DYNAMICS OF ELECTRIC POWER SUPPLY CHAIN NETWORKS UNDER RISK AND UNCERTAINTY

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1 DYNAMICS OF ELECTRIC POWER SUPPLY CHAIN NETWORKS UNDER RISK AND UNCERTAINTY Dmytro Matsypura and Anna Nagurney International Conference on Computational Management Science, March 31 - April 3, 2005 University of Florida, Gainesville

2 Motivation The transformation of the electric power industry from a regulated to a competitive industry - In the US, - In the EU, - And in many other countries The dramatic increase in the number of market participants - By the end of 2000, approximately 20% of all electric utility generating capacity had been sold or transferred to unregulated companies - In the US alone as of February 1, 2005, more than 1200 companies were eligible to sell wholesale power at market-based rates (source: Changes in electricity trading patterns

3 [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 (North American Electric Reliability Council (1998)) There is a critical need to be sure that reliability is not taken for granted as the industry restructures, and thus does not fall through the cracks (Secretary of Energy Advisory Board s (SEAB) Task Force on Electric System Reliability (1998))

4 Schweppe et al., 1988 Hogan, 1992 Chao and Peck, 1996 Wu et al., 1996 Kahn, 1998 Singh, 1999 These Concerns have Stimulated Much Research Activity Jing-Yuan and Smeers, 1999 Hobbs et al., 2000 Day et al.,

5 Despite all the Analytical Efforts... August 14, The biggest power outage in US history occurred - Approximately 50 million people were left without electricity - 61,800 megawatts of electric load were affected - Estimates of total associated cost in the US range between $4 and $10 billion - In Canada, the gross domestic product was down 0.7%, there was a net loss of 18.9 million work hours September A major outage occurred in England - Significant outage initiated in Switzerland and cascaded over much of Italy

6 Dynamic model A Novel Approach Based on a supply chain network framework Several different types of decision-makers: Power Generators Power Suppliers Consumers with Multiple Transmission Service Providers Multicriteria decision-making Explicit modeling of the behavior of decision-makers Computation of dynamic trajectories of the evolution of the electric power flows, as well as the prices associated with various transactions

7 References Dupuis, P. and A. Nagurney (1993), Dynamical Systems and Variational Inequalities, Annals of Operations Research 44, 9-42 Nagurney, A. and D. Zhang (1996), Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers, Boston, Massachusetts Nagurney, A. and D. Matsypura (2005), A Supply Chain Network Perspective for Electric Power Generation, Supply, Transmission, and Consumption, to appear in Advances in Computational Economics, Finance and Management Science, E. J. Kontoghiorghes and C. Gatu, editors, Kluwer Academic Publishers, Dordrecht, The Netherlands.

8 The Electric Power Supply Chain Network Power Suppliers Power Generators 1 g G 3 1 s S Transmission Service Providers 1 t T 1 t T 1 k K Demand Markets

9 Notation Flow between generator g and supplier s Flow between supplier s and demand market k through TSP t Generating cost function of generator g Transaction cost function of generator g Risk function of generator g Operating cost function of supplier s Transaction cost function of supplier s Transaction cost function of supplier s Risk function of supplier s Demand function Expected demand function Price at power generator g for supplier s Price at power supplier s for demand market k Price at demand market k Clearing price at supplier s Speed of adjustment q gs qsk t f g c gs r g c s ĉ gs c t sk r s ˆdk d k ρ 1gs ρ t 2sk ρ 3k γ s φ

10 Demand Market Price Dynamics ρ 3k = { φk (d k (ρ 3 ) S T s=1 t=1 qt sk), if ρ 3k > 0 max{0, φ k (d k (ρ 3 ) S T s=1 t=1 qt sk)} if ρ 3k = 0

11 The Dynamics of the Prices at the Power Suppliers γ s = { φ s ( K T k=1 t=1 qt sk G q g=1 gs), if γ s > 0 max{0, φ s ( K T k=1 t=1 qt sk G q g=1 gs)}, if γ s = 0

12 Multicriteria Decision-Making Behavior of the Power Generators Maximize U g = subject to: S ρ 1gs q gs f g (Q 1 ) s=1 q gs 0, s S c gs (Q 1 ) α g r g (Q 1 ) s=1

13 Multicriteria Decision-Making Behavior of the Power Suppliers Maximize U s = K T ρ t 2skq t sk c s (Q 1, Q 2 ) G ρ 1gs q gs G ĉ gs (Q 1 ) k=1 t=1 g=1 g=1 K T c t sk(q 2 ) β s r s (Q 1, Q 2 ) subject to: k=1 t=1 K T q t sk G q gs k=1 t=1 g=1 q gs 0, g q t sk 0, k, t

