Lecture Notes EEE 360 - Warsaw University of Technology
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Transcript Lecture Notes EEE 360 - Warsaw University of Technology
EEE 360
Energy Conversion and
Transport
George G. Karady & Keith Holbert
Chapter 6
Synchronous Machine
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Lecture 15
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Synchronous Machines
• The DC excitation current in the rotor
generates a flux.
• The turbine drives the rotor and produces a
rotating flux
• The rotation cause flux changes in the windings
• The rotating flux induce a ac three phase
voltage in the stator winding.
• This is the rotor induced voltage in the stator.
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Synchronous Machines
• The generator is loaded
• The load current produces a rotating flux
• This rotating flux induces a ac three phase voltage in
the stator winding.
• This voltage is
– subtracted from the induced voltage.
– represented by a voltage drop on the synchronous reactance
• The equivalent circuit of a synchronous generator is a
voltage source and a reactance connected in series
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Synchronous Machines
6.3 Generator Application
–
Power angle:
Angle between the dc
excitation current generated induced voltage and
the terminal voltage
Xsyn
Rsta
Flux
Esta
Ista
Vt
DC
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Synchronous Machines
6.3 Generator Application
–
6.3.1 Loading: power is less than angle 90 deg
•
•
•
All generators in the system are connected in parallel
All generators rotates with the synchronous speed
The load can be increased by increasing the input mechanical power by
regulating the turbine impute power
The speed does not change, the power angle increases
Maximum power angle is 90 degree, beyond that operation is unstable
•
•
–
–
6.3.2 Reactive power regulation
When the excitation is:
•
•
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Increased, the generator reactive power also increases;
Decreased, the generator reactive power also decreases
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Synchronous Machines
6.3.3 Synchronization
–
Verify that the phase sequences of the two systems are the
same.
–
Adjust the machine speed with the turbine that drives the
generator until the generator voltage frequency is nearly the
same as the frequency of the network voltage.
–
Adjust the terminal voltage of the generator by changing the dc
field (rotor) current until the generator terminal voltage is almost
equal to the network voltage. Acceptable limit is 5%.
–
Adjust the phase angle of the generator terminal voltage by
regulating the input power until it is nearly equal with the
phase angle of the network voltage. Acceptable limits are
about 15°.
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Synchronous Machines
6.3.4 Static stability
Generator
Sgen 150kV A
Vgen 12.47kV
xgen 128%
Transmission line
Lline 48mi
ZL ( 0.5 j 0.67)
mi
Network
Vnet 12.47kV
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Inet_short 2kA
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Synchronous Machines
Generator
Figure 6.19
Distribution line
One-line diagram of a simple power system
Xsyn
Esta
Network
Zline
Xnet
Isyt
Vnet_ln
Figure 6.20 Single-phase equivalent circuit of the network in Figure 6.19 .
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System Data
The system data are:
Generator
Sgen 150kV A
Vgen 12.47kV
Network
Vnet 12.47kV
Inet_short 2kA
Distribution line
Lline 48mi
ZL ( 0.5 j 0.67)
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xgen 128%
mi
10
The first step of the study is to calculate the impedances. The generator snchronous reactance
is :
Xsyn xgen
Vgen
2
Sgen
Xsyn 1.3269 k
The line impedance is:
Zline ZL Lline
Zline 24 32.16j
The network line -to-netral voltage and reactance is:
Vnet_ln
Xnet
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Vnet
3
Vnet_ln
Inet_short
Vnet_ln 7.2 kV
Xnet 3.6
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The generator induced voltage is:
j
Esta 2 Vnet_ln e
1
Esta ( 60deg) 7.2 12.47j kV
The equivalent circuit shows that the current in this sytem is the voltage
difference divided by the total system impedance
2
Isyt
Esta Vnet_ln
j Xsyn Zline j Xnet
Isyt ( 60deg) 9.148 0.161jA
The complex power of the generator and network is
Sg 3 Esta Isyt
Sg ( 60deg) 203.614 338.751j kV A
Snet 3 Vnet_ln Isyt
Snet ( 60deg) 197.587 3.48j kV A
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The power of the generator and network is
Pg Re Sg
Pg ( 60deg) 203.614 kW
Pnet Re Snet
Pnet ( 60deg) 197.587 kW
The generator terminal voltage is:
Vt Esta jXsyn Isyt
Vt ( 60deg) 7.413 0.331j kV
Vt_ll
Vt_ll ( 60deg) 12.853 kV
Reg
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3 Vt
Vt_ll Vnet
Vnet
Reg ( 60deg) 3.072 %
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The generator and network power vs power angle
0deg 1deg 180deg
250
Pg
kW
200
Stable
operation
Unstable
operation
Maximum
power
150
Pnet
kW
100
110
Operation
point
50
0
0
30
60
90
120
150
180
deg
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Maximum power transmitted
60deg
max Maximize Pg
max 91.009 deg
Pg_max Pg max
Pg_max 236.226 kW
110kW root Pg 110kW
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110kW 27.55 deg
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6.4 Induced Voltage
Synchronous Reactance
Calculation
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Synchronous Machines
6.4 Induced Voltage and Synchronous Reactance
Calculation
E sta
2
X syn
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0 N sta N rot
3
0 N
2
sta
sta rot
If
gap
sta rot
If
gap
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Synchronous Machines
Lgap
0
C-
Lstator
(Drotor/2) d
B+
+
-
N
-/2
A+
d
-
+
+
-
Lrotor
/2
A-
Lstator
S
Lgap
-
B-
+
C+
Figure 6.26 Rotor-generated magnetic field in the simplified generator.
