NL2005400C2 - Method and system for wind gust detection in a wind turbine. - Google Patents
Method and system for wind gust detection in a wind turbine. Download PDFInfo
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- NL2005400C2 NL2005400C2 NL2005400A NL2005400A NL2005400C2 NL 2005400 C2 NL2005400 C2 NL 2005400C2 NL 2005400 A NL2005400 A NL 2005400A NL 2005400 A NL2005400 A NL 2005400A NL 2005400 C2 NL2005400 C2 NL 2005400C2
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- 238000000034 method Methods 0.000 title claims description 95
- 238000001514 detection method Methods 0.000 title claims description 39
- 238000012545 processing Methods 0.000 claims description 36
- 238000012360 testing method Methods 0.000 claims description 36
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- 238000007476 Maximum Likelihood Methods 0.000 description 11
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Classifications
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F03—MACHINES OR ENGINES FOR LIQUIDS; WIND, SPRING, OR WEIGHT MOTORS; PRODUCING MECHANICAL POWER OR A REACTIVE PROPULSIVE THRUST, NOT OTHERWISE PROVIDED FOR
- F03D—WIND MOTORS
- F03D7/00—Controlling wind motors
- F03D7/02—Controlling wind motors the wind motors having rotation axis substantially parallel to the air flow entering the rotor
- F03D7/04—Automatic control; Regulation
- F03D7/042—Automatic control; Regulation by means of an electrical or electronic controller
- F03D7/043—Automatic control; Regulation by means of an electrical or electronic controller characterised by the type of control logic
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F03—MACHINES OR ENGINES FOR LIQUIDS; WIND, SPRING, OR WEIGHT MOTORS; PRODUCING MECHANICAL POWER OR A REACTIVE PROPULSIVE THRUST, NOT OTHERWISE PROVIDED FOR
- F03D—WIND MOTORS
- F03D17/00—Monitoring or testing of wind motors, e.g. diagnostics
-
- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F05—INDEXING SCHEMES RELATING TO ENGINES OR PUMPS IN VARIOUS SUBCLASSES OF CLASSES F01-F04
- F05B—INDEXING SCHEME RELATING TO WIND, SPRING, WEIGHT, INERTIA OR LIKE MOTORS, TO MACHINES OR ENGINES FOR LIQUIDS COVERED BY SUBCLASSES F03B, F03D AND F03G
- F05B2260/00—Function
- F05B2260/84—Modelling or simulation
-
- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F05—INDEXING SCHEMES RELATING TO ENGINES OR PUMPS IN VARIOUS SUBCLASSES OF CLASSES F01-F04
- F05B—INDEXING SCHEME RELATING TO WIND, SPRING, WEIGHT, INERTIA OR LIKE MOTORS, TO MACHINES OR ENGINES FOR LIQUIDS COVERED BY SUBCLASSES F03B, F03D AND F03G
- F05B2270/00—Control
- F05B2270/30—Control parameters, e.g. input parameters
- F05B2270/322—Control parameters, e.g. input parameters the detection or prediction of a wind gust
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- F—MECHANICAL ENGINEERING; LIGHTING; HEATING; WEAPONS; BLASTING
- F05—INDEXING SCHEMES RELATING TO ENGINES OR PUMPS IN VARIOUS SUBCLASSES OF CLASSES F01-F04
- F05B—INDEXING SCHEME RELATING TO WIND, SPRING, WEIGHT, INERTIA OR LIKE MOTORS, TO MACHINES OR ENGINES FOR LIQUIDS COVERED BY SUBCLASSES F03B, F03D AND F03G
- F05B2270/00—Control
- F05B2270/30—Control parameters, e.g. input parameters
- F05B2270/331—Mechanical loads
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- Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
- Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
- Y02E—REDUCTION OF GREENHOUSE GAS [GHG] EMISSIONS, RELATED TO ENERGY GENERATION, TRANSMISSION OR DISTRIBUTION
- Y02E10/00—Energy generation through renewable energy sources
- Y02E10/70—Wind energy
- Y02E10/72—Wind turbines with rotation axis in wind direction
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- Engineering & Computer Science (AREA)
- Life Sciences & Earth Sciences (AREA)
- Sustainable Development (AREA)
- Sustainable Energy (AREA)
- Chemical & Material Sciences (AREA)
- Combustion & Propulsion (AREA)
- Mechanical Engineering (AREA)
- General Engineering & Computer Science (AREA)
- Wind Motors (AREA)
Description
Method and system for wind gust detection in a wind
TURBINE
Field 5
The present invention relates to a method for gust detection in a wind turbine according to the preamble of claim 1. Additionally, the present invention relates to a system for gust detection in a wind turbine. Moreover, the present invention relates to a computer program product for gust detection in a wind turbine. The present invention 10 also relates to a method and system for control of a wind turbine using gust detection.
Background
The control of a wind turbine is typically arranged for control under steady wind 15 conditions, i.e., under a substantially constant or slowly varying wind velocity and/or a substantially constant or slowly varying wind direction. Variations under steady wind conditions are typically described by a model of turbulence. Under this type of conditions the load(s), i.e. forces and moments on the wind turbine and its components such as the rotor and rotor blades is (are) substantially constant with stationary 20 stochastic variations from turbulence. The substantially constant part of the load(s) uses to change on a time scale of typically 10 minutes, due to the so called variability of the wind. This variability includes a slowly varying mean longitudinal wind speed (Uvlf). However, in unstable atmospheric conditions, extreme conditions much larger than the stationary stochastic variations may occur: wind conditions may vary strongly and 25 rapidly over time.
These unsteady wind variations are further referred to as wind condition variations or simply gusts. A class of wind condition variations (gust class) concerns rotor coherent gusts. Another gust class concerns rotor coherent fast wind direction changes. Yet other gust classes concern fast establishing asymmetric wind velocity 30 distributions over the rotor plane.
Under such extreme conditions as occurring gusts, the loading of the wind turbine and the stress on its components may be very high, which may cause damage or may contribute to damage to the wind turbine and its components. To avoid such damage 2 the control of the wind turbine is arranged to stop operation of the wind turbine, i.e., the rotation of the rotor is stopped. Since in most cases, the reason for stopping operation may be unclear, an inspection of the wind turbine may be required before resuming operation, or in some cases, operation may be resumed after a pre determined waiting 5 period. During the stop of the wind turbine, no power is generated.
Extreme conditions relate to relatively strong and rapid variations of wind conditions which cause loading of the rotor, such as rotor-coherent gusts, rotor-coherent wind direction changes and asymmetric distributions of wind velocity. Rotor coherent is defined as a condition in which the wind condition variation has a spatial 10 full coherence over the rotor plane.
Such extreme conditions occur frequently at high wind velocity over broken ground, rough terrain and in wind farms. At sea asymmetric wind velocity distributions in vertical direction can occur, which can affect the operation of the wind turbine when the top of the rotor is struck by a so called low level jet, a layer of air with high wind 15 velocity.
Also, asymmetric wind velocity distributions in horizontal direction can occur in wind turbine farms when a rotor of one wind turbine is partially in the lee of another wind turbine.
EP 2 025 929 discloses a system and a method for loads reduction in a horizontal-20 axis wind turbine using upwind information. US 7 281 891 discloses a wind turbine control having a LIDAR wind velocity measurement apparatus. Both patent publications relate to the use of a LIDAR system (laser radar) to identify an upcoming gust front at some distance before the rotor blade. Such systems allow detection of gusts over a portion of the wind field.
25 There is a need for a system and method for gust detection to detect rapid and strong variations of wind velocity and wind direction on the magnitude scale of the rotor plane to avoid a stop of a wind turbine due to impermissible rotational velocity or other undesirable conditions of operation.
Further there is a need to have a system and method for gust detection that is 30 capable to discriminate between loads caused by rotor-coherent gusts and loads due to turbulence. The latter include moderate components in both low frequencies, typically between 0.01 Hz and 0.1 Hz, and relatively high frequencies that are centered around multiples of the rotational frequency, typically around ca 1Hz and multiples. The high 3 frequency load components are caused when blades of the rotor during rotation traverse a turbulent wind field.
Summary of the invention 5
The objective of the invention is achieved by a method for wind gust detection in a wind turbine, the wind turbine comprising a tower and a rotor provided with a number of rotor blades on a rotor axis, the rotor being arranged on the tower; the method comprising 10 a) collecting data of a mechanical loading condition of the wind turbine, the data of the mechanical loading condition being associated with at least one load exerted on the construction of the wind turbine in a non-rotating reference frame; b) monitoring a temporal evolution of the data of the mechanical loading condition, comprising: 15 bl) determining a steady state component of each of the data associated with mechanical load on the wind turbine; b2) obtaining residuals of the data of the mechanical loading condition by removing the respective steady state components from the respective data, the temporal evolution being determined for the obtained residuals; 20 c) detecting by a test condition if the temporal evolution indicates that a wind gust is evolving; d) based on the result of the test condition, generating signals for adjusting the operation of the wind turbine.
According to an aspect there is provided a method as described above, wherein 25 the residuals are obtained by performing a Kalman filter procedure based on a stochastic turbulence model, comprising: determining a residue vector, the residue vector comprising the residuals of the data of the mechanical loading condition.
According to an aspect there is provided a method as described above, 30 comprising: providing sensors arranged on a non rotating part of the construction of the wind turbine for measuring at least one load ; wherein the collection of data of the mechanical loading condition of the wind turbine comprises measuring said data by the sensors arranged on the construction.
4
According to an aspect there is provided a method as described above, wherein each rotor blade comprises a sensor for measuring data associated with a mechanical load of the rotor blade, and the collection of data of the mechanical loading condition of the wind turbine comprises, in sequence: 5 al) measuring by the respective sensor on the rotor blade data associated with the mechanical load of each rotor blade; a2) transforming the data associated with the mechanical load of each rotor blade to the data of the mechanical loading condition.
