Download Analysis of Transient Power-angle Characteristic for Synchronous

International Journal of Science, Engineering and Technology Research (IJSETR)
Volume 1, Issue 1, July 2012
Analysis of Transient Power-angle Characteristic
for Synchronous Generator
Naing Naing Htun

Abstract— Transient condition may lead to instability
where the machines in the power system fall out of synchronism.
The electrical air-gap power depends on the generator loading
and varies depending on the generator parameters and the
power angle. It also depends on the operating state of the
generator. Mechanical movement of the generator rotor is
influenced by electromagnetic effects and this movement varies
depending on the operating state of the generator. An important
advantage of the classical model is that the generator reactance
may be treated in a similar way to the reactance of the
transmission lines and other network elements.
the d-axis. Generators operating at lower speed, such as
driven by hydro turbines, need many magnetic poles in order
to operate 50 or 60 Hz. As the centrifugal forces experienced
by the rotors of these low-speed machines are lower than in
the corresponding turbo generators, salient-poles can be used
and the rotor diameter increased. Normally salient-pole rotors
have more than two poles and the angles and speed expressed
in electrical units. The main problem with modeling a
salient-pole machine is that the width of the air gap varies
around the generator with the narrowest gap being along the
d-axis and the widest along the q-axis.
Index Terms—Synchronous Generator, Steady State
Stability, Transient Stability, Constant Flux Linkage Model,
Classical Model
I. INTRODUCTION
An electrical power system consists of many individual
elements connected together to form a large, complex and
dynamic system capable of generating, transmitting and
distributing electrical energy over a large geographical area.
Based on their physical character, the different power system
dynamics may be divided into four groups such as wave,
electromagnetic, electromechanical and thermodynamic. The
electromagnetic phenomena mainly involve the generator
armature and damper windings and partly the network. These
electromechanical phenomena, namely the rotor oscillations
and accompanying network power swings, mainly involve the
rotor field and damper windings and the rotor inertia. As the
power system network connects the generators together, this
enables interactions between swinging generator rotors to
take place.
II. SYNCHRONOUS GENERATOR AND ITS EQUIVALENT
NETWORK
A. Salient-Pole Machine
A synchronous generator consists of a stator, on which the
three-phase armature winding is normally wound, and a rotor,
on which the DC field winding is wound. The stator has three
axes A, B and C each corresponding to one of the phase
windings. The rotor has two axes such as the direct axis
(d-axis), which is the main magnetic axis of the field winding,
and the quadrature axis (q-axis), π/2 electrical radians behind
Manuscript received Oct 15, 2011.
Ma Naing Naing Htun, Electrical Power Engineering Department,
Mandalay Technological University, (e-mail: [email protected]).
Mandalay, Myanmar, 09401573078.
Fig.1 A simplified salient-pole generator
B. Stability Problem of Synchronous Machine
The stability problem is concerned with the behavior of the
synchronous machines after they have been perturbed. If the
perturbation does not involve any net change in power, the
machines should return to their original state. If an unbalance
between the supply and demand is created by a change in
load, in generation, or in network conditions, a new operating
state is necessary. In any case all the
interconnected
synchronous machines should remain in synchronism if the
system is stable. The transient following a system
perturbation is oscillatory in nature, but if the system is stable,
these oscillations will be damped toward a new quiescent
operation condition. These oscillations are reflected as
fluctuations in the power flow over the transmission lines. If a
certain line connecting two groups of machines undergoes
excessive power fluctuations, it may be tripped out by its
protective equipment thereby disconnecting the two groups of
machines. This problem is termed the stability of the tie line,
even though in reality it reflects the stability of the two groups
of machines. A statement declaring a power system to be
stable is rather ambiguous unless the conditions under which
this stability has been examined are clearly stated. This
includes the operating conditions as well as the type of
perturbation given to the system. The same thing can be said
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International Journal of Science, Engineering and Technology Research (IJSETR)
Volume 1, Issue 1, July 2012
about tie-line stability. Since, there is concerned with the
tripping of the line, the power fluctuation that can be tolerated
depends on the initial operating condition of the system
including the line loading and the nature of the impacts to
which it is subjected.
