Extended Kalman Filter for a Monitoring System of the Guyed Towers

Abstract

The Kalman filter is used in a wide range of systems. Due to its efficiency to estimate the state variables in structures and mechanisms, its usage is already well known in Control Systems and in addition to smooth measured and unmeasured signals, it allows the sensor fusion technique to consider the best characteristics of each type of sensor. In this Chapter, we will present the application of the Extended Kalman Filter (EKF) to predict the movement of the upper top center of a guyed tower of a power line, based on its dynamic model and considering the indirect measurement of the force in the stay cables through accelerometers. In a mockup tower, built in Dynamic Testing Laboratory (LabEDin) at University of Campinas (UNICAMP), it is possible to apply external loads and simulate failures, such as degradation of the stay cable foundation. Variations in the cable forces are used as inputs in the EKF algorithm and as estimate, we obtain the amplitudes and directions of the tower’s top movements, which will be considered to predict the health of its structure, indicating the need for maintenance intervention. All the theories considered in this proposed methodology are validated with the experiments carried out on the mockup tower in the laboratory.

1. Introduction

The objective of this Chapter is to present a practical application of the Kalman filter considering a dynamic model of a steel tower structure, which is not usual for a Bayesian filtering utilization. Usually this filtering technique is considered to estimate time-dependent physical quantities, such as in Global Positioning Systems (GPS), target tracking, satellite control, etc.

The origin of Bayesian analysis belongs to the field of optimal linear filtering and the concept of building mathematically optimal recursive estimators, was first presented for linear systems due to their mathematical simplicity [1]. The success of the Bayesian filtering applied to engineering problems, started with an article presented in 1960 by the Hungarian-American mathematician and engineer, Rudolf Emil Kálmán. He developed a mathematical method for linear filtering which does not requires powerful computing machinery and several applications of his method led to further application of the named Kalman filter in Apollo program, NASA Space Shuttle, Navy submarines, in Unmanned Aerospace Vehicles (UAV) and also in weapons, such as cruise missiles.

In our studied case, the guyed towers of a transmission power line need to be inspected along the year, to check the inclination and the forces at the stay cables to assure its structural integrity. In order to reduce the preventive maintenance costs, there is a proposal for a remote monitoring system to estimate the position of the upper part of the tower based on the indirect measurements of stay cable forces. The natural frequencies of the cables can be determined from time-domain accelerations, obtained from accelerometers connected to the stay cables. Based on an existing relation between the cable force and its natural frequency, it is possible to calculate the instantaneous force at every acceleration measurement. As redundancy, the signal from an inclinometer will be fused into the algorithm to provide the direction of the tower movement. Another important characteristic of the Kalman filter to be highlighted, is that in addition to fuse different types of sensors, it is possible to estimate variables which are not being measured directly.

The proposed monitoring system will transfer (via RF) the accelerations measured in time-domain from the leaf nodes positioned in each stay cables, to a central router in the tower. These signals will be transferred to a border router responsible to organize the acquisitions of several towers to a main server, where the post processing will be carried out. The EKF algorithm and all calculations are performed in a MATLAB® (Mathworks, EUA) code developed at the University of Campinas (UNICAMP).

2. Kalman filter concepts

When any signal is collected through a sensor, generally it presents some level of noise which can directly affect the evaluation of the measured quantity. The usage of the Kalman filter, in addition to smooth the measured signal, allows to estimate quantities which are not being measured directly. And as also mentioned before, Kalman filtering presents the possibility to fuse signals from different sensor types to achieve the desired objectives [2]. Let us consider an instant $\mathit{k}$ of the measured signal. At that time, the prediction of the estimated signal ${\widehat{\mathit{x}}}_{\overline{\mathit{k}}}$ can be calculated considering the Eq. (1), as follow:

$${\widehat{x}}_{\overline{k}}=A{\widehat{x}}_{k-1}$$E1

And the error covariance for the predicted instant $\overline{\mathit{k}}$ will be calculated as shown in Eq. (2).

$$P_{\overline{k}}=A{P}_{k-1}{A}^T+Q$$E2

Where $\mathit{A}$ is the state transition matrix that contains the characteristics of the physical model with dimensions (n x n) and $\mathit{Q}$ is the noise covariance matrix of the state transition (n x n). The next step within the algorithm will be the determination of Kalman gain, which will be recursively calculated within the code for the entire signal length. The Kalman gain can be obtained via Eq. (3).

