Research Article  Open Access
Filippo Ubertini, Fabio Giuliano, "Computer Simulation of Stochastic Wind Velocity Fields for Structural Response Analysis: Comparisons and Applications", Advances in Civil Engineering, vol. 2010, Article ID 749578, 20 pages, 2010. https://doi.org/10.1155/2010/749578
Computer Simulation of Stochastic Wind Velocity Fields for Structural Response Analysis: Comparisons and Applications
Abstract
The digital simulation of wind velocity fields, modeled as multivariate stationary Gaussian processes, is a widely adopted tool to generate the external input for response analysis of windsensitive nonlinear structures. The problem does not entail any theoretical difficulty, existing already a large number of wellestablished techniques, such as the accurate weighted amplitude wave superposition (WAWS) method. However, reducing the computational effort required by the WAWS method is sometimes necessary, especially when dealing with complex structures and highdimensional simulation domains. In these cases, approximate formulas must be adopted, which however require an appropriate tuning of some fundamental parameters in such a way to achieve an acceptable level of accuracy if compared to that obtained using the WAWS method. Among the different techniques available for this purpose, autoregressive (AR) filters and algorithms exploiting the proper orthogonal decomposition (POD) of the spectral matrix deserve a special attention. In this paper, a properly organized way for implementing stochastic wind simulation algorithms is outlined at first. Then, taking the WAWS method as a reference from the viewpoint of the accuracy of the simulated samples, a comparative study between PODbased and AR techniques is proposed, with a particular attention to computational effort and memory requirements.
1. Introduction
The simulation of wind velocity fields has been one of the main topics of wind engineering for the last decades. In this framework, wind velocity is usually idealized as the sum of a mean part, assumed as constant within a conventional time interval, and a fluctuating part representing the atmospheric turbulence. This last is usually modeled as a stationary zeromean Gaussian random process [1, 2].
Several techniques were proposed in the literature in order to simulate Gaussian wind velocity fields to be employed in structural analysis [3–5]. Among those, the classic WAWS method, based on the pioneering work by Shinozuka and Jan [6] and then modified by Deodatis [1] in such a way to achieve ergodic realizations and to be efficiently implemented through fast Fourier transform (FFT) algorithms, has proved to guarantee the best quality of the obtained results [7]. Nevertheless, such a procedure requires the Cholesky factorization of the spectral matrix, which unfortunately leads to high computational expenses, especially when dealing with complex structures and highdimensional simulation domains. These difficulties are mainly related to memory allocation and time consuming operations, thus requiring, on the one hand, the reduction of the problem size. On the other hand, an accurate wind simulation is essential for predicting the windinduced response of flexible structures, such as transmission power lines, tall buildings, suspension, and cablestayed bridges. Less demanding, yet approximate, procedures may be obtained by exploiting the properties of the POD decomposition of the spectral matrix, proposed in the papers by Li and Kareem [8], Di Paola [9], and Solari and Carassale [10]. A PODbased technique, in particular, was recently applied to simulate the wind velocity field on a domain representing a longspan suspension bridge [11] with significantly low computational efforts. A third wellestablished class of simulation formulas is represented by autoregressive (AR) and autoregressive moving average (ARMA) filters, early introduced in the paper by Samaras et al. [12] and applied to wind simulation by Mignolet and Spanos [13], by Li and Kareem [8] and, more recently, by Di Paola and Gullo [14].
This paper addresses, at first, the problem of the efficient implementation of wind simulation algorithms by proposing a twosteps approach which allows to properly handle large numbers of simulations, which is usually the case for performing structural response analysis. Then, the WAWS method is adopted as a reference for calibrating the parameters of PODbased and AR techniques. These last methods are finally compared in terms of computational effort and memory requirements. To this end, two numerical examples are presented. The former, represented by the tower of a suspension bridge, is considered for models calibration. The latter, represented by an entire suspension bridge, is a more demanding case, which is here chosen as a benchmark in order to test the capability of AR and PODbased techniques in handling complex simulations and to compare their computational efficiency. Finally, the convergence rates of the accuracies of the two simplified methods are also investigated, in order to provide indications for tuning the parameters that affect their results.
