Optimal Sensor Placement and Minimum Number Selection of Sensors for Health Monitoring of Transmission Towers

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Introduction
Transmission towers often operate in adverse and harsh environments with complex and varying loads and may experience failure, including collapse, from extreme conditions or vibration fatigue damage [1,2]. To prevent such failures and ensure consistent, high-quality, and safe power provision to both the grid and dependent loads, health monitoring of transmission towers must be an integral consideration in the power ecosystem.
Structure health monitoring for transmission towers is increasingly attracting attention. In current literature [3][4][5], the prevailing method for structural health monitoring of transmission towers is based on Structural Dynamic Characteristics Identification. In this method, the health of the transmission tower can be monitored by measuring the vibration response of the tower structure and then identifying the parameters of its structure modal. It is important to point out here that the placement of vibration sensors directly affects the accuracy and completeness of the structural information of the tower structure. To maximize the valueadded of sensors, sensor placement should strive to make transparent damage information, reduce monitoring costs, and improve detection efficiency [6].
ere are many methods to choose from to optimize sensor placement, some of which include the Effective Independent Method (EIM), Energy Method, and Modal Assurance Criterion. e EIM [7] can be used to obtain the optimal placement scheme through removing the degrees of freedom with the greatest effect on the error of the estimation of the modal coordinate from the Fisher information matrix. is method was proposed by Kammer in 1991 and was originally applied in the selection of the best test points for experimental modal analysis. It was only later that it was extended to the field of structural health monitoring for optimizing sensor placements. e health monitoring applications [8][9][10][11][12] used EIM to optimize the sensor placement of the plates, dams, houses, and other structures and achieved good results. However, the EIM has some shortcomings such as information redundancy and avoidance of large energy measurement points. In response, some scholars [13][14][15][16][17][18] have proposed various methods to improve EIM. Since also sensors can more easily collect information of a measuring point with high energy output, some literatures reinforce the energy criterion when evaluating sensor placement. e Energy Method depends on the modal kinetic energy of the structure formed to optimize sensor placement [19][20][21]. e relationship between the Energy Method and EIM has also been studied [22]. e Modal Assurance Criterion is a method that can be used for the optimal sensor placement and evaluating the results of optimum sensor placement [23,24].
ere are only a few literatures that deal with the placement of sensors for tower structure. Yin et al. [5] present a methodology for identifying the "optimal" locations to install a given number of sensors on a structure and have brief inferences on using this methodology on transmission towers. Vergara et al. [25] compared the effects of a transmission tower and a transmission tower-line system and optimal sensor placement. Benny and Sai [26] evaluated a methodology for sensor placement in transmission line towers based on the concepts of entropy and model falsification.
ese researches on sensor arrangement mainly focus on optimizing the arrangement of sensors after giving the number of sensors, but there are few researches on the number of sensors that should be placed. e number of sensors is the prerequisite for optimal sensor placement, which in turn affects accuracy, sensitivity, and economy of acquiring the structural health information.
With these problems in mind, this paper proposes several sensor placement schemes of a cat-head-type transmission tower and a method for determining the number of sensors to be used for health monitoring. To solve this problem, the optimal sensor placement should first be performed by the EIM under different numbers of sensors. en, the corresponding MAC matrices are calculated and the relationship of the maximum value of the nondiagonal elements of the MAC matrices with the sensor numbers is analyzed. ese inputs are then used to fit the curve of the maximum nondiagonal element of the MAC matrix with the number of sensors. Finally, the point corresponding to the minimum value of the curve is taken as the best number of placed sensors.

