CN104573249A - Time-variant ARMA model based non-stable wind speed simulation method - Google Patents
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Abstract
The invention discloses a time-variant ARMA model based non-stable wind speed simulation method. The time-variant ARMA model based non-stable wind speed simulation method comprises the following steps of 1 determining parameters of a target non-stable wind speed simulated wind speed model and a time-variant ARMA model; 2 dispersing approximate non-stable fluctuating wind speed into a plurality of stable fluctuating wind speeds within a short time interval, and meanwhile adopting a non-uniform modulation function to modulate a power spectrum of each section of stable fluctuating wind speeds into an evolution spectrum; 3 establishing the time-variant ARMA model of non-stable wind speed and performing non-stable wind speed simulation; 4 outputting and displaying a simulation result and comparing the power spectrum of simulation point wind speeds and respective and a self-correlation function and a cross-correlation function with a target power spectrum, a target self-correlation function and a target cross-correlation function. The time-variant ARMA model based non-stable wind speed simulation method is used for fluctuating wind speed non-stability simulation and ensuring that the frequency component of the fluctuating wind speed changes with time, namely a time-varying characteristic of the power spectrum.
Description
Technical Field
The invention relates to a method for simulating non-stationary wind speed by adopting time series analysis, in particular to a non-stationary wind speed simulation method based on a time-varying ARMA model.
Background
For buildings (structures) such as large-span space structures, large-span bridges, high-rise (super) building structures, high-rise structures (such as guyed masts, television towers, chimneys and the like), wind load is one of the control loads of the wind-resistant design of the structure. The wind resistance analysis of the structure firstly needs to obtain sample data of wind load, and the current main means for determining the wind load comprises wind tunnel test, field actual measurement, numerical simulation and the like. With the rapid development of computer technology and the deep research of people on random process numerical simulation technology, the wind speed time curve obtained by adopting a numerical simulation method can consider the arbitrariness of conditions such as fields, wind spectrum characteristics, building characteristics and the like, so that the load obtained by simulation is as close to the actual wind power of a structure as possible, can meet the arbitrariness of certain statistical characteristics, is more representative than actual records, and is widely applied to actual engineering.
The non-stationary characteristic is a phenomenon (such as turbulent flow of an atmospheric boundary layer, thunderstorm strong wind, earthquake and the like) commonly existing in various random loads in the nature, and the amplitude and the frequency of the phenomenon change along with time, so that the non-stationary characteristic of the wind is a factor which needs to be considered when the numerical simulation is carried out on pulsating wind under certain specific environments. Particularly in downburst (i.e., a very sudden and destructive strong wind that is violently hit the ground in a violent downdraft in thunderstorm weather and propagates all around the ground from the point of impact), its very strong non-stationarity is likely to generate a greater dynamic response to the structure. Numerical simulation of the current stochastic process is mainly classified into two categories: the method is a spectral representation method based on trigonometric series superposition, and is a regression method based on a linear filtering technology. These conventional wind speed simulation methods generally approximate natural wind as a stationary gaussian process that experiences each state and perform the simulation based on the existing fluctuating wind speed power spectrum. For benign weather wind with good stability on an open and flat field, the stability assumption can be basically satisfied. However, analysis of a large number of actual test data shows that many wind speed records in a complex terrain in a strong wind environment do not meet the stability requirement. Particularly, when steady wind speed is assumed, non-steady data needs to be discarded in the case of non-steady pulsating wind in a complex terrain and strong wind environment, which can cause large analysis errors, for example, turbulence intensity values can be overestimated, and further accuracy of subsequent analysis is influenced.
