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.
Efficiency and Marginal Cost Pricing in Dynamic Competitive Markets with Friction
1. Dynamics of Prices in Electric Power Networks
Sean Meyn Prices
Normalized demand
Reserve
Department of ECE
and 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
DOE Support: http://www.sc.doe.gov/grants/FAPN08-13.html
Extending the Realm of Optimization for Complex Systems: Uncertainty, Competition and Dynamics
PIs: Uday V. Shanbhag, Tamer Basar, Sean P. Meyn and Prashant G. Mehta
2. OREGON
California’s 25,000
Mile Electron Highway COAL
LEGEND
GEOTHERMAL
HYDROELECTRIC
NUCLEAR
OIL/GAS
BIOMASS
MUNICIPAL SOLID
WASTE (MSW)
SOLAR
WIND AREAS
SAN
FRANCISCO
What is the value
of improved transmission? NEVADA
More responsive ancillary service?
How does a centralized planner
optimize capacity?
MEXICO
Is there an efficient decentralized solution?
3. Meeting Calendar | OASIS | Employment | Site Map | Contact Us
HOME | Search
http://www.caiso.com/outlook/outlook.html
September 28, 2008
40,000 MW
4,000
Megawatts
Available Resources
The current forecast of generating and import
resources available to serve the demand for energy 30,000 MW
within the California ISO service area
Forecast Demand
Forecast of the demand expected today.
The procurement of energy resources for the day
is based on this forecast 20,000 MW Day Ahead Demand Forecast Revised Demand Forecast Actual Demand Hour Beginning
Available Resources Forecast
Actual Demand
Today's actual system demand
Revised Demand Forecast
The current forecast of the system demand expected throughout the remainder of the day.
This forecast is updated hourly.
4. Meeting Calendar | OASIS | Employment | Site Map | Contact Us
HOME | Search http://www.caiso.com
Emergency Notices
Generating
Reserves
7.0% Stage 1
Stage 1 Generating reserves less than requirements
Emergency
Emergency (Continuously recalculated. Between 6.0% & 7.0%)
6.0%
5.0%
Stage 2
Stage 2 Generating reserves less than 5.0%
4.0% Emergency
Emergency
3.0%
Stage 3
Stage 3 Generating reserves less than largest contingency
2.0%
Emergency
Emergency (Continuously recalculated. Between 1.5% & 3.0%)
1.0%
0.0%
7. First Impressions:
July 1998: first signs of "serious market dysfunction" in California
Lessons From the California “Apocalypse:”
Jurisdiction Over Electric Utilities
Nicholas W. Fels and Frank R. Lindh
Energy Law Journal, Vol 22, No. 1, 2001
FERC ....authorized the ISO to "[reject]...bids in excess
of whatever price levels it believes are appropriate
... file additional market-monitoring reports".
8. APX Europe, June 2003
Prices (Eur/MWh) Week 25
400 Week 26
350
300
250
200
150
100
50
0
Mon Tues Weds Thurs Fri Sat Sun
9. APX Europe, October 2005
4000
3000 Previous week
2000 Current week (10/24/05)
1000
Volume (MWh)
0
800
Price (Euro)
600
400
200
0
vr za zo ma di wo do
10. Ontario, November 2005
Ancillary service
contract clause:
Minimum overall
ramp rate of 50 MW/min.
Projected power prices
reached $2000/MWh
11. Ontario, November 2005
Ancillary service
contract clause:
Minimum overall
ramp rate of 50 MW/min.
Projected power prices
reached $2000/MWh
12. Ontario, November 2005
Ancillary service
contract clause:
Minimum overall
ramp rate of 50 MW/min.
Projected power prices
reached $2000/MWh
16. Dynamic model
On-line capacity
Forecast
Reserve options for services Actual demand
based on forecast statistics Revised forecast
Centered demand:
D(t) = demand - forecast
t
17. Dynamic Single-Commodity Model
Stochastic model: G Goods available at time t
D Normalized demand
Excess/shortfall: Q(t) = G(t) − D(t)
Normalized cost as a function of Q:
c−
Excess production
c+
Shortfall
q
Scarf, Arrow, Bellman, ...
