SIMPLIFICATION OF WATER SUPPLY NETWORK MODELS THROUGH LINEARISATION

Transcription

1 SIMPLIFICATIO OF WATER SUPPLY ETWORK MODELS THROUGH LIEARISATIO Tobias Maschler and Draan A. Savic Report umber:99/ 999 Centre for Water Systems, University of Exeter, orth Park Road, Exeter, EX4 4QF, Devon, United Kindom.

2 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Abstract This work enhances a simplification alorithm for water network models with routines to identify the simplification rane and to classify the importance of pipes in the reduced network model. To areate a water network model, information about its control components is necessary. The variables used to optimise the operational schedules of the water supply network with the iven model have also to be known. Basin on this, a simplification rane is identified in the water network model. Afterwards, the model is linearised around a iven workin point and all redundant nodes are eliminated with Gauss-Jordan elimination. The remainin nodes are re-linked with pipes accordin to the structure of the simplified model. on-important links will be deleted to keep the simplified model compact and with as less loops as possible. The simplification alorithm is presented in detail in theory as in practise with an example network. Finally, it is applied in case studies to two water network models and the results are discussed. The necessary information about the implementation of the developed computer proram code is described alonside as well as essential information about the use of existin software.

3 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Table of Contents Abstract... Table of Contents... 2 List of Fiures... 5 List of Tables... 7 I Introduction...8 II Problem Formulation... II.. Introduction to Hydraulic etwork Models... II... Purpose of Water etwork Models... II.. 2. etwork Topoloy... 2 II.. 2. a. odes... 3 II.. 2. b. Pipes... 4 II. 2. Improvin the Simulation Time Problem... 6 II. 2.. The Simulation Time Problem... 6 II Simplification Objectives... 7 II Simplification of etwork Model Components... 9 II a. odes... 2 II b. Parallel Pipes... 2 II c. Series Pipes... 2 II d. odes with Similar Head... 2 II e. Tree Structures... 2 II f. Low Conductance Pipes... 2 II Sinle Input Subsystems II Black Box Simplification II a. Static Simplification II b. eural etwork Approach II. 3. Conclusions III etwork Simplification...25 III.. Theoretical Development III... The Alorithm for Static Simplification III... a. The Mathematical Water etwork Model III... b. The Linear Model III... c. Linear Model Reduction III... d. The Reduced on-linear Model... 3 III... e. Requirements... 3 III.. 2. Identification of the Simplification Rane... 3 III.. 3. Pipe Attributes III. 2. Alebraic Simplification of etwork Model Components III. 2.. odes III. 2.. a. Mathematics III. 2.. b. Interpretation... 4 III Pipes in Series

4 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. III a. Mathematics III b. Interpretation III Parallel Pipes III a. Mathematics III b. Interpretation III Trees III a. Addin the Demands III b. Static Simplification III Low Conductance Pipes III. 3. Implementation III. 3.. Modellin Environment III. 3.. a. StruMap... 5 III. 3.. b. C++ Interface to StruMap... 5 III. 3.. c. Matrix Library... 5 III. 3.. d. Prorammin Environment... 5 III The Alorithm... 5 III a. Simplification Preparation... 5 III b. Static Simplification III. 4. Example III. 4.. Procedure III Proram Test... 6 III a. Preparations... 6 III b. Simplification III. 5. Summary IV Case Studies...65 IV.. Skipton Water etwork Model IV... Brief Description IV.. 2. Preparations IV.. 3. Simplification IV.. 4. Errors at the 8: Snapshot... 7 IV.. 4. a. Simplification with all Pipes... 7 IV.. 4. b. Relative Writeback Criterion IV.. 4. c. Absolute Writeback Criterion IV.. 4. d. Comparison of Maximum and Standard Deviation of the Absolute Value of the Absolute Errors IV.. 4. e. Comparison of Relative Errors IV.. 4. f. Conclusions IV Hour Simulation IV.. 5. a. Absolute Errors IV.. 5. b. Relative Errors IV.. 5. c. Conclusions IV.. 6. Benchmarkin IV.. 6. a. Conclusions IV.. 7. Summary IV. 2. Hawksworth Lane Water etwork Model in Guiseley IV. 2.. Brief Description IV Preparations IV Simplification... 9 IV Errors IV a. Errors in the on-removable odes IV b. Errors in the Untouchable Pipes IV Errors in the 24-Hour-Simulations IV Benchmarkin

5 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. IV Summary IV. 3. Conclusions V Summary and Conclusions...97 V.. Summary V. 2. Recommendations for Future Work VI References...99 VII Acknowledements...2 VIII Appendix...3 VIII.. Calculation of the c Constant in the Hazen-Williams Equation for StruMap. 3 VIII. 2. The umber of Loops in a Water etwork... 4 VIII. 3. Determination of the Maximum umber of Pipes in a Water etwork... 4 VIII. 3.. The Maximum umber of Links in a General etwork... 4 VIII The Maximum umber of Pipes in a Water etwork... 5 VIII. 4. Output of the Simplification Routine for the Skipton Water etwork... 6 VIII. 4.. Skipton Water etwork... 6 VIII. 4.. a. Case... 6 VIII. 4.. b. Case VIII. 4.. c. Case VIII. 4.. d. Case VIII. 4.. e. Case 5... VIII. 4.. f. Case 6... VIII Case 7... VIII. 4.. h. Case VIII Hawksworth Lane, Guiseley Water etwork Model... 3 VIII a. Case... 3 VIII b. Case 2 and VIII c. Case VIII d. Case VIII. 5. User Manual

6 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. List of Fiures Fiure. The Skipton Water etwork Model... Fiure 2. Input and Output for a Water etwork Model... Fiure 3. Example of a Tree (left) and a Loop (riht) etwork Structure....3 Fiure 4. Simplification of etwork Model Components...9 Fiure 5. A Water etwork Model is split into Two Sinle Input Subsystems...22 Fiure 6. Example Water etwork Model with Trees, Loops, Parallel Pipes, a Fixed Head Reservoir and an Open Valve...3 Fiure 7. The Example etwork with the Interface Components outside the Simplification Rane...33 Fiure 8. The etwork Model with the Simplification Boundary after performin the Identification Alorithm Fiure 9. ode Replacement Fiure. The new etwork Structure after removin....4 Fiure. etwork Model of Pipes in Series....4 Fiure 2. etwork Model of Pipes in Series, one ode was removed...43 Fiure 3. etwork Model with Parallel Pipes Fiure 4. etwork Model with Unified Parallel Pipes Fiure 5. A Water etwork Model with a Tree Fiure 6. Water etwork Model with removed Tree Fiure 7. The Modellin Environment...5 Fiure 8. Activity Diaram for the Simplification Preparation...52 Fiure 9. Structure of the Jacobian matrix A...55 Fiure 2. Part of Activity Diaram for the Static Simplification Fiure 2. Part 2 of Activity Diaram for the Static Simplification Fiure 22. The Example etwork with marked Simplification Areas...59 Fiure 23. The simplified example network...6 Fiure 24. Screenshot of the Oriinal Example etwork....6 Fiure 25. The Simplified Example etwork Fiure 26. Skipton Fiure 27. umbers of etwork Elements after Simplification compared to the ones before for the Skipton Water etwork Model

7 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Fiure 28. The network after Simplification, all linear pipe conductances were written back. 7 Fiure 29. Absolute Error of Head, Simplification with all Pipes....7 Fiure 3. Absolute Error of Flow, Simplification with all Pipes...72 Fiure 3. Relative Error of Flow, Simplification with all Pipes...72 Fiure 32. Absolute Error of the Available Head in the Demand odes for the relative cases. 73 Fiure 33. Absolute Error of Flow in the Pipes...74 Fiure 34. Absolute Error of the Available Head in the Demand odes...75 Fiure 35. Absolute Error of Flow in the Pipes...75 Fiure 36. Comparison of the Relative Error of the Head in the odes: Maximal Value, Standard Deviation Fiure 37. Comparison of Relative Error of the Flow in the Pipes: Maximal Value, Standard Deviation...78 Fiure 38. Available Head at ode 723 durin 24 hours...8 Fiure 39. Absolute Error of Available Head in ode 723 durin 24 Hours....8 Fiure 4. Available Head at ode 924 durin 24 hours...8 Fiure 4. Absolute Error of Available Head in ode 924 durin 24 Hours Fiure 42. Available Head at ode 4 durin 24 hours...83 Fiure 43. Absolute Error of Available Head in ode 4 durin 24 Hours...83 Fiure 44. Flow in Pipe 246 durin 24 hours...84 Fiure 45. Absolute Error of Flow in Pipe 246 durin 24 Hours...84 Fiure 46. Comparison of Benchmark Results of Simplified etworks...86 Fiure 47. Guiseley...88 Fiure 48. Screenshot of the oriinal Hawksworth Lane water network model....9 Fiure 49. Screenshot of the simplified Hawksworth Lane water network model....9 Fiure 5. umbers of etwork Elements after Simplification compared to the ones before for the Hawksworth Lane water network model...92 Fiure 5. Relative Solvin Time of the Simplified Hawksworth Lane Water etwork Models

8 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. List of Tables Table. Attributes of a ode...3 Table 2. Attributes of a Pipe...4 Table 3. Similarities between Electrical etwork Models and Water etwork Models. 23 Table 4. Attributes of Interface Components...34 Table 5. Simplifications Steps in the Example etwork Table 6. Comparison of the umber of etwork Elements...68 Table 7. Pipes with the hihest relative error...73 Table 8. Maximal Errors and Std. Deviation for the Heads in the Simplifications Table 9. Maximal Relative Error and Maximal Relative Std. Deviation for the Flows of the simplified etwork Models...78 Table. Pipes with very low Flow in the Simplified etwork Models...78 Table. Absolute maximal Errors and Standard Deviation of the Skipton Water etwork Model in a 24h Simulation. 79 Table 2. Maximal Relative Errors for the Simplification Cases, 5 and Table 3. Overview over the total solvin time...85 Table 4. The solvin time per run:...86 Table 5. Comparison of the number of network elements....9 Table 6. Errors in the non-removable nodes Table 7. Errors in the Untouchable pipes...93 Table 8. Maximal Relative Errors and Standard Deviation of the Relative Errors of the Hawksworth Lane etwork Model...94 Table 9. Solvin Time for different umbers of Runs for the Hawksworth Lane Water etwork Model Table 2. Solvin Time per Run for the Hawksworth Lane Water etwork Model. 94 Table 2. Calculation of the c Constant for the Hazen-Williams-Formula

9 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. I ITRODUCTIO The water industry in the United Kindom spends approximately 7,, per annum on electricity for pumpin water supply. Similarly, almost 7% of the electricity consumed in the United States is used by the municipal water utilities. Since treated water pumpin compromises the major fraction of the total enery budet, optimised operational schedules can improve the enery efficiency of a water distribution system. System operators make these operational schedules with the aid of special decision support software. This software bases on mathematical models, which contain in some cases more than 2 components. Durin an optimisation process for the operational schedules, these water network models are simulated many times for different schedules. However, only very small part of the simulation results is normally necessary, so an areated model is adequate. It should contain all control components of the oriinal model and the variables, which are used to calculate the quality of the simulation. Areated simplified models will require less computation time and provide all information needed for the optimisation. Hence they will speed up the optimisation process and allow bier models to be calculated. This work uses a simplification alorithm Ulanicki et al (996) desinated as static simplification later to derive simplified models. Input components, as sources, tanks and reservoirs and valves, will be considered as well as those variables, which are needed for the optimisation process. The first chapter in this report will start with an introduction into hydraulic network models and their components, mainly nodes and pipes. Afterwards, the simulation time 8

10 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. problem will be discussed, followed by an overview of simplification approaches in literature. Then, the simplification alorithm of Ulanicki et al (996) will be presented. It linearises the network equations in an eleant way around a iven workin point and reduces the linear model with Gauss-Jordan elimination. From the reduced linearised model, a simplified non-linear model is recovered. Afterwards, an approach for the identification of the simplification area in the network model will be formulated. ext, it will be shown with brief alebraic examples, that the static simplification carries out network model component based simplification approaches like unification of nodes, parallel pipes and pipes in series and deletion of trees. In addition, the static simplification alorithm will be enhanced with the facility to delete pipes with very little influence on the simulation results, low conductance pipes. After this theoretical part, the implementation of the static simplification alorithm will be discussed. It will be implemented as a module in the eoraphical information System StruMap Structural Technoloies Ltd. (996). The routines will be coded in C++. As compiler, the Borland C++-Builder Calvert (997) will be used. Followin, a network model will be simplified and its simplification procedure discussed in detail. As case studies, the models of two water networks in Yorkshire, United Kindom, will be simplified. One of them is very lare, with around 2 network components. The errors of the simplified models at the workin point will be analysed, as well as in 24h simulations. Finally, the resultin reduction in solvin time will be estimated. 9

11 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. II PROBLEM FORMULATIO II.. Introduction to Hydraulic etwork Models This section introduces water network models. Firstly, the purpose of modellin water networks will be discussed. Then the macroscopic structure of water network models will be explained, followed by a short description of pipes and nodes, the main components of water network models. Afterwards, different simplification methods are presented. Finally, the aim of this report, to reduce the simulation time of water network models, will be formulated. II... Purpose of Water etwork Models Fiure below shows a typical rural water network model with approximately nodes and 3 pipes.

12 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Fiure. The Skipton Water etwork Model. Water distribution systems supply water to households, companies, fire departments and farms. The aim of water supply companies is to do this with an available head between 2 and 5m, and with as little pipe leakae as possible. Pipe leakae increases with hiher pressure. Therefore, a suitable compromise must be found. This is achieved by optimisin the schedules of the control devices in the network. Control devices are mainly pumps and valves. Control Schedules, Demands Water etwork Model Flow in Pipes, Head in odes Cost Function Costs caused by the Input Vector Input Output Fiure 2. Input and Output for a Water etwork Model.

