Merge remote-tracking branch 'gitea/master'
This commit is contained in:
commit
2412966e41
4
.obsidian/plugins/copilot/data.json
vendored
4
.obsidian/plugins/copilot/data.json
vendored
@ -113,7 +113,7 @@
|
||||
"provider": "ollama",
|
||||
"enabled": true,
|
||||
"isBuiltIn": false,
|
||||
"baseUrl": "https://possibly-engaged-filly.ngrok-free.app/https://possibly-engaged-filly.ngrok-free.app/v1/",
|
||||
"baseUrl": "https://possibly-engaged-filly.ngrok-free.app",
|
||||
"apiKey": "",
|
||||
"isEmbeddingModel": false,
|
||||
"capabilities": [
|
||||
@ -122,7 +122,7 @@
|
||||
"websearch"
|
||||
],
|
||||
"stream": true,
|
||||
"displayName": "gemma3:12b_ngrok",
|
||||
"displayName": "gemma3:12",
|
||||
"enableCors": true
|
||||
}
|
||||
],
|
||||
|
||||
0
[steady控制算法.excalidraw.md
Normal file
0
[steady控制算法.excalidraw.md
Normal file
@ -224,7 +224,7 @@ However, the work going on with optimization of the wind turbines results inevit
|
||||
|
||||
The basis for the example is the schematic drawing in Fig.1 and a reduced expression for the inertia loads which are part of the mathematical model. The kinematic analysis in the model includes deformation at the shaft end, teeter and yaw but only the tower top deformation will be included in the following survey. The angular rotor velocity is assumed constant, in order to keep the expression for the blade inertia force as simple as possible.
|
||||
|
||||
|
||||
|
||||
Figure 1: Tower top elastic rotations.
|
||||
|
||||
Applying these simplifications the following formal expression for the inertia load at a blade point is obtained
|
||||
@ -369,7 +369,7 @@ The finite element used in the model is a simple two node prismatic beam element
|
||||
|
||||
The elastic stiffness matrix is derived by use of the constitutive relations and the principle of virtual displacements. A general expression for consistent transformation of the distributed loads to the nodes is derived. This expression is used for transformation of the inertia load to the nodes, resulting in element mass-, Coriolis-, and softening matrices as described in Sec. 4.11.
|
||||
|
||||
|
||||
|
||||
Figure 2: Derivation of blade substructure equations of motion.
|
||||
|
||||
Additional inertia loads, represented as vectors, result from this transformation. These vectors are composed of terms, which are functions of DOFs outside the blade substructure, and the angular velocity of the rotor, $\omega$ . The terms including DOFs can be extracted from the vectors, as shown for the example calculation in [Part 2, Sec. E], giving rise to additional mass- and Coriolis-matrices. The remaining terms in the inertia vectors are dominated by the centrifugal forces related to the square of the angular velocity of the rotor, $\omega^{2}$
|
||||
@ -390,7 +390,7 @@ Similar procedures are followed for the tower and the shaft substructures.
|
||||
|
||||
As shown schematically in Fig. 3, the substructure equations are assembled (Sec. 6) by imposing force equilibrium at the coupling nodes. Further, the boundary conditions at the tower foundation are introduced, thus removing the rigid body motion of the total structure. These conditions are assumed to be purely geometric, equivalent to zero displacement. The displacement compatibility between substructures is ensured through the kinematic analysis.
|
||||
|
||||
|
||||
|
||||
|
||||
# Figure 3: Assembly of substructure equations of motion.
|
||||
|
||||
@ -446,7 +446,7 @@ In [R2] and [R3] the Euler angles have been used for a geometric nonlinear model
|
||||
|
||||
If finite rotations should be allowed within a substructure a reasonable description should at the same time allow for an updating of the equilibrium equations in accordance with the change in geometry. This possibility is inherent in many methods, especially those developed for the finite element models.
|
||||
|
||||
|
||||
|
||||
Figure 4: Substructures and coordinate systems. Undeformed state.
|
||||
|
||||
Another way of treating finite rotations, especially well suited for numerical solution on a computer, is described in the by now comprehensive litterature dealing with solution of nonlinear structural problems in the finite element method.