Effective Independence Method (EIM).
e idea of the EIM is to rank each degree of freedom based on its contribution to the linear independence of the target modality (Fisher Information Matrix). e degrees of freedom with lesser contribution degree are then deleted and the degrees of freedom with greater contribution are retained. ereby, the optimal sensor placement is realized. e dynamic equation of a multi-degree-of-freedom linear time-invariant system is where M is the mass matrix, C is the damping matrix, K is the stiffness matrix, u (t) is the displacement response vector, and F (t) is the outer excitation vector. According to the modal analysis theory, the solution of equation (1) can be expressed as where the matrix Φ is the modal shape matrix and q (t) is the modal coordinate vector. For m sensors (m < s) arranged at s positions, the degree of freedom of the system is s. If the influence of noise is considered, the output response of the sensor U s can be expressed as where the s × N matrix Φ s is the modal shape matrix, in which s is the number of degrees of freedom and N is the number of modal orders. ψ is a Gaussian white noise whose variance is σ 2 ; and q is a generalized modal coordinate vector. e estimated value of the generalized modal coordinate vector q is For a complex structure with s degrees of freedom, if m sensors need to be arranged for structure health monitoring to maximize the linear independence of the modal information, it is necessary to obtain the best estimate of the generalized coordinate q. e covariance matrix of the estimated deviation is where E is the mathematical expectation and Q is the Fisher Information Matrix. To minimize the covariance matrix to get the best estimation, it is necessary to maximize the determinant of Q. To simplify the analysis, it is assumed that the noise is uncorrelated and has the same statistical properties in each sensor. Combined with equation (4), the Fisher Information Matrix Q can be expressed as Obviously, the Fisher Information Matrix Q can be represented by the matrix A 0 . en A 0 can be represented in a matrix format: 2 Shock and Vibration where Φ s i is the ith row of the matrix Φ s . A 0 can be maximized by removing the degree of freedom that contributes less to the linear independence of target modes.
In order to reduce the degree of freedom A 0 , the contribution value of each degree of freedom of the target modes should be calculated. e determination of this value is given by the following procedure. e characteristic equation of matrix A 0 is where λ is the eigenvalue of A 0 and Ψ is the eigenvector of A 0 . It can be derived from equation (8) that with F set as a contribution matrix of each degree of freedom to target modes; and F can be expressed as where the symbol ⊗ represents a term-by-term matrix multiplication. erefore, F is an s × N matrix. e jth item of the ith row of the matrix F represents the contribution of the ith degree of freedom to the jth eigenvalue. If each row of F is added, the column vector E D can be obtained. e ith row of E D represents the contribution of the ith degree of freedom to the linear independence of the entire mode shape matrix.
In addition, the contribution matrix of each degree of freedom to the linear independence of the target modes can also be represented by another matrix E: Using equation (7), the following equation can be noted: where E is an idempotent matrix and the ith element E ii on the diagonal represents the contribution of the ith degree of freedom to the rank of the mode shape matrix Φ s . erefore, the elements on the diagonal of E represent the contribution of the linear independence of the target modes. After the matrix E is obtained, the degrees of freedom that have more contribution to the linear independence of the target modes are selected and the locations of the sensors can be obtained.
For the finite element model, the process of using EIM for optimal sensor placement is shown in Figure 1. e mode shape matrix of the finite element modal can be extracted after modal analysis. e form of the mode shape matrix is arranged as shown in Figure 2. e mode shape matrix is an n×N matrix. e number of nodes is s. us, n � 3 × s if the mode shapes of X, Y, and Z directions are considered. During the process of OSP, the row in the mode shape matrix of the degree of freedom related to the minimum value of the diagonal in the E matrix will be removed at each time, which means that the position on this degree of freedom is not selected. After n-m times of iterations, m rows are left, which are the optimal sensor positions.

Evaluation Criterion.
e Modal Assurance Criterion (MAC) [27] is an effective tool for evaluating the intersection angle of modal vectors. It can reflect the correlation of two space vectors. e calculation formula of MAC is where Φ i is the ith modal vector and Φ j is the jth modal vector and the values of MAC ij fall between 0 and 1. e MAC ij value will be 0 if the two vectors are orthogonal, while the value equals 1 if the two vectors are completely correlated. e smaller the value of MAC ij is, the stronger the structural characteristics are reflected. It is generally believed that when MAC ij is greater than 0.9, the two modal vectors are completely related and cannot be distinguished.

