Transcript No Slide Title

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
[email protected]
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?


Scope of economic impact
• ~$1000/person/yr in US (~petroleum use)
• Almost half of US energy use
Ongoing restructuring and reforms Finland`s 1995
Elect. Market Act
• Vertical disintegration
– separate generation, transmission,
distribution
X
• Competition in bulk generation
– grant access to transmission
– creation of regional spot & forward
markets
•
•
•
•
X
EL-EX, Nord Pool
Competition in retail sales
Horizontal disintegration, mergers
Privatization
Emissions trading
X
Fingrid
None
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


What are bottom-up/engineering-economic
models? And how can they be used for
policy analysis?
D
= Explicit representation & optimization of
individual elements and processes based
on physical relationships
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 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

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

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


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

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




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
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
Conclusion
What can process-based policy models do
well?
• Exploratory modeling: examining implications
of assumptions/scenarios upon
impacts/decisions
Exploratory modeling is becoming easier
because of increasingly nimble models,
and is becoming more important because
of increased uncertainty/complexity
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
(Geoffrion)
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
MIN Variable Cost = Si,f,t Ht CYift yift
subject to:
Si,f yift = LOADt
Sf yift < (1- FORi)Xi
Sf,t Ht yift < CFi 8760Xi
yift > 0
t
i,t
i
i,f,t
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 + VOC +

 NOx
O 
O3
Power plants emit ~1/3 of anthropogenic
NOx in USA
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
Framework
We want less cost and less NOx
Cost
Inefficient
Efficient
=Alternative
dispatch
order
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)
How To Generate Alternatives
Solve the following model for alternative
levels of the regulatory constraint:
MIN
Si CYi yi
s.t. 1. MRi < yi < Xi
(note nonzero LB)
2. Si yi > LOAD (MW)
3. Regulatory caps: either
Si Ei yi < MASS CAP (tons) or
(Si Ei yi )/(Si 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
Results



11,400 MW peak and 12,050 MW of
capacity, mostly gas and some coal.
Most of capacity has same fuel
cost/MBTU. Plant emission rates vary by
order of magnitude (0.06 - 0.50 lb/MBTU)
With single tonnage cap, the cost of
reducing emissions by 20% is $60M (a
5% increase in fuel cost).
Emissions rate cap raises control cost
by $1M due to “dilution” effect (increase
BTU rather than decrease NOx). More
diverse system results in larger penalty.
Cost of Inefficient Energy Trading
Higher than Dilution Effect
Two area analysis: energy trading for
compliance purposes discouraged by
rate limits
1260
Cost ($M/year)
1250
1240
1230
1220
Cost ($M), Ton Cap
Cost ($M), Rate Cap
1210
1200
1190
1180
1170
75000
80000
85000
90000
95000 100000
Tons NOx / year
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
Node or “bus” m
Current Imn
Bus n
Voltage Vn

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
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
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 + Sif yifmt)] < Tk+
Transmission Constraints in
Operations LP
MIN Variable Cost = Si,f,t Ht CYift yifmt
subject to:
Si,f,m yifmt = Sm LOADtm
t
Sf yifmt < (1- FORi)Xi m
i,m,t
Sf,t Ht yifmt < CFi 8760Xim
i,m
Tk- < [Sm PTDFmk(-LOADmt + Sif yifmt)] < Tk+
k,t
yifmt > 0
i,f,m,t
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 CYit yit + Si,t CUi uit
s.t. Si yit = LOADt
MRi uit < yi < Xi uit
i,t
-RRi < (yit - yi,t-1) < RRi i,t
St yit < CFi T Xi
i
yit > 0 i,t; uit {0,1}
i,t
t
IV. Deterministic Investment
Analysis: LP Snap Shot Analysis

Let generation capacity xi now be a
variable, with (annualized) cost CRF [1/yr]
CXi [$/MW], and upper bound XiMAX.

