Finite element model updating of a large structure using multi-setup stochastic subspace identification method and bees optimization algorithm

Abstract

In the present contribution, operational modal analysis in conjunction with bees optimization algorithm are utilized to update the finite element model of a solar power plant structure. The physical parameters which required to be updated are uncertain parameters including geometry, material properties and boundary conditions of the aforementioned structure. To determine these uncertain parameters, local and global sensitivity analyses are performed to increase the solution accuracy. An objective function is determined using the sum of the squared errors between the natural frequencies calculated by finite element method and operational modal analysis, which is optimized using bees optimization algorithm. The natural frequencies of the solar power plant structure are estimated by multi-setup stochastic subspace identification method which is considered as a strong and efficient method in operational modal analysis. The proposed algorithm is efficiently implemented on the solar power plant structure located in Shahid Chamran university of Ahvaz, Iran, to update parameters of its finite element model. Moreover, computed natural frequencies by numerical method are compared with those of the operational modal analysis. The results indicate that, bees optimization algorithm leads accurate results with fast convergence.

Keywords

operational modal analysis / solar power plant structure / multi-setup stochastic subspace / bees optimization algorithm / sensitivity analysis

Introduction

While the worldwide developments in science and technology have prompted the quality of lives, in contrast, the development impacts on the environment have been harmful in some instances, or the long-term effects give causes for serious concerns. These impacts include environmental pollution, ecological collapse, and global climate changes that are influencing sooner or later, all human beings. A current major trend toward pollution stemming from the consumption of fossil and nuclear fuels, as major energy resources, is influencing the focus and practice of science [¹]. In this regard, in recent two decades, worldwide research has been conducted by scientists into renewable energy resources, including solar, wind and ocean energy to replace fossil and nuclear fuels energy sources. The availability and stability of solar energy have led to the significant attention of scientists toward this source of energy, compared to other renewable energy resources. The development of various types of solar-thermal power plants, including parabolic troughs, linear concentrating systems, solar power towers and linear Fresnel reflectors, is the result of scientific efforts to convert solar power into electrical energy [²].