Method to Choose the Minimum Number of Sensors.
From the concept of the EIM, if the target number of sensors is m (m < n) and the degrees of freedom are n, then the dimension of modal matrix will change from n×N to m×N after EIM: e linear independence of the modal matrix after EIM will be worse than before, which means that the nondiagonal elements value of the MAC matrix gradually decreases. e maximum value of nondiagonal elements in the MAC matrix represents the worst correlation of two modal vectors. is paper therefore presents a method for calculating the minimum number of sensors for health monitoring of transmission tower structures. e steps are as follows: (1) Use the EIM to obtain a series of placement schemes of differing number of sensors. k schemes can be obtained: Shock and Vibration 3 where m 1 , m 2 , m 3 . . .m k represents the assumed number of sensors.
(2) Calculate the MAC matrix of each scheme separately, and extract the maximum nondiagonal element in the matrix: (3) Take the number of sensors in X as the variable x and the corresponding value in Y as the function value y, and use the sequence of the two variables to perform polynomial fitting to obtain the fitting curve: (4) Calculate the derivative of the fitting curve and set it to 0: Get the degree of freedom (the position and direction of sensors) Find the minimum value of the diagonal in the E matrix, and delete the row of the degree of freedom related to this value in the modal shape matrix Establish the finite element model with n degrees of freedom of s "nodes; assume" m sensors to place Conduct modal analysis and extract modal shape matrix Calculate E according to equation (13) m < n n = n -1 yes No Figure 1: e process of EIM for optimal sensor placement. and, with this method, upon setting the slope of the curve to 0, the curve reaches the extreme point, and the value of x at this point can be interpreted as the minimum number of target sensors required.

Finite Element Model.
is paper presents a case study on a cat-head-type transmission tower in order to verify the optimal sensor placement method for structural health monitoring. In this example, it is assumed that the transmission tower bears the gravity of the overhead lines and the insulator strings and the horizontal force caused by the wind.
To facilitate the subsequent test verification, a 1 : 36 scale model of the cat-head-type transmission tower is analyzed in this paper. e finite element model of the transmission tower is shown in Figure 3. ere are 194 nodes and 579 elements in the model. e element type of the finite element model is beam188 element, and the material parameters are set as follows: the elastic modulus is 7.5e7, Poisson's ratio is 0.03, and the density is 2.7e−6. e four nodes of the tower foot are fixed and the degrees of freedom of the remaining nodes are not constrained. e directions of three main axes are shown in Figure 3.
In this paper, the first 10 modes of the transmission tower are selected as the target modes. e Fisher Information Matrix 2-norm method is used to determine the target modes number. e detailed information of the method is introduced in literature [28]. e essence of the Fisher Information Matrix 2-norm method is to obtain the Fisher Information Matrix of the structure through the vibration mode information of each measuring point of the structure. e rate of change (ROC) of the 2-norm of the matrix on the i-th order and the i + 1-th order is taken as the index to measure the influence of the number of modes, and the mode is selected by the change rate curve. e function of the ROC is where Q is the 2-norm of Fisher Information Matrix. e ROC value changes with the increase of mode number i. When i reaches a certain value, the ROC value approaches zero, or i continues to increase but the ROC value has little change. en, the whole modal information is concluded in the first i mode shapes. e rate of change (ROC) of the Fisher Information Matrix 2-norm of the transmission tower is calculated through this method, as shown in Figure 4. It can be seen from the figure that the ROC will not change when the modal number is larger than 8. us, to ensure the integrity of the modal information, the first 10 modes are chosen as the target modes to conduct OSP.
rough the modal analysis of the finite element model of the transmission tower, the first 10 natural frequencies of the transmission tower can be obtained, as shown in Table 1. e results show that the first 10 natural frequencies range between 13.259 and 117.067 Hz. e first 4 modal shapes are shown in Figure 5. It can be seen that the first modal shape is the torsional vibrations in the Z direction, and the second mode is the X direction bending vibrations. e third and fourth modes are the bending vibrations in the Y direction.

