flow (Kang and Lansey, 2009; Zhang et al. 2019a; Yan et al. 2019). Nodal water demands are the parameters with the most uncertainty that can significantly affect the accuracy of model outputs for hydraulics and water quality (Zheng et al. 2016; Zhao et al., 2018; Qi et al., 2018b; Zhang et al. 2019b). These parameters can only be partly measured because of current technical and economic constraints; hence, these must be thoroughly calibrated for network modeling and operations (Walski, 1983). The WDS model calibration is a process of adjusting model parameters to reduce the deviation between model outputs and observed values (Ostfeld et al., 2012). Three commonly used methods have been developed—trial and error, analytical or explicit, and implicit methods. For the trial and error method, the model parameters are manually adjusted to make the model outputs match the field measurements (Walski, 1983). Although this traditional method can be easily implemented, the time consumption of large-scale networks can be intensive; hence, its application in real-time network models may be difficult. Alternatively, explicit methods have been proposed by solving an extended set of hydraulic equations (Ormsbee and Wood, 1986). Although this method consumes less time, it requires that the number of unknown parameters be equal to the number of measurements; moreover, it cannot consider measurement uncertainties (Savic et al., 2009). The implicit method considers the model calibration process as an optimization problem, and the model parameters, such as decision variables, are adjusted to minimize the difference between observed measurements and model outputs (Zhang et al., 2018b). The aforementioned method can be classified as an evolutionary algorithm (EA)-based method and a numerical method. The EA-based methods, such as genetic algorithms (Do et al., 2016), can solve optimization problems. However, because they are time-consuming, frequently, their requirements significantly exceed the available resources in practice (Zheng et al., 2017).

The numerical methods, such as weighted least squares (Reddy et al., 1996) and singular value decomposition (SVD) (Cheng and He, 2011; Sanz and Pérez, 2015), are more computationally efficient than the EA-based approaches. However, the ill-conditioning problem in the calibration process leads to no unique solution (Du et al., 2015) because the number of measurements is generally less than the number of nodal demands (Qi et al., 2018a). Accordingly, the calibration result is one of the numerous feasible solutions, and the calibration accuracy of nodal demands is frequently beyond the acceptable range even if it exhibits a good performance in matching the measurements. The essence of the ill-conditioning problem is the lack of sufficient information to estimate unique model parameters. This limitation has motivated several researchers to develop algorithms that can combine more information to calibrate nodal demands. Prior information has been used for offline (Kapelan et al., 2004; Kapelan et al., 2007) and online nodal demand calibrations (Kang and Lansey, 2009; Xie et al., 2017). A common method is to consider user characteristics (e.g., population served and customer billings) as prior information and group nodes with similar characteristics to reduce the number of unknown parameters; this makes the system solvable (Zhou et al., 2018). Thereafter, the estimated group demand is proportionally redistributed to individual nodes according to the base demands estimated from the population served or consumer billings (Kang and Lansey, 2009). However, improper grouping generates new errors among the estimated parameters. Moreover, the base demands estimated from the population served or consumer billings yield only offline information or the total amount of water demands over a long time step, which is typically for consumer billings. The Bayesian-type calibration techniques provide a more efficient approach to combine prior information with observed values for both offline and online calibration. For the offline calibration, prior information, such as population served or customer billings, is represented by a prior probability distribution (Hutton et al., 2014b; Kapelan et al., 2007, 2004; Kozelj et al., 2014). The water demand posterior probability distribution is proportional to the priori and

likelihood functions. Thereafter, the nodal demands are estimated by maximizing the posterior probability distribution. The real-time water calibration based on the Bayesian probability theory includes Kalman filter algorithm (Kang and Lansey, 2009; Zhou et al., 2018), extended Kalman filter algorithm (Shang et al., 2006), MCMC method(Qin and Boccelli 2019), and particle filter algorithm (Do et al., 2017; Xie et al., 2017). These approaches consist of two stages, where both the prior probability distribution and likelihood function are updated by the value observed realtime. These real-time calibration approaches can effectively solve the problem of ill-conditioning and avoid that of parameter grouping. However, nodal demands are indirectly estimated by matching the measurement time sequence data. Consequently, these approaches may lead to the overfitting of measurements and result in considerable calibration error. To avoid the overfitting of measurements, more information should be incorporated to govern the nodal demand adjustment domain. Some researchers develop the stochastic demand models at a high time resolution (down to 1 s) to estimate the real-time stochastic water demands directly. Thereafter, the spatial and temporal aggregation of these demands in time series enables the reconstruction of nodal demand (Creaco et al., 2016). The PRP (Buchberger and Wu, 1995) and end-use (Blokker et al., 2011) models are two widely used methods for modeling real-time water demands in time series obtained from the automatic water meter or by field survey information. In addition, the increasing utilization of automatic meter reading equipment affords a new means to directly estimate real-time water demand for single users (Arandia-Perez et al., 2014; Creaco et al., 2016). The foregoing approaches can efficiently and directly model the real-time demand in time series for a single user. After spatial and temporal aggregation, the nodal water demands are constructed and described by statistical parameters, such as mean, variance, and correlation in the form of probability distributions (Vertommen et al., 2015). Such probability distributions significantly represent the stochastic nature of nodal demand, particularly node demand fluctuation, which can be used to govern the feasible domain for demand adjustment. Considering that water demands