13 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Water network models are used to determine a convenient system input vector, which consists of pump, valve and source schedules as well as a demand vector for a iven water network. The quality of the input vector is measured with a cost function. It weihs the input and output of the water network model and ives a scalar value (mainly in a currency unit) for it. This scalar value expresses the costs caused by the input vector. II.. 2. etwork Topoloy Hydraulic water networks consist enerally of the followin components: Demand odes (nodes in network theory) Pipes (arcs in network theory) Valves Pumps Reservoirs Sources Pipes connect demand nodes, valves, pumps, sources, and reservoirs. One or more pipes can be connected to a demand node. A valve is connected two pipes to two nodes. A source is usually connected to one pipe. A Reservoir has two pipes, one to refill it and one for the outflow. One pipe connects exactly two of the other network components above. Generally, there are two types of pipe structures in a water network model: Trees Loops A tree is a non-closed chain of connected pipes in the water network model. It can have sub-trees. A loop is a closed chain of connected pipes. 2

14 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Fiure 3. Example of a Tree (left) and a Loop (riht) etwork Structure. The tree structure in Fiure 3 has one sub-tree. The sub-tree starts at the second topmost node from the left and oes to the riht hand side. The loop structure contains two independent loops, one at the top and one at the bottom. In a network consistin of nodes and P pipes, the number of loops L in a network can be determined usin the followin relation: () L = P + The derivation of this equation is explained in the Appendix, Chapter VIII. 2. II.. 2. a. odes A node is a termination point for one or more pipes. It has a certain demand (withdrawal) and a unique, fixed eoraphic position with an elevation above ordinance datum. The available head at a demand node is the difference between the total head and the elevation. Table. Attributes of a ode. ode x m Georaphic Location y m Georaphic Location z Elevation above m Ordinance Datum q l/s Demand H_T m Total Head H = H_T z m Available Head 3

15 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. The equilibrium of volumetric flow rates of all pipes connected to the node is calculated usin the followin equation: j P j (2) Q i = q i= j i k j Q = Q : flow in pipe k l/s q j : demand at node j l/s j P : umber of pipes connected to node j. A demand node is a node with a certain water withdrawal. Valves, sources, and reservoirs will be treated in this report as nodes because of their fixed eoraphic location. Sources have neative demands, as flow is put in the water network throuh them. The sin of the demand of reservoirs is neative, when it is bein filled with water; when water is withdrawn, it is positive. Valves have no demand. The headloss in a valve is described by additional equations. II.. 2. b. Pipes A pipe connects two nodes with each other. It has a start and an end node. Table 2. Attributes of a Pipe. Pipe _s Start ode _e End ode d mm Diameter l m Lenth C Rouhness Coefficient delta_h m Head Loss Q l/s Flow A pipe has a certain diameter and a lenth. The head loss between the start and the end node of the pipe is expressed in this report usin the Hazen-Williams equation (4). Other equations, which ive a relation between the flow and the head loss in a pipe, are the one of Colebrook-White and the Mannin-Equation (see Chadwick and Morfett (993)). The head loss between the start and the end node are determined usin the followin relationship: 4

16 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. (3) H i H j = ij = k k =,2, K, P H, H : total heads at nodes i,j m i j ij = k : head loss between nodes i,j m P : umber of pipes in the hydraulic network. The relationship between head loss k and flow Williams equation (Chadwick and Morfett (993)): Q k is expressed with the Hazen- e (4) k = r k Qk Qk k =,2, K, P r k : pipe resistance in pipe k k : head loss in pipe k m Q k : flow in pipe k l/s Constant: e :=.85 The Hazen-Williams pipe resistance is defined Savic and Walters (997) as: c lk (5) r k = k =,2, K, e e P 2 C d k k l k : lenth of pipe k m C k : Hazen-Williams coefficient of pipe k, dimensionless d k : diameter of pipe k mm Constants: c :=.25283E+ e :=.85 e2 := 4.87 The constant c was determined from StruMap as described in Appendix VIII.. The relationships (4) and (5) can also be expressed in terms of the pipe conductance k instead of the pipe resistance r k (Ulanicki et al (996)): 5

17 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. K (6) Q e k ( k ) = k k k k =,2,, P where the pipe conductance k is e k r k k =,2, K, = or e2 e k d k k =,2, K e e lk C k =, P. c P II. 2. Improvin the Simulation Time Problem II. 2.. The Simulation Time Problem The equations described in the previous Chapter define the equation system of a water network model. The equation system is solvable explicitly only, if the network has a tree structure. Then, the head at the nodes can be directly calculated in terms of flow. This will be discussed in chapter III If a water network model has loops, it must be solved iteratively. Several solvin approaches are described in Ellis and Simpson (996). The number of equations used when solvin water network model depends on the solver alorithm. HARP, the solver of StruMap (Structural Technoloies Ltd. (996)), is based on the freely available hydraulic network solver EPAET (Rossman (994)), which uses a node-based model. A water network model is mainly represented there with the equations (2) and (4) above. Rossman (994) writes that EPAET uses the radient alorithm Todini and Pilati (987) and sparse matrix techniques from Geore and Liu (98). Water network models may be simulated static and dynamic. As only the data at some time points is necessary for water network schedulin plannin, static, steady-state solutions are used in practice: a water network model is solved usin a set of hourly time steps (snapshot s) over a period of 24 hours. This allows the pronosis of chanin reservoir levels and the determination of convenient valve and pump schedules. Therefore, the solution time of a water network model depends on its size, the number of snapshots and the equation solver. Further, the optimisation method bein used to determine the cheapest input vector requires numerous simulation runs. This will be explained usin Genetic Alorithms as an example. 6

18 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. To determine an input vector, which minimises a iven cost function, different approaches are used. The Centre for Water Systems at the University of Exeter uses Genetic Alorithms (De Schaetzen and Walters (998)). Genetic Alorithms have the ability to deal easily with non-linear, discrete, multimodal and non-differentiable functions, for example the Heavyside Step Function (CRC Press and Weisstein (996)). The Heavyside Step Function is used to describe the on-off schedulin of pumps and valves. Genetic Alorithms solve the network model for a rane of different input vectors. Then they select the ones with the best cost function values and derive from these a new eneration of input vectors which will be evaluated. This is done until the population of input vectors has convered. A further stop criterion is, for example, a fixed number of iterations or similar input vectors. Michalewicz (992) ives a eneral introduction about Genetic Alorithms. Therefore, enetic alorithms require the followin run time: solution time := number of enerations number of individuals per eneration simulation time of the water network model. The simulation time for the Skipton water network model with approximately nodes and 3 pipes is.25 sec. For, enerations of individuals, the enetic alorithm may require 25, sec, a little bit less than 7 h. So reducin the simulation time whilst maintainin the model accuracy will sinificantly speed up the input vector optimisation process without touchin the optimisation alorithm. II Simplification Objectives As described above, the critical item when determinin convenient input vectors for water networks is the simulation time. To cut down the simulation time, the solvin time of the network can be reduced. Therefore, the number of network components and especially of the loops in the network must be minimised in a way which uarantees a solution accuracy as close as possible to that of the oriinal water network model. 7

19 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Further, the simplified model must react in the same way to chanin boundary conditions (i.e.: demands) (Anderson and AL-Jamal (995)). There are two main ways to simplify a network model:. Simplification of etwork Model Components. The basic types of network components are maintained, but the individual network components are combined and replaced. An example for this would be to remove intermediate nodes from pipes in series. 2. Black Box Simplification. The network model is replaced with an abstract model, which provides the same functionality with less solution time. This can be done for the whole network or for parts of it. Examples are neural networks (Swiercz (995)) or eneral models, whose parameters are fitted (Anderson and Al-Jamal (995)). The first method allows the network model to be simplified directly usin the network simulation software. The second simplification technique can only be carried out in prorams, which allow the interation of the black box model. The followin sections ives an overview of a number of simplification approaches described in literature. Firstly, approaches to simplifyin water network models based on the model components will be presented; then approaches that replace the network model with black boxes will be described. 8

20 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. II Simplification of etwork Model Components odes Parallel Pipes Pipes in Series odes with Similar Head Tree Structures Low Conductance Pipes Fiure 4. Simplification of etwork Model Components. 9

21 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. II a. odes odes, that are eoraphically closely located can be unified to eliminate converence problems of the solver (Martínez and García-Serra (992)). odes are understood to be close, if the head loss between them is very low or in more abstract terms, the pipe resistance between them is very low. This is the same as unifyin pipes in series (described below) which are of a very low pipe resistance. II b. Parallel Pipes Pipes, which have the same start and end nodes, can be replaced by an equivalent one with an appropriate diameter and lenth (Martínez and García-Serra (992)). The pipe conductance of the new pipe can be calculated by summin up the conductances of the old pipes Anderson and Al-Jamal (995): (7) = +, 2. new old, old Afterwards, the pipe diameter can be obtained usin Equation (6) with an appropriate pipe lenth and Hazen-Williams coefficient. This is an exact substitution as lon as the head loss for the pipe does not chane, but it leads to some error otherwise. II c. Series Pipes Two pipes, connected by a sinle node, can be replaced by one pipe (Martínez and García-Serra (992)). The demand of the node which is removed has to be redistributed to the start and end node of the new pipe. The resistance of the new pipe can be calculated by summin up the old pipe resistances Anderson and Al-Jamal (995): (8) r = r + r, 2. new old, old ow, the new diameter can be calculated usin equation (5) and a suitable pipe lenth and Hazen-Williams coefficient. As this method concatenates two parallel pipes, this is Pipes with hih resistance/ low conductance pose bi problems when solvin. They require far more iterations to meet the accuracy criterion of the solver than other pipes. 2

22 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. only an exact replacement as lon as the head loss between the start and end nodes stays the same. The redistribution of the demand of the node, which is removed, will be discussed in Chapter III... d. II d. odes with Similar Head Anderson and Al-Jamal (995) write that a roup of nodes with similar head can be reduced to a sinle node with the total demand of the node roup. II e. Tree Structures Tree structures in a network can be replaced by addin the summed demand of all nodes in the tree to the node, which connects this structure Martínez and García-Serra (992). This will be described in detail in Chapter III below. II f. Low Conductance Pipes Martínez and García-Serra (992) emphasise that pipes below a certain diameter do not contribute sinificantly to the carryin capacity of a water network, so they can be nelected. They ive as rule of thumb, that in small networks (with pipe diameters up to 2-25mm), pipes with less than 8mm diameter are neliible, in lare networks (with pipe diameters bier than 8mm) pipes with less than 2mm can be eliminated. Swiercz (995) describes the same procedure as skeletonisation. As the flow is the decision criterion for nelectin a pipe, it appears to be more exact to use the pipe resistance or the pipe conductance instead of the pipe diameter to decide. These terms reflect all the pipe attributes in the equations (5) and (6). Therefore, the pipe rouhness with the Hazen-Williams coefficient and the pipe lenth will be taken into account, as well. 2

23 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. II Sinle Input Subsystems Fiure 5. A Water etwork Model is split into Two Sinle Input Subsystems. A lare water network model with multiple input points (sources and reservoirs) can be separated in multiple independent networks, when pipes, which are under the influence of two input points are bein split Swarnee and Sharma (99). This breaks the oriinal network model down to a number of models correspondin to the number of input points. These models can be simulated with less computational effort because of their smaller size. II Black Box Simplification The followin approaches replace the network model partially or fully with a system, which provides the same function with less complexity. II a. Static Simplification Basin on the relationships between electrical network models and water network models, Ulanicki et al (996) present an approach to simplify latter models. The followin table ives a brief overview of these similarities. 22

24 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Table 3. Similarities between Electrical etwork Models and Water etwork Models. Electrical network models water network models current I flow Q voltae U head H non-linear resistor Pipe I = at a node, Kirchhoff s Law I U = for a loop, Kirchhoff s Law II = Q q nod at a node Continuity Equation H = for a loop Enery Equation The approach is based on formulatin the full non-linear model, linearisin it around a iven workin point and reducin it with the Gaussian Elimination Alorithm. A nonlinear model with fewer components can then be retrieved. The case studies (Ulanicki et al (996)) show that the larest relative errors in terms of heads are mostly less than % for water network models with up to 797 nodes, 88 elements and 3 devices. The full alorithm will be described in detail in chapter III... This approach is very simple and fast and allows a direct physical interpretation: the resultin simplified model also consists of pipes and nodes. It is valid for a wide rane of operatin conditions, but shows sliht errors when the workin conditions are outside the vicinity of the linearisation point. II b. eural etwork Approach Swiercz (995) replaces the water network model with a neural network. eural networks are trained with iven input and output vectors to mimic the behaviour of the water network model. Swiercz example network contained 6 nodes, 24 elements and 4 reservoirs. Dependin of the complexity of the neural networks, the averae relative error for the heads was below 2%, while the trainin covered epochs (snapshots). Swiercz (995) points out that neural networks require far less calculation time than their oriinal water network models without a loss of accuracy. In addition to the approaches described above, Anderson and Al-Jamal (995) use parameter-fittin to equalise the results of the simplified and the oriinal network model. This is an additional step durin the model simplification procedure to enhance the accuracy of the simplified model. 23

25 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. II. 3. Conclusions In this work, the static simplification method of Ulanicki et al (996) was chosen for network simplification. It combines the advantae of providin an abstract model, which can be expressed with pipes and nodes with the ability to reduce the water network models to a minimum without sinificant accuracy loss. Combinin nodes with similar head was not done in this work, as the heads of nodes may vary drastically over a period althouh they appear to be similar at a specific time. In addition, splittin networks with multiple sources into sinle input systems was not carried out here. 24

