D-optimal sensor placement for load identification in wind turbine rotor blades

Abstract

Accurate load identification is a prerequisite for monitoring damage processes such as fatigue accumulation. This work approaches the load recognition problem in an inverse inferential setting by processing readings from a strain sensor grid mounted on the operating wind turbine rotor blade. The aeroelastic pressure field is considered an equivalent lumped load vector applied at given stations along the blade’s length, and its magnitude is the quantity of inferential interest. The technical challenge of optical sensor placement is addressed through D-optimal designs that promise sensor architectures, that is, locations and features, which offer a minimal uncertainty propagation of the sensor readings to the load inferences. Synthetic data are generated through finite element simulations based on an actual composite material geometry to demonstrate and quantitatively assess the effectiveness of the process. D-optimal sensor grid designs are obtained through the employment of Genetic Algorithms. Further reduction of the involved epistemic uncertainty due to the problem’s inherent ill-conditioning is assessed by evaluating sensor grids with increasing sensor numbers. The proposed inferential scheme presents a robust way to approach the inverse load identification problem.

Keywords

Structural Health Monitoring load identification Genetic Algorithms optimal sensor placement (OSP)composite structures

1. Introduction

The current industrial practice on the inspection, monitoring, and maintenance planning for most wind turbine rotor blades is based on preventive maintenance schemes (Badihi et al., 2022). Potential damages and their corresponding locations (hot spots) are nominally identified during the design phase. Having available a theoretical load spectrum, designers estimate the expected fatigue life of the blade based on these nominal hot spots. Considering this information and other historical data (e.g. field experience), wind farm operators accordingly plan the maintenance scheme (Song et al., 2022). However, each installation site is unique as the environmental conditions (e.g. area statistical wind speed) are uncertain during the design phase. Knowledge of the exact load spectrum that applies on a wind turbine is one of the main key enablers that will allow for the transformation from preventive to condition-based or predictive maintenance schemes (Dai et al., 2022). Fatigue accumulation is a typical damage case that is controlled by the number of stress cycles and the corresponding stress range along with other factors that exhibit variability. Field inspections reveal that fatigue generated cracks along the blade’s bondline, that might turn out to be critical, are frequently encountered prior to a planned inspection.

The field of Structural Health Monitoring (SHM) involves methods for diagnostics and prognostics (Liangou et al., 2023; Silionis and Anyfantis, 2022; Taher et al., 2022) and finds applications in wind turbines (Malekimoghadam et al., 2021; McGugan and Mishnaevsky, 2020). A major field of research and application in SHM is related to load identification, where one may solicit for methods that are used for recognizing the exact or more practically, the equivalent loads applied to a blade during operation. There does exist an important literature body that has worked either theoretically or/and experimentally on methods that promise load identification under dynamic or static operational and environmental conditions (Colombo et al., 2021; Gupta and Dhingra, 2013; Liu et al., 2022b; Schroeder et al., 2006; Wang et al., 2013, 2021; Zhang et al., 2019a, 2019b, 2019c; Zhang and Xu, 2016; Zhou et al., 2023a, 2023b). In cases where dynamic effects (inertia loads) are small or negligible or even compensated with the application of weighting factors applied on quasi-static loads, static load identification methods are preferred due to their straight forward applicability (Colombo et al., 2021; Kam et al., 2017; Qin et al., 2022; Zhang et al., 2019a, 2019b, 2019c). The problem of load identification is an inverse problem per se, since we can seldom find cases where the load may be directly measured. Therefore, we rely on the registration of some kind of measurable load effect (response) and we back-calculate the loads that are considered responsible for exhibiting the measured response magnitudes. One typical class of indirect load identification methods relies upon strain measurements for the estimation of the unknown load quantities (Gupta and Dhingra, 2013; Schroeder et al., 2006; Wang et al., 2013, 2021; Zhang et al., 2019a, 2019b, 2019c; Zhang and Xu, 2016). It has been demonstrated in the real world that a strain sensing system is durable and reliable in the long run and may serve the purpose of load identification. Schroeder et al. (2006) present the case study where a network of six sensors was mounted along the span of a wind turbine rotor blade in order to monitor the actual operational load. The authors demonstrated a continuous load monitoring process for two operational years of the sensing/structural system under realistic environmental conditions. Moreover, the authors envisage the use of the recorded data for the determination of the fatigue load footprint as a diagnostic tool for the blade material and also for wind park site characterization. Although the system allows for counting the number of stress cycles, the localized strain measures do not allow for a full field stress reconstruction. This is considered to be the core technical challenge in the development of a structural digital twin.