are increasingly monitored with the automatic water metering technology, how such data can be used and fused with other conventional measurements (such as pipe flow or nodal pressure field measurements) to improve the accuracy of WDS hydraulic models becomes a problem. In fact, the demand stochastic characters are affected by several factors, such as geographic, climatic, and socioeconomic conditions as well as air temperature (Vertommen et al. 2012). These lead to various forms of probability distributions, such as normal, lognormal, and uniform distributions (Gargano et al. 2017; Surendran et al. 2005). The simultaneous merging of these various probability distributions in the same demand estimation framework is a bigger problem. As previously stated, existing Bayesian-type calibration approaches provide an efficient means to combine prior information with measurements. However, these approaches are still insufficient to couple various probability distributions simultaneously. One reason is that existing methods rely on the assumption of normal distribution of parameters (Kapelan et al., 2004; Kang and Lansey, 2009), which are difficult to couple with other forms of probability distribution simultaneously. Another reason is that the methods involve parameter sampling from the prior or posterior probability distribution (Kapelan et al. 2007; Kozelj et al., 2014; Hutton et al. 2014a; Do et al., 2017; Xie et al. 2017), which may be time consuming and may require intensive computational effort. The foregoing restricts the application of the methods in real-time network modeling. This study provides an efficient framework to couple various nodal water demand probability distributions with field measurements simultaneously and thereby improve the accuracy of the WDS model. The method that directly estimates nodal water demand probability distribution from the automatic water meter has been studied by several research works and is beyond the scope of this study. In this research, the nodal water demand distributions are assumed to be estimated in advance. A Bayesian framework is developed; the various nodal demand distributions are treated as prior probability distributions and combined with the likelihood function to calibrate nodal water demands. This approach avoids parameter sampling

and provides analytic solutions for the nodal water demands. The results show that this approach can efficiently fuse various prior probability distributions and is expected to improve the WDS model accuracy in view of the increasing use of automatic water metering technology. 2 Materials and Methods 2.1 Bayesian theory-based real-time calibration process In the first stage, the nodal water demand from the previous time step, 𝑡 ―1, is employed to forecast the nodal water demand at the current time step, 𝑡, which obeys normal distribution. Its probability distribution is expressed as follows: 𝑃𝑝𝑟𝑖𝑜𝑟(𝑿𝑡│𝑿𝑡 ― 1) = 𝑁[𝐹(𝑿𝑡 ― 1),𝜎] (1) where 𝑿𝑡 ― 1 and 𝑿𝑡 are the nodal water demand vectors at times 𝑡 ―1 and 𝑡, respectively; 𝐹() is the forecast function; 𝑁() represents the normal distribution; 𝜎 is the standard deviation. In the second stage, prediction and measurement data are fused. The forecasted probability distribution, also called prior distribution, is combined with the likelihood function of measurements to formulate the posterior probability distribution. Thereafter, the nodal water demand is estimated by maximizing the posterior distribution, as follows: 𝑚𝑎𝑥𝑃(𝑿𝑡) ∝ 𝑃𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑(𝒀𝑡│𝑿𝑡)𝑃𝑝𝑟𝑖𝑜𝑟(𝑿𝑡│𝑿𝑡 ― 1) (2) 𝑃𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑(𝒀𝑡│𝑿𝑡) = [(2𝜋)𝑛𝑜𝑏𝑠𝑑𝑒𝑡𝑪𝑫] 𝑪𝑫 = 𝑪𝑻 + 𝑪𝑴

1

―2

[

1

]

exp ― 2(𝒈(𝑿𝑡) ― 𝒀𝑡)𝑇𝑪𝑫―1(𝒈(𝑿𝑡) ― 𝒀𝑡)

(3) (4)

where 𝑃(𝑿𝑡) is the posterior probability distribution for nodal water demand, 𝑿𝑡; 𝒀𝑡 is the field measured value; 𝑛𝑜𝑏𝑠 is the number of measurements; 𝒈() is the nonlinear function (mass continuity and energy conservation equations) that describes the hydraulic steady-state flows and

pressure in a WDS; 𝑃𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑(𝒀𝑡│𝑿𝑡) is the normal distributed likelihood; 𝑪𝑴 is the covariance matrix of the measured value; 𝑪𝑻 is the covariance matrix that represents model structure uncertainty, which is assumed to be negligible in several calibration studies (Du et al., 2015; Kang and Lansey 2011). However, certain studies (e.g., Piller et al. 2011) indicate that the model structure uncertainty can be sufficiently significant to reduce the calibration accuracy. The quantification of the WDS model uncertainty is not only important but also difficult. Temporarily, the model uncertainty is ignored in this paper, and we focus on the uncertainties of field measurements and prior water demand. Equations (1) and (2) provide a general approach for real-time demand estimation; the process is driven by real-time prediction and measurements. When the considerable noise among the measurements exists, the overfitting of inaccurate measured values may result in calibration error. The increasing use of automatic water meters provides a direct way to get the more accurate prior distribution than other ways. To resolve the overfitting problem, the strategy is to utilize the nodal water demand probability distribution estimated from the automatic water meter as the prior distribution and combine it with the measurements to calibrate the nodal water demand. However, this is problematic because the forms of prior probability distribution vary with user types; moreover, fusing these various prior probability distributions simultaneously in Eq. (2) is a crucial problem that has to be resolved. Generally, it is quite difficult to obtain enough prior information from the automatic water meter because only a subset of users in China has installed automatic water meters. The proposed approach is based on Bayesian framework and can also utilize the prior demand estimated by the conventional forecast model (Eq. (1)) or consumer billings when no automatic water meters are installed. For example, based on water usage proportion calculated from months of billing, total real-time water consumption for a DMA can be decomposed and re-distributed to each node as prior water demands.

