AckermannTh. (ed) Wind Power in Power Systems
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So far, the external AC power system, and in many cases even the generator itself, has
traditionally not been represented very accurately, probably because of the fact that the
mechanical and aerodynamic properties have been much more decisive for the design.
For traditional fixed-speed wi nd turbines with induction generators it has been an
acceptable approximation to represent the generator and, consequently, the external
AC power system by a simple speed-dependent generator air gap torque. However, with
the controllability of variable-speed wind turbines and the available choices of generator
control strategies (power or speed control; see Section 24.4.5.2), the option of choosing a
control strategy will have to be taken into consideration as well.
24.6.2.5 Flicker investigations
In the everyday operation of all electric power systems there are all kinds of small
disturbances, whenever loads are switched on or off, or whenever the operating point of
any equipment is altered. All such small disturbances result in small changes in the
voltage amplitude. If the short-circuit capacity of the power system is sufficiently high
compared with the disturbance, or if the frequencies of the disturbances are outside the
human visual susceptibility range, the disturbances are mostly without significance.
However, in many cases they are not, and these disturbances are referred to as flicker.
There are standardised methods of measuring flicker (IEC 61000-4-15; see IEC, 1997)
as well as recomm endations for maximum allowed flicker levels (IEC 61000-3-7; see
IEC, 1996).
Most flicker is caused by discrete events, such as equipment being switched on or off
(e.g. when motors start and capacitor banks switch in or out). But flicker can also be
caused by periodic disturbances of any kind, and the tower shadow (i.e. the brief
reduction in the shaft torque every time a blade is in the lee in front of the tower) is
exactly such a periodic disturbance. This is often denoted as the 3p effect, because it is
three times higher than the rotational speed of the wind turbine. If many wi nd turbines
are grid-connected close to each other the cumulative effect may become large enough to
cause flicker problems. Section 25.5.3 describes how the tower shadow can be included
in a wind turbine model, and an empirical model of this periodic nature of the wind
turbine output is described in Akhmatov and Knudsen (1999a), in which it is also
outlined how dynamic stability simulations with a customised wind turbine flicker
model can be used to evaluate the expected flicker contribution from wind turbines in
areas with high wind power penetration.
It may be added that as wind turbines gro w large r and , consequently, have a lower
rotational speed [see Equation (24.8)] the flicker contribution from the 3p effect will
occur at lower and lower frequencies, where the recommended flicker limits are
higher. This fact will in itself mitigate possible flicker problems caused by the tower
shadow.
Wind power also contributes to flicker through wind gusts, turbulence and random
variations in incoming wind. These changes in the wind cause changes in the instant-
aneous power production of a wind turbine, which, in turn, will alter the instantaneous
voltage and thereby contribute to the flicker level. A possible way of representing this is
shown in Section 25.5.2.
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24.6.2.6 Load flow and short-circuit calculations
Load-flow (LF) calculations require knowledge of the basic electrical properties of the
power system, such as line and transformer impeda nces and transformer tap-changer
data, in addition to operational data of the power system, such as location and size of
momentary power consumption and production, voltage (or reactive power) set points
of voltage controlling devices, and transformer settings. Short-circuit (SC) calculations
require basically the same information but must also include generator impedances. All
these data are comparatively simple to obtain. However, the amount of data required in
the case of larger networks may be substantial.
LF and SC calculations are usually performed with use of specia l programs, that are
customised for the task. There is a large number of programs available for that purpose
(e.g. NEPLAN, Integral, PSS/ E
TM
, Simpow
TM
, PowerFactory and CYMFLOW and
CYMFAULT, to mention just a few). LF and SC calculations both assume a steady
state (i.e. all time derivatives are considered zero). LF calculations are used to calculate
the voltages in the nodes and the flow of active and reactive power in the lines between the
nodes of the power system in various points of operation. SC calculations are used to
calculate the level of fault currents in the case of short circuits anywhere in the power
system.
In the case of LF calculations, it is necessary only to represent the input of active and
reactive power, P and Q. Wind turbines with induction generators (Type A) must be
represented with an induction machine, with the active power P set to the momentary
value. Based on the machi ne impedance, the LF program will then calculate the reactive
power consumption Q corresponding to the terminal voltage. (A few LF programs do
not have an inbuilt induction generator model, in which case it is necessary to apply
a PQ representation with the Q value adjusted to an appropriate value. A user-defined
macro may be very convenient for that, if such an option is available in the program.)