|
||||
@ -617,7 +617,7 @@ $$
|
||||
|
||||
and the angular velocity of the $N$ -system relative to the $\pmb{T}^{\prime}.$ system in $N$ -coordinates is
|
||||
|
||||
|
||||
|
||||
Figure 5: Elastic rotation at tower top, $\left\{\theta_{T\ell}^{T}\right\}$
|
||||
|
||||
$$
|
||||
@ -634,7 +634,7 @@ $$
|
||||
\left[T_{N R}\right]=\left[\begin{array}{c c c}{{1}}&{{0}}&{{0}}\\ {{0}}&{{\cos\left(\theta_{1R}^{R}\right)}}&{{\sin\left(\theta_{1R}^{R}\right)}}\\ {{0}}&{{-\sin\left(\theta_{1R}^{R}\right)}}&{{\cos\left(\theta_{1R}^{R}\right)}}\end{array}\right]
|
||||
$$
|
||||
|
||||
|
||||
|
||||
Figure 7: Tilt rotation, $\pmb{\theta}_{1R}^{R}$
|
||||
|
||||
The tilt angle $\pmb{\theta}_{1R}^{R}$ is constant and therefore the angular velocity of the $\pmb{R}$ system relative to the $N$ -system is zero
|
||||
@ -655,7 +655,7 @@ $$
|
||||
|
||||
and the angular velocity of the $\pmb{A}$ -system relative to the $\pmb{R}$ -system in A-coordinates is
|
||||
|
||||
|
||||
|
||||
|
||||
# Figure 8: Azimuthal rotation, $\pmb{\theta_{2A}^{A}}=\pmb{\theta}.$
|
||||
|
||||
@ -689,7 +689,7 @@ $$
|
||||
\left[T_{S^{\prime}B}\left(t\right)\right]=\left[\begin{array}{c c c}{{1}}&{{0}}&{{0}}\\ {{0}}&{{\cos\left(\theta_{1H}^{H}\right)}}&{{\sin\left(\theta_{1H}^{H}\right)}}\\ {{0}}&{{-\sin\left(\theta_{1H}^{H}\right)}}&{{\cos\left(\theta_{1H}^{H}\right)}}\end{array}\right]
|
||||
$$
|
||||
|
||||
|
||||
|
||||
Figure 9: Teeter rotation, $\pmb{\theta}_{1H}^{H}$
|
||||
|
||||
and the angular velocity of the $B\!\cdot$ system relative to the $S^{\prime}.$ system in $B\cdot$ coordinates is
|
||||
@ -857,10 +857,10 @@ The element length is denoted by &.
|
||||
|
||||
The following distributed forces may act on the beam element. They may arise from surface forces or body forces and are resolved after the coordinate axes
|
||||
|
||||
|
||||
|
||||
Figure 10: Element coordinate system.
|
||||
|
||||
|
||||
|
||||
Figure 11: Cross section of beam element.
|
||||
|
||||
$f_{x},\,f_{y},\,f_{z}$ : Components after coordinate axis of distributed force, (force per unit length)
|
||||
@ -1144,7 +1144,7 @@ $$
|
||||
[N(z)]=\left[\begin{array}{c c c c c c c c}{f_{4}^{y}}&{0}&{0}&{\left|\begin{array}{c c c c c}{0}&{-f_{6}^{y}}&{0}\\ {0}&{f_{4}^{x}}&{0}\\ {0}&{0}&{f_{2}}\end{array}\right|\begin{array}{c c c c c c c}{0}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right|\begin{array}{c c c c c c}{f_{3}^{y}}&{0}&{0}\\ {0}&{f_{3}^{x}}&{0}\\ {0}&{0}&{f_{1}}\end{array}\right|\begin{array}{c c c c c c}{0}&{f_{5}^{y}}&{0}\\ {-f_{5}^{x}}&{0}&{0}\\ {0}&{f_{7}^{x}}&{0}\\ {-f_{7}^{y}}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right]_{0}^{x}\!\!\!\!\!=\!\!\!0\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
|
||||
$$
|
||||
|
||||
|
||||
|
||||
Figure 12: Interpolation functions.
|
||||
|
||||
Now the local deformations resulting from node deformations can be found as the first term on the right hand side of Eq. 4.3.16. The local deformations resulting from distributed forces and moments, as represented by the vector $\{d(z)\}$ in Eq. 4.3.16, may be derived from the inhomogeneous set of differential equations $4.3.9-4.3.14$ with homogeneous boundary conditions, $\{q\}=\{0\}$ . However, this solution is only of interest when the exact expression for the totai deformations is needed, e.g. in a stress calculation. In the present context we are primarily concerned with the node deformations. They can be derived by a transformation of distributed forces and moments to the nodes, consistent with the principle of virtual displacements, and by solving the equilibrium equations, when the stiffness matrix is known.
|
||||
@ -1227,7 +1227,7 @@ $$
|
||||
\mathbf{[}K\mathbf{]=}
|
||||
$$
|
||||
|
||||
|
||||
|
||||
|
||||
In the matrix above the following new symbols have been introduced
|
||||
|
||||
@ -1262,7 +1262,7 @@ $$
|
||||
\left.{\begin{array}{l l l}{u_{x}}&{=}&{u_{x}^{*}-\theta_{z}^{*}e_{s2}}\\ {u_{y}}&{=}&{u_{y}^{*}+\theta_{z}^{*}e_{s1}}\\ {u_{z}}&{=}&{u_{z}^{*}}\\ {\theta_{x}}&{=}&{\theta_{x}^{*}}\\ {\theta_{y}}&{=}&{\theta_{y}^{*}}\\ {\theta_{z}}&{=}&{\theta_{z}^{*}}\end{array}}\right\}
|
||||
$$
|
||||
|
||||
|
||||
|
||||
Figure 13: Position of shear center.
|
||||
|
||||
which in a matrix equation is expressed as
|
||||
@ -1325,7 +1325,7 @@ $$
|
||||
|
||||
Carrying out the matrix multiplication we find
|
||||
|
||||
|
||||
|
||||
|
||||
In the following the star index will be omitted, and unless otherwise stated, the deformations are referred to the elastic axis coordinate system.