This study does not focus on the manner of updating the prior distribution (Eq. (1)); instead, it attempts to employ numerical methods to incorporate various prior probability distributions into a single calibration framework (Eq. (2)). It is found that Eq. (2) is the critical step for offline calibration. Therefore, the proposed approach can be adapted for both online and offline demand calibrations. In order to keep the notation uncluttered, subscript 𝑡 is omitted from the variables because the interest is not on updating the variables over time. 2.2 Prior probability distribution The nodal demands that are aggregated from the data of automatic water meters are represented by certain forms of probability distributions. The derivative among different distribution forms is very complicated and hardly to be solved by numerical approach. To avoid the complicated derivative, the nodal water demands with the same probability distribution form are assigned to the same group. For certain nodes without installed automatic water meters, the prior demand distribution should be estimated by the prediction function (Eq. (1)) or estimated from consumer billings. Together with the aggregated nodal demand, these nodal water demands are grouped based on their prior probability distribution types. As shown in Fig. 1, the water demands are divided into several groups based on distribution forms (normal, lognormal, and uniform distributions). Each group consists of a number of nodes, and their demands compose an n-dimensional vector (𝑿𝑖): 𝑿 = [𝑿1,𝑿2…𝑿𝑖…𝑿𝑚]𝑇

（5）

𝑿𝑖 = [𝑿𝑖(1),𝑿𝑖(2)…𝑿𝑖(𝑛)]𝑇

（6）

where 𝑿 is the nodal water demand vector (for all nodes), which is divided into m number of grouped demand vectors, 𝑿𝑖; 𝑿𝑖 is the n-dimension nodal water demand vector, which consists of n number of nodal water demands that obey the same distribution form described by a

multivariate probability density function, 𝜌𝑿𝑖; 𝑿𝑖(𝑗) is the jth nodal water demand in group i; m is the number of demand groups; n is the number of nodes in group i. The nodal water demands that belong to different groups are assumed to be independent of each other. The correlations among the nodal demands are typically relevant to the user types. The users belonging to the same type usually have a similar water using behavior, leading to the same form of probability distribution. The spatial locations may be another factor influencing the correlations among users. However, users with close spatial locations may also be irrelevant if they have different behaviors and weakly correlated, e.g., resident and hospital. Ignoring the correlations between different distributions will not significantly change the overall correlation relationship among the nodal water demands. Thus, the correlations between different distributions are not considered. For the independent random variable, the probability density function is the product of its marginal probability density function. Accordingly, for all nodal water demands, the prior probability distribution, X, has the following form. 𝑚 𝜌 (𝑿𝑖) = 1 𝑿𝑖

（

𝑃𝑝𝑟𝑖𝑜𝑟(𝑿) = ∏𝑖 7）

2.3 Posterior probability distribution By substituting Eqs. (3) and (7) into Eq. (2), the posterior distribution has the following form. 𝑚 𝜌 (𝑿𝑖) = 1 𝑿𝑖

𝑃(𝑿) ∝ ∏𝑖

[(2𝜋)𝑛

𝑑𝑒𝑡𝑪𝑫]

𝑜𝑏𝑠

1

―2

[

1

exp ― 2(𝒈(𝑿) ― 𝒀)𝑇𝑪𝑫―1(𝒈(𝑿) ― 𝒀)

]

(8)

The nodal water demand is calculated by maximizing the posterior distribution, 𝑃(𝑿), or the logarithm of 𝑃(𝑿), as shown in Eq. (9). Therefore, Eq. (10) may be the objective function. 1

𝑚 𝐼𝑛(𝜌𝑿𝑖) =1

𝑚𝑎𝑥: 𝐼𝑛[𝑃(𝑿)] ∝ ― 2[𝒈(𝑿) ― 𝒀]𝑇𝑪𝑫―1[𝒈(𝑿) ― 𝒀] + ∑𝑖

(9)

1

𝑇

𝑚 𝐼𝑛(𝜌𝑿𝑖) =1

𝑚𝑖𝑛:𝐽(𝑿) = 2[𝒈(𝑿) ― 𝒀] 𝑪𝑫―1[𝒈(𝑿) ― 𝒀] ― ∑𝑖

(10)

The right hand side of Eq. (10) consists of two parts: the first part minimizes the difference between the model output, 𝒈(𝑿), and the measured value, 𝒀; the second part, which minimizes ― 𝑚 𝐼𝑛(𝜌𝑿𝑖), =1

∑𝑖

𝑚 𝐼𝑛(𝜌𝑿𝑖). =1

means maximizing the prior distribution, ∑𝑖

In other words, the

calibrated nodal water demand should not only match the measurements but also maximize its prior probability distribution. 2.4 Solution for nonlinear systems The WDS hydraulic model is determined by a set of nonlinear functions of continuity equations for nodes and energy equations for pipes. The nonlinear function, 𝒈(𝑋), is approximated by the first-order Taylor’s expansion to obtain the following: (11)

