Use of network theory and reliability indexes for the validation of synthetic water distribution systems case studies

Sustainable Energy Technologies and Assessments xxx (2016) xxx–xxx

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Original article

Use of network theory and reliability indexes for the validation of synthetic water distribution systems case studies D. Paez ⇑, Y. Filion Department of Civil Engineering, Queen’s University, Kingston, ON, Canada

a r t i c l e

i n f o

Article history: Received 25 August 2016 Revised 29 November 2016 Accepted 5 December 2016 Available online xxxx Keywords: Water distribution systems Graph theory Resilience Synthetic case studies

a b s t r a c t The examination of the relationship between energy use, reliability and network characteristics in water distribution systems (WDS) requires a set of sufficient case studies to give statistical significance to the conclusions. Considering the difficulty of finding real-world systems that can be used in this kind of research, synthetic (virtual) distribution systems are the next available alternative to use as case studies. This paper describes the use of a new method for the generation of synthetic distribution systems and its subsequent comparison against real-world systems to validate the suitability of the synthetic set to drive the conclusions of future research. The algorithm for the generation of synthetic WDSs is based on the work, methods and software presented by Mair et al. (2014). For the validation procedure, five metrics or indexes that account for the network’s connectivity (link density, average node degree, meshedness coefficient) and the system’s reliability (Resilience Index, Network Resilience) are evaluated; the ranges in each of the metrics are then compared. The generation of synthetic WDS required an enhancement of the connectivity of the networks and a pipe sizing that accounted for practical system design. An acceptable degree of similarity between the synthetic and the real-life sets of WDSs was achieved, although some modifications to the networks may be required in the future. Ó 2016 Elsevier Ltd. All rights reserved.

Introduction Currently, there is a lack of realistic data to drive research on water distribution systems (WDS), based especially on safety reasons [2], and on time and costs associated with data collection [3]. These restrictions have meant that realistic case studies typically are detailed but representative of small neighbours and rural settlements (e.g. [4,5]) or larger but highly skeletonized (e.g. [6– 8]). To overcome that difficulty, virtual or synthetic systems can be used given the smaller effort required to generate them and the small or null risk that its publication can imply. Brumbelow et al. [2] made one of the first attempts to propose virtual WDSs with realistic data, with two virtual cities named ‘‘Micropolis” and ‘‘Mesopolis” based on some real data available to the authors. A more automated approach was proposed by Sitzenfrei et al. [3] who developed the software DynaVIBe-Web [1] for the generation of synthetic WDSs based on street networks patterns and some GIS data described in Section ‘‘Topography and layout” of this paper. Murano et al. [9] also proposed and developed a tool for the generation of synthetic networks based on some ⇑ Corresponding author at: 58 University Ave, Kingston, ON K7L 3N9, Canada. E-mail address: [email protected] (D. Paez).

user-defined parameters and constraints. Finally, Trifunovic et al. [10] presented another algorithm which makes use of graph theory algorithms that allow the randomized and non-randomized generation of networks. Given the novelty of the algorithms for the generation of synthetic WDSs, and the small use they have had so far, a validation of its results is required before they can be used for further research, especially to ensure that the models are representative. This paper presents the use of the methodology and software developed by [1] to generate 45 synthetic WDSs and the posterior comparison against 15 real-world systems gathered by [4]. The comparison was made to validate the synthetic set using networks’ connectivity metrics [11] and systems’ reliability indexes based on their energy allocation [12]; which are considered two main aspects for further research on WDSs.

Indexes description A WDS can be represented as a graph G ¼ ðV; EÞ where V is the set of vertices and E is the set of edges within G. All junctions, demand nodes and sources of a WDS are considered elements of V, while all pipes and link-represented elements (e.g., pumps and

http://dx.doi.org/10.1016/j.seta.2016.12.002 2213-1388/Ó 2016 Elsevier Ltd. All rights reserved.