Variable-speed wind turbines, however, have reactive power controllability. Therefore,
the most suitable representat ion is a fixed PQ representation or, if the wind turbine is set
to voltage control, a PV representation.
In the case of SC calculations, the necessary wind turbine representation will depend
on the specific generator technology applied. In general, the SC representation must
include the parts that will provide support to the SC current, if there is a short circuit
somewhere in the vicinity of the wind turbine. For fixed-speed wind turbines with
induction generators (Types A and B) the machine itself is an appropriate representa-
tion. For variable-speed wind turbines with doubly-fed induction generators (Type C)
the machine itself must be represented, whereas the smaller partial load converter often
can be neglected. For variable-speed wind turbines with full-load converters (Type D)
the generator is decoupled from the AC network and only the converter is directly
connected to the grid; consequently, the appropriate SC representation must be a
converter of suitable type and ratin g.
24.7 Conclusions
In this chapter we provided a general overview of aspects related to wind turbine
modelling and computer simulations of electrical systems with wind turbines.
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We provided an overview of wind turbine aerodynamic modelling, together with basic
descriptions of how to represent the physical properties of a wind turbine in different
mathematical models.
In order to arrive at a better understanding of wind turbine models in general, a
generic wind turbine model was shown, where the various independent elements of a
wind turbine and their interaction with each other were explained.
We presented a general overview of p.u. systems, with special emphasis on p.u.
systems for the mechanical system of a wind turbine. Typical mechanical data for a
contemporary sized wind turbine were given, and the conversion of these data from
physical units to p.u. was presented in detail.
Finally, we gave an overview of various types of simulation and discussed what to
include in a wind turbine model for each specific type of simulation.
References
[1] Aagaard Madsen, H. (1991) ‘Aerodynamics and Structural Dynamics of a Horizontal Axis Wind
Turbine: Raw Data Overview’, Risø National Laboratory, Roskilde, Denmark.
[2] Akhmatov, V. (1999) ‘Development of Dynamic Wind Turbine Models in Electric Power Supply’,
MSc thesis, Department of Electric Power Engineering, Technical University of Denmark, [in Danish].
[3] Akhmatov, V. (2002) ‘Variable Speed Wind Turbines with Doubly-fed Induction Generators, Part I:
Modelling in Dynamic Simulation Tools’, Wind Engineering 26(2) 85–107.
[4] Akhmatov, V. (2003) Analysis of Dynamic Behaviour of Electric Power Systems with Large Amount of
Wind Power, PhD thesis, Electric Power Engineering, Ørsted-DTU, Technical University of Denmark,
Denmark.
[5] Akhmatov, V., Knudsen, H. (1999a) ‘Dynamic modelling of windmills’, conference paper, IPST’99 –
International Power System Transients, Budapest, Hungary.
[6] Akhmatov, V., Knudsen, H. (1999b) ‘Modelling of Windmill Induction Generators in Dynamic Simula-
tion Programs’, conference paper BPT99-243, IEEE Budapest Power Tech ’99, Budapest, Hungary.
[7] Bachmann, U., Erlich, I., Grebe, E. (1999) ‘Analysis of Interarea Oscillations in the European Electric
Power System in Synchronous Parallel Operation with the Central-European Networks’, conference
paper BPT99-070, IEEE Budapest Power Tech ’99, Budapest, Hungary.
[8] Bruntt, M., Havsager, J., Knudsen, H. (1999) ‘Incorporation of Wind Power in the East Danish Power
System’, conference paper BPT99-202, IEEE Budapest Power Tech ’99, Budapest, Hungary.
[9] Christensen, T., Pedersen, J., Plougmand, L. B., Mogensen, S., Sterndorf, H. (2001) ‘FLEX5 for Offshore
Environments’, paper PG 5.8, EWEC European Wind Energy Conference 2001, Copenhagen, Denmark.
[10] Freris, L. (1990) Wind Energy Conversion Systems, Prentice Hall, New York.
[11] Gasch, R., Twele, J. (2002) Wind Power Plants, James & James, London, UK.
[12] Hansen, M. O. L. (2000) Aerodynamics of Wind Turbines: Rotors, Loads and Structure, James & James,
London, UK.
[13] Heier, S. (1996) Windkraftanlagen im Netzbetrieb, 2nd edn, Teubner, Stuttgart, Germany.
[14] Hinrichsen, E. N. (1984) ‘Controls for Variable Pitch Wind Turbine Generators’, IEEE Transactions on
Power Apparatus and Systems PAS-103(4) 886–892.
