Advanced simulation of windmills in the electric power supply

Electrical Power and Energy Systems 22 (2000) 421–434 www.elsevier.com/locate/ijepes

Advanced simulation of windmills in the electric power supply V. Akhmatov a,*, H. Knudsen a, A.H. Nielsen b a

b

Transmission Planning, NESA A/S, Hagedornsvej 4, DK-2820 Gentofte, Denmark Department of Electric Power Engineering, Technical University of Denmark, Build. 325, DK-2800 Lyngby, Denmark

Abstract An advanced model of a grid-connected windmill is set up where the windmill is a complex electro-mechanical system. The windmill model is implemented as a standardised component in the dynamic simulation tool, PSS/E, which makes it possible to investigate dynamic behaviour of grid-connected windmills as a part of realistic electrical grid models. This means an arbitrary number of wind farms or single windmills within an arbitrary network configuration. The windmill model may be applied to the study of electric power system stability and of power quality as well. It is found that a grid-connected windmill operates as a low-pass filter, whereby the two following observations are made: (1) interaction between the electrical grid and the mechanical systems of grid-connected windmills is given by a low frequency oscillation as the result of disturbances in the electric grid; (2) flicker, which is commonly explained by the dynamic wind variation, may also be caused by mechanical eigenswings in the windmill mechanical construction. 䉷 2000 Elsevier Science Ltd. All rights reserved. Keywords: Modelling of grid-connected windmills; Frequency-response characteristics; Mechanical eigenswings; Flicker; Electric power system stability; Power oscillations; Dynamic simulation tool PSS/E

1. Introduction The implementation of wind power in Denmark is developing faster than expected. The amount of energy produced by wind power in Denmark is still increasing, and by the end of 1999 it is 10%, a level which was not expected to be reached before 2005. Until now a few medium-size wind farms (for example Rejsby Hede of 24 MW) and offshore wind farms (sea settings just south of the island of Lolland, Tunø Knob etc.) have been established in Denmark. The rest of the wind power comes from single windmills or small groups of them. On Zealand the construction of three large offshore wind farms each of 150 MW capacity has been announced. These will be connected to the power grid, and the 450 MW will supply 1/6 of the power consumption on Zealand at maximum load (on a winter day). There is a wish to supply 50% of the total electric energy consumption in Denmark from wind power by the year 2030. In several countries, ambitious plans for development of wind power in the electric power system have been announced or work has begun on them [1]. Due to the environmental and economic aspects, wind power is especially promising in the developing countries where the electric energy consumption is still increasing [2]. The location of windmills is almost exclusively based on wind properties, and matters concerning the power grid are * Corresponding author.

given little consideration. Frequently, areas with good wind resources are geographically far from the consumers, and are in regions where the power grid is relatively weak. The fact that the power grid is weak in the areas where the windmills are placed leads to two technical problems: 1. The reliability of electric power delivery from the windmills to the customers when a fault occurs in the grid. 2. Achieving delivery of quality power. The reliability of electric power supply is no major problem today because most of the electric power comes from large conventional thermal power plants. When a fault occurs in the power grid the windmills are disconnected and the delivery of electric power is left to the conventional power plants. As the amount of electric power supply coming from windmills increases, the need for the conventional power plants will reduce and some of them will be removed from the grid. Then the power reserve, which is necessary for compensating the power loss in case of tripping the windmills when a fault occurs, will no longer be available. The islanding of windmills during handling of faults in the power grid can, for example, result in the loss of 450 MW on Zealand, every time the offshore wind farms have to be disconnected during a fault in the grid. Problems with power quality have already appeared with wind power. Voltage flicker has been observed at customers connected to the grid close to the windmills, e.g. on the island Lolland south of Zealand. As the amount of electric

0142-0615/00/$ - see front matter 䉷 2000 Elsevier Science Ltd. All rights reserved. PII: S0142-061 5(00)00007-7

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Fig. 1. Physical model of a stall controlled windmill.

power derived from wind power increases and that from conventional power plants decreases, we expect that the flicker problem will increase. We believe that the solution of these technical problems, which is a great challenge for the electric power sector, depends on the development of realistic physical models of windmills. The models must be based upon a relevant physical description of the windmill, where the windmill is looked upon as a complex electromechanical system. We present below a physical windmill model that can be used both for investigation of flicker in a grid with windmills in normal operation and for stability analysis when faults occur in the gird. We compare our model with the conventional ones currently used in electric power analysis. The conventional windmill models commonly consist of a dynamic induction generator model, in which the windmill mechanical system is given by a lumped mass. These models do not include the same detailed description of the dynamic behaviour inside the windmill and, therefore, lead to less accurate results for electric power analysis of grids that use wind power. The simulation results as well the measurements presented below are given for a specific Danish mediumsized windmill. Appendix A contains the technical data for this windmill.

2. The windmill model in electric power supply studies The basic components of the physical windmill model are shown in Fig. 1. The windmill model consists of only the components that are necessary and sufficient for the reasonable description of windmills in electric power system studies. At NESA Transmission Planning, where the stability studies for the electric power system of Zealand with grid-connected wind farms are taking place, the physical windmill model has been implemented in the dynamic simulation tool, PSS/E.

