Extended modulating functions for simulation of wind

Renewable Energy 83 (2015) 384e397

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Extended modulating functions for simulation of wind velocities with weak and strong nonstationarity Jinhua Li a, Chunxiang Li b, *, Liang He b, Jianhong Shen c a

Department of Civil Engineering, East China Jiaotong University, No.808 Shuanggang St., Nanchang 330013, PR China Department of Civil Engineering, Shanghai University, No.149 Yanchang Rd., Shanghai 200072, PR China c School of Civil Engineering, Qingdao Technological University, No. 11 Fushun Rd., Qingdao 266520, PR China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 15 May 2014 Accepted 21 April 2015 Available online

Following the theory of evolutionary power spectral density (EPSD) for nonstationary processes, it is anticipated that the nonstationary wind velocities can be generated through modulating the stationary wind velocities by resorting to desirable modulating functions. Naturally, the key to the simulation of the nonstationary wind velocities lies in seeking out desirable modulating functions. The purpose of this study thus is how to obtain the modulating functions suitable for modulating stationary wind velocities. This work begins with systematically deducing the modulating function according to Kaimal power spectrum. The modulating functions corresponding respectively to other power spectra are subsequently presented. Derived modulating functions herein are collectively referred to as the extended modulating functions (EMFs). Employing the recent research by Li et al. (2009) that integration of both the spline interpolation algorithm (SIA) and spectral representation (SR) method, then forming the SIA-SR method, is able to remarkably reduce the increasing number of Cholesky decomposition of the time-varying spectral density matrix, the simulation of the nonstationary wind velocities has been carried out with an EMF. It can be inferred in terms of simulation results that the EMFs can fully capture the nonstationarity of wind velocities, including both the weak and strong nonstationary wind velocities. © 2015 Elsevier Ltd. All rights reserved.

Keywords: Simulation Nonstationary wind velocities Weak and strong nonstationarity Extended modulating function Spectral representation Spline interpolation algorithm

1. Introduction Extreme wind events in many districts around the world are caused by severe convective thunderstorm winds and typhoon (or hurricane)-induced winds, which are responsible for significant structural damage and failures. This sort of wind flow is significantly different from the traditional atmospheric boundary layer wind flows in light of its unique mean wind speed vertical profile, rapid time-varying mean wind speed, and spatially strongly correlated wind fluctuations. Extreme winds exhibit the transient nonstationary features which may remarkably influence windstructure interaction and wind load effects on buildings [1]. More specifically, not only does the load quantity change with timefrequency, but the direction in which these loads impinge on structures also changes, thus further complicating the load and performance assessment process. Apparently, in order to obtain more reliable wind-resistant designs as well as rapid performance

* Corresponding author. E-mail addresses: [email protected], [email protected] (C. Li). http://dx.doi.org/10.1016/j.renene.2015.04.044 0960-1481/© 2015 Elsevier Ltd. All rights reserved.

assessment, it is imperative and of practical interest to capture the transient nonstationary phenomena in thunderstorm downbursts and typhoon (or hurricane)-induced winds by resorting to the numerical simulation techniques. Naturally, it is envisaged to develop an efficient yet accurate simulation method suitable for transient nonstationary winds. Monte Carlo-based simulation (MCS) approaches, which may generate the sample functions with the probabilistic characteristics of target stochastic process, have been widely employed in simulating the stochastic processes [2]. The scope of simulations spans uni-dimensional to multi-dimensional fields; univariate to multivariate processes; homogeneous to non-homogeneous fields; stationary to nonstationary; Gaussian to non-Gaussian; and conditional to unconditional cases. To date, the following approaches are available for the simulation of stochastic processes: (1) auto-regressive (AR) model in the white noise filtration method (WNFM), such as [3e7]; (2) moving average (MA) model in the WNFM, e.g. Refs. [8,9]; (3) auto-regressive moving average (ARMA) model in the WNFM, for example, [10e13]; and (4) spectral representation method, referred thereinafter to as the SR method, for instance, [14e18]. Both the AR and MA can be included in ARMA

