Parameter uncertainty and stochastic optimization ofcantilever piezoelectric energy harvesters

V.R. Franco 1, P.S. Varoto1

1 Laboratory of Dynamics, University of Sao Paulo - School of Engineering of Sao Carlos,Av.Trabalhador Sao-Carlense, 400,ZIP CODE 13566-590, Sao Carlos - SP - Brazile-mail: vrfranco@sc.usp.br, varoto@sc.usp.br

AbstractIncreasing demand for sustainable energy sources has attracted considerable attention to the issue of har-vesting electrical energy from structural vibration signals through the use of piezoelectric materials. In thiscontext, structural optimization techniques play an important role in the design of a given energy harvest-ing device. The goal of the present article is to compare deterministic and stochastic optimization methodsapplied to the design of energy harvesters. Firstly, a comprehensive discussion on the effects of aleatory un-certainties on the dynamics of a beam type piezoelectric energy harvesters carrying a tip mass is performedusing the well known Monte Carlo Simulation (MCS) method. Following, a multi-parameter SequentialQuadratic Programming (SQP) optimization technique is employed in either the deterministic and stochasticproblems in order to obtain a set of optimum geometric parameters for the harvester. Results from the numer-ical simulations simulations revealed that the superior performance of the stochastic programming approachin terms of the harvested electrical power. Additionally the improved results obtained from the stochasticprograming approach also reinforces the importance of accounting for uncertainties in the design of energyharvesting systems.

1 Introduction

In the past years considerable research efforts have been dedicated to the issue of piezoelectric energy har-vesting through the conversion of vibration signals into usable electrical energy using the well known Euler-Bernoulli beam model [1], [2]. A crucial issue in the design of beam type piezoelectric energy harvestersis to properly tune the device’s first natural frequency to the frequency of the input motion thus maximizingthe electrical output generated in the piezo layers. Generally speaking, this tunning process raises someimportant questions as for example: (i) what is the nature of the input motion ? (ii) what frequency rangeis covered by the input base motion ? (iii) What geometric scaling requirements must be met in the designof a given harvester ? In order to properly address these issues and aiming to maximize the mechanical toelectrical energy conversion process several optimization strategies have been recently proposed.

Optimization of piezoelectric energy converters are generally based on variation of design geometrical pa-rameters [3], [4] and on the properties of the electrical circuits used in the mechanical to electrical transduc-tion process [5]. In either case, the effects of specific mechanical or electrical parameters on the device outputelectrical power are investigated as well as comparative analysis between several geometrical or electricalconfigurations are made in the sense that the optimum performance of the device can be achieved. Addi-tional work has been reported on the issue of generating the largest electrical power by changing the shapeof the beam and piezo layers [6], [7], [8]. Similarly, performance improvements based on the optimizationof nonlinear properties and frequency range of operation have been considered [9], [10], [11].

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A common characteristic among many of these optimization strategies is that they often seek for a singleoptimal parameter (mechanical, electrical or electromechanical) at a time, where in practical applications,several parameters can vary simultaneously thus causing a combined effect on the harvester’s performance.The importance of accounting for uncertainties (Epistemic or Aleatory) in improving the predictability of agiven mean model has been emphasized in several recent works [12], [13]. In the context of piezoelectric en-ergy harvesting most of the reported literature presents a simplified assessment of the problem of propagationof uncertainties, with the exception of a few important contributions [14], [15].

The goal of the present article is to perform a comparative analysis between deterministic and stochasticoptimization methods applied to energy harvesting systems. The deterministic Sequential Quadratic Program(SQP) and the nonlinear stochastic programing techniques are used with the electromechanical model of acantilever piezoelectric harvester. First, the multi-parameter SQP optimization technique is used to producea set of optimum parameters for the harvester. These parameters are used to calculate a series of optimumelectromechanical Frequency Response Function (FRF) mean model covering the 0−100Hz test frequencyrange. Next, Random perturbations are introduced in each design parameter in order to investigate howthe aleatory uncertainties affect the mean output electrical. The Monte Carlos Simulation (MCS) methodis used to perform the uncertainty analysis. Finally, the perturbed model is used in the nonlinear stochasticprogramming method and the optimization results from these two techniques are compared. Numericalresults indicate a superior performance of the nonlinear stochastic technique in terms of the output electricalpower generated by the harvester.

