MASTER'S THESIS Uncertainties in Kaplan Cam Curve Hanna Isaksson 2015 Master of Science in Engineering Technology Sustainable Energy Technology Luleå University of Technology Department of Engineering Sciences and Mathematics
The water enters the plant from the headwater through the intake gate, goes through the pressure
conduit, through the turbine and is then discharged through the draft tube after which it flows out to
the tailwater (Krivchenko, 2.1 Turbine head, 1993).
The difference in elevation between the headwater and the tailwater area is called the static head
and is calculated through equation 1.1, same notation as in Figure 1.1.
Where and denotes the elevation at the headwater and tailwater water level (Krivchenko,
2.1 Turbine head, 1993) .
The head of a turbine however is the difference in mechanical energy between the inlet and outlet. It
is also called the net head and is specified by equation 1.2, same notation as in Figure 1.1
(Krivchenko, 2.1 Turbine head, 1993).
Where the static head, are the hydraulic losses, the free fall acceleration the velocity
at a point , and is the correction factor for kinetic energy (Krivchenko, 1.2 Energy of fluid (water),
1993)at a point , “0” denotes the intake (Krivchenko, 2.1 Turbine head, 1993).
The power that the turbine removes from the moving water, is expressed by
Where Q is the volume flow and H is the head (Krivchenko, 2.2 Turbine capacity (Power), 1993).
There are losses between in the turbine accounted in an efficiency η, defined in equation 1.4
Where is the net capacity of the turbine. By combining equations 1.3 and 1.4, an expression for the
net capacity can be defined as seen in 1.5 (Krivchenko, 2.2 Turbine capacity (Power), 1993)
The specific speed is used to evaluate the kind of turbine suited for a head and volume flow rate. It is
independent of the turbine diameter and is a characteristic of the turbine type. It is calculated by
equation 1.6 (Krivchenko, 5.8 Specific speed, 1993)
This means, the larger the head the lower the specific speed.
In Sweden no new hydropower plants are built due to regulations and the environmental impact of
establishing new hydropower plants (Vattenfall AB, 2013). Sweden and Norway also have a joint
system to stimulate renewable electricity production consisting in awarding “electricity certificates”
to producers of renewable electricity and by declaring that energy intensive industries and energy
producers have an obligation quota. They must then procure a quota of their
production/consumption in renewable electricity. A new plant or a plant that has increased its
production is eligible for green certificates for 15 years. However not longer than until 2035 when
the system will expire (Energimyndigheten; Norges vassdrags- og energidirektorat, 2013).
In order for a hydropower plant in Sweden or Norway to be eligible for a green certificate, the
production increase has to be due to an increase of the average flow rate through the station,
reduced losses in the waterways excluding the turbine, and having reduced the losses in the energy
transformation system. Efficiency improvements due to routine maintenance are not eligible for
certificates (Statens energimyndighets författningssamling, 2011). Therefore, it is of importance to
verify that at least one of the three requirements previously mentioned is fulfilled. The verification is
performed with index tests.
1.1 Problem description The Swedish hydropower is to a large extent used in the frequency control of the power-grid (Svensk
Energi, 2014). When a turbine is used for frequency control it is constantly making small changes to
the power output by changing the flow rate of the water going through the turbine, both increasing
and decreasing the flow. The index test performed today only measures in one direction, with
increasing flow in order to avoid hysteresis in the mechanism affecting the result (Bard, Att utföra
indexprov och utvärdera resultaten, 1993).
Kaplan turbines are often used for this frequency regulation in Sweden since they have a high
efficiency over a larger interval compared to other turbine types. The constant change of angles due
to frequency regulation creates wear and tear which may either create or increase the hysteresis
phenomenon (JR. & KAWASAKA, 2014).
Kaplan turbines are run according to a cam curve (Krivchenko, 8.3. Plotting the characteristics of the
adjustable blade turbines, 1994). The question is how the sensitive is the cam curve to the guide
vane and runner blade angles.
1.2 State of the Art Research about uncertainties in efficiency measurements for Kaplan turbines is being and has been
performed during the last couple of years; several articles have been made regarding the scope of
the measurement error for different relative flow measurement methods.
Cervantes, Andrée, Klason and Sundström have investigated available flow rate measurement
methods for low head turbines and their advantages and disadvantages (Cervantes, Andrée, Klason,
& Sundström, 2012).
Nicolle and Proulx have made investigations into the uncertainties of the Winter-Kennedy flow rate
measurement method, where it was showed that the k-coefficient varies with the guide vane angle
and that the measurements are sensitive to changes in the inlet boundary conditions (Nicolle &
Directives regarding how to perform index tests to obtain the best results exist as complements to
IEC 60041. One such example is “Index test and best cam curves design procedure for Kaplan
turbines” (JR. & KAWASAKA, 2014).
1.3 Aim of thesis This thesis aims to investigate the sensitivity of the index test with focus on the behavior of the guide
vanes and runner blades. This is to be done by comparing the cam curves from an index test and
from an experiment with measurements performed in a random order. The opening and closing of
the guide vanes and runner blades are also investigated closely.
2 Kaplan turbines There are two main types of turbines used for hydroelectric power plants, reaction and impulse
(active) turbines. Differences are aside from the way they work the heads and specific speeds at
which they operate. The most common types are Pelton, Francis, Kaplan and Mixed-flow turbines
(Krivchenko, 2.3 Principal turbine kinds, 1993).
Table 2.1 and Table 2.2 indicate the head and specific speed range for different turbines.
Table 2.1 Heads for different types of hydraulic turbines. For turbines with low capacity (100 – 3000 kW) the limits are not corresponding to the data in this table (Krivchenko, 2.4 Applications of turbines of various kinds, 1993).
Table 2.2 – Specific speeds for selected turbine types, specific speed calculated with equation 1.6 (Krivchenko, 5.8 Specific speed, 1993).
Turbine type Specific speed, [rev/min]
Adjustable blade axial flow (Kaplan) 1200 – 450
Adjustable blade mixed flow 500 – 300
Radial-axial flow (Francis) 400 – 80
Impulse (Pelton) 50 – 10
Radial-axial, axial, impulse and mixed flow are designation describing how the water enters and
leaves the turbine in question. Figure 2.1 presents the different types.
Figure 2.1 – How water enters and leaves different turbines, a) is an axial flow turbine, b) is a mixed (diagonal) flow turbine, c) radial-axial flow turbine and d) an impulse turbine (Pelton) (Krivchenko, 2.4 Applications of turbines of various kinds, 1993).
