Dynamic Electric Power Supply Chains andTransportation Networks:

an Evolutionary Variational Inequality Formulation (To appear in Transportation Research E)

Anna NagurneyRadcliffe Institute for Advanced Study, Harvard University andIsenberg School of Management, University of Massachusetts, Amherst Zugang LiuDepartment of Finance and Operations Management, Isenberg School ofManagement, University of Massachusetts,Amherst Monica-Gabriela CojocaruDepartment of Mathematics and Statistics, University of GuelphGuelph, Ontario, Canada Patrizia DanieleDepartment of Mathematics and Computer Sciences,University of Catania, Catania, Italy

21st European Conference on Operational ResearchIceland, July 4, 2006

Acknowledgements

This research was supported by NSF Grant No. IIS - 002647.

The first author also gratefully acknowledges support from theRadcliffe Institute for Advanced Study at Harvard Universityunder its 2005 – 2006 Fellowship Program.

The New Book

Supply Chain Network Economics(New Dimensions in Networks)

Anna Nagurney

Edward Elgar Publishing Inc.

Available July 2006!

Electricity is Modernity

tv.gsfc.nasa.gov

Motivation

In US: half a trillion dollars worth of net assetsConsumes almost 40% of domestic primary energyElectric power supply chains, provide the foundations for thefunctioning of our modern economies and societies.

Communication, transportation, heating, lighting, cooling,computers and electronics.August 14, 2003, blackout in the Midwest, the Northeastern UnitedStates, and Ontario, Canada.Two significant power outages during the month of September2003 – one in England and one in Switzerland and Italy.

Deregulation: from vertically integrated to competitive marketsIn US, Europe and many other countries

Inelastic, seasonal demand.

Literature

Kahn (1998)Day et al. (2002)Schweppe et al. (1988)Hogan (1992)Chao and Peck (1996)Wu et al. (1996)Willems (2002)Casazza and Delea (2003)Zaccour (1998) and Singh (1999).

Objectives

The objective of this research was to develop a dynamic electricpower supply chain network equilibrium model with exogenoustime-varying demand

The theory that has originated from the study of transportationnetworks was utilized to construct this time-dependentequilibrium modeling framework for electric power supply chainnetworks

The new dynamic electric power supply chain network modelthat we developed in this research is also motivated by theunification of projected dynamical systems theory andevolutionary (infinite-dimensional) variational inequalities

Outline

The static electric power network model with fixed demands

The supernetwork equivalence of the electric power supplychain networks and the transportation networks

Overview of the transportation network equilibrium modelsThe supernetwork equivalence of the transportation networks andthe electric power supply chain networks with fixed demands

The electric power supply chain network model with time-varying demands

Evolutionary variational inequalities and projected dynamicalsystems; Applications to transportation network equilibriumThe computation of the electric power supply chain networkequilibrium model with time-varying demands.

Some of the Related LiteratureBeckmann, M. J., McGuire, C. B., and Winsten, C. B. (1956),Studies in the Economics of Transportation. Yale UniversityPress, New Haven, Connecticut.

Nagurney, A (1999), Network Economics: A VariationalInequality Approach, Second and Revised Edition, KluwerAcademic Publishers, Dordrecht, The Netherlands.

Nagurney, A., Dong, J., and Zhang, D. (2002), A Supply ChainNetwork Equilibrium Model, Transportation Research E 38, 281-303.

Nagurney, A (2005), On the Relationship Between Supply Chainand Transportation Network Equilibria: A SupernetworkEquivalence with Computations, Appears in TransportationResearch E 42: (2006) pp 293-316.

Some of the Related Literature(Cont’d )

Nagurney, A. and Matsypura, D. (2004), A Supply ChainNetwork Perspective for Electric Power Generation, Supply,Transmission, and Consumption, Proceedings of theInternational Conference on Computing, Communications andControl Technologies, Austin, Texas, Volume VI: (2004) pp 127-134.

Nagurney, A. and Liu, Z (2005), Transportation NetworkEquilibrium Reformulations of Electric Power Networks withComputations.

Wu, K., Nagurney, A., Liu, Z. and Stranlund, J. (2006), ModelingGenerator Power Plant Portfolios and Pollution Taxes in ElectricPower Supply Chain Networks: A Transportation NetworkEquilibrium Transformation, Transportation Research D 11:(2006) pp 171-190.

The Electric Power Supply Chain NetworkEquilibrium Model with Fixed Demands

Conservation of flow equations must hold for each generator

Power generator’s optimization problem

The optimality conditions of the generators

The Behavior of Power Generator and TheirOptimality Conditions

The Behavior of Power Suppliers

Supplier’s optimization problem

For notational convenience, we let

The Optimality Conditions of the PowerSuppliers

The optimality conditions of the suppliers

The Equilibrium Conditionsat the Demand Markets

Conservation of flow equations must hold

The vector (Q2*, ρ3*) is an equilibrium vector if for each s, k, vcombination:

Electric Power Supply Chain NetworkEquilibrium (For Fixed Demands at the Markets)

Definition: The equilibrium state of the electric power supplychain network is one where the electric power flows between thetiers of the network coincide and the electric power flows satisfythe sum of the optimality conditions of the power generators andthe suppliers, and the equilibrium conditions at the demandmarkets.

