SIMPLIFICATION OF WATER SUPPLY NETWORK MODELS THROUGH LINEARISATION Tobias Maschler and Dragan A. Savic [email protected] Report Number:99/01 1999 Centre for Water Systems, University of Exeter, North Park Road, Exeter, EX4 4QF, Devon, United Kingdom. 1 Maschler, T. and D.A. Savic, (1999) Simplification of Water Supply Network Models through Linearisation, Centre for Water Systems, Report No.99/01, School of Engineering, University of Exeter, Exeter, United Kingdom, p.119. Abstract This work enhances a simplification algorithm for water network models with routines to identify the simplification range and to classify the importance of pipes in the reduced network model. To aggregate 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 given model have also to be known. Basing on this, a simplification range is identified in the water network model. Afterwards, the model is linearised around a given working point and all redundant nodes are eliminated with Gauss-Jordan elimination. The remaining nodes are re-linked with pipes according to the structure of the simplified model. Non-important links will be deleted to keep the simplified model compact and with as less loops as possible. The simplification algorithm 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 program code is described alongside as well as essential information about the use of existing software. 1 Maschler, T. and D.A. Savic, (1999) Simplification of Water Supply Network Models through Linearisation, Centre for Water Systems, Report No.99/01, School of Engineering, University of Exeter, Exeter, United Kingdom, p.119. Table of Contents Abstract ................................................................................................................................... 1 Table of Contents.................................................................................................................... 2 List of Figures ......................................................................................................................... 5 List of Tables........................................................................................................................... 7 I — Introduction ..............................................................................................................8 II — Problem Formulation............................................................................................10 II. 1. Introduction to Hydraulic Network Models.............................................................. 10 II. 1. 1. Purpose of Water Network Models......................................................................................10 II. 1. 2. Network Topology ...............................................................................................................12 II. 1. 2. a. Nodes...........................................................................................................................13 II. 1. 2. b. Pipes ............................................................................................................................14 II. 2. Improving the Simulation Time Problem.................................................................. 16 II. 2. 1. The Simulation Time Problem.............................................................................................16 II. 2. 2. Simplification Objectives.....................................................................................................17 II. 2. 3. Simplification of Network Model Components ...................................................................19 II. 2. 3. a. Nodes...........................................................................................................................20 II. 2. 3. b. Parallel Pipes...............................................................................................................20 II. 2. 3. c. Series Pipes..................................................................................................................20 II. 2. 3. d. Nodes with Similar Head ............................................................................................21 II. 2. 3. e. Tree Structures.............................................................................................................21 II. 2. 3. f. Low Conductance Pipes...............................................................................................21 II. 2. 3. g. Single Input Subsystems .............................................................................................22 II. 2. 4. Black Box Simplification.....................................................................................................22 II. 2. 4. a. Static Simplification ....................................................................................................22 II. 2. 4. b. Neural Network Approach ..........................................................................................23 II. 3. Conclusions .................................................................................................................. 24 III — Network Simplification........................................................................................25 III. 1. Theoretical Development........................................................................................... 25 III. 1. 1. The Algorithm for Static Simplification.............................................................................25 III. 1. 1. a. The Mathematical Water Network Model..................................................................25 III. 1. 1. b. The Linear Model.......................................................................................................27 III. 1. 1. c. Linear Model Reduction.............................................................................................29 III. 1. 1. d. The Reduced Non-Linear Model ...............................................................................30 III. 1. 1. e. Requirements..............................................................................................................30 III. 1. 2. Identification of the Simplification Range..........................................................................31 III. 1. 3. Pipe Attributes ....................................................................................................................35 III. 2. Algebraic Simplification of Network Model Components...................................... 36 III. 2. 1. Nodes..................................................................................................................................37 III. 2. 1. a. Mathematics ...............................................................................................................37 III. 2. 1. b. Interpretation..............................................................................................................41 III. 2. 2. Pipes in Series.....................................................................................................................41 2 Maschler, T. and D.A. Savic, (1999) Simplification of Water Supply Network Models through Linearisation, Centre for Water Systems, Report No.99/01, School of Engineering, University of Exeter, Exeter, United Kingdom, p.119. III. 2. 2. a. Mathematics ...............................................................................................................42 III. 2. 2. b. Interpretation..............................................................................................................43 III. 2. 3. Parallel Pipes ......................................................................................................................43 III. 2. 3. a. Mathematics ...............................................................................................................44 III. 2. 3. b. Interpretation..............................................................................................................45 III. 2. 4. Trees ...................................................................................................................................45 III. 2. 4. a. Adding the Demands..................................................................................................46 III. 2. 4. b. Static Simplification...................................................................................................47 III. 2. 5. Low Conductance Pipes .....................................................................................................48 III. 3. Implementation .......................................................................................................... 49 III. 3. 1. Modelling Environment......................................................................................................49 III. 3. 1. a. StruMap......................................................................................................................50 III. 3. 1. b. C++ Interface to StruMap ..........................................................................................51 III. 3. 1. c. Matrix Library............................................................................................................51