Proceedings of the Institution of
Civil Engineers
Water Management 160
June 2007 Issue WM2
Pages 115–121
doi: 10.1680/wama.2007.160.2.115
Paper 14617
Received 25/01/2006
Accepted 27/11/2006
Keywords:
field testing & monitoring/
infrastructure planning/water supply
Carla Tricarico
Lecturer, Dipartiment di Meccanica,
Strutture, Ambiente e Territorio,
University of Cassino, Italy
Giovanni de Marinis
Professor, Dipartiment di Meccanica,
Strutture, Ambiente e Territorio,
University of Cassino, Italy
Rudy Gargano
Associate Professor, Dipartiment di
Meccanica, Strutture, Ambiente e
Territorio, University of Cassino, Italy
Angelo Leopardi
Assistant Professor, Dipartiment di
Meccanica, Strutture, Ambiente e
Territorio, University of Cassino, Italy
Peak residential water demand
C. Tricarico
PhD,
G. de Marinis
PhD,
R. Gargano
PhD
Residential water consumption has been analysed by
monitoring a water distribution system in a small town of
about 1200 inhabitants, Piedimonte San Germano, in
southern Italy. The design of a water distribution system is
usually undertaken with reference to the maximum water
required by customers—one of the most onerous
operating conditions to which an hydraulic network is
exposed. The aim of the present work has been to
contribute to the characterisation of the peak water
demand through statistical inferences on a large data
sample collected from the system under consideration.
Specifically, the data have been analysed for the effect of
resampling the raw data with respect to time interval on
the estimate of peak demand factor. Formulae are
suggested to estimate the maximum flow demand for
small towns, in relation to the number of users. In
addition, statistical inferences have shown that the
stochastic, maximum flow demand is described by the
log-normal and Gumbel models. With reference to small
residential areas, the parameters of such statistical
distributions have been estimated. These have shown that
the coefficient of variation (CV) of the peak water demand
is a function of the number of users. Although these
results are only directly applicable to the specific context
from which they have been obtained, the comparison with
the sparse data available in the technical literature leads
to the belief that the proposed relationships could be
extended to other small residential areas.
1. INTRODUCTION
The maximum residential water requirement assumes a
noteworthy importance in the study of water distribution systems
(WDSs) because the peak flow demand represents one of the most
onerous operating conditions of the network. The maximum
residential demand is therefore taken into account when
designing or rehabilitating hydraulic networks.
Diverse and complex factors influence water consumption;
therefore, some researchers (e.g. Garcia et al.,1 Buchberger and
Wells,2 Alvisi et al.3 and de Marinis et al.4) have studied these
using experimental approaches. Indeed, through statistical
inference on data samples, these researchers have proposed some
models to represent residential water demand.
In the technical literature, copious relationships exist to evaluate
the peak phenomenon (e.g. Babbit5 and Rich6) but they are
Water Management 160 Issue WM2
and A. Leopardi
PhD
usually with reference to residential areas with a minimum of
5000–10 000 inhabitants. Less attention is given to the case of a
small number of users for which, moreover, the peak
phenomenon is often more important if compared with the
average flow requirement.
In addition, it should be noted that the study of small town
hydraulic networks has relevance in the case of larger
conurbations. In these cases, indeed, the larger network may be
seen as being segregated into smaller sections, of which each
single node will supply a small number of inhabitants.
With the aim of contributing to the characterisation of water
demand models, a WDS of a small town in southern Italy,
Piedimonte San Germano (PSG), has been monitored. The
redundancy of this system allows the consideration of the flow
measurements being equal to the water demand required by
users. Even if this study refers to the specific small town
analysed, the results obtained are in agreement with those
obtained from the study of similar real-life WDS (Alvisi et al.3
and LATibi7).
The study of maximum water demand has been undertaken
considering both a deterministic approach and a more accurate,
probabilistic approach. Several researchers, indeed, have
highlighted the need to tackle flow demand with a specific
probabilistic approach as a stochastic variable (e.g. Bao and Mays8
and Gargano and Pianese9). Some formulae to describe the
maximum flow demand are suggested.
Probabilistic models to describe the peak flow requested and the
relative parameters (mean and standard deviation) have been
obtained through statistical inferences on experimental data
collected from the network under consideration. These results
could be useful in a probabilistic approach to generate
homogeneous data samples of the maximum flow demand using a
sampling technique such as the Monte Carlo method.