Indirect load identification has always been problematic due to the ill-conditioning exhibited in the inverse problem. Within the inverse estimation of an ill-conditioned problem, a small error or perturbation in the input data can easily scale up in a significant error or perturbation in the load that is of estimation interest (Zhang et al., 2020). A forward predictive model, for example, analytical formula, statistical model, finite element model, is required in order to construct the relationship between the loads that are of estimation interest and the strains at the sensor locations (Gupta and Dhingra, 2013). Modifying the sensor locations will end up at a different representation of the forward predictive model. Consistently, a variation in sensor placement will result in a change in the model ill-condition state. Therefore, sensor locations may methodically be defined such that ill-conditioning effect is alleviated. The D-optimality criterion, which is used for parameter estimation, can serve as a promising criterion in deciding/guiding the optimal sensor placement (OSP) locations (Gupta and Dhingra, 2013; Schroeder et al., 2006; Wang et al., 2013, 2021; Zhang et al., 2019a, 2019b, 2019c; Zhang and Xu, 2016). However, to the authors knowledge, the D-optimal criterion has been demonstrated in simplified laboratory geometries/designs that involve a limited design exploration space and a small size load vector. In practice, the number of potential locations for sensor placement increases exponentially with the number of available sensors due to the complexity and dimensions associated with a real structure. We undertake the challenge to develop an optimization scheme that is able to provide solutions to the OSP problem based on the well-recognized features of the D-optimality criterion. The application case presented in this paper deals with a structure made of composite material. Composite materials, due to their heterogenous and anisotropic nature, increase the complexity and the induced uncertainties (e.g. material properties) of the structural system, thus rendering the OSP task more challenging as opposed to conventional materials (such as steel).

The extraction of D-optimal designs poses a complex optimization challenge, characterized by multimodality, non-linearity, high dimensionality, and often, constraints. This necessitates the use of a versatile optimizer capable of global or near-optimal solution search within the parameter space, while preserving efficiency and numerical precision in D-optimal design extraction.

The selection of Genetic Algorithms (GAs) for extracting D-optimal designs has emerged as a robust and dependable choice for optimization (Limmun et al., 2013). The preference for achieving a highly efficient design over the globally optimal one in experimental design arises from practical considerations and trade-offs. Factors such as computational costs, inherent modeling uncertainties, and the risk of overfitting the model underscore the importance of prioritizing efficiency over achieving a global optimum. GAs have proven their effectiveness in tackling complex, multimodal optimization problems due to their global exploration capabilities.

Furthermore, many experimental design problems incorporate constraints. For instance, in our specific application, the inclusion of duplicate sensor positions, although consistent with Design of Experiments (DoE) theory, is impractical. Therefore, a versatile optimizer that can effectively handle constraints becomes imperative. Various constraint-handling techniques have been developed, implemented, and tested for GAs across diverse engineering applications. However, the flexibility offered by GAs, which allows users to adjust several parameters to suit the specific problem at hand (e.g. selection, crossover, mutation), increases complexity and entails additional time costs for parameter tuning.

In addition, GAs, despite their effectiveness, are associated with high computational costs. Although they are easily parallelized, their convergence speed remains comparatively lower when compared to other optimizers, such as those from the Swarm Intelligence (SI) family. Recent studies have brought metaheuristic algorithms from the SI family, notably the Grey Wolf Optimizer (GWO; Mirjalili et al., 2014) to the forefront of various applications. GWO’s strength lies in its simplicity and rapid convergence, although it may lag in terms of exploration potential and constraint handling flexibility compared to GAs. Nevertheless, recent efforts to enhance the versatility of this algorithm (Hou et al., 2022; Knypińskia, 2021; Rodrigues, 2023) indicate its growing potential in solving complex engineering problems both effectively and efficiently (Pace et al., 2022).

Lastly, the deployment of hybridization techniques, combining the strengths of GAs and GWO, represents an effective and efficient optimization approach. This fusion can harness the merits of both algorithms, offering a promising strategy for optimizing D-optimal experimental designs while addressing the intricacies associated with constraints, computational costs, and trade-offs in design efficiency.