Please cite this article in press as: Paez D, Filion Y. Use of network theory and reliability indexes for the validation of synthetic water distribution systems case studies. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.12.002

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valves in EPANET) are elements of E. Defining n as the number of elements in V and m as the number of elements in E, the following connectivity indexes can be defined [11]:

q¼

2m nðn  1Þ

on the attributes of real WDSs from the state of Kentucky, which have been slightly modified to prevent the identification of their location. Table 1 presents the ranges of some properties of the database systems (column ‘‘Real set”).

ð1Þ Generation of synthetic study cases

2m < k >¼ n

ð2Þ

mnþ1 2n  5

Rm ¼

ð3Þ

where q is the ‘‘link density” that represents the fraction between the current number of links and the maximum number of links in a non-multigraph graph; < k > is the ‘‘average node degree” defined as the average number of links connected to a node; and Rm is the ‘‘meshedness coefficient” that represents the fraction between the current number of loops and the maximum number of loops in a planar graph. On the other hand, to quantify the reliability associated with a WDS it has been proposed to use indexes that do not require a stochastic analysis of hydraulic or mechanical perturbations that a WDS could have (e.g., [13], [14] and [15]), but that represent in different ways the impact and response of the network to those uncertain perturbations. For this research, two energy related indexes were used: the Resilience Index ðRIÞ proposed by [13]; the Network Resilience ðNRÞ proposed by [14]: ðminÞ

RI ¼

EU  EU

ðminÞ

E0  EU

NR ¼

  ðreqÞ Hi  Hi ¼ Pnr Pmp Pnn ðreqÞ k¼1 Doutk H k þ i¼1 Di Hi j¼1 P j =c  Pnn

i¼1 Di

ð4Þ

  ðminÞ UN  EU  EU ðminÞ

E0  EU Pnn

  ðreqÞ  Di H i  H i ¼ Pn r Pmp Pnn ðreqÞ k¼1 Dout k H k þ i¼1 Di Hi j¼1 P j =c  i¼1 UN i

ð5Þ

where E0 is the total energy per unit weight and per unit time supplied to the system by the water sources and the pumps (input energy) in m4 =s; EU in the energy per unit weight and per unit time ðminÞ

delivered to the end-users (output energy) in m4 =s; EU is the minimum EU required to fulfill the end-users’ requirements of demand and pressure, computed as the product of the required demand and minimum required head for each demand node; Di is the demand in node i in m3 =s; Hi is the computed (delivered) head in node i in m; ðreqÞ

is the required head in node i in m, usually calculated as the Hi node’s elevation plus the minimum allowable pressure; Doutk is the outflow from reservoir k in m3 =s; Hk is the head in reservoir k in m; Pj is the power of pump j in W, c is the specific weight of water in N=m3 ; UNi is the diameter uniformity coefficient of node i calculated as the ratio of the diameters connected to the node and the maximum of those diameters; UN is the diameter uniformity coefficient for the whole network computed as the weighted average of UNi ; nn is the number of demand nodes, nr is the number of reservoir or water sources (may include some tanks); and mp is the number of pumps.