A Kaplan turbine is a double-regulated axial-flow turbine, working at heads between 1 and 60 m
(Alvarez, 2006) and at specific speeds between 450-1200 rev/s. Kaplan machines are used at low
heads, their popularity stems from that they maintain a high efficiency over a wide operation
compared to single-regulated turbines. In a Francis turbine, only the guide vane opening is
adjustable. In a Kaplan turbine, both the guide vane opening and the runner blade angle are
adjustable; doubly regulated (Krivchenko, 2.3 Principal turbine kinds, 1993). The main components of
a Kaplan turbine are viewed in Figure 2.2.
Figure 2.2 – Main components of a Kaplan turbine (Krivchenko, 2.3 Principal turbine kinds, 1993).
1. Runner blades
2. Runner cone (casing for the runner blade)
4. Turbine case
6. Wicket gate, guide vanes
7. Top support band of the stayring
8. Lower support band of the stayring
9. Draft tube
The runner diameter characterizes the size of the turbine. There are usually four to eight runner
blades, depending on the head. Larger head leads to more blades. Figure 2.3 shows a picture of a
section of a runner from above. The stayrings purpose is to strength the structure. The wicket gate
consists of 20-32 guide vanes arranged in an annular cascade, as can be seen in the B-B section in
Figure 2.2. The guide vanes direct the water flow; the velocity vector v0 has the same angle relative
to the radius as the guide vanes. In Figure 2.3, guide vanes and runner blades installed at a
hydropower plant are presented.
Figure 2.3 – To the left: Photography of the runner and runner blades of a Kaplan turbine. To the right: Photography of a section of the guide vanes and stayring vanes in a Kaplan turbine. Both pictures are taken at Laxede kraftstation in the Lule Älv River by the author.
The operating gear of the guide vanes serves to adjust the angle of the guide vanes and make sure
that they all have the same angle . There are different technical solutions, the most common
(Krivchenko, 4.1 Guide vane operating gear, 1993) is through a regulation ring, see Figure 2.4.
Figure 2.4 - Common operating gear for guide vanes in hydropower plants, regulating ring. (1) is levers set upon the upper pivot point of the guide vanes, (2) is the shackles and pull rods connecting the levers (1) to the regulating ring (3) (Krivchenko, 4.1 Guide vane operating gear, 1993).
The regulation ring ensures that all the levers are turned the same angle. To avoid solid objects in the
waterways damaging the guide vanes or the operating gear, weakened links are incorporated in the
construction. The regulation ring is controlled by hydraulic servomotors that are capable of
developing the forces required to move the wicket gate at smooth and even pace to the right
position (Krivchenko, 4.1 Guide vane operating gear, 1993).
Individual servomotors for each guide vane are occasionally used for large high-power Francis
turbines. When using this method, the regulator ring and the weakened links are not necessary, the
construction is thus lighter. The control of the guide vanes becomes more complicated and special
2.1 Cam curves In order for Kaplan turbines to have a high efficiency, the adjustable vanes and blades needs to be
run according a cam curve (Krivchenko, 8.3. Plotting the characteristics of the adjustable blade
turbines, 1994). The cam curve shows the efficiency depending on the flow rate and the
corresponding guide vane angle for each flow rate.
Figure 2.9 – Cam curve for a Kaplan turbine. The upper curves are the efficiency depending on the flow rate. The lower curves are the guide vane angle depending on the flow. (Krivchenko, 8.3. Plotting the characteristics of the adjustable blade turbines, 1994)
The cam curve is derived by interpolating between the best efficiency points for propeller curves; see
Figure 2.9, which are explained closely in the following chapter.
A propeller curve is the plot of the efficiency for a certain runner blade angle as a function of the flow
rate, changing with the guide vane opening. It is derived by holding the runner blade angle ϕ at a
constant level while testing different guide vane openings a0 until the combination with the best
efficiency is found (Krivchenko, 8.3. Plotting the characteristics of the adjustable blade turbines,
1994). More about how such measurements is presented later on and in “Field acceptance tests to
determine the hydraulic performance of hydraulic turbines, storage pumps and pump-turbines” (IEC
The efficiency for a certain runner blade angle varies with the guide vane opening. As a matter of
fact, the circulation between the guide vanes trailing edge and the runner blades leading edge is
created by the wicket gate (Krivchenko, 5.1 Water flow created by the wicket gate of the reaction
turbines, 1993). The shape of the propeller curve is explained by analyzing the forces affecting a
runner blade. In Figure 2.10, the forces acting upon a runner blade are presented.
Figure 2.10 – Vector components of forces acting on a runner blade (Cervantes, Propellerkurvor, 2015).
The drag force is defined as
And the lift force as
Where is the average relative velocity of and , the average velocity of the water at the
leading and trailing edge of the blade. and are the drag respectively lift coefficients. l is the
length of the blade and is the angle between and the peripheral velocity , see Figure 2.10
(Cervantes, Propellerkurvor, 2015).
The angle at which the sum of the forces and , is directed at is then given by
The portion of the force acting in the rotational direction is
) ( ) 2.4
( ) 2.5
The power then developed by the runner blade is
( ) 2.6
The maximum power available per airfoil of thickness is given by a version of Equation 1.5
( ) 2.7
The efficiency can then be expressed as follows
( ) 2.8
is the number of runner blades, is the axial velocity, is the radius of the turbine and is the
blade pitch (Cervantes, Propellerkurvor, 2015).
The lift coefficient varies with the attack angle according to
The relation is valid for attack angles of which can be verified by looking at Figure 2.11.
Figure 2.11 - and derived from experimental data (Cervantes, Propellerkurvor, 2015).
In Figure 2.11, is at its lowest for attack angles in the range ; .
From Equation 2.8 it can be read that CL multiplied with a function of the quota between the drag-
and lift coefficient affects the efficiency, an estimation of the combined effect can be seen in Figure
Figure 2.12 – Estimation of the combined effect of the drag and lift coefficient depending on the attack angle.
The combined effects curve in Figure 2.12 displays a clear peak, just as the propeller curve
(Cervantes, Propellerkurvor, 2015).
2.2 Index tests Kaplan turbines require experiments to determine and/or verify the best combination of guide vane
opening and runner blade angle. There is also a need to verify that a turbine fulfills the specification
given by the manufacturer. These tests are generally called “Field acceptance tests to determine the
hydraulic performance of hydraulic turbines, storage pumps and pump-turbines” and the method for
performing these tests is described in detail in IEC 60041 (International Standard IEC 41, 1991).
An index method test is such a test performed using a relative method for measuring the discharge.