26 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. III ETWORK SIMPLIFICATIO This chapter describes the theory behind static simplification based on Ulanicki et al (996) and illustrates that this method carries out a simplification based on network model components. In addition, the functionin of the method is shown by an example network model. Finally, details of the implementation, like an Activity-Diaram of the Alorithm, are discussed. III.. Theoretical Development In this section, the alorithm for static simplification will be described in detail. Afterwards, the treatment of network components will be discussed. III... The Alorithm for Static Simplification The stratey described here is based on the work by Ulanicki et al (996). Alebraic examples are iven in Section III. 2. below. III... a. The Mathematical Water etwork Model It is very convenient for alebraic manipulations to express the network model with a branch incidence matrix: 25

27 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. (9) r r = Λ nod q q Kirchhoff s law I for nodes Λ q r nod q r branch incidence matrix vector of flows in the pipes demands in the nodes. This matrix has one row for each node and one column for each branch. Each column has exactly one entry - in the row correspondin to the node, at which the water oes into the pipe and one entry at the row correspondin to the node, at which the water leaves the pipe. All other entries are. The matrix Λ can be also used to determine the head loss in the pipes: r r T () h = Λ h Kirchhoff s law II h = M r 2 H H h = M r 2 H P vector of head losses vector of total heads in the nodes. The relationship between the head loss and the flow in the pipes can be expressed with the pipe component law usin the Hazen-Williams coefficient (6): () r q = Q( h) = P h h e P 3 e sin( h ) M 3 sin( h P ) pipe component law m conductance of the pipe m; m=() p e : = e 3 The relationship uses the followin definition of the sinum function: for x sin ( x) =. for x < 26

28 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. The equations (9) - () describe the network model completely. With these equations, the mass balance for a node n can be expressed with e3 nod (2) Λ n, j( n, m) j( n, m) h j( n, m) sin( h j( m, n) ) = qn n, m =,2, K, m, n j(n,m) branch j connectin nodes m and n; j P, number of nodes connected to node n. n Insertin () in () and () in (9) leads to a compact expression for (2): r T r (3) ΛQ( Λ{ h) = q nod. r h This expression has one redundant equation as the sum of all demands the network model: nod q i is zero in (4) nod q n n= =. Hence, the simplification can only cover nodes. From now on, will represent the number of nodes covered by the simplification. III... b. The Linear Model r r, Linearisin (3) around the operatin point ( ) nod h, q leads to: of the water network model (5) r dq( h) Λ r d h r h Λ T r r δ h = δq nod with the Jacobian r dq( h) T A = Λ r Λ d h r h and r r r δ h = h h and r nod r nod δ q = q r nod, q and r T h = Λ r h, correspondin to Kirchhoff s Law II (). The matrix with the linearised pipe conductances is determined as follows: 27

29 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. (6) r dq( h) r d h r h = dia e P 3 j h e4 e 4 : = e3 = 2. e j P j= number of branches. This leads to the followin form of the Jacobian matrix: e4 e3 m, n h h n m for m, n e4 (7) Am, n = e3 k, n h h for m = n m, n =,2, K, n k k n, k n for m, n The Jacobian matrix is a symmetric one. A non-diaonal entry on position (n, m) indicates a pipe between the nodes n and m. Ulanicki et al (996) introduce the notion of the linearised branch conductance : (8) e4 j( n, m) ˆ e3 j( n, m) hn hm = j=,2,, P, m, n. This notion will be called linearised pipe conductance in this report. It reflects the pipe attributes such as diameter, lenth and Hazen-Williams coefficient and, in addition, the head loss in the pipe. It will be used to transform between the non-linear and linear network model. Ulanicki et al (996) also introduce the notion of the linearised node conductance : (9) n = ˆ k, n n,2, K, k n, k n =. The linearised node conductance is the sum of the linearised pipe conductances of the pipes which are connected to the node. This allows the presentation of the linearised model (5) as: 28

30 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. (2) M 2,, M,2 2,2 L L O L M, 2, δh δh2 M δh δq = δq 2 M δq nod nod nod with the linearised branch conductances on the non-diaonal elements and the linearised node conductances on the diaonal elements. III... c. Linear Model Reduction This linearised model (2) describes the chanes of heads and flows at a default workin point. The model can be reduced in the followin way: the rows of the nodes that shall be removed have to be put in the first r row positions of the matrix A. Afterwards, the Gauss-elimination (Hammerlin and Hoffman (99) (referenced by Ulanicki et al (996))) has to be applied r times. This yields the reduced matrix A(r). A(r) has ( - r) rows and columns. It is a submatrix of A after applyin the Gauss-elimination. Its rane in A oes from position (r +, r + ) to position (, ). The Gauss-elimination does the followin: The pipes are restructured around the nodes, which are removed. The demand of a replaced node is redistributed between nodes connected to this node. This is done proportionally to the conductance of each branch. The conductances of the pipes are chaned as well. A non-linear model can be recovered from the reduced linear model by readin the topoloy from the reduced network matrix A(r). on-zero-entries indicate branches between nodes. Ulanicki et al (996) introduce the new node-branch incidence matrix (r ) Λ with (r ) number of nodes in the reduced model and (r ) P number of branches in the reduced model. The new linear model can be represented in a form equivalent to (5): 29

31 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. (2) Λ r ( r r r Λ δ h = δq ( r) d h ( r ) ( r ) ( r) dq h ) ( r ) T ( r) ( r) nod, where r q r q ( r ) nod ( r ) nod δ δ dq = P [ ( r) ] ( r) ( r ) ( r ) ( h ) r = dia ( r ) n, m j= d h r and is a ( r ) ( r ) diaonal matrix. ( r ) n, m is the new linear branch conductance between the nodes n and m and δ new nodal demand vector. r r nod q ( ) is the III... d. The Reduced on-linear Model This yields a new non-linear model: (22) r ( r) ( r) ( r ) T ( r ) ( r ) r nod Λ Q ( Λ h ) = P q r ( r ) ( r) T ( r ) where Q ( Λ h ) is the function for the new nonlinear branch law This model has the new branch conductance (23) e h h 4 = r ( r ), m, n ( : = r). ( r) n, m 3 n e m ( ) n, m These steps wre done by Ulanicki et al (996). III... e. Requirements To reduce a model, the followin requirements are necessary:. A full hydraulic network model. 2. All values of h r nod and q r in the workin point have to be known. 3. The knowlede of which nodes shall be removed and which not. The vector nod q r is known, because it is part of the input vector, while h r can be obtained by solvin the water network model at nod q r. As Ulanicki et al (996) ive only the alorithm for static simplification, the third point has to be discussed in detail. This will be done in the followin section. 3

32 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. III.. 2. Identification of the Simplification Rane In this section, a method will be developed to separate the part of the model which can be simplified from the rest. This model part will be denoted simplification rane from now on. The simplification rane is the part of the full model, which can be replaced durin simplification without chanin the behaviour of the water network model. The simplification rane a black box has with nodes as interfaces to the in- and output components of the water network model. Generally, it is not necessary to know all heads and flows in the system since only a subset of all system output is needed for the cost function. A simplified model with fewer details should of course have the same input vector, so valves and pumps, etc. shall be kept. To maintain their functionality, flows in selected pipes and heads at selected nodes shall not be chaned by the simplification alorithm. CF Reservoir, H=m CF Valve, opened Demand ode, q nod =5 l/s CF Pipe, l=5m, d=mm C= CF: Head/ Flow monitored for Cost Function Fiure 6. Example Water etwork Model with Trees, Loops, Parallel Pipes, a Fixed Head Reservoir and an Open Valve. 3

33 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. The Water etwork Model in Fiure 6 will be used as an example to identify the simplification rane. It contains a fixed head reservoir, an opened valve, 2 demand nodes and 7 pipes. Therefore, the number of loops is 4 (see equation ()). All demand nodes have a demand of 5 l/s. The reservoir level is fixed at m. All pipes have a lenth of 5m, a diameter of mm and a Hazen-Williams coefficient of C= (see i.e. Chadwick and Morfett (993)). The head at two nodes (those ones marked with CF 2 in Fiure 6) and the flow in one pipe (marked with CF, as well) are inputs for a cost function. The aim of the identification is to find all nodes which form the boundary of the simplification rane. This boundary depends on the input and output components of the water network model. Input components are all valves, reservoirs and sources. Output components are the pipes, whose flow is used for the cost function and the nodes, whose heads are input to the cost function. The simplification rane without these interface components is drawn in Fiure 7. 2 CF stands for cost function. 32

34 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. CF Reservoir, H=m CF Valve, opened CF Demand ode, q nod =5 l/s Simplification Rane Pipe, l=5m, d=mm C= CF: Head/ Flow monitored for Cost Function Fiure 7. The Example etwork with the Interface Components outside the Simplification Rane. So far, this boundary is not clearly defined. To do this, two new network component attributes will be introduced: on-removable etwork model nodes with this attribute have to stay durin simplification. Their attributes may be chaned. Therefore, they form the boundary of the simplification rane. Untouchable etwork model components with this attribute have to stay durin simplification, but their attributes are not permitted to chane. Hence, they have to stay outside the simplification rane. Input Components have to stay unchaned durin simplification. Their attributes are not allowed to chane, for example to avoid a situation where a valve could et a demand associated with it. Hence, the valves have to be marked with as untouchable. Output components are nodes, where the head is needed for the cost function. Output pipes are 33

35 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. pipes, whose flow is needed for the cost function. Output nodes have to stay unchaned durin simplification, but one of their attributes, the demand, is allowed to chane because the pressure at the node should stay the same after simplification. Therefore, they have to be marked as non-removable. The flow in output pipes would be chaned, when the pipe conductance is varied, hence they have to be marked as untouchable. The followin table ives a brief overview over the assinment of attributes to network components. This has to be done manually. Table 4. Attributes of Interface Components. Water etwork Model Component Source Reservoir Valve Pump ode Pipe Sinal Flow Direction Input Output Attribute Untouchable Untouchable Untouchable Untouchable on-removable Untouchable The static simplification alorithm in this work is based on nodes. Therefore, the identification alorithm must clearly identify the nodes, which shall stay and those, which can be simplified. All nodes, which are non-removable, will stay durin the simplification. However, the attribute untouchable can be assined to nodes and pipes. If a node is untouchable, its attributes have to be protected from bein chaned. This can be achieved by protectin the connections, i.e. the pipes connected to the node. So the pipes which are connected to an untouchable node have to be marked as untouchable as well. Similarly, preventin their nodes from bein removed will protect untouchable pipes. Hence, the start and the end node of a pipe have to be marked as non-removable. The simplification rane identification alorithm will have to perform the followin preparation steps:. Mark all pipes, which are connected to input components (nodes with the attribute untouchable ) as untouchable as well. 2. Mark the start and end nodes of untouchable pipes as non-removable. 34

36 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. After performin these steps, the boundary of the simplification rane is defined by the non-removable nodes. This is illustrated in Fiure 8. CF Reservoir, H=m CF Valve, opened Simplification Rane CF Demand ode, q nod =5 l/s on-removable Demand ode Pipe, l=5m, d=mm C= CF: Head/ Flow monitored for Cost Function Fiure 8. The etwork Model with the Simplification Boundary after performin the Identification Alorithm. Output Pipes may eventually have ot assined the linear pipe conductance of other pipes durin the simplification process. This may be corrected by parallel pipes. The steps of the identification alorithm above uarantee that no demand is assined to valves, pumps, sources and reservoirs and that new pipes are not connected to them. III.. 3. Pipe Attributes This part explains how the pipe attribute values like the lenth, the diameter or the Hazen-Williams coefficient will be assined when the simplification alorithm creates new pipes. 35

37 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. A pipe has to transport water from one node to another accordin to its conductance. When the simplified water network model is constructed from the reduced branch incidence matrix A (r), the pipe attributes will be specified as follows: The new pipe lenth is the eoraphic distance between the start and the end node. The Hazen-Williams coefficient is set to. The diameter of a pipe k will be calculated for the new pipes durin simplification usin the followin relationship: (24) d k e e2 k c lk ( r) = k =,2, K, e P C. k The equation above is based on the formula for the non-linear pipe conductance (equation (6)). In remainin pipes, the oriinal Hazen-Williams coefficient, the diameter and the lenth cannot be kept since in the simplified model, these pipes may represent a conlomeration of several others. III. 2. Alebraic Simplification of etwork Model Components This section ives some brief examples of the static simplification based on componentby-component simplification approaches. 36

38 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p III. 2.. odes P 3 P 2 P P 4 P 5 P 6 Fiure 9. ode Replacement. The water network model in Fiure 9 has sources in the nodes 3, 4 and 5, therefore their demand is neative. Water flows from 3 and 4 to 2, from 2 and 5 to, from to 6 and from 6 to 7. The network has 6 pipes and 7 nodes. ode will be removed from the water network model. III. 2.. a. Mathematics The network model can be represented usin the enery and the continuity equation and the pipe component law as described on pae 26. The continuity equation for nodes: (25) nod q q r r = Λ Λ = p p p p p p ; = 6 q q q M r ; = nod nod nod q q q 6 M r. The row for 7 was excluded because of the dependency

39 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. (26) nod nod q = q with =7. 7 i= i The enery equation ives the relationship between the total head at the nodes and the head loss in the pipes: (27) h r = Λ T h r h r h = M ; h 6 h r h = M. h 6 The pipe component law ives the relationship between the head loss and the flow in the pipes: (28) h r v q = Q( h) = 6 h e3 e3 6 sin( h ) M. sin( h ) 6 r r, Linearisation of (26) with (28) around the workin point ( ) nod h, q linearised water network model leads to the (29) r r dq( h) Λ r d h 442 r 4h 43 A r T r Λ δ h = δq nod with r r r δ h = h h and r r r nod nod, δ q = q q. The JacobianA of the linearised model requires the linearised pipe conductance, equation (8): 38

40 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p (3) = = ) ( h e h e h e h d h dq e e e h L O O M M O L L O O M M O L r r r r. This yields A from equation (25) as (3) = A. The linearised network model: (32) nod q h A r r δ δ = nod q h r r δ = δ It can be seen that the matrix is symmetric with a dominant diaonal. The diaonal elements are the neative sum of all other row or column entries. ow the node will be removed by Gauss-Elimination. This means, the first row and column of the equation system (32) will be eliminated. So the matrix part in the bottom riht of (32) will remain. With (33) 5 4 f + + = this solvin aainst h δ results to:

41 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. 4 (34) ( ) r r nod r nod nod nod nod nod nod nod nod r r q q f q q f q q q q f q h A f f f f f f f f f ), ( ) ( ) ( δ δ δ δ δ δ δ δ δ δ = with (r) A and nod r q ), r ( δ as marked above. The properties of (r ) A are summarised in Ulanicki et al (996). ow a non-linear model is recovered from the reduced model (34). Ulanicki et al (996) writes, that a non-zero entry at a position (i,j) in the matrix (r ) A indicates a pipe between the nodes i and j. The new linear pipe conductance for this pipe is ) (, ) (, r j i r j i A =. The new non-linear pipe conductance ) (, r j i can be calculated usin the relation (35) ) (, 3 ) (, 4 r j i e j i r j i h h e =, correspondin to (8). The followin new network topoloy corresponds to (r ) A (equation (34)): 3 ' 5 4 ' 6 7 ' 2 P 3 P 2 P 8 P 9 P 6 P Fiure. The new etwork Structure after removin.

