efficiency and marginal cost pricing in dynamic competitive markets with friction
DESCRIPTION
https://netfiles.uiuc.edu/meyn/www/spm_files/Market06/Market06.html Abstract: This paper examines a dynamic general equilibrium model with supply friction. With or without friction, the competitive equilibrium is efficient. Without friction, the market price is completely determined by the marginal production cost and the consumers gain positive surplus from trading. If friction is present, no matter how small, then the market price fluctuates between zero and the ``choke-up'' price, without any tendency to converge to the marginal production cost, exhibiting considerable volatility. The gains from trading can deviate significantly from the prediction of the static model in the efficient market outcome. Also considered is a monopolistic market model in which a single firm determines market prices as a function of time. The market outcome is identical in the case of a continuum of consumers. In a model with a single consumer the market prices increase, and the supplier extracts the entire gain from trading.TRANSCRIPT
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Dynamics of Prices in Electric Power Networks
Sean Meyn
Department of ECEand the Coordinated Science Laboratory University of Illinois
Joint work with M. Chen and I-K. Cho
NSF support: ECS 02-17836 & 05-23620 Control Techniques for Complex Networks
Prices
Normalized demand
Reserve
DOE Support: http://www.sc.doe.gov/grants/FAPN08-13.htmlExtending the Realm of Optimization for Complex Systems: Uncertainty, Competition and DynamicsPIs: Uday V. Shanbhag, Tamer Basar, Sean P. Meyn and Prashant G. Mehta
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What is the valueof improved transmission?More responsive ancillary service?
How does a centralized planneroptimize capacity?
Is there an efficient decentralized solution?
OREGON
NEVADA
MEXICO
SANFRANCISCO
LEGEND
COAL
GEOTHERMAL
HYDROELECTRIC
NUCLEAR
OIL/GAS
BIOMASS
MUNICIPAL SOLID WASTE (MSW)
SOLAR
WIND AREAS
California’s 25,000Mile Electron Highway
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Forecast DemandForecast of the demand expected today. The procurement of energy resources for the day is based on this forecast
Actual DemandToday's actual system demand
Revised Demand ForecastThe current forecast of the system demand expected throughout the remainder of the day.This forecast is updated hourly.
Available ResourcesThe current forecast of generating and import resources available to serve the demand for energy within the California ISO service area
Meeting Calendar | OASIS | Employment | Site Map | Contact Us
HOME | Search
Hour BeginningDay Ahead Demand Forecast
Available Resources Forecast
Revised Demand Forecast Actual Demand
http://www.caiso.com/outlook/outlook.html September 28, 2008
4,000Megawatts
40,000 MW
30,000 MW
20,000 MW
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http://www.caiso.com
Emergency Notices
Generating reserves less than requirements
(Continuously recalculated. Between 6.0% & 7.0%)
Generating reserves less than 5.0%
Generating reserves less than largest contingency
(Continuously recalculated. Between 1.5% & 3.0%)
Generating Reserves
7.0%
6.0%
5.0%
4.0%
3.0%
2.0%
1.0%
0.0%
Stage 1EmergencyStage 1
Emergency
Stage 2EmergencyStage 2
Emergency
Stage 3EmergencyStage 3
Emergency
Meeting Calendar | OASIS | Employment | Site Map | Contact Us
HOME | Search
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One Hot Week in Urbana ...
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Spinning Reserve Prices PX Prices
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First Impressions:
July 1998: first signs of "serious market dysfunction" in California
FERC ....authorized the ISO to "[reject]...bids in excess of whatever price levels it believes are appropriate ... file additional market-monitoring reports".
Lessons From the California “Apocalypse:” Jurisdiction Over Electric UtilitiesNicholas W. Fels and Frank R. LindhEnergy Law Journal, Vol 22, No. 1, 2001
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vr za zo ma di wo do
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Ancillary service contract clause:Minimum overall ramp rate of 50 MW/min.
Projected power pricesreached $2000/MWh
Ontario, November 2005
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Ancillary service contract clause:Minimum overall ramp rate of 50 MW/min.
Projected power pricesreached $2000/MWh
Ontario, November 2005
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Ancillary service contract clause:Minimum overall ramp rate of 50 MW/min.