The physical windmill model is set up using a modular representation where each block relates to a physical phenomenon or to a physical component in a windmill. 2.1. Wind power available In this block, the total mechanical power available from the wind at the speed v on the swept area of the rotating mill A ˆ pR2 is given in accordance with [1]: PWIND ˆ

1 2

rAIR Av3 ;

…1†

where the air density, r AIR, is 1.225 kg/m 3. It is possible to implement a dynamic model for the wind speed applied to the rotating mill if it is desired. 2.2. Power coefficient versus tip speed ratio The power coefficient of the rotating mill, Cp, is given by the Cp – l -curve, where l ˆ …vMILL R=v† is the tip speed ratio. The dependency of the power coefficient on the tip speed ratio, Cp(l ), defines the aerodynamic, mechanical efficiency of the wind turbine and establishes the feedback in the physical windmill model through the rotational mill speed v MILL. Where the pitch or the active stall control is wished, the Cp – l -curve will be replaced with the dependency of Cp – l – a , where a denotes the pitch angle. 2.3. Transients in wind turbine blades The transients in the mechanical system of a stall controlled wind turbine are caused by a number of phenomena that are due to the aerodynamic behaviour of the air at the surface of the blades. The transients are due to: 1. the building-up of the boundary layer around the blades as the wind speed changes; 2. the rotational mill speed delaying with respect to the changing wind speed;

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The contributions from the mechanical eigenswings are systematic and, therefore, they may be separated from the stochastic contributions from the varying wind speed. The mechanical power of the wind turbine disturbed by the mechanical eigenswings with the systematic contributions may be expressed by: ! ! 3 2 X X PMECH …t† ˆ PMILL 1 ⫹ An anm gnm …t† hn …t† ; …2† nˆ1

gnm ˆ sin

Zt 0

mˆ1

 mvn …t 0 † dt 0 ⫹ wnm ;

where the average value of the mechanical power is PMILL ˆ PWIND CP …l†; Fig. 2. The windmill response to a stepwise change of the wind speed.

3. reaching the flapping moment, when the blades have bent to the correct position in accordance with the changing rotational mill speed [3], etc. as an iterative process. With respect to the description of the transients caused by the aerodynamic phenomena around the blades of a stall controlled wind turbine, one may take into account that there is a 1st-order differential equation between the dynamic wind speed applied to the stall controlled wind turbine and its mechanical power. We show the simulated responses of the mechanical mill power, the mill speed, the generator rotor speed and the electric power of a medium-sized windmill to a stepwise change of the wind speed in Fig. 2. The windmill response to the change in wind speed is given by an overshoot and a slow decrease in the mill power, and by a torsional swing in the shaft as described in Section 2.5. Since we assume the windmills use induction generators, it is clear that corresponding transient behaviours will also be observed when a disturbance is applied to the electric terminals of the windmill generator, causing the rotational mill speed to change. The magnitude of overshoot and the decrease time constant in the mill power depend on the wind speed and on the blade configuration and size. 2.4. Mechanical eigenswings The description of the eigenswings in the windmill mechanical construction is set up as an empirical model based on the analysis of measured spectra of the electric power from a Danish wind farm; this is illustrated in Fig. 3. The description is based on the consideration that some of the eigenswings excited by the vortex tower interaction [4] in the rotating mill are transferred through the shaft to the windmill generator and result in disturbances of the electric power supplied from the windmill.

which has to be supplied to transients in accordance with the description in Section 2.3, PWIND is given by Eq. (1), n is the kind of the mechanical eigenswing excited in the rotating mill and m is the harmonic of the given eigenswing, An,, v n, gnm(t), anm, w nm and hn(t) are the magnitude, the eigenfrequency, the distribution between the harmonics in the eigenswing n, and its normalised magnitude, the phase of the harmonic, and the modulation of the eigenswing, respectively. The frequency range of the windmill model with mechanical eigenswings is from 0.1 to 10 Hz. Due to the wind turbine construction, it is only possible to excite the eigenswing in the mill itself due to asymmetry in the mill (the 1P frequency), to the vortex tower interaction (the 3P frequency) and to the eigenswing in the blades. Other mechanical swings that it is possible to measure in the spectra of the electric power supplied from windmills are excited in the shaft and in the gearbox. It is shown below that flicker in the electric power supplied from windmills may be related to the mechanical eigenswings. Thus, this block is of interest when power quality studies have to be done. The mechanical eigenswings described in Eq. (2) have no influence on the stability of an electric power system with grid-connected windmills. 2.5. Shaft model In all the stability studies of electric power systems with grid-connected windmills performed so far, only a lumped element representation of the inertia of the rotating system has been considered. However, all the rotating shaft systems are divided into sections. In case of windmills, it is the mill itself which is quite heavy, whereas the machine rotor is light; the gearbox may also be represented by its small inertia. Since the shaft, which serves as the mechanical connection between the rotating masses with quite different inertias, is relatively soft, it is necessary to include a representation of the shaft. The gearbox, which is the element connecting the low rotational mill speed and the high generator speed, also