2. THE PIEDIMONTE SAN GERMANO HYDRAULIC
NETWORK
In order to obtain realistic demand data, a monitoring system has
been installed on a real hydraulic network. The WDS under
consideration here comprises a section of the PSG hydraulic
network.4,10,11 The network topology is composed of 45 pipes
(comprising 12 loops), 33 junction nodes and a reservoir (node 0),
as illustrated in Fig. 1.
Peak residential water demand
Tricarico et al.
115
0
3a
1
7
8
42
31
1
29
2
2
43
3
30
22
26
34
3
24
4
4
35
28
23
27
30
31
21
26
32
25
20
32
5
24
19
40
25
36
5
23
33
22
28
29
45
41
7
40
27
Monitoring point
10
39
7
Data logger
8
17
16
11
17
16
12
8
9
22b
Node
9
13
44
12
14
15
13
33
50m
18
18
10
38
Pipe
0
20
11
6
37
8
15
19
6
14
100m
Fig. 1. Topology and monitoring arrangements of the PSG hydraulic network
All of the existing pipes are cast iron, with the exception of two
newer branch lines (43 and 45) which are high-density
polyethylene (HDPE). The measurement system consists of four
pressure loggers and four bidirectional electromagnetic flow
meters, each connected to a data logger. The location of the
meters is shown in Fig. 1. The measurements could thus be taken
continuously with an acquisition frequency of up to 1 Hz.
An example of data collected by one of the meters in a day is
illustrated in Fig. 2.
In addition to obtaining a detailed knowledge of the physical
system, a census of consumers was undertaken with the aim of
determining the principal characteristics of the users and their
sanitary fittings. The portion of the hydraulic network considered
supplies 400 customers (i.e. properties) under contract, with
2·5
Cd = Q(t)/μq
2
Using these data, it was further possible to estimate accurately the
exact number of properties (Nc) and consequently the number of
people (NAb) that can be considered to be supplied from each
measurement point (Table 1). Through the use of mass balance
equations, it has been possible to investigate the water demand
not only at the measurement points but also for other numbers of
inhabitants derived from the flows taken at the measurement
points, as shown in the last two rows of Table 1.
The study of water demand, using flow data collected from a
real-life WDS, cannot ignore leakage estimation. Neglecting this
could lead to the consideration of water demand higher than
reality. In the case of small towns, the minimum night flow datum
could be interpreted as the value of possible leakage.12–15 This
approach has been used to estimate the leakage value for the data
analysed herein. The resulting values have been compared with
1·5
Measurement point
Nc
NAb
3a
7
8
22b
7–(8 þ 22b)
3a–7
400
320
48
20
252
80
1220
981
144
60
777
239
1
0·5
0
00:00
06:00
12:00
Time
18:00
00:00
Fig. 2. Demand data collected over 24 h from meter 7 (data
collected each second and averaged over 1 min intervals)
116
around 1220 inhabitants. The dominant contract type for this
section, accounting for some 96.8% of contracts, is for domestic
usage.
Water Management 160 Issue WM2
Table 1. Number of properties and inhabitants supplied at each
measurement point
Peak residential water demand
Tricarico et al.
those obtained from the methodology of Buchberger and
Nadimpalli16 and are seen to have produced concordant results.
Moreover, the PSG system is redundant, having pressures in all
nodes of the network above the required minimum and little
piezometric head variation during the day.10,11 For this reason, the
leakage has been considered constant throughout the day. In
addition, the redundancy of this system allows the use of data
collected from the monitoring system to characterise the water
demand, once the leakage has been estimated.
Number of inhabitants
Babbitt5
Funel18
Fair and Geyer19
Rich6
100
250
750
1000
1250
7.9
9.8
5.2
7.3
6.6
—
5.1
6.3
5 .3
—
4 .9
5 .3
5.0
4.4
4.8
5.0
4.8
—
4.7
4.8
Table 2. Peak coefficient for small residential areas
3. PEAK DEMAND COEFFICIENT
Data collected by the PSG monitoring system over a period of
almost two years show that the average flow does not change
significantly with seasons and years. This is owing mainly to two
factors. First, in the PSG network the type of usage is especially
indoor water use. Second, the number of inhabitants supplied is
constant throughout the year because the seasonal fluctuation is
balanced. The seasonal component, moreover, being strictly
dependent on the specific characteristics of the WDS under
consideration, cannot be generalised and its analysis could lead to
results that can be considered as being valid only in the specific
context in which they have been obtained. These considerations
meant that the focus of the study was on the daily demand
variation, neglecting the seasonal component.