J. Li et al. / Renewable Energy 83 (2015) 384e397

model [19,20], which is a counterpart to the SR method. It is noted that the computational efficiency of the SR method is rather low. A possible remedy to this issue is to introduce the fast Fourier transform (FFT) algorithm into the SR method. Yang [21] showed that the FFT algorithm can remarkably enhance the computational efficiency of the SR method, and furthermore developed a formula to simulate the random envelop processes. Wittig and Sinhat [22] demonstrated that the SR method combined with the FFT algorithm appears to be more than an order of magnitude faster than other simulation methods. Further, Li and Kareem [23] used a computationally efficient FFT-based approach to simulate the multivariate Gaussian nonstationary random processes (regarding the reproduction of seismic ground motions) with the prescribed evolutionary spectral description. Here, it should be pointed out that the FFT algorithm is not directly applicable to the nonstationary case. However, when the modulating function is a deterministic time function, the FFT algorithm may be directly employed for the simulation of the nonstationary stochastic process. The SR method has very high demands on both the computer memory and computational speed but does not have the problem of model selection. Likewise, it is easy to implement and moreover, render high accuracy. Therefore, the SR method has yet been receiving increasing attention in simulating the multivariate stochastic processes. In view of these, the SR method will be taken into consideration in this paper. It is known that the simulation of nonstationary stochastic processes may be simplified via representing nonstationary stochastic processes in terms of stationary processes modulated by slowly time-varying deterministic functions, referred to as the uniformly modulated (UM) stationary processes. Saragoni and Hart [24] partitioned the nonstationary stochastic process into several segments and then proposed a piecewise stationary model. Later, an extension of the Saragoni-Hart model was made by Der Kiureghian and Crempien [25]. They represented the model as a summation of modulated banded white noise. Li and Kareem [15] also used the modulated stationary stochastic process concept. A nonstationary process was expressed as a sum of mutually correlated stationary processes modulated by a deterministic time function. The spectrum of the stationary processes and the deterministic modulating time function can be derived through matching a prescribed evolutionary spectrum. Liang et al. [26] presented a rigorous derivation of a previously known formula for simulation of one-dimensional, univariate, nonstationary stochastic processes through integrating Priestley's evolutionary SR theory. The utilization of the SR method thus is believed to be possible in simulating the nonstationary stochastic process. It is worth pointing herein out that most simulations of the nonstationary stochastic processes have involved around the reproduction of seismic ground motions known to be highly nonstationary [27]. The literature [27] presented an overview of the simulation of nonstationary processes including seismic ground motions. The recent research works on nonstationary winds mainly consist of the contributions by the references [1,28e39]. Notwithstanding, the numerical simulation of transient nonstationary wind velocities has been rather limited due to the lack of the valid modulating functions. Evidently, it is significant to find the satisfactory modulating functions with the change of time and frequency for the simulation of nonstationary fluctuating winds. The purpose of this study thus is how to obtain the modulating functions suitable for non-uniformly modulating stationary wind velocities. The modulating functions established herein are collectively referred to as the extended modulating functions (EMFs) to generate the wind velocities with weak and strong nonstationarity. In order to examine the capability that the EMFs capture the nonstationarity of wind velocities, the simulation of the

385

nonstationary wind velocities will be carried out with an EMF, furthermore making use of integration of both the spline interpolation algorithm (SIA) and SR method, then constituting the SIA-SR method, proposed by Li et al. [40] for the reduction of the increasing number of Cholesky decomposition of the time-varying spectral density matrix.

2. Extended modulating functions (EMFs) The boundary layer longitudinal wind velocity f(t) at a given height is generally assumed to be an ergodic random process, which consists of a constant mean wind speed component U and a longitudinal fluctuating wind velocity component u(t) in the following form.

f ðtÞ ¼ U þ uðtÞ

(1)

It is worth pointing herein out that the term “mean” refers to an average over a time interval T, which generally is taken as 1 h or 10 min with respect to wind effects on structures, thereupon leading to the so-called hourly or 10 min mean wind speed which is given by the following equation.

U¼

1 T

ZT f ðtÞdt

(2)

0

However, the boundary layer wind measured during both thunderstorm and typhoon (or hurricane) may not comply with the assumption of the ergodic or stationary random process [27]. A recent preliminary study [30] of nonstationary wind data recorded in the field during a nearby typhoon (or hurricane) reveals that the mean wind speed over 1 h often takes on a significant temporal (slowly time-varying) trend. On the basis of this situation, a nonstationary wind velocity model has been put forward by Xu and Chen [30]. This model, which is taken into consideration in the present paper, means that the nonstationary wind velocity is ~ viewed as a deterministic time-varying mean wind speed UðtÞ plus a fluctuating wind velocity u(t) that can be modelled as a zeromean stationary or nonstationary random process, which has the form:

~ f ðtÞ ¼ UðtÞ þ uðtÞ

(3)

It is known that the fluctuation, which may be regarded to be the stationary random process, admits the well-known spectral representation below.