Topography and layout DynaVIBe-Web is a software that generates synthetic case studies at a city scale using the algorithm proposed by [3]. The software generates WDSs based on street patterns of real cities by following 3 steps: 1. Location of demand nodes, 2. Generation of the layout based on an initial tree layout and a posterior generation of loops with a defined probability, and 3. Pipe sizing that accounts for non-optimal design and even some installation errors using the concept of economic flow velocity (maximum velocity for economic purposes). Using DynaVIBe-Web, 5 real cities were selected as baselines for the generation of 45 WDSs (9 WDSs per city). Two of the cities are located in Europe, two in North America and one in Asia, and they cover a range of area between 41 and 974 km2, and a population densities between 684 and 8453 PD./km2 (People Density per square kilometer), as seen on Table 1 (column ‘‘Synthetic Sets”). The main attributes of the systems were assigned based on data publically available online such as the population, the consumption per capita and the location some elements as treatment plants and tanks. The execution of the software DynaVIBe-Web was made varying the parameters for demand distribution, spanning tree for the layout, the cycle indicator, and the main pipe offset. The demand distribution can be either uniform or normal depending on how the demand per area varies in the system. The spanning tree for the layout can be generated using either a random tree on a minimum spanning tree that minimizes the length of the pipes. The cycle indicator defines which loops are closed by selecting only the ones with a total length higher than the product of the cycle indicator and the length of the added pipe. Finally, the main pipe offset is the offset distance used to locate the trunk pipes of the network. After varying each of these parameters, an enhancement of the models was required to cover a wider range for the connectivity indexes. This need was evidenced after computing q, < k >, and Rm for the real WDSs set and for the five sets of synthetic WDSs (e.g., Table 2 column ‘‘Before increasing connectivity”), and noting the differences in the maximum, average and standard deviation values with the real set. In order to increase the connectivity, another option recently implemented in DynaVIBe-Web called ‘‘maximum possible graph” was used. With this option, all possible pipes that lay under a street are drawn and included in the resulting network. Using this maximum-connectivity network, randomly-selected pipes were added to the existing models making sure to connect junctions already present in the network and without including parallel pipes. The results of the procedure can be seen in Table 2 by comparing the columns ‘‘Before increasing connectivity” and ‘‘After increasing connectivity”. Pipe sizing

Real-world study cases Even though the use of synthetic systems is one of the alternatives to solve the scarcity of realistic data, some researchers (e.g. [4,16]) have assembled databases of real-world models for their own research. In the case of [4] they presented and made public 12 systems (although the current database has 15 systems), based

Expert criteria is more likely to have driven (and probably continue to drive) the design of WDSs, than optimization methods [17]. Therefore, to make the synthetic systems more realistic, different expert criteria was used. Table 3 presents values and constraints recommended for different expert sources. It includes values for the minimum diameter ðdmin Þ, the maximum flow

Please cite this article in press as: Paez D, Filion Y. Use of network theory and reliability indexes for the validation of synthetic water distribution systems case studies. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.12.002

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D. Paez, Y. Filion / Sustainable Energy Technologies and Assessments xxx (2016) xxx–xxx Table 1 Summary of the real set, and the sets used for the generation of synthetic WDS.

Population (Peo.) Area (km2) Pop. Density (PD./km2) Mean daily consumption (L/s) Consumption per capita (L/day/hab.)

Real Set

Synthetic Sets

KY

City 01

City 02

City 03

City 04

City 05

(5545–25,165) (5–978) (18–3598) (46–108) (288–1404)

666,058 974 684 2532 328

808,515 114 7092 5100 545

278,475 188 1481 1400 434

297,943 263 1134 1609 467

346,574 41 8453 2060 514

Table 2 Statistics for the meshedness coefficient Rm for the real set and for the synthetic sets before and after increasing the connectivity. Real set

No. of WDSs Min. Max. Average Std. Dev.

15 0.024 0.232 0.103 0.059

Before increasing connectivity

After increasing connectivity

City 01

City 02

City 03

City 04

City 05

City 01

City 02

City 03

City 04

City 05

9 0.084 0.093 0.087 0.003

9 0.024 0.034 0.027 0.003

9 0.019 0.022 0.021 0.001

9 0.039 0.045 0.042 0.002

9 0.045 0.047 0.046 0.001

9 0.151 0.219 0.188 0.023

9 0.057 0.159 0.115 0.032

9 0.045 0.168 0.115 0.037

9 0.091 0.132 0.118 0.014

9 0.107 0.173 0.153 0.023

Table 3 Recommendations and constraints for pipe sizing. (Values converted to SI units.) Reference

dmin

[18] Cesario (1995) Surveya

200 mmb

Recommended

[19] Trifunovic (2006)

