Optimal operation of a hydropower plant in a stochastic environment

Abstract

Given the currently changing climate conditions it is of primary importance to optimise the management of hydropower resources. This paper proposes a framework in a dynamic setting to determine the water outflow that maximises the value of a water resource for a given reservoir. The model includes two sources of uncertainty, the water inventory determined mainly by the water inflow and the electricity prices. It is implemented under the stochastic optimal control approach and calibrated using monthly data of reservoir characteristics from ResOpsUs. The results indicate that the inventory dynamics are specially important in valuing reservoir resources. The application of optimal management policies guarantees the long-term sustainability of the reservoir. The possible effects of climate change are considered in a sensitivity analysis to changes in the price and water inventory dynamics.

Keywords:

• Water management
• Climate change
• Hydro power
• Sustainable reservoir

Open Scholarship

This article has earned the Center for Open Science badge for Open Data. The data are openly accessible at https://figshare.com/s/5bdb7bfcf0b399b9b049.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 See article ‘What is a deadpool: A water expert explains' available at: The Conversation US, 1st June 2022.

2 See The Economist Podcast ‘The World Ahead 2023’ 19th Dec 2022.

3 The work of Uria-Martinez et al. is published in the 2021 US hydropower market report.

4 Note that these three papers are closely related. Indeed (Zhao and Davison 2009b) analyse a pumped storage facility with the same engineering parameters as presented in Thompson et al. (2004) and Zhao and Davison (2009a).

5 Note that the authors claim that they use daily Spanish market data for the 1998–2007 period but they do not give details about the source or specific electricity prices used.

6 The ResOpsus database includes the identifier of the facility. The GRandD data base allows identification of the geographical area that is used to download the corresponding monthly electricity prices. The full code will be provided upon request.

7a Chen and Forsyth (2008) have used Poisson processes to represent hourly spot electricity prices under optimal control approaches for hydroelectric power plants. However, the proposed pricing scheme is not calibrated with real price data.

7 This correlation is used when solving the entire system using the value function iteration approach.

8 Note that the inventory level t + 1 is not known at time t as there are random factors that determine future outcomes. In order to estimate this equation empirically the expected level of inventory (estimated from historical data) is used.

9 https://www.duke-energy.com/our-company/environment/renewable-energy/hydroelectric-energyhttps://www.duke-energy.com/our-company/environment/renewable-energy/hydroelectric-energy.

10 https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.htmlhttps://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

11 https://www.federalreserve.gov/faqs/economy_14400.htm.

12 All data series that have been used in this paper are available at (https://doi.org/10.6084/m9.figshare.23257712.v1) with corresponding python code and the readme files.

13 Note that the data used in this manuscript is provided as a supplementary file following the Journal's data sharing guidance.

14 We observed that daily correlations where low but they exhibited strong dependence. For example, a dam that exhibits an inflow in time (t) and the same quantity of outflow in time (t + 1) indicates high dependence (even if there is a low correlation). For this reason the correlation measure was done by grouping the data by months.

15 The first 4 ranked dams had missing the evaporation data while the fifth only presents 0.01% of missing data.

16 https://www.eia.gov/.

17 https://www.epa.gov/greenpower/understanding-electricity-market-frameworks-policieshttps://www.epa.gov/greenpower/understanding-electricity-market-frameworks-policies.

18 Model results show a high goodness of fit for model estimates. They suggest that the process is highly predictable. This was not expected ex ante as weather-related variables are regarded as highly noisy. This raises the possibility that the data may be artificially generated.

19 We thank an anonymous referee for making this point.

20 We assume that the cost function takes a value of 0 due to a lack of data availability. Reservoir operation costs are typically very low see (Latorre et al. 2014).

21 Note that electricity production depends on the elevation of the reservoir since this variable determines the pressure of the water flowing into the turbine, as modelled in section 2.3.

22 We thank an anonymous referee for making this point.

23 Water pricing is the key to equitable access, 12th March 2023.

24 See article ‘Climate graphic of the week: flood risk replaces drought across western US states’ Financial Times, 1st April 2023.

25 See the FT article ‘The trickle-down effect of empty rivers and reservoirs’, 22nd August 2022.

26 The analysis of confidence intervals (which can be provided upon request) confirms that, as expected ex-ante the 5% and 95% bands are significantly wider for the high volatility case than under the benchmark case.

27 $e^{\mathrm{ix}}=\cos \left(x\right)+i\sin \left(x\right)$.

Additional information

Funding

This work was supported by the Ministerio de Ciencia, Innovación y Universidades [grant number PID2019-104960GB-I00]; Ryan Beedie Chair in Finance; and Center for low Carbon Hydrogen Studies, ICAI-ICADE, Universidad Pontificia Comillas.