Piezoelectric energy harvester lifetime prognostics by novel Gaidai multi-modal reliability scheme

Abstract

Green Energy Harvesters (EH) are nowadays quite an important component of the contemporary green energy sector; thus, experimental data is required for their robust design validation. Existing reliability concepts are not applicable to dynamic systems possessing dimensionality above bivariate, except Monte Carlo-based methods, which are not applicable to measured data, or FORM, SORM methods that require prior knowledge of the underlying distribution. Existing system reliability methods are not always well suited to treat the latter problems, especially given the high-dimensional spatiotemporal nature of the nonlinearity of both system and environmental loadings. Presented case study proposes generic, state-of-the-art multivariate reliability methodology, assessing complex nonlinear system’s lifetime distribution, based on even a limited underlying dataset, extracted from an empirically recorded dynamic system time history. A comprehensive state-of-the-art multivariate reliability assessment along with methodological benchmarking was carried out. Methodology advocated in this study is not only applicable to energy harvesting devices; it can also be utilized for a range of engineering complex sustainable systems, subjected to environmental loads during designed service life. Novelty: presented case study benchmarks state-of-the-art, Gaidai multivariate reliability approach for evaluating an energy system’s operational lifetime, that is especially well suited for complex engineering systems, possessing high dimensionality, accompanied by complex inter-correlations between system primary dimensions or components.

Keywords

Gaidai multivariate reliability method piezoelectric energy harvesting sustainable system environment first passage probability green energy

Introduction

Low-frequency EH technology, harvesting wind energy from the ambient environment received extensive research attention in recent decades. EH innovations target energy supply for various low-power, low-cost gadgets, including Micro-Electro-Mechanical Systems (i.e., MEMS), along with Wireless Sensor Networks (i.e., WSN). Piezoelectric Vibration EH (PVEH) has demonstrated that it can transform mechanical vibro-energy into piezo-electrical energy. Unfortunately, mechanical vibrations are often irregular and intermittent. Aero-instability causes resonant vibrations; hence, Flow-Induced Vibrations (FIV) may be utilized to generate green energy. Vortex-Induced Vibration (VIV), galloping, wake galloping, as well as other phenomena constitute examples of FIV examples.^1–8

This study benchmarks novel structural reliability method for a multi-dimensional energy harvesting system’s service LifeTime Distribution (LTD) assessment, which is critical for EH design. Recent research has demonstrated the benefits of capturing low-frequency energy from the environment.^9–11 Recently, research has been also conducted on developing micro- and nano-scale wind EH. Macroscale wind energy is harvested by an electromagnetic EH, whereas piezoelectric, electrostatic, and triboelectric EH harvest microscale wind energy.¹² These technological advances fuel several low-cost, low-power devices, including MEMS and WSN. Triboelectric generators are based on triboelectric effects, which transform mechanical energy into electrical energy through coupling effects between triboelectrification and electrostatic induction. Through piezoelectric vibration energy harvesting, mechanical vibrational energy from ambient environment may be transformed into piezo-electrical energy. FIVs cause aero-instability manifesting itself as resonant vibrations. Galloping, wake galloping, and VIV are a few examples of phenomena categorized as FIVs. Numerous studies were undertaken during the past decade, utilizing experimental, numerical, and theoretical methods. In.¹ an aero-instability coupled mechanical vibration model, based on an electrical circuit, was presented. In.4, the authors have presented the Computational Fluid Dynamics (i.e., CFD) approach for VIV Piezoelectric EH (i.e., VIVPEH). In,⁵ authors have reported series of non-linear tests, optimizing VIVPEH design, using Galerkin’s method to study theoretical VIV piezoelectric EH properties. In recent research,⁷ authors reported switching from VIVPEH towards GalloPing EH (GPEH) by adding 2 different Y-shaped connectors. In⁸ and,⁹ authors utilized bistable, as well as tristable features to examine non-linear GPEH responses.

