HR: 1330h
AN: H12B-0998    [PDF]
TI: Runge Kutta Algorithm applied to a Hydrology Problem
AU: * Narayanan, M
EM: narayam@muohio.edu
AF: DR. MYSORE NARAYANAN, ENGINEERING, MIAMI UNIVERSITY, HAMILTON, OH 45011 United States
AB: In this paper, the author utilizes a fourth order Runge Kutta Algorithm technique to solve a design problem in Hydrology and Fluid Mechanics. Principles of Fuzzy Logic Design methodologies were utilized to analyze the problem and arrive at an appropriate solution. The problem posed was to examine the depletion of water from a reservoir. A suitable model was to be created to represent different parameters that contributed to the depletion, such as evaporation, drainage and seepage, irrigation channels, city water supply pipes, etc. The reservoir was being fed via natural resources such as rain, streams, rivers, etc. A model of a catchment area and a reservoir lake is simulated as a tank and exit discharge is represented as fluid output via a long pipe. The Input to the reservoir is assumed to be continuous-time and time varying. In other words, the flow rate of fluid input is presumed to change with time. The required objective is to maintain a predetermined level of water in the reservoir, regardless of input conditions. This is accomplished by adjusting the depletion rate. This means that some of the Irrigation channels may have to be closed or some of the city water supply lines need to be shut off. The differential equation governing the system can be easily derived using Bernoulli's' equation. If hd is the desired height of water in the reservoir and h(t) represents the height of water in the reservoir at any given time, K represents a positive constant. (dh/dt) + K [ h(t) - hd ] = 0 The closed loop system is simulated by using fourth-order Runge-Kutta algorithm. The controller output u(t) can be calculated using the above equation. The Runge-Kutta algorithm is a very popular method, which is widely used for obtaining a numerical solution to a given differential equation. The Runge-Kutta algorithm is considered to be quite accurate for a broad range of scientific and engineering applications, and as such, the method is heavily used by many scholars and researchers. In summary, Runge-Kutta is a common method of solving ordinary differential equations using numerical integration techniques. The principle is to use a trial step at the midpoint of an interval to cancel out lower-order error terms. Suppose that hn is the value of the variable at time tn. The Runge-Kutta formula takes hn and tn and calculates an approximation for hn+1 at a brief time later, tn+„. It uses a weighted average of approximated values of f(t, h) at several times within the interval (tn, tn+„). hn+1 = hn + (1/6) [ k1 + 2k2 + 2k3 + k4 ] k1, k2, k3 \& k4 are four gradient terms. Fuzzy logic FLC rule base can be developed based on the above derivations and equations. Further, a graphical representation of water level over a time step period can be obtained.\\ References :\\ Nguyen, Hung T.; Prasad, Nadipuram R.; Walker, Carol L. and Walker, Elbert A. (2003). A First Course in Fuzzy and Neural Control. Boca Raton, Florida : Chapman \& Hall / CRC.\\ Yager, R. R., and Zadeh, L. A. (1991). An Introduction to Fuzzy Logic Applications in Intelligent Systems. New York : Kluwer Academic Publishers
DE: 1818 Evapotranspiration
DE: 1854 Precipitation (3354)
DE: 1857 Reservoirs (surface)
DE: 1860 Runoff and streamflow
DE: 1899 General or miscellaneous
SC: Hydrology [H]
MN: 2003 Fall Meeting