UPTEC20 028
Examensarbete 30 hp Juni 2020
Model based optimization of
flow sensor locations in drinking water networks
Cristopher Stedt
Teknisk- naturvetenskaplig fakultet UTH-enheten
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Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0
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Box 536 751 21 Uppsala
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018 – 471 30 03
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018 – 471 30 00
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http://www.teknat.uu.se/student
Abstract
Model based optimization of flow sensor locations in drinking water networks
Cristopher Stedt
This thesis uses a model-based approach to finding optimal positions of flow data sensors in a water distribution network (WDN). Flow data sensors can be used to identify and localize leaks in a network that lead to significant losses of both water and energy.
A simple hydraulic model of a WDN was constructed as well as a graph theoretic description of the network. A framework for describing the information gained by a specific sensor position was developed and used to formulate the sensor placement as an optimization problem.
Two algorithms were constructed for finding the optimal placements under different assumptions of the network. Three smaller networks of different complexity were constructed and on which the two algorithms were tested.
The results show that a model-based approach indeed can be used to find optimal sensor placement under different assumptions of the system at hand and that the framework can be used not only for placement, but also for localization of leaks present in the network.
ISSN: 1650-8300, UPTEC ES20 028 Examinator: Petra Jönsson
Ämnesgranskare: Steffi Knorn Handledare: Roxanne Jackson
Popul¨ arvetenskaplig sammanfattning
Rent vatten utg¨or en resurs som inte bara ¨ar essentiell som dricksvatten, utan ¨aven som komponent i produktframst¨allning, energiproduktion och livsmedelsproduktion.
De tekniska system som omfattar framst¨allningen och transporten av vatten utg¨or d¨arf¨or en kritisk del av samh¨allets infrastruktur. I takt med att ledningsr¨or och andra komponenter ˚aldras samt yttre p˚averkan uppst˚ar skador som leder till att vatten l¨acker fr˚an systemen. I vissa fall kan l¨ackorna enkelt uppt¨ackas d˚a vattnet som l¨acker ut n˚ar markytan och rapporteras d˚a ofta direkt till det ansvariga f¨oretaget.
I m˚anga andra fall kan dock inte l¨ackorna identifieras och kan under l˚ang tid bidra till stora f¨orluster fr˚an systemet.
Att hitta l¨ackorna i systemen ¨ar en tid- och resurskr¨avande process. Bland de metoder som anv¨ands finns bland annat manuell inspektion av r¨oren och metoder d¨ar man skickar en ljudsignal genom r¨oren som sedan analyseras. Gemensammt f¨or dessa metoder ¨ar att de ¨ar resurskr¨avande och kr¨aver att man redan har en god id´e om var l¨ackan kan t¨ankas finnas f¨or att kunna begr¨ansa s¨okomr˚adet.
Ett alternativ till dessa metoder ¨ar att installera ett antal sensorer som m¨ater vattenfl¨odet i r¨oren. Genom att m¨ata vattenfl¨odet kan man dra slutsatser om hu- ruvida det finns l¨ackor i systemet eller inte. Att installera och underh˚alla de h¨ar sensorerna ¨ar dock en omfattande investering och kr¨aver stora ingrepp i n¨atverket varf¨or endast ett begr¨ansat antal sensorer i praktiken kan anv¨andas. Givet ett visst antal sensorer f¨oljer d˚a problemet var dessa ska placeras.
I det h¨ar arbetet har en modellbaserad metod tagits fram f¨or att unders¨oka var i ledningsn¨aten man g¨or b¨ast i att placera permanenta fl¨odessensorer. Arbetet inled- des med att ta fram en enkel modell som beskriver vattnets v¨ag genom ledningsn¨atet och fortsattes sedan med att, utifr˚an modellen, ta fram metoder som kan anv¨andas f¨or att hitta de optimala sensorplaceringarna. Ett antal n¨atverk av olika komplexitet konstruerades och anv¨andes tillsammans med olika kriterier f¨or att utv¨ardera b˚ade metoderna och de f¨oreslagna sensorplaceringarna.
Projektet mynnade ut i ett antal koncept som kan anv¨andas f¨or analys av den typ av modell som anv¨ants samt tv˚a olika metoder f¨or att hitta l¨ampliga sensorplac- eringar. Den ena metoden testar varje t¨ankbar kombination av sensorpositioner och utv¨arderar den utifr˚an olika kriterier medan den andra metoden testar endast en del av alla positioner. Resultaten visade att i en del fall presterar de tv˚a metoderna likv¨ardigt, men f¨or mer komplicerade kriterier kan det vara stor skillnad mellan olika placeringar.
F¨orutom placering kan en del av metoderna ¨aven anv¨andas f¨or att lokalisera,
dvs. hitta, l¨ackor i n¨atverket. Genom att kombinera m¨atv¨arden fr˚an sensorerna som
¨
ar placerade i n¨atverket kan de omr˚aden som potentiellt kan ha l¨ackor begr¨ansas.
Genom detta kan de resurser och den tid som kr¨avs f¨or att hitta och laga l¨ackor i n¨atverket reduceras.
Det h¨ar arbetet har visat att det finns en potential i att anv¨anda modellbaserade metoder f¨or att ta fram optimala sensorplaceringar men ¨aven pekat p˚a en del av utmaningarna med detta tillv¨agag˚angss¨att.
Executive summary
With global warming in mind and the associated changes in weather and climate, the availability of fresh water is predicted to decline in the coming decades at the same time as the demand for it is expected to increase heavily due to population growth, urbanization and an increased energy demand. This makes the efficiency of the water systems a pressing matter and loss reduction in the distribution chain is one step in this direction.
Losses occur in water distribution system for multiple reasons and in many dif- ferent ways. Some damage happen that over time wear down pipes, valves and joints which in turn can lead to leakage in the system. Damage can also be contributed to external forces from for instance ground movement. While some leaks are large enough to cause water reaching the ground surface and thus be immediately visible, a large part of the leaks in the systems are not as easily detectable. Common meth- ods for finding and localizing leaks include manual surveys, acoustic methods and ground penetrating radar methods. While these methods often are effective, they are time and cost intensive and require some knowledge about where the leaks may potentially be to narrow down the search perimeters.
A novel alternative to the aforementioned methods is to install a permanent net- work of flow sensors in the system which continually measure the flow rate in the pipes. The flow data can be used for both leak detection as well as leak localization, where the first term refers to actually determining if a leak is present or not and the second term refers to finding the position of a leak.