Time series analysis (Time series analysis) is a statistical method of dynamic data processing. The invention is based on the random process theory and the mathematical statistics method, and researches the statistical rule followed by the random data sequence to solve the practical problem. An Autoregressive moving average model (ARMA model for short) is an important method for researching time series, and is formed by mixing an Autoregressive model (AR model for short) and a moving average model (MA model for short) as a basis. The method is based on a non-stationary wind speed model, adopts a Time series autoregressive moving average model and considers the Time-Varying property of ARMA model coefficients, namely a TARMA (Time-Varying ARMA) method, and carries out numerical simulation on the random process of non-stationary pulsating wind.
Disclosure of Invention
The invention aims to provide a non-stationary wind speed simulation method based on a time-varying ARMA model, which simulates the non-stationary of a pulsating wind speed and ensures that the frequency component of the non-stationary wind speed simulation method is time-varying, namely the time-varying property of a power spectrum; and simultaneously comparing the power spectrum and the auto-correlation function of the wind speed at the simulation point with the target power spectrum and the target auto-correlation function to ensure the effectiveness of the simulation.
In order to achieve the purpose, the invention has the following conception: the method comprises the steps of dispersing the unstable pulsating wind speed into a plurality of short-time sequences which can be approximated to the stable pulsating wind speed within a short enough time interval delta t, modulating the power spectrum through a non-uniform modulation function according to the 'evolutionary spectrum' theory to obtain a time-varying power spectrum, namely an evolutionary spectrum, generating the unstable pulsating wind speed through an established TARMA time-varying model of the unstable pulsating wind speed and based on the modulated evolutionary spectrum, and generating the wind speed while considering the spatial correlation of the pulsating wind.
According to the inventive concept, the invention adopts the following technical scheme: a non-stationary wind speed simulation method based on a time-varying ARMA model is characterized by comprising the following steps:
firstly, determining parameters of a wind speed model for simulating a target non-steady wind speed and a time varying ARMA model;
secondly, approximately dispersing the non-steady pulsating wind speed into a plurality of steady pulsating wind speeds in a short time interval, and simultaneously modulating the power spectrum of each section of steady pulsating wind speed into an evolutionary spectrum by adopting a non-uniform modulation function;
thirdly, establishing a time-varying ARMA model of the unsteady fluctuating wind speed, and carrying out unsteady wind speed simulation;
and fourthly, outputting and displaying a simulation result, and comparing the power spectrum and the auto-correlation function of the wind speed at the simulation point with the target power spectrum and the target auto-correlation function.
Preferably, the first step represents the downwind non-stationary wind speed as two parts, a deterministic time-varying mean wind and a zero mean stationary pulsating wind.
Preferably, the non-uniform modulation function in the second step is expressed by the following formula:
in the formula: omega is the circular frequency; z is a radical ofjIs the height of a certain point in space vertical to the ground;the time-varying average wind speed at a certain point in space;and counting the average wind speed for the non-steady wind speed at a certain point in space.
Preferably, the wind speed model of the non-stationary wind speed simulation is represented by the following formula:
in the formula,is the time-varying mean wind, u, at space (x, y, z)*(t) is the zero mean smooth pulsating wind process.
Preferably, the time-varying ARMA model employs the following equation:
wherein U (t) is a zero-mean non-stationary random process vector, Ai(t) is a time-varying autoregressive coefficient matrix, Bj(t) is a time-varying sliding regression coefficient matrix, p is an autoregressive order, q is a sliding regression order, and X (t) is a normally distributed white noise sequence with a variance of 1.
The non-stationary wind speed simulation method based on the time-varying ARMA model has the following advantages: the invention simulates the power spectrum of the unsteady fluctuating wind speed and the target average power spectrum of each pointComparing the wind speeds and simulating the auto-cross correlation function and the target auto-cross correlation function of the wind speeds at each pointAnd comparing and matching the results. PulseThe amplitude of the dynamic wind speed is related to the time-varying average wind speed, and the larger the time-varying average wind speed is, the larger the amplitude of the pulsating wind speed is, which is consistent with the actual wind field characteristics.