14 Dynamics of the Electricity Transactions between Power Generators and Power Suppliers q gs = φ gs (γ s f g(q 1 ) q gs c gs(q 1 ) q gs α g r g (Q 1 ) q gs c s(q 1,Q 2 ) q gs ĉ gs(q 1 ) q gs β s r s (Q 1,Q 2 ) q gs ), if q gs > 0 max{0, φ gs (γ s f g(q 1 ) q gs c gs(q 1 ) q gs α g r g (Q 1 ) q gs c s(q 1,Q 2 ) q gs ĉ gs(q 1 ) q gs β s r s (Q 1,Q 2 ) q gs )}, if q gs = 0

15 Dynamics of the Electricity Transactions between Power Suppliers and Demand Markets φ t sk(ρ 3k ĉ t sk(q 2 ) c s(q 1,Q 2 ) q t sk ct sk (Q2 ) q t sk q t sk = β s r s (Q 1,Q 2 ) q t sk γ s ), if q t sk > 0 max{0, φ t sk(ρ 3k ĉ t sk(q 2 ) c s(q 1,Q 2 ) q t sk β s r s (Q 1,Q 2 ) q t sk ct sk (Q2 ) q t sk γ s )}, if q t sk = 0

16 Projected Dynamical System where Ẋ = Π K (X, F (X)), X(0) = X 0, F (X) (F gs, F t sk, F s, F k ) g=1,...,g;s=1,...,s;t=1,...,t ;k=1,...,k, Π K is the projection operator of F (X) onto K at X, K R GS+SKT +S+K +, and X 0 = (Q 10, Q 20, γ 0, ρ 0 3) is the initial point

17 VI Formulation Since the feasible set K is a convex polyhedron, the set of stationary points of the projected dynamical system coincides with the set of solutions to the variational inequality problem given by: determine X K, such that + + S s=1 G S g=1 s=1 S K s=1 k=1 t=1 φ s φ gs [ fg (Q 1 ) q gs + c gs(q 1 ) q gs + α g r g (Q 1 ) q gs + c s(q 1, Q 2 ) q gs + ĉ gs(q 1 ) q gs T φ t sk G qgs g=1 [ cs (Q 1, Q 2 ) K T k=1 t=1 q t sk qsk t +β s r s (Q 1, Q 2 ) q gs + ct sk (Q2 ) q t sk [γ s γ s] + ] γs [q gs qgs] [q t sk qt sk ] + ĉ t sk (Q2 ) + β s r s (Q 1, Q 2 ) q t sk [ K S φ k k=1 (Q 1, Q 2, γ, ρ 3 ) K T s=1 t=1 q t sk d k(ρ 3) ] + γ s ρ 3k ] [ρ 3k ρ 3k ] 0,

18 The Euler Method The Algorithm Step 0: Initialization Set X 0 = (Q 10, Q 20, γ 0, ρ 0 3) K. Let T denote an iteration counter and set T = 1. Set the sequence {α T } so that T =1 α T =, α T > 0, and α T 0, as T Step 1: Computation Compute X T = (Q 1T, Q 2T, γ T, ρ T 3 ) 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

19 Theoretical Result The set of equilibria of the electric power supply chain network are independent of the speeds of adjustment Theorem: Assume that K is the convex polyhedron given by R+ N and that φ (φ 1,..., φ N ) is a vector of positive terms. Then, the set of stationary points of the projected dynamical system given above is equivalent to the set of solutions to the variational inequality VI(F, K) where: F (F 1,..., F N) and F (φ 1 F 1,..., φ n F N)

20 Numerical Examples: Network Topology Power Suppliers Transmission Service Provider Power Generators Demand Markets

21 Values of φ Associated Variable Parameter Experiment 1 Experiment 2 Experiment 3 Experiment 4 q 11 φ q 12 φ q 21 φ q 22 φ q 31 φ q 32 φ q11 1 φ q12 1 φ q13 1 φ q21 1 φ q22 1 φ q23 1 φ γ 1 φ γ 2 φ ρ 31 φ ρ 32 φ ρ 33 φ

22 Example 1 Power generating cost functions f 1 (q) = 2.5q q 1q 2 + 2q 1 f 2 (q) = 2.5q q 1q 2 + 2q 2 f 3 (q) =.5q q 1q 3 + 2q 3 Transaction cost functions c 11 (Q 1 ) =.5q q 11 c 12 (Q 1 ) =.5q q 12 c 21 (Q 1 ) =.5q q 21 c 22 (Q 1 ) =.5q q 22 c 31 (Q 1 ) =.5q q 31 c 32 (Q 1 ) =.5q q 32 Operating costs of power suppliers c 1 (Q 1, Q 2 ) =.5( 3 i=1 q i1) 2 c 2 (Q 1, Q 2 ) =.5( 3 i=1 q i2) 2 Expected demand functions d k (ρ 3k ) = E( ˆd k ) = (ρ 3k +1)