The considered formulas for wind simulation are only some of those currently available in the literature and this work does not attempt in any way to give a comprehensive literature review on this large topic. In particular, it must be underlined that nonGaussian techniques were also studied and applied in the literature with the aim of directly simulating wind pressure fields on structures. As examples, this problem was analyzed by Grigoriu [15, 16], by Popescu et al. [17], by Gioffrè et al. [18] and by Borri and Facchini [19], among others. NonGaussian responses of windexcited structures were also studied (see, e.g., Gusella and Materazzi [20, 21]). A quite exhaustive review of all the approaches currently available for simulating wind velocity fields and structural responses was recently proposed by Kareem [22] also considering the nonstationary and nonGaussian cases.
2. Stochastic Wind Modeling
Let be the global Cartesian reference system with the origin lying on the ground. In the following developments, without loss of generality, it is assumed that the axis is parallel to the gravity direction and the mean wind velocity is parallel to the x_{1} axis. Following these definitions, the wind velocity field is idealized as the sum of a mean value , function of the elevation from the ground , and a stationary zeromean fluctuation , that depends on the position and varies in time . The three components , , of the vector represent therefore the longitudinal, lateral and vertical components of turbulence, respectively. As customary in wind engineering, the classic logarithmic law [4] is assumed to represent the variability of the mean wind velocity with .
The atmospheric turbulence is usually modeled as a zeromean, Gaussian, stationary random field that depends on time. If and denote two points located at and , then, from a probabilistic point of view, the complete characterization of this field is ensured by the knowledge of the correlation function for every pair of turbulence components being the time lag. Assuming that is ergodic, is given by the Fourier transform of the cross power spectral density (CPSD) function between and , being the circular frequency. Most of the theoretical models adopted in wind engineering express the CPSD in terms of autospectra , and coherence function , for , where denotes the frequency [9]. Several models were developed in the literature to give an analytical representation to , for . Here, the model by Solari and Piccardo [23] is adopted for this purpose and is expressed by means of the classic exponential law [11]. The imaginary part of the CPSD (called quadrature spectrum) introduces a time lag between the simulated velocities, which may become significant for points placed in the alongwind direction . Without loss of generality, the quadrature spectrum is here neglected, since the considered examples focus on simulation domains located on the plane. This, however, does not limit the generality of the presented results, which apply also to the case in which the CPSD has a nonnegative imaginary part.
For simulation purposes, the spatial domain is discretized into points which usually represent significant nodes of the case study structure. The position of the th point is identified by the vector , with . This spatial discretization allows to represent the wind field as the sum of the mean velocity vectors and the turbulence vectors . Following this approach, the timedependent random field is transformed into a 3Nvariate stationary random process , where is a 3Norder vector containing the components , , of the turbulence vectors for . The complete characterization of the process is given by its power spectral density (PSD) matrix containing PSD and CPSD functions between turbulent velocities. Equivalently, the twoside PSD matrix can also be utilized [11]. Having neglected the quadrature spectrum, both matrices and are real and positive definite.
3. Digital Simulation Formulas
Realizations of the process are generated along a sequence for , of time instants with constant time step . The circular frequency domain is limited within the interval where denotes the cutoff circular frequency. Such an interval is discretized by means of an equally spaced sequence for of circular frequencies, with step amplitude , being thereby .