According to an aspect there is provided a method as described above, wherein 10 the test condition is a generalized likelihood ratio test, which comprises a test on a null hypothesis for testing if the temporal evolution is within a stochastic turbulence range not involving a wind gust condition, and on an alternative hypothesis for testing if the temporal evolution is outside the stochastic turbulence range, the test being indicative that a wind gust condition is evolving.
15 According to an aspect there is provided a method as described above, wherein the test condition comprises a test that the temporal evolution exceeds a pre-determined threshold value.
According to an aspect there is provided a method as described above, wherein the test condition comprises a matching between the temporal evolution and a matching 20 function.
According to an aspect there is provided a method as described above, wherein each steady state component is determined as simple moving average of the respective data associated with a mechanical loading condition.
According to an aspect there is provided a method as described above, wherein 25 the at least one mechanical loading condition is selected from a group comprising a thrust force Fa parallel to the rotor axis, a horizontal force Fh perpendicular to the rotor axis, a vertical force Fy perpendicular to the rotor axis, a driving moment Md parallel to the thrust force Fa, a tilt moment Mt parallel to the horizontal force and a yaw moment My parallel to the vertical force.
30 According to an aspect there is provided a method as described above, wherein the detecting by the test condition if the temporal evolution indicates that a wind gust is evolving comprises: 5 dl) associating the temporal evolution of the data of the mechanical loading condition with a wind gust class.
According to an aspect there is provided a method as described above, comprising: 5 classifying the wind gust in one of six wind gust classes; the first wind gust class being based on a uniform wind velocity variation perpendicular to the rotor plane; the second wind gust class being based on a wind condition variation by a backing and veering wind; 10 the third wind gust class being based on a wind condition variation by a jet stream; the fourth wind gust class being based on a wind direction variation; the fifth wind gust class being based on a wind condition variation by a wake condition for a side portion of the rotor plane of the wind turbine, and the sixth wind gust class being based on a wind condition variation by a sloping wind. 15 According to an aspect there is provided a method as described above, wherein the classification of the first and second wind gust class is based on the temporal evolutions of the thrust force and the driving moment, and their ratio.
According to an aspect there is provided a method as described above, wherein the classification of the third and fourth wind gust class is based on the temporal 20 evolutions of the horizontal force and the tilt moment, and their ratio.
According to an aspect there is provided a method as described above, wherein the classification of the fifth and sixth wind gust class is based on the temporal evolutions of the vertical force and the yaw moment, and their ratio.
According to an aspect there is provided a method as described above, wherein 25 the generated signals for adjusting the operation of the wind turbine are selected based on the wind gust class classification.
According to an aspect there is provided a method as described above, wherein adjusting the operation of the wind turbine comprises adjustment of at least one of a pitch angle of one or more of the rotor blades, an electro-magnetic counter torque by 30 the generator, a yaw angle of the rotor as set by the yaw system KS of the wind turbine, an aerodynamic conversion factor of the rotor, settings of a tip vane or flap on one or more of the rotor blades.
6
According to an aspect there is provided a method as described above, wherein the wind turbine further comprises a controller for control of operation of the wind turbine; the method comprising: 5 providing the generated signals for adjusting the operation of the wind turbine to the controller of the wind turbine.
The present invention also relates to a computer system for wind gust detection in a wind turbine, the wind turbine comprising a tower and a rotor provided with a number of rotor blades on a rotor axis, the rotor being arranged on the tower; 10 wherein the computer is provided with a central processing unit, memory, and sensors, the memory and sensors being connected to the central processing unit, and the central processing unit is arranged, when the wind turbine is in use, to carry out the following actions: a) collecting data of a mechanical loading condition of the wind turbine by the 15 sensors, the data of the mechanical loading condition being associated with at least one load exerted on the construction of the wind turbine in a non-rotating reference frame; b) monitoring a temporal evolution of the data of the mechanical loading condition, comprising: bl) determining a steady state component of each of the data associated with 20 mechanical load on the wind turbine; b2) obtaining residuals of each of the data of the mechanical loading condition by removing the respective steady state components for the respective data, the temporal evolution being determined for the obtained residuals; c) detecting by a test condition if the temporal evolution indicates that a wind gust is 25 evolving; d) based on the result of the test condition, generating signals for adjusting the operation of the wind turbine.
According to an aspect there is provided a computer system as described above, wherein the computer system is arranged with a Kalman filter, wherein the residuals 30 are obtained by using the Kalman filter based on a stochastic turbulence model, comprising: determining a residue vector, the residue vector comprising the residuals of the data of the mechanical loading condition.
7
According to an aspect there is provided a computer system as described above, wherein the sensors are arranged on a non rotating part of the construction of the wind turbine for measuring the at least one load; wherein the central processing unit is connected to the sensors, and the collection of 5 data of the mechanical loading condition of the wind turbine comprises measuring said data by the sensors arranged on the construction.
According to an aspect there is provided a computer system as described above, wherein each rotor blade comprises a sensor for measuring data associated with a mechanical load of the rotor blade, 10 the central processing unit is connected to the sensors on the rotor blades, and the collection of data of the mechanical loading condition of the wind turbine comprises, in sequence: al) measuring by the respective sensor at the rotor blade data associated with the mechanical load of each rotor blade; 15 a2) transforming the data associated with the mechanical load of each rotor blade to the data of the mechanical loading condition.
According to an aspect there is provided a computer system as described above, wherein the wind turbine further comprises a controller for control of operation of the wind turbine, wherein the central processing unit is linked to the controller, and the 20 computer system is arranged for providing the generated signals for adjusting the operation of the wind turbine to the controller.
According to an aspect there is provided a computer system as described above, having a capability as controller of the wind turbine for control of operation of the wind turbine, the controller being arranged for receiving the generated signals for adjusting 25 the operation of the wind turbine.
Also, the present invention relates to computer software stored on a computer-readable medium for a computer system for wind gust detection in a wind turbine, the wind turbine comprising a tower and a rotor provided with a number of rotor blades on a rotor axis, the rotor being arranged on the tower; 30 wherein the computer system is provided with a central processing unit, memory and sensors, wherein the memory and sensors are connected to the central processing unit, wherein the computer software comprises executable code which, when loaded on the computer system, enables the central processing unit to execute the following 8 operations when the wind turbine is in use: a) collecting data of a mechanical loading condition of the wind turbine by the sensors, the data of the mechanical loading condition being associated with at least one load exerted on the construction of the wind turbine in a non-rotating reference frame; 5 b) monitoring a temporal evolution of the data of the mechanical loading condition, comprising: bl) determining a steady state component of each of the data associated with mechanical load on the wind turbine; b2) obtaining residuals of each of the data of the mechanical loading condition by 10 removing the respective steady state components from the respective data, the temporal evolution being determined for the obtained residuals; c) detecting by a test condition if the temporal evolution indicates that a wind gust is evolving; d) based on the result of the test condition, generating signals for adjusting the 15 operation of the wind turbine.
According to an aspect there is provided computer software as described above, wherein the residuals are obtained by performing a Kalman filter procedure based on a stochastic turbulence model, comprising: determining a residue vector, the residue vector comprising the residuals of the data of 20 the mechanical loading condition.
Additionally the present invention relates to a computer-readable medium that comprises computer executable code which, when loaded on the computer system as described above, enables the computer system to execute the method as described above.
25 Further the present invention relates to a wind turbine comprising a tower and a rotor provided with a number of rotor blades on a rotor axis, the rotor being arranged on the tower, further comprising a computer system as described above.
Brief description of drawings 30
The invention will be explained in more detail below on the basis of a number of drawings, illustrating exemplary embodiments of the invention. The drawings are intended exclusively for illustrative purposes and not to restrict the inventive concept, which is defined by the claims.
9
In the drawings:
Figure 1 shows schematically a wind turbine;
Figure 2 shows schematically a front view of the rotor of the wind turbine;
Figure 3 shows schematically a relationship between the orientation of wind, rotor 5 plane and mechanical loads exerted on the wind turbine;
Figure 4a, 4b show a relationship between mechanical loads on the rotor plane for a first and second class of detectable wind variations respectively;
Figure 5a, 5b show a relationship between mechanical loads on the rotor plane for a third and fourth class of detectable wind variations respectively; 10 Figure 6a, 6b show a relationship between mechanical loads on the rotor plane for a fifth and sixth class of detectable wind variations respectively;
Figure 7 shows a flow diagram of a method for gust detection according to a first embodiment of the present invention;
Figure 8 shows a flow diagram of a method for gust detection according to a second 15 embodiment of the present invention;
Figure 9 shows schematically a computer system for carrying out the method of gust detection according to an embodiment of the invention;
Figures 10a, 10b show graphical representation of a method for gust detection in accordance with an embodiment of the present invention.
20 In the following figures, the same reference numerals refer to corresponding components in each of the figures.
Detailed description 25 The system and method for gust detection according to the present invention are based on the application of the wind turbine itself as a detector of the characteristics of the wind field. Basically, the system and method of the present invention are based on the detection of unsteady load components exerted on the rotor and/or the rotor blades to identify the presence of a wind gust.
30 Figure 1 shows a schematic view of a wind turbine WT. Wind turbine WT
comprises a rotor R with a rotor shaft RH provided with a number of rotor blades Bl, B2, B3. The rotor R is coupled to a transmission C for attaching the rotor shaft RH to an electric generator G by means of the rotor shaft. The electric generator G is provided with an output for the output of electrical energy, for example through a power 10 converter (not shown) to an electricity network. The assembly of rotor R, transmission C and generator G is located in a nacelle L on a tower T. The assembly of rotor R, transmission C and generator G is pivotally connected to the tower T by means of a yaw system KS.
5 The transmission C may be excluded when the rotor shaft RH is coupled directly to the generator, i.e. in the case of a direct-drive generator.
