Research Project

16 November 2015

Determination of the Impact of Fire

Demand on Water Distribution

System Performance Using Stochastic

Modelling Techniques

Research Project

16 November 2015

Faculty of Engineering & the Built Environment

DEPARTMENT OF CIVIL ENGINEERING

CIV4044S RESEARCH PROJECT

Determination of the Impact of

Fire Demand on Water

Distribution System Performance

Using Stochastic Modelling

Techniques

Prepared For:

Prof. Kobus van Zyl

Prepared By:

Niel Claassens

Date of Submission:

Monday 16 November 2015

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Plagiarism Declaration

1. I know that plagiarism is wrong. Plagiarism is to use another’s work and to pretend that

it is one’s own.

2. I have used the Harvard Convention for citation and referencing. Each significant

contribution to and quotation in this report form the work or works of other people has

been attributed and has been cited and referenced.

3. This report is my own work

4. I have not allowed and will not allow anyone to copy my work with the intension of

passing it as his or her own work.

Signature ______________________________

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Abstract

Stochastic analysis of water distribution systems enables more realistic system models and

thus enables the performance of a system to be evaluated under more realistic conditions.

@RISK is a software package used in industry to conduct stochastic analysis. @RISK is a

plug-in for Microsoft Excel and enables risk analysis using Monte Carlo simulation.

In a stochastic analysis of a water supply system the factors which influence the reliability of

the water distribution system such as water demand, pipe failures, fire occurrence, fire

duration and fire demand (“Key System Inputs”) are modelled according to appropriate

probability distributions. The system is then simulated over a chosen period of time. The

relationships between system Failure Rate and storage capacity of the reservoir as well as the

Key System Inputs are analysed. The data generated for this analysis is utilised to assess the

impact of fire demand specifically on system Failure Rate.

Generally fire demand is not a significant input variable for Failure Rate, except for cases of

extreme fire demand.

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Acknowledgements

First and foremost I would like to thank prof. Kobus van Zyl for the opportunity to do

this research project under his supervision.

My completion of this project could not have been accomplished without the support

of my friends and digs mates. To Lloyd, Steven, Jethro, Raymond, Herman and Bradley –

thank you for all the support and necessary beers after a busy week, not only during this

research project but also over the past few years.

Thanks to my parents as well, Mr. and Mrs. Claassens for all the love and support

over the past 4 years, I will forever be grateful for the opportunity that you have given me to

study at a world class institution such as UCT.

To my bursar of the past year, Hatch Goba: Thank you for the support throughout my

final year.

Finally I would like to give special thanks to my grandmother, Alta, for all the love

and support over the past 4 years. I love you very much.

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Table of Contents

1. Introduction 8

1.1 Background 8

1.2 Goals and Objectives 9

1.3 Structure of the Report 10

2. Literature Review 11

2.1 Statistical Principles 11

2.1.1 Uniform Distributions 15

2.1.2 Poisson Distribution 16

2.1.3 Normal Distribution 16

2.1.4 Log-normal Distribution 18

2.2 Modelling techniques 19

2.2.1 Deterministic models 19

2.2.2 Stochastic models 19

2.2.3 Basic modelling 19

2.2.4 Monte Carlo Simulation 21

2.3 Modelling water distribution systems 23

2.3.1 Components of a Water Distribution System 23

2.3.2 Bulk Water Supply Systems 24

2.3.3 Reliability of Water Supply Systems 26

2.3.4 The traditional modelling approach 28

2.3.5 The stochastic approach 28

2.4 Fire Demand 30

2.4.1 Current design guidelines 30

2.4.2 Comparison with international codes 33

2.4.3 Comparison with actual fire data 36

2.4.4 The need for new design guidelines 37

2.4.5 Probabilistic Fire Demand 37

2.4.5 Probability of Fire Occurring 41

3. Design of storage tanks from first principles 46

3.1 Introduction 46

3.2 Deterministic Design 46

3.3 Stochastic Design 47

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4. Methodology 50

4.1 Considerations 50

4.2 Input Parameters 51

4.2.1 Consumer Demand 51

4.2.2 Supply System 54

4.2.3 Fire Demand 55

4.4 Model Description 58

4.5 The Monte Carlo Simulation 58

5. Results 60

5.1 Results from Analysis 60

5.2 Comparison of results with previous research 62

5.3 Sensitivity Analysis 64

5.3.1 Introduction 64

5.3.2 Sensitivity Analysis used in previous studies 64

5.3.3 Methodology for Sensitivity Analysis used in this project 65

6. Discussion and Conclusion 67

7. References 68

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List of Figures

Figure 1: Relative Frequency Histogram of Daily Demand 12

Figure 2: Discrete Uniform Distribution (Johnson et al., 2011) 15

Figure 3: The Normal Curve (Walpole et al., 1987) 16

Figure 4: Normal curves with µ1 = µ2 and σ1 < σ2 (Walpole et al., 1987). 17

Figure 5: Normal curves with µ1 < µ2 and σ1 = σ2 (Walpole et al., 1987). 17

Figure 6: Log-normal distribution (Johnson et al., 2011) 18

Figure 7: Modelling Regimes 20

Figure 8: Owens' framework for a deterministic-dynamic model 21

Figure 9: Water Distribution System (Nel, 1993) 23

Figure 10: Cost sensitivity to fire public peak demand (Van Zyl & HAarhoff, 1997) 34

Figure 11: Comparison of the South African fire storage volume standard, European

standards and actual volumes used in Johannesburg in 90% of cases (Van Zyl & Haarhoff,

1997) 35

Figure 12: Johannesburg fire duration 38

Figure 13: Johannesburg fire flow 38

Figure 14: Pie Chart of the original Data Set Constitution (Davy, 2010) 43

Figure 15: Pie Chart of Data set after cleaning & re‐categorisation (Davy, 2010) 45

Figure 16: Deterministic Assurance of Supply 47

Figure 17: Probability Distribution of Failure Rate for 12 hour AADD storage tank 48

Figure 18: Stochastic Assurance of Supply 49

Figure 19: Simple Water Distribution System (Van Zyl et al., 2008) 50

Figure 20: Probability density function of white noise component 54

Figure 21: Failure rate vs. Tank Capacity 60

Figure 22: Variation of data 61

Figure 23: Probability Distribution of Failure Rate for 12 hour AADD storage tank 62

Figure 24: Annual average number of tank failures as a function of the tank capacity (Van

Zyl et al., 2008) 63

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List of Tables

Table 1: Raw Data of Daily Demand (kl/d) 11

Table 2: Relative Frequency Distribution of Daily Demand 12

Table 3: Typical South African guidelines for sizing reservoirs and feeder pipes

(Kretzman, 2004) 25

Table 4: Red Book Fire Risk Categories (CSIR, 2000) 30

Table 5: Red Book design fire flow (CSIR, 2000) 31

Table 6: Red Book fire duration and storage (CSIR, 2000) 32

Table 7: Fire flow design criteria for reticulation mains (CSIR, 2000) 32

Table 8: Comparison of fire standards (Van Zyl, 1993) 33

Table 9: Comparison of fire demands (Van Zyl & Haarhoff, 1997) 34

Table 10: Descriptive Statistics for Johannesburg Fire Duration 38

Table 11: Descriptive Statistics for Johannesburg Fire Flow 39

Table 12: Duration descriptive statistics and percentile values (Davy, 2010) 40

Table 13: Flow descriptive statistics and percentile values (Davy, 2010) 40

Table 14: Volume descriptive statistics and percentile values (Davy, 2010) 41

Table 15: Category numbers and percentages for the original data set (Davy, 2010) 42

Table 16: Category numbers & Percentages for the cleaned and verified data set (Davy,

2010) 44

Table 17: Seasonal Factors 52

Table 18: Day-of-the-week factors 52

Table 19: Hour factors Factors 53

Table 20: Summary of Input Parameters 55

Table 21: Input Parameter values used for sensitivity analysis (Van Zyl & Haarhoff, 2002)

64

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1. Introduction

1.1 Background

A water distribution system consists of a network of components which will typically

include a source, pumping station (if required), pipeline and storage facility (municipal

storage tank/reservoir). A bulk water supply system should ensure a reliable supply of water

to the consumer. A failure of the reservoir thus equates to a failure of the supply system. The

aim of the designer is to avoid failures from occurring. The risk of a failure occurring is

dependent on a number of factors such as a supply pipe failure, sudden increase in consumer

demand and a big fire occurring in the supply area. These factors are random as the instance

of occurrence is not known and the size of impact is not known. In this report such random

factors or variables are referred to as stochastic variables.

Traditionally guidelines used for designing water supply systems have been based on

deterministic analysis (Van Zyl et al., 2008). Deterministic analysis is when a single-point

value is assumed for a stochastic variable. This has ensured reliable water supply systems, but

not necessarily the optimal solution (Vlok, 2010).

Locally the guidelines for sizing municipal storage tanks are still based on such

deterministic analysis. In South Africa the “CSIR Guidelines for human settlement and

design” (also known as the “Red Book”) serves as a design guideline for the design of water

distribution- and storage systems. Typically the inflow (supply) and outflow (demand) are

assumed to be constant deterministic variables. For this purpose, Average Annual Daily

Demand (AADD) is a key design input variable.

The study conducted by Vlok (2010) concluded that risk-based analysis led to the design

of smaller reservoir sizes without jeopardising reliability. This has a financial benefit.

According to Vlok, risk-based techniques refer to methods that accommodate the events that

impose risk on the system under consideration. The probability of these risk-inducing events

having an effect on the system is also taken into account (Vlok, 2010). The term “risk-based

analysis” used by Vlok (2010) is another term for stochastic analysis as outlined in this

project (refer to section 2).

Such risk-based design techniques have seldom been used in the past due to the lack of

available computational power. Lack of computational power is no longer a restricting factor.

Van Zyl & Haarhoff (2002) proposed a theoretical framework for a probabilistic design

model of water distribution systems which includes user demand, pipe failures and fire

demand. The term “probabilistic design” also refers to stochastic analysis as outlined in this

project. The authors developed a software package (MOCASIM) specifically for analysing

water supply systems stochastically. Subsequently various authors have used this method in

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their research. Kretzman (2004) refined the software developed by Van Zyl & Haarhoff

(2002), now called MOCASIM II. Kretzman then analysed a simple network using

MOCASIM II and gave the findings of the analysis. Vlok (2010) then went further to

investigate the cost implications of risk-based design approaches on bulk water supply system

design with size and configuration used as primary design variables.

When investigating the impact of fire demand (i.e. the water demand from the system to

combat one or more fires) on the performance of a water supply system, it can be argued that

deterministic modelling techniques are not optimal because the occurrence of a new fire in

the area of the supply system is a random event, the water demand to combat to combat a

new fire is not known (thus a random variable) and the duration of any fire is not known and

thus also a random variable.

From the above it becomes clear that the stochastic modelling technique is more suitable

to model the occurrence of fires as well as the fire demand.

This research project is concerned with determining the impact of fire demand on the

performance of a water distribution system using stochastic modelling techniques. In

particular the report focuses on the stochastic modelling technique. Stochastic analysis

conducted for this project utilised commercial software, @RISK, for the stochastic analysis.