[15] IEC (International Electrotechnical Commission) (1996) ‘EMC-Part 3: Limits – Section 7: Assessment of
Emission Limits for Fluctuating Loads in MV and HV Power Systems – Basic EMC Publication.
(Technical Report)’, (IEC 61000-3-7, IEC), Geneva, Switzerland.
[16] IEC (International Electrotechnical Commission) (1997) ‘EMC, Part 4: Testing and Measurement Tech-
niques – Section 15: Flickermeter – Functional and Design Specifications’, IEC 61000-4-15, IEC, Geneva,
Switzerland.
[17] Johansen, J. (1999) ‘Unsteady Airfoil Flows with Application to Aeroelastic Stability’, Risø National
Laboratory, Roskilde, Denmark.
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[18] Knudsen, H., Akhmatov, V. (1999) ‘Induction Generator Models in Dynamic Simulation Tools’,
conference paper, IPST’99, International Power System Transients, Budapest, Hungary.
[19] Kundur, P. (1967) Digital Simulation and Analysis of Power System Dynamic Performance, PhD thesis,
Department of Electrical Engineering, University of Toronto, Canada.
[20] Kundur, P. (1994) Power System Stability and Control, EPRI, McGraw-Hill, New York.
[21] Noroozian, M., Knudsen, H., Bruntt, M. (2000) ‘Improving a Wind Farm Performance by Reactive
Power Compensation’, in Proceedings of the IASTED International Conference on Power and Energy
Systems, September 2000, Marbella, Spain, pp. 437–443.
[22] Øye, S. (1986) ‘Unsteady Wake Effects Caused by Pitch-angle Changes’, IEA R&D WECS Joint Action
on Aerodynamics of Wind Turbines, 1st symposium, London, UK, pp. 58–79.
[23] Paserba, J. (1996) ‘Analysis and Control of Power System Oscillations’, technical brochure, July 1996,
Cigr
´
e, Paris, France.
[24] Persson, J., Slootweg, J. G., Rouco, L., So
¨
der, L., Kling, W. L. (2003) ‘A Comparison of Eigenvalues
Obtained with Two Dynamic Simulation Software Packages’, conference paper BPT03-254, IEEE
Bologna Power Tech, Bologna, Italy, 23–26 June 2003.
[25] Slootweg J. G., Polinder H., Kling W. L. (2002) ‘Reduced Order Models of Actual Wind Turbine
Concepts’, presented at IEEE Young Researchers Symposium, 7–8 February 2002, Leuven, Belgium.
[26] Snel, H., Lindenburg, C. (1990) ‘Aeroelastic Rotor System Code for Horizontal Axis Wind Turbines:
Phatas II, European Community Wind Energy Conference, Madrid, Spain, pp. 284–290.
[27] Snel, H., Schepers, J. G. (1995) ‘Joint Investigation of Dynamic Inflow Effects and Implementation of an
Engineering Method’, Netherlands Energy Research Foundation, ECN, Petten, The Netherlands.
[28] Sørensen, J. N., Kock, C. W. (1995) ‘A Model for Unsteady Rotor Aerodynamics’, Wind Engineering and
Industrial Aerodynamics Vol. 58, 259–275.
[29] Walker, J. F., Jenkins, N. (1997) Wind Energy Technology, John Wiley & Sons Ltd, London, UK.
554 The Modelling of Wind Turbines
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25
Reduced-order Modelling of
Wind Turbines
J. G. Slootweg, H. Polinder and W. L. Kling
25.1 Introduction
In most countries and power systems, wind power generation covers only a small part of
the total power system load. There is, however, a tendency towards increasing the
amount of electricity generated from wind turbines. Therefore, wind turbines will start
to replace the output of conventional synchronous generators and thus the penetration
level of wind power in electrical power systems will increase. As a result, it may begin to
affect the overall behaviour of the power system. The impact of wind power on the
dynamics of power systems should therefore be investigated thoroughly in order to
identify potential problems and to develop measures to mitigate such problems.
In this chapter a specific simulation approach is used to study the dynamics of large-
scale power systems. This approach will hereafter be referred to as power system
dynamics simulation (PSDS). It was discussed in detai l in Chapter 24; therefore this
chapter will include only a short introduction. It is necessary to incorporate models of
wind turbine generating systems into the software packages that are used for PSDS in
order to analyse the impact of high wind power penetration on electrical power systems.