𝒈(𝑿 + 𝛥𝑿) = 𝒈(𝑿) + 𝑮(𝑿)𝛥𝑿 ∂𝒈

(12)

𝑮(𝑿) = ∂𝑿

where G is the Jacoby matrix for the WDS model that can be obtained by adopting the approach proposed by (Cheng and He, 2011): 𝑇 𝑮 = [𝑨sh 𝑨sq 𝑨sqt]

(13)

where 𝑨sh, 𝑨sq, and 𝑨sqt are the Jacoby matrices related to the monitored nodal pressure, pipe flow, and water station supplement. By substituting Eq. (11) into (10), we can write the following. 𝟏

𝒎

𝑱(𝑿 + 𝜟𝑿) = 𝟐[𝒈(𝑿) ― 𝒀 + 𝑮𝜟𝑿]𝑻𝑪𝑫―𝟏[𝒈(𝑿) ― 𝒀 + 𝑮𝜟𝑿] ― ∑𝒊 = 𝟏𝑰𝒏[𝝆𝒎𝒊(𝑿𝒊 +𝜟𝑿𝒊)]

(14)

The nodal water demand can be calculated by the method adopted by (Kang and Lansey, 2009): ∂𝐽(𝑿 + 𝜟𝑿) ∂𝑿𝑗

∂ 𝑚 [𝐼𝑛𝜌𝑿𝑖(𝑿𝑖 = 1∂𝑿𝑗

= 𝑮𝑇𝑗𝑪𝑫―1[𝒈(𝑿) ― 𝒀 + 𝑮𝑗𝛥𝑿𝑗] ― ∑𝑖

+ 𝛥𝑿𝑖)] = 0

(15)

where 𝑮𝑗 is a submatrix of 𝑮 and represents the sensitivity matrix. As previously mentioned, the nodal water demands that belong to different groups are assumed to be independent of each other. For the independent variables, we can write the following. ∂

∂𝑿𝑗[

(16)

𝐼𝑛𝜌𝑿𝑖(𝑿𝑖 +𝛥𝑿𝑖)] = 0, if i ≠ j

∂ 𝑚 [𝐼𝑛𝜌𝑿𝑖(𝑿𝑖 = 1∂𝑿𝑗

∑𝑖

∂

(17)

+ 𝛥𝑿𝑖)] = ∂𝑿 [𝐼𝑛𝜌𝑿𝑗(𝑿𝑗 +𝛥𝑿𝑗)] 𝑗

Equation (16) indicates that the water demand vector belongs to two different groups that can be linearized independently. This property is important for simultaneously coupling various distributions because it avoids differentiation among the different groups. As can be observed in Eq. (17), the linearization of the prior probability distribution becomes considerably simple and easy to implement. By substituting Eq. (17) into (15), the correction of the nodal water demand, 𝛥 𝑿𝑗,

can be independently calculated from Eq. (18). ∂

𝒅 = 𝑮𝑇𝑗𝑪𝑫―1[𝒈(𝑿) ― 𝒀 + 𝑮𝑗𝛥𝑿𝑗] ― ∂𝑿𝑗[𝐼𝑛𝜌𝑿𝑗(𝑿𝑗 +𝛥𝑿𝑗)] = 0 (18) where, 𝒅 is the gradient direction; 𝒅𝑻𝛥𝑿𝑗 is used to ensure that the demand adjustment 𝛥𝑿𝑗 is the correct direction to reduce the objective function. If 𝒅𝑻𝛥𝑿𝑗 > 0, 𝛥𝑿𝑗 = ― 𝛥𝑿𝑗. The optimal solution can be achieved by iteratively updating Eq. (19): 𝑿𝑘𝑗

+1

= 𝑿𝑘𝑗 + 𝜀𝛥𝑿𝑘𝑗

(19)

where 𝜀 is the step size and 𝜀 < 1. Generally 𝜀 = 0.5 is recommended; 𝛥𝑿𝑘𝑗 is the demand corrector for kth iteration. Equations (18) and (19) provide a framework for estimating the consumer demand with the various prior stochastic characteristics considered simultaneously. However, the derivative with respect to the nodal water demand,

∂

∂𝑿𝑗[

𝐼𝑛𝜌𝑿𝑗(𝑿𝑗 +𝛥𝑿𝑗)],

is problem-

dependent, and the derivation process varies with the forms of the prior probability distribution, 𝜌𝑿𝑗. The exponential family of probability distributions has been widely used to model the

stochastic nodal water demand. The general derivation method of the exponential family of probability distributions is discussed in the following section. 2.5 Application of exponential family of probability distributions The exponential distribution family, given 𝜼, is defined as a distribution set with the following form (Bishop, 2006): 𝜌(𝑿) = ℎ(𝑿)𝑡(𝜼)exp [𝜼𝑇𝒖(𝑿)]

(20)

where 𝜼 is called the natural parameter of the distribution, and 𝒖(𝑿) is a function with 𝑿 as argument. Function 𝑡(𝜼) is the coefficient that ensures that the distribution is normalized. Therefore, the logarithm of 𝜌(𝑿) has the following form. 𝐼𝑛[𝜌(𝑿)] = 𝐼𝑛[ℎ(𝑿)] +𝐼𝑛[𝑡(𝜼)] + 𝜼𝑇𝒖(𝑿)

(21)