It only gives relative values of the discharge and the efficiency. Their main use is during the
commissioning and operation of the machines. In order for an index test to be deemed acceptable to
use in a field acceptance test, the relative discharge needs to be calibrated by a method accepted in
IEC 60041. They are however accepted for usage in finding the correct relationship between runner
blade angle and guide vane opening in double-regulated machines (IEC 41, 1991).
In Sweden, index tests are used for
I Improvement quantification after refurbishment
Cam curve adjustment due to wear and tear of the machine.
Finding the cam curve for the operation of the turbine
An index test on a Kaplan turbine usually requires three days; one day for installing and controlling
measurement equipment, one day for the test, and one day for dismantling the equipment. The
workforce needed is one technician for setting up the equipment, collecting and treating the data, an
operator for changing the angles and someone from the power plant to control that prescribed
water levels are kept and to maintain communications with the operations center. A schedule has to
be prepared beforehand with the planned discharges and the time to plan the river discharge (Bard,
Att utföra indexprov och utvärdera resultaten, 1993).
-5 0 5 10 15
Attack angle [°]
The settings should be made based on the fixed scales or other fixed references. This since dials and
displays may change over time and it is important to use the same angles when comparing tests
(Bard, Att utföra indexprov och utvärdera resultaten, 1993).
The changes of the guide vane angle should be made consequently in one direction to avoid
hysteresis. Four to six guide vane angles should be tested for each runner blade angle. Usually the
direction is increasing the guide vane angle. The angle should also be controlled to regulate any drift
before moving on to the next setting. Five to eight runner blade angles should be tested. (Bard, Att
utföra indexprov och utvärdera resultaten, 1993)
The following variables need to be measured according to an “old” Swedish instruction (Bard, Att
utföra indexprov och utvärdera resultaten, 1993);
The relative flow measured with an accepted method, the Winter Kennedy method is one
such method. The use of transparent tubes is recommended so that possible bubbles of air
can be observed and eliminated.
The head, the sensors should be placed at the same place as station sensors.
o Headwater level with a submersible sensor, if only one sensor, after trash rack.
o Tailwater level, if possible with four pressure taps in the draft tube, otherwise with a
submersible sensor placed somewhere where the level is as close as possible to the
level in the draft tube outlet.
Generator power, make sure to measure only the power from the turbine
In practice index test today includes the following measurements (Sendelius, 2015);
Guide vane angle, fixed scales
Guide vane angle, station sensor
Runner blade angle, fixed scales
Runner blade angle, station sensor
Headwater level before trash rack
Headwater level after trash rack
Headwater level, station sensor
Tailwater level, station sensor
The station displays for power, current and voltage are also noted along with reactive power
and adjacent turbines power.
The relative flow estimated is described in section 2.3 and thus will not be explained here.
The relative efficiency is calculated by combining Equations 1.4 and 1.5
The net head is calculated by simplifying Equation 1.2 to
Where zhw is the headwater level after trash rack and ztw is the tailwater level according to either
station sensors or placed sensors.
Using where Ai is the cross-section area of the turbine inlet, respectively the outlet. The
gravity acceleration g is assumed constant. Sometimes, the gross head is used. The only difference
from the net head is that the hydraulic losses in the inlet and the outlet of the turbine are not taken
The turbine power is calculated as
Where Pgen is the generator power and Pgen-f the generator losses which in turn is calculated as
Where Ptom is the idle running losses, Pbel the operation losses, I the generator current and Im the
generator marked current. However, if there is a magnetic generator on the turbine- or the
generator axis the magnetic loss has to be taken into account too (Bard, Att utföra indexprov och
utvärdera resultaten, 1993). Then equation 2.14 becomes as below
( ) (
Where the subtext “mag” denotes the magnetic losses during operation and during idle running.
The magnetic losses are usually stated as one number and then 1/3 of this number is the magnetic
idle running losses and 2/3 magnetic operation losses (Bard, Att utföra indexprov och utvärdera
In order to compare the results, the flow, the guide vane opening and the power are normalized to
the same head using the similarity laws; (Krivchenko, Turbines of one type and similarity laws for
their modes of operation, 1994)
Traditionally, the efficiency is assumed unchanged η1/η2 = 1
For statistical calculations a significance level of 95 % should be assumed (IEC 41, 1991).
2.3 Winter-Kennedy flow rate measurement The Winter-Kennedy method measures the discharge with the differential pressure in the turbine
spiral case. The discharge is proportional to the differential pressure, see equation 2.19 (IEC 41,
Where is the flow constant, is the reading from a manometer, or differential pressure sensor,
and (IEC 41, 1991) n is between 0.48-52 depending on the shape of the spiral case (Lövgren, 2015).
The location of the taps is important. In a concrete semi-spiral, they are placed in the same radial
section. In Figure 2.13 the location of the pressure taps can be seen, the numbers in the following
section refers to this figure. (IEC 41, 1991)
The outer tap, “1” or “1’”needs to be positioned sufficiently far from the corners to avoid
disturbances. The inner tap “2” or “2’” should be located at the inner side, outside the stay vanes and
midway between two adjacent stay vanes and in a flow line. A third tap “3” can also be placed on
one of the stay vanes at the same elevation as the guide vane centerline or at the roof between two
stay vanes. Another set of taps is recommended to be placed in another radial section (IEC 41, 1991).
If the spiral case is made of steel, the same directives apply. However, the taps should not be placed
close to a geometry variation or welded joints. For horizontal spiral cases the taps should be located
in the upper half to simplify the pipes purging (IEC 41, 1991).
Figure 2.13-Placement of pressure taps in a concrete semi-spiral case for discharge measurement in turbines using the Winter-Kennedy method (IEC 41, 1991).
The Winter Kennedy method is based upon the conservation of momentum. When a fluid is flowing
through a curved pipe it is subject to a centrifugal force. This force can, when assuming negligible
radial (with respect to the runner) and vertical components, laminar, steady and non-viscous flow, be
expressed as below
Where uθ, r, p and ρ are the tangential velocity, the radial position relative to the spiral casing center,
the pressure, and the density, respectively.
Expressing the flow as
Inserting 2.21 into 2.20 and assuming uθ constant over the radial section the resulting equation is
Then by integrating equation 2.22 with respect to P and r from point “1” to point “2” in the spiral the
flow rate is given by
And thus the k and h in equation 2.19 are
√ ( )
( ) √ 2.25
The flow coefficient k is typically determined by a model test. By scaling up the model efficiency at
the best efficiency point, the BEP, the prototype efficiency at BEP is determined. The flow rate is
estimated based upon the prototype efficiency, the prototype power output and head at BEP. The k
coefficient can then be determined from the differential pressure at BEP and the theoretical flow
rate. It is not necessary to use the BEP to determine k. Any point can be used, for example the
maximum power output (Cervantes, Andrée, Klason, & Sundström, 2012).
