Electric Power Markets: Process and Strategic Modeling

1
Electric Power MarketsProcess and Strategic
Modeling

• HUT (Systems Analysis Laboratory)
• Helsinki School of Economics
• Graduate School in
• Systems Analysis, Decision Making, and Risk
  Management
• Mat-2.194 Summer School on Systems Sciences
• Prof. Benjamin F. Hobbs
• Dept. of Geography Environmental Engineering
• The Johns Hopkins University
• Baltimore, MD 21218 USA
• bhobbs_at_jhu.edu

2
I. Overview of Course
Multifirm Models with Strategic Interaction
Single Firm Models
Single Firm Models
Design/ Investment Models
Design/ Investment Models
Operations/ Control Models
Operations/ Control Models
Demand Models
Market Clearing Conditions/Constraints
3
Why The Power Sector?

• Scope of economic impact
• 1000/person/yr in US (petroleum use)
• Almost half of US energy use

• Ongoing restructuring and reforms
• Vertical disintegration
• separate generation, transmission, distribution
• Competition in bulk generation
• grant access to transmission
• creation of regional spot forward markets
• Competition in retail sales
• Horizontal disintegration, mergers
• Privatization
• Emissions trading

Finlands 1995 Elect. Market Act
X X EL-EX, Nord Pool
X Fingrid None
4
Why Power? (Continued)

• Scope of environmental impact
• Transmission lines landscapes
• 3/4 of SO2, 1/3 of NOx, 3/8 of CO2 in US CO2
  increasing (50 by 2020 despite goals?)
• Power A horse of a different color
• Difficult to store ? must balance supply demand
  in real time
• Physics of networks
• North America consists of three synchronized
  machines
• What you do affects everyone else ? pervasive
  externalities must carefully control to maintain
  security. Example of externalities parallel
  flows resulting from Kirchhoffs laws

5
II. Process or Bottom-Up AnalysisCompany
Market Models

• What are bottom-up/engineering-economic models?
  And how can they be used for policy analysis?
• Explicit representation optimization of
  individual elements and processes based on
  physical relationships

D
6
Process Optimization Models

• Elements
• Decision variables. E.g.,
• Design MW of new combustion turbine capacity
• Operation MWh generation from existing coal
  units
• Objective(s). E.g.,
• Maximize profit or minimize total cost
• Constraints. E.g.,
• S Generation Demand
• Respect generation transmission capacity limits
• Comply with environmental regulations
• Invest in sufficient capacity to maintain
  reliability
• Traditional uses
• Evaluate investments under alternative scenarios
  (e.g., demand, fuel prices) (3-40 yrs)
• Operations Planning (8 hrs - 5 yrs)
• Real time operations (lt1 second - 1 hr)

7
Bottom-Up/Process Models vs. Top-Down Models

• Bottom-up models simulate investment operating
  decisions by an individual firm, (usually)
  assuming that that the firm cant affect prices
  for its outputs (power) or inputs (mainly fuel)
• Examples capacity expansion, production costing
  models
• Individual firm models can be assembled into
  market models
• Top-down models start with an aggregate market
  representation (e.g., supply curve for power,
  rather than outputs of individual plants).
• Often consider interactions of multiple markets
• Examples National energy models

8
Functions of Process Model Firm Level Decisions

• Real time operations
• Automatic protection (lt1 second) auto. generator
  control (AGC) methods to protect equipment,
  prevent service interruptions. (Responsibility
  of Independent System Operator ISO)
• Dispatch (1-10 minutes) optimization programs
  (convex) min. fuel cost, s.t. voltage, frequency
  constraints (ISO or generating companies GENCOs)
• Operations Planning
• Unit commitment (8-168 hours). Integer NLPs
  choose which generators to be on line to min.
  cost, s.t. operating reserve constraints (ISO
  or GENCOs)
• Maintenance production scheduling (1-5 yrs)
  schedule fuel deliveries storage and
  maintenance outages (GENCOs)

9
Firm Decisions Made Using Process Models,
Continued

• Investment Planning
• Demand-side planning (3-15 yrs) implement
  programs to modify loads to lower energy costs
  (consumer, energy services cos. ESCOs,
  distribution cos. DISCOs)
• Transmission distribution planning (5-15 yrs)
  add circuits to maintain reliability and minimize
  costs/ environmental effects (Regional
  Transmission Organization RTO)
• Resource planning (10 - 40 yrs) define most
  profitable mix of supply sources and D-S programs
  using LP, DP, and risk analysis methods for
  projected prices, demands, fuel prices (GENCOs)
• Pricing Decisions
• Bidding (1 day - 5 yrs) optimize offers to
  provide power, subject to fuel and power price
  risks (suppliers)
• Market clearing price determination (0.5- 168
  hours) maximize social surplus/match offers
  (Power Exchange PX, marketers)

10
Emerging Uses

• Profit maximization rather than cost minimization
  guides firms decisions
• Market simulation
• Use model of firms decisions to simulate market.
  Paul Samuelson
• MAX consumer producer surplus
• ? Marginal Cost Supply Marg. Benefit
  Consumption
• ? Competitive market outcome
• Other formulations for imperfect markets
• Price forecasts (averages, volatility)
• Effects of environmental policies on market
  outcomes (costs, prices, emissions impacts,
  income distribution)
• Effects of market design structure upon market
  outcomes

