# Optimal Sensor Placement in Hydraulic Conduit Networks: A State-Space Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**D**=

**0**. The dynamics of the system are determined by the system specific parameters ${\mathcal{X}}_{ij}$, ${\mathcal{Y}}_{ij}$, and ${\mathcal{Z}}_{ij}$, which make up the elements of the $n\text{}\times \text{}n$ matrix $\mathit{A}$. The positions of the sensors are specified via the $p\text{}\times \text{}n$ matrix $\mathit{C}$. For applications of optimal sensor placement, only the system dynamics (matrix $\mathit{A}$) and the sensor locations (matrix $\mathit{C}$) are required. For model simulations, however, the $n\text{}\times \text{}m$ input matrix $\mathit{B}$ would also be required and would contain inputs such as height differences between junctions, minor losses (valves, pumps), storage junctions (tanks), and the set values of flow or pressure at water sources and sinks (demand or reservoir junctions), and thus, at the boundaries of the system. Thus, for the intended goal of optimal sensor placement, based on state-space methodology, matrices $\mathit{B}$ and $\mathit{D}$ do not need to be specified and no temporal discretization of the system, Equations (7) and (8), is required.

## 3. Results and Discussion

#### 3.1. Example 1: Triangular Network

#### 3.2. Example 2: Net1 Case Study

#### 3.3. Hanoi Network

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notation

Pressure | $p$ | $\mathrm{P}\mathrm{a}$ |

Flow velocity | $V$ | ${\mathrm{ms}}^{-1}$ |

Mass density of transported fluid | $\rho $ | ${\mathrm{kgm}}^{-3}$ |

Elastic wave velocity | $c$ | ${\mathrm{ms}}^{-1}$ |

Gravitational acceleration | $g$ | ${\mathrm{ms}}^{-2}$ |

Angle of conduit versus horizontal | $\theta $ | $\mathrm{rad}$ |

Darcy–Weisbach friction factor | $f$ | |

Inside diameter of conduit | $D$ | $\mathrm{m}$ |

Elevation | $z$ | $\mathrm{m}$ |

Cross-sectional area of conduit | $A$ | ${\mathrm{m}}^{2}$ |

Piezometric head | $H$ | $\mathrm{m}$ |

Volumetric flowrate | $Q$ | ${\mathrm{m}}^{3}{\mathrm{s}}^{-1}$ |

Hazen–Williams roughness coefficient | $C$ | |

Relative flow gradient | $\epsilon $ | ${\mathrm{m}}^{-1}$ |

Observability matrix $pn\times n$ | $\mathcal{O}$ | |

observability Gramian $n\times n$ | ${W}_{\mathcal{O}}$ | |

Eigenvalue optimality output energy | $\mathcal{E}$ | |

Eigenvalues of the observability Gramian $n\times 1$ | ${\lambda}_{{W}_{\mathcal{O}}}$ | |

State vector $n\times 1$ | $x$ | |

Output vector $p\times 1$ | $y$ | |

Output vector associated with the optimal sensor configuration $p\times 1$ | ${y}_{opt}$ | |

Input vector | $u$ | |

State matrix $n\times n$ | $\mathit{A}$ | |

Input matrix | $\mathit{B}$ | |

Output matrix $p\times n$ | $\mathit{C}$ | |

Feedthrough matrix | $\mathit{D}$ | |

Network junction | $i$ | |

Network conduit connecting junction $i$ and $j$ | $ij$ | |

Total number of states | $n$ | |

Number of junction head states | ${n}_{i}$ | |

Number of conduit flow states | ${n}_{ij}$ | |

Total number of states whose corresponding asset contains a sensor | $p$ |

## Appendix A

## Appendix B

^{−6},1], the sensor configuration with a pressure sensor at node 2 or 3 maximizes the output energy. Therefore, in the case studies we chose $\epsilon ={10}^{-3}{\mathrm{m}}^{-1}$, a good estimate regarding the application of optimal sensor placement.