The derivative of 𝐼𝑛[𝜌(𝑿)] with respect to 𝑿 has the following form. ∂

1 ∂ℎ(𝑿 + 𝛥𝑿) ∂𝑿

∂𝑿[𝐼𝑛[𝜌(𝑿 + 𝛥𝑿)] = ℎ(𝑿)

+

∂[𝜼𝑇𝒖(𝑿 + 𝛥𝑿)]

(22)

∂𝑿

Equation (22) yields the general form of linearization for the exponential distribution family; it is easy to implement because derivatives

∂ℎ(𝑿 + 𝛥𝑿) ∂𝑿

and

∂[𝜼𝑇𝒖(𝑿 + 𝛥𝑿)] ∂𝑿

are simple. In particular,

the linearization of the three commonly used distributions (normal, lognormal, and uniform distributions) is discussed in S-1. The other forms of probability distributions can be linearized in a similar manner. 3 Results The proposed approach is applied to a hypothetical simple network and a mid-scale network. For the simple network, we focus on the modeling attributes for fusing various prior probability distributions together; for the mid-scale network, it is designed to show the approach’s performance in avoiding the overfitting of measurement noise. The estimation of various prior demand distributions from the automatic water meter or forecast model necessitates intensive

work and is beyond the scope of this study. In the two foregoing cases, it is assumed that the prior water demand has been pre-acquired and is available. 3.1 Case Study 1: Hypothetical Simple Network The hypothetical simple network consists of 8 nodes, 11 pipes, and 1 reservoir (Fig. 2). The nodes are at a 0-m elevation, and the reservoir head is at 50 m. The pressure at nodes N3 and N7 are selected as the measured values, and the prior demands are assumed to be independent of each other. These nodal demands are divided into three groups: normal, lognormal, and uniform distributions. The stochastic parameters of the prior water demand and measurements are listed in Tables 1 and 2. This case illustrates each step of the calibration process. Measurements noises are assumed to be normal distribution, with its mean value of zero and its standard deviation of σ by several studies. Existing studies show that σ of pressure measurements ranges from 0.1 m to 1 m (Do et al. 2017; Kozelj et al. 2014; Zhang et al. 2018a). In this case, the measured value is generated by the sampling values from the measurement probability distribution (Table 2), with σ = 0.4 m. The detailed calibration process for simultaneously fusing various prior probability distributions is discussed in S-2. To investigate the impact of prior distribution on calibration, the above process is repeated for 15 000 times by sampling 15 000 different values from the measurement probability distribution. The comparison between calibrated and prior demand distributions is shown in Fig. 3. For the normal (N1 and N4) and lognormal (N2, N5 and N7) distribution demands, the mode (value with the largest probability density) for the calibrated and prior probability distributions are practically the same. The range of calibrated water demand is concentrated in the region with a high prior probability density. This observation definitely suggests that values with a high prior

probability density (typically those approach the prior mode value) have a high probability of acceptance. The uniform distribution can strictly confine the nodal water demands (N3, N6, and N8) within the feasible domain determined by the upper and lower boundaries. Nevertheless, the calibrated distribution of N6 is significantly different from those of N3 and N8. For N6, the calibrated value appears to be normally distributed around the median value of the prior uniform distribution. The constraining effect of the prior distribution of N6 is coincidently lower than that of the measurements applied. The nodal demand (N6) is mainly determined by the measurements, and the prior distribution performs a minor function in the demand calibration process. In a Bayesian framework, a posteriori depends on the “competing forces” of the likelihood (driven by the measurements) and the priori (assumed demand distributions). The prior distribution attempts to keep the parameter approach to its mode value, whereas the measurements tend to make an adjustment, leading to deviation from the prior mode value. For the uniform distribution, when the parameters are within the feasible domain (far away from the boundaries), they have the same prior probability density. Thus, the nodal demand can be adjusted freely to meet the monitoring values. However, once the parameters approach to the boundaries, the prior probability density will decrease rapidly, leading to the decrease of the posterior probability density. Therefore, the prior distribution confinement will keep the value within the feasible domain, e.g., the calibrated demand distributions of N3 and N8 are truncated by their upper and lower boundaries, respectively. The variance of the calibrated nodal demand is smaller than that of the prior demand distribution of all nodes (Fig. 3). This indicates that the nodal demand uncertainty is reduced by the measurements. Similar to the nodal demand, the variance of the calibrated nodal pressure is also smaller than that of the measurements (Fig. 4). For the nodes with measurements (N3 and N7), the calibrated distributions are practically the same as the measurement distribution with a standard deviation of std = 0.4 m. This indicates that the calculated pressure in the measured node

is mainly governed by the field measurement; this makes the model outputs agree with the measurements. The standard deviations of the calibrated pressure at the other nodes, except for N6, are less than 0.4 m (i.e., standard deviation of measurements), which indicates that the calibrated nodal pressure uncertainty is reduced. The synergistic effect of measurements and prior information in this method can efficiently reduce the model uncertainty. As for N6, as previously mentioned, its water demand is mainly determined by the measurements. The calibrated pressure uncertainty shows a certain degree of deterioration caused by the measurement noise. The prior probability distribution has the ability to keep the nodal water demand adjusted within the domain that has a higher prior probability density. However, this ability is affected by the certainty of prior values and measurements. To further investigate the competitive relationship between prior values and measurements, prior distributions with standard deviations in the range 0.05μ–3.5μ, where μ is the prior mode value of the prior, are adapted to calibrate the nodal water demand. As shown in Fig. 5, with the increase in prior variance, the average deviation between the calibrated nodal water demands and its prior mode values increases, whereas the deviation between the calibrated nodal pressures and measurements decreases. 3.2 Case Study 2: Mid-scale Network The mid-scale network, as shown in Fig. 6, has been investigated in public studies (Bentley Systems, 2013; Zhang et al., 2016). This network consists of 375 junctions, 469 pipes, 3 pumps, and 1 reservoir. The location and number of pressure sensors are shown in Fig. 6. The nodes are divided into two groups and obey the normal and lognormal distributions. The water demands in all nodes are known and treated as real water demands, which fluctuate over time according to the demand pattern (Fig. A1). Real water demands are employed to calculate the nodal pressure, which is treated as the true pressure value. The calibrated results and true values are compared and thereafter used to evaluate the calibration accuracy.