This work treats OSP as a constrained optimization problem with the D-optimal criterion acting as the fitness function. We employ Genetic Algorithms (GA) for an efficient design space exploration. The proposed scheme is generic and may be applied in other structures and material combinations, where both the design space (potential locations for sensor placement) and the load vector of inferential interest are characterized by high-dimensionality. The procedure is demonstrated for load identification in a wind turbine rotor blade. Initially the OSP design process is discussed along with brief presentation of basic theoretical aspects, in section 2. A particular geometry has been selected for demonstrating the proposed method and its particulars are provided in section 3. A comparison between engineering and D-optimal grids is presented in section 4. Results in an increasing number of load unknowns are provided in section 5 followed by concluding remarks on the research study.

2. Theoretical background

This section encapsulates the fundamentals associated with the least square estimation problem and the D-optimal design (sub-section 2.1), Genetic Algorithms (sub-section 2.2) and eventually the proposed OSP design framework (sub-section 2.3).

2.1. Least squares and D-optimal designs

The followed methodology is based on the Sensitivity Response Method (SRM), described in detail in Doyle (2004) and its numerical implementation was first introduced by Wickham et al. (1995). Intrinsic to this method are the following assumptions:

The applied loads do not lead to non-linear response of the structure, the material response remains within its elastic regime and thus the principle of superposition can be applied.

The structure is loaded at a finite number of known locations. The loading direction is also given, leaving their magnitude as the sole unknown variable for evaluation.

The imposed loads can be considered as static or quasi-static in nature.

It naturally follows that, based on the principle of superposition, given the linear elastic response of a structure, the strain at any location, $ε_{i}, i = 1, 2, \dots, d$ , may be expressed as the sum of the individual strains developed at that specific location when each load, $l_{j}, j = 1, 2, \dots, m$ , is applied separately, as shown below:

E [ε_{i}] = a_{i, 1} l_{1} + a_{i, 2} l_{2} + \dots + a_{i, j} l_{j} + \dots + a_{i, m} l_{m}

(1)

where $E [\cdot]$ is the expected value operator. It follows that each strain reading is regarded as a realization drawn from random vector $ε = {\begin{matrix} ε_{1} & ε_{2} & \dots & ε_{d} \end{matrix}}^{T}$ with $ε \in D_{ε}$ , $D_{ε} \subseteq R^{d}$ and $d \in Z^{+}$ . Each set of weighting factors $a_{i, j}$ , which is collected in matrix A with dimensions $d \times m$ , is regarded as a representation of the structure to be monitored. Each element $a_{i, j}$ is equal to the individual contribution of the strain component $ε_{i}$ when a unit load $l_{j}$ is applied on the structure,

a_{i, j} = ε_{i} | (l_{j = j} = 1, l_{j \neq j} = 0)

(2)

The load components are as well collected in the load vector $l = {\begin{matrix} l_{1} & l_{2} & \dots & l_{m} \end{matrix}}^{T}$ with $l \in D_{l}$ , $D_{l} \subseteq R^{m}$ and $m \in Z^{+}$ . Therefore equation (1) may be rewritten as:

ε = A l + e

(3)

where $e$ is the random vector that models the systems aleatoric uncertainty described by a white gaussian noise probabilistic model, $e ~ N (E [e] = 0, Σ [e] = I_{d x d} \cdot σ^{2})$ . A residual vector $e \in D_{e}$ , $D_{e} \subseteq R^{d}$ and $d \in Z^{+}$ is defined as:

r = ε - A l

(4)

The principle of least squares chooses as estimates of $l$ those values $\hat{l}$ which minimize the Euclidean distance $S = ‖ r (l) ‖_{2}^{2}$ . In the high dimensional space, quantity S is minimized from the equation below:

\nabla_{l} [S (l)] = 0

(5)

The solution of equation (5) yields:

\hat{l} = {(A^{T} A)}^{- 1} A^{T} ε

(6)

The preceding equations are the normal equations and reflect the normal solution of the least-squares problem and are provided that ${(A^{T} A)}^{- 1}$ exists. Accordingly, the covariance matrix for the load estimates is defined as:

var (\hat{l}) = var (ε) {(A^{T} A)}^{- 1} = σ^{2} {(A^{T} A)}^{- 1}

(7)

where the diagonal terms of the covariance matrix of $\hat{l}$ correspond to the variance of each individual load estimate, while each of the off-diagonal term is the covariance between the load estimates $l_{k}$ and $l_{p}$ for $k \neq p$ .