## Appendix C

${\mathit{\lambda}}_{\mathit{A}}$ | ${\mathit{v}}_{\mathit{A}}$ | ||||||
---|---|---|---|---|---|---|---|

Link 12 | Link 13 | Link 23 | Link 41 | Node 1 | Node 2 | Node 3 | |

−0.003 + 0.112j | −0 + 0.001j | 0.001 − 0.001j | 0 − 0.037j | −0.001 − 0.001j | −0.018 + 0.046j | 0.709 | −0.702 − 0.032j |

−0.003 − 0.112j | −0 − 0.001j | 0.001 + 0.001j | 0 + 0.037j | −0.001 + 0.001j | −0.018 − 0.046j | 0.709 | −0.702 + 0.032j |

−0.025 + 0.08j | 0.001 − 0.003j | 0.001 − 0.001j | −0.002 − 0.002j | −0.004 + 0.028j | 0.971 | −0.116 − 0.073j | −0.191 − 0.029j |

−0.025 − 0.08j | 0.001 + 0.003j | 0.001 + 0.001j | −0.002 + 0.002j | −0.004 − 0.028j | 0.971 | −0.116 + 0.073j | −0.191 + 0.029j |

−0.091 | 0.004 | −0.014 | 0.019 | −0.023 | −0.534 | 0.23 | 0.813 |

−0.025 + 0.003j | −0.005 + 0.001j | −0.001 | −0.003 + 0.001j | −0.019 + 0.002j | 0.102 + 0.012j | 0.716 | 0.69 + 0.008j |

−0.025−0.003j | −0.005 − 0.001j | −0.001 | −0.003 − 0.001j | −0.019 − 0.002j | 0.102 − 0.012j | 0.716 | 0.69 − 0.008j |