42 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. III. 2.. b. Interpretation Removin of ode resulted in the new pipes p 8 between 2 and 5, p 9 between 5 and 6 and p between 2 and 6. The nodal demands in the odes 2, 5 and 6 were chaned. So the star pipe structure around was exchaned with a trianle pipe structure between 2, 5 and 6. The simplified model has 4 pipes and 5 nodes. III Pipes in Series ow, two pipes in series will be unified. In Fiure, the source is 5. The flow direction is from 5 throuh 3, and 2 to 4. 5 P 3 3 P P 2 P Fiure. etwork Model of Pipes in Series. The node will be removed from the system. Therefore, the pipes P and P 2 will be unified. 4

43 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p III a. Mathematics As the stratey of static simplification was described in detail in the last section, from now on only sinificant steps and results will be explained. The branch incidence matrix Λ : (36) Λ = 4 4 p p M M L L. The row of node 5 was excluded because of the dependency, equation (4) above. The Jacobian matrix A evaluates to (37) = A. Removin leads with 2 f + = to (38) + + = ) ( ) ( ) ( ) ( ) ( f f f f A r and + + = nod nod nod nod nod nod r q q f q q f q q ), ( δ δ δ δ δ δ r. The followin new network topoloy can be read from A (r) (38):

44 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. 5 P 3 ' 3 P 5 P 4 ' 2 4 Fiure 2. etwork Model of Pipes in Series, with one ode removed. III b. Interpretation After removin the node, the new pipe P 5 ets the linear conductance 2 5 =. f P 5 is the replaces P and P 2. The demand of is redistributed to the nodes 2 and 3. III Parallel Pipes ow, two parallel pipes will be unified. The source in the network model is 4, from there, the water flows to and 5. From, the flow passes throuh two parallel pipes to 2 ; from 2, the flow oes to 3. 43

45 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p P 3 P P P 4 P 5 Fiure 3. etwork Model with Parallel Pipes. III a. Mathematics The node branch incidence matrix Λ : (39) Λ = 4 5 p p M M L L L. The demand of 5 will be calculated by the dependcency for the demand, similar to equation (4). P and P 2 have the same entries in Λ. This denotes from their identical start and end nodes. The Jacobian matrix A results to: (4) = A.

46 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. The linear pipe conductances of P and P 2 were simply added in A, so no Gaussian Elimination is needed. Therefore, the new network topoloy can be read directly from the Jacobian matrix A: 5 P 5 3 P 4 P 6 2 P 3 4 Fiure 4. etwork Model with Unified Parallel Pipes. III b. Interpretation The linear pipe conductances and 2 are added when the Jacobian matrix of the model is calculated. The sinle pipe P 6 linkin and 2 has the linear conductance = III Trees In this section, a tree in a network model will be removed with the static simplification method. The source in the network model illustrated below is the node 4. From there, the flow oes to 3 and from 3 to 5 and to via 2. The network model has 5 nodes and 4 pipes. The pipes P and P 2, which form the tree, will be removed. 45

47 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p P 3 P 4 5 P 2 2 P Fiure 5. A Water etwork Model with a Tree. There are two different methods to do this: addin the demands directly with static simplification Firstly, the alorithm for addin the demands directly will be explained. III a. Addin the Demands The head in is a function of the head in 2 and the flow in P, which equals to the demand in (see equation (2)): nod (4) h ( h q ) h 2, =. The head H 2 is a function of the head in 3 and the flow throuh P 2, equal to nod nod q q2 + : nod nod (42) h ( h q q ) h2 2 3, + 2 =. The head h 3 in 3 does not depend on the head of 2. The only influence from 2 and results from the demands nod q and nod nod nod (43) h ( q + q q ) h3 3..., =. nod q 2 : So, the demands of and 2 can be added to the demand of 3 : 46

48 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p (44) nod nod nod new nod q q q q 3 2, = without loosin accuracy. ow, and 2 can be deleted. The network topoloy after removin the branch: P 3 5 ' 3 4 P 4 Fiure 6. Water etwork Model with removed Tree. III b. Static Simplification In this section, the tree of the network model in Fiure 5 will be removed with the static simplification alorithm. The node branch incidence matrix Λ evaluates to: (45) Λ = - P P P P The row for 5 is excluded from Λ because of the dependcency (4). The Jacobian matrix A results to: (46) = A. ow, the nodes and 2 will be removed. Hence, the Jacobian matrix A (r) of the reduced model is: (47) + = ) ( A r.

49 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. The reduced demand vector evaluates to: nod nod nod r nod,( r ) q + + q2 q3 (48) q = nod. q4 As shown above, the demands are added without bein weihtin with the pipe conductances to the start node of the tree, here 3. The network structure, which can be derived from the Jacobian matrix of the reduced model (47), is identical to the one in Fiure 6. III Low Conductance Pipes Elimination of low conductance pipes is not directly possible with the static simplification approach that is carried out here, because the approach is based on relinkin selected nodes. So, these pipes will be simply deleted from the network model. Low conductance pipes can be removed from the network model before linearisin as well as after removin nodes with the Gaussian elimination procedure. Static simplification splits the connections of a node to connections between the nodes, to which it is coupled. These new pipes conductances between the remainin nodes may be much lower than the lowest pipe conductance in the oriinal network. Therefore, the conductance of a low conductance pipe miht increase as well, when the connections around a node are redistributed. Hence, it makes more sense to delete low conductance pipes after the simplification, so as much as possible from the structure of the oriinal network will remain and new low conductance pipes will be eliminated. Low flow in a pipe is not only the consequence of a low pipe conductance, but also of a bi head loss. Therefore, the head loss should be rearded as well. The linearised pipe conductance i reflects the head loss, so it will be the classification variable in this work, instead of the non-linear pipe conductance i. With the linearised pipe conductance of the static simplification alorithm, there are two possible decision criteria to identify low conductance pipes: 48

50 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. A fixed absolute linear pipe conductance level for the whole water network model. A linear pipe conductance level relative to the smallest linearised node conductance of start and end nodes of the pipes. Both classification criteria were implemented in this work. They were interated in the proram module, which links the remainin nodes after simplifyin. The second decision criterion did sometimes not remove all pipes, which caused solver errors. These solver errors appeared unpredictable in the simulations and were sometimes half of the order of the available head in the nodes. Further, the normal errors caused by this criterion were enerally much hiher than the ones of the first classification criterion. So the fixed absolute linear pipe conductance level as decision criterion for low conductance pipes is much better than the relative one. III. 3. Implementation In this section, the modellin environment will be briefly explained. Afterwards, the activities performed in the alorithm to identify the simplification rane and in the static simplification alorithm will be discussed. III. 3.. Modellin Environment The application for the static simplification is called SpeedUp in this report. It is interated with the eoraphic information system StruMap (Structural Technoloies LTD. (996)). StruMap and SpeedUp communicate via the application prorammer s interface, API, of StruMap via an interface in C++. SpeedUp was developed with the C++ Builder (i.e. Calvert (997)). The C++ Builder compiles the source code of SpeedUp and the interface to the API of StruMap. Further, for representin matrices in SpeedUp, a matrix library is used. 49

51 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. C++ C++ Proram Code C++ Interface Proram Code C++ Builder SpeedUp interated in StruMap StruMap API Fiure 7. The Modellin Environment. The parts of the modellin environment will be briefly explained now. III. 3.. a. StruMap The Centre for Water Systems of the School of Enineerin and Computer Science uses the modellin environment StruMap from Structural Technoloies Limited. It provides an environment to work with data contained in a map: it is a eoraphic information system. To simulate water networks, StruMap comes with a solver for water network models, HARP. Attributes of Map items can be manipulated with a builtin expression evaluator, an interpreter lanuae prorammin interface. StruMap comes as a set of library files. The functions in them are accessible via an application prorammer s interface, API, to C. So external prorammers can add custom functions to StruMap. The API is described in Structural Technoloies (996). The API itself is written in C. The StruMap version, which is used for this work, is StruMap 2 Release for Microsoft Windows 95. 5

52 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. III. 3.. b. C++ Interface to StruMap Roer Atkinson of the Centre of Water Systems has written an excellent interface class for the StruMap API. This class encapsulates the functions of the API as methods. It was extended in this work. III. 3.. c. Matrix Library To represent the Jacobian matrix A in the proram, the data structures of Press et al (992) were used. The used functions allocate memory for the matrix and free it. III. 3.. d. Prorammin Environment The alorithms for this work were coded in C++ with the Borland C++ Builder Professional Version 3. (Build 3.7) from Borland International. As C++ dialect, the Borland version was used. It provides a rane of functions, classes and methods specifically desined for Microsoft Windows. For documentation about the C++ Builder, see Calvert (997). III The Alorithm This section illustrates the alorithms for the simplification preparation and static simplification, which were developed in this work. III a. Simplification Preparation Fiure 8 shows the activity diaram for the simplification preparation. The simplification preparation consists of three main sections: the deletion of closed valves, the protection of untouchable nodes with untouchable pipes and the followin protection of untouchable pipes with non-removable nodes. 5

53 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Start Is there a closed valve in the model? Yes o Delete the pipes of the closed valve. Delete the closed valve. Is there a non-touchable node with not yet marked pipes? Yes o Mark all its pipes as non-touchable. Is there a non-touchable pipe with not yet marked nodes? Yes o Mark all its nodes as non-removable. Stop Fiure 8. Activity Diaram for the Simplification Preparation. 52

54 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. The simplification preparation will delete closed valves, because there is no flow throuh them and therefore no headloss in their pipes. Pipes without headloss have a zero linear pipe conductance (see equation (8) above). Pipes with a zero linear pipe conductance cannot represent a link in the Jacobian matrix A, as a zero entry in A denotes no connection between the nodes. So network components without flow will be excluded from simplification. The protection of untouchable nodes and afterwards the protection of untouchable pipes is carried out as described in section III.. 2. above. III b. Static Simplification The activity diaram for the static simplification alorithm is plotted in Fiure 2 and Fiure 2 below. Several activities are independent from each other, so they can be carried out in parallel. These activities are placed between two horizontal black bars. Firstly, a list of all nodes of the water network model has to be made. All nonremovable nodes, which will remain after simplifyin, are put at the end of the list. This is advantaeous, because the rows of the Jacobian matrix A will then not need to be sorted. Also, the node index for the rows and columns of A will be the same. Likewise, a list of all pipes is required. Their start and end nodes need to be identified and their linear pipe conductance needs to be calculated as well. To bypass possible flow errors of the solver, the linear pipe conductance is not calculated in terms of flow and head loss, but in terms of the pipe attributes lenth, diameter, Hazen-Williams coefficient and head loss. The Jacobian matrix needs to be initialised with a size of ( ) ( ), because of the dependcency for the demands as explained in III... a. on pae 25. All matrix elements have to be initialised with zero. ow, the linear pipe conductances can be placed in the matrix A. This is done by substractin them from their matrix elements: 53

55 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. (49) A n, m : = An, m n, m and = with n, m. A m, n : Am, n n, m Thereafter, the linear node conductances, which form the diaonal band of the matrix, can be calculated. This can be done with equation (9): n = ˆ k, n n ( ) k n, k n =. Before the Gaussian elimination can be performed, the demand vector has to be built accordin to the node index for the matrix A. ow, the Gaussian elimination can be applied on the Jacobian matrix A and the demand vector. The Gaussian elimination was implemented in this work as follows: // SIMPLIFY: Gauss-Elimination ======================================== for (int i=; i< numodestoremove; i++) { // iterate throuh all remainin rows, jacii!= for all rows for (int j=i+; j<=jac_size; j++) // is there somethin to do in the current row? if (jacji!= ) { double mult= -jacji / jacii; // mult is a multiplication constant, which is used at least // three times jacji = ; // iterate throuh all columns for (int k=i+; k<=jac_size; k++) // is there somethin to do in the current column? if (jacik!= ) { // add (mult * jacik) to jacjk =============== jacjk += mult * jacik; }; // the followin lines allocate the demand objects in memory demandtype demand = (demandtype)demandlist->itemsi; demandtype demand = (demandtype)demandlist->itemsj; // add the demands ========================================= demand->adddemand(demand, mult); }; // end of row iteration }; // end of column iteration The Jacobian matrix A is in the code above simply called jac. i is the index of the current diaonal element and j is the index for the actual row in jac. k is an index for the column, in which additions are carried out. The syntax of C++ is described in Calvert (997). 54

