Strain Virtual Sensing for Structural Health Monitoring under Variable Loads

Citation: Mora, B.; Basurko, J.; Sabahi, I.; Leturiondo, U.; Albizuri, J. Strain Virtual Sensing for Structural Health Monitoring under Variable Loads. Sensors 2023,23, 4706. https://doi.org/10.3390/s23104706 Academic Editor: Aldo Minardo Received: 4 April 2023 Revised: 8 May 2023 Accepted: 9 May 2023 Published: 12 May 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). sensors Article Strain Virtual Sensing for Structural Health Monitoring under Variable Loads Bartomeu Mora 1,2,* , Jon Basurko 1, Iman Sabahi 3, Urko Leturiondo 1and Joseba Albizuri 2 1Ikerlan Technology Research Centre, Basque Research and Technology Alliance (BRTA), 20500 Arrasate-Mondragon, Spain; [email protected] (J.B.); [email protected] (U.L.) 2Faculty of Engineering in Bilbao, University of the Basque Country (UPV/EHU), 48013 Bilbao, Spain; [email protected] 3KU Leuven, Department of Mechanical Engineering, B-3001 Leuven, Belgium; [email protected] *Correspondence: [email protected] Abstract: Virtual sensing is the process of using available data from real sensors in combination with a model of the system to obtain estimated data from unmeasured points. In this article, different strain virtual sensing algorithms are tested using real sensor data, under unmeasured different forces applied in different directions. Stochastic algorithms (Kalman filter and augmented Kalman filter) and deterministic algorithms (least-squares strain estimation) are tested with different input sensor configurations. A wind turbine prototype is used to apply the virtual sensing algorithms and evaluate the obtained estimations. An inertial shaker is installed on the top of the prototype, with a rotational base, to generate different external forces in different directions. The results obtained in the performed tests are analyzed to determine the most efficient sensor configurations capable of obtaining accurate estimates. Results show that it is possible to obtain accurate strain estimations at unmeasured points of a structure under an unknown loading condition, using measured strain data from a set of points and a sufficiently accurate FE model as input and applying the augmented Kalman filter or the least-squares strain estimation in combination with modal truncation and expansion techniques. Keywords: structural health monitoring; virtual sensing; Kalman filter; augmented Kalman filter; least squares estimation; strain virtual sensor 1. Introduction Structure health monitoring (SHM) involves monitoring structures to determine their current condition. The use of SHM systems increases the safety of structural facilities and allows the optimization of the maintenance actions, predicting the remaining useful life of critical components and detecting anomalies that may indicate the presence of damage [ 1 ]. SHM systems require measured data from the structure using sensors, but it is not always possible to install all the necessary sensors at all the points of interest, either for technical or economic reasons. Virtual sensing (VS) allows obtaining measures from a system, not directly from physical sensors, but using data inference from other sensors [ 2 ]. The use of vs. in SHM systems results of interest takes place when it is necessary to obtain measurement data at points where it is not technically feasible to locate a real sensor, or when it is necessary to obtain measurements at a large number of locations, requiring a sensor network that is too extensive [ 3 ]. In conclusion, the use of vs. offers technical and economic advantages. VS techniques can be classified into two main groups: data-driven techniques and model-based techniques [ 4 ]. Model-based techniques require a physics-based model capable of replicating the behavior of the monitored system. The model-based methods can be further classified into two groups: stochastic, where the system uncertainties are considered; and deterministic, where the system uncertainties are not considered [5,6]. Sensors 2023,23, 4706. https://doi.org/10.3390/s23104706 https://www.mdpi.com/journal/sensors Sensors 2023,23, 4706 2 of 19 Neural networks (NN) are commonly used in the data-driven vs. approaches. The NN are artificial intelligence algorithms that consist of complex networks of nodes (neurons) adjusted using training data, which relates the provided input data with the desired outputs. Artificial neural networks (ANNs) [ 7 ] and Convolutional neural networks (CNNs) [ 8 ] have been used for vs. applications. ANNs are simpler, because inputs are processed only in the forward direction, while CNNs are more complex because they use multiple types of layers for processing the provided input data. Stochastic vs. algorithms have been commonly used for the estimation of states. One of the most known stochastic estimation algorithms is the Kalman filter (KF), a physics model-based algorithm proposed by R. Kalman in 1960 [ 9 ]. The KF uses a state-space model of the system to make state predictions (mean and covariance) based on information from the previous states. The KF is then a Bayesian estimator [ 10 ]. Input data from real sensors are used to correct the predictions and to update the algorithm parameters. Some examples of the use of the KF for strain estimations are found in [ 11 , 12 ]. An implementation of the KF with an augmented state-space model (which estimates the inputs together with