## References

- Chaudhry, M.H. Applied Hydraulic Transients, 3rd ed.; Springer: New York, NY, USA, 2014. [Google Scholar]
- Díaz, S.; González, J.; Mínguez, R. Observability Analysis in Water Transport Networks: Algebraic Approach. J. Water Resour. Plan. Manag.
**2016**, 142, 04015071. [Google Scholar] [CrossRef] - di Nardo, A.; di Natale, M.; di Mauro, A.; Santonastaso, G.F.; Palomba, A.; Locoratolo, S. Calibration of a water distribution network with limited field measures: The case study of Castellammare di Stabia (Naples, Italy). In Learning and Intelligent Optimization; LION 12 2018: Lecture Notes in Computer Science; Springer: Cham, Switzerland, 2019; Volume 11353, pp. 433–436. [Google Scholar] [CrossRef]
- Sophocleous, S.; Savić, D.A.; Kapelan, Z.; Giustolisi, O. A Two-stage Calibration for Detection of Leakage Hotspots in a Real Water Distribution Network. Procedia Eng.
**2017**, 186, 168–176. [Google Scholar] [CrossRef] - Fuertes, P.C.; Alzamora, F.M.; Carot, M.H.; Campos, J.C.A. Building and exploiting a Digital Twin for the management of drinking water distribution networks. Urban Water J.
**2020**, 17, 704–713. [Google Scholar] [CrossRef] - Qi, Z.; Zheng, F.; Guo, D.; Maier, H.R.; Zhang, T.; Yu, T.; Shao, Y. Better Understanding of the Capacity of Pressure Sensor Systems to Detect Pipe Burst within Water Distribution Networks. J. Water Resour. Plan. Manag.
**2018**, 144, 04018035. [Google Scholar] [CrossRef] - Steffelbauer, D.B.; Fuchs-Hanusch, D. Efficient Sensor Placement for Leak Localization Considering Uncertainties. Water Resour. Manag.
**2016**, 30, 5517–5533. [Google Scholar] [CrossRef][Green Version] - Sarrate, R.; Nejjari, F.; Rosich, A. Sensor placement for fault diagnosis performance maximization in Distribution Networks. In Proceedings of the 2012 20th Mediterranean Conference on Control & Automation (MED), Barcelona, Spain, 3–6 July 2012; pp. 110–115. [Google Scholar] [CrossRef][Green Version]
- Farley, B.; Mounce, S.R.; Boxall, J.B. Field testing of an optimal sensor placement methodology for event detection in an urban water distribution network. Urban Water J.
**2010**, 7, 345–356. [Google Scholar] [CrossRef] - Bonada, E.; Meseguer, J.; Tur, J.M.M. Practical-Oriented Pressure Sensor Placement for Model-Based Leakage Location in Water Distribution Networks. In Proceedings of the International Conference on Hydroinformatics, New York, NY, USA, 17–21 August 2014. [Google Scholar]
- Boatwright, S.; Romano, M.; Mounce, S.; Woodward, K.; Boxall, J. Optimal Sensor Placement and Leak/Burst Localisation in a Water Distribution System Using Spatially-Constrained Inverse-Distance Weighted Interpolation. In Proceedings of the 13th International Conference on Hydroinformatics, Palermo, Italy, 1–6 July 2018; Volume 3, pp. 282–289. [Google Scholar]
- Nagar, A.K.; Powell, R.S. Observability analysis of water distribution systems under parametric and measurement uncertainty. In Proceedings of the Joint Conference on Water Resource Engineering and Water Resources Planning and Management, Minneapolis, MN, USA, 30 July–2 August 2000; Volume 104, p. 55. [Google Scholar] [CrossRef]
- Marchi, A.; Dandy, G.C.; Boccelli, D.L.; Rana, S.M.M. Assessing the Observability of Demand Pattern Multipliers in Water Distribution Systems Using Algebraic and Numerical Derivatives. J. Water Resour. Plan. Manag.
**2018**, 144, 04018014. [Google Scholar] [CrossRef] - Sarrate, R.; Blesa, J.; Nejjari, F.; Quevedo, J. Sensor placement for leak detection and location in water distribution networks. Water Sci. Technol. Water Supply
**2014**, 14, 795–803. [Google Scholar] [CrossRef][Green Version] - Quintiliani, C.; Vertommen, I.; van Laarhoven, K.; van der Vliet, J.; Van Thienen, P. Optimal Pressure Sensor Locations for Leak Detection in a Dutch Water Distribution Network. Environ. Sci. Proc.
**2020**, 2, 40. [Google Scholar] [CrossRef] - Pudar, R.S.; Liggett, J.A. Leaks in Pipe Networks. J. Hydraul. Eng.
**1992**, 118, 1031–1046. [Google Scholar] [CrossRef] - Cugueró-Escofet, M.; Puig, V.; Quevedo, J. Optimal pressure sensor placement and assessment for leak location using a relaxed isolation index: Application to the Barcelona water network. Control Eng. Pract.