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Fig. 3. (a) Measured spectra of the electric power from a Danish wind farm; vMILL is close to 18 rpm. The eigenswings found in the measured spectra: (b) asymmetry in the mill; (c) first harmonic and (d) second harmonic of vortex tower interaction; (e) torsional swing in the shaft, (f) considered to be excited in the blades, and (g) considered to be excited in the gearbox.

contributes to the effective shaft stiffness between mill and generator. The stiffness is relatively small when viewed from the generator. As mentioned in Ref. [5], typical values of the eigenfrequencies of such systems are around 1 Hz, and for the specific Danish windmills the eigenfrequency is known to be 1.67 Hz, see Fig. 3a. Using the values of the eigenfrequencies of the shaft torsional swing together with the known inertia constants, one may calculate the shaft stiffness, which typically is around 0.15–0.40 p.u. The shaft model with three rotating masses, i.e. the mill, the generator rotor, and the gearwheels, and two shafts is applied in the physical windmill model, where the description is analogous to Ref. [6].

2.6. Transient induction generator model Since the majority of windmills are equipped with induction generators it is necessary to represent the induction generator by a model in which transient behaviours are taken into account. In electric power system simulation tools it is common to use dynamic models for the representation of synchronous as well as induction generators. As is known from the theory of synchronous machinery [6], the dynamic description of generators corresponds to neglect of the dc-offset in the stator. Neglecting the dc-offset leads to an oversimplified model of the generator speed behaviour [7]. This simplification seems to be correct for synchronous generators. But it is not adequate for induction generators

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with low inertia constants due to the strong coupling between rotor speed and the electrical properties of the generator such as electric power and reactive power consumption [7].

3. Application of windmill model blocks We mention the advantages of the physical windmill model due to its modular structure:

425

4. Windmill operation as a low-pass filter In order to determine the frequency-response of the windmills and to clarify the nature of the voltage flicker from typical Danish medium-sized windmills, a small signal analysis of the physical windmill model has been carried out. The physical windmill model is subjected to the disturbance given be the wind model ! X An sin …vn t† ; …4† v…t† ˆ v0 1 ⫹ n

1. The model is universal in that it can be implemented in several electric power simulation tools. 2. Only the components that are relevant for the investigation being done need to be included in the windmill model used for the simulation. 3. It is possible to mix components of our physical windmill model and the built-in models in a given simulation tool if it is desired. 4. The windmill model can be expanded with more relevant blocks as the windmill construction and control system complexity increases. For example the wind dynamic model, the model of pitch control, or the model of windmill (induction) generator control can be included. The physical windmill model can be applied in two kinds of investigations of electric power systems with gridconnected windmills: 1. In flicker investigations on the electric grid at normal operation with respect to improvement of the power quality. 2. In stability studies of the systems when subjected to faults. The complete physical windmill model can be used in both situations. However, it is preferable to use only the components that are relevant for the kind of investigation. When investigating the flicker and the power quality in the grid at normal operation, one might apply the simplification of the windmill model so that the aerodynamic behaviour around the wind turbine is neglected, and the windmill induction generator is given by its dynamic model [8]. But when a control system is applied to the windmill itself in order to reduce the flicker level, the complete physical windmill model has to be used. All the simulation results for analysis of flicker in this paper have applied the complete windmill model. When making a stability study of such a system subjected to faults, the complete physical windmill model, except for the block corresponding to the mechanical eigenswings, should be used. (The eigenswings do not affect the stability.) The windmill induction generator has to be represented by its transient model. The aerodynamic mill model and the shaft model also have to be taken into account.

where the frequency range is 0.1–10 Hz. We considered that the rotating mill transfers all the disturbance frequencies introduced in the wind speed as an ideal media so that HMW …v† ˆ …PMECH …v†=v…v†† / 1; where v(v ) and PMECH(v ) are the spectra of the disturbance v(t) and the mill shaft mechanical power PMECH(t) excited by the wind variations. This consideration fits in with basic knowledge about wind turbine dynamic behaviour and does not lead to misleading conclusions in our study. For more detailed investigations, it is necessary, however, to take into account the wind turbine transient behaviour. The outputs are the frequency responses HEM …v† ˆ …PELEC …v†=PMECH …v†† and HUM …v† ˆ …U…v†=PMECH …v††; where PELEC(v ) and U(v ) denote the spectra of the active electric power from the windmills, PELEC, and the windmill generator terminal voltage, U, respectively. Both are excited by the disturbance v(t). In Fig. 4 we show the frequencyresponse characteristics HEM(v ) and HUM(v ). The characteristics are simulated for two operation points of windmill, namely, at PELEC ˆ 100 kW (corresponding to 7 m/s) and at PELEC ˆ 500 kW (nominal operation at 14 m/s). The behaviour of the frequency-response characteristic HEM(v ) depends on the windmill operation point which is expressed by reduction of the resonance at increasing PELEC. In any situation, the behaviour of the HEM(v ) implies, however, that the windmill behaves as a low-pass filter with the resonance frequency at about fd ˆ 0:6 Hz for the given windmill construction. The resonance frequency is dependent on the inertia of the mill and the generator and on the stiffness of the axis, but in normal operational conditions it is always below the natural frequency of the mechanical torsion oscillation in the shaft. For the given windmill construction, the natural frequency of the mechanical torsion oscillation in the shaft is ftorsion ˆ 1:67 Hz: This difference is caused by the system attenuation, D, according p  to the expression fd ˆ ftorsion 1 ⫺ j 2 ; where the damping factor, j , is proportional to the system attenuation [6]. We point out that the system attenuation, D, relates to the electric connection between the windmill and the grid as D / …2T ELEC =2vR †; where TELEC and v R are the electric torque and the speed of the windmill generator, respectively. We note that the damped resonance frequency fd is close to the 2P value (the 3P frequency is 0.9 Hz at a low wind speed). A local resonance at 7 Hz, which presumably comes from

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Fig. 5. The computed behaviour of the mill shaft power, PMECH, and the electric power from the windmill, PELEC, at the rotational mill speed 18 rpm.

