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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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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Special Issues
1 Hydroelectric dam highlights
D. Lopez Prime Movers
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Special Issues
1 Hydroelectric dam highlights
2 Hydraulic turbines
D. Lopez Prime Movers
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Special Issues
1 Hydroelectric dam highlights
2 Hydraulic turbines
3 Hydraulic turbine transfer function-Linear systems
D. Lopez Prime Movers
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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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Special Issues
1 Hydroelectric dam highlights
2 Hydraulic turbines
3 Hydraulic turbine transfer function-Linear systems
4 Special characteristics
D. Lopez Prime Movers
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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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Special Issues
1 Hydroelectric dam highlights
2 Hydraulic turbines
3 Hydraulic turbine transfer function-Linear systems
4 Special characteristics
5 Electrical analog
D. Lopez Prime Movers
Agenda Head Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Special Issues
1 Hydroelectric dam highlights
2 Hydraulic turbines
3 Hydraulic turbine transfer function-Linear systems
4 Special characteristics
5 Electrical analog
6 Simulink/Block diagram
D. Lopez Prime Movers
Agenda Head Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Special Issues
1 Hydroelectric dam highlights
2 Hydraulic turbines
3 Hydraulic turbine transfer function-Linear systems
4 Special characteristics
5 Electrical analog
6 Simulink/Block diagram
7 Conclusions
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Dam Scheme
Figure: Schematic of a Hydrolectric Dam.
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Agenda Head Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Head-Penstock
Variables
Figure: Schematic of a hydrolectric plant with its main variables.
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Agenda Head Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Pelton Wheel
Figure: Pelton wheel with five nozzles.D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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g y p g
Pelton Wheel
Description
1 Impulse-type turbine
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Pelton Wheel
Description
1 Impulse-type turbine
2
The supplied energy is enterly Kinetic
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Pelton Wheel
Description
1 Impulse-type turbine
2
The supplied energy is enterly Kinetic3 For high heads, up to 300 m or more
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Pelton Wheel
Description
1 Impulse-type turbine
2
The supplied energy is enterly Kinetic3 For high heads, up to 300 m or more
4 It works under atmospheric pressure because there is no wayto confine water
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Pelton Wheel
Description
1 Impulse-type turbine
2
The supplied energy is enterly Kinetic3 For high heads, up to 300 m or more
4 It works under atmospheric pressure because there is no wayto confine water
5 Ease of control-(Governors)
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Francis Turbine
Figure: Francis turbine and a synchronous generator.
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Francis Turbine
Description
1 Reaction-type turbine
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Francis Turbine
Description
1 Reaction-type turbine
2 Hybrid condition-Potential plus Kinetic energies
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Francis Turbine
Description
1 Reaction-type turbine
2 Hybrid condition-Potential plus Kinetic energies
3 For heads up to 360 m
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Francis Turbine
Description
1 Reaction-type turbine
2 Hybrid condition-Potential plus Kinetic energies
3 For heads up to 360 m4 Given that the lower head, the higher water flow
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Francis Turbine
Description
1 Reaction-type turbine
2 Hybrid condition-Potential plus Kinetic energies
3 For heads up to 360 m4 Given that the lower head, the higher water flow
5 The pressure is above the atmospheric because of waterimmersion
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Francis Turbine
Description
1 Reaction-type turbine
2 Hybrid condition-Potential plus Kinetic energies
3 For heads up to 360 m4 Given that the lower head, the higher water flow
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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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Kaplan Turbine
Figure: Kaplan turbine with its helix.D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Kaplan Turbine
Description
1 Reaction-type categorie-Propeller
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Kaplan Turbine
Description
1 Reaction-type categorie-Propeller
2 Hybrid condition-Potential plus Kinetic energies
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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Kaplan Turbine
Description
1 Reaction-type categorie-Propeller
2 Hybrid condition-Potential plus Kinetic energies3 It works with a high pressure due to high water flow volume
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
K l T bi
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Kaplan Turbine
Description
1 Reaction-type categorie-Propeller
2 Hybrid condition-Potential plus Kinetic energies3 It works with a high pressure due to high water flow volume
4 For heads up to 100 m
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
K l T bi
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Kaplan Turbine
Description
1 Reaction-type categorie-Propeller
2 Hybrid condition-Potential plus Kinetic energies3 It works with a high pressure due to high water flow volume
4 For heads up to 100 m
5 The water can impact axially or tangentially
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Hydraulic Turbines
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Hydraulic Turbines
Selection Chart
Figure: Turbine selection chart.
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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)
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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)
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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)
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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)
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Special Characteristics
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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
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Special Characteristics
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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
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
Special Characteristics
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Simulation
Figure: Simulink diagram block to evaluate the transfer functionresponses.
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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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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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Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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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.
D. Lopez Prime Movers
Agenda Head-Penstock Hydraulic Turbines Transfer Function Special Characteristics Electrical Analog Conclusions
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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