18. Dynamic Single-Commodity Model
Stochastic model: G Goods available at time t
D Normalized demand
Excess/shortfall: Q(t) = G(t) − D(t)
Generation is rate-constrained:
-ζ-
q
Q(t)
ζ+
t
High cost
21. Ancillary service
G(t)
a
G (t)
The two goods are substitutable, but
1. primary service is available at a lower price
2. ancillary service can be ramped up more rapidly
22. Ancillary service
G(t)
a
G (t)
The two goods are substitutable, but
1. primary service is available at a lower price
2. ancillary service can be ramped up more rapidly
23. Ancillary service
G(t)
a
G (t)
The two goods are substitutable, but
1. primary service is available at a lower price
2. ancillary service can be ramped up more rapidly
24. Ancillary service
a G(t )
G (t )
K
Excess capacity:
Q(t) = G(t) + G a(t) − D (t), t≥ 0.
Power flow subject to peak and rate constraints:
d a d
−ζ a− ≤ G (t) ≤ ζ a+ −ζ − ≤ G(t) ≤ ζ +
dt dt
25. Ancillary service
a G(t )
G (t )
K
Policy: hedging policy with multiple thresholds
Q(t) = G(t) + G a(t) − D (t) -ζ- Downward trend: Ga(t) = 0
d
G (t) = - ζ -
dt
q
ζ+
q
ζ ++ ζ a+ t
Blackout
26. Q(t)
Diffusion model & control X(t) =
G a(t)
Relaxations: instantaneous ramp-down rates:
d d a
−∞ ≤ G (t) ≤ ζ +, −∞ ≤ G (t) ≤ ζ a+.
dt dt
Cost structure:
c(X(t)) = c1G (t) + c2G a(t) + c3|Q(t)|1{Q(t) < 0}
Control: design hedging points to minimize average-cost,
min Eπ [c(Q(t))] .
27. Diffusion model & control
Markov model: Hedging-point policy:
Q(t)
X(t) =
G a(t)
Ancillary service
is ramped-up when
excess capacity falls
below q2
¯
q2
¯ q1
¯
32. Texas model
D1 E1 Gp
1
Line 1
Ga
2
Line 2 Ga
3
D2 D3
Line 3
E2 E3
Resource pooling from San Antonio to Houston?
33. Aggregate model
Line 1 Line 2
Line 3
QA(t) = extraction - demand = Ei(t) − Di(t)
a a
GA(t) = aggregate ancillary = Gi (t)
Assume Brownian demand, rate constraints as before
Provided there are no transmission constraints,
a
XA = (QA ,GA ) ≡ single producer/consumer model
34. Effective cost ¯(xA, d)
c
Line 1 Line 2
Given demand and aggregate state Line 3
find the cheapest consistent
network configuration subject to transmission constraints
p p −
min (ci gi + cagi + cboqi )
i
a
i
s.t. qA = (ei − di) consistency
a
gA = a
gi
p a
0 = (gi + gi − ei ) extraction = generation
q = e−d vector reserves
f = ∆p power flow equations
f ∈F transmission constraints
35. Effective cost ¯(xA, d)
c
Line 1 Line 2
Line 3
a
gA
50
40 xA
1
What do these
30
xA
2 + aggregate states say
X R about the network?
20
10 xA
3
xA
4
qA
0
- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50
36. Effective cost ¯(xA, d)
c
Line 1 Line 2
Line 3
a
gA
50
City 1 in blackout:
q1 = −46,
46, q2 = 3.0564,
3.0564, q3 = 12.9436
12
40 xA
1
Insufficient primary generation:
a a
g1 = −71, g2 = 33.0564,
33 0564, g3 = 6.9436
6.
30
Transmission constraints binding:
20 f1 = 13,
13, f2 = −5, f 3 = −8
10
qA
0
- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50
40. Supply & Demand
Cost of generation depends on source
Price
Supply Curve
Gas Turbine ($20-$30)
Coal ($10 -$15)
Nuclear ($6)
Quantity
41. Supply & Demand
Demand for power is not flexible
High Priority Customers $5,000/MW?
Price Customers with interruptible services
Demand Curve
Quantity
44. www.apx.nl
Main Page Market Results
Welcome to APX!
APX is the first electronic energy trading Prices (Eur/MWh)
400
platform in continental Europe. The daily spot 350
Week 25
market has been operational since May 1999. 300
Week 26
250
The spot market enables distributors, 200
producers, traders, brokers and industrial 150
end-users to buy and sell electricity on a 100
50
day-ahead basis. 0
Volumes (MWh)
The APX-index will be published daily around 2500 MWh
12h00 (GMT +01:00) to provide transparency 2000 MWh
in the market. Prices can be used as a
1500 MWh
benchmark.