|
||||
|
||||
@ -1379,13 +1379,13 @@ where the $6\times6$ submatrices are achieved after multiplication in Eq. 4.7.4
|
||||
|
||||
[Ku] = (4.7.6)
|
||||
|
||||
|
||||
|
||||
|
||||
[K12] = [K22] = (4.7.8)
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
4.8 Transformation between element- and substructure coordinates.
|
||||
|
||||
@ -1425,7 +1425,7 @@ The derivation of the geometric stiffness matrix is now carried out.
|
||||
|
||||
Consider an infinitesimal segment, $\pmb{d z}$ . of a beam finite element with No. i, influenced by an axial force $\left\{F_{a}^{E i}\right\}$ as shown in Fig.14.
|
||||
|
||||
|
||||
|
||||
Figure 14: Derivation of geometric stiffness.
|
||||
|
||||
The element has been deformed in the tranverse direction resulting in a slope $\frac{d u_{z}}{d z}$ .For an element which can be assumed not to extend during the transverse deformation, the projection of the element upon the $\pmb{z}$ -axis will be shortened by an amount of ds. The local work done by the axial force during the transverse deformation is expressed as
|
||||
@ -1526,7 +1526,7 @@ The geometric stiffness matrix for an element is in symbolic form given by
|
||||
|
||||
[K]=
|
||||
|
||||
|
||||
|
||||
|
||||
(4.9.12)
|
||||
|
||||
@ -1594,7 +1594,7 @@ The body forces arise from inertia forces and gravity forces, while the forces a
|
||||
|
||||
The displacements at a point of the element, with coordinates $(x,y,z)$ , are expressed by the relations (refer to Fig. 15)
|
||||
|
||||
|
||||
|
||||
Figure 15: Position of infinitesimal volume.
|
||||
|
||||
$$
|
||||
@ -1918,7 +1918,7 @@ $$
|
||||
[M]=
|
||||
$$
|
||||
|
||||
|
||||
|
||||
|
||||
where
|
||||
|
||||
@ -2018,7 +2018,7 @@ $$
|
||||
\left[I_{3L}\right]=
|
||||
$$
|
||||
|
||||
|
||||
|
||||
|
||||
The final result of the integrations in line 5 is achieved by multiplication of the matrices $[I_{3L}]$ from Eq. 4.11.36 and $[I_{3A}]$ from Eq. 4.11.29. We get the $12\times3$ matrix( $M=m\ell$
|
||||
|
||||
@ -2026,7 +2026,7 @@ $$
|
||||
\left[_{i}F_{5}\right]=\left[_{i}I_{3L}\right]\left[_{i}I_{3A}\right]=
|
||||
$$
|
||||
|
||||
|
||||
|
||||
|
||||
# Integration of the $\mathbf{\delta}\mathbf{\delta}^{t h}$ line in Eq. 4.11.10.
|
||||
|
||||
@ -2088,7 +2088,7 @@ This simple example may be used without any loss of generality, because only the
|
||||
|
||||
Only elastic stiffnesses and external node forces are taken into account.
|
||||
|
||||
|
||||
|
||||
Figure 16: Coupling of substructures.
|
||||
|
||||
# 5.1 Coupling procedure in usual FEM formulation.
|
||||
@ -2377,7 +2377,7 @@ Shaft: Nodes are numbered from 1 to $^m$ Shaft node A1 is at the joint to the to
|
||||
|
||||
Blade: Nodes are numbered from 1 to $\pmb{n}$ Bladenode $\pmb{B1}$ is at the joint to the shaft, and common with shaft node Am Further details concerning the node numbering for the blade substructure is given in sec.6.3
|
||||
|
||||
|
||||
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||||
Figure 17: Node numbering for boundary nodes of the substructures.
|
||||
|
||||
The details of the derivation and evaluation of the respective substructure accelerations are given in Sec. 3. Below, inertia matrices and vectors are used, which result from rewriting the acceleration expressions. The rewriting has character of a decomposition, based on sorting of the acceleration expressions according to common factors of degrees of freedom and their order with respect to time derivation. A complete listing of these matrices and vectors can be found in [Part 2, Sec. E], where also the introduced linearizations are described. A description of their origin is found there as well.
|
||||
@ -2782,7 +2782,7 @@ is composed of the externally applied loads, inciuding gravity and aerodynamic l
|
||||
|
||||
When the blade EOM is assembled, the node numbering must be considered, because the structure is no longer a simple chain, as was the case for the tower and the shaft. To illustrate the preferred systematics, a general example with 3 blades is used. The difference between global node numbers for one element determines the band width of the coefficient matrices. Therefore, the following way of assigning node numbers is used.
|
||||
|
||||
|
||||
|
||||
Figure 18: Node numbering for blades
|
||||
|
||||
As illustrated in fig. 18, the node which couples the blades to the shaft is given number 1. The remaining nodes are numbered sequentially by chosing the node closest to the center node on one blade as node 2, and then step from blade to blade in the same direction, e.g. clock-wise, always taking the unnumbered node closest to the center as the next one in the sequence. This procedure results in a maximum node difference equal to the number of blades, and this number is also equal to the half band-width of the coefficient matrices.