11
Advantages of Process Models for Policy Analysis

• Explicitness
• changes in technology, policies, prices,
  objectives can be modeled by altering
• decision variables
• objective function coefficients
• constraints
• assumptions can be laid bare
• Descriptive uses
• show detailed cost, emission, technology choice
  impacts of policy changes
• show changes in market prices, consumer welfare
• Normative
• identify better solutions through use of
  optimization
• show tradeoffs among policy objectives

12
Dangers of Process Models for Policy Analysis

• GIGO
• Uncertainty disregarded, or misrepresented
• Ignore intangibles, behavior (people adapt, and
  are not profit maximizers)
• Basic models overlook market interactions
• price elasticity
• power markets
• multimarket interactions
• Optimistic bias--overestimate performance of
  selected solutions

13
The Overoptimism of Optimization (e.g., B. Hobbs
A. Hepenstal, Water Resources Research, 1988
J. Kangas, Silvia Fennica, 1999)

• Auctions The winning bidder is cursed
• If there are many bidders, the lowest bidder is
  likely to have underestimated its cost--even if,
  on average, cost estimates are unbiased.
• Further, bidders whose estimates are error prone
  are more likely to win.
• Process models if many decision variables, and
  if their objective function coefficients are
  uncertain
• the cost of the winning (optimal) solution is
  underestimated (in expectation)
• investments whose costs/benefits are poorly
  understood are more likely to be chosen by the
  model
• E.g.
• this results in a downward bias in the long run
  cost of CO2 reductions
• there will be a bias towards choosing supply
  resources with more uncertain costs

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Conclusion

• What can process-based policy models do well?
• Exploratory modeling examining implications of
  assumptions/scenarios upon impacts/decisions
• Exploratory modeling is becoming easier
  because of increasingly nimble models, and is
  becoming more important because of increased
  uncertainty/complexity

15
Conclusions, continued

• What dont process-based models do well?
• Consolidative modeling assembling the best/most
  defensible data/assumptions to derive a single
  best answer
• Although computer technology makes
  comprehensive models more practical than ever,
  increased complexity and diverse perspectives
  makes consensus difficult
• Models are for insight, not numbers
• (Geoffrion)

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III. Operations Model System Dispatch LP

• Basic model (cost minimization, no transmission,
  pure thermal system, no storage, deterministic,
  no 0/1 commitment variables, no combined
  heat/power). In words
• Choose level of operation of each generator to
  minimize total system cost subject to demand
  level
• Decision variable
• yift megawatt MW output of generating unit i
  (i1,..,I) during period t (t1,..,T) using fuel
  f (f1,,F(i))
• Coefficients
• CYift variable operating cost /MWh for yift
• Ht length of period t h/yr. (Note in pure
  thermal system, periods do not need to be
  sequential)
• Xi MW capacity of generating unit i. (Note may
  be derated for random forced outages FORi
  )
• CFi maximum capacity factor for unit i
• LOADt MW demand to be met in period t

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Operations LP

• MIN Variable Cost Si,f,t Ht CYift yift
• subject to
• Si,f yift LOADt ?t
• Sf yift lt (1- FORi)Xi ?i,t
• Sf,t Ht yift lt CFi 8760Xi ?i
• yift gt 0 ?i,f,t

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Using Operating Models to Assess NOx
RegulationThe Inefficiency of Rate-Based
Regulation(Leppitsch Hobbs, IEEE Trans. Power
Systems, 1996)

• NOx an ozone precursor
• N2 O2 heat ? NOx
• NOx VOC O ? O3
• Power plants emit 1/3 of anthropogenic NOx in
  USA

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Policy Question Addressed

• How effectively can NOx limits be met by changed
  operations (emissions dispatch)?
• What is the relative efficiency of
• Regulation based on tonnage caps
• Total emissions tons lt Tonnage cap
• Regulation based on emission rate limits
  (tons/GJ)?
• (Total Emissionstons/Total Fuel Use GJ)
• lt Rate Limit

20
Framework

• We want less cost and less NOx
• Cost
• 
  Inefficient
• Efficient
• NOx
• Why might rate-based policies be inefficient?
• Dilution effect Increase denominator rather than
  decrease numerator of (NOx/Fuel Input)
• Discourage imports of clean energy (since they
  would lower both numerator denominator--even
  though they lower total emissions)

Alternative dispatch order
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How To Generate Alternatives

• Solve the following model for alternative
  levels of the regulatory constraint
• MIN Si CYi yi
• s.t. 1. MRi lt yi lt Xi
  (note nonzero LB)
• 2. Si yi gt LOAD (MW)
• 3. Regulatory caps either
• Si Ei yi lt MASS CAP (tons) or
• (Si Ei yi )/(Si HRi yi) lt RATE CAP (tons/GJ)
• Notes 1. MR, X, LOAD vary (used a stochastic
  programming method probabilistic production
  costing with side constraints)
• 2. Separate caps can apply to subsets of
  units

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Results

• 11,400 MW peak and 12,050 MW of capacity, mostly
  gas and some coal. Most of capacity has same
  fuel cost/MBTU. Plant emission rates vary by
  order of magnitude (0.06 - 0.50 lb/MBTU)
• With single tonnage cap, the cost of reducing
  emissions by 20 is 60M (a 5 increase in fuel
  cost).
• Emissions rate cap raises control cost by 1M due
  to dilution effect (increase BTU rather than
  decrease NOx). More diverse system results in
  larger penalty.