1000 MWh
500 MWh
Mon Tue Wed Thu Fri Sat Sun
45. Second Welfare Theorem
Each player independently optimizes ...
Consumer: value of consumption
5000
minus prices paid
4000
minus disaster
WD (t) := v min D(t),G p (t) + Ga (t) − pp Gp (t) + pa Ga (t) + cbo Q− (t)
3000
2000
Supplier: profits from two sources of generation
1000
WS (t) := pp − cp Gp (t) + pa − ca )Ga (t)
13
17
21
13
17
21
13
17
21
13
17
21
13
17
21
1
5
9
1
5
9
1
5
9
1
5
9
1
5
9
46. Second Welfare Theorem
Is there an equilibrium price functional?
Is the equilibrium efficient??
5000
4000
3000
2000
1000
13
17
21
13
17
21
13
17
21
13
17
21
13
17
21
1
5
9
1
5
9
1
5
9
1
5
9
1
5
9
47. Second Welfare Theorem
Is there an equilibrium price functional?
Is the equilibrium efficient??
5000
4000
Yes to all !
3000
2000 p (r , d) = (v + c )I{r < 0}
e e bo e
1000
13
17
21
13
17
21
13
17
21
13
17
21
13
17
21
1
5
9
1
5
9
1
5
9
1
5
9
1
5
9
cbo cost of insufficient service reserve Q(t) = q
v value of consumption demand D(t) = d
50. Conclusions
Spinning Reserve Prices PX Prices
70
250
60
200
50
150 40
30
100
20
50
10
0
Weds Thurs Fri Sat Sun Mon Tues Weds
The hedging point (affine) policy
is average cost optimal
Amazing solidarity between CRW and CBM models
Deregulation is a disaster!
Future work?
51. Extensions and future work 10000
8000
6000
4000
2000
0
Complex models:
Workload or aggregate relaxations
Price caps: No!
Responsive demand: Yes!
Is ENRON off the hook: ?
52. Extensions and future work 10000
8000
6000
4000
2000
0
Complex models:
Workload or aggregate relaxations
Price caps: No!
Responsive demand: Yes!
Is ENRON off the hook: ? 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
53. Epilogue 10000
8000
6000
4000
2000
0
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
54. Epilogue 10000
8000
6000
4000
2000
0
Justification: 1. Economic systems are complex
2. Regulators cannot be trusted
55. Epilogue 10000
8000
6000
4000
2000
0
Justification: 1. Economic systems are complex
2. Regulators cannot be trusted
Airplanes, highway networks, cell phones... all complex
56. Epilogue 10000
8000
6000
4000
2000
0
Justification: 1. Economic systems are complex
2. Regulators cannot be trusted
Airplanes, highway networks, cell phones... all complex
Complexity is only inherent in the uncontrolled
system: In each of these examples, the
behavior of the closed loop system is very simple,
provided appropriate rules of use, and
appropriate feeback mechanisms are adopted.
57. References
q1 − q2
¯ ¯ 16
3 q2
¯
1 c
¯∗ ¯∗
q1 − q2 = log 2
γ1 c1
1 c
¯∗
q2 = log 3
γ0 c2
• 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
58. Poisson’s Equation
First reflection times,
τp :=inf{t ≥ 0 : Q(t) = q p },
¯ τa :=inf{t ≥ 0 : Q(t) ≥ q a }
¯
τp
h(x) = Ex c(X(s)) − φ ds
0
return
59. Poisson’s Equation
First reflection times,
τp :=inf{t ≥ 0 : Q(t) = q p },
¯ τa :=inf{t ≥ 0 : Q(t) ≥ q a }
¯
τp
h(x) = Ex c(X(s)) − φ ds
0
Solves martingale problem,
t
M (t) = h(X(t)) + c(X(s)) − φ ds
0
63. Derivative Representations
τa
1
∇h(x), 1 = Ex λa (X(t)) dt
0
τa
= ca E[τa ] − cbo Ex I{Q(t) ≤ 0} dt
0
Computable based
on one-dimensional height (ladder) process,
H a (t) = q a − Q(t)
¯
64. Dynamic Programming Equations
If q p = q p∗ and ¯ a = ¯ a∗
¯ ¯ q q
Then h solves the dynamic programming equations,
1. Poisson's equation
1
2. ∇h(x), 1 < 0, x ∈ Ra
¯ a∗
q q p∗
1
3. ∇h(x), 0 < 0, x ∈ Rp return