|
||||
@ -2905,7 +2905,7 @@ $$
|
||||
\left[M_{S}\right]=
|
||||
$$
|
||||
|
||||
|
||||
|
||||
|
||||
# the system damping matrix
|
||||
|
||||
@ -2913,7 +2913,7 @@ $$
|
||||
\left[C_{S}\right]=
|
||||
$$
|
||||
|
||||
|
||||
|
||||
|
||||
# the system stiffness matrix
|
||||
|
||||
@ -2921,7 +2921,7 @@ $$
|
||||
\left[K s\right]=
|
||||
$$
|
||||
|
||||
|
||||
|
||||
|
||||
# and the system geometric stiffness matrix
|
||||
|
||||
@ -2929,7 +2929,7 @@ $$
|
||||
\left[K_{g S}\right]=
|
||||
$$
|
||||
|
||||
|
||||
|
||||
|
||||
The geometric stiffness submatrices are not listed. Their derivation is based on the considerations in Sec. 5 and specifically Eq. 5.3.18.
|
||||
|
||||
@ -2949,7 +2949,7 @@ The eigensolutions are found by a two step procedure. The first step involves ca
|
||||
|
||||
The time integration is performed by use of the Newmark implicit integration scheme [B1, pp. 549-552]. The sequence in the integration is shown in the diagram of Fig. 19, where also the step concerning updating of the geometry is shown. Due to the time dependent coefficient matrices, it is necessary to perform iterations in order to obtain equilibrium at each time step.
|
||||
|
||||
|
||||
|
||||
Figure 19: Solution of equations of motion.
|
||||
|
||||
The Newton-Raphson iteration scheme is used [B1, pp. 490-491]. In genral no convergence problems arise when just an appropriate time step is chosen.
|
||||
@ -2980,7 +2980,7 @@ The rotor azimuthal position is with blade No. 1 vertical upward, i.e. $\pmb\the
|
||||
|
||||
Table 1: Main data for Danwin $180~\mathbf{kW}.$
|
||||
|
||||
|
||||
|
||||
|
||||
Below, the number following the eigenfrequencies in parenthesis is the percentage increase due to the centrifugal stiffening. The mode shapes have been normalized. The units are $[m]$
|
||||
|
||||
@ -2996,23 +2996,23 @@ It is characteristic for the mode that the deformations are mainly in the $\pmb{
|
||||
|
||||
Table 2: Finite element model.
|
||||
|
||||
|
||||
|
||||
|
||||
# Mode 2.
|
||||
|
||||
Eignefrequency $=0.888~{\mathsf{H z}}~(+1.85\%)$ Mode shape displacements at tower top and blade tips:
|
||||
|
||||
|
||||
|
||||
|
||||
The deformations are mainly in the $\pmb{x}$ -direction and the tower top and the blade tips are in phase. This mode is tower dominated as it is the case with mode 1. This fact is also reflected in the almost identical eigenfrequencies.
|
||||
|
||||
Mode 3. Eignefrequency $=2.67\ H z\ (+7.66\%).$ Mode shape displacements at tower top and blade tips:
|
||||
|
||||
|
||||
|
||||
|
||||
The deformations are mainly in the $\pmb{x}$ -direction and the tower top and the blade tips are in
|
||||
|
||||
|
||||
|
||||
Figure 20: Coordinates for fundamental mode shapes.
|
||||
|
||||
counter phase in this direction. The tower is in a torsional motion (yaw). Blade 1 is almost at rest while blade 2 and 3 are in counter phase with respect to displacement in the $\pmb{y}_{}.$ direction. The rotor mode is often referred to as the asymmetrical mode.
|
||||
@ -3021,7 +3021,7 @@ counter phase in this direction. The tower is in a torsional motion (yaw). Blade
|
||||
|
||||
Eignefrequency $=2.78\ H z\ (+8.59\%).$ Mode shape displacements at tower top and blade tips:
|
||||
|
||||
|
||||
|
||||
|
||||
The deformations are mainly in the $\pmb{y}_{}$ -direction. The tip ends of blade 2 and blade 3 are in phase with the tower top and blade 1 is in counter phase. The tower is in a bending motion (tilt). This rotor mode is also referred to as an asymmetrical mode. In fact, if the rotor was considered as an isolated structure, the rotor mode from mode shape 3 and this one correspond to a double eigenfrequency of the rotor.
|
||||
|
||||
@ -3049,7 +3049,7 @@ A similar analysis has been carried through for the blade moment in the lead lag
|
||||
|
||||
The present treatment of the simulated deterministic response is concluded with presentation of some time series showing deformations and resulting loads corresponding to important degrees of freedom.
|
||||
|
||||
|
||||
|
||||
Figure 21: Blade fap wise bending moment. Deterministic.
|
||||
|
||||
The deflection of the blade tip in the $\pmb{x}-$ and $\pmb{y}_{}$ -direction is shown in Fig. 28. The shape and the dynamic characteristics are very similar to the blade root moments described initially. In Fig. 29 some of the tower top deformations are shown. Generally, the transient response has died out after approximately 10 rotor revolutions. For deformations in the wind direction even before due to the aerodynamic damping. The tower top displacement in the across wind direction $(u_{x T\ell}^{T})$ is clearly lower damped than the displacement in the along wind direction $(u_{y r t}^{x})$ . The 3P harmonic is observed in all the tower top signals. The DFT of the across wind displacement in Fig. 27 is presented as an example. Further, in this signal the lowest mode eigenfrequency is clearly present $\mathbf{\mu_{(0.88}\ H z)}$ due to the low aerodynamic damping in this direction and the harmonic input from ${\tt1P}=0.79~{\tt H z}$ The lowest peak corresponds to the eigenfrequency and not 1P. The signicant, higher harmonics are multiples of 3P. The rotor lads, thrust $(F_{y T\ell}^{T})_{\mathrm{~\,~}}$ tilt moment $(\bar{M}_{x T\ell}^{T})$ and yaw moment $(M_{z T\ell}^{T})$ , and further the power $(P)$ presented in Fig. 30, show the same characteristic 3P content. These observations are in good agreement with the measurements.
