MASTER'S THESIS

Uncertainties in Kaplan Cam Curve

Hanna Isaksson2015

Master of Science in Engineering TechnologySustainable Energy Technology

Luleå University of TechnologyDepartment of Engineering Sciences and Mathematics

Abstract The aim of this project was to investigate the sensitivity of the index test with focus on the behavior

of the guide vane and runner blade angles. This was achieved by comparing the cam curves from an

index test with a design of experiment. Index tests are a popular method for determining cam curves

for Kaplan turbines. Index tests, as they are performed today, do not take into account a suspected

hysteresis phenomenon in the guide vane and runner blade mechanism. If and how this affects the

cam curve is today unknown which this project has investigated.

An experiment was designed based on the “design of experiment” method. The difference from a

traditional index testing is that the measurements were performed in a random order and only two

propeller curves were plotted, but with eight replicates. Ten different guide vane angles were used;

five for each runner blade angle and a total of 80 measurements were performed over the span of

two days. Then, the measurements were compared with the results from an index test performed

the day before on the same machine, a full-scale turbine in the Lule Älv-river using the same

equipment.

When analyzing the distribution of the efficiency, it has a larger spread on the left hand side of the

propeller curve compared to the best efficiency point and negligible variation around the best

efficiency point. The difference in the cam curves between the two methods was smaller than the

estimated measurement error at the best efficiency point. The peak efficiency in the experiment was

found for a slightly higher flow rate at one of the runner blade angles but this point was between two

settings at the index test. The variations in efficiency are at their largest before the peak and may

vary between a high efficiency and significantly lower efficiency at this point. Additional

measurements may reduce the risk for choosing a guide vane angle at which such behavior occurs.

A constant difference in the readings between the fixed scale and the station sensor were observed

for the runner blade mechanism. This difference varied with the runner blade angle and amounted to

0.26° for 1° and 0.70° for 4° runner blade angle.

For the guide vane mechanism no hysteresis could be validated. A difference between the fixed scale

and the station sensor which varied depending on the direction was spotted and could be validated

with 95% significance. The difference in the readings from the fixed scale and the station sensor was

in average 0.15° larger when increasing the guide vane angle than when decreasing the angle.

Sammanfattning Målet med detta projekt var att undersöka känsligheten hos indexprov med fokus på löphjulets och

ledskenornas beteende. Detta skulle uppnås genom att jämföra kombineringskurvorna från ett

indexprov och från ett ”design of experiment”. Indexprov är en populär metod för att bestämma

kombineringskurvor till Kaplan turbiner i Sverige. Det sätt som indexprov utförs på idag tar inte

hänsyn till eventuell hysteres i maskineriet för löphjulet och ledskenorna. Om och hur detta påverkar

kombineringskurvan är idag okänt vilket detta projekt undersökt.

Ett experiment utformades efter ”design of experiment” metoden och det skiljde sig från ett

traditionellt indexprov i att mätningarna planerades i en slumpvis ordning och att endast två

propellerkurvor, men med åtta repetitioner istället uppmättes. Tio olika ledskenevinklar användes,

fem för varje löphjulsvinkel och totalt uppmättes 80 mätpunkter under loppet av två dagar. Dessa

jämfördes därefter med mätvärdena från ett indexprov som utfördes dagen innan experimentet på

samma maskin, en fullskalig turbin i Luleälven och med samma utrustning.

Vid analys av fördelningen av mätpunkterna för verkningsgraden så konstaterades en större

spridning vid de låga volymflödena på propellerkurvorna och försumbar variation vid bästa

verkningsgrads-punkten. Skillnaden i kombineringskurvorna mellan de två genomförandemetoderna

var vid bästa kombineringspunkten mindre än det normala mätfelet. Toppen på en av

propellerkurvorna återfanns för ett lite högre flöde än vid indexpunkten men denna punkt befann sig

mellan två mätpunkter för indexprovet. Repetitionspunkter för bästa verkningsgradpunkterna på en

propellerkurva bör tas för att undvika att välja en för låg ledskenevinkel då variationen hos

verkningsgraden är som störst före toppen av propellerkurvan.

En konstant skillnad i avläsningarna mellan fasta skalan och stationens givare återfanns för

löphjulsmekanismen. Denna skillnad varierade med löphjulsvinkeln och uppgick till 0,26° för 1° och

0,70° för 4° löphjulsvinkel.