C. The Equivalent Network Impedance
Synchronous generators are rarely used to supply
individual loads but are connected to a power system that
consists of a large number of other synchronous generators
and loads linked by the transmission network. The power
ratings of all remaining generators in the system can be treated
as one equivalent, very large, generating unit with an infinite
power rating. This equivalent generating unit is referred to as
the infinite busbar and can be represented on a circuit diagram
as an ideal voltage source behind equivalent system
impedance. The infinite busbar maintains a constant terminal
voltage and is capable of absorbing all the active and reactive
power output of the generator. This concept is illustrated in
figure (2) where the generator is connected to the system via a
step-up transformer represented by the series impedance. It is
assumed that the ideal transformer has been eliminated from
the circuit diagram by either using per units or recalculating
all the quantities using a common voltage level. The rest of
power system is represented by the infinite busbar that is by
the ideal voltage source Vs behind the equivalent system
impedance. The infinite busbar is assumed to have constant
voltage and frequency, neither of which is influenced by the
action of an individual generator. This means that the voltage
Vs can be used as reference and the phase angles of all the
other voltages and currents in the circuit measured with
respect to it. The impedance of the power angle δ is defined as
the phase shift between Er and Vs. As all the angles in the
phasor diagram have a dual time/space meaning, δ is also the
spatial angle between the two synchronously rotating rotors.
This spatial angle is referred to as the rotor angle and has the
same numerical value in electrical radians as the power angle.
As the rotor of the infinite busbar is not affected by an
individual generator, this rotor also provides a synchronously
rotating reference axis with respect to which the space
position of all the rotors may be defined. The elements of the
equivalent circuit may be combined to give the total
parameters as
x d =Xd +XT +Xs
(1)
x q =X q +X T +Xs
(2)
r=R+R T +R s
(3)
The salient-pole generator can be described as
E f = Vs + rI + jx d Id + jx q Iq
(4)
Fig.2 Equivalent circuit of the generator operating on an
infinite busbar
III. ARMATURE FLUX PATHS AND THE EQUIVALENT
REACTANCE
Figure (3) shows three characteristic states that correspond
to three different stages of rotor screening. Immediately after
the fault, the current induced in both the rotor field and
damper windings forces the armature reaction flux completely
out of the rotor to keep the rotor flux linkages constant as
shown in figure (3a), and the generator is said to be in the
sub-transient state. As energy is dissipated in the resistance of
Fig.3 patch of the armature flux in (a) the sub-transient state
(b) the transient state (c) the steady state
the rotor windings, the currents maintaining constant rotor
flux linkages decay with time allowing flux to enter the
windings. As the rotor damper winding resistance is the
largest, the damper current is the first to decay, allowing the
armature flux to enter the rotor pole face. However, it is still
forced out of the field winding itself, Figure (3b), and the
generator is said to be in the transient state. The field current
then decays with time to its steady-state value allowing the
armature reaction flux eventually to enter the whole rotor and
assume the minimum reluctance path. This steady state is
illustrated in Figure (3c). It is convenient to analyze the
dynamics of the generator separately when it is in the
sub-transient, transient and steady states. This is
accomplished by assigning a different equivalent circuit to the
generator when it is in each of the above states, but in order to
do this it is first necessary to consider the generator reactances
in each of the characteristic states.
Fig.4 Three-step approximation of the generator model: (a)
rms value of the AC component of the armature current (b)
generator reactances
If a generator is in the sub-transient state, and the armature
mmf is directed along the rotor d-axis, then the armature
reaction flux will be forced out of the rotor by the currents
induced in the field winding, the damper winding and the
rotor core. This flux path corresponds to the direct-axis
sub-transient reactance Xd''. On the other hand, if the armature
mmf is directed along the rotor q-axis, then the only currents
forcing the armature reaction flux out of the rotor are the rotor
core eddy currents and the currents in the q-axis damper
winding. If a generator only has a d-axis damper winding,
then the q-axis screening effect is much weaker than that for
the d-axis, and the corresponding quadrature axis
sub-transient reactance Xq''is greater than Xd''. This difference
between Xq''and Xd'' is called sub-transient saliency. For a
generator with a damper winding in both the d-axis and the
q-axis, the screening effect in both axes is similar,
sub-transient saliency is negligible and Xq'' ≈ Xd''.
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All Rights Reserved © 2012 IJSETR
International Journal of Science, Engineering and Technology Research (IJSETR)
Volume 1, Issue 1, July 2012
When the generator is in the transient state, screening is
provided by the field winding which is only in the d-axis.
However, in the round-rotor generator some q-axis screening
will be produced by eddy currents in the rotor iron with the
effect that Xq' > Xd'. The actual value of Xq' is somewhere
between Xd' and Xq with typically Xq' ≈ 2Xd'. In the
salient-pole generator the laminated rotor construction
prevents eddy currents flowing in the rotor body, there is no
screening in the q-axis and Xq' = Xq. Because of the absence
of a field winding in the q-axis, there is some degree of
transient saliency in all types of generator.
