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Electric Power Markets: Process 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 - PowerPoint PPT Presentation
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Electric Power Markets:Electric Power Markets:Process and Strategic ModelingProcess 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. HobbsDept. of Geography & Environmental Engineering
The Johns Hopkins University
Baltimore, MD 21218 USA
bhobbs@jhu.edu
I.I. Overview of Course Overview of Course
Operations/ Operations/ Control ModelsControl Models
Design/ Design/ Investment Investment
ModelsModels
Single Firm ModelsSingle Firm Models
Demand ModelsDemand Models
Multifirm Models with Strategic InteractionMultifirm Models with Strategic Interaction
Operations/ Operations/ Control ModelsControl Models
Design/ Design/ Investment Investment
ModelsModels
Single Firm ModelsSingle Firm Models
Market Clearing Conditions/ConstraintsMarket Clearing Conditions/Constraints
Why The Power Sector?Why The Power Sector?
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
Finland`s 1995 Finland`s 1995 Elect. Market ActElect. Market Act XX
XX
EL-EX, Nord PoolEL-EX, Nord Pool
XX
FingridFingrid
NoneNone
Scope of economic impact• ~$1000/person/yr in US (~petroleum use)• Almost half of US energy use
Why Power? (Continued)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 Kirchhoff`s laws
II. II. “Process” or “Bottom-Up” Analysis:“Process” or “Bottom-Up” Analysis:Company & Market ModelsCompany & 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
Process Optimization ModelsProcess 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.,– 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 (<1 second - 1 hr)
Bottom-Up/Process Models Bottom-Up/Process Models vs. Top-Down Modelsvs. Top-Down Models
Bottom-up models simulate investment & operating decisions by an individual firm, (usually) assuming that that the firm can’t 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
Functions of Process Model: Functions of Process Model: Firm Level DecisionsFirm Level Decisions
Real time operations:• Automatic protection (<1 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)
Firm Decisions Made Using Firm Decisions Made Using Process Models, Continued 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)
Emerging UsesEmerging Uses Profit maximization rather than cost minimization
guides firm’s decisions Market simulation:
• Use model of firm’s 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
Advantages of Process Models Advantages of Process Models for Policy Analysisfor 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
Dangers of Process Models Dangers of Process Models for Policy Analysisfor 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 Optimistic bias--overestimate performance of selected solutions of selected solutions
The Overoptimism of Optimization 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
ConclusionConclusion 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
Conclusions, continuedConclusions, continued What don´t 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)
III. III. Operations Model: Operations Model: System Dispatch LPSystem 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 (i=1,..,I) during period t
(t=1,..,T) using fuel f (f=1,…,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 iLOADt = MW demand to be met in period t
Operations LPOperations LP
MIN Variable Cost = i,f,t Ht CYift yift
subject to:
i,f yift = LOADt t
f yift < (1- FORi)Xi i,t
f,t Ht yift < CFi 8760Xi i
yift > 0 i,f,t
Using Operating Models to Assess NOUsing Operating Models to Assess NOx x Regulation:Regulation:
The Inefficiency of Rate-Based RegulationThe Inefficiency of Rate-Based Regulation(Leppitsch & Hobbs, IEEE Trans. Power Systems, 1996)(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
Policy Question AddressedPolicy Question Addressed How effectively can NOx limits be met by
changed operations (“emissions dispatch”)? What is the relative efficiency of:
• Regulation based on tonnage capsTotal emissions [tons] < Tonnage cap
• Regulation based on emission rate limits (tons/GJ)?
(Total Emissions[tons]/Total Fuel Use [GJ])
< Rate Limit
FrameworkFramework 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 =Alternative dispatch dispatch orderorder
How To Generate AlternativesHow To Generate Alternatives Solve the following model for alternative levels of the
regulatory constraint:
MIN i CYi yi
s.t. 1. MRi < yi < Xi (note nonzero LB)
2. i yi > LOAD (MW)
3. Regulatory caps: eitheri Ei yi < MASS CAP (tons) or
(i Ei yi )/(i HRi yi) < 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
ResultsResults 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 $1M due to “dilution” effect (increase BTU rather than decrease NOx). More diverse system results in larger penalty.