$$K_k=P_{\overline{k}}{H}^T{\left(H{P}_{\overline{k}}{H}^T+R\right)}^{-1}$$E3

Where $\mathit{R}$ is the covariance matrix of the measurement noise (m x m) and $\mathit{H}$ is the state matrix for the measurement (m x n). With these data, the estimated values and the covariance of the signal error will be finally calculated. The estimate signal $\widehat{\mathit{x}}$ (n x 1) can be determined by Eq. (4) from the measured signal $\mathit{z}$ (m x 1).

$${\widehat{x}}_k={\widehat{x}}_{\overline{k}}+K_k\left(z_k-H{\widehat{x}}_{\overline{k}}\right)$$E4

And the error covariance, through Eq. (5):

$$P_k=P_{\overline{k}}-K_{k}H{P}_{\overline{k}}$$E5

Behind this formulation, there are sophisticated statistical concepts that allow the Kalman filter not only to smooth the signals, but also to estimate the values of different quantities in regions of the structure where we do not have access for instrumentation. It is important to highlight that the matrix $\mathit{H}$ is the responsible to correlate measurements and estimations of different quantities within the algorithm.

So far, the application of the Kalman filter in linear systems has been presented. This methodology cannot be applied when there are time-dependent variations in the state transition matrix $\mathit{A}$ and/or in the state matrix for the measurement $\mathit{H}$, i.e. nonlinear systems. In these cases, the Extended Kalman filter is used to estimate the system outputs. In a simple manner, the next equations present the difference between the conventional (or linear) Kalman filter and the Extended Kalman filter that shall be considered for nonlinear systems.

Thus, the algorithm for the application of the Extended Kalman filter, shall be able to introduce the non-linearities presented by Eqs. (6) and (7). Figure 1 illustrates the diagram used to develop the computational code considering the Extended Kalman filter for the tower model. It is possible to consider a linearization technique through the Jacobian matrix, as follow:

Figure 1.

Representation of the extended Kalman filter algorithm.

$${\mathit{Ax}}_k\Rightarrow f\left(x_k\right);A\equiv \frac{\partial f}{\partial x}\mid {\widehat{x}}_k$$E6

$${\mathit{Hx}}_k\Rightarrow h\left(x_k\right);H\equiv \frac{\partial h}{\partial x}\mid {\widehat{x}}_{\overline{k}}$$E7

The objective of this Chapter is to give a brief explanation about Kalman filter theory and focus on the application of the algorithm in an engineering problem. There are several good references for a deeper explanation about linear and nonlinear Kalman filtering methodology, such as presented in [3, 4].

3. Determination of the natural frequencies of the stay cables

Based on the proposed methodology to estimate the displacement of the tower using EKF, it is first necessary to define a relation between the actual stay cable force and the natural frequencies of this component. This mathematical relation is based on the equation of the uni-dimensional wave propagation in a cable fixed in both extremities. In [5], Steidel proposed an equation for cables with different lengths and cross sections. The steel cables used in the mockup tower have a core fiber, which presents a nonlinear behavior and different dynamic responses when compared to the all-metal cable of the theory presented in [5]. Thus, a verification of these cables was proposed in the Dynamic Testing Laboratory (LabEDin) at UNICAMP, to determine the actual relationship between cable force and natural frequencies.

Figure 2 shows the devices used in LabEDin to apply the forces and determine the natural frequencies of the cable with the same equivalent length of the stay cables of the mockup tower. The load was applied in the cable through a hydraulic cylinder and the acceleration signals were measured from both sensors at the positions of ${\mathit{L}}_{\mathit{e}}/\mathbf{2}$ and ${\mathit{L}}_{\mathit{e}}/\mathbf{4}$ of the cable equivalent length ${\mathit{L}}_{\mathit{e}}\mathbf{=}\mathbf{4.3}\mathit{m}$. It was observed that there are no significant differences to determine the natural frequencies from the PSDs, with or without impacts on the cable.

Figure 2.

Device for testing the stay cable of the mockup tower.