IV. STEADY-STATE AND TRANSIENT CHARACTERISTICS ON
THE POWER-ANGLE DIAGRAM
A. Steady-state Power-Angle Characteristics
Figure (5) shows the phasor diagram of a generator
operating under a particular load as defined by the length and
direction of the phasor I. Obviously the diagram will change if
the load is changed. The excitation is constant, Ef = Eq =
constant, and any change in load will change the generator
terminal voltage Vg , the angle δg and the power angle δ. If the
resistance r is neglected the real power supplied by the
generator to the system is given by the following equation.
PsEq =
E q Vs
xd
sinδ +
Vs2 x d -x q
sin 2δ
2 xd xq
(5)
Where PsEq is a function of the power angle δ and function
PsEq(δ) is referred to as the power-angle characteristic of the
generator operating on the busbar. The reluctance power term
deforms the sinusoidal characteristic so that the maximum of
PsEq(δ) occurs at δ < π/2.
winding (transient period). Consequently, changes in the
armature flux can penetrate the damper winding.
C. Constant Flux Linkage Model
Assume that the generator is connected to the infinite busbar
as shown in figure (6) and that all the resistances and shunt
impedances is associated with the transformer and network
can be neglected. The corresponding equivalent circuit and
phasor diagram of salient-pole generator in the transient state
is as shown in figure (6). The fictitious rotor of the rotor of the
infinite busbar serves as synchronously rotating reference
axis. The reactances of the step-up transformer and the
connecting network can be combined with that of the
generator to give
x'd = X'd +X , x'q = X'q +X , X=XT +Xs
Where Xd' and Xq' are the d- and q-axis transient reactance
of the generator. The voltage equations can be constructed as
E'd = Vsd +x q' Iq , Eq' = Vsq +x d' Id
(7)
Where Vsd and Vsq are the d- and q-components of the infinite
busbar voltage Vs given by Vsd = −Vs sin δ and Vsq = Vs cos δ.
As all resistances are neglected, the air-gap power is
Pe = Vsd Id Vsq Iq
(8)
Assuming constant rotor flux linkages, then the values of the
emfs Ed' and Eq' are constant, implying that both E' = constant
and α = constant. The generator power-angle characteristic
Pe(E', δ') can be described in terms of the transient emf and the
transient power angle and is valid for any type of generator. A
generator with a laminated salient-pole rotor cannot produce
effective screening in the q-axis with the effect that xq'= xq.
Inspection of the phasor diagram in Figure (6b) shows that in
this case E lies along the q-axis so that α = 0 and δ' = δ.
Pe = PE'  δ'  =
Xd’
Eq’
Ed
’
E q ' Vs
xd'
sinδ ' -
'
'
Vs2 x q - x d
sin2δ '
'
'
2 xd xq
(9)
Id
Vs
X q’
Fig.5 Power-angle characteristic
B. Transient Power-angle Characteristics
The disturbance acting on a generator will produce a
sudden change in armature current and flux. This flux change
induces additional currents in the rotor filed and damper
windings that expel the armature flux into high-reluctance
paths around the rotor so as to screen the rotor and keep the
rotor flux linkage constant. As the emf Eq is proportional to
the filed current, the additional induced field current will
cause changes in Eq so that the assumption of constant Eq used
to drive the static power angle characteristic. The induced
rotor currents decay with time as the armature flux penetrates
first the damper windings (sub-transient period) then the field
(6)
q
I
Iq
Vs
Vs
d
Figure.6 Generator-infinite busbar system in the transient
state (a) circuit diagram (b) phasor diagram of the salient-pole
generator (xq' =xq)
D. Classical Model
In this model, all the voltages, emfs and currents are
phasors in the network reference frame rather than their
components resolved along the d- and q-axes. The classical
model can be expressed by ignoring transient saliency, that is
assuming xd' ≈ xq'.
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International Journal of Science, Engineering and Technology Research (IJSETR)
Volume 1, Issue 1, July 2012
Pe = PE'  δ  
'
Eq ' Vs
xd'
sin δ
'
(10)
Consequently, when the network reactance is large the
classical model, and the constant flux linkage model give very
similar results even for a generator with a laminated
salient-pole rotor. It is important to note that δ is the angle
between Vs and E and not the angle between Vs and the q-axis.
