Mathematical Problems in Engineering

Volume 2015, Article ID 350328, 9 pages

http://dx.doi.org/10.1155/2015/350328

## A Novel Statistical Model for Water Age Estimation in Water Distribution Networks

College of Architecture and Civil Engineering, Zhejiang University, 866 Yuhantan Street, Hangzhou 310058, China

Received 3 May 2015; Revised 9 August 2015; Accepted 18 August 2015

Academic Editor: Jian Guo Zhou

Copyright © 2015 Wei-ping Cheng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The water retention time in the water distribution network is an important indicator for water quality. The water age fluctuates with the system demand. The residual chlorine concentration varies with the water age. In general, the concentration of residual chlorine is linearly dependent on the water demand. A novel statistical model using monitoring data of residual chlorine to estimate the nodal water age in water distribution networks is put forward in the present paper. A simplified two-step procedure is proposed to solve this statistical model. It is verified by two virtual systems and a practical application to analyze the water distribution system of Hangzhou city, China. The results agree well with that from EPANET. The model provides a low-cost and reliable solution to evaluate the water retention time.

#### 1. Introduction

Water quality will deteriorate with the increment of retention time in the water distribution system, leading to malfunctions such as disinfection by-product formation, disinfectant decay, corrosion, taste, and odor. Water age is very important for the water quality of water distribution system. The water age primarily depends on the water distribution system design and its demands. Although Brandt et al. [1] reviewed some tools to estimate the retention time and several examples presented, they conclude that there are no low-cost, effective, and reliable ways to estimate it in any circumstances. In some circumstances, these tools may be appropriate, but this is not always the case.

There are two types of tools to estimate the water age: tracer studies and numerical models. Tracer studies involve injecting chemical into the water distribution system for a fixed period, and sensors are set up at downstream nodes to determine the duration before the water containing the chemicals passes the monitoring stations. This method has been applied to calculate the water age throughout the water distribution system and calibrate the water quality and hydraulic models [2–4]. The tracer study is useful in validating hydraulic and water quality models. However, it is seldom applied in water distribution networks for its disadvantage. Some reports [1, 5] have shown its disadvantages, that is, the tracer chemical stability, continuous regulatory compliance, customer perceptions, lack of studies on the larger distribution systems, and high operational cost. Numerical models give the other way to estimate the water age in water distribution systems. The steady traveling time models were proposed by Males et al. [6]. These models were subsequently extended to dynamical representations that determine varying water age throughout the distribution systems [7]. A simplified model of water age in tanks and reservoirs was developed in the early 1990s [8]. Many hydraulic network modeling packages incorporate certain algorithms to calculate the water age at any node in the network [9–14]. Water quality is directly related to water distribution system operation conditions. Thus, a careful hydraulic calibration is necessary under varying demand assumed for the accurate estimation of water age. Unfortunately, numerical models may have some limitations in the capability of accurately predicting the water age for the following reasons [1, 5]: (1) skeletonization: the skeletonization is necessary if the water distribution system contains more pipe segments than the model can handle, and in almost all cases, the skeletonization is inevitable. The effect of skeletonization on the accuracy of water age estimation differs from system to system; (2) insufficient calibration: in most cases, the roughness cannot be estimated accurately. If the overall demand is miscalculated, it will result in more or less source and reservoir operation than what actually occurs. Misestimating the demand allocation might lead to error flow direction; (3) water storage tanks: tanks are modeled as completely mixed reactors in most models, which will lead to misestimation of the water age.

You et al. [15] have shown that the residual chlorine decay ratio per unit length is different at different time. For example, it is 0.175 mg/(LKm) at peak-demand time, that is, 0.459 mg/(LKm) at the minimum-demand time in Shenzhen city. You et al. [15] said that the retention time is one of the most important elements of the residual chlorine fluctuation for long distance water distribution systems. Our monitoring data in Hangzhou city is the same as theirs. Some water distribution systems have SCADA (supervisory control and data acquisition). The concentration of residual chlorine in the water system can be monitored by SCADA. In the present paper, a novel statistic model is proposed to estimate the water age in water distribution systems according to the monitoring data serials of the residual chlorine concentration from SCADA. The model is discussed theoretically and numerically. And it is also applied to predict the water age of water distribution system in Hangzhou city. The statistic model is in good agreement with EPANET 2.0 numerical results.

#### 2. Governing Equations for Water Quality

A water distribution system consists of pipes, pumps, valves, fittings, and storage facilities that are used to convey water from the source to consumers. The dissolved substance travels along the pipe with the same average velocity as the carrier fluid while reacting (either growing or decaying) at certain rates. The equations governing the water quality are based on the principle of conservation of mass coupled with reaction kinetics [10–12]. Usually, the role of longitudinal dispersion is neglectable. The conservation of mass during transport within a pipe is described by the classical one-dimensional advection-reaction equation. The advection transport within a pipe is represented aswhere is the concentration (mass/volume) in pipe as a function of distance and time , is the flow velocity (m/s) in pipe , and denotes the rate of reaction (mass/volume/time) as a function of concentration.

