# Identifying Critical Elements in Sewer Networks Using Graph-Theory

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Studied Sewer Systems

#### 2.2. Used Storm Events in the Simulations

#### 2.3. The Achilles Approach

c = Catchment | (-) |

F = Probability of damage cause by flooding | (-) |

#J = Nr of Junctions | (-) |

N = Node | (-) |

PI = Performance indicator | (-) |

V_{P} = Ponded Volume of each node | (m^{3}) |

V_{R} = Total rainfall runoff volume | (m^{3}) |

x = Flooding volume | (m^{3}) |

#### Application of the Hydrodynamic Modelling Method as Reference Method

#### 2.4. Introduction to Graph Theory

#### 2.5. The Graph Theory Applied on Sewer Systems

A_{i} = the surface connected to the source node | (ha) |

C_{graph} = costs of the graph | (-) |

Cv_{i}_{−}v_{n} = the costs of the shortest path from v_{i} to target v_{n} | (-) |

_{x},v

_{y}and v

_{y},v

_{x}are deleted. When all nodes are connected the new graph remains a strongly connected digraph (Figure 4 lower left). Otherwise the result is two strongly connected sub-digraphs (Figure 4, lower right). Two situations are possible. First, all (sub-) digraph(s) contain at least one target node. Second, only one of the sub-digraphs contains at least one target node. If the first situation, the total costs of the (sub)digraph(s) are determined. If the second situation, the runoff surface is summed of the nodes that are not connected to a target (see Section 2.4).

#### 2.5.1. Costs of Links Sewer System

A = area of pipe | (m^{2}) |

C = Chézy coefficient | (m^{1/2}/s) |

ΔH = head loss | (m) |

k = wall roughness | (m) |

L = length | (m) |

q = discharge | (m^{3}/s) |

R = hydraulic radius | (m) |

#### 2.6. Comparison of Criticality between Hydrodynamic Model Method (HMM) and Graph Theory Method (GTM)

_{b}) is used to determine the relationship between the outcomes of the HMM and the GTM (see Formula (6)). τ

_{b}is a nonparametric measure of association based on the number of concordances and discordances in paired observations. τ

_{b}is used to compare the relationship of datasets and not of individual conduits. Minus one (−1) implies a 100% negative association one (1) is a 100% positive association.

τ_{b} = Kendall’s tau b coefficient | (-) |

P = the number of concordant pairs | (-) |

Q = the number of discordant pairs | (-) |

X_{0} = the number of pairs tied only on the X variable | (-) |

Y_{0} = the number of pairs tied only on the Y variable | (-) |

#### 2.7. Software and Hardware

^{®}Core™ i5-3380M CPU @ 2.90 GHz processor and 8.00 Mb RAM and a Windows 7 operating system (Microsoft, Washington, DC, USA).

## 3. Results

#### 3.1. Effect of Small Opening Instead of Fully Blocked Pipe

_{b}value is 0.98. The degree of criticality is ranked from most important (1) to least important (778). If the dots are at situated at one line, the ranking of the conduits in both methods is the same. The more the points deviate from the line, the greater the difference between the two methods.

#### 3.2. The Degree of Criticality Based on Hydrodynamic Model Method

_{b}is determined. If the degree of criticality is independent of the storm event the τ

_{b}value is 1.

_{b}≈ 1 and the degree of criticality of the individual elements is almost the same. If the differences between the storm events become larger τ

_{b}drops below 0.6 (see Appendix A Table A1, Table A2, Table A3, Table A4 and Table A5) and the degree of criticality of the individual elements changes. This can be both an increase and a decrease of the degree of criticality.

#### 3.3. Comparison Degree of Criticality Based on a Hydrodynamic Model and on the Graph Theory

_{b}. For the GTM, the costs of the conduits are decisive for the outcomes. As described in Section 2.5.1 the costs of the conduits depend on the parameters discharge, water level and crest difference. The impact of the value of the parameters is described in more detail in Section 4.1.

_{b}for Loenen-1. τ

_{b}varies between 0.97–0.90. That implies that for all combinations of parameters and storm events there is a strong relation in the outcomes of the HMM and the GTM.

_{b}varies between 0.80–0.96: that is 0.01–0.1 less than the τ

_{b}of Loenen-1. A τ

_{b}of 0.80–0.96 implies that also for Loenen-2 there is a strong relation in the outcomes of the HMM and the GTM. Figure 9 shows the results of the comparison with the dynamic and stationary storm events with the highest and lowest τ

_{b}.