These models have to match the assumptions and simplifications applied in this type of
simulation.
This chapter presents models that fulfil these requirements and can be used to
represent wind turbines in PSDSs. After briefly introducing PSDS, the three main wi nd
turbine types will be described. Then, the modelling assumptions are given. The models
were required to be applicab le in PSDSs. Many of the presented assumptions are a result
of this requirement. Then, models of the various subsystems of each of the most
Wind Power in Power Systems Edited by T. Ackermann
Ó 2005 John Wiley & Sons, Ltd ISBN: 0-470-85508-8 (HB)
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important current wind turbine types are given. Finally, the model’s response to a
measured wind speed sequence will be compared with actual measurements.
25.2 Power System Dynamics Simulation
PSDS software is used to investigate the dynamic behaviour and small signal stability of
power systems. A large power system can easily have hundreds or even thousands of state
variables. These are associated with branches, with generators and their controllers and, if
dynamic rather than static load models are used, with loads. If the relevant aspect of power
system behaviour is characterised by rather low frequencies or long time constants, long
simulation runs are required. A simulation run of sufficient length would, however, take a
substantial amount of computation time if all fast, high-frequency, phenomena were
included in the applied models. If the investigated power system is large it will be difficult
and time consuming to study different scenarios and setups.
To solve this problem, only the fundamental frequency component of voltages and
currents is taken into account when studying low-frequency phenomena. In this chapter,
this approach will hereafter be referred to as PSDS. It is also known as fundamental
frequency or electromechanical transient simulation . First, this approach makes it
possible to represent the network by a constant impedance or admittance matrix, similar
to load-flow calculations. The equations associated with the network can thus be solved
by using load-flow solution methods, the optimisation of which has been studied
extensively and which can hence be implemented very efficiently, resulting in short
computation times. Second, this approach reduces the computation time by cancelling
a number of differential equations, because no diff erential equations are associated with
the network and only a few are associated with generating equipment. This approach
also makes it possible to us e a larger simulation time step (Kundur, 1994). This way,
reduced-order models of the components of a power system are developed.
Examples of PSDS packages are PSS/E
TM
(Power System Simulator for Engineers)
and Eurostag.
(1)
This type of so ftware can be used when the phenomena of interest have
a frequency of about 0.1–10 Hz. The typical problems that are studied when using these
programs are voltage and angle stability. If the relevant frequencies are higher, instant-
aneous value simulation (IVS) software, such as the Alternative Transient Program
(ATP), an electromagnetic transient program (EMTP) or SimPowerSystems from
Matlab must be used.
(2)
These packages contain more de tailed and higher-order equip-
ment models than do PSDS software. However, their time step is much smaller and it is
hardly feasible to simulate a large-scale power system in these programs.
Some advanced software packages, such as PowerFactory, Netomac
TM
and
Simpow
TM
, offer both dynamic and instantaneous value modes of simulation.
(3)
They
may even be able automatically to switch between the dynamic and instantaneous value
domains, each with their own models, depending on the simulated event and/or the
user’s preferences. IVSs are also referred to as electromagnetic transient simulations.
(1)
PSS/E
TM
is a trademark of Shaw Power Technologies, Inc.
(2)
Matlab
TM
is a registered trademark.
(3)
Netomac
TM
is a trademark of Siemens; Simpow
TM
is a trademark of STRI AB.
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For more information on the different types of power system simulation and the type of
problems that can be studied with them, see Chapter 24.
25.3 Current Wind Turbine Types
The vast majority of wind turbines that are currently installed uses one of the three main
types of electromechanical conversion system. The first type uses an (asynchronous)
squirrel cage induction generator to convert the mechanical energy into electricity. Owing
to the different operating speeds of the wind turbine rotor and the generator, a gearbox is
necessary to match these speeds (wind turbine Type A0).
(4)
The generator slip varies slightly
with the amount of generated power and is therefore not entirely constant. However,
because these speed variations are in the order of 1 %, this wind turbine type is normally
referred to as a constant-speed or fixed-speed turbine. Today, the Danish, or constant-
speed, design is nearly always combined with a stall control of the aerodynamic power,
although in the past pitch-controlled constant-speed wind turbine types were also built.