Most of the previous works were primarily concerned with improving EH performance, paying attention to several critical design elements, including EH severe reaction and fatigue life. The analysis of the EH’s extreme performance and prognostics of hazardous operating conditions constitute crucial steps within the design process,^13–18 For instance, writers in^19–21 investigated durability and reliability of the P1 MFC transducer, being subjected to forced EH base excitations to forecast degradation and damage rates of the piezoelectric EH. According to recently reported experimental results,¹⁴ EH was damaged when acceleration reached 0.5 g. In¹³, authors studied EH performance with a DuraAct P-876, piezo-sheet being mounted on a horizontal railway, reporting EH failure after about 100 cycles under acceleration of 1g. In¹², authors investigated long-term EH fatigue behaviour using a P-2 MFC sheet and Finite Element Method (FEM), supported by laboratory experiments.¹⁸ EH steel beam was reinforced with shims that could sustain a flow rate 16 L/min for $\approx$ 40 minutes, and after 9 hours of testing, it was reported that a flow rate $\approx$ 9 L/min did not result in any damage.

Areal (geometrically spread) extremes may be accurately modelled by utilizing e.g., Max-Stable Processes (i.e., MSP) approach.²² To assess geometrically spread exceedances, accounting for spatial inter-dependencies in extremes, MSP model may be adjusted to the underlying data, gathered over in situ geographical, geometrical or structural geometrical domains. The Extreme Value Theory (EVT) paradigm provides a solid MSP theoretical foundation. The advocated multivariate Gaidai reliability concept may be combined with the MSP, essentially replacing MSP EVT part with sub-asymptotic functional class of distributions.

Reliability of flow-based EHs, including piezoelectric EH/GPEH, is an important operational issue and should be further studied. The analysis of EH’s extreme dynamics and, hence its operational safety has to be examined at the design stage. To assess the durable EH life span, recent research has frequently focused on the dynamic behaviour of EH beams under fatigue loading, caused by certain excitations. The dynamic responsiveness of a GPEH system is investigated for extreme value statistics in this research. A specific galloping energy harvester’s dynamic performance has been studied theoretically and empirically to achieve the second objective. The details of GPEH bluff body and the experimental setup are presented in Figure 1. A laboratory wind tunnel with circular cross-section has been utilized to assess GPEH performance. As seen in Figure 1, a honeycomb EH structure with a 400 mm diameter can be placed within the settling chamber of a wind tunnel to provide a steady incoming flow. The windspeed that was produced fluctuated between 1 and 6 m/s. For physical and geometrical parameters of the EH experimental prototype, ref. Table 1, with $C_{p}$ being EH piezoelectric transducer clamping capacitance; $M$ being EH bluff body effective mass; $ζ_{1}$ being damping ratio; $ω_{1}$ – being natural frequency of vibration; $θ$ - being equivalent electro-mechanical coefficient, for details see.⁷ The circular cross-sectioned GPEH bluff body was attached to GPEH piezoelectric cantilever, as seen in Figure 1.

Figure 1.

Wind tunnel and experiment setup.¹¹⁰

Table 1.

EH physical and geometrical parameters.

Properties	Values
$M$ (gram)	4.2
$ζ_{1}$	1.3 $\cdot 10^{- 2}$
$ω_{1}$ (rad/s)	48.0
$θ$ (N/V)	5.0 $\cdot 10^{- 5}$
$C_{p}$ (nF)	30.5

The bluff body was composed of rigid foam and was 0.12 m long, 0.03 m in diameter. The logarithmic decrement approach has been used to test the prototype piezoelectric cantilever damping ratio. Using a hot-wire anemometer, the approaching windspeed U was determined. Utilizing a digital oscilloscope, the voltage $V$ output was measured.