The Winter-Kennedy method is the most widely used method for flow rate measurements in low-
head machines in Sweden because it is simple and cheap.
However, the results may present variation and the reasons are somewhat unclear. The results may
for example show that the efficiency of a turbine has dropped after a refurbishment. This may be due
to old pressure taps, the surfaces of the guide vanes and runner which have been changed between
the measurements. The geometry in the spiral case may be modified, the neighboring turbines
operated at different conditions during the tests. But no systematic error analysis of the method
3 Design of Experiment The purpose of this work is to estimate the uncertainty in the determination of a combination curve
on a Kaplan. Special attention will be given to hysteresis phenomenon. The results will be put in
perspective with the conventional index method. The following chapter describes the theoretical
basis for the planning of the measurement procedure.
The first step is to specify the object of the investigation and the factors of interest to be controlled.
In the present case, the efficiency and difference between sensors and fixed scales are the responses
under investigation. The factors affecting the efficiency to be investigated are the runner blade angle,
guide vane angle, and their respective direction of movement, i.e., decreasing or increasing angle to
achieve the desired value. The last parameter in the following sections is referred as “direction of
3.1 Basic statistical terms The design of experiment is a method to plan experiments using statistics. A description of statistical
terms used in this report follows in this section, more information may be found in “Design and
Analysis of Experiments (Montgomery, Design and Analysis of Experiments, 2005).
One of the main elements for a properly designed experiment is randomization. This is a necessary
condition in order to be able to make the assumption that the measurement errors are
independently distributed random variables. This can be achieved by using a computer program with
a random number generator to plan the order of the measurements (Montgomery, 2-3 Sampling and
sampling distributions, 2005).
Table 3.1 – Short descriptions of certain statistical terms and associated relationships. The letter “y” denotes a random variable (Montgomery, Simple comparative experiments; 2-2 Basic statistical concepts, 2-3 Sampling and distributions, 2005).
Term Symbol Formula Description Equation number
∫ ( )
∑ ( )
A measure of a probability distributions central tendency or location.
∫ ( ) ( )
∑( ) ( )
The variability or dispersion of a probability distribution
Degrees of Freedom
The number of independent elements in a sum of squares, n is the number of samples
The average value of the random variables
∑ ( )
Dispersion of sample with n measurements
Sample standard deviation
The standard deviation of the sample, more commonly used since it has the same units as y
Corrected sum of squares
SS ∑( )
Unit used in ANOVA tables.
From Table 3.1 it can be derived that and are unbiased estimations of µ and , meaning that
they are the average value that the point estimator will assume in the long run. For more information
see (Montgomery, Simple comparative experiments; 2-2 Basic statistical concepts, 2-3 Sampling and
One method used for proving or discarding a theory regarding a certain relationship between test
parameters is “hypothesis testing”. The “null hypothesis”, is tested against the alternative
hypothesis . The difference in the mean efficiency of a machine between two different days is an
The hypothesis is tested through a random sample and then rejected or confirmed by computing the
test statistic. The set values in the test statistic for rejecting or failing to reject the null hypothesis are
called the critical region or rejection region for the test. There are two types of error related to
hypothesis testing. Type I error, when the null hypothesis is rejected even though it is true. The
second is type II error, when the null hypothesis is not rejected when it is false. The chance that each
of these errors were to be made is denoted as in equations 3.9 and 3.10
( ) ( | ) 3.9
( ) ( | ) 3.10
Usually, the “power” of a test is utilized to decide whether to reject or not, see equation 3.11
( | ) 3.11
The normal procedure is to specify the probability for type I error α, the significance level. Then
design the test procedure in a manner such as that the probability of type II error β is suitably small.
Thus if you have a -value larger than your selected you cannot reject the null hypothesis . A
common level for α is 0.05, thus giving a significance level of 95 %. The -value is typically calculated
using statistical software (Montgomery, Simple comparative experiments; 2-4.1 Hypothesis testing,
The excel add-in “Analysis Toolpak” can be used for statistical analysis of data in Excel, for example to
generate Analysis-of-variance- tables, or ANOVA-tables as the short name is. The commands that
have been used in this project are Anova-Two factors with replicates and Anova – One factor.
3.1.1 Theory of “design of experiment” – factorials
For experiments where the effect of several parameters and the interaction between them is of
interest thus a factorial experiment is advised. The parameters are then called factors (Montgomery,
Introduction to Factorial Designs, 2005).
In a factorial design of an experiment, all possible combinations of factors are tested. In the present
study, A is the runner blade angle and a is the number of levels for the blade angle, two. B denotes
the guide vanes angle and b the number of levels for the angle. One replicate of a factorial contains
ab treatments combinations see Table 3.2. The number of replicates is denoted by the letter n.
Table 3.2 - Possible combinations for a 2 factor experiment with 5 and 2 levels each with n replicates.
B, b=5 a1 a2
b1 a1b1-1, a1b1-2, …, a1b1-n
a2b1-1, a2b1-2, …, a2b1-n
b2 a1b2-1, a1b2-2, …, a1b2-n
a2b2-1, a2b2-2, …, a2b2-n
b3 a1b3-1, a1b3-2, …, a1b3-n
a2b3-1, a2b3-2, …, a2b3-n
b4 a1b4-1, a1b4-2, …, a1b4-n
a2b4-1, a2b4-2, …, a2b4-n
b5 a1b5-1, a1b5-2, …, a1b5-n
a2b5-1, a2b5-2, …, a2b5-n
How much each factor affects the response is defined as the change in the response when the factor
is changed. Interaction is when the level of one factor affects the result when changing the other
factor. Meaning that if we change A we get different magnitude of the change depending on whether
B is on a low or a high level (Montgomery, Introduction to Factorial Designs, 2005).
There are several advantages with factorials;
They require fewer experiments to determine the same thing as an experiment where each
factor is changed one at the time,
When interactions are present they are considered as necessary in order to prevent
Since each factor can be used at several levels the range of the experimental conditions for
which valid conclusions about the effects of a factor can be made is increased.
The analysis of variance table for a two-factor factorial is shown in Table 3.3.
Table 3.3 – Analysis of variance table- ANOVA, for a two-factor factorial with the fixed effects model (Montgomery, Introduction to Factorial Designs, 2005).
