Modeling of Electric Power Supply Chain Networks with Fuel Suppliers via Variational Inequalities

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1 Modeling of Electric Power Supply Chain Networks with Fuel Suppliers via Variational Inequalities Anna Nagurney Zugang Liu Faculty of Economics and Business, The University of Sydney Radcliffe Institute for Advanced Study, Harvard University Department of Finance and Operations Management Isenberg School of Management, University of Massachusetts Amherst February 2007

2 Acknowledgments Ts research was supported in part by NSF Grant No.:IIS Radcliffe Institute for Advanced Study at Harvard University John F. Smith Memorial Fund at the Isenberg School of Management Ts support is gratefully acknowledged

3 Motivation Introduction In US: half a trillion dollars worth of net assets Electric power supply chains provide the foundations of the functioning of our modern economies and societies Communication, transportation, heating, lighting, cooling, computing, entertainment, etc. Deregulation: from vertically integrated to competitive market In US, EU and many other countries Inelastic, seasonal demand

4 Motivation Introduction [In recent years] the adequacy of the bulk power transmission system has been challenged to support the movement of power in unprecedented amounts and in unexpected directions (NERC (1998)) Electricity transformed the past century, and it will be even more crucial in the years to come (EPRI (2003))

5 Earth at Night Introduction

6 Supply Chain Perspective Network equilibrium Several classes of decision-makers Optimization problem for every decision-maker is unique

7 Related Literature Introduction Beckmann, M. J., McGuire, C. B., and Winsten, C. B. (1956), Studies in the Economics of Transportation. Yale University Press, New Haven, Connecticut Nagurney, A (1999), Network Economics: A Variational Inequality Approach, Second and Revised Edition, Kluwer Academic Publishers, Dordrecht, The Netherlands Nagurney, A. and Matsypura, D. (2004), A Supply Chain Network Perspective for Electric Power Generation, Supply, Transmission, and Consumption, Proceedings of the International Conference in Computing, Communications and Control Technologies, Austin, Texas, Volume VI,

8 Related Literature cont d Nagurney, A., Dong, J., and Zhang, D. (2002), A supply chain network equilibrium model, Transportation Research E 38, Nagurney, A (2006), On the Relationsp Between Supply Chain and Transportation Network Equilibria: A Supernetwork Equivalence with Computations, Transportation Research E 42, Wu, K., Nagurney, A., Liu, Z., and Stranlund, J. (2006), Modeling Generator Power Plant Portfolios and Pollution Taxes in : A Transportation Network Equilibrium Transformation, Appears in Transportation Research D 11,

9 Characteristics of the Model Explicit modeling of fuel suppliers Spatially distributed generation plants owned by one company Self-supply generation Inelastic demand

10 Electric Power Supply Chain Network 11 1I H1 HI Fuel Supplier/Fuel Type Combinations A1 AI Alternative Uses 11 1n 1N G1 Gn GN 11 gm GM Power Generator/ Power Plant Combinations 1 S Power Suppliers 1 T 1 T 1 T 1 T Transmission Service Providers 1 K Demand Markets

11 Decision-Making Behavior of Fuel Supplier h Maximize Profit Maximize subject to: I G N i=1 g=1 n=1 ρ gn 0 G N q gn c (q ) g=1 n=1 c gn (q gn ) G N g=1 n=1 q gn q gn U, 0, i, g, n. i,

12 Electric Power Supply Chain Network 11 1I H1 HI Fuel Supplier/Fuel Type Combinations A1 AI Alternative Uses 11 1n 1N G1 Gn GN 11 gm GM Power Generator/ Power Plant Combinations 1 S Power Suppliers 1 T 1 T 1 T 1 T Transmission Service Providers 1 K Demand Markets

13 Decision-Making Behavior of Power Generator g Maximize Profit Maximize N S ρ 1gns qgns + K ρ 1gnk q gnk H I S K ρ gn q gn f 0 gn(q gn) c gns(q gns) c gnk (q gnk ) n=1 s=1 k=1 h=1 i=1 s=1 k=1 M S + ρ K 1gms qgms + ρ S 1gmk q gmk fgm(qgm) K c gms(q gms) c gmk (q gmk ) m=1 s=1 k=1 s=1 k=1 subject to: S K H I q gns + q gnk = s=1 k=1 h=1 i=1 α gn q gn, n, q gns 0, q gnk 0, q gms 0, q gmk 0, n, m, s, k.