The second type uses a doubly fed induction generator instead of a squirrel cage
induction generator (Type C). Similar to the first type, it needs a gearbox. The stator
winding of the generator is coupled to the grid, and the rotor winding to a power
electronics converter, which today usually is a back-to-back voltage source converter
(VSC) with current control loops. This way, the electrical and mechanical rotor fre-
quencies are decoupled, because the power electronics converter compensates the dif-
ference between mechanical and electrical frequency by injecting a rotor current with a
variable frequency. Thus, variable-speed operation becomes possible (i.e. control of the
mechanical rotor speed according to a certain goal function, such as energy yield
maximisation or noise minimisation is possible). The rotor speed is controlled by
changing the generator power in such a way that it equals the value as derived from
the goal function. In this type of conversion system, the required control of the aero-
dynamic power is normally performed by pitch control.
The third type is called a direct-drive wind turbine as it works without a gearbox (Type
D). A low-speed multipole synchronous ring generator with the same rotational speed as
the wind turbi ne rotor is used to convert the mechanical energy into electricity. The
generator can have either a wound rotor or a rotor with permanent magnets. The stator
is not coupled directly to the grid but to a power electronics converter. This may consist
of a back-to-back VSC or a diode rectifier with a single VSC. By using the electronic
converter, variable-speed operation becomes possible. Power limitation is again achieved
by pitch control, as in the previously mentioned type. The three main wind turbine types
are depicted in Figure 19.2 (page 421) and are discussed in more detail in Chapter 4.
25.4 Modelling Assumptions
This section discusses the assumptions on which the models in this chapter are based.
First, in order to put a reasonable limit on data requirements and computation time, a
(4)
For definitions of wind turbine Types A–D, see Section 4.2.3.
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quasistatic approach is used to describe the rotor of the wind turbine. An algebraic
relation between the wind speed at hub height and the mechanical power extracted from
the wind is assumed. More advanced methods, such as the blade element impulse
method, require a detailed knowledge of aerodynamics and of the turbine’s blade profile
characteristics (Heier, 1998; Patel, 2000). In many cases, these data will not be available
or, in preliminary studies, a decision regarding the wind turbine type may not have been
taken and, consequently, the blade profile will of course not be known. However, this is
not considered a problem , because the grid interaction is the main topic of interest in
power system dynamics studies, and the impact of the blade profile on the grid interaction
can be assumed to be rather limited.
This chapter presents models for representing wind turbines in PSDSs. The models
have to comply with the requirements that result from the principles on which PSDS
software is based, as described in Section 25.2. We first make the following assumptions
that ap ply to the model of each wind turbine type:
.
Assumption 1: magnetic saturation is neglected.
.
Assumption 2: flux distribution is sinusoidal.
.
Assumption 3: any losses, apart from copper losses, are neglected.
.
Assumption 4: stator voltages and currents are sinusoidal at the fundame ntal
frequency.
Furthermore, the following assumptions apply to some of the analysed systems:
.
Assumption (a): in both variable-speed systems (Types C and D), all rotating mass is
represented by one element, which means that a so-called ‘lumped-mass’ representa-
tion is used.
.
Assumption (b): in both variable-speed systems, the VSC s with current control loops
are modelled as current sources.
.
Assumption (c): in the system based on the doubly fed induction generator (Type C),
rotor voltages and currents are sinusoidal at the slip frequency.
.
Assumption (d): in the direct drive wind turbine (Type D), the synchronous generator
has no damper windings.
.
Assumption (e): when a diode rectifier is used in the direct drive wind turbine,
commutation is neglected.
Assumption (a) is made because the mechanical and electrical properties of variable-
speed wind turbines are decoupled by the power electronic converters. Therefore, the
shaft properties are hardly reflected in the grid interaction, which is the main point of
interest in power system studies (Kru
¨
ger and Andresen, 2001; Petru and Thiringer,
2000).
Assumptions (b) and (c) are made in order to model power electronics in PSD S
simulation software and are routinely applied in power system dynamics simulations
(Fujimitsu et al., 2000; Hatziargyriou, 2001; Kundur, 1994). A full converter model
would require a substantial reduction of the simulation time step and the incorporation
of higher harmonics in the network equations. This is not in agreement with the
intended use of the models described here.
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Assumption (d) is made because the damper windings in the synchronous generator
do not have to be taken into account. When a back-to-back VSC is used, generator
speed is controlled by the power electronic converter, which will prevent oscillations.
When a dio de rectifier is used, damper windings are essential for commutation. How-
ever, commutation is neglected according to Assumption (e), and therefore the damper
windings will be neglected as well.