1.2 Goals and Objectives

This research project is aimed at demonstrating the relationship between storage capacity

and water distribution system performance using dynamic stochastic analysis. In particular,

the research project focuses specifically on the impact of fire demand on water distribution

system performance. Emphasis is placed on the thorough understanding of the process and

techniques of stochastic modelling as practiced in industry.

In order to reach these goals it is vital to carry out a literature review on basic statistics,

stochastic modelling and fire demand. The author will build a dynamic stochastic model of a

“typical” water distribution system. This model will be used to investigate the relationship

between tank storage capacity and system performance. The data from the stochastic model

will be utilised to assess the impact of fire demand specifically on the performance of the

water distribution system modelled.

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1.3 Structure of the Report

Chapter 2 contains the literature review. The literature review starts by giving a brief

explanation of some basic statistical principles that are of importance in a stochastic model. A

description of different modelling regimes is given, explaining the difference between

stochastic models and deterministic models. Water distribution systems are described, in

particular looking at the components that make up a water distribution system and the

guidelines for designing the components of a water distribution system as well as defining the

reliability of bulk supply systems. Finally an overview of fire demand is given, also

explaining why it is important to describe fire demand in a statistical sense.

Chapter 3 does not form part of the literature review but rather serves as a discussion that

highlights how reservoirs are designed from first principles in industry using both

deterministic – and stochastic design techniques. This discussion will also introduce some

key aspects of stochastic modelling as used in this research project as well as in industry.

Chapter 4 outlines the methodology for this research project. The model is discussed and

the various input parameters are given and summarized in a table.

Chapter 5 contains results of the stochastic analysis. After the results are given the chapter

explains the sensitivity analysis used in this project. Firstly sensitivity analyses as used in

other research is discussed, thereafter the sensitivity analysis used in this project discussed

and the results given.

Chapter 6 is the discussion and conclusion of this research project.

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2. Literature Review

2.1 Statistical Principles

This section gives a basic overview of some statistical principles as well as properties of

the various probability distributions used in this research project.

Walpole & Myers (1978) defines raw data as: “any recorded information in its original

collected form, whether it is counts or measurements” (Walpole & Myers, 1978). Raw data

can be presented in many different ways. The simplest manner to present data is through a

frequency distribution. A frequency distribution is a table that divides the raw data into

different categories, showing also how many items belongs to each category. The histogram

is a common graphical representation of a frequency distribution and often serves as a first

estimate of the probability distribution of a set of data. A histogram of a given frequency

distribution is constructed of adjacent rectangles, with the height of each rectangle

representing the frequency of the category. The bases of each rectangle extend between

successive categories. An example of a frequency distribution and a histogram is shown

below in figure 1.

The data given in the table below represents the daily demand (kl/d) at a specific node in a

water distribution system:

Table 1: Raw Data of Daily Demand (kl/d)

2.2 4.1 3.5 4.5 3.2 3.7 3.0 2.6

3.4 1.6 3.1 3.3 3.8 3.1 4.7 3.7

2.5 4.3 3.4 3.6 2.9 3.3 3.9 3.1

3.3 3.1 3.7 4.4 3.2 4.1 1.9 3.4

4.7 3.8 3.2 2.6 3.9 3.0 4.2 3.5

From a visual inspection of the data, the variability of the data becomes evident and

already indicates the need not to represent the data by a single value, as would be the case in

deterministic analysis. Raw data is then ordered into a relative frequency distribution:

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Table 2: Relative Frequency Distribution of Daily Demand

Class Interval Class midpoint Frequency (f) Relative Frequency

1.5-1.9 1.7 2 0.050

2.0-2.4 2.2 1 0.025

2.5-2.9 2.7 4 0.100

3.0-3.4 3.2 15 0.375

3.5-3.9 3.7 10 0.250

4.0-4.4 4.2 5 0.125

4.5-4.9 4.7 3 0.075

Figure 1: Relative Frequency Histogram of Daily Demand

Revisiting the raw data set, it is clear that 40 readings were taken during a certain period

of time. The mean of the data set is defined by the formula:

Where xi represents the ith

reading, thus x1 is the first reading, x2 is the second reading and

so forth. In the formula n represents the total number of readings, hence n=40. The mean is an

important and commonly used statistic to describe the center of a set of data.

A second important statistic used to describe the center of a set of data is the median. The

median of a data set can be roughly defined as the middle value of the data set, once the

values have been ordered according to size. The median is defined by the following formula:

�̅� =

1

𝑛∑ 𝑥𝑖

𝑛

𝑖=1

(1)

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A further statistic for describing the center of a data set is the mode. The mode is defined as

the value of the sample which occurs most often or with the greatest frequency. The mode

does not always exist and when it does it is not necessarily unique (Walpole & Myers, 1978).

It is unlikely that the values in a data set are all equal. Measures of spread are used to

express the variability of a set of data such as data showed in table 1. For example data sets

where all values are close to the mean have a small spread and data sets where values are

scattered widely about the mean have a large spread. A key measure of the spread of a data

set is the variance. The variance of a data set is defined by the following formula:

From equation 4 it is clear that the variance calculates the sum of the squared differences

between each data value and the mean of the data set, with the sum being divided by one less

than the number of terms in the sum. The standard deviation of the data set, s, is the square

root of the variance. The standard deviation is the easier of the two measures of spread to use,

because it is measured in the same units as the original data set. The variance is measured in

“squared units” which makes quantifying it awkward. For example the data set above would

have units of (kl/d)2.

A random experiment is an experiment whose outcome can’t be predicted with certainty

before the experiment is completed. Although it is impossible to predict the outcome of any

single repetition of the experiment one has to be able to list the set of all possible outcomes of

the random experiment. An example in the context of this project would be the measurement

of fire occurrence over a period of time in a given area.

Theoretically, random experiments must be capable of unlimited repetition and it must be

possible to view the outcome of each repetition of the experiment. The set of all possible

outcomes of a random experiment is called the sample space, denoted by S, of the random

experiment. Each repetition of the random experiment is called a trial and gives rise to only

one of the possible outcomes.

�̃� = 𝑋(

𝑛+1

2) if n is odd numbered.

(2)

�̃� =

𝑋(

𝑛2

)+𝑋

(𝑛2

)+1

2 if n is even numbered.

(3)

𝑠2 =

1

𝑛 − 1∑(𝑥𝑖 − �̅�)2

𝑛

𝑖=1

(4)

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If X denotes a variable to be measured in a random experiment, the value of X will vary

depending on the outcome of a random experiment. X is called a random variable whose

domain is a sample space (Introstat, 2014).

Random variables can either be classified as discrete- or continuous variables. Discrete

random variables usually take on natural numbers. The function f(x) is called a probability

distribution function or a probability distribution of the discrete random variable X if, for

each possible outcome of x,

1. F(x) ≥ 0.

2. ∑ 𝑓(𝑥) = 1.𝑥

3. P(X=x) = f(x).

A continuous random variable has a probability of zero of assuming exactly any of its

values and there is a probability density function, f(x), such that:

1. 𝑓(𝑥) ≥ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ∈ 𝑅

2. ∫ 𝑓(𝑥)𝑑𝑥 = 1∞

−∞

3. 𝑃(𝑎 < 𝑋 < 𝑏) = ∫ 𝑓(𝑥)𝑑𝑥.𝑏

𝑎

The cumulative distribution function (CDF), or distribution function as it is also known,

describes the probability that a random variable, X, with a given probability distribution will

have a value less than or equal to X.

The cumulative distribution F(x) of a discrete random variable X with probability

distribution f(x) is given by:

The cumulative distribution F(x) of a continuous random variable, X, with density function

f(x) is given by:

𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) = ∑ 𝑓(𝑡).

𝑡≤𝑥

(5)

𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) = ∫ 𝑓(𝑥)𝑑𝑡.

𝑥

−∞

(6)

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i.e. the formula gives the area under the probability density function from minus infinity to

x (Walpole & Myers, 1978).

2.1.1 Uniform Distributions

The uniform distribution is the simplest possible discrete distribution. In a uniform

distribution all values in the interval (a, b) have equal probability of occurrence. The most

common example of a uniform distribution is the throw of a fair die where the probability of

obtaining any one of the six possible outcomes is 1/6. All outcomes are equally possible,

hence the distribution is uniform (Introstat, 2014).

The uniform distribution with parameters a and b can be described by the following

probability density function:

The uniform distribution is often used as a continuous distribution in stochastic analysis

when little experimental data is available, but where extreme values are known or can be

estimated.

𝑓(𝑥) = {

1

𝑏 − 𝑎 𝑎 ≤ 𝑥 ≤ 𝑏

𝑓(𝑥) = 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(7)

Figure 2: Discrete Uniform Distribution (Johnson et al., 2011)

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2.1.2 Poisson Distribution

The Poisson distribution is a discrete probability distribution named after French

mathematician Simeon Denis Poisson. The Poisson distribution describes the probability of a

given number of events occurring during a fixed time. Alternatively the Poisson distribution

can also be used to determine the number of occurrences of an event in a fixed amount of

“space”. The conditions for a “Poisson process” are that events occur at random. This means

that an event is equally likely to occur at any instant of time.

Thus the Poisson distribution only has one parameter, λ which is the average rate at which

events occur during a period of time. Note that the time period referred to in the rate must be

the same as the time period during which events are counted.

Let the random variable X represent the number of events that occur during the time

period. X can be described by the Poisson distribution with parameter λ, i.e. X~P(λ), and has

probability mass function:

(Introstat, 2014).

2.1.3 Normal Distribution

The normal distribution was discovered by Abraham de Moivre in 1733 and is by far the

most important continuous probability distribution in the entire field of statistics. Its graph is

called the normal curve but is also referred to as the bell curve due to its bell shape. Figure 3

below represents the normal distribution.

Figure 3: The Normal Curve (Walpole et al., 1987)

𝑝(𝑥) =

𝑒−λλ𝑥

𝑥! 𝑥 = 0,1,2, …

𝑝(𝑥) = 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

(8)

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The distribution of many sets of data in nature, industry and research can be described by

the normal distribution. Normal curves may differ in how spread out they are, but the area

under any probability distribution curve will always equal 1.

The constant µ (the mean) is an indication of the location of the graph and marks the

center of the graph. The constant σ (standard deviation) indicates how spread out the

distribution is; as σ becomes larger the distribution becomes flatter.

Let X denote a random variable that can be described by the normal distribution. X is then

referred to as a normal random variable. The mathematical equation for the probability

distribution of X only depends on two parameters: µ (mean) and σ (standard deviation).

The normal distribution has probability density function

(Introstat, 2014).

𝑓(𝑥) =

1

√2𝜋𝜎2𝑒−

12

(𝑥−µ

𝜎)

2

− ∞ < 𝑥 < ∞ (9)

Figure 5: Normal curves with µ1 < µ2 and σ1 = σ2 (Walpole et al., 1987).

Figure 4: Normal curves with µ1 = µ2 and σ1 < σ2 (Walpole et al., 1987).

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2.1.4 Log-normal Distribution

A log-normal distribution is a continuous probability distribution whose logarithm is

normally distributed. Let X denote a random variable which is log-normally distributed, then

it follows that Y = ln(X) has a normal distribution. Figure 6 below is a graph of a log-normal

distribution with a mean (µ) of 0 and standard deviation (σ) of 1.

