- Email: [email protected]
Effect of Uncertainty Demand Location on Transient behavior of WDS
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 89 (2014) 1321 – 1328
16th Conference on Water Distribution System Analysis, WDSA 2014
Effect of Uncertainty Demand Location on Transient Behavior of WDS M. Ferrantea , C. Capponia,∗, R. Collinsb , J. Edwardsb , B. Brunonea , S. Meniconia a Dipartimento b Pennine
di Ingegneria Civile ed Ambientale, University of Perugia, Via G. Duranti 93, Perugia 06125, Italy Water Group, Department of Civil and Structural Engineering, University of Sheffield, Sheffield, S1 3JD, United Kingdom
Abstract The effect of demand uncertainty in water distribution systems (WDS) is a key problem, especially for water company assets. The presence or absence of demands, their number, size, and locations significantly affect the functioning of a pipe system and its response to a transient. The validity and the effectiveness of the Lagrangian model is firstly investigated by means of a comparison with a Method of Characteristics model in frictionless systems. Demands locations are stochastically varied to assess the impact of the uncertainty of the geometry of the network on transient propagation. A normal distribution has been used to model the uncertainty of demand location. c 2014 2014 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license © (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Organizing Committee of WDSA 2014. Peer-review under responsibility of the Organizing Committee of WDSA 2014 Keywords: customer demand; water distribution system; pressure transient; water hammer; pipe network; lagrangian model
1. Introduction The effect of demand uncertainty in water distribution systems (WDS) is a key problem, especially for water company assets [2,3]. The presence or absence of demands, their number, size, and locations significantly affect the functioning of a pipe system and its response to a transient. The uncertainty in the number of demands placed along a pipe has been analyzed by Edwards and Collins[4] by means of a Method of Characteristics (MOC)–based model [6] in the case of a straight reservoir–pipe–valve system. Furthermore the uncertainty in location and distribution of demands has been modeled with customer connection randomly either on or off. However this study has been bound by the constrains of the MOC model, i.e., the spatial and temporal discretizations. As a matter of fact, when the location of a singularity is varied in a MOC model, that singularity can be placed only at the points of the spatial discretization. Then it is not possible to vary demand locations at any point of the pipe. This constrain suggests the use of a Lagrangian Model (LM). In fact, the LM does not need a fixed time step and nor it follows a definite, regular, time–space grid [5]. The LM is based on the D’Alembert’s solution of the differential equations governing unsteady flow in pressurized pipes when friction is neglected. The D’Alembert’s solution allows to estimate the pressure at any section in the system by means of the cumulative sum of all the waves that have passed that section until that instant. ∗
Corresponding author. Tel.: +(39)-075-585-3893 ; fax: +(39)-075-585-3892. E-mail address: [email protected]
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Organizing Committee of WDSA 2014 doi:10.1016/j.proeng.2014.11.447
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M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328
The LM is able to store the amplitude of each wave and the instant of time when it passes the singularities (e.g. leaks, partial blockages, junctions,...). Thus it is possible to simulate the time-history of the pressure at any section of the system without choosing it in advance. A wave that meets a singularity which implies an external flow (demand, leak) is reflected and this phenomenon may be utilized for leak detection [1]. In the first part of this paper, an assessment of the LM comparing it with the MOC in a simple frictionless system is given. Then the LM is applied to the same system analyzed by Edwards and Collins[4], and the demand location are randomly varied in order to evaluate the effect of such an uncertainty. The results are provided for two case studies. 2. Comparison between LM and MOC A simple reservoir–horizontal pipe–valve system with a leak (Fig. 1) has been used to test the LM. The smooth pipe has an internal diameter D = 0.0933 m, and a length L = 166.32 m. Pressure signals obtained by means of LM and MOC have been compared for different values of the piezometric head at the leak, HL . In Fig. 2 the comparison between LM and MOC is shown for HL = 7 m (Fig. 2a), HL = 40 m (Fig. 2b), and HL = 100 m (Fig. 2c). In this figure, ΔH = H - H0 , with the subscript 0 indicating the steady-state conditions, is plotted vs. the dimensionless time, t∗ = t/τ, with τ = 2 L / c (assuming the pressure wave speed, c, equal to 377.15 m/s). The differences between pressure signals generated by LM (continuous line) and by MOC (dashed line) are very close to zero in the short period while they increase slightly in the long period; moreover it can be seen that the larger the HL the smaller the differences. The comparison of pressure signal has been shown over 20 characteristic time length, to check the role played by the friction. Friction was not modelled in the original study by Edwards and Collins[4] to allow clarity in the exploration of the reflections and damping due to the random demands. The lack of friction also allows easy comparison to the LM model where friction is difficult to incorporate.