För ledskenorna kunde ingen hysteres valideras. Mellan fasta skalan och stationens sensor

upptäcktes en skillnad som varierade med förändringsriktningen, som kunde valideras med 95 %

signifikans. Skillnaden i avläsningarna från fasta skalan och stationens givare var i snitt 0.15° större

när ledskenevinkeln ökades mot när den minskades.

Preface The work within this report presents my Master Thesis in Sustainable Energy Engineering with

specialization in Energy Optimization and Biofuels at Luleå University of Technology (LTU). The

project was done for Vattenfall with assistance from LTU and Sweco Energuide.

I would like to thank my main supervisors; Prof. Michel Cervantes at LTU, Mr. Mikael Sendelius at

Sweco, Mr. Stefan Sandgren at Vattenfall, and Mr. Magnus Lövgren at Vattenfall. Special thanks to

Prof. Michel Cervantes for introducing me to hydropower and this project. I would also like to

express my gratitude to Ms. Jennie Molin at Sweco for the invaluable help during the experiments

and for practical information about Index tests. Great thanks also to the operators at the station at

which the experiment was performed, and to all employees of Vattenfall, Sweco and LTU that I have

met on different occasions for their friendly reception.

I would also like to thank my family and my friends for their support.

Luleå, September 2015

Hanna Isaksson

Content 1 Introduction ..................................................................................................................................... 1

1.1 Problem description ................................................................................................................ 3

1.2 State of the Art ........................................................................................................................ 3

1.3 Aim of thesis ............................................................................................................................ 3

2 Kaplan turbines ................................................................................................................................ 4

1.1 Cam curves .............................................................................................................................. 9

1.2 Index tests ............................................................................................................................. 13

1.3 Winter-Kennedy flow rate measurement ............................................................................. 16

3 Design of Experiment .................................................................................................................... 18

3.1 Basic statistical terms ............................................................................................................ 18

3.1.1 Theory of “design of experiment” – factorials .............................................................. 20

3.2 Choice of sample size ............................................................................................................ 22

3.3 Measurement error ............................................................................................................... 23

4 Materials and methods ................................................................................................................. 24

4.1 Experimental Setup ............................................................................................................... 24

4.2 Measurement program ......................................................................................................... 26

4.2.1 Index test ....................................................................................................................... 27

4.2.2 Experiment .................................................................................................................... 27

4.3 Measurement procedure ...................................................................................................... 28

5 Results ........................................................................................................................................... 29

5.1 Statistical analysis .................................................................................................................. 29

5.1.1 Comparability of the results and measurement error .................................................. 29

5.1.2 Sensor comparison ........................................................................................................ 30

5.1.3 Sensitivity to deviations in guide vane- and runner blade angles ................................. 36

5.2 Comparison with index test................................................................................................... 39

5.2.1 Index test curve ............................................................................................................. 39

5.2.2 Experiment average curve compared to index test curve ............................................ 40

6 Recommendations......................................................................................................................... 44

7 Conclusions .................................................................................................................................... 45

8 References ..................................................................................................................................... 46

Variable list Variable Symbol Unit

Static head [m]

Headwater level [möh]1

Tailwater level [möh]

Head [m]

Hydraulic losses [m]

Gravity acceleration [m/s2]

Velocity at a point i in the turbine waterways

[m/s]

Correction factor for kinetic energy

-

The power removed from the water by the turbine

[W]

Density (of water) [kg/m3]

Flow rate [m3/s]

Efficiency -

Net capacity of the turbine [W]

Specific speed [rev/s]

Rotational speed of runner [rev/s]

Power in watt for one horsepower

[W]

Drag force [N]

Drag coefficient -

Average absolute velocity over the runner blade

[m/s]

Length of runner blade [m]

Derivative of the radial position in comparison to the turbine

Radius of the turbine

Lift coefficient -

Lift force [N]

Resulting force acting upon the runner blade

[N]

Pi -

The angle between the average relative velocity and the peripheral velocity over the runner blade

[°]

Peripheral velocity [m/s]

Power developed by the runner blade

[N]

Axial velocity [m/s]

Number of runner blades -

Relationship between number of runner blades and turbine radius

[m]

Attack angle of water flow relative to runner blade

[°]

1 Swedish for meters above sea level

Relative efficiency -

Relative flow rate [m3/s]

Cross-section area at a point i in the waterways

[m2]

Generator power [W]

Generator losses [W]

no load losses [W]

Generator load losses [W]

Generator current [A]

Designed generator current [A]

Magnetic no load losses [W]

Magnetic losses depending on load

[W]

Rotational speeds, two similar turbines or same turbine.

