Dynamic Electric Power Supply Chains andTransportation Networks:

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

Anna Anna NagurneyNagurneyRadcliffeRadcliffe Institute for Advanced Study, Harvard University and Institute for Advanced Study, Harvard University andIsenberg School of Management, University of Massachusetts, AmherstIsenberg School of Management, University of Massachusetts, Amherst ZugangZugang Liu LiuDepartment of Finance and Operations Management, Isenberg School of Management,Department of Finance and Operations Management, Isenberg School of Management,University of Massachusetts,AmherstUniversity of Massachusetts,Amherst Monica-Gabriela Monica-Gabriela CojocaruCojocaruDepartment of Mathematics and Statistics, University of Department of Mathematics and Statistics, University of GuelphGuelphGuelphGuelph, Ontario, Canada, Ontario, Canada PatriziaPatrizia DanieleDanieleDepartment of Mathematics and Computer Sciences,Department of Mathematics and Computer Sciences,University of University of CataniaCatania, , CataniaCatania, Italy, Italy

INFORMS Annual Meeting Pittsburgh, PA, November 5-8, 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.

Electricity is Modernity

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Motivation

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

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

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

Inelastic, seasonal demand.Inelastic, seasonal demand.

Literature

Kahn (1998)Kahn (1998)Day et al. (2002)Day et al. (2002)SchweppeSchweppe et al. (1988) et al. (1988)Hogan (1992)Hogan (1992)ChaoChao and Peck (1996) and Peck (1996)Wu et al. (1996)Wu et al. (1996)WillemsWillems (2002) (2002)CasazzaCasazza and and DeleaDelea (2003) (2003)ZaccourZaccour (1998) and Singh (1999). (1998) and Singh (1999).

Objectives

The objective of this research was to develop a dynamic electric powerThe objective of this research was to develop a dynamic electric powersupply chain network equilibrium model with exogenous time-varyingsupply chain network equilibrium model with exogenous time-varyingdemanddemand

The theory that has originated from the study of transportationThe theory that has originated from the study of transportationnetworks was utilized to construct this time-dependent equilibriumnetworks was utilized to construct this time-dependent equilibriummodeling framework for electric power supply chain networksmodeling framework for electric power supply chain networks

The new dynamic electric power supply chain network model that weThe new dynamic electric power supply chain network model that wedeveloped in this research is also motivated by the unification ofdeveloped in this research is also motivated by the unification ofprojected dynamical systems theory and evolutionary (infinite-projected dynamical systems theory and evolutionary (infinite-dimensional) dimensional) variationalvariational inequalities inequalities

Outline

The static electric power network model with fixed demandsThe static electric power network model with fixed demands

The The supernetworksupernetwork equivalence of the electric power supply chain equivalence of the electric power supply chainnetworks and the transportation networksnetworks and the transportation networks

Overview of the transportation network equilibrium modelsOverview of the transportation network equilibrium modelsThe The supernetworksupernetwork equivalence of the transportation networks and equivalence of the transportation networks andthe electric power supply chain networks with fixed demandsthe electric power supply chain networks with fixed demands

The electric power supply chain network model with time-varyingThe electric power supply chain network model with time-varyingdemandsdemands

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

Some of the Related Literature

Beckmann, M. J., McGuire, C. B., and Beckmann, M. J., McGuire, C. B., and WinstenWinsten, C. B. (1956), Studies, C. B. (1956), Studiesin the Economics of Transportation. Yale University Press, Newin the Economics of Transportation. Yale University Press, NewHaven, Connecticut.Haven, Connecticut.

NagurneyNagurney, A (1999), Network Economics: A , A (1999), Network Economics: A VariationalVariational Inequality InequalityApproach, Second and Revised Edition, Approach, Second and Revised Edition, KluwerKluwer Academic Publishers, Academic Publishers,DordrechtDordrecht, The Netherlands., The Netherlands.

NagurneyNagurney, A., Dong, J., and Zhang, D. (2002), A Supply Chain, A., Dong, J., and Zhang, D. (2002), A Supply ChainNetwork Equilibrium Model, Network Equilibrium Model, Transportation Research E Transportation Research E 38, 281-303.38, 281-303.