Fig. 4. Frequency-response of a medium-sized windmill of the given construction: (a) HEM …v† at PELEC ˆ 100 kW: (b) HEM …v† at PELEC ˆ PRATED ˆ 500 kW: (c) HUM …v† for one windmill at PELEC ˆ 100 kW: (d) HUM …v† for five windmills at PELEC ˆ 100 kW each. (e) HUM …v† for one windmill at PELEC ˆ PRATED ˆ 500 kW:

the inertia of the gear, has influence on flicker and is discussed in Section 5. The fact that the frequency-response between mechanical oscillations in the windmill construction and disturbances in the electric power is a low-pass filter leads to the following conclusions: 1. The magnitudes of the disturbances in the electric power caused by the eigenswings in the rotating mill and, there-

fore, their contributions to the voltage flicker depend on the rotational mill speed. The closer the disturbances are to the resonance frequency, fd, the larger their contributions at the electric power spectra are. Voltage flicker is discussed in Section 5. 2. When using the measured spectra of the electric power together with the frequency-response HEM(v ), it is possible to define the behaviour of the mechanical disturbances subjected to the rotating mill that is the mechanical eigenswings. 3. Since the windmill resonance frequency, fd, is so low, there is a risk that the frequency may be close to the electric power oscillation frequency in the utility grid. Such an oscillation frequency is usually around or below 1 Hz. We discuss this subject in Sections 6 and 7. In Fig. 3a the measured spectrum of the electric power supplied from a Danish wind farm is shown. When the electric power was measured, the wind speed was relatively low so that the rotational mill speed was around 18 rpm. The simulated behaviours of the electric power and the mechanical power in the mill shaft at a similar value of wind speed are given in Fig. 5. The computation is made for a grid-connected windmill with a construction which is similar to the windmills in the wind farm. The simulated

Table 1 Mechanical eigenswings excited in the mill; An and v3 correspond to the given windmill construction Kind, n

Classification

Magnitude An

Eigenfrequency, vn

Modulation, hn …t†

Harmonics, m

Distribution, anm

Phase, wnm

1

Asymmetry

0.01

vMILL (1P)

1

2

Vortex tower interaction

0.08

3vMIL (3P)

1

3

Blades

0.15

2p 4.5 (Hz)

1/2g11(t) ⫹ 1/2g21(t)

1 2 1 2 1

4/5 1/5 1/2 1/2 1

0 p=2 0 p=2 0

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Fig. 6. The simulated spectra of the electric power PELEC: (a) at mill speed 18 rpm; and (b) at 27 rpm.

behaviour of the mechanical power in the mill shaft of the grid-connected windmill is calculated so that the behaviour of the computed electric power is the same as the measured electric power behaviour. When computing the mill shaft power, PMECH, we have applied Eq. (2), and the values used for the calculation of PMECH are given in Table 1. In Fig. 6 we introduce the computed spectra of the electric power from the grid-connected windmill, where the rotational mill speeds are 18 and 27 rpm, respectively. In both the simulations, the disturbances in the mechanical power in the mill shaft are given in Eq. (2) and the magnitudes, An, given in Table 1. From the simulation results we see that the largest contribution to the disturbances in the mill itself is the eigenswings in the blades at the eigenfrequency 4.5 Hz. Their total contribution is around 15% taken with respect to the average value of the mechanical power of the mill shaft. However, these swings will be efficiently damped in the shaft before they reach the generator; this is because of the low-pass filter operation of the windmill construction. In the disturbances in the mill, there is an equal distribution between first and second harmonics of the vortex tower interaction, and the respective angle between the harmonics is p=2: The behaviour of the vortex tower interaction presented in the results is in accordance with the blade– tower interaction behaviour derived in Ref. [9] with use of the fluid mechanics theory. From the computed spectra of the electric power shown in Fig. 6, we see the contribution of the first harmonic of vortex tower interaction is around