Figure 6: Log-normal distribution (Johnson et al., 2011)

From the figure it is clear that this distribution is positively skewed, meaning that it has a

long right-hand tail. The log-normal distribution has probability density function:

𝑓(𝑥) =

1

√2𝜋𝜎2𝑒

−12

(ln𝑥−µ

2𝜎2 )2

𝑥 > 0 (10)

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2.2 Modelling techniques

2.2.1 Deterministic models

The approach in the deterministic modelling technique is to eliminate uncertainties by

breaking variables up into smaller parts or components in order to “isolate” the uncertain

variables or elements from the “certain” variables or elements. The modelling approach then

assumes certainty by assigning a single value for each such uncertain variable. In order to

investigate the impact of the assumed values one has to conduct sensitivity-analyses for every

variable for which such a value has been assumed. This modelling technique often leads to

large, complex and inefficient models. Such models may be difficult to review and any

coding errors may remain undetected. The deterministic approach often leads to over-designs

of engineering systems, which means that unnecessary capital is spent on infrastructure

(Claassens, 2015).

2.2.2 Stochastic models

In the stochastic modelling technique the approach is to model the uncertainty that one

tries to isolate in the deterministic approach. It is thus not necessary to break the known

variables up into smaller parts. Instead of assuming a single point value for uncertain

variables, a range of values is used to describe the uncertain variable. The range of values can

be described by a probability distribution. By simulating different scenarios, every possible

value of the uncertain value (described by the probability distribution) is used in the

simulation.

This process is referred to as Monte Carlo simulation (see section 2.2.4). By definition the

“output” variables of the model will also be defined by probability distributions and not by

single point values. The benefit of stochastic modelling is that it is possible to model the risk

associated with each “uncertain” input and thus it is also possible to model the risk associated

with each “output”. This makes it possible to make a design decision based on pre-

determined risk limits (Claassens, 2015).

2.2.3 Basic modelling

Mathematical modelling is the description of real-life situations/events/changes using

mathematics (Quarteroni, 2009). A mathematical model allows one to understand the

interaction between different variables of a system. Mathematical models can further be

broken up into static or dynamic models. Static and dynamic models can then further be

classified as deterministic or stochastic. Figure 7 is a representation of the 4 different

modelling regimes that may be used to model engineering systems.

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From the diagram it is clear that models can either be Deterministic-Static, Deterministic-

Dynamic, Stochastic-Static or Stochastic-Dynamic. A dynamic model describes time-varying

relationships whereas a static model describes relationships, which stay constant over time.

Owens provides a general framework for the development of deterministic-dynamic

models of engineering systems. Such models are based on a set of system inputs (denoted

U), a set of system outputs (denoted Y) as well as a set of initial conditions.

The most commonly used method of expressing the relationship between the set of inputs

and the set of outputs is through an nth

order ordinary differential equation of the general

(non-linear) form expressing the time derivative(s) of the output set as a function of previous

values of the output set, and values of the input set. Owens also demonstrates that the static-

deterministic model is a special case of this general dynamic model where all time derivatives

are equal to zero (Owens, 1982).

Stochastic

Deterministic

Sta

tic

Dy

na

mic

Figure 7: Modelling Regimes

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Goodwin & Sing provides a general framework for the development of a dynamic-

stochastic model for engineering systems (Goodwin & Sing, 1984).

2.2.4 Monte Carlo Simulation

Palisade Corporation, the authors of @RISK, the software used in this project to conduct

stochastic analysis, gives the following definition of Monte Carlo simulation: “Monte Carlo

simulation (also known as the Monte Carlo Method) gives an overview of all the possible

outcomes of a decision and assesses the impact of risk on the decision, thus allowing for

better decision making.

The technique was first used by scientists working on the atom bomb and was named after

Monte Carlo, the Monaco resort town renowned for its casinos and gambling. Since its

introduction in World War II, Monte Carlo simulation has been used to model a variety of

physical and conceptual systems.

Today Monte Carlo simulation is a computerized, mathematical technique that enables

accounting for risk in quantitative analysis and decision making. The technique is used by

professionals in a wide variety of fields such as finance, project management, energy,

manufacturing, engineering, research and development, insurance, oil & gas, transportation,

and the environment.

Monte Carlo simulation enables risk analysis by building models of possible results by

substituting a range of values (a probability distribution) for any factor or input parameter

that has inherent uncertainty. It then calculates results over and over, each time using a

different set of random values from the probability functions. Depending upon the number of

uncertainties and the ranges specified for them, a Monte Carlo simulation could involve

thousands or tens of thousands of recalculations before it is complete. Monte Carlo

simulation produces probability distributions of possible outcome values.

System Input (U) Output (Y)

Initial conditions

Figure 8: Owens' framework for a deterministic-dynamic model

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During a Monte Carlo simulation, values are sampled at random from the input probability

distributions. Each set of samples is called an iteration, and the resulting outcome from that

sample is recorded. Through an appropriate number of iterations the probability distribution

of each output can be constructed. In this way, Monte Carlo simulation provides a more

comprehensive analysis of possible outcomes” (Palisade, 2015).

With abundance of computing power available to engineers and through software that

enables Monte Carlo simulation becoming readily available it is likely that stochastic analysis

of engineering systems will become more common, due to the benefits that will derive from

such analysis.

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2.3 Modelling water distribution systems

2.3.1 Components of a Water Distribution System

A water distribution system describes the facilities used to produce and supply potable

water from a source to the consumer. A water distribution system consists of a network of

components which will typically include:

1. A source;

2. Water Treatment Works;

3. Pump station (if needed);

4. Feeder Pipe;

5. Municipal storage tanks/reservoirs (or any other water storage facility);

6. A network of pipelines to carry water between different components and to consumers

and fire hydrants (Distribution Network).

Figure 9 is a schematic representation of a typical water distribution system, with a short

description of each component given below.

A river or a dam serves as a source of water. A dam is usually built if a river can’t deliver

a reliable supply of water, but the average supply exceeds the average demand and losses.

Water is then stored in the dam when the supply of the river exceeds the average demand and

losses, and can be withdrawn once the river runs dry.

The raw water which is extracted from the source is seldom suitable for human consumption.

Raw water is fed to the water treatment works where it is treated and rid of impurities.

Water is treated until it is of an acceptable standard. The treated water must look, smell and

taste acceptable.

Figure 9: Water Distribution System (Nel, 1993)

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It is also important that the water does not have unnecessarily high mineral concentrations

which may act aggressively on the infrastructure which transport it through the water

distribution system.

The feeder pipe transports water from the water treatment works to the municipal storage

tank.

If water can’t be supplied using gravity, or if an inadequate amount of water is supplied, it

has to be pumped through the system.

The reservoir (municipal storage tank) is filled with water from the feeder pipe. Water is

usually supplied at a constant rate over a certain period of time. Stored water is distributed to

consumers based on their immediate needs.

The distribution network consists of pipes from the reservoir to the consumer. Pipes must be

sized in such a way that the required amount of water can be provided during peak demand at

an acceptable pressure (Nel, 1993).

2.3.2 Bulk Water Supply Systems

Water Distribution Systems are usually divided into two components:

1. A distribution system and;

2. A bulk water system.

The distribution system is defined as that section which conveys water from the reservoir

to the consumers. The bulk water system is the section which delivers water from the source

to the reservoir. Reservoirs will now be discussed in more detail.

Reservoirs are storage containers for water and may also be referred to as municipal

storage tanks. In this project there is no difference in meaning between reservoirs and

municipal storage tanks. Reservoirs play an important role in the water distribution system as

they allow for the source to produce water at a constant rate and the consumers to extract

water at a varying rate. Put in other words, reservoirs balance the difference between supply

and demand (Van Zyl et al., 2008). Reservoirs are also able to supply consumers with potable

water should a supply interruption occur. Traditionally reservoirs have been sized based on

relatively simple guidelines, usually as a function of the average volume of water drawn by

consumers over a certain period. Kretzman (2004) summarised typical South African

guidelines for the sizing of reservoirs in terms of AADD:

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Table 3: Typical South African guidelines for sizing reservoirs and feeder pipes (Kretzman, 2004)

Authority Nature of Supply Feeder Capacity Reservoir Capacity

Department of Water

Affairs

Gravity feed

Pumped main

1.5 x AADD

1.5 x AADD

24h of AADD

48h of AADD

Co-operation &

Development

Gravity feed

Pumped main

1.5 x AADD

1.5 x AADD

24h of AADD

48h of AADD

National Building

Institute

One source

Two sources

1.5 x AADD

1.5 x AADD

48h of AADD

36h of AADD

National Housing

Gravity feed

Pumped main

Two sources

1.5 x AADD

1.5 x AADD

1.5 x AADD

20h of AADD

30h of AADD

66% of capacity with

one source

The guidelines are not only restricted to those listed in table 3. As early as 1952 this matter

had been internationally discussed at an IWSA conference. A survey at this conference

indicated that reservoir capacity varied from below 50% to more than 200% of the maximum

daily capacity of the water treatment plant feeding the reservoir, depending on which

guideline was used.

Besides the inconsistency, Kretzman (2004) found that there are further problems with

inflexible guidelines, which include the following:

1. No allowance is made for the size, character or unique features of the supply area;

2. A fixed feeder pipe capacity into the reservoir means that the designer doesn’t have the

freedom to exploit the optimal combination of feeder pipe and reservoir capacity;

3. No allowance is made in the guideline for the design of a reservoir according to

predetermined reliability.

Kretzman also noted that in South Africa reservoirs have to provide for one or more of the

following:

1. Emergency storage;

2. Fire storage;

3. Demand storage;

4. Operational requirements

2.3.2.1 Emergency Storage

Volume must be provided to guarantee water supply to consumers, even when the supply

to the reservoir is partially or completely discontinued. These events may be scheduled

maintenance, which is not a stochastic variable, or unscheduled events such as power failures,

pipe failures or source failures. The volume required for these unscheduled events is thus a

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stochastic variable, as neither time of occurrence nor the duration of the interruption can be

predicted (Kretzman, 2004).

2.3.2.2 Fire Storage

It is important to have an adequate water supply available for firefighting. Water for

firefighting is usually supplied through the water distribution system and is thus drawn from

the reservoir. Guidelines usually specify an additional volume of water for which allowance

must be made in the reservoir. This is based on the conservative assumption that the fire

demand will coincide with a period of maximum consumer demand and emergency use of

water. The required fire storage is not stochastic, as the time of occurrence nor actual fire

demand required can be predicted (Kretzman, 2004).

2.3.2.3 Demand Storage

Consumers draw water from the reservoir at a variable rate, while the supply to the

reservoir is delivered at a constant rate. The reservoir has to balance the difference between

inflow and outflow. Inflow of water to the reservoir is easily determined and usually well

controlled. Outflow is highly variable and usually determined by the cumulative effect of a

multitude of stochastic variables and is therefore itself a stochastic variable (Kretzman,

2004).

2.3.2.4 Operational Requirements

There could be additional requirements for service reservoir volume, such as freeboard

(dependent on the sophistication of level sensing and control equipment), bottom storage

(dependent on the potential of air or sediment entrainment at the outlet), or a pump control

band (required for automatic switching of pumps if water is being pumped to or from the

service reservoir).