GPEHs being typically designed to scavenge a low-speed range of wind energy, e.g., U ≤6 m/s, within the so-called “Fresh breeze” region, see Table 2. EH operated as follows: periodic vortex generation and shed occurred when a continuous wind acted on the cylinder bluff body, and the alternating pressure differences induced horizontal lift force to act on the EH bluff body. Since the EH cantilever’s beam provided elastic support, the aerodynamic force caused bluff body vibrations along the horizontal y-axis. Specifically, the vibrational frequency may be locked on the EH resonant frequency (known as lock-in) and the EH system will create a large amplitude vibration when the VIV shedding frequency nears the EH system’s natural resonance frequency. Piezoelectric sheets can convert EH beam vibration into electrical power/voltage. Conversely, the horizontal aerodynamic force exerted on GPEH bluff body stimulated the dielectric elastomer generator, allowing vibrational energy to be further transformed into electrical energy. Previous evaluations have revealed that the suggested EH system may provide comparatively high output power within a specific windspeed range, allowing it to function well within a wind environment at the proper wind speed range. Another benefit of the suggested setup is that a dielectric elastomer generator integrated into the EH cylinder bluff body may significantly improve the energy-collecting capabilities of a comparable VIV-based piezoelectric generator.

Table 2.

Windspeed ranges.

Level	Regime	Windspeed range (m/s)
0	Calm	0 - 0.2
1	Light airflow	0.3 - 1.5
2	Breeze, light	1.6 - 3.3
3	Breeze, gentle	3.4 - 5.4
4	Breeze, moderate	5.5 - 7.9
5	Breeze, fresh	8.0 - 10.7
6	Breeze, strong	10.8 - 13.8

When incoming wind speeds are high for EH (above level 6 m/s), traditional propeller generators perform better since the motor size better fits in situ windspeeds. It’s also significant to note the other EH kinds that currently exist.

Enhanced GPEH, based on cooperative collision and vibration modes²³

Self-regulation for triboelectric nano-generator with wind-speed self-powered sensor²⁴

EH, hybrid water-proof magnetically with piezoelectric-electromagnetic coupling²⁵

Figure 2 presents a schematic flowchart for advocated long-term experimental-based multimodal EH reliability analysis. The suggested multimodal survival assessment scheme offers a consistent solution to data de-clustering, for alternative reliability/statistical techniques to account for the clustering effects see.²⁶ For a comprehensive review reflecting modern reliability methods within green energy harvesting see.²⁷ From the management standpoint, this study advocates roust, novel approach for structural lifetime prognostics, and thus related to economic viability.

Figure 2.

Flowchart for described long-term multivariate reliability assessment.

Application to multi-dimensional dynamic control

The failure of a building typically occurs typically due to its incapacity to fulfill specified functional requirements. Structural failure is a comprehensive concept that includes various phenomena, such as loss of stability, elevated reaction levels in displacements, velocities, accelerations, and plastic deformations or fractures caused by overload or fatigue. The consequences of different types of failures also differ substantially. The malfunction of a single sub-component does not necessarily imply that the entire structure becomes incapable of withstanding the applied loads. A sudden loss of stability frequently leads to a complete and catastrophic structural failure. Failure may stem from a convoluted sequence of detrimental events, potentially caused by a combination of unlikely external or human-induced acts and internal deficiencies. Control approaches are often based on the minimization of objective functions (or loss functions), which differ in character depending on the specific control algorithm employed. Loss functions are frequently expressed in terms of the costs associated with response and control mechanisms. Structural dependability criteria and the associated costs of structural failure are rarely explicitly addressed. The failure of a structural multidimensional system can be clearly defined as comparable to the objective function, serving as input for real-time control algorithms, which are categorized as either online or pre-calibrated.²⁸