_{b}between the results of the hydraulic model and the graph method are less than for the Loenen cases. The Kendalls’ τ

_{b}varies between 0.46–0.78. Although the Kendalls’ τ

_{b}is reduced, it is still possible to identify the 250–300 (30–40%) most important conduits with the GTM.

## 4. Discussion

#### 4.1. Sensitivity of Parameters in the Graph Methodology

^{3}/s is used, the dynamic part of the costs of the conduits is zero. With an increasing discharge, the relative importance of the dynamic part of the costs of the conduits increases. The ratio of the dynamic part of the costs between the conduits remains the same because for all conduits the same discharge is used.

^{3}/s, the water level between 0–1 m above the lowest crest level, the costs for the differences in crest level between 0–0.5 for the Tuindorp case and between 0–1 for the Loenen case. For each value the degree of criticality is determined. The outcomes of the graph method are compared with the outcomes of the hydraulic model. For the comparison of the degree of criticality the Kendall’s τ

_{b}has been used.

#### 4.1.1. Discharge

_{b}. The variation in τ

_{b}is limited to ≈0.05. τ

_{b}is small if the discharge is 0, and after a peak around 0.02 m

^{3}/s τ

_{b}decreases with increasing discharge. As mentioned before, the discharge influences the dynamic costs. With a discharge of 0 m

^{3}/s, the dynamic costs are ignored and with a higher discharge the dynamic costs become relative high in relation to the static costs. It is important that the dynamic costs have the same order of magnitude as the static costs.

#### 4.1.2. Water Level Sewer System

#### 4.1.3. Difference in Crest Levels Sewer System

_{b}but the effect is limited (<0.05) (see Figure 13). The degree of criticality can change strongly if another crest height is used. Conduits with a high degree of criticality can get a low degree of criticality when another difference in crest height is used and vice versa. This means that it is important to select the additional costs for overflows with different crest heights carefully in flat areas.

#### 4.2. Performance

#### 4.3. Criticality of the Conduits

## 5. Conclusions

_{b}> 0.72) between GTM and HMM results.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

List of Symbols | ||

A | area of pipe | (m^{2}) |

A i | the surface connected to the source node (ha) | (ha) |

c | Catchment | (-) |

C | Chézy coefficient | (m^{1/2}/s) |

Cgraph | costs of the graph | (-) |

Cv
i_{−}vn | the costs of the shortest path from vi to target vn | (-) |

F | Probability of damage cause by flooding | (-) |

I | the source node | (-) |

J | Junctions | (-) |

k | wall roughness | (m) |

L | length | (m) |

N | Node | (-) |

P | the number of concordant pairs | (-) |

PI | Performance indicator | (-) |

q | discharge | (m^{3}/s) |

Q | the number of discordant pairs | (-) |

R | hydraulic radius | (m) |

Vp | Ponded Volume of each node | (m^{3}) |

Vr | Total rainfall runoff volume | (m^{3}) |

W | width of the pipe | (m) |

x | Threshold flooding volume | (m^{3}) |

X_{0} | the number of pairs tied only on the X variable | (-) |

Y_{0} | the number of pairs tied only on the Y variable | (-) |

Greek symbols | ||

ΔH | head loss | (m) |

τ_{b} | Kendall’s tau b coefficient | (-) |

## Appendix A

**Table A1.**Loenen-1, τ

_{b}value of the comparison of the degree of criticality of various dynamic storm events.

Storm Event | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 1.00 | 0.97 | 0.96 | 0.95 | 0.95 | 0.93 | 0.93 | 0.91 | 0.91 | 0.91 |

2 | 1.00 | 0.95 | 0.96 | 0.94 | 0.94 | 0.93 | 0.91 | 0.91 | 0.91 | |

3 | 1.00 | 0.96 | 0.98 | 0.95 | 0.95 | 0.93 | 0.93 | 0.92 | ||

4 | 1.00 | 0.96 | 0.96 | 0.95 | 0.93 | 0.93 | 0.93 | |||

5 | 1.00 | 0.96 | 0.96 | 0.94 | 0.94 | 0.94 | ||||

6 | 1.00 | 0.96 | 0.95 | 0.94 | 0.93 | |||||

7 | 1.00 | 0.95 | 0.95 | 0.94 | ||||||

8 | 1.00 | 0.95 | 0.94 | |||||||

9 | 1.00 | 0.98 | ||||||||

10 | 1.00 |

**Table A2.**Loenen-2, τ

_{b}value of the comparison of the degree of criticality of various dynamic storm events.