Cost of Inefficient Energy Trading Cost of Inefficient Energy Trading Higher than Dilution EffectHigher than Dilution Effect
Two area analysis: energy trading for compliance purposes discouraged by rate limits
1170
1180
1190
1200
1210
1220
1230
1240
1250
1260
75000 80000 85000 90000 95000 100000
Tons NOx / year
Co
st
($M
/ye
ar)
Cost ($M), Ton Cap
Cost ($M), Rate Cap
Operating Model Formulation, Operating Model Formulation, Continued: ComplicationsContinued: 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)
Including Transmission:Including Transmission:oror Why Power Transport is Not Like Why Power Transport is Not Like
Hauling Apples in a CartHauling Apples in a Cart
Ohm`s law• Voltage drop m to n = Vmn = Vm-Vn = ImnZmn
– DC: mn = current from m to n, Zmn = resistance r
– AC: Imn = complex current, Zmn = reactance = r + -1x
• Power loss = I2R = I V Kirchhoff`s Laws:
• Net inflow of current at any bus = 0• voltage drops around any loop in a circuit = 0
Node or “bus” mNode or “bus” m
Bus nBus n
Current Imn
Voltage Vn
Some Consequences of Some Consequences of Transmission LawsTransmission Laws
Power from different sources intermingled: moves from seller to buyer by “displacement”
Can`t direct power flow: “unvalved network.” Power follows many paths (“parallel flow”)
Flows are determined by all buyers/sellers simultaneously. One`s actions affect everyone, implying externalities:• 1 sells to 2 -- but this transaction congests 3`s
transmission lines and increases 3`s costs• One line owner can restrict capacity & affect entire
system Adding transmission line can worsen
transmission capability of system
Modeling Transmission FlowsModeling Transmission Flows (See Wood & Wollenburg or F. Schweppe et al., Spot Pricing of Electricity, Kluwer, 1988)
Linearized DC approximation assumes:• r << 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- < [m PTDFmk(-LOADmt + if yifmt)] < Tk+
Transmission Constraints in Transmission Constraints in Operations LPOperations LP
MIN Variable Cost = i,f,t Ht CYift yifmt
subject to:
i,f,m yifmt = m LOADtm t
f yifmt < (1- FORi)Xi m i,m,t
f,t Ht yifmt < CFi 8760Xim i,m
Tk- < [m PTDFmk(-LOADmt + if yifmt)] < Tk+
k,t
yifmt > 0 i,f,m,t
Unit Commitment:Unit Commitment:A Mixed Integer ProgramA 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 Ht=1• RRi = Max allowed hourly change in output
MIN i,t CYit yit + i,t CUi uit
s.t.i yit = LOADt tMRi uit < yi < Xi uit i,t
-RRi < (yit - yi,t-1) < RRi i,t
t yit < CFi T Xi i
yit > 0 i,t; uit {0,1} i,t
IV. IV. Deterministic Investment Deterministic Investment Analysis: LP Snap Shot AnalysisAnalysis: LP Snap Shot Analysis
Let generation capacity xi now be a variable, with (annualized) cost CRF [1/yr] CXi [$/MW], and upper bound XiMAX.
MIN i,f,t Ht CYift yift + i CRF CXi xi
s.t. i,f yift = LOADt t
f yift - (1- FORi)xi < 0 i,t
f,t Ht yift - CFi 8760xi < 0 i
i xi > LOAD1 (1+M) (“reserve margin” constraint)
xi < XiMAX (Note: equality for existing plants) i
yift > 0 i,f,t; xi > 0 i
Some ComplicationsSome Complications Dynamics (timing of investment) Plants available only in certain sizes Retrofit of pollution control equipment Construction of transmission lines “Demand-side management” investments Uncertain future (demands, fuel prices) Other objectives (profit)
Demand-side investmentsDemand-side investments Let zk = 1 if DSM program k is fully implemented, at
cost CZk [$/yr]. Impact on load in t = SAVkt [MW]
MIN i,f,t Ht CYift yift + i CRF CXi xi + k CZk zk
s.t. i,f yift + k SAVkt zk = LOADt t
f yift - (1- FORi)xi < 0 i,t
f,t Ht yift - CFi 8760xi < 0 i
i xi + (1+M) k SAVkt zk > LOAD1 (1+M)
xi < XiMAX i
yift > 0 i,f,t; xi > 0 i; zk > 0 k
V. Pure Competition AnalysisV. Pure Competition AnalysisSimulating Purely Competitive Commodity Simulating Purely Competitive Commodity
Markets: An Equivalency ResultMarkets: 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
KKT ConditionsKKT 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,λ} to the KKT conditions below is an optimal solution to the above problem, and vice versa. I.e., KKTs are necessary & sufficient for optimality.