Research gap

The research gap can be succinctly delineated as follows: Recent years have witnessed substantial advancements in data-driven artificial intelligence methodologies addressing uncertainty quantification.²⁹; physics-based frameworks for multi-failure-mode system reliability and optimal design, both at the sample level and at the probability density level.³⁰; approaches that integrate uncertainty quantification with increasingly refined deterministic models and rich monitoring datasets.³¹ Those above-mentioned approaches are however mostly suitable for numerical MCS models, where multivariate system dynamics can be simulated in a certain way, thus improving underlying data sample. Current reliability approaches, with the exception of those based on Monte Carlo simulations, are constrained in dimensionality, specifically NDOF ≤ 2D. When only a single data sample is available, it is infeasible to employ widely-used Monte Carlo techniques, such as significance sampling, because the underlying dataset cannot be resampled. In the latter scenario, the proposed multivariate dependability approach offered the designer a robust and unparalleled solution.

Method

This Section introduces the novel Life Time Distribution (i.e., LTD) estimation approach for complex environmental dynamic systems, prone to multiple failure or damage modes during planned service duration. Consecutive time-lapses between component-wise maxima indicate crossing threshold, are denoted as $L$ , indicating the dynamic system’s consecutive life spas $L_{i}$ , $i = 1, 2, \dots$ . Thus $L$ is a Random Variable (RV), that represents dynamic system’s lifetime. In particular, when the number of system’s dimensions (or components) as well as failure modes is high, modern engineering reliability methodologies do not provide the ready-to-use formula to analyse LTD of complex engineering systems.^29–38 In theory, it is eligible to directly assess the target Cumulative Distribution Function (CDF)

LTD (L) = Prob (Lifetime \leq L)

(1)

Complex dynamic system’s reliability can be assessing either by employing e.g., enough measurement data or by carrying out extensive direct Monte Carlo Simulations (MCS).^{19,20,39–45} The cost of computation and experimentation may be practically prohibitive for many complex dynamic energy systems. In the following LTD evaluation method for EH systems will be outlined, which enables the reduction of measurement and MCS costs at the design stage. Taking into consideration EH structural dynamic Multi-Degree Of Freedom (i.e., MDOF) load (or response) system vector: $X = (X_{1} (t), X_{2} (t), \dots, X_{NDOF} (t))$ , being either measured or MCS over a representative (sufficiently extended) timelapse $(0, T)$ with NDOF being number of system dimensions/components. EH system’s 1-Dimensional (1D) component-wise global maxima, being then denoted as ${X_{1}}_{T}^{\max} = \max_{0 \leq t \leq T} X_{1} (t)$ , …, ${X_{NDOF}}_{T}^{\max} = \max_{0 \leq t \leq T} X_{NDOF} (t)$ . By a sufficiently extended observational timelapse $T$ one means a large enough $T$ compared to dynamic system auto-correlation, auto-regression, as well as relaxation times. ${X_{1}}_{1}, \dots, {X_{1}}_{N_{X_{1}}}$ being temporally consequent local maxima of the system’s critical component random process $X_{1} = X_{1} (t)$ occurring at discrete, temporally increasing time instants $t_{1}^{X_{1}} < \dots < t_{N_{X_{1}}}^{X_{1}}$ within $(0, T)$ . Similar definitions are valid for the rest of MDOF EH principal components: $X_{2} (t), \dots, X_{NDOF} (t)$ . All series-type dynamic system’s principal components were assumed here to be non-negative, holding survival probability $P$

P = ∭_{(0, \dots, 0)}^{(η_{X_{1}}, \dots, η_{X_{NDOF}})} p_{{X_{1}}_{T}^{\max}, \dots, {X_{NDOF}}_{T}^{\max}} ({x_{1}}_{T}^{\max}, \dots, {x_{NDOF}}_{T}^{\max}) d {x_{1}}_{T}^{\max} \dots d {x_{NDOF}}_{T}^{\max}

(2)

being target survival probability of dynamic EH system, relying on component-wise failure limits, denoted here as