The experiment consists of applying constant loads from 1000 to 2600 N with an increment of 100 N and determine the first three natural frequencies of the cable from the measured signals. The variation of the cable length was also taken into account, as presented in [6], to define the equation that relates the natural frequency as a function of the cable force. The Eq. (8) shows this relationship for the $1/{4^{\prime }}^{\prime }$ diameter steel wire ropes used to clamp the tower.

$${\omega }_n=\frac{0.3027n\sqrt{\frac{F}{\mu }}}{{L_e}^{0.718}}$$E8

where:

${\omega }_n$	Natural frequency of the cable [Hz]
$n$	Mode of vibration [−]
$F$	Tensile force on the cable [N]
$\mu$	Linear mass density [kg/m]
$L_e$	Equivalent cable length [m]

Figure 3 presents the measured natural frequencies of the first three vibration modes of the cable, compared with the values calculated using Eq. (8) as a function of the applied force. It is possible to notice that an accurate approximation of the calculated values was achieved using the equation, which allows considering this procedure to determine the cable forces via accelerometers as input to the EKF algorithm.

Figure 3.

Comparison between the measured and calculated natural frequencies of the first three modes, as a function of the cable load.

4. Dynamic model of the guyed tower

A successful application of the Extended Kalman filter, depends on a well defined system model and the reliability of the outputs is directly related to the ability of the model to reproduce the physics of the actual mechanical structure. The dynamic model of the guyed tower was developed to provide fast responses as function of the force variation on the stay cables.

Figure 4 shows the picture of the tower assembled in LabEDin, indicating the position of the inclinometer, accelerometers and load cells. There is one accelerometer (triaxial) and one load cell (S-type) per stay cable. The load cells are used just to validate the calculation of the cable force using Eq. (8). The structure of the mockup tower was designed based on a 1:5 scale of a power line tower with 22.5 meters height.

Figure 4.

Structure of the mockup tower assembled in LabEDin, with the indication of the sensor positioning.

The dynamic equations of the tower model were defined considering the Lagrange function [7], as presented in Eq. (9). The angles ${\mathit{\theta }}_{\mathbf{x}},{\mathit{\theta }}_{\mathbf{y}},{\mathit{\theta }}_{\mathbf{z}}$ at the bottom point of the tower are considered as generalized coordinates, with $\mathit{i}\mathbf{=}\mathbf{3}$, defining the number of degrees of freedom (DOF) of the model:

$$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial \mathcal{L}}{\partial {\dot{q}}_i}\right)-\frac{\partial \mathcal{L}}{\partial q_i}={\mathcal{Q}}_i$$E9

with:

$$\mathcal{L}=T-U=T_v+T_{\omega }-\left(U_g+U_K+U_{K_t}\right)$$E10

Figure 5 shows the proposed model of the guyed tower. In this figure are presented the main dimensions of the mockup tower and the global system of coordinates. The structure can rotates freely in the three direction at the pivot point represented by the green circle.

Figure 5.

Simplified structure considered to build the dynamic model of the tower for the EKF application.

The kinetic energy shown in Eq. (10) is determined as follows:

$$T_v+T_{\omega }=\frac{1}{2}m{v}_{\mathit{cg}}^2+\frac{1}{2}\left(I_{\mathit{xx}}{\dot{\theta }}_x^2+I_{\mathit{yy}}{\dot{\theta }}_y^2+I_{\mathit{zz}}{\dot{\theta }}_z^2\right)$$E11

Considering small oscillations of the tower, it is possible to define the partial derivatives of the kinetic energy of Eq. (11).

$$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial T}{\partial {\dot{\theta }}_x}\right)=\left(m{z}_{\mathit{cg}}+I_{\mathit{xx}}\right){\ddot{\theta }}_x$$E12

$$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial T}{\partial {\dot{\theta }}_y}\right)=\left(m{z}_{\mathit{cg}}+I_{\mathit{yy}}\right){\ddot{\theta }}_y$$E13

$$\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial T}{\partial {\dot{\theta }}_z}\right)=I_{\mathit{zz}}{\ddot{\theta }}_z$$E14

With the moments of inertia referring to the center of gravity of the model, the total potential energy can be defined according to the Eq. (15).

