The hydraulic analysis of water distribution networks (WDNs) is divided into two approaches: namely, a demand-driven analysis (DDA) and a pressure-driven analysis (PDA). In the DDA, the basic assumption is that the nodal demand is fully supplied irrespective of the nodal pressure, which is mainly suitable for normal operating conditions. However, in abnormal conditions, such as pipe failures or unexpected increase in demand, the DDA approach may cause unrealistic results, such as negative pressure. To address the shortcomings of DDA, PDA has been considered in a number of studies. For PDA, however, the head-outflow relation (HOR) should be given, which is known to contain a high degree of uncertainty. Here, the DDA-based simulator, EPANET2 was modified to develop a PDA model simulating pressure deficient conditions and a Monte Carlo simulation (MCS) was performed to consider the quantitative uncertainty in HOR. The developed PDA model was applied to two networks (a well-known benchmark system and a real-life WDN) and the results showed that the proposed model is superior to other reported models when dealing with negative pressure under abnormal conditions. In addition, the MCS-based sensitivity analysis presents the ranges of pressure and available discharge, quantifying service reliability of water networks.

## INTRODUCTION

### Water distribution network analysis

Water distribution networks (WDNs) are essential infrastructures for human life. The main purpose of WDNs is to supply an adequate quantity of fresh water with sufficient pressure. The hydraulic analysis models for WDNs are useful tools for checking water quantity, pressure, and quality of the systems. A WDN analysis model is developed by using a link–node formulation governed by two conservation laws: mass balance at the nodes, and energy conservation in the hydraulic loops of the systems.

As shown in Table 1, a WDN analysis can be classified into two approaches: one is a demand-driven analysis (DDA), which mainly applies to normal operating conditions; and the other is a pressure-driven analysis (PDA), which can be applied to abnormal conditions. The DDA is based on the assumption that nodal demands are fully supplied regardless of the pressure at nodes. From this assumption, the DDA models often produce unrealistic results (e.g., negative nodal pressure) while nodal demands are fully supplied. In actual situations, pressure-dependent demand (PDD) is more realistic, and the PDA provides better results under abnormal conditions. For PDA, however, the head-outflow relation (HOR) at nodes must be provided beforehand to determine the nodal pressure and discharge simultaneously. That is, if the nodal head drops below the desirable head, the nodal demand can only be supplied partially, and if the nodal head further reduces to minimum required head, no demand can be supplied. However, the relationship between the nodal head and outflow could be system specific, and no generally accepted relation is known. Although a number of HORs have been proposed, all of the relations include uncertainty in the parameters.

Model . | DDA . | PDA . |
---|---|---|

Application | Normal operating condition | Abnormal operating condition (pipe breakage, pump failure, etc.) |

Reliability for abnormal condition analysis | Low | High |

Assumption | Required demand is always satisfied at demand junction | Available demand is dependent on nodal pressure head |

Weakness | Produces unrealistic results, such as negative pressure, while satisfying the required demand | HOR of demand nodes should be provided, which contains uncertainty |

Model . | DDA . | PDA . |
---|---|---|

Application | Normal operating condition | Abnormal operating condition (pipe breakage, pump failure, etc.) |

Reliability for abnormal condition analysis | Low | High |

Assumption | Required demand is always satisfied at demand junction | Available demand is dependent on nodal pressure head |

Weakness | Produces unrealistic results, such as negative pressure, while satisfying the required demand | HOR of demand nodes should be provided, which contains uncertainty |

### Pressure-driven analysis

There are a number of methods for obtaining the available nodal flow for PDA in the literature. These methods can generally be divided into two categories (Siew & Tanyimboh 2012).

The first category consists of methods that incorporate DDA. Ang & Jowitt (2006) proposed an algorithm that progressively adds artificial reservoirs to pressure deficient nodes. The approach used is similar to that of Bhave (1991). Rossman (2007) and Yoon *et al.* (2012) calculated PDDs by using emitters. Ozger (2003) described a semi-pressure-driven framework. The technique called the demand-driven-available-demand-fraction (DD-ADF) method was used for more realistic reliability assessment of WDNs compared with pure DDA. Kalungi & Tanyimboh (2003) considered zero and partial flow with a DDA model. Gupta & Bhave (1996) developed an iterative method that revised available nodal demands using DDA runs. All of the methods in the first category involve the iterative use of DDA, and have low applicability and reliability for actual WDN analysis.

