1
Wind rotor inertia and variable efficiency:
fundamental limits on their exploitation for
inertial response and power system damping
Barry Rawn
Energy Systems Group, Edward S. Rogers Sr. Dept of Electrical and Computer Engineering
University of Toronto
Toronto, Ontario, Canada M5S 3G4
office: 416 978 2613, fax: 416 971-2325
e-mail: barry.rawn@utoronto.ca
Abstract
Participation in frequency regulation by wind farms becomes necessary as they displace other types
of generation. It has been shown that wind turbines can employ their variable pitch and speed to provide
power variations that stabilize system frequency. However, limits on the exploitation of rotor inertia have
not been characterized. Also, most studies examine a single operating point and employ either a constant
wind input, or a single time series.
This paper examines the dependence of stabilizing capability on operating point, and the sensitivity of
this capability to wind variations. Operation below rated wind speed is evaluated. Two services are studied:
inertial response, and power system damping. A single mass model for a full-converter wind turbine is
simulated to determine how maximum and minimum limits on rotor speed and power constrain the delivery
of these services. The outcome of the study is a graph for each service showing the largest power variation
available as a function of the mean operating wind speed. Constant wind speed is compared with a set of
random wind time series. The tradeoffs of de-loading the wind turbine to create a surplus of kinetic energy
are evaluated.
Large contributions to inertial response and power systems damping are possible, but vary with operating point. Contributions are limited to below a certain wind speed due to rotor over-speed. De-loading
can be used to maximize the contribution over the viable range. Wind variations reduce the magnitude of
the contribution, and make rotor speed instability possible for some control references.
2
I. I NTRODUCTION
It has been suggested that modern variable speed wind turbines should contribute to frequency regulation by exploiting their ability to dynamically control power output. It becomes crucial to maintain this
functionality when wind turbines displace a large proportion of conventional generators. Modern variable
speed wind turbines have a rotational speed that is decoupled from grid frequency. Consequently, reduction
in system inertia is becoming a concern for small island power systems [1]. Studies of other jurisdictions
have indicated that when DFIG and full-converter wind turbines replace other forms of generation, their
influence on the damping of electromechanical modes can range from neutral to undesirable [2], [3].
Several papers have demonstrated through simulation how the introduction of a corrective signal in the
turbine power reference can produce useful damping [4],[5]. Contributions to primary frequency response
by endowing turbines with emulated inertia and frequency droop have also been considered [6],[7], [8].
Controllers have been implemented on realistic models, and these case studies have been performed
using realistic power system models. Most of these focus on one or two operating points, and assume a
constant wind speed. However, wind variations may introduce limitations. Also, stability contributions must
be bounded by the inertia and aerodynamic characteristics of the turbine rotor, and by safety constraints.
They are therefore strongly dependent on operating point.
This study assesses how the capacity available for power system stabilization varies over the range
of operating wind speeds, and how it may be degraded by wind variations. Two services are assessed:
inertial response, and damping of inter-area modes. It is shown these services can be provided, but to a
degree less than what would be expected from a constant-wind analysis.
II. W IND
PLANT CHARACTERISTICS
For this study, a 1.5MW machine interfaced through a full-converter has been employed. Parameters
given in [9] were used to obtain aerodynamic and rotor characteristics suitable for a machine this size. It
is assumed that closed-loop control of the current allows generator torque to be assigned. Because only
the upper bounds on performance due to the rotor inertia and aerodynamics are of interest, a single mass
model with the inputs of aerodynamic torque and generator torque is modeled.
A. Operation below rated wind speed
The controlled power-wind speed curve of a wind turbine is shown in Fig 1(a). Above or near rated wind
speed, pitch control activates to regulate speed and power to the rated values. At wind speeds below
rated, more of the machine’s rated converter capacity becomes available as the available wind power
decreases. Operation in this mode is largely governed by the generator torque as set by the machine-side
converter. Fig 1(b) shows rotor aerodynamic torque and generator load curves for a family of wind speeds.
Usually an optimal load curve is set to ensure that the operating point at every wind speed extracts power
with maximum efficiency.
Tgen = kload ωh 2
1
kload = ρACp (λ∗ )
2
(1)
R
λ∗
3
(2)
where λ∗ is chosen to be that value of the tip-speed ratio that for which Cp (λ) has a maximum. Setting
λ∗ to a value λ∗ > λopt sets a different quadratic curve which causes the steady state operating points to
be higher than optimal, extracting a lesser, ”de-loaded” power (high speed load curve in Fig 1(b)). There
exists another value λu for which the same power is obtained at operating speeds lower than optimal.