|
||||
@ -3058,7 +3058,7 @@ The deflection of the blade tip in the $\pmb{x}-$ and $\pmb{y}_{}$ -direction is
|
||||
|
||||
A simulation has been carried out with the purpose of comparing with measurements and of illustrating the influence of turbulence on the response. The simulation parameters are chosen in agreement with the immediately available corresponding parameters for a measurement carried out in a mean wind speed of ${8.1~\mathsf{m/s}}$ . Representative response characteristics of the simulation results are compared with the measured. The turbulence length scale has not been adjusted to the actual measured but is chosen to ${150~\mathfrak{m}}$ . which is known to be typical for the test site for similar conditions. The standard deviation of the measured wind speed is $\mathbf{1.04}\;\mathbf{m/s}$ at hub height, corresponding to a turbulence intensity of $12.5\%$ . Only flap wise blade bending moment at radius $\mathbf{0.47~m}$ is presented in the comparison.
|
||||
|
||||
|
||||
|
||||
Figure 22: DFT of blade fap wise bending moment. Deterministic.
|
||||
|
||||
The measured signal is sampled at a frequency of $\pmb{25}\ \forall\pmb{2}.$ and 8192 samples covering a time period of 328 secs are used. As mentioned previously, only the relative distribiution of energy on the frequencies and not the energy amount can be expected to agree when adjustment of the wind field model and the aerodynamic model has not been accomplished, and the actual energy content is not considered.
|
||||
@ -3071,7 +3071,7 @@ Fig. 31 and 33 show a 12 secs interval of the simulated and the measured blade r
|
||||
|
||||
The infuence of the rotational sampling of the turbulence on the wind speed felt by the blade is shown in Fig. 36, where the PSD of the sampled turbulence at the blade tip is presented. The energy is shifted from lower frequencies to harmonics of the rotational frequency. A time signal of the sampled turbulence is shown in Fig. 35. Some wind speed variance is lost during the simulation. The target spectrum has a variance of 1.1 $(\mathsf{m}/\mathsf{s})^{2}$ while the sampled turbulence variance has decreased to $0.6\:(\mathsf{m}/\mathsf{s})^{2}$ . The explanation for this is partly the limited simulation time but also inherent in the turbulence simulation model as mentioned in [Part 2, Sec. F].
|
||||
|
||||
|
||||
|
||||
Figure 23: Fourier series resolution of measured signal.
|
||||
|
||||
Other simulation variables are presented below in order to give a more complete picture of the dynamic response. In Fig. 37 some tower top deformations are shown. The PSD of the across wind tower displacement $(u_{x T\ell}^{T})$ is shown in Fig. 40. The lowest eigenfrequency (0.88 Hz) is again very significant in this signal. But generally, the 3P content is dominating for these deformations as it was the case for the deterministic response. Again it is observed that the damping in the along wind direction is higher than in the across wind direction.
|
||||
@ -3082,61 +3082,61 @@ Simulated signals corresponding to rotor loads and power are shown in Fig. 39. I
|
||||
|
||||
The preliminary simulation results presented above show in general a reasonable agreement with corresponding measured results and experience. Allthough the material is rather limited and covers a wind turbine of the "rigid type" it is concluded that satisfactory model representation of the most important dynamic effects is achieved.
|
||||
|
||||
|
||||
|
||||
Figure 24: DFT of blade fap wise bending moment. Deterministic without tower shadow.
|
||||
|
||||
|
||||
|
||||
Figure 25: Blade lead-lag wise bending moment. Deterministic.
|
||||
|
||||
|
||||
|
||||
Figure 26: DFT of blade lead-lag wise bending moment. Deterministic
|
||||
|
||||
|
||||
|
||||
Figure 27: DFT of tower across wind displacement. Deterministic
|
||||
|
||||
|
||||
|
||||
Figure 28: Blade tip deformation. Deterministic.
|
||||
|
||||
|
||||
|
||||
Figure 29: Tower top deformation. Deterministic
|
||||
|
||||
|
||||
|
||||
Figure 30: Tower top node loads (rotor loads). Deterministic.
|
||||
|
||||
|
||||
|
||||
Figure 31: Blade fap wise bending moment. Simulated, stochastic.
|
||||
|
||||
|
||||
|
||||
Figure 32: PSD of blade fap wise bending moment. Simulated, stochastic.
|
||||
|
||||
|
||||
|
||||
Figure 33: Blade fap wise bending moment. Measured.
|
||||
|
||||
|
||||
|
||||
Figure 34: PSD of blade fap wise bending moment. Measured.
|
||||
|
||||
|
||||
|
||||
Figure 35: Sampled turbulence at blade tip.
|
||||
|
||||
|
||||
|
||||
Figure 36: PSD of simulated turbulence, sampled at blade tip.