The main structure of these solar power plants is in direct exposure to sunlight without any protection. Therefore, these structures are always subjected to external forces, including wind. Structural vibration occurs when dynamic forces generated by the aforementioned effects causes the power plant structure to vibrate. These vibrations affect the load bearing capacity of the structure. In some instances of the wind speeds, this effect might cause fatigue or failure of the structure, depending on the oscillations amplitude [³–¹¹]. Finding dynamic parameters of the solar power plant structure is the first step to prevent catastrophic effects of the mentioned external loads. In the strategies for structural design, it is very important to keep structures working safe during their lifetime that requires gaining a fundamental understanding of the material failure [¹²–²¹]. Furthermore, well organized techniques to computationally model fracture consist of extended finite element method, phantom node methods [²²–²⁴], meshfree methods [²⁵] the extended meshfree methods [²⁶–³⁴], phase-field modeling method and efficient remeshing techniques [³⁵–³⁹], and cracking particles methods [⁴⁰–⁴⁶], to name a few. Developing an understanding of the structure modal parameters will result in its safe operation under dynamic and environmental loads. Furthermore, it will be possible to update the finite element model of the structure if its dynamic parameters to be specified carefully. Having accurate and updated finite element model of the structure enables us to perform various analyses on it and to observe its dynamic behavior and response to various dynamic loads [⁴⁷]. Experimental modal analysis is one of the strongest methods to estimate dynamic characteristics of structures. This analysis identifies dynamic properties of structures, such as natural frequencies, mode shapes and damping ratios, from the response of the structures and by measuring applied forces on the systems [⁴⁸]. However, as mentioned previously one of the causes of solar power plant structure excitation is wind energy, where the calculation of wind-related applied loads on the system in real conditions, are cumbersome. As a solution for the aforementioned problem it is possible to employ operational modal analysis that is a branch of the experimental modal analysis to identify dynamic characteristics of structures. This method is only based on the ambient responses of the system and therefore no tool and equipment are required for the system excitation that consequently is resulted in the decrease of the analysis expenses and additionally will not affect the normal operation of the system [⁴⁹]. Operational modal analysis methods are investigated and categorized in time and frequency domains. Frequency and time domain techniques are based on power spectral density function and correlation function, respectively [⁵⁰]. In frequency domain category, frequency domain decomposition technique and transmissibility are among the mostly demanded methods. White noise input assumption and leakage error are among the main problems in using frequency domain methods [⁵¹]. Stochastic subspace identification (SSI) method, ARMA model and eigen system realization algorithm (ERA), are among the most popular and accurate time-domain parametric models used for system identification purposes [⁵²,⁵³]. These robust methods additionally have the ability to detect modal parameters in time domain strategies too, that eliminates any leakage error problem [⁵⁴]. Based on the provided evidences, operational modal analysis is one of the most ideal and effective tools to assess the structural condition and to identify dynamic parameters of large and complex structures including solar power plants, dams and bridges [⁵⁵]. The best and most accurate strategy to collect modal data of a large structure is to place a large number of sensors at different locations on the structure that enable us to record dynamic properties of the system completely [⁵⁶]. The drawback of this procedure is that, employing a large number of sensors incurs higher expenses [⁵⁷]. To alleviate the shortcomings of these techniques, sensor-integrating methods are employed so that it would be possible to collect data from the whole large structure using only a limited number of sensors [⁵⁸]. The application of aforementioned approaches makes it possible to update dynamic characteristics of large structures with acceptable accuracy and low cost. Additionally, special care should be taken with respect to uncertainties which could arise in the input parameters to determine how much the model outputs are changing by the variation in these inputs. Uncertainty and sensitivity analysis are great helps for these purposes [⁵⁹–⁶⁴]. Updating finite element model is assumed as an inverse approach that decreases the difference between empirical data and finite element model results. Gradient based approaches are broadly employed in finite element model updating [⁶⁵]. Although, these approaches propose some benefits, they have some disadvantages, too. The inability to acquire system global optimal point is a drawback of these methods. The development of intelligence optimization methods, including bees’ optimization algorithm, ant colony optimization and so on is the solution to overcome the mentioned difficulties [⁶⁶]. In spite of their ability to solve the disadvantages of classical optimization methods, their simplicity and initial guess independency are among the unique and significant characteristics of these methods. Performance and simplicity of the bees algorithm in comparison with other intelligent algorithms in inverse problems, encourage scientists to utilize bees algorithm instead of other optimization methods in updating problems [⁶⁶]. Moreover, according to [⁴⁷], the convergence of bees algorithm is faster than other optimization algorithms and its objective function has the lowest value. Therefore, in this article, bees’ algorithm is applied in order to update finite element model of the large structures.

This research is performed to investigate modal parameters and model updating of a 20 KW capacity solar power plant, located at Shahid Chamran university of Ahvaz, Iran, using SSI method. First of all, initial finite element model of the whole structure is prepared in Ansys software and the modal parameters are calculated by eigenvalue analysis module of this commercial finite element package. Additionally, by employing modal parameters obtained from finite element method, the best locations are found to install the sensors in experimental model. In the following, the actual structure is subjected to stochastic excitation and the related responses detected by sensors are recorded for further investigation. With respect to structural free vibration responses identification and noise omission, random decrement technique (RDT) is employed. Results obtained with RDT are used in SSI method as input data and therefore the computation accuracy of the modal parameters of the system increases significantly. In the following, by defining an objective function as the sum of the square errors between the natural frequencies which are calculated by both FEM and OMA, the finite element model is being updated using bees optimization algorithm and therefore, design variables are being optimized too. To reduce the updating time of the structure, using bees optimization algorithm, sensitivity analysis is employed to determine design variables. A scheme showing this research procedure is presented in Fig. 1.

Theory

As depicted in Fig. 1, this research methodology consists of three main steps described in the following sections. In the first part, finite element model of the solar structure is constructed using Ansys software where this model is used to obtain structural modal parameters, effective design parameters and reference sensors positions required for experimental modal analysis. The required steps to acquire structural dynamic parameters, using operational modal analysis are stated in the second part. In the third step of this research, in order to update finite element modeling of the solar structure, natural frequencies calculated by operational modal analysis, finite element method and bees optimization algorithm are applied. In the following, these three steps are described in details.