23
Cost of Inefficient Energy Trading Higher than
Dilution Effect

• Two area analysis energy trading for
  compliance purposes discouraged by rate limits

24
Operating Model Formulation, Continued
Complications

• Other objectives (Max Profit? Min Health Effect
  of Emissions?)
• Energy storage (pumped storage, batteries),
  hydropower
• Explicitly stochastic (usual assumption forced
  outages are random and independent)
• Including transmission constraints
• Including commitment variables (with fixed
  commitment costs, minimum MW run levels, ramp
  rates)
• Cogeneration (combined heat-power)

25
Including Transmissionor Why Power Transport is
Not Like Hauling Apples in a Cart
Node or bus m
Current Imn
Bus n
Voltage Vn

• Ohms law
• Voltage drop m to n DVmn Vm-Vn ImnZmn
• DC Imn current from m to n, Zmn resistance r
• AC Imn complex current, Zmn reactance r
  ?-1x
• Power loss I2R I ?DV?
• Kirchhoffs Laws
• Net inflow of current at any bus 0
• S voltage drops around any loop in a circuit 0

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Some Consequences of Transmission Laws

• Power from different sources intermingled moves
  from seller to buyer by displacement
• Cant direct power flow unvalved network.
  Power follows many paths (parallel flow)
• Flows are determined by all buyers/sellers
  simultaneously. Ones actions affect everyone,
  implying externalities
• 1 sells to 2 -- but this transaction congests 3s
  transmission lines and increases 3s costs
• One line owner can restrict capacity affect
  entire system
• Adding transmission line can worsen transmission
  capability of system

27
Modeling Transmission Flows (See Wood
Wollenburg or F. Schweppe et al., Spot Pricing of
Electricity, Kluwer, 1988)

• Linearized DC approximation assumes
• r ltlt x (capacitance/inductance dominates)
• Voltage angle differences between nodes small
• Voltage magnitude ?Vm? same all busses
• ? an injection yifmt or withdrawal LOADmt at a
  node m has a linear effect on power (? I) flowing
  through interface k
• Let Power Transmission Distribution Factor
  PTDFmk MW flow through k induced by a 1 MW
  injection at m
• assumes a 1 MW withdrawal at a hub bus
• Then total flow through k in period t is
  calculated and constrained as follows
• Tk- lt Sm PTDFmk(-LOADmt Sif yifmt) lt
  Tk

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Transmission Constraints in Operations LP

• MIN Variable Cost Si,f,t Ht CYift yifmt
• subject to
• Si,f,m yifmt Sm LOADtm ?t
• Sf yifmt lt (1- FORi)Xi m ?i,m,t
• Sf,t Ht yifmt lt CFi 8760Xim ?i,m
• Tk- lt Sm PTDFmk(-LOADmt Sif yifmt) lt Tk
• ?k,t
• yifmt gt 0 ?i,f,m,t

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Unit CommitmentA Mixed Integer Program

• Disregard forced outages fuels assume
• uit 1 if unit i is committed in t (0 o.w.)
• CUi fixed running cost of i if committed
• MRi must run (minimum MW) if committed
• Periods t 1,..,T are consecutive, and Ht1
• RRi Max allowed hourly change in output
• MIN Si,t CYit yit Si,t CUi uit
• s.t. Si yit LOADt ?t
• MRi uit lt yi lt Xi uit ?i,t
• -RRi lt (yit - yi,t-1) lt RRi ?i,t
• St yit lt CFi T Xi ?i
• yit gt 0 ?i,t uit ?0,1 ?i,t

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IV. Deterministic Investment Analysis LP Snap
Shot Analysis

• Let generation capacity xi now be a variable,
  with (annualized) cost CRF 1/yr CXi /MW, and
  upper bound XiMAX.
• MIN Si,f,t Ht CYift yift Si CRF CXi xi
• s.t. Si,f yift LOADt ?t
• Sf yift - (1- FORi)xi lt 0 ?i,t
• Sf,t Ht yift - CFi 8760xi lt 0 ?i
• Si xi gt LOAD1 (1M) (reserve margin
  constraint)
• xi lt XiMAX (Note equality for existing
  plants) ?i
• yift gt 0 ?i,f,t xi gt 0 ?i

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Some Complications

• Dynamics (timing of investment)
• Plants available only in certain sizes
• Retrofit of pollution control equipment
• Construction of transmission lines
• Demand-side management investments
• Uncertain future (demands, fuel prices)
• Other objectives (profit)

32
Demand-side investments

• Let zk 1 if DSM program k is fully implemented,
  at cost CZk /yr.
• Impact on load in t SAVkt MW
• MIN Si,f,t Ht CYift yift Si CRF CXi xi Sk CZk
  zk
• s.t. Si,f yift Sk SAVkt zk LOADt ?t
• Sf yift - (1- FORi)xi lt 0 ?i,t
• Sf,t Ht yift - CFi 8760xi lt 0 ?i
• Si xi (1M) Sk SAVkt zk gt LOAD1 (1M)
• xi lt XiMAX ?i
• yift gt 0 ?i,f,t xi gt 0 ?i zk gt 0
  ?k

33
V. Pure Competition AnalysisSimulating Purely
Competitive Commodity Markets An Equivalency
Result

• Background Kuhn-Karesh-Tucker conditions for
  optimality
• Definition of purely competitive market
  equilibrium
• Each player is maximizing their net benefits,
  subject to fixed prices (no market power)
• Market clears (supply demand)
• KKT conditions for players market clearing
  yields set of simultaneous equations
• Same set of equations are KKTs for a single
  optimization model (MAX net social welfare)
• Widely used in energy policy analysis

34
KKT Conditions

• Let an optimization problem be
• MAX F(X)
• X
• s.t. G(X) 0
• X 0
• with X Xi, G(X) Gj(X). Assume F(X)
  is smooth and concave, G(X) is smooth and convex.
  