The most accurate and wellestablished algorithm for simulating samples of Gaussian processes is the spectral representation method, early proposed by Shinozuka and Jan [6]. This last was significantly improved by Deodatis [1] who proposed a simulation formula that ensures the generation of ergodic sample functions and which is suitable to be implemented using efficient FFT algorithms. The ShinozukaDeodatis formula is based on the following frequencydependent decomposition of : where * denotes the complex conjugate operator. It is worth noting that the decomposition (1) is not unique. To this regard, in the ShinozukaDeodatis method the Cholesky factorization algorithm is adopted which provides in lower triangular form. Based on this approach, the following digital simulation formula was obtained [1]: where , are the elements of matrix and are independentrandom phases uniformly distributed in the interval . Without loss of generality, (2) is written under the assumption of neglecting the quadrature spectrum, therefore being a real matrix. This method is sometimes called “weighted amplitude waves superposition” (WAWS) method. As observed by Di Paola [9], the central limit theorem ensures that the process simulated by means of (2) is asymptotically Gaussian as becomes large. Equation (2) can be efficiently implemented through a fast Fourier transform (FFT) algorithm, as discussed in [1].
An alternative approach to the WAWS formula is represented by the PODbased technique proposed by Carassale and Solari [11]. With such an approach the matrices , in (1), are calculated as where , for , are the eigenvalues of and are the corresponding eigenvectors. These last assume the wellknown physical meaning of wind blowing mode shapes [9]. The use of (3) leads to the following simulation formula: where is the imaginary unit and are complexvalued Gaussian random numbers with unit variance. Likewise (2), (4) can be efficiently implemented through a fast Fourier transform (FFT) algorithm as discussed in [11], assuming again and .
Equation (4) generates as the superposition of 3N independent fully coherent stochastic processes, which represent the contributions of the different wind modes. A very convenient way for reducing both computer time and allocated memory using (4) is to calculate eigenvectors and eigenvalues of only for a small number of circular frequencies distributed along a reduced sequence , for , and then use interpolation formulas elsewhere. This approach, proposed by Carassale and Solari [11], is here adopted and briefly recalled in Appendix A.
Autoregressive methods generate the wind velocity field by filtering a 3Norder vector of uncorrelated bandlimited Gaussian white noises , for , with unit variance. The turbulent wind process is thus expressed in the following form: where and are convenient matrices that can be easily calculated by imposing that the simulated process satisfies the covariance structure of the target one [12]. Equation (5) represents a general ARMA method, in which denotes the order of autoregression and is the order of the moving average component. As it is well known [7] an ARMA() can be approximated by an AR(), with , where the AR() filter is readily obtained by assuming in (5). Regarding this point, it must be mentioned that AR() and ARMA() provided good results for turbulent wind simulations in many different environmental conditions [7]. The necessary derivations for calculating the coefficient matrices of an AR(p_{1}) filter are reported in Appendix A.
4. Efficient Implementation of Wind Simulation Methods
The three simulation algorithms presented in Section 3 are implemented in the MATLAB R2009a [24] environment to generate samples of alongwind and acrosswind turbulence velocities and , respectively. Without loss of generality, the turbulent wind field is thus reduced to a 2Nvariate stationary Gaussian process.
Equation (2) is implemented as the simulation formula of the WAWS method, while (4) is adopted in the PODbased technique. Both in the WAWS and the POD methods, FFT algorithms are employed, as discussed in [1, 11], to improve their computational efficiency. An AR () filter, given by (5) with , is considered to represent autoregressive methods assuming a sufficiently large . A twostep approach is adopted in the implemented codes, following the diagram reported in Figure 1. Accordingly, an initial phase is established (indicated as “phase 0”), which must be run only once, after which a general number n of simulations (Sim) can be performed (phase 1). Typically, in the “phase 0”, all the data which are needed for the successive simulations (e.g., spectral matrix, factorizations, coefficients, etc.) are calculated and collected. This allows minimizing the number of operations when simulating several wind realizations, which is usually the case for performing Monte Carlo simulations of the structural response. The results of the “phase 0” are deterministic and must be stored in the computer memory.
The two main aspects that should be considered when comparing different wind simulation techniques, from the computational point of view, are the allocated memory in the “phase 0” and the time needed for each single simulation in the “phase 1”. The former aspect is crucial since collecting a lot of data in the computer memory may cause overflows and thus it may preclude the use of a certain procedure when the dimension of the simulation domain becomes large. The latter aspect is even more important since it gives the measure of the computational effort required by the considered algorithm. On the other hand, when the accuracy of the different methods is concerned, the most relevant points that should be considered are the agreement between target and simulated characteristics of the stochastic process, mainly spectra and correlation functions, and peak values.