Source of Variation
Sum of Squares Degrees of Freedom
Mean Square F0
( )( )
( )( )
Error ( )
Experiments are run sometimes in “blocks” to mitigate the effect of nuisance factors such as
different batches of raw material, different operators or different days. The differences from a
regular factorial are both in how the experiment is designed and in how the data is treated
(Montgomery, Introduction to Factorial Designs, 2005).
The ANOVA in Table 3.4 is for the case of randomized complete block. In a randomized complete
block design, each block contains one complete replicate; the experiment is designed with 4
replicates per block.
Table 3.4 - ANOVA for a two-factor factorial in a randomized complete block (Montgomery, Introduction to Factorial Designs, 2005).
Source of Variation
Sum of Squares Degrees of Freedom
Expected Mean Square
( )( )
∑∑ ( )( )
Error ( )( )
For more information on factorials see for example “Design of Experiments” by Douglas C.
3.2 Choice of sample size The choice of sample size, i.e., how many measurements and the probability of failing to reject a
false null-hypothesis are closely connected. In most cases, the error decreases as the sample size
increases. The difference between the two samples also affects the sample size, the larger the
difference, the less samples are needed to detect it. (Montgomery, 2-4.2 Choice of sample size, 2005)
By using “Operating Characteristic Curves for the Fixed Effects Model Analysis of Variance”
(Montgomery, Appendix V. Operating Characteristic Curves for the Fixed Effects Model Analysis of
Variance, 2005), assuming that the two populations under investigation have equal variance and
sample size equation 3.12, the necessary number of samples to achieve a specific β can be
estimated. This method is used for results with more than one variable and multiple levels.
In equation 3.12, D is the difference in treatments, i.e., between two treatments.
| | 3.13
For comparing a single variable with two levels figure 2-12 in (Montgomery, 2-4.2 Choice of sample
size, 2005), the operating characteristic curves for the two-sided t-test with α = 0.05 can be used.
The probability for β error is read on the vertical axis and the parameter d on the horizontal axis, see
Equation 3.14. The number of replicates for a specified difference or the reversed can then be read
from where the two conditions intersect. This method also assumes equal variance and sample size
(Montgomery, 2-4.2 Choice of sample size, 2005).
When several aspects are under investigation, the necessary amount of samples for each aspect
under investigation needs to be calculated separately if they have different means and/or variance.
If the sample variances were to differ the following equation has been used to calculate the average
estimated standard deviation
3.3 Measurement error Index test measurements typically have a measurement error around 0.2 - 0.3 % on the efficiency
(Sendelius, 2015) (Bard, Att utföra indexprov och utvärdera resultaten, 1993). This error is calculated
( ) (
( ) (
( ( ))
For the experiment the measurement error is as follows; by randomizing the experiment the error
can be approximated as random, and it is then calculated according to the Error row in Table 3.3 or
Table 3.4 depending on if blocking was deemed necessary or not. The error is then the mean square
of the sum of squares error for a fixed effects factorial,
( ) 3.17
For a randomized block design the error then becomes
4 Materials and methods
4.1 Experimental Setup The experiment was performed at one turbine at a full-scale hydropower plant in the Lule Älv-river
with the assistance of Mikael Sendelius and Jennie Molin from Sweco Energuide and their
equipment. The changes in guide vane and runner blade angles were performed by two operators at
The measurement equipment was connected and controlled according to standard procedures
during index tests. The same equipment and placement of the sensors was used during both the
experiment and the index test. The data acquisition system was a 16-channel measurement
computer/data logger from Damill AB, the sensors are listed in Table 4.1.
Table 4.1 – The sensors used and their name, serial number and accuracy (Sendelius, 2015).
The Yokogawa power meter had variable settings and the range stated in Table 4.1 were the settings used during the test. The relation between primary and secondary current and voltage were
Table 4.2 – Station signals (Sendelius, 2015).
Signal Measurement range Range signal
Station sensor headwater level 74.5 – 78.5 möh 0 – 10 mA
Station sensor tailwater level 43 – 47 möh 4 – 20 mA
Runner blade angle -13° – +11° 4 – 20 mA
Guide vane opening 0 – 100 % 4 – 20 mA
The sampling frequency were 2500 Hz during the test, each 250th sample were registered and saved
in a log file. In order to have stable measuring values a low pas filer of 0.500 Hz l were used.
Two types of cables were used, for longer distances screened cables and for shorter distances
laboratory cables. (Sendelius, 2015).
The headwater level sensor before the trash rack was placed next to the fixed scale. The same
location of the sensor was used during the index test made the day before. (Sendelius, 2015).
The headwater level sensor after the trash rack was placed on a ladder in a maintenance pit/shaft of
the intake floor. This location was chosen to measure the head water level according to IEC standard
and also to be able to measure the head losses of the trash rack. The positions of both the headwater
level sensors can be seen in Figure 4.1.
Figure 4.1 – To the left: Location of headwater level sensor before the trash rack, placed next to the fixed scale. Red arrow points to the sensor, orange arrow towards the fixed scale that is partly hidden by the ladder. To the right: Headwater after trash rack-sensor, visible on the picture is the hatch, the ladder and the cable for the sensor (Sendelius, 2015).
The sensor for the tailwater level was placed at the outlet of the turbine, behind a breakwater pillar,
see Figure 4.2. The sensor was weighted down with an anchor which in turn was secured with two
lines. The sensor was placed behind the breakwater to minimize fluctuations in the readings due to
Figure 4.2 – Placement of tailwater sensor (Sendelius, 2015).
In order to estimate the discharge using the Winter-Kennedy method, the differential pressure across
the spiral needs to be measured. This was done using two pressure taps in the turbine spiral case.
The same taps were used for the index test performed at the station. The pressure taps, the
differential pressure sensor and some control equipment are presented in Figure 4.3 below.
Figure 4.3 – The equipment used for measuring the differential pressure in the turbine spiral case. The pressure taps are marked with the blue arrow, the differential pressure sensor with the red arrow and the control equipment with the orange arrow (Sendelius, 2015).
The data acquisition system was connected to the plant/unit governing system to register the
stations reading of the generator power, head- and tailwater level, generator current, generator
voltage, runner blade angle and guide vane opening, see Figure 4.4.
Figure 4.4 – Connections to station readings of generator power, generator voltage, generator current, water levels, runner blade angle and guide vane opening (Sendelius, 2015).
4.2 Measurement program This section presents the measurement program for both the experiment and the index test.
Before starting the measurements, the sensor signals were checked and adjusted. The water level
readings from the sensors were first controlled and checked with respect to fixed scales and cross-
checked with station readings.