In each rotor blade Bl, B2, B3 a deflection moment of the respective rotor blade can be measured by means of a sensor. The deflection moment of a rotor blade is defined as the moment of force in the rotor blade, the vector of which is directed 10 perpendicular to the plane in which the rotor shaft and the longitudinal axis of the rotor blade lie. This will be explained in more detail with reference to figure 3.
In an embodiment of the system, each rotor blade Bl, B2, B3 is provided with a respective sensor SI, S2, S3 attached to the blade root in close proximity to the rotor shaft RH for monitoring a value of the deflection moment in the respective rotor blade. 15 The blade root is the portion of the blade that is attached to the rotor shaft RH.
Additionally, in a further embodiment the radial and tangential force in the root of the respective rotor blade may also be measured.
Furthermore, in an embodiment, the deflection force in the root of the respective rotor blade may also be measured.
20 Below, the wind turbine will be described in terms of a fixed part, i.e. a non rotating part, which will be referred to as the construction of the wind turbine and a rotating part which relates to the rotor and its parts, inter alia, the rotor blades and rotor axis.
Figure 2 shows schematically a front view of the rotor R of the wind turbine WT. 25 In this embodiment the wind turbine has three rotor blades, in other embodiments the number of rotor blades may be different (i.e., at least two or more than three).
The dashed circle RP indicates the outer circle of the rotor plane with a rotor shaft RH as axial center.
Figure 3 shows schematically a relationship between the orientation of wind, 30 rotor plane and mechanical loads exerted on the wind turbine.
On the wind turbine WT that is directed towards the wind a number of forces and moments are exerted due to the wind which has an average wind velocity WY. The average wind velocity may determine the rotor speed RS.
11
On the average wind velocity a number of 3D stochastic wind velocity variations are superimposed: a longitudinal turbulence u in the direction along the rotor axis RH, a lateral (horizontal) turbulence v and a vertical turbulence w.
As a result of the wind that interacts with the rotor on the rotor plane RP, a thrust 5 force Fa on the wind turbine WT along the direction of the rotor axis RH is generated. Additionally, perpendicular to the rotor axis, on the wind turbine a horizontal force Fh and a vertical force Fv are generated. Furthermore, a driving moment Md on the wind turbine along the direction of the rotor axis RH is generated. A tilt moment Mt (horizontal) and a yaw moment My (vertical) are generated, perpendicular to the rotor 10 axis R. Note that these forces and moments are in a fixed (non-rotating) reference frame.
The generated forces and moments Fa, Fv, Fh and Md, Mt, My will show variations due to the turbulence variations u, v, w, and also due to wind condition variations when they are occurring . The variations are denoted as AFa, AFv, AFh and 15 AMd, AMt, AMy, respectively.
As a next step, the values of the mechanical parameters need to be related to the characteristics of the wind field. It is observed that the six variations relate to six classes of wind variations (gust classes). Below, the gust classes are described in more detail with reference to figures 4a, 4b, 5a, 5b, 6a, 6b.
20 Figure 4a, 4b show a relationship between mechanical loads on the rotor plane for a first and second class of detectable unsteady wind condition variations respectively.
Figure 4a shows schematically the rotor plane RP of the rotor of the wind turbine. A vertical axis y and horizontal axis x are shown.
The average wind is directed in a main wind direction WD substantially parallel 25 along the longitudinal direction of the rotor axis RH. The first class of wind velocity variation is defined by a rotor coherent variation of the wind velocity AWVU , i.e. a variation of the wind velocity with full spatial coherence over the rotor plane RP (subscript u is used for identifying the unsteady part of a variation A) . Due to the exposure of the rotor to the variation of the wind AWVU, the thrust force Fa will vary 30 by AFau. Also the driving moment Md will vary by AMdu. The other forces and moments Fv, Fh and Mt, My are substantially not affected by the wind velocity variation AWVU.
12
In Figure 4b, the effect is shown for a second class of wind condition variation that is referred to as a backing and veering wind (a wind direction that turns counterclockwise relative to the main wind direction with height) which in an upper or lower portion of the rotor plane RP veers in a veering direction different from the main 5 direction WD, while in the opposite portion the wind backs to a backing direction different from the main direction WD and the veering direction.
As result of this second class wind condition variation AWVU, the thrust force Fa will vary by AFau, and the driving moment Md will vary by AMdu.
Thus, the first and second classes of wind variations are detectable but can not be 10 isolated from each other. However, it will be appreciated by the skilled in the art that due to the aerodynamic conversion of the wind flow on the rotor blade, the axial and tangential forces on each rotor blade develop differently for different type of wind variations. As a result, the ratio of AFau/AMdu will be relatively low for a rotor coherent variation of the longitudinal wind velocity and relatively high for a backing 15 and veering wind condition.
This ratio for a backing and veering wind condition is usually at least 10 times higher than for a coherent gust condition (typically larger than 1.0 and smaller than 0.1 respectively); like a paradox, tangential wind speed variations hardly influence the tangential blade forces and therefore hardly the driving moment AMdu.
20 Figure 5a, 5b show a relationship between mechanical loads on the rotor plane for a third and fourth class of detectable wind variations respectively.
In figure 5a as a third class of wind condition variation, a wind condition variation known as a jet stream is shown. The jet stream condition relates to a variation of the wind velocity AWVU at only the upper part of the rotor plane RP. This may occur 25 when the top of the rotor is struck by a so called low level jet. The variation AWVU is in the substantially same direction as the main wind direction WD.
As a result, the horizontal force Fh shows a variation AFhu and tilt moment Mt will vary by a variation AMtu. The other forces and moments Fa, Fv and Md, My are substantially not affected.
30 As shown in figure 5b, a fourth class of wind condition variation relates to a change of the wind direction: the wind changes from main direction WD to a different wind direction WD2.
13
As a result, the horizontal force Fh shows a variation AFhu and tilt moment Mt will vary by a variation AMtu.
Thus, the third and fourth classes of wind variations are detectable but can not be isolated from each other. However, it will be appreciated that the ratio of AFhu/AMtu 5 will be relatively low for the variation of the jet stream and relatively high for the wind direction change condition.
Figure 6a, 6b show a relationship between mechanical loads on the rotor plane for a fifth and sixth class of detectable wind variations respectively.
In figure 6a a fifth class of wind condition variation is shown that relates to a 10 partial wake condition for a side portion of the rotor plane of the wind turbine. This variation occurs when a portion of the rotor of the wind turbine is in the lee of an object such as another wind turbine. The variation can be modeled by a counteracting wind WF that acts on the portion of the rotor that is in the wake. The counteracting wind WF is opposite to the main wind direction WD.
15 Asa result of the partial wake, a variation of the vertical force AFvu is generated.
Also, a variation of the yaw moment AMyu is generated. The other forces and moments Fa, Fh and Md, Mt are substantially not affected.
In figure 6b as a sixth class wind condition variation the effect of a sloping wind is shown. At the rotor plane RP, the wind direction changes from the incoming main 20 wind direction WD to an upward or downward direction. This sixth class wind variation condition results in a variation of the vertical force AFvu and a variation of the yaw moment AMyu. The other forces and moments Fa, Fh and Md, Mt are substantially not affected.
Again, the fifth and sixth classes of wind variations are detectable but can not be 25 isolated from each other. However, it will be appreciated that the ratio of AFvu/AMyu will be relatively low for the partial wake condition and relatively high for the sloping wind condition.
Figure 7 shows a flow diagram of a method for gust detection according to a first embodiment of the present invention.
30 According to the method of the present invention a wind condition variation of one of the above mentioned six classes is detected from deflection force/deflection moment related data measured on the construction of the wind turbine. The incoming wind interacts with the wind turbine rotor by exerting mechanical loads on the rotor.
14
Since the wind may vary over time, the mechanical load on the construction of the wind turbine varies more or proportionally.
Basically, in this embodiment the wind turbine comprises on its construction sensors for measuring signals that are associated with the thrust force Fa, the horizontal 5 force Fh, and the vertical force Fv and further sensors for measuring signals that are associated with the driving moment Md, the tilt moment Mt, and the yaw moment My.
In this embodiment, the present invention relates to a procedure 100 for detection of wind condition variations to be carried out by a monitoring and control system of a wind turbine.
10 Asa first optional action 101, the procedure may comprise some initializations, such as a system initialization.
Next, in action 102, the procedure comprises measurement of data associated with the mechanical load (Fa, Fh, Fv, Md, Mt, My) on the construction of the wind turbine These data can be collected in a mechanical load column vector Q.
15 Then in action 103, the procedure comprises determining steady state components and residuals for the forces Fa, Fh, Fv and moments Md, Mt, My by isolating stationary stochastic variations that are caused by turbulence. Steady state components and residues of the mechanical loads in the mentioned order can be collected in respective column vectors (Qs, r).
20 The steady state components of the loads Qs may be defined as a simple moving average of each of the measured loads Q over a fixed time period (Qlf).
The fixed time period is longer than a time period over which a wind condition variation occurs but shorter than the time scale of the variability of the wind, which typically amounts to 10 minutes. It is noted that the moving average for each steady 25 state component may be determined in any conceivable manner known to the skilled person.
The steady component can also be estimated as the sum of moving average mechanical load values on the time scale of the variability (Qvlf) and the outputs of a Kalman filter (Aq) that is driven by the difference Q-Qvlf (q).
30 On the time scale of the variability of the wind, both the working conditions for the Kalman filter and the turbulence properties use to change. The parameterization of the Kalman filter may be adapted on that time scale.
15
After establishing the steady state components the procedure determines residuals for the forces Fa, Fh, Fv and moments Md, Mt, My by a removal of the respective steady state component from the thrust force Fa data, the horizontal force Fh data, the vertical force Fv data, the driving moment Md data, the tilt moment Mt data and the 5 yaw moment My data. As a result, data for a residual value thrust force 8Fa, a residual horizontal force 5Fh, a residual vertical force 5Fv, a residual driving moment 6Md, a residual tilt moment 8Mt and a residual yaw moment óMy are obtained.