η_{X_{1}}

η_{X_{2}}

,… with

p_{{X_{1}}_{T}^{\max}, \dots, {X_{NDOF}}_{T}^{\max}}

being system’s Joint Probability Density Function (JPDF) of component-wise global maxima. If a dynamic EH system’s NDOF being large, it is often impractical to directly assess JPDF

p_{{X_{1}}_{T}^{\max}, \dots, {X_{NDOF}}_{T}^{\max}}

and thus target system’s failure probability

P_{failure} = 1 - P

.^21,46–51 where survival probability

P

has to be estimated following Equation (1). Next,

L_{expected} \equiv E [L] = \int_{0}^{\infty} L \cdot \frac{T}{P_{failure}} d LTD (L)

(3)

having failure/damage risk/probability

P_{failure}

. Series-type dynamic system is regarded as damaged or failed when any EH system’s critical component

X_{1} (t)

out-crosses

η_{X_{1}}

; or

X_{2} (t)

out-crosses

η_{X_{2}}

, … Fixed failure/hazard/risk levels

η_{X_{1}}

η_{X_{2}}

… Being individually designer pre-set for each system’s 1D key part or component.

{X_{1}}_{N_{X_{1}}}^{\max} = \max {{X_{1}}_{j}; j = 1, \dots, N_{X_{1}}} = {X_{1}}_{T}^{\max}

{X_{2}}_{N_{X_{2}}}^{\max} = \max {{X_{2}}_{j}; j = 1, \dots, N_{X_{2}}} = {X_{2}}_{T}^{\max}

, , … Dynamic EH system’s critical 1D components

X_{1}, X_{2}, \dots

being

λ

-scaled, and then non-dimensionalized

X_{1} \to \frac{X}{λ η_{X_{1}}}, X_{2} \to \frac{X_{2}}{{λ η}_{X_{2}}}, \dots

(4)

making all scaled EH system’s principal parts (or components) non-dimensional, possessing the same failure (risk/hazard/damage) limit equal to

λ \leq 1

, with

λ = 1

corresponding to the initial (unscaled, non-dimensional) system. Dynamic EH system 1D component’s local maxima, merged into a temporally increasing synthetic vector:

R \equiv \vec{R} = (R_{1}, R_{2}, \dots, R_{N})

according to the merged temporal vector

t_{1} < \dots < t_{N}

N \leq N_{X_{1}} + N_{X_{2}} + \dots

. Each component-wise local maxima

R_{j}

being encountered at an instant

t_{j}

, and it corresponds to either

X_{1} (t)

X_{2} (t)

, or other dynamic EH system’s critical components. Synthetic

\vec{R}

-vector possesses 0 net data loss, see Figure 3.^52–58

Figure 3.

Illustration of 3 component-processes X, Y, Z combined to form the synthetic nondimensional vector R.

The lifetime RV $L$ distributed by LTD, is defined as follows:

L_{i} = t_{i} - t_{i - 1}

(5)

for

i = 2, \dots, N

. Synthetic system 1D process (vector)

R ≜ R (t)

holding principal information about target LTD PDF.^59–64 Survival probability:

P = P (1)

to be expressed via the Mean Up-crossing Rate (MUR) function

P (λ) \approx \exp (- ν^{+} (λ) T); ν^{+} (λ) = \int_{0}^{\infty} ζ p_{R \dot{R}} (λ, ζ) d ζ

(6)

holding $ν^{+} (λ)$ as MUR of a damage, failure or hazard threshold level $λ$ related to synthetic non-dimensional vector: $R (t)$ . MUR function in Equation (6) is known as Rice formulation with $p_{R \dot{R}}$ being JPDF for $(R, \dot{R})$ with $\dot{R} = R^{'} (t)$ being the time-derivative.⁶⁵ Note that Rice formula is mentioned here only from historical perspective, in this case study MUR was extracted directly from raw measured data, without utilizing Rice formula. Dynamic EH system approaches its pre-set failure (damage) limit $λ \to 1$

