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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions

Hydraulic Prime MoversElectrical Engineering

Power Systems Analysis

Universidad Nacional de Colombia

Diego Nicolas Lopez [email protected]

April 21th, 2012

D. Lopez Prime Movers
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Special Issues

1 Hydroelectric dam highlights

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Special Issues


2 Hydraulic turbines

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Special Issues



3 Hydraulic turbine transfer function-Linear systems


A d H d P k H d li T bi T f F i S i l Ch i i El i l A l C l i
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Special Issues




4 Special characteristics


A d H d P t k H d li T bi T f F ti S i l Ch t i ti El t i l A l C l i
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Special Issues





5 Electrical analog


Agenda Head Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Special Issues





5 Electrical analog

6 Simulink/Block diagram


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Special Issues





5 Electrical analog

6 Simulink/Block diagram

7 Conclusions


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Dam Scheme

Figure: Schematic of a Hydrolectric Dam.


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Head-Penstock

Variables

Figure: Schematic of a hydrolectric plant with its main variables.




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Pelton Wheel

Figure: Pelton wheel with five nozzles.D. Lopez Prime Movers

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g y p g

Pelton Wheel

Description

1 Impulse-type turbine


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Pelton Wheel

Description


2

The supplied energy is enterly Kinetic




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Pelton Wheel

Description


2

The supplied energy is enterly Kinetic3 For high heads, up to 300 m or more


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Pelton Wheel

Description


2


4 It works under atmospheric pressure because there is no wayto confine water


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Pelton Wheel

Description


2


4 It works under atmospheric pressure because there is no wayto confine water

5 Ease of control-(Governors)




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Francis Turbine

Figure: Francis turbine and a synchronous generator.


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Francis Turbine

Description

1 Reaction-type turbine


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Francis Turbine

Description


2 Hybrid condition-Potential plus Kinetic energies




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Francis Turbine

Description



3 For heads up to 360 m


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Francis Turbine

Description



3 For heads up to 360 m4 Given that the lower head, the higher water flow




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Francis Turbine

Description




5 The pressure is above the atmospheric because of waterimmersion


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Francis Turbine

Description




5 The pressure is above the atmospheric because of waterimmersion

6 The water impacts the runner through the vane guidestangentially and exits axially


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Francis Turbine

Figure: Francis turbine in Three Gorges Dam, China.




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Kaplan Turbine

Figure: Kaplan turbine with its helix.D. Lopez Prime Movers



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Kaplan Turbine

Description

1 Reaction-type categorie-Propeller


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Kaplan Turbine

Description






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Kaplan Turbine

Description


2 Hybrid condition-Potential plus Kinetic energies3 It works with a high pressure due to high water flow volume



K l T bi


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Kaplan Turbine

Description






K l T bi


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Kaplan Turbine

Description




5 The water can impact axially or tangentially



Hydraulic Turbines


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Hydraulic Turbines

Selection Chart

Figure: Turbine selection chart.



Hydraulic turbines
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Hydraulic turbines

Some hydroplants

Guavio, 5 230 MW Pelton turbines, Q=25 m3/s

Chivor, 8 125 MW Pelton turbines, Q=20.25 m3

/sSan Carlos, 8 155 MW Pelton turbines, Q=32.7 m3/s

Itaipu, 20 715 MW Francis turbines, Q=645 m3/s

T. Gorges, 26 700 MW Francis turbines, Q=600 940 m3/s


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Turbine Transfer Function-Introduction

In order to relate the variables defined before, it is necessary toneglect and idealize the dynamics in the system, therefore, we havethe following assumptions

1 The water resistance is negligible2 The penstock pipe is inelastic

3 The velocity of water varies direcltly with the gate openingand the square root of the head

4 The turbine output power is proportional to the product ofhead and the water volume flow



Transfer Function


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Transfer Function

Velocity

The turbine and penstock and turbine characteristics aredetermined by

1 The velocity of water in the penstock V

2 Turbine mechanical power Pm3 Acceleration of the water column

The velocity of the water in the penstock is given by

V = KuG

H (1)

By taking the total differential of V and normalizing with steadystate operation values V0 = KuG0

H0

V = G +1

2H (2)



Transfer Function
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Mechanical Power

As we defined before, the mechanical power is given by

Pm = KmHV (3)

Again, by taking the total differential and normalizing, for smallchanges around the steady state condition, the change in Pm isgiven by

Pm = H + V (4)



Transfer Function


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Mechanical Power

Substituting equation 1 in equation 2 to obtain

Pm =

3

2 H +

G (5)

Incorporating H again from equation 1 in 5, it yields

Pm = 3V 2G (6)



Transfer Function


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Newtons Second Law

From fundamental fluid mechanics the acceleration in the watercolumn is given from the Newtons Second Law

m

dV

dt = F = LAdV

dt = AgH (7)Where: is the mass density (fluid)L is the length of the penstock

A is the penstock cross areag is the acceleration due to gravity



Transfer Function


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Newtons Second Law

If we take the steady condition for power, it is, Pm0 = A gH0V0and dividing equation 7 for normalizing we have

LV0

gH0

dU

dt

=

H =

Tw

V

dt

=

H (8)

where Tw is the water starting time. Finally, by making the pastsubstitutions and taking the Laplace transform in 8, the turbinetransfer function is given by

Pm

G=

1 Tws1 + 1

2Tws

(9)


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Unit Step Response

By regarding the the initial and final vaules for the turbine transferfunction, it yields

P(0) = lims

sU(s)T(s) = lims

s1

s

1 Tws

1 +

1

2

Tws= 2 (10)

P() = lims0

sU(s)T(s) = lims

s1

s

1 Tws1 + 1

2Tws

= 1 (11)

The inverse Laplace Transform of P(s) with a step change isgiven by

P(t) =

1 3e2

Twt

G u(t) (12)



Special Characteristics
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Unit Step Response

0 2 4 6 8 10 12 14 16 182.5

2

1.5

1

0.5

0

0.5

1

1.5

2

Time [s]

Powerpu

Step

Response

Figure: Power response due to U(t) gate opening.D. Lopez Prime Movers




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U(t) Responses

0 0.5 1 1.5 2 2.5 3 3.5 43

2

1

0

1

2

3

Time [s]

Responsepu

Step

Power

Velocity

Head

Figure: All of the relevant variations during the turbine gate positions.D. Lopez Prime Movers




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Ramp Response

5 6 7 8 9 10

0.5

1

1.5

Time [s]

Responsepu

Ramp

Power

Velocity

Figure: Variations due to ramp gate closing.D. Lopez Prime Movers




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Simulation

Figure: Simulink diagram block to evaluate the transfer functionresponses.


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Electric circuit

In understanding the performance of a hydraulic turbine, it isworthwhile to visualize a RL circuit comparison

+

E0

i

L

G

Figure: Electrical Analog of a hidraulic turbine




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Conclusions

Given that ideal transfer function is an approximate model toinvestigate the mechanical power response in a generation systemdue to changes in the water column, it permits to calculate a useful

design for a hydro turbine installation and at the other hand, tocreate control systems to prevent a malfunction in the synchronousmachine operation in means of voltage and frequency disturbancesat the generation buses. By regarding that the past model was notthe most precise, we can include the water elasticity, the waterhammer effect, the water resistance etc. in the initial model.



References
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References

1 Kundur, P. S. Power System Stability and Control. ElectricPower Reasearch Intitute. McGrawHill. 1994.

2 Suescun, I. Centrales Electricas. Universidad de Antioquia.

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