Each residual is an estimate for the unsteady variation of the respective force or moment. In this method 8Fa and 8Md are estimated contributions by wind condition 10 variations to the overall variations AFax and AMd, 5Mt and 5Fh are estimated contributions by wind condition variations to the overall variations AMt and AFh, and 8My and óFv are estimated contributions by wind condition variations to the overall variations AMy and AFv.
An embodiment of a Kalman filter is derived in Appendix A from a model in 15 which three pairs of wind speed coordinates ([ua,va], [ut,vt] and [uy,vy]) are the sole channels for the affection of the mechanical loads from the turbulence. These wind speed coordinate pairs strongly relate to before mentioned mechanical load pairs [Fa,Md], [Mt,Fh] and [My,Fv], while the latter relate to wind condition variation pairs [rotor coherent gust, backing and veering wind], [jet stream, wind direction change] 20 and [partial wake condition, sloping wind] respectively, as per figures 4a up to 6b.
The points of departure of a model formulation for the channeling of the influence of turbulence through only six wind speed coordinates are: - assuming a cylinder with wind velocity variations from turbulence that moves with the very slowly varying mean longitudinal wind speed (Uvlf) through the rotor 25 plane in which blades that rotate with a constant angular speed are slicing helix-like shapes (rotationally sampled turbulence); - rotation angle dependent multi-blade transformation of the rotationally sampled turbulence towards the coordinate system of the loads Q; - assuming time-invariancy of the relative distribution of wind speed variations 30 over the radial coordinate of the rotor plane (s).
A person skilled in the art can appreciate a power spectrum matrix for the turbulence driven variations of the six wind speed coordinates (ws) and subsequently derive a linear model with six Gaussian distributed zero-mean purely random variables 16 as inputs (e) and stationary stochastic variations in ws as outputs. Let e have the 6x6 covariance matrix E. The evolution ws can be formulated through a recurrent relationship between two subsequent time points (n, n+1): [EqMwindst]: 5 xw(n + l) = Aw-xw{n) + Bw-e(n) ws(n) = Cw· xw(n) + Dw· e(n)
This equation set represents a linear time-discrete state space representation of the steady wind model with state vector xw and parameter matrices Aw, Bw, Cw and Dw. 10 The affection of mechanical loads q by a wind speed variation (w) is also modelled by a linear state space representation. Let w be the sum of stationary stochastic variations ws and a potential unsteady variation in any of the six wind speed coordinates that arises from a wind condition variation (wu). Because of linearity, the wind speed coordinate variations ws and wu can be considered as separate input vectors 15 to the wind turbine model: [EqMturb]: xt(n+\) = At-xt(n) + Bt- ws(n) + Br wu(n) q(n) = Cr xt(n) + Dt· ws(n) + Dt- wu(n) 20 The related Kalman filter provides load signal estimations on time point n (Aq(n)) from load signal measurements up to time point n-1 ({q}(n-l)) as per the following state space representation: [EqKf]: x{n +1) = (A -K- C)· x{n) + K- q(n) 25 q(n) = C x{n) with κ=Μ=ί Bt'D;l {bw- (Dt Dw) j
The Kalman filter state vector Λχ is an estimation of the stacked state vectors of 30 the wind model and turbine model ([xw' xt']').
The evolution of the Kalman filter residues r depends on the occurrence of a wind condition variation (wu!=0): [EqKfr]: 17 r(n) = q(n) -q(n) = (D,· Dw)· e(«) if w„ = 0, r(n) = q(n)-q(n) = 5 =(A· flw)· e(«) + C· ξ(η) + ΰ(· wu(n) ifwM*0 with state estimation error ξ(η) evolving as f o ) ξ(η + 1) = (Λ-Κ· C> ξ(η)+ i · wu(n) V-öw· vw j
Alternative Kalman filter embodiments can be obtained through any alternative 10 wind speed coordinates as long as the six gust classes can be uniquely identified from combinations of these wind speed coordinates. Also, the Kalman gain matrix K may deviate from the expression in [EqKf], e.g. through just a scale factor smaller than 1 in order to achieve a more smooth behavior.
In a next action 104, the method comprises monitoring the evolution of residues 15 r. The occurrence of an unsteady mechanical load variation is searched for in a history of stored residue data ({r}). In steady wind conditions a residue will evolve with modest changes and with short-term random changes of sign whereas during a wind condition variation a residue will mainly evolve with large changes that show a definite trend or even a pattern. The residue vector elements that relate to the thrust force Fa 20 (r[ 1 ]), tilt moment Mt (r[3]) and yaw moment My (r[5]) provide the most significant information on the occurrence of a wind condition variation in the first or second, the third or fourth, or the fifth or sixth class.
The aerodynamic behavior of rotor blades makes that the occurrence of a wind condition variation in the second, fourth and sixth class has very little effect on the 25 driving moment Md, horizontal force Fh and vertical force Fv respectively, which are related to residues r[2], r[4] and r[6].
The method may be performed for any subset of the six residues with consequent reach of wind condition variation classes.
Subsequently, in action 105 if an unsteady load variation is detected in a history 30 {r} the class of the wind condition variation is determined. This action may comprise the appreciation of the ratio between the residues that relate to a pair of mechanical loads that are both affected by a same wind condition variation, viz. the thrust force and 18 driving moment, the tilt moment and horizontal force, and the yaw moment and vertical force for the first and second, the third and fourth and the fifth and sixth class of wind condition variation classes, respectively. The size of the ratio between the residues identifies which wind condition variation class applies, just as the ratio between the 5 related mechanical load variations does as per figure pairs [4.a, 4.b], [5.a, 5.b] and [6.a, 6b], A generalized maximum likelihood ratio test (GLRT) is a feasible option for the monitoring of the evolution of a residue that is obtained with a Kalman filter. A GLRT provides on a time instance (t) an upper limit for the likelihood ratio between two types 10 of wind evolution (L(t)). A large value for this upper limit implies wind evolution in unsteady conditions, that is to say a gust is occurring; a small value implies wind evolution in steady conditions, that is to say only turbulence applies. The likelihood ratio concerns two conditional probability density functions of the residue history in a time span W preceding time instance t ({r}(t, t-W), short form (r}(t)). The numerator in 15 this ratio (p({r}(t)|Hl)) is appreciated conditional to the validity of an alternative hypothesis (HI), in which the residue is assumed to be affected by unsteady wind variations from any time instance in the interval t-W up to t-1.
The denominator (p({r}(t)|H0)) is appreciated conditional to the validity of a null hypothesis (HO), in which the residue is assumed to only represent effects of 20 turbulence.
GLRT's for six classes of the alternative hypothesis HI allow for the detection of gusts from all six distinctive classes of unsteady wind speed variations as per fig 4.a, 4.b, 5.a, 5.b, 6.a, 6.b.
An embodiment of a GLRT is derived in Appendix B for the residue vector as per 25 the included embodiment of the Kalman filter as per Appendix A . Let alternative hypothesis class HI [i] for this embodiment of a GLRT belong to the unsteady wind speed variations in the i'th element of wu (wu[i]) for the included embodiment of the Kalman filter. The performance of a GLRT for HI [i] requires the assumption that wu[i] evolves as a known amplitude-normalized unsteady wind speed variation that is scaled 30 with an unknown amplitude (a[i]), that is to say as a gust class evolution prototype GCP[i] multiplied by a[i].
The six prototypes (GPC[i],i=l, 2, 3, 4, 5, 6} thus provide an amplitude-normalized change in time of one of the respective 19 wind speed coordinates: ua (thrust oriented axial) for a rotor coherent gust, va (thrust oriented tangential) for a backing and veering wind, ut (tilt oriented axial) for a jet stream, 5 vt (tilt oriented tangential) for a wind direction change, uy (yaw oriented axial) for a partial wake condition, vy (yaw oriented tangential) for a sloping wind
So the GLRT for a gust class prototype GCP[i] provides on a time instance (t) the 10 generalized maximum likelihood ratio (L[i](t)) between the presence and absence of a gust that evolves as per GCP[i] with scaled amplitude from the gust class belonging to wind speed coordinate wu[i]. The starting time instance of the gust lies anywhere in time span W preceding time instance t.
The generalized maximum L[i](t) is obtained as the maximum of specific 15 maximum likelihood ratio's. Each specific maximum (L[i|k](t)) is derived for a fixed value of the beginning time instance of the gust (k), which ranges fromt-W upto t-1. The assumed unsteady wind speed coordinate then agrees with the amplitude-scaled gust class prototype evolution with starting time instance k, denoted as a[i|k]*{GCP[i|k]}(t).
20 Thus |wu[i]}(t)= a[i|k]* (GCP[i|k]}(t) is the assumption on unsteadiness for the specific alternative hypothesis (HI [i|k]).
The specific likelihood ratio L[i|k](t) is then derived from the residue history (r}(t) and amplitude-normalized gust evolution (GCP[i|k]}(t), while a[i|k] is eliminated through maximizing the ratio between the conditional probability density function 25 p( {r} (t)|H 1 [i|k]) and p( {r} (t)|H0).
The natural logarithm of L[i|k](t), which actually is a ratio upper limit, includes a modeled evolution of the residues that relates to the correlation with a predictive function. This concerns the correlation of the residues (r}(t) with the assumed contribution history (~μ }(t|i,k) to (r}(t) from the amplitude-normalized gust class 30 evolution prototype realization {GCP[i|k]}(t} for a gust in wind speed coordinate wu[i] that starts on time instance k: [EqLik] 20 ί ' Υ Σμ(η |/',£))'· R l-r(n) log(I[i I *](()) = -— Σβ(η \ hk)y· R~1· μ(η \ i,k)
n=t—W
The amplitude ami[i|k] that belongs to the ratio upper limit L[i|k](t) is given by: [Eqaik]: t Σμ(η I i,k)y- R~[- r(n) 5 aml\i\k]=-afX·- Σμ(η I i,k)y- R~ · μ(η i,k)
n=t—W
The interval between subsequent time instance n and n+1 (ΔΤ) and the window size W*AT are design parameters. A window size of a few seconds usually suffices while the time interval ΔΤ is to agree with a sample frequency (β=1/ΔΤ) that amply captures six times the rotational frequency (6p) of a 3 bladed wind turbine. For a 10 rotational speed of 15 rpm, the 6p frequency amounts to 1.5 Hz.
