Estimation of Ideal Configuration and Dimension of Pico Hydropower using Euler-Lagrange Equation and Runge-Kutta Method

Abstract

Pico hydropower is a type of hydropower that can produce electric power under or equal to 5000 Watt. Consisting of one turbine to rotate one generator, it has been used in remote areas to provide for electricity supply. Although the system serves quite well, sometimes the necessity for electricity supply is larger than the system can supply. The solution would be to build two pico hydropower systems. However, the cost of such a solution will also double. In addition to that, the water source and landscape may not sustain more than one pico hydropower. An alternative solution that is being proposed to resolve the problem is by using one turbine to rotate two electric generators in a unity pico hydropower system. The objective of the study concerning the proposed solution is to obtain a configuration and dimension of the energy converter. They are sought by determining the equation of motion for energy converter which is configured in several wheels. The equation of motion, which is in differential form, is obtained using the Euler-Lagrange procedure in classical mechanics. Both configuration and dimension of energy converter are solved using numerical methods. The numerical method used in this study is the fourth-order of the Runge-Kutta method for analyzing some physical parameters i.e.: angular velocity of the energy converter, friction coefficient, driving force, radius and inertia of the wheels. By employing varying numbers in the computational process using the fourth-order of the Runge-Kutta method in Matlab, we seek that the pico hydropower system with ideal configuration and dimension of energy converter based on the radius of the wheels are configured as r 1> r 3>r 2>r 4>r 5=r 6. The configuration yield an optimum rotation of the coupled wheels (r 5 and r 6) as a representation of the electric generator's axis although the thrust is assumed lower. The design expected to be a solution for optimizing a pico hydropower, especially in remote areas with lower water flow.