Random measurement error, which is normally distributed at N (0,σ), is added to the true nodal pressure measurements as white noise; the standard deviation is σ = 0.5 𝑚. The prior demand distribution is given as follows: the prior mode value is provided by adding a uniformly distributed random value with a boundary of 0–200% of the real water demand, and the standard deviation is 150% of the mode value. For comparison, the modeling results that are estimated by the SVD calibration method (Cheng and He, 2011), in which the nodal demand is only determined by measurements without using prior information, are presented. In addition, directly using the mode value of the synthetic prior water demand as the model input, the model simulating results by are also examined. The calibrated pressure always approaches the real pressure in the 10 pressure measurements (Fig. 7). The deviation between calibrated pressure and real value is less than that between the measured value (or prior value) and real value. The calibration accuracy is improved compared with techniques that merely use measurements or prior information. This method also exhibits good robustness when large errors/uncertainties exist in the measurements and/or prior probabilities. For example, although considerable noise exists in both measured and prior values for sensors 1 and 4 at t=17 h, the calibrated value still closely approaches the real value (Fig. 7). For sensor 5 at t=8 h and sensor 7 at t=17 h, both the prior and measured values deviate from the real value in the same direction; however, the calibrated pressure remains close to the real value (Fig. 7). This is because the nodal water demands are determined by the measurements and prior probability distributions combined. Even though the calibrated pressure at the node where sensor installed is the most sensitive to the pressure sensor at that node, still less than the sum of the other sensors and priors. Thus the wrong adjustment caused by the pressure noise is corrected by other sensors and priors. The calibration accuracy is not considerably reduced even if a small number of measurements or its prior probability has errors. This highlights the synergistic effect of measurements and/or prior probability distributions on demand calibration.

In most cases, the calibrated pressure is within the area between prior and measured values (Fig. 7); this indicates a compromise between the prior and measured values. The measurements adjust the nodal water demand by minimizing the difference between the calibrated and measured values. However, such a demand adjustment is confined by the prior probability to a certain extent, which typically makes the nodal water demand approach the prior mode value. The cumulative probability distribution of the nodal demand absolute errors for the 375 nodes over 24 h is shown in Fig. 8. The relative errors are not presented here as the relative error can be extremely large for the node with a small demand and may provide misleading information. As shown in Fig.8, for most time steps, the errors of the calibrated nodal water demand are less than 5 L/s for 80% and 10 L/s for 90% of the nodes; such are also the errors of the prior mode value (Fig. 8). The error of the calibrated nodal demand is close to but slightly smaller than that of the prior mode value; this indicates that the nodal water demand is slightly adjusted within the higher prior probability domain. It is interesting to note that although the nodal demand compared with the prior mode value is slightly adjusted, the calibrated nodal pressure performs significantly better than that of the prior mode in the cumulative probability distribution of the nodal pressure error, as shown in Fig. 9. The calibrated nodal pressure error is less than 0.1 m for approximately 75% of the nodes, whereas that of the calculated pressure of the prior demand (prior mode value) is greater than 0.1 m for approximate 90% of the nodes for most time steps. The calibrated nodal pressure error is evidently smaller than the prior error (Fig. 9). This significant improvement in pressure accuracy is attributed to the nodal demand adjustment made according to the pressure measurements, although such an adjustment is small. This indicates that the prior demand may be more competitive to make the parameter approach the prior mode value.

For the demand calibration, the ratio value

𝑚 𝑛

affects the "competing forces" between the

likelihood and prior, where 𝑚 is the number of measurements and 𝑛 is the number of nodal water demands. Given the measurement residual (∆𝑌𝑖) for sensor i and the demand adjustment (∆𝑋𝑗) for node j, ∆𝑋𝑗 ∝