• A solution X,? to the KKT conditions
  below is an optimal solution to the above
  problem, and vice versa. I.e., KKTs are
  necessary sufficient for optimality.
• MF/MXi - Sj ?j MGj/MXi 0
• œ Xi Xi 0
• Xi(MF/MXi - Sj ?j MGj/MXi) 0
• œ ?j Gj 0 ?j 0
• ?j Gj 0

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Notation Each node i is a separate commodity
(type, location, timing)
Consumer Buys QDi
QDi
i
j
TEij
TIij
Transporter/Transformer Uses exports TEij from
i to provide imports TIij to j
h
QSi
Supplier Uses inputs Xi to produce sell QSi
36
Players Profit Maximization Problems
Consumer at i MAX Ii(QDi) - Pi QDi
QDi s.t. QDi 0
j
Transporter for nodes i,j MAX Pj TIij - Pi TEij
- Cij(TEij,TIij) TEij,TIij s.t.
Gij(TEij,TIij) 0 (dual ?ij) TEij,
TIij 0
i
Supplier at i MAX PiQSi - Ci(Xi) QSi,Xi s.t.
Gi(QSi,Xi) 0 (µi) Xi , QSi 0
37
Suppliers Optimization Problem and KKT Conditions

• Supplier at i
• MAX PiQSi - Ci(Xi)
• QSi,Xi
• s.t. Gi(QSi,Xi) 0 (dual mi)
• Xi , QSi 0
• KKTs
• QSi (Pi - µi MGi/MQSi) 0 QSi 0
• QSi (Pi - µi MGi/MQsi) 0
• Xi (-MCi/MXi - µi MGi/MXi) 0 Xi 0
• Xi (-MCi/MXi - µi MGi/MXi) 0
• µi Gi 0 µi 0
• µi Gi 0

38
KKTs for All Players in Market Game Market
Clearing Condition
Consumer KKTs, œ i QDi (MB(QDi) - Pi) 0
QDi 0 QDi (MB(QDi) - Pi) 0
Market Clearing, œ i Pi QSi Sj
0 I(i) TEji - Sj 0 E(i) TIij - QDi
0
Supplier KKTs, œ i QSi (Pi - µi MGi/MQSi)
0 QSi 0 QSi (Pi - µi MGi/MQsi) 0 Xi
(-MCi/MXi - µi MGi/MXi) 0 Xi 0 Xi
(-MCi/MXi - µi MGi/MXi) 0 µi Gi 0 µi
0 µi Gi 0
Transporter/Transformer KKTs, œ ij TEij (-Pi -
MCij/MTEij - ?ij MGij/MTEij) 0 TEij 0
TEij(-Pi - MCij/MTEij - ?ij MGij/MTEij)
0 TIij (Pj - MCij/MTIij - ?ij MGij/MTIij) 0
TIij 0 TIij(Pi - MCij/MTIij - ?ij
MGij/MTIij) 0 ?ij Gij 0 ?ij 0 Gij ?ij 0
N conditions N unknowns!
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An Optimization Model for Simulating a Commodity
Market

• MAX Si Ii(QDi) - Si Ci(Xi) - Sij
  Cij(TEij,TIij)
• QDi, QSi, Xi, TEij, TIij
• s.t. QSi Sj 0 I(i) TIji - Sj 0 E(i) TEij -
  QDi 0 (dual Pi) œ i
• Gi(QSi,Xi) 0 (µi) œ
  i
• Gij(TEij,TIij) 0 (?ij) œ
  ij
• QDi, Xi, QSi 0 œ i
• TEij, TIij 0 œ
  ij
• Its KKT conditions are precisely the same as the
  market equilibrium conditions for the purely
  competitive commodities market! Thus
• a single NLP can be used to simulate a market
• a purely competitive market maximizes social
  surplus

40
Applications of the Pure Competition Equivalency
Principle

• MARKAL Used by Intl. Energy Agency countries for
  analyzing national energy policy, especially CO2
  policies
• Similar to EFOM used by VTT Finland (A. Lehtilä
  P. Pirilä, Reducing Energy Related Emissions,
  Energy Policy, 24(9), 805819, 1996)
• US Project Independence Evaluation System (PIES)
  successors (W. Hogan, "Energy Policy Models for
  Project Independence," Computers and Operations
  Research, 2, 251-271, 1975 F. Murphy and S.
  Shaw, "The Evolution of Energy Modeling at the
  Federal Energy Administration and the Energy
  Information Administration," Interfaces, 25,
  173-193, 1995.)
• 1975 Feasibility of energy independence
• Late 1970s Nuclear power licensing reform
• Early 1980s Natural gas deregulation
• US Natl. Energy Modeling System (C. Andrews, ed.,
  Regulating Regional Power Systems, Quorum Press,
  1995, Ch. 12, M.J. Hutzler, "Top-Down The
  National Energy Modeling System".)
• Numerous energy environmental policies
• ICF Coal and Electric Utility Model
  (http//www.epa.gov/capi/capi/frcst.html)
• Acid rain and smog policy
• POEMS (http//www.retailenergy.com/articles/cecasu
  m.htm)
• Economic environmental benefits of US
  restructuring
• Some of these modified to model imperfect
  competition (price regulation, market power)