$$\begin{array}{ll}{U}_g+U_K+U_{K_t}=& mg{z}_{\mathit{cg}}cos{\theta }_{x}cos{\theta }_y+\sum _{j=1}^4\frac{1}{2}{k}_j{\left(L_c-L_{\mathit{cn}}\right)}_j^2+\cdots \\ & \cdots +\frac{1}{2}{k}_{t_x}{\theta }_x^2+\frac{1}{2}{k}_{t_y}{\theta }_y^2+\frac{1}{2}{k}_{t_z}{\theta }_z^2\end{array}$$E15

In our case, it is possible to consider only small oscillations of the tower and disregard the torsional stiffness around the base pivot point, without loss of accuracy in the results. Thus, the total potential energy is equivalent to the elastic potential energy and its partial derivatives are shown in Eq. (16):

$$\frac{\partial U}{\partial {\theta }_{x,y,z}}=\frac{\partial U_K}{\partial {\theta }_{x,y,z}}=\sum _{j=1}^4{k}_j{\left(L_c-L_{\mathit{cn}}\right)}_j\frac{\partial L_{c_j}}{\partial {\theta }_{x,y,z}}$$E16

Finally, based on the definition of Eq. (9), it is possible to determine the system of second order differential equations for the tower motion according to Eqs. (17)-(19):

$$\left(m{z}_{\mathit{cg}}+I_{\mathit{xx}}\right){\ddot{\theta }}_x-\sum _{j=1}^4{k}_j{\left(L_c-L_{\mathit{cn}}\right)}_j\frac{\partial L_{c_j}}{\partial {\theta }_x}=Q_{{\theta }_x}$$E17

$$\left(m{z}_{\mathit{cg}}+I_{\mathit{yy}}\right){\ddot{\theta }}_y-\sum _{j=1}^4{k}_j{\left(L_c-L_{\mathit{cn}}\right)}_j\frac{\partial L_{c_j}}{\partial {\theta }_y}=Q_{{\theta }_y}$$E18

$$I_{\mathit{zz}}{\ddot{\theta }}_z-\sum _{j=1}^4{k}_j{\left(L_c-L_{\mathit{cn}}\right)}_j\frac{\partial L_{c_j}}{\partial {\theta }_z}=Q_{{\theta }_z}$$E19

The numerical values of the tower properties are: $m=90.7\mathit{kg}$, $I_{\mathit{xx}}=269.53\mathit{kg}{m}^2$, $I_{\mathit{yy}}=194.95\mathit{kg}{m}^2$, $I_{\mathit{zz}}=76.53\mathit{kg}{m}^2$. Considering the free vibrations of the tower, the dynamic equations of the movement can be written as shown in Eq. (20):

$$\left\{\begin{array}{c}{M}_x{\ddot{\theta }}_x=-C_x{\dot{\theta }}_x-K_{x}f\left({\theta }_x,{\theta }_y,{\theta }_z\right)\\ \\ M_y{\ddot{\theta }}_y=-C_y{\dot{\theta }}_y-K_{y}f\left({\theta }_x,{\theta }_y,{\theta }_z\right)\\ \\ M_z{\ddot{\theta }}_z=-C_z{\dot{\theta }}_z-K_{z}f\left({\theta }_x,{\theta }_y,{\theta }_z\right)\end{array}\right.$$E20

It is important to highlight that the values in $\mathit{M}$, $\mathit{K}$ and $\mathit{C}$ are dependent on the variations in all DOF of the corresponding axes (${\mathit{\theta }}_{\mathbf{x}},{\mathit{\theta }}_{\mathbf{y}},{\mathit{\theta }}_{\mathbf{z}}$). The Eq. (21) expresses the dynamic behavior of tower, through the first order differential state equations for free vibrations.

$$\dot{x}\left(t\right)=Ax\left(t\right)$$E21

With:

$$x\left(t\right)={\left[{\theta }_x{\theta }_y{\theta }_z{\dot{\theta }}_x{\dot{\theta }}_y{\dot{\theta }}_z\right]}^T$$E22

