# Greedy Algorithms for Sensor Location in Sewer Systems

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{1}, y

_{2}, … y

_{i}, … y

_{N}], where y

_{i}is the original node index of sensor i. It is also assumed that a node can accommodate only one sensor.

#### 2.1. Design Objectives

#### 2.1.1. Detection Time (D)

_{i}is exceeded. The detection time of the monitoring network, ${D}_{s}\left(Y\right)$, is defined as the shortest time among the detection times of the N monitoring sensors. It is mathematically expressed as

_{sim}, obtaining

_{sp}(Y) over all possible scenarios:

#### 2.1.2. Reliability (R)

#### 2.1.3. Joint Entropy (JH)

_{1}, x

_{2}, …, x

_{n}and corresponding probabilities of occurrence p(x

_{1}), p(x

_{2}), …, p(x

_{n}), the entropy is expressed as

_{i}at a node X. The amount of information available within two variables (nodes equipped with a sensor) X

_{1}and X

_{2}is given by the joint entropy, JH, expressed by

_{1i}, x

_{2j}) is the joint probability of the variables X

_{1}and X

_{2}, and n and m are the number of elementary events (measurements) in X

_{1}and X

_{2}, respectively. This definition is similarly extended to the N nodes.

#### 2.1.4. Total Correlation (TC)

#### 2.2. The Proposed Procedures

_{max}and D

_{min}are the maximum and the minimum detection time, assumed to be equal to the total simulation time and the reporting time step of the hydraulic simulation, respectively. Similarly, R

_{max}and R

_{min}are the maximum and minimum reliability of the system, respectively. The first sensor is chosen as that with the maximum reliability.

_{max}and TC

_{min}are the maximum and minimum total correlation of the system, respectively, while JH

_{max}and JH

_{min}are the maximum and minimum joint entropy of the system, respectively. In this formulation, the most informative sensor is chosen as the starting sensor.

#### 2.3. Fitness Function Evaluation

_{q}:

## 3. Results and Discussion

#### 3.1. Case Study

^{2}and serving a population of 14,087 (2011) with an approximate volume of yearly produced wastewater of $1.13\times {10}^{6}$ m

^{3}. The scheme consists of 1909 circular conduits connecting 1902 junctions, 14 pumps, 14 storage units and 1 treatment plant. All geometric data, not reported herein, can be requested from the authors. The calibration of the input file was previously performed using discharge measurements, obtaining a good agreement between simulated results and measured data adopted; for all conduits, Manning’s roughness coefficient is equal to 0.016 m

^{−1/3}·s.

_{sys}and TC

_{sys}, and they depend on the detection threshold. Table 3 reports JH

_{sys}and TC

_{sys}for the different considered thresholds. The minimum JH and TC are assumed to be equal to 1 and 0 bits, respectively. We note that TC = 0 means that the locations do not give redundant data from the information theory perspective.

#### 3.2. Procedures’ Comparison

_{1}, M

_{2}and M

_{3}, which consider all four objectives. These are estimated as the mean of the parameters W

_{i}(with i = 1, …, 4) computed for each objective:

_{i}are evaluated in the three different ways described in the following, and the index j = 1, …, 3 represents the criterion adopted. The first, used for computing M

_{1}, is

_{i_M}and O

_{i_MN}are the maximum and minimum values of the objective among all the selected solutions, respectively.

_{2}is calculated considering the following parameter:

_{3}, considers the following parameter:

_{max}and O

_{min}are the maximum and minimum possible values of the objective i. As mentioned above, for the considered case study, the maximum R value is 97.39%, while the minimum values of D and TC are taken as 5 min and 1 bit, respectively. The tests being realized with 0.0001 mg/L as the detection threshold, the maximum value of JH is 16.71 bits (Table 3).

_{1}, M

_{2}and M

_{3}are in the range [0, 1], and a higher score indicates a better solution. Figure 3 and Table 4 report the values of M

_{1}, M

_{2}and M

_{3}for all the procedures obtained with 8, 12 and 14 sensors.

_{2}indicator is estimated with 12 sensors. Procedures GR3, GR5 and GR6 have similar performances with eight sensors, while procedure GR6, which considers all objectives, is third in the list with a higher number of monitoring stations. These results indicate detection time to be the more suitable objective.