NagurneyNagurney, A (2005), , A (2005), On the Relationship Between Supply Chain andOn the Relationship Between Supply Chain andTransportation Network Transportation Network EquilibriaEquilibria: A : A SupernetworkSupernetwork Equivalence with Equivalence withComputationsComputations, Appears in , Appears in Transportation Research ETransportation Research E 42: (2006) pp 42: (2006) pp293-316.293-316.

Some of the Related Literature(Cont’d )

NagurneyNagurney, A. and , A. and MatsypuraMatsypura, D. (2004), , D. (2004), A Supply Chain NetworkA Supply Chain NetworkPerspective for Electric Power Generation, Supply, Transmission, andPerspective for Electric Power Generation, Supply, Transmission, andConsumptionConsumption, , Proceedings of the International Conference onProceedings of the International Conference onComputing, Communications and Control TechnologiesComputing, Communications and Control Technologies, Austin, Texas,, Austin, Texas,Volume VI: (2004) pp 127-134.Volume VI: (2004) pp 127-134.

NagurneyNagurney, A. and Liu, Z (2005), , A. and Liu, Z (2005), Transportation Network EquilibriumTransportation Network EquilibriumReformulations of Electric Power Networks with ComputationsReformulations of Electric Power Networks with Computations..

Wu, K., Wu, K., NagurneyNagurney, A., Liu, Z. and , A., Liu, Z. and StranlundStranlund, J. (2006), , J. (2006), ModelingModelingGenerator Power Plant Portfolios and Pollution Taxes in ElectricGenerator Power Plant Portfolios and Pollution Taxes in ElectricPower Supply Chain Networks: A Transportation NetworkPower Supply Chain Networks: A Transportation NetworkEquilibrium TransformationEquilibrium Transformation, , Transportation ResearchTransportation Research DD 11: (2006) pp 11: (2006) pp171-190.171-190.

The Electric Power Supply Chain NetworkEquilibrium Model with Fixed Demands

Conservation of flow equations must hold for each generatorConservation of flow equations must hold for each generator

Power generator’s optimization problemPower generator’s optimization problem

The optimality conditions of the generatorsThe optimality conditions of the generators

The Behavior of Power Generator and TheirOptimality Conditions

The Behavior of Power Suppliers

Supplier’s optimization problemSupplier’s optimization problem

For notational convenience, we letFor notational convenience, we let

The Optimality Conditions of the Power Suppliers

The optimality conditions of the suppliersThe optimality conditions of the suppliers

Conservation of flow equations must holdConservation of flow equations must hold

The vector (The vector (QQ22*, *, ρρ33**) is an equilibrium vector if for each) is an equilibrium vector if for each s, k, v s, k, vcombination:combination:

The Equilibrium Conditionsat the Demand Markets

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

Definition 1Definition 1:: The equilibrium state of the electric power supply chain The equilibrium state of the electric power supply chainnetwork is one where the electric power flows between the tiers of thenetwork is one where the electric power flows between the tiers of thenetwork coincide and the electric power flows satisfy the sum of thenetwork coincide and the electric power flows satisfy the sum of theoptimality conditions of the power generators and the suppliers, andoptimality conditions of the power generators and the suppliers, andthe equilibrium conditions at the demand markets.the equilibrium conditions at the demand markets.

Variational Inequality Formulation

The Supernetwork Equivalence of Supply ChainNetwork Equilibrium

and Transportation Network Equilibrium

NagurneyNagurney, A. (2006), , A. (2006), On the Relationship Between Supply Chain andOn the Relationship Between Supply Chain andTransportation Network Transportation Network EquilibriaEquilibria: A : A SupernetworkSupernetwork Equivalence with Equivalence withComputationsComputations, , Transportation Research E (2006) Transportation Research E (2006) 42: (2006) pp 293-42: (2006) pp 293-316316

Smith, M. J. (1979), Existence, uniqueness, and stability of trafficSmith, M. J. (1979), Existence, uniqueness, and stability of trafficequilibriaequilibria. . Transportation Research 13BTransportation Research 13B, 259-304., 259-304.

DafermosDafermos, S. (1980), Traffic equilibrium and , S. (1980), Traffic equilibrium and variationalvariational inequalities. inequalities.Transportation ScienceTransportation Science 14, 42-54. 14, 42-54.

In equilibrium, the following conditions must hold for each O/D pairIn equilibrium, the following conditions must hold for each O/D pairand each path.and each path.