v max

Sfmax (1/1000)

pmin

pmax


x ¼ 2:0 m/s x 2 ½1  6 m/s 1:5 m/s @d < 600 mm


x ¼ 6:2 x 2 ½1  15 12 @d P 600 mm 25 @d < 600 mm


x ¼ 23 m x 2 ½14  42 m 28  35 m


x ¼ 77 m x 2 ½42  151 m 63  77 m

1:0 m/s @large d 1:5 m/s @small d

12 @large d 25 @midrange d 5  10 @small d

20  30 m

60  70 m @flat zones 100  120 m @hilly zones


x ¼ 28:5 m x 2 ½20  40 m


x ¼ 93:6 m x 2 ½70  160 m

14 m @ Dmax 24.5 m @ Dav erage

70 m

14 m

56 m

[20] Kujundzic (1996)a

a b

[21] GLUMR (2012)

75 mm @no fire protection 1:0 mm @with fire protection

[22] Mississippi Dept. of Health (2001)

1:0 mm

1:5 m/s

Values reported for surveys on different cities and/or utilities (
x and x 2 ½  are respectively the mean and the range of values for the variable within the evaluated sample). Value reported for at least one utility of the sample.

velocity ðv max Þ, the maximum unit head loss ðSfmax Þ, the minimum and maximum allowable pressures ðpmin Þ and ðpmax Þ. Sometimes these parameters are defined for different demand conditions where Dmax is the maximum demand condition (commonly defined as the peak hour demand plus some fire scenario), and Dav e is the average demand condition (commonly defined as the mean daily demand). Therefore, the design of the synthetic WDSs was based on the values of Table 3 by using a demand’s peak factor of 2:0 and taking the original flow distribution for each DynaVIBe-Web model [1], and then resizing the diameters to fulfill three conditions: 1. dmin ¼ 100 mm; 2. v max ¼ 2:0 m/s; and 3. Sfmax ¼ 2 m/km, for d > 500 mm, Sfmax ¼ 3:5 m/km, for 500 > d > 200 mm, and Sfmax ¼ 5 m/km for d 6 200 mm. However an additional consideration was made to account for realistic pipe replacement, and in case a pipe already sized by DynaVIBe-Web fulfilled the criteria within a ±10% range, it maintained its original diameter.

Results and discussion Connectivity indexes The computed connectivity indexes for the set of real WDSs (KY) and the set of synthetic WDSs (C01 to C05) are presented in Fig. 1 as a box-and-whisker plot displaying minimum and maximum values as well as the 25th, 50th and 75th percentiles of the sample. These results were computed for the networks after the connectivity of the models generated with DynaVIBe-Web [1] was increased as described previously. The results show that, for Rm and < k >, the synthetic sets cover most of the range of the real set, with cities C02, C03 and C04 covering the two central quartiles, and cities C01 and C05 covering the highest quartile. For q the values reached by the synthetic sets are always in or below the lower quartile of the real set. This is explained by the dependence of this index q to the scale of the graph as described by [11]. In this case the size of the synthetic

Please cite this article in press as: Paez D, Filion Y. Use of network theory and reliability indexes for the validation of synthetic water distribution systems case studies. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.12.002

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Fig. 1. Comparison of connectivity indexes between the set of real WDSs (KY) and the set of synthetic ones (C01 to C05).