In this project a model-based approach for finding the optimal placements of sensors in a water distribution network was developed. A framework for describing the information gained by sensors was worked out together with two algorithms for actually finding the optimal placements given different networks and criteria. The first algorithm finds the global optima for any given number of sensors and criteria while the second algorithm finds at least local optima. While outside the scope of this thesis, a short example is included to demonstrate that the concepts used in this thesis also can be used for leak localization.
The results show that indeed a model-based approach can successfully be used to optimize the positions under the assumptions made. While the methods developed here are general, the computational complexity of some of the methods require further adaption to be applicable to large scale networks. The results show that there are many ways to build upon the work presented in this thesis and some of the possibilities are discussed in more depth at the end of the report.
Acknowledgements
First of all, I would like to thank all the people who have supported me and pro- vided feedback on the project. A special thanks goes out to the people at IFAT at Otto von Guericke Universiy Magdeburg for making my stay in Magdeburg both educative and enjoyable.
Of course, this list of benefactors would not be complete without any mention of my family, both immediate and extended. No names mentioned and none forgotten.
Nomenclature
Abbreviations AL Apparent losses
DAG Directed Acyclic Graph
IWA International Water Association RL Real losses
WDN Water Distribution Network WSS Water Supply System
General
ci Consequence of a leak on edgei li Leak flow from edge i
pi Probability of a leak on edgei yi Measured flow on edgei Matrices and vectors
C Sensor matrix L Leak matrix
S Flow matrix s Flow vector
S∗ Conjugate trasnpose of S S† Pseudo-inverse of S y Measurement vector Sets
B Basin
L Leak zone
Lˆ Complete leak zone F Flow signature P Parallel set
SU i Subset with upper bound SLi Subset with lower bound U Universe
Contents
1 Introduction 1
1.1 Problem definition . . . 1
1.2 Purpose and goal . . . 2
1.3 Project boundaries . . . 2
1.4 Approach . . . 2
2 Background 3 2.1 Water Supply Systems . . . 3
2.1.1 System overview . . . 3
2.1.2 Water loss . . . 4
2.1.3 Leakage management . . . 5
2.2 Mathematical prerequisites . . . 6
2.2.1 Set theory . . . 6
2.2.2 Graph theory . . . 7
2.2.3 Linear algebra . . . 10
3 System setup 12 3.1 Model and assumptions . . . 12
3.1.1 Assumptions . . . 12
3.1.2 The model . . . 13
3.1.3 Flow measurements in the model . . . 14
3.2 Simulating flow . . . 15
4 Method 17 4.1 Analysis of the network . . . 17
4.1.1 Partitioning of the network . . . 17
4.1.2 Basins and leak zones . . . 18
4.1.3 Relations between basins and leak zones . . . 20
4.1.4 Bounded subsets . . . 25
4.1.5 Global structure of the sets . . . 27
4.2 Optimization problem . . . 29
4.2.1 Cost functions . . . 30
4.2.2 Global optimum . . . 32
4.2.3 Local optimum . . . 32
4.3 Test cases . . . 34
4.3.1 Performance of Algorithm 1 and 2 . . . 34
4.3.2 Using prior knowledge . . . 34
4.3.3 Simulating flow and localizing leaks . . . 35
4.3.4 Networks . . . 35
Chapter 0 CONTENTS
5 Results 37
5.1 Algorithm 1 and 2 . . . 37
5.1.1 Unbiased cost . . . 37
5.1.2 Probability and risk . . . 39
5.2 Flow simulations and localizing . . . 41
6 Discussion 45 7 Conclusions and outlook 49 Bibliography 51 A Tables 52 B Code 57 B.1 MATLAB code . . . 57
B.2 Python code . . . 58
C Proofs 79
Chapter 1 Introduction
Water plays an important role in all parts of society; the availability of drinking water is essential for sustaining life and clean water is needed for agriculture, industry and energy production among other things. While not strictly a limited resource, the availability of fresh water is expected to be heavily impacted in the coming years by urbanization, population growth and climate change [1]. Some research point to scenarios where the global fresh water demand increase with around 55 % between the years 2000 and 2050 due to increasing manufacturing, electricity production and domestic usage [2].
The relationship between water and energy is sometimes called the water-energy nexus where the term refers to the fact that water production requires energy at the same time as energy production requires water. This inter-dependency has the implication that any losses in the water system will eventually lead to energy losses as well. The fact that water losses can make up for as much as 30-40 % of the produced water [3] coupled with the predicted water scarcity makes loss-reduction a pressing issue for the water sector and for society in general.
1.1 Problem definition
Some leaks are easily identified and localized, as they lead to water appearing on the ground, and can be managed relatively easy by the water company. The leaks that do not result in visible changes however often remain hidden and can continue to contribute to the losses over a long time until they either lead to a visible loss or until found for some other reason. The methods available for handling the leaks range from manual inspection of individual pipes to acoustic methods and ground penetrating radar techniques, all of which are time and cost intensive options. An alternative method made possible due to increasingly cheaper technology is installing a permanent set flow sensors inside the WDN.
However, since the installation and maintenance of the sensor may constitute a major investment, the number of sensors put in use in the network is likely to be limited. Given a number of sensors, the problem still remain however where they should be placed in the network to maximize their ability to provide useful information.
Chapter 1 1.2. PURPOSE AND GOAL
1.2 Purpose and goal
The purpose of this thesis was to develop a method for finding optimal flow sensor placements using a model-based approach. The results should include a method for finding these positions as well as an outlook of possible extensions on the results obtained and implications of the results.
1.3 Project boundaries
• This project only considered flow sensors for both modeling and analysis.
• This project only considered the distribution part of a water supply system
• Dynamic changes were not considered
• The project is theoretical in its nature and the model was not validated, but rather physical principles were used concepts developed.
1.4 Approach
Introductory survey The project was initiated by a survey of relevant literature to broaden the understanding of the problem posed and the field itself. Following the survey, different ways of framing and formulating the problem was investigated.
Model Following the survey of relevant literature, a model of the studied system was constructed. The model needed to capture some essential aspects of the move- ment of water in the network since this is what the flow sensors intercept. Together with this description, the topology of the network also needed to be taken into consideration since this is a key difference between any two distribution networks.
Analysis Given the model, the information gained by a sensor placed on a specific position then needed to be described in order to evaluate the position itself. This was then generalized to encompass the information gained by multiple sensors and how this could be used to guide the placement of the sensors.
