Evaluation of Rotor Speed Stability Margin
of a Constant Speed Wind Turbine Generator
Mital G. Kanabar, Student Member, IEEE, and S. A. Khaparde, Senior Member, IEEE
Abstract-Many developing countries like India have installed
large number of constant speed wind turbine generators
(WTGs) as they are robust and economical. Most of the
countries have their own grid codes (rules and regulations) to
integrate WTGs into the utility grid. One of the primary grid
code for WTGs is low voltage ride through (LVRT) capability.
The regulations of grid integration are likely to make LVRT
requirements mandatory for WTGs with high penetration
level. To demonstrate LVRT capability, a WTG has to remain
connected to the grid at a specific low voltage and for a specific
duration. Generally, a constant speed WTG does not satisfy
LVRT requirements. This is because during a nearby fault,
the rotor accelerates to a very high speed, and hence the
WTG becomes unstable. This phenomenon is referred to as
rotor speed instability. To satisfy LVRT requirements, the rotor
speed stability margin of a constant speed WTG has to be
improved. One of the methods to meet the LVRT requirements
is by providing additional reactive power support which can
improve the terminal voltage during a disturbance. This, in
turn, will increase the electromagnetic torque and hence, the
rotor acceleration can be reduced. This paper presents an
evaluation of rotor speed stability margin to obtain critical slip
and critical clearing time of a constant speed wind turbine
generator. For this analysis, analytical formulae have been
presented to determine the exact amount of additional reactive
power support required to meet the LVRT capabilities. For
a sample system, using simulation of the dynamic model in
SIMULINK, it has been shown that the analytical value of
reactive power is indeed able to make the WTG comply with
the LVRT requirements.
Index Terms- Constant speed (squirrel cage) induction generators, grid codes, rotor speed stability, wind turbine generating
system.
I. INTRODUCTION
MONG all renewable resources, wind power is the
/A most
booming renewable technology all over the
world. Further, India ranks fourth in the world with a total
installed capacity of more than 7, 311 MW by the end
of March, 2007 [1]. Most of the wind turbine generators
(WTGs) installed in India are constant speed (squirrel cage)
induction generators. This is because of their robustness,
mechanical simplicity and low price. However, a constant
speed WTG always demands reactive power, hence reactive
Mital G. Kanabar is with the Department of Electrical Engineering,
Indian Institute of Technology Bombay, Mumbai, India - 400076 e-mail:
(mital.kanabar@gmail.com).
S. A. Khaparde is with the Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai, India - 400076 e-mail:
(sak@ee.iitb.ac.in).
978-1-4244-1762-9/08/$25.00
2008 IEEE
1.1
1.1
.S.
[
Wind plant required to remain Online
0
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a)
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0.7
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a)
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0
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0.5
1
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0.3
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0.
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a)
.ot
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;t.0
Wind plan t Not required to remain Online
0
0.1
-0.1
0.0 0.15
0.625
-
-
1.0
-
-
2.0
Time (sec)
WECC Standard
AWEA Standard
3.0
4.0
Fig. 1. LVRT requirements for wind generation facilities.
power compensation is needed.
During a grid disturbance near to a WTG, severe voltage
sag in the connecting network may cause a significant
reduction in active power generation and rise in rotor speed.
After voltage recovery, the rotor speed of the induction
generator may be so high that it does not return to the prefault value. This may lead to rotor speed stability problem [2].
According to [2], rotor speed stability refers to the
ability of an induction (asynchronous) machine to remain
connected to the electric power system and running at a
mechanical speed close to the speed corresponding to the
actual system frequency after being subjected to a disturbance.
As the penetration level of constant speed WTGs increases,
it is critical to maintain the rotor speed stability during low
voltage at point of common coupling (PCC). This is referred
to as low voltage ride through (LVRT) capability of a WTG.