A time interval of 0.1 seconds, which agrees with 10 Hz sample frequency, then will suffice. For the skilled in the art it is known that the wind speed coordinates in the coordinate system of the mechanical loads Q include variations in low frequencies, that is to say up to ca 0.1 Hz, and variations around multiples of B times the rotational 15 frequency (number of rotor blades B) due to turbulence. Practically, the variations around the Bp- and 2Bp-frequency are relevant as concerns gust detection, i.e. 3p and 6p for a 3 blade rotor.
GFRT's for more than one gust class evolution prototype can be performed for an unsteady wind speed coordinate wu[i]. For example the well known '1 - cosine' 20 function and so called 'mexican hat', which are common evolutions of rotor coherent gusts through wind speed coordinate ua, can be applied as normalized gust class evolution prototype GCP[i].
Both positive and negative evolutions can be applied, that is to say in order to detect a fast rising wind and fast falling wind respectively. For other gust classes 25 deviating evolution prototype can be applied. For example, a partial wake condition uses to grow linearly in time through 'alignment rotation' (yawing) of an upwind turbine in a wind array assemblage.
21 A skilled in the art can appreciate a valid threshold value for accepting the null or alternative hypothesis for the generalized maximum L[i](t) as the maximum over k of (L[i|k](t) for k between t-W and t-1. The appreciation of a threshold value T[i] for an alternative hypotheses class HI [i] may include wind turbine simulation in the time-5 domain in which turbulence and relevant wind condition variations are simulated. In order to achieve robust gust detection, the skilled in the art will perform simulations with wind condition variations of which the amplitude-normalized evolution intentionally differs from used types of gust class evolution prototypes (GCP[i]).
A gust as per wu[l] or wu[2] affects both load coordinates Q[l] and Q[2], so also 10 affects both residues r[l] and r[2]. As a consequence, the generalized maximum likelihood ratio's L[l](t) and L[2](t) may both sharply rise when t passes the beginning time instance of a gust as per wu[l] or wu[2]. When a gust is detected from L[l](t) or L[2](t), the ratio between r[l] and r[2] determines which unsteady wind speed coordinate applies (wu[l] or wu[2]).
15 This also holds for sharply rising ratio pairs (L[3](t), L[4](t)} and {L[5](t), L[6](t)}.
Alternatively, the monitoring of a residue that is obtained otherwise than from a Kalman filter may comprise a matching of its evolution with a modeled evolution that relates to the correlation with a predictive function as per [EqLikt], The threshold 20 exceeding of the maximum of the right hand side of expression [EqLikt] over k (k from t-W to t-1) can yet be used for gust detection but will not agree anymore with the natural logarithm of a generalized maximum likelihood ratio.
As yet other alternatives, the monitoring of a residue that is obtained either from a Kalman filter or as the difference between measured loads Q and simply determined 25 moving average load values Qlf may comprise a matching of its evolution with any other modeled evolution of the wind condition variation for the monitored residual. Such a modeled evolution may relate to e.g. a cumulative sum determination or a correlation with a predictive function other than by [EqLikt], Also a parameter fitting procedure to any predictive function may be used.
30 In an action 106, the method may comprise a control of the operation of the wind turbine by adjusting settings of the wind turbine, such as a pitch angle of the rotor blades, an electro-magnetic counter torque of the generator, a yaw angle of the rotor as set by the yaw system KS of the wind turbine, or by other ways of changing the 22 aerodynamic conversion factor of the rotor, e.g., by changing settings of a tip vane or flap on one or more of the rotor blades.
This adjustment is carried out to anticipate the gust that is evolving.
The control allows a reduction of the effect of the gust on the wind turbine and its 5 operation. Depending on the gust class and the magnitude of the gust, the method may advantageously provide that a full stop of the wind turbine can be avoided. For a relatively low magnitude of the gust, the operation of the wind turbine can be continued possibly with some adjustments of the settings of the wind turbine. Only under severe conditions, a full stop may still be necessary. Also, the method has the advantage that if 10 a full stop is executed, the method provides information on the wind conditions that caused the full stop. The availability of this information may be helpful during the assessment of the event and during the inspection before resuming operation.
Since the method relates to monitoring and controlling the operation of the wind turbine, the above actions are typically executed in either a continuous mode or in a 15 consecutive looping mode. As will be appreciated by the skilled person, some actions may be carried out in parallel.
The procedure 100 - 106 as per fig. 7 is valid for any alternative set of measurements associated with wind characteristics in the non rotating reference frame, that is to say from loads with other orientations or movements in the non-rotating 20 reference frame, as long as a non-singular relationship with the above mentioned six gust classes exists.
Figure 8 shows a flow diagram of a method for gust detection according to a second embodiment of the present invention.
According to yet another embodiment of the method of the present invention a 25 wind condition variation of one of the above mentioned six classes is detected from deflection force/deflection moment related data measured on the rotor blades. The incoming wind interacts with the rotor by exerting mechanical loads on the rotor blades. Since the wind may vary over time, the mechanical load on each rotor blade varies accordingly. The present invention recognizes that the variation of the 30 mechanical loads on the rotor blades can be used for detection of wind condition variations. In an embodiment, the present invention relates to a procedure 200 for detection of wind condition variations to be carried out by a monitoring and control system of a wind turbine.
23
As a first optional action 201, the procedure may comprise some initializations, such as a system initialization.
Next, in action 202, the procedure comprises measurement of data associated with the mechanical load on each rotor blade Bl, B2, B3 as a function of the rotor 5 blade azimuth angle by using the respective mechanical load sensor S1, S2, S3. Thus, for each rotor blade, data associated with the mechanical load in a rotating reference frame (i.e. on the rotor) and the azimuth angle are determined.
Then in action 203, the procedure comprises transforming the mechanical load data for each rotor blade at given azimuth to data associated with forces Fa, Fv, Fh and 10 moments Md, Mt, My in the non-rotating reference frame. This transformation may be done by using the so-called Coleman transformation. The Coleman transformation is capable of transforming all quantities (deflection force, blade moment, etc.), measured for each rotor blade on the rotating reference frame to parameters associated with forces and moments in the non rotating reference frame.
15 The data for blade root flapwise bending moments, for instance, are transformed into data for the rotor tilt moment Mt and yaw moment My.
The data for blade leadwise bending moments are transformed into data for the rotor driving moment Md.
The data for the blade flapwise deflection forces are transformed to the rotor 20 thrust force Fa. The data for the blade leadwise deflection forces are transformed to the rotor horizontal force Fh and to the rotor vertical force Fv. Subsequently, the procedure is continued by steps 204 - 207 which are identical to steps 103 - 106 as described above with reference to figure 7.
It is noted that the wind condition variations as described above can also be 25 detected from alternative measurements in a rotating reference frame associated with wind characteristics in the non rotating reference frame, that is to say blade loads with other orientations, or loads on the rotor shaft, or movements of the blades and or rotor shaft as long as a non-singular relationship with the above mentioned six gust classes exists.
30 The blade loads or movements can also be transformed through other types of data transformations than the Coleman transformation. The use of loads on the rotor shaft implies another type of transformation to the non-rotating frame, that is to say a transformation that maps two non-coinciding rotating orientations that are 24 perpendicular to the rotation axis of the shaft to fixed orientations. For example these alternative measurement signals include the acceleration of a radial station of the rotor blades or moments and forces on the rotor shaft.
Figure 9 shows schematically a computer system for carrying out the method of 5 gust detection according to an embodiment of the invention.
The computer 8 comprises a central processing unit 21 with peripheral equipment. The central processing unit 21 is connected to memory means 18, 19, 22, 23, 24 which store instructions and data. Furthermore, the computer may have one or more reading units 30 (for example, to read floppy disks, CDROMs, DVDs, portable 10 non-volatile memories, etc.), a keyboard 26, and a mouse 27 as input devices and output devices, a display 28 and a printer 29. Other input units, such as a trackball, a scanner and a touch screen, as well as other output devices, can be provided.
Furthermore, the central processing unit 21 can be provided with a network 32 for data communications with a network 33. The network adaptor 32 is connected to 15 the network 33. The network can be any network suitable for data communications. For example, the network can be a Local Area Network (LAN) or a Wide Area Network (WAN). Other computer systems can be linked to the network 33, which can communicate with the computer 8 via that network connection 32.
The memory means shown in Figure 8 comprise one or more means 20 selected from RAM 22 (E) EPROM 23, ROM 24, tape unit 19, and hard disk 18.
However, there may be more and/or other storage units provided, as will be clear to an expert in the field. Moreover, if necessary, one or more of the memory resources may be placed at a distance from the central processing unit 21.
The central processing unit 21 is shown as a single unit, but may also comprise 25 various secondary processing units that operate in parallel or are controlled by a single central processing unit. These secondary processing units can be arranged at some distance from each other, as will be known to those skilled in this field.
The computer 8 comprises an interface 34 for receiving signals from the load measuring sensors SI, S2, S3 in each of the rotor blades, which sensors are arranged 30 for measuring signals of the deflection moment and possibly the radial and tangential force in the root of each rotor blade. The interface 34 is connected to the central processing unit 21.