Methods By using the model and the concepts developed during the analysis, concrete methods were constructed that were used to generate positions optimal with respect to chosen criteria. Tests were done on networks of varying complexity and the results analyzed.
Chapter 2 Background
This chapter provides an overview of water supply systems and how losses arise in the distribution parts of the system. The report then goes on describing how such leaks can be detected with an emphasis on model based techniques. The chapter concludes with an overview of relevant theory in graph theory, set theory, linear algebra, optimization and probability theory.
2.1 Water Supply Systems
2.1.1 System overview
Drinking water is the end product of a water supply system, WSS, which has the sole purpose of supplying drinking water to customers, water for industrial use as well as water for fire mains. The main components of a WSS can be thought of as sources, reservoirs, pipelines and controlling equipment like valves. Much like the electric power system we can group some of these components together based on function in the system. The transmission system is the part of the system whose function is to produce and transport large quantities of water over large distances.
The components in this part of the system are typically treatment plants, reservoirs and pipelines. While the transmission system may provide water directly to some customers like large industry sites, the most common use is to provide a distribution system with water. The function of the distribution system is to deliver water in residential areas and while the distances involved may not be as long as in the transmission system the structure and topology of this system is much more complex.
The components in this part of the system include pipelines, pressurizing and control components as well as reservoirs like water towers.
The topology of the distribution often follows the layout of streets and buildings and can therefore inherit complex patterns and connections. The structure can be both branched and looped; both of which have their merits in the system. A branched system is less expensive since it requires fewer pipes and connections but is also less robust against disturbances in the network. The looped structure allows for multiple paths that the water can travel as well as it gives the operators the ability to close certain parts of the network while maintaining supply to a larger part of the network.
Chapter 2 2.1. WATER SUPPLY SYSTEMS
2.1.2 Water loss
Much like in other technical systems, the losses in a WSS can be grouped together based on different criteria. An aid in this process is the International Standard Water Balance developed by the International Water Association, IWA [4]. This balance and corresponding terminology is shown in Figure 2.1. With this terminology the water losses are grouped into real losses and apparent losses. This balance can be thought of as providing an overview of the origins to monetary losses in a WSS. While apparent losses and unbilled authorized consumptions do not necessarily constitute water leaving the system in an uncontrolled way, they still contribute to the mass of non-revenue water in the system.
Figure 2.1: IWA Standard Water Balance [5]
The real losses may also be called physical losses as the term refers to water leaving the systems as leakage in different parts of the system. This leakage occurs in all parts of the system and in the IWA balance it is broken down into leakages on transmission and distribution mains, on storage tanks and on service connections up to customer metering. In [6] the real losses are broken down even further by distin- guishing between background leakage, reported bursts, unreported bursts and leakage or overflow from reservoirs. Background leakages include very small losses, typically with low flow rates that are hard or impossible to detect. If the leakage instead has a higher flow rate it is considered a burst that may or may not be reported to the water supplier. The difference between background leakage and bursts is not clearly defined but in practice a threshold value is used to distinguish one from the other [7].
The background leakage is an umbrella term for a wide array of losses that typically have a very low flowrate. Among these are leaks from bad pipe connections, microscopic to small holes and even cracks in the pipes. Bursts on the other hand are generally contributed to a partial or complete failure of a pipe in the network[8].
Chapter 2 2.1. WATER SUPPLY SYSTEMS
2.1.3 Leakage management
Leakage management is the overall strategy for estimating, detecting and handling the losses in the system. When estimating (or assessing) the losses, the goal is to quantify the size of the losses while not being concerned with their location.
Depending on whether the water supplier has a proactive or a reactive management strategy losses are handled and mitigated in different ways.
A proactive strategy involves condition assessment of pipes and other assets which is used both for prediction and prevention of failures as well as for the reha- bilitation of assets in the network. Most of these methods depend on either physical or statistical methods, both of which describe the deterioration process of parts in the network that lead to failures. A reactive strategy on the other hand consists of methods for detecting, locating and repairing leaks in the system. These strategies can be used together and even if no proactive measures are taken, should a failure in the system occur reactive measures must still be taken to minimize the losses [7].
Leakage detection The term leakage detection often refers to the problems of finding evidence of the size and location of the leaks as well as if there are any leaks at all. In some cases these are separate problems, but for other methods they are one and the same. There are a wide range of methods in use for leakage detection and they can be classified in a number of ways. In [1] Yuan et al. categorise methods based on the data-source that the methods require while [9] distinguishing between localization and pin-pointing methods. In [8] the class of leakage awareness methods is added to this list.
The main objective of localization methods is to narrow down the vicinity of a leak in the network while pin-pointing methods aimed to estimate the exact position of a leak. Using these definitions, the leakage awareness methods could easily be fitted into the localization framework, but the use of hydraulic models for anomaly detection in these methods makes them slightly different.
Common methods in the localization framework include visual surveys, step testing and leak noise mapping. Pin-pointing methods on the other hand include acoustic sensing and gas injection. For a more elaborate description see [8] and [9].
The leakage awareness methods can be either steady-state or transient based and depending on which they are used on single pipelines or a network segment.
Leakage detection using a sensor network Finding and handling leaks is a multi-objective problem whose preferred solution is a trade-off between the mone- tary costs of leaks and the costs of repairing them. Even though leaks continually contribute to monetary losses some of the methods described above are both time and labour intensive and can not be justified in some cases. While most localization and pin-pointing methods rely on field surveys and sometimes intrusive measures in the system, the leak awareness methods offers a low-cost alternative to narrow down the search area. A recent development made possible by cheaper technology is the use of a permanent network of sensors installed in the system. This approach reduces the time and labour cost involved with setting up temporary measurement points; used in for example acoustic sensing.
Chapter 2 2.2. MATHEMATICAL PREREQUISITES
2.2 Mathematical prerequisites
2.2.1 Set theory
The following section presents a summary of the set theory that is used in this work.
The notation and definitions are taken from [10] which can be consulted for further reference. Set theory is the branch of mathematics concerned with collections, usu- ally called sets, and their elements. The most fundamental concept in set theory is that of belonging. If x is an element of the set A we say that “...x belongs to A”
and write
x∈ A.
A common way to denote sets is by enclosing its elements inside{} and in order to specify a set, set-builder notation can be used. To specify a setB with elements x from another setA that meet a certain condition S(x), the corresponding set-builder notation is
B ={x ∈ A : S(x)},
which is read The set B is all elements x in A such that S(x) is true.
The number of elements in a set is referred to as its cardinality and is written
|A| for a set A.