The matrix ${(A^{T} A)}^{- 1}$ is known as the sensitivity matrix of $A$ and for a given variance in strain measurements, $σ^{2}$ , its minimization can essentially increase the precision of the load estimates. The sensitivity matrix of $A$ is a function of locations and angular orientations of the strain gauges attached to the structure. Hence, the optimal set of locations (together with their number and orientations) for installing strain gauges is the one that minimizes the error between predicted loads and strain measurements, which essentially corresponds to the minimization of the sensitivity matrix of $A$ . Let the information related to sensor number (n), topology $(x_{n}, y_{n}, z_{n})$ , and strain component $(x, y, xy)$ be collected in parameter vector θ as:

θ = {n, (x_{n}, y_{n}, z_{n}), (component)}^{T}

(8)

In the theory of optimal experimental design, there exist various statistical criteria that lead to the optimal design. Among them, one of the most widely employed is the D-optimal criterion—with D representing determinant—according to which one seeks to find the matrix $A$ that minimizes the quantity:

det | {(A^{T} \cdot A)}^{- 1} |

(9)

This is equivalent to maximizing the determinant of the information matrix, that is, $det | (A^{T} \cdot A) |$ and such design is called D-optimal design (Galil and Kiefer, 1980; Mitchell, 1974). The optimization problem that needs to be tackled is formally given as:

{\hat{θ}}_{D - optimal} = \arg \max_{θ} (\det | (A {(θ)}^{T} \cdot A (θ)) |)

(10)

2.2. Genetic algorithms

According to Pradubsri et al. (2019) the Genetic Algorithms (GAs) were described for first time by Holland (1992) and subsequently they became popular by Goldberg (1989). Since then, they have been utilized in a variety of application that have to do with optimization (Liangou et al., 2023; Liangou and Dentsoras, 2021; Liu et al., 2023; Pradubsri et al., 2019). The idea behind the function of GAs is based on the Darwin’s theory of natural evolution. GAs perform a search in the space of candidate solutions, with the aim of finding acceptable, according to some criterion, solutions. They start from an initial population, and they generate future populations of individuals by simulating fundamental laws of natural selection using genetic operators such as crossover, mating, and mutation to optimize an objective function that assesses the fitness of each individual in the population (Pradubsri et al., 2019). The GAs seems to be extremely attractive since it is easy to be implemented and unlike other optimization algorithms, GAs does not require continuity, differentiability, and convexity of the objective function (Pradubsri et al., 2019). In cases where the size of a problem makes it prohibitive to use classical search methods to solve it, GAs are able to minimize the probability of being trapped in local optima and give an optimal or sub-optimal solution in a short time. However, it is well known that depending on the complexity of the design space they might require a significant number of generations before they converge.

2.3. OSP design framework

The entire proposed sensor placement design process is encapsulated in Figure 1, where two main sequential groups of actions are schematically depicted. Initially, a direct structural analysis method is required that is able to provide spatial distributions of strains given a load condition. Here we have employed the Finite Element method to serve this purpose. In accordance with equation (2), the strain distribution over the analyzed domain gives directly the corresponding sensitivity distribution. Having run the FE model as many times as the number of unknown loads then the repository with the sensitivities is considered fully populated. This process is happening only once. Then the GA optimization process is employed for a requested number of sensors, which quantity sets the size of the design vector. The GA recursively explores the design space, that is, searching the sensitivity repository, for solutions (equation (8)) that exhibit comparatively high fitness levels, in accordance to equation (10), until a predefined number of generations is reached. For each design vector θ, the corresponding A matrix is fetched and then its fitness is computed. The output of this process is ${\hat{θ}}_{D - optimal}$ and its conjugate sensitivity matrix $A_{D - optimal}$ .

3. Demonstrative case study

The applicability and effectiveness of the proposed OSP design framework for load identification is demonstrated on a realistic wind turbine rotor blade geometry made of composite materials in a computational setting.

