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

Abstract

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