The methods in the second category use an HOR to determine PDDs. HORs determined in previous studies include those by Tanyimboh & Templeman (2010), Udo & Ozawa (2001), Germanopoulos (1985), Gupta & Bhave (1996), Fujiwara & Ganesharajah (1993), Cullinane *et al.* (1992), and Wagner *et al.* (1988). In these methods, the nodal demand is satisfied fully only if the nodal pressure head is equal to or greater than the desired pressure head. Tanyimboh & Templeman (2010) developed a robust algorithm based on the Newton–Raphson method for analysis of WDN with pumps and valves. In Giustolisi *et al.* (2008a), the extended period simulation with the PDA is used to calculate the hydraulic performance of a WDN with a single reservoir. Giustolisi *et al.* (2008b) added the Wagner *et al.* (1988) equation to the global gradient algorithm. However, the performance of the PDA simulator for analyzing networks with pumps and valves was not included. Wu *et al.* (2009) revised the global gradient algorithm to calculate PDDs. Baek *et al.* (2010) proposed a PDA model that included the harmony search algorithm and applied it to evaluate the reliability of a WDN. Tanyimboh & Templeman (2010) proposed a differentiable HOR to solve discontinuous problems in previous studies and by using differentiable HOR, the convergence of PDA model was improved. Laucelli *et al.* (2012) applied a PDA for WDNs and considered climate change and deterioration impacts on the WDNs. Giustolisi & Walski (2012) classified demand categories of WDNs (human-based, volume-based, uncontrolled orifice-based, and leakage-based demands) and also considered multilevel orifice model for demand components. More recently, Liserra *et al.* (2014) applied a PDA model for the evaluation of a reliability indicator.

**is the head at node**

*j*.

**is the minimum required head at node**

*j*, so that if the head at a node is equal to or less than this value, there will be no discharge.

**is the desirable head at node**

*j*, and if the available head is less than this value, the required demand is only partially served.

**is the available discharge at node**

*j*, and

**is the required discharge at node**

*j*. Parameter

*n*is an exponent value of the orifice function.

In fact, the nodes of WDN in the simulation model represent a number of users in the demand areas of the real network. To determine the parameter *n* value, which is system-specific, with accuracy sufficient amount of field data for outlets, such as elevations, pressure and discharge of each demand area should be provided. However, observation and analysis of every point in each demand area of real WDN are impractical. Due to lack of such data sets in practice, the *n* value is frequently assumed, leading to uncertain simulation results (Shirzad & Tabesh 2012). Thus, the quantitative uncertainty of the HOR should be carefully considered when interpreting the PDA results.

In this study, the gradient algorithm used for DDA in EPANET2 model was modified to develop a PDA model based on the global gradient algorithm. In addition, a Monte Carlo simulation (MCS) was conducted to quantify the available discharge at nodes considering the uncertainty in HOR. The developed model was applied to a well-known benchmark network for illustration of the approach. Further, the model was utilized for emergency water supply planning of a real-life WDN in South Korea.

## METHODOLOGY

### Development of PDA model

*et al.*2009). A unified formulation to analyze a WDN is generalized as a gradient algorithm (Todini & Pilati 1988). Equation (2) shows the constitution of the gradient algorithm for DDA. where

**= [**

*n*,

_{l}*n*] is a diagonal matrix of the elements that represent head losses.

_{l}**= [**

*n*,

_{l}*n*] and

_{n}**= [**

*n*,

_{n}*n*] are incidence matrices that define the pipe and node connectivity.

_{l}*Q*= [

*n*,1] and

_{l}*H*= [

*n*,1] are state variable vectors (unknown pipe discharge and nodal head, respectively).

_{n}**= [**

*n*−

_{t}*n*,1] is the reduced incidence matrix to the tank nodes, and

_{n}**= [**

*n*−

_{t}*n*,1] is the known head vector.

_{n}**= [**

*n*,

_{n}*n*] is the zero matrix, and

_{n}**= [**

*n*,1] is the required discharge, which is a fixed value of nodal demand.

_{n}*n*is the total number of links,

_{l}*n*is the total number of unknown head nodes and

_{n}*n*is the total number of nodes.

_{t}*Q*and

^{k}*H*are assumed at initial step, and updated values (

^{k}*Q*

^{k+}^{1}and

*H*

^{k+}^{1}) are calculated by solving Equation (2). Two iterative equations are obtained as given in Equation (3) and Equation (4). where

*G*=

*NA*

_{11},

*N*is a diagonal matrix with head loss exponent elements,

*I*is an identity matrix and

*k*is the iteration number.

In the global gradient algorithm, ** represents a diagonal matrix whose elements consist of an HOR. In addition, **** is replaced with **** = [***n _{n}*,1] representing the unknown value of PDD.

The additional input and output variables of the proposed PDA model are shown in Table 2. The minimum pressure, required pressure, and parameter *n* value are new input variables for nodes. Newly obtained output results for nodes are the PDD and the pressure mode. Here, the pressure mode denotes the status of nodal pressure and discharge (no flow, partial flow or adequate flow). Note that no additional entries for links are needed for PDA.

