1. ©GoldSim Technology Group LLC., 2012 Network Flows and Linear Programming The Mathematical Madness behind the Magic GoldSim Technology Group

2. ©GoldSim Technology Group LLC., 2012 Objective of the Flow Module Given a system of discrete locations connected by conduits of flowing material… …determine the “optimal” flow of material through that network.

3. ©GoldSim Technology Group LLC., 2012

7. Flow SolverGoldSim 10.5: Solve using iteration

8. ©GoldSim Technology Group LLC., 2012 Benefits Optimal allocation of material Mass conservation Integrates handling of flow and transport Built-in storage functions Integrated handling of priorities and costs Influence lines represent flows

9. ©GoldSim Technology Group LLC., 2012 What do we mean by “Optimal”? Meaning 1: Maximal Profit (e.g., commodity distribution) –If the network is controlled by a single operator selling to multiple customers, then the goal is to maximize profit. –Example: Natural gas distributor (PSE) Meaning 2: Prioritized Flow (e.g., water distribution) –In this case water is divided up based on various users’ priorities:  Priority 1 users get first dibs on water until all their demands are met…  …and so on until the lowest priority (farmers) get what’s left over.  The prioritized flow method uses the same underlying functions as maximal profit

10. ©GoldSim Technology Group LLC., 2012 Flowing “Media” The quantity of liquids and/or solids that flow from one discrete location to another. Examples: water, CO 2, rocks, sediment in water. Assume incompressible and volume is additive, taking porosity of any solid media into account. –For example, 1 gallon of water dumped into a tank containing 1 gallon of sediment whose porosity is 0.3 would consume a total volume of 1.7 gallons (1 gal of Water + (1 – 0.3)*1 gal of Sediment).

11. ©GoldSim Technology Group LLC., 2012 Cells (Any Flow Network Elements) Model elements that produce, consume, store, or route fluid. Examples: –Pump –Evaporation –Detention pond –A city –Stockpile

12. ©GoldSim Technology Group LLC., 2012 Flows (Influence Lines)

13. ©GoldSim Technology Group LLC., 2012 Flow Capacity and Costs

14. ©GoldSim Technology Group LLC., 2012 Source Cells A source cell feeds fluid into the system. Source cells have infinite supply, but their outflow rate(s) may be limited. Examples: –Rainfall in a particular geographic area –CO2 from a power plant –Sediments from erosion

15. ©GoldSim Technology Group LLC., 2012 Sink Cells A sink cell removes fluid from the system. The capacity of sink to absorb fluid is infinite, but the inflow rate may be limited. Examples: –Evaporation –Outflow from a river (model boundary) –Consumers

16. ©GoldSim Technology Group LLC., 2012 Zero-Volume Cells (Routers)

17. ©GoldSim Technology Group LLC., 2012 User-Specified Cells Cells found in current GoldSim version (CT module) Distinct from Routers, which have zero volume Implications on CT models (need volume for concentration to make sense)

18. ©GoldSim Technology Group LLC., 2012 Dynamic Volume Cells (Stores)

19. ©GoldSim Technology Group LLC., 2012 Store Cells Attributes

20. ©GoldSim Technology Group LLC., 2012 Demand Priorities and Revenues

21. ©GoldSim Technology Group LLC., 2012 Simple Example…