Modeling Infrastructure and Network Industries: Theory and Applications * Steven A. Gabriel Steven A. Gabriel Project Management Program, Dept. of Civil & Project Management Program, Dept. of Civil & Env Env. Engineering, University of Maryland, . Engineering, University of Maryland, College Park Maryland, 20742 USA College Park Maryland, 20742 USA Applied Mathematics and Scientific Computation Program, University of Maryland, College ty of Maryland, College Applied Mathematics and Scientific Computation Program, Universi Park, Maryland 20742 USA Park, Maryland 20742 USA Gilbert F. White Fellow, Resources for the Future, Washington, DC USA (2007 C USA (2007- -2008) 2008) Gilbert F. White Fellow, Resources for the Future, Washington, D Visiting Scholar, LMI Research Institute, McLean, Virginia, USA (2007 (2007- -2008) 2008) Visiting Scholar, LMI Research Institute, McLean, Virginia, USA Presented at Presented at Infraday 2007 2007 Infraday Berlin, Germany Berlin, Germany October 6, 2007 October 6, 2007 *National Science Foundation Funding, Division of Mathematical Sciences, Awards ciences, Awards 0106880, 0408943 *National Science Foundation Funding, Division of Mathematical S
Outline of Presentation � Briefly, My Background � From Optimization to Complementarity Problems then on to MPECs and EPECs: Why All the Fuss? – Complementarity Problem Application: Natural Gas Market Equilibrium � Stochastic Optimization Models – Stochastic Multiobjective Optimization Application: Telecommunications Network Reconfiguration � Conclusions and Future Work � General invitation to Trans-Atlantic Critical Infrastructure Modeling Conference at Univ. of Maryland, Nov. 2, 2007 2 2
My Background 3 3
University of Maryland � My Affiliations – Department of Civil & Environmental Engineering – Applied Mathematics and Scientific Computation Program – Engineering and Public Policy Program (joint between Engineering School and Public Policy School) 4 4
Overview of Research � Research: Main Topics – Mathematical modeling in engineering-economic systems usually involving critical infrastructure using optimization and equilibrium analysis • energy market models ( natural gas and electricity) • transportation/traffic • land development (Multiobjective optimization for “Smart Growth” in land development) • wastewater treatment (Optimization and statistical modeling in biosolids) • telecommunications (Optimization) – Development of algorithms for solving equilibria in energy & transportation systems and other planning problems – Development of general purpose algorithms for equilibrium models (using the nonlinear complementarity format) 5 5
Overview of Research Design of Mathematical Optimization/ Modeling Complementarity of Algorithms Critical Infrastructure Analysis of Public Policy Issues 6 6
From Optimization to Complementarity Problems then on to MPECs and EPECs: Why All the Fuss? 7 7
Example of an Equilibrium Problem A Variation on a Transportation Problem c ij 10 i j S 1 =20 D 1 =10 1 1 5 4 2 D 2 =10 6 2 S 2 =20 2 3 D 3 =10 10 Supplies Demands 8 8
Example of an Equilibrium Problem A Variation on a Transportation Problem θ 1 = 9 ψ 1 = 0 Solution: 0 S 1 =20 D 1 =10 1 1 • flow on arcs • dual prices at 0 nodes θ 2 = 5 10 2 D 2 =10 10 ψ 2 = 3 10 θ 3 = 4 S 2 =20 2 3 D 3 =10 0 Supplies Demands 9 9
Example of an Equilibrium Problem A Variation on a Transportation Problem Optimality conditions are of the form ψ + ≥ θ = = c , 1,2, 1,2,3 i j ij i j > ⇒ ψ = θ 0 +c ,(+ other conditions) x ij ij i j Example: ψ = + ≥ θ = > +c 0 4 4 and 10 x 1 13 3 13 ψ = + > θ = = +c 3 10 4 and 0 x 2 23 3 23 10 10
Example of an Equilibrium Problem A Variation on a Transportation Problem Re marks : 1 . The supply and demand quantities were given as constants, this is less realistic than allowing them to vary as a function ψ = θ = of the appropriat e prices ( , i 1,2 for supply, , j 1 , 2 , 3 for demand) i j why? 2. Can generalize the optimality conditions stated before using price - dependent supply and demand 11 11
Example of an Equilibrium Problem A Variation on a Transportation Problem Assume the following (inverse) supply and demand functions : Supply ( ) ψ i S ( ) i ψ = − S S 20 1 1 1 ( ) ψ = − S 0 . 2 S 1 S 2 2 2 ( ) i ψ S i i ( ) θ D ( ) Demand j j θ j D ( ) j θ = − D 19 D 1 1 1 ( ) θ = − D 10 0 . 5 D D 2 2 2 j ( ) θ = − D 14 D 3 3 3 12 12