Drawings
FIG. 1 is a schematic view of non-stationary wind speed simulation points along the vertical direction of the ground;
FIGS. 2(a) and 2(b) are schematic diagrams of downburst models illustrating different stages of downburst movement toward the building;
FIG. 3 is a design framework diagram of a non-stationary wind speed simulation method based on a time-varying ARMA model;
FIG. 4 is a flowchart of a non-stationary wind speed simulation method routine based on a time-varying ARMA model;
FIGS. 5(a) to 5(c) are schematic diagrams of non-stationary and pulsating wind speeds at 800s at respective simulated spatial points along the ground vertical heights 10m, 25m and 80 m; FIGS. 5(d) to 5(f) are schematic views of a non-stationary pulsating wind speed at 800s at simulated spatial points along the ground vertical height 10m, 25m and 80 m;
6(a) through 6(c) are schematic diagrams of non-uniform modulation functions for various simulated spatial points; 6(d) to 6(f) are schematic diagrams of time-varying power spectra for various simulated spatial points; FIGS. 6(g) to 6(i) are schematic diagrams comparing simulated power spectra at various simulated spatial points with target spectra;
fig. 7(a) to 7(f) are schematic diagrams comparing the auto and cross correlation function target with the target auto and cross correlation function at 10m, 25m, 80m, 10m and 25m, 25m and 80m, 10m and 80m, and 80m of each simulated spatial point along the vertical height of the ground.
Detailed Description
The simulation procedure for non-stationary pulsating wind speeds (hereinafter storm flow as an example) according to the present invention is described in further detail below with reference to the accompanying drawings.
The invention relates to a non-stationary wind speed simulation method based on a time-varying ARMA model, which comprises the following steps:
firstly, determining a wind speed model (comprising a wind speed model, a simulation space point, a terrain parameter, simulation duration and the like) of a target non-stationary wind speed simulation, a target power spectrum and each parameter (an autoregressive order and a sliding regression order) of a time-varying ARMA model. The downwind non-stationary wind speed is expressed as two parts of deterministic time-varying mean wind and zero mean stationary pulsating wind.
In the first step, the wind speed model for non-stationary wind speed simulation is expressed as the following formula (1):
in the formula,is the time-varying mean wind, u, at space (x, y, z)*(t) is the zero mean smooth pulsating wind process. From the above stationary and non-stationary wind speed models, it can be seen that if the total downwind direction record U (t) itself is strictly stationaryAnd in a random process, the time-varying mean wind U (t) is degraded into a constant mean wind in a traditional steady wind model. In fact, the definition in the non-stationary wind speed model is the simplest model to describe the non-stationary process: stationary process + trend term.
As a non-stationary wind speed simulation was performed on the three points shown in fig. 1 using the downburst model shown in fig. 2(a) and 2(b), the autoregressive order p of the time-varying ARMA model was 16, and the slip regression order q was 1. The simulation point is located along the downburst moving direction and is 3500m away from the downburst thunderstorm center. The downdraft storm flow wind speed model adopts an Oseguera and Bowles average wind speed model and a Vicroy vertical distribution model, and the maximum wind speed V in the vertical distribution wind speedmax80m/s at a height position Zmax67 m; radial maximum wind speed V at a certain height in wind speed fieldr,max47m/s, horizontal distance r from downburst centermax1000m, radial length scaling factor Rr700 m; the time-dependent change in the intensity of a thunderstorm is represented by the following formula (2):
downburst translation velocity Vo8 m/s. The upper limit cut-off frequency is 4 pi rad, N is 2^11,meanwhile, the movement of the downburst is considered, the simulation time interval delta t is 0.1s, and the simulation time duration is 800 s.
Secondly, approximately dispersing the non-stable pulsating wind speed into a plurality of stable pulsating wind speeds within a short time interval delta t, and simultaneously modulating the power spectrum of each section of stable pulsating wind speed into an evolutionary spectrum by adopting a non-uniform modulation function; the existing power spectrum (such as a Kaimal spectrum) of the pulsating wind speed is selected, the Kaimal spectrum is taken as an example below, and the power spectrum is modulated into a Kaimal evolutionary spectrum by adopting a Kaimal non-uniform modulation function.