F/Xi - Σj λj Gj/Xi 0;œ Xi: Xi 0;
Xi(F/Xi - Σj λj Gj/Xi) = 0
œ λj: Gj 0; λj 0;λj Gj = 0{{
Notation: Each node Notation: Each node ii is a separate is a separate commodity (type, location, timing)commodity (type, location, timing)
ii jj
hh
Consumer: Buys QDi
Supplier: Uses inputs Xi toproduce & sell QSi
Transporter/Transformer: Uses exports TEij from i toprovide imports TIij to j
QDi
QSi
TEij TIij
Players’ Profit Maximization Players’ Profit Maximization ProblemsProblems
Supplier at i:MAX PiQSi - Ci(Xi){QSi,Xi}
s.t. Gi(QSi,Xi) 0 (μi)Xi , QSi 0
Consumer at i:MAX Ii(QDi) - Pi QDi {QDi}
s.t. QDi 0
Transporter for nodes i,j:MAX Pj TIij - Pi TEij - Cij(TEij,TIij){TEij,TIij}
s.t. Gij(TEij,TIij) 0 (dual θij) TEij, TIij 0
ii
jj
Supplier’s Optimization Supplier’s Optimization Problem and KKT ConditionsProblem and KKT Conditions
Supplier at i: MAX PiQSi - Ci(Xi) {QSi,Xi}
s.t. Gi(QSi,Xi) 0 (dual i)
Xi , QSi 0
KKTs:QSi: (Pi - μi Gi/QSi) 0; QSi 0;
QSi (Pi - μi Gi/Qsi) = 0
Xi: (-Ci/Xi - μi Gi/Xi) 0; Xi 0;
Xi (-Ci/Xi - μi Gi/Xi) = 0
μi: Gi 0; μi 0; μi Gi = 0
KKTs for All Players in Market Game KKTs for All Players in Market Game + Market Clearing Condition+ Market Clearing Condition
Supplier KKTs, œ i :QSi: (Pi - μi Gi/QSi) 0; QSi 0;
QSi (Pi - μi Gi/Qsi) = 0Xi: (-Ci/Xi - μi Gi/Xi) 0; Xi 0;
Xi (-Ci/Xi - μi Gi/Xi) = 0μi: Gi 0; μi 0; μi Gi = 0
Consumer KKTs, œ i:QDi: (MB(QDi) - Pi) 0; QDi 0; QDi (MB(QDi) - Pi) = 0
Transporter/Transformer KKTs, œ ij:TEij: (-Pi - Cij/TEij - θij Gij/TEij) 0; TEij 0; TEij(-Pi - Cij/TEij - θij Gij/TEij) = 0TIij: (+Pj - Cij/TIij - θij Gij/TIij) 0; TIij 0; TIij(+Pi - Cij/TIij - θij Gij/TIij) = 0θij: Gij 0; θij 0; Gij θij = 0
MarketClearing, œ i:
Pi: QSi + Σj I(i) TEji
- Σj E(i) TIij
- QDi = 0
N conditionsN conditions& N unknowns!& N unknowns!
An Optimization Model for An Optimization Model for Simulating a Commodity MarketSimulating a Commodity Market
MAX Σi Ii(QDi) - Σi Ci(Xi) - Σij Cij(TEij,TIij) {QDi, QSi, Xi, TEij, TIij}
s.t.: QSi + Σj I(i) TIji - Σj E(i) TEij - QDi = 0 (dual Pi) œ i
Gi(QSi,Xi) 0 (μi) œ i
Gij(TEij,TIij) 0 (θij) 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
Applications of the Pure Applications of the Pure Competition Equivalency PrincipleCompetition 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), 805+819, 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/cecasum.htm)
• Economic & environmental benefits of US restructuring Some of these modified to model imperfect competition (price
regulation, market power)
VI. VI. Analyzing Strategic Analyzing Strategic Behavior of Power GeneratorsBehavior of Power GeneratorsPart 1. Overview of ApproachesPart 1. Overview of Approaches
((Utilities PolicyUtilities Policy, 2000), 2000)
Benjamin F. HobbsDept. Geography & Environmental Engineering
The Johns Hopkins University
Carolyn A. BerryWilliam A. MeroneyRichard P. O’Neill
Office of Economic PolicyFederal Energy Regulatory Commission
William R. Stewart, Jr.School of Business
William & Mary College
Questions Addressed by Questions Addressed by Strategic ModelingStrategic 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?