25.5 Model of a Constant-speed Wind Turbine
25.5.1 Model structure
Figure 25.1 depicts the general structure of a model of a constant-spe ed wind turbine.
This general struc ture consists of models of the most important subsystems of this wind
turbine type, namely, the rotor, the drive train and the generator, combined with a wind
speed model. Each of the blocks in Figure 25.1 will now be discussed separately, except
for the grid model. The grid model is a conventional load-flow model of a network . The
topic of load-flow mo dels and calculations is outside the scope of this chapter and the
grid model is therefore not treated any further. Instead, the reader is referred to text-
books on this topic (e.g. Grainger and Stevenson, 1994).
25.5.2 Wind speed model
The output of the first block in Figure 25.1 is a wind speed sequence. One approach to
model a wind speed sequence is to use measurements. The advantage of this is that a
‘real’ wind speed is used to simulate the performance of the wind turbine. The dis-
advantage is, however, that only wind speed sequences that have already been measured
can be simulated. If a wind speed sequence with a certain wind speed range or turbu-
lence intensity is to be simulated, but no measurements that meet the required char-
acteristics are available, it is not possible to carry out the simulation.
A more flexible approach is to use a wind speed model that can generate wind speed
sequences with characteristics to be chosen by the user. This makes it possible to
simulate a wind speed sequence with the desired characteristics, by setting the value of
the corresponding parameters to an appropriate value. In the literature concerning the
simulation of wind power in electrical power systems, it is often assumed that the wind
Wind
speed
model or
measured
sequence
Wind
speed
Rotor
model
Mechanical
power
Mechanical
power
Shaft
model
Rotational
speed
Squirrel
cage
induction
generator
model
Active
and
reactive
power
Voltage
and
frequency
Funda-
mental
frequency
grid
model
Figure 25.1 General structure of constant-speed wind turbine model
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speed is made up by the sum of the following four components (Anderson and Bose,
1983; Wasynczuk, Man and Sullivan, 1981):
.
the average value;
.
a ramp component, representing a steady increase in wind speed;
.
a gust component, representing a gust;
.
a component repres enting turbulence.
This leads to the following equation:
w
ðtÞ¼
wa
þ
wr
ðtÞþ
wg
ðtÞþ
wt
ðtÞ; ð25:1Þ
in which v
w
(t) is the wind speed at time t; v
wa
is the average value of the wind speed;
v
wr
(t) is the ramp component; v
wg
(t) is the gust component; and v
wt
(t) is the turbulence
component. The wind speed components are all in ‘metres per second’, and time t is in
seconds. We will now discuss the equations used in this chapter to describe the various
components of Equation (25.1).
The average value of the wind speed, v
wa
, is calculated from the power generated in
the load-flow case combined with the nominal power of the turbine. Hence, the user
does not need to specify this value (Slootweg, Polinder and Kling, 2001). An exception is
a variable-speed wind turbine with pitch control at nominal power. In this case, there is
no unique relation between the generated power, indicated in the load-flow case, and the
wind speed. Therefore, the user must give an initial value either of the wind speed or of
the pitch angle. The equations describing the rotor, which will be discussed below, can
then be used to calculate the pitch angle or the average wind speed, respectively.
The wind speed ramp is characterised by three parameters – the amplitude of the wind
speed ramp,
^
AA
r
(in m/s), the starting time of the wind speed ramp T
sr
(in seconds), and
the end time of the wind speed ramp, T
er
(in seconds). The wind speed ramp is described
by the following equation:
t < T
sr
; for
wr
¼ 0;
T
sr
t T
er
; for
wr
¼
^
AA
r
ðt T
sr
Þ
ðT
er
T
sr
Þ
;
T
eg
< t; for
wg
¼
^
AA
r
:
9
>
>
>
=
>
>
>
;
ð25:2Þ
The wind speed gust is characterised by three parameters – the amplitude of the wind
speed gust,
^
AA
g
(in m/s), the starting time of the wind speed gust, T
sg
(in seconds), and the
end time of the wind speed gust T
eg
(in seconds). The wind gust is modelled using the
following equation (Anderson and Bose, 1983; Wasynczuk, Man and Sullivan, 1981):
t < T
sg
; for
wg
¼ 0;
T
sg
t T
eg
; for
wg
¼
^
AA
g
1 cos 2
t T
sg
T
eg
T
sg
;
T
eg
< t; for
wg
¼ 0:
9
>
>
>
=
>
>
>
;
ð25:3Þ
560 Reduced-order Modelling of Wind Turbines