Fig. 1: Reservoir (R)–pipe–operated valve (OV) system used to compare LM and MOC results changing the piezometric head at the leak, HL (L = leak, M = measurement section).
The LM is much less computationally cumbersome with respect to the MOC. As a matter of fact, for each simulation of Fig. 2, carried out with the same computer, the ratio between the time-machine required by LM and by MOC is approximately 1 to 50. Furthermore the LM does neither require a fixed spatial nor temporal discretization, so the study of the wave propagation is more precise and does not depends on model constraints. 3. Frictionless system with demands The LM has then been used to extend the analysis of the frictionless system studied by Edwards and Collins[4], shown in Fig. 3. It consists of a reservoir at the upstream end, that is assumed to maintain a constant head of 100 m through out the simulation, a straight section of pipe (D = 0.1 m) of length 180 m links the reservoir to a 120 m length of pipe (D = 0.1 m) — hereafter referred to as the middle pipe — along which several demands are placed, according to the modeled scenario. Finally there is a further 240 m of pipe (D = 0.1 m), which has no demand before the operated end valve, which is closed fastly to generate transients. The wave speed is 1000 m/s for all pipes. Two measurement sections, s1 and s2, are provided, upstream and downstream of the middle pipe, respectively. The middle pipe is split into 30 sections, at the end of each of them an external flow is placed, based on the orifice equation. Assuming the
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M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328 10 a)
0 −10 0
2
4
6
8
10
12
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16
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Δ H (m)
10 b)
0 −10 0
2
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10 c)
0 −10 0
2
4
6
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10
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dimensionless time, t* Fig. 2: Time-history of ΔH for Hor = 7 m (a), 40 m (b), and 100 m (c), obtained by means of LM (continuous line) and by MOC (dashed line).
Fig. 3: Reservoir (R)–pipe–operated valve (OV) system studied by Edwards and Collins[4] (s1 and s2 are the measurement sections).
total flow out of all costumer demand equal to 2 l/s, and considering that this total flow is divided equally between the 31 demands, the total effective area (i.e., the product of area and discharge coefficient) is maintained constant during the transient. The overpressure due to the fast closure of the operated valve has been evaluated by means of the Allievi–Joukowsky formula: ΔHA−J =
cΔV = 26 m g
(1)
where ΔV is the change in flow velocity in the pipe, and g is the acceleration due to gravity. The transient solution obtained by LM in presented in Fig. 4, where the head over the dimensionless time is shown in correspondence to the measurement sections s1and s2 of Fig. 3. The results of the two simulation approaches are very similar, however the LM model produces a slightly higher amount of damping than the MOC and critically a different shape of response for s1. Whilst in the MOC the peak pressure wave recorded at s1 is increasingly rounded with repeated oscillations in the LM this peak remains perfectly square (Fig. 2 [4])
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M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328
s1 s2
25 20 15 10
Δ H (m)
5 0 −5 −10 −15 −20 −25 0
2
4
6
8
10
12
14
16
18
dimensionless time, t*
Fig. 4: Head at the measurement sections s1 and s2 of Fig. 3, for the case of 31 demands along the middle pipe.