The residential water requirement is a function of multiple factors,
many of them specific to the conurbation under consideration.
This explains the numerous relations and tables available in the
technical literature, which suggest means for estimating the
maximum water consumption (e.g. Babbit5 and Rich6).
These relations can often lead to extremely different estimates of
the peak factor and, usually, they do not take into consideration
the random nature of the peak water demand. A common element
to most of them, however, is the expression of the peak factor as a
function of the number of inhabitants NAb, as for example
suggested by Babbitt5 in the relationship
2
Figure 2 illustrates the typical daily demand pattern of a small
Italian residential area (e.g. Molino et al.17 and Alvisi et al.3), in
which the maximum value of water demand is reached early in the
morning and two further peaks of lower intensity are present,
corresponding with lunch and dinner time. Fig. 3 shows the same
daily pattern but takes into account three different measurement
sections with a different number of users. The flow required by users
has been represented dimensionlessly (Cd) considering the mean
demand requested by users every 15 min of the day over the daily
average flow, q . Results show that the daily Cd pattern does not
vary its shape during the day with the number of users supplied.11
A dimensionless expression of the maximum water required is
thus given by the peak demand coefficient Cp, a ratio of the
maximum flow, QM, against daily average flow, q
Cp ¼
1
QM
q
2·50
1220 users
981 users
777 users
Cd = Q(t)/μq
2·00
1·50
:
NAb 0 2
0:2
:
ffi 20:NAb
Cp ¼ 5
1000
Even if equation (2) originally refers to Cp evaluation for sewer
systems, it has been widely applied in the peak drinking water
studies for conurbations with at least 5000–10 000 inhabitants.
Table 2, referring to some of the well-established relations and
tables in the technical literature,5,6,18,19 shows the Cp values
obtained when considering a number of users similar to the case
study presented in the current research.
4. EFFECT OF THE SAMPLING INTERVAL ON THE
PEAK FACTOR
In the technical literature, the maximum water demand is
usually related to the hour of the maximum demand. Equation (1)
could be obtained as the volume of water required at the peak
hour over the average, hourly flow demand volume.
Assuming a time interval of 1 h could result in a lower
estimation of demand than is correct. Indeed, taking the average
may neglect major peaks that could arise during the peak hour.4
On the other hand, considering finer time scales (e.g. 1 s) would
produce more information than can be justified by the quality
of the hydraulic models, which operate at coarser time intervals,
and could lead to the over-fitting of the model.20 Researchers de
Marinis et al.4 demonstrated that 1 min intervals can be
considered to be a good compromise.
1·00
0·50
0·00
00:00
06:00
12:00
Time
18:00
00:00
Fig. 3. Cd values varying the number of users (data collected each
second and averaged over 15 min intervals)
Water Management 160 Issue WM2
The effect of sampling resolution has been investigated in the
present study. From one set of flow data collected at 1 min
intervals, several further data sets have been derived
(t 2 f10 ; 20 ; 50 ; 100 ; 200 ; 300 ; 600 g). For each of these, the
corresponding maximum Cp values (MCp) have been estimated.
Results obtained for one of the measurement sections (termed ‘3a’)
are shown in Fig. 4, from which it can be seen that when
considering t ¼ 1 h, instead of 1 min, leads to an
underestimation of about 28% of demand during the peak.
Peak residential water demand
Tricarico et al.
117
2·8
6
2·6
4
Cp
MCp
2·4
2·2
2
2·0
1·8
0
MCp = 2·6 – 0·14 ln Δt
0
250
500
750
1·6
0
20
40
1000
1250
1500
NAb
60
Δt: min
Fig. 6. Curve of average Cp and confidence intervals with 99%,
98%, 95% and 90% levels
Fig. 4. The effect of sampling interval (t) on Cp
Consequently, this analysis of demand has been undertaken using
1 min interval data.