Zþ∞ uðtÞ ¼

ei ut dZðuÞ

(4)

∞

whereas the fluctuation characterized by a nonstationary random process can be formulated as follows:

Zþ∞ uðtÞ ¼

Aðu; tÞ ei ut dZðuÞ

(5)

∞

where A(u,t) is a deterministic modulating function with two variables u and t; and Z(u) represents a spectral process with orthogonal increments which has a distribution function in the interval (∞, þ∞). From here we see that unlike the simulation of stationary stochastic processes, a desirable deterministic modulating function

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needs to be predetermined in simulating the nonstationary fluctuating wind velocities. 2.1. Time-varying power spectral density (TVPSD) function Arbitrary nonstationary wind velocity during the time [0, T] may be divided into n ¼ T/Dt segments. For each segment with [t1, ~ z ðtÞzU ~ z ðt Þ, and the t1 þ Dt], the time-varying mean wind speed U 1 j j TVSPD function Gzj zj ðu; tÞzGzj zj ðu; t1 Þ when the interval Dt is very small. Therefore, each segment is considered as stationary wind ~ z ðt Þ and PSD function velocity with mean wind speed Uzj ¼ U 1 j Szj zj ðuÞ ¼ Gzj zj ðu; t1 Þ. In order to model the two-sided PSD function Gzj zj ðu; t1 Þ of the approximately stationary wind velocity fluctuations with transitory (small) interval [t1, t1 þ Dt], the expression proposed by Kaimal et al. [41] is herein taken into consideration, which is given by

Gzj zj ðu; t1 Þ ¼

¼

1 200 zj 2 2p Uzj "

u2* 1þ

1 200 zj ~ z ðt Þ " 2 2p U 1 j

#5=3

uz 50 2pUjz j

#5=3

ln

ðj;kÞ

ya p p ¼

  p U zk þ U zj Cq

(10)

In the preceding Eq. (10), Cq refers to an appropriate coefficient that has to be determined from experimental data. In the present paper, Cq ¼ 5.5 [45] is taken into account. Employing Eqs. (9) and (10), we compute that

  u zj  zk ðj;kÞ

ya p p   Cq zj  zk u ¼ ~ z ðt Þ þ U ~ z ðt Þ pU 1 1 j k

  u Cq z j  z k ¼ p U zk þ U zj (11)

Hence, the cross PSD function Gzjzk ðu; t1 Þ of the stationary fluctuation wind velocities in [t1, t1 þ Dt] can then be determined as follows:

~ Z ðt Þ. in which u* ðt1 Þ ¼ k  UZj ¼ k  U 1 j zj zj z0

(9)

ðj;kÞ

ya p p

Likewise, the apparent speed of waves can be assumed to be the following form given by Simiu and Scanlan [43].

(6)

uz 50 ~ j 2pU z ðt1 Þ j

ln

qj k ðuÞ ¼

  u zj  zk

qj k ðu; t1 Þ ¼ qj k ðuÞ ¼

u2* ðt1 Þ 1þ

in which Cz is a constant that may generally be set to be equal to 10 for structural design purposes [43]. In regard of the stationary fluctuating wind velocity, the phase angle qjk(u) can be represented in the form suggested by Di Paola [44] as

z0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi  Gzj zj ðu; t1 Þ Gzk zk ðu; t1 Þ ei qj k ðu;t1 Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   #v u Cz zk  zj  zj u1 200 2 u 1 u u ðt Þ ¼ exp  #5=3 ~ z ðt Þ u2 2p * 1 U ~ z ðt Þ þ U ~ z ðt Þ " pU 1 1 u 1 j j k uz t 1 þ 50 ~ j

Gzj zk ðu; t1 Þ ¼ Gðu; t1 Þ "

2pU zj ðt1 Þ

(12)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "   # u Cq zj  zk u1 200 2 zk 1 u u u u* ðt1 Þ #5=3 exp  i p ~ ~ z ðt Þ ~ z ðt Þ " U U zk ðt1 Þ þ U u2 2p 1 1 j k u zk t 1 þ 50 ~ 2pU zk ðt1 Þ

According to the logarithmic law, Uz k at the height zk can be calculated as

# # " lnðzk =z0 Þ lnðzk =z0 Þ ~ ~ z ðt Þ    U zj ¼    U zj ðt1 Þ ¼ U ¼ 1 k ln zj z0 ln zj z0

For the nonstationary fluctuating wind velocity during the time [0, T], its TVPSD function can be obtained when Dt/0. Accordingly, the TVPSD function is expressed as follows:

"

U zk

(7)

