# "Take-Back-Half" Convergence Algorithm Stabilizes Microhydro Turbine Controller

Oct. 2, 2008
THE SOARING COSTS OF fossil fuels, combined with environmental concerns like climate change, are driving the increased interest in renewable (“green”) energy. No energy source is “greener” than hydroelectric power, particularly small (“microhydro

THE SOARING COSTS OF fossil fuels, combined with environmental concerns like climate change, are driving the increased interest in renewable (“green”) energy. No energy source is “greener” than hydroelectric power, particularly small (“microhydro”) installations that involve minimal artificial water impoundment and associated environmental impact. Even with an adequate water source, however, efficient and cost-effective implementation of appropriate turbine, generation, and control systems can be problematic. One of the major challenges facing the designer of microhydro control systems is how to accommodate variable electrical loads while operating the turbine at its constant maximum-power specific speed. The control circuit in the figure accomplishes this by servoing the alternator field excitation current (IX) with output voltage (VOUT) feedback.

Conversion efficiency is always key to successful power generation. Efficient hydroelectric generation depends on a good interface between the hydrodynamics of the water-driven turbine (traditionally called a “runner”) and its electromechanical (alternator) load. A variety of turbine types is employed in microhydro installations, but “impulse” turbines (e.g., Pelton and Turgo) as opposed to “reaction” types (e.g., Francis and Kaplan), are popular because of the relative simplicity of their unpressurized housings.

In impulse turbines, the hydrostatic energy (i.e., volume × pressure = VP = m3 × Pa) of the water input is first converted into kinetic energy (MV2/2) by one or more high-velocity open-air jets, which then impinge upon and drive the turbine.

All impulse turbines are characterized by a “specific-speed.” Loosely defined, specificspeed is the speed of rotation (rpm) that, for a given jet velocity, provides the most efficient conversion of water power to mechanical power and therefore produces maximum output for any given hydrostatic input. Water jet velocity is determined by the source pressure (hydrostatic “head”) of the microhydro water supply: (m/s) = (Pa/500)1/2.

Thus, the impulse turbine achieves maximumpower- point (MPP) tracking implicitly, without the need for a separate tachometer or other explicit means of sensing turbine speed. It works by using a universal property of alternators: the output voltage equation:

VOUT = K × rpm × IX

This implicit property of excited-field alternators makes output voltage proportional to the product of rotational speed and excitation current. If a feedback loop is established that regulates IX according to the constant ratiometric relationship:

IX = VOUT/K

then at equilibrium, a constant rpm will be implicitly maintained. So, if K is manually adjusted so that the resulting constant rpm is equal to the maximum-power specific speed of the turbine, MPP tracking will be automatically achieved.

Unfortunately, there’s a fly in this regulatory ointment. IX = VOUT/K feedback inherently comprises positive feedback. Therefore, any such connection must be expected to be unstable and even oscillate violently without ever converging to a stable setpoint. The circuit handles this instability using “take back half,” a simple and robust feedback algorithm that’s described in an earlier IFD (see “Take Back Half: A Novel Integrating Temperature-Control Algorithm,” Electronic Design, Dec. 4, 2000, ED Online 4994). Although used for temperature regulation in that application, the principle is more general.

Error integrator U5a continuously compares the ratio of alternator stator voltage VS to the sensed amplitude of excitation current VI. If VS/VI is greater than the setpoint, the rpm is too fast. If VS/VI is less than the setpoint, the rpm is too slow. The excitation current, supplied by switching regulator U2, is adjusted appropriately. The otherwise inevitable oscillation is suppressed by the convergence-forcing of the iterated bisection of take back half (TBH).

The setpoint error voltage (VS - VI) is input to the TBH integrator (U5a). The integrator output is buffered by U5c and input to regulator U2. Therefore, if VS/VI is greater than the setpoint (indicating alternator rpm > MPP), IX will ramp up. This causes the alternator rpm to slow and the VS/VI ratio to ramp down toward setpoint. If VS/ VI is less than the setpoint (i.e., alternator rpm < MPP), IX will ramp down, causing the alternator to accelerate toward MPP.

Meanwhile, crossed-diodes U3/U4 and comparator U5d track the sign of the VS/VI - setpoint difference. U5d’s output goes high when VS/VI < setpoint and low when VS/VI > setpoint. Setpoint crossings and the associated toggling of U5d will cause the U1c/U1b cross-connected CMOS switches to merge the charges on the 0.1-F integrator capacitors. This allows the TBH convergence-forcing bisection (described in the TBH article mentioned above) to go into effect.

For further details about TBH, see “Use IFDs To Develop And Showcase Your Design Concepts,” p. 42.