2 Electromechanical FRF Model

This section briefly presents some highlights of the electromechanical FRF model of the harvesting systemused in the numerical simulations. A more comprehensive discussion of the theoretical modeling can befound in [20]. The harvester model used in this work is shown in Figure1. It consists of a cantilever beampartially covered by PZT piezoelectric layers on both the top and bottom surfaces subjected to the base driveninput displacement ub(t). The harvester’s electromechanical model consists of the FRF relating the relative

!

Mt

ub(t) x

y

!

!

!

!

A-A B-B

!

bS

!

hp

!

hS

!

hp!

hpc!

bp

z

y

z

y

!

Rl

!

vser(t)

!

L1

!

L2

Poling

direction A-A B-B

(a) (b)

x = LS

Figure 1: Bimorph cantilever structural model

displacement coordinate ur (x, t) and the output voltage to the input base motion ub(t) and for the seriesconnection of the piezoelectric layers the expressions for these FRF are expressed as [20]

β (x, ω) =∞∑r=1

σr − χr

∞∑r=1

jωκrσr

ω2r−ω2+j2ςrωrω

1Rl

+ jωCp

2 +∞∑r=1

jωκrχr

ω2r−ω2+j2ςrωrω

φr (x)ω2r − ω2 + j2ςrωrω

(1)

α (ω) =

∞∑r=1

j ωκrσr

ω2r−ω2+j 2ςrωrω

1Rl

+ j ωCp

2 +∞∑r=1

j ωκrχr

ω2r−ω2+j 2ςrωrω

(2)

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Equations (1) and (2) are the frequency response funtions relating two different outputs to the same input.The first, Eq. 1 is the displacement transmissibility FRF since it relates the output relative displacement tothe input base motion. The second, Eq. (2) relates the output electrical voltage produced in the piezo layersto the input base motion. It is important to notice that Eqs. (1) and (2) are multi-modes FRF, i.e., they containan infinite number of mode shapes of the harvesting system. Simpler expressions for single mode FRFs canbe obtained from Eqs. (1) and (2) when r = 1 in the summations in the numerator and denominator.Thevoltage FRF from Eq. (2) can be further used in order to estimate of the electrical power FRF that canbe generated by the harvester. A simple way to calculate the electrical power FRF is to use the followingexpression

γ (ω) =[α (ω)]2

Rl=

1Rl

∞∑r=1

j ωκrσr

ω2r−ω2+j 2ςrωrω

1Rl

+ j ωCp

2 +∞∑r=1

j ωκrχr

ω2r−ω2+j 2ςrωrω

2

(3)

3 Deterministic Optimization Method

The main goal here is to obtain a set of geometric parameters such that the peak amplitude of the voltageFRF and electrical power FRF given by Eqs. (2) and (3) can be maximized . The parameters to be optimizedare: the length of the substructure, Ls, the length of the piezoelectric layer, Lp, the height of the tip mass,ht, and the resistive load, Rl. A initial step in the optimization process is to define a target natural frequencyrange (fmin ≤ fn ≤ fmax). A suitable choice for the limiting values of the project natural frequency rangedepends primarily on the frequency content of the excitation signal and that in turn will vary according tothe nature of the specific field vibration environment in which the harvester is exposed to. For the case ofambient vibration signals it is generally accepted that the range covering the 0− 100Hz covers most of thepractical applications. Independently of the specific limits of the frequency range for a given application, afirst requirement consists in optimizing the harverster’s parameters such that its natural frequency will fallwithin the specified range.