5. PEAK FLOW DATA ANALYSIS
The analysis has been performed taking into account weekly data,
collected over a period of almost two years. The data collected
show that on weekdays the peak phenomenon is more marked
than for weekends. Indeed, the average daily water volume
supplied on Saturdays or Sundays is greater than the average daily
water volume supplied on a weekday (Monday–Friday).
Analysis of the data has shown that the maximum water
requirement can be represented as a function of the number of
users. Fig. 5 illustrates the regression curve of maximum values of
mean Cp, that are fitted by the curve following the relationship in
equation (3).
:
0 2
Cp ¼ 11NAb
3
This expression can thus allow the estimation of the maximum
flow required using a deterministic approach.
It should be noted that just the outlier point (T) in Fig. 5 is not
fitted by the curve. This point relates to the measurement point 8
of the network and may be explained considering that in the
6
branched pipes downstream of point 8, the users are equipped with
small tanks with pumps. This results in the truncation of the water
demand during the peak requirement, reducing its effective value.
The presence of pumps is testament to the inadequacy of the
private links to the hydraulic network as the measured pressures at
the corresponding node are fully satisfactory.
The proposed expression (3) is similar to that related by
Babbitt,5 equation (2), that refers to Cp evaluation for sewer
systems, but with a lower coefficient. If this expression
represents a useful relationship for estimating the maximum
flow demand with a deterministic approach, the description of a
stochastic variable—such as the water demand—should be
studied with a probabilistic approach (e.g. Bao and Mays8 and
Gargano and Pianese9). From this point of view, it is more
appropriate to define confidence intervals of Cp, with a
predefined failure probability, than to use a specific value. In
Fig. 6, the experimental points (g) refer to average values of Cp
estimation, ^Cp ðNAb Þ, and the relative confidence intervals of
90%, 95%, 98% and 99% are plotted. In addition, the mean
peak demand values have been fitted by a curve with a power
relationship as follows
:
0 2
^Cp ¼ 8NAb
4
The plot in Fig. 6 has been prepared considering standard
deviation as being variable with respect to the number of
inhabitants. Reducing the number of users leads to an increase
of the confidence intervals. The trend of the coefficient of
variation CV (=) when varying the number of inhabitants is
shown in Fig. 7.
4
Cp
5
CV ¼ 0:1 þ
2
:
½1 þ ðNAb =0:4Þ0 55
2
0
0
250
500
750
1000
1250
NAb
Fig. 5. Regression curve of maximum measured Cp
118
Water Management 160 Issue WM2
1500
The relationship proposed in equation (5) shows a good fit with the
data collected from the monitoring system and could be assumed
valid for a range of inhabitants from 60 to 1220. When decreasing
the number of users, however, this relationship gives results
comparable with some existing studies based on field data
(e.g. Buchberger and Wells2). For a greater number of customers
the CV, according to its expression, tends to 0.1—the value largely
used in the literature for large networks (e.g. Kapelan et al.21 and
Peak residential water demand
Tricarico et al.
0·25
Alvisi et al.3
LATibi7
0·20
0·15
Inhabitants
Cp
600–2000
2805
2.87–2.12
1.30
CV
Table 4. Peak demand coefficient in similar experimental research
0·10
sensible to use other experimental data relating to other real WDS
in order to validate these relationships further.
0·05
0·00
0
200
400
600
800
1000
1200
1400
NAb
Fig. 7. Variation of CV in relation to the number of users
Babayan et al.22). However, the results presented in the present
study are still under investigation.
Table 3 shows the maximum peak demand coefficient (from the
deterministic approach) and the upper bounds of confidence
intervals, obtained for different confidence levels (from the
probabilistic approach). Comparing, for the same number of users,
the results shown in Table 3 with the corresponding peak demand
coefficients reported in Table 2, it is evident how the classical
relations available in the technical literature lead to a significant
overestimation of the peak factor. Such overestimations are
unjustified because the classical relations refer to the hourly peak,
which should be lower with respect to that estimated in this paper,
based on data averaged to a 1 min interval.
It is probable that, in the past, the knowledge that the water
demand is an uncertain variable might have induced practitioners
to employ wide precautions in estimating the Cp, that could have
been used in WDS studies—representing a safety margin.
Nevertheless, the lower peak demand values observed should lead
to a more accurate design of the hydraulic networks, avoiding
excessive additional costs, reducing the size of the pipes and,
consequently, allowing for a reduction of the pressures with the
consequent reduction of pressure-driven leakages.