41
VI. Analyzing Strategic Behavior of Power
GeneratorsPart 1. Overview of
Approaches(Utilities Policy, 2000)

• Benjamin F. Hobbs
• Dept. Geography Environmental Engineering
• The Johns Hopkins University
• Carolyn A. Berry
• William A. Meroney
• Richard P. ONeill
• Office of Economic Policy
• Federal Energy Regulatory Commission
• William R. Stewart, Jr.
• School of Business
• William Mary College

42
Questions Addressed by Strategic Modeling

• Regulators and Consumer Advocates
• How do particular market structures (, size,
  roles of firms) and mechanisms (e.g., bidding
  rules) affect prices, distribution of benefits?
• Will workable competition emerge? If not, what
  actions if any should be taken?
• approval of market-based pricing
• approval of access
• approval of mergers
• vertical or horizontal divestiture
• price regulation
• Market players What opportunities might be
  taken advantage of?

43
Market Power The ability to manipulate prices
persistently to ones advantage, independently of
the actions of others

• Generators The ability to raise prices above
  marginal cost by restricting output
• Consumers The ability to decrease prices below
  marginal benefit by restricting purchases
• Generators may be able to exercise market power
  because of
• economies of scale
• large existing firms
• transmission costs, constraints
• siting constraints, long lead time for generation
  construction

44
Projecting Prices Assessing Market Power
Approaches

• Empirical analyses of existing markets
• Market concentration (Herfindahl indices)
• HHI Si Si2 Si market share of firm i
• But market power is not just a f(concentration)
• Experimental
• Laboratory (live subjects)
• Computer simulation of adaptive automata
• Can be realistic, but are costly and difficult to
  replicate, generalize, or do sensitivity analyses

45
Projecting Prices Assessing Market Power
Approaches

• Equilibrium models. Differ in terms of
  representation of
• Market mechanisms
• Electrical network
• Interactions among players
• The principal result of theory is to show
  that nearly anything can happen, Fisher (1991)

46
Price Models for Oligopolistic Markets Elements

• 1. Market structure
• Participants, possible decision variables each
  controls
• Generators (bid prices generation)
• Grid operator (wheeling prices network flows,
  injections withdrawals)
• Consumers (purchases)
• Arbitrageurs/marketers (amounts to buy and
  resell)
• (Assume that each maximizes profit or
• follows some other clear rule)
• Bilateral transactions vs. POOLCO
• Vertical integration

47
Model Elements (Continued)

• 2. Market mechanism
• bid frequency, updating, confidentiality,
  acceptance
• price determination (congestion, spatial
  differentiation, price discrimination, residual
  regulation)
• 3. Transmission constraint model. Options
• ignore!
• transshipment (Kirchhoffs current law only)
• DC linearization (the voltage law too)
• full AC load flow

48
Model Elements (Continued)

• 4. Types of Games
• Noncooperative Games (Symmetric) Each player has
  same strategic variable
• Each player implicitly assumes that other players
  wont react.
• Nash Equilibrium no player believes it can do
  better by a unilateral move
• No market participant wishes to change its
  decisions, given those of rivals (Nash). Let
• Xi the strategic variables for player i.
  Xic Xj, j ? i
• Gi the feasible set of Xi
• pi(Xi,Xic) profit of i, given everyones
  strategy
• Xj, ?i is a Nash Equilibrium iff
• pi(Xi,Xic) gt pi(Xi,Xic), ? i, Xi?Gi
• Price Quantity at each bus stable

49
Model Elements (Continued)

• 4. Types of Games, Continued
• Examples of Nash Games
• Bertrand (Game in Prices). Implicit You believe
  that market prices wont be affected by your
  actions, so by cutting prices, you gain sales at
  expense of competitors
• Cournot (Game in Quantities) Implicit You
  believe that if you change your output, your
  competitors will maintain sales by cutting or
  raising their prices.
• Supply function (Game in Bid Schedule) Implicit
  You believe that competitors wont alter supply
  functions they bid

Bidi
Qi
50
Model Elements (Continued)

• 4. Types of Games, Cont.
• Noncooperative Game (Asymmetric/Leader-Follower)
  Leader knows how followers will react.
• E.g. strategic generators anticipate
• how a passive ISO prices transmission
• competitive fringe of small generators, consumers
• Stackelberg Equilibrium
• Cooperative Game (Exchangable Utility/Collusion)
  Max joint profit.
• E.g., competitors match your changes in prices or
  output

51
Model Elements (Continued)

• 5. Computation methods
• Payoff Matrix Enumerate all combinations of
  player strategies look for stable equilibrium
• Iteration/Diagonalization Simulate player
  reactions to each other until no player wants to
  change
• Direct Solution of Equilibrium Conditions
  Collect profit max (KKT) conditions for all
  players add market clearing conditions solve
  resulting system of conditions directly
• Usually involves complementarity conditions
• Optimization Model KKT conditions for maximum
  are same as equilibrium conditions

52
Simple Cournot Example

• Each firm i's marginal cost function QSi , i
  1,2
• Demand function P 100 - QD/2 /MWh

100 P
MCi
1
1/2
1
1
QSi
QD
53
Example of Nonexistence of Pure Strategy
Equilibria

• Definitions
• Pure strategy equilibrium A firm i chooses Xi
  with probability 1
• Mixed strategy Let the strategy space be
  discretized Xih, h 1,..,H. In a mixed
  strategy, a firm i chooses Xih with probability
  Pih lt 1. The strategy can be designated as the
  vector Pi
• Can also define mixed strategies using continuous
  strategy space and probability densities
• Let Pic Pj, ? j ?i
• Mixed strategy equilibrium Pi, ?i is mixed
  strategy Nash Equilibrium iff
• pi(Pi,Pic) gt pi(Pi,Pic), ? i ? Pi Sh
  Pih 1, Pihgt0
• By Nashs theorem, a mixed strategy equilibrium
  always exists (perhaps in degenerate pure
  strategy form) if strategy space finite.

