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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

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Electric power markets process and strategic modeling

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

Dept. of Geography & Environmental Engineering

The Johns Hopkins University

Baltimore, MD 21218 USA


I overview of course

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

Why the power sector

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

Finland`s 1995

Elect. Market Act



EL-EX, Nord Pool




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 process or bottom up analysis company market models

II. “Process” or “Bottom-Up” Analysis:Company & 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 models

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 (<1 second - 1 hr)

Bottom up process models vs top down models

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 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 firm level decisions

Functions of Process Model: Firm 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 process models continued

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)

Emerging uses

Emerging 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 for policy analysis

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

Dangers of process models for policy analysis

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

Electric power markets process and strategic modeling

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



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 continued

Conclusions, 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


Iii operations model system dispatch lp

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 (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 i

    LOADt = MW demand to be met in period t

Operations lp

Operations LP

MIN Variable Cost = Si,f,t Ht CYiftyift

subject to:


Sfyift<(1- FORi)Xi i,t

Sf,tHtyift<CFi 8760Xi i

yift> 0i,f,t

Electric power markets process and strategic modeling

Using Operating Models to Assess NOx Regulation:The 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

Policy question addressed

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] < Tonnage cap

    • Regulation based on emission rate limits (tons/GJ)?

      (Total Emissions[tons]/Total Fuel Use [GJ])

      < Rate Limit



We want less cost and less 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

How to generate alternatives

How To Generate Alternatives

Solve the following model for alternative levels of the regulatory constraint:


s.t. 1. MRi<yi< Xi(note nonzero LB)

2. Siyi> LOAD (MW)

3. Regulatory caps: either

Si Eiyi< MASS CAP (tons)or

(Si Eiyi)/(Si HRiyi)< 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



  • 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 higher than dilution effect

Cost of Inefficient Energy Trading Higher than Dilution Effect

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

Operating model formulation continued complications

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)

Including transmission or why power transport is not like hauling apples in a cart

Including Transmission:or Why Power Transport is Not Like Hauling Apples in a Cart

Node or “bus” m

Current Imn

  • Ohm`s 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

  • Kirchhoff`s Laws:

    • Net inflow of current at any bus = 0

    • S voltage drops around any loop in a circuit = 0

Bus n

Voltage Vn

Some consequences of transmission laws

Some Consequences of Transmission 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

Electric power markets process and strategic modeling

Modeling 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- < [Sm PTDFmk(-LOADmt + Sifyifmt)] < Tk+

Transmission constraints in operations lp

Transmission Constraints in Operations LP

MIN Variable Cost = Si,f,t Ht CYiftyifmt

subject to:


Sfyifmt<(1- FORi)Ximi,m,t

Sf,tHtyifmt<CFi 8760Ximi,m

Tk-< [Sm PTDFmk(-LOADmt + Sifyifmt)] < Tk+


yifmt> 0i,f,m,t

Unit commitment a mixed integer program

Unit Commitment:A 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 Si,t CYityit+ Si,t CUiuit

    s.t. Siyit=LOADtt

    MRi uit<yi<Xi uiti,t

    -RRi < (yit - yi,t-1)<RRi i,t

    Styit<CFi T Xi i

    yit> 0 i,t;uit {0,1} i,t

Iv deterministic investment analysis lp snap shot analysis

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 CYiftyift + Si CRF CXi xi

    s.t. Si,fyift=LOADtt

    Sfyift- (1- FORi)xi< 0i,t

    Sf,tHtyift- CFi 8760xi < 0i

    Si xi> LOAD1 (1+M) (“reserve margin” constraint)

    xi<XiMAX (Note: equality for existing plants) i

    yift> 0 i,f,t; xi > 0 i

Some complications

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)

Demand side investments

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 CYiftyift + Si CRF CXi xi+ Sk CZkzk

    s.t. Si,fyift+ Sk SAVktzk=LOADtt

    Sfyift- (1- FORi)xi< 0i,t

    Sf,tHtyift- CFi 8760xi < 0i

    Si xi+ (1+M) Sk SAVktzk> LOAD1 (1+M)

    xi<XiMAX i

    yift> 0 i,f,t; xi > 0 i; zk> 0 k

V pure competition analysis simulating purely competitive commodity markets an equivalency result

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

Kkt conditions

KKT Conditions

Let an optimization problem be:



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.