56 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. A = j O L L jac[ i ][ i ] M M k jac[ i][ i] M jac[ j][ i] L L jac[ i][ k] M jac[ j][ k] Fiure 9. Structure of the Jacobian matrix A. As the matrix A is a sparse one, the alorithm always tests firstly, if there is somethin to do: when jac[j][i] is zero, the alorithm inores this row, because there is nothin to eliminate. If jac[i][k] is zero, the alorithm inores the column k. The solvin alorithm was not further optimised, because the runnin time spent with Gaussian elimination is very small compared to the hard disk access time, which is needed to write the chaned demand vectors. After the Gaussian elimination is finished, all pipes, except of the untouchable ones are dispensable and can be deleted. ow, the untouchable pipes need to be removed from the matrix A. This is done by addin their linear pipe conductance to the correspondin matrix position in A: (5) A n, m : = An, m + n, m and = with n, m. A m, n : Am, n + n, m If the matrix element A, is still lower than zero, an additional pipe parallel to the n m untouchable one will be inserted automatically later on to correct the flow. ow, all nodes, except of the non-removable ones can be deleted, as they are no loner required. The demand of the non-removable nodes can be written back. 55

57 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Start Make a list of all nodes. Put the non-removable at the end. Make a list of all pipes. Identify the start and end nodes, calculate the linear pipe conductance. Initialise the jacobian matrix A with - columns and rows. Place the linear pipe conductances in the jacobian matrix A. Build up the demand vector. Calculate the diaonal band of A (node conductances). Apply Gaussian elimination r times on A and on the demand vector. Fiure 2. Part of Activity Diaram for the Static Simplification. Finally, the remainin nodes need to be reconnected with the pipe structure in A (r). A (r) is a submatrix in A from the position ( r + ) ( r + ) to ( ) ( ). If there is a non-zero linear pipe conductance entry in A at the position (m, n), which meets the writeback criteria, a pipe need to be inserted in the reduced model between the node m and n. There are two writeback criteria: 56

58 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Relative writeback criterion. The linear pipe conductance A n,m is bier than its lowest linear node conductance multiplied by a factor rfact: (5) A > rfact min( A,, A, ) n, m ( r + ) K ( ). n, m n n m m Absolute writeback criterion. The linear pipe conductance A n,m is bier than the lowest linear pipe conductance of the oriinal network model multiplied by a factor afact: (52) A > afact min({ A, }) n, m ( r + ) K ( ) and n, m i j i,2, K,( ) and = j =,2, K,i. The lowest linear pipe conductance of the oriinal network model was chosen as a scale. This allows a comparison relative to the linear pipe conductance of the oriinal network model. After simplification, some statistics are displayed to compare the simplified with the oriinal network model. 57

59 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Delete all pipes, except of the non-touchable ones. Delete all nodes, except of the non-removable ones. Iterate throuh the non-touchable pipes. Add their linear pipe conductance to the correspondin matrix pos. in A. Write all demands. Iterate throuh the linear pipe conductances in A (r). Iteration finished o Fulfils the linear pipe conductance at the current matrix position the writeback criterions? Yes Insert a new pipe in the reduced model correspondin to the matrix pos. and the lin. pipe conductance. Display statistics. Stop Fiure 2. Part 2 of Activity Diaram for the Static Simplification. 58

60 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. III. 4. Example In this section, the example network from above with the identified simplification boundary (Fiure 8, pae 35) in Chapter III.. 2. will be simplified with the static simplification alorithm. Low conductance pipes will not be eliminated, as the main intention of the example is to illustrate static simplification. III. 4.. Procedure E CF Reservoir, H=m CF D A, B Valve, opened Simplification Rane CF F C Demand ode, q nod =5 l/s on-removable Demand ode Pipe, l=5m, d=mm C= CF: Head/ Flow monitored for Cost Function Fiure 22. The Example etwork with marked Simplification Areas. In this network model, the static simplification will carry out the followin simplification steps: Table 5. Simplifications Steps in the Example etwork. Simplification Step Marked Area in Fiure 22 above Unification of Series Pipes A Deletion of a Tree Structure B Unification of Parallel Pipes C ode Replacement D, E, F 59

61 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. The simplification areas of these steps are illustrated in Fiure 22. The resultin network is shown in Fiure 23. The static simplification replaces the network structures in the area A, B, C and F with one direct pipe. The demand of the removed nodes is added to the start and end node of this pipe. The star structures in the areas D and E are replaced with two trianle structures. The demand of the removed nodes is redistributed to the 4 corner nodes of the trianle structures. CF Reservoir, H=m CF Valve, opened Simplification Rane CF on-removable Demand ode Pipe, C= CF: Head/ Flow monitored for Cost Function Fiure 23. The simplified example network. The simplified network model has 2 pipes and 9 nodes. Its number of loops is 4. The oriinal network model has 7 pipes and 4 nodes. So the simplification will reduce the number of nodes to 9 and the number of pipes to 2, but the number of loops will stay the same. 6

62 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. III Proram Test As a test, the simplification of the example network will be carried out with the static simplification alorithm in StruMap. The screenshot below shows the heads and flows in the example water network model. A neative flow in a pipe indicates that the end node of a pipe has a hiher head than its start node. Fiure 24. Screenshot of the Oriinal Example etwork 3. III a. Preparations The simplification preparation identifies the boundary of the simplification area. Its output: SIMPLIFICATIO PREPARATIO STATISTICS BEFORE =============================================== odes: 4 non-removable: 2 Pipes: 7 Loops: 4 3 The numbers without variable name and unit are the flows l/s. 6

63 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9.. of W 4 Components: 3 The pipes represent 8 % of all possible links. The loops represent 5 o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 4 non-removable: 9 Pipes: 7 Loops: 4. of W Components: 3 The pipes represent 8 % of all possible links. The loops represent 5 o/oo of all possible loops. The network had before the simplification preparation only 2 non-removable nodes, whose head is output for the cost function. After the preparation, it has 9 non-removable nodes. Untouchable nodes are included in this number. III b. Simplification ow the example network model is solved at the workin point and the calculated heads and flows are assined to the network model components. Afterwards, the network model is simplified. The output of the static simplification alorithm: LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 3x3. Readin pipes back. had a linear branch conductance lower than the draw-back-criteria and were not written back. STATISTICS BEFORE =============================================== odes: 4 non-removable: 9 Pipes: 7 Loops: 4. of W Components: 3 The pipes represent 8 % of all possible links. The loops represent 5 o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 9 non-removable: 9 4 W stands for network. 62

64 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Pipes: 2 Loops: 4. of W Components: 2 The pipes represent 32 % of all possible links. The loops represent 37 o/oo of all possible loops. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: 4 4 % W Components: % The Jacobian matrix A has the dimensions 3 3. The number of nodes after simplification is nine, so only the non-removable nodes have stayed. The new water network model has 5 pipes less than the oriinal one. The number of loops has not chaned there are still 4 loops. These results are the same as predicted in the section above. Fiure 25. The Simplified Example etwork 5. The structure of the simplified network model in Fiure 25 is similar to the proposed one in Fiure 23. The heads in all the nodes are the same than the ones in the oriinal 5 The black point in the bottom left ede in the screenshot is the cursor. 63

65 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. network model in Fiure 24 above. The flows in the untouchable pipes have stayed the same as well. III. 5. Summary This chapter introduced the static simplification alorithm of Ulanicki et al (996) and showed, that the network model component based simplification approaches node unification, parallel and series pipes unification and tree elimination are carried out by the static simplification alorithm. The static simplification alorithm was enhanced with two methods to eliminate low conductance pipes after simplifyin the network model. As well, a method was presented to identify the simplification rane in a network model. These are the main extensions of Ulanicki et al (996). Then, the alorithm for the proram was presented in form of activity diarams. Finally, a simplification example was iven with the static simplification alorithm in this work. 64

66 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. IV CASE STUDIES This chapter describes the simplification procedure and the results for two lare water network models. The model simplifications will be carried out for several cases with the absolute and the relative writeback criterion. Further, the solvin time of the oriinal water network models and the simplified ones will be benchmarked. IV.. Skipton Water etwork Model IV... Brief Description Skipton is a rural town situated in orth Yorkshire, United Kindom. Yorkshire is an Enlish shire next to the boundary of Scotland. Skipton is situated on the south boundary of the Yorkshire Dales, approximately 35km north-west of Leeds. 65

67 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Fiure 26. Skipton. The Skipton water supply network model consists of about nodes and approximately 3 pipes. The number of loops is rouhly 2. A screenshot of the Skipton water network model can be found in chapter II... on pae. The network contains the followin special network components: 2 Variable Head Reservoirs (StruMap Property R_VHR ). Floatin Valve (StruMap Property V_FL ). 42 Motorised Throttlin Valves (StruMap Property V_MTV ). 36 of them are totally closed and six are totally opened. For a cost function, the flow in some pipes, as well as the heads of some nodes, will be chosen. As this report covers only the simplification alorithm to minimise the simulation time, the network components above will be chosen randomly. IV.. 2. Preparations All special network such as reservoirs, valves and sources were marked as untouchable, as well as those pipes, whose flow will be used in the cost function: pipe 246 and 862. The nodes for the cost function were marked as non-removable : node 723, 4 and 924. The number corresponds to their attribute UID in StruMap. 66

68 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. ow the simplification preparation routine was started. Its output: SIMPLIFICATIO PREPARATIO STATISTICS BEFORE =============================================== odes: 9 non-removable: 3 Pipes: 295 Loops: 25. of W Components: 2386 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 9 non-removable: 34 Pipes: 295 Loops: 25. of W Components: 2386 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. The number of non-removable nodes increases from 3 to 34 as the simplification area is identified. This hih number denotes of the numerous valves in the network model. After the simplification preparation, the network was solved for a snapshot at 8 o clock and the heads and flows were imported as the linearisation point for the network. The network was saved and SpeedUp restarted. IV.. 3. Simplification The simplification procedure was run for the followin cases:. without writeback criterion Three times with the followin parameters with the relative writeback criterion: And four times with the absolute writeback criterion: 67

69 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p The output of the static simplification routine can be found in the appendix, chapter VIII. 4., which starts on pae 6. Dependin on the level of the writeback criterion, the simplification procedure reduces the number of network components differently: Case Table 6. Comparison of the umber of etwork Components. odes Pipes Loops W Components Pipes/ possible links Ori. W % % % % % % % % % 32 loops/ possible loops The first column ives the case number, the next four columns contain the absolute numbers of nodes, pipes, loops and network components. The prefinal column refers the number of pipes to the possible number of network links as a percentae. The last column refers the number of loops to the possible number of loops as parts per thousand. Hence, the last two columns reflect the link density of the network model. The maximum number of links in a water network model was calculated as described in Appendix VIII. 3. on pae 4. Fiure 27 shows the new numbers of network components relative to the number of network components in the oriinal water network model: 6 The lowest linear pipe conductance of the oriinal network is kept in the simplified one. It is seen as a lower limit to which small chanes are allowed. Therefore, if the solver has no problems with the oriinal network, it will have no problems with the simplified network, also. 68

70 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p % 6.% 3.% 4.% 25.% 2.% 2.%.% 8.% 5.% 6.%.% 4.% 5.% 2.%.% odes 2.28% 2.28% 2.28% 2.28% 2.28% 2.28% 2.28% 2.28% Pipes 3.58% 23.94% 2.93% 2.62% 23.7% 22.47% 22.% 2.46% W Elements 22.76% 8.6% 7.52% 6.8% 8.9% 7.8% 7.56% 6.72% Loops 34.63% 86.34% 73.66% 65.37% 8.46% 77.7% 74.5% 64.39% Case.% Fiure 27. umbers of etwork Components after Simplification 7 compared to the ones before for the Skipton Water etwork Model. As expected, the number of nodes is reduced to the same value in each simplification. The number of pipes in the simplified network model varies, their number depends on the heiht of the relative and absolute writeback criteria. All simplification approaches with writeback criteria reduce the total number of pipes more than without writeback criterion. In the simplification with all pipes (case ), the number of loops is about / 3 hiher than in the oriinal network model. With writeback criteria, the number of loops is between ¼ and / 3 lower than in the oriinal network model. The numbers of network components et reduced to 23% for all pipes and down to 7% for the hihest relative and absolute writeback criteria. 7 The percentae of loops refers to the secondary value axis, all others to the first one. 69

71 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Fiure 28. The network after Simplification, all linear pipe conductances were written back. The simplified network model with all pipes in Fiure 28 has a totally different appearance than the oriinal network model. The models, which were simplified with writeback criteria, have the same appearance, but their pipe networkin is less dense. IV.. 4. Errors at the 8: Snapshot ow, the errors of the simplified models at the 8: snapshot will be discussed. After simplification, only the non-removable nodes of a network model and its untouchable pipes will stay the same. Therefore, the head in the non-removable demand nodes and flows in the untouchable pipes will be analysed here. To facilitate the mappin between the heads and flows, the nodes and pipes are sorted in ascendin order accordin to their label ( UID ) in StruMap. This results in a roupin effect, because network components, which are eoraphically close, have also close numbers. Hence, the errors of flow in both pipes, which are connectin a valve, appear toether in the followin fiures. 7

72 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Firstly, the results of the simplification with all pipes will be compared. Then, the absolute errors produced by the relative writeback criterion will be viewed, followed by the ones of the absolute writeback criterion. Finally, the relative errors of all performed simplifications will be compared. IV.. 4. a. Simplification with all Pipes Absolute Error of Head, Simplification with all Pipes Absolute Error of Head [m] on-removable Demand ode Index Fiure 29. Absolute Error of Head, Simplification with all Pipes. The maximum absolute error of the head is.63m and the standard deviation is.3m. Hence, these results are very acceptable. 7