𝑚 ∂𝑋𝑗

, and ∆𝑌𝑖. In real network, because 𝑚 ≪ n, and 𝑛 , ∂𝑌𝑖

∂𝑋𝑗 ∂𝑌𝑖

decreases with

increase of network scale. Therefore, the demand adjustments made by the measurements decreases with increase of network scale. Comparatively, for the large network with 𝑚 ≪ n, the demand adjusted by the measurements may be slight and the prior demand is more competitive to make the parameter approach the prior mode value. This highlights the importance of using an accuracy prior information. Figures 7, 8, and 9 show the calibrated results by the SVD method (Cheng and He, 2011). The SVD calibrated values perfectly match the measured values (Fig. 7). Unfortunately, the matching at measurements may be considerably misleading for real applications because a calibration that is based on limited field measurements is an ill-conditioned problem. Without nodal demand confinement, the nodal water demands can be freely adjusted to match the calibrated value with the measured value. Moreover, because of the measurement noise, this matching usually means overfitting; in this case, the calibrated value matches with the inaccurate measurements, but actually deviates from the real value. The measurement noise propagates to the calibrated model parameters (i.e., water demands) and introduces considerable calibration error. As a result, the calibrated nodal demand errors for the SVD method may be considerable (Fig. 8); some nodal water demands are even negative, which is an infeasible condition in practice. Consequently, the calibrated pressure errors at the nodes without installed measurement sensors are also large, some of which are more than 2 m (Fig. 9), which far exceeds the measurement noise. Compared with the SVD method, the proposed method is an efficient approach to avoid the overfitting of measurement noise by using prior distribution; this is regarded as a compromise between matching measurements and maintaining high prior

probability. Additionally, the parameter-grouping methods (Kang and Lansey, 2009; Zhang et al., 2018b), which transform the ill-conditioned problem into a well-conditioned problem, remain unable to eliminate the overfitting of measurement noise and improper parameter grouping problem. In contrast, the proposed method has no grouping dilemma and has the strength of flexibly employing data and other information. 3.3 Discussion This paper presents a flexible nodal demand calibration framework, which allows the use of demand information estimated from different methods, such as automatic water meter, demand forecast model, or consumer billings, to improve the WDS hydraulic model accuracy. Based on the two cases presented above, a discussion on the features of the approach is elaborated here. This approach can fuse various prior information as prior distribution to calibrate the nodal water demand. In this technique, the synergistic effect of measurements and prior probability distributions efficiently improves the calibration accuracy. To a certain extent, this approach shows good robustness when uncertainty exists in the measurements or prior demand. The nodal water demand is not over-adjusted by the measurement noise; this may be attributed to the use of prior distribution. Consequently, the overfitting of measurements is avoided. Moreover, the nodal water demand is adjusted in the domain with a high prior probability. This means that the model accuracy can be considerably improved when a more precise prior information, such as the prior demand aggregated from the automatic water meter, is used. The calibration results obtained using the proposed method are better than those of the conventional SVD method, which does not employ any prior information. The calibration accuracy of a Bayesian-based approach depends on the accuracy of initially assumed prior probability density functions (PDFs) and measurements used. Even if a small number of measurements or demand PDFs has errors, the calibration accuracy is not significantly reduced. The foregoing highlights the synergistic effect of measurements and demand PDFs on demand calibration. On the other hand, the negative

effects of erroneous/inaccurate initial demand PDFs on demand calibration exist; erroneous initial demand PDFs can make calibrated results even worse compared with those without any prior information. If no credible estimates of nodal demands are available, then allowing them to vary within a larger range is recommended. To describe the stochastic characteristics of water demand, a large number of probability density functions, such as normal distribution (Filion et al., 2007; Xu and Goulter, 1998), lognormal distribution (Creaco et al., 2016; Surendran et al., 2005; Tricarico et al., 2007), Poisson distribution (Blokker et al. 2011), and exponential distribution (Yang and Boccelli, 2013), have been used. These distributions are typically classified as an exponential family; they are convenient, widely used, and contains most of the standard discrete and continuous distributions for practical modeling. Other distribution forms can be fitted or approximated by exponential family distributions. The developed approach provides the general derivation method of the exponential family of probability distributions by explicit analytic solutions (S-1). Based on these analytic solutions, the numerical iterative equations for the nodal demand are given as Eqs. (18) and (19). This approach is expected to be more computationally efficient than other sampling-based Bayesian approaches (Kapelan et al., 2007; Do et al., 2017; Xie et al., 2017; Qin and Boccelli 2019). Sampling-based approaches are usually time-consuming because their performances depend on the number of samples and can only converge after several iterations. Each sample should be evaluated independently by its posterior probability density; this involves the simulation of the hydraulic model, in which the time complexity is 𝑂(𝑁3), where N is the number of nodal water demands. Therefore, the time complexity for each iteration is 𝑀 × 𝑂(𝑁3), where M is the number of samples. In contrast, the proposed numerical approach considers only one hydraulic model simulation for each iteration in the time complexity 𝑂(𝑁3). In fact, the bottleneck of computational efficiency for the proposed approach lies in solving Eq. (18), which involves the inversion of Hessian matrix (Eqs. (S-8), (S-20), and (S-33)) for each grouped demand. The Hessian matrix is an 𝑛 × n matrix, and its time complexity for its inversion is 𝑂(𝑛3)