MIN Si,f,t Ht CYift yift + Si CRF CXi xi
s.t. Si,f yift = LOADt
Sf yift - (1- FORi)xi
t
i,t
<0
Sf,t Ht yift - CFi 8760xi < 0
Si xi > LOAD1 (1+M)
xi < XiMAX
yift > 0
i
(“reserve margin” constraint)
(Note: equality for existing plants)
i,f,t;
xi > 0
i
i
Some Complications







Dynamics (timing of investment)
Plants available only in certain sizes
Retrofit of pollution control equipment
Construction of transmission lines
“Demand-side management” investments
Uncertain future (demands, fuel prices)
Other objectives (profit)
Demand-side investments

Let zk = 1 if DSM program k is fully
implemented, at cost CZk [$/yr].
Impact on load in t = SAVkt [MW]

MIN Si,f,t Ht CYift yift + Si CRF CXi xi + Sk CZk zk

s.t. Si,f yift + Sk SAVkt zk = LOADt
Sf yift - (1- FORi)xi
t
i,t
<0
Sf,t Ht yift - CFi 8760xi < 0
i
Si xi + (1+M) Sk SAVkt zk > LOAD1 (1+M)
i
xi < XiMAX
yift > 0
i,f,t;
xi > 0
i;
zk > 0
k
V. Pure Competition Analysis
Simulating 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
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.
MF/MXi - Σj λj MGj/MXi # 0;
 Xi:
Xi $ 0;
Xi(MF/MXi - Σj λj MGj/MXi) = 0
{
 λj:
{
Gj # 0; λj $ 0;
λj Gj = 0
Notation: Each node i is a separate
commodity (type, location, timing)
Consumer: Buys QDi
QDi
i
h
QSi
TEij
TIij
Transporter/Transformer:
Uses exports TEij from i to
provide imports TIij to j
Supplier: Uses inputs Xi to
produce & sell QSi
j
Players’ Profit Maximization
Problems
Consumer at i:
MAX Ii(QDi) - Pi QDi
j
{QDi}
s.t. QDi $ 0
Transporter for nodes i,j:
MAX Pj TIij - Pi TEij - Cij(TEij,TIij)
i
{TEij,TIij}
s.t. Gij(TEij,TIij) # 0 (dual θij)
TEij, TIij $ 0
Supplier at i:
MAX PiQSi - Ci(Xi)
{QSi,Xi}
s.t. Gi(QSi,Xi) # 0
Xi , QSi $ 0
(μi)
Supplier’s Optimization
Problem and KKT Conditions
Supplier at i:
MAX
PiQSi - Ci(Xi)
{QSi,Xi}
s.t. Gi(QSi,Xi) # 0 (dual mi)
Xi , QSi $ 0
KKTs:
QSi:
X i:
μ i:
(Pi - μi MGi/MQSi)# 0; QSi $ 0;
QSi (Pi - μi MGi/MQsi) = 0
(-MCi/MXi - μi MGi/MXi) # 0; Xi $ 0;
Xi (-MCi/MXi - μi MGi/MXi) = 0
Gi # 0; μi $ 0;
μi Gi = 0
KKTs for All Players in Market Game
+ Market Clearing Condition
Consumer KKTs,  i:
QDi: (MB(QDi) - Pi) # 0; QDi $ 0;
QDi (MB(QDi) - Pi) = 0
Supplier KKTs,  i :
Market
QSi: (Pi - μi MGi/MQSi)# 0; QSi $ 0;
Clearing,  i:
QSi (Pi - μi MGi/MQsi) = 0
Xi:
(-MCi/MXi - μi MGi/MXi) # 0; Xi $ P0;:
QSi
i
Xi (-MCi/MXi - μi MGi/MXi) = 0 + Σ
j 0 I(i) TEji
μ i:
Gi # 0; μi $ 0; μi Gi = 0
- Σj 0 E(i) TIij
- QDi
Transporter/Transformer KKTs,  ij:
TEij: (-Pi - MCij/MTEij - θij MGij/MTEij) # 0; TEij $ 0; = 0
TEij(-Pi - MCij/MTEij - θij MGij/MTEij) = 0
TIij: (+Pj - MCij/MTIij - θij MGij/MTIij) # 0; TIij $ 0;
TIij(+Pi - MCij/MTIij - θij MGij/MTIij) = 0
N conditions
θij: Gij # 0; θij $ 0; Gij θij = 0
& N unknowns!
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
Gi(QSi,Xi) # 0
Gij(TEij,TIij) # 0
QDi, Xi, QSi $ 0
TEij, TIij $ 0

(dual Pi)  i
(μi)
i
(θij)  ij
i
 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

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. Analyzing Strategic
Behavior of Power Generators
Part 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

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
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


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: 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
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)
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 (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 (Continued)
4. Types of Games, Continued:
• Examples of Nash Games:
Bidi
– 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
Qi
Model Elements (Continued)
4. Types of Games, Cont.:
• Noncooperative Game (Asymmetric/LeaderFollower): 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 (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