[rev/s]

Turbine (Runner) diameter, if same turbine equal to 1

[m]

Head, same turbine, different heads

[m]

Efficiency -

Flow rate [m3/s]

Power output [W]

Flow coefficient [m7/2/kg1/2]

Differential pressure [mmvp]2

Pressure exponent -

Tangential velocity [m/s]

Radial position relative to spiral casing center

[m]

Pressure [Pa]

Pressure at a position i in the spiral case

[Pa]

Random sample variable -

Mean value -

Variance -

Degrees of freedom DOF -

Sample mean -

Sample variance -

Estimated sample standard deviation

-

Sum of squares -

Null hypothesis -

Alternative hypothesis -

Type I error -

Type II error -

Power of a test -

Number of replicates -

Number of levels for factor A -

Number of levels for factor B

2 1 mmvp = 9.81 Pa (Alvarez, 2006)

Mean sum of squares for factor i -

Impact of factor divided by the mean sum of squares for the error

-

Absolute difference between two sample means (two-factor factorial)

-

Variable used for estimating risk for β error in a two-sided t-test

-

Mean standard deviation -

1

1 Introduction Hydropower or hydroelectric power plants utilizes the kinetic and potential energy in flowing water

to generate electricity through the use of hydraulic turbines. The water makes the turbine rotate and

this energy is then converted into electricity by a generator.

A sketch of a typical hydropower plant is presented in Figure 1.1.

Figure 1.1 – Turbine installation at hydroelectric plant (Krivchenko, 2.1 Turbine head, 1993).

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.

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).

2

1.2

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

1.3

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

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)

1.5

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)

√

√ 1.6

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

3

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 &

Proulx, 2010).

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.

4

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).

Turbine type Head (m)

Active Pelton 400-1500

Reaction Axial – flow 1-65 Mixed – flow 40-200 Radial – Axial – flow (Francis) 40-700

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

5

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)

3. Shaft

4. Turbine case

5. Stayring

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.

6

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).

7

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

synchronization devices are necessary (Krivchenko, 4.1 Guide vane operating gear, 1993).

One of the most common servomotors used in combination with a regulating ring can be seen in

Figure 2.5

Figure 2.5 – Two servomotor solution for operation of guide vane regulation ring (Krivchenko, 4.1 Guide vane operating gear, 1993).

Servomotors use high pressure oil to create the desired movement by supplying high-pressure oil to

either pipe A or pipe B (see Figure 2.5) and connecting the other pipe to the return line. The supplied

oil forces the rods and pistons in the servomotor to move which forces the regulating ring to turn. To

turn the regulating ring in Figure 2.5 clockwise, pipe A should be connected to the high-pressure oil

while pipe B should be connected to the return line.

Other configurations are four servomotors; see Figure 2.6 a) and anchor-ring servomotors Figure 2.6

b). The four servomotors configuration, or “two twinned” as it is also called, requires smaller

diameter on the servomotors and shorter pullrods compared to the two servomotors solution.

The anchor-ring configurations works in a slightly different fashion from the two previously described

solutions; the moving element has the shape of an anchor ring. The center of the anchor ring

coincides with the regulation ring’s axis of revolution which necessitates a large size. No intermediate

links are necessary to connect the servomotor to the regulation ring (Krivchenko, 4.1 Guide vane

operating gear, 1993).

8

Figure 2.6 – Guide vane operating gear solutions, in a) using four hydraulic servomotors in a two-twinned configuration. b) anchor-ring servomotors (Krivchenko, 4.1 Guide vane operating gear, 1993).

The operating gear of the runner blades is responsible for setting and maintaining the runner blade

angles which varies in a 30-40° interval. The operating rings are the mechanism responsible for this

action. The specifications that the operating rings must meet are high; the blades must be able to

operate despite water pressure, centrifugal and frictional forces, the space in the runner hub is

limited. As repairs require disassembling all of the component units, high reliability is expected.

