Mathematical Optimization of Water Networks (International Series of Numerical Mathematics Book 162)

Part I Optimization of Water Supply Networks.- Modelling and Numerical Simulation of Pipe Flow Problems in Water Supply Systems.- Simulation and Continuous Optimization.- Mixed Integer Optimizationof Water Supply Networks.- Nonlinear and Mixed Integer Linear Programming.- Part II Optimal Control of Sewer Networks.- Optimal Control of Sewer Networks Problem Description.- Modelling of Channel Flows with Transition Interface Separating Free Surface and Pressurized Channel Flows.- Optimal Control of Sewer Networks Engineers View.- Real-Time Control of Urban Drainage Systems.- Performance and Comparison of BlueM.MPC and Lamatto.- Multicriteria Optimization in Wastewater Management Cover 1 Mathematical Optimization of Water Networks 4 Preface 6 Acknowledgements 11 Contents 12 Contributors 13 Chapter 1: Modeling and Numerical Simulation of Pipe Flow Problems in Water Supply Systems 15 1.1 Introduction 15 1.2 Example of a Water Supply System 15 1.3 Modeling Equations 17 1.3.1 Free Surface Flow 17 1.3.2 Pressure Flow 18 1.3.3 Storage Tanks, Pumps and Valves 19 1.4 Numerical Solution 20 1.4.1 Method of Lines 20 1.4.2 Space Discretization 21 1.4.3 Implementation of Boundary and Coupling Conditions 23 1.4.4 Solution of the Differential Algebraic Equations 24 1.5 Simulation Results and Conclusions 25 References 26 Chapter 2: Simulation and Continuous Optimization 28 2.1 Numerical Solution of the Model Equations 28 2.1.1 Network Equations 28 2.1.2 Properties of the Water Hammer Equations 29 2.1.3 Implicit Box Scheme 30 2.2 Adjoint Calculus 33 2.2.1 The First-Discretize Approach 33 2.2.2 Application to Time-Dependent Problems 35 2.3 Singularities 37 2.3.1 Introduction 37 2.3.2 Theoretical Analysis-Forward Direction 39 2.3.3 Theoretical Analysis-Backward Direction 42 References 43 Chapter 3: Mixed Integer Optimization of Water Supply Networks 45 3.1 Introduction 45 3.2 Basic Network Model 46 3.3 Flow in Pipelines 46 3.4 A Model for Dynamic Water Supply Network Optimization 47 3.4.1 Pipes 48 3.4.2 Tanks 49 3.4.3 Pumps 49 3.4.4 Valves 51 3.4.5 Flow Conservation 52 3.4.6 Further Transient Conditions 52 3.4.7 Optimization Task 54 3.5 Piecewise Linearization 54 3.5.1 Mixed Integer Model of a Univariate Piecewise Linearization 56 3.5.2 Mixed Integer Model of a Multivariate Piecewise Linearization 57 3.6 Computational Results 60 3.6.1 Network 1 60 3.6.2 Network 2 61 3.7 Conclusion 62 References 64 Chapter 4: Nonlinear and Mixed Integer Linear Programming 65 4.1 Introduction 65 4.2 Heuristic Approach 66 4.3 Results for the Meso Network 68 4.4 Results for a Municipal Water Supply Network 72 4.5 Conclusion 74 References 74 Chapter 5: Optimal Control of Sewer Networks Problem Description 76 5.1 Introduction 76 5.2 Technical Principles 78 5.2.1 Dynamic Flow Routing Modeling 78 5.2.2 Model Predictive Control 78 Receding Horizon 79 Process Model 79 Optimization 79 Setup of MPC Software 80 5.3 Applications of MPC 82 5.4 An Industrial Viewpoint 83 5.4.1 SIWA Sewer Management System 83 5.4.2 Industrial Requirements and Mathematical Challenges 84 5.5 Practical Relevance and Research Demand 86 References 88 Chapter 6: Modeling of Channel Flows with Transition Interface Separating Free Surface and Pressurized Channel Flows 90 6.1 Introduction 90 6.2 Basic Equations 91 6.2.1 Free Surface Flow 91 6.2.2 Shock Waves 94 6.2.3 Pressurized Flow 94 6.3 Review of Existing Flow Regime Transition Models 95 6.3.1 Rigid Column Technique 95 6.3.2 Preissmann Slot Technique 96 6.3.3 Shock Fitting Method 97 6.4 A New Flow Regime Transition Model 97 6.5 Discontinuous Galerkin Scheme for Numerical Simulation of the Shallow Water Equations 100 6.6 Numerical Formulation of Fluxes 102 6.7 Numerical Stability and Limiters 104 6.8 Test Problems and Numerical Results 105 6.8.1 Error Analysis for the Linearized Shallow-Water Equations with Smooth Initial Conditions 106 6.8.2 Numerical Stability for the Linearized Shallow-Water Equations with Discontinuous Initial Conditions 107 6.8.3 Dam Breaking in a Rectangular Channel 108 6.8.4 Channel Flow with the Moving Transition Separating Free-Surface and Pressurized Flow Regions 112 6.9 Concluding Remarks 113 References 115 Chapter 7: Optimal Control of Sewer Networks Engineers View 117 7.1 Prerequisites 117 7.2 Modeling Approach 118 7.2.1 MPC Controller 118 7.2.2 Process Model 120 7.2.3 Optimization 123 PES 124 Hooke and Jeeves 124 7.2.4 Summary 127 7.3 Numerical Results 127 References 132 Chapter 8: Real-Time Control of Urban Drainage Systems 134 8.1 Introduction 134 8.2 Shallow Water Equations on Networks 134 8.3 Finite Volume