Projected power pricesreached $2000/MWh
Ontario, November 2005
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Australia January 16 2007
Tasmania
Victoria
0
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Australia January 16 2007
Tasmania
Victoria
Volu
me
(MW
h)
Pric
e (A
us $
/MW
h)Pr
ice
(Aus
$/M
Wh)
Volu
me
(MW
h)
- 1,000
- 500
0
+ 500
- 5000
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6,000
8,000
800
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+ 10,000
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ICentralized Control
3
16
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D(t
t
) = demand - forecast
Centered demand:
Reserve options for servicesbased on forecast statistics
On-line capacity
Forecast
Actual demand
Revised forecast
Dynamic model
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Stochastic model:
Normalized cost as a function of Q:
G Goods available at time t
D Normalized demand
Excess/shortfall:
c−
c+
Shortfall
Excess production
q
Q(t) = G(t) − D(t)
Dynamic Single-Commodity Model
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Stochastic model:
Generation is rate-constrained:
G Goods available at time t
D Normalized demand
Excess/shortfall: Q(t) = G(t) − D(t)
Q
q
(t
t
)
High cost
ζ +
- ζ -
Dynamic Single-Commodity Model
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Average cost: density when
Optimal hedging-point: solves
q
c−
c+
pQ(q )
c−P{ Q ≤ 0} + c+P{ Q ≥ 0} = 0
∗q
∗q
− ∗q
∗q
∫c( q−q ) p
Q(dq)=
=
E[c(Q)]
pQ 0
¯
−
Dynamic Single-Commodity Model
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Average cost: density when
Optimal hedging-point: solves
q
c−
c+
ccc−
ccc+c−
c+
pQ(q )
c−P{ Q ≤ 0} + c+P{ Q ≥ 0} = 0
∗q
∗q
− ∗q
∗q
∫c( q−q ) p
Q(dq)=
=
E[c(Q)]
pQ RBM model:
exponentialpQ
0
¯
∗ = 12
σ2
ζ+ logq−
Dynamic Single-Commodity Model
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The two goods are substitutable, but
1. primary service is available at a lower price
2. ancillary service can be ramped up more rapidly
G(t)
G (t)a
Ancillary service
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The two goods are substitutable, but
1. primary service is available at a lower price
2. ancillary service can be ramped up more rapidly
G(t)
G (t)a
Ancillary service
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The two goods are substitutable, but
1. primary service is available at a lower price
2. ancillary service can be ramped up more rapidly
G(t)
G (t)a
Ancillary service
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Excess capacity:
Power flow subject to peak and rate constraints:
K
G(t )G (t )a
Q(t) = G(t) + Ga (t) − D (t), t≥ 0 .
−ζa− ≤ d
dtGa (t) ≤ ζa + −ζ− ≤ d
dtG(t) ≤ ζ +
Ancillary service
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Policy: hedging policy with multiple thresholds
K
G(t )G (t )a
Q(t) = G(t) + Ga (t) − D (t)
q
t
G (t ) = 0
Downward trend:
Blackout
a
G (t ) = - ζ
q
-ddt
ζ +
ζ +
ζ a ++
- ζ -
Ancillary service
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Relaxations: instantaneous ramp-down rates:
Cost structure:
Control: design hedging points to minimize average-cost,
−∞ ≤ d
dtG (t) ≤ ζ +, −∞ ≤ d
dtGa (t) ≤ ζa +.
c(X(t)) = c1G (t) + c2Ga (t) + c3 |Q(t)| 1 {Q(t) < 0}
minEπ [c(Q(t))] .
( )X(t) =
( Q(t)
Ga (t)
)Diffusion model & control
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( )X(t) =
( Q(t)
G a (t)
)Markov model:Markov model: Hedging-point policy:Hedging-point policy:
q2 q1
Ancillary serviceis ramped-up whenexcess capacity fallsbelow
Ancillary serviceis ramped-up whenexcess capacity fallsbelow q2
Diffusion model & control
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( )X(t) =
( Q(t)
Ga (t)
)Markov model:Markov model: Hedging-point policy:Hedging-point policy:
q2 q1
γ0 = 2ζ++ζa+
σ2D
, γ1 = 2 ζ+
σ2D
.
Optimal parameters:Optimal parameters:
q∗2 =1
γ0log
c3c2
q∗1 − q∗2 =1
γ1log
c2c1
Markov model & control
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Discrete Markov model: Optimal hedging-pointsfor RBM:
q1 − q2 = 14.978
q2 = 2.996E(k) i.i.d. Bernoulli.
Q(k + 1) − Q(k)
= ζ(k) + ζa(k) + E(k + 1)
ζ(k), ζa(k) allocation increments.
Simulation
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Discrete Markov model:
q1 − q2
q1 − q2q2 q2
Optimal hedging-pointsfor RBM:
q1 − q2 = 14.978
q2 = 2.996
Q(k + 1) − Q(k)
= ζ(k) + ζa(k) + E(k + 1)
18
20
22
24
7
18
20
22
24
3
16
Average cost: CRW Average cost: Diffusion model
Simulation
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IIRelaxations
(skip to market)
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Line 1 Line 2
Line 3
E1D1
E2
D2E3
D3
Gp1
Ga2
Ga3
Resource pooling from San Antonio to Houston?