Variational Inequality Formulation

Determine satisfying

The Supernetwork Equivalence of Supply ChainNetwork Equilibrium

and Transportation Network Equilibrium

Nagurney, A. (2006), On the Relationship Between SupplyChain and Transportation Network Equilibria: A SupernetworkEquivalence with Computations, Transportation Research E(2006) 42: (2006) pp 293-316

Overview of the Transportation NetworkEquilibrium Model with Fixed Demands

Smith, M. J. (1979), Existence, uniqueness, and stability oftraffic equilibria. Transportation Research 13B, 259-304.

Dafermos, S. (1980), Traffic equilibrium and variationalinequalities. Transportation Science 14, 42-54.

In equilibrium, the following conditions must hold for each O/Dpair and each path.

A path flow pattern is a transportation network equilibrium if andonly if it satisfies the variational inequality:

The Supernetwork Equivalence of the ElectricPower Network and the Transportation Network

The fifth chapter of the Beckmann, McGuire, and Winsten’sclassic book, “Studies in the Economics of Transportation”(1956), described some “unsolved problems” including a singlecommodity network equilibrium problem that the authors intuitedcould be generalized to capture electric power networks.

We took up this challenge of establishing the relationship andapplication of transportation network equilibrium models toelectric power networks.

Transportation Network EquilibriumReformulation of the Electric Power Network

Model with Fixed Demands

Transportation Network EquilibriumReformulation of the Electric Power Network

Model with Fixed DemandsThe following conservation of flow equations must hold on theequivalent transportation network:

Transportation Network EquilibriumReformulation of the Electric Power Network

Model with Fixed DemandsWe can construct a feasible link flow pattern for the equivalenttransportation network based on the corresponding feasibleelectric power flow pattern in the electric power supply chainnetwork model in the following way:

Transportation Network EquilibriumReformulation of the Electric Power Network

Model with Fixed DemandsWe assign user (travel) costs on the links of the transportationnetwork as follows:

Transportation Network EquilibriumReformulation of the Electric Power Network

Model with Fixed DemandsPath cost

We assign the (travel) demands associated with the O/D pairsas follows:

The (travel) disutilities:

Substitute into the equilibrium condition (27)

Transportation Network EquilibriumReformulation of the Electric Power Network

Model with Fixed Demands

Finite-Dimentional Variational Inequalities andProjected Dynamical Systems Literature

Dupuis, P., Nagurney, A., (1993). Dynamical systems andvariational inequalities. Annals of Operations Research 44, 9-42.

Nagurney, A., Zhang, D., (1996). Projected Dynamical Systemsand Variational Inequalities with Applications. Kluwer AcademicPublishers, Boston, Massachusetts.

Nagurney, A., Zhang, D., (1997). Projected dynamical systemsin the formulation, stability analysis, and computation of fixeddemand traffic network equilibria. Transportation Science 31,147-158.

More Finite-Dimentional Variational InequalitiesLiterature

Smith, M. J. (1979), Existence, uniqueness, and stability oftraffic equilibria. Transportation Research 13B, 259-304.

Dafermos, S. (1980), Traffic equilibrium and variationalinequalities. Transportation Science 14, 42-54.

Nagurney, A. (1999), Network Economics: A VariationalInequality Approach, Second and Revised Edition, KluwerAcademic Publishers, Dordrecht, The Netherlands.

Patriksson, M. (1994), The Traffic Assignment Problem, Modelsand Methods, VSP Utrecht.

The Evolutionary Variational Inequalities andProjected Dynamical Systems Literature

Cojocaru, M.-G., Jonker, L. B., (2004). Existence of solutions toprojected differential equations in Hilbert spaces. Proceedings ofthe American Mathematical Society 132, 183–193.

Cojocaru, M.-G., Daniele, P., Nagurney, A., (2005a). Projecteddynamical systems and evolutionary variational inequalities viaHilbert spaces with applications. Journal of Optimization Theoryand Applications 27, no. 3, 1-15.

Cojocaru, M.-G., Daniele, P., Nagurney, A., (2005b). Double-layered dynamics: A unified theory of projected dynamicalsystems and evolutionary variational inequalities. EuropeanJournal of Operational Research.

Cojocaru, M.-G., Daniele, P., Nagurney, A. (2005c). Projecteddynamical systems, evolutionary variational inequalities,applications, and a computational procedure. Pareto Optimality,Game Theory and Equilibria. A. Migdalas, P. M. Pardalos, and L.Pitsoulis, editors, Springer Verlag.

Barbagallo, A., (2005). Regularity results for time-dependentvariational and quasivariational inequalities and computationalprocedures. To appear in Mathematical Models and Methods inApplied Sciences.

