Water Distribution System Aggregation for Water Quality Analysis

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Water Distribution System Aggregationfor Water Quality Analysis

Lina Perelman1 and Avi Ostfeld, M.ASCE2

Abstract: Control and design problems of water distribution systems rely on simulation models for predicting the system behavior underdynamic boundary conditions. A detailed network model may result in thousands to tens of thousands of pipelines and nodes, making thesystem hydraulics and water quality analysis a complicated task. In this manuscript a methodology and application of a conjunctivehydraulic and water quality model for water distribution systems aggregation is presented. The model outcome provides a reducednetwork with fewer nodes and links which resembles the original system performance in both the hydraulics �i.e., quantities and pressures�and water quality �i.e., concentrations� at high accuracy. The proposed methodology is demonstrated using the three example applicationsof EPANET.

DOI: 10.1061/�ASCE�0733-9496�2008�134:3�303�

CE Database subject headings: Optimization; Water distribution systems; Aggregation; Water quality.

Introduction

A water distribution system consists of hydraulic components—sources, pipes, pumps, and control elements. The system elementsand their physical behavior can be represented and modeled usinga simulation model, such as EPANET �USEPA 2002�, for ex-tended period water quantity and quality conditions. The systemlayout, the typically large number of consumer nodes, the systempipes, and the dynamic loading conditions, make the modeling ofa water distribution system a complicated task. The water distri-bution system aggregation problem is to construct a simplifiedreduced model, i.e., with fewer nodes and links, which resemblethe original system performance.

Aggregation of a water distribution system is not unique. Itdepends on the aggregation objectives and on the differentadopted levels and methods of aggregation. Only few methodshave been developed to simplify water distribution systems, fo-cusing only on the system’s hydraulic behavior. Hamberg andShamir �1988� presented a simplification methodology for the de-sign of water distribution systems. The simplification was devel-oped through using a step-wise combination of system elements,and through a nonlinear continuum representation of the systemas a bundle. Anderson and Al-Jamal �1995� introduced a method-ology for the simplification of complicated hydraulic networksusing a parameter-fitting approach. The layout of the final simpli-

1Ph.D. Student, Faculty of Civil and Environmental Engineering,Technion–Israel Institute of Technology, Haifa 32000, Israel. E-mail:[email protected]

2Senior Lecturer, Faculty of Civil and Environmental Engineering,Technion–Israel Institute of Technology, Haifa 32000, Israel. E-mail:[email protected]

Note. Discussion open until October 1, 2008. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this paper was submitted for review and pos-sible publication on February 12, 2007; approved on June 15, 2007. Thispaper is part of the Journal of Water Resources Planning and Manage-ment, Vol. 134, No. 3, May 1, 2008. ©ASCE, ISSN 0733-9496/2008/3-
303–309/$25.00.
JOURNAL OF WATER RESOURCE

J. Water Resour. Plann. Mana

fied network is specified a priori; then, pipe conductances anddemand distribution are determined by maximizing the fitnessbetween the simplified and the full system performances. Ulanickiet al. �1996� presented a mathematical model for the hydraulicaggregation of water distribution systems. The approach com-putes a network model equivalent to the original system withfewer components by analytical elimination of system compo-nents. The reduced nonlinear model preserves the nonlinearity ofthe original model and approximates the original system within awide range of operating conditions.

All previously proposed methods compare the full and reducedmodels based on their hydraulic performance, i.e., pressure atnodes, water level at tanks, pump operations, etc. As concernsrelated to water quality in drinking water distribution systemsgrow, it is imperative to include water quality considerations inhydraulic-network aggregation objectives. With this respect awater-quality aggregation problem is to establish a reduceddistribution system which preserves both the hydraulics �i.e.,quantities and pressures� and water quality �i.e., concentrations�behavior of the original system.

The proposed method suggests a water-quality aggregation al-gorithm for constructing an equivalent reduced system, contain-ing fewer numbers of nodes and links which resembles both thehydraulics and the water quality system performance. The pro-posed methodology extends the existing hydraulic-aggregationtechnique of Ulanicki et al. �1996�, and incorporates two graph-search algorithms �Tarjan 1972�. The method is demonstrated onthree water distribution systems of increasing complexity.