Finite element method

Modal parameters determination

Figure 2 shows the finite element model of the investigated solar power plant. Three different types of elements including Beam18, Solid186, and Mass21 are employed to model solar power plant structure, solar panels and power transformers, respectively. The whole finite element model is composed of 17312 nodes and 16926 elements. Additionally, structure sub-legs are modeled using clamped boundary condition. Table 1 represents five first natural frequencies obtained by modal analysis module of Ansys finite element software.

Figure 3 illustrates mode shapes related to the five first frequencies of the solar structure.

Sensitivity analysis

Physical parameters of the structure that affect the most influence on system dynamic parameters including natural frequencies are identified through a sensitivity analysis. A small change in effective parameters causes significant changes in the outputs. Finding these physical parameters simplifies updating finite element model process. Once-at-time (OAT) index is one of the most practical approaches utilized to determine sensitive parameters to which model outputs are affected by the changes in each input parameter. This approach to sensitivity analysis uses local and global sensitivity methods [⁶⁷]. OAT coefficient, in local sensitivity method is defined using,

(1)\mathrm{O}\mathrm{A}\mathrm{T}={\displaystyle \frac{\mathrm{\Delta }Y}{\mathrm{\Delta }X}}{\displaystyle \frac{X}{Y}},

where, X and Y represents input and output parameters respectively. Additionally, the changes in input parameters are assumed to be a factor of the parameter standard deviation in its defined limit. To estimate overall sensitivity, global sensitivity index (GSI) is calculated using Eq. (2),

(2)\mathrm{G}\mathrm{S}\mathrm{I}={\displaystyle \frac{Y_{\max }-Y_{\min }}{Y_{\max }}},

where, Ymin and Ymax are the minimum and maximum output values of the model, respectively resulting from upper and lower limit bonds of the input parameter. Based on Eqs. (1) and (2), it is possible to identify the parameters which play important role on natural frequencies variations. The investigated parameters in this research and their related upper and lower limits with respect to sensitivity analysis are tabulated in Table 2.

Figures 4 and 5 depict local and global sensitivity index, respectively, using five first natural frequencies of the solar structure. All presented data on these two figures are normalized to the maximum values. As can be inferred from these two figures, sensitivity index related to some of structure physical parameters have more effect on five first natural frequencies and can be considered as sensitive parameters. The determination of sensitive parameters increases problem accuracy and decreases the required time to update structure finite element model. Based on Figs. 4 and 5, sensitive parameters can be shown in Table 3.

Determination of sensors installation places

The determination of sensors installation locations is a very important subject that increases the accuracy of experimental modal tests. To find the structure modes properly, it is required that the sensors to be installed at proper places on the structure. When there are only a limited number of accelerometers and measuring channels in operational modal analysis for large structures, it is not possible to measure structure response in all points, simultaneously. Thus, for data acquisition in these structures, multi-setup measurement strategies are often performed. In this type of sampling strategy, a number of sensors are selected as reference sensors and the remaining available sensors are then moved over the structure in each step, in order to assess all data sets in the whole structure.

It worth mentioning that, sensors installation places should not be located on the nodes of the mode shapes and on the other hand, sampling points must have significant contribution for all mode shapes. Several methods have been proposed to determine sensor locations, where this research uses the strategy presented in reference [⁶⁸]. In this strategy, modal information matrix, F is obtained by:

(3)\mathit{F}={({\varphi }_{k\times l})}^{\mathrm{T}}{\varphi }_{{}_{k\times l}},

where, in the above equation, f is the mode shape matrix. Additionally, k is the number of degrees of freedom and l is the number of investigated modes. Matrix Q which its diagonal elements determine the share of each degree of freedom in the structural response, can be obtained by Eq. (4) as below:

(4)\mathit{Q}={({\varphi }_{k\times l})}^{\mathrm{T}}{(\mathit{F})}^{-\mathrm{1}}({\varphi }_{k\times l})\mathrm{.}

Additionally, in order to contribute kinetic energy in the system degrees of freedom, Eq. (4) is converted into Eq. (5) as below:

(5){\mathit{Q}}^{\ast }=\mathrm{D}\mathrm{i}\mathrm{a}\mathrm{g}(\mathit{Q})\cdot \mathit{E},

where, · sign indicates dot product of the elements of two vectors. Additionally, in this equation E is modal energy and is calculated as below:

(6)E_i={\displaystyle \sum _{r=\mathrm{1}}^l\frac{{({\varphi }_{ir})}^{\mathrm{2}}}{{\omega }_r^{\mathrm{2}}}},

where, w_r is the natural frequency of mode r and f_ir is the ith element of the structure rth mode shape. The maximum values of Q* vector indicate the appropriate locations of sensors installation. The estimated places with respect to sensor installation on solar structure are shown in Fig. 6.