Figure 2.7 – Runner blade operating gear (Krivchenko, 4.2 Runner blade operating gear of adjustable blade turbines, 1993).

The main components of the operating ring can be seen in Figure 2.7, the components are the

following;

1. Blade flange 2. Blade pivot 3. Housing 4. Housing

5. Lever 6. Pull rod 7. Piston connected to runner servomotor

9

The blade pivot is supported by the two housings. It is moved by the lever and the pull rod by the

piston connected to the runner servomotor. A downward motion of the piston results in a greater

blade angle. A high-pressure oil system is responsible for moving and maintaining the desired

position (Krivchenko, 4.2 Runner blade operating gear of adjustable blade turbines, 1993).

Figure 2.8 – Detailed schematic of a solution for the operating gear of the runner blades in a low-head machine (Krivchenko, 4.2 Runner blade operating gear of adjustable blade turbines, 1993).

There are many different solutions for how the operating gear is constructed; Figure 2.8 shows one

such solution used for low-head machines. The numbering and the different parts are explained in

the following sentences. (1) is the runner hub to which the runner blades (2) are connected by the

pivot flange (3). The pivot flange (3) is in turn connected to the bearings (4) and (5). Hinge blande

levers (6) and pullrods (9) are connected to the piston (7), that all the blades are connected to the

same piston ensures that all the blades will be turned equally. (8) are cups rigidly connected to the

servomotor piston (7). (10) and (11) are oil supply pipes, (10) to the rod end and (11) to the head end

(Krivchenko, 4.2 Runner blade operating gear of adjustable blade turbines, 1993).

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.

10

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

41, 1991).

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.

11

Figure 2.10 – Vector components of forces acting on a runner blade (Cervantes, Propellerkurvor, 2015).

The drag force is defined as

2.1

And the lift force as

2.2

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

2.3

The portion of the force acting in the rotational direction is

(

) ( ) 2.4

Where

( )

( ) 2.5

12

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

2.9

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

2.12.

13

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).

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

-5 0 5 10 15

CL·f(λ)

Attack angle [°]

14

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

Tailwater level, station sensor

Differential Pressure

Generator Power

Generator Current

Generator Voltage

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

2.10

15

The net head is calculated by simplifying Equation 1.2 to

2.11

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

into account.

2.12

The turbine power is calculated as

2.13

Where Pgen is the generator power and Pgen-f the generator losses which in turn is calculated as

(

)

2.14

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

( ) (

)

2.15

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

resultaten, 1993).

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)

√

√ √

√ 2.16

(

)

√

√ √

√ 2.17

(

)

√

√ √

√ 2.18

16

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,

1991).

2.19

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).

17

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

2.20

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

2.21

Inserting 2.21 into 2.20 and assuming uθ constant over the radial section the resulting equation is

( )

2.22

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

√

( )

2.23

And thus the k and h in equation 2.19 are

18

√ ( )

2.24

( ) √ 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

exists (Cervantes, Andrée, Klason, & Sundström, 2012).

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

movement”.

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).

19

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

Mean µ

{

∫ ( )

∑ ( )

A measure of a probability distributions central tendency or location.

3.1

Variance

{

∫ ( ) ( )

∑( ) ( )

The variability or dispersion of a probability distribution

3.2

Degrees of Freedom

DOF

The number of independent elements in a sum of squares, n is the number of samples

3.3

Sample mean

∑

The average value of the random variables

3.4

Sample variance

∑ ( )

Dispersion of sample with n measurements

3.5

Sample standard deviation

√

The standard deviation of the sample, more commonly used since it has the same units as y

3.6

Corrected sum of squares

SS ∑( )

Unit used in ANOVA tables.

3.7

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

distributions, 2005).

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

example.

20

3.8

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,

2005).

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.

21

Table 3.2 - Possible combinations for a 2 factor experiment with 5 and 2 levels each with n replicates.

A, a=2

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

misleading conclusions

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

A treatments

∑

B treatments

∑

AB Interaction

∑∑

( )( )

( )( )

Error ( )

( )

Total ∑∑∑

22

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

F0

Blocks

∑

A treatments

∑

∑

B treatments

∑

∑

AB Interaction

∑∑

( )( )

∑∑ ( )( )

Error ( )( )

Total ∑∑∑

For more information on factorials see for example “Design of Experiments” by Douglas C.