Problem Statement

A real-world water distribution system may contain thousands ofnodes and connecting links, subject to dynamic loading condi-tions, for providing prescribed demands at desired pressures andqualities. The complexity of a water distribution system makesthe hydraulics and the water quality analysis a complicated task.The objective of water-quality aggregation is to construct an
equivalent reduced system �i.e., fewer nodes and links�, which
S PLANNING AND MANAGEMENT © ASCE / MAY/JUNE 2008 / 303

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resembles both the hydraulics and the water quality performancesof the original system.

The proposed algorithm lets the user select a set of nodeswithin the system to remain in the aggregated model �e.g., moni-toring station locations�, after which a new topology and compo-nent characteristics of the reduced system are calculated. Themain stages of the algorithm are provided in Fig. 1 and are furtherdescribed in subsequent sections. The final result of the qualityaggregation model is a simplified model which preserves pres-sures and concentrations of the full network at the reducedsystem.

EPANET is used as the simulation model for hydraulic andwater quality analysis for extended period simulations. The one-dimensional advective-reactive transport equation is used to pre-dict the changes in constituent concentrations along pipes. Theconstituent mixing at nodes is assumed to be complete and instan-taneous. Water is assumed to be completely mixed in storagefacilities �i.e., tanks and reservoirs�. The concentration leaving anode is assumed to follow advective transport, and is computed asa flow-weighted average of the incoming pipe concentrations.

The results of EPANET simulation runs are taken as referenceparameters of the system hydraulics and water quality perfor-mances, for comparing the original and aggregated systemoutcomes.

Water Quality Aggregation

A water distribution system can be intuitively modeled as a graphG�V ,E� with the set of nodes V representing the sources andconsumer nodes, and the set of edges E—the connecting pipes,pumping units, and valves.

The first step of the algorithm is to represent the water distri-bution system as a directed graph. This is accomplished byperforming hydraulic simulations to establish all possible flowdirections in the network pipes.

The next step is to reduce the original directed graph to itsstrongly connected components. A strongly connected componentis a set of nodes such that for any two elements u and v within the

Fig. 1. Stages of water

component there exists at least one directed path from u to v and

304 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT


a directed path from v to u. In other words, the concentration at anode depends on the concentration of any other node within itsstrongly connected component. The directed graph �i.e., the out-come of the first step� is reduced to its strongly connected com-ponents using the Depth-First Search �DFS� algorithm �Tarjan1972� which is described in the next section.

Depth-First SearchThe DFS �Tarjan 1972� algorithm starts at some node of the graphand explores as far as possible before backtracking. The searchbegins by selecting a source node, an outgoing edge from thatnode, and progresses in the direction of that edge. The procedurecontinues by choosing an unexplored edge of the recently reachednode. When no more unexplored edges are left the algorithmtracks back to a previously reached node which still has unex-plored edges. The search completes when all nodes of the graphhave been visited. Examples of DFS applications are: testing forconnectivity in a graph, finding a path between two nodes of agraph or equivalently reporting that no such path exists, comput-ing a cycle in a graph or reporting that no cycles exist, etc. TheDFS algorithm is highly efficient, having a linear computationaltime proportional to the number of vertices plus the number ofedges.

Further, all nodes of a strongly connected component arejoined together into a single metanode, yielding a directed acyclic�or meta� graph �DAG�, which contains no circular paths and itsnodes comprise the strongly connected components. Thus, theconnectivity structure of the system is two-layered: the top layeris the DAG �or metagraph�, and the second layer is the full de-tailed system structure.

Next, the metagraph can be topologically sorted and all up-stream nodes which can contribute to the concentration of somecritical nodes can be identified. This is achieved by reversing thedirections of all links in the metagraph, resulting in a reverseDAG, and searching the reverse DAG using the Breadth-FirstSearch �BFS� algorithm. This stage is described in the following.

Breadth-First SearchBFS is another type of graph search algorithm. It begins at a root

aggregation algorithm
quality
node and explores all the nodes which are adjacent to the current

© ASCE / MAY/JUNE 2008

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node before visiting other nodes. The traversal goes a level at atime, defining each level in terms of the number of edges in thepath from the root node. Nodes not reachable from the root nodeare ignored. Examples of BFS applications are: finding all nodeswithin one connected component, finding the shortest path of anunweighted graph, etc.