|
||||
|
||||
|
||||
|
||||
Figure 37: Tower top deformation. Simulated, stochastic.
|
||||
|
||||
|
||||
|
||||
|
||||
Figure 38: Blade tip deformation. Simulated, stochastic.
|
||||
|
||||
|
||||
|
||||
|
||||
Figure 39: Tower top node loads (rotor loads) and power. Simulated, stochastic.
|
||||
|
||||
|
||||
|
||||
|
||||
Figure 40: DFT of tower across wind displacement. Simulated, stochastic.
|
||||
|
||||
|
||||
|
||||
Figure 41: PSD of yaw moment. Simulated, stochastic
|
||||
|
||||
# 9 Conclusion.
|
||||
@ -3386,7 +3386,7 @@ $$
|
||||
|
||||
Further, the following symbols are used below, apart from the DOFs listed above $\begin{array}{l}{\theta=\theta\left(t\right)}\\ {\omega=\dot{\theta}}\end{array}$ is the azimuthal position of the rotor. is the angular velocity of the rotor, which is assumed constant.
|
||||
|
||||
|
||||
|
||||
|
||||
In the example calculation in Sec. 6 a limited linearization has been introduced in order to limit the number of terms in the integrated model, which also includes shaft and teeter DOFs. A detailed description of this linearization is given in Sec. 6. In the terms below this linearization has been omitted.
|
||||
|
||||
@ -3501,7 +3501,7 @@ where the abbreviations s for sin and $\pmb{c}$ for cos have been introduced to
|
||||
|
||||
The total transformation from $\pmb{A}$ to $D$ is expressed by the matrix product
|
||||
|
||||
|
||||
|
||||
Figure 42: Finite transformation angles
|
||||
|
||||
$$
|
||||
|
||||
@ -0,0 +1,92 @@
|
||||
# 8.3.2 Control of variable-speed, pitch-regulated turbines
|
||||
|
||||
A variable-speed generator is decoupled from the grid frequency by a power converter, which can control the load torque at the generator directly, so that the speed of the turbine rotor can be allowed to vary between certain limits. An often-quoted advantage of variable-speed operation is that below rated wind speed, the rotor speed can be adjusted in proportion to the wind speed so that the optimum tip speed ratio is maintained. At this tip speed ratio the power coefficient, $C_{p}$ , is a maximum, which means that the aerodynamic power captured by the rotor is maximised. This is often used to suggest that a variable-speed turbine can capture much more energy than a fixed-speed turbine of the same diameter. In practice it may not be possible to realise all of this gain, partly because of losses in the power converter and partly because it is not possible to track optimum $C_{p}$ perfectly.
|
||||
变速发电机通过功率转换器与电网频率解耦，该转换器可以直接控制发电机上的负载转矩，从而允许涡轮机转子的转速在一定范围内变化。变速运行的一个常被引用的优点是，在额定风速以下，转子转速可以按比例调整以适应风速，从而保持最佳叶尖速度比。在该叶尖速度比下，功率系数 $C_{p}$ 达到最大值，这意味着涡轮机叶片捕获的空气动力学功率被最大化。这常被用来表明，与相同直径的定速涡轮机相比，变速涡轮机可以捕获更多的能量。然而，在实践中，由于功率转换器中的损耗以及无法完美跟踪最佳 $C_{p}$，可能无法实现全部的增益。
|
||||
|
||||
Maximum aerodynamic efficiency is achieved at the optimum tip speed ratio $\lambda=\lambda_{\mathrm{opt}}$ , at which the power coefficient $C_{p}$ has its maximum value $C_{p(\mathrm{max})}$ . Because the rotor speed $\varOmega$ is then proportional to wind speed $U.$ , the power increases with $U^{3}$ and $\varOmega^{3}$ , and the torque with $U^{2}$ and $\varOmega^{2}$ . The aerodynamic torque is given by
|
||||
最大空气动力学效率在最佳叶尖速度比 $\lambda=\lambda_{\mathrm{opt}}$ 时实现，此时功率系数 $C_{p}$ 达到其最大值 $C_{p(\mathrm{max})}$ 。由于转子转速 $\varOmega$ 此时与风速 $U$ 成正比，因此功率随 $U^{3}$ 和 $\varOmega^{3}$ 增大，而扭矩随 $U^{2}$ 和 $\varOmega^{2}$ 增大。空气动力学扭矩的表达式为
|
||||
|
||||
$$
|
||||
Q_{a}=\frac{1}{2}\rho A C_{q}U^{2}R=\frac{1}{2}\rho\pi R^{3}\frac{C_{p}}{\lambda}U^{2}
|
||||
$$
|
||||
|
||||
Since $U\!=\!\varOmega R/\lambda$ we have
|
||||
|
||||
$$
|
||||
Q_{a}=\frac{1}{2}\rho\pi R^{5}\frac{C_{p}}{\lambda^{3}}\varOmega^{2}
|
||||
$$
|
||||
|
||||
In the steady state therefore, the optimum tip speed ratio can be maintained by setting the load torque at the generator, $Q_{g}$ , to balance the aerodynamic torque, that is,
|
||||
稳态条件下，可以通过在发电机处设置负载转矩 $Q_{g}$，以平衡气动转矩，从而维持最佳叶片尖速比，即：
|
||||
|
||||
$$
|
||||
{\it Q}_{g}=\frac{1}{2}\frac{\pi\rho R^{5}C_{p}}{\lambda^{3}G^{3}}\omega_{g}^{2}-{\it Q}_{L}
|
||||
$$
|
||||
|
||||
Here $Q_{L}$ represents the mechanical torque loss in the drive train (which may itself be a function of rotational speed and torque), referred to the high-speed shaft. The generator speed is $\omega_{\mathrm{g}}=G2$ , where $G$ is the gearbox ratio.