A reference pressure from the differential pressure transducer was taken with the turbine stopped;
this reading was used in the calculation as an offset. The pressure transducer was cross-checked with
The station readings of the guide vane opening and runner blade angle were compared with the
values at the operation control center, this to see if there were any fault in the communication
between the power plant and the control center. This type of check was primarily made for
evaluating the index test.
The measurement time was determined by performing a measurement of 10 min. The mean value of
the differential pressure was then computed after 1 min, 2 min etc. When the mean value did not
change by more than 0.1% between two calculations it was assumed that the measuring time was
sufficient. It was observed that after 4 min the results of the differential pressure did not change by
more than 0.1%. This is the reason why 4 min was chosen as a measuring time.
A deviation between the station readings of the runner blade angle and fixed scales was discerned.
Difference was also observed between the station power readings, the dial in the control room and
the power reading from the operations center. The control room dial showed the highest reading,
followed by the stations sensor and the operations center with the lowest reading. An attempt at
adjusting the station display was made before the start of the index test.
After all measurements were completed for the index test and the experiment, the turbine was
stopped. The differential pressure sensor reference pressure was checked. The water levels and the
dry-value for the sensors were also checked.
4.2.1 Index test
Two of the runner blade openings from the index test performed the day before were chosen. The
guide vane angles were chosen to roughly match the guide vane angles from the previously
performed index test.
The following order was used for the settings during the test:
1) Start with the lowest runner blade angle and the lowest guide vane angle
2) Increase the guide vane angle by 1.5-3 % until you have a clear peak.
3) Then change the runner blade angle to the second lowest angle and repeat.
4) If the peak is “missed” the guide vane angle is first lowered and then increased again. This was
performed for 6 runner blade angles during the index test.
The experiment consisted of 80 measurements with 10 different settings varied according a pre-
planned schedule, see Appendix A. The ten different settings consisted of two runner blade angles
and 5 guide vane angles for each runner blade angle.
A decision for not using the “direction of movement” as a factor in the planning of the experiment
was made due to the increased measurement time.
The 10 different settings were chosen based on the index test and wishes from headquarters. The
two runner blade angles were chosen to match angles from the index test. Five guide vane angles
were chosen for each runner blade angle. The best efficiency point and two points “before” and two
points after with roughly 1-1.5% difference between them. The difference between the guide vane
angles was 2-3 % in the index test; a smaller interval was chosen for the experiments.
Using different guide vane angles for each of the blade angles increases the random error. However,
10 guide vane angles would have resulted in uninteresting combinations and doubling the number of
The experiment was planned in two blocks; one for each day with 40 measurements, or 4 replicates.
The design was made to avoid the risk for nuisance factors between the days.
Using two runner blade angles was necessary in order to investigate joint effects for the guide vane
and the runner blade mechanism
The changes were made manually. Some measurement points were supposed to have a positive
change in the angle and ended up with a negative change and vice versa. The precision to set the
angles affects the analysis of the results.
The measurement schedules for day 1 and day 2 can be seen Appendix A
4.3 Measurement procedure The measurements for the index test and the experiment were performed as follows.
1. The operator sets the guide vane or runner blade angle
2. Wait 4 minutes for the flow to settle and reach steady state
3. Start the measurements lasting for 4 minutes (240 seconds). The data acquisition system
measured the following: Water levels from sensors; headwater before trash rack, after trash
rack and tailwater, water levels from station sensors; headwater after trash rack and
tailwater level. Differential pressure in turbine spiral case. Guide vane and runner blade
angles. Generator current, voltage and power.
4. Note the following on sheets
a. Guide vane angle according to fixed scale (both in mm and degrees), see Figure 4.5
b. Runner blade opening according to fixed scale, see Figure 4.5
c. Time the measurements was started
d. Measurement time
e. Measurement number
f. Generator Power according to control room dial
g. Generator Current according to control room dial
h. Generator Voltage according to control room dial
i. The reactive power according to control room dial
j. Adjacent turbine generator power according to control room dial
5. Repeat from point 1.
Figure 4.5 – Fixed scales. Picture to the left: Runner blade angle [°]. Center picture: Guide vane angle [°]. Picture to the right: Guide vane angle [mm] (Sendelius, 2015) .
5.1 Statistical analysis
5.1.1 Comparability of the results and measurement error
A statistical comparison of the results from the two days was performed for each blade angle in
order to investigate whether or not the resulting efficiencies from day 1 and day 2 could be
evaluated together or would have to be evaluated separately.
The results were evaluated separately for the different blade angles, to avoid any effect on the
The measurement data was ordered according to intended guide vane setting and day for each of
the runner blade angles. Using the “two-factor factorial with reproduction” command in MS Excel, an
ANOVA-table was generated for each of the blade angles.
Table 5.1 – ANOVA-table for comparing differences between days for runner blade angle 1°.
From Table 5.3 it is found that the β error for both runner blade angles is less than 0.01. Therefore,
the difference in the efficiency function of the day is insignificant.
The random measurement error for the efficiency can be found for each runner blade in Table 5.1
and Table 5.2 to be 1.16E-05 for 1° runner blade angle and 1.13E-05 for 4° runner blade angle.
5.1.2 Sensor comparison
The readings were summarized as a function of the angle change for both the runner blade and guide
vanes. The fixed scales readings were subtracted with the station sensor readings. In order to
compare the resulting differences statistically, an equal number of measurement values were
necessary. Therefore, a number of values were removed, depending on if the guide vanes or the
runner blades were under investigation. This removal was done by sorting the measurement values
after a column with random values generated through the command “=slump()” and then remove
the bottom values until an equal amount of measurements was achieved.
Figure 5.1 – The difference between the fixed scale and the station sensor plotted for each of the two runner blade angles, the measurements are ordered in the internal order that they were performed for each angle.
There is a difference between the magnitudes of the scales for the two runner blade angles
investigated, see Figure 5.1.
The p-value for a one-factor factorial using the command in MS Excel for the difference between the
different angles is 1.9·10^-52. It is thus highly probable that this difference in readings is not only due
to faulty readings of the fixed scale. There is a difference in the angle readings for the station
function of the angle.
This deviation between fixed scale and station sensor for the runner blade is a known problem for
some machines and the ambition is that this deviation is at its worst at the time of the index test.
In order to investigate the runner blade mechanism, the responses for the two sensor outputs
depending on direction of movement were investigated. Since only two runner blade angles were
used for the lower angle, only decrease can be investigated from the lower angle and only increase
from the higher angle. The lower blade angle difference was thus adjusted with the ratio of the
average difference at 4° increase and the average at 1° decrease.