The three methods described above are implemented as schematically outlined in the pseudocodes provided in Appendix B. Accordingly, the main computational steps are summarized in Table 1. The comparison between the operations required by the considered algorithms immediately outlines that the WAWS technique is the heaviest one from the computational viewpoint. On the contrary, both POD and AR techniques significantly reduce the computational effort with respect to the WAWS method. Table 1 also summarizes the memory that is allocated in the “phase 0” in each of the three implemented codes. It is worth underlying that the ratio between the memory allocated in the PODbased method and the one allocated in the WAWS method is almost equal to . In the next section it will be shown that the PODbased algorithm gives sufficiently accurate results with , if compared to the corresponding WAWS method with . The PODbased technique allows therefore drastically reducing the memory required by the WAWS method. This practically eliminates the risk of memory overflows in many technical conditions. Nonetheless, in the most demanding cases, an AR(p_{1}) filter can be adopted as it further reduces memory allocation even with respect to the PODbased method. Indeed, the ratio between the memory allocated by the AR method and the one allocated by the POD method is equal to and, usually, .

5. Numerical Tests and Discussion
5.1. The Case Studies
The above described wind simulation techniques are commonly employed for the definition of the external inputs for time domain response analyses of slender structures, as described in [25–28], among others. Time domain response analysis are essential for evaluating the structural safety of wind sensitive structures, whose dynamic behavior is strongly nonlinear, for instance, in the case of structural cables and longspan bridges [29, 30].
Two numerical examples are considered in order to evaluate the performances of the simplified models, when their parameters are chosen in such a way to obtain a similar accuracy with respect to the WAWS method. In the former example the simulation domain is composed by 9 significant points disposed along the height of the tower of a suspension bridge (see Figure 2(a)). This case is here adopted in order to calibrate the parameters of PODbased and AR techniques. In the latter example the simulation domain is represented by an entire suspension bridge. In this case, the spatial domain is composed by 83 nodes located on the mean plane of the bridge (see Figure 2(b)). In both cases, the direction of the mean wind velocity is orthogonal to the plane of the simulation domain and its modulus is assigned by means of the classic logarithmic profile. In particular, a mean velocity of 48.0 m/s is assumed at the top of the tower in the former example (node number 9 in Figure 2(a)) and a mean velocity of 40.1 m/s is assumed at the midspan of the bridge in the latter example (node 70 in Figure 2(b)). The exponential decay coefficients that appear in the expression of the coherence function are assumed to be equal to those reported in [11], while the variances of , for , are calculated following [23]. A nil coherence is assumed between orthogonal turbulence components and . The following parameters are adopted in the simulations: , , rad/s, rad/s. The simulations have been performed with the aid of a Core2 Duo Intel processor.
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5.2. Calibration of PODBased and AR Methods
Before discussing the performances of the simplified formulas, the parameters of PODbased and AR techniques must be properly calibrated in order to achieve similar accuracies with respect to the WAWS method. This task is here accomplished in the case of the first example. Literature studies [7, 11] suggest to choose, in similar cases, a reduced number of factorizations (for PODbased technique only) and an order of autoregression (for AR filter only) . After performing several tests, it was found that similar values could be also adopted here, as it will be clearer in the rest of the paper by comparing the quality of the results obtained by means of the three methods. Therefore, the values and are chosen here, thus also allowing a direct comparability with the results presented in other literature works. As an example, a realization of the turbulent wind field generated by means of the WAWS method is shown in Figure 3.
