1. Modeling and Computing Oligopolistic Strategic Forward Market Equilibrium in a Congested Electricity Network Shmuel Oren Earl J. Isaac Chair Professor Department of IEOR, UC Berkeley (Joint work with Jian Yao and Ilan Adler) Presented at The Electric Power Optimization Center University of Auckland, New Zealand October 14, 2008

2. All Rights Reserved to Shmuel Oren Scale of Regional Transmission Organizations (Cover half the states and 70% of load ) CAISO (1998) Population: 30 Million Peak Load: 45,500MW Annual Total Energy: 236,450GWh Generation Capacity: 52,000MW Average Net Imports: 6,500MW (Peak 8,300MW) Wholesale Average price: $56.7/MWh Annual Wholesale Market: $14 Billion PJM (1999) Population: 51 million Peak load: 144,644 MW Annual Total Energy: 729,000GWh Generating capacity: 164,905 MW Transmission lines - 56,250 miles Members/customers - 450+ Annual Wholesale Market: $40 Billion ERCOT (2001) (not under FERC) Population: 18 Million (85% of Texas) Peak Load: 63,000MW Annual Total Energy: 300,000GWh Generation Capacity: 80,000MW Average Net Imports: Non Wholesale Average price: ~$70/MWh Annual Wholesale Market: $20 Billion

3. All Rights Reserved to Shmuel Oren POWER INDUSTRY RESTRUCTURING Power Generation Micro-Grids Retail Providers Competitive State Regulated FERC Regulated Customers Demand Management Transmission Distribution SC - Scheduling Coordinator PX - Power Exchange ISO - Indep. System Operator TO - Transmission Owner LSE - Load Serving Entity SC - Scheduling Coordinator PX - Power Exchange ISO - Indep. System Operator TO - Transmission Owner LSE - Load Serving Entity Generation SC PX LSE TO ISO

4. All Rights Reserved to Shmuel Oren Market Organization

5. All Rights Reserved to Shmuel Oren Market Mechanics Day-Ahead Market closes Day-ahead Results Posted & Balancing Market Bid period opens Balancing Market Bid period closes Day-ahead Market  determines commitment profile that satisfies fixed demand, price sensitive demand bids, virtual bids and PJM Operating Reserve Objectives  minimizes total production cost Reserve Adequacy Assessment  focus is reliability  updated unit offers and availability  Based on PJM load forecast  minimizes startup and cost to run units at minimum Transmission Security Assessment  focus is reliability  performed as necessary starting two days prior to the operating day  Based on PJM Load Forecast

6. All Rights Reserved to Shmuel Oren Locational Real Time Marginal Prices at PJM and MISO

7. All Rights Reserved to Shmuel Oren Dispatch is reoptimized every five minutes and LMP updated to reflect shadow prices on transmission constraints

8. All Rights Reserved to Shmuel Oren Generation Resource Stack in WECC Demand shocks Supply Curve

9. All Rights Reserved to Shmuel Oren PJM Real-Time Market Prices Show Volatility

10. All Rights Reserved to Shmuel Oren

11. Features of Electricity Market Model Generators (Cournot Players) Load nodes (Demand functions) Transmission network (Lossless DC approximation of Kirchhoff laws) System Operator (Set import and export quantities and nodal prices so as to meet transmission constraints) Probabilistic Contingencies (Load uncertainty, Gen outages, Line outages)

12. All Rights Reserved to Shmuel Oren Oligopolistic Two Settlement Market Equilibrium  Generators enter into forward contracts to supply specified quantities at agreed upon prices (forward markets) and decide in real time (spot market) how much to produce.  Forward and spot markets may have different granularly of settlement points.  Nodal spot market  Forward contracts are settled at clusters of nodes (Hubs) based on a weighted average of the nodal spot prices.

13. All Rights Reserved to Shmuel Oren Motivation  Forward contracting  mitigates generators’ horizontal market power,  protects market participants against spot price volatility,  increases social welfare.  Cournot producers have incentives to forward contract in a single node system due to a prisoner’s dilemma effect (Allaz and Vila 1992, 1993). Is that conclusion valid in more realistic settings with:  network constraints,  demand uncertainty and  generation/transmission contingencies?  Can we develop a realistic computational tool for simulating market outcomes with endogenous contracting as needed for various applications.  market power analysis  assessing the economic value of transmission expansion projects

14. All Rights Reserved to Shmuel Oren Forward prices Two-settlement Model Structure Firms sign forward contracts (Financial CFDs) Nature picks a state c. Firms produce & ISO re-dispatches Nodal prices & Congestion rents Zonal settlement prices Expected zonal settlement prices Forward market (upper level) Spot market (Lower level) Solution Concept: Subgame Perfect Nash equilibrium

15. All Rights Reserved to Shmuel Oren The ISO Problem  Conducts the energy redispatch  Sets locational prices and transmission charges  Maximizes social welfare Import/Export quantity Balancing constraint Flows constraints Local production Inverse demand function

