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

**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 (I_{X}) with output voltage (V_{OUT})
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 = m^{3} × Pa) of
the water input is first converted into kinetic
energy (MV^{2}/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:

V_{OUT} = K × rpm × I_{X}

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 I_{X} according to the constant ratiometric
relationship:

I_{X} = V_{OUT}/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. I_{X} = V_{OUT}/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 V_{S} to the
sensed amplitude of excitation current V_{I}. If V_{S}/V_{I}
is greater than the setpoint, the rpm is too fast.
If V_{S}/V_{I} 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 (V_{S} - V_{I}) is input to
the TBH integrator (U5a). The integrator output
is buffered by U5c and input to regulator U2.
Therefore, if V_{S}/V_{I} is greater than the setpoint
(indicating alternator rpm > MPP), I_{X} will ramp up.
This causes the alternator rpm to slow and the
V_{S}/V_{I} ratio to ramp down toward setpoint. If V_{S}/
VI is less than the setpoint (i.e., alternator rpm
< MPP), I_{X} will ramp down, causing the alternator
to accelerate toward MPP.

Meanwhile, crossed-diodes U3/U4 and
comparator U5d track the sign of the V_{S}/V_{I} -
setpoint difference. U5d’s output goes high
when V_{S}/V_{I} < setpoint and low when V_{S}/V_{I} >
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.