Texas model
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Line 1 Line 2
Line 3
single producer/consumer model
Assume Brownian demand, rate constraints as before
Provided there are no transmission constraints,
QA(t) =∑
Ei(t) − Di(t)
GaA(t) =
=
=∑
Gai (t)
extraction - demand
aggregate ancillary
XA = (QA GaA) ≡,
Aggregate model
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Line 1 Line 2
Line 3Given demand and aggregate statefind the cheapest consistentnetwork configuration subject to transmission constraints
min
s.t.
∑(cp
i gpi + ca
i gai + cbo
i q−i )
qA =∑
(ei − di)gaA =
∑gai
0 =∑
(gpi + ga
i − ei)
q = e d−f = ∆p
f ∈ F
consistency
vector reserves
extraction = generation
power flow equations
transmission constraints
c(xA, d)Effective cost
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Line 1 Line 2
Line 3
c(xA, d)
- 50- 50 - 40- 40 - 30- 30 - 20- 20 - 10- 10 00 1010 2020 3030 4040 505000
1010
2020
3030
4040
5050
1
2
3 4
ggaaAA
qqAA
XX++
xxAA
xxAA
xxAA xxAA
RRWhat do theseaggregate states sayabout the network?
Effective cost
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qq11 == −−4646, q, q22 = 3= 3..05640564, q, q33 = 1= 122..94369436
gg11 == −−7711, g, gaa22 = 3= 333..05640564, g, gaa
33 = 6= 6..94369436
ff11 == 1313, f, f22 == −−55,, ff33 == −−88
City 1 in blackout:City 1 in blackout:
Insufficient primary generation:Insufficient primary generation:
Transmission constraints binding:Transmission constraints binding:
Line 1 Line 2
Line 3
c(xA, d)
- 50- 50 - 40- 40 - 30- 30 - 20- 20 - 10- 10 00 1010 2020 3030 4040 505000
1010
2020
3030
4040
5050
ggaaAA
qqAA
1xxAA
Effective cost
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Controlled work-release, controlled routing,uncertain demand.
demand 1
demand 2
Inventory model
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−(1 − ρ)−(1 − ρ)
WW++Resource 1is idle
Resource 2is idle
w2
w1
∗= 12
σ2
ζ+q
Asymptotes:
c−
c+log
Inventory model:Workload relaxation
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IIIDecentralized Control
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Supply & Demand
Cost of generation depends on source
Nuclear ($6)
Coal ($10 -$15)
Gas Turbine ($20-$30)
Supply Curve
Price
Quantity
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Supply & Demand
Demand for power is not flexible
High Priority Customers $5,000/MW?
Customers with interruptible services
Demand Curve
Price
Quantity
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Supply & Demand
Equilibrium Price & Quantity: Intersection of supply & demand curves
High Priority Customers
Consumer Surplus
Supplier Surplus
Equilibrium Price$20 - $30
Equilibrium Quantity 45,000MW
Customers with interruptible services
Demand Curve
Price
Quantity
Nuclear ($6)
Coal ($10 -$15)
Gas Turbine ($20-$30)
Supply Curve
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Supply & Demand
Equilibrium Price & Quantity: Intersection of supply & demand curves
High Priority Customers
Consumer Surplus
Supplier Surplus
Equilibrium Price$20 - $30
Equilibrium Quantity 45,000MW
Customers with interruptible services
Demand Curve
Price
Quantity
Nuclear ($6)
Coal ($10 -$15)
Gas Turbine ($20-$30)
Supply Curve
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Where do ramp constraints appear?Variability?Where is hedging?
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Volumes (MWh)
Week 25
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Welcome to APX!
APX is the first electronic energy tradingplatform in continental Europe. The daily spotmarket has been operational since May 1999.The spot market enables distributors,producers, traders, brokers and industrialend-users to buy and sell electricity on aday-ahead basis.
The APX-index will be published daily around12h00 (GMT +01:00) to provide transparencyin the market. Prices can be used as abenchmark.
www.apx.nl
Main Page Market Results
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Second Welfare Theorem
Each player independently optimizes ...
Supplier: profits from two sources of generation
Consumer: value of consumption minus prices paid minus disaster
WD(t) := v min D(t),Gp(t) + Ga(t) − ppGp(t) + paGa(t) + cboQ−(t)
WS(t) := pp − cp Gp(t) + pa − ca)Ga(t)
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Is there an equilibrium price functional?
Is the equilibrium efficient??
Second Welfare Theorem
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Is there an equilibrium price functional?
Is the equilibrium efficient??
Yes to all !