**2017**, 63, 1–12. [Google Scholar] [CrossRef][Green Version] - Qi, J.; Sun, K.; Kang, W. Optimal PMU Placement for Power System Dynamic State Estimation by Using Empirical Observability Gramian. IEEE Trans. Power Syst.
**2015**, 30, 2041–2054. [Google Scholar] [CrossRef][Green Version] - Xu, B.; Abur, A. Observability analysis and measurement placement for systems with PMUs. In Proceedings of the IEEE PES Power Systems Conference and Exposition, New York, NY, USA, 10–13 October 2004; Volume 2, pp. 943–946. [Google Scholar] [CrossRef][Green Version]
- Johnson, T.; Moger, T. A critical review of methods for optimal placement of phasor measurement units. Int. Trans. Electr. Energy Syst.
**2021**, 31, e12698. [Google Scholar] [CrossRef] - Quiñones-Grueiro, M.; Bernal-De-Lázaro, J.M.; Verde, C.; Prieto-Moreno, A.; Llanes-Santiago, O. Comparison of Classifiers for Leak Location in Water Distribution Networks. IFAC-PapersOnLine
**2018**, 51, 407–413. [Google Scholar] [CrossRef] - Casillas, M.V.; Puig, V.; Garza-Castañón, L.E.; Rosich, A. Optimal sensor placement for leak location in water distribution networks using genetic algorithms. Sensors
**2013**, 13, 14984–15005. [Google Scholar] [CrossRef] [PubMed][Green Version] - Giustolisi, O.; Savic, D.; Kapelan, Z. Pressure-Driven Demand and Leakage Simulation for Water Distribution Networks. J. Hydraul. Eng.
**2008**, 134, 626–635. [Google Scholar] [CrossRef][Green Version] - Kwakernaak, H.; Sivan, R. Linear Optimal Control Systems; Wiley-Interscience: New York, NY, USA, 1972; Volume 1. [Google Scholar]
- Kalman, R.E. Mathematical Description of Linear Dynamical Systems. J. Soc. Ind. Appl. Math. Ser. A Control
**1963**, 1, 152–192. [Google Scholar] [CrossRef] - Georges, D. Use of observability and controllability gramians or functions for optimal sensor and actuator location in finite-dimensional systems. In Proceedings of the IEEE Conference on Decision and Control, New Orleans, LA, USA, 13–15 December 1995; Volume 4. [Google Scholar] [CrossRef]
- Rossman, L.A. EPANET 2; U.S. Environmental Protection Agency: Washington, DC, USA, 2000. [Google Scholar]
- Watters, G.Z. Analysis and Control of Unsteady Flow in Pipelines, 2nd ed.; Butterworths: Waltham, MA, USA, 1984. [Google Scholar]
- Dager, R.; Zuazua, E. Wave propagation, observation and control in 1-d flexible multi-structures. Math. Appl.
**2006**, 50, 227. [Google Scholar] - Izquierdo, J.; Pérez, R.; Iglesias, P.L. Mathematical models and methods in the water industry. Math. Comput. Model.
**2004**, 39, 1353–1374. [Google Scholar] [CrossRef] - Ramos, H.; Covas, D.; Borga, A.; Loureiro, D. Surge damping analysis in pipe systems: Modelling and experiments. J. Hydraul. Res.
**2004**, 42, 413–425. [Google Scholar] [CrossRef] - Zhang, Z. Hydraulic Transients and Computations; Springer: Cham, Switzerland, 2020. [Google Scholar]
- Grubben, N.L.M.; Keesman, K.J. Controllability and observability of 2D thermal flow in bulk storage facilities using sensitivity fields. Int. J. Control
**2018**, 91, 1554–1566. [Google Scholar] [CrossRef][Green Version] - Keesman, K.J. Sytem Identification, an Introduction, Advanced Textbooks in Control and Signal Processing; Springer: Cham, Switzerland, 2011. [Google Scholar]
- Pronzato, L.; Pázman, A. Design of Experiments in Nonlinear Models: Asymptotic Normality, Optimality Criteria and Small-Sample Properties. Linear Notes Stat.
**2013**, 212, 404. [Google Scholar] - Wald, A. On the Efficient Design of Statistical Investigations. Ann. Math. Stat.
**1943**, 14, 134–140. [Google Scholar] [CrossRef] - Fujiwara, O.; Khang, D.B. A two-phase decomposition method for optimal design of looped water distribution networks. Water Resour. Res.
**1990**, 26, 539–549. [Google Scholar] [CrossRef] - Bi, W.; Dandy, G.C.; Maier, H.R. Improved genetic algorithm optimization of water distribution system design by incorporating domain knowledge. Environ. Model. Softw.
**2015**, 69, 370–381. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**): schematic overview of the triangular network, where