23 Solutions to Example 1 Computed Equilibrium Values Experiment 1 Experiment 2 Experiment 3 Experiment 4 Number of Iterations q q q q q q q q q q q q γ γ ρ ρ ρ

24 Dynamics of Electric Power Transaction q 11

25 Dynamics of Electric Power Transaction q 12

26 Dynamics of Electric Power Transaction q 1 11

27 Dynamics of Price γ 1

28 Dynamics of Price ρ 31

29 Example 2 Power generating cost functions f 1 (q) = 2.5q q 1q 2 + 2q 1 f 2 (q) = 2.5q q 1q 2 + 2q 2 f 3 (q) =.5q q 1q 3 + 2q 3 Transaction cost functions c 11 (Q 1 ) =.5q q 11 c 12 (Q 1 ) =.5q q 12 c 21 (Q 1 ) =.5q q 21 c 22 (Q 1 ) =.5q q 22 c 31 (Q 1 ) =.5q q 31 c 32 (Q 1 ) =.5q q 32 Operating costs of power suppliers c 1 (Q 1, Q 2 ) =.5( 3 i=1 q i1) 2 c 2 (Q 1, Q 2 ) =.5( 3 i=1 q i2) 2 Expected demand functions d k (ρ 3k ) = E( ˆd k ) = (ρ 3k +1) Risk function for generator 1 r 1 = ( 2 s=1 q 1s 2) 2 Associated weight α 1 = 1

30 Solutions to Example 2 Computed Equilibrium Values Experiment 1 Experiment 2 Experiment 3 Experiment 4 Number of Iterations q q q q q q q q q q q q γ γ ρ ρ ρ

31 Example 3 Power generating cost functions f 1 (q) = 2.5q q 1q 2 + 2q 1 f 2 (q) = 2.5q q 1q 2 + 2q 2 f 3 (q) =.5q q 1q 3 + 2q 3 Transaction cost functions c 11 (Q 1 ) =.5q q 11 c 12 (Q 1 ) =.5q q 12 c 21 (Q 1 ) =.5q q 21 c 22 (Q 1 ) =.5q q 22 c 31 (Q 1 ) =.5q q 31 c 32 (Q 1 ) =.5q q 32 Operating costs of power suppliers c 1 (Q 1, Q 2 ) =.5( 3 i=1 q i1) 2 c 2 (Q 1, Q 2 ) =.5( 3 i=1 q i2) 2 Expected demand functions d k (ρ 3k ) = E( ˆd k ) = (ρ 3k +1) Risk function for generator 1 r 1 = ( 2 s=1 q 1s 2) 2 Associated weight α 1 = 1 Risk function for supplier 2 r 2 = 3 g=1 q g2 Associated weight β 2 = 0.5

32 Solutions to Example 3 Computed Equilibrium Values Experiment 1 Experiment 2 Experiment 3 Experiment 4 Number of Iterations q q q q q q q q q q q q γ γ ρ ρ ρ

33 Example 4 Power generating cost functions f 1 (q) = 2.5q q 1q 2 + 2q 1 f 2 (q) = 2.5q q 1q 2 + 2q 2 f 3 (q) =.5q q 1q 3 + 2q 3 Transaction cost functions c 11 (Q 1 ) =.5q q 11 c 12 (Q 1 ) =.5q q 12 c 21 (Q 1 ) =.5q q 21 c 22 (Q 1 ) =.5q q 22 c 31 (Q 1 ) =.5q q 31 c 32 (Q 1 ) =.5q q 32 Operating costs of power suppliers c 1 (Q 1, Q 2 ) =.5( 3 i=1 q i1) 2 c 2 (Q 1, Q 2 ) =.5( 3 i=1 q i2) 2 Expected demand functions d 1 (ρ 31 ) = E( ˆd 1 ) = (ρ 31+1) d 2 (ρ 32 ) = E( ˆd 2 ) = (ρ 32+1) d 3 (ρ 33 ) = E( ˆd 3 ) = (ρ 33+1) Risk function for generator 1 r 1 = ( 2 s=1 q 1s 2) 2 Associated weight α 1 = 1 Risk function for supplier 2 r 2 = 3 g=1 q g2 Associated weight β 2 = 0.5

34 Solutions to Example 4 Computed Equilibrium Values Experiment 1 Experiment 2 Experiment 3 Experiment 4 Number of Iterations q q q q q q q q q q q q γ γ ρ ρ ρ