Discretization 136 8.4 Finite Volume Junctions 139 8.5 The Finite Volume Network Problem 141 8.6 The Optimal Control Problem 143 8.7 Discrete Optimal Control Problem 145 8.8 Design Variables 149 8.9 Numerical Results 149 References 155 Chapter 9: Performance and Comparison of BlueM.MPC and Lamatto 156 9.1 Comparison and Conclusion 156 9.1.1 Realistic Network Case Study 156 9.1.2 Influence of Time Horizons 167 9.1.3 Influence of Optimization Algorithm 167 9.1.4 Conclusion 168 Chapter 10: Multicriteria Optimization in Wastewater Management 171 10.1 Introduction 171 10.1.1 Literature Overview and Goals in Wastewater Management 172 10.1.2 Terminology and Definitions 174 10.2 Methods 175 10.2.1 Scalarizations 176 The Weighted-Sum Approach 176 The epsilon-Constraint Method and Lexicographic Minimization 177 The Weighted and the Augmented Weighted Tchebycheff Approach 178 10.2.2 Trade-off 179 10.2.3 Approximation of the Nondominated Set 185 10.3 Computational Results 187 10.3.1 Implementational Details 187 Solving the Single Criterion Scalarized problems 187 Approximation 188 Scalarizations 189 Parameter Update 189 10.3.2 Evaluation I: Interdependencies of the Considered Objectives 190 Test Network 191 Total Release Versus Total Pollution Mass 191 Total Release Versus Constant Inflow to the Wastewater Treatment Plant 194 10.3.3 Evaluation I: Discussion and Challenges 195 Comparison of the Parameter Selection Rules 195 Comparison of the Scalarizations 195 10.3.4 Evaluation II: Academic Test Network with Three Criteria 196 10.4 Conclusions 197 References 198 Cover......Page 1 Mathematical Optimization of Water Networks......Page 4 Preface......Page 6 Acknowledgements......Page 11 Contents......Page 12 Contributors......Page 13 1.2 Example of a Water Supply System......Page 15 1.3.1 Free Surface Flow......Page 17 1.3.2 Pressure Flow......Page 18 1.3.3 Storage Tanks, Pumps and Valves......Page 19 1.4.1 Method of Lines......Page 20 1.4.2 Space Discretization......Page 21 1.4.3 Implementation of Boundary and Coupling Conditions......Page 23 1.4.4 Solution of the Differential Algebraic Equations......Page 24 1.5 Simulation Results and Conclusions......Page 25 References......Page 26 2.1.1 Network Equations......Page 28 2.1.2 Properties of the Water Hammer Equations......Page 29 2.1.3 Implicit Box Scheme......Page 30 2.2.1 The First-Discretize Approach......Page 33 2.2.2 Application to Time-Dependent Problems......Page 35 2.3.1 Introduction......Page 37 2.3.2 Theoretical Analysis-Forward Direction......Page 39 2.3.3 Theoretical Analysis-Backward Direction......Page 42 References......Page 43 3.1 Introduction......Page 45 3.3 Flow in Pipelines......Page 46 3.4 A Model for Dynamic Water Supply Network Optimization......Page 47 3.4.1 Pipes......Page 48 3.4.3 Pumps......Page 49 3.4.4 Valves......Page 51 3.4.6 Further Transient Conditions......Page 52 3.5 Piecewise Linearization......Page 54 3.5.1 Mixed Integer Model of a Univariate Piecewise Linearization......Page 56 3.5.2 Mixed Integer Model of a Multivariate Piecewise Linearization......Page 57 3.6.1 Network 1......Page 60 3.6.2 Network 2......Page 61 3.7 Conclusion......Page 62 References......Page 64 4.1 Introduction......Page 65 4.2 Heuristic Approach......Page 66 4.3 Results for the Meso Network......Page 68 4.4 Results for a Municipal Water Supply Network......Page 72 References......Page 74 5.1 Introduction......Page 76 5.2.2 Model Predictive Control......Page 78 Optimization......Page 79 Setup of MPC Software......Page 80 5.3 Applications of MPC......Page 82 5.4.1 SIWA Sewer Management System......Page 83 5.4.2 Industrial Requirements and Mathematical Challenges......Page 84 5.5 Practical Relevance and Research Demand......Page 86 References......Page 88 6.1 Introduction......Page 90 6.2.1 Free Surface Flow......Page 91 6.2.3 Pressurized Flow......Page 94 6.3.1 Rigid Column Technique......Page 95 6.3.2 Preissmann Slot Technique......Page 96 6.4 A New Flow Regime Transition Model......Page 97 6.5 Discontinuous Galerkin Scheme for Numerical Simulation of the Shallow Water Equations......Page 100 6.6 Numerical Formulation of Fluxes......Page 102 6.7 Numerical Stability and Limiters......Page 104 6.8 Test Problems and Numerical Results......Page 105 6.8.1 Error Analysis for the Linearized Shallow-Water Equations with Smooth Initial Conditions......Page 106 6.8.2 Numerical Stability for the Linearized Shallow-Water Equations with Discontinuous Initial Conditions......Page 107 6.8.3 Dam Breaking in a Rectangular Channel......Page 108 6.8.4 Channel Flow with the Moving Transition Separating Free-Surface and Pressurized Flow Regions......Page 112 6.9 Concluding Remarks......Page 113 