The circles depicted on the structure indicate proper installation places for accelerometers. Additionally, dotted circles shown on the Fig. 6 have the maximum value of Q* vectors and they are selected as the installation locations of the reference accelerometers in this research.

Experimental modal analysis

The measurement of structure response

The investigated solar structure is located at north side of Shahid Chamran University, Ahvaz, Iran, which has 20 KW capacity. Figure 7 illustrates a summary of the executed experimental work. The solar structure is excited randomly, and the mode responses were sampled at a rate of 16328 Hz and were recorded in a measurement time of 20 s. Figure 8 shows the obtained response of one of the accelerometers.

The experimentally measured structural responses always have noise content and these disturbances must be deleted before starting to determine the modal parameters. Random Decrement method that is described as follow is one of the most effective strategies to omit these disturbances.

RDT

The RDT was introduced by Cole [⁶⁹]. The aim of this strategy is the identification of the system free response from measured ambient vibration random signals. In this technique it is assumed that the response signal of a system to the exciting input loads is composed of a deterministic component and a random component. The main purpose of the mentioned method is to make the random part disappear from the obtained signal. By averaging a sufficient number of the selected system responses with a fixed initial condition, (threshold “a”), the random part of the response will have a tendency to disappear from the raw signals. Consequently, the remaining result has only the deterministic part associated with the free vibration response of the system that can be used to extract modal parameters. Generally, the main equation in RDT is defined as below:

(7)\delta (\tau )={\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum _{i=\mathrm{1}}^{N}y(\tau +t_i)}|{}_{y(t_i)=a}

where in the above equation, y(t_i) is a sample of the calculated response of signal x(t_i) at the time instant t_i, N is the number of triggering point (measured distances) and t is the time variable. One important aspect of the application of this technique is the selection of “a” in Eq. (7). Establishing appropriate value of “a” will result in achieving a good estimation of the system free response. There are a large number of criterias proposed by scientists in the papers for the mentioned value, but it has been proved that the value proposed by Carlos, et al. [⁵¹] is the optimum point for this method which is stated as follows:

(8)a=\sqrt{\mathrm{2}}{\sigma }_x

where s_x is the signal standard deviation. By inserting the row signals obtained from the response of the system into the RDT, the disturbances in the signals can be reduced significantly and then these signals will be ready to be entered into modal parameters identification process. In this research, multi-setup SSI method is used to determine the natural frequencies and mode shapes of the structure that is described in details in the following section.

Multi-Setup subspace identification method

Dynamic behavior of a linear vibrating structure can be mathematically modeled by a set of linear second-order vector differential equations with constant coefficients such as the following expression:

(9)\mathit{M}\ddot{y}(t)+\mathit{D}\dot{y}(t)+\mathit{K}y(t)=\mathit{f}(t)\mathrm{.}

In Eq. (9), M, D, and K denote the mass, damping ratio and stiffness matrices respectively. Additionally, t is continuous time, vector y(t) is the displacement vector of the structure degrees of freedom and f(t) presents vector input forces applied on the structure. Various methods can be used to rewrite Eq. (9) as a system of first order differential equations. As measurement data are time-discrete, it is possible to discretize Eq. (9), similar to the steady-space model described in the following equation:

(10)\begin{array}{l}{x}_{k+\mathrm{1}}=\mathit{A}{x}_k+v_k\\ y_k=\mathit{C}{x}_k+w_k\end{array},

In Eq. (10), x is state vector, y denotes output vector, vector v and w are input and output perturbations, respectively. Additionally, A is state transition matrix and C is the system output matrix. Natural frequencies and mode shapes of the system can be determined by eigenvalue and eigenvector analysis of matrix A and matrix C as follow:

(11)(\mathit{A}-{\lambda }_i\mathit{I}){\varphi }_i=\mathrm{0},\begin{array}{ll}& {\phi }_i\end{array}=\mathit{C}{\varphi }_i\mathrm{.}

In Eq. (11), l is the dimensionless natural frequency and f is stated as mode shape of the system. Furthermore, natural frequencies and damping ratios are determined by Eqs. (12) and (13) as follow:

(12) a=\left|\arctan {\displaystyle \frac{\mathrm{Im}(\lambda )}{\mathrm{Re}(\lambda )}}\right|,\begin{array}{lll}& & b=\ln \left|\lambda \right|\end{array},

(13)\omega ={\displaystyle \frac{\sqrt{a^{\mathrm{2}}+b^{\mathrm{2}}}}{\tau }}\begin{array}{lll}& & \zeta ={\displaystyle \frac{-\mathrm{100}\left|b\right|}{\sqrt{a^{\mathrm{2}}+b^{\mathrm{2}}}}}\end{array},

where the dimension of natural frequency is rad/s and t is the sampling rate. By the way as mentioned above vibration equation of the system is transformed to a structural identification problem. The solution of the mentioned problem gives the interested dynamic parameters of the system. The problem of multiple sensor setups SSI using non-simultaneously data recording, initially was proposed by Döhler et al[⁷⁰]. In this method N_s output data records of the system are collected, instead of a single record of the outputs. Equation (14) describes the multi-setup method of output data recording.

(14)\begin{array}{llll}\left(\begin{array}{l}{y}_k^{(\mathrm{1},\mathrm{r}\mathrm{e}\mathrm{f})}\\ y_k^{(\mathrm{1},\mathrm{m}\mathrm{o}\mathrm{v})}\end{array}\right)& \left(\begin{array}{l}{y}_k^{(\mathrm{2},\mathrm{r}\mathrm{e}\mathrm{f})}\\ y_k^{(\mathrm{2},\mathrm{m}\mathrm{o}\mathrm{v})}\end{array}\right)& \dots & \left(\begin{array}{l}{y}_k^{(N_s,\mathrm{r}\mathrm{e}\mathrm{f})}\\ y_k^{(N_s,\mathrm{m}\mathrm{o}\mathrm{v})}\end{array}\right)\end{array}\mathrm{.}

As the reference sensors are fixed throughout the measurement, there are a fixed number of them in the experiment and therefore in every record of Eq. (14) they are fixed, in contrast the number of moving sensors can be different in each setup. For every record, j ₌ 1,2,3,…,N_s corresponds a state-space realization which can be stated in the form of Eq. (15):

(15)\{\begin{array}{l}{x}_{k+\mathrm{1}}^{(j)}=\mathit{A}{x}_k^{(j)}+v_k\\ y_k^{(j,\mathrm{r}\mathrm{e}\mathrm{f})}={\mathit{C}}^{(\mathrm{r}\mathrm{e}\mathrm{f})}{x}_k^{(j)}+w_k^{(j)}\\ y_k^{(j,\mathrm{m}\mathrm{o}\mathrm{v})}={\mathit{C}}^{(j,\mathrm{m}\mathrm{o}\mathrm{v})}{x}_k^{(j)}+w_k^{(j)}\end{array}\mathrm{.}

As can be implied from Eq. (15) matrices A and C^(ref) have constant values for each record but matrix C^{(j, move)} is variable in each setup. Generally stating, observable matrix C which contains information about the location of sensors can be stated as Eq. (16) as below:

(16)\mathit{C}=\left[\begin{array}{l}{\mathit{C}}^{(\mathrm{r}\mathrm{e}\mathrm{f})}\\ {\mathit{C}}^{(\mathrm{1},\mathrm{m}\mathrm{o}\mathrm{v})}\\ ⋮\\ {\mathit{C}}^{(N_s,\mathrm{m}\mathrm{o}\mathrm{v})}\end{array}\right]\mathrm{.}

The most basic step in SSI method is the construction of Hankel matrix from the output data. Besides, the most important property of Hankel matrix is that it can be decomposed into controllable and observable matrices as per Eq. (17):