Normally, LVRT requirements are stringent in regions with
high penetration of wind power [3], [4]. AWEA recommended
adoption of an LVRT requirement developed by E.ON Netz as
shown in Fig. 1, if required, on a case-by-case basis. WECC
(Western Electricity Coordinating Council) has put effort to
lenient this stringent requirement in May 2005, and revised
that machine should stay connected for 15% of nominal per
unit of terminal voltage for approximately 0.15 s [3].
To understand rotor speed stability for a constant speed
WTG, it is necessary to model the dynamics of the wind
turbine and generating system connected to the grid. Detailed
dynamic modeling equations for a constant speed (squirrel
jX1
Infinite Bus
JX,
jXtr
Constant
jXs
jxtr
Speed WTG
IG
Capacitor Banks
rrI
s
-jxc
Fig. 3. Equivalent circuit diagram of a sample system.
Fig. 2.
Single line diagram of a sample system.
cage) induction generation have been described in [5].
References [6], [7] have presented a dynamic model of a
wind turbine for power system analysis. An analysis of rotor
speed stability has been carried out in [8], in which, active
stall control strategy has been presented to enhance the
B
rotor speed stability of a constant speed WTG. The study of
transient stability of a constant speed WTG using dynamic Fig. 4. Thevenin's equivalent circuit.
simulation has been presented in [9], [10].
rr
S
Reference [12] examines the response of a constant
speed WTG during faults and the possible impacts on the
system stability when the percentage of wind generation
increases. In [13], Salman etal. have investigated the factors
that influence the dynamic behavior of a WTG following
network fault conditions. Using this technique, torque-time
characteristic of the WTG has been obtained to determine
the critical clearing time. In [14], [15], Senjyu etal. have
demonstrated the control of terminal voltage and power
factor of a constant speed WTG with shunt capacitor banks.
The dynamic behavior of a constant speed WTG with shunt
capacitor banks has been reported in [16].
To calculate the exact amount of reactive power support
required to satisfy the LVRT requirement, it is necessary
to obtain a relation between the critical slip (scr) and the
reactive power support in steady state
Let us consider the steady-state equivalent model of a constant
speed induction generator as shown in Fig. 3. In this figure, rl
and Xl are the line resistance and reactance respectively; Xtr
is the transformer reactance; Xc is the capacitive reactance;
rs and Xs are the stator resistance and reactance respectively;
Xm is the magnetizing reactance; r, and Xr are the rotor
resistance and reactance referred to the stator side respectively
and s is the rotor slip. All these above quantities are in per unit.
This paper contributes towards quantification of additional
reactive power support necessary to satisfy LVRT requirements
for the existing constant speed WTGs. To satisfy LVRT
requirements of a constant speed WTG, the rotor speed
stability margin has to be improved. To evaluate rotor speed
stability margin, the formulae for critical slip and critical
clearing time have been obtained analytically.
To obtain the torque-slip characteristics of a constant speed
WTG, a Thevenin equivalent that has been derived across
points A and B for the circuit in Fig. 3 is shown in Fig. 4.
The organization of this paper is as follows. In section
2, steady state analysis of a constant speed WTG has been
described. In this section, formula for critical slip has been
obtained from torque-speed characteristic of a constant speed
WTG. The calculations of critical clearing time has been
detailed in section 3. Simulation results of a constant speed
WTG with capacitor bank connected to the sample system
have been discussed in section 4. Section 5 concludes the
paper.
The formula to calculate the Thevenin's voltage is indicated
below:
VXCXTm +
Vt h (rl + jXt)[rs + j(Xsm
-Xc)1 XCXsm -jXcrs
where, Xlt = Xl + Xt,
Xsm
=
Xs + Xm
The values of the Thevenin resistance (rth) and Thevenin
reactance (Xth) are obtained as follows:
'rth =
II. STEADY STATE ANALYSIS OF A CONSTANT SPEED
WTG
Fig. 2 shows a single line diagram of a sample system with
a constant speed WTG and capacitor banks connected to an
infinite bus through a step-up power transformer.
Xth=
where,
a c+b d
2+
bc-ad
(2)
(3)
A. Evaluation of Critical Slip
As shown in Fig. 5, the critical slip can be obtained by
equating the electrical torque of the WTG with its mechanical
torque.
