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