The time-variant state transition matrix $\mathit{A}$ has the following aspect:

$$A=\left[\begin{array}{cc}0& I\\ -M^{-1}K& -M^{-1}C\end{array}\right]=\left[\begin{array}{cc}{A}_{\left(1,1\right)}& A_{\left(1,2\right)}\\ A_{\left(2,1\right)}& A_{\left(2,2\right)}\end{array}\right]$$E23

And the sub-matrices ${\mathit{A}}_{\left(2,1\right)}$ and ${\mathit{A}}_{\left(2,2\right)}$ can be defined respectively according to Eqs. (24) and (25).

$$A_{\left(2,1\right)}=\left\{\begin{array}{ccc}-\frac{K_{1,1}}{M_x}& -\frac{K_{1,2}}{M_x}& -\frac{K_{1,3}}{M_x}\\ -\frac{K_{2,1}}{M_y}& -\frac{K_{2,2}}{M_y}& -\frac{K_{2,3}}{M_y}\\ -\frac{K_{3,1}}{M_z}& -\frac{K_{3,2}}{M_z}& -\frac{K_{3,3}}{M_z}\end{array}\right\}$$E24

Where ${\mathit{K}}_{\mathit{i}\mathit{,}\mathit{j}}$ depends on the cable characteristics and disposition.

$$A_{\left(2,2\right)}=\left\{\begin{array}{lll}-\frac{C_x}{M_x}& 0& 0\\ 0& -\frac{C_y}{M_y}& 0\\ 0& 0& -\frac{C_z}{M_z}\end{array}\right\}$$E25

5. Tower movements estimate based on EKF algorithm

During 1 year, several tests were carried out in the UNICAMP’s laboratory to simulate the foundation failures of the guyed tower. Systematic relaxation of the stay cables was introduced by unscrewing the tensioners close to each base. Figure 6 presents the cable assembly and the position of the devices. The measurements of the accelerometers, load cells and inclinometer are made in time-domain every 15 minutes. Table 1 shows the specification of the sensors used in the tower monitoring.

Figure 6.

Devices for the force measurements and load application at the stay cable.

Sensor	Model/Characteristics
Accelerometera	PCB Piezotronics® 356A15, ICP®, Triaxial
Load cellb	Alfa Instruments®, S-type, Stainless Steel, Capacity of 500 kg
Inclinometera	Fredericks Company®, 0729-1760-04, Interface RS485 USB, Dual-axis

Table 1.

Specification of the sensors used for the measurements.

^a

EKF estimations consider the sensor fusion from the signals of the accelerometers and inclinometer.

^b

Load cells are used only to validate the forces calculated indirectly by the accelerometers.

The signal acquisition of all sensors was performed using LMS SCADAS Mobile SCR05® manufactured by Siemens, which has a total of 40 ADC channels. The system was configured with a frequency sampling rate of 256 Hz and an acquisition time of 64 seconds for each measurement. Every sensor generates signals with 16,384 points per channel and Figure 7 presents the picture of the data acquisition system.

Figure 7.

Simens LMS SCADAS® used for the signal acquisition of the mockup tower.

5.1 Definition of the extended Kalman filter parameters

The variation of the stay cable force promote strain variations, which are converted into displacements projected on plane $\left(X,Y\right)$ at the connecting points of two cables to the tower, i.e. Points 5 and 5′ presented in Figures 5 and 8. The Eqs. (26) and (27) present the formula to calculate the displacements based on the cable force determined by the measured natural frequency.

Figure 8.

Dynamic model of the tower. Top view details.

$${\delta }_5=\left[\left(\frac{F_2}{k_2}+L_{{\mathit{cn}}_2}\right)-L_{f_2}-\left(\frac{F_1}{k_1}+L_{{\mathit{cn}}_1}\right)+L_{f_1}\right]÷2$$E26

$${\delta }_{5^{\prime }}=\left[\left(\frac{F_3}{k_3}+L_{{\mathit{cn}}_3}\right)-L_{f_3}-\left(\frac{F_4}{k_4}+L_{{\mathit{cn}}_4}\right)+L_{f_4}\right]÷2$$E27

Where: $F_j$ = Force at cable $j$.

