Editor's note: Kaplan turbines were designed in the 1910s by Austrian professor Viktor Kaplan to provide a turbine technology that could more efficiently generate power at low-head sites. Kaplans are propeller-type turbines with adjustable blades that allow for efficient power production over head ranges from as low as 2 m to as high as 70 m.

Although the method discussed in this article was developed in the USA, it is applicable worldwide. In fact, Kaplan turbines are installed in a variety of countries, including Argentina, Austria, Brazil, China, Czech Republic, Germany, Honduras, Iceland, Italy, Norway, Panama, Poland, Russia, South Korea, Sweden, Turkey, Uganda, Ukraine, the USA and Zambia.

One of the basic purposes for performing index testing on a Kaplan turbine is to derive the cam curve, or blade angle to wicket gate relation, to program into the governor for turbine control. However, cam profiles change as head changes. This means either index tests need to be performed at various heads or the cam profile from an index test of a single head needs to be extrapolated to other heads. The commonly used methods for extrapolating this data involve a shift or translation along the wicket gate or horizontal axis based on comparison with model test data. However, the results are known to have a significant degree of uncertainty, often resulting in a loss of efficiency of a couple of percent.

The Tennessee Valley Authority in the USA developed a graphical extrapolation method using a homologous model hill curve. This method involves extrapolating the single head index test cam curve by both translation and rotation, while also retaining the shape relationships of the model hill curve. This results in a more accurate family of cam curves and reduces the need for additional index tests, saving time and money.

Deciding to develop the method

An index test is a relative efficiency test that serves two purposes. First, the relative efficiency profile of the turbine-generator can be derived. A relative curve provides information on the power at real best efficiency and the decrease in real efficiency at other power levels. In other words, it reveals how to run the unit for optimum generation, longest service life and minimum mortality to downstream migrating fish. Second, for Kaplan turbines it can provide the cam curve to program into the governor for unit control. Because an index test is performed on a constant head basis, both the resulting relative efficiency profile and cam curve are valid only for the constant head used for the test.

As the head changes, the profile of the cam curve also changes. This is due to the fact that changes in head or wicket gate opening result in a change in the velocity vector of the fluid entering the wheel case. The blade angle needs to change to match the angle of the entering fluid. Specifically, as the head increases or the wicket gates open, the blade angle should go steeper to maintain optimum efficiency.

This variation with head usually necessitates performing several index tests over the head range of the unit to obtain the most accurate cam curves. Due to the expense associated with index testing (US$10,000 to $50,000), project owners often opt to extrapolate a cam curve derived from a single index test into a family of cam curves covering the full head range. This extrapolation generally is done by comparison with the cam curves derived from a prior homologous model test. Specifically, the whole family of model cam curves is shifted horizontally along the wicket gate axis until the model cam curve corresponding to the head of the index test most closely matches the cam curve from the index test.

This method of extrapolation has a significant degree of uncertainty. It does not account for differences in the shapes of the model and prototype cam curves. When the shapes differ, the extrapolation shift usually is accomplished by matching points of peak efficiency. It also does not account for differences between the model laboratory and prototype powerhouse instrumentation. Further, it usually does not account for differences in head loss between the model and prototype. It may not even differentiate between net and gross heads.

Beginning in 1992, TVA undertook a Hydro Modernization program. The goal of the program, to be completed in 2018, is to increase capacity and efficiency at most of TVA's 31 hydro plants, which have a total capacity of more than 5,400 MW. As of July 2011, this program had increased capacity by 564 MW and efficiency by 5%.

As part of this program, TVA personnel performed index tests on all Kaplan turbines before and after their individual runner replacements. To provide the most accurate family of cam curves at the lowest cost, TVA developed a graphical cam curve extrapolation procedure.

Basics of the method

First, the index test data is reduced in the conventional manner, wherein the relative efficiency profiles of a series of fixed blade angles are plotted and a tangent curve is drawn connecting the peaks. The corresponding wicket gate opening for each point of tangency to a fixed blade angle is then determined by interpolation. Head used is the constant gross head as measured by the powerhouse instrumentation. Figure 1 shows relative efficiency profiles and a tangent curve from a U.S. Army Corps of Engineers unit.

This new method of cam curve extrapolation involves cross plotting the model hill curves so that their shape relationships may be retained in the extrapolation. In addition, differences in losses are graphically accounted for while the actual points of tangency from the index test remain unaltered.

Specific example

The below example of this method uses the results of an index test performed on Unit 5 at TVA's 160 MW Pickwick plant. This unit is rated at 48,000 hp at 43 ft of net head and has a synchronous speed of 81.8 rpm. Figure 2 shows the cam curve derived at a gross head of 49 ft.