Simultaneously, the coherence function between the velocity fluctuations at two different heights zj and zk proposed by Davenport [42] is taken into consideration in the present paper, which is given by

Gzj zj ðu; tÞ ¼

zj 1 200 2 1 u ðtÞ #5=3 ~ z ðtÞ " 2 2p * U j uz 1 þ 50 ~ j 2pU zj ðtÞ

zj 1 200 2 u* ðtÞ ¼ 2 2p U zj " 2

2

3   u Cz zk  zj  7 6 Gðu; t1 Þ ¼ CohðuÞ ¼ exp4   5 2p 12 Uzk þ Uzj " ¼ exp 

  # Cz zk  zj  u ~ z ðt Þ ~ z ðt Þ þ U pU 1 1 j k



(8)

U zj #5=3 ~ U zj ðtÞ

1 uz

1 þ 50 2pUjz

(13)

j

35=3

uz 6 1 þ 50 2pUjz 7 6 j 7 7 6 41 þ 50 u~ zj 5 2pU zj ðtÞ

Simultaneously, the time-varying cross PSD (TVCPSD) function can be determined using the following expression:

J. Li et al. / Renewable Energy 83 (2015) 384e397

387

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    Gzj zk ðu; tÞ ¼ Gðu; tÞ Gzj zj ðu; tÞ Gzk zk ðu; tÞ ei qj k ðu;tÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "   #v u zj u1 200 2 u Cz zk  zj  1 u u ðtÞ ¼ exp  #5=3 ~ z ðtÞ u2 2p * U ~ z ðtÞ þ U ~ z ðtÞ " pU u j j k uz t 1 þ 50 ~ j 2pU zj ðtÞ

(14)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi "   # u u1 200 2 zk 1 u Cq zj  zk u u u* ðtÞ #5=3 exp  i p ~ ~ z ðtÞ ~ z ðtÞ " U zk ðtÞ þ U U u2 2p j k u zk t 1 þ 50 ~ 2pU zk ðtÞ

Evidently, the nonstationary fluctuating wind velocity with both the TVPSD and TVCPSD functions can be simulated by resorting to modulating stationary fluctuating wind velocity. In order to generate the nonstationary fluctuating wind velocity, a satisfactory deterministic modulating function for modulating the stationary fluctuating wind velocity in accordance with Kaimal spectrum will be derived in details in the next subsection. 2.2. EMFs for wind velocities with weak and strong nonstationarity Based on the EPSD [46], the nonstationary stochastic process u(t) with a zero-mean value admits the preceding Eq. (5), i.e.

Zþ∞ uðtÞ ¼

Taking into account that G 0z j z j ðu; tÞ in Eq. (18) is equivalent to Gzjzj ðu; tÞ in Eq. (13), then Eq. (18) can be further recast in the following form:

G0zj zj ðu; tÞ ¼ Gzj zj ðu; tÞ ¼

zj 1 200 2 u ðtÞ ~ z ðtÞ " 2 2p * U j

35=3 u zj ~ z ðtÞ 6 1 þ 50 2pUz 7 1 U 6 j 7 ¼ j 6 7 Uzj 41 þ 50 u~ zj 5 2 2pU zj ðtÞ

1

#5=3

uz 1 þ 50 ~ j 2pU zj ðtÞ

2

200 2 zj 1 u # 5=3 2p * Uzj " u zj 1 þ 50 2pUz j

Aðu; tÞ ei ut dZðuÞ

(15)

∞

2

¼ jAðu; tÞj SðuÞ (19)

Likewise, Z(u) satisfies

E½dZðuÞ ¼ 0

(16)



 E dZ * ðu1 Þ dZðu2 Þ ¼ Sðu1 Þ dðu1  u2 Þ du1 du2

(17)

in which S(u) represents the PSD function of the corresponding stationary stochastic process. Now, it is assumed that the fluctuating wind velocity is modelled as a nonstationary stochastic process and furthermore its PSD function of stationary fluctuating wind velocity to be modulated is in obedience to the Kaimal power spectrum. It follows that the EPSD (The EPSD is similar to the PSD for stationary excitations.) function G 0z j z j ðu; tÞ of the nonstationary fluctuating wind velocity can be expressed in the following form:

G0zj zj ðu; tÞ ¼ jAðu; tÞj2 SðuÞ ¼ jAðu; tÞj2

1 200 2 zj u 2 2p * Uzj "

1 1þ

uz 50 2pUjz j

#5=3

It is known that the TVPSD being founded on a deterministic power spectrum can be represented by EPSD. Hence, we derive that