25
The computer 8 may also comprise an interface 35 for transmitting control signals to each of the rotor blades for defining the respective blade pitch or for controlling other devices for influencing the aerodynamic conversion of each rotor blade, such as micro-tabs, flaps and synthetic jets. The interface 35 may also allow for 5 setting an electro-magnetic counter torque of the generator or a yaw angle of the rotor as set by the yaw system KS of the wind turbine. The interface 35 is connected to the central processing unit 21.
The computer 8 comprises functionality in hardware and/or software in order to execute a procedure according to the above method. The computer is equipped, or is 10 operable in the way of computer software, to perform calculations in accordance with one or more of the aforementioned methods. Such computer software stored in or on a computer-readable medium, enables the computer, after being loaded from the computer-readable storage into the memory of the computer, to carry out the method of gust detection according to the present invention.
15 In an embodiment the computer is a SCADA system (SCADA: supervisory command and data acquisition) that is suitable for data processing and analysis.
In an embodiment the computer is arranged to execute the procedure 200 as described above in accordance with the method for gust detection in a wind turbine.
The processing unit of the computer is arranged for receiving signals with the aid 20 of the measuring sensors to obtain data associated with the deflection force/deflection moment in each rotor blade.
Next the processing unit is arranged for transforming the data associated with the mechanical load for each rotor blade at given azimuth to data associated with forces Fa, Fv, Fh and moments Md, Mt, My in the non-rotating reference frame. This 25 transformation may be done by using the so-called Coleman transformation.
Then, the processing unit is arranged for determining steady state components and residuals for Fa, Md, Mt, Fh, My, Fv in the non-rotating reference frame (as described above with reference to action 204 of procedure 200 in figure 8). Subsequently, the processing unit is arranged for monitoring the (temporal) evolution 30 of each residual. The processing unit is capable of detecting if a gust is evolving based on the evolution of the residual. The detection is based on a test criterion, such as exceeding a threshold value or a matching test in which a correlation between a matching function and the temporal evolution of the residual.
26
Additionally, the processing unit is arranged to determine a gust class for the wind condition variation that relates to the evolution of the residual.
Finally, the processing unit is arranged for controlling the operation of the wind turbine by adjusting settings of the wind turbine based on the detected evolving gust.
5 Alternatively, the processing unit is arranged to transmit signals to a further controller that operates the wind turbine, in which the transmitted signals comprise information to the further controller to adjust the operation of the wind turbine, in anticipation of the evolving gust.
In an alternative embodiment, the computer is arranged to execute the procedure 10 100 as described above in accordance with the method for gust detection in a wind turbine, by using data associated with the loads Fa, Fh, Fv and moments Md, Mt, My that act on the construction of the wind turbine.
Figure 10a, 10b show graphical representations of a method for gust detection in accordance with a respective embodiment of the present invention.
15 In figure 10a, the monitoring of one residual is graphically represented when using a threshold value detection. The evolution of one residual is depicted by curve R1 as a function of time t. A horizontal dashed line represents the threshold value. At given time tl, the residual exceeds the threshold value. At time tl, the probable occurrence of a relevant wind condition variation that indicates an evolving gust is detected.
20 In figure 10b, the monitoring of one residual is graphically represented when using a matching function M. The evolution of one residual is depicted by curve R1 as a function of time t. The matching function M is schematically depicted at various time instances. At given time t2, the residual occurs to evolve along the matching function. At time t2, the probable occurrence of a relevant wind condition variation is detected. If 25 a GLRT is applied for detection, the evolution of a residue (r[i]}(t) along a matching function coincides with a sharp rise of the generalized maximum likelihood ratio L[i](t).
It is noted that in an embodiment, the invention relates to a method and system in which a single load is being measured, a steady state component is determined, a 30 residual is determined and subsequently the temporal evolution of the residual for that single load is monitored.
Next, the temporal evolution of the residual for this single load is tested against the test condition. The exceeding of a threshold value by this residue or the evolution 27 along a matching function enables to detect whether a gust out of only one pair of gust classes is evolving but not which pair member applies. Based on this incomplete observation, it yet can be decided to adjust the operation conditions of the wind turbine. In this embodiment, no wind gust class can be established but still an improvement of 5 the wind turbine control is obtained in comparison to the prior art.
The invention has been described with reference to some embodiments. Obvious modifications and alterations will occur to the skilled in the art upon reading and understanding the preceding detailed description. Other embodiments of the invention can be conceived and reduced to practice without departing from the spirit of the 10 invention, the scope of the invention being limited only by the appended claims. The above description is not intended to limit the scope of the invention.
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Appendix A: Kalman filter embodiment with turbulence modelling through six wind speed coordinates
An embodiment of a Kalman filter is derived from a model in which three pairs of wind speed coordinates ([ua,va], [ut,Vt] and [uy,vy]) are the sole channels for the 5 affection of the mechanical loads from the turbulence. These wind speed coordinate pairs strongly relate to before mentioned mechanical load pairs [Fa,Md], [Mt,Fh] and [My,Fv], while the latter relate to wind condition variation pairs [rotor coherent gust, backing and veering wind], [jet stream, wind direction change] and [partial wake condition, sloping wind] respectively, as per figures 4a up to 6b.
10 The points of departure of a model formulation for the channeling of the influence of turbulence through only six wind speed coordinates are: - assuming a cylinder with wind velocity variations from turbulence that moves with Uvlf through the rotor plane in which blades that rotate with a constant angular speed are slicing helix-like shapes (rotationally sampled turbulence); 15 - rotation angle dependent multi-blade transformation of the rotationally sampled turbulence towards the coordinate system of the loads Q; - assuming time-invariancy of the relative distribution of wind speed variations over the radial coordinate of the rotor plane (s).
Usually, the model parameters are adapted on the time scale of the variability of 20 the wind, which is typically 10 minutes.
Rotationally sampled turbulence by a rotor blade with ranking number b comprises an axial wind speed distribution (waxb) over the radial coordinate(s) as well as a tangential wind speed distribution (wtgb) over s.
25 These wind speed variation distributions in the rotating frame relate to the longitudinal, lateral and vertical turbulence components (u, v, w) through the azimuth position of the rotor blade (ψι,) as per: waxj-, (a0 — A 1///,,0 30 w(gb (s,t) = v(.s·, 1///,,0- sin( 1///,) - w(s,\^b,t)· cos(i///,) where ψι, relates to the rotor azimuth (ψ) and the overall number of rotor blades (B) as per: 29
Vb = ψ + φ -Ϊ)/Β· 2π A multi-blade transformation to the coordinate system of the mechanical load 5 measurements provides a pair of thrust oriented wind speed coordinate distributions (dua, dva) that relates to axial and tangential wind speed variations in such a way that a time-constant value of dua or dva coincides with a constant variation in the thrust force (Fa) and also in the driving torque (Md). The belong relationships are:
B
dua(s,t) = YB- Y,waXb(s,t); b=1
B
10 dva (s, t) = yB-Hvtgb (s, 0; b=1
Second, the multi-blade transformation provides a pair of tilt oriented wind speed coordinate distributions (dut, dvt) that relates to axial and tangential wind speed variations in such a way that a time-constant value of dut or dvt coincides with a constant variation in the tilt moment (Mt) and also in the horizontal force (Fh). The 15 belong relationships are:
B
dut(s,t) = 2 B- Hwax/) (s,t)· sin(i//è); b=1 B
dvt(s,t) = yB- sin(wb); b= 1
Third, the multi-blade transformation provides a pair of yaw oriented wind speed coordinate distributions (duy, dy) that relates to axial and tangential wind speed 20 variations in such a way that a time-constant value of duy or dvy coincides with a constant variation in the yaw moment (My) and also in the vertical force (Fv). The belong relationships are:
B
duy{s,t) = 2/β· (λ0· cos(vb); b=1
B
dvy(s,t) = 2/b- Y,wtgb (s,t)· cos(vb); b=l 25 The skilled in the art can verify the stated coincidences of time-constant wind speed coordinate values and constant mechanic load variations.
30
The scaling factors of the (harmonic) sums in the multi-blade transformation above are arbitrary. The above chosen values of 1/B, 2/B and 2/B match with the well known Coleman transformation for three-bladed rotors.
Finally, time-invariancy of the above mentioned wind speed distributions over 5 the radial coordinate s enables to lumped wind speed coordinates. These can be obtained by integration over s of the wind speed distributions with an arbitrary weighting function (d(s)). The turbulence induced longitudinal, lateral and vertical wind speed variations (u, v, w) now relate through the blade azimuth positions to pairs of rotor effective thrust oriented wind speed coordinates ([ua, va]), tilt oriented wind 10 speed coordinates ([ut, vt]), and yaw oriented wind speed coordinates ([%, vy]) as per:
B [S
ua(t) = \/B· ZJ0<i(s)· u(s,y/b,t)· ds b= 1
B
να(ί) = 1/β· Σ J0 d{s)· [v(s,wb,t)· sm(y/b)-w(s,y/b,t)· cos(y/fc)]· ds b=1 B .s
ut(t)=2/B· ZJ0^> u(s,wb,t)· d**· sin(i/^) b=1 B (S
15 vt(t) = 2/5· Σ J0 d(s)-[v(s,y/b,t)· sin(y^)- w(s^b,t)· cos(yAè)]· ds- sin(y^) b=1 B.s uy(t)=2/B 2J0 d(s)· u(s,\i/b,t)· ds· cos(y^,) b=1
B {S
vy(t) = 2/B- Σ J0 d{s)· [v(s,y/b,t)· sin(y/è) - w(s,\yb,t)· cos(y/b)]· ds· cos(y/b) b=l
Just as for the wind speed coordinate distributions it holds that a time-constant 20 value Δ of a rotor effective wind speed coordinate coincides with a constant variation Δ of a pair of mechanical loads as per:
Aua or Δνα coincides with AFa and AMd;
Aut or Avt coincides with AMt and AFh;
Auy or Avy coincides with AMy and AFv.