\log_{λ \to 1} P_{failure} = \frac{T}{L_{expected}} = ν^{+} (λ) T \Rightarrow L_{expected} = \frac{1}{ν^{+} (1)}

(7)

according to Equation (3). In the above, the MDOF dynamic EH system was pre-assumed to be jointly quasi-stationary. Environmental in situ scatter-diagram typically comprises

M

environmental states, each environmental short-term environmental state holding probability:

q_{m}

, with

\sum_{m = 1}^{M} q_{m} = 1

m = 1, . ., M

. According to long-term reliability formulation

LTD (L) \equiv \sum_{m = 1}^{M} {LTD}_{m} (L) q_{m}

(8)

with

{LTD}_{m} (L)

being LTD, corresponding to a short-term quasi-stationary environmental state, indexed with

m

.^65–73 The following Section will illustrate how LTD

q -

quantiles of interest

L_{q} = {LTD}^{- 1} (q)

(9)

may be derived from available system dynamics, physically measured or numerically MCS underlying timeseries, with

q \in (0, 1)

, and

{LTD}^{- 1}

being inverse function of LTD, i.e.,

{LTD}^{- 1} ° LTD = 1

, with

1

standing for identity operator. Note that at high damage (failure or hazard) levels, EH dynamic system damage (failure or hazard) events become nearly independent; thus, LTD will follow a Poisson-type PDF holding parameter

ν^{+} (λ) T

. The Poisson-type PDF is given by:

f_{Poisson} (x) = e^{- λ_{P}} λ_{P}^{x} / x!

with Poisson parameter

λ_{P} = N \cdot p (1)

.^74–80

Note that for multifaceted series-type system its failure is determined by a first passage event, and every parallel-type system can be equivalently restructured as a series-type system.

Results

Piezoelectric EHs scavenge energy from low windspeeds as EH efficiency may dramatically decline as the windspeed hits more significant levels. Therefore, since the stopper would lessen the substantial vibration amplitude that high wind creates, it can be added to EH prototype to avoid damage when windspeeds exceeds a certain threshold. The following equation was utilized to compute the aerodynamic force, which acts on GPEH bluff body, utilizing recorded output voltage $V$ time series

F = \frac{C_{p}}{θ} (M \ddot{V} + C \dot{V} + K V) + θ V

(10)

Equation (10) contains experimental setup parameters: $M = 4.2 {\cdot 10}^{- 3} kg$ ; $C_{p} = 30.5 {\cdot 10}^{- 9}$ F; $C = 5.3 {\cdot 10}^{- 3} N / (m / s)$ ; $k = 9.7 N / s$ $θ = {5.0 \cdot 10}^{- 3} N / V$ , for details see Table 1. Both variables (force, voltage) measured synchronously, hence they constitute one bivariate RV. In this study, a bivariate (2D) random process $X = (X (t), Y (t))$ to be considered, consisting of EH output voltage and horizontal total aerodynamic force records $X (t), Y (t)$ , having 1^st component (voltage $X = V$ ) measured, along with 2^nd component (horizontal aerodynamic force $Y = F$ ) being computed based on Equation (10), temporally synchronously, over the same period $(0, T)$ .^81–87 Assuming that samples: $(X_{1}, Y_{1}), \dots, (X_{N}, Y_{N})$ being taken at $N$ equidistant, discrete temporal instants $t_{1}, \dots, t_{N} \in (0, T) .$ In the following 2D Joint Cumulative Distribution Function (JCDF) $P (ξ, η) : = Prob ({\hat{X}}_{N} \leq ξ, {\hat{Y}}_{N} \leq η)$ will be considered, corresponding to the 2D vector $({\hat{X}}_{N}, {\hat{Y}}_{N})$ , having principal components ${\hat{X}}_{N} = \max {X_{j}; j = 1, \dots, N}$ , and ${\hat{Y}}_{N} = \max {Y_{j}; j = 1, \dots, N}$ . In the current study, $ξ$ and $η$ are output voltage $V$ and horizontal aerodynamic force $F$ values, respectively, synchronously recorded in time.^88–94 Note that the presented multi-variate reliability method possesses no limit on system number of dimensions or/and components. An analyzed example was limited to NDOF = 2D due to experimental limitations.^95–102