(17)\mathit{H}=\mathit{O}\mathit{Z}

where, in the above equation, H is the Hankel matrix, O and Z are observable and controllable matrices, respectively. If, the controllable matrices Z(j) are equal for all setups j, general Hankel matrix can be computed and then the calculation of modal parameters of the system can be performed, similar to the traditional SSI [⁷⁰]. In contrast, for multi-setup measurements the properties of ambient excitation change between the records and therefore the controllable matrices Z are not equal for different records and thus it is not possible to build Hankel matrix from Eq. (17). The solution for this problem is often achieved by normalizing and merging the results obtained for records corresponding to different sensor setups [⁵⁷]. The aim of this method is obtaining general observable matrix O^all = O(C,A) from all delivered data. A scheme showing this procedure is described in Fig. 9.

Similar to single setup stochastic subspace identification method, this analysis depends on the order parameter. Underestimating the value of the system order may delete the contribution of some of the modes by SSI method, while overestimating the values of model order may result in the computation of extra modes than what the system has [⁵⁰]. As a solution to this problem, the obtained results of the stochastic subspace identification method are plotted for different orders. The points which deployed next to each other in rows for a specified order are an approximation of natural frequencies. This diagram is known as stabilization diagram. The main problem of the stabilization diagram is that when the order of the system increases, spurious modes appear in the mentioned diagram. Therefore, distinguishing between real modes and spurious modes will be difficult. To solve this problem, some constrains are defined (Eqs. (18–20)). Additionally, by eliminating high damping modes and negative frequencies, almost half of spurious modes existing in the stabilization diagram will be deleted automatically [⁵⁰]. This technique will result in the increase of the stabilization diagram contrast and also more accurate identification of the system physical modes.

(18){\displaystyle \frac{\left|{\omega }_{pi}-{\omega }_{qi}\right|}{{\omega }_{pi}}}\le {\epsilon }_f,

(19) {\displaystyle \frac{\left|{\zeta }_{pi}-{\zeta }_{qi}\right|}{{\zeta }_{pi}}}\le {\epsilon }_{\zeta },

(20)\mathrm{1}-\mathrm{M}\mathrm{A}\mathrm{C}({\varphi }_{pi}-{\varphi }_{qi})\le {\epsilon }_{\mathrm{M}\mathrm{A}\mathrm{C}}\mathrm{.}

In the aforementioned equations, ϵ_f, ϵ_z and ϵ_MAC are three tolerance limits which state that ith real mode obtained from order p is slightly different from the same mode of the order q (in most cases, p and q are consecutive numbers, i.e., q = p + 1). Therefore, by implementing these conditions in SSI technique, spurious modes can be realized and vanished from the real modes. The values for tolerance limit are chosen by trial and error method which in this work they are assumed to be 0.02, 0.05, and 0.05, respectively. The stabilization diagram obtained with respect to the solar structure in y direction is depicted in Fig. 10.

According to Fig. 10, the system order varies between 40 and 200. As can be seen from Fig. 10, most of the spurious modes have been omitted from the stabilization diagram which resulted in the increased accuracy to choose physical modes. Additionally, five first mode shapes obtained from multi-setup stochastic subspace identification method are depicted on Fig. 11. Table 4 shows five first mode shapes obtained by multi-setup SSI method along with their relative error obtained from finite element method results. It can be implied from Table 4 that natural frequencies calculated by finite element method differs obviously from those obtained by operational modal analysis which this result shows that the created finite element models are imperfect.

To solve this problem and reduce the discrepancy between the computed and measured natural frequencies, an objective function is defined and minimized by the bees’ optimization algorithm that is used to update finite element model. This process is discussed in the following sections.

Finite element model updating

To update the finite element model of the structure, an objective function is defined based on the summation of the squared errors between the natural frequencies obtained by the operational modal analysis and the finite element method. This objective function can be presented as Eq. (21):

(21)\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}{\mathit{Z}}_t={\displaystyle \sum _{j=\mathrm{1}}^s{({\mathrm{\Omega }}_{c,j}^{}-{\mathrm{\Omega }}_{uc,j}^{})}^{\mathrm{2}}}

where s is the number of natural frequencies, Ω_c are natural frequencies obtained from operational modal analysis and Ω_uc denote the natural frequencies calculated from the finite element method. Furthermore, design parameters identified by sensitivity analysis which was discussed in section 2-1-2 are included in vector Z.

