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

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

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

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

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

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

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

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

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 

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

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

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

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

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

Descriptive Statistics

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

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

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

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

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

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

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

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

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

Thanks!