|
||||
这里，$Q_{L}$ 表示传递到高速轴上的驱动系机械扭矩损耗（本身可能旋转速度和扭矩的函数）。发电机转速为 $\omega_{\mathrm{g}}=G\varOmega$，其中 $G$ 为齿轮箱传动比。
|
||||
|
||||
This torque-speed relationship is shown schematically in Figure 8.3 as the curve B1–C1. Although it represents the steady-state solution for optimum $C_{p}$ , it can also be used dynamically to control generator torque demand as a function of measured generator speed. In many cases, this is a very benign and satisfactory way of controlling generator torque below rated wind speed.
|
||||
图 8.3 示意性地显示了转矩-转速关系曲线 B1–C1。虽然它代表了最佳 $C_{p}$ 的稳态解，但也可以动态地用于控制与测得的发电机转速相关的发电机转矩需求。在许多情况下，这是一种在额定风速以下控制发电机转矩的温和且令人满意的方案。
|
||||
|
||||
For tracking peak $C_{p}$ below rated in a variable-speed turbine, the quadratic algorithm of Eq. (8.4) works well and gives smooth, stable control. However, in turbulent winds, the large rotor inertia prevents it from changing speed fast enough to follow the wind, so rather than staying on the peak of the $C_{p}$ curve it will constantly fall off either side, resulting in a lower mean $C_{p}$ . This problem is clearly worse for heavy rotors, and also if the $C_{p}-\lambda$ curve has a sharp peak. Thus, in optimising a blade design for variable-speed operation, it is not only important to try to maximise the peak $C_{p}$ , but also to ensure that the $C_{p}-\lambda$ curve is reasonably flat-topped.
|
||||
为了跟踪额定转速以下的最大 $C_{p}$ 值，公式(8.4)中的二次算法效果良好，能提供平稳、稳定的控制。然而，在湍流风况下，大型转子的惯性阻碍了其快速改变转速以跟踪风速，因此它无法始终保持在 $C_{p}$ 曲线的峰值上，而是会不断偏离两侧，导致平均 $C_{p}$ 值降低。这种问题对于重型转子尤其明显，并且当 $C_{p}-\lambda$ 曲线具有尖锐峰值时也会加剧。因此，在优化用于可变速运行的叶片设计时，不仅要努力最大化峰值 $C_{p}$，还要确保 $C_{p}-\lambda$ 曲线具有较为平坦的峰顶。
|
||||
|
||||
It is possible to manipulate the generator torque to cause the rotor speed to change faster when required, so staying closer to the peak of the $C_{p}$ curve. One way to do this is to modify the torque demand by a term proportional to rotor acceleration (Bossanyi 1994):
|
||||
|
||||
|
||||
Figure 8.3 Schematic torque-speed curve for a variable-speed pitch-regulated turbine
|
||||
|
||||
$$
|
||||
\mathcal{Q}_{g}=\frac{1}{2}\frac{\pi\rho R^{5}C_{p}}{\lambda^{3}G^{3}}\omega_{g}^{2}-Q_{L}-B\dot{\omega}_{g}
|
||||
$$
|
||||
|
||||
where $B$ is a gain that determines the amount of inertia compensation. For a stiff drive train, and ignoring frequency converter dynamics, the torque balance gives
|
||||
|
||||
$$
|
||||
I\dot{\boldsymbol\Omega}=Q_{a}-G Q_{g}
|
||||
$$
|
||||
|
||||
where $I$ is the total inertia (of rotor, drive train and generator, referred to the low-speed shaft) and $\varOmega$ is the rotational speed of the rotor. Hence
|
||||
|
||||
$$
|
||||
(I-G^{2}B)\dot{\Omega}=Q_{a}-\frac{1}{2}\frac{\pi\rho R^{5}C_{p}}{\lambda^{3}G^{2}}\omega_{g}^{2}+G Q_{L}
|
||||
$$
|
||||
|
||||
Thus, the effective inertia is reduced from $I$ to $I-G^{2}B$ , allowing the rotor speed to respond more rapidly to changes in wind speed. The gain $B$ should remain significantly smaller than $I/G^{2}$ otherwise the effective inertia will approach zero, requiring huge power swings to force the rotor speed to track closely the changes in wind speed.