the states of the system) was first proposed in 1969 by B. Friedland, to perform a state estimation with unknown inputs [ 13 ]. In 2010, E. Lourens et al. used the KF with the augmented state-space for dynamic force identification, and the following year they consolidated the so-called augmented Kalman filter (AKF), which has been used in later publications [ 14 ]. Other variants of the KF for nonlinear systems have been proposed: for example, the Extended Kalman filter (EKF) [15], that performs a linearization of the estimated mean and covariance for each time step, or the Unscented Kalman filter (UKF) [ 16 ], that avoids linearization by applying an unscented transformation to the estimated mean and covariance. An alternative to the Kalman filter is the Particle filter (PF) [ 17 ], which are also stochastic Bayesian estimators. For each time step, the PF generates multiple random estimations (particles) using Monte Carlo simulations. A weight is assigned to each particle, and the closest particles to the observation measurements are more weighted for the following time-steps. Deterministic vs. algorithms have been used for force and strain estimation [ 3 , 18 , 19 ]. In this article, the least-squares state estimation (LSSE) is used. This method uses a MoorePenrose pseudoinverse (a generalization of the matrix inverse which allows to obtain the pseudo-inverse matrix of a non-squared matrix [ 20 ]) to obtain the least-squares solution of the unknown strains [ 21 ]. Unlike the probabilistic methods, such as the previously described Kalman filters, the LSSE does not use the information of previous states and does not update its internal parameters to improve the estimation. In this article, both stochastic and deterministic model-based algorithms are tested. The classic KF and the AKF are used as examples of stochastic methods and the LSSE is used as example of deterministic method. The AKF has been chosen because it is specifically designed to work without information of the external forces (which is of great interest for the work developed in this article), while the KF has been chosen to compare it with the AKF under the unknown external forces condition. On the other hand, the LSSE has been chosen due to its simplicity. EKF and UKF have been discarded because it is not intended to work with nonlinear models; meanwhile, the PF has been discarded due to its much higher computational complexity. NN algorithms are not used in this work because it is intended to avoid providing substantial amounts of training data. In recent years, Kalman filter and variants [ 22 , 23 ] and deterministic algorithms [ 24 , 25 ] have been used in SHM systems applied to structural facilities, such as wind turbines or bridges. In [ 22 ], the AKF is applied in a wind turbine to estimate the state of the structure and the external wind forces, using the turbine speed and the generator torque, together with accelerometer data, as input sensors. In [ 23 ], the KF algorithm is used for damage detection in aircraft frames and bridges, using accelerometers as input sensors. In [ 24 ], modal expansion is used for stress and strain estimation in an offshore structure prototype, using accelerometers and strain gauges as input sensors. In [ 25 ], modal expansion is used for strain estimation in a monopile offshore wind turbine, using accelerometers and strain gauges as input sensors too. Sensors 2023,23, 4706 3 of 19 In order to work with strain measurements using the mentioned algorithms, as well as to obtain strain estimates from them, it is necessary to use the modal expansion/reduction method [ 18 , 26 ]. This method allows a number of strain measurements to be related to displacements in a model, and vice versa. Model reduction methods are also used to obtain lighter models from complex FE models, capable of being used by the vs. algorithms. Strain estimation is of interest due to its relationship with fatigue: by estimating the strain at critical points, the remaining useful life of a structure due to the accumulated fatigue can be estimated. The main contribution of this article is to test different vs. algorithms (stochastic and deterministic) using real data obtained from a wide variety of experimental tests, obtained from an offshore wind turbine scaled prototype. For each selected algorithm, different input sensor configurations have been tested under different types of external forces applied in different directions (using an electromagnetic shaker installed on a rotating base on the top of the prototype), simulating variable loads on the prototype. The vs. algorithms are tested without measuring the applied forces, increasing the difficulty of the study. This article is organized as follows: in Section 2, the modeling processes, the virtual sensing algorithms used, and the use case are described. In Section 3, the obtained experimental results are shown. In Section 4, the results are discussed and in Section 5, the final conclusions are presented. 2. Materials and Methods 2.1. System Modeling In this subsection, the theoretical bases used on system modeling are described: massdamper-spring equation, state-space formulation, and model discretization. The selected model reduction method, modal truncation, is also described. 