Moreover, the results achieved in this work are in agreement with
those obtained by similar experimental analysis on residential
areas as reported in Table 4 (e.g. Alvisi et al.3 and LATibi7).
The results could therefore be generalized and the abovementioned relations could be used to study the maximum flow
demands for other case studies. Nevertheless, it would also be
Number of inhabitants
Deterministic approach
Probabilistic approach
90%
95%
98%
99%
100
250
750
4.4
3.8
3.9
4.1
4.2
3.7
2.9
3.0
3.1
3.1
2.9
2.3
2.3
2.3
2.3
Table 3. Peak demand coefficient
Water Management 160 Issue WM2
1000 1250
2.8
2.1
2.1
2.2
2.2
2.6
2.0
2.0
2.1
2.1
6. MAXIMUM WATER DEMAND MODELS
Probabilistic models have been studied in this paper to describe
the maximum water demand. Optimal rehabilitation/design and
reliability studies on WDS use probabilistic models to generate
data samples of the maximum flow demand. This is usually done
through the application of the Monte Carlo sampling scheme or a
similar sampling approach, for example Latin hypercube (McKay
et al.23), modelling the demand with the normal probability
density function. In the case of a small number of users, it has
been seen that other models could be better suited to representing
the maximum water requirement. Statistical inferences on a wide
range of data samples, indeed, have shown (Fig. 8) that both
log-normal and Gumbel models represent well the peak
requirement for a small number of users. Both distributions satisfy
the Kolmogorov–Smirnov test. The parameters required for the
presented models are the mean value, , and the standard
deviation, . These have both been estimated from the data
collected and their values are shown in Fig. 8 in relation to the
number of users.
7. CONCLUSIONS
A better knowledge of the maximum water demand allows for the
more effective design or management of a WDS. With the water
demand being a random variable, in the technical literature there
is a general agreement to tackle WDS studies with a probabilistic
approach. The lack of empirical research on the topic, however,
means that there are few practical applications. The present work,
therefore, has had the aim of contributing to a deeper knowledge
of the maximum water demand. Indeed, new formulae to estimate
the peak flow demand for a small number of users have been
proposed. This has been achieved through analysing data collected
from a real WDS. Gumbel and log-normal distributions represent
well the peak water requirement—at least for the range of users
investigated here. Moreover, in relation to the number of users, the
parameters of the models have been estimated and it has
subsequently been shown that the CV decreases, with the
increasing number of inhabitants. The above-mentioned models
and the corresponding relations for the estimation of the
parameters (e.g. , ) can be used for generating random demand
data and, thus, could allow the application of probabilistic
approaches.
For a better data analysis, the effect of the resolution for
resampling the collected data, with respect to the time interval, on
the peak rate estimation has been investigated. The results
obtained show that a more accurate estimation of water demand
at the peak can be achieved by using 1 min interval data.
Moreover, it has been highlighted that the classical relations
available in the technical literature lead to an overestimation of
the maximum water demand and consequently to an over-design
of the WDS components. The results obtained are, however, in
Peak residential water demand
Tricarico et al.
119
1·00
1·00
Measurement point 7
Log-normal
Gumbel
Measurement point 3a
Log-normal
Gumbel
0·75
Φ(Cp)
Φ(Cp)
0·75
0·50
0·50
0·25
0·25
0·00
0·00
1
1·5
2
Cp
2·5
3
1
1·5
2·5
3
(b)
(a)
1·00
2
Cp
Measurement point 7–(8 + 22b)
Log-normal
Gumbel
Φ(Cp)
0·75
0·50
0·25
μ
σ
0·00
1
1·5
2
Cp
2·5
Number of inhabitants
1220
981
777
1·99
2·04
2·21
0·23
0·24
0·33
3
(c)
Fig. 8. Comparison of experimental data, log-normal and Gumbel distributions; relative estimates of and parameters for the
corresponding number of inhabitants: (a) 1220 inhabitants; (b) 981 inhabitants; (c) 777 inhabitants
agreement with the few available studies based on experimental
data.
The useful information obtained by this analysis of the PSG
network could encourage the installation of new monitoring
systems, which could then lead to the verification of the
relationships suggested and result in an increased availability of
usable data for future studies.
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