The measurements were sorted into two columns depending on the runner blade angle and then
into two subcategories depending on whether the angle was changed or left unchanged.
Table 5.4 - Average differences between fixed scales and station sensors function of the movement direction of the runner blade angle.
By reading the p-value from Table 5.7 to 0.0024 < 0.05 it is possible reject the null-hypothesis. The
probability for the β error was estimated by calculating the d-value through Equation 3.14, using the
test statistics from Table 5.8. Since the variances differed a mean variance was calculated through
From the operating characteristic curve for a two sided t-test, with 34 replicates, the β error is below
0.05. The minimum number of replicates is according to the same operating characteristic curve 15.
Therefore, it is likely to be a difference in the response of the guide vane sensors depending on the
direction of movement. The different readings vary roughly 0.15° as a function of the guide vanes
This sensor difference when to the right of the propeller curve does not have a significant effect as
the efficiency drop is insignificant.
A difference in the sensor readings has been spotted. To determine if this difference is or not just
due to different placement of the sensors, the square root of the differential pressure has been
compared with the readings from both sensors. The square root of the differential pressure is used
since the flow rate is calculated by taking this multiplied with a constant, see Equation 2.19.
Figure 5.3 – Square root of differential pressure versus guide vane angle for fixed scale (black) and station sensor (grey) for 1° runner blade angle.
In Figure 5.3, the difference between the sensors can be spotted. Furthermore, the same guide vane
angle may result in different differential pressures and vice versa. For reference, a difference of 0.3 in
the square root of the differential pressure represents a difference of 2.1 m3/s in flow rate.
37 37,5 38 38,5 39 39,5
Square root of differential pressure reading
diffp vs guide vanes fixed, alfa 1
diffp vs guide vanes station, alfa 1
Figure 5.4 – The square root of the differential pressure versus the fixed scale guide vane angle readings. Light grey squares and dark grey rhombs are readings when decreasing and increasing the guide vane angle, respectively.
In Figure 5.4 no clear trend between differential pressure and direction of movement is found.
Figure 5.5 - The square root of the differential pressure versus the station sensor guide vane angle readings. Light grey rhombs and dark grey are readings when decreasing and increasing the guide vane angle, respectively.
In Figure 5.5 no clear trend of behavior due to the guide vanes movement direction can be spotted.
37 38 39 40 41 42 43 44 45 46
Square root of differential pressure reading
diffp vs guide vanes fixed +
diffp vs guide vanes fixed -
37 38 39 40 41 42 43 44 45 46
Square root of differential pressure reading
diffp vs guide vanes station +
diffp vs guide vanes station -
Both Figure 5.4 and Figure 5.5 display variation in differential pressure for the same guide vane
angle, especially for high guide vane angles. This variation can be seen for low guide vane angles too
(25° and 27.5°) for the station sensor but not in the same extent for the fixed scale readings. Worth
notice is the clear separation between the two runner blade angles and the resulting difference in
discharge between runner blade angles.
Figure 5.6 – Fixed scale readings on the vertical axis and station sensor readings on the horizontal axis. Light grey and dark grey are decreasing and increasing movement of the guide vane angle, respectively. The reference line marks the line at which the station sensor and the fixed scale display the same value.
In Figure 5.6 almost all readings follow a linear pattern. For a majority of the measurements where
the vane angle was decreased the two readings match. This is not true for the measurements where
the angle was increased.
5.1.3 Sensitivity to deviations in guide vane- and runner blade angles
The intention was to repeat the same measurement point several times. However, the
measurements were performed with deviation in the settings. The results can however be used to
evaluate the distribution of the efficiency during repeated measurements.
The efficiency was calculated according to Equation 2.10
The density ρ was assumed to 998.98 kg/m3 and the gravity acceleration constant g to 9.81 m/s2. The
turbine power, head and flow rate were calculated as follows.
The flow rate for each measurement point was calculated according to Equation 2.19
24 25 26 27 28 29 30 31
Station sensor [°]
The flow coefficient k and the exponential factor n were decided by the consultants from Sweco and
h was the measured differential pressure in the spiral case.
The head used for calculating the efficiency was the gross head, Equation 2.12
Where A0 and Atw were obtained from the blueprints, the flow rate calculated as in Equation 2.19, zhw
and ztw were measured at the headwater after trash rack and tailwater, respectively.
The turbine power was calculated according to Equation 2.13
Where Pgen was the measured generator power. Pgen-f is the generator loss calculated by Equation
( ) (
Where Ptom, Ptom-mag, Pbel, Pbel-mag and Im were known. The generator current I was measured.
The head was normalized to the rated head. The flow rate was normalized according to the similarity
laws, Equation 2.18. Since A turbine diameter ratio of 1 was assumed because the flow rate was
The resulting efficiency for the runner blade angle read on the fixed scale is presented in Figure 5.7.
All efficiencies have been normalized for confidentiality reasons.
Figure 5.7 - Normalized efficiency function of the normalized flow rate (left vertical axis) for two blade angles: triangles for 1° and squares for 4° blade angle. The fixed scales readings of the runner blade angle (right vertical axis) have the rhomb symbol.
In Figure 5.7, the efficiencies vary more for low flow rates as the runner blade angle varies with
about 0.2 degrees. The variation of the efficiency around the peak of the propeller curve is barely
larger than the usual measurement error. No difference in the variation of the runner blade angle at
the peak of the propeller curve compared to the rest of the curve can be discerned. The variation in
the runner blade angle does not seem to be the major factor for the efficiency.
The distribution of the efficiency appears to be dependent to a large extent on the flow rate. The
guide vane angle, controlling the flow rate, is now investigated.
250 260 270 280 290 300 310 320 330
Blade angle 1
Blade angle 2
Runner blade angle
Figure 5.8 – Normalized efficiency as a function of the normalized flow rate (left vertical axis) for each of the blade angles: triangles for 1° and squares for 4° blade angle. On the right vertical axis are the guide vane angle readings of the fixed scales marked with the rhomb symbol.
The major factor affecting the variation in the guide vane angle appears to be the position on the
propeller curve, see Figure 5.. For the lowest flow rates at both runner blade angles, the extreme
values differ more than 1 % in some cases. At the other side of the curves, at flow rates of 270 m3/s
and 310 m3/s the difference is instead on the scale of 0.3 %.
When comparing the variation of the guide vanes and the variation of the efficiency in Figure 5. with
the results regarding the variation of the response of the guide vane angle this variation is smaller
than the usual measurement error at index tests.