(low speed load curve). For a fixed wind speed, two points A and B are marked in Fig 1(b). The power
delivered at points A and B is the same. Operation using the high-speed curve makes a greater stored
kinetic energy available during operation, and a reduction in speed increases available power. Thus, the
variable efficiency of a wind turbine’s blades can be used to operate with surplus power and energy.
3
(a) Operating ranges of wind turbine below rated wind (b) Aerodynamic and generator torque curves below rated
speed
Fig. 1.
Wind turbine operating characteristics.
(a) Static constraints on speed.
(b) Static constraints on power.
Fig. 2.
Constraints on operation. Arrows indicate speed and power margins available for contributions to power system
stabilization.
B. Limits on operation
There exist limits on the speed and power of the wind turbine. Some typical values for safety limits are
proposed in [10]. Rotor speed can exceed rated, but limiting systems activate above 1.2p.u. to prevent
excessive tip noise and to ensure safety. For a full converter system, the converter power limit can be
briefly exceeded by a factor of 1.14, and both positive and negative power can in principle be commanded.
For a given de-loading, the maximum speed 1.2ωrated determines a maximum operating wind speed
vhigh
vhigh = 1.2ωrated /λ∗
(3)
Operation ceases below the cut-in wind speed. For a given de-loading, this wind speed determines a
lower speed limit.
ωmin = λ∗ vcut−in /R
(4)
These constraints are depicted in Fig 2(a).
There is a difference between the allowable excess and the power associated with the steady-state
operating point at a constant wind speed. This constraint is depicted in Fig 2(b). This defines a static limit
on power variations ∆P that can be demanded
1
∆P ≤ 1.14Prated − ρACp (λ∗ ) vw 3
(5)
2
4
vcut−in < vw < vhigh
(6)
Some margin of power and speed is available, as indicated by the thick arrows in Fig 2. This static
margin defines the maximum operating space available for providing stabilizing services. The margin is
reduced when wind variations are introduced, since they cause speed and power variations.
III. S TABILIZING
SERVICES
A. Inertial response
The loss of a large generator or connection of a large load can cause system frequency to drop suddenly.
Other generators are enlisted to support the frequency by injecting inertial energy and increasing their
power output. Requirements vary depending on the utility concerned, but contribution to frequency support
is increasingly spelled out in grid codes. A detailed discussion of requirements is given in [8]. A simple
representation of the frequency signal involved used in [7] and [11] is an exponential drop, shown in Fig 3
[7],[11]. It is characterized by a magnitude ∆fdrop and time constant τ . The power of a generator supplying
inertial energy in response is shown below the frequency signal.
(a) Frequency drop and desired generator behaviour
(b) Simple derivative reference
Fig. 3. Inertial response to a drop in frequency. Fig 3(a) shows a response ∆P from the setpoint power. Fig 3(b) shows a simple
controller used to produce an inertial response from variable speed turbine.
The kinetic energy of a variable speed wind turbine can be contributed in an inertial response using a
derivative torque term in the control reference, as shown in Fig 3(b). For a practical implementation, such
a reference would have to be shaped to limit the derivative of torque and prevent excessive shaft stress.
However, it is an appropriate simplification for studying limits due to rotor inertia.
The effect of employing such a reference is showed in Fig 4(a). During the drop in frequency, this
derivative term produces a brief decelerating torque. A spike of power from the wind turbine is followed
by a dip as the rotor speed returns to its setpoint, as shown in the dashed line of Fig 4(a) and pointed
out in [7], [8]. For a constant wind speed and operating point, the initial height of the resulting power can
be used to quantify inertial response capability. For an exponential frequency drop the height is directly
related to the constant KD and is given by
∆P = ∆Tgen ωh
(7)
∗
∆fdrop λ vw
= KD
(8)
τ
R
A dip in power is undesirable, as the energy delivered to the system is demanded back. A description of
the effect can be found in [8], where the compensating reference shown in Fig 4(b) was proposed. Such
a reference adjusts the generator torque so that the post-event operating point rests at the point B shown
in Fig 1(b), delivering the same power at reduced speed ωlow . For an expected worst case frequency drop
∆f , the constant KP can be computed as noted at the end of this paper. There is a limit on the magnitude
5
ω
P
h
of KD that can be compensated; it must not cause a reduction in rotor speed to less than ωlow , or a drop
in power output will still occur.
Thus, two types of inertial response are possible. Using the reference in Fig 3(b), the energy associated
with slowing the rotor to ωmin could be delivered and then re-absorbed. Using the reference in Fig 4(b)
it is possible to deliver to the system the difference in kinetic energy between the speeds ωhigh and ωlow
without re-absorbing it.