35 Example 5 Power generating cost functions f 1 (q) = 2.5q q 1q 2 + 2q 1 f 2 (q) = 2.5q q 1q 2 + 2q 2 f 3 (q) =.5q q 1q 3 + 2q 3 Transaction cost functions c 11 (Q 1 ) =.5q q 11 c 12 (Q 1 ) =.5q q 12 c 21 (Q 1 ) =.5q q 21 c 22 (Q 1 ) =.5q q 22 c 31 (Q 1 ) =.5q q 31 c 32 (Q 1 ) =.5q q 32 Operating costs of power suppliers c 1 (Q 1, Q 2 ) =.5( 3 i=1 q i1) 2 c 2 (Q 1, Q 2 ) =.5( 3 i=1 q i2) 2 Expected demand functions d k (ρ 3k ) = E( ˆd k ) = (ρ 3k +1)

36 Example 6 Power generating cost functions f 1 (q) = 2.5q q 1q 2 + 2q 1 f 2 (q) = 2.5q q 1q 2 + 2q 2 f 3 (q) =.5q q 1q 3 + 2q 3 Transaction cost functions c 11 (Q 1 ) =.5q q 11 c 12 (Q 1 ) =.5q q 12 c 21 (Q 1 ) =.5q q 21 c 22 (Q 1 ) =.5q q 22 c 31 (Q 1 ) =.5q q 31 c 32 (Q 1 ) =.5q q 32 Operating costs of power suppliers c 1 (Q 1, Q 2 ) =.5( 3 i=1 q i1) 2 c 2 (Q 1, Q 2 ) =.5( 3 i=1 q i2) 2 Expected demand functions d k (ρ 3k ) = E( ˆd k ) = (ρ 3k +1) Risk function for generator 1 r 1 = ( 2 s=1 q 1s 2) 2 Associated weight α 1 = 1

37 Example 7 Power generating cost functions f 1 (q) = 2.5q q 1q 2 + 2q 1 f 2 (q) = 2.5q q 1q 2 + 2q 2 f 3 (q) =.5q q 1q 3 + 2q 3 Transaction cost functions c 11 (Q 1 ) =.5q q 11 c 12 (Q 1 ) =.5q q 12 c 21 (Q 1 ) =.5q q 21 c 22 (Q 1 ) =.5q q 22 c 31 (Q 1 ) =.5q q 31 c 32 (Q 1 ) =.5q q 32 Operating costs of power suppliers c 1 (Q 1, Q 2 ) =.5( 3 i=1 q i1) 2 c 2 (Q 1, Q 2 ) =.5( 3 i=1 q i2) 2 Expected demand functions d k (ρ 3k ) = E( ˆd k ) = (ρ 3k +1) Risk function for generator 1 r 1 = ( 2 s=1 q 1s 2) 2 Associated weight α 1 = 1 Risk function for supplier 2 r 2 = 3 g=1 q g2 Associated weight β 2 = 0.5

38 Example 8 Power generating cost functions f 1 (q) = 2.5q q 1q 2 + 2q 1 f 2 (q) = 2.5q q 1q 2 + 2q 2 f 3 (q) =.5q q 1q 3 + 2q 3 Transaction cost functions c 11 (Q 1 ) =.5q q 11 c 12 (Q 1 ) =.5q q 12 c 21 (Q 1 ) =.5q q 21 c 22 (Q 1 ) =.5q q 22 c 31 (Q 1 ) =.5q q 31 c 32 (Q 1 ) =.5q q 32 Operating costs of power suppliers c 1 (Q 1, Q 2 ) =.5( 3 i=1 q i1) 2 c 2 (Q 1, Q 2 ) =.5( 3 i=1 q i2) 2 Expected demand functions d 1 (ρ 31 ) = E( ˆd 1 ) = (ρ 31+1) d 2 (ρ 32 ) = E( ˆd 2 ) = (ρ 32+1) d 3 (ρ 33 ) = E( ˆd 3 ) = (ρ 33+1) Risk function for generator 1 r 1 = ( 2 s=1 q 1s 2) 2 Associated weight α 1 = 1 Risk function for supplier 2 r 2 = 3 g=1 q g2 Associated weight β 2 = 0.5

39 Solutions to Examples 5 8 Computed Equilibrium Values Example 5 Example 6 Example 7 Example 8 Number of Iterations q q q q q q q q q q q q γ γ ρ ρ ρ

40 Conclusions We proposed a novel, supply chain network framework for rigorous: modeling, qualitative analysis, computation of trajectories of dynamic evolutions of the electricity transactions between tiers of the electric power supply chain network as well as prices associated with the electric power at different distribution levels We have generalized the recent work of Nagurney and Matsypura (2004) to include supply-side risk and demand uncertainty We have established and theoretically proven that the set of equilibria are independent of the speeds of adjustment

41 Thank you! The full paper is available upon request

Dynamics of Electric Power Supply Chain Networks under Risk and Uncertainty

Dynamics of Electric Power Supply Chain Networks under Risk and Uncertainty Dmytro Matsypura and Anna Nagurney Department of Finance and Operations Management Isenberg School of Management University of