3.1. Assumptions and design basis

Design and maintenance of the entire wind turbine depends significantly on the loading profile that is associated with the rotor blades. Blade loads are characterized by a high degree of complexity are stochastic per se. They may be decomposed to aerodynamic, gravitational, centrifugal, Coriolis, and inertia loads. Computational fluid dynamics or coupled aeroelastic analysis, that captures the fluid-structure interaction, have revealed that there does exist a quite complex spatial distribution in the air velocity and pressure field around the airfoils (see Figure 2). In principle, we are interested in inferring the actual pressure distribution that develops at the boundary of the rotor blade at each time instance when focusing on aerodynamic loads. Having such knowledge available, we are able to simulate the blade’s response in a forward setting, for example, by performing FE simulation. Direct pressure profile recognition would require a rather dense grid of pressure gauges mounted on the external surface of the blade. However, the cables and sensors might downgrade the aerodynamic characteristics of the blade and as such this solution is disregarded. An alternative approach, which is undertaken in this work, is the indirect load identification scheme, where from strain readings, taken by sensors mounted in the internal surfaces of the blade, the corresponding loads are back-estimated. In this setting, the loads of inferential interest are the effective lumped sectional forces and moments that yield an equivalent structural response when compared to a simulation that involve the corresponding pressure loads. This is considered as the design basis of the selected case study. Preliminary dimensioning or final structural assessment of rotor blades is based on the employment of beam aeroelastic codes (e.g. FAST) where the load effects are solely based on sectional forces/moments at a predefined number of stations. Such an approach is founded on the principle of force decomposition, where all the loads on the blades are represented by three forces and three moments defined on a cartesian triad. This approach allows for the estimation of all related load generation mechanism and not only the aerodynamic related loads.

Moreover, the D-optimal design requires linearity in the system. Material behavior of composite materials is characterized by linear elasticity in the load range of operation. If we consider that the non-linear geometric effects that are related to large deflections are minimal in the selected geometry and that no degradation effects are present in the system, then it may safely be assumed that linearity holds.

As aforementioned, load is dynamic and it remains to be decided whether inertia effects are significant or not. In small scale wind turbine rotor blades, as the demonstrative case study of this work, inertia effects are usually negligible in steady-state operational profiles, that are mainly of interest for fatigue accumulation estimation. This allows us to assume that a time series response may well be reproduced by and is equivalent to sequential static responses. In any case, it is highly convenient to consider that the load effect is solely based on static solutions in order to avoid any error accumulation arising from dynamic simulations (damping quantification) that adds negatively to the already ill-posed problem. Dynamic Amplification Factors may be used as a compensation measure for disregarding inertia effects.

3.2. Geometry and load discretization

The demonstration structure revolves around a typical wind turbine blade with a relatively thin skin and two shear webs made of Glass Fiber Reinforced Polymer (GRP) composite materials similar to those mentioned in Liu et al. (2022a), Raman et al. (2016), and Ullah et al. (2022). The blade’s effective length is equal to 7 m and the internal diameter at the root is equal to 0.55 m. The chord length at the tip is taken as 0.1 m. The thickness varies from 45.6 mm at the root to 9.6 mm at the tip, as illustrated in Figure 3. The shear webs have a constant thickness equal to 14.4 mm and as fabricated by a sandwich material with PVC core. The spar beam sections, are placed at 60% of the chord length of the airfoil at the root and between 34% of the chord length of the airfoil from the trailing edge and 30% of the chord length of the airfoil from the leading edge at the tip. The layup angles of skin and webs from internal surface to external surface are listed in Table 1 (taken from Liu et al., 2022a; Raman et al., 2016). By using this sequence of layers, the blade is able to withstand the bending load along its length and the shear forces on the webs, as these forces are dominated at each region. Material parameters for the skin and spar are listed in Table 2 (taken from Liu et al., 2022a). For simplification, the thickness of 0.3 mm of each layer is the taken the same for both the skin and the two shear webs. First, a sequence of 32 layers is set for the entire skin. On top 120 layers are set at the blade root and the number of layers reduce smoothly until the thickness becomes equal to 9.6 mm. Finally, eight layers are set for the web, on either side of the foam.

Table 1.

Ply orientation and thickness distribution according to Figure 3.

Region identifier	Lay-up
A	[0₄ / 45₄ / 90₄ / −45₄]₂+ [0₄ / 45₄]₁₅
B	[0₄ / 45₄ / 90₄ /−45₄]₂+ [0₄ / 45₄]₁₂
C	[0₄ / 45₄ / 90₄ / −45₄]₂+ [0₄ / 45₄]₉
D	[0₄ / 45₄ / 90₄ / −45₄]₂+ [0₄ / 45₄]₆
E	[0₄ / 45₄ / 90₄ / −45₄]₂+ [0₄ / 45₄]₃
F	[0₄ / 45₄ / 90₄ / −45₄]₂
W	[+45₄ / −45₄ / core /+45₄ / −45₄]

Table 2.

Material properties for skin and spar.