427

3% at the rotational mill speed 18 rpm and it is less than 1% at the mill speed 27 rpm. The reason for that behaviour is the low-pass filter operation of the windmill construction, see Fig. 4. A similar behaviour with respect to the first harmonic of vortex tower interaction has been observed in the spectra of PELEC measured at 18 rpm, when the wind is light, and at 27 rpm, when the wind is strong. We make the following interesting observation: the relatively small contribution in the simulated spectra of the electric power at 7 Hz is related to the local resonance in the frequency-response characteristic HEM(v ) at the same frequency. The investigations have shown that the local resonance at 7 Hz in HEM(v ) is caused by the representation of the gearwheels with their inertia in the windmill shaft model. This observation is of interest because the contribution at 7 Hz is not represented in the mechanical eigenswings excited in the mill, see Table 1, but it is excited in the gearbox and then transferred to the generator. When transferred to the generator, it results in the corresponding disturbance at 7 Hz in the electric power and in the flickers. We mention that a contribution around 7 Hz in all the measured spectra of the electric power from the wind farm is identified. Hence, it is considered that the 7 Hz swings in the measured spectra are excited in the gearbox, and the frequency depends on the windmill construction. 5. Flicker and its nature Flicker is a technical expression for the RMS voltage variations in the frequency range above 1 Hz which occur in electric power systems and may cause light bulbs to flicker. When windmills feed into a relatively weak electric system, flicker is often observed close to the locations of the windmills. The cause of flicker has commonly been attributed to wind speed variations [10]. It has been thought that wind speed variations lead to corresponding variations in the mechanical power, and hence result in disturbances in the electric power from grid-connected windmills producing flicker [10]. This mechanism for explaining flicker has been discussed for relatively small windmills [10]. This mechanism of the flicker excitation seems to be partly correct. Let us discuss this further. Since all the disturbances in the wind speed can be transferred and, then observed in the mill shaft power so that …PMECH …v†=v…v†† / 1; it is reasonable to explain the voltage flicker as being due to spontaneous, stochastic wind speed variations. However, this point of view needs two following corrections: 1. In the measured Fourier-spectra of the wind farm electric power we have seen systematic contributions, which we separated into a number of wavelets. This behaviour cannot correspond to a stochastic wind variation, but this presumably is in accordance with the mechanical eigenswings excited in the windmill mechanical construction.

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2. The windmill mechanical construction operates as a lowpass filter, as a result of which the disturbances in the mill shaft power will be attenuated until they reach the windmill generator. We see the agreement between the shape of the windmill transfer function in Fig. 4a and the shape of measured Fourier-spectrum given in Fig. 2a. Both results are for the given windmill construction and light wind. We have to assume that the flicker has two causes. Firstly the flicker may be excited by wind speed variations, when wind is strongly turbulent. Secondly, the eigenswings excited in the windmill mechanical construction itself are always represented in the voltage flicker. In Denmark, where the landscape is relatively flat, we may assume the mechanical eigenswings to be the major mechanism in the voltage flicker. A similar conclusion may be found in case of offshore wind farms—there are usually no obstacles, which could produce turbulence in wind, on the sea surface close to windmills. When studying the frequency-response characteristics HUM(v ) shown in Fig. 4c–e, we conclude that the actual flicker level for a given windmill depends on: 1. The mechanical construction of windmill that is represented by the disturbances excited (strongly turbulent wind and mechanical eigenswings) and its transfer function. 2. The operational point of the windmill generator and the electrical grid strength. With respect to the influence of the mechanical eigenswings on the flicker, we mention in particular the local resonance at 7 Hz that is presumably excited in the gearbox. We note that even small disturbances in the range from 5 to 10 Hz will result in flicker that is extremely bothering to the human eye. Local resonance in the windmill frequency-response at 7 Hz is, therefore, of interest. On the other hand, the voltage flicker level is dependent on the value of the electric power supplied from gridconnected windmills in accordance with: DU ˆ RDPELEC ⫺ XDQELEC ;

…5†

written in p.u. DU, DPELEC and DQELEC denote variations in the terminal voltage, the electric power and the reactive consumption of the windmill induction generator, respectively. R ⫹ jX are the line impedance to the grid. Furthermore, we need to apply the windmill generator QELEC versus PELEC characteristic when estimating the voltage flicker level. This characteristic is schematically shown in the insertion in Fig. 4. Our physical windmill model indicates that flicker may be caused by mechanical eigenswings in the windmill construction. The model shows that the closer the disturbances in the mill shaft power, PMECH, to the resonance frequency, fd, the larger their contributions are to the windmill electric power, PELEC. These contributions will be

present in the voltage flicker. However, the voltage flicker level will be higher when the electric power amount supplied from windmills is larger (compare Fig. 4c and d), and, especially, when the sloop of the QELEC versus PELEC characteristic of the windmill induction generator is larger (compare Fig. 4d and e). Since the disturbances appear to be caused by the windmill mechanical construction it is possible to predict their nature and to find ways to damp them. Possible solutions include adding mechanical damping to the design, for example, so that the windmill resonance, fd, is not at the 3P frequency. On the electric side of the windmill, one of several control systems for the windmill generator, net reinforcements, or voltage control systems such as SVC can be used for the voltage flicker reduction.