2.1.1. Finite Element Model A Finite Element (FE) linear model of the monitored structure is created. Geometry, construction details and boundary conditions must be taken into account during the model creation. Mathematically, a FE model is defined by the mass-damper-spring second order differential Equation (MCK) Equation (1), that is able to describe the dynamical behavior of the model over time. M.. q(t)+CD . q(t)+Kq(t)=f(t)(1) With n being the number of degrees of freedom (DoFs) of the model, q (t) is the displacement vector (with n × 1 dimension), M , CD and K are the stiffness, damping and mass matrices, respectively (with n × ndimension), and f (t) is the external forces vector (with n×1 dimension). The FE models of complex structures contain a large number of degrees of freedom (DoF), which implies that a big processing capacity and large amounts of time are needed to work with them. To remedy this issue, reduction methods need to be applied. 2.1.2. Model Reduction By applying model reduction methods to a full FE model, it is possible to obtain models with a much smaller number of DoFs, which are much lighter in terms of computation. It is a necessity when it is intended to work with FE models that represent complex structures (usually made of thousands or even millions of DoFs) and it is required to perform a high number of calculations over time (for example, a transient simulation) [ 27 ]. The reduced models can reproduce the dynamic behavior of the structure in limited ranges of use. Several model reduction methods can be found in the bibliography. Some examples are the Guyan static condensation [ 28 ], the improved reduced system (IRS) [ 29 ], the CraigBampton component mode synthesis [ 30 ] and the modal truncation [ 31 ]. In this article, the modal truncation is selected as model reduction method because it is a method that allows to maintain a great precision from the full model, within a defined range of use, and due to its simplicity of application [32]. Sensors 2023,23, 4706 4 of 19 To introduce the modal truncation method, first it must be explained that a dynamic model can be described through its mode shapes, using the mode-shapes matrix ( Φ ). Each column of Φ corresponds to an eigenvector ( ϕi ), associated to an eigenvalue ( λi ). The square root of every eigenvalue corresponds to a natural frequency of the system ( ωi ). The Φ -transformation implies a change of domain for the model, from the physical domain (with cartesian base) to the modal domain. Φ can be obtained solving the undamped Equation (2), discarding the trivial solution Φ = 0 . Φ is considered mass-normalized when expression (3) is satisfied. (K−λM)Φ=K−ω2MΦ=0(2) ΦTMΦ=I(3) In its full form, Φ contains as many mode shapes as DoFs of the full model, but it is possible to reduce the model removing the modes out of the frequency range of interest (modal truncation). For a knumber of modes of interest, Φ is reduced to ΦK (4), with its dimension reduced to n×k. Φ(n,k)=[ϕ1,ϕ2. . . ϕk](4) Through the Φ -transformation, the dynamic Equation (1) can be transformed into the generalized dynamical Equation (5), where z (t) is the vector of modal displacements (also known as generalized displacements), obtained with the transformation q (t) = ΦK z (t). Equation (5) can also be expressed as (6), ΦKTMΦK being an identity matrix, Σ a diagonal matrix containing the damping ratios ( ξ ) associated with each frequency, and Ωthe diagonal matrix with the natural frequencies of the model (ω). ΦkTMΦk .. z(t)+ΦkTCDΦk . z(t)+ΦkTKΦkz(t)=ΦkTf(t)(5) .. z(t)+2ΣΩ . z(t)+Ω2z(t)=ΦkTf(t)(6) 2.1.3. State-Space Model A MCK model can be described as a state-space system (7), that consists of two equations: the state Equation (above) and the output Equation (below). x is the state vector, with 2n × 1 dimension. As shown in (8), the state vector contains the displacements and the velocities of each DoF. u is the input vector, and with n × 1 dimension, contains the possible external input for each DoF. A and B are the state and input matrices, respectively. As it seen in (9) and (10), the dimensions of these matrices are 2n × 2nand 2n × n, respectively. The elements of the output equation, the output vector yand the output and feedthrough matrices C and D , change according to the desired output variables. State-space notation is required to implement the model in Kalman filters and variants. . x=Ax +Bu y=Cx +Du (7) x=q . q(8) A=0 I −M−1K−M−1CD(9) B=0 M−1(10) To use the state-space model in a discrete-time approach, the Aand Bmatrices must be discretized. Ad (11) and Bd (12) are the discretized versions of the state-space model matrices. Ad=eA∆t(11) Sensors 2023,23, 4706 5 of 19 Bd=A−1(Ad−I)B(12) 2.2. Virtual Sensing Algorithms In this subsection, the selected vs. algorithms in this article are described: the Kalman filter, the Augmented Kalman filter and the least-squares strains estimation. The observability conditions of each algorithm are also described. 