Nodes . | Links . |
---|---|

Input variables | Input variables |

- ID | - ID |

- Base demand | - Length |

- Initial quality | - Diameter |

- Minimum pressure | - Roughness |

- Required pressure | - Bulk coefficient |

- Parameter n | - Wall coefficient |

- Status | |

Output variables | Output variables |

- Demand | - Flow |

- Quality | - Velocity |

- Pressure | - Unit head loss |

- Head | - Friction factor |

- PDD (pressure-dependent demand) | - Reaction rate |

- Pressure mode | - Quality |

Nodes . | Links . |
---|---|

Input variables | Input variables |

- ID | - ID |

- Base demand | - Length |

- Initial quality | - Diameter |

- Minimum pressure | - Roughness |

- Required pressure | - Bulk coefficient |

- Parameter n | - Wall coefficient |

- Status | |

Output variables | Output variables |

- Demand | - Flow |

- Quality | - Velocity |

- Pressure | - Unit head loss |

- Head | - Friction factor |

- PDD (pressure-dependent demand) | - Reaction rate |

- Pressure mode | - Quality |

### Considering uncertainty in HOR

To consider the uncertainty in the HOR, MCS is used. MCS is a broad class of computational algorithm that uses repeated random sampling to obtain numerical results. MCS is often used in physical and mathematical problems and is most suitably applied when it is impossible to obtain a closed-form expression or infeasible to apply a deterministic algorithm.

In this study, the exponent value (*n*) of the orifice function (Equation (1)), is considered uncertain and randomly generated within a range, and the corresponding PDA results are estimated using MCS. The simulation starts by creating a scenario with an abnormal condition. Next, the *n* values of each node are generated in random following a statistical distribution function. The PDA simulation is conducted using the configured *n* values. The parameter *n* values are changed at each iteration of the simulation. Once the defined iterations are completed, the MCS results are collected and demonstrated.

Several studies have applied MCS for uncertainty quantification of WDN. Kang *et al.* (2009) and Kang & Lansey (2011) carried out hydraulic and water quality simulations with considering uncertainties in pipe diameter, roughness, decay coefficients, and spatial/temporal nodal demands. In those studies, the input parameters were assumed to follow the normal distribution to generate sets of random input parameters. However, the research for considering the uncertainty in HOR of the PDA modeling is first attempted in this study.

## APPLICATIONS AND RESULTS

### Hypothetical network

^{3}/h. The minimum required head at a node

**is assigned the nodal elevation, and the desirable head**

**is assumed to be 15 m greater than the nodal elevation (Ozger 2003).**

The pressure-deficient condition is simulated by assuming the pipe failure. As a simulation example, Table 3 lists the DDA and PDA results under the Pipe 3 failure scenario. For the DDA results, the available discharges are fixed, despite the pressure heads being decreased below the desirable head. In contrast, the PDA models calculate a PDD that is lower than the system required demand. The total available discharge is 2,749.65 m^{3}/h from the PDA option of the WaterGEMS model (Bentley Systems 2006) and the proposed PDA model with an *n* value of 0.5. Note that the WaterGEMS is a commercial software with a PDA option included and is widely used for WDN analyses. As seen in Table 3, the simulation results are identical for both PDA models, ensuring the accuracy of the developed PDA model.

No failure condition . | Pipe 3 failure condition . | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

. | . | . | . | EPANET (DDA) . | WaterGEMS (PDA) . | Developed PDA model (n = 0.5). | ||||||

Node . | Required discharge . | Total head . | Pres. head . | Available discharge . | Total head . | Pres. head . | Available discharge . | Total head . | Pres. head . | Available discharge . | Total head . | Pres. head . |

ID . | (m^{3}/h)
. | (m) . | (m) . | (m^{3}/h)
. | (m) . | (m) . | (m^{3}/h)
. | (m) . | (m) . | (m^{3}/h)
. | (m) . | (m) . |