Market Power = The ability to manipulate Market Power = The ability to manipulate prices persistently to one’s advantage, prices persistently to one’s advantage, independently of the actions of othersindependently 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
Projecting Prices & Assessing Projecting Prices & Assessing Market Power: Market Power: ApproachesApproaches
Empirical analyses of existing markets Market concentration (Herfindahl indices)
• HHI = i 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
Projecting Prices & Assessing Projecting Prices & Assessing Market Power: Market Power: ApproachesApproaches
Equilibrium models. Differ in terms of representation of:• Market mechanisms• Electrical network• Interactions among players
“ “The principal result of theory is to show that The principal result of theory is to show that nearly anything can happen”, Fisher (1991)nearly anything can happen”, Fisher (1991)
Price Models for Oligopolistic Price Models for Oligopolistic Markets: Markets: ElementsElements
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
Model Elements (Model Elements (Continued)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 (Kirchhoff’s current law only)• DC linearization (the voltage law too)• full AC load flow
Model Elements (Model Elements (Continued)Continued)4. Types of Games:
• Noncooperative Games (Symmetric): Each player has same “strategic variable”
– Each player implicitly assumes that other players won’t 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
i(Xi,Xic) = profit of i, given everyone`s strategy
{Xj*, i} is a Nash Equilibrium iff:i(Xi*,Xi
c*) > i(Xi,Xic*), i, XiGi
• Price & Quantity at each bus stable
Model Elements (Model Elements (Continued)Continued)
4. Types of Games, Continued:• Examples of Nash Games:
– Bertrand (Game in Prices). Implicit: You believe that market prices won`t 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 won’t alter supply functions they bid
QQii
BidBidii
Model Elements (Model Elements (Continued)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
Model Elements (Model Elements (Continued)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
Simple Cournot ExampleSimple Cournot Example Each firm i's marginal cost function = QSi , i
= 1,2 Demand function : P = 100 - QD/2 [$/MWh]
QQSiSi
MCMCii
1111
QQDD
100100
PP1/21/2
11
Example of Nonexistence of Example of Nonexistence of Pure Strategy EquilibriaPure 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 < 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:i(Pi*,Pi
c*) > i(Pi,Pic*), i; Pi: h Pih =1, Pih>0
By Nash’s theorem, a mixed strategy equilibrium always exists (perhaps in degenerate pure strategy form) if strategy space finite.
Approaches to Calculating Mixed EquilibriaApproaches to Calculating Mixed Equilibria(See S. Stoft, “Using Game Theory to Study Market Power in Simple Networks,” (See S. Stoft, “Using Game Theory to Study Market Power in Simple Networks,”
in H. Singh, ed., in H. Singh, ed., Game Theory TutorialGame Theory Tutorial, IEEEE Winter Power Meeting, NY, Feb. 1, 1999), IEEEE Winter Power Meeting, NY, Feb. 1, 1999)
Repeated Play• Initialization: Assume initial Pih, i: h Pih =1, Pih>0.
