i BASIC OPTIMIZATION MODELS FOR WATER AND ENERGY MANAGEMENT By Daene C. McKinney And Andre G. Savitsky June 1999 (revision 1, October 2000) (revision 5, February 2002) (revision 6, February 2003)ii Contents ACKNOWLEDGEMENTS..........................................................................................................................................V 1. INTRODUCTION ....................................................................................................................................................1 1.1 SYSTEMS APPROACH...........................................................................................................................................1 1.2 SIMULATION AND OPTIMIZATION MODELS .........................................................................................................2 1.3 MODEL BUILDING PROCESS ................................................................................................................................3 1.4 BOOK ORGANIZATION.........................................................................................................................................4 PART 1: RESOURCE MANAGEMENT MODELS ...................................................................................................6 2. NONLINEAR ALGEBRAIC EQUATIONS............................................................................................................6 2.1 SOLUTION OF ALGEBRAIC EQUATIONS................................................................................................................6 2.2 METHOD OF LEAST SQUARES FOR ARBITRARY FUNCTIONS ................................................................................8 2.3 EXERCISES ........................................................................................................................................................10 3. WATER MANAGEMENT IN RIVER SYSTEMS................................................................................................10 3.1 OPTIMAL ALLOCATION OF WATER TO USERS ...................................................................................................10 3.2 OPTIMAL MANAGEMENT OF A SINGLE RESERVOIR ...........................................................................................13 3.3 OPTIMAL MANAGEMENT OF A RIVER SYSTEM ..................................................................................................24 3.4 ONSTREAM AND OFFSTREAM RESERVOIR MANAGEMENT.................................................................................31 3.5 WATER RIGHTS AND MARKETS.........................................................................................................................35 3.6 MANAGING WATER QUALITY IN STREAMS .......................................................................................................47 3.7 EXERCISES ........................................................................................................................................................52 4. OPTIMAL MANAGEMENT OF GROUNDWATER............................................................................................59 4.1 FINITE-DIFFERENCE METHOD ...........................................................................................................................59 4.2 NUMERICAL DIFFERENTIATION .........................................................................................................................59 4.3 GRIDS AND DISCRETRIZATION ..........................................................................................................................61 4.4 ONE-DIMENSIONAL FLOW.................................................................................................................................62 4.5 TWO-DIMENSIONAL FLOW ................................................................................................................................65 4.6 TRANSIENT PROBLEMS......................................................................................................................................76 4.7 EXERCISES ........................................................................................................................................................82 5. OPTIMAL SELECTION OF AGRICULTURAL CROPS......................................................................................85 6. OPTIMAL DEVELOPMENT OF CANALS ..........................................................................................................87 7. PROBLEMS OF POWER NETWORKS ...............................................................................................................91 7.1 PROBLEM OF POWER GENERATION, DISTRIBUTION, AND CONSUMPTION ..........................................................91 7.2 DEFINING VOLTAGE AND STRENGTH OF CURRENT IN DIRECT-CURRENT CIRCUIT ............................................94 7.3 PASSING A COMPLEX ELECTRICAL SIGNAL .......................................................................................................98 7.4 ALTERNATING CURRENT CIRCUITS .................................................................................................................106 8. OPTIMAL SOLUTION OF HEAT TRANFER PROBLEMS ..............................................................................114 8.1 BACKGROUND.................................................................................................................................................114 8.2 STATIONARY TEMPERATURE FIELD IN A RECTANGULAR AREA.........................................................................116 8.3 STATIONARY TEMPERATURE FIELD IN A RECTANGULAR AREA WITH BORDER HEAT FLOW ...........................121 8.4 TIME DEPENDENT TEMPERATURE FIELD .........................................................................................................126 9. OPTIMAL SOLUTION OF FLUID FLOW PROBLEMS....................................................................................129iii 9.1 BACKGROUND.................................................................................................................................................129 9.2 STATIONARY FLOW OF AN INCOMPRESSIBLE FLUID IN A RECTANGULAR AREA ..............................................130 9.3 STATIONARY FLOW OF WATER IN A RECTANGULAR AREA IN THE PRESENCE OF AN OBSTACLE ........................136 PART 2: THE GAMS