systems was n 2 ½1151; 6847 and m 2 ½1254; 9211 while in the real set only three networks have over 1000 nodes, and only two have more than 1500 pipes. This difference in sizes and the nature of the link density index that increases its numerator proportional to m but increases its denominator proportional to n2 , explain the low values of q found for the synthetic systems. Considering that the generated models are meant to represent whole cities rather than independent sectors, this difference in sizes is intended; meaning that the observed behaviour of q is consistent with the expectations. On the other hand, the results for the meshedness coefficient show the performance of the algorithm to increase the connectivity of the network, producing networks with similar ranges and values for this index and moreover with the same magnitude of variability in the two mid quartiles. As previously mentioned, the models based on Cities 01 and 05 have higher values of < k > and Rm than the others, for both their medians and their central quartiles. This preliminary observation may be explained by the presence of suburban and rural areas in the edges of the Cities 02, 03 and 04 which are mainly low connectivity zones that, when connected to the main WDS, decrease its overall connectivity indexes. In the case of Cities 02 and 03, low connectivity zones are present mainly due to urban sprawl, while in City 04 is due to residential suburbs and rural settlements.

Reliability-related indexes The reliability-related indexes in this study were energy indexes that relate the surplus output energy with the input energy and the minimum required energy. To visualize the distribution of the output energy, Fig. 2 presents the cumulative distribution of pressures in the networks as a percentage of the total demand (i.e., a point ðp; PTÞ in a curve, represents the percentage of the total outflow of the network PT that is delivered with a pressure equal or lower than p). Taking into account that the pipe sizing procedure implemented only the recommendations and constraints at a pipe level, and did not considered the pressures in the system for any feedback, many networks presented some percentage of its demand nodes having negative pressures for the evaluated scenario (peak factor of 2.0). Considering that some real systems also presented negative pressures in some of its nodes, and that the percentages were in most cases relatively small (less than 10%), these values were considered for the calculation of the indexes including its effect of reduction on the overall indexes. To ensure that the comparison was fair, a minimum pressure of ðreqÞ

0 m was selected for all the networks, meaning that Hi was set equal to the elevation of the node. Fig. 3 presents the box-andwhisker plot (with the same descriptive statistics as in Fig. 1) for the reliability indexes.

The results from Fig. 3 show that the reliability indexes of the synthetic WDSs are in the ranges of the real set. However, it is also evident that the variability of the synthetic systems is considerably lower than the one from the real set. This can be explained by looking at the nature of the indexes used. The RI is a relation between the surplus output energy and the difference between the input energy and the minimum required energy, while the NR is the same relationship but considering diameter uniformity weights for each demand node. On one side, as the surplus output energy is the input energy minus the energy expended in friction and minor losses and minus the minimum required energy; in a set of WDSs with the same topography, the same head in its sources and the same total demand, the variation in the numerator is only due to the differences in the total energy losses on the network. But, as the networks were designed using the same recommendations and constraints for all pipes, the differences in the sum of losses was relatively small. This implies that the variation on the numerators was small for networks based on the same city. On the other side, the denominators of the indexes are a function of the input energy, which does not vary significantly if the sources in all the systems have the same head (variations are due to different values of Doutr ), and of the minimum required energy, which does not vary at all if the systems have the same topography. Therefore, the denominators were even less likely than the numerators to vary among systems of the same set and that produced the low variability in the synthetic WDSs sets. To verify this observation, the coefficients of variation for the numerators and denominators of the RI were computed. For the set of WDSs based on city C02, the coefficient of variation of the numerators was CV ¼ 7:6% and for the denominator was CV ¼ 1:7%. Similarly for the set based on C03 city the coefficient of variation of the numerators was CV ¼ 4:4% and for the denominator was CV ¼ 4:7%, which evidence the small variability in both factors of the equation. It is worth to notice that those results do not imply that the WDSs are similar in every sense. It is a known fact that the exact same RI value can be achieved in a network by an infinite combination of continuous diameters in its pipes [23]. In this case, this fact explains how different WDSs with different connectivity and flow rate distributions among their pipes, still produce very similar RI (and likewise similar NR). Regarding the differences between the synthetic sets, only the Cities 03, 04 and 05 presented models with RI or NR below 0.6. Moreover, the curves from Fig. 2 allow a qualitative comparison on how much energy is being delivered to the end-users in terms of water pressure (based on the horizontal position of the curve), and on how unequal is that energy being distributed among the end-users (based on the slope of the curve). It can be seen that the methodology is able to acceptably emulate the amount of

Please cite this article in press as: Paez D, Filion Y. Use of network theory and reliability indexes for the validation of synthetic water distribution systems case studies. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.12.002

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Fig. 2. Pressure distribution along the network for: a. the set of real WDSs (KY), b. the set of synthetic WDSs based on City 01, c. the set of synthetic WDSs based on City 02, d. the set of synthetic WDSs based on City 03, e. the set of synthetic WDSs based on City 04, and f. the set of synthetic WDSs based on City 05.