Relations between sets
Much like an element can have the relation of ‘belonging’ to a set, a set can have a relation to another set. A set is uniquely defined by its element and two sets are equal if and only if their elements are exactly the same. If so, we writeA = B for two sets A and B. If all elements of a set B also belong to another set A we say thatB is a subset of, or is included in, A. The corresponding notation is B ⊆ A. If B ⊆ A and B 6= A, then B is a proper subset of A and we write B ⊂ A. If B is a subset ofA, proper or not, then A is a superset of B.
Operations on sets
The following are fundamental operations and corresponding notation that can be performed on any number of sets in order to construct new sets. These operations can be visualized using Venn diagrams, with an example being the diagrams in Figure 2.2
Union A∪ B
The union of two sets are all elements belonging to either of the operands A∪ B = {x : x ∈ A or x ∈ B}
Intersection. A∩ B
The intersection of two sets are all elements belonging to both of the operands A∩ B = {x : x ∈ A and x ∈ B}
Chapter 2 2.2. MATHEMATICAL PREREQUISITES
Difference. A\ B
The difference between two sets are all elements belonging to one set but not the other. The difference is sometimes called the relative complement of one set in another.
A\ B = {x : x ∈ A and x /∈ B}
A B
A∪ B
(a)
A B
A∩ B
(b)
A B
A\ B
(c)
Figure 2.2: From left to right: The union, the intersection and the set difference of sets A and B. The area colored in dark blue designates the resulting set.
Complement. A0
The complement or absolute complement of a set A is all elements belonging to a universe U containing A, but not to A itself.
A0 ={x ∈ U : x /∈ A}
Special sets
Sets are ubiquitous across all of mathematics but some sets have special importance, names and notation associated with them. Some of these are
The empty set: ∅ = {}
A singleton: {x}
The natural numbers: N ={0, 1, 2, . . .}
The integers: Z ={. . . , −2, −1, 0, 1, 2, . . .}
2.2.2 Graph theory
This section present important definitions and concepts of graph theory used in this work. Most names and nomenclature used is standard to the field and [11],[12] and [12] have been used as reference. A graph G is a pair of sets (V, E) whose elements are called vertices and edges respectively. An edgee in E is a pair of vertices in V , which givesE ⊆ V × V . If a vertex u belongs to an edge e we say that u is incident with e.
The basic definition of a graph can be extended to a directed graph where each edgee has a direction. A directed edge has a initial vertex and a terminal vertex and we say that e is directed from the initial vertex to the terminal vertex. A directed graph is said to be an orientation of the corresponding undirected graph on the same vertex- and edge sets .
Chapter 2 2.2. MATHEMATICAL PREREQUISITES
A graph is usually drawn with its vertices (also called nodes) as points and its edges as lines between the points if e = (u, v) = uv is an edge in E. Figure 2.3 shows an undirected graph together with a directed graph.
a
b
c
d e
f g
(a)
1 2
3
4 5
6 7 8
(b)
Figure 2.3: On the left in (a) showing an undirected graph with vertex set V = {a, b, c, d, e, f, g}. On the right in (b) showing a directed graph with vertex set V ={1, 2, 3, 4, 5, 6, 7, 8}
Important definitions
Degree For a given vertexv in G we say that its degree is the number of edges at v. For a directed graph it is also meaningful to distinguish between in-degree and out-degree where the terms refer to whetherv is the initial or terminal vertex of an edge at v. The vertex b in Figure 2.3 a) has degree 3 while the vertex 2 in Figure 2.3 b) has in-degree 1 and out-degree 2.
Path A path P = (V, E) is a graph of the form V ={x0, x1, . . . , xm},
E ={x0x1, x1x2, . . . , xm−1xm},
where the vertices xi are all distinct. The first and last vertices of P are reffered to as its ends. For any two vertices x, y in P it is common to write xP y which refers to the subpath ofP between x and y. Figure 2.4 shows an example of a path.
Cycle A cycle C is a path such that the ends of C are one and the same. In a directed graph, a directed cycle is a directed path such that it starts and ends at the same vertex. The graph in Figure 2.3 a) has a cycle on the vertices {b, c, d} while the graph in Figure 2.3 b) does not have a directed cycle.
Connectivity An undirected graph is said to be connected if it is non-empty and there is a path between any two vertices in the graph. A component is a maximal connected subgraph on a graph. In directed graphs, an out-component is the set of vertices that are reachable with directed paths starting in a specific vertex. The in-component is defined in a similar way, but consists of all vertices such that there are directed paths from them to a specific vertex. The graph in Figure 2.3 a) has two components and the graph in Figure 2.3 b) has one.
Chapter 2 2.2. MATHEMATICAL PREREQUISITES
Trees A tree is an acyclic and connected graph. The vertices of degree 1 are called leaves while all other nodes are referred to as inner vertices. Figure 2.4 is an example of a tree where the vertices x, y, z and u are examples of inner vertices while the vertex v is a leaf.
x y u z v
Figure 2.4: A tree graph. The bold edges denote a path P with ends x and v.
Representation of graphs
A graph can be represented in many ways besides its visual depiction. Some of the most useful representations of a graph are its adjacency matrix and incidence matrix. An example of these matrices for a directed graph is shown in Figure 2.5 below.
Adjacency matrix The adjacency matrix A of a directed graph is constructed by taking each element as
aij =
(1, if there is an edge from i to j
0, else .
The adjacency matrix has dimensions m× m where m is the number of vertices in the graph.
Incidence matrix The incidence matrix Q of a directed graph is defined as [12]
qij =
1, if vertex i is the terminal vertex of edge j
−1, if vertex i is the initial vertex of edge j 0, else
.
The incidence matrix has dimensions m × n where m and n are the numbers of vertices and edges in the graph respectively. The adjacency and incidence matrices for the graph in Figure 2.5 are
A =
0 1 0 0 1 0 1 1 0 1 0 0 0 1 0 0
, Q =
−1 0 0
1 −1 −1
0 0 1
0 −1 0
(2.1)
.
Chapter 2 2.2. MATHEMATICAL PREREQUISITES
1
2 3
e1 4 e3
e2
Figure 2.5: Sample graph for the adjacency and incidence matrices in equation 2.1
2.2.3 Linear algebra
The last section in this chapter is a short summary of important concepts found in linear algebra. There is an extensive amount of literature on the subject and for notation and definitions [12] has been used as a reference.
Linear algebra is the study of linear maps and the most fundamental unit is the vector. A vector is an element in a vector space that in turn is defined on a field.