Assume initial n.• Play: For i = 1,…, I, find pure strategy Xih’ = Arg MAX{Xih} E(i(Xih,Pi
c))• Update: Set:
Pih’ = Pih’ + 1/(n+1)Pih = Pih [1- 1/(n+1)](1/hh’ Pih)n = n+1
• If n>N, 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
Linear Complementarity Problem Linear Complementarity Problem
Approach for Two Player GamesApproach for Two Player Games The “Bimatrix” problem:
• Player 1: MAX h,k P1hP2k 1(X1h,X2k) {P1h}
s.t. h P1h = 1 P1h > 0, h
• Player 2: MAX h,k P1hP2k 2(X1h,X2k) {P2k}
s.t. k P2k = 1 P2k > 0, k
Solution approach:• Define KKTs for the two problems• Solve the two sets of KKTs simultaneously by LCP algorithm
(e.g., Lemke’s algorithm, PATH) Limitations: Yields NCP for >2 players; strategy space must be
small
Example POOLCO Supply Example POOLCO Supply Function Competition AnalysisFunction Competition Analysis
DC Electric Network
A
B
C
D
high costgenerator
low cost generator
high value load
low value load
30 MWflow limit
Market Assumptions:Market Assumptions:POOLCO Supply Function ModelPOOLCO 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)
Computational ApproachesComputational 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)
A Simple Model of a POOLCO SystemA Simple Model of a POOLCO System
(from Berry et al., (from Berry et al., Utilities PolicyUtilities Policy, 2000), 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
ISO Maximizes Total WelfareISO Maximizes Total Welfare
Quantity
Price
P*
Q*
D
S
Maximize consumer valueminus production costs
ISO Maximizes Total WelfareISO Maximizes Total Welfare
Quantity
Price
DC
SB
SA
mcA = mcB = mvC = mvD
P*
DD
4 NodesNo Transmission Constraints
- same price everywhere
- DC + DD = SA + SB
ISO Maximizes Total WelfareISO Maximizes Total Welfare
Quantity
Price
PS*
Q*
D
S
Transmission ConstraintsS D
Transmission Constraint at Q*
PD*CongestionRevenues
Price Dispersion: Prices are Duals Price Dispersion: Prices are Duals of Nodal Energy Balancesof Nodal Energy Balances
No TransmissionConstraints1 Price
Transmission Constraint(s) 4 Prices
AGen
BGen
CLoad
DLoad
AC=30AC=30
Two Types of CompetitionTwo Types of Competition
Perfect CompetitionPerfect Competition
Generators bid cost functions.
ISO uses demand functions and cost functions to find prices and quantities that maximize total welfare.
Imperfect CompetitionImperfect Competition
Generators bid supply functions that maximize profits.
ISO uses demand functions and supply functions to find prices and quantities that maximize total welfare.
Choice of Supply FunctionChoice of Supply Function
Quantity
Supply
Price
p=mq+b
slope intercept
Complete bid:(m,b)
AlternativelyFix m, choose borFix b, choose m
Choice Variable and EquilibriumChoice 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
Imperfect Competition with No Imperfect Competition with No Transmission ConstraintsTransmission Constraints
Perfect Competition Imperfect Competition
QA=104
P=46 everywhere
QD=82
QB=81
QC=103
P=54 everywhere
QA=85
QB=73
QD=70
QC=88
Bids: bA=10, bB=10Profits: A= 1901, B= 1478
Bids: bA=25, bB=22Profits: A= 2506, B= 2044
NoSurprise
Imperfect Competition with Imperfect Competition with Transmission Constraint (AC=30)Transmission Constraint (AC=30)
Perfect Competition Imperfect Competition
PA=18
QA=24
PD=61
PB=50
PC=72
QD=60
QB=90
QC=54
P=67 everywhere
QA=20
QB=94
QD=51
QC=63
Bids: bA=10, bB=10Profits: A=101, B=1819
Bids: bA=60, bB=25Profits: A=1087, B=3373
Surprise!
Imperfect Competition Eliminates Transmission Constraint
30
Gen B betteroff with constraint
Imperfect Competition and Imperfect Competition and Multiple Generators Multiple Generators (3 at A, 1 at B)(3 at A, 1 at B)
PA=24
QA=30
PD=65
PB=55
PC=75
QD=54
QB=73
QC=48
P=67 everywhere
QA=20
QB=94
QD=51
QC=63
1 Gen at A1 Gen at B
3 Gen at A1 Gen at B
Surprise!