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The quality of the generated samples is analyzed in Figures 4–10. To this end, the turbulence velocities simulated in two points (number 5 and 9 in Figure 2(a)) of the spatial domain are considered. First of all, by virtue of the ergodicity of the process simulated by means of (2), it is expected that the WAWS method provides a perfect accuracy when one entire period (for a 2Nvariate process) of the process is simulated. This result has been confirmed here as shown, for instance, in the results presented in Figure 4, which have been obtained by assuming and . In this case, as expected, target and computed spectra and correlation functions are overlapped (some meaningless residual differences are related to numeric FFT calculations). Figures 5 and 6 refer to alongwind and acrosswind (vertical) velocities, respectively, simulated by means of the WAWS method assuming , that is, . The corresponding results obtained by means of PODbased and AR techniques, for the chosen parameters values, are shown in Figures 7, 8, 9, and 10. From the presented results it points out that the WAWS method provides a very good match between target and computed spectra and correlation functions, even for . The chosen PODbased and AR methods are obviously less accurate than the WAWS formula, which is especially apparent by looking at the errors on auto and crosscorrelation functions. In particular, the analysis of the results, also extended to other points of the spatial domain, confirms that PODbased and AR techniques produce errors in the correlation functions, although, in general, the agreement with the target functions is improved as the coherence is decreased (as, e.g., in the acrosswind components). Clearly, the desired levels of accuracy could be obtained in both cases of PODbased and AR algorithms by choosing larger values of either the reduced number of factorizations , adopted in the former case, or the finite order chosen in the latter one. For the purposes of this study, the parameters and are considered as good compromises between accuracy and computational efficiency. Indeed, with such choices, the accuracies of the samples generated using PODbased and AR techniques are essentially comparable to each other and the errors with respect to the WAWS method can be regarded as acceptable in view of a structural response analysis [28]. In any case, the comments reported in the following developments of the work could be readily extended to other possible choices of the models’ parameters.
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5.3. Discussion
The results presented so far have permitted to choose the parameters of PODbased and AR methods that guarantee efficient simulations with an accuracy that is essentially comparable to that of the WAWS method. The second example is now worth considering to better compare POD and AR methods from the computational viewpoint.
The accuracy of the simulated samples is analyzed in Figures 11, 12, 13, and 14. To this end, to the alongwind and acrosswind velocities at the points number 1 (top of one tower) and 70 (bridge midspan), indicated in Figure 2(b), are considered. The presented results are analogous to those obtained in the first example, thus indicating that the quality of the simulated process is, as expected, independent on the dimension of the simulation domain. In particular, the results confirm that PODbased and AR methods, for the chosen values of and p_{1}, essentially guarantee similar accuracies. The samples generated by means of the three algorithms are also in good agreement as it concerns the peak values and of and . As examples, some relevant results are summarized and compared in Table 2 for the two numerical examples.

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The computer times required by the considered methods in the two numerical examples are summarized in Table 3. These results emphasize that, as expected, POD and AR methods are significantly more computationally efficient than the WAWS formula. Although it is implicit that these results strongly depend on the environment of simulation, they still allow to derive some conclusions. Particularly, the PODbased method seems to be more computationally efficient than the AR filter, both in performing preliminary calculations (“phase 0”), which could become relevant when computing only a few wind simulations, and wind simulations (“phase 1”). Moreover, the computational improvement in using the PODbased technique instead of the AR filter seems to become more relevant as long as the dimension of the simulation domain becomes larger. On the other hand, as expected, the AR filter is the method that mostly reduces memory allocation, at least in the case of highdimensional simulation domains, which might be greatly beneficial in practical cases.

It is also worth mentioning that the eigenvalue analysis of the spectral matrix, performed in the PODbased method, reveals interesting properties of the wind field in view of a structural analysis. In particular, as it is wellknown, eigenvectors and eigenvalues of the spectral matrix represent respectively the shapes and the associated powers of the 3N stochastic processes summed in (4). The eigenvalue analysis usually reveals that only a few of these spectral modes have large associated powers and that the eigenvectors corresponding to the largest eigenvalues are very similar to the fundamental mode shapes of the structure. Thus, the response of a slender structure is expected to be dominated by the contribution of these waves [9].