Storm Event | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 1.00 | 0.90 | 0.84 | 0.79 | 0.78 | 0.72 | 0.69 | 0.65 | 0.56 | 0.48 |

2 | 1.00 | 0.86 | 0.86 | 0.81 | 0.77 | 0.72 | 0.68 | 0.60 | 0.52 | |

3 | 1.00 | 0.91 | 0.93 | 0.83 | 0.81 | 0.76 | 0.66 | 0.57 | ||

4 | 1.00 | 0.91 | 0.88 | 0.83 | 0.77 | 0.69 | 0.60 | |||

5 | 1.00 | 0.88 | 0.86 | 0.80 | 0.71 | 0.62 | ||||

6 | 1.00 | 0.91 | 0.86 | 0.76 | 0.66 | |||||

7 | 1.00 | 0.91 | 0.80 | 0.69 | ||||||

8 | 1.00 | 0.83 | 0.72 | |||||||

9 | 1.00 | 0.85 | ||||||||

10 | 1.00 |

**Figure A1.**Overview of the degree of criticality of the conduits based on the hydraulic model method of the sewer systems Loenen-1 based on dynamic storm events, Loenen-2 based on stationary storm events and Tuindorp, based on dynamic storm events of the long-time rainfall series.

**Table A3.**Loenen-2, τ

_{b}value of the comparison of the degree of criticality of various stationary storm events.

Rainfall Intensity | 10 L/s.ha | 20 L/s.ha | 30 L/s.ha | 40 L/s.ha | 50 L/s.ha | 60 L/s.ha | 70 L/s.ha | 80 L/s.ha | 90 L/s.ha | 100 L/s.ha |
---|---|---|---|---|---|---|---|---|---|---|

10 L/s.ha | 1.00 | 0.90 | 0.86 | 0.86 | 0.86 | 0.87 | 0.87 | 0.85 | 0.84 | 0.82 |

20 L/s.ha | 1.00 | 0.90 | 0.89 | 0.89 | 0.90 | 0.90 | 0.88 | 0.86 | 0.83 | |

30 L/s.ha | 1.00 | 0.95 | 0.94 | 0.93 | 0.92 | 0.90 | 0.88 | 0.84 | ||

40 L/s.ha | 1.00 | 0.96 | 0.92 | 0.92 | 0.90 | 0.88 | 0.84 | |||

50 L/s.ha | 1.00 | 0.93 | 0.92 | 0.91 | 0.88 | 0.85 | ||||

60 L/s.ha | 1.00 | 0.98 | 0.94 | 0.92 | 0.84 | |||||

70 L/s.ha | 1.00 | 0.94 | 0.92 | 0.84 | ||||||

80 L/s.ha | 1.00 | 0.96 | 0.86 | |||||||

90 L/s.ha | 1.00 | 0.88 | ||||||||

100 L/s.ha | 1.00 |

**Table A4.**Tuindorp, τ

_{b}value of the comparison of the degree of criticality of various dynamic storm events.

Storm Event | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

1 | 1.00 | 0.80 | 0.49 | 0.53 | 0.22 | 0.31 | 0.53 | 0.55 | 0.53 | 0.55 |

2 | 1.00 | 0.47 | 0.55 | 0.17 | 0.30 | 0.53 | 0.56 | 0.56 | 0.55 | |

3 | 1.00 | 0.58 | 0.22 | 0.29 | 0.48 | 0.48 | 0.45 | 0.47 | ||

4 | 1.00 | 0.28 | 0.36 | 0.60 | 0.59 | 0.56 | 0.56 | |||

5 | 1.00 | 0.18 | 0.29 | 0.26 | 0.21 | 0.23 | ||||

6 | 1.00 | 0.45 | 0.40 | 0.33 | 0.33 | |||||

7 | 1.00 | 0.85 | 0.72 | 0.66 | ||||||

8 | 1.00 | 0.77 | 0.73 | |||||||

9 | 1.00 | 0.77 | ||||||||

10 | 1.00 |

**Table A5.**Tuindorp, τ

_{b}value of the comparison of the degree of criticality of various storm events from the rainfall series.