4. Random demand location Since LM allows to run simulations without the constrains of spatial discretization, it is possible to vary randomly and continuously the location of demands. According to the scheme of Fig. 5, there are three demands: out of these, two are placed at a given distance from the valve (constant demands), while the other demand is placed between these demands and its location varies randomly according to a normal distribution. The probability density function (pdf) of the normal distribution is specified by two parameters: the mean, μ (i.e., the location parameter) and the standard deviation, σ (i.e., the scale parameter). The effective area is calculated on the basis of the orifice equation, equally dividing the total flow in the three demands.
Fig. 5: Reservoir (R)–pipe–operated valve (OV) system. Two demands are placed at a given distance from the valve, while one demand between these is characterized by a randomly varied location (M = measurement section).
For μ = 60 m (i.e., the middle point of the middle pipe) and σ = 20 m, the 99.7 % of the demand positions are expected to fall in the range of the length of middle pipe. Around 500 simulations corresponding to the demand position that fall in the middle pipe have been used. The obtained pdf is shown in Fig. 6. The LM has been used to evaluate the mean arrival times, ti , corresponding of the waves at the measurement section (M in Fig. 5) and their
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M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328 0.025
0.02
fx(x)
0.015
0.01
0.005
0
0
20
40
60
xi (m)
80
100
120
Fig. 6: Normal probability density function, f x (x), of the locations population generated for the demand of Fig. 5. 30
t*
m
t*m+3σ t*m−3σ
20
Δ H (m)
10
0
−10
−20
−30 0
1
2
3
4
5
6
dimensionless time, t* ∗ , t∗ − 3σ and t∗ + 3σ for the case of Fig. 5. Fig. 7: Time-histories of ΔH for tm m m
standard deviation, σ. In Fig. 7 the time-histories of ΔH for the expected dimensionless time, tm∗ , are shown. On the same figure, for the sake of comparison, the time-histories corresponding to tm∗ − 3σ and tm∗ + 3σ are shown. In Fig. 8 the system with the 31 demands used by Edwards and Collins[4] is shown. The location of the demands at the ends of the middle pipe has been again maintained constant, while the other 29 demand locations have been varied according to normal distributions. Since the middle pipe can be split in 30 sections of 4 m length each, each demand position can vary around the end of each of these sections; the chosen standard deviation value is σ = 0.6667 m. The pdf of the generated random location for each demand within the middle pipe is plotted in Fig. 9. Around 500 simulations have been run by means of LM and again the standard deviation, σ, of the time instances, ti , corresponding
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M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328
Fig. 8: Reservoir (R)–pipe–operated valve (OV) system. Two demands are placed at a given distance from the valve, while other 29 demands between these are characterized by randomly varied locations (M = measurement section). 0.7
0.6
f i,x (x)
0.5
0.4
0.3
0.2
0.1
0 0
20
40
60
xi (m)
80
100
120
Fig. 9: Normal probability density functions, fi,x (x), of the location populations generated for the demands of Fig. 8.
to the waves crossing the measurement section (Fig. 8) has been calculated. In Fig. 10 the time-histories of ΔH for tm∗ , tm∗ − 3σ and tm∗ + 3σ are shown. Since in this case σ is smaller than in the previous one, the differences between the three time-histories of ΔH are smaller. In order to show more clearly these differences, the first characteristic time length is highlighted in Fig. 11. 5. Conclusions This paper shows the excellent performance of the Lagrangian Model (LM) — characterized by significantly smaller computational effort — compared to the Method of Characteristics (MOC) when investigating frictionless systems. The cases of a system with a leak or with several demands have been studied in order to compare the two models. Since the LM allows to vary randomly and continuously demand locations, it lends itself to exploring uncertain systems using Monte-Carlo approaches due to the ability to run large numbers of trials. For real systems
1327
M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328 30
20
Δ H (m)
10
0
−10
−20
−30 0
0.5
1
1.5
2
2.5
3
3.5
4
dimensionless time, t* ∗ , t∗ − 3σ and t∗ + 3σ for the case of Fig. 8. Fig. 10: Time-histories of ΔH for tm m m
26
25.5
Δ H (m)
25
24.5
24
23.5
23
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
dimensionless time, t* Fig. 11: The first characteristic time of Fig. 10 highlighted.
which include non-linearities like unsteady friction and visco-elasticity further work is required to incorporate these phenomenon into LM. Results show that uncertainty in location of the system demands manifested in the transient pressures passing through the system. It has been shown that the larger the number of demands in the same link, the smaller the standard deviation of the time instances corresponding to the wave crossing the measurement section, and the effect of random variation of demand locations.