Montgomery.

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.

3.12

In equation 3.12, D is the difference in treatments, i.e., between two treatments.

23

| | 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).

| |

3.14

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

√

3.15

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

according to

√(

√ )

( ) (

√ )

( ) (

( )

)

( ( ))

3.16

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

3.18

24

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 station.

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).

Measurement parameter

Product Serial number Range

Headwater level before trash rack

JUMO 4 APT-26-242 4404 4121 0 – 10 bar

Headwater level after trash rack

DRUCK PTX-1730 3657131 0 – 7.0 mvp

Tailwater level DRUCK PTX-1730 2159721 0 – 7.0 mvp

Differential pressure in the spiral case

Rosemount 3051S1CD2A2A12A1BD1L4M5Q4

9591060 0 – 4000 mmvp, 4 -20 mA

Generator power Yokogawa WT 230

(760503-C2-F/DA12/EC)

91H62988 0 – 1125 W, 1 – 5 V

Generator current 0 – 2.5 A, 1 – 5 V

Generator voltage 0 – 150 V, 1 – 5 V

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

and

√

√

respectively.

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).

25

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

surface oscillations.

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.

26

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

known inputs.

27

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.

4.2.2 Experiment

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.

28

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

measurements.

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) .

29

5 Results

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

results.

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°.

Source of variation

Sum of squares

Degrees of freedom

Mean sum of squares

F P-value F-Crit

Guide vane setting 0.000645777 4 0.000161444 13.91605 1.57E-06 2.689628

Day 7.12228E-07 1 7.12228E-07 0.061392 0.805997 4.170877

Interaction 9.64059E-06 4 2.41015E-06 0.207748 0.932106 2.689628

Error 0.000348039 30 1.16013E-05

Total 0.001004168 39

The null hypothesis concerning the 1° runner blade angle cannot be rejected as the p-value in Table

5.1 is 0.806, which is above 0.05. The null hypothesis means no significant difference between the

efficiencies from the two days. The measurements from day 1 and day 2 can then be evaluated

together for a runner blade angle 1°.

Table 5.2 – ANOVA-table for comparing differences between days for runner blade angle 4°.

Source of variation

Sum of squares

Degrees of freedom

Mean sum of squares

F P-value F-Crit

Guide vane setting 0.000416249 4 0.000104 9.186414 5.71E-05 2.689628

Day 1.7153E-07 1 1.72E-07 0.015142 0.902885 4.170877

Interaction 4.71266E-06 4 1.18E-06 0.104006 0.980233 2.689628

Error 0.000339835 30 1.13E-05

Total 0.000760969 39

For the 4° runner blade angle, the p-value in Table 5.2 is found to be 0.903 > 0.05. We cannot reject

the null hypothesis for this runner blade angle either.

To check the probability for the β error for both runner blade angles, the Φ2 value is calculated for

both cases according to Equations 3.12 and 3.13 with the values from the operating characteristic

curve (Montgomery, Appendix V. Operating Characteristic Curves for the Fixed Effects Model

30

Analysis of Variance, 2005) This method assumes equal variance of the samples; this was not the case

during the experiment. The variance was different for each of the days for both runner blade angles.

The variance chosen for this analysis was calculated according to Equation 3.15 to avoid optimistic

estimation of β.

5.1

| | 5.2

√

5.3

Table 5.3 - Probability of β error for both runner blade test statistics when deciding whether or not all measurements can be evaluated together.

Angle n a b µ1 µ2 σ D Φ2 Φ v1 v2 β 1° 4 5 2 0.9433 0.9435 3.80E-05 2.7E-04 39.47 6.3 4 30 <0.01

4° 4 5 2 0.9425 0.9423 2.90E-05 1.3E-04 14.41 4.1 4 30 <0.01

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.

31

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.

5.4

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.

Diff 1° decrease Diff 1° neutral Diff 4° neutral Diff 4° increase

Unadjusted 0.270 0.259 0.704 0.735

1° differences Adjusted

0.735 0.704 0.704 0.735

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Difference between fixed scales and station sensor

Diff 4°

Diff 1°

32

A statistical analysis was performed, a two-factor factorial anova for the adjusted differences, which

can be seen in Table 5.5.

