Structural Estimation Analysis of
Hydropower Scheduling
Maren Boger, Stein-Erik Fleten,
Jussi Keppo, Alois Pichler and Einar Midttun Vestbøstad
IAEE 2017
Goals
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We are interested in how hydropower production planners form
expectations regarding future prices.
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Want to establish an empirical model for hydropower operations
– Based on observed time series
– Assuming operators were acting rationally
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Develop a method for estimating water values from time series of
production, inflow and prices, and technical hydropower plant data
Method: Structural Estimation
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We develop an estimable dynamic programming approach to a hydropower
planning problem
Maximum likelihood estimation with an SDP as a constraint
Use observed decisions to estimate economic primitives: managers beliefs
– Producer price expectations
• Forward information emphasis
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Bellman Equation
Value of future profits
Can write it as
Assuming stationarity
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Idiosyncratic shock, ε(d), observed by decision maker, but not by the analyst
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Define value function
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Gumbel distribution
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Value function becomes
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Define operator to be used as constraint in maximum likelihood estimation
Structural Estimation Problem
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Maximum likelihood estimation problem
– Based on original algorithm (NFXP) by Rust(1987)
– We use NLP approach suggested by Su and Judd (2012)
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Likelihood function
Estimating Conditional Expectation
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Need to evaluate conditional expectation in the value function
Use a set of ARX(1) models, i.e a linear time series approach
Given state space transition
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Define function
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Then value function becomes
Hydropower
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Kolsvik, Helgelandskraft
– Hydropower producer in Norway
– One reservoir approximation
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Hydropower planning assumptions:
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One reservoir approximation
Constant head assumption
Sufficient reservoir flexibility
Sufficient production capacity
Price taker
No marginal production cost
Insignificant start-up and shutdown cost
State Space
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Have six state variables
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Inflow, I
Deviation from normal cumulative local inflow, C
Deviation from normal overall reservoir level in Norway, R
Forward price, F
Spot price, P
Storage (reservoir level), S
– Connection between inflow and price!
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Weekly resolution
Inflow Process
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Seasonal and base process
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AR base process
– Only one lag, since Markovian
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Cumulative Inflow and Overall Reservoir Level
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Cumulative inflow
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Deviation from normal cumulative inflow
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Deviation from normal overall reservoir level
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Autoregressive process, also dependent on C
Forward Price Process
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Write forward with time to maturity T, Ft,T as Ft
– (we only deal with one maturity at a time)
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Seasonal and base process
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Autoregressive base process, also dependent on R
Price Process
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Seasonal and base process, also a factor, ζ, to include forward price
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ζ is the parameter we want to estimate!
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Base process has mean reverting level depending on R
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Underlying autoregressive process
Descriptive Statistics
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Structural Estimation for a Hydropower Producer
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Structural parameter we want to estimate
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ζ – a factor in the price process
– To what degree the price process depends on the forward process, instead of the
price seasonal and base process
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Release function
– Discrete decisions
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Profit function: price times production
– Price taker
– No cost
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Hydropower Specific Value Function
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Need to include time of the year as a state variable
– Approximate stationarity – stationary between years
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Have four random error terms in the state space
Value function for a single agent hydropower producer
Preliminary results – Water Values
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Able to calculate water values!
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Similar shape
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Do not capture the extremes
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Good indication that our model works
Preliminary results - Forward
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Testing for forward contracts with different time to maturity
– 2 months, 6 months and 1 year
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Highest likelihood for 6 months
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Max likelihood closer to ζ=1 than ζ=0, for all
Producer likely to use forward information when planning
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Max likelihood for lower interest rate
– As expected
– Industry uses low rate
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Further Studies
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Validate model further by simulating decision process and use as input to
the model
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Apply model to a general sample of hydropower producers
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Reduce memory usage
Conclusion
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Have developed a working structural estimation model for a hydropower
producer
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Able to calculate water values
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Preliminary results indicate that the producer uses forward price with 6
months to maturity to form expectations for the price, when planning
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Work in progress. Need further studies to validate and improve model
Thanks!