1. Introduction
Energy harvesting (EH) from environmental energy sources such as vibrations, wind, heat, solar power etc., has attracted significant research interest due to the growing demand for energy. Vibration energy harvesting is the conversion of the ambient vibrations into electrical energy and provides a viable solution for powering small electronic components. Among the principles of energy conversion, piezoelectricity is known as one of the most efficient and practical way for conversion of mechanical vibration energy into electrical energy [
1,
2].
Piezoelectric energy harvesters (PEHs) have attracted research interest because of the high conversion efficiency compared to electromagnetic and electrostatic based harvesters [
2]. However, efficiency of PEH systems depends on several parameters such as material properties, geometric dimensions, electric circuit components, etc. Conventional piezoelectric materials, such as lead-based piezoceramics, are brittle in nature and difficult to manufacture. Despite the high electromechanical response of lead-based piezoelectric ceramics, the problem of brittle nature has not been effectively solved for a long time. To overcome these drawbacks, new flexible piezoelectric composite materials have attracted attention of the research interest [
3]. These new type of piezoelectrics are suitable for wearable energy harvester applications. Wearable energy harvesting focuses on the fabrication of reasonable-cost smart garments using special nanostructure piezoelectric fibers that make use of the vibrations due to the natural movements of the body [
4,
5]. Various materials are proposed, e.g., barium titanate and polyvinylidene fluoride, with satisfactory results, indicating the potential for industrial mass production of commercial devices. However, this approach seems more appropriate for personalized sensoring and monitoring applications in everyday life, such as patient monitoring, robotic assistance, etc.
Piezoelectric energy harvesters are typically designed as cantilever beams or plates with one or two piezoceramic layers covering the structure either entirely or partially. Several methods have been employed to model the electromechanical behavior of a PEH. The works of Erturk and Inman [
6,
7] present analytical distributed electromechanical models for unimorph and bimorph cantilever beam which provides closed form expressions for harmonic behavior of PEHs. Based on classical laminated theory, a distributed parameter electroelastic model was developed in [
8] for piezoelectric energy harvester structurally integrated to cantilever composite beam. Electrical and mechanical closed form steady state solution response have been obtained by harmonic base excitation.
On the other hand, the finite element method has proven to be very useful in modeling the dynamics of PEHs [
9,
10,
11]. A coupled electromechanical finite element (FE) model for predicting the electrical power output of piezoelectric energy harvester plates was presented in [
9]. The FE formulation is based on the Kirchhoff plate assumptions which is suitable for modelling thin structures. Additionally in this paper, an optimization problem for aluminum wing spar generator of an unmanned air vehicle (UAV) was solved for the maximum electrical power without exceeding a prescribed mass addition limit.
Most analytical and FE models of PEHs are based on classical beam/plate theories which ignore shear stresses and are suitable for modelling thin structures. However, for accurate modeling of thick PEH for various applications, such as aircraft wing structure or wind turbine blade, higher order shear deformation theories are needed. Recently, Khazaee et al. [
11] developed a coupled electromechanical model for non-uniform piezoelectric energy harvesting composite laminates based on third-order shear deformation theory. The presented high-order shear FE model also considers the contact layer thickness in the harvester beams, non-uniformity in the piezoelectric sheet, non-constant thickness of the piezoelectric sheet and is suitable for analysis of a wider range of problems in piezoelectric harvesting.
On the other hand, several studies have been carried out on design optimization of PE harvester to improve the energy harvesting efficiency by optimizing the dimensions of the piezoelectric energy harvesters [
10,
12,
13]. The performance of few important piezoelectric materials has been simulated by Kumar et al. [
10] for unimorph-type cantilever piezoelectric energy harvester. The genetic algorithm (GA) optimization approach is used to optimize the structural parameters of mechanical energy-based energy harvester for maximum power density. In [
12], a new design of piezoelectric energy harvester subject to tip excitation is proposed. The mechanical and electrical behaviors of piezoelectric materials are solved by coupled analysis using ANSYS, and the design optimization is performed for power maximization using Sequential Quadratic Programming (SQP) algorithm.
Most studies on design optimization of PEH are limited to single objective optimization techniques using the maximization of power output as the main performance criterion. However, not much research has been carried out on the optimization of the parameters of the vibration-based piezoelectric harvester based on multicriteria.
The concept of energy harvesting has received much attention in recent years to enhance operational autonomy of low-power electronic applications (biosensors, microelectronics etc.) as well as for aerospace applications (e.g., unmanned aerial vehicle (UAV)), as it can offer a sustainable solution for power supply from ambient vibrations. However, a crucial aspect of the design of such kind of systems is the potential effect that the additional mass of the piezoelectric energy harvesting system might have on the performance of the initial structure. Since mass densities of typical piezoceramics used in energy harvesting are considerably large compared to typical substrate materials such as steel, aluminum or graphite/epoxy material, the minimization of the mass of the system should consider as an additional performance criterion in optimal design of PEH.
A design optimization problem for UAV applications has been studied in [
9]. The aluminum wing spar of a UAV is modeled using a FE plate model appropriately modified to design a generator wing spar. In order to take into consideration the mass added by the piezoelectric layers, an upper limit for mass addition is imposed as a design constraint. The resistor load
and the geometric dimensions of the embedded piezoceramics have been determined to maximize the generator spar’s output power by varying their values in a reasonable range without applying any optimizationtechnique.