The power spectrum using Kaimal is represented by the following formula (3):
the Kaimal non-uniform modulation function is expressed as the following formula (4):
in the formula: omega is the circular frequency; z is a radical ofjIs the height of a certain point in space vertical to the ground;the time-varying average wind speed at a certain point in space;and counting the average wind speed for the non-steady wind speed at a certain point in space.
And thirdly, establishing a time-varying ARMA model of the unsteady fluctuating wind speed, and carrying out unsteady wind speed simulation according to the frame diagram shown in FIG. 3 and the flow diagram shown in FIG. 4, namely, considering the time-varying property of an ARMA coefficient matrix and the spatial correlation of fluctuating wind, wherein the Matlab language can be adopted in the third step.
The third step comprises the following specific steps:
assume that the following equation (5) is used for the simulated M-point non-stationary fluctuating wind speed:
U(t)＝[u1(t) u2(t) … uM(t)]T (5)
the formula of the time-varying ARMA model is shown in the following formula (6):
wherein U (t) is a zero-mean non-stationary random process vector, Ai(t) is a time-varying autoregressive coefficient matrix (MxM order), Bj(t) is the moment of the time-varying slip regression coefficientThe matrix (MxM order), p is an autoregressive order, q is a sliding regression order, and X (t) is a white noise sequence with a variance of 1 and normal distribution, and satisfies the following formula (7):
as can be seen from equation (7), the key to calculating the pulsating wind speed U (t) is to determine Ai(t)、Bj(t) a coefficient matrix.
When t is equal to t0When the formula (7) is represented by the following formula (8):
two sides simultaneously right ride UT(t0-k Δ t) (k ═ 1, 2.. p), and mathematically expected to yield the following formula (9):
according to the definition of the correlation function, i.e. the following formula (10):
pair (8) then two sides multiply X right simultaneouslyT(t0-l Δ t) (l ═ 1, 2.. q), and taking mathematical expectations, we obtain the following formula (11):
namely, the following formula (12):
in addition, the current output U (t) can be seen from equation (9)0) Dependent only on the current input X (t)0) And past input X (t)0τ) with future input X (t)0+ τ) is irrelevant (τ > 0), i.e., U (t)0) And X (t)0+ τ) are independent of each other, then: rux(. tau.) -. 0,. tau. > 0 and Rux(-τ)＝Rxu(τ)。
When j is 0, k-j > 0, then Rxu[(k-j)Δt]＝0。
Combining formula (11) with formula (12) and developing the following formula (13):
wherein,
according to the formula of Vena-Sinkian, the following formula (14) is obtained:
can find [ Ruu(jΔt)]In this case, it is also necessary to know [ R ]ux(lΔt)]The equation can be found outi(t)、Bj(t) a coefficient matrix.
When q in formula (10) is 0, the following formula (15) is obtained:
two sides simultaneously right multiply XT(t0-l Δ t) (l ═ 0,1, 2.. q) and mathematically expected, yielding the following formula (16):
pair (15) right-handed at two sidesT(t0-k Δ t) (k ═ 0,1,2,. p), mathematically expected to yield the following formula (17):
expansion type <math>
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</math> Obtaining the following formula (18):
from the foregoing [ R ]uu(jΔt)]And equationCan find out
And a second formula from formulae (16), (17), andcan obtain the product <math>
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</math> Namely, the following formula (19):
at this point, the matrix at the right end of the above equation is completely solved, and Cholesky decomposition is performed on the right equation to obtain B0(t0) Then, the matrix [ R ] is obtained by substituting formula (16) in a recursion mannerux(lΔt)]. The matrix [ R ] obtaineduu(jΔt)]And [ R ]ux(lΔt)]By substituting formula (13), a TARMA time-varying model coefficient matrix A can be determinedi(t)、Bj(t), finally obtaining t0The wind speed at that moment.