2.2.1. Kalman Filter The KF is a Bayesian recursive algorithm used to estimate the hidden states of a system. A state-space model of the system is used to make predictions of the states, and information coming from a limited number of real sensors is used to correct the predictions. The KF is an algorithm of stochastic nature that manages gaussian uncertainties associated with the used model and with the measurements. Q (13) is the covariance matrix of the model (with 2n × 2n dimension) and R (14) is the covariance matrix of the input sensors (with r × r dimension, being r the number of input sensors). Assuming that the states and the measurements are not correlated with each other, the matrices Q and R are simplified to diagonal matrices, where each value of the diagonal corresponds to the uncertainty associated with each state (q) and with each sensor input (r), respectively. The Qmatrix must be discretized when used in a discrete-time Kalman filter (15). Q=diag(q1, q2. . . , q2n)(13) R=diag(r1, r2. . . , rr)(14) Qd=(AdQAdT∆t (15) In absence of external force measurements, the KF is implemented as follows: states prediction (16), covariance prediction (17), Kalman gain determination (18), states prediction update (19) and covariance prediction update (20). xt=Axt−1(16) Pt=APt−1AT+Q(17) Kt=PtHTHPtHT+R−1(18) xtupdated =xt+Kt(zt−Hxt)(19) Ptupdated =Pt−KtHPt(20) The incorporation of the real sensor measurements into the filter (described in states prediction update step) is performed with the measurement matrix ( H ). This matrix relates each measurement with their corresponding states. It has r × 2ndimension, being 2nthe number of states of the system and r the number of input sensors. With a measurement data vector z (t) containing xnumber of strain gauges and ynumber of accelerometers, the H matrix is built as seen in (21). To relate the strain gauge data to the modal states, the modal strains are obtained from the FE model. These can be obtained from a modal analysis of the FE model of the structure, compiling the strain value ( ε ) obtained in each gauge (1 to x) for each mode (1 to n). To relate the accelerometer data to the modal states, the corresponding rows of modal M,Cand Kmatrices are used. Because of the external force measurements are not available, no relation between the force and acceleration measurements is implemented (in the case that external force measurements were available, these would be related to the acceleration measurements through a J matrix (22)). Because of this, the uncertainty of accelerometer measurements is expected to be greater. Sensors 2023,23, 4706 6 of 19 H=           ε1,1· · · ε1,n . . ..... . . εx,1· · · εx,n 0· · · 0 . . ..... . . 0· · · 0 −M1,1 −1K1,1· · · −M1,n −1K1,n . . ..... . . −My,1 −1Ky,1· · · −My,n−1Ky,n −M1,1 −1C1,1· · · −M1,n −1C1,n . . ..... . . −My,1 −1Cy,1· · · −My,n−1Cy,n           (21) J=           0· · · 0 . . ..... . . 0· · · 0 M1,1 −1· · · M1,n −1 . . ..... . . My,1 −1· · · My,n−1           (22) In a KF, observability can be defined as the capacity of the algorithm to obtain enough information from the real system (through the input sensors and the observation matrix) to be able to estimate all the states. To determine if a KF is observable, the observability matrix O (23) is calculated using the transpose of A . Only if the rank of O is equal to 2n (the number of states of the model) is the KF is fully observable. O=     ATH0 ATH1 . . . ATH2n−1      (23) 2.2.2. Augmented Kalman Filter The AKF is a variant of the KF in which the external forces over the system are considered additional states of the model. Thanks to this feature, this filter does not need the external force applied on the monitored system as input. The AKF uses an augmented state-space model of the system that combines the A and B matrices of the state-space model in a single matrix A* (24) with (2n+n f ) × (2n+n f ) dimension (n f being the number of expected external forces), and an augmented vector of states x * (25) that combines the displacements, their first derivatives and the external input forces (resulting in a 2n+n f dimension). The discretization of A* is shown in (26). A*=A B 0 0(24) x*=  q . q u  (25) Ad*=AdBd 0 I (26) The unknown input is modeled as a zero-mean random walk model, so the covariance matrix of the model Q must be augmented to (2n+n f ) × (2n+n f ) dimension by adding a term related to the uncertainty associated to the external forces (27). Q*=Qd0 0 Qu(27) An augmented observation matrix H* (28) must be defined by combining the observation matrix H (21) and the input observation matrix J (22), resulting in a matrix of r × (2n+n f ) dimension. Sensors 2023,23, 4706 7 of 19 H*=H J(28) In the AKF, observability has the same meaning as in the classical KF. To determine if an AKF is observable, the observability matrix O* (29) must be calculated. Only if the rank of O* is equal to 2n +nfis the AKF is fully observable. O*=     A*TH*0 A*TH*1 . . . A*TH*2n−1      (29) 2.2.3. Least-Squares Strain Estimation (LSSE) The LSSE is a deterministic virtual sensing algorithm that uses a matrix generalized inversion to obtain the least squares solution of the unknown strains. The Moore-Penrose pseudoinverse [ 20 ] and the Modal Expansion [ 24 ] are used for this purpose. This method allows obtaining strain estimates at unmeasured points both in the presence and absence of dynamic effects. The linear equation is stated by relating the measured strain and the modal displacements of the system (30). In a linear system, displacements x (t) and measured strains zi (t) are linearly related through the modal strain matrix Gi (with g × mdimension, gbeing the number of strain measurements, and m the number of modal displacements). zi(t)=Gix(t)(30) Using the same statement, strain virtual measurements zvs (t) can be obtained from the modal displacements, through the modal strain matrix Gvs (with o × mdimension, obeing the number of virtual strain sensors, and mthe number of modal displacements) (31). zvs(t)=Gvsx(t)(31) Using the pseudoinverse of Gi , both statements can be combined to obtain strain virtual measurements from a set of real strain measurements (32). zvs(t)=GvsGi+z(t)(32) If the number of strain measurements gis equal to the number of modal displacements m, the statement (31) is determined, and the solution is found by the LSSE. If gis higher than m, the statement is overdetermined. If, on the contrary, gis lower than m, the statement (31) is underdetermined. In both cases, the LSSE gives a best-fit approximation of the solution. To provide a good approximation of the solution, the condition number of the matrix Gi must be close to 1. If the condition number of Gi is high, the statement (31) is ill conditioned and significant errors can be expected in the solution. 