The stiffness of the cable presented in Eqs. (26) and (27) can be calculated according to Eq. (28) and considering the characteristics of the steel wire rope used in the mockup tower ($\mathrm{\Phi }$=$14$″, class 6×7, fiber core), the equivalent elasticity modulus and the cross section area are respectively 88,200 MPa and 16.1792 mm². This information can be verified in [8].

$$k_j=\frac{E_c{A}_c}{L_{f_j}}$$E28

With the displacements of these points it is possible to define the rotation angles of the structure in respect to $X,Y$ axes. The displacements not related to variations in the average cable force, e.g. moderate wind condition, can be quantified according to Eqs. (29)-(31). These angles correspond to the indirect measured elements of vector ${\mathit{z}}_{\mathit{\delta }}$, considering ${\overline{\mathit{\delta }}}_{\mathbf{5}}$ as the average displacement of Points 5 and 5′.

$$z_{{\delta }_x}=arcsin\left[\frac{{\overline{\delta }}_{5}cos\left(90-\gamma \right)cos\left(180-\alpha \right)}{h}\right]$$E29

$$z_{{\delta }_y}=arcsin\left[\frac{{\overline{\delta }}_{5}cos\left(90-\gamma \right)sin\left(180-\alpha \right)}{h}\right]$$E30

$$z_{{\delta }_z}=arcsin\left[\frac{{\overline{\delta }}_{5}cos\left(90-\gamma \right)sin\left(180-\alpha \right)}{a}\right]$$E31

On the other hand, the stay cable relaxation promotes a decrease in average force $\overline{\mathit{F}}$ and this reduction is also responsible to generate proportional rotations of the tower in respect to $X,Y$ axes. From the mathematical model of the tower we can define the Eqs. (32) and (33), which present these relations. It is possible to observe, that the equations indicate zero inclinations when the average force is approximately 2600 N, which is the assembly preload of the cables.

$$z_{F_x}=4.32\cdot {10}^{-3}-1.659\cdot {10}^{-6}\overline{F}$$E32

$$z_{F_y}=4.91\cdot {10}^{-3}-1.883\cdot {10}^{-6}\overline{F}$$E33

The sensor fusion technique is shown in Figure 9. It presents the input measurements consisted by the inclinometer angles ${\mathit{z}}_{{\overline{\mathit{\theta }}}_{\mathit{x}}},{\mathit{z}}_{{\overline{\mathit{\theta }}}_{\mathit{y}}}$ and the stay cable forces $\overline{\mathit{F}}$, which are calculated from the natural frequencies ${\mathit{\omega }}_{\mathbf{n}}$ determined by the accelerometer measurements.

Figure 9.

Sensor fusion considered to estimate the tower movements.

A question that often comes up is why not just measure angles using the inclinometer to monitor the structural health of the tower. The reason is that there may be a situation where two symmetrically opposite guyed cables exhibit approximately the same amount of relaxation. In this case, the position of the top center of the tower presents practically no angles measured by the inclinometer in $X,Y$ directions, but the average stay cable force will decrease. This situation can lead to an unstable structural condition with high-intensity winds affecting the tower. Thus, only inclination angles of the guyed tower cannot inform the actual integrity of its structure.

According to the presented methodology, the extended Kalman filter can only estimate the absolute value of the displacement. Fusing this information with the inclinometer data, it is possible to define the direction of this displacement and it also helps to indicate the cable that presented the failure. The tower inclinations due to the thermal expansion of the structure can be verified by the consideration of the inclinometer measurements.

The state transition matrix changes with rotation angles ${\mathit{\theta }}_{\mathbf{x}}$, ${\mathit{\theta }}_{\mathbf{y}}$ and ${\mathit{\theta }}_{\mathbf{z}}$, which are estimated for each interaction of the algorithm after the calculation of the cable forces from the measured natural frequencies. In our case, the estimate vector is defined according to:

$${\widehat{x}}_k={\left[\begin{array}{cccccc}{\widehat{\theta }}_x& {\widehat{\theta }}_y& {\widehat{\theta }}_z& {\widehat{\dot{\theta }}}_x& {\widehat{\dot{\theta }}}_y& {\widehat{\dot{\theta }}}_z\end{array}\right]}^T$$E34

In the sequence, the matrices considered in the EKF algorithm will be presented. These matrices are determined from known analytical results of the tower dynamic model, i.e., by applying forces at the tower structure and considering its responses as targets for EKF estimates. The state matrix for the measurements:

$$H=\left[\begin{array}{cccccc}0.50& 0& 0& 0& 0& 0\\ 0& 0.47& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\end{array}\right]$$E35

The noise covariance matrix of the state transition:

$$Q={10}^{-3}\cdot \left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right]$$E36

The covariance matrix of the measurement noise:

$$R=5\cdot \left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$E37

Note: Null values were adopted as initial conditions for EKF estimate and error covariance.