54
Approaches to Calculating Mixed Equilibria(See
S. Stoft, Using Game Theory to Study Market
Power in Simple Networks, in H. Singh, ed.,
Game Theory Tutorial, IEEEE Winter Power Meeting,
NY, Feb. 1, 1999)

• Repeated Play
• Initialization Assume initial Pih, ? i Sh Pih
  1, Pihgt0. Assume initial n.
• Play For i 1,, I, find pure strategy Xih
• Arg MAXXih E(pi(Xih,Pic))
• Update Set
• Pih Pih 1/(n1)
• Pih Pih 1- 1/(n1)(1/Sh?h Pih)
• n n1
• If ngtN, quit and report Pih else return to Play
• Repeated play works nicely for some simple cases
  (such as the simple two generator bidding game).
  But in general
• May not converge to the equilibrium
• Very slow if strategy space H large and/or many
  players

55
Linear Complementarity Problem Approach for Two
Player Games

• The Bimatrix problem
• Player 1 MAX Sh,k P1hP2k p1(X1h,X2k)
• P1h
• s.t. Sh P1h 1
• P1h gt 0, ? h
• Player 2 MAX Sh,k P1hP2k p2(X1h,X2k)
• P2k
• s.t. Sk P2k 1
• P2k gt 0, ? k
• Solution approach
• Define KKTs for the two problems
• Solve the two sets of KKTs simultaneously by LCP
  algorithm (e.g., Lemkes algorithm, PATH)
• Limitations Yields NCP for gt2 players strategy
  space must be small

56
Example POOLCO Supply Function Competition
Analysis
DC Electric Network
B
high cost generator
low value load
A
D
low cost generator
30 MW flow limit
high value load
C
57
Market AssumptionsPOOLCO Supply Function Model

• Market mechanism generators submit bid curves
  (price vs. quantity supplied) to system operator,
  who then chooses suppliers to maximize economic
  surplus
• Grid model Linearized DC
• Players
• Generators Decide what linear bid curves to
  submit (adjust intercept of slope) believe other
  generators hold bids constant correctly
  anticipate how grid calculates prices. Play
  game in supply functions
• Grid Solves OPF to determine prices and winning
  generators assumes bids are true
• Consumers Price takers
• Arbitrageurs None (no opportunity)

58
Computational Approaches

• Approach for 2 player, 2 plant game define all
  combinations of strategies payoff table, then
  calculate pure or mixed equilibria (Berry et al.,
  2000)
• For larger game (Hobbs, Metzler, Pang, 2000)
• Define bilevel quadratic programming model for
  each generation firm
• Objective Choose bid curve to maximize profits
• Constraints Bid curves of other players, optimal
  power flow (OPF) solution of grid (defined by 1st
  order conditions for OPF solution)
• This is a MPEC (math program with equilibrium
  constraints) yielding the optimal bid curves for
  one firm, given bid curves of others. Not an
  equilibrium
• Cardell/Hitt/Hogan diagonalization approach to
  finding an equilibrium iterate among firm models
  until solutions converge--if they do (no
  guarantee that pure strategy solution exists)

59
A Simple Model of a POOLCO System (from Berry et
al., Utilities Policy, 2000)

• The Independent System Operator (ISO) takes
  supply and demand information from market
  participants.
• ISO finds the dispatch (quantity and price at
  each node) that
• equates total supply and total demand
• is feasible (does violate any transmission
  constraints)
• maximizes total welfare

60
ISO Maximizes Total Welfare
Maximize consumer value minus production costs
Price
S
P
D
Q
Quantity
61
ISO Maximizes Total Welfare
4 Nodes No Transmission Constraints - same
price everywhere - DC DD SA SB
Price
SB
SA
P
DC
DD
mcA mcB mvC mvD
Quantity
62
ISO Maximizes Total Welfare
Transmission Constraints
S
D
Price
S
PD
Congestion Revenues
PS
D
Q
Quantity
Transmission Constraint at Q
63
Price Dispersion Prices are Duals of Nodal
Energy Balances
B Gen
No Transmission Constraints ?1 Price
A Gen
D Load
Transmission Constraint(s) ? 4 Prices
AC30
C Load
64
Two Types of Competition

• Perfect Competition
• Generators bid cost functions.
• ISO uses demand functions and cost functions to
  find prices and quantities that maximize total
  welfare.

• Imperfect Competition
• Generators bid supply functions that maximize
  profits.
• ISO uses demand functions and supply functions to
  find prices and quantities that maximize total
  welfare.