6. Windmill model in stability studies We have carried out dynamic simulations using a realistic network model, which is built upon an equivalent of the electric grid on the Danish island of Lolland, just south of Zealand. The wind power capacity of the island is around 85 MW [4] and is still increasing. In the electric grid equivalent, single grid-connected windmills as well as wind farms are grouped around two bigger farms Munkeby (MBY) and Rødbyhavn (RBH). The wind farms are connected to a 10 kV radial distribution network, which through a large 50 kV ring distribution system is connected to the 132 kV transmission network in Zealand. In the dynamic simulations we investigated the consequences of a transient three-phase short circuit of 100 ms duration in the 132 kV system, without disconnection of the wind farms in post-fault behaviour (see the explanation below). We compare the voltage behaviour of the wind farm terminals at post-fault operating situations in the following cases: 1. The classical approach—a dynamic induction generator model (of 3rd-order) and the inertia constant of 3 s are used. The inertia constant of the mill and the generator rotor are simply lumped together. 2. Our new approach—a physical windmill model (neglecting mechanical eigenswings in the windmill construction) is used. The inertia constant of the wind turbine is 2.5 s and that of the generator is 0.5 s, and the effective stiffness of the shaft is 0.29 p.u. [11]. The gearbox is given with its small inertia. We have compared the two models in two studies. In the first, the installed wind power capacity is chosen as 40 MW in each wind farm, which is a little below the installed wind power capacity on the island today. The actual electric power production is set at 30 MW in each farm, this corresponds to a wind speed of about 11 m/s. The 132 kVnetwork is subjected to a transient fault of 100 ms duration.

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reduce to a frequency around 0.6 Hz (the natural frequency of the windmill, fd). The oscillation frequency transformation from ftorsion to fd is due to the re-establishing of the electrical connection between wind farms and the electric grid, when the voltage is recovering, and the system attenuation, D, increases as well. This observation is interesting, since electric power oscillations in the transmission systems usually are below 1 Hz. In Fig. 8 we show the dynamic behaviour of the speeds for MBY simulated with the classical approach and with the physical windmill model. With the classical approach the speed shows a small deviation from the steady-state value followed by a smooth fall back to steady state level. With the physical windmill model we observe:

Fig. 7. Behaviours of the wind farm terminal voltages in post-fault operation, the electric power is 30 MW in each wind farm.

The post-fault behaviours of the terminal voltages in each wind farm for each model are shown in Fig. 7. With the classical approach, the wind farm terminal voltages recover quickly. With the physical windmill model, we see oscillations in the terminal voltages. The voltage oscillations start with the frequency around 1.7 Hz (the shaft torsional eigenfrequency, ftorsion) and eventually

Fig. 8. Speed behaviour at the wind farm MBY at post-fault operation, the electric power production in the farm is 30 MW.

1. the overspreading of windmills as a consequence of the fault is not high enough for the windmills to be disconnected and stopped by the protection system; 2. the generator rotor briefly decreases in speed, just when subjected to the fault, and thereafter increases in speed; 3. there are oscillations in the generator rotor speed as well as in the mill speed, where the oscillations start with the frequency of around 1.7 Hz, that is ftorsion. The later delayed oscillation is at 0.6 Hz, that is fd, because of the above-explained process; 4. the generator speed has a 7 Hz oscillation due to the oscillation in the gearbox during post-fault operation. The oscillation in the gearbox decays relatively slowly. In the second study the installed wind power capacity is increased to 50 MW in each wind farm. We mention that the wind power capacity of the island is expected to increase up to around 225 MW by the year 2008. Hence the level of 100 MW may be reached quite soon. The wind speed again is about 11 m/s, and with this assumption the actual value of the electric power supplied by each farm is 37.5 MW. The network model is subjected to the same type of fault in the 132 kV system, without disconnection of wind farms. In Fig. 9 we show the post-fault behaviour of the terminal voltage at the wind farm MBY; in Fig. 10 we give the speeds in the wind farm MBY. The wind farms are again simulated with the classical approach and with the physical windmill model. We mention that the wind farms should be disconnected and stopped by the windmill protection system in the case of high value of overspeeding. However, this is not represented in our simulations, since the differences between the conventional approach and our physical windmill model are of interest in this investigation. Simulation of the wind farms with a classical approach results in a slow recovery of the voltage profile; the speed deviation does not exceed 0.05 p.u. during the fault, and it returns smoothly to the steady state level. The protection system does not even react to such a low value of windmill overspeeding. The simulation with the physical windmill model, when the windmill protection system is ignored, shows that a fault

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Fig. 9. The post-fault behaviour of terminal voltage at the wind farm MBY; the power production before faulting is 37.5 MW in each farm. The conventional model shows a slow re-establishing of the voltage, whereas the physical windmill model predicts voltage collapse and oscillations.

leads to a voltage collapse in the grid. The collapse is characterised by remarkable voltage oscillations at the wind farm terminals (also in the 50 kV system and to a lesser extent in the 132 kV network) with the frequency 1.7 Hz (the natural frequency of torsional eigenswing in the shaft, ftorsion). Due to the voltage collapse in progress, the system attenuation, D, is close to zero, and the oscillations keep the frequency of ftorsion. We see that during a voltage collapse, where speedcontrolling effect in the electric connection is weakened, the windmill runs away and is exposed to large mechanical stress. Before we discuss the prevention of voltage collapses in an electrical network with grid-connected windmills, we point out: 1. It is not sufficient to represent grid-connected windmills by a dynamic induction generator model of 3rd-order with a large, lumped inertia. This results in too optimistic an estimation of the stability limit for the windmill generator and in this way also for the electric network. 2. Windmill shafts are relatively soft and, hence, oscillations due to the shaft torsion occur and must be taken into account. The strong coupling between the generator slip and the active and reactive electric power produced by an induction generator leads to shaft oscillations being observed in the terminal voltage and current. 3. It is suitable to use a windmill model with a transient

Fig. 10. The post-fault behaviour of the speed at the wind farm MBY; the electric power production before faulting is 37.5 MW in each park. A neat behaviour according to the conventional description of the windmill in electric power system simulation tools, but a large load of the mechanical system if the physical wind model is used.

generator model of 5th-order, a shaft model and a model of the aerodynamic state of the wind turbine given by CP(l ), when representing grid-connected windmills in the electric power system studies.

