Optimization of hydro storage systems and indifference pricing of power contracts

In this paper, we aim at a mid-term planning model for hydropower production based on multistage stochastic optimization. We decide about a production schedule for a horizon of one year from the point of view of a producer that owns pumped-storage hydropower plants. These consist of large seasonal connected reservoirs. We consider stochastic inflows, stochastic electricity prices and stochastic loads. The optimization of hydropower production is based on multistage stochastic programming with an aggregation using price levels to overcome the curse of dimensionality. The produced electricity is sold at the spot market. In addition, we follow an indifference pricing approach for non-standard power contracts to determine the price at which the producer is willing to deliver electricity to individual consumers.

The dynamics of electricity prices is described by a novel regime-switching approach where a base regime is distinguished from two spike regimes that reflect large price movements down- or upwards. A price is considered to be in one of the spike regimes if it is below or above some limit values which will be estimated simultaneously with the other model parameters. This allows for a more realistic fit to the data than the common approach in the literature where regime limits are set to three standard deviations.

We analyze historical data for the inflows in each reservoir from an existing system in the Swiss Alps. The inflows in each reservoir will be simulated jointly based on two driving factors that are identified by principal component analysis. The generated scenarios will be aggregated to a scenario tree using scenario reduction techniques. In each node of the scenario tree we will have information about the possible price levels for electricity in the upcoming week and about the level of inflows. Based on this information, we decide about generating, pumping or overflows at different price levels. These decisions will be updated in weekly steps.

Our objective is a mixture of expectation and average value of risk over the revenues at the end of the planning horizon of one year. In addition to the model for power dispatch optimization, we formulate a second multistage stochastic programming model to determine the price at which the producer is indifferent with respect to selling the produced electricity on the spot market or entering in individual power contracts. We take into account as well individual demand profiles of consumers. The indifference price is computed for different levels of risk aversion.

To our knowledge, this is the first study in the literature which proposes indifference pricing for multiple contracts in the context of hydropower and it is of particular relevance for the risk management and production planning of power plants holders.