#### 3.3. Detection Threshold Influence

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Falconer, R.A. Global water security: An introduction. Sci. Parliam.
**2011**, 68, 34–36. [Google Scholar] - Pouet, M.F.; Thomas, O.; Marcoux, G. Quality survey of wastewater discharges. In Wastewater Quality Monitoring and treAtment; Quevauviller, P., Thomas, O., Van der Beken, A., Eds.; J. Wiley & Sons: Chichester, UK, 2006; pp. 275–288. [Google Scholar]
- Field, R.; Pitt, R.; Lalor, M.; Brown, M.; Vilkelis, W.; Phackston, E. Investigation of dry-weather pollutant entries into storm-drainage systems. J. Environ. Eng.
**1994**, 120, 1044–1066. [Google Scholar] [CrossRef] - Irvine, K.; Rossi, M.C.; Vermette, S.; Bakert, J.; Kleinfelder, K. Illicit discharge detection and elimination: Low cost options for source identification and track down in stormwater systems. Urban Water J.
**2011**, 8, 379–395. [Google Scholar] [CrossRef] - Bourgeois, W.; Burgess, J.E.; Stuetz, R.M. On-line monitoring of wastewater quality: A review. J. Chem. Technol. Biotechnol.
**2001**, 76, 337–348. [Google Scholar] [CrossRef] - Llopart-Mascaró, A.; Gil, A.; Cros, J.; Alarcón, F. Guidelines for on-line monitoring of wastewater and stormwater quality. In Proceedings of the 11th International Conference on Urban Drainage, Edinburgh, Scotland, UK, 31 August–5 September 2008. [Google Scholar]
- Banik, B.K.; Di Cristo, C.; Leopardi, A. A pre-screening procedure for pollution source identification in sewer systems. Procedia Eng.
**2015**, 119C, 360–369. [Google Scholar] [CrossRef] - Banik, B.K.; Di Cristo, C.; Leopardi, A.; de Marinis, G. Illicit intrusion characterization in sewer systems. Urban Water J.
**2017**, 14, 416–426. [Google Scholar] [CrossRef] - Alfonso, L.; He, L.; Lobbrecht, A.; Price, R. Information theory applied to evaluate the discharge monitoring network of the Magdalena River. J. Hydoinform.
**2013**, 15, 211–228. [Google Scholar] [CrossRef] - Ridolfi, E.; Alfonso, L.; Di Baldassarre, G.; Dottori, F.; Russo, F.; Napolitano, F. An entropy approach for the optimization of cross-section spacing for river modelling. Hydrol. Sci. J.
**2013**, 59, 126–137. [Google Scholar] [CrossRef] - Alfonso, L.; Lobbrecht, A.; Price, R. Information theory based approach for location of monitoring water level gauges in polders. Water Resour. Res.
**2010**, 46, W03528. [Google Scholar] [CrossRef] - Aral, M.M.; Guan, J.; Maslia, M.L. Optimal Design of Sensor Placement in Water Distribution Networks. J. Water Resour. Plan. Manag.
**2010**, 136, 5–18. [Google Scholar] [CrossRef] - Moss, M.E.; Tasker, G.D. Intercomparison of hydrological network-design technologies. Hydrol. Sci. J.
**1991**, 36, 209–221. [Google Scholar] [CrossRef] - Davar, Z.K.; Brimley, W.A. Hydrometric network evaluation: Audit approach. J. Water Resour. Plan. Manag.
**1990**, 116, 134–146. [Google Scholar] [CrossRef] - Telci, I.T.; Nam, K.; Guan, J.; Aral, M.M. Optimal water quality monitoring network design for river systems. J. Environ. Manag.
**2009**, 90, 2987–2998. [Google Scholar] [CrossRef] [PubMed] - Chacon-Hurtado, J.C.; Alfonso, L.; Solomatine, D.P. Rainfall and streamflow sensor network design: A review of applications, classification, and a proposed framework. Hydrol. Earth Syst. Sci.
**2017**, 21, 3071–3091. [Google Scholar] [CrossRef] - Rathi, S.; Gupta, R. Sensor placement methods for contamination detection in water distribution networks: A review. Procedia Eng.
**2014**, 89, 181–188. [Google Scholar] [CrossRef] - Dorini, G.; Jonkergouw, P.; Kapelan, Z.; Savic, D. SLOTS: Effective algorithm for sensor placement in water distribution systems. J. Water Resour. Plan. Manag.