MF/MXi - Σj λj MGj/MXi# 0;

œ Xi:Xi$ 0;

Xi(MF/MXi - Σj λj MGj/MXi) = 0

œλj:Gj# 0; λj$ 0;

λj Gj = 0



Notation each node i is a separate commodity type location timing

Notation: Each node i is a separate commodity (type, location, timing)

Consumer: BuysQDi







Uses exportsTEijfrom i to

provide importsTIijto j



Supplier: Uses inputsXito

produce & sellQSi

Players profit maximization problems

Players’ Profit Maximization Problems

Consumer at i:

MAX Ii(QDi) - Pi QDi


s.t. QDi$ 0


Transporter for nodes i,j:

MAX Pj TIij - Pi TEij - Cij(TEij,TIij)


s.t. Gij(TEij,TIij) # 0 (dual θij)

TEij, TIij$ 0


Supplier at i:

MAX PiQSi - Ci(Xi)


s.t. Gi(QSi,Xi) # 0 (μi)

Xi , QSi$ 0

Supplier s optimization problem and kkt conditions

Supplier’s Optimization Problem and KKT Conditions

Supplier at i:

MAX PiQSi - Ci(Xi)


s.t. Gi(QSi,Xi) # 0 (dual mi)

Xi , QSi$ 0


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

Kkts for all players in market game market clearing condition

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


Clearing, œ i:

Pi: QSi

+ Σj 0 I(i) TEji

- Σj 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 - θijMGij/MTEij) # 0; TEij$ 0;

TEij(-Pi - MCij/MTEij - θijMGij/MTEij) = 0

TIij: (+Pj - MCij/MTIij - θijMGij/MTIij) # 0; TIij$ 0;

TIij(+Pi - MCij/MTIij - θijMGij/MTIij) = 0

θij: Gij# 0; θij$ 0; Gij θij = 0

N conditions

& N unknowns!

An optimization model for simulating a commodity market

An Optimization Model for Simulating a Commodity Market

  • MAX Σi Ii(QDi) - Σi Ci(Xi) - Σij Cij(TEij,TIij)

    {QDi, QSi, Xi, TEij, TIij}

    s.t.: QSi + Σj 0 I(i) TIji - Σj 0 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 competition equivalency principle

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), 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)

Electric power markets process and strategic modeling

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. O’Neill

Office of Economic Policy

Federal Energy Regulatory Commission

William R. Stewart, Jr.

School of Business

William & Mary College

Questions addressed by strategic modeling

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?

Electric power markets process and strategic modeling

Market Power = The ability to manipulate prices persistently to one’s 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

Projecting prices assessing market power approaches

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

Projecting prices assessing market power approaches1

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)

Price models for oligopolistic markets elements

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

Model elements continued

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 (Kirchhoff’s current law only)

  • DC linearization (the voltage law too)

  • full AC load flow

Model elements continued1

Model Elements (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

    pi(Xi,Xic) = profit of i, given everyone`s strategy

    {Xj*, i} is a Nash Equilibrium iff:

    pi(Xi*,Xic*) > pi(Xi,Xic*),  i, XiGi

  • Price & Quantity at each bus stable

Model elements continued2

Model Elements (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



Model elements continued3

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

Model elements continued4

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

Simple cournot example

Simple Cournot Example

  • Each firm i's marginal cost function = QSi , i = 1,2

  • Demand function : P = 100 - QD/2 [$/MWh]










Example of nonexistence of pure strategy equilibria

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 < 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*) >pi(Pi,Pic*),  i;  Pi: Sh Pih =1, Pih>0

  • By Nash’s theorem, a mixed strategy equilibrium always exists (perhaps in degenerate pure strategy form) if strategy space finite.

Electric power markets process and strategic modeling

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, Pih>0. Assume initial n.

    • Play: For i = 1,…, I, find pure strategy Xih’ =

      Arg MAX{Xih} E(pi(Xih,Pic))

    • Update: Set:

      Pih’ = Pih’ + 1/(n+1)

      Pih = Pih [1- 1/(n+1)](1/Shh’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 approach for two player games

Linear Complementarity Problem Approach for Two Player Games

  • The “Bimatrix” problem:

    • Player 1: MAX Sh,k P1hP2kp1(X1h,X2k)


      s.t. Sh P1h = 1

      P1h> 0,  h

    • Player 2: MAX Sh,k P1hP2kp2(X1h,X2k)


      s.t. Sk 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 function competition analysis

Example POOLCO Supply Function Competition Analysis

DC Electric Network

high cost



low value load



low cost generator

30 MW

flow limit

high value load


Market assumptions poolco supply function model

Market Assumptions:POOLCO 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 approaches

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)

A simple model of a poolco system from berry et al utilities policy 2000

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

Iso maximizes total welfare

ISO Maximizes Total Welfare

Maximize consumer value

minus production costs







Iso maximizes total welfare1

ISO Maximizes Total Welfare

4 Nodes

No Transmission Constraints

- same price everywhere

- DC + DD = SA + SB







mcA = mcB = mvC = mvD


Iso maximizes total welfare2

ISO Maximizes Total Welfare

Transmission Constraints













Constraint at Q*

Price dispersion prices are duals of nodal energy balances

Price Dispersion: Prices are Duals of Nodal Energy Balances



No Transmission


1 Price







 4 Prices




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.