A vector is an ordered list of elements, but is different from a set in that it might contain duplicate elements. The number of elements in a vector is referred to as its dimension. A matrix is another fundamental constituent of linear algebra and is a representation of a linear map.
Linear system of equations
An important application of linear algebra is that of solving linear systems of equa- tions. Any linear system can be written in vector-matrix form and can as such be the subject of several useful results in linear algebra. Consider the general linear system of m equations with n unknowns:
a11x1 + a12x2 +· · · + a1nxn = b1
a21x1 + a22x2 +· · · + a2nxn = b2
... ... . .. ... ... am1x1+am2x2+· · · + amnxn=bm
The corresponding matrix-vector equation becomes Ax = b,
where the coeffecient matrix A has dimension m× n and elments aij for i = 1, . . . n and j = 1, . . . m.
Finding a solution to this system is equivalent to finding the vectorx that satisfies the conditions posed by the system equations. It can be shown that every linear system has exactly one, zero or infinitely many solutions. A system that has at least one solution is called consistent and inconsistent if it has no solutions. A system with fewer unknowns than equations is said to be overdetermined and underdetermined if it has more unknowns than equations. If the number of unknowns n is exactly the number of equationsm, the system is said to be square.
Chapter 2 2.2. MATHEMATICAL PREREQUISITES
The inverse If the coeffecient matrix of a system is square, i.e has dimensions m× m, and the system is consistent we can express the unique solution x of the system as
x = A−1b,
where A−1 is called the inverse of A. If the system is inconsistent or not square, this matrix does not exist and so the system has no unique solution.
The pseudo-inverse If the system is not square, but consistent there exists in- finitely many solutions to the system. Since the inverse does not exist, any possible solutions cannot be expressed in terms of the same matrix. A generalization of the inverse that always exists however is the Moore-Penrose inverse A†, often called the pseudoinverse. This matrix satisifies the following criteria
AA†A = A A†AA† =A†, (AA†)∗ =AA†, (A†A)∗ =A†A,
where ‘∗’ denotes the conjugate transpose of a matrix. The pseudoinverse can be used to obtain the best solution to a system in a least squares sense. This means that if Ax = b is consistent then z = A†b is a solution and ||z||2 ≤ ||x||2 for all x.
Chapter 3
System setup
This chapter describes the model that was used and what assumptions were made related to the model. While rather short, the concepts and the model presented in this chapter were the foundation for the analysis in the successive chapters.
3.1 Model and assumptions
3.1.1 Assumptions
A hydraulic model of a WDN is a mathematical model of the movement of water in the network. The real behaviour of water inside a network of pipes is dependent on numerous physical parameters like the density of the water, pipe material, the number and shape of elbows and so on [13]. How and in what way the model is constructed is mainly dependent on the application of the model. While a model that accurately describes turbulent pipe flow is well motivated for a single pipe, it might not be computationally feasible to use it for a network of hundreds, perhaps thousands, of pipes.
The model developed here is not intended to describe the movement of water in the network as realistically as possible, but rather to capture some aspects of this movement. The model in this work is used as a foundation for further analysis of the network and while a model is essential to the analysis, this particular model can easily be replaced by a more sophisticated model.
The fundamental condition that must be satisfied by any model that describes the flow of water is that of conservation of mass and energy. As is common in fluid mechanics, the energy of a system is often expressed in terms of pressure.
The pressure essentially determines in which direction the water flows while the conservation of mass ensures that no water is created, lost or stored in the system.
Chapter 3 3.1. MODEL AND ASSUMPTIONS
These two conditions can be stated more formally as X
loop
h = 0, (3.1)
X
in
Q−X
out
Q = 0, (3.2)
where (3.1) says that the head loss around any closed path in the network must be zero and (3.2) says that the water that flows into a junction also must flow out. Head loss is a term commonly used in hydraulics that describes the decrease in pressure measured in fluid elevation over a given datum.
The approach used in this model is to exchange the pressure in the system for an assumed direction of the flow in each pipe. These two viewpoints are in fact not entirely equivalent since the pressure not only determines the direction but also influences the magnitude of the flow.
3.1.2 The model
The following model is an abstraction of a WDN which in reality consists of a large number of components among which are pumps, valves and storages etc. In this description the network consists of pipes and junctions. The way the pipes and junctions are configured is described by a directed acyclic graph G, with order
|G| = Nv and number of edges ||G|| = M. A vertex in this graph can represent a supply inlet uk, a pipe intersection or a customer (demand) dk in the network whereas and edge represents a pipe. The description of how the water moves in the network is based on the conservation of mass at every node. If constant density of the water is assumed, the conservation of mass can be extended to the conservation of volume yielding
0 =X
i∈in
si− X
j∈out
Qj, (3.3)
where si is a volumetric flow entering a pipe junction and Qj is a flow leaving the same intersection. A leakli occurring on pipe i is then considered an outflow from junction where the pipe begins.
Qj =sj +lj. (3.4)
Figure 3.1 shows an example of how the leak and flow at each pipe are considered to contribute to the balance equations. A supply or demand is considered an in- going or out-going flow respectively at a junction, but is separated in this model for clarity. Equations (3.3) and (3.4) must hold for every junction in the network and so for every vertexk in the graph there is a balance equation of the form
0 =
X
i∈k,in
si+uk
−
X
j∈k,out
(sj +lj) +dk
. (3.5)
The balance equations together constitutes a linear system of equations which can be written in matrix-vector form as
0 = Ss− Ll + u − d (3.6)
whose elements are discussed next.
Chapter 3 3.1. MODEL AND ASSUMPTIONS
l1 l2
l3
s1
s3
s2
u1
d3
d2
Figure 3.1: Small network with four junctions and three pipes. Each junction, or vertex, will require a corresponding balance equation where the terms consists of the flows and leaks present at the given vertex.
The system of balance equations
Flow matrix,S The matrixS is referred to here as the flow matrix and essentially describes the direction of the flows. By the definition given in Equation (2.2.2) this matrix is the incidence matrix of the graph representation of the system and so has dimension Nv× M and elements belonging to the set {−1, 0, 1}.
Flow vector, s. The vector s has dimension M × 1 and is constructed from the flowssi, i = 1, . . . , M in each pipe. Since water is assumed to flow in the direction given by the matrixS, each flow is considered to be positive, i.e si ≥ 0 for every i.
