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Scheduling Energy and Reserve in Systems with Stochastic Production
Antonio J. Conejo Univ. Castilla – La Mancha
2012
November 16, 2012 2
Expected behavior?
A. J. Conejo
10 pm 10 pm
2 “Nuclear Power” Plants
8 “Nuclear Power” Plants
Spain February 18, 2012 Wind power
Expected behavior?
• Unprecedented decrease in production capacity (equivalent to 6 nuclear power plants) over a 24-hour time span
• No such thing in the past! • Quite different from the failure of a large
production facility • Quite different from the failure of a large
transmission facility
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Expected behavior?
November 16, 2012 4
Market of the Iberian Peninsula (Spain) Day-ahead market prices, March 1, 2010
A. J. Conejo
zero price 25 GW
Prices
Expected behavior?
November 16, 2012 5
ERCOT Balancing Market prices: March 7, 2009 A. J. Conejo
Negative prices
-40
-30
-20
-10
0
10
20
30
40
50
60
0 10 20 30 40 50 60 70 80 90 100
South zone price
West zone price
Houston zone price
North zone price$/MWh
15-minute periods
Prices
Expected behavior?
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Simulation! Price variability
A. J. Conejo
0
2
4
6
8
10
12
0 10 20 30 40 50 60
Bus20
Bus8
Bus7
$/MWh
Wind penetration level (% system demand)
High wind penetration
• High uncertainty in production: This is a entirely new phenomenon!
• New solutions needed.
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?
Contents
• Further motivation • Dispatching/scheduling model • Pricing scheme • Computational considerations • Conclusions • Examples (dispatching and scheduling)
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Further Motivation
Uncertainty makes scheduling and dispatching electricity generation in advance a challenging task.
Anticipation is required for the optimal operation of the less flexible units (coal, oil).
Enough reserve needs to be ensured by flexible units (gas, hydro) to cope with system uncertainties in real time.
Appropriate prices need to be derived.
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Further Motivation
Day-ahead market
Balancing market
Time
Day d-1 Day d
Uncertainty ↑ (stochastic production ↑) Flexibility ↓ ⇒ Balancing costs ↑
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Further Motivation
Day-ahead market
Balancing market
Time
Day d-1 Day d
Adjustment markets
Adjustment markets allow redefining forward positions and trading with a lesser degree of uncertainty
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Further Motivation
Day-ahead market
Balancing market
Time
Day d-1 Day d
Adjustment markets
Adjustment markets allow redefining forward positions and trading with a lesser degree of uncertainty
Illiquid
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Further Motivation
Day-ahead market
Balancing market Reserve capacity
markets
Time Day d-1 Day d
• Guarantee balancing resources
• Promote flexible generation via capacity payments (price capped markets)
?
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Further Motivation
Day-ahead market
Balancing market Reserve capacity
markets
Time Day d-1 Day d
Energy-only market (no cap)
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Further Motivation
Day-ahead market
Time Day d-1 Day d
The day-ahead market is cleared by accounting for the projected impact on subsequent balancing
operation
Balancing market
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Further Motivation
Day-ahead market
Balancing market
Day-ahead market
Balancing market
Balancing prognosis
Decoupled (DAM and BM are cleared
independently)
Coupled (Day-ahead energy dispatch
decisions account for balancing operation)
Dx Dx
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Aim
Two-stage stochastic programming model for the
simultaneous scheduling of energy and reserve under
uncertainty.
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Remarks
The scheduling model provides
amount and allocation of reserve
energy dispatch
Prices
… and envisions the implementation of corrective actions such as reserve deployment, load shedding and wind spillage.
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Remark
DC load flow model for a linear representation of the network.
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"Everything should be made as simple as possible ... but not simpler." Einstein
Stochastic programming approach
Two-stage stochastic programming:
Scenario 1
Market decisions (here-and-now decisions)
Operation decisions (wait-and-see decisions)
Scenario 2
Scenario NΩ
Scheduled production and consumption.
Scheduled reserves.
Market prices.
Deployment of reserves.
Involuntary load shedding.
Wind power spillage.
Balancing prices.
Others (angles, power flows and power injections).
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Scheduling model Formulation
Minimize Expected cost
Subject to:
Scheduling constraints
Real-time operation constraints
Linking (scheduling-operation) constraints
MILP problem
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Scheduling model Simplifying assumptions
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Scheduling model Simplifying assumptions
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Scheduling model Objective function
• Minimize Energy production cost (conventional units) + Wind energy production cost (if any) + Unserved energy cost + Reserve cost
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Scheduling model Objective function
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Variable
Scheduling model Scheduling equilibrium
• Energy balance per bus at scheduling time (day-ahead): Power production + Wind power production – Power demand – Power leaving through transmission lines = 0
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Scheduling model Day-ahead market equilibrium
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Per bus First stage variables!
