A Fokker–Planck–Kolmogorov equation approach for the monthly affluence forecast of Betania hydropower reservoir

摘要

This paper presents a finite difference, time-layer-weighted, bidirectional algorithm that solvesnthe Fokker–Planck–Kolmogorov (FPK) equation in order to forecast the probability density curven(PDC) of the monthly affluences to the Betania hydropower reservoir in the upper part of thenMagdalena River in Colombia. First, we introduce a deterministic kernel to describe the basicndynamics of the rainfall–runoff process and show its optimisation using the S/sD performancencriterion as a goal function. Second, we introduce noisy parameters into this model, configuringna stochastic differential equation that leads to the corresponding FPK equation. We discuss thenset-up of suitable initial and boundary conditions for the FPK equation and the introduction ofnan appropriate Courant–Friederich–Levi condition for the proposed numerical scheme that usesntime-dependent drift and diffusion coefficients. A method is proposed to identify noise intensities.nThe suitability of the proposed numerical scheme is tested against an analytical solution and thengeneral performance of the stochastic model is analysed using a combination of the Kolmogorov,nPearson and Smirnov statistical criteria.