A path flow pattern is a transportation network equilibrium if and onlyA path flow pattern is a transportation network equilibrium if and onlyif it satisfies the if it satisfies the variationalvariational inequality: inequality:

Overview of the Transportation NetworkEquilibrium Model with Fixed Demands

Transportation Network Equilibrium Reformulationof the Electric Power Network Model with Fixed

Demands

The following conservation of flow equations must hold on theThe following conservation of flow equations must hold on theequivalent transportation network:equivalent transportation network:

Transportation Network Equilibrium Reformulationof the Electric Power Network Model with Fixed

Demands

Transportation Network Equilibrium Reformulationof the Electric Power Network Model with Fixed

DemandsWe can construct a feasible link flow pattern for the equivalentWe can construct a feasible link flow pattern for the equivalenttransportation network based on the corresponding feasible electrictransportation network based on the corresponding feasible electricpower flow pattern in the electric power supply chain network modelpower flow pattern in the electric power supply chain network modelin the following way:in the following way:

Transportation Network Equilibrium Reformulationof the Electric Power Network Model with Fixed

DemandsWe assign user (travel) costs on the links of the transportation networkWe assign user (travel) costs on the links of the transportation networkas follows:as follows:

Path costPath cost

We assign the (travel) demands associated with the O/D pairs asWe assign the (travel) demands associated with the O/D pairs asfollows:follows:

The (travel)The (travel) disutilities disutilities::

The equilibrium conditions:The equilibrium conditions:

Transportation Network Equilibrium Reformulationof the Electric Power Network Model with Fixed

Demands

Transportation Network Equilibrium Reformulationof the Electric Power Network Model with Fixed

Demands

The The variationalvariational inequality in link flow form inequality in link flow form

Transportation Network Equilibrium Reformulationof the Electric Power Network Model with Fixed

Demands

Finite-Dimentional Variational Inequalities andProjected Dynamical Systems Literature

Dupuis, P., Dupuis, P., NagurneyNagurney, A., (1993). Dynamical systems and , A., (1993). Dynamical systems and variationalvariationalinequalities. inequalities. Annals of Operations ResearchAnnals of Operations Research 44, 9-42. 44, 9-42.

NagurneyNagurney, A., Zhang, D., (1996). Projected Dynamical Systems and, A., Zhang, D., (1996). Projected Dynamical Systems andVariationalVariational Inequalities with Applications. Inequalities with Applications. KluwerKluwer Academic AcademicPublishers, Boston, Massachusetts.Publishers, Boston, Massachusetts.

NagurneyNagurney, A., Zhang, D., (1997). Projected dynamical systems in the, A., Zhang, D., (1997). Projected dynamical systems in theformulation, stability analysis, and computation of fixed demandformulation, stability analysis, and computation of fixed demandtraffic network traffic network equilibriaequilibria. . Transportation ScienceTransportation Science 31, 147-158. 31, 147-158.

More Finite-Dimentional Variational InequalitiesLiterature

Smith, M. J. (1979), Existence, uniqueness, and stability of trafficSmith, M. J. (1979), Existence, uniqueness, and stability of trafficequilibriaequilibria. . Transportation Research 13BTransportation Research 13B, 259-304., 259-304.

DafermosDafermos, S. (1980), Traffic equilibrium and, S. (1980), Traffic equilibrium and variational variational inequalities. inequalities.Transportation ScienceTransportation Science 14, 42-54. 14, 42-54.

NagurneyNagurney, A. (1999), Network Economics: A, A. (1999), Network Economics: A Variational Variational Inequality InequalityApproach, Second and Revised Edition,Approach, Second and Revised Edition, Kluwer Kluwer Academic Publishers, Academic Publishers,DordrechtDordrecht, The Netherlands., The Netherlands.

PatrikssonPatriksson, M. (1994), The Traffic Assignment Problem, Models and, M. (1994), The Traffic Assignment Problem, Models andMethods, VSP Utrecht.Methods, VSP Utrecht.

The Evolutionary Variational Inequalities andProjected Dynamical Systems Literature

CojocaruCojocaru, M.-G., , M.-G., JonkerJonker, L. B., (2004). Existence of solutions to, L. B., (2004). Existence of solutions toprojected differential equations in Hilbert spaces. projected differential equations in Hilbert spaces. Proceedings of theProceedings of theAmerican Mathematical SocietyAmerican Mathematical Society 132, 183–193. 132, 183–193.