Glass fiber/epoxy composites for skin and spar (Liu et al., 2022a)	E₁ = 55GPa, E₂ = E₃ = 9500 MPa, G₁₂ = G₁₃ = 5500 MPa, G₂₃ = 3000 MPa, ν₁₂ = 0.33, ν₁₃ = ν₂₃ = 0.3, Tensile Strength: X_T = 2500 MPa, Y_T = 50 MPa, Z_T = 75 MPa, Compressive Strength: X_C= 2000 MPa, Y_C = 150 MPa, Z_C = 400 MPa
PVC Foam for spar core	ρ_f = 60 kg/m³, E = 70 MPa, G = 27 MPa, ν = 0.3, Tensile/ Compressive Strength: X_T/C = Y_T/C = 1.5 MPa, Z_T/C = 0.95 MPa

With regards to the equivalent lumped beam loads, these are considered to be applied at discrete stations along the blade’s span, at locations presented in Figure 4. It is evident that the number of considered stations is proportional to the number of unknowns and subsequently to the minimum number of required sensors, since d ≥ m. For the sake of this demonstrator, we consider that four stations, i = 1, 2, 3, 4, along the span are sufficient to capture the structural response. At each station three section forces ( $F_{x or y or z}^{(i)}$ ) and three section moments ( $M_{x or y or z}^{(i)}$ ) are considered as unknown and are collected in vector $l_{i}$ as:

l_{i} = {\begin{matrix} F_{x}^{(i)} & F_{y}^{(i)} & F_{z}^{(i)} & M_{x}^{(i)} & M_{y}^{(i)} & M_{z}^{(i)} \end{matrix}}^{T}

(11)

The complete load identification vector contains the four sub-vectors as:

l = {\begin{matrix} l_{1} & l_{2} & l_{3} & l_{4} \end{matrix}}^{T}

(12)

The total number of unknowns in this setting is equal to 4 stations × 6 load components = 24 loads of inferential interest, which requires at least 24 virtual sensors.

3.3. Finite element modeling details

The finite element simulations were performed in ANSYS commercial finite element software. A fine mesh consisting of quadratic (8-node) layered shell elements was constructed, with an element size of 25 mm, ensuring both precision and computational efficiency. It shall be noted that the selected mesh size provides the model’s strain resolution in the sensor placement problem. The discretized numerical model consists of ∼17 k elements and ∼52 k nodes. The blade exhibits characteristics of a cantilever beam and therefore its root is rigidly fixed to emulate its fastening to the rotor hub, while the tip remains free. The blade’s tip is omitted from the analysis as it is considered that its role is mainly due to aerodynamic effects and not a load bearing member. With regards to the loading of the structure, the loads will be applied at four stations (longitudinal positions of the blade) that have a certain distance from the root (see Figure 4). The loads are applied at 0.4 of the chord length measuring from the leading edge and at 0.5 of the chord length when the cross-section is circular. Kinematic constraints in the form of rigid links are applied at each loading station. A reference point (master node) is placed at each station where the load vector (see equation (11)) is applied (Figure 5).

3.4. Response analysis

The equivalent lumped design loads as obtained from an aeroelastic beam analysis (FAST) are listed in Table 3. These values correspond to the design loads for extreme wind conditions, in compliance with the IEC 61400 standard that provides essential information for the design of small wind turbines. There are many references in literature where prove that the imposed loading follows reality. While synthetic data offers a viable alternative when experiments with real data are not feasible, it is crucial to acknowledge its limitations and consider the context in which it is applied. Understanding the assumptions, limitations, and validation steps associated with synthetic data is essential for ensuring its appropriate use and interpretation in research and practical applications. Lee et al. (2010) discuss experimental testing and structural analysis of small wind turbine blades, including insights into actual loading conditions. This study includes values of forces and moments on blades during operation for different wind speeds, providing a glimpse into how actual loading affects small turbine components. The moment and force distribution are in compliance with this work. Kong et al. (2005) propose a structural design for developing a medium scale composite wind turbine blade made of composite materials. They include both numerical and experimental data. In order to simulate the aerodynamic load, they apply three concentrated loads at points located a specific distance from the blade root. The same concept is followed in the present work.

Table 3.

Design load vector.