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Acknowledgements This research has been supported in part by the Italian Ministry of Education, University and Research (MIUR), under the Projects of Relevant National Interest “Advanced analysis tools for the management of water losses in urban aqueducts”, “Tools and procedures for an advanced and sustainable management of water distribution systems”, and Fondazione Cassa Risparmio Perugia under the project “Hydraulic characterization of innovative pipe materials (no. 2013.0050.021)”. References [1] Brunone, B., Ferrante, M., 2001. Detecting leaks in pressurised pipes by means of transients. Journal of hydraulic research 39, 539–547. [2] Collins, R., Beck, S., Boxall, J., 2011. Intrusion into water distribution systems through leaks and orifices: Initial experimental results. 11th Computing and Control in the Water Industry 2011. [3] Collins, R., Boxall, J., Karney, B., Brunone, B., Meniconi, S., 2012. How severe can transients be after a sudden depressurization? Journal AWWA 104. [4] Edwards, J., Collins, R., 2014. The effect of demand uncertainty on transient propagation in water distribution systems. Procedia Engineering 70, 592–601. [5] Ferrante, M., Brunone, B., Meniconi, S., Mar. 2009. Leak detection in branched pipe systems coupling wavelet analysis and a Lagrangian model. Journal of Water Supply: Research and Technology—AQUA 58 (2), 95. [6] Karney, B., McInnis, D., 1992. Efficient calculation of transient flow in simple pipe networks. Journal of Hydraulic Engineering 118, 1014– 1032.
ScienceDirect Procedia Engineering 89 (2014) 1321 – 1328
16th Conference on Water Distribution System Analysis, WDSA 2014
Effect of Uncertainty Demand Location on Transient Behavior of WDS M. Ferrantea , C. Capponia,∗, R. Collinsb , J. Edwardsb , B. Brunonea , S. Meniconia a Dipartimento b Pennine
di Ingegneria Civile ed Ambientale, University of Perugia, Via G. Duranti 93, Perugia 06125, Italy Water Group, Department of Civil and Structural Engineering, University of Sheffield, Sheffield, S1 3JD, United Kingdom
Abstract The effect of demand uncertainty in water distribution systems (WDS) is a key problem, especially for water company assets. The presence or absence of demands, their number, size, and locations significantly affect the functioning of a pipe system and its response to a transient. The validity and the effectiveness of the Lagrangian model is firstly investigated by means of a comparison with a Method of Characteristics model in frictionless systems. Demands locations are stochastically varied to assess the impact of the uncertainty of the geometry of the network on transient propagation. A normal distribution has been used to model the uncertainty of demand location. c 2014 2014 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license © (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Organizing Committee of WDSA 2014. Peer-review under responsibility of the Organizing Committee of WDSA 2014 Keywords: customer demand; water distribution system; pressure transient; water hammer; pipe network; lagrangian model
1. Introduction The effect of demand uncertainty in water distribution systems (WDS) is a key problem, especially for water company assets [2,3]. The presence or absence of demands, their number, size, and locations significantly affect the functioning of a pipe system and its response to a transient. The uncertainty in the number of demands placed along a pipe has been analyzed by Edwards and Collins[4] by means of a Method of Characteristics (MOC)–based model [6] in the case of a straight reservoir–pipe–valve system. Furthermore the uncertainty in location and distribution of demands has been modeled with customer connection randomly either on or off. However this study has been bound by the constrains of the MOC model, i.e., the spatial and temporal discretizations. As a matter of fact, when the location of a singularity is varied in a MOC model, that singularity can be placed only at the points of the spatial discretization. Then it is not possible to vary demand locations at any point of the pipe. This constrain suggests the use of a Lagrangian Model (LM). In fact, the LM does not need a fixed time step and nor it follows a definite, regular, time–space grid [5]. The LM is based on the D’Alembert’s solution of the differential equations governing unsteady flow in pressurized pipes when friction is neglected. The D’Alembert’s solution allows to estimate the pressure at any section in the system by means of the cumulative sum of all the waves that have passed that section until that instant. ∗