Measured windspeed PDF frequently follows a Weibull-type PDF with the following shape.¹¹

H (\bar{U}) = \frac{δ}{β} {(\frac{\bar{U}}{β})}^{δ - 1} \cdot e^{- {(\frac{\bar{U}}{β})}^{δ}}

(11)

where

β

and

δ

are fitted to a given PDF, known as scale & shape parameters.

\bar{U} = \frac{U}{ω_{n} D}

represents reduced windspeed with

ω_{n}

being natural EH frequency and

D

being EH bluff body cross-flow dimension. Weibull-type PDF, with a shape factor

δ \in (1, 3)

being often found in engineering. Weibull-type PDF is typically used to characterize how galloping EH responds to in situ wind load data.^103–107 Weibull-type PDF scale parameter

β

was chosen as equal to 5, allowing for lower wind speeds. Figure 4 left presents a cross-correlation between EH measured voltage versus corresponding EH total horizontal aerodynamic force. Next, Figure 4 left and Eq. (1) indicate that horizontal aerodynamic force is strongly linearly correlated with underlying EH voltage due to the smallness

o (1)

\frac{C_{p}}{θ}

. In extreme PDF tail’s non-linear relationship dominates, as seen from Figure 4 left, Eq. (9) due to an increase in values of

\dot{V}

\ddot{V}

. Figure 4 right presents practical example of a non-dimensional synthetic assembled vector

\vec{R}

. Synthetic vector

\vec{R}

may not have any particular physical meaning on its own, being assembled of various dynamic system’s principal components with different inherent physical dimensions. Index

j

represents running index of EH component-wise local maxima, recorded in a temporally increasing order.

Figure 4.

Left: measured output voltage V versus synchronous total horizontal force⁹⁰ Right: non-dimensional scaled synthetic system’s vector $\vec{R}$ .

Figure 5 left presents the expected EH system LTD $L_{expected}$ , along with system LTD $q -$ quantile $L_{q}$ for 90% quantile $q = 0.9$ . In Figure 5 right lower corner, the target LTPDF (LifeTime PDF) is expressed as LTPDF $(L) = \frac{d}{d L}$ LTD $(L)$ , according to Equation (1), represented in green colour. To prevent adding artificial effects, for example, due to sorting windspeeds in increasing order, the underlying windspeeds range utilized for EH dynamic response/loading measurements had been randomly permuted. Figure 5 left has $λ -$ parameter from Equation (6) on its horizontal axis, with the target $λ$ value being $1$ . Figure 5 right shows extrapolation to the target $λ = 1$ level, utilizing modified (namely, 4-parameter) Weibull-extrapolation scheme. Validation of extrapolation results, presented by Figure 5 right was carried out by tenfold reducing original measured dataset, i.e., retaining only each 10^th datapoint. Tenfold reduced (thinned) underlying dataset led to quite similar predictions, within predicted 95% CI, for both the expected system’s lifetime as well as the system’s LTD percentiles/quantiles.

Figure 5.

Left: Finding an expected LTD $L_{expected}$ (blue line), along with LTD $q -$ quantile $L_{q}$ (red line); LTD given on a decimal log scale. Right: extrapolation towards target (design) level, the star indicates the target nondimensional level with $λ = 1$ .

Discussion

Key design, engineering advantages of advocated multivariate Gaidai reliability method are summarized in Table 3. Multivariate Gaidai hypersurface methodology has no limit on dynamic GPEH system’s NDOF, whereas existing reliability methods are limited to NDOF

\leq 2

Table 3.