More Evolutionary Variational Inequalities andProjected Dynamical Systems Literature

Daniele, P., Maugeri, A., Oettli, W., (1998). Variationalinequalities and time-dependent traffic equilibria. ComptesRendue Academie des Science, Paris 326, serie I, 10591062.

Daniele, P., Maugeri, A., Oettli, W., (1999). Time-dependenttraffic equilibria. Journal of Optimization Theory and itsApplications 103, 543-555.

More Evolutionary Variational Inequalities andProjected Dynamical Systems Literature

Finite-Dimensional Projected DynamicalSystems

Finite-Dimensional Projected Dynamical Systems (PDSs)(Dupuis and Nagurney (1993))

PDSt describes how the state of the network system approaches anequilibrium point on the curve of equilibria at time t.

For almost every moment ‘t’ on the equilibria curve, there is a PDStassociated with it.

A PDSt is usually applied to study small scale time dynamics, i.e [t,t+τ]

Finite-Dimensional Projected DynamicalSystems

Definition:

Projected Dynamical Systemsand Finite-Dimensional Variational Inequalities

Definition:

where

with the projection operator given by

The feasible set is defined as follows

Infinite-Dimensional Projected DynamicalSystems

Evolutionary Variational Inequalities

Evolutionary Variational Inequalities (EVIs)

EVI provides a curve of equilibria of the network system over afinite time interval [0,T]

An EVI is usually used to model large scale time, i.e, [0, T]

EVIs have been applied to time-dependent equilibrium problems intransportation, and in economics and finance.

Evolutionary Variational Inequalities

Define

where

EVI:

Projected Dynamical Systemsand Evolutionary Variational Inequalities

Cojocaru, Daniele, and Nagurney (2005b) showed the following:

Projected Dynamical Systemsand Evolutionary Variational Inequalities

A Pictorial of EVIs and PDSs

x(t1)

t=T

t=0

x(t1,0)

x(t2, 0)

x(t2)

x(t1, τ)

x(t2, τ)

PDSt1

PDSt2EVI

Define

EVI Formulation:

Feasible set

The EVI Formulation of the TransportationNetwork Model with Time-Varying Demands

The Numerical Solution of EvolutionaryVariational Inequalities

(Cojocaru, Daniele, and Nagurney (2005 a, b, c))

The vector field F satisfies the requirement in the precedingTheorem.

We first discretize time horizon T. (Barbagallo, A., (2005) )

At each fixed time point, we solve the associated finitedimensional projected dynamical system PDSt

We use the Euler method to solve the finite dimensionalprojected dynamical system PDSt.

The Euler Method

The EVI Formulation of the Electric PowerNetwork Model with Time-Varying Demands

We know that the electric power supply chain networkequilibrium problem with fixed demands can be reformulated asa fixed demand transportation network equilibrium problem inpath flows over the equivalent transportation network.

Evolutionary variational inequality (61) provides us with adynamic version of the electric power supply chain networkproblem in which the demands vary over time, where the pathcosts are given by (44) but these are functions of path flows thatnow vary with time.

Evolutionary variational inequality (61) is a dynamic (and infinite-dimensional) version of variational inequality (50) with the pathcosts defined in (44).

Solving Electric Power Supply Chain NetworkModel with Time-Varying Demands

First, construct the equivalent transportation network equilibriummodel

Solve the transportation network equilibrium model with time-varying demands

Convert the solution of the transportation network into the time-dependent electric power supply chain network equilibriummodel

Dynamic Electric Power Supply Chain NetworkExamples with Computations

Example 1

Numerical Example 1

Generating cost functions

Transaction cost functions of the products

Numerical Example 1

Operating cost functions of the suppliers

Unit transaction cost between the suppliers and the demandmarkets

Numerical Example 1

Three paths

The time-varying demand function

Numerical Example 1

The Solution of Numerical Example 1

Explicit SolutionPath flows

Travel disutility

Time-Dependent Equilibrium Path Flows forNumerical Example 1

The Solution of Numerical Example 1

t=0

The Solution of Numerical Example 1

t=1/2

The Solution of Numerical Example 1

t=1/2t=1

Numerical Example 2

The network structure and the cost functions are the same asthe first example.

The demand function is the step function:

The explicit solution:

Time-Dependent Equilibrium Path Flows forNumerical Example 2

Numerical Example 3

Generating cost functions

Transaction cost functions of the products

Numerical Example 3

Operating cost function of the suppliers

Unit transaction costs between the suppliers and the demandmarkets

Numerical Example 3

Four paths

The time-varying demand functions

Numerical Example 3

The Solution of Numerical Example 3

Numerical Solutiont=0

t=1/2

t=1

t=0

The Solution of Numerical Example 3

The Solution of Numerical Example 3

t=1/2

The Solution of Numerical Example 3

t=1t=1

ConclusionsWe established the supernetwork equivalence of the electricpower supply chain networks with transportation networkswith fixed demands.

This identification provided a new interpretation ofequilibrium in electric power supply chain networks in termsof path flows.

We utilized this isomorphism in the computation of theelectric power supply chain network equilibrium with time-varying demands.

Thank You!

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