J1 | 0 | 59.71 | 32.28 | 0 | 60.39 | 32.96 | 0 | 60.5 | 33.07 | 0 | 60.5 | 33.07 |

J2 | 212.4 | 59.2 | 25.67 | 212.4 | 60.15 | 26.62 | 212.4 | 60.31 | 26.78 | 212.4 | 60.31 | 26.78 |

J3 | 212.4 | 56.08 | 27.12 | 212.4 | 34.73 | 5.77 | 193.06 | 41.35 | 12.39 | 193.06 | 41.35 | 12.39 |

J4 | 640.8 | 54.99 | 22.99 | 640.8 | 34.76 | 2.76 | 506.92 | 41.39 | 9.39 | 506.92 | 41.39 | 9.39 |

J5 | 212.4 | 55.08 | 24.6 | 212.4 | 42.31 | 11.83 | 212.4 | 46.82 | 16.34 | 212.4 | 46.82 | 16.34 |

J6 | 684 | 49.85 | 18.46 | 684 | 34.79 | 3.4 | 558.58 | 41.39 | 10 | 558.58 | 41.39 | 10 |

J7 | 640.8 | 49.95 | 20.39 | 640.8 | 36.23 | 6.67 | 588.01 | 42.19 | 12.63 | 588.01 | 42.19 | 12.63 |

J8 | 327.6 | 48.95 | 17.56 | 327.6 | 36.16 | 4.77 | 277.52 | 42.15 | 10.76 | 277.52 | 42.15 | 10.76 |

J9 | 0 | 52.23 | 19.62 | 0 | 49 | 16.39 | 0 | 51.36 | 18.75 | 0 | 51.36 | 18.75 |

J10 | 0 | 53.54 | 19.4 | 0 | 51.48 | 17.34 | 0 | 53.29 | 19.15 | 0 | 53.29 | 19.15 |

J11 | 108 | 48.98 | 13.93 | 108 | 46.45 | 11.4 | 103.87 | 48.92 | 13.87 | 103.87 | 48.92 | 13.87 |

J12 | 108 | 48.75 | 12.17 | 108 | 45.93 | 9.35 | 96.89 | 48.65 | 12.07 | 96.89 | 48.65 | 12.07 |

J13 | 0 | 52.14 | 18.61 | 0 | 38.85 | 5.32 | 0 | 44.25 | 10.72 | 0 | 44.25 | 10.72 |

Total | 3,146.40 | 3,146.40 | 2,749.65 | 2,749.65 |

No failure condition . | Pipe 3 failure condition . | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

. | . | . | . | EPANET (DDA) . | WaterGEMS (PDA) . | Developed PDA model (n = 0.5). | ||||||

Node . | Required discharge . | Total head . | Pres. head . | Available discharge . | Total head . | Pres. head . | Available discharge . | Total head . | Pres. head . | Available discharge . | Total head . | Pres. head . |

ID . | (m^{3}/h)
. | (m) . | (m) . | (m^{3}/h)
. | (m) . | (m) . | (m^{3}/h)
. | (m) . | (m) . | (m^{3}/h)
. | (m) . | (m) . |

J1 | 0 | 59.71 | 32.28 | 0 | 60.39 | 32.96 | 0 | 60.5 | 33.07 | 0 | 60.5 | 33.07 |

J2 | 212.4 | 59.2 | 25.67 | 212.4 | 60.15 | 26.62 | 212.4 | 60.31 | 26.78 | 212.4 | 60.31 | 26.78 |

J3 | 212.4 | 56.08 | 27.12 | 212.4 | 34.73 | 5.77 | 193.06 | 41.35 | 12.39 | 193.06 | 41.35 | 12.39 |

J4 | 640.8 | 54.99 | 22.99 | 640.8 | 34.76 | 2.76 | 506.92 | 41.39 | 9.39 | 506.92 | 41.39 | 9.39 |

J5 | 212.4 | 55.08 | 24.6 | 212.4 | 42.31 | 11.83 | 212.4 | 46.82 | 16.34 | 212.4 | 46.82 | 16.34 |

J6 | 684 | 49.85 | 18.46 | 684 | 34.79 | 3.4 | 558.58 | 41.39 | 10 | 558.58 | 41.39 | 10 |

J7 | 640.8 | 49.95 | 20.39 | 640.8 | 36.23 | 6.67 | 588.01 | 42.19 | 12.63 | 588.01 | 42.19 | 12.63 |

J8 | 327.6 | 48.95 | 17.56 | 327.6 | 36.16 | 4.77 | 277.52 | 42.15 | 10.76 | 277.52 | 42.15 | 10.76 |

J9 | 0 | 52.23 | 19.62 | 0 | 49 | 16.39 | 0 | 51.36 | 18.75 | 0 | 51.36 | 18.75 |

J10 | 0 | 53.54 | 19.4 | 0 | 51.48 | 17.34 | 0 | 53.29 | 19.15 | 0 | 53.29 | 19.15 |

J11 | 108 | 48.98 | 13.93 | 108 | 46.45 | 11.4 | 103.87 | 48.92 | 13.87 | 103.87 | 48.92 | 13.87 |

J12 | 108 | 48.75 | 12.17 | 108 | 45.93 | 9.35 | 96.89 | 48.65 | 12.07 | 96.89 | 48.65 | 12.07 |

J13 | 0 | 52.14 | 18.61 | 0 | 38.85 | 5.32 | 0 | 44.25 | 10.72 | 0 | 44.25 | 10.72 |

Total | 3,146.40 | 3,146.40 | 2,749.65 | 2,749.65 |

*n*values were randomly generated and applied under the same pipe failure scenario. It was assumed that the random

*n*values follow a normal distribution with a mean value of 0.5 and a standard deviation of 0.025 (

*n*∼

*N*(0.5, 0.025)). Table 4 and Figure 2 show the MCS results considering the uncertainty.