Corresponding author. Tel.: +(39)-075-585-3893 ; fax: +(39)-075-585-3892. E-mail address: [email protected]
1877-7058 © 2014 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Organizing Committee of WDSA 2014 doi:10.1016/j.proeng.2014.11.447
1322
M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328
The LM is able to store the amplitude of each wave and the instant of time when it passes the singularities (e.g. leaks, partial blockages, junctions,...). Thus it is possible to simulate the time-history of the pressure at any section of the system without choosing it in advance. A wave that meets a singularity which implies an external flow (demand, leak) is reflected and this phenomenon may be utilized for leak detection [1]. In the first part of this paper, an assessment of the LM comparing it with the MOC in a simple frictionless system is given. Then the LM is applied to the same system analyzed by Edwards and Collins[4], and the demand location are randomly varied in order to evaluate the effect of such an uncertainty. The results are provided for two case studies. 2. Comparison between LM and MOC A simple reservoir–horizontal pipe–valve system with a leak (Fig. 1) has been used to test the LM. The smooth pipe has an internal diameter D = 0.0933 m, and a length L = 166.32 m. Pressure signals obtained by means of LM and MOC have been compared for different values of the piezometric head at the leak, HL . In Fig. 2 the comparison between LM and MOC is shown for HL = 7 m (Fig. 2a), HL = 40 m (Fig. 2b), and HL = 100 m (Fig. 2c). In this figure, ΔH = H - H0 , with the subscript 0 indicating the steady-state conditions, is plotted vs. the dimensionless time, t∗ = t/τ, with τ = 2 L / c (assuming the pressure wave speed, c, equal to 377.15 m/s). The differences between pressure signals generated by LM (continuous line) and by MOC (dashed line) are very close to zero in the short period while they increase slightly in the long period; moreover it can be seen that the larger the HL the smaller the differences. The comparison of pressure signal has been shown over 20 characteristic time length, to check the role played by the friction. Friction was not modelled in the original study by Edwards and Collins[4] to allow clarity in the exploration of the reflections and damping due to the random demands. The lack of friction also allows easy comparison to the LM model where friction is difficult to incorporate.
Fig. 1: Reservoir (R)–pipe–operated valve (OV) system used to compare LM and MOC results changing the piezometric head at the leak, HL (L = leak, M = measurement section).
The LM is much less computationally cumbersome with respect to the MOC. As a matter of fact, for each simulation of Fig. 2, carried out with the same computer, the ratio between the time-machine required by LM and by MOC is approximately 1 to 50. Furthermore the LM does neither require a fixed spatial nor temporal discretization, so the study of the wave propagation is more precise and does not depends on model constraints. 3. Frictionless system with demands The LM has then been used to extend the analysis of the frictionless system studied by Edwards and Collins[4], shown in Fig. 3. It consists of a reservoir at the upstream end, that is assumed to maintain a constant head of 100 m through out the simulation, a straight section of pipe (D = 0.1 m) of length 180 m links the reservoir to a 120 m length of pipe (D = 0.1 m) — hereafter referred to as the middle pipe — along which several demands are placed, according to the modeled scenario. Finally there is a further 240 m of pipe (D = 0.1 m), which has no demand before the operated end valve, which is closed fastly to generate transients. The wave speed is 1000 m/s for all pipes. Two measurement sections, s1 and s2, are provided, upstream and downstream of the middle pipe, respectively. The middle pipe is split into 30 sections, at the end of each of them an external flow is placed, based on the orifice equation. Assuming the
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M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328 10 a)
0 −10 0
2
4
6
8
10
12
14
16
18
20
Δ H (m)
10 b)
0 −10 0
2
4
6
8
10
12
14
16
18
20
10 c)
0 −10 0
2
4
6
8
10
12
14
16
18
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dimensionless time, t* Fig. 2: Time-history of ΔH for Hor = 7 m (a), 40 m (b), and 100 m (c), obtained by means of LM (continuous line) and by MOC (dashed line).