These components are all deterministic, i.e. they can be calculated once and simply added

to the volume required for the stochastic components described above (Kretzman, 2004).

2.3.3 Reliability of Water Supply Systems

As discussed in the previous section, a water distribution system consists of various

components. A supply system must be able to deliver potable water to consumers in

prescribed quantities under a desired pressure. The reliability of a supply system is a measure

of the ability of the system to meet consumer demands in terms of quantity and quality under

normal and emergency conditions.

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The required water quantities and qualities are defined in terms of the flows to be supplied

within given ranges of pressure and concentrations (e.g. residual chlorine, salinity).

Water distribution systems play a vital role in preserving and providing a desirable quality

of life to consumers, of which the reliability of the supply system is a critical component.

Answering the question of whether a system is reliable or not is not straightforward, as it

requires both the quantification and calculation of reliability measures.

Reliability has traditionally been defined by empirical guidelines, such as ensuring two

alternative paths to each consumer node from at least one source, or having all pipe diameters

bigger than a minimum prescribed value.

By using guidelines such as these it is implicitly assumed that reliability will be assured.

The level of reliability that is provided is, however, not quantified or measured. This means

that limited confidence can be placed on such guidelines because reliability has not been

explicitly quantified

Recently there has been a growing interest in simulation approaches with more emphasis

put on explicit incorporation of reliability in the design and operation phases (Kretzman,

2004).

Lewis (1996) gives an accurate definition of reliability: “In the broadest sense, reliability

is associated with dependability, with successful operation, and with absence of breakdowns

or failures. It is necessary for engineering analysis however, to define reliability

quantitatively as a probability. Thus reliability is defined as the probability that a system will

perform its intended function for a specified period of time under a given set of conditions. A

product or system is said to have failed when it ceases to perform its intended function.”

The main function of a bulk water supply system is to supply water to reservoirs, and not

consumers. The reliability of supply systems can thus be defined in terms of their ability to

maintain water in the reservoir. A reservoir that runs dry would equate to a failure of the bulk

water supply system. The reliability of a bulk water supply system can thus be described in

terms of the failure behaviour of its reservoir(s). The failure behaviour can be described in

terms of the annual number of failure events, the total annual fail time, or the maximum

duration and variation in failure duration.

There is a clear relationship between the reliability of a bulk water supply system and the

capacity of its reservoir(s). Larger reservoirs would fail less often, thus providing a higher

level of reliability. The higher reliability has a higher associated capital cost and the potential

for water quality problems due to longer retention times. Reliability can also be improved by

increasing the capacity of the supply pipelines, changing pipe configurations or reducing the

time taken for repairing burst pipes (Kretzman, 2004).

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In a stochastic analysis of a water supply system the factors which influence the reliability

of the water distribution system such as water demand, pipe failures, fire occurrence, fire

duration and fire demand (“Key System Inputs”) are modelled according to appropriate

probability distributions.

2.3.4 The traditional modelling approach

Traditionally water distribution systems have been designed using the deterministic

approach. In South Africa the “CSIR Guidelines for human settlement and design” (also

known as the “Red Book”) serves as a design guideline for the design of a water distribution

and storage systems.

As discussed in the previous sections, municipal storage tanks play an important role in

the performance of a water distribution system. As supply and demand fluctuate throughout

the day the storage tank has to provide a suitable buffer to ensure delivery of water under

these differing conditions. A water distribution system fails when its storage tank runs dry.

Municipal storage tanks have traditionally also been sized using the deterministic

approach. Supply to a storage tank is usually fixed in order to minimize capital cost. Demand

on the other hand is highly variable. A stochastic approach is thus more suitable to design the

storage tank than a deterministic approach (Van Zyl et al., 2008).

2.3.5 The stochastic approach

A water distribution system is a highly variable engineering system with little

deterministic characteristics. It is thus more realistic to model a water distribution system

using the stochastic approach to design an optimized system (Van Zyl et al., 2008).

In their research Van Zyl et al. (2002) found that current design standards for bulk water

supply systems do not allow much design flexibility:

1. Current guidelines don’t allow the designer to differentiate meaningfully between

urban and rural systems and;

2. They don’t allow the designer to assume different levels of reliability.

For these reasons the authors deemed it necessary to provide a methodology for the

analysis of water distribution systems which couples reliability with system capacity. The

authors also commented that similar methods are commonplace in many other fields of civil

engineering, e.g. hydrology, however no such tools are generally available for designers of

bulk water supply systems.

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Since the research done by Van Zyl et al. (2002), stochastic analysis has been used in

subsequent research to analyze water distribution systems.

Van Zyl et al. (2008) used a stochastic model to analyze consumer demand, fire demand

and pipe failures in water distribution systems in the most critical time of the year (seasonal

peak).

From the analysis they were able to size the storage tank based on user-defined reliability-

criteria. The authors proposed that tanks should be sized for a failure rate of 1 in 10 years for

the peak seasonal demand.

Van Zyl et al. (2012) then used the stochastic model to investigate only the effect of

different user demand parameters on the reliability of the storage tank. From this analysis it

was found that tank reliability varies greatly throughout the year. The authors recommend

that municipalities do everything possible to ensure that their water distribution systems run

smoothly for the peak period.

Finally Van Zyl et al. (2014) used the stochastic model to analyze different configurations

of pipes to find the optimal combination feeder pipe configurations, the feeder pipe capacity

and the size of the tank for a given risk of failure. From their analysis the authors found that

the most optimal pipe configuration is a single-feeder pipe in most cases, but that two parallel

pipes are desirable for shorter feeder pipes. The authors also concluded that it is often cost-

effective to trade off smaller tank size with larger feeder pipe capacity.

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2.4 Fire Demand

2.4.1 Current design guidelines

Most countries around the world make provision for fire demand through their water

distribution systems. Most water distribution systems are sized according to the determined

fire demands. This is because fire demands require a high flow rate and volume of water in

order to combat big fires. The water distribution system should be able to cope with such

high demands. National design codes provide guidelines for the determination of fire

demand. In South Africa the Red Book serves as a guideline for the determination of fire

demand. It is important to note that the Red Book is based on SABS 090-1972 and has not

been updated to the changes made in the current SANS 10090:2003 design code (Davy,

2010). The red book specifies different fire risk categories based on building size and

building zoning. Table 4 below is an extract from the red book which shows the different fire

risk categories. Table 4: Red Book Fire Risk Categories (CSIR, 2000)

Fire Risk Category Description

High-Risk

Congested industrial and commercial

areas, warehouse districts, central

business districts and general

residential areas where buildings are

more than 4 storeys in height.

Moderate-Risk

Industrial, areas zoned "general

residential" where buildings are not

more than 3 storeys in height and

commercial areas normally occurring

in residential areas.

Low-

Risk

Group 1

Residential areas where gross floor

area of the dwelling is likely to be

more than 200 m2.

Group 2

Residential areas where gross floor

area of the dwelling is likely to be

between 100 m2 and 200 m

2.

Group 3

Residential areas where gross floor

area of the dwelling is likely to be

between 55 m2 and 100 m

2.

Group 4 Residential areas where gross floor

area of the dwelling is likely to be less

than 55 m2.

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The following section contains extracts from the Red Book which explain the design process

of a water distribution system for fire demand:

The elements in a water distribution system that is used to supply water for firefighting

are:

1. Trunk main: the pipeline used for bulk water supply

2. Water storage: reservoir and elevated storage

3. Reticulation mains: the pipelines in the water distribution system to which hydrants

are connected

4. Fire hydrants (any kind)

The applicable fire risk category determines the capacity of the above mentioned elements.

The fire flow and hydrant flow for which the water reticulation is designed should be

available to the firefighting team at all times. Close liaison between the water department of

the local authority and the fire service should be maintained at all times, so that the water

department can be of assistance in times of emergency – for example, isolating sections of the

reticulation in order to increase the quantity of water available from the hydrants at the scene

of the fire.

2.4.1.1 Design of trunk mains

The mains supplying fire areas should be designed so that the supply is assured at all times.

Trunk mains serving fire areas should be sized for a design flow equivalent to the sum of the

design instantaneous peak domestic demand for the area served by it, and the fire flow given

in table 5.

Table 5: Red Book design fire flow (CSIR, 2000)

Risk Category

Minimum

design fire flow

(l/min)

High-Risk 12 000

Moderate-Risk 6 000

Low-Risk - Group 1 900

Low-Risk - Group 2 500

Low-Risk - Group 3 350

Low-Risk - Group 4 N/A

Where an area served by the trunk main incorporates more than one risk category, then the

fire flow adopted should be for the highest risk category pertaining to the area.

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2.4.1.2 Water Storage

The storage capacity of reservoirs serving fire areas should, over and above the allowance

for domestic demand, include for the design fire flow obtained from figure 11 for duration at

least equal to that given in table 6.

Table 6: Red Book fire duration and storage (CSIR, 2000)

Fire-Risk Category

Minimum

design fire flow

(l/min)

Duration of

design fire flow

(h)

Storage required

for fire flow (kl)

High-Risk 12 000 6 4320

Moderate-Risk 6 000 4 1440

Low-Risk - Group 1 900 2 108

Low-Risk - Group 2 500 1 30

Low-Risk - Group 3 350 1 21

Low-Risk - Group 4 N/A N/A N/A

Where an area served incorporates more than one risk category, than the design fire flow

and duration used should be for the highest risk category pertaining to the area served by the

reservoir.

2.4.1.3 Reticulation mains

Reticulation mains in fire areas should be designed according to the design domestic

demand required. The mains should, however, have sufficient capacity to satisfy the criteria

given in table 7.

Table 7: Fire flow design criteria for reticulation mains (CSIR, 2000)

Fire-Risk Category

Minimum Hydrant Flow

Rate (for each hydrant)

(l/min)

Minimum Residential

Head (m)

High-Risk 1 500* 15

Moderate-Risk 1 500* 15

Low-Risk - Group 1 900 7

Low-Risk - Group 2 500 6

Low-Risk - Group 3 350 6

Low-Risk - Group 4 N/A N/A

*With a design maximum of 1 600 l per hydrant

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The minimum residual head should be obtained with the hydrant discharging at the

minimum hydrant flow rate, assuming the reticulation is operating under a condition of

instantaneous peak domestic demand at the time.

2.4.1.4 Estimation of Total Fire Demand

To estimate the effect of firefighting on the reliability of municipal storage tanks, the total

volume of water used for each fire is required. The code only specifies the maximum fire

duration and maximum fire flow rate. The fire volume can then be determined by multiplying

the maximum fire duration with the maximum fire flow rate (Kretzman, 2004).

2.4.2 Comparison with international codes

Despite the fact that organizations in the UK, USA, Canada, New Zealand and Germany

assisted South Africa in the creation of its first fire water provision code, SABS 090-1966,

the South African design code has remained much the same since its inception, whilst other

countries have significantly lowered their standards (Van Zyl & Haarhoff, 1993).

An international review of different fire codes by Van Zyl (1993) showed that wide

discrepancies exist amongst international codes, in terms of their underlying philosophy as

well as their numerical guidelines. Table 8 shows a selection of such values from different

fire codes.