From (4),
01)
Vth
2 . rr
-TM
[>r +rth]2 + [XI+Xth]2
(5)
This leads to,
-Scr
-So
Slip (p.u.)
Vth
2
____
Fig. 5.
Tm
Torque-slip characteristic of a constant speed WTG.
/I
'
T.r
S
=
(r
S
I
)22
+ 2rth
rth +(Xr
+ Xth)
(6)
S
Finally,
Bc
a
=
=
1= Total
susceptance of the capacitor banks
[r2h+(X+Xth)2].s2+[2rthrr-rr'VtTmh2
I
Bc[XsXtXm - rsrXm] - Xm(Xs + Xlt)
=-Bc[riXsXm
c
=-Bc[r,X,m + r,Xlt] + rs + r,
=
0 (7)
This is a quadratic equation in 's', and following are its roots
(scr,s3) (as shown in Fig. 5)
+ rsXitXm] + Xm(rs + ri)
b
s+(r;)2
Equation (8) (in next page) shows that s,. is mainly a
function of the parameters such as rth, Xth and Vth. The
value of these parameters also depends on Bc (the amount
of reactive power compensation in per unit).
d = Bc [rsrl - XitXsm] + Xs +X +
X+lt
From Fig. 4, the electromagnetic torque (in p.u.) of a
constant speed WTG can be determined as follows:
III. CALCULATION OF CRITICAL CLEARING TIME
Vh 12
[>r
+ rth ]2 +
.
rr
S
[XI +Xth1]2
(4)
Using (4), the torque versus slip characteristic has been
obtained, and is shown in Fig. 5.
In the normal operating condition, the electromagnetic
and mechanical torques will be equal; hence, the WTG will
operate at slip so (point Q). When a severe fault occurs
close to the WTG, the terminal voltage of the WTG falls
drastically. This will reduce the electrical torque to almost
zero. Consequently, the rotor will oscillate, and the slip of the
WTG will increase gradually. Once the fault is cleared, the
terminal voltage and electrical torque will again increase to
its nominal value and thereby, the rotor will decelerate. If the
fault is cleared after the critical clearing time (ter), the rotor
may accelerate to a higher than critical slip (scr) value. In this
case, although the fault is cleared and the terminal voltage is
recovered, the rotor will continue to accelerate (beyond scr),
and therefore, the WTG will enter the unstable region. This
phenomenon implies that if the rotor slip crosses the point P
(as shown in Fig. 5), the WTG will get disconnected from
the grid due to over-speed protection. In practice, over-speed
protection circuit disconnects the WTG from the grid when
the speed of the WTG exceeds 1.2 p.u.
Let us consider rotor dynamics to obtain the critical clearing
time,
Tm-Te
ds
dt
(10)
2H
where, s is the slip in p.u., Tm is the mechanical torque
in p.u., Te is the electromagnetic torque in p.u., H is the
combined inertia constant of the WTG system in sec.
Integration of (10) leads to,
Jter dt
2H
dt
m
-
e
Jscrd
s
(1 1)
It has been assumed that during the fault Tm - Te remains
approximately constant.
Finally,
tcr
=
IH
(Scr - So)
- Te
m
(12)
From (12), it can be observed that the critical clearing time is
directly proportional to the inertia constant and the difference
of the critical and initial slip, and inversely proportional to the
difference between the mechanical and electromagnetic torque.
Ilr
- rIr
-[2rthrr-rr
Scr
2[rt2h + (XI
and,
-[It 1
sO
+hl2
] + -[2rthr-r
=
-[2rthT-/
r.
2122 4(r )2 [r2h + (XI + Xth)21
+
Xth)2]
rliV ,2 -4r)2r
]'-[2rthrrrTl ]2-4r)2[rh+
2[rt2h +
(XI
+ Xth)2]
(8)
2
(XI +Xth)21
(9)
TABLE I
PARAMETERS OF A CONSTANT SPEED WTG
Parameters
Value
Rated Power
Rated Phase Voltage
Rated Frequency
Number of Poles
Stator Resistance (r,)
Stator Leakage Reactance (X18)
Rotor Resistance (re)
Rotor Leakage Reactance (X;r)
Magnetizing Reactance (Xm)
Inertia constant (J)
600 kW
690 V
50 Hz
4
0.016 p.u.
