The picture shows the fluctuations in  water level of Vessingsjøen in Tydal, an artificial water reservoir. Photo: Knut Andreas Eikland

There are a number of magazines in Norway that have a lower magazine limit in the summer season for environmental reasons. The associated power plant is not allowed to draw more water from the reservoir if the water level is below this limit.

These are state-dependent restrictions, i.e. restrictions that depend on the state, which in this case is the filling level in the magazine. Such a restriction is "soft", in the sense that tapping from the reservoir is not permitted if the filling is lower than the specified limit.

Illustration of soft magazine restrictions. When the water level of the reservoir is less than the lower reservoir limit, the waterway is closed for operational water flow.

Not high enough precision in the current system

ProdRisk is today a widely used program for long-term planning and seasonal planning of hydropower plants used in the Nordic market. The ProdRisk model is based on "Stochastic Dual Dynamic Programming" or SDDP, as a suitable well-tested method, but which is known to have problems representing state-dependent/soft restrictions with high precision. 

New method for soft restrictions

Researchers at HydroCen have developed a new method to treat such soft reservoir restrictions. In this way, with ProdRisk, production can be better planned for reservoirs where the maximum tapping capacity depends on the degree of reservoir filling.

The solution consists in linearizing the non-convex relationship between reservoir condition and maximum permitted tapping capacity which is described by a soft reservoir restriction. As a supplement, an artificial lower reservoir limit is calculated during the restriction period. This is based on incoming inflow statistics and is calculated as the minimum accumulated inflow from the start of the restriction period. Even with zero reservoir filling at the start of the restriction period, one will therefore be able to meet this artificial minimum requirement with certainty. The linearization is based on the artificial lower reservoir limit.

Tested and ready for implementation

A prototype of the method has been tested on the hydropower systems in Bergsdalen and Rjukan in Norway, and parts of the functionality have been tested in a prototype by ProdRisk.

Based on good results from the tests, the method is considered suitable for implementation in operational versions of the program ProdRisk.

The new method defines what we know as best practice for modeling soft restrictions in production planning models for water value calculation. By adopting the new method, producers will be able to represent soft reservoir restrictions in the water value calculation with higher precision without compromising calculation time.