No failure condition . | Pipe 3 failure condition . | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

. | . | . | Developed PDA model (n = 0.5). | Developed PDA model with uncertainty (n ∼ N (0.5, 0.025)). | ||||||||

. | Required discharge . | Pres. head . | . | Pres. head . | Available discharge . | Pressure head . | ||||||

Node . | . | . | Available . | . | Minimum . | Median . | Maximum . | . | Minimum . | Median . | Maximum . | . |

ID . | (m^{3}/h)
. | (m) . | discharge . | (m) . | (m^{3}/h)
. | (m^{3}/h)
. | (m^{3}/h)
. | COV* . | (m) . | (m) . | (m) . | COV* . |

J1 | 0 | 32.28 | 0 | 33.07 | 0 | 0 | 0 | 0 | 33.07 | 33.07 | 33.07 | 0 |

J2 | 212.4 | 25.67 | 212.4 | 26.78 | 212.4 | 212.4 | 212.4 | 0 | 26.78 | 26.78 | 26.78 | 0 |

J3 | 212.4 | 27.12 | 193.06 | 12.39 | 190.75 | 193.11 | 195.54 | 0.0053 | 12.21 | 12.39 | 12.62 | 0.0053 |

J4 | 640.8 | 22.99 | 506.92 | 9.39 | 495.75 | 506.71 | 517.69 | 0.0091 | 9.2 | 9.38 | 9.61 | 0.0069 |

J5 | 212.4 | 24.6 | 212.4 | 16.34 | 212.4 | 212.4 | 212.4 | 0 | 16.22 | 16.34 | 16.5 | 0.0027 |

J6 | 684 | 18.46 | 558.58 | 10 | 550.67 | 558.92 | 570.58 | 0.0074 | 9.82 | 9.99 | 10.23 | 0.0065 |

J7 | 640.8 | 20.39 | 588.01 | 12.63 | 581.97 | 588.33 | 594.7 | 0.0041 | 12.51 | 12.63 | 12.82 | 0.0042 |

J8 | 327.6 | 17.56 | 277.52 | 10.76 | 272.62 | 277.53 | 282.4 | 0.0077 | 10.64 | 10.76 | 10.97 | 0.005 |

J9 | 0 | 19.62 | 0 | 18.75 | 0 | 0 | 0 | 0 | 18.71 | 18.74 | 18.8 | 0.001 |

J10 | 0 | 19.4 | 0 | 19.15 | 0 | 0 | 0 | 0 | 19.12 | 19.14 | 19.19 | 0.0008 |

J11 | 108 | 13.93 | 103.87 | 13.87 | 103.37 | 103.88 | 104.31 | 0.0016 | 13.82 | 13.87 | 13.94 | 0.0017 |

J12 | 108 | 12.17 | 96.89 | 12.07 | 95.66 | 96.93 | 98.09 | 0.0047 | 12.01 | 12.07 | 12.15 | 0.0025 |

J13 | 0 | 18.61 | 0 | 10.72 | 0 | 0 | 0 | 0 | 10.59 | 10.72 | 10.91 | 0.0047 |

Total | 3,146.40 | 2,749.65 | 2,736.56 | 2,750.00 | 2,759.15 | 0.0014 |

No failure condition . | Pipe 3 failure condition . | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

. | . | . | Developed PDA model (n = 0.5). | Developed PDA model with uncertainty (n ∼ N (0.5, 0.025)). | ||||||||

. | Required discharge . | Pres. head . | . | Pres. head . | Available discharge . | Pressure head . | ||||||

Node . | . | . | Available . | . | Minimum . | Median . | Maximum . | . | Minimum . | Median . | Maximum . | . |

ID . | (m^{3}/h)
. | (m) . | discharge . | (m) . | (m^{3}/h)
. | (m^{3}/h)
. | (m^{3}/h)
. | COV* . | (m) . | (m) . | (m) . | COV* . |