(18)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 35=3 u u u zj u uU ~ z ðtÞ 6 1 þ 50 2pUz 7 6 j 7 j A ðu; tÞ ¼ u 6 7 u t Uzj 41 þ 50 u~ zj 5

(20)

2pU zj ðtÞ

For other power spectra, the modulating functions can be derived in the same way. Derived modulating functions herein are collectively referred to as the extended modulating functions (EMFs). Likewise, the EMFs are presented in tabular forms as shown in Table 1 to facilitate their applications. Based on Kaimal power spectrum, the evolutionary cross power spectrum density (ECPSD) function G 0z j z k ðu; tÞ can be derived through a deterministic modulating function as Eq. (20).

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J. Li et al. / Renewable Energy 83 (2015) 384e397

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   G0zj zk ðu; tÞ ¼ Aj ðu; tÞAk ðu; tÞ Sj ðuÞ ½Sk ðuÞ Gðu; tÞ ei qj k ðu;tÞ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 35=3 u u 2 35=3 u u vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffivffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u zj u u zk u u u 1 þ 50 uU ~ z ðtÞ 6 ~ z ðtÞ 6 1 þ 50 2pUz 7 2pUzj 7 u1 200 2 zj u1 200 2 zk 1 1 U 6 7 u j u k k 7 6 u u u u u  ¼u 6 7 #

5=3 " 5=3 u2 2p * U u2 2p * Uzj t Uzj 41 þ 50 u~ zj 5 t Uzk 41 þ 50 u~ zk 5 zk u t u zk 2pU zk ðtÞ 2pU zj ðtÞ u zj 1 þ 50 t 1 þ 50 2pUz 2pUzj

k

"

"   #   # u Cz zk  zj  u Cq z j  z k  exp  exp  i ~ z ðtÞ ~ z ðtÞ ~ z ðtÞ þ U ~ z ðtÞ þ U pU pU j j k k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " "   #v u u   u1 200 2 zj u1 200 2 zk u Cz zk  zj 1 1 u u u ¼ exp  #5=3  u2 2p u* ~ #5=3 exp " ~ z ðtÞ u2 2p * U ~ z ðtÞ " ~ z ðtÞ þ U pU U zk ðtÞ u u j j k u zj u zk t t 1 þ 50 ~ 1 þ 50 ~ 2pU zj ðtÞ

2pU zk ðtÞ

  # u Cq z j  z k i ~ z ðtÞ þ U ~ z ðtÞ pU j k (21)

Seeing that Eq. (21) is equivalent to Eq. (14), the ECPSD function G0zj zk ðu; tÞ is also equal to the TVCPSD function Gzj zk ðu; tÞ. From here we see that the derivation of the EMFs is effective for the non-uniform modulation of stationary fluctuating wind velocities. The EMFs are the time-frequency functions and can describe the nonstationarity embedded in the wind fluctuations, effectively meaning that the strict mathematical derivation based EMFs are available for modelling the nonstationary wind velocities.

3. Examination of ability for EMFs to catch nonstationarity of wind velocities 3.1. Simulation of wind velocities with weak nonstationarity In order to scrutinize the capability that the EMFs capture the nonstationarity of wind velocities, the numerical simulation of the nonstationary wind velocities will be carried out with an EMF using

Table 1 EMFs corresponding respectively to the Kaimal, Davenport, Harris, and Simiu power spectra. Name

Power spectra

Kaimal

SðuÞ ¼ 12

EMFs

200u2 zj 2p * Uzj

1

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 15=3ffi u u u u zj B uU~ z ðtÞ B 1þ50 2pUz C j C j Aðu; tÞ ¼ u C u z u Uz B t j @1þ50 ~ j A

!5=3

u zj

1þ50 2pUz

j

2pU z ðtÞ j



Davenport

1200u 2pU10

2 SðuÞ ¼ 4KU10  u

Harris



1þ



2þ

0 B B Aðu; tÞ ¼ B @

2 4=3

1200u 2pU10

1800u 2pU10

SðuÞ ¼ 4u2*  u

2

2 12=3

 1þ

1þ

1200u 2pU10

C C !2 C A

1200u ~ ðtÞ 2pU 10

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0  u 2 15=6 u u 1800u B 2þ 2pU C u~ 10 C U 10 ðtÞ B Aðu; tÞ ¼ u !2 C u U10 B @ A u t 1800u 2þ

