POWER January 2020 - 37

HYDROPOWER
entiation to the basic formula for fluid
power:
HP = Q γ H E / 550
This resulted in:
dQ / dHP = [550 / 62.4 H E] x [ 1 - (HP /
E) x (dE / dHP) - (HP / H) x (dH / dHP)]
It was immediately noted that head
(H) appears in the denominator of the
entire equation. This explains why the
lowest head tends to be the least stable.
Also, efficiency droop appears as: dE/
dHP. Therefore, at any power higher than
peak efficiency, the value of dE/dHP becomes
negative, making the entire term
positive, and contributing to instability.
The derivative of the rate of change of
head with respect to power (dH/dHP)
had not been previously noted as a factor
in surge tank stability, but it appears
to be a stabilizing influence.
3. The U.S. Army Corps of Engineers began building New Melones in 1966, completing the
dam in 1978, and the spillway and powerhouse in 1979. New Melones' surge tank was excavated
into the hillside. The large open pipe seen in the foreground of this image vents the surge
tank to atmosphere. Source: Bureau of Reclamation
riser surge tank, with the Thoma crosssectional
area sized using a 50% safety
margin multiplier.
The Hydroelectric Design Branch
(HEDB) of the Corps was tasked with the
powerhouse design, including a computer
simulation of the entire hydraulic
system. This simulation was done with
a recently developed computer program
called WHAMO, which stands for water
hammer and mass oscillation.
Two different hydraulic Francis turbines
(Figure 4) of the proper specific
speed were selected to be used in the
simulations. After checking maximum
and minimum surge elevations for maximum
and minimum head losses, a fourth
order partial differential equation was
used to represent a mechanical governor
to check stability, as if the project were
serving an isolated load.
The first turbine showed no area of instability.
However, the second did show
instability and this was in an area where
it would be most likely to occur, that is,
at minimum head, near full gate, with
minimum head losses in the penstock,
and maximum head losses in the conduit
between the surge tank a nd the turbine.
Because the surge tank had already
been built, the Corps was directed to
find the turbine performance characteristic
that initiated instability and prepare
a specification for turbine procurement
January 2020 | POWER
that eliminated any such possibility.
It
took
dozens
upon
dozens
of
simulations before that unique characteristic
could be identified. It was a
derivative: the rate of change of the
volumetric flow rate with respect to
generated power (dQ / dHP). Whenever
a simulation was started with a small
load change causing that derivative to
exceed a value of 0.03 cubic feet per
second per horsepower, the surge tank
became unstable.
This derivative was investigated further
by applying the chain rule of differSpecifications
for Turbine
Procurement
For turbine procurement specifications,
the Corps of Engineers inverted this
discovered derivative, that is, dHP/dQ,
or the rate of change of power with respect
to flow rate, and gave it the name
" demand rate. " The specification was
written, " The turbine shall be capable of
increasing its output by 33.3 horsepower
per cubic foot per second at any head. "
A check was performed of the other
turbine used in the simulations, which
did not show evidence of instability. It
was found that its peak efficiency on the
model hill curve was closer to the peripheral
speed coefficient for minimum
head. When converted to prototype performance
values, a demand rate for that
turbine did not occur at less than 33.3
horsepower per cubic foot per second
anywhere within its operating envelope.
In view of the foregoing, it is less
likely that today's hydropower projects
will serve an isolated load (or a grid
that has less than twice its power from
other surge tank sources). If it does, today's
computer programs can simulate
any worst-case scenario to identify the
demand rate to be specified in turbine
specifications to avoid the possibility of
4. This cutaway view shows the downstream
side of a Francis turbine runner with
its wicket gates at the full-discharge setting.
Source: Creative Commons / Stahlkocher
www.powermag.com
surge tank instability. ■
-Lee Sheldon, PE is a consulting engineer
in hydropower and an adjunct professor
teaching hydropower engineering,
fluid mechanics, and hydraulic laboratory
at the Oregon Institute of Technology in
Wilsonville, Oregon.
37

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POWER January 2020

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