2.3. Virtual Sensing Implementation The selected vs. algorithms are tested on a use case defined in Section 2.4. First, an FE model of the use case is created. This model is used to choose the location of the sensors (strain gauges and accelerometers) in the real prototype. Measurement data obtained from the sensors is first used to adjust and validate the model, and then to feed the vs. algorithms. The obtained estimations are compared to the equivalent measurement data to evaluate the performance of the vs. algorithms under the different conditions. The entire process is summarized in the flowchart shown in Figure 1. Sensors 2023,23, 4706 8 of 19 Sensors 2023, 23, x FOR PEER REVIEW 8 of 20 evaluate the performance of the vs. algorithms under the different conditions. The entire process is summarized in the flowchart shown in Figure 1. Figure 1. Flowchart of the process followed to implement and test vs. algorithms. 2.4. Use Case In this subsection, the use case, the installed sensors, and the modelling process are described. 2.4.1. Prototype Description The use case is a scaled wind turbine tower prototype installed on a jacket-type structure, which is fixed to the ground (Figure 2). The main specifications of the prototype can be seen in Table 1. An electromagnetic inertial shaker, considered as part of the system, is placed on top of the prototype attached to a rotating platform to excite the structure in different directions (Figure 3) and frequency components (<25 Hz). The specifications of the shaker used can also be seen in Table 1. Figure 1. Flowchart of the process followed to implement and test vs. algorithms. 2.4. Use Case In this subsection, the use case, the installed sensors, and the modelling process are described. 2.4.1. Prototype Description The use case is a scaled wind turbine tower prototype installed on a jacket-type structure, which is fixed to the ground (Figure 2). The main specifications of the prototype can be seen in Table 1. An electromagnetic inertial shaker, considered as part of the system, is placed on top of the prototype attached to a rotating platform to excite the structure in different directions (Figure 3) and frequency components (<25 Hz). The specifications of the shaker used can also be seen in Table 1. Sensors 2023, 23, x FOR PEER REVIEW 8 of 20 evaluate the performance of the vs. algorithms under the different conditions. The entire process is summarized in the flowchart shown in Figure 1. Figure 1. Flowchart of the process followed to implement and test vs. algorithms. 2.4. Use Case In this subsection, the use case, the installed sensors, and the modelling process are described. 2.4.1. Prototype Description The use case is a scaled wind turbine tower prototype installed on a jacket-type structure, which is fixed to the ground (Figure 2). The main specifications of the prototype can be seen in Table 1. An electromagnetic inertial shaker, considered as part of the system, is placed on top of the prototype attached to a rotating platform to excite the structure in different directions (Figure 3) and frequency components (<25 Hz). The specifications of the shaker used can also be seen in Table 1. Figure 2. General view of the prototype and the concrete foundation. Sensors 2023,23, 4706 9 of 19 Sensors 2023, 23, x FOR PEER REVIEW 9 of 20 Figure 2. General view of the prototype and the concrete foundation. Figure 3. Inertial shaker Data Physics IV47 attached on top of the prototype. Table 1. Main specs of the prototype and the shaker. Feature Value Tower + nacelle weight 42 kg Jacket weight 13.5 kg Shaker + support weight 27 kg Total weigh t 82.5 kg Tower height 1790 mm Jacket height 1300 mm Total height 3090 mm Material Steel Supports Fixed to a concrete base Shaker model Data Physics IV47 Inertial mass 14.5 kg Max sinus force (peak) 250 N Total shaker mass 21 kg Shaker main mode freq. 20 Hz 2.4.2. FE Model and Model Reduction A FE model of the prototype is built based on the 3D CAD of the structure. The shaker and its support, including the bushing, are simplified to an equivalent point mass located at the mass center of the replaced components and attached to the structure. The behavior of the bushing has been tested in the frequency range of interest (0 to 25Hz), verifying its linearity. The platform has been designed to keep the mass center of the rotating components in the rotating axis, so the system can be considered invariant. The bolted joints present in the prototype are also simplified using bonded contacts. Due to the tower and nacelle of the wind turbine prototype being thin steel profile components, shell-type elements have been used to reduce the total number of elements in the mesh. In the jacket support