Fig. 3: Reservoir (R)–pipe–operated valve (OV) system studied by Edwards and Collins[4] (s1 and s2 are the measurement sections).
total flow out of all costumer demand equal to 2 l/s, and considering that this total flow is divided equally between the 31 demands, the total effective area (i.e., the product of area and discharge coefficient) is maintained constant during the transient. The overpressure due to the fast closure of the operated valve has been evaluated by means of the Allievi–Joukowsky formula: ΔHA−J =
cΔV = 26 m g
(1)
where ΔV is the change in flow velocity in the pipe, and g is the acceleration due to gravity. The transient solution obtained by LM in presented in Fig. 4, where the head over the dimensionless time is shown in correspondence to the measurement sections s1and s2 of Fig. 3. The results of the two simulation approaches are very similar, however the LM model produces a slightly higher amount of damping than the MOC and critically a different shape of response for s1. Whilst in the MOC the peak pressure wave recorded at s1 is increasingly rounded with repeated oscillations in the LM this peak remains perfectly square (Fig. 2 [4])
1324
M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328
s1 s2
25 20 15 10
Δ H (m)
5 0 −5 −10 −15 −20 −25 0
2
4
6
8
10
12
14
16
18
dimensionless time, t*
Fig. 4: Head at the measurement sections s1 and s2 of Fig. 3, for the case of 31 demands along the middle pipe.
4. Random demand location Since LM allows to run simulations without the constrains of spatial discretization, it is possible to vary randomly and continuously the location of demands. According to the scheme of Fig. 5, there are three demands: out of these, two are placed at a given distance from the valve (constant demands), while the other demand is placed between these demands and its location varies randomly according to a normal distribution. The probability density function (pdf) of the normal distribution is specified by two parameters: the mean, μ (i.e., the location parameter) and the standard deviation, σ (i.e., the scale parameter). The effective area is calculated on the basis of the orifice equation, equally dividing the total flow in the three demands.
Fig. 5: Reservoir (R)–pipe–operated valve (OV) system. Two demands are placed at a given distance from the valve, while one demand between these is characterized by a randomly varied location (M = measurement section).
For μ = 60 m (i.e., the middle point of the middle pipe) and σ = 20 m, the 99.7 % of the demand positions are expected to fall in the range of the length of middle pipe. Around 500 simulations corresponding to the demand position that fall in the middle pipe have been used. The obtained pdf is shown in Fig. 6. The LM has been used to evaluate the mean arrival times, ti , corresponding of the waves at the measurement section (M in Fig. 5) and their
1325
M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328 0.025
0.02
fx(x)
0.015
0.01
0.005
0
0
20
40
60
xi (m)
80
100
120
Fig. 6: Normal probability density function, f x (x), of the locations population generated for the demand of Fig. 5. 30
t*
m
t*m+3σ t*m−3σ
20
Δ H (m)
10
0
−10
−20
−30 0
1
2
3
4
5
6
dimensionless time, t* ∗ , t∗ − 3σ and t∗ + 3σ for the case of Fig. 5. Fig. 7: Time-histories of ΔH for tm m m
standard deviation, σ. In Fig. 7 the time-histories of ΔH for the expected dimensionless time, tm∗ , are shown. On the same figure, for the sake of comparison, the time-histories corresponding to tm∗ − 3σ and tm∗ + 3σ are shown. In Fig. 8 the system with the 31 demands used by Edwards and Collins[4] is shown. The location of the demands at the ends of the middle pipe has been again maintained constant, while the other 29 demand locations have been varied according to normal distributions. Since the middle pipe can be split in 30 sections of 4 m length each, each demand position can vary around the end of each of these sections; the chosen standard deviation value is σ = 0.6667 m. The pdf of the generated random location for each demand within the middle pipe is plotted in Fig. 9. Around 500 simulations have been run by means of LM and again the standard deviation, σ, of the time instances, ti , corresponding
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M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328
Fig. 8: Reservoir (R)–pipe–operated valve (OV) system. Two demands are placed at a given distance from the valve, while other 29 demands between these are characterized by randomly varied locations (M = measurement section). 0.7
0.6
f i,x (x)
0.5
0.4
0.3
0.2
0.1
0 0
20
40
60
xi (m)
80
100
120
Fig. 9: Normal probability density functions, fi,x (x), of the location populations generated for the demands of Fig. 8.