Leak matrix, L The matrix L has dimension Nv × M and describes from which node the water exiting the system as a leak originates from. The way the system is defined makes L have as many zero rows as the number demand vertices, Nd. This is due to the fact that the demand vertices have no pipes (edges) leaving them which makes the leak term on these nodes zero. In contrast to S , the elements in L belong to {0, 1} which is due to the fact that a leak can only leave a junction whereas a flow can both enter and exit a junction.
Leak, demand and supply vectors l, d, u The leak vector has dimension M× 1 and is constructed from the leaks li on the pipes. The demand- and supply vectors have dimensionNv×1 and are constructed from the flows entering or leaving supply or demand vertices in the network. The input,u, to the system is entirely determined by the leaks and demands in the network. This can be described as
kuk1 =
M
X
i=1
(di+li) =kdk1+klk1. (3.7)
3.1.3 Flow measurements in the model
Equation (3.6) describes a system with directed flows, demands, leaks and inputs.
If a numberp≤ M of sensors are introduced into the system to measure flows, this
Chapter 3 3.2. SIMULATING FLOW
is equivalent to observing only a subset y of the flow vector s. This subset can be described by
y = Cs, (3.8)
where the matrix C has dimension p× Nv and y p× 1.
3.2 Simulating flow
To simulate flow in the network (3.6) can be used to solve for the flow vector s.
Without loss of generality and for S invertible, combining (3.6) and (3.8) yields
y = Cs = CS−1 Ll− u + d. (3.9)
However, since S generally will not be a square matrix, the inverse S−1 will not exist for most cases. To obtain the best least-squares approximation of the solution to (3.9) the inverse can be replaced with the pseudo-inverse S† yielding
y = Cs = CS† Ll− u + d = C(S∗S)−1S∗ Ll− u + d. (3.10) Computing the flow vector s from (3.10) ensures that the balance equations (3.6) are satisfied, but it does not guarantee that the elementssi are all positive. To this end the following optimization scheme can be employed:
s∗ = arg min
s kSs − Ll + u − dk2, s.t s0 =S†(Ll− u + d),
si ≥ 0, i = 1, . . . , M,
where the solution of (3.10) can be used as an initial point of an optimization algorithm.
During some hours in the night and under normal circumstances the water de- mand is so low that any flows observed in the system are due to leaks in the system.
This flow is often referred to as the minimum night flow and is usually observed sometime between midnight and 5 am [7]. If no demands are assumed to be present, (3.6)-(3.10) can be simplified in the following way: If there are no demands present, there will be no water flowing in the pipes leading to the demand nodes. This re- duces the dimensions of the flow vector s and flow matrix S but leaves l and L unchanged. This is due to the fact that even though a pipe may transport no water there can still be leaks present on the given pipe. This is equivalent to saying that for a pipe i the corresponding leak flow li can be nonzero even though the flow in the pipe,si, is zero. These assumptions lead to the reduced model
0 = Sdimsdim− Ll + u (3.11)
whereSdim andsdim have dimensions Nv× (M − Nd) and (M− Nd)× 1 respectively where Nd is the number of demand nodes in the network. Under this assumption, the flows can be computed according to
s∗ = arg min
s kSdimsdim− Ll + uk2, s.t s0 =S†(Ll− u),
si ≥ 0,
Chapter 3 3.2. SIMULATING FLOW
where no i corresponds to an edge leading to a demand node.
Chapter 4 Method
This chapter gives a summary of the analysis done on the model in order to obtain the results. The material builds on the model and concepts presented in Chapter 3 as well as on theory presented in Chapter 2. Short proofs related to this chapter can be found in Appendix B. The chapter concludes with a formulation of the sensor placement problem and a presentation of what was implemented and tested.
4.1 Analysis of the network
4.1.1 Partitioning of the network
The information gained by a measurement in the network is readily described in terms of the graph representation of the system. Since the direction of the flow in any pipe is assumed not to change, a distinction can be made between the pipes that can supply a specific pipe p with water, those that can be supplied with water passing through p and those for which none of the previous apply. In the rest of this text, any pipe in the first and second set is referred to as being upstream and parallel of given pipe p while the third is referred to as downstream of p. A sensor placed on an edge ei partitions the network into an upstream part, a parallel part and a downstream part. These sets are specified further below and for each of the sets, the edge ei is said to induce them. In the rest of this work, these sets will be referred to as the network sets.
Upstream Given an edge in the network, the set of all edges that are upstream from it can be divided into those that can contribute to a flow through the given edge and those that cannot.
Definition 4.1.1: Flow signature
For the edgeei =xy, let Fi be the set of all edges in G such that they belong to any directed path inG that ends in x.
Downstream The set of edges downstream from an edge can be divided into those that can only be supplied with water that passes through the given edge and
Chapter 4 4.1. ANALYSIS OF THE NETWORK
those for which this is not necessarily the case. The first category is referred to here as the leak zone of and edge and the second as its basin.
Definition 4.1.2: Basin
For a given edge ei =xy let the basin Bi be the set of all edgesej in G such that ej lies on a directed path starting fromy.
Definition 4.1.3: Leak zone
For a given edgeei =xy let the corresponding leak zoneLi ⊆ Bi ofei be the set of edges such that the starting vertex u of every edge e = uv in Li only has incoming edges that belong to Li∪ {ei}.
It is useful to define the sets above also for multiple inducing edges.
Definition 4.1.4: Basin of a set of edges
Given any set S = {e1, e2,· · · , ek} of edges in G, the basin BS of this set is defined as
BS =B1∪ B2∪ · · · ∪ Bk (4.1) Definition 4.1.5: Leak zone of a set of edges
Given any set S = {e1, e2,· · · , ek} of edges in G, the corresponding leakzone LS is defined as
LS ={e = xy ∈ BS :
every in-going edge to the start vertex x belongs to LS∪ S}
Parallel The parallel edges induced by the edgee are exactly the edges that do not belong to any of the upstream or downstream sets. This leads to the first definition
Definition 4.1.6: Parallel edges
For a given edgeei the set of parallel edges is denoted Pi and are exactly the edges inG\ {ei} that do not belong Fi∪ Li.
In contrast to both basins and leak zones, the set of parallel edges Pi of an inducing edge need not all be connected. IfG0 is the subgraph induced by vertices incident with the edges inPi then every component of G0 is connected.
By definition, the sets Bi\ Li, Li, Fi and Pi are all disjoint and is together with {ei} a partition of the edgeset E of G. Figure 4.1 shows an example of the different network sets induced by the edge ei.