0.15 p.u.
0.01 p.u.
0. 11 p.U.
7.28 p.u.
18.029 kg.m2
IV. SIMULATION RESULTS AND DISCUSSIONS
Modeling of the constant speed WTG, capacitor banks
and grid has been carried out using MATLAB/SIMULINK
software tool. Currents from the WTG have been added to
the currents from the capacitor banks. The total current is
then injected into the grid. From this injected current, the
terminal voltage is calculated which is then given as an input
to the WTG and capacitor banks.
As shown in Fig. 2, a 600 kW constant speed WTG with
capacitor bank is simulated using MATLAB/SIMULINK.
Dynamic model of this 600kW constant speed WTG is
same as given in reference [5]. The network transients are
neglected. The machine parameters are listed in table I [15].
This WTG is stall controlled (Type Ao) and hence, it does
not possess blade-pitch control.
-
A. Effect of Additional Reactive Power Support
With the help of simulation, it has been shown that this
600 kW constant speed WTG does not comply with the
LVRT requirements (as shown in Fig. 8). Hence, the WTG
has to be disconnected from the grid due to rotor speed
instability whenever a fault occurs in its vicinity. However,
by providing additional reactive power support, rotor speed
stability of the WTG can be enhanced such that it can satisfy
the LVRT requirements. This phenomenon has been discussed
in this section with the help of simulation results. Further, the
exact quantification of reactive support required to achieve
particular values of critical slip and critical clearing time has
been obtained theoretically. Finally, using the dynamic model
of the system, it has been shown that dynamic simulation
results (critical slip and time) match with the analytically
calculated results using (8) and (12).
Fig. 6. Torque-slip characteristics of the WTG with nominal and additional
reactive power.
Fig. 6 shows the torque-slip characteristics of a 600 kW
WTG with two different value of reactive power injection.
Equation (4) shows that the electromagnetic torque is a
function of Vth, rth and Xth. For a given set of machine
parameters, Vth, rth and Xth are functions of BC (the value
of reactive power compensation) as shown in (1), (2) and
(3) respectively. For nominal reactive power support, the
value of BC is 0.23 p.u., and with an additional capacitor
bank of 0.22 p.u., the value of BC will be 0.45 p.u. As
indicated in Fig. 6, an additional value of BC will shift the
torque-slip characteristic upwards. Consequently, the value
of critical slip will increase from -0.118 p.u. to -0.15 p.u.
Improvement of sc, because of additional BC can also be
obtained numerically using (8). Similarly, using (12), t, has
been calculated as 0.12 s with the nominal capacitor bank,
and 0.155 s with an additional capacitor bank.
The equations for s, and t, (as a function of Bc) have
been verified using a dynamic-simulation model of the sample
system. A severe three phase-to-ground fault has been created
on the system such that the terminal voltage at the constant
speed WTG should remain as per LVRT requirements. To
consider the worst condition, the wind velocity has been kept
constant at its rated value in the simulation. Therefore, the
mechanical torque of the turbine will remain at 1 p.u. during
the fault. As per LVRT requirements (refer Fig. 1), a WTG
should remain stable for 0.15 s with a terminal voltage of
0.15 p.u. This means that to satisfy the LVRT requirement, a
WTG should have a critical clearing time of at least 0.15 s.
CD=3
2
X-.
0
E -4
E
2
,2
aD
-22
-
LL]
2.5
3
3.5
4 4.12
4.5
5
5.5
aD
LL]
2
0.9i
2
Time (Sec.)