; hence, the total time complexity for each iteration is 𝑚 × 𝑂(𝑛3), where m is the number of groups, n is the number of nodal demands in the group, and 𝑚 × n ≈ 𝑁. Usually, 𝑚 × 𝑂(𝑛3) ≪ 𝑀 × 𝑂(𝑁3) because 𝑚 ≪ 𝑀 and n ≪ N. Thus, this approach will be more computationally efficient than other sampling-based approaches. This approach is considerably flexible in fusing various prior information. For nodes installed with automatic water meters, their prior demands can be fitted as an exponential family distribution. Otherwise, prior distributions are calculated by the forecast model, and the approach will be similar with (or the same as) the existing two-stage Bayesian calibration approach (Kang and Lansey, 2009; Zhou et al., 2018). The calibration accuracy of nodal demands relies on the accurate prior demand distribution (PDF) obtained from automatic meters or consumers billings. However, acquiring these accurate PDFs are still difficult for all node demands because only a subset of users in China has installed automatic water meters. The flexible attribute in fusing various prior distribution highlights its usefulness in real applications. In real applications, the forms of prior probability distributions rely on their estimation methods. For the automatic water meter, the distribution forms are calculated by stochastic demand models and aggregation methods. For the forecast model, the prior demands are typically assumed as normally distributed (Eq. (1)). For the prior demand estimated by information that is not exactly precise, such as customer billing or population size served, the uniform distribution is recommended. The proposed methodology is applied to two artificial examples without real measurements. Real nodal demands are assumed, and real pressures are generated by model simulation. Thereafter, artificial pressure measurements are created by adding white noise to model-generated pressures; the assumed prior demand PDFs are given according to different PDF types. The advantage of using artificially designed examples is that the influence of other factors (e.g., presence of various errors other than measurement noise and their impact on results) is eliminated. For real applications, the model uncertainty (e.g., node elevation and valve states) and the data from SCADA should be thoroughly checked to eliminate their errors/uncertainty. The

application of the proposed methodology should be further extended to resolve more complex problems associated with model calibration. 4 Conclusions The accurate calibration of nodal demand is critical to WDS modeling and is thus frequently a critical step towards WDS control and operations. A numerical method that can simultaneously fuse various prior probability distributions and measurements in one calibration framework is presented in this paper. The flexible calibration framework allows the use of automatic water meter information and merges it with other conventional information, such as field measurements or consumer billings, to improve the WDS hydraulic model accuracy. The developed method eliminates the normal assumption of prior distribution and avoids the time-consuming sampling from prior or posterior probability distributions. This method shows a compromise between matching measurements and maintaining high prior probability. The two case studies demonstrate that the nodal water demand is adjusted in the domain with high prior probability; this allows the calibrated nodal demand to retain the prior stochastic nature to a considerable extent as possible. The variance of calibrated nodal demand is smaller than that of prior distribution. This highlights the contribution made by measurements on the improvement of calibration accuracy and the reduction of model uncertainty. The method performs better than the calibration that merely utilizes measurements or prior information; it is also robust when significant error or uncertainty exists in the prior demand distribution or measurements. Compared with existing methods, the proposed method avoids the overfitting of measurement noise and improper parameter grouping problem; it is flexible in the use of data and information. The confidence of prior demand distribution can greatly influence the calibration accuracy of nodal demands. Acquiring the accurate prior demand are still difficult for all node demands. The method to get accurate prior demand should be further explored. The stochastic nature of nodal water demand is described as a probability distribution. In particular, the exponential

family linearization of prior distribution has been well-discussed in this paper. However, prior distribution can be beyond the exponential family probability distribution, and the strategies to handle other forms of prior probability distribution should be further explored in future studies. The two case studies used synthetic data, whereas in real measurements, various errors other than measurement noise may exist. Therefore, in a future work, the approach should be further validated by real data. Acknowledgments, Samples, and Data This work was supported by the National Key Research and Development Program of China (No. 2016YFC0400600); the National Science and Technology Major Projects for Water Pollution Control and Treatment (2017ZX07502003-05); the Science and Technology Program of Zhejiang Province (Nos.2017C33174 and 2015C33007); the National Natural Science Foundation of China (No. 51761145022); and the Fundamental Research Funds for the Central Universities (No. 2019FZA4019). The EPANET input file of the model and data in the two cases may be downloaded from the link indicated in S-4. References Arandia-Perez, E., Uber, J. G., Boccelli, D. L., Janke, R., Hartman, D., & Lee, Y. (2014). Modeling automatic meter reading water demands as nonhomogeneous point processes. Journal of Water Resources Planning and Management, 140(1), 55-64. Bentley Systems. (2013). WaterGEMS V8i users’ manual, Watertown, CT Bishop, C. M. (2006). Pattern recognition and machine learning. Information Science and Statistics. (4), 049901.

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Figure 1 Aggregation of water demand

Figure 2 Schematic of hypothetical network

Figure 3 Comparison between prior and calibrated distributions for nodal water demand

Figure 4 Calibrated distribution for nodal pressure

Figure 5 Deviation from the calibrated results to mode of prior values and measurements

Figure 6 Network layout

Figure 7 Comparison of modeling results in 10 pressure sensors

Figure 8 Cumulative probability of absolute errors of nodal demand

Figure 9 Cumulative probability of absolute errors of nodal pressure

Table 1 Parameters of prior nodal water demand probability distribution (L/s) Node

Distribution forms

mode

N1

Normal

0.87

1.04

N2

Lognormal

6.24

7.49

N3

Uniform

17.64

14.11, 21.17*

N4

Normal

4.25

5.10

N5

Lognormal

12.92

15.50

N6

Uniform

7.05

5.64, 8.46*

N7

Lognormal

8.66

10.39

N8

Uniform

11.90

9.52, 14.28*

* boundaries for uniform probability distribution

Standard deviation (std)

Table 2 Parameters of pressure measurement probability distribution (m) Measured nodes

Distribution form

mode

Standard deviation (std)

N3

Normal

34.25

0.4

N7

Normal

27.54

0.4



The approach fuses prior information and measurements for demand calibration



The synergistic effect of prior and measurements improves the calibration accuracy



The approach can avoid over-fitting on the measurement noise