to the waves crossing the measurement section (Fig. 8) has been calculated. In Fig. 10 the time-histories of ΔH for tm∗ , tm∗ − 3σ and tm∗ + 3σ are shown. Since in this case σ is smaller than in the previous one, the differences between the three time-histories of ΔH are smaller. In order to show more clearly these differences, the first characteristic time length is highlighted in Fig. 11. 5. Conclusions This paper shows the excellent performance of the Lagrangian Model (LM) — characterized by significantly smaller computational effort — compared to the Method of Characteristics (MOC) when investigating frictionless systems. The cases of a system with a leak or with several demands have been studied in order to compare the two models. Since the LM allows to vary randomly and continuously demand locations, it lends itself to exploring uncertain systems using Monte-Carlo approaches due to the ability to run large numbers of trials. For real systems
1327
M. Ferrante et al. / Procedia Engineering 89 (2014) 1321 – 1328 30
20
Δ H (m)
10
0
−10
−20
−30 0
0.5
1
1.5
2
2.5
3
3.5
4
dimensionless time, t* ∗ , t∗ − 3σ and t∗ + 3σ for the case of Fig. 8. Fig. 10: Time-histories of ΔH for tm m m
26
25.5
Δ H (m)
25
24.5
24
23.5
23
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
dimensionless time, t* Fig. 11: The first characteristic time of Fig. 10 highlighted.
which include non-linearities like unsteady friction and visco-elasticity further work is required to incorporate these phenomenon into LM. Results show that uncertainty in location of the system demands manifested in the transient pressures passing through the system. It has been shown that the larger the number of demands in the same link, the smaller the standard deviation of the time instances corresponding to the wave crossing the measurement section, and the effect of random variation of demand locations.
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Acknowledgements This research has been supported in part by the Italian Ministry of Education, University and Research (MIUR), under the Projects of Relevant National Interest “Advanced analysis tools for the management of water losses in urban aqueducts”, “Tools and procedures for an advanced and sustainable management of water distribution systems”, and Fondazione Cassa Risparmio Perugia under the project “Hydraulic characterization of innovative pipe materials (no. 2013.0050.021)”. References [1] Brunone, B., Ferrante, M., 2001. Detecting leaks in pressurised pipes by means of transients. Journal of hydraulic research 39, 539–547. [2] Collins, R., Beck, S., Boxall, J., 2011. Intrusion into water distribution systems through leaks and orifices: Initial experimental results. 11th Computing and Control in the Water Industry 2011. [3] Collins, R., Boxall, J., Karney, B., Brunone, B., Meniconi, S., 2012. How severe can transients be after a sudden depressurization? Journal AWWA 104. [4] Edwards, J., Collins, R., 2014. The effect of demand uncertainty on transient propagation in water distribution systems. Procedia Engineering 70, 592–601. [5] Ferrante, M., Brunone, B., Meniconi, S., Mar. 2009. Leak detection in branched pipe systems coupling wavelet analysis and a Lagrangian model. Journal of Water Supply: Research and Technology—AQUA 58 (2), 95. [6] Karney, B., McInnis, D., 1992. Efficient calculation of transient flow in simple pipe networks. Journal of Hydraulic Engineering 118, 1014– 1032.