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Synchronous Machines
6.4.1 Induced Voltage
Ampere’s circuital law for this magnetic loop yields
2 Hgap Lgap Hrotor Lrotor Hstator Lstator
2 Hgap Lgap
Idc_rotor Nrotor
Hgap Idc_rotor
Idc_rotor Nrotor
If iron core is neglected
Idc_rotor Nrotor
2 Lgap
Bgap Idc_rotor o Hgap Idc_rotor
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Synchronous Machines
6.4.1 Induced Voltage
Bbase Bbase_max cos
2
Figure 6.27 Rotor
generated flux
density distribution
along the rotor
surface
Brotor Idc_rotor
1
T
0
0
Bbase( )
T
1
2
90
60
30
0
30
60
120 150 180 210 240 270
Square wave flux equation
90
deg
Brotor Idc_rotor if 90 deg Bgap Idc_rotor Bgap Idc_rotor
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Synchronous Machines
6.4.1 Induced Voltage
The base component is calculated using the Fourier series
3
2
1
Bbase_max
Brotor Idc_rotor cos d
4
Bgap Idc_rotor
2
Bbase_max Idc_rotor
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Bgap Idc_rotor
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Synchronous Machines
6.4.1 Induced Voltage
Substitution of current and flux density results in:
Bbase_max Idc_rotor
4
o
Idc_rotor Nrotor
2 Lgap
2
Drotor
rotor Idc_rotor
Bbase
Lstator d
2
Flux integral
2
rotor Idc_rotor Bbase_max Drotor Lstator
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Synchronous Machines
6.4.1 Induced Voltage
Substitution of current and flux density results in:
A Idc_rotor t rotor Idc_rotor cos t
Flux integral
rotor Idc_rotor
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o
Idc_rotor Nrotor
2 Lgap
360 Chapter 6 Synchronous Machine
Drotor Lstator
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Rotating Flux Generation
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Synchronous Machines
Lgap
0
C+
Lstator
(Drotor/2) d
BN
-/2
A-
-
d
-
+
+
Lrotor
/2
A+
Lstator
S
Lgap
B+
C-
Figure 6.29 Arrangement for calculation of load current generated flux.
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Synchronous Machines
0.7
A_ac(t)
0.5
C+
B-
A-
0.3
+
A+
A_ac(t) 0.1
0
B+
C-
0.1
0.3
0.5
0.7
0
60
120
180
240
300
360
t
deg
Figure 6.30 Phase A load current generated ac flux.
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Synchronous Machines
B(t)
ABC(t)
C(t)
A(t)
B(t)
C+
+
B-
C(t)
-
+
A-
B+
+
-
A+
C-
Figure 6.31 AC flux generated by the phase currents.
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Synchronous Machines
0.7
A_ac ( t ) 0.5
Wb
B_ac ( t )
0.3
0.1
Wb
C_ac ( t ) 0.1
Wb
0.3
0
0.5
0.7
0
60
120
180
240
300
360
t
deg
t = 0°
t = 45°
C(t)
B(t)
ABC(0)
B(t)
C(t)
ABC(t)
A(t)
B(t)
C+
+
B-
Figure 6.32 Rotating
flux generated by the
phase currents
A(t)
C+
C(t)
-
+
A-
AA+
B+
-
C(t)
-
+
A+
B(t)
B+
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+
-
C-
B+
+
-
C-
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Synchronous Machines
t 0ms 0.1ms
1
60
s
1
ABC ( t ) 0.75
Wb
0.5
AA ( t )
Wb
BA ( t )
0.25
0
Wb
CA ( t )
Wb
0.25
0.5
0
0.75
1
0
60
120
180
240
300
360
t
deg
Figure 6.33 Flux linkages with phase A.
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Exercises using MATCAD
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Synchronous Machines
Transformer
Network
Transmission line
Figure 6.34 One-line
diagram of synchronous
generator network
Generator
Xsy
Eg_ln
Ig
Eg_s
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Xtr
Vg_ln
Vtrs_ln
Xsy_s
Xtr
ZLine
Vnet_ln
Isy
Figure 6.35 Single-phase
equivalent circuit of
synchronous generator
network.
ZLine
Isy
Vnet_ln
360 Chapter 6 Synchronous Machine
Figure 6.36 Simplified singlephase equivalent circuit of
synchronous generator network.
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Synchronous Machines
0deg 1deg 180deg
600
500
Pnet
400
M W
Pnetwork
MW
300
200
100
0
0
30
60
90
120
150
180
deg
Figure 6.37 Power delivered to the network as a function of power angle.
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