73 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Absolute Error of Flow, Simplification with all Pipes,6,4,2 Absolute Error of Flow [l/s] -,2 -, ,6 -,8 Untouchable Pipe Index Fiure 3. Absolute Error of Flow, Simplification with all Pipes. The maximum flow error is.55 l/s, the standard deviation is.3 l/s. The values above will be further analysed relative to the oriinal flow in the untouchable pipes to identify hih relative errors. Relative Error of Flow, Simplification with all Pipes 3,% 2,%,% Relative Error,% ,% -2,% -3,% Untouchable Pipe Index Fiure 3. Relative Error of Flow, Simplification with all Pipes. 72

74 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. The relative errors of flow are mainly beyond ±%, there is one peak with 23.7% in the left third of the fiure above, one in the riht third with %. These extreme peaks appear in the untouchable pipes with the lowest flow of all: Table 7. Pipes with the hihest relative error. Label (UID) Oriinal Simplified Abs. Error Relative Error Flow l/s Flow l/s Flow l/s % % % % It is very likely, that solver causes these errors. Except for these errors, the accuracy of the flows is very acceptable also. IV.. 4. b. Relative Writeback Criterion The followin fiures illustrate the absolute error of the simplified models with the relative writeback criterion. Firstly, the absolute error of the heads will be viewed. Absolute Error of the Available Head in the Demand odes Absolute Error [m] Demand ode Index Fiure 32. Absolute Error of the Available Head in the Demand odes for the relative cases. As expected, the models with a hiher relative writeback criterion show a hiher absolute error. The error appears unpredictable, as the peak of case 2 in the left hand side and the sequence of case 4 in the left middle of the fiure illustrate. These peaks are 73

75 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. extraordinary lare, compared to the all other absolute errors. Their size is about ten times bier. ow, the absolute errors of the flow will be rearded. Absolute Error of Flow in the Pipes 2,5 2,5,5 Flow [l/s] -, ,5-2 -2,5-3 Untouchable Pipe Index Fiure 33. Absolute Error of Flow in the Pipes. The absolute errors of flows for hih relative writeback levels are partially of the same size as the flows in the pipes themselves. Aain, the errors seem to appear randomly, as it can be seen when case 2 is compared to case 4. 74

76 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. IV.. 4. c. Absolute Writeback Criterion Absolute Error of the Available Head in the Demand odes.5..5 Absolute Error [m] Demand ode Index Fiure 34. Absolute Error of the Available Head in the Demand odes. With the absolute writeback criterion, the absolute errors of the heads are of manitudes smaller than with the relative writeback criterion. Except for case 8, the errors are enerally all within ± 5 cm. The latter writeback criterion shows errors, which are mainly below ± cm. The larest ones are around ±. m, a very ood value. The errors seem to increase slihtly with the level of the absolute writeback criterion. Absolute Error of Flow in the Pipes,3,2, -, -, Flow [l/s] ,3 -,4 -,5 Untouchable Pipe Index Fiure 35. Absolute Error of Flow in the Pipes. 75

77 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. The absolute error of flow lies mainly beyond ± 5 ml/s, except for the errors of case 8, where the absolute error is mostly beyond ± 25 ml/s. As the absolute errors of the available heads, the absolute errors of flow increase with the level of the writeback criterion. IV.. 4. d. Comparison of Maximum and Standard Deviation of the Absolute Value of the Absolute Errors The followin table shows the maximal values of the absolute values of the absolute errors for the simplifications. Table 8. Maximal Errors and Std. Deviation for the Heads in the Simplifications Demand odes ( Absolute Error of Available Head, m) Max Std Dev Pipes ( Absolute Error of Flow, l/s) Max Std Dev The maximal errors are enerally much hiher for the relative writeback criterion then for the absolute one. The maximal absolute error of flow is lower than the absolute error of flow (for case 5, 6 and 7) in the simplified network with all pipes. This may result from the absence of the pipes with the lowest conductances in the simplified network with the mentioned writeback criteria. The same phenomenon is true for the standard deviation of the absolute error in the pipes of case 6. IV.. 4. e. Comparison of Relative Errors Firstly, the relative errors of the heads will be compared. 76

78 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. 3,5% 3,% 2,5% 2,%,5%,%,5%,% Max,7% 4,6% 7,39% 95,78%,2%,37%,4%,5% Std Dev,%,55% 2,92% 34,%,3%,6%,7%,8% Simplification Case Fiure 36. Comparison of the Relative Error of the Head in the odes: Maximal Value, Standard Deviation. For the relative writeback criterion, the maximal relative errors are all hiher than %, their standard deviations are enerally hiher than %. For the absolute writeback criterion, the maximal absolute errors are except for the one of case 8 with.5%, lower than.5%, which is very ood. The standard deviation for the absolute errors of the absolute writeback criterion is for all simplifications below.%, an excellent result. ow, the relative error of the flow will be compared. 77

79 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Table 9. Maximal Relative Error and Maximal Relative Std. Deviation for the Flows of the simplified etwork Models. case Max % % % % % % 563.4% % Std Dev 98.4% 99.7% 242.7% % 98.4% 83.6% 82.98% 98.3% The hih maximum values and the hih standard deviations result from the four pipes (see Table 7), which have a very low conductance and therefore a very low flow: Table. Pipes with very low Flow in the Simplified etwork Models. case Or. W Label Flow l/s This cause could derive from a solvin error. In further analyses, these four pipes will not be included. The followin chart was enerated without the mentioned pipes: 4,% 35,% 3,% 25,% 2,% 5,%,% 5,%,% max 2,93% 66,48% 5,77% 97,7% 3,75%,%,7% 9,38% std dev,48% 3,% 25,45% 49,%,75%,74%,6% 2,7% Simplification Case Fiure 37. Comparison of Relative Error of the Flow in the Pipes: Maximal Value, Standard Deviation. 78

80 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. It can be seen that the maximal relative errors of flow are more than ten times as worse with the relative writeback criterion as with the absolute one. The maximum relative errors with the absolute writeback criterion are around %, the standard deviation is about 3 to 4 times as hih as in the simplified model with all pipes. IV.. 4. f. Conclusions As the errors were much worse usin the relative writeback criterion, rather than usin the absolute one, the absolute writeback criterion shall be iven preference for this water network model. With the absolute writeback criterion, there is a minimal rowth of error in the head, the error of flow increases more. IV Hour Simulation IV.. 5. a. Absolute Errors The followin table ives a brief overview over the maximal errors and the standard deviation of the Skipton water network model in a 24h simulation. Table. Absolute maximal Errors and Standard Deviation of the Skipton Water etwork Model in a 24h Simulation. maximal Errors Standard Deviation Case Head in the odes m Flow in the Pipes l/s P P All the maximal absolute errors in terms of heads for the first node, node 723 are all below.8m. ode 924 has maximal absolute errors below.5m. ode 4 has the smallest ones: the absolute errors are smaller than cm. Case 5 has the same maximal absolute errors than case. Case 8 is aainst case still acceptable, the errors did not et much bier with the hihest absolute writeback level. 79

81 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. The pipe 862 shows no errors at all. This pipe supplies the whole area on the riht hand side in Fiure on its own. Therefore, all demand of its supply area is summed up at its end. ow, the absolute error of the heads 723, 924 and 4 and the absolute error of the flow in pipe 246 will be investiated. Available Head at ode Available Head [m] Hour Oriinal etwork 5 8 Fiure 38. Available Head at ode 723 durin 24 hours. The available head is the same at 8: for all cases. It differs the more, the available head is away from the available head at the workin point. The curves for the cases and 5 are identical. 8

82 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Absolute Error of the Available Head in ode Available Head [m] Hour 5 8 Fiure 39. Absolute Error of Available Head in ode 723 durin 24 Hours. The absolute error of available head vanishes for all cases at 7: and 8:. As seen above, the errors of the cases and 5 have exactly the same pattern. The error of case 8 is only a little bit hiher than the ones of case and 5. Available Head at ode Available Head [m] Hour Oriinal etwork 5 8 Fiure 4. Available Head at ode 924 durin 24 hours. 8

83 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. The available head at node 924 follows from : to 8: same curve, but at 9: and 2:, the available heads of the simplified network are all hih above the available head of the oriinal network model. Absolute Error of the Available Head in ode Available Head [m] Hour 5 8 Fiure 4. Absolute Error of Available Head in ode 924 durin 24 Hours. The absolute error of available head follows the same pattern for all three cases. There is a stron peak at 9:, the absolute error peak is less than 5% of the demand at 9: at node 924. This peak may result from different demands in the evenin in the northwest of the water network compared with its rest. 82

84 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Available Head at ode Available Head [m] Hour Oriinal etwork 5 8 Fiure 42. Available Head at ode 4 durin 24 hours. The available heads at node 4 are identical with the available heads for all three cases. Absolute Error of the Available Head in ode Available Head [m] Hour 5 8 Fiure 43. Absolute Error of Available Head in ode 4 durin 24 Hours. As seen in Table above, the absolute error of ±7cm is the lowest one in the three observed nodes. The absolute error of case 8 is far bier than the ones of the other two cases. 83

85 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Flow in Pipe Flow [l/s] Hour Oriinal etwork 5 8 Fiure 44. Flow in Pipe 246 durin 24 hours. The flow in pipe 246 is the same as the one of the oriinal network model for the cases and 5, the flow of case 8 lies always a little bit beyond the other flows. Absolute Error of the Flow in Pipe Flow [l/s] Hour 5 8 Fiure 45. Absolute Error of Flow in Pipe 246 durin 24 Hours. The absolute error in pipe 246 is almost zero for the cases and 5, but the absolute error of case 8 shows an offset between 25 and 3 ml/s. 84

86 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. IV.. 5. b. Relative Errors In this subsection, the relative errors in 24h simulations of the simplification cases, 5 and 8 will be viewed. Table 2. Maximal Relative Errors for the Simplification Cases, 5 and 8. Maximal Relative Errors Standard Deviation Case odes %.4%.46%.36%.36%.36% % 4.4% 4.35%.%.%.99% 4.5%.4%.8%.%.%.2% Pipes P 246.3%.3%.64%.%.%.4% P 862.%.%.%.%.%.% Althouh all relative errors lie beyond 5%, only the maximal relative errors for node 924 are bier than.5%. The standard deviation is mainly between and % for the investiated network components. The relative error in pipe 862 is zero, as seen above. IV.. 5. c. Conclusions IV.. 6. Benchmarkin To determine the solvin time reduction when simplifyin, the followin models were run 5, and 2 times at the 8 o clock snapshot: The full model. The simplification cases, 5 and 8. Due to their comparatively hih errors, a simplification case with a relative writeback criterion was not analysed here. The benchmarkin was done by determinin the difference between start and end time for the numbers of runs mentioned above. The solvin process includes writin the results to the hard disk. So the actual solvin time alone would be slihtly smaller than determined here. Table 3. Overview over the total solvin time. runs Full / s Case / s Case 5 / s Case 8 / s

87 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p Table 4. The solvin time per run: runs Full / s Case / s Case 5 / s Case 8/ s The relative solvin time is illustrated in Fiure 46 below. Comparison of Benchmark Results of Simplified etworks 3.% 29.% 27.% Percentae of Oriinal Run Time 25.% 23.% 2.% 9.% 7.% 5.% Case Case 5 Case % 23.8% 22.33% 28.7% 24.7% 22.3% % 24.7% 23.39% Fiure 46. Comparison of Benchmark Results of Simplified etworks. The total solvin time seems to increase slihtly with the number of runs. This may be due to the rowin output file on the hard disk. Over all, it can be said that the solvin time decreases to approximately one third of the oriinal one, when usin the simplified model with all pipes. With the writeback criterion, the solvin times decreases even further, to around one fourth and slihtly lower, dependin on the set limit. 86

88 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. IV.. 6. a. Conclusions A writeback criterion is a ood choice to increase solvin speed by further 5%. The solvin speed does not increase sinificantly for a hiher level of the absolute writeback criterion. IV.. 7. Summary In terms of errors in the available head, a moderate absolute writeback criterion increases the relative error only very slihtly, in terms of errors in the flows a little bit more. As well, the solvin time decreases further with an absolute writeback criterion. The relative writeback criterion was not able to remove low conductance pipes from the simplified water network models, which caused solver errors. IV. 2. Hawksworth Lane Water etwork Model in Guiseley IV. 2.. Brief Description Guiseley is a small town in Yorkshire, 4km north-west of Leeds. The Hawksworth Lane water supply network is situated in the part of the city west of the railway. 87

89 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Fiure 47. Guiseley. The Hawksworth Lane water network model has approximately 5 pipes and 45 demand nodes. It contains the followin special network components: Source (StruMap Property V_Source ). 2 Pressure Reducin Valves (StruMap Property V_PRV ). Closed Valves (StruMap Property V_SLUICE_CLOSED ). For a cost function, the heads in some nodes and the flow in some pipes will be chosen. This will be done randomly, as in section IV.. IV Preparations The nodes 722, 558 and 675 were marked as non-removable and the pipes 443 and 236 were marked as untouchable. The number corresponds to the attribute UID in StruMap. The mentioned special network components from above were marked as untouchable, as well. The simplification preparation routine returned the followin output: SIMPLIFICATIO PREPARATIO 88

90 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. closed valves have been deleted. STATISTICS BEFORE =============================================== odes: 46 non-removable: 3 Pipes: 59 Loops: 59. of W Components: 98 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 45 non-removable: 5 Pipes: 497 Loops: 48. of W Components: 947 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. The simplification preparation routine removed the closed valves, because their connections cannot be represented by the static simplification. This was discussed in chapter III a. above. The number of non-removable nodes increased from the 3 nodes for the cost function to 5. 4 of the new non-removable nodes are start and end nodes for the 2 untouchable pipes, 3 represent the special network components source and pressure reducin valves. The remainin 5 non-removable nodes protect the special network components from bein removed. A screenshot of the Hawksworth Lane water network model can be found in Fiure 48 below. 89

91 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Fiure 48. Screenshot of the oriinal Hawksworth Lane water network model. IV Simplification The simplification was done for the followin cases:. without writeback criterion 2 times with the absolute writeback criterion: 2. level: level: And 2 times with the relative writeback criterion: 4. level:.2 5. level:. 9