Increased competition leadsto higher prices for consumers
30 30
Counterintuitive Result: Counterintuitive Result: Increased Increased Competition Worsens PricesCompetition Worsens Prices
Compare one and three generators located at Node A:
Run PC
($/MWh)PD
($/MWh)
ConsumerSurplus
($)
TotalSurplus
($)1 Generator
at A67 67 1873 6322
3 Generators at A
75 65 1559 6315
Strategic Modeling Part 2. Large Scale Market ModelsStrategic Modeling Part 2. Large Scale Market ModelsA Large Scale Cournot Bilateral & POOLCO ModelA Large Scale Cournot Bilateral & POOLCO Model
(Hobbs, IEEE Transactions on Power Systems, in press)(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
Generating Firm Model:Generating Firm Model:No ArbitrageNo Arbitrage
Assume generation and sales routed through hub bus Firm f’s 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
f’s problem:
MAX i{[Pi(sif +gf sig)-Wi]sif [Cif(gif) Wi gif ]} s.t.: gif CAPif , i
isif = igif
sif , gif 0, i
• In Cournot model, f sees wheeling fees Wi and rivals’ sales gf sig as fixed
Its first-order (KKT) conditions define a set of complementarity conditions in
the dv’s & duals xf : CPf: xf 0; Hf(xf ,W) 0; xf Hf(xf ,W)=0
ISO’s Optimization ProblemISO’s Optimization Problem ISO’s decision variable:
yiH = transmission service to hub from i
ISO’s value of services maximization problem:
MAX ISO(y) = iWi yiH
s.t.: Tk iPTDFiHk yiH Tk+ , interfaces k
iyiH = 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 model’s KKT conditions define complementarity conditions in the
decision variables & duals xISO : CPISO: xISO 0; HISO(xISO,W) 0; xISO HISO(xISO,W)=0
Equilibrium CalculationEquilibrium 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 0; Hf(xf ,W) 0; xf Hf(xf ,W)=0
• CPISO: xISO 0; HISO(xISO ,W) 0; xISO HISO(xISO ,W)=0
• Market clearing: yiH = f (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
Generating Firm Model:Generating Firm Model:Variation on a ThemeVariation 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 i(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 don’t change
Three Node-Two Generator Three Node-Two Generator SystemSystem
33
11 22Constrained
Interface
Q
P Elastic Demand
Q
P Inelastic Demand
Q
P Inelastic Demand
HubHub
MC = 15 $/MWh MC = 20 $/MWh
Unconstrained TransmissionUnconstrained Transmission
33
11 22
P3 = 15 $/MWh
W3R = 0 $/MWh
P1 = 15 $/MWh
W1R = 0 $/MWh
G1 = 954 MW
P2 = 15W2R = 0G2 = 0
33
11 22
P3 = 22.3W3R = 0
P1 = 25W1R = 0G1 = 392
P2 = 25W2R = 0G2 = 170
33
11 22
P3 = 23.8W3R = 0
P1 = 23.8W1R = 0G1 = 392
P2 = 23.8W2R = 0G2 = 170
PerfectPerfectCompetitionCompetition
Cournot,Cournot,No ArbitrageNo Arbitrage
CournotCournotWithWith Arbitrage Arbitrage
Net Benefits = $10,614/hr Net Benefits = 7992 Net Benefits = 8031
318 MW 74 MW 74 MW
Constrained TransmissionConstrained Transmission
33
11 22
P3 = 17.5 $/MWh
W3 = 0 $/MWh
P1 = 15 $/MWh
W1 = +2.5$/MWh
G1 = 491 MW
P2 =20
W2 = -2.5G2 = 353
33
11 22
P3 = 22.3W3 = 0
P1 = 24.1W1 = +1.4G1 = 330
P2 = 25.9
W2 = -1.4G2 = 232
33
11 22
P3 = 23.8W3 = 0
P1 = 22.4W1 = +1.3G1 = 335
P2 =25.1
W2 = -1.3G2 = 228
PerfectPerfectCompetitionCompetition
Cournot,Cournot,No ArbitrageNo Arbitrage
CournotCournotWithWith Arbitrage Arbitrage
Net Benefits = $8632/hr Net Benefits = 7672 Net Benefits = 7723
T =30 MW
Eastern Interconnection ModelEastern Interconnection Model developed by Judith Cardell, Thanh Luong and Michael Wander, OEP/FERC; Cournot developed by Judith Cardell, Thanh Luong and Michael Wander, OEP/FERC; Cournot
development & application by Udi Helman, OEP/FERC & Ben Hobbs, JHUdevelopment & 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
Eastern Interconnection ModelEastern 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
Equilibrium Prices with Demand Elasticity = 0.4Equilibrium Prices with Demand Elasticity = 0.4 (Load-Weighted Average System Prices)(Load-Weighted Average System Prices)