7. Prevention of voltage collapse in the network and protection of windmills In order to protect the windmill mechanical system against overloading and prevent the collapse in the electric grid, windmills are equipped with a protection system. When windmills are overspeeding (typically above 0.05 p.u.) as a consequence of a fault in the electric power grid, the protection system will disconnect and stop the windmills. This procedure is standardised today, when the amount of the grid-connected wind power is relatively low (around 10%). In Fig. 11 we show the voltages in the electrical network model, when the 132 kV system is subjected to a transient fault of 100 ms duration, when both wind farms are disconnected 0.5 s after the fault has been cleared (when overspeeding of windmills is above 0.05 p.u.). Before the fault, the wind farms are supplying 37.5 MW each. After disconnection of the farms, the voltages recover rapidly in the whole electric power system, and a voltage collapse is avoided. We point out, however, that the conventional power plants cover the necessity of the electric power which corresponds to the 37.5 MW of electric power supplied from each wind farm before the faulting and disconnection. In Fig. 12 we give the speeds, the mechanical power,

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Fig. 11. The voltage behaviours in the electric networks; disconnection of both wind farms occurs 0.5 s after the fault is cleared.

PMECH, and the mechanical torque, TMECH, in the wind turbine after disconnection of the farms, but without stopping. This simulation is made for verification of the model of windmill mechanical system, when the wind turbine runs away. After disconnection without stopping, the mill speed and the generator speed follow each other. However, there is a minor steady torsion oscillation in the shaft. The wind turbine accelerates drastically after disconnection, whereas the mechanical torque decreases. This phenomenon is explained due to loss of the air boundary layer around the wind turbine blades that can lead to breaking of the blades. In general, the behaviour of TMECH (increasing or decreasing) may be found from the CP …l†=l versus l curve. The behaviour found with our windmill model, where the speeds and the mill mechanical power increase and the mill mechanical torque decreases at overspeeding, is in accordance with the known experimental results obtained for overspeed wind turbines. This confirms that the feedback from the speed to the mechanical power implemented in the model is as in a physical stall controlled windmill. After the wind turbines are stopped and the electric network is in stable operation again, the windmills are reconnected to the network. The strategy with disconnection of the windmills at a fault in the electric network has up to now been a— technologically seen—correct way of handling this kind of fault. In the future, when a large amount of wind power capacity is installed and the conventional power plants capacity is decreased, it will be unacceptable to disconnect all the

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Fig. 12. The post-fault behaviours of the speeds, PMECH and TMECH in the wind farm MBY at disconnecting both the farms 0.5 s after the fault is cleared.

windmills, even by the reason to prevent a voltage collapse as the result of a transient short circuit faults. The disconnection will be unacceptable because the power loss in the network will be too large. The power reserve of the conventional power plants in the grid will not be sufficient to compensate for the loss. In order to avoid both disconnection of the windmills and voltage collapse, solutions other than just disconnection of all the windmills should be considered. The solution could be reinforcement of the electric network, or installation of SVC, STATCOM, VSCHVDC or other equipment that can control the voltage in an electric network. The above-mentioned solution with application of voltage control equipment could be combined with the disconnection of a number of windmills, which is just enough to maintain voltage stability. When only some of the windmills are disconnected, the electric power supply from the grid-connected windmills to the grid can be kept at a sufficient level. As an illustration of this thinking, we show the post-fault behaviours of the voltages where only one wind farm has been disconnected 0.5 s after the transient fault on the 132 kV system is cleared. The other wind farm remains grid-connected throughout the fault and post-fault behaviour in the grid. In Fig. 13 we give the post-fault behaviour of the voltages, when the wind farm MBY is disconnected, and in Fig. 14—when the wind farm RBH is disconnected. In both cases, the electric power supplied to the grid was 37.5 MW from each farm, before the fault occurs. As can be seen, it is of importance which one of the two farms will be disconnected—the voltage is recovering in

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Fig. 13. The post-fault behaviours of voltages, when the wind farm MBY is disconnected; the electric power production is 37.5 MW in each farm before faulting.