Example of an Equilibrium Problem A Variation on a Transportation Problem Complete Optimality Conditions ( ) ( ) + ψ ≥ θ ≥ = = c , 0, 1,2, 1,2,3 S D x i j ij i i j j ij ( ) ( ) > ⇒ + ψ = θ 0 c x S D ij ij i i j j 3 ∑ = = , 1,2 S x i i ij = 1 j 2 ∑ = = , 1,2,3 D x j j ij = 1 i This is an example of a complementarity problem (Spatial Price Equilibrium) 13 13
Complementarity Problems vis-à-vis Optimization and Game Theory Problems Complementarity Problems NLP Other non-optimization based problems e.g., spatial price LP equilibria, traffic QP equilibria, Nash- Cournot games, zero- finding problems 14 14
Complementarity Problems and Variational Inequalities Variational Inequality Problems But, when polyhedral constraints, VI is Complementarity a special case of Problems the mixed complementarity problem 15 15
Optimization vs. Complementarity Problems � Complementarity problems are more general covering: – Zero-finding problems – Optimization problems (via Karush-Kuhn-Tucker conditions) – Game Theory problems (e.g., Bimatrix or Nash-Cournot games) – Host of other interesting problems in engineering and economics � Thus, theorems and algorithms designed for CPs can be applied to a wide variety of applications � Some problems have no natural optimization counterpart (e.g., via Principle of Symmetry), therefore, can only use CPs in this context � CPs very useful for solving policy-related network infrastructure problems (cf. SPE) – Can include some network participants having market power – Can include other players as price-takers 16 16
Optimization vs. Complementarity Problems (con’t) � Complementarity problems can also include problems in which prices (Lagrange multipliers) appear in the primal formulation – PIES energy infrastructure model of the 1970s – More generally infrastructure models whose modules might represent a detailed sector (e.g., power production) and for which subsets of prices and quantities (and other variables) are passed between these modules, e.g., National Energy Modeling System S. A. Gabriel, A. S. Kydes, P. Whitman, 2001. "The National Energy Modeling System: A Large-Scale Energy-Economic Equilibrium Model," Operations Research , 49 (1), 14-25. Source: http://enduse.lbl.gov/Projects/NEMS.gif Source: http:// enduse.lbl.gov/Projects/NEMS.gif 17 17
Extensions of Complementarity Problems: MPECs and EPECs � Stackelberg Games or More Generally MPECs – What if two-level problem where top level is a dominant company or the government and bottom level is the rest of the market – This is no longer a complementarity problem since all the players are not at the same level – Instead it’s an example of a mathematical program with equilibrium constraints (MPEC) 18 18
Extensions of Complementarity Problems: MPECs and EPECs � Stackelberg Games or More Generally MPECs – x upper-level planning variables, y lower-level variables, S(x) solution set of lower-level problem (e.g., Nash-Cournot game or optimization) – Lately a number of research papers on MPECs in energy infrastructure planning, transportation planning, etc. min ( , ) f x y ∈Ω . . s t x ∈ ( ) y S x 19 19
Extensions of Complementarity Problems: MPECs and EPECs � EPECs – Can also make the top level a game to get equilibrium problems with equilibriuim constraints (EPEC) � MPECs and EPECs are hard problems for several reasons – Feasible region not generally known in closed form (can use KKT conditions though) – Instance of global optimization problem � Advantages for regulators – Can more accurately reflect market behaviors when both strategic players exist in combination with non-strategic ones – Can allow regulators to see what effects for certain potential regulations or policies might be on the market with better feedback mechanisms 20 20
Example of Complementarity Problem for Natural Gas Infrastructure Planning 21 21
The Natural Gas Supply Chain DISTRIBUTION SYSTEM CITY GATE STATION INDUSTRIAL COMMERCIAL TRANSMISSION SYSTEM ELECTRIC POWER RESIDENTIAL GAS PROCESSING PLANT UNDERGROUND STORAGE Impurities Liquid Gaseous Compressor Station Products Products GAS PRODUCTION From well-head Cleaner Associated Gas and Oil Well Gas Well to burner-tip 22 22
Recommend
More recommend