structure, solid elements have been used. The primary features of the FE model can be seen in Table 2, and the FE model of the prototype can be seen in Figure 4. Figure 3. Inertial shaker Data Physics IV47 attached on top of the prototype. Table 1. Main specs of the prototype and the shaker. Feature Value Tower + nacelle weight 42 kg Jacket weight 13.5 kg Shaker + support weight 27 kg Total weight 82.5 kg Tower height 1790 mm Jacket height 1300 mm Total height 3090 mm Material Steel Supports Fixed to a concrete base Shaker model Data Physics IV47 Inertial mass 14.5 kg Max sinus force (peak) 250 N Total shaker mass 21 kg Shaker main mode freq. 20 Hz 2.4.2. FE Model and Model Reduction A FE model of the prototype is built based on the 3D CAD of the structure. The shaker and its support, including the bushing, are simplified to an equivalent point mass located at the mass center of the replaced components and attached to the structure. The behavior of the bushing has been tested in the frequency range of interest (0 to 25Hz), verifying its linearity. The platform has been designed to keep the mass center of the rotating components in the rotating axis, so the system can be considered invariant. The bolted joints present in the prototype are also simplified using bonded contacts. Due to the tower and nacelle of the wind turbine prototype being thin steel profile components, shell-type elements have been used to reduce the total number of elements in the mesh. In the jacket support structure, solid elements have been used. The primary features of the FE model can be seen in Table 2, and the FE model of the prototype can be seen in Figure 4. Sensors 2023,23, 4706 16 of 19 Table 14. Evaluation of the results shown in Figure 8. F XY Noise X-1-90 Y-1-90 KF 19.8/97.4 13.7/99.1 AKF 20.0/97.4 13.9/99.0 LSSE 18.2/97.3 15.9/98.9 Table 15. Evaluation of the results shown in Figure 9. F VAR 5 Hz X-1-90 Y-1-90 KF 12.3/92.8 8.6/99.0 AKF 19.5/99.0 4.9/98.9 LSSE 16.8/99.3 4.1/98.7 Table 16. Evaluation of the results shown in Figure 10. F X HIT X-1-90 Y-1-90 KF 11.6/99.5 2.1/93.7 AKF 11.3/99.5 1.8/93.3 LSSE 11.6/99.6 3.2/92.4 Sensors 2023, 23, x FOR PEER REVIEW 16 of 20 system behavior, as is predicted by sensor location methods (such as the Modal Kinetic Energy method, mentioned in Section 2.4.4). According to the obtained results in the virtual sensors when forces in X direction are applied, the estimates obtained in the virtual sensors are generally better than when forces in the Y direction or in combined directions are applied. This difference can be explained by the certain lack of precision in the behavior of the FE model (from which the reduced model has been obtained) in the Y-direction bending and the Z-direction torsion. Of the sensor configurations tested in the different vs. algorithms, the best-performing one is configuration 3. This configuration uses the 3X, 3Y, 4X, 4Y, 5X and 5Y gauges as input sensors. Configuration 8, that uses the same input gauges but adding an accelerometer, also performs well when is used with the AKF. From the data obtained from the experiments carried out in this article, it can be concluded that, in terms of robustness, the LSSE is preferable because, unlike Kalman filters, it does not depend on tuning parameters. Among the Kalman filters, the AKF can be considered more robust than the KF because, under conditions of unmeasured forces, it has a stable performance when accelerometers are used as input sensors. Some examples of the obtained results applying different input forces, using configuration 3 (input gauges X-2-90, Y-2-90, X-3-90, Y-3-90, X-4-90, Y-4-90, X-5-90, Y-5-90), are provided in Figures 6–10. The values corresponding to these results are summarized in Tables 12–16. The virtual sensors are the gauges 1X (first column of the tables) and 1Y (second column of the tables). The estimations obtained with the KF, AKF and LSSE are compared with real strain data at the same location (indicated as REF). The left values in the tables correspond to the percentage error of the estimation, and the right values correspond to the PCC error. Figure 6. Results obtained applying a 5Hz sinusoidal force in X direction. Figure 6. Results obtained applying a 5Hz sinusoidal force in X direction. Sensors 2023, 23, x FOR PEER REVIEW 17 of 20 Figure 7. Results obtained applying a 15 Hz sinusoidal force in Y direction. Figure 8. Results obtained applying a zero-mean white noise force in 45° direction. Figure 9. Results obtained applying a 5 Hz sinus force in variable direction along time. Figure 7. Results obtained applying a 15 Hz sinusoidal force in Y direction. Sensors 2023,23, 4706 17 of 19 Sensors 2023, 23, x FOR PEER REVIEW 17 of 20 Figure 7. Results obtained applying a 15 Hz sinusoidal force in Y direction. Figure 8. Results obtained applying a zero-mean white noise force in 45° direction. Figure 9. Results obtained applying a 5 Hz sinus force in variable direction along time. Figure 8. Results obtained applying a zero-mean white noise force in 45◦direction. Sensors 2023, 23, x FOR PEER REVIEW 17 of 20 Figure 7. Results obtained applying a 15 Hz sinusoidal force in Y direction. Figure 8. Results obtained applying a zero-mean white noise force in 45° direction. Figure 9. Results obtained applying a 5 Hz sinus force in variable direction along time. Figure 9. Results obtained applying a 5 Hz sinus force in variable direction along time. Sensors 2023, 23, x