Variable
Scheduling model Real time balance
• Energy balance at operation time per bus and scenario (real-time): Reserve deployment + Unserved load + Wind power deviation = Line flow adjustments
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Scheduling model Real time balance
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Per bus and scenario Second stage variable
First stage Variable
Scheduling model Bounds (i)
• Thermal production capacity • Wind production capacity • Production needs to be non-negative • Production needs to be below capacity • Transmission capacity limits at scheduling time • Transmission capacity limits at operation time
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Scheduling model Bounds (i)
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Scheduling model Bounds (ii)
• Unserved energy bounds • Wind spillage bounds • Reference angle setting at scheduling time • Reference angle setting at operation time
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Scheduling model Bounds (ii)
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Scheduling model Bounds (iv)
• Scheduled reserve bounds, up • Scheduled reserve bounds, down • Deployed reserve bounds, up • Deployed reserve bounds, down
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Scheduling model Bounds (iv)
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Scheduling model Variable declarations
• Non-negativity declarations (production, deployed reserves, unserved load, wind production, spilled wind)
• Free variables declarations (angles at scheduling and operation times)
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Scheduling model Variable declarations
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Pricing Scheme
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Pricing Scheme Scheduling (first stage)
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Pricing Scheme Operation (second stage)
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Pricing Scheme Operation (second stage)
Revenue adequacy in expectation
• Revenue Adequacy in Expectation: the payments that the system/market operator must make to and receive from the participants do not cause it to incur a financial deficit.
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Cost recovery in expectation
• Cost Recovery in Expectation: the proposed market pricing guarantees that the expected profit of each generating unit (including wind farms) is greater than or equal to its operating costs.
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44
Math structure
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45
Math structure
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Decomposition techniques welcome!
Conclusions Scheduling (market clearing) procedure able to
cope with major uncertainties.
Scheduling procedure that resolves the tradeoff security vs. economic efficiency.
Two-stage stochastic programming model that reproduces real-world (market) operation.
Energy and reserve co-optimization!
Appropriate pricing scheme proposal: it ensures cost recovery and revenue adequacy.
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Conclusions
Wind generation decreases the expected operation costs, but increases the costs of reserves.
The reserve cost due to uncertainty is relevant with respect to the energy production cost.
Network congestion may seriously hinder the cost reduction achievable by integrating wind.
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Future work
Modeling uncertainty in a compact manner: robust optimization?
Pricing if nonconvexities are present
Pricing if robust optimization is used
November 16, 2012 A. J. Conejo 48
Further information
• A. J. Conejo, M. Carrión, J. M. Morales, “Decision Making Under Uncertainty in Electricity Markets” International Series in Operations Research & Management Science, Springer, New York. 2010.
• S. Gabriel, A. J. Conejo, B. Hobbs, D. Fuller, C. Ruiz, “Complementarity Modeling in Energy Markets” International Series in Operations Research & Management Science, Springer, New York. 2012.
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Thanks!
November 16, 2012 A. J. Conejo 50
Example Single-period Dispatching
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Example Network
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Example Network
• Line reactances: 0.13 p.u. • Capacities: 100 MW
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Physical characteristics of the lines
Example Wind scenarios
• High: 50 MW, with probability 0.2 • Medium: 35 MW, with probability 0.5 • Low: 10 MW, with probability 0.3
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Wind scenario characterization
Example Load
• Load: 200 MW • Unserved energy cost: $1000/MWh
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High cost of unserved load
Example Generator data
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Cheap but inflexible Expensive but flexible
Example Scheduling outcome
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Scheduled power
At maximum
Example Prices (scheduling and balancing)
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Scheduling Balancing
Example Profits
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Example Outcomes with reserve capacity offers
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100 50 30 At maximum
No reserve capacity offers
Example Prices with reserve capacity offers
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29 30 25 30
No reserve capacity offers
Example Profits with reserve capacity offers
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Different!
Case study: Multi-period Scheduling
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Case study: data Multi-period Unit Commitment
Based on the IEEE RTS 24-bus system.
A 24-hour market horizon.
Peak load: 2650.5 MW.
Wind power at bus 7.
Cardinality of initial wind scenario set: 3018.
Cardinality of reduced wind scenario set: 20.
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Case study: Problem size
Constraints: 228,562
Continuous variables: 153,409
Binary variables: 4,536
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No that big
Case study: Results 1. Per-unit expected energy cost =
Total expected energy cost / Installed non-wind capacity
≈ 4.7 – 4.8 ($/MWh)
2. Per-unit expected reserve cost = Total expected reserve cost / Installed wind capacity
≈ 60 – 76% per-unit expected energy cost
November 16, 2012 A. J. Conejo 66
Case study: Computational issues
1. Excluding non-spinning reserves.
−∆Time (%) ≈ 99% (∆Cost (%) < 0.21%)
2. MIP gap = 1%.
−∆Time (%) ≈ 88% (∆Cost (%) < 0.66%)
3. Warm start.
−∆Time (%) ≈ 48% at no cost
up to 81.3%!
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Notation
November 16, 2012 A. J. Conejo 68
Notation Indices and Numbers
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Notation Variables
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Notation Variables
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Notation Variables
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Notation Random Variables
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Notation Constants
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Notation Constants
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Notation Constants
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Notation Sets
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Thank you!
November 16, 2012 A. J. Conejo 78