The deterministic optimization problem can be formulated as a minimization problem and can be stated as:

Find X =

LShtLpRl

which minimizes f (X), subject to the constraints

g1(X) : ωr ≥ ωmin = 2πfmin

g2(X) : ωr ≤ ωmax = 2πfmax

g3(X) : LS ≥ LLSg4(X) : LS ≤ ULSg5(X) : ht ≥ Lhtg6(X) : ht ≤ Uhtg7(X) : Lp ≥ LLpg8(X) : Lp ≤ Lsg9(X) : Rl ≥ LRlg10(X) : Rl ≤ URl

(4)

where the fixed terms LLS (lower length of substructure), ULS (upper length of substructure), Lht (lowerheight of tip mass), Uht (upper height of tip mass), LLp (lower length of piezoelectric layer), LRl (lowerload resistance) and URl (upper load resistance) are the pre-assigned parameters and their values are setaccording to the design characteristics of the harvesting system. X is a vector containing the design variables.The objective function f (X) is the inverse of the electrical voltage or power FRF for series connection ofthe piezoelectric layers, given by Eq. (3). Constraints g1(X) to g10(X) depend on the design parameters

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and the optimization problem attempts to find the value of the peak FRF electrical power. In this study,the previously described constraints g1(X) and g2(X) are nonlinear constraints since the modal undampednatural frequency ωr relates the design variables according to a nonlinear relation; g3(X), g4(X), g5(X),g6(X), g9(X) whereas g10(X) are bounds; g7(X) and g8(X) are linear constraints. Further details on thelinear and nonlinear relations among the optimization variables will be provided later. Implementation ofthe deterministic optimization problem defined by relationships 4 is performed by employing the classicalSequential Quadratic Programming (SQP) technique [17] [20].

4 Uncertainty Analysis

The uncertainty analysis of the energy harvesting device shown in Fig. 1 employs the MCS technique andit can be divided in three major steps. First, a mean model in terms of the voltage and power FRFs shownin Eqs. (2) and (3) is generated by employing the SQP multiparameter optimization technique described inthe previous section. The FRF mean model is generated according to a specified target frequency range ofinterest, as pointed out in the previous section. Second, a set of key parameters is defined in order to generaterandom samples according to a prescribed probability density function (pdf) distribution and the perturbedparameters are used in the MCS. The parameters that will be perturbed are:the length of substructure, Ls, theheight of tip mass, ht, the length of piezoelectric layer, Lp, the load resistance, Rl and the modal dampingratio, ζ. For each one of these parameters, a vector of random numbers following a Gaussian distribution isgenerated.

The mean values of the random vectors generated are the values obtained through the SQP optimizationmethod. A Monte Carlo Simulation (MCS) with 2500 realizations (ns = 2500) is then performed in orderto obtain the FRFs confidence intervals. To that end, the vectors of random realizations for each parameterwere combined in ns sets of (Ls,ht,Lp,Rl,ζ) random numbers (rn), which were then used with Eqs. (2) and(3). Therefore the effects of all uncertain parameters are first evaluated simultaneously. The convergencewith respect to the optimum mean model and to the total number ns of realizations used in the Monte Carlonumerical method, can then be studied by constructing the following function [12]:

Conv(ns) =1ns

ns∑j=1

∫||α1(rnj , ω)− αopt1 (ω)||2dω (5)

where αopt1 (ω) is the FRF generated using of optimum parameters obtained through the SQP optimizationmethod. The third and final step in the uncertainty analysis is to consider the effects of perturbations in eachparameter individually. As it would be expected that the effects of uncertainties can present variations fordifferent frequencies. In order to accommodate this variation in the uncertainty analysis of the harvester ofFig. 1, the analysis was performed for five different frequency ranges within the 0−100 Hz frequency range:5− 20, 20− 35, 35− 50, 50− 65 and 65− 80 Hz.

5 Numerical Application

The geometric and material parameters used to generate the deterministic mean model of the harvester shownin Fig. 1 are shown in Tab. 1. The permittivity at constant strain is given in terms of the permittivity in freespace, ε0.

Equation (6) shows the optimum parameters obtained from the application of SQP technique to the parame-ters of Table 1. The columns represent each frequency ranges analyzed. These values were used to generatethe vectors with the random numbers in order to perform the stochastic analysis. For each one of the optimumparameters, the optimum value corresponds to the mean value of the vector formed by the random samples.For the damping ratio, the mean value used is ζ = 0.0027 and corresponds to an equivalent damping ratio

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obtained from actual experimental data [16]. Generation of the vector with random sample according to aGaussian PDF, it is necessary to define the standard deviation for each uncertain parameter and these valuesare presented in Tab. 2. The assumptions of the standard deviation was based in the fact that the three firstparameters (LS , ht and Lp) are dimensions and for the Rl parameter is due to the fabrication issues and thepercentage is explained by the change of the Rl value for each design frequency range. The same reason isapplied for the zeta parameter, since an error of 10% is acceptable for data obtained experimentally.