**R**is the reservoir and conduit 41 has a flow sensor. (

**b**): Eigenvalue decomposition of the corresponding observability Gramian ${\mathit{W}}_{\mathcal{O}}$ of the triangular network, with only a flow sensor in conduit 41, where each column represents an eigenvector of ${\mathit{W}}_{\mathcal{O}}$ and each column label at the bottom has the corresponding eigenvalue ${\mathit{\lambda}}_{{\mathit{W}}_{\mathcal{O}}}$.

**Figure 2.**Triangular network, where

**R**represents a Reservoir, squares are junctions, and lines are conduits. Black dashed lines indicate a conduit with a flow sensor. Each additional possible sensor junction and conduit is colored based on the square root of the output energy (Equation (11)), corresponding with sensor placement in that specific junction or conduit.

**Figure 3.**Optimal sensor placement for Net1 network with one tank (

**T**), Reservoir (

**R**), and pump (

**P**) at 08:00 (

**a**) and 20:00 (

**b**). Black dashed lines indicate a conduit with a flow sensor. Each additional possible sensor junction and conduit is colored based on the square root of the energy corresponding with sensor placement in that specific state. At 20:00, conduit 9 is closed, thus decoupling reservoir 9 from the network.

**Figure 4.**Comparison of possible pressure sensor placements in the Hanoi network. Possible pressure sensor junctions (squares) are colored based on 10-log output energy associated with placement of a sensor in that junction. Those conduits (black lines) attached to a reservoir (R) are already metered (dashed black lines).

Conduit | Length [m] | Diameter [m] | Roughness | Flow [m^{3}/s] | ${\mathcal{X}}_{\mathit{i}\mathit{j}}[1/{\mathbf{m}}^{2}/\mathbf{s}]$ | ${\mathcal{Y}}_{\mathit{i}\mathit{j}}[{\mathbf{m}}^{2}/{\mathbf{s}}^{2}]$ | ${\mathcal{Z}}_{\mathit{i}\mathit{j}}\phantom{\rule{0ex}{0ex}}\left[{\mathbf{s}}^{-1}\right]$ |
---|---|---|---|---|---|---|---|

12 | 1524 | 0.2032 | 120 | 2.50 × 10^{−2} | 4.53 × 10^{3} | 2.09 × 10^{−4} | −4.85 × 10^{−2} |

13 | 914.4 | 0.1524 | 80 | 1.10 × 10^{−2} | 8.05 × 10^{3} | 1.96 × 10^{−4} | −1.10 × 10^{−1} |

23 | 243.8 | 0.3048 | 200 | −1.48 × 10^{−3} | 2.01 × 10^{3} | 2.93 × 10^{−3} | −5.29 × 10_{-4} |

41 | 304.8 | 0.3048 | 100 | 4.86 × 10^{−2} | 2.01 × 10^{3} | 2.35 × 10^{−3} | −3.74 × 10^{−2} |

State | Conduit | Junction | ||||||
---|---|---|---|---|---|---|---|---|

State | 12 | 13 | 23 | 41 | 1 | 2 | 3 | |

Conduit | 12 | −4.85 × 10^{−2} | 0 | 0 | 0 | 2.09 × 10^{−4} | −2.09 × 10^{−4} | 0 |

13 | 0 | −1.00 × 10^{−1} | 0 | 0 | 1.96 × 10^{−4} | 0 | −1.96 × 10^{−4} | |

23 | 0 | 0 | −5.29 × 10^{−4} | 0 | 0 | 2.93 × 10^{−3} | −2.93 × 10^{−3} | |

41 | 0 | 0 | 0 | −3.74 × 10^{−2} | −2.35 × 10^{−3} | 0 | 0 | |

Junction | 1 | −4.53 × 10^{3} | −8.05 × 10^{3} | 0 | 2.01 × 10^{3} | 0 | 0 | 0 |

2 | 4.53 × 10^{3} | 0 | −2.01 × 10^{3} | 0 | 0 | 0 | 0 | |

3 | 0 | 8.05 × 10^{3} | 2.01 × 10^{3} | 0 | 0 | 0 | 0 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Geelen, C.V.C.; Yntema, D.R.; Molenaar, J.; Keesman, K.J.
Optimal Sensor Placement in Hydraulic Conduit Networks: A State-Space Approach. *Water* **2021**, *13*, 3105.
https://doi.org/10.3390/w13213105

**AMA Style**

Geelen CVC, Yntema DR, Molenaar J, Keesman KJ.
Optimal Sensor Placement in Hydraulic Conduit Networks: A State-Space Approach. *Water*. 2021; 13(21):3105.
https://doi.org/10.3390/w13213105

**Chicago/Turabian Style**

Geelen, Caspar V. C., Doekle R. Yntema, Jaap Molenaar, and Karel J. Keesman.
2021. "Optimal Sensor Placement in Hydraulic Conduit Networks: A State-Space Approach" *Water* 13, no. 21: 3105.
https://doi.org/10.3390/w13213105