**2010**, 136, 620–628. [Google Scholar] [CrossRef] - Rathi, S.; Gupta, R. A simple sensor placement approach for regular monitoring and contamination detection in in water distribution networks. KSCE J. Civ. Eng.
**2016**, 20, 597–608. [Google Scholar] [CrossRef] - Preis, A.; Ostfeld, A. Multiobjective contaminant sensor network design for water distribution systems. J. Water Resour. Plan. Manag.
**2008**, 134, 366–377. [Google Scholar] [CrossRef] - Weickgenannt, M.; Kapelan, Z.; Blokker, M.; Savic, D.A. Risk based sensor placement for contaminant detection in water distribution systems. J. Water Resour. Plan. Manag.
**2010**, 136, 629–636. [Google Scholar] [CrossRef] - Shen, H.; McBean, E. Pareto optimality for sensor placements in a water distribution system. J. Water Resour. Plan. Manag.
**2011**, 137, 243–248. [Google Scholar] [CrossRef] - Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evolut. Comput.
**2002**, 6, 182–197. [Google Scholar] [CrossRef] - Banik, B.K.; Alfonso, L.; Torres, A.S.; Mynett, A.; Di Cristo, C.; Leopardi, A. Optimal placement of water quality monitoring stations in sewer systems: An information theory approach. Procedia Eng.
**2015**, 119, 1308–1317. [Google Scholar] [CrossRef] - Banik, B.K.; Alfonso, L.; Di Cristo, C.; Leopardi, A.L.; Mynett, A. Evaluation of different formulations to optimally locate pollution sensors in sewer systems. J. Water Resour. Plan. Manag.
**2017**, 143. Available online: http://ascelibrary.org/doi/abs/10.1061/%28ASCE%29WR.1943-5452.0000778 (accessed on 31 March 2017). [CrossRef] - Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Ostfeld, A.; Salomons, E. Optimal layout of early warning detection stations for water distribution systems security. J. Water Resour. Plan. Manag.
**2004**, 130, 377–385. [Google Scholar] [CrossRef] - International Electrotechnical Commission. International Electrotechnical Commission; IEC 80000–13:2008; IEC: Geneva, Switzerland, 2013. [Google Scholar]
- Markus, M.; Vernon, K.H.; Tasker, G.D. Entropy and generalized least square methods in assessment of the regional value of stream gages. J. Hydrol.
**2003**, 283, 107–121. [Google Scholar] [CrossRef] - Alfonso, L.; Lobbrecht, A.; Price, R. Optimization of water level monitoring network in polder systems using information theory. Water Resour. Res.
**2010**, 46, W12553. [Google Scholar] [CrossRef] - McGill, W.J. Multivariate information transmission. Psychometrika
**1954**, 19, 97–116. [Google Scholar] [CrossRef] - Watanabe, S. Information theoretical analysis of multivariate correlation. IBM J. Res. Dev.
**1960**, 4, 66–82. [Google Scholar] [CrossRef] - Greco, S.; Zaniolo, C. Greedy algorithms in Datalog. Theor. Pract. Log. Prog.
**2001**, 1, 381–407. [Google Scholar] [CrossRef] - Tallam, S.; Gupta, N. A concept analysis inspired greedy algorithm for test suite minimization. In CM SIGSOFT Software Engineering Notes; ACM: New York, NY, USA, 2005; Volume 31, pp. 35–42. [Google Scholar]
- Kumar, R.; Moseley, B.; Vassilvitskii, S.; Vattani, A. Fast greedy algorithms in mapreduce and streaming. In Proceedings of the Twenty-Fifth Annual ACM Symposium on Parallelism in Algorithms and Architectures, Montreal, QC, Canada, 23–25 July 2013; pp. 1–10. [Google Scholar]
- Rieckermann, J.L.; Neumann, M.; Ort, C.; Huisman, J.L.; Gujer, W. Dispersion coefficients of sewers from tracer experiments. Water Sci. Technol.
**2005**, 52, 123–133. [Google Scholar] [PubMed] - Banik, B.K.; Di Cristo, C.; Leopardi, A. SWMM5 toolkit development for pollution source identification in sewer systems. Procedia Eng.
**2014**, 89, 750–757. [Google Scholar] [CrossRef] - Cozzolino, L.; Della Morte, R.; Palumbo, A.; Pianese, D. Stochastic approaches for sensors placement against intentional contaminations in water distribution systems. Civ. Eng. Environ. Syst.
**2011**, 28, 75–98. [Google Scholar] [CrossRef]