It is possible that there is an efficiency loss during the operation of the turbine due to the difference
in readings between fixed scales and station sensors for the runner blades. The difference and
subsequent control error results in a lower flow rate than the one intended to be used for 4° runner
blade angle, thus the efficiency may be significantly lowered. How much and if this assumption is
correct cannot be determined from the data of this experiment.
5.2 Comparison with index test
5.2.1 Index test curve
The efficiency and the flow rate were calculated in the same way as the efficiencies from the
experiment in the previous chapter. The difference is that the values used were measured in a
sequential order during the index test. Only the two angles also measured during the experiment are
In Figure 5.9 the index test propeller curves are presented.
250 260 270 280 290 300 310 320 330
Blade angle 1
Blade angle 2
Guide vane angle
Figure 5.9 – Propeller- and guide vane versus flow curves from the index test. The propeller curve depicting the efficiency is marked by rhombs and the curve showing the guide vane angle corresponding to each flow is depicted with squares.
In Figure 5.9, a clear peak for the efficiency and approximately linear relationship between the flow
and the guide vane angle are observed.
5.2.2 Experiment average curve compared to index test curve
The propeller curves from the index test are compared to the average value obtained from the
The average value curve was derived by taking the average values for each intended setting. An
average cam curve for the experiment measurements can be derived. The settings were however not
exact and some of the measurements ended up in another setting interval than the one intended.
The cam curve derived from the experimental values is only intended to be used for comparison with
the index curve.
250 260 270 280 290 300 310 320 330
Guide vane index
Figure 5.10 – Cam curve comparison between index test and experiment. Rhombs for the efficiency and squares for the guide vane angles at the index test. The results from the experiment; the average efficiency is marked by crosses and the average guide vane angle by dots. The guide vane angle reading is from the station sensor for both index test and experiment.
In Figure 5.10, for a runner blade angle of 1°, the curves have almost the same shape, except the
peak efficiency appears for a slightly higher flow rate. For a runner blade angle of 4°, the peaks
coincide, although the experiment curve has two points with almost the exact same efficiency. A
sharper efficiency slope for low flow rates is observed; slope smaller for higher flow rates compared
to the index test. The flow rates and guide vane angles appears to coincide fairly well for both runner
blade angles but better for the lower runner blade angle. For the second runner blade angle, the
guide vane angles are lower than during the index test, except for the first one. The two methods
yielded no difference in efficiency. The measurement error for the efficiency is typically considered
to be around 0.2 % and the difference between the peak efficiencies is smaller than this.
The index test curves are now plotted with all resulting efficiencies and guide vane angles from the
250 260 270 280 290 300 310 320 330
average eff exp
Guide vane index
average guide vanes exp
Figure 5.11 – Index test curve compared to all measurements from experiment.
Comparing Figure 5.10 with Figure 5.11, the efficiency at the high flow rates for the second runner
blade angle are low, and the efficiency at the lowest flow rate were among the larger values. This is
confirmed further by looking at Figure 5.12, where black bars represent the standard deviation in
flow rate, efficiency and guide vane opening.
Figure 5.12 – Same curves as in Figure 5.10 but with black bars to represent the standard deviation. For the efficiency curve the vertical error bar is the standard deviation in efficiency and the horizontal error bar is the standard deviation for the flow rate. For the guide vanes the vertical error bar is the standard deviation in guide vane angle and the horizontal bar is the standard deviation for the different flows used at that point.
250 260 270 280 290 300 310 320 330
Guide vane index
guide vanes exp
260 270 280 290 300 310 320
average eff exp
Guide vane index
average guide vanes exp
The standard deviation in guide vane opening is small since the black bars are difficult to discern in
the figure. The standard deviation in flow rate and efficiency is however larger.
In Figure 5.12, the shape of the experiment test propeller curve is closer to the theoretical shape
than the index test propeller curve because the slope is flatter after the best efficiency point.
The variation in head on the efficiency has not been investigated. Today the efficiency is assumed to
be constant when normalizing the flow with respect to the head variation. It is possible that some of
the variation in the distribution of the efficiency is due to this variation.
6 Recommendations Regarding the difference in the guide vane sensors more investigations are necessary for two
reasons; the station sensors calibration was not taken into account, and if this phenomenon is
present in other machines.
As for the difference between fixed scales and stations sensors for the runner blade, the station
sensors need to be recalibrated. Furthermore, a test of 22 replicates is necessary for the angles
variation to determine with certitude if the sensor response varies with the direction of movement.
Also of interest, is to investigate the relationship of the difference in the readings with respect to the
runner blade. The number of measurements necessary to assess this question is possible to find
using my measurement data. Index test data can be used for deriving this relationship if required
number of measurements is less than five.
More measurement points around the expected BEP during index tests may be of use to avoid
choosing a guide vane angle and flow rate at which the efficiency varies significantly.
A final recommendation is to investigate the relation between the distribution of the actual head and
the efficiency. This may allow to validate the constant efficiency assumption made when normalizing
the flow rate to one head using the measurement data collected during the experiment.
7 Conclusions The station sensor for the runner blade angle is in need of an adjustment. By looking at the sensor
outputs, no hysteresis phenomenon resulting in different responses function of the blade movement
(increased or decreased angle) could be seen; the results are inconclusive.
A sensor difference exists for the guide vane mechanisms. The difference between the fixed scale
and the station sensor was larger when the guide vane angle was increased, resulting in an average
of 0.1° difference between closure and opening.
The variation in the efficiency around the best efficiency point was negligible. The result from the
different measurement methods appears to give roughly the same result and therefore it shouldn’t
be necessary to change the method for the index test. An option may however be to make repetition
points if the initial values for a propeller curve are higher than expected since they seem to vary
more for low flows.
Furthermore, it is worse if the angle deviation for the guide vanes result in smaller than intended
flow rates since the peak is sharper for low flows.
8 References Alvarez, H. (2006). Energiteknik Del 1. Lund: Studentlitteratur.
Bard, A. (1993). Att utföra indexprov och utvärdera resultaten. Stockholm : STF ingenjörsutbildning
Bard, A. (1993). Att utföra indexprov och utvärdera resultaten. Stockholm: STF Ingenjörsutbildning
Cervantes, M. (2015, March). Propellerkurvor. (H. Isaksson, Interviewer)
Cervantes, M., Andrée, G., Klason, P., & Sundström, J. (2012). Flow Measurements in Low-Head Hydro
Power Plants. Stockholm: Svenskt Vattenkraftcentrum / Elforsk.
Energimyndigheten; Norges vassdrags- og energidirektorat. (2013). The Norwegian-Swedish
Electricity Certificate Market- Annual report 2013. Retrieved June 1, 2015, from