To determine design parameters and to design a precise finite element model, the objective function described by Eq. (21) is minimized using bees optimization algorithm. In the following section bees optimization algorithm method is described along with the method of determination modal parameters.

The bees’ algorithm

Honey bees swarm optimization algorithm might be deemed as belonging to the classification of evolutionary algorithms. An organized social behavior has been seen among honey bee swarms which can be taken to solve complex optimization problems. Each bees swarm has some scout bees that seek in a random fasion to find food sources for their hives. When they come back to the hive and assess various found gardens, those scout bees that found a patch which is related above a certain quality threshold, deposit their nectar or pollen and perform a dance called the “waggle dance” which gives the information of the direction, distance and amount of nectar in these gardens for worker bees [⁴⁷,⁷⁰]. In the next stage the worker bees fly to the located site of flower patch. Additionally, the worker bees’ quantity sent to promising patches is proportional to gardens which have more nectars and shorter distance to the hive. Therefore, the mentioned strategy enables bees swarm to earn food sources in an optimal way. Following every iteration the new population has two parts: representative from neighborhood searches having the best fitness value in their neighborhood space and randomly elected solutions. The iteration goes on and finally, there are a series of optimum fitness values; the best of them would represent the global optimum. Figure 12 indicates the flowchart of the bees algorithm. The number of worker bees is proportional with the nectar amount and is inversely proportion to the distance. From a set of random solutions N_t, N_t1 solutions with the highest fitness values are selected as the best ones. Among the best answers, N_t2 solutions with highest reliability will be selected as elite solutions. To find better solutions, neighborhood searching around the best and elite solutions is done. n_t1 and n_t2 indicate the number of neighborhoods searched around the best and elite solutions, respectively (n_t1<n_t2).

Figure 13 depicts convergence diagram for the problem of updating solar structure by bees’ algorithm using sensitivity analysis and also without this analysis. As can be seen from Fig. 13, by implementing sensitivity analysis to select design parameters, the best value for objective function reaches to 0.05, after 1200 iterations, while this value can be obtained after almost 3200 iteration if sensitivity analysis does not perform. However, it is illustrated that sensitivity analysis could significantly reduce the time and computational cost of updating finite element model. Control parameters which were used for bees’ algorithm were tabulated at Table 5. This value obtained after performing some trial and error.

Table 6 shows the natural frequencies obtained by finite element model updating of the solar structure. In addition, the relative errors based on natural frequencies calculated by experiment and numerical model are presented in the mentioned table. It is noticeable from Tables 4 and 6 that the relative errors among experiment and numerical model reduces significantly after updating finite element model.

The summation of the errors before model updating was 77.708 percent which is reduced to only 0.32 percent after finite element model updating. The amount of optimum design parameters of solar structure which were obtained by bees algorithm and sensitivity analysis are shown in Table 7. The behavior of the updated finite element model has the most similarity to the experimental model and can be used as the actual model. By using the aforementioned updated software model it can be possible to perform various modeling analysis on the solar structure.

Conclusions

In this research an applied method was stated in order to investigate finite element model updating of a solar power plant with 20 KW capacity located at Shahid Chamran university of Ahvaz, Iran, using operational modal analysis, multi-setup stochastic subspace and bees optimization algorithm. In this method, first of all, finite element model of the solar power plant was created in Ansys software and by applying sensitivity analysis; effective parameters in updating finite element model were determined using sensitivity method. Furthermore, the best locations for the installation of reference and movable accelerometers on the solar structure were determined. In the following vibration tests were conducted to obtain raw vibration signals of the solar structure. In the aim of deleting the noises, random decrement analysis was performed on the raw signals and the signals obtained by this method were considered as input data for multi-setup SSI method and the natural frequencies of the system were determined. By implementing bees optimization algorithm and defining an error function which is based on the differences between the natural frequencies obtained by finite element modal analysis and the multi-setup SSI method, the finite element model of the system was updated and then optimum values of model parameters were calculated. The results show that the error summation of the five first natural frequencies is reduced from 77.708% to only 0.32%.

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