Two Types of Competition

Choice of supply function

Choice of Supply Function



Complete bid:



Fix m, choose b


Fix b, choose m


slope intercept


Choice variable and equilibrium

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

Imperfect competition with no transmission constraints

Imperfect Competition with No Transmission Constraints





P=46 everywhere

P=54 everywhere







Perfect Competition

Imperfect Competition

Bids: bA=10, bB=10

Profits: pA= 1901, pB= 1478

Bids: bA=25, bB=22

Profits: pA= 2506, pB= 2044

Imperfect competition with transmission constraint ac 30

Imperfect Competition with Transmission Constraint (AC=30)


Gen B better

off with constraint






P=67 everywhere









Imperfect Competition

Perfect Competition

Bids: bA=60, bB=25

Profits: pA=1087, pB=3373

Bids: bA=10, bB=10

Profits: pA=101, pB=1819

Imperfect Competition Eliminates Transmission Constraint

Imperfect competition and multiple generators 3 at a 1 at b

Imperfect Competition and Multiple Generators (3 at A, 1 at B)


1 Gen at A

1 Gen at B

3 Gen at A

1 Gen at B






P=67 everywhere










Increased competition leads

to higher prices for consumers

Counterintuitive result increased competition worsens prices

Counterintuitive Result: Increased Competition Worsens Prices

  • Compare one and three generators located at Node A:

Electric power markets process and strategic modeling

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

Generating firm model no arbitrage

Generating Firm Model:No 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 Si{[Pi(sif +Sg¹f sig)-Wi]sif-[Cif(gif) -Wi gif ]}

    s.t.: gif £ CAPif , "i

    Sisif = Sigif

    sif , gif ³ 0, "i

    • In Cournot model, f sees wheeling fees Wi and rivals’ sales Sg¹f sigas fixed

  • Its first-order (KKT) conditions define a set of complementarity conditions in the dv’s & duals xf :

    CPf:xf ³ 0; Hf(xf ,W) £ 0; xf Hf(xf ,W)=0

Iso s optimization problem

ISO’s Optimization Problem

  • ISO’s decision variable:

    yiH = transmission service to hub from i

  • ISO’s value of services maximization problem:

    MAX pISO(y) = SiWi yiH

    s.t.: Tk-£SiPTDFiHk yiH£ Tk+ , "interfaces k

    SiyiH = 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 ³ 0; HISO(xISO,W) £ 0; xISO HISO(xISO,W)=0

Equilibrium calculation

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

Generating firm model variation on a theme

Generating Firm Model:Variation 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 don’t change

Three node two generator system

Three Node-Two Generator System


Elastic Demand






Inelastic Demand

Inelastic Demand







MC = 15 $/MWh

MC = 20 $/MWh

Unconstrained transmission

Unconstrained Transmission




No Arbitrage


With Arbitrage

P3 = 15 $/MWh

W3R = 0 $/MWh

P3 = 22.3

W3R = 0

P3 = 23.8

W3R = 0




318 MW

74 MW

74 MW







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

Constrained transmission

Constrained Transmission




No Arbitrage


With Arbitrage

P3 = 17.5 $/MWh

W3 = 0 $/MWh

P3 = 22.3

W3 = 0

P3 = 23.8

W3 = 0




T =

30 MW







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 = 7723

Net Benefits = 7672

Electric power markets process and strategic modeling

Eastern Interconnection Modeldeveloped 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

Eastern interconnection model

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

Equilibrium prices with demand elasticity 0 4 load weighted average system prices

Equilibrium Prices with Demand Elasticity = 0.4(Load-Weighted Average System Prices)

Merger example firm a at node a firm b at node b

Merger Example(Firm A at Node A, Firm B at Node B)



  • 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

Electric power markets process and strategic modeling

Competition in Markets for Electricity: A Conjectured Supply Function Approach

Christopher J. Day*


Benjamin F. Hobbs

DOGEE, Johns Hopkins

* Work performed while first author was at the University of California Energy Institute, Berkeley, California



  • 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?

Electric power markets process and strategic modeling

Supply Function Conjecture:Each firm f anticipates that rival suppliers follow a linear supply function

Electric power markets process and strategic modeling

Supply Function Conjecture(fixed intercept)

(Rival supply assumed to

follow line through intercept

and present solution)

Present solution

Assumed intercept

Electric power markets process and strategic modeling

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)

England wales analysis














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

Fixed intercept solution with no transmission constraints

Fixed Intercept Solution withNo Transmission Constraints



Pre-divestment (1995)


After 1st divestment (1996)


Price (£/MWh)




After 2nd divestment (1999)








Demand (MW)

Results with a constrained network

Results - with a Constrained Network

E w results competition a la cournot supply function competition and pure competition

E&W Results - Competition a la Cournot, Supply Function Competition, and Pure Competition



  • 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 leader follower model

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 < 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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