Fig. 7. Electromagnetic torque and rotor speed without additional capacitor
bank for a fault of duration 0.12 s.
i 4
l
lA
. -2
0
-4
-r
2
2.5
3
3.5
4 4.15
4.5
5
2j
a 1.8
-o
CD
CD
3
3.5
4 4.15
4.5
5
5.5
6
4.5
5
5.5
6
,
i~~~~
"/
(X)1.4-
3.5
4
.1
Tim(Sec.)
45
3
3.5
4 4.15
Time (Sec.)
Fig. 9. Electromagnetic torque and rotor speed with additional capacitor
bank for a fault of duration 0.15 s.
6
~~~/
i
1.6
° 1.2
5.5.5
2.5
executing the dynamic model again, it has been observed that
the WTG remains stable even for 0.15 s. The rotor accelerates
to -0.15 p.u. slip, and after the fault is cleared, the rotor
comes back to its nominal value of -0.005 p.u. slip as shown
in Fig. 9. Additional capacitor bank remains connected only
during transient condition. Further, capacitor bank should be
switched on before the critical clearing time of the WTG.
2
0
aD)
T
-o
CD
o5 1.00
75
LL]
2.5
.15
CD
CD
E
r
-4
0 -b
6
,1.1
C3
_X
r~
H0
TABLE II
SUMMARY OF THE RESULTS
/
Cases
212
Fig. 8. Electromagnetic torque and rotor speed without additional capacitor
bank for a fault of duration 0.15 s.
Fig. 7 shows the electromagnetic torque and rotor speed
in per unit. The fault has been simulated at 4 s such that the
terminal voltage of the WTG remains 0.15 p.u (as per LVRT
requirements). During the fault, the value of Te will reduce
to almost zero. Hence, the rotor will accelerate to -0.11 p.u.
slip within 0.12 s. The numerical values of the critical slip
and time are calculated to be -0.118 p.u. and 0.12 s from (8)
and (12) respectively.
If the fault persist for more than the critical clearing time
(0.12 s), the rotor will continue to accelerate and the WTG
becomes unstable. Fig. 8 shows the electromagnetic torque
and rotor speed for a fault of duration 0.15 s. The speed of the
WTG ramps up gradually and it has to be disconnected from
the grid before the mechanical constraint on the rotor speed
(1.2 p.u.) is reached. However, as per LVRT requirements,
a constant speed WTG should remain connected for at least
0.15 s, which is not satisfied for this 600 kW WTG with
nominal value of reactive power compensation.
a
To meet the LVRT requirements, an additional capacitor
bank has been employed in parallel with the WTG. With this,
Nominal reactive power
Additional reactive power
Analytically
scr
dynamic
simulation
(p.u.)
(p.u.)
-0.118
-0.150
Cases
Analytically
Nominal reactive power
Additional reactive power
(sec.)
0.120
0.155
From
-0.110
-0.150
tcr
From
dynamic
simulation
(sec.)
0.120
0.150
V. CONCLUSIONS
To obtain rotor speed stability margin of a constant speed
WTG, the analytical formulae for s, and t, are presented.
Enhancement of the rotor speed stability margin has been
achieved with the help of additional reactive power compensation. Steady state analysis has been carried out to calculate
the exact amount of additional reactive power required to
satisfy LVRT requirements. This extra reactive power would
recover the voltage and hence, the electromagnetic torque.
Therefore, the rotor acceleration reduces and the value of
tcr increases. Proposed analytical model for the evaluation of
Scr and tcr are validated using dynamic simulation results in
MATLAB/SIMULINK. Further, it has been shown that with
this nominal value of reactive power support (BC=0.23 p.u.),
tc, is 0.12 s. However, according to the LVRT requirements,
t, should be at least 0.15 s. Therefore, analytically calculated
reactive power support of (BC=0.22 p.u.) has been connected
additionally. This extra reactive power increases the value
of t, to 0.15 s, which is in compliance with the LVRT
requirements. From Table II, it can be inferred that analytically
calculated values of s, and t, match with those obtained by
dynamic simulation. Though the requirements vary according
to severity and location of fault, this evaluation would still be
applicable to any generic WTG system.
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