Time
Time
(a) Simple (dashed) versus compensated (solid) response
(b) Proportional compensation
Fig. 4.
Compensation of simple inertial response. Fig 4(a) shows natural and compensated response. Fig 4(b) shows
compensating control reference proposed in [8].
B. Power system damping
The damping of oscillatory modes in the power system can be altered by adding control loops that alter
the terminal voltage or output power of a generator based on a measurement of network frequency or
voltage. Inter area modes involving the swinging of groups of generators against each other can range
from a frequency of 0.1 to 0.7 Hz [12], while higher frequency modes occur between single generators
and the system. Compensating the latter modes must be treated with great caution as they will interact
with a wind turbine’s own torsional modes and active dampers, which respond at frequencies between 2-5
Hz and higher. This study focuses on the slower inter-area modes. Only the size and frequency of power
demand is of interest to this study, so a simplified representation of such a service shown in Fig 5(b) was
used. In this case, damping capability is quantified as
∆P = A
IV. S TUDY
(9)
OF DYNAMIC CONSTRAINTS AND WIND VARIABILITY
To study dynamic limitations on operation, simulations are used to study operation at several wind speeds
in the operating range defined in by the static power limit 5. Power demands from the control references
shown in Fig 3(b), 4(b) and 5 are imposed separately on the single mass model. These references produce
speed and power variations according to the controlled wind turbine’s dynamics. The magnitude of the
demand is increased to the point where a power or rotor speed constraint is violated. In the case of the
inertial controllers, the value KD is altered. The frequency drop used was characterized by ∆f = 0.01p.u.
and τ = 6.3s. In the case of the power system damping reference, the amplitude A of the imposed sinusoid
is altered. Frequencies of 0.1, 0.2 and 0.5 were studied.
This procedure was first carried out for constant wind speeds to obtain a dynamic constraint. This
produced constrained results that reflect speed and power limits. The same procedure was then carried
6
(a) Excitation of inter-area mode and generator response
Fig. 5.
(b) Structure used to represent power system stabilizer
Damping of inter-area mode
out for an set of synthetic wind series centered on the mean speed. This produced constrained results
that also reflect the effect of wind variations. For each wind series, a different limiting value of KP or A
is found, giving rise to an ensemble of limiting values represented by the error bars in the results that
follow. In the case of inertial response, a frequency drop and recovery is triggered successively over the
period of simulation. In the case of power systems damping, the sinusoidal power demand is demanded
throughout the duration of simulation.
Synthetic wind series are generated as the superposition of a constant mean speed v w and N sinusoidal
components having amplitudes Ai and random phases ψi . The frequencies ωi and amplitudes Ai of these
N components are given by an empirically determined spectral density function [13], and each selection
of random phases ψi yields a unique series that possesses appropriate frequency content. One hundred
series were used, each having a duration of 60 minutes.
V. R ESULTS
A. Inertial response: Uncompensated
Fig 6 compares the static limit 5 on available power with the limit obtained from dynamic simulation for
a constant wind input. It is evident that at the lowest wind speeds, minimum rotor speed limits the largest
value of KD and consequently ∆P , through 8. At higher wind speeds, KD is instead limited by converter
rating. The relatively low power level associated with low wind speeds (e.g. approximately 0.1 p.u. at 6
m/s) means that much of the converter rating can be utilized.
Fig 6(b) shows the ∆P corresponding to the largest values of KD found when realistic wind variations
are allowed. Error bars indicate the standard deviation of values around a mean KD . They are bounded by
the constant wind result from Fig 6(a). The dependable use of inertial response is significantly degraded
before the static limit vhigh (for this case 8.8 m/s), where it shrinks to zero due to upper speed limit
violation. Figure 7 shows variable wind results for de-loadings of 2% and 10%. Larger de-loadings lose
viability at higher wind speeds due to their higher operating speed. However, higher speeds offer more
inertial energy. This suggests that de-loading should be adjusted to maximize capacity for inertial response
over the range of operating wind speeds.
B. Inertial response: Compensated
Fig 8(a) shows the ∆P corresponding to the largest KD that can be compensated without reducing
the rotor speed below ωlow , for a 2% de-loading. For wind speeds between 5 and 8 m/s, this KD was
achievable, but for a wind speed of 9 m/s, the static power limit was encountered.
7
dynamic
static
1
1
constant wind
variable wind
0.8
0.6
∆P
∆P
0.8
0.4
0.4
0.2
0
3
0.6
0.2
4
5
6
v (m/s)
7
8
9
0
3
4
w
(a) Limitations: static vs dynamic
5
6
vw (m/s)
7
8
9
(b) Dynamic constraints: constant vs variable wind
Inertial response for a de-loading of 2%. Largest ∆P for derivative reference.