The nodes which were not identified by the BFS can be elimi-nated from the network. These nodes do not have direct influenceon the concentration of the chosen critical nodes �e.g., nodes withmonitoring stations� and can be removed from the network byaggregating their properties with the remaining nodes and links ofthe network. This is accomplished by using the hydraulic-aggregation algorithm of Ulanicki et al. �1996�, described brieflyin the following.

Hydraulic Aggregation

The purpose of hydraulic aggregation is to construct an equivalentreduced system �i.e., fewer nodes and links� which retains, forexample, only critical nodes at which pressure is monitored. Inaddition, the connectivity of the simplified �aggregated� systemshould resemble the connectivity of the original system andits hydraulic performance �i.e., pressure at nodes, water levelsat tanks, pumps operation� over a wide range of operatingconditions.

As described previously, there are several approaches forhydraulic water distribution systems aggregation. Of those, themethod proposed by Ulanicki et al. �1996� was chosen forhydraulic aggregation. The method of Ulanicki et al. �1996� isbased on reducing the algebraic system of equations by eliminat-ing some of the variables using Gauss-elimination �Hammerlinand Hoffman 1991�. The method involves four stages, as de-scribed in the following.

Create a Full Nonlinear ModelThe complete nonlinear mathematical description of the physicalbehavior of the system can be obtained by formulating mass con-servation equation for each node of the network �Kirchoff’s law1� and energy balance equation for all basic loops of the network�Kirchoff’s law 2�

AQ = q �1�

ATh = �h �2�

where A=A�V ,E��directed incidence matrix of the networkgraph G�V ,E� defined as nodes �V� on edges �E�; Q�vector ofunknown flows in the links, where Qi��hi�=gi�Di ,CHWi ,Li��hi

1/e1 where gi�ith pipe conductance�a function of the pipe diameter Di, the Hazen-Williams head-losscoefficient CHWi, and the pipe length Li, with e1=1.852�;q�vector of known demands, �h�vector of the head-lossesalong the pipes, and h�vector of unknown nodal heads.

Linearize the Established ModelFor a given operating point defined by the nodal head h0 anddemand q0, the linearized approximation describes the relation-ships between small changes in nodal head �h and demand �q
around the chosen operating point.


After linearization, Eqs. �1� and �2� take the following form:

G��h = �q �3�

where G=A�dQ��h� /d��h��AT�Jacobian symmetric matrixwhose elements are linear pipe conductances g̃ which can beevaluated using Eq. �4�, and ��h and �q�fluctuations in thenodal head and demand, respectively.

The elements of the Jacobian matrix are computed using

g̃i,j �1

e1gi,j��hi,j��1/e1−1�, g̃i,i � �

j

1

e1gi,j��hi,j��1/e1−1� ∀ i, j

�4�

where g̃i,j and gi,j�linear and the nonlinear conductances of apipe connecting nodes i and j, respectively, and �hi,j�pipe’s headloss.

The linear system of equations �Eq. �3�� describes head/flowrelationship around an operating point. Eq. �3� corresponds toOhm’s law—the flow in the pipe is equal to the linear conduc-tance of the pipe multiplied by its head loss.

Reduce the Linear Model Using Gauss-EliminationProcedureFollowing linearization of the full nonlinear model, the linearmodel is further reduced by applying the Gauss-elimination pro-cess �Hammerlin and Hoffman 1991�.

For example, to remove one node, ni, from the network it isnecessary to apply one step of Gauss elimination, which elimi-nates the corresponding ith equation. The demand of that node,�qi, is redistributed among other nodes connected to node ni,proportionally to the conductance of the connecting pipes. Theconnecting pipes of the removed node are removed as well, andnew linear conductances and nodal demands are calculated for theremaining elements of the network. The reduced linear modelpreserves all the hydraulic properties of the full linear model.

Retrieve a Reduced Nonlinear Model from the ReducedLinear ModelThe last stage of the hydraulic aggregation is to recover thereduced nonlinear model from the reduced linear model. The re-

Fig. 2. Network 1—Layout


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duced nonlinear model can be retrieved using the relationshipsformulated in Eq. �4�. The aggregated model contains less numberof nodes and links, forms a new network topology, and resemblesthe hydraulic performance of the original system. The methodpresented by Ulanicki et al. �1996� showed to be very simple,robust, and reflecting closely the hydraulics behavior of the origi-nal full-sized network.