As an example, Figure 15(a) shows the first ten eigenvalues () multiplied by the frequency and plotted versus the circular frequency , in the case of the second example. As inferable from this figure, the first few eigenvalues are, as expected, much greater than the others. It is also worth mentioning that, as shown in Figure 15(b), the eigenvalues of the spectral matrix are distinct and thus the lines versus do not cross. The turbulence eigenvectors corresponding to the first six eigenvalues, calculated for a fixed value of the circular frequency ( rad/s), are shown in Figure 16. The figure evidences that the first turbulence eigenvector involves the whole bridge in the outofplane (alongwind) direction and it is very similar to the typical first structural mode shape of a suspension bridge. Similar observations can be made for the remaining eigenvectors shown in Figure 16. It is noteworthy that hybrid horizontal/vertical eigenvectors are not detected since a nil coherence has been assumed between alongwind and acrosswind turbulence components.
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The presented results have indicated that the PODbased technique exhibits a significant computational efficiency, while the AR filter is especially convenient for preserving the allocated memory. Another important aspect that needs to be addressed in the viewpoint of practical applications is the rate of convergence of the accuracies of the AR filter and the PODbased technique towards the target, when increasing the relevant control parameters, that is, the order of autoregression, p_{1}, in the case of the AR filter, and the reduced number of factorizations, , in the case of the PODbased technique. This aspect is analyzed here, by considering the error, ε, between correlation functions of generated samples, and , and the corresponding targets, and . This error is here defined as: and calculated in the time lag interval 0–10 sec. The error defined in (6) is realization dependent and, so, an averaging of ε among a conveniently large number of generations is sought, where is here taken as 200.
Following the previous definitions, Figure 17 shows plots of the average errors versus p_{1} and , for the AR filter and the PODbased technique, in the case of the first numerical example. For clarity of presentation, the results in Figure 17 are normalized with respect to the residual error of the WAWS method with . The results presented in Figure 17 confirm the expected trends that the errors decrease with increasing and . Moreover, they show that, in the case of the PODbased technique, the rate of convergence appears to be very fast for small values of . Namely, already for the system reaches an accuracy which is only slightly worse than that of the WAWS method with (the normalized average error is about equal to 1.3), while for larger values of the convergence rate becomes smaller with some erratic fluctuations of less than . This result indicates that the interpolation expressions of eigenvectors and eigenvalues along the unequally spaced sequence of circular frequencies, proposed by Carassale and Solari [11] and recalled in Appendix A, is very accurate when applied in the case of turbulence characteristics similar to those considered here. The results presented in Figure 17 might also suggest to choose values of that are smaller than the one adopted by Carassale and Solari [11] in a similar application and chosen in the numerical examples presented above (), without substantially decreasing the resulting accuracy. Nonetheless, calculations performed aside have shown that the average error in predicting peak values, calculated with respect to the WAWS method and considering 200 wind generations, rapidly decreases for up to about 50, where it is almost equal to 0.5%, while remaining essentially constant for larger values of . This result shows that, in the present case, choosing is convenient in terms of accuracy in predicting peak values.
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In the case of the AR filter, the results presented in Figure 17 show that for large values of p_{1} the error is still slightly larger than the error of the WAWS method with (the ratio between the two is about 2.5). On this respect, a value of p_{1} equal to 200 can be regarded as large if compared to the value recommended in the literature in similar conditions [7] and adopted in the numerical examples presented above. Looking at the results presented in Figure 17, the ratio between the average error of the AR filter and the residual error of the WAWS method, for , is about equal to 3.2. Considering the high accuracy of the WAWS method, this result can be regarded as acceptable in many practical applications. It is also worth mentioning that, for , the average error in predicting peak values is also small and approximately equal to 2.5%.
It is important to note that the results presented in Figure 17 do not allow to derive conclusions on the relative accuracy of the two simplified methods one with respect to the other. Indeed, for a fair and complete comparison between the two methods, different turbulence characteristics should be considered, which, however, goes beyond the purposes of the present investigation.