XSQP =

0.1904 0.1150 0.1001 0.0623 0.05990.0050 0.0051 0.0050 0.0062 0.00590.1863 0.0993 0.0800 0.0205 0.02418568.0 22473 27991 31274 36251

(6)

Once all vectors of random numbers are generated for all perturbed parameters, a MCS with 2500 samplesis then performed. Figure 2 depicts the confidence region (yellow) predictions for the FRFs of the stochasticsystem and the comparison between the deterministic response of the mean analytical model (correspondingto the optimum FRF) and the mean of the random response for the stochastic model, obtained throughMCS (corresponding to MCS FRF). In this figure, it can be noticed that at the lower frequency ranges theresponse is typically described by fairly well-defined natural frequencies and the FRFs for the ensemblespread around that of the deterministic system. As frequency increases, the FRF peak regions become widerand the difference between the peak voltages of the optimum and the mean MCS FRFs becomes larger. Thistrend is also verified by [18]. In this work authors pointed out that the effects of the uncertainties depend onthe frequency and the level of uncertainties. For low frequency ranges, the numerical mean model is relativelyrobust with respect to data uncertainties. As frequency increases, the robustness of the mean model tendsto decrease and the effects of uncertainties in the system parameter become more evident. Furthermore, forall excitation frequencies, the response of the mean model falls within the corresponding confidence region.This fact is commonly verified in other contributions that consider parametric uncertainties (see for instance[19]. For the case of non-parametric probabilistic approach, there are excitation frequencies for which theresponse of the mean model falls outside the confidence region [12], [13], [19].

Next, the the influence of uncertainties in each individual parameter was considered. The goal of this simu-lation is to investigate which parameter or parameters will have larger influence in the harvested power. Thissimulation is particularly important in the design process of the energy harvesting system since the device’snatural frequency is to be tunned to match the frequency of the excitation signal. Hence, if the value ofthe natural frequency becomes sensitive to fluctuations in one or more design parameters, this effect willbecome evident on the voltage and power FRF. Figure 3 depicts the region of peak voltage FRF variationwhen uncertainties in a single design parameter. For convenience the earlier shown result for uncertaintiespresent in all parameters is shown again for comparison purposes. It can be seen that the regions when un-

Geometric ParametersStructure/Structural Part

Substructure (Spring Steel) Piezo. (PZT-5A) Tip MassLength, L (mm) 50.8 - 203.2 0 - LS 5 - 20Width, b (mm) 25.4 25.4 35.4

Thickness, h (mm) 2.54 2.54 5 - 20Material Parameters

Mass density, ρ (kg/m3) 7860 7800 7860Young’s Modulus, Y (GPa) 207 67 207

Piezo. Constant, d31, (pm/V) — -190 —Permittivity, ε33, (F/m) — 830ε0 * —

* ε0 = 8.854 pF/m

Table 1: Geometric and material parameters used to construct the mean model of the harvester

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Parameter Standard DeviationLS 0.0001 (m)ht 0.0001 (m)Lp 0.0001 (m)Rl 10% Rlζ1 10% ζ1

Table 2: Standard deviation for each parameter

0 10 2010

−3

10−2

10−1

100

Ampl

itude

[V/g

]

20 30 4010

−3

10−2

10−1

100

30 40 5010

−3

10−2

10−1

100

Excitation Frequency [Hz]40 60

10−3

10−2

10−1

100

60 8010

−3

10−2

10−1

100

Optimum FRFMCS FRF

Figure 2: Envelopes of the confidence region considering uncertainty in all parameters.

certainties in all parameters are considered are wider than the regions for uncertainties in a single parametertherefore indicating clearly the combined effects when uncertainties are considered in all design parameterson the harvester electromechanical response. However, for the case where uncertainties in ht parameter isconsidered, uncertainty effects become larger. The variation in the peak power is practically the same for theLS , ht and Lp parameters, but the variation in the natural frequency of the harvester is considerably different.The variation of the natural frequency when uncertainties in ht parameters are considered is larger than forthe other parameters. For the two higher frequency ranges result in harvester’s parameter whose resultingnatural frequencies does not fall within the design frequency range. This can certainly affect the device’sperformance as far as electrical energy generation is concerned. When the peak voltage FRFs for uncer-tainties in Rl and ζ are compared, the natural frequency of the harvester almost does not exhibit significantvariations, but the amplitude variation for the Rl is larger than the corresponding for ζ.