Advantages of presented multi-modal risk assessment approach.

	Multi-modal structural Gaidai’s risk assessment approach	Existing 1D & 2D reliability/risks evaluation methods
Potentials to investigate nonlinear multi-variate systems	NDOF = $\infty$	NDOF $\leq 2$
Potentials to accommodate multiple dynamic system’s nonlinearities	Full	Partial–to–Full
Potentials to assess non-stationary multi-modal dynamic system, possessing underlying trend	Full	Partial

A potential limitation of presented multivariate reliability method is its reliance on dynamic system joint quasi-stationarity presumption. For non-stationary dynamic systems with underlying multidimensional trends, system trends should be first identified, prior to the advocated reliability methodology being applied. Future studies will include temporally-varying components critical (e.g., damage or failure) thresholds $η_{i} (t)$ , $i = 1, \dots,$ NDOF.

Advocated multimodal reliability methodology can assist in devising dynamic control systems, aiming at the reduction of damage or failure risks.

Presented reliability methodology is mathematically exact, i.e., synthetic vector $R (t)$ fully represents system’s joint dynamics and the only source of inaccuracies may originate from either selected extrapolation scheme or the underlying data itself. Note that the presented methodology is generic, it can be applied to a range of complex multidimensional dynamic systems, provided representative input, e.g., sufficiently long-time system dynamics series. Authors are not aware of existing alternative reliability methods, applicable to systems with dimensionality above 2D, except MCS-based methods, which do not apply to measured data. Key advantage of advocated multidimensional reliability method lies within its applicability to raw measured multidimensional data, thus offering solutions where multivariate MCS-based methods are not applicable.

Conclusions

Existing reliability approaches lack the capacity to handle high-dimensional systems, given nonlinear inter-correlations between the system’s parts (components). The capacity to assess LTD for non-linear, high dimensional dynamic systems constitutes major strength of the presented state-of-the-art multivariate reliability method. The current study investigated nonlinear EH dynamic behaviour under realistic windspeed conditions. LTD throughout EH’s specified design lifespan was predicted, utilizing a novel multivariate reliability approach. The theoretical basis of the proposed strategy was extensively analysed. Dynamic EH systems are complex and often highly dimensional, necessitating the development of new, accurate, yet reliable methods, utilizing efficiently even limited underlying datasets. Extensive direct MCS, as well as lab./field measurements, are often unaffordable. The novel multivariate reliability method, advocated in this work has already shown to be efficient, if applied to a wide range of dynamic structural systems – forecasts that were made had been overall quite accurate. Primary objective of this case study was to create a generic high-dimensional EH system’s reliability approach that is multivariate, multi-purpose, reliable, yet easy to use. As can be observed, the proposed multivariate reliability methodology generated reasonably narrow CIs. As the example of measured structural EH response in this case study has shown – it is well feasible to forecast LTD of complex EH systems, thus improving their sustainability at the design stage. Regarding further research direction, authors have previously applied modification of presented method to mixed system types, where failure is defined not by crossing 1D threshold by one of critical components, but by out-crossing failure hypersurface in multidimensional space.^108,109

Results verification

Even though full scale measured data does not require verification, the applied statistical analysis does.

I. Data-based verification: the underlying dataset was reduced 10-fold by retaining only each 10th consecutive data-point within multidimensional timeseries dataset. Forecasts, based on the reduced dataset was found to lie within 95% CI, estimated from the full dataset.

II. Method-based verification: for reduced dimensionality, i.e., for 2D system, forecast, based on the proposed multidimensional methodology was compared with Gumbel Logistic copula-based method, found to lie within 95% CI, estimated from the full dataset.

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 iD

Oleg Gaidai

Data Availability Statement

Availability of data & material–measured data will be made available on request from the corresponding author. For source code see https://github.com/OlegGaidai/ (MATLAB code) and (R programming language).

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