Table 5.5 – Anova-table for comparing differences between days for runner blade angle 4°.

Source of variation

Sum of squares

Degrees of freedom

Mean sum of squares

F P-value F-Crit

Neutral or changed position 0.02996 1 0.029959 3.959664 0.050626 3.981896

Runner blade angle 0.00167 1 0.001669 0.22063 0.640063 3.981896

Interaction 0.00097 1 0.000973 0.128602 0.720996 3.981896

Error 0.51449 68 0.007566

Total 0.5741 71

The p-value for the difference depending on the runner blade angle is larger than the pre-decided

significance α = 0.05. Thus no significant difference between the adjusted difference for 1° and 4°

runner blade angle can be discerned. There is a difference in the readings between when the runner

blade was changed and when it was not which is on the verge of being significant, P=0.0506. This

may be due to the angle which drifted during the measurements and was closer to the read value on

the fixed scales after one or more measurements.

The probability for β error was investigated for the difference between the angles, once again using

Equations 3.12 and 3.13 and assuming equal variance. The variance was calculated through Equation

3.15.

5.5

| | | | 5.6

√

√ 5.7

Table 5.6 - Probability of β error when evaluating differences in readings from fixed scales and station sensor for the runner blade.

Aspect under investigation

n a b µ1 µ2 Φ2 Φ v1 v2 β

Difference between 1° and 4°

18 2 2 0.7105 0.7202 0.0120 5.76 2.4 1 68 0.07

From Table 5.6 the β error probability derived from the experiment was 0.07 for 18 replicates. This is

slightly high to draw any conclusions about the adjusted difference. In summary, 18 replicates are

too few to investigate the adjusted difference, 22 replicates are necessary in this case.

33

The difference in the readings between the fixed scales and the station sensor for the guide vanes is

presented, see Figure 5.2.

Figure 5.2 – Fixed scale and station sensor readings for the guide vanes arranged after internal order of measurement for each angle each day.

For the guide vanes, the measurements with no change were not considered because only five

measurement points were available. The major interest was in the vanes response to an input.

The differences were sorted into two columns based upon the blade angle increase “+” or decrease

“-“. The resulting two columns were not equal in length. Four measurement values were selected

randomly and removed in the same way as for the runner blades. A one-factor ANOVA was executed,

and yielded the results in Table 5.7 and Table 5.8.

Table 5.7 – One factor ANOVA table for comparing the differences between the fixed scale and the station sensor depending on if the guide vane angle was increased or decreased.

Source of variation

Sum of squares

Degrees of freedom

Mean sum of squares

F P-value F-Crit

Increase or decrease 0.377134 1 0.377134 9.959529 0.002411 3.986269

Error 2.499198 66 0.037867

Total 2.876332 67

Table 5.8 – Summary of test-statistics from the data used in the ANOVA in Table 5.7.

SUMMARY Groups Number Sum Mean value Variance

+ 34 11.37856 0.334664 0.042709

- 34 6.31446 0.185719 0.033024

24

25

26

27

28

29

30

31

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77

Guide vane angle [°]

Order, arranged after day and position

Fixed scale

Sensor

34

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

Equation 3.15.

| |

5.8

√

√ 5.9

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

movement direction.

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.

24,5

25

25,5

26

26,5

27

27,5

28

37 37,5 38 38,5 39 39,5

Gu

ide

van

e an

gle

[°]

Square root of differential pressure reading

diffp vs guide vanes fixed, alfa 1

diffp vs guide vanes station, alfa 1

35

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.

24

25

26

27

28

29

30

31

37 38 39 40 41 42 43 44 45 46

Gu

ide

van

e an

gle

[°]

Square root of differential pressure reading

diffp vs guide vanes fixed +

diffp vs guide vanes fixed -

24

25

26

27

28

29

30

31

37 38 39 40 41 42 43 44 45 46

Gu

ide

van

e an

gle

[°]

Square root of differential pressure reading

diffp vs guide vanes station +

diffp vs guide vanes station -

36

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

5.10

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

24 25 26 27 28 29 30 31

Fixe

d s

cale

[°]

Station sensor [°]

+

-

Reference

37

5.11

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

(( )

(

)

) 5.12

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

5.13

Where Pgen was the measured generator power. Pgen-f is the generator loss calculated by Equation

2.15

( ) (

)

5.14

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

normalized.