Table 8: Comparison of fire standards (Van Zyl, 1993)

Parameter Germany Netherlands USA South Africa

Fire flow (l/min)

High-Risk 3 200 6 000 17 700 12 000

Moderate-Risk 1 600 3 000 11 800 6 000

Low-Risk 800 1 500 3 800 900

Pressure (m)

High-Risk 15 20 14 15

Moderate-Risk 15 20 14 15

Low-Risk] 15 20 14 7

Fire duration (h)

High-Risk 2 2 4 6

Moderate-Risk 2 2 3 4

Low-Risk 2 2 2 2

Code DVGW- KIWA #50 AWWA M31 SABS 090

W405 (1977) (1989) (1972)

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A water distribution system is usually evaluated under two separate loading conditions:

1. Demand under peak flow conditions, called peak demand,

2. A reasonable peak demand assumed to occur at the time of a major fire happening in

the supply area (referred to as fire public peak demand) plus the water required to

combat a fire (referred to as fire demand).

Internationally the fire public demand is lower than the peak demand for a specific supply

area. The South African code, however, does not differentiate between the fire public demand

and the peak demand used in the two loading cases (Van Zyl & Haarhoff, 1997). Table 9

below compares fire demands from various international codes:

Table 9: Comparison of fire demands (Van Zyl & Haarhoff, 1997)

Country Fire Demand used Approximate factor of Peak

Demand

South Africa Instantaneous peak demand 1,00

USA Daily peak demand 0,35

Germany Hourly peak demand of a day

with average water use 0,45

The Netherlands Hourly peak demand 0,63

From the table it is clear that the South African standard for fire public peak demand is

considerably higher than that of other countries. Van Zyl & Haarhoff (1997) put into

perspective the effect of fire public peak demand on network cost by redesigning actual water

distribution systems with different levels of fire public peak demand. The result of this cost

analysis is shown in figure 10 below.

Figure 10: Cost sensitivity to fire public peak demand (Van Zyl & HAarhoff, 1997)

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From the cost analysis it is clear that a small but significant saving in network cost can be

obtained if a fire public demand lower than the peak demand is used. A literature review by

the authors established the reasoning behind the use of lower fire demands in other countries:

1. Peak demand occurs over a small interval in a year, and the chance of a simultaneous

major fire, although it exists, is small. An analysis of the fire data for the

Johannesburg area showed that a chance of a major fire is the highest in mid-winter,

when the water demand is also the lowest.

2. In the case of a major fire, public water usage will be reduced owing to the fire

demand (decrease in pressure due to the increased fire demand) and public interest in

the fire.

3. A small fraction of fires are classified as major fires (those requiring more than

5 000 l of water to extinguish). In Johannesburg only 0,56 per cent of fires are

classified as major fires and on average only 12 major fires occur annually in the

Johannesburg municipal area.

The fact that fire flow is added to the peak demand in South Africa means that a situation

is analysed where a major fire occurs during the peak demand. The probability of this

happening, even though it does exist, is small. It thus becomes evident that the South African

design standard is overly conservative.

Furthermore, the European standard for fire water storage volume is 2 hours for all fire

risk categories. This, in combination with their lower fire flow requirements results in lower

storage volume requirements when compared to those in South Africa. Figure 11 compares

the storage of the South African standard with European standards and the volumes used in

90% of cases in Johannesburg (Van Zyl & Haarhoff, 1997).

Figure 11: Comparison of the South African fire storage volume standard, European

standards and actual volumes used in Johannesburg in 90% of cases (Van Zyl & Haarhoff, 1997)

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2.4.3 Comparison with actual fire data

Van Zyl & Haarhoff (1997) analysed fire data from Johannesburg spanning 12

consecutive years from 1980 to 1991. This study indicated that 90% of major fires in high-

risk areas were extinguished in 2 hours or less (where the Red Book specifies 6 hours), even

though the fire flows were significantly lower than the high-risk fire flow standard of 12 000

l/min (90% of the fires were extinguished using 3 100 l/min or less). The study also indicated

that 90% of major fires in Johannesburg high-risk areas were extinguished using water

volumes of 440 Kl or less. This volume is in stark contrast to the water volume requirement

of 4 320 Kl as specified in the Red Book.

Davy (2010) analysed data regarding fire events from the City of Cape Town’s fire

department. The data set was used to model the fire demands and durations of fires in Cape

Town. The modelled data was then compared to the South African design guidelines as given

in the Red Book. From this comparison Davy was able to show that fire flow requirements

for high-risk areas were unnecessarily high whereas the requirements for low-risk areas were

found to be inadequate (Davy, 2010).

The comparison clearly indicated that the South African design guidelines are overly

conservative. The design guidelines for fire demand are conservative in nature as they have to

cater for a vast range of water distribution systems. As a consequence most water distribution

systems are not efficiently designed. It is a well-established fact that engineering overdesign

can be costly. Knowing how much water is needed for fire demand would result in the design

of more efficient water distribution systems (Davy, 2010).

Similar research was done by Jacobs et al. (2014). In their research fire demand

requirements for 5 towns (from 3 different municipalities) in proximity to Stellenbosch was

analysed.

The data included duration of fires, method used for extinguishing the fire and whether the

water distribution system was used to extinguish the fire. From the research the authors were

able to determine fire flow volume and fire flow rate.

Jacobs et al. (2014) found that only 1.4 % of the data analysed represented fires which

were extinguished using water directly from the water distribution system during the

firefighting process. This does not mean that the water distribution system is not used at some

stage, but it indicates that the water distribution system is often not used during the

firefighting process.

The research done by Jacobs et al. also showed that fire flow requirements for high-risk

areas were unnecessarily high and the requirements for low-risk areas were inadequate, thus

confirming the research done by Davy. Jacobs et al. also found that flow rate for fighting fires

were much lower than what was required in the South African standards.

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2.4.4 The need for new design guidelines

The work done by Van Zyl & Haarhoff, Davy and Jacobs et al. (previous sections) clearly

indicate that there is a need for new basic design guidelines pertaining to fire demand.

Data for fire events, like the ones analysed by van Zyl & Haarhoff, Davy and Jacobs et al.,

is scarce. When the data is available it is not always possible to compare it to other data. The

research done by the authors is however a clear indication that there is a need for new design

guidelines concerning fire demand. This will invariably lead to the design of more efficient

water distribution systems.

2.4.5 Probabilistic Fire Demand

In order to create a stochastic model for fire demand it is necessary to describe the

occurrence of a fire, fire duration as well as the fire flow rate in a statistical sense. National

guidelines only provide deterministic information concerning fire demand and fire duration

and are thus not useful for stochastic analysis.

In order to obtain a probabilistic estimate for these parameters it is necessary to analyse

fire data for the region where the water distribution system is to be built. These analyses can

be cumbersome as fire data obtained from the local fire department is not always complete

and requires filtering which can take a lot of time. Another problem that occurs is that fire

departments may only keep fire records for a certain period of time and these fire records

may also contain significant gaps in the data. Major fires, which are of interest in a stochastic

analysis, occur infrequently and thus might not be represented in fire records.

Van Zyl and Haarhoff (1997) conducted one of the few studies in this regard. In the study

fire flow records of 12 consecutive years (1980-1991) of Johannesburg were statistically

analysed. From this database the “large” fire events (those using more than 5 000 litres of

water) were isolated and subjected to frequency analysis. Figures 12 and 13 summarises the

results from this analysis. From data such as this, the mean, the appropriate statistical

distribution and the standard deviation can be obtained.

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From these graphs simple descriptive statistics for both fire duration and fire flow in

Johannesburg could be determined. Table 10 and table 11 below is a summary of these

descriptive statistics.

Table 10: Descriptive Statistics for Johannesburg Fire Duration

Fire Duration

Sample Size 149

Mean 1.18

Mode 0.34

Std. Dev. 1.30

Skewness 3.88

Kurtosis 38.42

Percentile Duration (hrs)

5% 0.1

10% 0.19

25% 0.38

Median 0.8

75% 1.4

90% 2

95% 3.5

Max. 5

Figure 12: Johannesburg fire duration

(Van Zyl & Haarhoff, 1997)

Figure 13: Johannesburg fire flow

(Van Zyl & Haarhoff, 1997)

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Table 11: Descriptive Statistics for Johannesburg Fire Flow

Fire Flow

Sample Size 149

Mean 1 627.61

Mode 485.34

Median 1 115.00

Std. Dev 1 759.70

Skewness 3.85

Kurtosis 37.78

Percentile Fire Flow (l/min)

5% 160

10% 200

25% 600

Median 1 115

75% 1800

90% 3000

95% 4 780

Max. 10 000

A similar study was conducted by Davy (2010). Data regarding fire events received from

the City of Cape Town Fire Department was analysed to model the water demands and

durations of fires in the Cape Town area. The data set analysed by Davy spanned a period of

just over 5 years.

From the data it was found that commercial fires have duration of 3 hours and require a flow

of 1160 litres per minute (at the 95th

percentile). Industrial fires have duration of 4 hours and

20 minutes and require a flow rate of 1720 litres per minute (at the 95th

percentile).

Residential fires have duration of 1 hour and 20 minutes and require a flow rate of 830 litres

per minute. Descriptive statistics of the analysed data is given below in tables 12-14. Data

such as this is scarce.

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Table 12: Duration descriptive statistics and percentile values (Davy, 2010)

Table 13: Flow descriptive statistics and percentile values (Davy, 2010)

Duration Industrial Commercial Residential

Sample Size 284 594 7129

Range 65.95 45 59.98

Mean 1.52 0.67 0.55

Variance 33.39 4.82 3.5

Std. Dev. 5.78 2.19 1.87

Skewness 9.46 15.43 17.32

Excess Kurtosis 98.01 291.82 366.7

Percentile Value Value Value

Min 0.05 0 0.02

5% 0.08 0 0.08

10% 0.08 0.08 0.12

25% (Q1) 0.17 0.12 0.17

50 % (Median) 0.41 0.25 0.33

75% (Q3) 1 0.53 0.5

90% 2.95 1.34 0.9

95% 4.33 2 1.27

Max 66 45 60

Flow Industrial Commercial Residential

Sample Size 284 594 7129

Range 22401 18060 1.08E+05

Mean 441.82 342.11 276.58

Variance 2.08E+06 9.40E+05 1.82E+06

Std. Dev. 1440.6 969.26 1350.91

Skewness 12.9 12.62 72.33

Excess Kurtosis 192.85 205.26 5755.21

Percentile Value Value Value

Min 7 1 0.01

5% 30 20 33

10% 50 33 50

25% (Q1) 95.5 75 100

50 % (Median) 120 120 120

75% (Q3) 444 300 278

90% 865 602 602

95% 1388.8 1200.5 722

Max 22408 18061 108360

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Table 14: Volume descriptive statistics and percentile values (Davy, 2010)

It is interesting to note that Davy classified fires as being industrial, commercial or

residential whereas Van Zyl & Haarhoff do not distinguish between different fire categories.

When comparing the two data sets it is evident that they are very different. Observation of the

descriptive statistics for both sets of data clearly shows that the data collected by Van Zyl &

Haarhoff is more descriptive of big fires. This can be expected as the data analysed by Van

Zyl & Haarhoff spans 12 consecutive years. The data analysed by Davy on the other hand

spans a period of just over 5 years. It is important to note that big fires (which are of

importance to designers of water distribution systems) occur infrequently and might thus not

be reflected in the data analysed by Davy.