25 31
The weighting function d(s) for the radial distribution can be a constant scaling factor like 1/S but can also be used for assigning a different weight over the radius, for example s/S. Any weighting function can be used and they may differ for the different wind speed coordinates.
5 A person skilled in the art can appreciate a power spectrum matrix for the stationary stochastic (steady) part of the six wind speed coordinates in the left hand side of the expressions above for given spectral properties of the spatially distributed turbulent wind velocity components u, v and w, under the conditions of the first and third point of departure: 10 · the rotor azimuth evolves as per a constant angular speed while the moving average wind speed (Uvlf), the carrier of the turbulence, varies on a much larger time scale than that of the turbulence-induced wind speed variations.
• assuming time-invariancy of the relative distribution of wind speed variations over the radial coordinate of the rotor plane (s).
15 Further, the skilled in the art can derive a linear model from this power spectrum matrix with six Gaussian distributed zero-mean purely random variables as inputs and steady state variations of the six wind speed coordinates as outputs. The steady wind speed variations can be collected in a column vector (ws) as well as the purely random variables (e); let e have the 6x6 covariance matrix E. This steady wind model can be 20 formulated through the two vector equations below. The first equation is a recurrent relationship between two subsequent time points (n, n+1). The equations govern the evolution of the effect by turbulence on the lumped wind speed coordinates: [EqMwindst]: 25 xw(n + \) = Aw-xw(n) + Bw-e(n) ws(n) = Cw· xw(n) + Dw· e(n)
This equation set represents a linear time-discrete state space representation of the steady wind model. The state transition equation is parameterized by a state 30 transition matrix (Aw) and input matrix (Bw); the output equation by an output matrix (Cw) and feed-through matrix (Dw). The variables are the input vector (e), state vector (xw) and output vector (ws). Since the steady wind model by [EqMwindstr] provides the creation of time-correlated random variables from purely random variables, which 32 thus have a power spectrum that definitely deviates from unity all-over, it is typed as a noise coloring filter.
On time scales below that of the variability of the wind (changes in Uvlf), the affection of mechanical loads q by any, not too large wind speed variation (w) can also 5 be modeled by a linear state space representation. Let w contain a steady variation ws and also an unsteady variation (wu) that arises from a wind condition variation. Because of linearity, the wind speed coordinate variations ws and wu can be considered as separate input variables to the wind turbine model: [EqMturb]: 10 xt(n +1) = At- xt(n) + Bt ws(n) + Bt wu(n) q(n) = Ct-xt(n) + Dr ws(n) + Dt wu(n) A Kalman filter provides load signal estimations on time point n (Aq(n)) from 15 load signal measurements up to time point n-1 ((q}(n-l)). The Kalman filter is derived from the combination of the steady wind model and the wind turbine model (augmented model): [EqMaugstus]: (BrDw) (Bt) χ(η + ΐ) = Α· x(n)+ -e(n)+ · wu(n) \ J νυ/ q(n) = C- x{n) + Dt Dw e{n) + Dt- wu(n) 20 with χί(η)1 ( At Bt· Cw x(n)= tK/ ; A = ‘ ' w ; C = (Ct Dr Cw) yxw(nV vO aw )
The point of departure of the Kalman filter design is residue minimization in case of no wind condition variation (wu=0). This is very transparent for the adopted model structure in which all turbulence influences are channeled through six wind speed 25 coordinates.
The number of random variables that drive the steady wind speed signals in the model [EqMaugstus] agrees with the number of measured load signals if wu=0.
The person skilled in the art immediately recognizes the innovations model, which allows for the asymptotically exact reconstruction of the purely random inputs e 30 on time point n from the measurements q on time point n as per 33 e(n) = (Dt-Dwyl-(q(n)-C· x(n))
The feed-in of Ae(n) in the right hand side of the state transition equation of [EqMaugstus] with wu(n) set to 0 yields the update equation for the Kalman filter state 5 Λχ(η). The load estimation Aq(n) in the output equation of the Kalman filter is based on history (q(n-l)}. Up to time point n-1 the estimation Ae(n) is not yet available and so cannot be included in the output equation.
The state space representation of the Kalman filter thus becomes: [EqKF]: 10 x(n +1) = (A -K C)· x(n) + K· q(n) q(n) = C x{n) with KJK< If β,-dj1 yKJ \BW- (Dt- Dw) 1// 15
For the residues r then holds: [Eqrst]: r(n) = q(n) - q(n) = (Dt Dw)· e(n) + C· (x(n) - x(n)) with (use [EqKF] and feed in e(n) = q(n) — C x(n) into [EqMaugstus] while wu=0): 20 x(n) - x(n) evolving as per x(n + 1)- x(n +1 ) = {A-K- C)· (x(n) - x(n))
Since |A-K*C| provides a stable Kalman filter, the state estimation error x(n)-25 Ax(n) will fade to zero. So r(n) converges to Dt*Dw*e(n) when n increases, which uses to occur within a few seconds. Since e is a zero-mean Gaussian process with covariance matrix E, the residue r will also be a zero-mean Gaussian process (rs), but with 6x6 covariance matrix Dt*Dw*E*Dw'*Dt'.
If a wind condition variation does occur (wu != 0), it holds for the residues r: 30 [EqrstusO]: r(n) = q(n) - q(n) 34 = (A' Αν)' e(n) + C- (x(n)-x(n)) + Dr wu(n)
The state estimation error χ(η)-Λχ(η) now relates to the influence of the unsteady wind conditions on the mechanical loads. The load measurements q that are now fed in 5 the state transition equation of the Kalman filter obey q(n) = C· x(n) + Dt-Dw· e(n) + Dt-wu (n) and make the estimated state vector to evolve by: 10 x(n + Y) = (A -K- C)· x(n) + K- C x(n) + K· Dt- Dw e(n) + K Dt wu(n) with K by EqKF it holds (Kw-DrDw) (BrDw) 15 K-DrDw= w ( w = * w and
? w [KrDrDwJ { Bw J
fKw-Dt) ( Bt λ K- Dt = w = _i
IA · A J · Av J
so that the state estimation error now evolves as per:
{ 0 N
x(n + \)-x(n+\) = (A-K-C)-(x(n)-x(n))+ i · wu(n)
\-Dw- Lfw J
20
The state estimation error is now 'sustained' by the unsteady wind speed variation wu while the influence of the initial state error χ(0)-Λχ(0), which will fade to zero in a few seconds. After the faded influence of χ(0)-Λχ(0), the state estimation error relates to wu(n) as per (time-shift operator z: z(x(n))=x(n+l)): _if 0 Ϊ 25 x(n) — x(n) = (zI — (A—K C)) · i wu(n)
\aw' uw J
So, after a few seconds, the residue r as per [Eqrstus] behaves as the sum of a part that solely relates to zero-mean Gaussian purely random variables e (rs) and a part that solely relates to the unsteady wind speed variation wu(n) (ru): [Eqrstus]: 35 r(n) = rs(n) + ru(n) with rs(n) = (E>t· Dw)· e(n) r 1(0)) r(n) = C- (z/-(^ - * O)"1· n_! + A · w„(«) V v^w' uw y y 5
Further, the load estimation Aq appears a biased estimation of the load variations caused by steady wind speed variations (qs).
q(n) - qs{n)-Dt - Dw - e(n) + Aq{n) 10 with bias Aq(n) given by: f f 0 ) ίβ))
Aq(n) = -C- (zI-(A-K-C))-1· 1 -(zI-Ay1· ‘ · wu(n)
V V--°w · uw J K'JJJ
This follows from splitting up the augmented model [EqMaugstus] in model parts 15 driven by e and by wu, which is allowed because of its linearity: x(n) = xs(n) + xu(n) with (BrDw) (Bt) xs(n + V) = A- xs(n)+ · e(«); xu{n + 1) = A- xu(n)+ · wu{n)
\ J
q(n) = qs(n) + qu(n) with 20 qs(n) = C- xs(n) + Dt- Dw- e(n); qu(n) = C- xu(n) + Dr wu(n)
This allows to rewrite the residue definition r(n) = q{n) - q(n) as: q(n) = qs(n) + qu(n)-r(n) with ( (βλ ) 25 <7M(«) = C (zI-A) l- * +A · wu(n) y so that with residues r(n) as per [Eqrstus]: 36 -1 ί 0 1 q(n)-r(n) = -Dt Dw e(n)-C ((zI-(A-KC)) · ι \-üw· L)w ^ (β ^ (zI-A)~1· 1 -wu(n) vOy which implies bias Δς(η) as per 5 (qu(n)-r(n))-(~Dt-Dw)· e(n)
Because load estimations Aq are affected by both steady and unsteady wind speed variations, they are not suited for the detection of a wind condition variation whereas residues r are very well suited.
10 Alternative Kalman filter embodiments can be obtained through wind speed coordinates as any functions of wind speed coordinate distributions that relate to wind condition variations.
37
Appendix B: GLRT embodiment for gust detection in six wind speed coordinates with residues from a Kalman filter embodiment in which influences of turbulence are channeled through these six wind speed coordinates.
A generalized maximum likelihood ratio test (GLRT) is a feasible option for the 5 monitoring of the evolution of a residue that is obtained with a Kalman filter.
A GLRT provides on a time instance (t) an upper limit for the likelihood ratio between two types of wind evolution (L(t)). A large value for this upper limit implies wind evolution in unsteady conditions, that is to say a gust is occurring; a small value implies wind evolution in steady conditions, that is to say only turbulence applies.
10 The likelihood ratio concerns two conditional probability density functions of the residue history in a time span W preceding time instance t ({r}(t, t-W), short form (r}(t)). The numerator in this ratio (p({r}(t)|Hl)) is appreciated conditional to the validity of an alternative hypothesis (HI), in which the residue is assumed to be affected by unsteady wind variations from any time instance in the interval t-W up to t-15 1.
The denominator (p({r}(t)|H0)) is appreciated conditional to the validity of a null hypothesis (HO), in which the residue is assumed to only represent effects of turbulence.