92 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. A screenshot of the Hawksworth Lane water network model simplified with case is shown below. Fiure 49. Screenshot of the simplified Hawksworth Lane water network model. The output of the static simplification routine can be found in the appendix, chapter VIII on pae 8. The absolute writeback criterion did not delete any pipe in the Hawksworth Lane network model in the cases 2 and 3. This is due to the hih reduction of network components. The relative writeback criterion removed in the 4 th case 8 pipes from the 3 remainin after simplification, in the 5 th case pipes, / rd 3 of the remainin ones. The followin table ives a brief overview over the reduction of the number of network components: Table 5. Comparison of the number of network components. Case odes Pipes Loops W Components Pipes/ loops/ possible loops 9

93 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. possible links Ori. W %,2, % % % 84 The followin chart ives an overview over the new numbers of network components relative to the ones of the oriinal water network model: 7.% 45.% 6.% 4.% 35.% 5.% 3.% 4.% 25.% 3.% 2.% 2.% 5.%.%.% 5.%.%,2,3 4 5 odes 3.33% 3.33% 3.33% Pipes 6.4% 4.43% 4.2% W Elements 4.75% 3.9% 3.7% Loops 33.33% 6.67% 2.5% Case.% Fiure 5. umbers of etwork Components after Simplification compared to the ones before for the Hawksworth Lane water network model 8. The number of nodes ets reduced to approximately 3% of the oriinal number of nodes for all cases. The number of pipes ets reduced to 6% in case,2 and 3 and to 4% in case 5. The number of loops ets reduced to 33% for the cases,2 and 3 and by further 7% to 7% for case 4 and by 2% to 2.5% in case 5. IV Errors In this section, the errors of the cases, 4 and 5 at the 8: will be compared. 8 The number of loops refers to the secondary value axis on the riht hand side of the fiure. 92

94 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. IV a. Errors in the on-removable odes The errors in the non-removable nodes in terms of head at the 8: snapshot are iven below. Table 6. Errors in the non-removable nodes. abs. Err. rel. Err. Label %.%.% 522 -E %.%.5% %.%.4% %.%.3% %.%.4% %.%.3% 679.%.%.% 69.%.%.% 69.%.%.% 722.%.%.% %.%.3% %.%.3% The hihest absolute errors are lower than one millimetre for case one, for case 4 and 5 lower than 3cm. The relative errors are zero for case and 4 and for case 5 beyond.5%. IV b. Errors in the Untouchable Pipes The followin table ives the flow errors at the 8: snapshot in the untouchable pipes. Table 7. Errors in the Untouchable Pipes. abs. Err. rel. Err. Label %.%.% 236.%.%.% %.% -.98% 955.%.%.% 956.%.%.% 958.%.%.% 959.%.%.% The relative errors are mainly zero, except of the relative error in the pipe 443 in case 5. Its absolute value is beyond %, which is still very ood. 93

95 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. IV Errors in the 24-Hour-Simulations In this section, the maximal relative errors and the standard deviation of the relative errors will be compared for 24-hour simulations of the resultin network models for the cases, 4 and 5. Table 8. Maximal Relative Errors and Standard Deviation of the Relative Errors of the Hawksworth Lane etwork Model. max. rel. error Std. Dev. Case odes 722.%.%.%.%.%.% 558.%.%.3%.%.%.% 675.%.%.4%.%.%.% Pipes 443.5%.8%.7%.%.%.2% 236.%.%.%.%.%.% The maximal relative error is for the cases, 5 and 5 well beyond.%, except for the pipe 443 in case 5. The standard deviation is beyond.5% for all nodes and cases. IV Benchmarkin The oriinal network and the model case studies, 4 and 5 were used in this section. As above in chapter IV.. 6., the solvin time is slihtly smaller than determined here, because of the writin to the hard disk. The followin table ives an overview over the total solvin time for 5, and 2 runs. Table 9. Solvin Time for different umbers of Runs for the Hawksworth Lane Water etwork Model. Runs Ori. W Model Case Case 4 Case From the table above, the relative solvin time in table Table 2 below was calculated: Table 2. Solvin Time per Run for the Hawksworth Lane Water etwork Model. Runs Ori. W Model Case Case 4 Case 5 94

96 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p The followin fiure illustrates the solvin times of the simplified models compared to the one of the oriinal model. Relative Solvin Time of the Simplified Hawksworth Lane Water etwork Models Solvin Time compared to the Solvin Time of the oriinal Water etwork 4.6% 4.4% 4.2% 4.% 3.8% 3.6% 3.4% 3.2% 3.% Case Case 4 Case 5 5 Runs 4.5% 3.73% 3.5% Runs 4.24% 3.86% 3.75% 2 Runs 4.34% 3.83% 3.78% Fiure 5. Relative Solvin Time of the Simplified Hawksworth Lane Water etwork Models. The solvin time does not decrease sinificantly for the cases 4 and 5 compared to case. It can be said, that the solvin time decreases to about 4 per cent for all runs and to slihtly less with the absolute writeback criterion. IV Summary The Hawksworth Lane water network model has in the current confiuration only 3 special network components. Therefore, the simplified water network models are very small. The absolute writeback criterion did not remove pipes with the iven decision level. The relative writeback criterion resulted with the tested levels with about / rd 3 less pipes than the simplified model with all pipes. The solvin time went down beyond 5% 95

97 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. of the one of the oriinal network model. It did not decrease much more with the absolute writeback criterion. IV. 3. Conclusions The solvin time ets reduced drastically in all cases for both water network models: it sank to / 3 and less for the Skipton Water network model and to approximately /25 for the Hawksworth Lane water network model. The number of network components of the simplified network models and therefore their solvin time is depends on the number of special network components and on the output vector for the cost function. The relative writeback criterion did not remove all pipes in the simplifications of the Hawksworth Lane network model, which caused solver errors, but the absolute writeback criterion did so. The 24h simulations were except for the worst cases all very well. To ensure that possible errors will not affect the cost function, the errors of the heads and flows for the output vector should be rearded for the worst settins of the input vector. 96

98 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. V SUMMARY AD COCLUSIOS V.. Summary This report has described and analysed methodoloies to simplify water network models for simulation purposes. To do so, it was structured in three major chapters, the problem formulation, the development of a simplification alorithm and case studies. The problem formulation introduced briefly the purpose of water network models and their network topoloy. It explained the equations that describe the behaviour of the main water network model components, nodes and pipes. ext, it defined the simulation time problem and discussed simplification objectives. Then, it ave a brief overview over simplification alorithms found in recent literature, component-based approaches as well as those, where the network model is substituted with an adequate, quicker model. The second chapter, dealin with the development of a simplification alorithm, elucidates the static simplification alorithm of Ulanicki et al (996) and defined requirements for it. Based on these requirements and on the interfaces for the water network schedulin optimisation process, an alorithm was developed to identify the simplification rane in the water network model. As well, the treatment of pipe attributes was discussed. Followin, it was shown that the static simplification alorithm carries out the network model component based simplification approaches unification of nodes, parallel pipes and pipes in series and tree elimination. In addition, the static simplification alorithm was improved with the facility to delete low conductance pipes before eneratin the simplified model. Then, the implementation of the alorithm was discussed. The static simplification alorithm was included as a module in the eoraphic information system StruMap. The codin was done in C++. 97

99 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Then, the alorithm was explained in detail with activity diarams. Finally, an example for the alorithm was iven and tested with the proram. The case studies dealt with two network models, one of the water supply network in Skipton, Yorkshire, and another one with a model on the water supply network of the Hawksworth Lane Area in Guiseley, Yorkshire. Both networks were simplified with a number of different writeback criterion levels to eliminate low conductance pipes. The Skipton water network model with about 5 special network components showed extreme solvin errors with relative writeback criteria, but very low errors with absolute writeback criteria. The Hawksworth Lane water network model with only 3 special network components did not lose any pipe with a moderate absolute writeback criterion, because the simplified model was very compact. The relative writeback criterion deleted up to / rd 3 of the pipes in the simplified model with an almost nelectable error. The simplified models were benchmarked, it turned out that the simplified Skipton water network model is solved about 3-4 times faster, the simplified Hawksworth Lane water network model around 25 times faster. Both models were checked in 24h simulations, as well. V. 2. Recommendations for Future Work The eoraphical information system StruMap displays after usin the simplification modules sometimes memory manaement error messaes. To avoid the dependency on StruMap, the static simplification alorithm could be coupled to a database or be made an application on its own. Further, special network components like pumps, valves, etc. could be identified automatically. In addition, the absolute writeback criterion should be reformulated, so that it relates to the total sum of linear pipe conductances in the network model. For various tasks of water network models, component-based simplification is very useful to maintain as much as possible of the network structure. The component-based simplifications, which are discussed in this report, could be implemented for these tasks. As s final suestion, the static simplification alorithm could be mered with a solver. Therefore, redundant heads and flows in the output vector of the network model would be avoided when optimisin schedules, for example with enetic alorithms. 98

100 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. VI REFERECES Savic, D.A. and Walters, G.A. (997): GEETIC ALGORITHMS FOR LEAST-COST DESIG OF WATER DISTRIBUTIO ETWORKS. Journal of Water Resources Plannin and Manaement, March/ April 997, p. 67ff. Ulanicki, B., Zehnpfund, A., Martínez, F. (996): SIMPLIFICATIO OF WATER DISTRIBUTIO ETWORK MODELS. Hydroinformatics 996: Rotterdam, pp Michalewicz, Z. (992): GEETIC ALGORITHMS + DATA STRUCTURES = EVOLUTIO PROGRAMS. Spriner-Verla: Berlin, Heidelber, 992. CRC Press and Weisstein E.W.(996): CRC COCISE ECYCLOPEDIA OF MATHEMATICS. Internet Version: ( ). Anderson, E.J., Al-Jamal, K.H. (995): HYDRAULIC ETWORK SIMPLIFICATIO. Journal of Water Resources Plannin and Manaement, May/ June 995, pp Swarnee, P.K., Sharma, A.K. (99): DECOMPOSITIO OF LARGE WATER DISTRIBUTIO SYSTEMS. Journal of Environmental Enineerin, Vol. 6, o. 2, March/April 99, pp Martínez, F., García-Serra, J. (992): MATHEMATICAL MODELLIG OF WATER DISTRIBUTIO SYSTEMS I SERVICE. Conference Paper presented at the International Course on Water Supply Systems. State of the Art and Future Trends. In Valencia (Spain) from 9-23 October 992. Computational Mechanics Publications: Southampton,

101 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Hammerlin, G. and Hoffman, K.H. (99): UMERICAL MATHEMATICS. Graduate Tests in Mathematics. Spriner-Verla: ew York, Berlin, Heidelber, London, 99. Structural Technoloies Ltd (996): STRUMAP: GEOGRAPHIC IFORMATIO SYSTEM. Proram documentation. Software version n9. Calvert, C. (997): BORLAD C++BUILDER ULEASHED. SAMS Publishin: Indianapolis, 997. Press, W.H., Teukolsky, S.A., Vetterlin, W.T., Flannery, B.P. (994): UMERICAL RECIPES I C. THE ART OF SCIETIFIC COMPUTIG. Second Edition. Cambride University Press: 988, 992. Water Software Systems (999): FUDAMETAL WATER ETWORK MODEL OTOLOGY. On the Internet: (2-7-99). Swiercz, M. (995): A EURAL ETWORK APPROACH TO SIMPLIFY MATHEMATICAL MODELS OF URBA WATER DISTRIBUTIO SYSTEMS. 7 th IFAC/ IFORS/ IMACS Symposium on Lare Scale Systems: Theory and Applications LSS 95, London, -3 July 995. Ellis, D.J., Simpson A.R. (996): COVERGECE OF ITERATIVE SOLVERS FOR A SIMULATIO OF A WATER DISTRIBUTIO PIPE ETWORK. University of Adelaide, Department of Civil and Environmental Enineerin, Research Report o. R38, Au Rossman L.A. (994): EPAET USERS MAUAL. VERSIO.. Risk Reduction Enineerin Laboratory, Office of Research and Development, U.S. Environmental Protection Aency Cincinnati (Jan 994). On the Internet: (3-7-99). Todini, E. and Pilati, S. (987): A GRADIET METHOD FOR THE AALYSIS OF PIPE ETWORKS. International Conference on Computer Applications for Water Supply and Distribution, Leicester Polytechnic, UK, September 8-, 987. Geore, A. and Liu, J. W-H. (98): COMPUTER SOLUTIO OF LARGE SPARSE POSITIVE DEFIITE SYSTEMS. Prentice-Hall, Inc., Enlewood Cliffs, J, 98.

102 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. De Schaetzen W., Savic D.A., Walters G.A. (998): A GEETIC ALGORITHM APPROACH FOR PUMP SCHEDULIG I WATER SUPPLY SYSTEMS. 3rd Hydroinformatics conference of the International Association for Hydraulic Research. Copenhaen, Denmark. Auust 998. Chadwick A. and Morfett J. (993): HYDAULICS I CIVIL AD EVIROMETAL EGIEERIG. Second Edition 993. London (UK): Chapman and Hall.