2.4.5 Probability of Fire Occurring

Historical fire data gives an insight into the probability of a fire occurring for a specific

region. Most of the time this is the only usable data for determining the occurrence of future

fires in a specific region. As mentioned previously, fire records are seldom analysed. This

means that not much information is available to describe the occurrence of a fire in a

statistical sense. The studies conducted by Van Zyl & Haarhoff (1997) and Davy (2010) are

two of the few studies that have been conducted to analyse fire data statistically.

Volume Industrial Commercial Residential

Sample Size 284 594 7129

Range 2276.7 3250.9 3.58E+04

Mean 63.72 28.73 15.92

Variance 6.40E+04 3.68E+04 1.86E+05

Std. Dev. 252.02 191.82 431

Skewness 6.28 13.08 80.25

Excess Kurtosis 43.16 189.44 6639.96

Percentile Value Value Value

Min 0.1 0.01 0.01

5% 0.35 0.1 0.5

10% 0.55 0.3 0.6

25% (Q1) 1.2 0.6 1.2

50 % (Median) 3.55 2 2.5

75% (Q3) 18 9 6.6

90% 98 36.12 17.8

95% 233.73 72 32

Max 2276.8 3250.9 35758.8

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In the case of Johannesburg, all the large fires (refer to previous paragraph) amounted to

149 fire events. The total Johannesburg supply area was divided into approximately 30 zones,

each with its own service reservoir(s). This indicates a historic "large fire frequency" of 0,4

fires/year for each zone (Kretzman, 2004).

Davy (2010) analysed fire data for the City of Cape Town Fire Department for a period

spanning just over 5 years (61 months). The data set contained 72 589 entries which were

separated into 10 spreadsheets namely:

1. Vegetation;

2. Commercial;

3. Hazardous Material;

4. Industrial;

5. Transport;

6. Institutional;

7. Public Assembly;

8. Residential;

9. Outside Storage;

10. Miscellaneous.

The distribution of data is shown in table 15 and the pie chart below.

Table 15: Category numbers and percentages for the original data set (Davy, 2010)

Nr. of Fires % Total

Commercial 1 758 2.4%

Hazmat 609 0.8%

Industrial 735 1.0%

Institutional 391 0.5%

Miscellaneous 389 0.5%

Outside

Storage 381 0.5%

Public

Assembly 168 0.2%

Residential 14 762 20.3%

Transport 3 955 5.4%

Vegetation 49 441 68.1%

Total 72 589

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Figure 14: Pie Chart of the original Data Set Constitution (Davy, 2010)

Data such as this is valuable in determining the probability of a fire occurring, especially if

fire categories are of interest. Davy decided to exclude the vegetation, hazardous material,

transport and miscellaneous categories from the analysis. These categories were excluded

because there is no specification for these types of fires in the design codes and because there

was no fire events logged for the miscellaneous category. Furthermore the data set contained

a few incomplete records which had to be cleaned.

The data set as received from the City of Cape Town Fire Department was already

categorized into the categories as shown in table 15. The data set, however described the

categories by a further field, subcategory. Davy decided that in order to gain a true

perspective and understanding of the fire behaviours of each more common fire type it was

essential to observe each of the fire types in isolation of the others, which could then be

compared to others to draw out similarities or differences. Each category was broken down

into its respective sub categories:

2.4% 0.8%

1.0%

0.5% 0.5%

0.5%

0.2%

20.3%

5.4%

68.1%

Original Data Set Constitution

Commercial Hazmat Industrial Institutional Miscellaneous

Outside Storage Public Assembly Residential Transport Vegetation

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Table 16: Category numbers & Percentages for the cleaned and verified data set (Davy, 2010)

Nr.of Fires % Total

Commercial 533 6.7%

Churches and

Halls 38 0.5%

Educational 113 1.4%

Flats 190 2.4%

Formal 2 273 28.4%

Hotels 28 0.3%

Industrial 244 3.0%

Informal 4 515 56.4%

Museums 2 0.0%

Night Clubs 12 0.1%

Medical 19 0.2%

Warehouses 40 0.5%

Total 8 007

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Figure 15: Pie Chart of Data set after cleaning & re‐categorisation (Davy, 2010)

Van Zyl et al. (2008) proposed a fire occurrence of 2 fires per year in their model based on

their research. Vlok (2010) used these same parameters in his model.

6.7%

0.5% 1.4%

2.4%

28.4%

0.3%

3.0%

56.4%

0.0% 0.1% 0.2%

0.5%

Data Set Constituition

Commercial Churches and Halls Educational Flats

Formal Hotels Industrial Informal

Museums Night Clubs Medical Warehouses

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3. Design of storage tanks from first principles

3.1 Introduction

When designing a storage tank from first principles the key design objective is to ensure

that the storage tank will meet the required assurance of supply. In South Africa, the required

assurance of supply for domestic water distribution is 98%. This means that during any

period of time a storage tank should be able to deliver a water supply to the consumers 98%

of the time, while there is an allowance for the storage tank to fail 2% of the time. In other

words no supply interruptions will occur for 359 days of a year, or alternatively a system

should fail no more than 175 hours in a year. The discussion below will highlight how this

key design objective is achieved with both deterministic and stochastic design techniques.

This discussion will also introduce some key aspects of stochastic modelling as used in this

research project as well as in industry.

3.2 Deterministic Design

Under a deterministic design approach, the designer will typically fix (by assuming single

point values) consumer demand patterns, inflow and outflow while adding fixed volumes at

regular intervals for emergency usage and fire demand and similarly modelling supply

interruptions of fixed duration at regular intervals. The designer will typically arrive at a

relationship between the annual Failure Rate and the storage tank capacity (typically

expressed in hours of AADD). Expressing the annual Failure Rate as a percentage would

guide the designer towards the required storage tank capacity to ensure the required assurance

of supply (98%).

Figure 16 below is an example of the relationship between Failure Rate and storage tank

capacity. From figure 16 it is evident that in order to ensure a 98% assurance of supply a

storage tank with a capacity of 11.2 hours AADD would be needed, this is shown by arrow A

in the diagram. Ensuring a 100% assurance of supply would require a storage tank with a

capacity of more than 32 hours AADD.

Given the uncertainty created by the use of single point values, how does the designer

know that his design will work? The answer is that he doesn’t. This is typically managed in

the following ways:

1. The designer can test how sensitive the performance of the design is to the single

point values through a sensitivity analysis. With a relatively large number of point

values assumed (as in this case) such a sensitivity analysis is a complex exercise

in its own right. Without sufficient statistical data for each input for which a single

point value was assumed, it is not possible to ensure that the sensitivity analysis

truly tests the robustness of the design;

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2. The designer can apply a safety factor to the model in order to compensate for the

uncertainty. The question is: what should the safety factor be? As in the case with

the sensitivity analysis the lack of sufficient statistical data effectively reduces this

process to guesswork. The likelihood is that this often leads to overdesign;

3. The designer can model a “worst case scenario”. This is similar to applying a

safety factor with the same likelihood of overdesign

These problems can largely be overcome by using a stochastic design technique.

3.3 Stochastic Design

To conduct a stochastic design, the designer would use a stochastic model as described in

the following section. From the stochastic model a probability distribution of Failure Rate is

obtained for a given storage tank capacity. A typical probability distribution of Failure Rate is

presented in figure 17 below.

Figure 16: Deterministic Assurance of Supply

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

8 10 12 14 16 18 20 22 24 26 28 30 32

Fail

ure

s p

er a

nn

um

as

%

Storage Capacity (hours AADD)

Deterministic Assurance of Supply

Mean

2%

Worst Case

A B

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To size a storage tank stochastically the designer would like to arrive at the same

relationship between annual Failure Rate and storage tank capacity as in the deterministic

case. The question is: which value from the probability distribution, figure 17, should be used

for this purpose? The design objective is a 98% assurance of supply and to achieve this

objective the value of the 98th

percentile (i.e. 0.0799%).

Figure 18 below illustrates the 98th

percentile values of Failure Rate for different storage

tank capacities. In this case it is clear that a storage tank capacity of just larger than 26 hours

of AADD would yield the design objective. Figure 18 also shows the relationship for lower

assurance of supply figures, clearly illustrating the reducing storage tank capacity for a

reduced design objective. Provided that sufficiently reliable statistical data was used for the

input variables in the stochastic model the designer does not have any of the uncertainty

faced by the designer in the deterministic approach as discussed above. At the same time the

stochastic methodology yields the optimum design meeting the design objective.

Figure 17: Probability Distribution of Failure Rate for 12 hour AADD storage tank

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Figure 18: Stochastic Assurance of Supply

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4. Methodology

4.1 Considerations

The purpose of this research project is to determine the impact of fire demand on the

performance of a water distribution system. In order to achieve this it is necessary to carry

out a stochastic analysis on a typical water distribution system. A typical water distribution

system is shown in figure 19 below. The system consists of a source, feeder pipe and storage

tank which delivers water to the users. This water distribution system was modelled on

Microsoft Excel and @RISK was used for the stochastic analysis.

Van Zyl et al. (2008) proposed such a stochastic analysis method to model both the

deterministic and stochastic components of consumer demand, fire demand and pipe failures

in a water distribution system. The same input parameters were used for this research project.

A detailed discussion of the input parameters is given in the following section.

Failure of the water source was assumed to be outside the scope of this study, the

behaviour of the storage tank was thus of importance for this study. The main purpose of a

municipal storage tank is to balance the difference between supply and demand in the most

economical way. A storage tank is said to have failed if it runs dry. The reliability of a

storage tank can thus be described through its failure behaviour. Increasing the tank capacity

will increase the tank reliability and decrease the costs incurred due to pumping. This

however comes with an increase in capital cost and an increase in the time that the water is

retained in the storage tank which may lead to lower water quality. It is thus important to

determine the optimal storage tank size in order to ensure tank reliability and lower capital

cost.

The stochastic model can be used to evaluate the behavior of the storage tank. Bulk supply

to the storage tank is delivered at a fixed flow rate in order to minimize capital costs and

allow water treatment plants and pumps to operate at maximum efficiency.

Supply

Storage

Users

Figure 19: Simple Water Distribution System (Van Zyl et al., 2008)

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The water distribution system is generally the main source of water for firefighting

purposes. The stochastic model will thus be used to determine the effect of fire demand on

the capacity of the storage tank (Van Zyl et al., 2008).

4.2 Input Parameters

4.2.1 Consumer Demand

Based on a review of water demand literature, Van Zyl et al. (2008) identified 4 generic

components for the water demand unit model: average demand, cyclic patterns, persistence

and randomness. The average demand is the average water consumption for the modelled

period. Within a year a number of cyclical patterns can be identified: seasonal patterns, day-

of-the-week and hourly patterns. After the deterministic factors have been identified and

removed from the data, it is possible to characterize the remaining white noise component

using a statistical distribution. In a good model the remaining white noise components should

have a mean of zero and a constant variance.