4.1.2 Basins and leak zones
Since a leak zone of a given edge is defined as a subset of the corresponding basin it is of interest to determine what conditions determine the size of this subset. For brevity, a vertex with in-degree higher than 1 is referred to as a multiple-supply vertex.
Chapter 4 4.1. ANALYSIS OF THE NETWORK
F i
P i
L i
B i
ei
Figure 4.1: The network sets induced by the edgeei which is displayed in red. The parallel setPi is spread across three components.
Being a subset, a leak zone can be empty, a proper subset of or equal to the corresponding basin. Which of the above that is true for a given edge e cannot be determined from properties of e alone. For an edge ei = xy, let Vi be the set of vertices incident with the edges in Bi. The following three conditions hold for any leak zone and basin induced by single edges.
1. Li =∅: The end vertex y in ei =xy is a multiple-supply vertex or a leaf.
2. Li ⊂ Bi: There is at least one multiple-supply node inVi that is not a leaf and that has in-going edges that do not belong to Bi
3. Li =Bi: No vertex in Vi that is not a leaf is a multiple-supply node and has incident edges not belonging to Bi
Leak zones satisfying the conditions of the bullet point 3 above are of special im- portance and for this reason such sets are given a specific name and notation
Chapter 4 4.1. ANALYSIS OF THE NETWORK
Definition 4.1.7: Complete leakzone
A complete leak zone ˆLi is a leakzone which is equal to the corresponding basin Bi and where no leaf is a multiple supply vertex with incoming edges that are not part of the leak zone.
Magnitude of basins and leak zones By using the framework of basins and leak zones, a measurement on an edgeei can be connected to the leaks and demands downstream from the given edge. Since each edge is assumed to potentially have one leak, summing a set of edges also means summing up the potential leaks. This is equivalent to assigning each edge a weight and then summing these weights. For brevity, this operation is given a special notation
Definition 4.1.8: Sums of leaks on a set in the network
Let X be a basin or a leak zone, then sum of the leaks on this set is defined as
|X|l = X
ei∈X
li (4.2)
By combining (4.2) with the definitions of leak zones and basins, a measurement can be used to put bounds on the sums of the leaks in these sets. Let yi be the measured flow on edgeei and Li and Bi are the leak zone and basin induced by the same edge. The bounds on the downstream sets then become
|Bi|l ≥ yi (4.3)
|Li|l ≤ yi (4.4)
where equality holds only if the leak zone is complete and therefore is equal to the basin.
4.1.3 Relations between basins and leak zones
Since leak zones, by definition, are a set of edges that are uniquely supplied with water from a certain edge, there can be no general intersection between them. If there would be such an intersection, the edges in this intersection would not be uniquely supplied through one edge only which is a contradiction by definition.
Basins on the other hand can be disjoint, have a non-empty intersection, be proper subsets or be equal. Since complete leak zones are equal to their corresponding basins, these set may have the same relations as basins.
Below is a description of the possible configurations of two basins and what con- clusions that can be drawn about the corresponding leak zones. For all cases that follow,ei inducesBiand Li while the edgeej inducesBj andLj. Since all sets in the network are subsets of the network itself, the edge setE of the DAG that describes the network can itself be considered a complete leak zone. In the rest of this text, this leak zone will be referred to as ˆU with | ˆU|l =kuk1.
For each case it is of interest to determine what conclusions can be drawn of the magnitude of potential leaks given the different configurations and measured flow values yi and yj. By using (4.3) and (4.4) the magnitude of the leaks can be found.
Chapter 4 4.1. ANALYSIS OF THE NETWORK
Disjoint basins
Case 1 For disjoint basins, the leak zones are by necessity also disjoint. None of the edges that induce the basins and leak zones can be part of the other pair of set, meaningei ∈ B/ j,Lj and vice versa for ej. The following equations describe the
Figure 4.2: Disjoint basins and corresponding leak zones. The red arrows represent the inducing edges.
magnitude of the leaks in the two basins, their union and intersection respectively.
yi ≤|Bi|l ≤ | ˆU \ Bj|l ≤kuk1 − yj (4.5) yj ≤|Bj|l≤ | ˆU \ Bi|l ≤kuk1 − yi (4.6)
yi+yj ≤|Bi∪ Bj|l ≤ | ˆU|l (4.7)
0≤| ˆU \ (Bi∪ Bj)|l ≤kuk1 − (yi+yj) (4.8)
|Bi∩ Bj| = 0 (4.9)
Intersecting basins: non-subsets
Case 2 For basins with a non-empty intersection a distinction can be made be- tween when one basin is a subset of the other or not. Consider first the case when Bi 6⊆ Bj and Bj 6⊆ Bi. . It is also true that ei ∈ B/ j,Lj and vice versa for ej. This can be proved using the fact that ifei would have belonged to Bj, everything in Bi
could have been reached from ej and so Bi ⊂ Bj which is a contradiction.
yi ≤|Bi|l ≤ | ˆU|l (4.10)
yj ≤|Bj|l≤ | ˆU|l (4.11)
yi+yj ≤|Bi∪ Bj|l ≤ | ˆU|l (4.12)
0≤| ˆU \ (Bi∪ Bj)|l ≤kuk1 − (yi+yj) (4.13)
|Bi∩ Bj|l ≤ |Bi∪ Bj|l (4.14)
|Bi∩ Lj|l =|Bj ∩ Li|l = 0 (4.15) Intersecting basins: subsets
For this case, let Bj ⊂ Bi. When one basin is a subset of another there are four distinct possible cases depending on where the inducing edge of the “subset-basin”
Chapter 4 4.1. ANALYSIS OF THE NETWORK
Figure 4.3: Intersecting basins and corresponding leak zones. The red arrows rep- resent the inducing edges
is located. If one of the edges, ej, is a member of the other basin Bi a distinction can be made between ifej belongs toLi or not.
(a) (b)
Figure 4.4: The inducing edgeej, shown in red, belongs to the leak zoneLi. On the left in (a) both the induced basin and leak zone of ej are contained in Li. On the right in (b), the basin is not contained in Li.