Fig. 3. Comparison of reliability-related indexes between the set of real WDSs (KY) and the set of synthetic ones (C01 to C05).

energy delivered to the end-users as most of the curves have their core within the region delimited by the real set. However, the uniformity on the perceived energy by the end-users is not similar to the real set for these same three cities. These are also the cities with higher variation in topography due to some peripheral high elevations. This shows the need to enhance the models using pumps and valves that allow a better control of the pressures in these cases. Additionally, the higher dispersion between systems from Cities 04 and 05 is due to the presence of models that rely dif-

ferently on tanks to fulfill the demand, showing also the importance of tank sizing and location criteria.

Conclusions A new set of synthetic WDSs was generated using the methodology and the software presented by [1] coupled with a method to increase the connectivity of the networks and a set of criteria to

Please cite this article in press as: Paez D, Filion Y. Use of network theory and reliability indexes for the validation of synthetic water distribution systems case studies. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.12.002

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size pipes in a realistic manner. The increasing of connectivity presented successful results as it generated networks with good variety of connectivity (measured by 3 graph theory indexes) and always with consistent values. The criteria to size pipes, although realistic, do not have an effectiveness of 100% in solving the pressure needs for the nodes of the network. Moreover, it also leads to WDSs that may be different in their hydraulic behaviour, but have not much variation on the reliability indexes evaluated. Nevertheless, the generated set presents a good performance in 3 connectivity indexes (link density, node average degree and meshedness coefficient) and 2 reliability-related indexes (resilience index and network resilience) when compared with the values of a set of 15 real WDSs. A further enhancement of the models and of the pipe sizing criteria is recommended to ensure more variability in the reliability-related indexes, and therefore in the representativeness of the synthetic set for future research. Also, for the networks with high percentages of the demand outside the range of allowable pressures, it is recommended the evaluation of operational rules for pumps and valves that allow a better control of the pressures (e.g. [24]) and a consequent impact in the energy use of the system (e.g. [25,26]). Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council of Canada. References [1] Mair M, Rauch W, Sitzenfrei R, Spanning tree-based algorithm for generating water distribution network sets by using street network data sets. In: World environmental and water resources congress 2014, 2014. p. 465–74. [2] Brumbelow K, Torres J, Guikema S, Bristow E, Kanta L, Virtual cities for water distribution and infrastructure system research. In: World environmental and water resources congress 2007, 2007. p. 1–7. [3] Sitzenfrei R, Möderl M, Rauch W. Automatic generation of water distribution systems based on GIS data. Environ Model Softw 2013;47:138–47. [4] Jolly MD, Lothes AD, Sebastian Bryson L, Ormsbee L. Research database of water distribution system models. J Water Resour Planning Manage 2013;140 (4):410–6. [5] Sitzenfrei R, von Leon J. Long-time simulation of water distribution systems for the design of small hydropower systems. Renewable Energy 2014;72:182–7. [6] Fujiwara O, Khang DB. A two-phase decomposition method for optimal design of looped water distribution networks. Water Resour Res 1990;26(4):539–49.

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Please cite this article in press as: Paez D, Filion Y. Use of network theory and reliability indexes for the validation of synthetic water distribution systems case studies. Sustainable Energy Technologies and Assessments (2016), http://dx.doi.org/10.1016/j.seta.2016.12.002