6. Conclusions
Simulating the wind velocity field in the case of highdimensional domains may become a very demanding computational task due to memory occupation and time consuming simulations. In the present paper, a properly organized way for implementing wind simulation methods is presented at first. Then, the widely adopted method using waves superposition (WAWS method) is taken as a reference for calibrating the parameters of two wellknown simplified methods (PODbased and AR formulas). These last are then compared, by devoting a special care to algorithm structure and computational expense.
Two numerical examples, with increasing complexity, are considered in the comparative study. The results indicate that PODbased and AR techniques significantly reduce both computer time and memory allocation with respect to the WAWS method, at the expense of smaller accuracies. The PODbased method appears to be more computationally efficient than the AR filter when the relevant parameters affecting their results are tuned in such a way to achieve similar accuracies. On the other hand, the AR filter is greatly preferable in terms of reduction of allocated memory.
Concerning the accuracy of the simulated samples, the WAWS method is known to guarantee the ergodicity of the generations. As shown in the paper, this means that if an entire period of the process is simulated, the match between target and simulated samples is perfect. Even when considering shorter simulations, the accuracy of the WAWS method is seen to be very high, namely the differences between target and computed correlation functions are, for the considered parameters values, almost unnoticeable. The results presented in this paper also confirm that, although less accurate, both PODbased and AR techniques can provide good results if the models’ parameters are properly chosen. It must be also mentioned that, regardless the chosen simulation formula, the POD decomposition of the spectral matrix is always a worth task to be tackled, as it provides useful information about the stochastic wind process in view of a structural analysis. Finally, the rates of convergence of the accuracies of the two simplified approaches with increasing tuning parameters have also been investigated in order to give preliminary indications for choosing these parameters in practical cases.
Appendices
A. Additional Details on the Adopted Simulation Formulas
The rules proposed by Carassale and Solari [11] for reducing the number of eigenvectors and eigenvalues of matrix allocated in the computer memory, are described, at first, in this section. Then, the derivations for calculating the coefficient matrices of an AR () filter are briefly recalled.
Concerning the PODbased technique, the following sequence of circular frequencies must be introduced: Eigenvectors and eigenvalues of matrix , calculated for the circular frequency values reported in (A.1), are denoted by and , respectively. Then, by defining the sequence : the eigenvalues of matrix are interpolated as: where and provide the upper and lower integer rounds of the argument respectively. Finally, the eigenvectors are approximated stepwise as: where returns the closest integer to the argument.
The simulation formula of an AR(p_{1}) filter is obtained, by assuming in (5), as: In order to calculate the coefficient matrices and , the correlation matrix must be defined as: Assuming the shorter notation , the following hypersystem of algebraic equations can be obtained [12]: Once the correlation matrices are calculated by means of inverse FFT algorithms applied to CPSD functions, the coefficient matrices can be easily determined by solving the algebraic hypersystem (A.7). Then it is possible to calculate matrix , by postmultiplying (A.1) by and taking the average, namely: By definition of correlation matrix, (A.8) may be rewritten in the following form: From (A.9), it follows that the covariance structure of the process is preserved if the following condition is satisfied: Thus, the choice of matrix is not unique and a possible strategy is to assume and to obtain it through the Cholesky decomposition.
B. Implemented Algorithms
The wind simulation algorithms implemented in this study are schematically outlined in this section, for a 2Nvariate process. The simulated wind field is denoted by the matrix variable field_u. Variables definitions and adopted functions are implicit from the presented algorithms.
B.1. WAWS Algorithm
Step 0. initialize wind simulation parametersfor for calculate as a function of r and k for for end end endendsave wind simulation parameterssave .
Step 1. load wind simulation parametersload for for for end for end endendsave .
B.2. PODBased Algorithm
Step 0. initialize wind simulation parametersfor for for end end endsave wind simulation parameterssave and
Step 1. load wind simulation parametersload and for if end if end for end endfor endsave
B.3. AR Algorithm
Step 0. for for for end endendfor endarrange and from solve hypersystem (A.7) for endsave and
Step 1. load wind simulation parametersload and for for end endsave .
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Copyright © 2010 Filippo Ubertini and Fabio Giuliano. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.