65
Choice of Supply Function
Price
Supply
Complete bid (m,b) Alternatively Fix m, choose
b or Fix b, choose m
pmqb
slope intercept
Quantity
66
Choice Variable and Equilibrium

• A firm chooses the intercept of its supply
  function (fixed slope) that maximizes its profits
• Given that supply functions bid by rivals are
  fixed (Nash)
• Given that the ISO will maximize total welfare
  subject to the system constraints
• Nash Equilibrium
• The set of bids (intercepts) such that no firm
  can increase profits by changing its bid
• We used an payoff matrix/grid search to find the
  solution
• In general, a pure strategy equilibrium may not
  exist! (Edgeworth-like cycling). Generally, a
  mixed equilibrium will exist, but is difficult to
  calculate

67
Imperfect Competition with No Transmission
Constraints
No Surprise
QB81
QB73
P46 everywhere
P54 everywhere
QD82
QA85
QD70
QA104
QC103
QC88
Perfect Competition
Imperfect Competition
Bids bA10, bB10 Profits pA 1901, pB 1478
Bids bA25, bB22 Profits pA 2506, pB 2044
68
Imperfect Competition with Transmission
Constraint (AC30)
Surprise!
Gen B better off with constraint
PB50
QB90
QB94
PA18
PD61
P67 everywhere
QA24
QD60
QA20
QD51
30
PC72
QC54
QC63
Imperfect Competition
Perfect Competition
Bids bA10, bB10 Profits pA101, pB1819
Bids bA60, bB25 Profits pA1087, pB3373
Imperfect Competition Eliminates Transmission
Constraint
69
Imperfect Competition and Multiple Generators (3
at A, 1 at B)
Surprise!
1 Gen at A 1 Gen at B
3 Gen at A 1 Gen at B
QB94
PB55
QB73
PA24
PD65
P67 everywhere
30
30
QA30
QA20
QD51
QD54
QC63
PC75
QC48
Increased competition leads to higher prices for
consumers
70
Counterintuitive Result Increased Competition
Worsens Prices

• Compare one and three generators located at Node
  A

71
Strategic Modeling Part 2. Large Scale Market
ModelsA Large Scale Cournot Bilateral POOLCO
Model(Hobbs, IEEE Transactions on Power Systems,
in press)

• Features
• Bilateral market (generators sell to customers,
  buy transmission services from ISO)
• Cournot in power sales
• Generators assume transmission fees fixed
  linearized DC load flow formulation
• If there are arbitragers, then same as POOLCO
  Cournot model
• Mixed LCP formulation allows for solution of
  very large problems
• Being Implemented by US Federal Energy Regulatory
  Commission staff
• Spatial market power issues (congestion, addition
  of transmission constraints)
• Effects of mergers

72
Generating Firm ModelNo Arbitrage

• Assume generation and sales routed through hub
  bus
• Firm fs decision variables
• gif MW generation at bus i by f--NET cost at
  system hub is Cif( ) - Wi (wheeling fee Wi
  charged by ISO)
• sif MW sales to bus i by f--NET revenue
  received is Pi( )-Wi
• fs problem
• MAX Si Pi(sif Sg¹f sig)-Wisif
  -Cif(gif) -Wi gif
• s.t. gif CAPif , "i
• Si sif Si gif
• sif , gif ³ 0, "i
• In Cournot model, f sees wheeling fees Wi and
  rivals sales Sg¹f sig as fixed
• Its first-order (KKT) conditions define a set of
  complementarity conditions in the dvs duals xf
  
• CPf xf ³ 0 Hf(xf ,W) 0 xf Hf(xf ,W)0

73
ISOs Optimization Problem

• ISOs decision variable
• yiH transmission service to hub from i
• ISOs value of services maximization problem
• MAX pISO(y) Si Wi yiH
• s.t. Tk- Si PTDFiHk yiH Tk , "
  interfaces k
• Si yiH 0
• Solution allocates interface capacity to most
  valuable transactions (a la Chao-Peck)
• Tk- , Tk transmission capabilities for
  interface k
• PTDFiHk power distribution factor (assumes
  DC model)
• The models KKT conditions define complementarity
  conditions in the decision variables duals xISO
  
• CPISO xISO ³ 0 HISO(xISO,W) 0 xISO
  HISO(xISO,W)0

74
Equilibrium Calculation

• First order conditions for each player together
  with market clearing conditions determines an
  equilibrium
• Find xf , " f xISO W that satisfy
• CPf ," f xf ³ 0 Hf(xf ,W) 0 xf Hf(xf
  ,W)0
• CPISO xISO ³ 0 HISO(xISO ,W) 0 xISO
  HISO(xISO ,W)0
• Market clearing yiH Sf (sif -gif)
  " i
• Solution approaches
• Mixed LCP solver (PATH or MILES, in GAMS)
• Under certain conditions, can solve as a single
  quadratic program (as in Hashimoto, 1985)
• Solution characteristics For linear demand,
  supply, solution exists, and prices profits
  unique

75
Generating Firm ModelVariation on a Theme

• With Arbitrage Additional player
• ai Net MW sales by arbitrageurs at i
  (purchased at hub, sold at i)
• PH /MWh price at hub
• Model
• MAX Si (Pi - PH - Wi)ai
• Add the following KKTs to the original model
• ai Pi PH Wi , "i
• Equivalent to POOLCO Cournot model, in which
  generators assume that other generators keep
  outputs constant and ISO adjusts bus prices
  maintain equilibrium and bus price differences
  dont change