Optimal Sensor Placement.
e optimal sensor placement is carried out by EIM and the results are shown in Table 2.
e positions of the sensors are represented by "node number-direction." For example, "1701x" means that the sensor is arranged in the X direction of the node 1701. In this paper, the placement results of different target sensor numbers from 10 to 50 are calculated. Table 2 lists the placement positions when the number of sensors is 50, 30, 20, and 10. As can be seen in Table 2, the placement directions of the sensors are all horizontal (X and Y), which indicates that the modal information in the X direction and Y direction contributes more to the Fisher Information Matrix. Also, it reflects that the main vibration directions of the transmission tower are in the X direction and Y direction.
To show the positions of the sensors more clearly, the placement positions when the number of sensors is 20 and 10 are shown in Figure 6. e sensors are mainly located at the sharp corner and the two ends of the tower top. ere are more sensors at the middle of the tower and fewer sensors at the tower leg. At the same time, some positions of the sensors are symmetrical at the middle and top of the tower and as the number of sensors is gradually reduced, some symmetric points are gradually deleted. In addition,

Effect Evaluation.
e modal matrices calculated by the EIM are verified by the Modal Assurance Criterion. e MAC matrixes are shown in Figure 7. e horizontal axis is the modal order and the vertical axis is the MAC value. As can be seen in Figure 7 is result indicates that when the number of sensors is large, the linear correlation between the various modes is small, and the modal information of the original structure can be well reflected. When the number of sensors is small, the correlation between some modes will be degraded, which has a negative impact on the acquisition of modal information.      Sensor position  50  30  20  10  50  30  20  10  1701x •

Application Examples of Calculating the Minimum
Shock and Vibration numbers of sensors are 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, and 60. e polynomial fitting regression method can then be subsequently used to fit the curve of the number of sensors and the maximum nondiagonal elements of the MAC matrix. e curve is fitted using the curve fitting tool in MATLAB software. e curve fitting modal is a polynomial linear modal with 3 degrees, which is as follows: By using the curve fitting tool, the parameters of the function and the goodness of fit are calculated, as shown in Table 3. e fitting curve is shown in Figure 8. e goodness of fit shows that R-squared is near 1, which means that the fitting modal is acceptable. e scatter diagram and fitting curve are shown in Figure 9. It can be seen that the maximum values of the nondiagonal elements of the MAC matrix gradually decrease as the number of sensors increases. When this number of sensors exceeds 40, the maximum values of nondiagonal elements of the MAC matrices no longer change significantly. According to steps (3) and (4) mentioned above, an extreme point of the fitting curve can be obtained, and the number of sensors corresponding to the value of the extreme point is 46, which indicates that when the number of sensors exceeds 46, the maximum values of the nondiagonal elements of the MAC matrix will not change noticeably. In other words, the minimum number of sensors required to obtain the most comprehensive structural modal information is 46. Figure 10 shows the sensor positions when the number of sensors is 46. e position of sensors is arranged mainly at the middle and the top of the tower. ere are still some symmetric measurement points, similar to the position when the number of sensors is less than 46. In addition, as indicated by the circle in Figure 10, the position of the tower     leg and the middle of the tower is increased by some points compared with the position when the number of sensors is less than 46. e average value of nondiagonal elements is below 0.1, which shows that the optimal sensor placement can guarantee the acquisition of the modal information.

Conclusion
(3) A method for choosing the number of sensors is proposed based on the EIM and MAC. A cat-head-type transmission tower modal is used to verify this method. e curve by which MAC values change with the number of sensors is obtained. e result shows that the minimum number of sensors required to obtain the most comprehensive structural modal information is 46.
Data Availability e data supporting the results of this study can be obtained upon request to the corresponding author.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this paper.