Storm Event | 550717 | 560823 | 570920 | 580818 | 600623 | 601007 | 601201 | 610605 | 620725 | 621001 | 630802 | 630817 | 640817 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

550717 | 1.00 | 0.62 | 0.66 | 0.67 | 0.68 | 0.66 | 0.67 | 0.66 | 0.58 | 0.69 | 0.67 | 0.60 | 0.66 |

560823 | 1.00 | 0.64 | 0.67 | 0.66 | 0.60 | 0.62 | 0.65 | 0.57 | 0.65 | 0.68 | 0.58 | 0.61 | |

570920 | 1.00 | 0.74 | 0.72 | 0.66 | 0.68 | 0.72 | 0.61 | 0.73 | 0.73 | 0.61 | 0.68 | ||

580818 | 1.00 | 0.91 | 0.74 | 0.79 | 0.84 | 0.68 | 0.89 | 0.87 | 0.65 | 0.75 | |||

600623 | 1.00 | 0.74 | 0.79 | 0.84 | 0.67 | 0.86 | 0.89 | 0.64 | 0.75 | ||||

601007 | 1.00 | 0.81 | 0.74 | 0.67 | 0.75 | 0.73 | 0.64 | 0.79 | |||||

601201 | 1.00 | 0.83 | 0.70 | 0.80 | 0.79 | 0.64 | 0.84 | ||||||

610605 | 1.00 | 0.65 | 0.80 | 0.88 | 0.61 | 0.77 | |||||||

620725 | 1.00 | 0.70 | 0.65 | 0.64 | 0.65 | ||||||||

621001 | 1.00 | 0.83 | 0.63 | 0.76 | |||||||||

630802 | 1.00 | 0.63 | 0.75 | ||||||||||

630817 | 1.00 | 0.63 | |||||||||||

640817 | 1.00 |

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**Figure 4.**Brief overview of some basic principles of the graph theory. The numbers along the edges show the costs of the edges. The

**upper left**figure represents a graph in which v1, v2 = v2, v1; The

**upper right**figure represent a digraph in which v1, v2 ≠ v2, v1; The

**lower left**graph shows a strongly connected digraph after one connection is removed; The

**lower right**figure shows two strongly connected sub-digraphs after a connection is removed.

**Figure 6.**Static costs of conduits for an empty system (

**top**) and a system with a certain water level (

**bottom**).

**Figure 7.**Comparison between the degree of criticality based on the HMM where conduits are completely blocked (closed) and where the conduit diameter is reduced to 10 mm (open).

**Figure 8.**Comparison of the degree of criticality of the conduits based on the graph theory method (GTM) and the HMM. The Figures show the results for Loenen-1. The graphs at the left side show the ranking with the highest τ

_{b}of both the stationary storm events (

**upper graphs**) and the dynamic storm events (

**lower graphs**); The graphs at the right side show the ranking with the lowest τ

_{b}of both the stationary storm events and the dynamic storm events.

**Figure 9.**Comparison of the degree of criticality of the conduits based on the GTM and the HMM. The Figures show the results for Loenen-2. The graphs at the left side show the ranking with the highest τ

_{b}of both the stationary storm events (

**upper graphs**) and the dynamic storm events (

**lower graphs**); The graphs at the right side show the ranking with the lowest τ

_{b}.

**Figure 10.**Comparison of the rank of importance of the conduits based on the GTM and HMM. The Figures show the results for Tuindorp. The graphs at the left side show the ranking based on the graph and hydraulic model outcomes with the largest τ

_{b}of the stationary storm events (

**upper graphs**) and the dynamic storm events (

**middle graphs**) and the event of the rainfall series (

**lower graphs**); The graphs at the right side show the ranking based on the graph and hydraulic model outcomes with the smallest values for τ

_{b}.

**Figure 11.**The results of the sensitivity analysis of the graph theory method. The x-axis shows the discharge and the y-axis Kendall’s τ

_{b}. Please note that the scale of the y-axis varies.

**Figure 12.**The results of the sensitivity analysis of the graph theory method. The x-axis shows the water level and the y-axis Kendall’s τ

_{b}.

**Figure 13.**The results of the sensitivity analysis of the graph theory method. The x-axis shows the difference in crest levels and the y-axis Kendall’s τ

_{b}.

**Figure 14.**Degree of criticality for Loenen-1 (

**left**) and Loenen-2 (

**right**): the darker the line, the more critical the conduits. A part of the conduits has the same degree of criticality, but the conduits south of the high overflow structure have a different degree of criticality. In Loenen-2, these conduits are less important because in the case these conduits are blocked the water can flow to the high overflow structure which is not possible in the Loenen-1 system.

**Figure 17.**Difference in criticality rank between the HMM and the GTM for Tuindorp. The classification of the groups is based on the equal count method.