**Figure 2.**Objective function values obtained with the considered procedures (

**a**) JH and TC objectives; (

**b**) R and D objectives (12 sensors; detection threshold equal to 0.0001 mg/L).

**Figure 3.**Indicator values of the different procedures with 8, 12 and 14 sensors. (

**a**) M

_{1}: Equations (15) and (16); (

**b**) M

_{2}: Equations (15)–(17); (

**c**) M

_{3}: Equations (15)–(18).

**Figure 4.**Procedure GR1 for different detection thresholds: (

**a**) D values as a function of the number of sensors; (

**b**) sensor placement.

**Figure 5.**Procedure GR4 for different detection thresholds: (

**a**) D and R values as a function of the number of sensors; (

**b**) sensor placement.

**Table 1.**The procedures and the required computational time (C-Time) for the test with 14 sensors and detection threshold (0.0001 mg/L).

Procedure | GR1 | GR2 | GR3 | GR4 | GR5 | GR6 | B_IT | B_DR |
---|---|---|---|---|---|---|---|---|

Algorithm | GR_S | GR_S | GR_S | GR_M | GR_M | GR_M | NSGA-II | NSGA-II |

Objectives | D | R | JH | D, R | JH, TC | D, R, JH, TC | JH, TC | D, R |

C-Time (s) | 1.4 | 2.6 | 5200.4 | 3.8 | 2460.7 | 5205.3 | 143,415.2 | 1812.0 |

Procedures | All |
---|---|

Number of sensors | From 1 to 14 |

Detection threshold (mg/L) | 0.1, 0.01, 0.001, 0.0001, 0.00001 |

Detection Threshold (mg/L) | 0.00001 | 0.0001 | 0.001 | 0.01 | 0.1 |
---|---|---|---|---|---|

JH_{sys} (bits) | 16.74 | 16.71 | 16.64 | 16.40 | 15.70 |

TC_{sys} (bits) | 1895.16 | 1601.84 | 1270.35 | 948.65 | 685.80 |

Procedure | 8 Sensors | 12 Sensors | 14 Sensors | ||||||
---|---|---|---|---|---|---|---|---|---|

M_{1} | M_{2} | M_{3} | M_{1} | M_{2} | M_{3} | M_{1} | M_{2} | M_{3} | |

B_IT | 0.4552 | 0.6476 | 0.5237 | 0.1811 | 0.5757 | 0.5165 | 0.4715 | 0.6380 | 0.5212 |

B_DR | 0.4000 | 0.6236 | 0.5172 | 0.5795 | 0.6690 | 0.5356 | 0.5000 | 0.5960 | 0.5013 |

GR1 | 0.8777 | 0.7067 | 0.5429 | 0.7772 | 0.6566 | 0.5458 | 0.9293 | 0.7303 | 0.5540 |

GR3 | 0.7073 | 0.5816 | 0.5292 | 0.7167 | 0.5785 | 0.5433 | 0.7301 | 0.6134 | 0.5476 |

GR4 | 0.8867 | 0.7077 | 0.5433 | 0.7883 | 0.6578 | 0.5463 | 0.9378 | 0.7314 | 0.5547 |

GR5 | 0.7132 | 0.5732 | 0.5309 | 0.7167 | 0.5785 | 0.5433 | 0.8277 | 0.6640 | 0.5501 |

GR6 | 0.7057 | 0.5819 | 0.5289 | 0.7550 | 0.6033 | 0.5462 | 0.8509 | 0.6793 | 0.5523 |

© 2017 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/).

## Share and Cite

**MDPI and ACS Style**

Banik, B.K.; Alfonso, L.; Di Cristo, C.; Leopardi, A.
Greedy Algorithms for Sensor Location in Sewer Systems. *Water* **2017**, *9*, 856.
https://doi.org/10.3390/w9110856

**AMA Style**

Banik BK, Alfonso L, Di Cristo C, Leopardi A.
Greedy Algorithms for Sensor Location in Sewer Systems. *Water*. 2017; 9(11):856.
https://doi.org/10.3390/w9110856

**Chicago/Turabian Style**

Banik, Bijit K., Leonardo Alfonso, Cristiana Di Cristo, and Angelo Leopardi.
2017. "Greedy Algorithms for Sensor Location in Sewer Systems" *Water* 9, no. 11: 856.
https://doi.org/10.3390/w9110856