2 5=6

1800u 2pU10

~ ðtÞ 2pU 10

Simiu

when SðuÞ ¼

when SðuÞ ¼

u zj  0:2 2pUzj 1 200 2 zj u 2 2p * Uzj

1 1 þ 50

u zj > 0:2 2pUzj 1

0:26u2* u

u zj 2pUzj

u zj 2pUzj

!5=3

u zj  0:2 when ~ z ðtÞ 2pU j vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 15=3 u u u zj u ~ z ðtÞ B 1 þ 50 2pUz C uU B j C j Aðu; tÞ ¼ u B C u t Uzj @1 þ 50 u~ zj A 2pU zj ðtÞ

u zj > 0:2 ~ z ðtÞ 2pU j ! ~ z ðtÞ 4=3 U j AðtÞ ¼ Uzj when

!2=3

J. Li et al. / Renewable Energy 83 (2015) 384e397

Fig. 1. Locations of points 1, 2, and 3 along a vertical line.

integration of both the spline interpolation algorithm (SIA) and spectral representation (SR) method, then forming the SIA-SR method, due to reducing the increasing number of Cholesky decomposition of the time-varying spectral density matrix [40]. To be specific, the work of this section is to numerically simulate the

389

nonstationary fluctuating wind velocity time series at three dissimilar points 1, 2, and 3 envisaged, shown in Fig. 1 [40]. The variables u1(t), u2(t), u3(t) are, respectively, employed to represent the three components of this one-dimensional trivariate stochastic process. The Kaimal power spectrum and corresponding EMF listed in Table 1 are herein taken into consideration. It is hypothesized that the timevarying mean wind speed at the first point 1 (height z ¼ 35 m) sat~ ¼ U ½1 þ g cosðu0 tÞ ¼ 45 ½1 þ 0:3 cosð0:0052tÞ ðm=sÞ. It isfies U 35 35 is worth pointing out that the time-varying mean wind speed can be also generated by using a sequence of independent random numbers from normal distribution and Weibull distribution [47,48]. For the downburst wind velocities, the time-varying mean wind speed, namely the non-turbulent wind velocity of downburst can be simulated by either empirical model or analytical model [31,34]. Likewise, the surface rough roughness length is taken as z0 ¼ 0.001266 m, which corresponds to the shear velocity of the flow u* ¼ 1.76 m/s. It is pointed herein out that the values z0 ¼ 0.001266 m and u* ¼ 1.76 m/s are taken from Simiu and Scanlan [43]. Then, the time-varying mean wind ~ ¼ speeds at the second and third points are calculated in terms of U 45 ~ 46:1 ½1 þ 0:3 cosð0:0052tÞ and U ¼ 51:3 ½1 þ 0:3 cosð0:0052tÞ, 145 respectively. In the present simulation, the upper cutoff circle frequency is set equal to be uup ¼ 4 rad/s, the dividing number of circle frequency is selected to be N ¼ 2048, and Dt ¼ 0.785 s is the time step of numerical simulation with a time length equal to T ¼ 3216.99 s. By

Fig. 2. Generated sample functions for longitudinal nonstationary fluctuating wind velocities at three different heights.

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Fig. 3. Enlarged views of the initial 600 s of the simulated sample functions for longitudinal nonstationary fluctuating wind velocities at three different heights, displayed in Fig. 2.

resorting to the short-time Fourier transform (STFT), the EPSD function can be estimated from the generated nonstationary fluctuating wind velocity generated by use of the proposed method. The STFT, F(u,t), of stochastic process, u(t), is defined by the convolution integral below.

Zþ∞ Gðu; tÞ ¼

uðtÞ hðt  tÞ ei u t dt

(22)

∞

In the preceding equation, h(t) represents an appropriate time window. In the present paper, the time window squared function is pffiffiffiffiffiffi 2 2 selected to be h 2 ðtÞ ¼ 1= 2pset =2s ðs ¼ 0:25Þ. The square of the module for the EPSD function, namely G(u,t), can then be explicitly written as follows:

jFðu; tÞj2 ¼

Zþ∞ Zþ∞

uðt1 Þ uðt2 Þ hðt  t1 Þ hðt  t2 Þ ei u t1 ei u t2 dt

∞ ∞

(23) In order to estimate EPSD function of nonstationary processes, there are a number of methods available. For example, it is also a good choice to adopt wavelet transform (WT), which is studied as a