16. All Rights Reserved to Shmuel Oren KKT conditions for the ISO problem

17. All Rights Reserved to Shmuel Oren Modeling Choices  Cournot generation firms act as multi- Stackelberg leaders anticipating the outcome of the ISO redispatch  Cournot generation firms and ISO move simultaneously as Nash players taking each other’s strategic variables as parameters in their optimization problem  The ISO’s strategic variables are import/export quantities for each node (Cournot-Cournot)  The ISO’s strategic variables are the nodal injection charges = nodal premium relative to the slack bus price (Cournot-Bertrand)

18. All Rights Reserved to Shmuel Oren Generation Firms Anticipate ISO Redispatch

19. All Rights Reserved to Shmuel Oren Problem with Sequential Move Formulation  A two settlement model will result in a non- tractable three level optimization problem  Generation firms in spot market have incentive to induce degeneracy in ISO problem (Induce flow just below constraints to avoid congestion rents)  Non-uniqueness  Discontinuity in reaction functions  Possible non-existence of pure strategy equilibrium

20. All Rights Reserved to Shmuel Oren Simultaneous Move Model (generation firms do not account for impact on congestion) When demand function are linear and supply functions quadratic set of N+1 KKT conditions can be reduced to an LCP M is symmetric or bisymmetric PSD

21. All Rights Reserved to Shmuel Oren Implications  No effect of multiple ownership (problem is separable so that each unit is priced independently)  When there is no congestion import/exports variables are selected so as to equalize nodal prices across nodes.  The resulting market equilibrium is different than the Cournot equilibrium when nodal demand is aggregated (residual demands at each node retain the slope of the local demand function)

22. All Rights Reserved to Shmuel Oren Firms’ Optimization when ISO Strategic Variables are the Nodal Price Premiums Implicit residual demand function Firms do not see transmission constraints only nodal price premiums

23. All Rights Reserved to Shmuel Oren Market equilibrium for simultaneous move model where ISO strategic variables are the locational premiums (Cournot-Bertrand)  Can be reduced to an LCP when demand functions are linear and cost functions are quadratic M is a bisymmetric PSD matrix

24. All Rights Reserved to Shmuel Oren Implications  Ownership structure affects results  When there is no congestion nodal price premiums go to zero and market equilibrium is identical to the single node oligopoly solution.  The market equilibrium will reflect an oligopoly solution even when the market is separated (strategically decoupled) due to a thin line (zero capacity) or permanently congested line (fixed imports/exports)

25. All Rights Reserved to Shmuel Oren Firms’ programs with forward contracts for the Cournot-Bertrand Case Forward contract price Forward settlement price Forward quantity

26. All Rights Reserved to Shmuel Oren Forward Market Decisions and Complementarity conditions characterizing spot market EPEC Formulation – Each firm solves an MPEC

27. All Rights Reserved to Shmuel Oren The EPEC problem structure  The MPECs for each firm x g : decision variable, y,w : state variables, x -g : parameters  The EPEC …

28. All Rights Reserved to Shmuel Oren The EPEC Algorithm Round 0 Round 1 Round 2

29. All Rights Reserved to Shmuel Oren Stylized Belgian System 53 nodes, 71 lines, 19 generators, 6 contingency states

30. All Rights Reserved to Shmuel Oren Contingency states StateProbDescription 1.2On-peak state: All demands are on the peak. 2.5Normal state: Demands are at shoulder. 3.03Shoulder demands with line breakdown: Line [31,52] goes down. 4.03Shoulder demands with generation outage: Plant at node 10 goes down. 5.04Shoulder demands with generation outage: Plant at node 41 goes down. 6.2Off-peak state: All demands are off-peak.

31. All Rights Reserved to Shmuel Oren Impact of Forward Contracting on Spot Prices (in normal state) Single Settlement Two Settlement

32. All Rights Reserved to Shmuel Oren Firm’s Forward Commitments 2 Firms 3 Firms

33. All Rights Reserved to Shmuel Oren Test Case – WECC Light Summer 2005 Total number of buses2161 Number of generation buses401 Number of consumption buses1205 Number braches/transformers3398 Number of firms16 (9 strategic players) Total demand22700 MW

34. All Rights Reserved to Shmuel Oren Spot and forward trading Total installed capacity (MW)* Spot output under two settlements (MW) Forward contracts (MW) Southern California Edison224074887.1444.2 SDG&E32052132.20 WAPA – SNR825.8 Bureau of Reclamation (PG&E)14391208.60 PG&E customer owned facilities167204770.6508 Department of Water Resources914.3 Sacramento Utility District2119.51893.498.8 PG&E79213320.276.1 Northern California Power Agency633.4 Total5618520847.33500.5  Small units (with capacities less than 10MW) are ignored.  The total number of iterations is 79 (stopped with a relative error of 1e-5).

35. All Rights Reserved to Shmuel Oren Single settlementTwo settlement Range of nodal prices($/MWh) 27.9 ~ 179.628.0 ~ 117.4 % Change in nodal prices -56.5 ~ 29.9 Average nodal prices 87.965.6 Nodal Prices

36. All Rights Reserved to Shmuel Oren Questions?