J1 | 0 | 32.28 | 0 | 33.07 | 0 | 0 | 0 | 0 | 33.07 | 33.07 | 33.07 | 0 |

J2 | 212.4 | 25.67 | 212.4 | 26.78 | 212.4 | 212.4 | 212.4 | 0 | 26.78 | 26.78 | 26.78 | 0 |

J3 | 212.4 | 27.12 | 193.06 | 12.39 | 190.75 | 193.11 | 195.54 | 0.0053 | 12.21 | 12.39 | 12.62 | 0.0053 |

J4 | 640.8 | 22.99 | 506.92 | 9.39 | 495.75 | 506.71 | 517.69 | 0.0091 | 9.2 | 9.38 | 9.61 | 0.0069 |

J5 | 212.4 | 24.6 | 212.4 | 16.34 | 212.4 | 212.4 | 212.4 | 0 | 16.22 | 16.34 | 16.5 | 0.0027 |

J6 | 684 | 18.46 | 558.58 | 10 | 550.67 | 558.92 | 570.58 | 0.0074 | 9.82 | 9.99 | 10.23 | 0.0065 |

J7 | 640.8 | 20.39 | 588.01 | 12.63 | 581.97 | 588.33 | 594.7 | 0.0041 | 12.51 | 12.63 | 12.82 | 0.0042 |

J8 | 327.6 | 17.56 | 277.52 | 10.76 | 272.62 | 277.53 | 282.4 | 0.0077 | 10.64 | 10.76 | 10.97 | 0.005 |

J9 | 0 | 19.62 | 0 | 18.75 | 0 | 0 | 0 | 0 | 18.71 | 18.74 | 18.8 | 0.001 |

J10 | 0 | 19.4 | 0 | 19.15 | 0 | 0 | 0 | 0 | 19.12 | 19.14 | 19.19 | 0.0008 |

J11 | 108 | 13.93 | 103.87 | 13.87 | 103.37 | 103.88 | 104.31 | 0.0016 | 13.82 | 13.87 | 13.94 | 0.0017 |

J12 | 108 | 12.17 | 96.89 | 12.07 | 95.66 | 96.93 | 98.09 | 0.0047 | 12.01 | 12.07 | 12.15 | 0.0025 |

J13 | 0 | 18.61 | 0 | 10.72 | 0 | 0 | 0 | 0 | 10.59 | 10.72 | 10.91 | 0.0047 |

Total | 3,146.40 | 2,749.65 | 2,736.56 | 2,750.00 | 2,759.15 | 0.0014 |

*COV (coefficient of variation) = standard deviation (σ)/mean (μ).

The COV (coefficient of variation) is a measure of the dispersion, and is defined as the ratio of the standard deviation to the mean. The COV can also be used as a normalized index to evaluate the system reliability.

Nodes J2 and J5 have low COVs, whereas nodes J4, J6, and J8 show relatively high COVs. This indicates that the sensitivities of J4, J6, and J8 are greater than those of the other nodes, and the sensitivities of J2 and J5 are less than those of the others under the Pipe 3 failure scenario. The range of the total supplied water is from 2,736.56 m^{3}/h as the minimum flow to 2,759.15 m^{3}/h as the maximum flow – a difference of 22.59 m^{3}/h. The reasons for the different sensitivity analysis results for nodes are the nodal pressure and position of the nodes. Demand nodes that are positioned near the reservoir have small COVs. Additionally, the nodes that have a mean pressure head less than the desirable pressure head vary within a large range.

^{3}/h as the minimum flow to 3,146.17 m

^{3}/h as the maximum flow – a large difference of approximately 1,500 m

^{3}/h. It is nearly one-half of the total required demand in the system. Moreover, like the results under the Pipe 3 failure scenario, the nodes that have a mean pressure head less than desirable pressure head vary within a large range. The available discharge and pressure head of nodes J11 and J12 have a large variance, because they have a low pressure head and are located far from the water sources.

No failure condition . | Overall single pipe failure conditions (n ∼ N (0.5, 0.025)). | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

. | . | . | Available discharge . | Pres. head . | ||||||

. | Required discharge . | Pres. head . | Minimum . | Median . | Maximum . | COV . | Minimum . | Median . | Maximum . | . |

Node ID . | (m^{3}/h)
. | (m) . | (m^{3}/h)
. | (m^{3}/h)
. | (m^{3}/h)
. | . | (m) . | (m) . | (m) . | COV . |

J1 | 0 | 32.28 | 0 | 0 | 0 | 0 | 7.96 | 32.3 | 33.53 | 0.17 |

J2 | 212.4 | 25.67 | 66.99 | 212.4 | 212.4 | 0.2 | 1.86 | 25.69 | 26.78 | 0.3 |

J3 | 212.4 | 27.12 | 135.18 | 212.4 | 212.4 | 0.1 | 6.51 | 27.13 | 30.33 | 0.27 |

J4 | 640.8 | 22.99 | 309.35 | 640.8 | 640.8 | 0.16 | 3.75 | 22.93 | 23.37 | 0.32 |

J5 | 212.4 | 24.6 | 192.11 | 212.4 | 212.4 | 0.03 | 12.44 | 24.57 | 25.48 | 0.2 |

J6 | 684 | 18.46 | 362.53 | 684 | 684 | 0.14 | 4.35 | 17.38 | 18.76 | 0.28 |

J7 | 640.8 | 20.39 | 364.12 | 640.8 | 640.8 | 0.12 | 5.12 | 19.32 | 21.03 | 0.26 |

J8 | 327.6 | 17.56 | 158.56 | 327.6 | 327.6 | 0.15 | 3.76 | 15.07 | 18.03 | 0.28 |

J9 | 0 | 19.62 | 0 | 0 | 0 | 0 | 2.95 | 19.32 | 22.68 | 0.32 |

J10 | 0 | 19.4 | 0 | 0 | 0 | 0 | 1.37 | 19.36 | 23.21 | 0.35 |

J11 | 108 | 13.93 | 16.57 | 104.45 | 108 | 0.29 | 0.47 | 14.03 | 16.54 | 0.4 |

J12 | 108 | 12.17 | 0 | 97.08 | 107.77 | 0.37 | 0 | 12.08 | 14.93 | 0.43 |

J13 | 0 | 18.61 | 0 | 0 | 0 | 0 | 5.55 | 17.78 | 20.92 | 0.27 |

Total | 3,146.40 | 1,619.58 | 3,115.89 | 3,146.17 | 0.15 |

No failure condition . | Overall single pipe failure conditions (n ∼ N (0.5, 0.025)). | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