CojocaruCojocaru, M.-G., , M.-G., DanieleDaniele, P., , P., NagurneyNagurney, A., (2005a). Projected, A., (2005a). Projecteddynamical systems and evolutionary dynamical systems and evolutionary variationalvariational inequalities via Hilbert inequalities via Hilbertspaces with applications. spaces with applications. Journal of Optimization Theory andJournal of Optimization Theory andApplicationsApplications 27, no. 3, 1-15. 27, no. 3, 1-15.

CojocaruCojocaru, M.-G., , M.-G., DanieleDaniele, P., , P., NagurneyNagurney, A., (2005b). Double-layered, A., (2005b). Double-layereddynamics: A unified theory of projected dynamical systems anddynamics: A unified theory of projected dynamical systems andevolutionary evolutionary variationalvariational inequalities. inequalities. European Journal of OperationalEuropean Journal of OperationalResearchResearch..

CojocaruCojocaru, M.-G., , M.-G., DanieleDaniele, P., , P., NagurneyNagurney, A. (2005c). Projected, A. (2005c). Projecteddynamical systems, evolutionary dynamical systems, evolutionary variationalvariational inequalities, applications, inequalities, applications,and a computational procedure. and a computational procedure. Pareto Optimality, Game Theory andPareto Optimality, Game Theory andEquilibriaEquilibria. A. . A. MigdalasMigdalas, P. M. , P. M. PardalosPardalos, and L. , and L. PitsoulisPitsoulis, editors,, editors,Springer Springer VerlagVerlag..

BarbagalloBarbagallo, A., (2005). Regularity results for time-dependent, A., (2005). Regularity results for time-dependentvariationalvariational and and quasivariationalquasivariational inequalities and computational inequalities and computationalprocedures. procedures. To appear in Mathematical Models and Methods inTo appear in Mathematical Models and Methods inApplied SciencesApplied Sciences..

More Evolutionary Variational Inequalities andProjected Dynamical Systems Literature

DanieleDaniele, P., , P., MaugeriMaugeri, A., , A., OettliOettli, W., (1998). , W., (1998). VariationalVariational inequalities inequalitiesand time-dependent traffic and time-dependent traffic equilibriaequilibria. . ComptesComptes RendueRendue AcademieAcademie des desScienceScience, Paris 326, , Paris 326, serieserie I, 10591062. I, 10591062.

DanieleDaniele, P., , P., MaugeriMaugeri, A., , A., OettliOettli, W., (1999). Time-dependent traffic, W., (1999). Time-dependent trafficequilibriaequilibria. . Journal of Optimization Theory and its ApplicationsJournal of Optimization Theory and its Applications 103, 103,543-555.543-555.

More Evolutionary Variational Inequalities andProjected Dynamical Systems Literature

Finite-Dimensional Projected Dynamical Systems

Finite-Dimensional Projected Dynamical Systems (Finite-Dimensional Projected Dynamical Systems (PDSsPDSs) (Dupuis and) (Dupuis andNagurneyNagurney (1993)) (1993))

PDSPDStt describes how the state of the network system approaches an describes how the state of the network system approaches anequilibrium point on the curve of equilibrium point on the curve of equilibriaequilibria at time t. at time t.

For almost every moment ‘t’ on the For almost every moment ‘t’ on the equilibriaequilibria curve, there is a curve, there is aPDSPDStt associated with it. associated with it.

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

Finite-Dimensional Projected Dynamical Systems

Definition:

Projected Dynamical Systemsand Finite-Dimensional Variational Inequalities

Definition:Definition:

wherewhere

with the projection operator with the projection operator given bygiven by

The feasible set is defined as followsThe feasible set is defined as follows

Infinite-Dimensional Projected Dynamical Systems

Evolutionary Variational Inequalities

Evolutionary Evolutionary VariationalVariational Inequalities ( Inequalities (EVIsEVIs))

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

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

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

Evolutionary Variational Inequalities

Define

where

EVI:

Projected Dynamical Systemsand Evolutionary Variational Inequalities

CojocaruCojocaru, , DanieleDaniele, and , and NagurneyNagurney (2005b) showed the following: (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

DefineDefine

EVI Formulation:EVI Formulation:

Feasible setFeasible set

The EVI Formulation of the Transportation NetworkModel with Time-Varying Demands

The Numerical Solution of Evolutionary VariationalInequalities

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

The vector field The vector field FF satisfies the requirement in the preceding Theorem. satisfies the requirement in the preceding Theorem.