FOR PEER REVIEW 18 of 20 Figure 10. Results obtained applying a hammer impact in X direction. Table 12. Evaluation of the results shown in Figure 6. F X 5 Hz X-1-90 Y-1-90 KF 3.6/99.4 178.6/61.6 AKF 7.7/99.4 187.3/60.2 LSSE 4.1/99.8 197.1/58.3 Table 13. Evaluation of the results shown in Figure 7. F Y 15 Hz X-1-90 Y-1-90 KF 195.6/95.4 7.3/99.9 AKF 209.3/95.2 7.0/99.9 LSSE 217.3/98.5 6.9/99.9 Table 14. Evaluation of the results shown in Figure 8. F XY Noise X-1-90 Y-1-90 KF 19.8/97.4 13.7/99.1 AKF 20.0/97.4 13.9/99.0 LSSE 18.2/97.3 15.9/98.9 Table 15. Evaluation of the results shown in Figure 9. F VAR 5 Hz X-1-90 Y-1-90 KF 12.3/92.8 8.6/99.0 AKF 19.5/99.0 4.9/98.9 LSSE 16.8/99.3 4.1/98.7 Table 16. Evaluation of the results shown in Figure 10. F X HIT X-1-90 Y-1-90 KF 11.6/99.5 2.1/93.7 AKF 11.3/99.5 1.8/93.3 LSSE 11.6/99.6 3.2/92.4 5. Conclusions In this article, three different vs. algorithms have been tested to obtain virtual strain estimations under unknown forces. Twelve different sensor configurations have been used under 15 different dynamic loads. Two virtual strain gauges have been implemented Figure 10. Results obtained applying a hammer impact in X direction. 5. Conclusions In this article, three different vs. algorithms have been tested to obtain virtual strain estimations under unknown forces. Twelve different sensor configurations have been Sensors 2023,23, 4706 18 of 19 used under 15 different dynamic loads. Two virtual strain gauges have been implemented in the base of the tower of the prototype, and two real strain gauges have been used as reference sensors, to compare the estimated data obtained from the virtual sensors with the equivalent real sensor data. It has been verified that, through the modal truncation, a reduced model can be used to obtain the response of a much more complex FE model (in a limited range of frequencies) using a limited number of modes (which implies a significative reduction in the number of DoFs used). The AKF shows itself to be better than the classical KF in absence external force measurements, especially when strain and acceleration measurements are available. If only strain measurements are available, the AKF and the LSSE perform similarly, so, to obtain strain virtual measurements, the LSSE may be preferable due to its simplicity. The experience and results obtained with the experiments presented in this article can be useful when implementing strain virtual sensors. Examples of application can range from wind turbines (as in the case of this article) to many other types of complex structural assets, for example distinct types of offshore structures, bridges, communication towers or even large industrial frames (such as industrial presses). Several future lines of inquiry may continue the work presented in this article. On one hand, it would be interesting to add real-time force measurements to allow comparison vs. results with known forces and with unknown forces. Forces can also be estimated using vs. algorithms. On the other hand, it would also be interesting to install more accelerometers to the use case, in order to be able to test more sensor configurations in the different vs. algorithms. It would be also interesting to install gauges with an orientation of 45 º with respect to the tower axis, with the aim of measuring torsional strain. Furthermore, it would be interesting to apply the tested vs. algorithms in other use cases of a different nature, to check if vs. estimations of comparable quality can be obtained in other types of structures. Author Contributions: B.M. and J.B. conceived the conceptual ideas and carried out the experimental work. I.S. and J.A. helped with the methodology and with the improvement of the performance of the vs. algorithms. B.M. built the necessary software and managed the experimental data. B.M., J.B., I.S., U.L. and J.A. participated in the writing of the article, and I.S., U.L. and J.A. reviewed the final manuscript and made style suggestions. All authors have read and agreed to the published version of the manuscript. Funding: The research presented in this work has been carried out by Ikerlan Research Center, a center certificated as “Centro de Excelencia Cervera”. This work has been funded by CDTI, dependent on the Spanish Ministerio de Ciencia e Innovación, through the “Ayudas Cervera para centros tecnológicos 2019” program, project MIRAGED with expedient number CER-20190001. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to privacy restrictions. Conflicts of Interest: The authors declare no conflict of interest in this work. References 1. Rih-Teng, W.; Mohammad Reza, J. Data Fusion Approaches for Structural Health Monitoring and System Identification: Past, Present, and Future. SAGE Struct. Health Monit. 2018,19, 552–586. [CrossRef] 2. Liu, L.; Kuo, S.M.; Zhou, M.C. Virtual Sensing Techniques and Their Applications. In Proceedings of the IEEE International Conference on Networking, Sensing and Control, Okayama, Japan, 26–29 March 2009; pp. 31–36. 3. Maes, K.; Iliopoulos, A.; Weijtjens, W.; Devriendt, C.; Lombaert, G. Dynamic Strain Estimation for Fatigue Assessment of an Offshore Monopile Wind Turbine Using Filtering and Modal Expansion Algorithms. Mech. Syst. Signal. Process. 2016 ,76–77, 592–611. [CrossRef] 4. Ilyas, E.B.; Fischer, M.; Iggena, T.; Tonjes, R. Virtual Sensor Creation to Replace Faulty Sensors Using Automated Machine Learning Techniques. In Proceedings of the GIoTS 2020—Global Internet of Things Summit, Online, 3 June 2020; pp. 3–8. 