76
Three Node-Two Generator System
P
Elastic Demand
Hub
3
Q
P
P
Inelastic Demand
Inelastic Demand
2
1
Q
Q
Constrained Interface
MC 15 /MWh
MC 20 /MWh
77
Unconstrained Transmission
Perfect Competition
Cournot, No Arbitrage
Cournot With Arbitrage
P3 15 /MWh W3R 0 /MWh
P3 22.3 W3R 0
P3 23.8 W3R 0
3
3
3
318 MW
74 MW
74 MW
2
2
2
1
1
1
P1 15 /MWh W1R 0 /MWh G1 954 MW
P2 15 W2R 0 G2 0
P1 25 W1R 0 G1 392
P2 25 W2R 0 G2 170
P1 23.8 W1R 0 G1 392
P2 23.8 W2R 0 G2 170
Net Benefits 10,614/hr
Net Benefits 7992
Net Benefits 8031
78
Constrained Transmission
Perfect Competition
Cournot, No Arbitrage
Cournot With Arbitrage
P3 17.5 /MWh W3 0 /MWh
P3 22.3 W3 0
P3 23.8 W3 0
3
3
3
T 30 MW
2
2
2
1
1
1
P1 15 /MWh W1 2.5/MWh G1 491 MW
P2 20 W2 -2.5 G2 353
P1 24.1 W1 1.4 G1 330
P2 25.9 W2 -1.4 G2 232
P1 22.4 W1 1.3 G1 335
P2 25.1 W2 -1.3 G2 228
Net Benefits 8632/hr
Net Benefits 7672
Net Benefits 7723
79
Eastern Interconnection Model developed by
Judith Cardell, Thanh Luong and Michael Wander,
OEP/FERC Cournot development application by
Udi Helman, OEP/FERC Ben Hobbs, JHU

• 100 nodes representing control areas and 15
  interconnections with ERCOT, WSSC, and Canada
• 829 firms (of which 528 are NUGs)
• 2725 generating plants (in some cases aggregated
  by prime mover/fuel type/costs) approximately
  600,000 MW capacity

80
Eastern Interconnection Model

• 814 flowgates, each with PTDFs for each node
  (most flowgates and PTDFs defined by NERC a mix
  of physical and contingency flowgate limits)
• Four load scenarios modeled from NERC winter 1998
  assessment superpeak, 5 peak, shoulder,
  off-peak
• 68 firms represented as Cournot players (with
  capacity above 1000 MW). Remainder is
  competitive fringe

81
Equilibrium Prices with Demand Elasticity 0.4
(Load-Weighted Average System Prices)
82
Merger Example(Firm A at Node A, Firm B at Node
B)
83
Challenges

• Prices cant be predicted precisely because games
  are repeated, and conjectural variations are
  fluid and more complex than can be modeled.
  Models most useful for exploring issues/gaining
  insight--thus, simpler models preferred
• Apply to merger evaluation, market design, and
  strategic pricing
• Formulate practical market models that capture
  key features of the market Current voltage
  laws, transmission pricing, generator strategic
  behavior
• Need comparisons of model results with each
  other, and with actual experience

84
Competition in Markets for Electricity A
Conjectured Supply Function Approach
Christopher J. Day ENRON UK Benjamin F.
Hobbs DOGEE, Johns Hopkins Work performed
while first author was at the University of
California Energy Institute, Berkeley, California
85
Introduction

• Cournot (game in quantities) seems descriptively
  unappealing
• Is it valid to believe that rivals wont adjust
  quantities?
• Miniscule demand elasticities yield absurd
  results
• Instead supply/response function conjectures?
• Formulate model for firm f with
• Assumed rest-of-market supply response to price
  changes
• Conjectural variations (change in rest-of-market
  output in response to change in fs output
• Can model richer set of interactions, inelastic
  demand
• Can we incorporate models of transmission
  networks?
• How do the results from Cournot differ from those
  from supply function models?

86
Supply Function ConjectureEach firm f
anticipates that rival suppliers follow a linear
supply function
87
Supply Function Conjecture(fixed intercept)
(Rival supply assumed to follow line through
intercept and present solution)
Present solution
Assumed intercept
88
Producers Models with Supply Function Conjectures
(fixed intercept ai)
subject to

• To obtain market equilibrium
• Define KKTs for producer
• Combine with ISO ( arbitrager) KKTs and market
  clearing constraints
• Solve with MCP algorithm (PATH in GAMS)

89
England Wales Analysis
Issue What were the competitive effects of the
1996 and 1999 divestitures? Approach Cournot
and supply-function equilibrium models of
competition on a DC grid
90
Fixed Intercept Solution withNo Transmission
Constraints
30
25
Pre-divestment (1995)
20
After 1st divestment (1996)
15
Price (/MWh)
10
Demand
5
After 2nd divestment (1999)
0
10000
20000
30000
40000
50000
60000
Demand (MW)
91
Results - with a Constrained Network
92
EW Results - Competition a la Cournot, Supply
Function Competition, and Pure Competition
93
Discussion

• Supply/response function conjecture
• Descriptively more appealing than Cournot
• Can model inelastic demand
• Can include transmission models
• Gives insights into locational prices
• Has the potential to analyze large systems
• 200 to 300 node networks
• Can gain insights into possible market power
  abuses in bilaterally traded and POOLCO markets

94
Example of a Stackelberg (Leader-Follower) Model

• Large supplier as leader, ISO other suppliers
  as followers in POOLCO market.
• Problem choose bids BLi to max pL
• MAX pL Si PigLi - Ci(gLi)
• s.t. 0 lt gLi lt Xi, "i
• KKTs for ISO (depend on BLis)
• KKTs for other suppliers (price
  takers)
• The Challenge the complementarity conditions in
  the leaders constraint set render the leaders
  problem non-convex (i.e., feasible region
  non-convex)
• Algorithms for math programs with equili-brium
  constraints (MPECs) are improving