Each firm i's marginal cost function = QSi , i
= 1,2
Demand function : P = 100 - QD/2 [$/MWh]
100
MCi
P
1
1/2
1
1
QSi
QD
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.
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

The “Bimatrix” problem:
• Player 1: MAX Sh,k P1hP2k p1(X1h,X2k)
{P1h}
s.t. Sh P1h = 1
P1h > 0,  h
• Player 2: MAX Sh,k P1hP2k p2(X1h,X2k)
{P2k}
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
DC Electric Network
high cost B
generator
low value
load
D
A
low cost
generator
30 MW
flow limit
C
high value load
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

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)
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
Maximize consumer value
minus production costs
Price
S
P*
D
Q*
Quantity
ISO Maximizes Total Welfare
Price
4 Nodes
No Transmission Constraints
- same price everywhere
- DC + DD = SA + SB
SB
SA
P*
DC
DD
mcA = mcB = mvC = mvD
Quantity
ISO Maximizes Total Welfare
Transmission Constraints
S
D
Price
S
PD*
Congestion
Revenues
PS*
D
Q*
Quantity
Transmission
Constraint at Q*
Price Dispersion: Prices are Duals
of Nodal Energy Balances
No Transmission
Constraints
B
Gen
1 Price
A
Gen
D
Load
AC=30
C
Load
Transmission
Constraint(s)
 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.
Choice of Supply Function
Price
Supply
Complete bid:
(m,b)
Alternatively
Fix m, choose b
or
Fix b, choose m
p=mq+b
slope intercept
Quantity
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
No
Transmission Constraints Surprise
QB=81
QB=73
P=46 everywhere
P=54 everywhere
QD=82
QA=104
QC=103
QA=85
QD=70
QC=88
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 Surprise!
Transmission Constraint (AC=30)
PB=50 QB=90 Gen B better
off with constraint
PA=18
QB=94
PD=61
P=67 everywhere
QA=24
QD=60
QA=20
QD=51
30
PC=72 QC=54
QC=63
Perfect Competition
Imperfect Competition
Bids: bA=10, bB=10
Profits: pA=101, pB=1819
Bids: bA=60, bB=25
Profits: pA=1087, pB=3373
Imperfect Competition Eliminates Transmission Constraint
Imperfect Competition and Surprise!
Multiple Generators (3 at A, 1 at B)
1 Gen at A
1 Gen at B
3 Gen at A
1 Gen at B
QB=94
P=67 everywhere
QA=20
30
PB=55 QB=73
PA=24
QD=51 QA=30
QC=63
PD=65
30
PC=75 QC=48
Increased competition leads
to higher prices for consumers
QD=54
Counterintuitive Result: Increased
Competition Worsens Prices

Compare one and three generators located
at Node A:
Consumer
Total
Run
PC
PD
Surplus Surplus
($/MWh) ($/MWh)
($)
($)
1 Generator
67
67
1873
6322
at A
3 Generators
75
65
1559
6315
at A
Strategic Modeling Part 2. Large Scale Market Models
A 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


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 +Sgf sig)-Wi]sif -[Cif(gif) -Wi gif ]}
gif  CAPif , i
Si sif = Si gif
sif , gif 0, i
• In Cournot model, f sees wheeling fees Wi and rivals’
sales Sgf sig as fixed
s.t.:

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 Problem

ISO’s decision variable:
yiH = transmission service to hub from i

ISO’s value of services maximization problem:
s.t.:
MAX pISO(y) = Si Wi yiH
Tk-  Si PTDFiHk yiH  Tk+ ,  interfaces k
Si yiH = 0
• Solution allocates interface capacity to most valuable
transactions (a la Chao-Peck)
• Tk- , Tk+ = transmission capabilities for interface k
PTDFiHk = power distribution factor (assumes DC model)

The 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 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 = 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