. | . | . | Available discharge . | Pres. head . | ||||||

. | Required discharge . | Pres. head . | Minimum . | Median . | Maximum . | COV . | Minimum . | Median . | Maximum . | . |

Node ID . | (m^{3}/h)
. | (m) . | (m^{3}/h)
. | (m^{3}/h)
. | (m^{3}/h)
. | . | (m) . | (m) . | (m) . | COV . |

J1 | 0 | 32.28 | 0 | 0 | 0 | 0 | 7.96 | 32.3 | 33.53 | 0.17 |

J2 | 212.4 | 25.67 | 66.99 | 212.4 | 212.4 | 0.2 | 1.86 | 25.69 | 26.78 | 0.3 |

J3 | 212.4 | 27.12 | 135.18 | 212.4 | 212.4 | 0.1 | 6.51 | 27.13 | 30.33 | 0.27 |

J4 | 640.8 | 22.99 | 309.35 | 640.8 | 640.8 | 0.16 | 3.75 | 22.93 | 23.37 | 0.32 |

J5 | 212.4 | 24.6 | 192.11 | 212.4 | 212.4 | 0.03 | 12.44 | 24.57 | 25.48 | 0.2 |

J6 | 684 | 18.46 | 362.53 | 684 | 684 | 0.14 | 4.35 | 17.38 | 18.76 | 0.28 |

J7 | 640.8 | 20.39 | 364.12 | 640.8 | 640.8 | 0.12 | 5.12 | 19.32 | 21.03 | 0.26 |

J8 | 327.6 | 17.56 | 158.56 | 327.6 | 327.6 | 0.15 | 3.76 | 15.07 | 18.03 | 0.28 |

J9 | 0 | 19.62 | 0 | 0 | 0 | 0 | 2.95 | 19.32 | 22.68 | 0.32 |

J10 | 0 | 19.4 | 0 | 0 | 0 | 0 | 1.37 | 19.36 | 23.21 | 0.35 |

J11 | 108 | 13.93 | 16.57 | 104.45 | 108 | 0.29 | 0.47 | 14.03 | 16.54 | 0.4 |

J12 | 108 | 12.17 | 0 | 97.08 | 107.77 | 0.37 | 0 | 12.08 | 14.93 | 0.43 |

J13 | 0 | 18.61 | 0 | 0 | 0 | 0 | 5.55 | 17.78 | 20.92 | 0.27 |

Total | 3,146.40 | 1,619.58 | 3,115.89 | 3,146.17 | 0.15 |

Additionally, to identify the link with the most impact on the nodal discharge and pressure variation, the PDA results were arranged with a link ID (Pipe 1–Pipe 21). The Pipe 1 and Pipe 2 failure scenarios have the largest COVs and cause a major shortage of total available discharge. The Pipe 10 and Pipe 20 failure scenarios have the opposite results. However, the Pipe 6 and Pipe 16 failure scenarios have a small amount of shortage, but large COVs. Thus, they should be considered when operating and managing the system. The results from uncertainty analysis by using developed model also can be used as an index for evaluation of WDNs. The developed PDA model with uncertainty quantification can be used for reliability and resilience analysis of an actual WDN considering abnormal system conditions.

### Yeong-Wol block system in South Korea

The total system demand is 8,391.66 m^{3}/day, and the service area is mainly residential. The network is composed of seven sub-blocks delineated based on the location of the reservoirs. The water network with sub-blocks is known to increase the operating flexibility compared with the single-source network. Under normal operating practice, each sub-block in the network is operated as a completely isolated system. However, if one of the sub-blocks is interrupted due to main pipeline breakage and pumping station suspension, neighboring sub-blocks can supply the interrupted sub-block based on the emergency water supply plan. The DDA approach is generally applied for the emergency supply plan in South Korea, but PDA should be conducted to determine the most appropriate supply strategy by estimating the pressure variations and available system demands.

In this study, the reservoir of the YW-2 sub-block is assumed to be closed, and the emergency water supplies from the neighboring sub-blocks, YW-1 and SN, were simulated to calculate the serviceable system demands by applying the proposed PDA model. Note that the YW-1 and YW-2 sub-blocks are highly dense with relatively high demands, while the SN sub-block shows a widely spread layout with lower water demand. The sub-blocks YW-1 and YW-2 can be connected by simply opening the shut-off valve connecting the two sub-blocks, while there exists a bypass pipeline directly connecting the SN reservoir to the YW-2 sub-block, which is closed during normal operation.