Station (i)	Load vector	F _x (kN)	F _y (kN)	F _z (kN)	M _x (kNm)	M _y (kNm)	M _z (kNm)
1	l ₁	10.88	6.6	24.48	34.06	41.4	0.94
2	l ₂	8.74	5.55	19.08	26.09	24.74	0.68
3	l ₃	4.69	3.63	8.68	5.12	6.69	0.2
4	l ₄	2.77	1.82	3.77	1.66	2.22	0.08

Figure 6 depicts the distribution of the minimum -through the plies- safety factor according to the Maximum Stress failure criterion. It is evident that design requirement is met. The maximum deflection is equal to 130.8 mm (Figure 7) and the first two mode shapes are depicted in Figure 8 which are within the allowable limits.

Figure 9 shows indicatively the axial strain response of the structure for a unit force out of the 24 applied to populate the repository with the sensitivities (Figure 1).

4. D-Optimal variance reduction

4.1. GA assisted D-optimality

This section showcases the effectiveness of employing the D-optimal criterion within the OSP problem in relation to two potential sensor networks that are based on common engineering judgment. For clarity, we have focused solely on the identification of the load components in one station, that is, i = 4, and as such vector l₄ is of inferential interest. The two different network types are also compared for increasing number of sensors, as it is expected that more sensors will provide with more information content and thus lead to further variance reduction. Identification of the full load vector is discussed in the next section.

Since the number of unknowns is equal to six in this case study, then the minimum number of required sensors is also six. A common engineering judgment suggests a span-wise equispaced sensor distribution on the skin (divided in the suction and pressure side, respectively) as presented in Figure 10(a) for a six-sensor grid instance. Since the cantilever is mainly subjected to combined flap-wise and edge-wise bending, the axial strain component (ε_z) was selected as the feature of interest.

With regards to the GA parameters used within the D-optimal setting, a population of 100 chromosomes was defined, the percentage of chromosomes for crossover was set equal to 80%, the mutation percentage per gene was set equal to 10 and the maximum number of generations until convergence was set equal to 10⁴. In this study uniform crossover is selected and as such each gene in the offspring is randomly selected from the genes of the parents. This method promotes genetic diversity in the offspring by allowing each gene to be independently inherited from either parent based on a specified probability. Additionally, random mutation with replacement is selected. It replaces the values of some genes from a randomly generated value within the gene space. The optimization process was completed once the predefined number of generations was reached. A sensitivity analysis over all parameters along with the generation number revealed that the optimizer convergence to the global minimum. An indicative convergence plot of the fitness function (see equation (10)) with respect to the number of generations is presented in Figure 11. These parameters remained unchangeable in all subsequent optimizations. Although the values of the parameters are case specific, the final selected settings compare well with the corresponding ones found in the work by Limmun et al. (2013). The resulting optimal sensor grid is shown in Figure 10(b) and it is evident that four sensors were identified as optimal near the load application point and two sensors further away from the load application point.

The quality of these two sensor grids is assessed through uncertainty quantification of the variability in the estimated loads. Since the mapping model is a linear representation, then in a forward uncertainty propagation setting, we may employ the analytical solution of the load variance, that is, $var [\hat{l}]$ and in particular its diagonal terms, in the presence noisy strain readings as derived by equation (7). Therefore, sensitivity matrix A is considered a property of the sensor topology and as such by controlling the sensor grid, we eventually control the amplification of the propagated uncertainty. Figure 12 presents the estimated root of the variance, that is, standard error, for each one of the six-components in the load vector under the presence of a 5 με noise level. This standard error attached to the virtual strains was maintained constant in the remaining paper. It is evident that the D-optimal sensor grid leads to a tremendous variance reduction when compared to the engineering judgment-based sensor grid.

A Monte Carlo (MC) Simulation has been performed in order to validate the normality and homogeneity assumption in the load identification as promised by the WGN model. The station four design load vector (Table 3) has been considered in the computational experimentation and the corresponding results are presented in Figure 13. It is evident that the error model is, as expected, unbiased, the joint plots exhibit very low statistical correlation and the MC variance load estimates are nearly identical to the analytical derived ones.

In an effort to examine whether a further variance reduction may be achieved, the number of sensors is increased fourfold resulting to a set of n = 24 sensors as shown in Figure 14(a), for sensor grids based on engineering judgment. The results of this high-dimensional sensor grid are compared in Figure 15 with the results of the GA for six-sensor grid (Figure 10(b)). Although the number of sensors was increased the 24-sensor grid based on the engineering judgment failed to reduce the variance as much as achieved with the D-optimal 6-sensor grid. As shown in Figure 15(a) the standard error of the load estimates in the six-sensor D-optimal design remains in lower levels with the exception of the force $F_{x}^{(4)}$ and the moment $M_{y}^{(4)}$ where the engineering judgment based grid produces a negligible better variance reduction.