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
Hub
P
Elastic Demand
3
Q
P
P
Inelastic Demand
Q
1
MC = 15 $/MWh
2
Constrained
Interface
Inelastic Demand
Q
MC = 20 $/MWh
Unconstrained Transmission
Perfect
Competition
Cournot,
No Arbitrage
P3 = 15 $/MWh
W3R = 0 $/MWh
3
1
318 MW
2
P1 = 15 $/MWh
P2 = 15
W1R = 0 $/MWh
W2R = 0
G1 = 954 MW
G2 = 0
Net Benefits = $10,614/hr
1
Cournot
With Arbitrage
P3 = 22.3
W3R = 0
P3 = 23.8
W3R = 0
3
3
74 MW
2
P1 = 25
P2 = 25
W1R = 0
W2R = 0
G1 = 392
G2 = 170
Net Benefits = 7992
1
74 MW
2
P1 = 23.8
P2 = 23.8
W1R = 0
W2R = 0
G1 = 392
G2 = 170
Net Benefits = 8031
Constrained Transmission
Perfect
Competition
Cournot,
No Arbitrage
P3 = 17.5 $/MWh
W3 = 0 $/MWh
P3 = 22.3
W3 = 0
P3 = 23.8
W3 = 0
3
3
3
T=
30 MW
1
2
P1 = 15 $/MWh
P2 =20
W1 = +2.5$/MWh W2 = -2.5
G1 = 491 MW
G2 = 353
Net Benefits = $8632/hr
Cournot
With Arbitrage
2
1
P1 = 24.1
W1 = +1.4
G1 = 330
P2 = 25.9
W2 = -1.4
G2 = 232
Net Benefits = 7672
2
1
P1 = 22.4
W1 = +1.3
G1 = 335
P2 =25.1
W2 = -1.3
G2 = 228
Net Benefits = 7723
Eastern Interconnection Model
developed by Judith Cardell, Thanh Luong and Michael Wander, OEP/FERC; Cournot
development & application by Udi Helman, OEP/FERC & Ben Hobbs, JHU
100 nodes representing control areas
and 15 interconnections with ERCOT,
WSSC, and Canada
 829 firms (of which 528 are NUGs)
 2725 generating plants (in some
cases aggregated by prime
mover/fuel type/costs);
approximately 600,000 MW capacity

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)
super peak 5% peak
shoulder off-peak
Competition
$30.99
$22.17
$19.89
$16.39
Cournot w/
arbitrage
Difference
$31.48
$22.85
$20.14
$16.74
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 where
difference > 5%
1.01%
15.12%
2.38%
13.27%
Merger Example
(Firm A at Node A, Firm B at Node B)
pre-merger
merger
Competition
$22.17
$22.17
Cournot w/
arbitrage
$22.85
$22.86
Difference
3.07%
3.11%
Cournot Price
Node A
$18.89
$18.92
Cournot Price
Node B
Profits
$27.65
$27.66
A+B=$162,723
AB=$162,400
Challenges




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 Electricity:
A Conjectured Supply Function Approach
Christopher J. Day*
ENRON UK
Benjamin F. Hobbs
DOGEE, Johns Hopkins
* Work performed while first author was at the University
of California Energy Institute, Berkeley, California
Introduction

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
s- f   sh
h f
Supply Function Conjecture
(fixed intercept)
p
p 
p
p* - 
*
-f
s
s- f  
(Rival supply assumed to
follow line through intercept
and present solution)
*

Present solution
Assumed intercept
*
-f
s
s- f
Producers Models with Supply Function
Conjectures (fixed intercept i)
max
g fi , s fi , p fi
 p
fi
- Wi s fi -  MC fi - Wi g fi
i
subject to
i
s fi  D p fi  - s- fi  p fi 
s- fi  p  
p fi s-* fi -  i s-* fi
pi* -  i
g fi  G fi
g
i
fi
  s fi
i
g fi , s fi  0
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
2
1
3
5
6
4
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
7
8
10
9
11
13
12
Fixed Intercept Solution with
No Transmission Constraints
30
Price (£/MWh)
25
Pre-divestment (1995)
20
After 1st divestment (1996)
15
10
Demand
5
After 2nd divestment (1999)
0
10000
20000
30000
40000
Demand (MW)
50000
60000
Results - with a Constrained Network
Demand Elasticity 0.0005
25
Price
20
15
10
5
SF 1995
SF 1996
SF 1999
P. Comp
0
N1 N2 N3 N4 N5 N6 N7 N8 N9 N10 N11 N12 N13
E&W Results Competition a la Cournot, Supply Function
Competition, and Pure Competition
Demand Elasticity 0.1
60
Cournot 1995
Cournot 1996
Cournot 1999
SF 1995
SF 1996
SF 1999
P. Comp
40
30
20
10
N13
N12
N11
N10
N9
N8
N7
N6
N5
N4
N3
N2
0
N1
Price
50
Discussion

Supply/response function conjecture
• Descriptively more appealing than Cournot
• Can model inelastic demand
• Can include transmission models
– Gives insights into locational prices

Has the potential to analyze large systems
• 200 to 300 node networks

Can gain insights into possible market
power abuses in bilaterally traded and
POOLCO markets
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 equilibrium constraints (MPECs) are improving