By inspecting Fig. (3), it can be concluded that LS , ht and Lp are the uncertain parameters that had largerinfluence in the harvested power. Thus, the next step of this work is to implement a stochastic programmingmethod that consider uncertainties in these 3 parameters in order to obtain an optimized Energy Harvestingsystem. Theory and implementation of stochastic programming method is present in the following sections.

6 Stochastic Programming Applied to the Energy Harvesting System

According to [17], the basic idea in the stochastic programming is to convert the optimization problem into anequivalent deterministic. This resulting deterministic problem is then solved by using a nonlinear program-ming technique such as the SQP method previously discussed. For the present purpose, it is assumed thatall the random variables are independent and follow a normal distribution. Hence, the harvester stochasticnonlinear programming problem can be stated in standard form as

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Figure 3: Region of peak electrical power variation: blue line: 05-20 Hz; red line: 20-35 Hz; green line:35-50 Hz; pink line: 50-65 Hz; black line: 65-80 Hz

Find X =

LShtLpRl

which minimizes f (Y =

LShtLp

), subject to the constraints

P [gj (Y) ≥ pj ] , j = 1, 2, ...,m (7)

where Y is the vector of N random and independent variables y1, y2, ..., yN and X includes the decisionvariables x1, x2, ..., xN . Equation (7 states that the probability of the realizing gj (Y) greater than or equal tozero must be greater than or equal to the specified probability pj , that in this case is 95%. The optimizationproblem stated can be converted into an equivalent deterministic nonlinear programming problem by apply-ing the Chance Constrained Programming (CCP) technique [17] This technique is applied in the objectivefunction, Eq. 3, along with the constraints of the proposed optimization problem. The constraints givenby Eq. 4 must be rearranged in such a way that they assume the form of Eq. 7. Once the CCP techniqueis applied, a new objective function in terms of the mean values and variance, F (Y), can be formulatedaccording to

F (Y) = k1ψ + k2σψ (8)

where the mean (ψ) and variance (σψ) of ψ are given by

ψ = γ(Y)

(9)

and

V ar (ψ) = σ2ψ =

N∑i=1

(∂γ

∂yi

∣∣∣∣Y)2

σ2yi (10)

since all yi (i = 1, 2, ..., N ) follow a normal distribution and if the standard deviations of yi, σyi, are small.

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05−20 20−35 35−50 50−65 65−800

0.05

0.1

0.15

0.2

0.25

Leng

th o

f the

Sub

stru

ctur

e −

Ls

[m]

Frequency Range [Hz]

05−20 20−35 35−50 50−65 65−800

2

4

6

8

10

Hei

ght o

f Tip

Mas

s −

ht [

mm

]

Frequency Range [Hz]

05−20 20−35 35−50 50−65 65−800

0.05

0.1

0.15

0.2

0.25

Frequency Range [Hz]

Leng

th o

f the

Pie

zo −

Lp

[m]

Nonl. Determ.Stoch. Prog.

05−20 20−35 35−50 50−65 65−800

50

100

150

Res

istiv

e Lo

ad −

Rl [

kOhm

s]

Frequency Range [Hz]

Figure 4: Comparison between the SQP and Stochastic Programming results

k1 ≥ 0 and k2 ≥ 0 are constants and their numerical values indicate the relative importance of ψ and σψ forminimization.

If some parameters are random in nature, the constraints will also be probabilistic and one would like to havethe probability that a given constraint is satisfied to be greater than a certain value. This is precisely what isstated in Eqs. (7) also. The constraint inequality after applying the CCP technique can be expressed as

gj − φj (pj)

[N∑i=1

(∂gj∂yi

∣∣∣∣Y)2

σ2yi

] 12

≥ 0, j = 1, 2, ...,m (11)

where φj (pj) is the value of the standard normal variate corresponding to the probability pj and

gj = gj(Y) (12)

Thus, the optimization problem stated first can be stated in its equivalent deterministic forma as: minimizeF (Y) given by Eq. (8) subject to the constraints given by Eq. (11).

