STI 2014 Schomaker & Noyons - methodological study to structure liquid biofue...
Niel Claassens-CLSNIE001
1. Research Project
16 November 2015
Determination of the Impact of Fire
Demand on Water Distribution
System Performance Using Stochastic
Modelling Techniques
2. 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:
𝑥̅ =
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.
𝑥̃ = 𝑋(
𝑛+1
2
)
if n is odd numbered. (2)
𝑥̃ =
𝑋
(
𝑛
2
)
+𝑋
(
𝑛
2
)+1
2
if n is even numbered.
(3)
𝑠2
=
1
𝑛 − 1
∑(𝑥𝑖 − 𝑥̅)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−λ
λ 𝑥
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
√2𝜋𝜎2
𝑒
−
1
2
(
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
√2𝜋𝜎2
𝑒
−
1
2
(
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
Static
Dynamic
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 m2
.
Group 3
Residential areas where gross floor
area of the dwelling is likely to be
between 55 m2
and 100 m2
.
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 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
Failuresperannumas%
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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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 =
𝑥̅ 𝑚
𝐴𝐴𝐷𝐷
Where:
1. 𝑥̅ 𝑚 = mean montly demand
2. AADD = Average Annual Daily Demand
(12)
𝐷𝐹𝑑 =
𝑌̅
𝐴𝐴𝐷𝐷
Where:
1. 𝑌̅ = 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 =
24𝐻̅
𝐴𝐴𝐷𝐷
Where:
1. 𝐻̅ = 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)
66. 64
Research Project
16 November 2015
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