We first We first discretizediscretize time horizon T. ( time horizon T. (BarbagalloBarbagallo, A., (2005) ), A., (2005) )

At each fixed time point, we solve the associated finite dimensionalAt each fixed time point, we solve the associated finite dimensionalprojected dynamical system projected dynamical system PDSPDStt

We use the Euler method to solve the We use the Euler method to solve the finite dimensional finite dimensional projectedprojecteddynamical system dynamical system PDSPDStt..

The Euler Method

The EVI Formulation of the Electric Power NetworkModel with Time-Varying Demands

We know that the electric power supply chain network equilibriumWe know that the electric power supply chain network equilibriumproblem with fixed demands can be reformulated as a fixed demandproblem with fixed demands can be reformulated as a fixed demandtransportation network equilibrium problem in path flows over thetransportation network equilibrium problem in path flows over theequivalent transportation network.equivalent transportation network.

Evolutionary Evolutionary variationalvariational inequality provides us with a dynamic version inequality provides us with a dynamic versionof the electric power supply chain network problem in which theof the electric power supply chain network problem in which thedemands and path costs vary over time.demands and path costs vary over time.

Evolutionary Evolutionary variationalvariational inequality is a dynamic (and infinite- inequality is a dynamic (and infinite-dimensional) version ofdimensional) version of variational variational inequality with the path costsinequality with the path costsdefined in (44).defined in (44).

Solving Electric Power Supply Chain NetworkModel with Time-Varying Demands

First, construct the equivalent transportation network equilibriumFirst, construct the equivalent transportation network equilibriummodelmodel

Solve the transportation network equilibrium model with time-varyingSolve the transportation network equilibrium model with time-varyingdemandsdemands

Convert the solution of the transportation network into the time-Convert the solution of the transportation network into the time-dependent electric power supply chain network equilibrium modeldependent electric power supply chain network equilibrium model

Dynamic Electric Power Supply Chain NetworkExamples with Computations

Example 1Example 1

Numerical Example 1

Generating cost functionsGenerating cost functions

Transaction cost functions of the productsTransaction cost functions of the products

Numerical Example 1

Operating cost functions of the suppliersOperating cost functions of the suppliers

Unit transaction cost between the suppliers and the demand marketsUnit transaction cost between the suppliers and the demand markets

Numerical Example 1

Three pathsThree paths

The time-varying demand functionThe time-varying demand function

Numerical Example 1

The Solution of Numerical Example 1

Explicit SolutionExplicit SolutionPath flowsPath flows

Travel disutilityTravel 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 as the firstThe network structure and the cost functions are the same as the firstexample.example.

The demand function is the step function:The demand function is the step function:

The explicit solution:The explicit solution:

Time-Dependent Equilibrium Path Flows forNumerical Example 2

Numerical Example 3

Generating cost functionsGenerating cost functions

Transaction cost functions of the productsTransaction cost functions of the products

Numerical Example 3

Operating cost function of the suppliersOperating cost function of the suppliers

Unit transaction costs between the suppliers and the demand marketsUnit transaction costs between the suppliers and the demand markets

Numerical Example 3

Four pathsFour paths

The time-varying demand functionsThe time-varying demand functions

Numerical Example 3

The Solution of Numerical Example 3

Numerical SolutionNumerical Solutiont=0t=0

t=1/2t=1/2

t=1t=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

Conclusions

We established the We established the supernetworksupernetwork equivalence of the electric power equivalence of the electric powersupply chain networks with transportation networks with fixedsupply chain networks with transportation networks with fixeddemands.demands.

This identification provided a new interpretation of equilibrium inThis identification provided a new interpretation of equilibrium inelectric power supply chain networks in terms of path flows.electric power supply chain networks in terms of path flows.

We utilized this isomorphism in the computation of the electricWe utilized this isomorphism in the computation of the electricpower supply chain network equilibrium with time-varyingpower supply chain network equilibrium with time-varyingdemands.demands.

Thank You!

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