5. Woodbury, T.; Srivastava, A.N. Analysis of Virtual Sensors for Predicting Aircraft Fuel Consumption. In Proceedings of the AIAA Infotech at Aerospace Conference and Exhibit, Pasadena, CA, USA, 11–13 September 2012; pp. 1–15. 6. Renard, P.; Alcolea, A.; Ginsbourger, D. Stochastic versus Deterministic Approaches. In Environmental Modelling: Finding Simplicity in Complexity, 2nd ed.; Wiley: Hoboken, NJ, USA, 2013; pp. 133–149, ISBN 9780470749111. Sensors 2023,23, 4706 19 of 19 7. Azzam, B.; Schelenz, R.; Roscher, B.; Baseer, A.; Jacobs, G. Development of a Wind Turbine Gearbox Virtual Load Sensor Using Multibody Simulation and Artificial Neural Networks. Forsch. Ing. Eng. Res. 2021,85, 241–250. [CrossRef] 8. Sun, S.; He, Y.; Zhou, S.; Yue, Z. A Data-Driven Response Virtual Sensor Technique with Partial Vibration Measurements Using Convolutional Neural Network. Sensors 2017,17, 2888. [CrossRef] [PubMed] 9. Kalman, R. A New Approach to Linear Filtering and Prediction Problems. Trans. ASME J. Basic Eng. 1960,82, 35–45. [CrossRef] 10. Chen, Z. Bayesian Filtering: From Kalman Filters to Particle Filters, and Beyond. Stat. J. Theor. Appl. Stat. 2003 ,182, 1–69. [CrossRef] 11. Maes, K.; De Roeck, G.; Iliopoulos, A.; Weijtjens, W.; Devriendt, C.; Lombaert, G. Kalman Filter Based Strain Estimation for Fatigue Assessment of an Offshore Monopile Wind Turbine. In Proceedings of the ISMA 2016, Leuven, Belgium, 19–21 September 2016; pp. 1649–1661. 12. Palanisamy, R.P.; Cho, S.; Kim, H.; Sim, S.H. Experimental Validation of Kalman Filter-Based Strain Estimation in Structures Subjected to Non-Zero Mean Input. Smart Struct. Syst. 2015,15, 489–503. [CrossRef] 13. Friedland, B. Treatment of Bias in Recursive Filtering. IEEE Trans. Automat. Contr. 1969,14, 359–367. [CrossRef] 14. Lourens, E.; Reynders, E.; De Roeck, G.; Degrande, G.; Lombaert, G. An Augmented Kalman Filter for Force Identification in Structural Dynamics. Mech. Syst. Signal. Process. 2011,27, 446–460. [CrossRef] 15. Julier, S.J.; Uhlmann, J.K. Unscented Filtering and Nonlinear Estimation. Proc. IEEE 2004,92, 401–422. [CrossRef] 16. Borges, G.A.; Vargas, A.N.; Ishihara, J.Y.; Menegaz, H.M.T. A Systematization of the Unscented Kalman Filter Theory. IEEE-TAC 2015,60, 2583–2598. [CrossRef] 17. Jouin, M.; Gouriveau, R.; Hissel, D.; Péra, M.C.; Zerhouni, N. Particle Filter-Based Prognostics: Review, Discussion and Perspectives. Mech. Syst. Signal. Process. 2016,72–73, 2–31. [CrossRef] 18. Henkel, M.; Weijtjens, W.; Devriendt, C. Fatigue Stress Estimation for Submerged and Sub-Soil Welds of Offshore Wind Turbines on Monopiles Using Modal Expansion. Energies 2021,14, 7576. [CrossRef] 19. Avitabile, P.; Pingle, P. Prediction of Full Field Dynamic Strain from Limited Sets of Measured Data. Shock Vib. 2012 ,19, 765–785. [CrossRef] 20. Penrose, R. A Generalized Inverse for Matrices. In Proceedings of the Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambridge, UK, 1955; Volume 51, pp. 406–413. 21. Elfving, T. A Stationary Iterative Pseudoinverse Algorithm. BIT Numer. Math. 1998,38, 275–282. [CrossRef] 22. Branlard, E.; Giardina, D.; Brown, C.S.D. Augmented Kalman Filter with a Reduced Mechanical Model to Estimate Tower Loads on a Land-Based Wind Turbine: A Step towards Digital-Twin Simulations. Wind Energy Sci. 2020,5, 1155–1167. [CrossRef] 23. Yan, A.; De Boe, P.; Golinval, J. Structural Damage Diagnosis by Kalman Model Based on Stochastic Subspace Identification. SAGE Struct. Health Monit. 2004,3, 103–119. [CrossRef] 24. Tarpø, M.; Nabuco, B.; Georgakis, C.; Brincker, R. Expansion of Experimental Mode Shape from Operational Modal Analysis and Virtual Sensing for Fatigue Analysis Using the Modal Expansion Method. Int. J. Fatigue 2020,130, 105280. [CrossRef] 25. Moghadam, F.K.; Nejad, A.R. Online Condition Monitoring of Floating Wind Turbines Drivetrain by Means of Digital Twin. Mech. Syst. Signal. Process. 2022,162, 108087. [CrossRef] 26. Baqersad, J.; Bharadwaj, K.; Poozesh, P. Modal Expansion Using Strain Mode Shapes. In Shock & Vibration, Aircraft/Aerospace, Energy Harvesting, Acoustics & Optics, Volume 9: Proceedings of the 35th IMAC, A Conference and Exposition on Structural Dynamics 2017; Springer: Berlin/Heidelberg, Germany, 2017; Volume 9, pp. 219–226. 27. Dorosti, M.; Fey, R.; Heertjes, M.; Van De Wal, M.; Nijmeijer, H. Finite Element Model Reduction and Model Updating of Structures for Control. IFAC Proc. Vol. 2014,47, 4517–4522. [CrossRef] 28. Guyan, R. Reduction of Stiffness and Mass Matrices. AIAA J. 1965,3, 380. [CrossRef] 29. Friswell, M.I.; Garvey, S.D.; Penny, J.E.T. Model Reduction Using Dynamic and Iterated IRS Techniques. J. Sound Vib. 1995 ,186, 311–323. [CrossRef] 30. Craig, R.; Bampton, M. Coupling of Substructures for Dynamic Analyses. AIAA J. 1968,6, 1313–1319. [CrossRef] 31. Tolani, S.K. Modal Truncation of Substructures Used in Vibration Analysis; University of Missouri-Rolla: Rolla, MO, USA, 1972. 32. Tarpø, M.; Vigsø, M. Modal Truncation in Experimental Modal Analysis. In Topics in Modal Analysis & Testing; Springer: Berlin/Heidelberg, Germany, 2019; Volume 9, pp. 143–152, ISBN 9783319747002. 33. Li, D.S.; Li, H.N.; Fritzen, C.P. The Connection between Effective Independence and Modal Kinetic Energy Methods for Sensor Placement. J. Sound Vib. 2007,305, 945–955. [CrossRef] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.