An annual average daily demand of 53.33 l/s was assumed for the model which is

equivalent to a low density suburban residential area of 3 000 to 5 000 dwellings (van Zyl et

al., 2008). The remaining parameters of the demand model were based on the measured

demand of 3 small residential towns located in the Moselle area, in the east of France. The

data consisted of hourly demands measured between September 1993 and December 1996. A

number of gaps were present in the data set and after removing all the incomplete records,

65% of the aforementioned period was covered. The data was provided by prof. Kobus van

Zyl in a Microsoft Excel spreadsheet, and the author decided to fit a simple model to the data.

The demand model is described by formula 11:

From the raw data (Y) it was possible to determine the AADD of the data set. This is done

by calculating the mean of all the daily water demands using equation 1 (section 2.1.1). The

AADD for the data set was 705.6 m3/d.

𝐷𝑡 = 𝐴𝐴𝐷𝐷 × 𝑆𝐹 × 𝐷𝐹 × 𝐻𝐹 + Ԑ

Where:

1. Dt = Demand at any time, t.

2. AADD = Average Annual Daily Demand

3. SF =Seasonal Factor

4. DF = Day Factor

5. HF = Hour Factor

6. Ԑ = white noise component

(11)

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Visual inspection of the data exhibited some seasonal variation. This seasonal variation

was removed from the data by using equation 12.

The seasonal factors are shown table 17 below.

After the seasonality had been extracted from the data it was able to determine the day-of-

the-week factors. From the de-seasonalised data each particular day (from Monday to

Sunday) is isolated and the average daily water demand calculated for that particular day over

the measured period. The average demand of every particular day-of-the-week was

determined by equation 13.

Day-of-the-week factors are given in table 18 below.

The new data set was removed of both the seasonal and daily patterns. The raw data

displayed readings from hour 1 to hour 24. The data was organised in columns from showing

the hourly demands from 1 to 24. In this way it was possible to determine the average

𝑆𝐹𝑚 =

�̅�𝑚

𝐴𝐴𝐷𝐷

Where:

1. �̅�𝑚 = mean montly demand

2. AADD = Average Annual Daily Demand

(12)

𝐷𝐹𝑑 =

�̅�

𝐴𝐴𝐷𝐷

Where:

1. �̅� = mean daily demand of a particular day

2. AADD = Average Annual Daily Demand

(13)

Table 17: Seasonal Factors

Table 18: Day-of-the-week factors

Month Jan Feb Mar Apr May Jun

Month Factor 1.02 0.90 0.88 1.07 0.96 1.04

Month Jul Aug Sep Oct Nov Des

Month Factor 1.21 1.15 0.91 0.86 0.82 0.99

Day Mon Tues Wed Thu Fri Sat Sun

Day Factor 1.00 0.93 0.96 0.95 0.97 1.14 1.06

Note that this data is from the Northern Hemisphere; therefore monthly

factors follow the opposite pattern to what would be expected in South Africa.

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demand of all the hour 1 readings and so forth. The hour factors were determined by using

equation 14.

The hour factors are summarized in table 19 below.

After all the factors had been determined it was tested whether the model is a good

representation of the raw data. This is done by comparing the actual water demand readings

of the raw data (Y) with the calculated water demands (Y*) as per Equation 15.

The mean of α was found to be zero with a standard deviation of 6.9. This confirmed that α is

a white noise component. Therefore α can be used as an estimate for Ԑ, as in equation 11.

The probability distribution function of α is shown in figure 20. The blue bars in the figure

show the actual data, from inspection it was evident that the actual data was close to a normal

distribution. The red line in the diagram is the data represented by a normal distribution. The

legend shows the minimum, maximum, standard deviation and mean of both the actual data

and the normally distributed data.

𝐻𝐹ℎ =

24�̅�

𝐴𝐴𝐷𝐷

Where:

1. �̅� = mean hourly demand of a particular hour

2. AADD = Average Annual Daily Demand

(14)

𝛼 = 𝑌 − 𝑌∗

Where:

1. Y =Raw data

2. Y* = AADD× SF× DF ×HF

3. α = modelling error

(15)

Table 19: Hour factors

Factors Hour 1 2 3 4 5 6 7 8

Hour Factor 0.02 0.02 0.02 0.02 0.02 0.02 0.03 0.04

Hour 9 10 11 12 13 14 15 16

Hour Factor 0.06 0.06 0.06 0.06 0.06 0.06 0.05 0.05

Hour 17 18 19 20 21 22 23 24

Hour Factor 0.05 0.05 0.06 0.06 0.05 0.04 0.03 0.02

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4.2.2 Supply System

A storage tank’s reliability depends on both the reliability and capacity of the system

supplying it. A municipal storage tank is supplied at a constant flow rate over an extended

part of the day, as alluded to before. Interruptions to the supply system can result from a

failure of a number of components including the water source, water treatment plant, pumps,

pipes or another storage tank.

From their literature review, Van Zyl et al. (2008) were able to identify that pipe failures

are most commonly dealt with and this was the only case considered in the model. The model

also assumed that the tank will be supplied by a single feeder pipe from the source.

For the purpose of this research project, 2 generic components of pipe failures were

identified: occurrence and duration. Pipe failures are random events and are thus best

modelled by a Poisson distribution. The Poisson distribution is described by equation 6 (see

section 2.1.2). Haarhoff & Van Zyl (2002) used a log-normal distribution to model the

duration of a supply failure, this approach was also adopted in the model.

Figure 20: Probability density function of white noise component

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In the model it was assumed that 2 pipe failures would occur every year. The rate

parameter for the Poisson distribution, λ, can then be calculated by dividing the number of

pipe failures per year by the number of hours in 1 year.

4.2.3 Fire Demand

Three generic components of fire demand were identified by van Zyl et al. (2008):

occurrence, duration and fire flow. The statistical values used for the fire demand model have

been determined from the fire study conducted by Van Zyl & Haarhoff (1997), see section

2.4.4. Just like pipe failures, the occurrence of a fire event is a random occurrence and was

modelled according to a Poisson distribution.

In the model it was assumed that 6 big fires would occur in a year. Similar to the pipe failure

model, the rate parameter for the Poisson distribution, λ, can be calculated by dividing the

number of fires per year by the number of hours in 1 year. In other words, there is a

probability of 0.0685% that a big fire can occur in any hour of the year (Van Zyl et al. 2008).

The input parameters used in the stochastic model is summarized in table 20 below.

Table 20: Summary of Input Parameters

Water Demand

Seasonal Peak

Factors

Month 1 2 3 4 5 6 7 8 9 10 11 12

PF 1.02 0.9 0.88 1.07 0.96 1.04 1.21 1.15 0.91 0.86 0.82 0.99

Hourly Peak

Factors

Hour 1 2 3 4 5 6 7 8 9 10 11 12

PF 0.44 0.39 0.38 0.41 0.45 0.52 0.77 1.05 1.34 1.47 1.49 1.47

Hour 13 14 15 16 17 18 19 20 21 22 23 24

PF 1.43 1.41 1.21 1.12 1.16 1.25 1.32 1.36 1.26 0.97 0.78 0.55

Day-of-the-week Peak Factors Day 1 2 3 4 5 6 7

Peak Factor 1.01 0.93 0.94 0.94 0.98 1.14 1.06

White noise distribution: Type Normal Distribution

White noise distribution: Mean 0

White noise distribution: Standard Deviation 6.905

Pipe Failure Characteristics

Failure Rate: Type Poisson Distribution

Failure Rate (failures/year) 2

Pipe Failure duration: Distribution Type Log-Normal Distribution

Pipe Failure Duration: Mean (hours) 1.49 (logarithm of value)

Pipe Failure Duration: Standard Deviation (hours) 0.48 (logarithm of value)

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Cumulative Frequency Plot: Pipe

Failure Duration

Probability Density Function: Pipe

Failure Duration

Fire Demand Characteristics

Fire Rate: Type Poisson Distribution

Fire Rate (fires/year) 6

Fire Duration: Distribution Type Log-Normal Distribution

Fire Duration: Mean (hours) -0.393 (logarithm of value)

Fire Duration: Standard Deviation

(hours)

0.66 (logarithm of value)

Cumulative Frequency Plot: Fire

Duration

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Probability Density Function: Fire

Duration

Fire Demand: Distribution Type Log-Normal Distribution

Fire Demand: Mean (l/s) 1.31 (logarithm of value)

Fire Demand: Std. Deviation (l/s) 1.31. (logarithm of value)

Cumulative Frequency Plot: Fire

Demand

Probability Density Function: Fire

Demand

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4.4 Model Description

Microsoft Excel was used to model the water distribution system. It was decided to model

the water distribution system on an hourly basis for one year, thus 8 760 hours. For this

reason the necessary inflows, consumer demands and fire demands were changed from l/s to

m3/hr (which is also equivalent to kl/hr). For every hourly interval the model describes

various conditions:

1. Time of day (from 1 to 24);

2. If a supply interruption occurs;

3. Duration of supply interruption;

4. Inflow;

5. Outflow;

6. If a fire event occurs;

7. Fire duration;

8. Fire demand;

9. Tank volume.

Various outputs that were of interest were also defined in the model. These outputs

displayed critical events that occur throughout the simulation:

1. Number of fires occurring throughout the year;

2. Average fire duration (hours);

3. Average fire demand (m3/hr);

4. A tank failure coinciding with a fire;

5. A fire coinciding with a supply pipe failure;

6. The number of hours in a year that the tank has failed (“Failure Rate”). The model is

based on hourly intervals and therefore duration of a failure modelled is a minimum

of 1 hour.

The inflow was assumed to be 1.2 times the AADD based on the model used by Van Zyl

et al. (2008).

4.5 The Monte Carlo Simulation

To study the impact of storage tank capacity (a key system design parameter) on the

number of annual failures the system was simulated over a 1 year period using Monte Carlo

simulation. Each simulation consisted of 10 000 iterations. Each iteration uses a different

sample for each of the input parameters from the probability distribution defined for each

input. The relative large number of 10 000 iterations was chosen in order to ensure that

virtually all possible combinations of input parameters that can occur in a year was in fact

modelled. In practice with more complex models even a greater number of iterations may be

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utilised. Note: this does not mean that the system was simulated for a continuous period of

10 000 years, but rather 10 000 possible instances of 1 year was simulated.

The simulation was repeated with different storage tank capacities specified for each

simulation. Tank capacities were specified in hours of AADD.

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5. Results

5.1 Results from Analysis

A key result of this research project is the failure characteristics for various user-specified

tank sizes as discussed above. This result is shown in figure 21. Note that the results given in

figure 21 are for the 98th

percentile of Failure Rate. From figure 21 it is evident that the

relationship between Failure Rate and tank capacity is that of a declining exponential. An

exponential curve was fitted to the data and it was found to be a good fit with R2 value of

0.9951.

From figure 21 it is clear that Failure Rate is sensitive to tank capacity. For instance, a

tank with 11.5 hours of storage will fail once or less a year (98th

percentile). This compared to

a tank of 8 hours storage which will fail 14 hours or less per annum (98th

percentile). Thus an

increase of 30.4% storage capacity decreases failure rate by 93.1%. To reduce the failure rate

of a tank with 11.5 hours storage to 1 in 10 and 1 in 100 years, respectively, the storage tank

capacity has to be increased by 41.64% and 80.14%.

It is important to note that the inflow is one of the determining factors of the number of

failures per year. The impact of the inflow on the reliability of the storage tank falls outside

the scope of this study, however it was noted that changing the inflow has a major impact on

the reliability of the storage tank. This confirms the findings of Van Zyl et al. (2008).