**is assigned a node elevation, and the desirable head**

**is set to 15 m above the node elevation. 100 sets of**

*n*values for HOR are randomly generated following the normal distribution with a mean of 0.5 and standard deviation of 0.025 (

*n*∼

*N*(0.5, 0.025)). Table 6 and Figure 5 show the results of the PDA for the emergency supply plans considering the uncertainty of the

*n*value.

. | Developed PDA model (n = 0.5). | Developed PDA model (n ∼ N (0.5, 0.025)). | |||||||
---|---|---|---|---|---|---|---|---|---|

. | Available discharge . | Available discharge . | |||||||

. | . | . | Minimum . | Median . | Maximum . | . | |||

Connected sub-block . | (m^{3}/day)
. | (%) . | (m^{3}/day)
. | (%) . | (m^{3}/day)
. | (%) . | (m^{3}/day)
. | (%) . | COV . |

YW-1 | 7,308.46 | 87.09 | 7,054.02 | 84.06 | 7,312.57 | 87.14 | 7,712.60 | 91.91 | 0.01429 |

SN | 6,449.48 | 76.86 | 5,431.30 | 64.72 | 6,449.92 | 76.86 | 7,006.41 | 83.49 | 0.05573 |

. | Developed PDA model (n = 0.5). | Developed PDA model (n ∼ N (0.5, 0.025)). | |||||||
---|---|---|---|---|---|---|---|---|---|

. | Available discharge . | Available discharge . | |||||||

. | . | . | Minimum . | Median . | Maximum . | . | |||

Connected sub-block . | (m^{3}/day)
. | (%) . | (m^{3}/day)
. | (%) . | (m^{3}/day)
. | (%) . | (m^{3}/day)
. | (%) . | COV . |

YW-1 | 7,308.46 | 87.09 | 7,054.02 | 84.06 | 7,312.57 | 87.14 | 7,712.60 | 91.91 | 0.01429 |

SN | 6,449.48 | 76.86 | 5,431.30 | 64.72 | 6,449.92 | 76.86 | 7,006.41 | 83.49 | 0.05573 |

When reservoir YW-1 serves as an alternative source for the YW-2 sub-block, 84.06–91.91% of the total system demand is supplied with a median value of 87.14%. When reservoir SN acts as the supply source for the YW-2 sub-block, the supplied demand is about 64.72–83.49% of the total demand, and the median value is 76.86%. Comparing the two emergency plans, the supply from the reservoir YW-1 is more reliable with a greater ability to fulfill the demand and less variation in the available discharge. It should be noted that the estimated available discharges are sensitive to the *n* value of the HOR equation; the emergency supply from reservoir SN shows a much wider variation in the available discharge.

## CONCLUSIONS

The hydraulic analysis of WDNs is divided into two types: DDA for normal conditions, and PDA for abnormal conditions. When calculating the pressure-dependent discharge of a WDN using PDA, the HOR is the most important information required with a high degree of uncertainty. In this study, the gradient algorithm used for the DDA in EPANET is modified to develop a PDA model based on the global gradient algorithm, and the MCS is applied to quantify the HOR uncertainty.

From the individual pipe failure simulations of the hypothetical network, the developed PDA model with HOR uncertainty quantification practice revealed the following advantages: (1) to be able to identify the nodes with the most variations in discharge and pressure; and (2) to determine the link with the greatest impact on the system discharge and pressure when failed. These nodes and links can be considered as critical elements for planning and management of systems. Furthermore, from the results of emergency water supply plan of the Yeong-Wol block system, the developed model derived the available discharge for each case and calculated the quantitative uncertainty in the HOR. The developed PDA model with uncertainty quantification can also be used for analysis of WDNs under pressure deficient conditions; that is, the unexpected demand variations due to fire hydrant operation and climate induced impact, and component failures, such as pump outage, and multiple pipe failure events.

The developed model can be applied for an extended period simulation by considering time-dependent hydraulic components, such as demand pattern, reservoir water level pattern, and status of pumps and valves. The hydraulic analysis results obtained by the developed model can effectively support decision-making by practitioners. Additionally, the model can be applied to the reinforcement and rehabilitation of WDNs and can be used in the reliability and resilience analyses of systems with consideration of the uncertainty.

It is expected that the developed model can be enhanced by applying various statistical distribution functions for uncertainty quantification and considering multiple demand categories of WDNs and multilevel orifice outlet points for demand components. This is fruitful area for research and should be accomplished in further studies.

## ACKNOWLEDGEMENT

This study was supported by the Korea Ministry of Environment as ‘The Eco-Innovation project (GT-11-G-02-001-2)’.