Q(t) = q
D(t) = d
c cost of insufficient service
v value of consumption
reserve
demand
bo
Second Welfare Theorem
pe(re, d) = (v + cbo) }{I re < 0
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Prices
Normalized demand
Reserve
Efficient Equilibrium
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Spinning Reserve Prices PX Prices
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Weds Thurs Fri Sat Sun Mon Tues Weds
Southern California, July 8-15, 1998 ...
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The hedging point (affine) policyis average cost optimal
Amazing solidarity between CRW and CBM models
Deregulation is a disaster!
Future work?
ConclusionsSpinning Reserve Prices PX Prices
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Complex models:Workload or aggregate relaxations
Price caps: No!
Responsive demand: Yes!
Is ENRON off the hook: ?
Extensions and future work
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Complex models:Workload or aggregate relaxations
Price caps: No!
Responsive demand: Yes!
Is ENRON off the hook: ?
Extensions and future work
What kind of society isn't structured on greed? The problem of social organization is how to set up an arrangement under which greed will do the least harm; capitalism is that kind of system. -M. Friedman
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Fundamentally, there are only two ways of coordinating the economic activities of millions. One is central direction involving the use of coercion - the technique of the army and of the modern totalitarian state. The other is voluntary cooperation of individuals - the technique of the marketplace.
-Milton Friedman
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Justification: 1. Economic systems are complex 2. Regulators cannot be trusted
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Justification: 1. Economic systems are complex 2. Regulators cannot be trusted
Airplanes, highway networks, cell phones... all complex
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Justification: 1. Economic systems are complex 2. Regulators cannot be trusted
Airplanes, highway networks, cell phones... all complex
Complexity is only inherent in the uncontrolledsystem: In each of these examples, thebehavior of the closed loop system is very simple, provided appropriate rules of use, and appropriate feeback mechanisms are adopted.
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3
16q1 − q2
q2
References
• M. Chen, I.-K. Cho, and S. Meyn. Reliability by design in a distributed power transmission network. Automatica 2006 (invited)
• I.-K. Cho and S. P. Meyn. The dynamics of the ancillary service prices in power networks. 42nd IEEE Conference on Decision and Control. De-cember 2003
• I.-K. Cho and S. P. Meyn. Efficiency and marginal cost pricing in dy-namic competitive markets. Under revision for J. Theo. Economics. 46th IEEE Conference on Decision and Control 2006
• P. Ruiz. Reserve Valuation in Electric Power Systems. PhD disserta-tion, ECE UIUC 2008
q ∗1 − q ∗
2 =1
γ1log
c2c1
q ∗2 =
1
γ0log
c3c2
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First reflection times,
τp:=inf{t ≥ 0 : Q(t) = qp}, τa:=inf{t ≥ 0 : Q(t) ≥ qa}
h(x) = Ex
[∫ τp
0
(c(X(s)) − φ
)ds
]
Poisson’s Equation
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First reflection times,
Solves martingale problem,
τp:=inf{t ≥ 0 : Q(t) = qp}, τa:=inf{t ≥ 0 : Q(t) ≥ qa}
h(x) = Ex
[∫ τp
0
(c(X(s)) − φ
)ds
]
M(t) = h(X(t)) +
∫ t
0
(c(X(s)) − φ
)ds
Poisson’s Equation
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X(t) =( Q(t)
Ga(t)
)
qa qp
h(x) = Ex
[h(X(τa))+
h(X(τa))
∫ τa
0
(c(X(s))−φ
)ds
]
Poisson’s Equation
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qa qp
h(x) = Ex
[h(X(τa))+
X(τa)
∫ τa
0
(c(X(s))−φ
)ds
]
〈∇h(x),(11
)〉 = =0, x
Derivative Representations
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qa qp
h(x) = Ex
[h(X(τa))+
X(τa)
∫ τa
0
(c(X(s))−φ
)ds
]
〈∇h(x),(11
)〉 = =0, x
λa(x) = 〈∇c(x),(11
)〉= ca − I{q ≤ 0}cbo
Derivative Representations
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〉 = Ex
[∫ τa
0
λa(X(t)) dt]
= caE[τa] − cboEx
[∫ τa
0
I{Q(t) ≤ 0} dt]
Computable basedon one-dimensional height (ladder) process,
Ha(t) = qa − Q(t)
〈∇h(x),(11
)
Derivative Representations
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qp∗qa∗
〈∇3.
If
Then h solves the dynamic programming equations,
qp = qp∗ and a = qq a∗
h(x),(10
)〉 < 0, x ∈ Rp
〈∇h(x),(11
)〉 < 0, x ∈ Ra
1. Poisson's equation
2.
Dynamic Programming Equations