And fourthly, outputting and displaying a simulation result, and comparing the power spectrum and the auto-correlation function of the wind speed at the simulation point with the target power spectrum and the target auto-correlation function as shown in figure 5 to ensure the effectiveness of the simulated wind speed.
The above steps are analyzed and verified based on a calculation program of a non-stationary wind speed simulation method which is compiled by a Matlab platform and is based on a time-varying ARMA model.
As can be seen from fig. 5(a) to 5(f), the amplitude of the pulsating wind speed is related to the magnitude of the time-varying average wind speed, and the amplitude of the pulsating wind speed is larger as the time-varying average wind speed is larger, which corresponds to the actual wind field characteristics. To further illustrate the effectiveness of the simulation, the power spectrum and the target average power spectrum are simulated for the non-stationary pulsating wind speed at each point(FIGS. 6(a) to 6 (i))Shown) were compared and the downburst Kaimal non-uniform modulation function at three different points is shown in fig. 6. And auto and cross correlation function and target auto and cross correlation functionComparison was made (shown in fig. 7(a) to 7 (f)), and it can be seen that the results are consistent. It can be seen from the cross-correlation function graph of fig. 7 that the cross-correlation of the pulsating wind speed at each point decreases with increasing distance.
Various modifications and changes may be made to the present invention by those skilled in the art. Thus, it is intended that the present invention cover the modifications and variations of this invention provided they come within the scope of the appended claims and their equivalents.
Claims (5)
1. A non-stationary wind speed simulation method based on a time-varying ARMA model is characterized by comprising the following steps:
firstly, determining parameters of a wind speed model for simulating a target non-stationary wind speed, a target power spectrum and a time-varying ARMA model, and simultaneously determining the time-varying average wind speed of each simulation space point;
secondly, modulating the target power spectrum into an evolutionary spectrum by adopting a non-uniform modulation function, and dispersing the unstable fluctuating wind speed within a plurality of short time intervals;
thirdly, establishing a time-varying ARMA model of the unsteady fluctuating wind speed, and simulating the unsteady fluctuating wind speed according to the previously determined time-varying ARMA model parameters and an evolutionary spectrum;
and fourthly, determining the final non-stationary wind speed according to the time-varying average wind speed and the generated non-stationary pulsating wind speed, outputting and displaying a simulation result, and simultaneously comparing the power spectrum and the auto-correlation function of the wind speed at the simulation point with the target power spectrum and the target auto-correlation function.
2. The time-varying ARMA model-based non-stationary wind speed simulation method according to claim 1, wherein the first step represents downwind non-stationary wind speed as two parts of deterministic time-varying mean wind and zero mean stationary pulsating wind.
3. The time-varying ARMA model-based non-stationary wind speed simulation method according to claim 1, wherein the non-uniform modulation function in the second step is represented by the following formula:
in the formula: omega is the circular frequency;is the height of a certain point in space vertical to the ground;the time-varying average wind speed at a certain point in space;and counting the average wind speed for the non-steady wind speed at a certain point in space.
4. The time-varying ARMA model-based non-stationary wind speed simulation method according to claim 1, wherein the non-stationary wind speed simulated wind speed model is represented by the following formula:
in the formula,is a spaceThe average wind of the wind is changed in time,is a zero mean smooth pulsating wind process.
5. The time-varying ARMA model-based non-stationary wind speed simulation method according to claim 1, characterized in that the time-varying ARMA model employs the following formula:
wherein,is a zero-mean non-stationary random process vector,is a time-varying matrix of auto-regressive coefficients,is a time-varying sliding regression coefficient matrix, p is the autoregressive order, q is the sliding regression order,is a variance of1. A normally distributed white noise sequence.
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