The Eq. (38) shows the measurement vector at instant $k$ considered as an input to the EKF, considering the weighting factors ${\mathrm{\Psi }}_{\delta },{\mathrm{\Psi }}_F,{\mathrm{\Psi }}_{\overline{\theta }}$ for each sensor quantity.

$$z_k={\mathrm{\Psi }}_{\delta }\left[\begin{array}{c}{z}_{{\delta }_x}\\ z_{{\delta }_y}\\ z_{{\delta }_z}\end{array}\right]+{\mathrm{\Psi }}_F\left[\begin{array}{c}{z}_{F_x}\\ z_{F_y}\\ 0\end{array}\right]+{\mathrm{\Psi }}_{\overline{\theta }}\left[\begin{array}{c}{z}_{{\overline{\theta }}_x}\\ z_{{\overline{\theta }}_y}\\ 0\end{array}\right]$$E38

In our case, the given weights to the cable force data are ${\mathrm{\Psi }}_{\delta }=0.48$; ${\mathrm{\Psi }}_F=0.10$ and the weight for the inclinometer measurements is ${\mathrm{\Psi }}_{\overline{\theta }}=0.42$.

5.2 Experimental procedure

Now will be presented some simulation of failures, which can occur in the real tower due to the degradation of the foundation, caused as an example, by the erosion of the soil. The tensioners of two symmetrically opposite cables were unscrewed, promoting the relaxation of the stays. Each turn of the tensioner generates a displacement of 4.20 mm along the direction of the cable.

Initially, the stay cables are fixed with an average force of 2600 N with the inclinometer indicating zero degree with respect to $X$ and $Y$ axes. Figure 10 presents the sequence of events to estimate the position of the top center of the tower. It consists of 96 measurements performed during every 24 h, when some manual intervention was made on the cable tensioners.

Figure 10.

Locations of the interventions to simulate the failures.

In this case, after the relaxation of Cable 1, the EKF estimates the position of the top center of the tower. Subsequently, the tensioner was re-tightened to its initial position and then, the tensioner of Cable 3 was released for new estimates of the tower position. In a real situation, this test represents two failures separated by one tower maintenance intervention.

After 4 days, the test was completed and Figure 11 compares the average force on the stay cables measured with the load cells, with the average force calculated from the natural frequencies obtained from the accelerometer measurements.

Figure 11.

Comparison between average cable forces: measured with the load cells and calculated with Eq. (8).

A top view of the tower showing the numbering of the support cables, the considered coordinate system and the estimated displacements in the central region based on Extended Kalman filter can be seen in the Figure 12.

Figure 12.

Tower top view showing the stay cable numbering and the considered coordinate system. In the central region there are the estimate displacements.

A detailed view of the estimated displacements for each different condition imposed to the mockup tower can be seen in Figure 13. It is possible to observe three groups of points. The central group of points (in a red ellipse) shows the top position for the stay cables in normal conditions, i.e. immediately after assembly with an average force of approx. 2600 N. On the left, it is possible to see a group of points inside of a dashed green ellipse, which presents the positions after the release of the Cable 1 tensioner by one turn. As expected for this case, the top center moves slightly to the location symmetrically opposite of the foundation of Cable 1.

Figure 13.

Detail of the estimate displacements of the tower top center based on the extended Kalman filter algorithm.

Starting from the second normal condition of the tower (after re-tightening the tensioner of Cable 1), there was a release of two turns in the tensioner of Cable 3, that moves slightly the top center position to the symmetrically opposite location of the Cable 3 foundation and these points are shown within a dashed cyan ellipse on the right side of the figure.