Example Applications

The proposed methodology is demonstrated on EPANET Ex-amples 1, 2, and 3. The first and most illustrative example isbased on Example 1 of EPANET �Fig. 2�. The system consists of12 pipes arranged in three loops, nine nodes, one source, onestorage tank, and one pump. The pipe data and consumer basedemands are listed in Table 1. The system is subject to variableloading conditions during an extended period simulation of 24 h.Water is chlorinated at the source at a constant concentration of 1�mg/L�.

Table 1. Network 1—Model Data

PipeLength

�m�Diameter

�mm� Node

Basedemand

�L/s�

10 3,210 457 10 0

11 1,609 356 11 9

12 1,609 254 12 9

21 1,609 254 13 6

22 1,609 305 21 9

31 1,609 152 22 13

110 61 457 23 9

111 1,609 254 31 6

112 1,609 305 32 6

113 1,609 203 — —

121 1,609 203 — —

122 1,609 152 — —

Fig. 3. Network 1—DFS tree example



Consider a monitoring station located at Node 23 �Fig. 2�. Theobjective is to find the equivalent reduced network using the pro-posed quality aggregation algorithm, with Node 23 in place, andwith minimum additional system nodes and pipes, while preserv-ing the water quantity, quality, and pressures of the original sys-tem. The first step is to represent the hydraulic network as adirected graph. Two hydraulic simulation runs were required todefine all possible flow directions in the network, resulting in twotypes of links in which flow can and cannot reverse �see Fig. 2�.

After establishing the connectivity of the graph �i.e., step one�,DFS is activated with any node of the network posed as a startingnode. The search of the graph, imposed by the direction of eachedge, defines a spanning tree. The edges which are not in thespanning tree constitute the remaining links of the network, thus,maintaining the connectivity of the original network. Fig. 3 dem-onstrates a possible DFS tree with node ten as the starting node.

Next, based on the results of the DFS, strongly connectedcomponents can be identified, and all nodes of each strongly con-nected component can be joined to a single metanode. The result-ing graph consists of metanodes which are connected by links inwhich flow cannot reverse, i.e., directed acyclic graph �DAG�.Fig. 4 shows the obtained DAG.

Now all the nodes which can contribute to the concentration atNode 23, which was selected by the user as the node of interest,can be identified. BFS is performed on the reverse DAG, startingthe search at Node 23. The attained BFS tree and the representa-tive nodes are shown in Fig. 5.

At the final step, only the nodes identified by the BFS remainin the network, the rest are eliminated from the network using thehydraulic aggregation algorithm �Ulanicki et al. 1996�. The finalaggregated network layout is given in Fig. 6, and the networkmodified component characteristics, in Table 2.

Fig. 7 shows a comparison between the original and reduced�aggregated� system for the pressures �marked A, top of Fig. 7�,

Fig. 4. Network 1—Directed acyclic metagraph

Fig. 5. Network 1—BFS tree


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and concentrations �marked B, bottom of Fig. 7� at Node 23. Itcan be seen from Fig. 7 that both the pressures and the concen-trations at Node 23 match very closely.

Network 2

Network 2 is Example 2 of EPANET. Its layout is given in Fig. 8.The system is comprised of 36 nodes, 41 pipes, one source, onepumping unit, and one storage tank. The system is subject tovariable loading and fluoride injection conditions at the pumpingstation during an extended period simulation of 55 h.

Consider, for instance, that a monitoring station is located atNode 19. The objective is to find a reduced network for that node.New network topology and characteristics were calculated usingthe quality aggregation algorithm. Eleven nodes and 13 pipes re-mained in the aggregated system, as shown in Fig. 9. Fig. 10provides a comparison of the concentrations at Node 19 betweenthe original and the aggregated system. It can be seen from Fig.10 that the concentrations match very closely.

Fig. 11 provides comparisons for concentrations at Node 19between the original and reduced system for different imposeddemands �top of Fig. 11, marked A� and fluoride injection patternsat the source �bottom of Fig. 11, marked B�. It can be seen fromFig. 11 that the concentrations match very closely for both cases,thus the aggregated model performs well.