Characteristics | Loenen | Tuindorp |
---|---|---|

Catchment area | Mildly sloping | Flat |

System type | Combined | Combined |

System structure | Partly branched | Looped |

Ground level/surface level (m) | 17.8–28.6 | 0.75–2.25 |

Contributing area (ha) | 20.5 | 56.2 |

Storage volume (m^{3}) | 900 (=4.39 mm) | 4669 (=8.3 mm) |

Number of combined sewer overflow (CSO) structures | 2 | 5 |

Number of pumping stations (-) | 1 | 1 |

Pumping capacity (m^{3}/h) | 209 | 540 |

Number of inhabitants (-) | 2100 | 10.656 |

Number of edges | 352 | 778 |

Number of nodes | 337 | 684 |

Number of conduits that, when deleted, lead to unconnected nodes * | 176 | 188 |

**Table 2.**Loenen-1, τ

_{b}value of the comparison of the degree of criticality based on the HMM of various stationary storm events.

Rainfall Intensity | 40 L/s.ha (14.4 mm/h) | 60 L/s.ha (21.6 mm/h) | 90 L/s.ha (32.4 mm/h) |
---|---|---|---|

40 L/s.ha (14.4 mm/h) | 1.00 | 0.89 | 0.77 |

60 L/s.ha (21.6 mm/h) | 1.00 | 0.85 | |

90 L/s.ha (32.4 mm/h) | 1.00 |

**Table 3.**Loenen-2, τ

_{b}value of the comparison of the degree of criticality based on the HMM of various stationary storm events.

Rainfall Intensity | 40 L/s.ha (14.4 mm/h) | 60 L/s.ha (21.6 mm/h) | 90 L/s.ha (32.4 mm/h) |
---|---|---|---|

40 L/s.ha (14.4 mm/h) | 1.00 | 0.92 | 0.88 |

60 L/s.ha (21.6 mm/h) | 1.00 | 0.92 | |

90 L/s.ha (32.4 mm/h) | 1.00 |

**Table 4.**Tuindorp, τ

_{b}value of the comparison of the degree of criticality based on the HMM of various stationary storm events.

Rainfall Intensity | 40 L/s.ha (14.4 mm/h) | 60 L/s.ha (21.6 mm/h) | 90 L/s.ha (32.4 mm/h) |
---|---|---|---|

40 L/s.ha (14.4 mm/h) | 1.00 | 0.97 | 0.90 |

60 L/s.ha (21.6 mm/h) | 1.00 | 0.90 | |

90 L/s.ha (32.4 mm/h) | 1.00 |

**Table 5.**Comparison of the performance of hydraulic model and the graph theory. The time of the hydraulic model is based on one storm event.

Network | Number Elements | Computer Time Hydraulic Model | Computer Time Graph Methodology | Computational Gain Factor |
---|---|---|---|---|

Loenen-1 stationary storm event | 337 | 2 h 45 min | 2 s | 4950 |

Loenen-1 Dynamic storm event | 337 | 2 h 24 min | 2 s | 4320 |

Loenen-2 stationary storm event | 337 | 2 h 45 min | 4 s | 2475 |

Loenen-2 Dynamic storm event | 337 | 2 h 24 min | 4 s | 2160 |

Tuindorp stationary storm event | 778 | 6 h 24 min | 38 s | 606 |

Tuindorp Dynamic storm events | 778 | 4 h 12 min | 38 s | 398 |

**Table 6.**Differences that influences the application of the GTM between sewer systems and water supply networks.

Characteristic | Sewer System | Water Supply Network |
---|---|---|

Driven by | Supply driven | Demand driven |

System type | Gravity | Pressurised |

Failing mechanism | Blockage | Leakage conduit burst |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Meijer, D.; Van Bijnen, M.; Langeveld, J.; Korving, H.; Post, J.; Clemens, F.
Identifying Critical Elements in Sewer Networks Using Graph-Theory. *Water* **2018**, *10*, 136.
https://doi.org/10.3390/w10020136

**AMA Style**

Meijer D, Van Bijnen M, Langeveld J, Korving H, Post J, Clemens F.
Identifying Critical Elements in Sewer Networks Using Graph-Theory. *Water*. 2018; 10(2):136.
https://doi.org/10.3390/w10020136

**Chicago/Turabian Style**

Meijer, Didrik, Marco Van Bijnen, Jeroen Langeveld, Hans Korving, Johan Post, and François Clemens.
2018. "Identifying Critical Elements in Sewer Networks Using Graph-Theory" *Water* 10, no. 2: 136.
https://doi.org/10.3390/w10020136