|
||||
|
||||
Another possible method is to use available measurements to make an estimate of the wind speed, calculate the rotor speed required for optimum $C_{p}$ , and then use the generator torque to achieve that speed as rapidly as possible. The aerodynamic torque can be expressed as
|
||||
|
||||
$$
|
||||
Q_{a}=\frac{1}{2}\rho A C_{q}R U^{2}=\frac{1}{2}\rho\pi R^{5}\varOmega^{2}C_{q}/\lambda^{2}
|
||||
$$
|
||||
|
||||
where $R$ is the turbine radius, $\varOmega$ the rotational speed, and $C_{q}$ the torque coefficient. If drive train torsional flexibility is ignored, a simple estimator for the aerodynamic torque is
|
||||
|
||||
$$
|
||||
{Q_{a}}^{*}=G Q_{g}+I\dot{\varOmega}=G Q_{g}+I\dot{\omega}_{g}/G
|
||||
$$
|
||||
|
||||
where $I$ is the total inertia. A more sophisticated estimator could take into account drive train torsion, etc. From this it is possible to estimate the value of the function $F(\lambda)=C_{q}(\lambda)/\lambda^{2}$ as
|
||||
|
||||
$$
|
||||
F^{*}(\lambda)=\frac{Q_{a}^{*}}{\frac{1}{2}\rho\pi R^{5}(\omega_{g}/G)^{2}}
|
||||
$$
|
||||
|
||||
Knowing the function $F(\lambda)$ from steady state aerodynamic analysis, one can then deduce the current estimated tip speed ratio $\lambda^{*}$ (see also Section 8.3.16 for a better estimation method). The desired generator speed for optimum tip speed ratio can then be calculated as
|
||||
|
||||
$$
|
||||
\omega_{d}=\omega_{g}\widehat{\lambda}/\lambda^{*}
|
||||
$$
|
||||
|
||||
where $\widehat{\lambda}$ is the optimum tip speed ratio to be tracked. A simple PI controller can then be used, acting on the speed error $\omega_{\mathrm{g}}-\omega_{\mathrm{d}}$ , to calculate a generator torque demand that will track $\omega_{\mathrm{d}}$ . The higher the gain of PI controller, the better will be the $C_{p}$ tracking, but at the expense of larger power variations. Simulations for a particular turbine showed that a below rated energy gain of almost $1\%$ could be achieved, with large but not unacceptable power variations.
|
||||
|
||||
Holley et al. (1999) demonstrated similar results with a more sophisticated scheme, and also showed that a perfect $C_{p}$ tracker could capture $3\%$ more energy below rated, but only by demanding huge power swings of plus and minus three to four times rated power, which is totally unacceptable.
|
||||
|
||||
Because such large torque variations are required to achieve only a modest increase in power output, it is usual to use the simple quadratic law, possibly augmented by some inertia compensation as in Eq. (8.5) if the rotor inertia is large enough to justify it.
|
||||
|
||||
As turbine diameters increase in relation to the lateral and vertical length scales of turbulence, it becomes more difficult to achieve peak $C_{p}$ anyway because of the non-uniformity of the wind speed over the rotor swept area. Thus if one part of a blade is at its optimum angle of attack at some instant, other parts will not be.
|
||||
|
||||
In most cases, it is actually not practical to maintain peak $C_{p}$ from cut-in all of the way to rated wind speed. Although some variable-speed systems can operate all of the way down to zero rotational speed, this is not the case with limited range variable-speed systems based on the widely used doubly fed induction generators. These systems only need a power converter rated to handle a fraction of the turbine power, which is a major cost saving. This means that in low wind speeds, just above cut-in, it may be necessary to operate at an essentially constant rotational speed, with the tip speed ratio above the optimum value.
|
||||
|
||||
At the other end of the range, it is usual to limit the rotational speed to some level, usually determined by aerodynamic noise constraints or blade leading-edge erosion, which is reached at a wind speed that is still some way below rated. It is then cost-effective to increase to torque demand further, at essentially constant rotational speed, until rated power is reached. Figure 8.3 illustrates some typical torque-speed trajectories, which are explained in more detail below. Turbines designed for noise-insensitive sites may be designed to operate along the optimum $C_{p}$ trajectory all of the way until rated power is reached. The higher rotational speed implies lower torque and in-plane loads, but higher out-of-plane loads, for the same rated power. This strategy might be of interest for offshore wind turbines.
|
||||
Binary file not shown.
Binary file not shown.
|
After Width: | Height: | Size: 103 KiB |
@ -148,4 +148,6 @@ def compute_delta_pitch(current_power, target_power, sensitivity):
|
||||
delta = (current_power - target_power) * sensitivity
|
||||
|
||||
return delta
|
||||
```
|
||||
```
|
||||
|
||||
[[[steady控制算法.excalidraw]]]
|
||||
950
多体+耦合求解器/steady控制算法.excalidraw.md
Normal file
950
多体+耦合求解器/steady控制算法.excalidraw.md
Normal file
@ -0,0 +1,950 @@
|
||||
---
|
||||
|
||||
excalidraw-plugin: parsed
|
||||
tags: [excalidraw]
|
||||
|
||||
---
|
||||
==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving'
|
||||
|
||||
|
||||
# Excalidraw Data
|
||||
|
||||
## Text Elements
|
||||
%%
|
||||
## Drawing
|
||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
|
||||
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|
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|
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|
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|
||||
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||||
|
||||
wDIBLsTBQ4ACwULMYT1DAAJrAIACawEAAA==
|
||||
```
|
||||
%%
|
||||
BIN
杂项/房屋资料/个人购房借款及担保合同.pdf
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杂项/房屋资料/个人购房借款及担保合同.pdf
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杂项/房屋资料/网签合同.pdf
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杂项/房屋资料/网签合同.pdf
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Loading…
x
Reference in New Issue
Block a user