6.1 Results of the Stochastic Programming Problem

The stochastic programming method was used along with the parameters obtained from the uncertaintyanalysis that showed higher influence in the harvested power: LS , ht and Lp. The standard deviations forthese parameters used here are the same shown in Table 2 and the value of the standard normal variatecorresponding to the probability of 95%, φj (0.95), is 1.645. The value of k1 and k2 used in the simulationsis 0.5 which reflects 50% of importance of ψ and σψ for minimization. Figure (4) compares the optimizationresults obtained in both the SQP and stochastic programming techniques.

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0 20 40 60 8010

−3

10−2

10−1

100

101

Excitation Frequency [Hz]

Ampli

tude [

V/g]

(a)

Nonl. Determ.Stoch. Prog.

05−20 20−35 35−50 50−65 65−800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Range [Hz]

Norm

alize

d Pea

k Pow

er

(b)

Nonl. Determ.Stoch. Prog.

Figure 5: (a) Voltage FRFs and (b) resulting normalized peak power for each optimization method.

Figure (4) reveals the difference between the parameters obtained for each method and frequency range.It should be recalled however that the values of the parameters meet the constraints requirements imposedto the optimization problem, including the resulting harvester natural frequencies falling within the desiredfrequency range. Aditionally, Figs. (5a) and (5b) show the peaks of the FRFs generated using the opti-mum obtained parameters and the comparison between the electrical power (normalized by the larger value)obtained with the parameters for both optimization methods, respectively.

Inspection of the FRFs curves confirms that all the natural frequencies fall within the frequency ranges andalso reveals that the peak voltages for the stochastic programming optimization problem are larger than thecorresponding values for the nonlinear programming method for the series connection of the piezo layers.It can be noticed in Fig. (5b) that the peak electrical powers obtained through the stochastic programmingmethod were larger than the corresponding values obtained from the deterministic method for all frequencyranges considered.

7 Final Remarks

The variability in the parameters during production, material degradation after long use associated to noisemeasurements are common sources of uncertainties and can alter the dynamic behavior of an energy harvest-ing system. This paper has analyzed vibration energy based piezoelectric energy harvesters under uncertain-ties in the design parameters (aleatory uncertainties). The harvester natural frequency of a bimorph cantileverbeam was set to fall within a prescribed design natural frequency range, that is, the range in which the sig-nificant components of environmental vibrations can occur. A classical optimization method was performedto obtain the parameters that get the peak electrical power. These parameters was then used to generate themean analytical model. This procedure was done for 5 different design frequency ranges.

A stochastic modeling and analysis using the Monte Carlo Simulation (MCS) was performed to evaluate theeffect of parametric uncertainties in the optimized energy harvester composed by a cantilever beam carrying atip mass and excited from its base. The random numbers followed a Gaussian probability density function. Inspite of the high computational cost attributed to the MCS solutions, the results presented a good understandof the influence of the data uncertainties in a energy harvesting device.

The results showed the influence of the frequency ranges in the electromechanical response under parametricuncertainties. It was observed that the effects of the uncertainties increase for the higher frequencies and thatthere are some uncertain parameters that have more influence in the predicability of the electromechanicalmodel. If this uncertain parameters can be controlled in the production, the predicability is also improved.Theparametric probabilistic approach was useful to estimate the robustness with respect to the data uncertainties.

After obtaining the parameters that present higher influence in the predicability of the electromechanicalmodel, a stochastic programming method was implemented to obtain the optimum configuration for the

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energy harvesting system. The method considered the uncertainties effects in the optimization procedure andthe results showed that the inclusion of the uncertain parameters in the optimization procedure is importantsince the efficiency of the resulting harvesting system improves in terms of the peak electrical power whencompared to deterministic optimization techniques.

Ackowledgements

The financial support provided by CAPES and FAPESP are gratefully acknowledged by the authors.

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