Figure 21: Failure rate vs. Tank Capacity

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Figure 21 shows the values of Failure Rate for the 98th

percentile. To give an idea of the

variability of the Failure Rate, Figure 22 shows the 25th

percentile, mean, and 75th

percentile

of Failure Rate for the different tank capacities. From the figure it is evident that a tank with

10 hours storage has a 50% probability of failure during the year.

To further illustrate the variability in the data, the probability distribution function is

shown in Figure 23 for the 12 hour AADD storage tank. From this probability distribution

function it is evident that a 12 hour AADD storage capacity tank has a probability of 85.3%

of failing once or less during the year. There is a 1% probability that the storage tank will fail

more than 9 times in a year.

There were no instances of a fire occurring during a storage tank failure or any instances

of a fire occurring during a supply pipe interruption. Even after simulating the model for

50 000 iterations no such events occurred. This does however not mean that these events will

not occur, but rather that the probability of such events is very small.

Even though the results given here is for a generic water distribution system, the results will

hold for any water distribution system with similar characteristics such as number of annual

fire events, number of annual supply pipe interruptions etc.

Figure 22: Variation of data

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5.2 Comparison of results with previous research

The input parameters used in this research project is the same as the input parameters used in

the study by Van Zyl et al. (2008) and it is thus necessary to compare the results of the two

projects. Van Zyl et al. (2008) analysed and sized a storage tank for seasonal peak conditions,

and thus for the minimum, rather than the annual, average tank reliability. For this reason the

seasonal pattern was not included in their stochastic model, but the simulation was run for a

day representing the seasonal peak in the network.

The authors determined the number of days to simulate by running the base model for

different number of days, varying between 1 000 and 10 000 000 days, and observing at what

duration the results stabilize. The authors tested the repeatability of the results by running the

simulation from ten different random seeds. The authors found that the tank failure properties

were consistently within 5% of the ultimate values when the number of tank failures exceed

2 000. All the results in their study were thus based on a minimum of 2 000 failure events.

The stochastic analysis done by Van Zyl et al. (2008) allowed the authors to calculate the

average number of failures per year for various user-specified tank sizes. The authors found

that the average number of annual failures can be described by a declining exponential (see

figure 24 below).

Figure 23: Probability Distribution of Failure Rate for 12 hour AADD storage tank

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The results of this analysis show that the average annual number of failures is very sensitive

to storage tank capacity. For example, a storage tank with 13.3 hours of storage will fail once

a year on average. To reduce the failure rate to one in 10 and one in 100 years, respectively,

the storage tank capacity has to be increased to 17.9 hours and 22.6 hours. Thus, increasing

the tank capacity by 35% and 70%, each increases the storage tank reliability by an order of

magnitude (Van Zyl et al., 2008).

The results of this research project are different to the results obtained by the stochastic

analysis of Van Zyl et al. (2008). Firstly the stochastic model used in this research project

models the system on an hourly basis for 1 year and not for a day representing the seasonal

peak in the network. The system was simulated for a 1 year period using Monte Carlo

simulation. Each simulation consisted of 10 000 iterations. This means that for every hour in

the stochastic model 10 000 different scenarios were simulated, this does not mean that the

system was simulated for 10 000 years. Van Zyl et al. (2008) simulated their system between

1 000 and 10 000 000 days, it is not clear whether the authors simulated their system between

1 000 and 10 000 000 iterations, or whether 10 000 000 days were actually simulated in their

analysis. There is a big difference between the two, and most probably the authors mean to

say that they simulated their model for between 1 000 and 10 000 000 iterations. By

simulating the system on an hourly basis for 1 year, it is possible to include sequential events

in the model such as a fire occurring soon after a supply pipe failure or two fires occurring

soon after one another. This is not possible when simulating the peak day. It is thus clear that

there is a difference between the two modelling approaches.

The results given by Van Zyl et al. (2008) (figure 24) are for the mean annual average

number of tank failures. The results of this research project are given for the 98th

percentile of

Failure Rate. See section 3.1 for a detailed description.

Figure 24: Annual average number of tank failures as a function of the tank capacity (Van Zyl et al., 2008)

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Finally different software packages were used for the stochastic analysis in both sets of

research. A commercial software package that is used in industry was utilised for the

stochastic analysis under this research project, while Van Zyl et al., (2008) developed their

own software to run the stochastic analysis. The author does not have any knowledge of the

software used by Van Zyl et al. (2008) and is thus not able to comment on the software.

5.3 Sensitivity Analysis

5.3.1 Introduction

In the previous chapter the results of the stochastic analysis modelling the Key Inputs were

presented. Each of these Key Inputs could influence the reliability of the water distribution

system as measured by Failure Rate. One of the objectives of this project, however, is to

assess the impact of fire demand on the reliability of the water distribution system modelled.

This section investigates the impact of fire demand on Failure Rate through sensitivity

analysis.

5.3.2 Sensitivity Analysis used in previous studies

Previous studies have made use of sensitivity analysis to determine the impact of fire

demand on storage tank capacity. Previous unpublished studies by the RAU water research

group have shown that fire demand has an almost unnoticeable effect on supply system

reliability. Based on this evidence, Van Zyl & Haarhoff (2002) used “extreme” fire

parameters in their model to check whether fire demand can be ignored even under extreme

circumstances. Table 21 below summarises the fire parameters used in their model.

Table 21: Input Parameter values used for sensitivity analysis (Van Zyl & Haarhoff, 2002)

Name Units Typical Extreme

Fire Frequency p.a. 0 52

Fire Duration Hours 2 4

Fire Demand

(% AADD)

70% 140%

From this analysis Van Zyl & Haarhoff (2002) concluded that fire demand has no impact

on the reliability of a water supply system.

Vlok (2010) used a 50% higher fire occurrence in his sensitivity analysis to determine the

effect that this might have on the size of the storage tank in his model. It was found that a

larger tank capacity is needed to obtain the same reliability, however the increase was not

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substantial, being below 1%. Vlok (2010) attributed this small increase to the small volume

of water required to combat a fire.

It is interesting to note that the aforementioned authors used deterministic methods to

conduct out sensitivity analyses.

5.3.3 Methodology for Sensitivity Analysis used in this project

During a Monte Carlo simulation an extensive volume of statistical data is generated. In

this project each simulation consisted of 10 000 iterations – this means 10 000 values were

generated for each input variable and each recorded output variable for each hour of a year.

One of the key benefits of using commercial software such as @RISK for stochastic

modelling, is the capability and functionality it provides to further analyse this volume of

data. In this research project, the Scenario Analyses functionality provided in @RISK was

employed to assess the impact of fire demand on the Failure Rate. This analysis was

conducted for storage tank capacities ranging from 12 hours AADD storage capacity to 24

hours AADD storage capacity.

The methodology for the Scenario Analysis as conducted is as follows:

1. Each “scenario” is defined by setting specific target values (in terms of the percentile

to be achieved or exceeded) for the Failure Rate. In this instance target values were

set at the 50th

, 75th, 80th

, 85th

and 90th

percentiles.

2. The median and standard deviation of each of the input variables for the whole

simulation (i.e. 10 000 iterations) are calculated.

3. A “subset” of data for each “scenario” defined as per 1. above is created containing

only the data for iterations in which the set target value for the Failure Rate is

achieved or exceeded. Thus a separate subset of data is created for those iterations in

which the 75th

percentile of Failure Rate is achieved or exceeded. Similar subsets of

data are created for the 80th

, 85th

and 90th

percentiles.

4. The median of each input variable is calculated for each subset of data.

5. For each input variable, the difference between the simulation median (as calculated

in step 2) and the subset median (as calculated in step 4) is calculated and compared

to the standard deviation of the input variable (as calculated in step 2). If the absolute

value of this difference is greater than ½ a standard deviation, then the input is

deemed to be “significant” in terms of achieving the set target value. Likewise if this

difference is less than ½ a standard deviation, then the input is deemed not to be

“significant” in terms of achieving the set target value.

6. Each significant input variable identified in step 5 is listed in the scenario report.

It is proposed for fire demand to be a significant contributor to Failure Rate it should test

as significant (in terms of the test outlined above) at the 50th

percentile.

From the analysis conducted, as described above, the following results were obtained:

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1. Generally fire demand is not a significant input variable for Failure Rate, except

for cases of extreme fire demand.

2. As storage tank capacity increases, fire demand is only a significant input for

Failure Rate at the percentiles higher than 80%. This trend is as follows:

a. For 12 hour AADD storage capacity, fire demand is significant for the 80th

percentile of Failure Rate or above.

b. For 14 hour AADD storage capacity, fire demand is a significant input for

the 90th

percentile of Failure Rate or above.

c. For 16 hour AADD storage capacity, fire demand is a significant input for

the 95th

percentile of Failure Rate or above.

d. For storage capacities greater than 22 hours AADD, fire demand is not a

significant input for Failure Rate.

The conclusion from this analysis is that for relatively low storage capacity (12 hours or

less) fire demand is a significant input variable for Failure Rate. At higher storage capacities,

as prescribed by the Red Book, fire demand is not a critical input variable for Failure Rate,

except possibly in certain extreme cases of fire demand.

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6. Discussion and Conclusion

The results obtained from this project confirm the previous results obtained from the research

of Van Zyl et al. (2008) and Van Zyl & Haarhoff (2002).

The results of this project has shown that for storage capacities below 22 hours AADD,

fire demand is only a significant input for storage tank Failure Rate above the 80th

percentile.

Above 22 hours AADD storage capacity fire demand is not a significant input for storage

tank Failure Rate.

South African design guidelines typically recommend storage tank capacities varying

between 24 hours AADD and 48 hours AADD (see section 2.3.2), with a certain amount of

volume reserved for fire demand. This is overly conservative as fire demand is not a

significant input for tank Failure Rate above 22 hour AADD storage capacity.

In municipal areas with ongoing growth and development the AADD of those supply areas

will increase annually, thus reducing the storage capacity (in terms of hours AADD) every

year. Storage capacity can also reduce due to poor operations and maintenance practice by

the relevant municipality. The results of this project demonstrate that as the storage capacity

decreases, fire demand becomes a significant input for the number of annual storage tank

failures. Based on the results of this project, storage capacities of less than 22 hours AADD

may be vulnerable in cases of extreme fire demand. It is recommended that municipalities

keep track of growth in supply areas, and if storage tank capacities fall below 22 hours

AADD storage alternatives should be provided for fire demand.

The probability of a fire occurring during a storage tank failure and a fire occurring during

a supply pipe interruption is very small. No such events occurred in the simulation, however

utilising a greater number of iterations in the simulation could possibly yield such events. The

author did not simulate the model for more than 50 000 iterations.

The results of this research project is given for a generic water distribution system,

however the results will hold for any water distribution system with similar characteristics

irrelevant of the layout of the water distribution system.

By building a dynamic stochastic model of a simple water distribution system, the

relationship between storage capacity and water distribution system performance could be

demonstrated. In doing this the impact of fire demand on water distribution system

performance could be determined. The author has gained a thorough understanding of

stochastic analysis and a commercial software package utilised for the stochastic analysis,

@RISK. The goals that were set out at the start of this project have thus been achieved.

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