14 Game Theory Introduction Competition game between decision-makers of the same class Coordination game between decision-makers of different classes

15 Example: Electric Power Supply Chain Network competition 1 h Fuel Suppliers cooperation competition 1 g Power Generators cooperation competition 1 s Power Suppliers

16 Example: Electric Power Supply Chain Network competition 1 h Fuel Suppliers cooperation competition 1 g Power Generators cooperation competition 1 s Power Suppliers

17 Example: Electric Power Supply Chain Network competition 1 h Fuel Suppliers cooperation competition 1 g Power Generators cooperation competition 1 s Power Suppliers

18 Decision-Making Behavior of Power Supplier s Maximize Profit Maximize K T G ρ t 2sk qt sk cs(q1, Q 2, Q 4 ) k=1 t=1 g=1 M ρ G 1gms qgms N ρ 1gns qgns m=1 g=1 m=1 g=1 n=1 G M G N K T ĉ gms(q gms) ĉ gns(q gns) c t sk (qt sk ) g=1 n=1 k=1 t=1 subject to: K T q t sk = G M G N q gms + q gns, k=1 t=1 g=1 m=1 g=1 n=1 q t sk 0, k, t, qgms 0, g, m, qgns 0, g, n.

19 Equilibrium Conditions for Demand Market k Prices { ρ t 2sk + ĉt sk (Q2 ) ρ 1gnk + ĉ gnk (Q3 ) = ρ 3k, if qt sk > 0 ρ 3k, if qt sk = 0 s, t { = ρ 3k, if q gnk > 0 ρ 3k, if q gnk = 0 g, n ρ 1gmk + ĉ gmk (Q5 ) { = ρ 3k, if q gmk > 0 ρ 3k, if q gmk = 0 g, m Flows S T d k = q t sk + G M G N q gmk + q gnk s=1 t=1 g=1 m=1 g=1 n=1

20 Variational Inequality Formulation Determine (Q 0, Q 1, Q 2, Q 3, Q 4, Q 5 ) K satisfying: G + g=1 m=1 s=1 H [ I G N c (q ) q gn + cgn (q gn ) q gn + fgn(q gn ) ] [ q gn ] h=1 i=1 g=1 n=1 [ G N S cgns(q gns + ) + cs(q 1, Q 2, Q 4 ) + ĉgns(q gns ) q g=1 n=1 s=1 gns q gns q gns [ G N K cgnk (qgnk + ) ] + ĉ gnk (Q 3 ) q g=1 n=1 k=1 gnk [ M S fgm(q gm ) + cgms(q gms ) + cs(q 1, Q 2, Q 4 ) + ĉgms(q gms ) ] q gms q gms q gms q gms [ G M K fgm(q gm + ) + c gmk (q gmk ) ] + ĉ gmk (Q 5 ) q g=1 m=1 k=1 gmk q gmk [ S K T cs(q 1, Q 2, Q 4 ) + q s=1 k=1 t=1 sk t + ct sk (qt sk ) ] q sk t + ĉ t [ sk (Q2 ) q gn q gn ] [ q gns q ] gns [ q gnk q ] gnk [ q gms q ] gms [ q gmk q ] gmk q t sk qt sk ] 0, (Q 0, Q 1, Q 2, Q 3, Q 4, Q 5 ) K

21 The Euler Method (Dupuis and Nagurney (1993)) Step 0: Initialization Set X 0 K. Let T denote an iteration counter. Let T = 1 and set the sequence {α T } so that T =1 α T =, α T > 0, α T 0, as T. Step 1: Computation Compute X T K by solving the variational inequality subproblem: X T + α T F (X T 1 ) X T 1, X X T 0, X K. Step 2: Convergence Verification If X T X T 1 ɛ, with ɛ > 0, a pre-specified tolerance, then stop; otherwise, set T := T + 1, and go to Step 1.

22 Questions? Comments? The full paper appears in International Journal of Emerging Electric Power Systems, 8(1). For related research please visit

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