1. Structural Estimation Analysis of Hydropower Scheduling
Maren Boger, Stein-Erik Fleten,
Jussi Keppo, Alois Pichler and Einar Midttun Vestbøstad
IAEE 2017

2. Goals
We are interested in how hydropower production planners form expectations regarding future prices.
Want to establish an empirical model for hydropower operations
Based on observed time series
Assuming operators were acting rationally
Develop a method for estimating water values from time series of production, inflow and prices, and technical hydropower plant data

3. Method: Structural Estimation
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

4. Bellman Equation
Value of future profits Can write it as Assuming stationarity

5. Idiosyncratic shock, ε(d), observed by decision maker, but not by the analyst
Define value function

6. Gumbel distribution
Value function becomes
Define operator to be used as constraint in maximum likelihood estimation

7. Structural Estimation Problem
Maximum likelihood estimation problem
Based on original algorithm (NFXP) by Rust(1987)
We use NLP approach suggested by Su and Judd (2012)
Likelihood function

8. Estimating Conditional Expectation
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
Define function
Then value function becomes

9. Hydropower Kolsvik, Helgelandskraft Hydropower planning assumptions:
Hydropower producer in Norway
One reservoir approximation
Hydropower planning assumptions:
Constant head assumption
Sufficient reservoir flexibility
Sufficient production capacity
Price taker
No marginal production cost
Insignificant start-up and shutdown cost 

10. State Space Have six state variables Weekly resolution 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!
Weekly resolution

11. Inflow Process Seasonal and base process AR base process
Only one lag, since Markovian

12. Cumulative Inflow and Overall Reservoir Level
Deviation from normal cumulative inflow
Deviation from normal overall reservoir level
Autoregressive process, also dependent on C

13. Forward Price Process
Write forward with time to maturity T, Ft,T as Ft
(we only deal with one maturity at a time)
Seasonal and base process
Autoregressive base process, also dependent on R

14. Price Process
Seasonal and base process, also a factor, ζ, to include forward price
ζ is the parameter we want to estimate!
Base process has mean reverting level depending on R
Underlying autoregressive process

15. Descriptive Statistics

16. Structural Estimation for a Hydropower Producer
Structural parameter we want to estimate
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

17. Profit function: price times production
Release function
Discrete decisions
Profit function: price times production
Price taker
No cost

18. Hydropower Specific Value Function
Need to include time of the year as a state variable
Approximate stationarity – stationary between years
Have four random error terms in the state space
Value function for a single agent hydropower producer

19. Preliminary results – Water Values
Able to calculate water values!
Similar shape
Do not capture the extremes
Good indication that our model works

20. Preliminary results - Forward
Testing for forward contracts with different time to maturity
2 months, 6 months and 1 year
Highest likelihood for 6 months

21. Max likelihood closer to ζ=1 than ζ=0, for all
Producer likely to use forward information when planning

22. Max likelihood for lower interest rate
As expected
Industry uses low rate

23. Further Studies
Validate model further by simulating decision process and use as input to the model
Apply model to a general sample of hydropower producers
Reduce memory usage

24. Conclusion
Have developed a working structural estimation model for a hydropower producer
Able to calculate water values
Preliminary results indicate that the producer uses forward price with 6 months to maturity to form expectations for the price, when planning
Work in progress. Need further studies to validate and improve model

25. Thanks!