When junctions receive inflow from two or more pipes, it is assumed that the complete mixing of fluid is accomplished simultaneously. Thus, the concentration of a substance in water when water leaves the junction is simply the flow-weighted sum of the concentrations from the inflowing pipes. For a specific node , the concentration is expressed as follows:where is the set of pipes with flow into node , is the flow (m^{3}/s) in pipe , is the external source flow entering the network at node , and is the concentration of the external flow entering at node ; the notation denotes the concentration at the start of node , while is the concentration of the tail of pipe at node .

#### 3. Water Age Estimation Model

Although more complicated models are available for modeling the decay of chlorine (e.g., [16]), the first-order decay model is popular for its convenience in implementation. The rate of reaction is as follows:where is the decay coefficient in pipe .

Traveling along with the water trace line, (1) can be rewritten as follows:

The solution of (4) iswhere is the concentration (mass/volume) in the source. denotes the decay coefficient on pipe .

Water travels from the water station to the consumer through many pipes. In most skeletal pipes, the influence of mixing is neglected. The solution of (5) at node is as follows:where denotes the concentration at node and is number of pipes through which water travels from the source to node .

The water age at node is . The average decay coefficient can be expressed as . Consequently, (6) is simplified as

The residual chlorine concentration varies with the water age. The variation of residual chlorine can show the fluctuations of water age. The first-order expansion of (7) near the average water age is as follows:where is the concentration at node at time , is the average concentration at node , is the water age of node at time and is the distance from the source to the node . denotes the average velocity from the source to node at time ; is the average velocity from the source to node . The relation between and is .

Assume that the average velocity is linearly dependent on the water demand in water distribution systems. Thus, where denotes the average water demand of the whole water distribution system during the water age of node at time and the formula is . is the mean value of , which can be expressed as .

The above derivation shows that the concentration of residual chlorine at node is linearly dependent on the average demand during the water retention time, which means the correlation coefficient between the water age and the average demand is close to unity. The length from the source to the monitoring point is const; so one gets . The average velocity is linearly dependent on the water demand; the above function can be rewritten as . Equation (9) can help us to estimate the water age at monitoring points.

Assuming that the standard deviations of and are constant and , respectively, a statistical model is built to estimate the water age at monitoring node according to the monitoring data from SCADA:where and . The objective function of the above model is the correlation coefficient between the residual chlorine and the water demand.

##### 3.1. Solution Procedure

Equation (10) gives out the model to estimate the water age at monitoring nodes. It is not easy to solve it directly, because this model involves too many variables and the constraint conditions are difficult to deal with. In this section, a solution procedure is put forward to solve this optimal problem. This solution procedure consists of two steps.

*Step 1. *Assuming , the objective function in (10) is expressed as follows:where and .

Equation (11) is an unconstrained optimization problem which involves only one variable, the average water age at monitor node . It is easy to estimate the average water age .

*Step 2. *Because the distance from the source to the monitor node is a constant, , the water age at node at any time can be calculated according to the following:Since the monitored data from SCADA is discrete, the above model is transformed to a discrete model. The sampling period is ; the water age at the monitoring node is , where is the number of sampling periods. The objective function of (11) can be expressed as According to (13), the average water age at node is . Equation (12) can be transformed to

#### 4. Verification of the Model

In order to verify the proposed model, two virtual water distribution systems, namely, the simplest system consisting of one pipeline and a complicated multisource water system, are modeled. They are also modeled using* EPANET 2.0* [14] for the purpose of comparisons.

*Scenario 1 (one pipeline system). *The simplest system consisting of one pipe is shown in Figure 1. The length of pipe is 10 km. Two demand patterns (Figure 2) are tested. Because the concentration of residual chlorine at the source fluctuates in the real conditions, white noise of the concentration of residual chlorine at levels of 5% or 20% is mixed in the simulations.

The correlation coefficient between the residual chlorine concentration and the mean demand is shown in Figures 3 and 4. The maximum objective function (correlation coefficient) is very close to 1.0. In the first demand pattern, the maximum objective function is 0.96, the average water age is 4.46 hours, and the maximum relation error is 3%. In pattern 2, the maximum objective function and the average water age are 0.97 and 4.7 h, respectively. Figures 5 and 6 show the water age modeled by the proposed model and by the EPANET 2.0, respectively. The maximum error is less than 0.3 hours for the first pattern and 0.5 hours for the second one, respectively. Figures 5 and 6 indicate that the statistic model agrees well with EPANET 2.0. The white noise has little influence on the results.