(

)

√

√ √

√ 5.15

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.

38

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.

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0,98

0,985

0,99

0,995

1

1,005

1,01

250 260 270 280 290 300 310 320 330

Fixe

d s

cale

ru

nn

er

bla

de

an

gle

[°]

Effi

cie

ncy

[%

]

Flow [m3/s]

Blade angle 1

Blade angle 2

Runner blade angle

39

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

analyzed.

In Figure 5.9 the index test propeller curves are presented.

24,7

25,7

26,7

27,7

28,7

29,7

30,7

0,98

0,985

0,99

0,995

1

1,005

1,01

250 260 270 280 290 300 310 320 330

Fixe

d s

cale

gu

ide

van

e a

ngl

e [

°]

Effi

cie

ncy

[%

]

Flow [m3/s]

Blade angle 1

Blade angle 2

Guide vane angle

40

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

experiment.

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.

60

65

70

75

80

85

0,986

0,988

0,99

0,992

0,994

0,996

0,998

1

1,002

1,004

1,006

250 260 270 280 290 300 310 320 330

Gu

ide

van

e a

ngl

e [

%]

Effi

cie

ncy

[%

]

Flow [m3/s]

Eff index

Guide vane index

41

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

experiment.

62

67

72

77

82

87

92

97

0,986

0,988

0,99

0,992

0,994

0,996

0,998

1

1,002

1,004

1,006

250 260 270 280 290 300 310 320 330

Gu

ide

van

e a

ngl

e [

%]

Effi

cie

ncy

[%

]

Flow [m3/s]

Eff index

average eff exp

Guide vane index

average guide vanes exp

42

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.

0

20

40

60

80

100

0,98

0,985

0,99

0,995

1

1,005

1,01

250 260 270 280 290 300 310 320 330

Gu

ide

van

e a

ngl

e [

%]

Effi

cie

ncy

[%

]

Flow [m3/s]

Eff index

eff exp

Guide vane index

guide vanes exp

63

68

73

78

83

88

93

98

103

108

0,985

0,987

0,989

0,991

0,993

0,995

0,997

0,999

1,001

1,003

1,005

260 270 280 290 300 310 320

Effi

cie

ncy

[%

]

Flow [m3/s]

Eff index

average eff exp

Guide vane index

average guide vanes exp

43

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.

44

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.

45

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.

46

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Vattenfall: http://corporate.vattenfall.se/om-energi/el-och-

varmeproduktion/vattenkraft/vattenkraft-i-framtiden/

Appendix A Measurement schedules for the experiment

Dag 1

Provpunkt Pådrag, position

Löphjul [°] Check

Provpunkt

Pådrag, position

Löphjul [°] Check

1 11 1

21 24 4

2 24 4

22 21 4

3 13 1

23 22 4

4 23 4

24 21 4

5 25 4

25 22 4

6 14 1

26 23 4

7 15 1

27 11 1

8 13 1

28 12 1

9 15 1

29 14 1

10 22 4

30 23 4

11 21 4

31 13 1

12 24 4

32 25 4

13 22 4

33 11 1

14 12 1

34 13 1

15 23 4

35 21 4

16 14 1

36 11 1

17 14 1

37 12 1

18 15 1

38 24 4

19 12 1

39 25 4

20 25 4

40 15 1

Dag 2

Provpunkt Pådrag, position

Löphjul [°] Check

Provpunkt

Pådrag, position

Löphjul [°] Check

1 22 4

21 14 1

2 14 1

22 13 1

3 15 1

23 12 1

4 11 1

24 22 4

5 23 4

25 22 4

6 21 4

26 25 4

7 15 1

27 14 1

8 15 1

28 23 4

9 13 1

29 13 1

10 11 1

30 21 4

11 22 4

31 25 4

12 23 4

32 25 4

13 12 1

33 11 1

14 11 1

34 14 1

15 12 1

35 24 4

16 21 4

36 13 1

17 15 1

37 24 4

18 25 4

38 23 4

19 21 4

39 24 4

20 12 1

40 24 4