Profit sharing mechanisms in multi-owned cascaded hydrosystems

We consider the optimal management of hydropower generated by a cascade of three interconnected reservoirs owned by different agents. In this setting, water availability at the downhill reservoirs depends on decisions taken by the agents upstream. This creates an opportunity for the hydroplant at the top to withhold water and take advantage of situations with higher selling prices, which makes the overall decisions of the agents deviate from what can be considered best for the cascade as whole. In order to mitigate the market power of the hydroplant uphill, we propose a mechanism to enforce some collaborative behavior among the agents. This is achieved by agents transferring upstream fractions of their profit in exchange for water released from the top. The corresponding mathematical models are trilevel deterministic and trilevel stochastic linear programming problems, with uncertainty in prices and streamflows (exogenous inflows). For the stochastic variants, analyzed both in two- and multi-stage formulations, we propose new solution methods, extending to the trilevel nested setting the well-known L-Shaped and Stochastic Dual Dynamic Programming methods. A successful implementation of the later depends on certain floating cuts, that represent symbolically Benders-like linearizations. Convergence properties are discussed for some of the procedures. Numerical experiments confirm the interest of the approach, because with the proposed mechanism the top owner as well as the downhill hydroplants earn more money than when acting in an individualistic manner.