more advanced approach to estimate EPSD functions of nonstationary stochastic processes [49,50]. Fig. 2 [including (a), (b), and (c)] displays the sample functions of longitudinal nonstationary fluctuating wind velocities at three dissimilar points 1, 2, and 3 generated by using the present method. In order to better visualize the differences and the similarities among these three time histories, Fig. 3 presents the enlarged views of the initial 600 s of simulated sample functions displayed in Fig. 2. As far as the degree of correlation among these three time histories is concerned, Fig. 3 clearly demonstrates that with the decreasing of the distance between arbitrary two points, the loss of coherence between these two samples, corresponding respectively to these two points, takes on a decreasing trend. For example, since the distance between the points 1 and 2 is only 10 m apart, there is a small loss of coherence between u1(t) and u2(t). On the other hand, there exists a considerable loss of coherence between u1(t) and u3(t) as well as between u2(t) and u3(t), as the point 3 is located at 110 m from the point 1 and 100 m from the point 2, respectively (see Fig. 1). The coherence between the generated wind fluctuations weakens along with the distance increases. Evidently, this phenomenon is controlled by the coherence function. Simultaneously, it can be clearly detected from Fig. 3 that there exists a phase angle between arbitrary two samples at the 100 s time instant. Figs. 4 and 5, respectively, provide both the temporal auto-

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Fig. 4. Temporal auto-correlation functions of the generated sample functions displayed in Fig. 2 with respect to the corresponding targets at the time instant t ¼ 100.53 s.

correlation and cross-correlation functions of the generated sample functions displayed in Fig. 2 with respect to the corresponding targets at the time instant t ¼ 100.53 s. Figs. 4 and 5 clearly demonstrate that the simulated Rjk(t, t þ t) is in reasonable agreement with the target, R 0j k ðt; t þ tÞ, while admitting that there exists a small degree of differences. Further, these differences tend to disappear when the auto-and cross-correlation functions, Rjk(t,t þ t) is computed in terms of the 20,000 sample functions at the time instant t ¼ 100.53 s. Under this scenario, the simulated Rjk(t,t þ t) is in remarkable agreement with the target, R 0j k ðt; t þ tÞ. Additionally, it can be detected from Fig. 5 that the peak values of auto-and cross-correlation functions exhibit the time shift. This phenomenon embodies that the sample possesses the phase difference. The EPSD can portray the energy distribution over both temporal and spectral domains. Likewise, in light of the EPSD, transient features of nonstationary winds may be identified.

Consequently, by means of the STFT, the EPSD function, G(u,t), is estimated from these sample time histories and plotted in Fig. 6. Indubitably, the simulated EPSD functions possess the time-varying features. Therefore, the present procedure is capable of fully capturing the nonstationarity. Moreover, Fig. 7 presents the EPSD functions of the generated sample functions displayed in Fig. 2 with reference to the corresponding targets at the time instant t ¼ 100.53 s. It is seen from Fig. 7 that the EPSD functions, G(u,t1), are generally in agreement with the targets at the time instant t1 ¼ 100.53 s. Furthermore, in order to highlight the superiority of the SIA-SR method in enhancing the computational efficiency of generating nonstationary stochastic processes by use of EMFs, the dividing number of circle frequency N ¼ 4096, 8192 and the length of simulation time T ¼ 4823.04 s, 6430.72 s are also taken into consideration. The Cholesky decomposition in the SIA-SR method is

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Fig. 5. Temporal cross-correlation functions of the generated sample functions displayed in Fig. 2 with reference to the corresponding targets at the time instant t ¼ 100.53 s.

implemented on only 128 circular frequency points at each instant of simulation. The comparisons among several different schemes are made and listed in Table 2. It can be seen from Case 1 in Table 2 that the elapsed time for the simulation with the SR method is 347; while that for the simulation with the SIA-SR method is only 58 s. The time consumption of the SIA-SR method reduces more than 80% the elapsed time of the SR method. With the further increasing of the length of simulation time, the proportion of the elapsed time between the SIA-SR and the SR methods keeps about 17%, as seen from Case 1 to Case 3. Apparently, in the calculation speed, the SIASR method has considerable advantage over the SR method. Furthermore, the superiority becomes more remarkable with the further increasing of N. For example, in Case 5, the elapsed time of the SIA-SR method is 8.9% of the time consumption of the SR method, and the percentage is the smallest in these cases. This is because, the number of the Cholesky decomposition in the SIA-SR method keeps constant with the value equal to 128 at each instant of simulation, but in the SR method the decomposition

number is also more and more as N increases. Therefore, the SIA-SR method is able to remarkably enhance the computational efficiency of simulating nonstationary stochastic processes. 3.2. Simulation of downburst wind velocities with strong nonstationarity It is noted that the above-mentioned fluctuating wind velocities shown in Fig. 2 take on weak nonstationarities. The main reason may be that the time-varying mean wind speed of the modulating function is slowly time-varying function with small-amplitude and long-period. For simulating strong nonstationarity, the turbulent wind velocity of downburst is taken into account. As displayed in Fig. 8, the modified OBV model [34] is considered as the nonturbulent wind velocity of downburst, namely the time-varying mean wind speed. In like manner, the Kaimal power spectrum and corresponding EMF listed in Table 1 are also taken into consideration herein. Fig. 9 presents the generated downburst, and

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Fig. 6. EPSD functions of the generated sample functions displayed in Fig. 2.