References......Page 115 7.1 Prerequisites......Page 117 7.2.1 MPC Controller......Page 118 7.2.2 Process Model......Page 120 7.2.3 Optimization......Page 123 Hooke and Jeeves......Page 124 7.3 Numerical Results......Page 127 References......Page 132 8.2 Shallow Water Equations on Networks......Page 134 8.3 Finite Volume Discretization......Page 136 8.4 Finite Volume Junctions......Page 139 8.5 The Finite Volume Network Problem......Page 141 8.6 The Optimal Control Problem......Page 143 8.7 Discrete Optimal Control Problem......Page 145 8.9 Numerical Results......Page 149 References......Page 155 9.1.1 Realistic Network Case Study......Page 156 9.1.3 Influence of Optimization Algorithm......Page 167 9.1.4 Conclusion......Page 168 10.1 Introduction......Page 171 10.1.1 Literature Overview and Goals in Wastewater Management......Page 172 10.1.2 Terminology and Definitions......Page 174 10.2 Methods......Page 175 The Weighted-Sum Approach......Page 176 The epsilon-Constraint Method and Lexicographic Minimization......Page 177 The Weighted and the Augmented Weighted Tchebycheff Approach......Page 178 10.2.2 Trade-off......Page 179 10.2.3 Approximation of the Nondominated Set......Page 185 Solving the Single Criterion Scalarized problems......Page 187 Approximation......Page 188 Parameter Update......Page 189 10.3.2 Evaluation I: Interdependencies of the Considered Objectives......Page 190 Total Release Versus Total Pollution Mass......Page 191 Total Release Versus Constant Inflow to the Wastewater Treatment Plant......Page 194 Comparison of the Scalarizations......Page 195 10.3.4 Evaluation II: Academic Test Network with Three Criteria......Page 196 10.4 Conclusions......Page 197 References......Page 198 Water supply- and drainage systems and mixed water channel systems are networks whose high dynamic is determined and/or affected by consumer habits on drinking water on the one hand and by climate conditions, in particular rainfall, on the other hand. According to their size, water networks consist of hundreds or thousands of system elements. Moreover, different types of decisions (continuous and discrete) have to be taken in the water management. The networks have to be optimized in terms of topology and operation by targeting a variety of criteria. Criteria may for example be economic, social or ecological ones and may compete with each other. The development of complex model systems and their use for deriving optimal decisions in water management is taking place at a rapid pace. Simulation and optimization methods originating in Operations Research have been used for several decades; usually with very limited direct cooperation with applied mathematics. The research presented here aims at bridging this gap, thereby opening up space for synergies and innovation. It is directly applicable for relevant practical problems and has been carried out in cooperation with utility and dumping companies, infrastructure providers and planning offices. A close and direct connection to the practice of water management has been established by involving application-oriented know-how from the field of civil engineering. On the mathematical side all necessary disciplines were involved, including mixed-integer optimization, multi-objective and facility location optimization, numerics for cross-linked dynamic transportation systems and optimization as well as control of hybrid systems. Most of the presented research has been supported by the joint project „Discret-continuous optimization of dynamic water systems“ of the federal ministry of education and research (BMBF). This book deals with the modeling, analysis and simulation of problems arising in the life sciences, and especially in biological processes. The models and findings presented result from intensive discussions with microbiologists, doctors and medical staff, physicists, chemists and industrial engineers and are based on experimental data. They lead to a new class of degenerate density-dependent nonlinear reaction-diffusion convective equations that simultaneously comprise two kinds of degeneracy: porous-medium and fast-diffusion type degeneracy. To date, this class is still not clearly understood in the mathematical literature and thus especially interesting. The author both derives realistic life science models and their above-mentioned governing equations of the degenerate types and systematically studies these classes of equations. In each concrete case well-posedness, the dependence of solutions on boundary conditions reflecting some properties of the environment, and the large-time behavior of solutions are investigated and in some instances also studied numerically.