Recently, a multi-objective design optimization of piezoelectric energy harvesting system for UAV has been presented in [
14]. In contrast with the previous approaches, this work considers the minimization of mass added by the embedded piezoceramics as an additional performance criterion along with the maximization of the power output as design optimization objectives. Non-dominated Sorting Genetic Algorithm II (NSGA-II), Non-dominated Sorting Genetic Algorithm III (NSGA-III) and Generalized Differential Evolution 3 (GDE3) algorithms are carried out to optimize the structural and the electric circuit parameters of vibration-based piezoelectric energy harvester. The results prove that Multi-Objective Genetic Algorithm (MOGA) approach is very promising for optimal design of PEH for aerospace applications.
So far, the literature review shows that the consideration of the material strength in optimal design of PEH is somewhat limited. In [
12], the design of piezoelectric energy harvester subject to tip excitation is addressed under the constraint of maximum bending stress. The work of [
15] focuses on nonlinear energy harvester design optimization with magnetic oscillator under the constraint that the maximum strain on piezoelectric material do exceed the allowable limits. However, the strength of the piezoelectric material is another crucial parameter in designing energy harvesters. This parameter is of major importance since the values of the strength of piezoelectric materials are much lower compared to the strength of substrate materials such as steel, aluminum, and brass [
9,
15]. Therefore, the stress generated in the energy harvesting process should be considered as a new design constraint in order to ensure the adequate mechanical behavior of the device.
Motivated by the above consideration, this study presents a multicriteria optimization approach to minimize mass and maximize power output in piezoelectric energy harvesting systems within the limits of allowable stress of the piezoelectric layers. A finite element model has been developed for modeling the behavior of the plate-type PEH under base excitation. The formulation is based on laminated plate theory combined with the first-order shear deformation theory (FSDT) for which each piezoelectric layer has one additional electrical degree of freedom. This paper extends the modeling and the optimization problem presented in [
14] by considering additional constraint on bending stress of the piezoelectric layers. NSGA-II, NSGA-III and GDE3 algorithms are applied in the optimization process and both trade-off Pareto optimal fronts and the respective optimal design are obtained. Finally, the results are analyzed and discussed.
5. Optimization Results
Experiments were carried out on a workstation running MATLAB 2018b on Windows 10 with an Intel Core i9 7960X @2.8 GHz CPU and 64 GB DDR4 RAM. The three multi-objective algorithms employed are those that are implemented in the PlatEMO v3.4 [
27] software, which is freely available for research purposes. It should be noted that PlatEMO implements numerous MOGA and other algorithms.
The optimization procedure is shown in
Figure 3. The MATLAB code that we have developed for the FE model implementation is executed at each iteration to calculate the objectives and the constraints required by the optimization procedure (e.g., the output power, the mass, the stress distribution). PlatEMO software provides the implementations of the multi-objective algorithms used (NSGA-II, NSGA-III, GDE3). Our MATLAB code is embedded in the procedure as a set of functions, which, when the values for the decision variables are given as input, return the fitness of the objective functions and the constraints violation amount. The convergence criterion is a predefined number of maximum generations allowed. As long as the criterion is not satisfied, the algorithm continues to evolve to population. Finally, the Pareto optimal solutions are obtained. A post-processing procedure is carried out with the aid of two extra python packages, pfevaluator [
28] that computes various Pareto front performance metrics, and OAPackage [
29] that can easily identify Pareto optimal solutions in a population of solutions.
The experiments entailed running each of the algorithms NSGA-II, NSGA-III, and GDE3 for 50 generations with a population of 50 individuals. Based on the collected findings of each run’s final population, a Base Pareto Front (BPF) is created, as shown in
Figure 4. The BPF comprises 493 points (135 from NSGA-II, 202 from NSGA-III, and 156 from GDE3). It is easily shown that GDE3 manages to extend the BPF to the top-right part of the graph. Each algorithm was run 10 times, each time taking roughly 5500 s. The indifference of the running times among the three algorithms can be attributed to the fact that the heavier parts of the approach, due to FEM, are the evaluation of the power objective and the assurance of the stress constraint which is common across the algorithms.
All three algorithms begin to converge, rather early, around the 10th generation.
Figure 5 shows the evolution of the Pareto fronts for generations 3, 10, and 50. In addition, the Pareto fronts for generation 5, 10, and 50, are shown in
Figure 6. BPF is also shown in each subfigure as a substitute for the unknown optimal Pareto front. GDE3 can achieve solutions that NSGA-II and NSGA-III were unable to attain for this problem and experiment setup. As a result, GDE3 achieves solutions that generate a maximum power of 322.2709
, whereas the maximum power generated by solutions produced by NSGA-II and NSGA-III is 284.3626
. By inspecting the evolution of solutions across the generations we observed that GDE3 attained solutions with power greater than 284.3626
for all 10 runs and this occurred on average at about the 33th generation. This occurred as early as at the 18th generation and as late as at the 45th generation. More experiments were undertaken for NSGA-II and NSGA-III, in an attempt to achieve solutions with greater values for the generated power. A population of 100 individuals and 100 generations is used for the optimization process. This time, each run took about 21,000 s. Nevertheless, the maximum power generated is again no greater than 284.3626
.