Continuing the analysis of Figure 13, it is also possible to observe the variations of the top position within each group of points, mainly along X-axis direction. These variations are due to the fluctuations in lab temperature during the 24 hours of each test. Basically, when the temperature increases the tower bends in the +X direction, while at night with the decrease in temperature the tower bends in the -X direction. The non-perfect symmetry of the mockup tower and the oblique incidence of sun rays during the day, contribute to the lateral movements in the Y-axis direction.

Figure 14 summarizes the adopted procedure, which starts by considering the accelerometer measurements in time-domain and ends with the EKF estimate for the tower top position. It is important to note that the code is able to evaluate the structural health of the guyed tower, not only for large movements of the top, but also for low average force on the stay cables.

Figure 14.

Procedure to estimate of the tower top center position starting from cable acceleration measurements.

6. Conclusions

The EKF was considered to estimate the displacements of the upper part of a mockup of a guyed tower in the laboratory. A MATLAB code was developed to execute the algorithm that indicates the behavior of the structure with a very good accuracy.

Stay cable relaxation is the type of failure that is judged to be more usual in the field. This situation was simulated extensively with good approximation between real and estimate values. The equation defined to establish the relation between the natural frequency and the cable force has indicated to be a reliable procedure, as an alternative substituting load cells to evaluate variations of the cable force over the time. The load cell is recommended only for initial measurements of the stay cable preload, for the adjustment of the cable equivalent length. After this step, the cable force can be monitored only by the accelerometers.

The influence of the transmission line conductors in the dynamic responses of the structure of the tower is considered negligible and due to this fact, the mockup tower in the laboratory does not have elements that simulate the influence of the conductors. According to [9], the electrical insulators promote some decoupling of the eolian vibrations that excite the transmission lines.

Sensor fusion considering the accelerometers and inclinometer, proved to be the best option to indicate the structural integrity of the mockup tower considering the stay cable relaxation, responses to external loads and tower thermal expansion. The individual consideration of those sensors cannot provide a reliable report about the guyed tower health. As explained previously, the inclinometer alone cannot indicate the inclination of the structure if two symmetrically opposite cables are losing preload. From the obtained results, the behavior of the real guyed towers is expected to be very similar to those simulated in the laboratory.

Conflict of interest

The authors declare no conflict of interest.

Acknowledgments

The authors are grateful to R&D Project PD-07130-0047 “Evaluation of Multivariable Modeling Methods for Monitoring the Health of Guyed Towers in Overhead Power Lines”, partially funded by: Transmissora Aliança de Energia Elétrica S.A. (TAESA), with resources from ANEEL R&D Program.

Nomenclature

x̂k	estimated value at instant k
Pk¯	error covariance at instant k
A	state transition matrix
Q	noise covariance matrix of the state transition
Kk	Kalman gain at instant k
R	covariance matrix of the measurement noise
H	state matrix for the measurement
zk	measured signal at instant k
zδ	rotation signals from the cable strain
zF	rotation signals from average the cable force
zθ¯	rotation signals from the inclinometer
ωn	natural frequency
n	mode of vibration
Le	equivalent cable length
F	cable tensile force
μ	cable linear mass density
Tv	translational kinetic energy
Tω	rotational kinetic energy
Ug	gravitational potential energy
UK	linear elastic potential energy
UKt	torsional elastic potential energy
Q	external load
vcg	velocity of the tower’s center of gravity
zcg	height of the tower’s center of gravity
m	mass of the tower structure
g	gravitational acceleration
Ixx,yy,zz	moments of inertia referred to the center of gravity
θ	tower angles (in x,y,z axes)
M	mass matrix
C	damping matrix
K	stiffness matrix
k	stay cable stiffness
kt	tower torsional stiffness at pivot base point
h	height of the tower
a	half length of the tower upper structure
Lc	stay cable instantaneous length
Lcn	stay cable assembly length with preload
Lf	stay cable free length
Ec	equivalent Young modulus of the cable
Ac	equivalent cross sectional area of the cable
Ψδ,F,θ¯	weighting factors for the measured quantities

Abbreviations

DOF	degrees of freedom
EKF	extended Kalman filter
IoT	Internet of Things
LabEDin	dynamic testing laboratory
PSD	power spectrum density
RF	radio frequency
UNICAMP	University of Campinas