During the transient period the emfs Ed and Eq are assumed to
be constant with respect to the rotor axes and α is also
constant with
δ= δ + α
'
(11)
E'd0 = Vgd +Xq ' Iq0
E 'd0 = (- 0.690) + 0.69×1.001 =0.00069 pu  0 pu
E'q0 = Vgq - Id0 Xd'
E'q0 = 0.737 -  -1.800 0.3=1.277 pu
E''d0 = Vgd +Xq '' Iq0
E ''d0 = -0.690 + (0.27) (1.001) = - 0.419pu
E''q0 = Vgq - Id0 Xd''
E''q0 = 0.737 -  -1.800 0.25 =1.187pu
I0 = 2.06  -17.8
φg0 = 17.8 deg
The total reactances are xd = 0.995pu, xq = 67.7pu, xd= 0.32pu
Taking Vg as reference, the phasor of the transient emf is
E' = 1.27743.13
Vs =Vg - j  XT +XL  I
Fig.7 Classical model of the generator in the transient state (a)
circuit diagram, (b) phasor diagram
V. CALCULATION OF STEADY-STATE AND TRANSIENT
POWER-ANGLE CHARACTERISTICS
A 230MVA salient-pole generator is referred to Yeywa
hydro generating station and which can be produced 790MW.
The generator real and reactive power are 1.98pu and 0.64pu.
The constant flux linkage and the classical generator model
are used to find and plot the steady-state and transient
characteristics.
*
 S  P - jQ 1.98-j0.64
I0 =   =
=
= 2.06  -17.8
V 
V
1.01
g
g
 
E Q =Vg +jXq I0
Vs = 1.01 - j  0.065×2.06 -17.8 = 1.06 - 6.92
δ0 = δ0' = 43.13 + 6.92 = 50.05°
δ +φ = 50.05 + 17.8 = 67.85°
0 0
The d- and q-components of the system voltage are
Vsd = -Vg sin δg0
Vsd = - 1.01 sin 50.05 = -0.774 pu
Vsq = Vg cos δg0
Vsq = 1.01 cos 50.05 = 0.65pu
PEq (δ) =
E q Vs
xd
sinδ +
Vs2 x d -x q
sin2δ
2 xd xq
E Q =1.01 + j (0.69× 2.06  -17.8) = 1.9843.13
PEq (δ) = 2.57 sin δ + 0.265sin 2δ
δg0 = 43.13°, φg0 = 17.8°
The transient characteristic can be calculated,
PE'  δ  =
Id0 = -I0 sin  φg0 + δg0 
Id0 = -I0 sin  φg0 + δg0 
I0 = - 2.06sin 17.8 +43.13 = -1.800 pu
E q ' Vs
xd'
'
'
Vs2 x q - x d
sinδ sin2δ '
'
'
2 xd xq
'
PE'  δ  = 4.13 sin δ - 0.883 sin 2δ
Iq0 = I0 cos  φg0 + δg0 
The approximated transient characteristic for the classical
model can be calculated assuming xd' = xq'. The transient emf,
calculated with respect to Vs is
Iq0 = 2.06cos(17.8 +43.13)= 1.001pu
E ' = Vs + j X d' I
Vgd = -Vg sin δg0
E' = 1.06 + j0.328 × 2.06 -17.8 = 1.421 26.93
Vgd = - 1.01 sin 43.13 = -0.690 pu
α = δ - δ' = 50.05 - 26.93 = 23.12°
E' = 1.421V, δ' = 26.93°
Vgq = Vg cos δg0
α = δ - δ' = 50.05 - 26.93 = 23.12°
Vgq = 1.01 cos 43.13 = 0.737pu
The approximated transient characteristic can be calculated as
E q0 = Vgq -Id0 X d
E q0 = 0.737- (- 1.800) (0.93) = 2.411pu
PE'  δ'  =
Eq ' Vs
xd
'
sinδ'
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International Journal of Science, Engineering and Technology Research (IJSETR)
Volume 1, Issue 1, July 2012
PE'  δ'  =
1.421×1.06
sin δ' = 4.952 sin δ'
0.328
Fig.8 Steady-state and transient characteristics of laminated
salient-pole machine
VI. CONCLUSION
The classical model gives a good approximation of the
constant flux linkage model. This characteristic is shifted with
respect to PEq(δ) by α = 23.12°. Figure (8) shows that by
neglecting transient saliency the classical model does not
generally significantly distort the transient characteristics. As
xd > xd', the amplitude of the transient characteristic is greater
than the amplitude of the steady-state characteristic.
ACKNOWLEDGMENT
The author would to express her special thanks to her
parents for their noble support and encouragement. The
author would like to express her gratitude to Dr. Khin Thuzar
Soe, Associate Professor, Head of Department of Electrical
Power Engineering, Mandalay Technological University, for
providing encouragement. The author especially appreciates
and thanks her teachers at MTU and corresponding people
who helped her directly or indirectly for this paper. The
author would like to thank U Nay Zar Win, Executive
Engineer from Ministry of Electric Power Engineering, for
providing the required data and information.
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