GLRT's for six classes of the alternative hypothesis HI allow for the detection of 20 gusts from all six distinguished classes of unsteady wind speed variations as per fig 4.a, 4.b, 5.a, 5.b, 6.a, 6.b.
The GLRT described here is derived for the residue vector as per the Kalman filter described in Appendix A.
Let alternative hypothesis class HI [i] belong to the unsteady wind speed 25 variations in the i'th element of wu (wu[i]) for the included embodiment of the Kalman filter.
The performance of a GLRT for HI [i] requires the assumption that wu[i] evolves as a known amplitude-normalized unsteady wind speed variation that is scaled with an unknown amplitude (a[i]), that is to say as a gust class evolution prototype GCP[i] 30 multiplied by a[i].
The six prototypes (GPC[i],i=l, 2, 3, 4, 5, 6} thus provide an amplitude-normalized change in time of one of the respective wind speed coordinates: ua (thrust oriented axial) for a rotor coherent gust, 38 va (thrust oriented tangential) for a backing and veering wind, ut (tilt oriented axial) for a jet stream, vt (tilt oriented tangential) for a wind direction change, uy (yaw oriented axial) for a partial wake condition, 5 vy (yaw oriented tangential) for a sloping wind
So the GLRT for a gust class prototype GCP[i] provides on a time instance (t) the generalized maximum likelihood ratio (L[i](t)) between the presence and absence of a gust that evolves as per GCP[i] with scaled amplitude from the gust class belonging to 10 wind speed coordinate wu[i]. The starting time instance of the gust lies anywhere in time span W preceding time instance t.
The generalized maximum L[i](t) is obtained as the maximum of specific maximum likelihood ratio's. Each specific maximum (L[i|k](t)) is derived for a fixed value of the beginning time instance of the gust (k), which ranges from t-W up to t-1.
15 The assumed unsteady wind speed coordinate then agrees with the amplitude-scaled gust class prototype evolution with starting time instance k, denoted as a[i]* {GCP[i|k]}(t). Thus (wu[i]}(t)= a[i]* (GCP[i|k]}(t) is the assumption on unsteadiness for the specific alternative hypothesis (HI [i|k]).
The specific likelihood ratio L[i|k](t) is then derived from the residue history 20 (r}(t) and amplitude-normalized gust evolution (GCP[i|k]}(t), while a[i] is eliminated through maximising the ratio between the conditional probability density funcntion P(W(t)|Hl[i|k]) and p({r}(t)|H0).
The application of a GLRT for gust detection will be clarified through the residues up to a time point t ({r}(t)) that are obtained with the Kalman filter as 25 described above in Appendix A. In this embodiment the residue r will approximately evolve as scaled zero-mean Gaussian purely random variables (Dt*Dw*e) in case of small scale turbulence only [Eqrst]: r(n) « Dt · Dw · e(n) [Eqapprst
Small differences with Dt*Dw*e will occur because of imperfect modelling of load variations from small scale turbulence ([EqMaugstus] for wu=0).
30 39
If a wind condition variation does occur, at least one residue will definitely deviate from the related element of Dt*Dw*e. The deviation will approximately evolve as per the contribution rus in [Eqrstus] through a history up to t of unsteady wind speed variations in an element i of wu ({wu[i]}(t)), so that then for the overall residues hold 5 (01 and Ot are leading and trailing zero-value column vectors relative to element i): [EqApprstus]: 0; r(n) *Dt Dw- e(n) + C- ξ(η) + Dt- wu [/](«) v 0, , with state error update equation f 0 ^ f °i ξ(η + ϊ)~(Α-Κ· Q- ξ(η)+ ,· wu\i](n) V vt y 10
Small differences with the right hand side occur because of imperfect modeling of load variations from steady and unsteady wind variations ([EqMaugstus]).
The null hypothesis HO for the GLRT is straightforward. It assumes that the 15 residues evolve as the right hand side of [EqApprst]:
Hq : r(n) = Dr Dw- e(n) [EqHO]
An alternative hypothesis HI [i] for the GLRT assumes residue evolution by the 20 right hand side of [EqApprstus] and the included state error update equation.
The generalized maximum L[i](t) is obtained as the maximum over k of specific maximum likelihood ratio's L[i|k](t). The chosen time instance k from which the unsteady wind speed coordinate wu[i] deviates from zero runs from t-W up to t-1.
In a specific alternative hypothesis Hl[i|k], it is assumed that wu[i] evolves as per 25 a gust class evolution prototype GCP[i] with amplitude a[i] from a starting time instance k (a[i]*GCP[i|k]).
The alternative hypothesis HI[i|k]) can be generically formulated as [EqHl]: H\[i I k]: r(n) = Dt-Dw e(n) + C· ξ(η) + Dt-wu(n\ i,k) 30 40 with state error update equation ( o 1 ξ(η +1 ) = (A -K Q ξ(η) + , · wu (n \ i,k) v "tv' uw y and assumed unsteady wind speed evolution by: 0/ ] [ 0, wu(n I i,k) = wu[i \ k](n) = a[i \ k\ GCP[i \ k](n) v 0, J [ 0? y 5
The linear expression in hypothesis HI allows for reformulation by - assuming amplitude-normalized unsteady wind speed input ~wu and related state error ~ξ instead of wu and ξ; - introduction of the contribution μ(η \ i,k) to the residues by the assumed 10 amplitude-normalized unsteady wind speed variation so that [EqHla]: H\ [i I k]: r(n) = Dt · Dw · e(n) + a[i \ k\ μ(η \ i,k) 15 with μ(η \ i,k) evolving as per state space model - ο ί i °' ξ(π + 1) = (Α-Κ·0·ξ(π)+ , · GCP[/| *](»)
L)w J
V Vf J
0/ μ(η I i,k) = C l(n) + Dt GCP[i \ k](n) l v
The defined hypotheses HO and HI [i|k] tell that r(n) conditional to HO evolves as 20 a zero-mean Gaussian distributed purely random variable with 6x6 covariance matrix R = Dt - Dw E (Dt Dw)' r(n) conditional to HI [i|k] evolves as a Gaussian distributed purely random variable with covariance matrix R and evolving mean value a[i|k]* ~u(n|i,k)
The random parts of the sequential residue vectors r(n-l), r(n), r(n+l) are 25 completely uncorrelated, both conditional to HO and conditional to HI.
41
Therefore, the probability density functions of the residue history {r}(t, t-W) are simply the products of those of the residue vectors for a single time instance, so that: t P(M(0 Itfo) = P(r(t-W) H„)· .., p(r(0 IH0) = ΠρΜ») l»o)
n=t-W
——τ’ R~l-r e 2 with p(r(n) \H0)= r— (-\/2· π· det(i?)) 5 /?(WO) I #lD' I ^]) = p(r(t - W) I Hl[i\ k])· ...· p(r(t) \ Hx[i \ k]) = t = Ylp(r{n)\Hl[i\k'\)
n=t—W
with P(K«) I m I £]) = ---===-- (λ]2· π· det(7?)) 10
The probability that r evolves in a delta-environment as per history {r}(t, t-W), with observed residue vectors r(n), conditional to the null hypothesis directly follows from the probability density function of (r}(t, t-W) conditional to HO: - /-(Oil < i Δ I //0) = Π fj===5-A6 2 n=,-w Φ *· det(P))6 15
The short-form probability notation AH'-wwhfw stands for the joint conditional probability as per: 20 Ρφ--Μ(()||<{Δ|/ί0) = P(|jfl] - r[l](< - w] < |ff0 AND... AND |r[6] - r[6](< - | H„
AND ... AND
Hi] - r[l](0l < ^ Δ IH0 AND... AND |r[6] - r[6](0| < ^ Δ | tf0) 42
The probability that r evolves in a delta-environment as per history {r}(t, t-W) conditional to the specific alternative hypothesis directly follows from the probability density function of {r}(t, t-W) conditional to Hi[i|k]: 5 - W(0|| < 2"Δ I Hi[i I k\) = t -\(r(n)-a[i\k]· jd(n\i,k))'-R~^ -(r(n)-a[i\k] ·μ(η\ί,1ί)) Π --, 6--Δ6 n=t-W (V2' π' det(7?)) with a similar short-form probability notation as 10 J’dl'-MMlkiAltfo)
The ratio between -p(lk - {ί-KOll < 1 /i, [/1 *]) and 15 J’dl'-MMlk^WI*]) yields the concerned probability density function ratio. The product operators in this ratio can be replaced by summation operators over the arguments of the exponents in the exponential functions. This yields: Σ -\(r{n)-a[i\k] ·μ{η\ί,Κ))'·]1~^ -(r(n)-a[i\k] jl(n\i,k)) p({r}(t) \Hl[i \k]) _e»=t-w_ P(WWI»o) Σ -lr(„y ,^ΚΦ)
gti=t-W
20
The specific maximum likelihood ratio L[i|k](t) is obtained by maximizing the exponent in the numerator to the unknown amplitude a[i]. The natural logarithm of the ratio upper limit L[i|k](t) includes a modeled evolution of the residues that relates to the correlation with a predictive function, viz. with the assumed contribution history 25 {~p}(t|i,k) to the residue by the amplitude-normalized gust class evolution prototype realization {GCP[i|k]}(t} for a gust in wind speed coordinate wu[i] that starts on time instance k: 43 ί ( Υ Σμ(η I i,k))' · i?-1· r{n) iog(i[/1 km={ -— Σμ(η \i,k))' · i?-1· μ(η | /, Ar))
n=t—W
with amplitude ami[i], that belongs to the ratio upper limit L[i|k](t) by: t Σμ(η I i,k))'· R r(n) 5 aml[i\k^ - ^ίΚn i.kIy 1 μ(η \i,k))
n=t-W
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WO2012044161A2 (en) | 2012-04-05 |
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