super peak 5% peak shoulder off-peak
Competition $30.99 $22.17 $19.89 $16.39
Cournot w/arbitrage
$31.48 $22.85 $20.14 $16.74
Difference 1.58% 3.07% 1.23% 2.13%
Max Difference 6.65% 10.4% 5.56% 9.07%
Min Difference 0.00% 0.00% 0.00% 0.00%
Load wheredifference > 5%
1.01% 15.12% 2.38% 13.27%
Merger ExampleMerger Example(Firm A at Node A, Firm B at Node B)(Firm A at Node A, Firm B at Node B)
pre-merger mergerCompetition $22.17 $22.17
Cournot w/arbitrage
$22.85 $22.86
Difference 3.07% 3.11%
Cournot PriceNode A
$18.89 $18.92
Cournot PriceNode B
$27.65 $27.66
Profits A+B=$162,723 AB=$162,400
ChallengesChallenges Prices can’t 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
Competition in Markets for ElectricityCompetition in Markets for Electricity: : A Conjectured Supply Function ApproachA Conjectured Supply Function Approach
Christopher J. Day*Christopher J. Day*
ENRON UKENRON UK
Benjamin F. HobbsBenjamin F. Hobbs
DOGEE, Johns HopkinsDOGEE, Johns Hopkins
* Work performed while first author was at the University of California Energy Institute, Berkeley, California
IntroductionIntroduction Cournot (game in quantities) seems descriptively unappealing
• Is it valid to believe that rivals won’t 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 f’s 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?
Supply Function Conjecture:Each firm f anticipates that rival
suppliers follow a linear supply function
p
fh
hf ss
Supply Function ConjectureSupply Function Conjecture(fixed intercept)(fixed intercept)
*p
p
f
f
ss
pp
*
*
*fs fs
Present solutionPresent solution
(Rival supply assumed to(Rival supply assumed tofollow line through interceptfollow line through interceptand present solution)and present solution)
Assumed interceptAssumed intercept
Producers Models with Supply Function Producers Models with Supply Function Conjectures Conjectures (fixed intercept (fixed intercept ii))
i
fii
ififiifipsg
gWMCsWpfififi ,,
max
0,
*
**
fifi
ifi
ifi
fifi
ii
fiifififi
fifififi
sg
sg
Gg
p
sspps
pspDs
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)
England & Wales AnalysisEngland & Wales Analysis
1
5
7
4
3
10
12
11
9
6
2
8
13
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
Fixed Intercept Solution withFixed Intercept Solution withNo Transmission ConstraintsNo Transmission Constraints
0
5
10
15
20
25
30
10000 20000 30000 40000 50000 60000
Pri
ce (
£/M
Wh)
Demand (MW)
Pre-divestment (1995)
Demand
After 2nd divestment (1999)
After 1st divestment (1996)
Results - with a Constrained NetworkResults - with a Constrained Network
Demand Elasticity 0.0005
0
5
10
15
20
25
N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N13
Pric
e
SF 1995
SF 1996
SF 1999
P. Comp
E&W Results -E&W Results - Competition Competition a la a la Cournot, Supply Function Cournot, Supply Function
Competition, and Pure CompetitionCompetition, and Pure Competition
Demand Elasticity 0.1
0
10
20
30
40
50
60
N1 N2 N3 N4 N5 N6 N7 N8 N9N1
0N1
1N1
2N1
3
Pric
e
Cournot 1995Cournot 1996Cournot 1999SF 1995SF 1996SF 1999P. Comp
DiscussionDiscussion 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
Example of a Stackelberg Example of a Stackelberg (Leader-Follower) Model(Leader-Follower) Model
Large supplier as leader, ISO & other suppliers as followers in POOLCO market.
Problem: choose bids BLi to max L
MAX L = i [PigLi - Ci(gLi)] s.t. 0 < gLi < Xi, i KKTs for ISO (depend on BLi’s) KKTs for other suppliers (price takers)
The Challenge: the complementarity conditions in the leader’s constraint set render the leader’s problem non-convex (i.e., feasible region non-convex)
Algorithms for math programs with equili-brium constraints (MPECs) are improving
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