Two important observations to note: �1� the aggregated model�in this and in all other examples� preserves the pressure andconcentrations at all other remaining nodes of the network; and�2� the aggregated model size �i.e., number of nodes and links�depends on the nodes selected to remain within the aggregatedsystem. For example if the monitoring station was located at

Table 2. Network 1—Reduced Model Data

PipeLength

�m�Diameter

�mm� NodeDemand

�L/s�

10 3,210 457 10 0

11 1,609 356 11 9

12 1,609 254 12 9

21 1,609 263 13 6

22 1,609 305 21 17

110 61 457 23 18

111 1,609 254 — —

112 1,609 305 — —

113 1,609 203 — —

Fig. 6. Network 1—Reduced network



Fig. 7. Network 1—Pressure and concentration comparisons forNode 23



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Node 16 �instead of Node 19�, then the aggregated model wouldprobably have less nodes and links.

Network 3

Network 3 is Example 3 of EPANET. Its layout is given in Fig.12. The system is comprised of 92 nodes, 117 pipes, two sources�a river and a lake�, two pumping units, and three storage tanks.


Fig. 10. Network 2—Fluoride concentration at Node 19



The system is subject to variable loading conditions during anextended period simulation of 24 h. A tracer at a variable patternwas injected at the outlet of the Lake.

Node 105 was selected as a monitoring station location forconstructing an aggregated system. The resulting aggregatedsystem is shown in Fig. 13. It consists of 36 nodes and 56 pipes.Fig. 14 shows a concentration comparison for Node 105 betweenthe full and aggregated model, showing close matching.

Fig. 11. Network 2—Fluoride concentration for modified demandand injection patterns



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Conclusions

In this manuscript an algorithm for water distribution systemsaggregation is developed and demonstrated for both the hydrau-lics �i.e., pressures and quantities, based on Ulanicki et al. �1996��and water quality.

The model was tested using the three example applicationsprovided within EPANET, and was found to be robust and reliablein that it preserved at high accuracy the water quantity and waterquality concentrations at the aggregated system.

As highly efficient well explored graph theory techniques areemployed �i.e., DFS and BFS� the model is capable of providingresults for much more complicated systems. This effort is thesubject of further undergoing research.

Acknowledgments

This Research was supported by the Technion Grand WaterResearch Institute �GWRI�, the Institute for Future Defense Tech-nologies Research Named for The Medvedi, Shwartzman andGensler families, and by NATO �Science for Peace �SfP� ProjectNo. CBD.MD.SFP 981456�.




Notation

The following symbols are used in this paper:A�V ,E� � directed incidence matrix of nodes on edges;

CHW � Hazen-Williams head-loss coefficient;D � pipe diameter;E � set of edges;e1 � Hazen-Williams head-loss power coefficient �i.e.,

1.852�;G � Jacobian matrix;

G�V ,E� � graph with a set of nodes V and a set of edges,E;

g � pipe conductance;g̃ � linear pipe conductance;h � vector of unknown heads;L � pipe length;Q � vector of unknown flows;q � vector of known demands;V � set of nodes;

�h � vector of head loss;��h � fluctuations in nodal head; and

�q � fluctuations in demand.

References

Anderson, E. J., and Al-Jamal, K. H. �1995� “Hydraulic-network simpli-fication.” J. Water Resour. Plann. Manage., 121�3�, 235–240.

Hamberg, D., and Shamir, U. �1988�. “Schematic models for distributionsystems design. I: Combination concept.” J. Water Resour. Plann.Manage., 114�2�, 129–140.

Hammerlin, G., and Hoffman, K. H. �1991�. “Numerical mathematics.”Graduate texts in mathematics, Springer, New York.

Tarjan, R. �1972� “Depth-first search and linear graph algorithms.” SIAMJ. Comput., 1�2�, 146–160.

Ulanicki, B., Zehnpfund, A., and Martinez, F. �1996�. “Simplification ofwater distribution network models.” Proc., 2nd Int. Conf. on Hydroin-formatics, Zurich, Switzerland, 493–500.

USEPA. �2002�. “EPANET 2.0.” �http://www.epa.gov/ORD/NRMRL/wswrd/epanet.html� �June 27 2007�.

Fig. 14. Network 3—Concentration comparison at Node 105


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Water Distribution System Aggregation for Water Quality Analysis