Case 3a Should ej ∈ Li, the two cases in Figure 4.4 arise. If Li does not contain any multiple-supply nodes that restrictLi, then bothBj and Lj are fully contained inside Li. If this is not the case Lj must still be a subset of Li, but Bj can contain edges downstream of any such multiple supply node and not necessarily be contained inside Li. This is the only case where a leak zone can have a general intersection
Chapter 4 4.1. ANALYSIS OF THE NETWORK
with a basin and is shown in Figure 4.4 (b).
yi ≤|Bi|l ≤ | ˆU|l (4.16)
yj ≤|Bj|l≤ |Li|l≤ |Bi|l (4.17)
yi ≤|Bi∪ Bj|l=|Bi|l (4.18)
0≤| ˆU \ (Bi∪ Bj)| ≤ | ˆU \ (Li∪ Lj)| ≤ | ˆU \ Li| ≤ | ˆU \ Lj| ≤kuk1− yj (4.19)
0≤|Li\ Bj|l ≤ yi− yj (4.20)
yj ≤|Bi∩ Bj|l=|Li∩ Bj|l =|Bj| ≤ yi (4.21)
|Bi∩ Lj|l =|Lj| (4.22)
3b For the case where Bj is not contained in Li, (4.21) is replaced with
yj ≤|Bi∩ Bj|l =|Bj|l (4.23)
|Li∩ Bj|l=|Lj| ≤ yj (4.24)
Case 3c If ej ∈ L/ i then the leak zones (and also Bj and Li) are disjoint. An example of this case is shown in Figure 4.5 (a) below.
(a) (b)
Figure 4.5: On the left in (a), the inducing edgeej does not belong to the leak zone Li. On the right in (b), ej does not belong to the basin Bi and the resulting leak zone Lj is empty.
max{yi, yj} ≤|Bi|l ≤ | ˆU|l (4.25)
yj ≤|Bj|l≤ |Bi|l (4.26)
max{yi, yj} ≤|Bi∪ Bj|l ≤ | ˆU|l (4.27) 0≤| ˆU \ (Bi∪ Bj)| ≤kuk1− max{yi, yj} (4.28)
|Bi∩ Bj|l =|Bj| (4.29)
|Bi∩ Lj|l =|Lj| (4.30)
The reason why the sumyi+yj is replaced by the maximum of the two is that since ej ∈ Bi it might be supplied with water from ei and hence the two flows cannot be added. The maximum follows from the fact that the magnitude of the basin must at least be equal to the largest flow that is observed flowing into or inside the basin.
Chapter 4 4.1. ANALYSIS OF THE NETWORK
For all cases whereej ∈ Bithere is the possibility of a subset ofBiwith a different lower bound. As long asLj ⊂ Bj the set difference Bi and ˆLj may have lower bound different from that ofBi
yi− yj ≤ |Bi\ Lj|l≤ |Bi|l. (4.31) These sets are further discussed in Section 4.1.4.
Case 3d If ej does not belong to Bi then Li = ∅. To prove this, assume that Li 6= ∅ and let ekbelong to this leak zone. By the definition of a leak zone, any edge in this set can only be reached by paths that includes edges in the leak zone or the inducing edgeej. SinceBj ⊂ Bi by assumption,ek ∈ Bi and so can be reached with a path starting in ei and not necessarily including edges in Li only. This violates the conditions of a leak zone and so Li = ∅. This case is shown in Figure 4.5 (b).
The resulting bounds for this case are almost as those for non-disjoint basins that are not subsets of each other:
yi ≤|Bi|l ≤ | ˆU|l (4.32)
yj ≤|Bj|l≤ |Bi|l ≤ | ˆU|l (4.33)
yi+yj ≤|Bi∪ Bj|l =|Bi|l (4.34)
0≤| ˆU \ (Bi∪ Bj)|l ≤kuk1 − (yi+yj) (4.35)
|Bi∩ Bj|l =|Bj|l (4.36)
Equal basins
Case 4 If the basin of two edgeseiandej are equal there are two possible outcomes for their leak zones; either they are both empty or they are also equal by using the definition of a leak zone of a set of edges. To see why this is the case, note that if two basins are equal then the target vertex v of the inducing edges must be the same. If the there are additional incoming edges to this vertex, then the resulting leak zones are empty by the definition of a leak zone. This is shown in Figure 4.6 (a). If there are no other incoming edges to v, or all edges are measured, then the corresponding leak zone may be non-empty and also equal for all the inducing edges.
This is shown in Figure 4.6 (b).
(a) (b)
Figure 4.6: Equal basins induced by multiple edges
Chapter 4 4.1. ANALYSIS OF THE NETWORK
yi+yj ≤|Bi|l =|Bj|l≤ | ˆU|l (4.37)
|Bi∪ Bj|l =|Bi∩ Bj|l≤ | ˆU|l (4.38) 0≤| ˆU \ (Bi∪ Bj)|l ≤kuk1 − (yi+yj) (4.39)
4.1.4 Bounded subsets
The results from the previous subsection hint that for some configurations of basins and leak zones, there exist smaller, possibly non-empty, subsets that also have bounds associated with them. These subsets are important as they allow for further localization of leaks even inside a basin or leak zone. Depending on how they are constructed, these subsets can be associated with an upper or lower bound.
Lower bounds To see how bounded subset can be constructed it is informative to consider some basinBi with corresponding lower bound yi ≤ |Bi|l. By (4.2),|Bi|l
is the sum of all leaks on the edges inBi. While this bound still holds if some edges are added to the basin, the opposite is not true. If some edges are removed from the basin the bound is no longer valid for the remaining set of edges. This is equivalent to
|Bi|l 6= |Bi \ S|l (4.40)
for some set S ⊆ Bi. As an example, consider the trivial case where Bi = {e1, e2} andS = {e2}. It is clear that if yi ≤ |Bi|l=l1+l2, then|Bi\ S|l =|{e1}|l=l1 does not necessarily share the same lower bound.
In order to remove some edges S from a basin and gain a lower bound on the remaining edges, the set S must have an upper bound. Consider the previous example, but letS have the upper bound|S|l=|{e2}|l =l2 ≤ yj. RemovingS from Bi now yields the following.
|Bi\ S|l=|{e1, e2} \ {e2}|l (4.41)
=|{e1}|l=l1 (4.42)
=(l1+l2)− l2 (4.43)
=|{e1, e2}|l− |{e2}|l (4.44)
=|Bi|l− |S|l ≤ yi− yj. (4.45) where the inequality in (4.45) hold only due toS having an upper bound.
Since the previous argument requires any set of edgesS to have an upper bound, the candidates for this set are reduced to leak zones and complete leak zones. This can be summarized as
For any basin Bi and a set S, the subset Bi \ S can be given the lower bound yi − yj if and only if S is a leak zone, complete or not, contained in Bi.
Upper bounds The subsets with upper bounds can be constructed following al- most exactly the same argumentation as for the lower bounds with only small ad- justments. First, if a subset with an upper bound is to be constructed, the subset