Fig. 14. The post-fault behaviours of voltages, when the wind farm RBH is disconnected; the electric power production is 37.5 MW in each farm before faulting.

both cases, but it recovers faster and surer if the wind farm MBY has been disconnected. Hence, the network configuration is an important parameter that must be taken into account when planning the stability of an electric grid with grid-connected windmills. When the voltage recovers after a fault and a significant number of windmills are kept grid-connected, the oscillations in the voltages as well as in the other electric behaviours in the grid will be present while the voltage is re-established. The oscillations in the voltages, as found in Section 6, are caused by torsion in the windmill shafts. We note that the low frequency voltage oscillations are also seen in the 50 and the 132 kV subsystem. This implies the solution that, when a significant number of windmills are operating, during the post-fault behaviour in the electric grid:

8. Summary

1. it is not necessary to disconnect all the windmills to prevent a voltage collapse; 2. it can be necessary to combine the solution above with net-reinforcements and application of the voltage control systems to reduce the voltage oscillations; 3. the electric behaviour in the grid is marked with low frequency oscillations for a few seconds after the fault is cleared, where the oscillations are excited by the windmill shafts. Anyway, the solution should be selected with careful reflection on the consequences by the personnel, with good knowledge about the electric transmission system and windmills.

The conventional representation of grid-connected windmills in the electric power supply systems consists, as usual, of a dynamic model of an induction generator equipped with a relatively large inertia. The large value of the induction generator inertia is because of the lumped mass consideration. This representation has been applied due to: 1. there being no better model of grid-connected windmills which could be applied in the electric power supply studies having been introduced; 2. underestimation of the windmill mechanical system importance and the induction generator transient behaviour. When using the lumped mass consideration in the representation of the windmill mechanical system, the stability of the windmill induction generators and then the whole electric power system with the grid-connected windmills will be increased unrealistically. As a result of the simplified description of the windmill mechanical system—the lumped mass consideration—the short-lived electric power oscillations will not be predictable when studying the post fault behaviours. Our study shows that the interaction between the windmill mechanical construction and the electric grid is an important property of the dynamics of the electric power system with grid-connected windmills. Therefore, the application of a physical description of grid-connected windmills—for example by use of the physical windmill

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Fig. 15. Developed standard model of a grid-connected windmill in the electric power supply.

model given here—is necessary in the stability investigation. The physical model developed here of grid-connected windmills in the electric power supply systems describes a windmill as a complex, electromechanical system, as shown in Fig. 15. One may recognise that the model of grid-connected windmills shown in Fig. 15 gives the standard model of windmills in the study of the electric power supply. The standard windmill model consists of three basic elements: 1. The transient induction generator model that is necessary due to the strong dependency of the electrical values in the induction generator terminals on the generator rotor speed. The windmill induction generator can be equipped with a control system, for example the widely-known variable rotor resistance control principle. Since the windmill generator is grid-connected, it is possible to influence its operation due to the net reinforcements. 2. The shaft model with three rotating masses—the mill, the generator rotor, and the gearwheels,—and two shafts. The windmill shaft is a closed system, which means that it is not possible to introduce any control system applied directly to the shaft. The twist behaviour in the shaft influences the electrical behaviours in the windmill induction generator terminals. 3. The model of the rotating mill with the aerodynamic behaviour around the blades, where it is possible to implement a model of the system—pitch or active stall control system—to control the mechanical power of the mill through the rotational mill speed. A dynamic wind model may be implemented, where the wind speed is applied to the rotating mill. The developed standard windmill model has the wind speed (stationary or dynamic) as the input and the exchanging power flow between the windmill generator and the grid. In accordance with the standard windmill model, a windmill absorbs the mechanical power from the incoming wind and the magnetising power from the electric power system, and it supplies the active electric power to the electrical grid.

The feedback through the speeds contributes to the complexity of the power flow inside the windmill. The developed standard windmill model introduced in this issue is suitable for the study of flicker from windmills connected to the utility grid at normal operation as well as for electric power systems stability investigations and the post faulting behaviour simulations. However, a simplified edition of the windmill model may be enough to represent windmills in some special situations as: 1. When studying flicker from windmills in the grid at normal operation and no control system is applied to the windmills themselves, the induction generators may be given with their dynamic (small signal) models. 2. When investigating the stability of the electric power system with grid-connected windmills, the representation of the mechanical eigenwings in the mill construction may be neglected. Besides, the shaft model may be given without the inertia of the gearwheels. When several kinds of control systems are applied to the windmill itself, or when the investigated situation in the electric power system with grid-connected windmills is complicated, the developed standard physical windmill model has to be applied. Acknowledgements The paper is the result of collaboration between Department of Electric Power Engineering, Technical University of Denmark (DTU) and NESA Transmission Planning, NESA A/S, Denmark. The authors thank gratefully Drs Jens N. Sørensen, Martin O. Laver Hansen and Stig Øye, Department of Fluid Mechanics DTU for useful discussions. Appendix The simulation results given in this paper are for windmills that are with two speed, pole-changing generators. The dual generator system implies that the wind turbine has two

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rated speeds and runs at lower rotational speed at lower wind speed and at higher rotational speed at higher wind speed; the data are as follows: Rated generator power (kW) Rated generator slip (%) Cut-in wind speed (m/s) Cut-off wind speed (m/s) Rated wind speed (m/s) Mill rotational speed (rpm) Mill rotor diameter (m) Number of blades Mill inertia seen from generator (s) Generator rotor inertia (s) Shaft effective Stiffness (p.u.)

500 2 3.5 25 14 27/18 41 3 2.5 0.5 0.29

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