Finally, an alternative 24-sensor grid shown in Figure 14(b) is assessed. The main difference is the enhancement of the first sensor design with new sensors placed on the webs that register the shear strain. In Figure 15(b) the results for this alternative are compared with the six-sensor D-optimal design and the latter remain a leading choice. It needs to be noted that as the generations increase, the GA converges to a better solution, in terms of standard error, but under higher computational cost. Given that 10,000 generations yield an acceptable solution compared to the computational time that is needed, the use of different topologies with sensor numbers ranging from 6 to 20 is investigated is assessed. Figure 16 represents the variance in the load identification. Given the variability in the optimization process it may well be said that a three- to four-fold number of sensors will lead to marginal enhancements.

4.2. Misplacement error

This section presents a sensitivity analysis on load identification error considering the sensor misplacement. As in the aforementioned subsection, we have focused solely on the identification of the load components in one station, that is, i = 4, and as such vector l4. Since the number of unknowns is equal to six in this case study, the number of required sensors is also six. The six-sensor D-optimal design was evaluated. The sensor misplacement is taken into consideration by assuming that there is the possibility of placing a sensor ± 40 mm away of the optimal position. Thus, considering the element size of the FE model, the eight surrounding finite elements with the optimal one in the center are considered as a set of candidate positions for sensor misplacement. Thus, 9⁶ different combinations are computed as different design vectors. Figure 17 shows the standard error in the load identification for the sensor misplacement. Boxes correspond to 75% prediction interval, error bars correspond to 95% prediction intervals, straight line to the median, square to the mean values. Thus, the variance of standard error in the load identification process is proven non-significant.

5. Results for a high dimensionality load vector

In this section the GA is implemented in order to identify a load vector in higher dimensions and showcase the strength of the approach in practical applications. Initially we consider a subproblem where the load components of stations 2 and 3 form the load vector of inferential interest. In this case we encounter 12 unknowns and as such an equal number of sensors is the minimum requirement so as the problem is definite. As already shown, additional sensors will reduce the epistemic uncertainty in the identification problem. Therefore, we consider three optimization problems each one associated with 12, 18, and 24 sensors. Corresponding results are presented in Figure 18. The trend is similar to the one depicted in Figure 16 and the axial force component includes the highest variance in the estimation. Figure 19 presents the corresponding optimal sensor topology for different numbers of sensors.

The final simulation is centered around the full problem that contains the four sub-vectors as given by equation (12). The GA hyperparameters, remained the same as described in Section 4. Figure 20 depicts the standard error of the load estimates for different sensor grid sizes ranging from 24 to 40 and the respective D-optimal designs are presented in Figure 21. It is evident that the sensor grid is sparsely distributed over the blade’s domain. This full coverage denotes that the optimization is not trapped around the load application points (four stations) or other highly stresses areas and as such the proposed process is robust.

The D-optimal results have an increased standard error at the first station and the error drops approaching station four. The dashed line shows the maximum standard error obtained from the six-sensor D-optimal design (see Figure 16) and is plotted as a baseline. The load identification process at stations two, three and four appear to have an acceptable behavior (Figure 22) and in particular when the force components are of inferential interest and the moments may be considered as latent. The latter occurs due to the fact that a larger pool of candidates for sensor placement is exhibited as one moves toward the tip, leaving behind a domain of increased size. Having available a higher dimensional design space for exploration, GA may effectively bring up combinations of sensitivities that assure D-optimality in the sensor design. It is evident, in this case as well that an increase in the numbers of sensors yields considerable reductions in the standard error.

6. Conclusions

Optimal sensor placement is crucial for accurate load identification and can greatly impact the performance and safety of wind turbine blades. The use of formal inferential statistics such as the D-optimal method with Genetic Algorithm optimization can significantly improve the accuracy (quantified through the standard error) of load identification and reduce the number of required sensors. Optimal sensor placement is highly dependent on the specific wind turbine blade design, and a customized approach based on the proposed method should be used to achieve the best results for each blade. The accuracy of the proposed methods has better results in terms standard error in the load estimate in low dimensional problems, but it is well performed in highly dimensional problems where it needs more computational time and an increased number of sensors. However, the proposed method is able to identify the most suitable sensor locations quickly (in less than 20min) and accurately in every instance, even when conventional engineering approaches fail to produce optimal results.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Theodora Liangou

Konstantinos N Anyfantis

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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