Fig. 6.
10%
2%
optimal
1
∆P
0.8
0.6
0.4
0.2
0
4
5
6
v (m/s)
7
8
9
w
Fig. 7.
Inertial response for family of de-loadings.
For some wind-speed, ωlow equals ωmin . This defines the lower limit of operation for a compensated
inertial response that returns to the set-point power. Below this wind speed, a steady state at the point B
after a frequency drop ∆fdrop would violate speed constraints. Fig 8(b) shows the range of capacity that
can be obtained for a family of de-loadings. The vertical line bounding each result on the left marks the
speed where ωlow = ωmin , and that on the right marks vhigh .
Fig 9(a) shows that in the set of wind time series studied, there was a wide range of outcomes for the
largest possible value of KD . For some winds, it was possible to set KD to a value equal or greater than
the limit of compensation. There also existed many cases where a small or zero value for KD still resulted
in lower speed limits being violated. The results thus indicate that stable operation can not be guaranteed
for any wind speed when the proportional compensation shown in Fig 4(b) is used. The sensitivity of this
reference to wind variations can be explained by observed by considering the torque speed curves of Fig
9(b).
Before a frequency event, the generator curve has an intersection with the aerodynamic torque curve
associated with any wind speed, and therefore also a stable operating point. After a frequency drop ∆f ,
the derivative term of the reference in Fig 4(b) is zero, while the proportional term provides a constant
offset of the load torque, as indicated by the upper generator torque curve in Fig 9(b). It can then be the
8
static
dynamic
1
1
0.8
∆P
∆P
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
3
10 %
4%
2%
4
5
6
7
vw (m/s)
8
9
10
(a) Limitations: static vs dynamic
0
4
5
6
7
vw (m/s)
8
9
10
(b) Dynamic constraints for constant wind, family of de-loadings
Fig. 8. Compensated inertial response for a de-loading of 2%. Fig 8(a) shows the ∆P corresponding to the largest KD that can
be compensated. Fig 8(b) shows the limit for a family of de-ratings. Vertical bars indicate bounds on viable range of compensation.
variable wind
constant wind
1
∆P
0.8
0.6
0.4
0.2
0
3
4
5
6
7
v (m/s)
8
9
10
w
(a) Dynamic constraints: constant vs variable wind
(b) Torque speed diagram showing potential for instability with
proportional compensation reference
Fig. 9. Compensated inertial response: Fig 9(a) compares the largest possible ∆P available for variable winds against the limit
of compensation. Fig 9(b) illustrates sensitivity of proportional compensation to wind variability.
case that during a sufficiently large drop ∆vw in wind speed, no intersection of generator and aerodynamic
torque. If the rotor speed dropped below the value ωstall , it would continue to drop even if wind speed
returned to the original value vw . Alterations of the proportional scheme could easily lead, however, to
a compensation that is less sensitive. The levels of compensated inertial response indicated in Fig 8(b)
could then be exploited.
C. Power system damping
Limits on the amplitude of sinusoidal power variations available shown in Fig 10(a) and Fig 10(b) follow
the same general trend as for inertial response. A notable exception is that at mean wind speed 9 m/s,
maximum rotor speed is more restrictive than the upper power limit for the constant wind case. It is been
assumed that negative power flow through the full-rated converter is permitted. The possible variations
are quite large, and more than the percentage usually requested from generators.
9
Fig 10(c) shows that the lower the frequency of the variation demanded, the more restricted the capability.
It is evident that the limited inertia of the rotor results in speed variations that violate the lower speed limit.
As in the case of inertial response, it appears that large de-loading maximizes capability at low wind
speeds, while no de-loading increases it for high wind speeds.
variable wind
constant wind
1
1
0.8
0.8
∆P
∆P
dynamic
static
0.6
0.4
0.4
0.2
0.2
0
2
4
6
vw (m/s)
8
10
0
3
12
(a) Limitations: static vs dynamic
0.1 Hz
0.2 Hz
0.5 Hz
5
6
7
8
vw (m/s)
9
10
10 %
2%
optimal
1
0.8
∆P
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
3
4
(b) Dynamic constraints: constant vs variable wind
1
∆P
0.6
4
5
6
v (m/s)
7
8
9
0
3
w
(c) Family of frequency farea , optimal power
4
5
6
vw (m/s)
7
8
9
(d) Family of de-loadings
Fig. 10. Power systems damping response: largest limit on amplitude ∆P of response. All results for no de-loading (optimal
power) and farea = 0.1Hz, except for Fig 10(c), which shows the family of frequencies studied.
