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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
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
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
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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Purchase Price $/MWh
One Hot Week in Urbana ...
Spinning Reserve Prices PX Prices
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Southern California, July 8-15, 1998 ...
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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APX Europe, June 2003
vr za zo ma di wo do
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Previous weekCurrent week (10/24/05)
Volume (MWh)
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APX Europe, October 2005
Ancillary service contract clause:Minimum overall ramp rate of 50 MW/min.
Projected power pricesreached $2000/MWh
Ontario, November 2005
Ancillary service contract clause:Minimum overall ramp rate of 50 MW/min.
Projected power pricesreached $2000/MWh
Ontario, November 2005
Ancillary service contract clause:Minimum overall ramp rate of 50 MW/min.
Projected power pricesreached $2000/MWh
Ontario, November 2005
Australia January 16 2007
Tasmania
Victoria
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Australia January 16 2007
Tasmania
Victoria
Volu
me
(MW
h)
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e (A
us $
/MW
h)Pr
ice
(Aus
$/M
Wh)
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me
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h)
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+ 10,000
ICentralized Control
3
16
D(t
t
) = demand - forecast
Centered demand:
Reserve options for servicesbased on forecast statistics
On-line capacity
Forecast
Actual demand
Revised forecast
Dynamic model
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
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
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
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
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
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
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
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
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
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
( )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
( )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
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
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)
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Average cost: CRW Average cost: Diffusion model
Simulation
IIRelaxations
(skip to market)
Line 1 Line 2
Line 3
E1D1
E2
D2E3
D3
Gp1
Ga2
Ga3
Resource pooling from San Antonio to Houston?
Texas model
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
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
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
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1
2
3 4
ggaaAA
qqAA
XX++
xxAA
xxAA
xxAA xxAA
RRWhat do theseaggregate states sayabout the network?
Effective cost
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)
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1010
2020
3030
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5050
ggaaAA
qqAA
1xxAA
Effective cost
Controlled work-release, controlled routing,uncertain demand.
demand 1
demand 2
Inventory model
−(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
IIIDecentralized Control
Supply & Demand
Cost of generation depends on source
Nuclear ($6)
Coal ($10 -$15)
Gas Turbine ($20-$30)
Supply Curve
Price
Quantity
Supply & Demand
Demand for power is not flexible
High Priority Customers $5,000/MW?
Customers with interruptible services
Demand Curve
Price
Quantity
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
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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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
Spinning Reserve Prices PX Prices
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Southern California, July 8-15, 1998 ...
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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10000Epilogue
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.
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
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
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
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
qa qp
h(x) = Ex
[h(X(τa))+
X(τa)
∫ τa
0
(c(X(s))−φ
)ds
]
〈∇h(x),(11
)〉 = =0, x
Derivative Representations
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
〉 = 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
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