Fig. 10 displays the simulated turbulent wind velocity which takes on strong nonstationarity. The nonstationarity is relevant to the EMF. It can be seen from Fig. 10 that the amplitude of fluctuating wind velocity is bigger when the non-turbulent wind velocity of downburst is greater, which is consistent with what expressed by the Eq. (20). By means of the STFT, the EPSD function is also estimated from the simulated turbulent wind velocity of downburst and plotted in Fig. 11. Undoubtedly, the simulated EPSD function possesses the time-varying features, and furthermore, the nonstationarity is strong. 4. Conclusions For the reproduction of nonstationary wind velocities, this paper has presented a novel methodology for obtaining the satisfactory non-uniform modulating functions. Firstly, according to the Kaimal power spectrum, a deterministic modulating function is deduced in detail. Next, including Davenport power spectrum,

Harris power spectrum, and Simiu power spectrum into consideration, the extended modulating functions (EMFs) are then presented in tabular forms to facilitate their applications. On the other hand, when digitally simulating the nonstationary stochastic processes using the spectral representation (SR) method, there is a need for reducing the increasing number of the Cholesky decomposition of the time-varying spectral density matrix. Distinguished from the FFT algorithm, the spline interpolation algorithm (SIA) can be used when the modulating function is a deterministic timefrequency function. In view of this problem, integration of both the SIA and SR method, then forming the SIA-SR method, is taken into consideration due to its ability to significantly reduce the increasing number of Cholesky decomposition of the time-varying spectral density matrix. An example application to reproducing the nonstationary fluctuating wind velocity time series at three dissimilar points envisaged (simulation of wind velocities with weak nonstationarity) shows that the EMFs are able to fully catch the nonstationarity of wind velocities. The simulation implementation

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Fig. 7. EPSD functions of the generated sample functions displayed in Fig. 2 with regard to the corresponding targets at the time instant t ¼ 100.53 s.

of downburst wind velocities with the strong nonstationarity further highlights the effectiveness and faithfulness of the EMFs. Therefore, we can conclude that the proposed EMFs, and combining with the use of SIA-SR procedure with high computational efficiency, can generate a library of weak and strong nonstationary fluctuating wind velocity for use in the structural wind-resistant analysis and design as well as rapid structural performance assessment. However, it should be noted that the EMFs are based on the nonuniform modulation of stationary fluctuating wind velocities and obtained via the strict mathematical derivations, meaning that the EMFs themselves are correct. Thus, the correctness of the final

simulation results for nonstationary fluctuating wind velocities rests with both the wind spectra as well as the time-varying mean wind speed models. Fortunately, the wind spectra may directly adopt the Kaimal, Davenport, Harris, and Simiu power spectra, while the current well-developed time-varying mean wind speed models, such as the modified OBV model (i.e. empirical vertical profile model) for downburst wind velocities with strong nonstationarity, may be taken into consideration. The present numerical simulation results have proven the aforementioned elucidations. Notwithstanding, in order to further improve the accuracy of nonstationary simulations, it would be certainly interesting to

Table 2 Comparisons of the computational efficiency between the SIA-SR and SR methods. Cases

1 2 3 4 5

Dividing number of frequency N

2048 2048 2048 4096 8192

Length of simulation time T (s)

3216.99 4823.04 6430.72 6430.72 6430.72

Time consumption TC (s) SR

SIA-SR

347 521 697 1486 2833

58 89 118 161 253

TC for SIASR

TC for SR 100 % 16.7 17.1 16.9 10.8 8.9

J. Li et al. / Renewable Energy 83 (2015) 384e397

Fig. 8. Time-varying mean wind speed of downburst generated using the modified OBV model.

Fig. 9. Wind velocity time history of the generated downburst.

Fig. 10. Turbulent wind velocity time history of the generated downburst.

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Fig. 11. EPSD function of the generated turbulent wind velocity displayed in Fig. 10.

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