VI. C ONCLUSIONS
The capability of variable speed wind turbines to provide power variations for inertial response and power
system damping was evaluated at different wind speeds, and for different percentages of de-loading. Static
limits on the viable range of wind speeds and power outputs were determined, and dynamic limits were
studied using simulation of a single-mass model and simple generator torque references. Capability was
studied for the case of constant and variable wind inputs.
A substantial fraction of unit power rating appears to be available for such power variations. The range of
wind speeds where such services are viable is limited but would still correspond to a substantial fraction
of operation hours depending on the wind statistics of a site. The percentage of de-loading produces
considerable variation in performance and should be taken into account to maximize stabilization capability.
10
Inertial response using a simple derivative reference was found to be stable for variable wind speeds
and capable of providing between 0.1 and 0.7 p.u. of response. The addition of a proportional reference
was found to cause instability for realistic wind variations. De-loading is not necessary to provide a simple
inertial response that delivers and then re-absorbs energy, but is necessary to provide a compensated
response that delivers net positive energy. Large variations in output power between 0.2 and 0.5 p.u.
appear to be possible for power system damping of frequencies in the inter area mode range. De-loading
is not necessary to provide damping, but it increases the largest possible amplitude. Both speed and power
constraints limited the size of such variations, with lower frequencies experiencing greater limitations.
VII. T ECHNICAL S PECIFICATIONS
Power rating: 1500kW
Blade diameter: 35.25m
Rotor inertia (low speed side): Jh : 4.2e6kgm2
Control Parameters
Proportional gain KP = 21 ρACp (λ∗ )R3 ωh2
1
λ∗
−
1
λu
/∆f
where ∆f is the worst case frequency drop, and λ∗ and λu are as previously defined in Section II.A.
R EFERENCES
[1] M. O. G. Lalor, A. Mullane, “Frequency control and wind turbine technologies,” IEEE Transactions on Power Systems, vol. 20,
no. 40, pp. 1905–1913, November 2005.
[2] E. Hagstrom, I. Norheim, and K. Uhlen, “Large-scale wind power integration in Norway and impact on damping in the Nordic
grid,” Wind Energy, vol. 8, no. 3, pp. 375–384, 2005.
[3] J. Slootweg and W. Kling, “The impact of large scale wind power generation on power system oscillations,” Electric Power
Systems Research, vol. 67, no. 1, pp. 9–20, 2003.
[4] F. Hughes, O. Anaya-Lara, N. Jenkins, and G. Strbac, “A power system stabilizer for DFIG-based wind generation,” IEEE
Transactions on Power Systems, vol. 21, no. 2, pp. 763–772, 2006.
[5] C. Jauch, “Transient and dynamic control of a variable speed wind turbine with synchronous generator,” Wind Energy, vol. 10,
no. 3, pp. 247–269, May/June 2007.
[6] F. Hughes, O. Anaya-Lara, N. Jenkins, and G. Strbac, “Control of DFIG-based wind generation for power network support,”
IEEE Transactions on Power Systems, vol. 20, no. 4, pp. 1958–1966, 2005.
[7] J. Morren, S. W. H. de Haan, W. L. Kling, and J. Ferreira, “Wind turbines emulating inertia and supporting primary frequency
control,” IEEE Transactions on Power Systems, vol. 21, no. 1, pp. 433–434, Februrary 2006.
[8] G. Ramtharan, J. Ekanayake, and N. Jenkins, “Frequency support from doubly fed induction generator wind turbines,” IET
Renewable Power Generation, vol. 1, no. 1, pp. 3–9, 2007.
[9] N. Miller, W. Price, and J. Sanchez-Gasca, “Dynamic modelling of ge 1.5 and 3.6 mw wind-turbine generators,” Version 3.0
Technical Report, 2003.
[10] R. Poore and T. Lettenmaier, “Alternative design study report: Windpact advanced wind turbine drive train designs study,”
Subcontractor Report, NREL/SR-500-33196, August 2003.
[11] J. Ekanayake and N. Jenkins, “Comparison of the response of doubly fed and fixed-speed induction generator wind turbines
to changes in network frequency,” IEEE Transactions on Energy Conversion, vol. 19, no. 4, pp. 800–802, 2004.
[12] P. Kundur, Power System Stability and Control. McGraw-Hill, 1997.
[13] C. Nichita, D. Luca, B. Dakyo, and E. Ceanga, “Large band simulation of the wind speed for real time wind turbine simulators,”
IEEE Transactions on Energy Conversion, vol. 17, no. 4, pp. 523–529, December 2002.