103 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. VII ACKOWLEDGEMETS This project was carried out at the University of Exeter, United Kindom, as part of the ERASMUS/ SOCRATES exchane proramme with the University of Stuttart, Germany, durin the academic year 998/99. I would like to thank the followin people, whose efforts made this project possible: Prof. D.H. Owens at the University of Exeter and Prof. M. Zeitz at the University of Stuttart. For the supervision of this interestin project and excellent support throuhout all steps of this work, I would like to thank my supervisors: Dr. Draan Savic and Dr. Godfrey Walters. The team spirit of the Centre for Water Systems ave me an excellent workin environment with motivation and support, for which I would like to thank its current and former members: Bernhard Alberto Alvarez Roer Atkinson Bonfilio Florian Bessler Jim Davidson Mark Enelhardt Zoran Kapelan Eduard Keedwell Mark Morley Martin Parker Marc Randall-Smith Werner de Schaetzen Jenny Thorne Jakobus van Zyl Finally I would like to thank all members of the staff of the University of Exeter and all British and International students, who made this year at the south cost of Devon an excellent, enrichin experience for me. 2

104 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. VIII APPEDIX VIII.. Calculation of the c Constant in the Hazen-Williams Equation for StruMap The constant c was calculated basin on the head difference, the pipe diameter, the Hazen-Williams-Coefficient, the pipe lenth and the flow of eiht pipes in a water network model in StruMap. Therefore, the followin relationship derived from (6) on pae 6: (53) C d = ; k=,2,, p Q e e2 k k k c e lk Qk k Difference in Total Head m Table 2. Calculation of the c Constant for the Hazen-Williams-Formula. Diameter mm H.W. coeff. Lenth mm Flow l/s calculated Constant c E E E E E E E+ In the proram code the averae of the calculated c values above is used: c :=.25283E+. 3

105 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. VIII. 2. The umber of Loops in a Water etwork Let P be the number of pipes in a network and the number of nodes. A network formed as a tree has one startin node, every other node is linked to this node or another one with exactly one pipe. So, for a tree network the followin relation can be found: (54) P =. When there are loops in a network, the number of pipes is reater than the number of nodes excludin the startin node: (55) P >. The supernumerary pipes form the loops of the network. Hence, the number of loops L can be introduced as: (56) L = P ( ) = P +. VIII. 3. Determination of the Maximum umber of Pipes in a Water etwork The maximum number of Pipes is used to express the pipe density and the loop density of a water network in SpeedUp. The output of the simplification modules of SpeedUp relates the number of pipes in the network model to the possible number of pipes. In addition, it relates the number of loops to the possible number of loops in the network model. Both will be derived via the maximum number of links in a eneral network. VIII. 3.. The Maximum umber of Links in a General etwork The maximum number of links M L in a network is the number of pipes in a network, where every node is connected throuh exactly one pipe with every other node 9. 9 A link can be expressed by one or more parallel pipes. 4

106 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Every network beins with a startin node. When addin a node to the network, it has to be connected to all other nodes. Therefore, the number of links increases by the number of already existin nodes: (57) M L ( ) = M L ( - ) + ( - ) with M L () = and M L () =. This recursive relationship is equal to the followin sum: = (58) M ( ) i ; > L i=. To prove this, (57) will be converted: (59) M L ( ) - M L ( - ) = ( - ). The same will be done with equation (58): 2 (6) M ( ) M ( ) = i i = ( ). L L i= i= VIII The Maximum umber of Pipes in a Water etwork Water networks possess a number of special components, which influence the maximum number of links in the network. It shall be assumed that reservoirs and sources, treated as nodes, are connected with exactly one pipe to the rest of the network. Valves and other special network components are treated as nodes as well. Characteristically they operate between exactly two nodes. Therefore, they require exactly two pipes. Therefore, the formula for the maximum number of links, equation (58), will be upraded: The maximum number of links in a eneral network can only be found between all normal nodes, their number is 5

107 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. (6) orm = R Spec with orm R : number of normal nodes, number of reservoirs and Spec : number of valves, etc.. Each reservoir requires one pipe, so it increases the maximum number of links by one. Each special network component increases the maximum number of links by two. This results in the followin upraded equation: R Spec (62) M L R Spec = (,, ) i + R + 2 Spec i= with > ; ; > +. R Spec R Spec VIII. 4. Output of the Simplification Routine for the Skipton Water etwork VIII. 4.. Skipton Water etwork The static simplification alorithm enerated the followin output. The static simplification alorithm used the heads and the flows at the 8 o clock snapshot. The snapshot was directly calculated, not as part of a 24h simulation. VIII. 4.. a. Case All new linear pipe conductances of the Jacobian matrix of the model were written back into StruMap. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 9x9. Readin pipes back. had a linear branch conductance lower than the draw-back-criteria and were not written back. Corrected hdiff. 6

108 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. STATISTICS BEFORE =============================================== odes: 9 non-removable: 34 Pipes: 295 Loops: 25. of W Components: 2386 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 34 non-removable: 34 Pipes: 49 Loops: 276. of W Components: 543 The pipes represent 9 % of all possible links. The loops represent 68 o/oo of all possible loops. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % VIII. 4.. b. Case 4 The relative writeback criterion was used with a writeback level of.2. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 9x9. Readin pipes back whose linear conductance is bier than 9/ of the summed conductances of its nodes. 42 had a linear branch conductance lower than the draw-back-criteria and were not written back. Corrected hdiff. STATISTICS BEFORE =============================================== odes: 9 non-removable: 34 Pipes: 295 Loops: 25. of W Components: 2386 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. 7

109 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. STATISTICS AFTER =============================================== odes: 34 non-removable: 34 Pipes: 267 Loops: 34. of W Components: 4 The pipes represent 6 % of all possible links. The loops represent 33 o/oo of all possible loops. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % VIII. 4.. c. Case 3 The relative writeback criterion was used with a writeback level of.. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 9x9. Readin pipes back whose linear conductance is bier than / of the summed conductances of its nodes. 25 had a linear branch conductance lower than the draw-back-criteria and were not written back. Corrected hdiff. STATISTICS BEFORE =============================================== odes: 9 non-removable: 34 Pipes: 295 Loops: 25. of W Components: 2386 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 34 non-removable: 34 Pipes: 284 Loops: 5. of W Components: 48 The pipes represent 6 % of all possible links. The loops represent 37 o/oo of all possible loops. 8

110 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % VIII. 4.. d. Case 2 The relative writeback criterion was used with a writeback level of.. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 9x9. Readin pipes back whose linear conductance is bier than / of the summed conductances of its nodes. 97 had a linear branch conductance lower than the draw-back-criteria and were not written back. Corrected hdiff. STATISTICS BEFORE =============================================== odes: 9 non-removable: 34 Pipes: 295 Loops: 25. of W Components: 2386 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 34 non-removable: 34 Pipes: 3 Loops: 77. of W Components: 444 The pipes represent 7 % of all possible links. The loops represent 44 o/oo of all possible loops. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % 9

111 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. VIII. 4.. e. Case 5 The absolute writeback criterion was used with a writeback level of.95. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 9x9. Readin pipes back whose absolute linear conductance is bier than.95 times the lowest linear conductance in the network. 9 had a linear branch conductance lower than the draw-back-criteria and were not written back. Corrected hdiff. STATISTICS BEFORE =============================================== odes: 9 non-removable: 34 Pipes: 295 Loops: 25. of W Components: 2386 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 34 non-removable: 34 Pipes: 3 Loops: 67. of W Components: 434 The pipes represent 7 % of all possible links. The loops represent 4 o/oo of all possible loops. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % VIII. 4.. f. Case 6 The absolute writeback criterion was used with a writeback level of 2. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 9x9.

112 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Readin pipes back whose absolute linear conductance is bier than 2 times the lowest linear conductance in the network. 8 had a linear branch conductance lower than the draw-back-criteria and were not written back. Corrected hdiff. STATISTICS BEFORE =============================================== odes: 9 non-removable: 34 Pipes: 295 Loops: 25. of W Components: 2386 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 34 non-removable: 34 Pipes: 29 Loops: 58. of W Components: 425 The pipes represent 7 % of all possible links. The loops represent 39 o/oo of all possible loops. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % VIII Case 7 The absolute writeback criterion was used with a writeback level of 3. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 9x9. Readin pipes back whose absolute linear conductance is bier than 3 times the lowest linear conductance in the network. 24 had a linear branch conductance lower than the draw-back-criteria and were not written back. Corrected hdiff. STATISTICS BEFORE

113 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. =============================================== odes: 9 non-removable: 34 Pipes: 295 Loops: 25. of W Components: 2386 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 34 non-removable: 34 Pipes: 285 Loops: 52. of W Components: 49 The pipes represent 6 % of all possible links. The loops represent 37 o/oo of all possible loops. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % VIII. 4.. h. Case 8 The absolute writeback criterion was used with a writeback level of. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 9x9. Readin pipes back whose whose absolute linear conductance is bier than times the lowest linear conductance in the network. 44 had a linear branch conductance lower than the draw-back-criteria and were not written back. Corrected hdiff. STATISTICS BEFORE =============================================== odes: 9 non-removable: 34 Pipes: 295 Loops: 25. of W Components: 2386 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== 2

114 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. odes: 34 non-removable: 34 Pipes: 265 Loops: 32. of W Components: 399 The pipes represent 6 % of all possible links. The loops represent 32 o/oo of all possible loops. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % VIII Hawksworth Lane, Guiseley Water etwork Model The static simplification alorithm enerated the followin output. The static simplification alorithm used the heads and the flows at the 8 o clock snapshot. The snapshot was directly calculated, not as part of a 24h simulation. VIII a. Case All pipes were written back to StruMap. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 449x449. Readin pipes back. had a linear branch conductance lower than the draw-back-criteria and were not written back. STATISTICS BEFORE =============================================== odes: 45 non-removable: 5 Pipes: 497 Loops: 48. of W Components: 947 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 5 non-removable: 5 3

115 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Pipes: 3 Loops: 6. of W Components: 45 The pipes represent 35 % of all possible links. The loops represent 225 o/oo of all possible loops. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % VIII b. Case 2 and 3 The output of the cases 2 and 3 is identical with case ; no pipes were deleted. Case 2 uses the absolute writeback criterion with a level of.95. Case 3 uses the absolute writeback criterion as well, but with a level of. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 449x449. Readin pipes back whose absolute linear conductance is bier than.95 times the lowest linear conductance in the network. had a linear branch conductance lower than the draw-back-criteria and were not written back. STATISTICS BEFORE =============================================== odes: 45 non-removable: 5 Pipes: 497 Loops: 48. of W Components: 947 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 5 non-removable: 5 Pipes: 3 Loops: 6. of W Components: 45 The pipes represent 35 % of all possible links. The loops represent 225 o/oo of all possible loops. COMPARISO 4

116 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % VIII c. Case 4 This case uses the relative writeback criterion with a level of.2. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 449x449. Readin pipes backwhose linear conductance is bier than 2/ of the summed conductances of its nodes. 8 had a linear branch conductance lower than the draw-back-criteria and were not written back. STATISTICS BEFORE =============================================== odes: 45 non-removable: 5 Pipes: 497 Loops: 48. of W Components: 947 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 5 non-removable: 5 Pipes: 22 Loops: 8. of W Components: 37 The pipes represent 25 % of all possible links. The loops represent 2 o/oo of all possible loops. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % 5

117 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. VIII d. Case 5 This last case study uses the relative writeback criterion with a level of.. LIEARISATIO & SIMPLIFICATIO Dimension of the Jacobian Matrix: 449x449. Readin pipes back, whose linear conductance is bier than / of the summed conductances of its nodes. had a linear branch conductance lower than the draw-back-criteria and were not written back. STATISTICS BEFORE =============================================== odes: 45 non-removable: 5 Pipes: 497 Loops: 48. of W Components: 947 The pipes represent % of all possible links. The loops represent o/oo of all possible loops. STATISTICS AFTER =============================================== odes: 5 non-removable: 5 Pipes: 2 Loops: 6. of W Components: 35 The pipes represent 23 % of all possible links. The loops represent 84 o/oo of all possible loops. COMPARISO =============================================== ew Old Compared odes: % Pipes: % Loops: % W Components: % VIII. 5. User Manual This section uides throuh the simplification procedure with SpeedUp. The attribute non-removable has to be set true for all nodes, whose head is necessary for the cost function, and false otherwise. The attribute untouchable has to be set true for all pipes, whose flow is needed for the cost function and for all input components, otherwise false. 6

118 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. ow, SpeedUp can be called. This can be done by callin the main menu entry SpeedUp etwork Simplification. Then, the followin dialo box appears: Simplification Preparation: Identification of the Simplification Rane Eliminate Trees Level of the relative writeback criterion Level of the absolute writeback criterion Start the Simplification Module Path, where the files for the new demands shall be written to Fiure 52. SpeedUp Dialo Box. It is very important to save the network model after every step and to restart SpeedUp. The path, where the current model is located on the hard disk, has to be inserted in the correspondin text-edit-box. It needs to finish with a backslash, \. Before any simplifications can be done, the simplification rane needs to be identified with the button Prepare Simplification. Afterwards, the either tree structures can be eliminated ( Eliminate Branches ) or the static simplification can be applied ( Linearise and Simplify ). If a writeback criterion is necessary, it should be ticked and an appropriate level entered. Both criterions can be combined. To simulate a network model multiple times, the HARP solver has to be called as usual. The pop-down-menu beyond Hazen-Williams / Colebrook-White has to be set to Genetic. A dialo box will appear and ask, how many times the model shall be run. It is very useful, to avoid savin the text output file. 7

119 Maschler, T. and D.A. Savic, (999) Simplification of Water Supply etwork Models throuh Linearisation, Centre for Water Systems, Report o.99/, School of Enineerin, University of Exeter, Exeter, United Kindom, p.9. Pipes with hih resistance/ low conductance pose bi problems when solvin. They require far more iterations to meet the accuracy criterion of the solver than other pipes. The term Static Simplification appears in Swiercz (995) and is used here for the approach of Ulanicki et al (996), as there is no term specified. The numbers without variable name and unit are the flows l/s. The black point in the bottom left ede in the screenshot is the cursor. The lowest linear pipe conductance of the oriinal network is kept in the simplified one. It is seen as a lower limit to which small chanes are allowed. Therefore, if the solver has no problems with the oriinal network, it will have no problems with the simplified network, also. The percentae of loops refers to the secondary value axis, all others to the first one. The number of loops refers to the secondary value axis on the riht hand side of the fiure. A link can be expressed by one or more parallel pipes. 8