Energy harvesting through backpacks employing piezoelectric stack systems optimized for users with different physiological features

Abstract

The physiological characteristics considered here are the weight carried by the user and the walking frequency. Since the input force is low frequency, the mechanical response of the piezoelectric stack may be neglected and a model based on a first order differential equation is developed. Direct optimal control techniques are used. Three objective functions are defined in case of the resistive load and one objective function is considered for the capacitive load. General conclusions: (1) the maximum harvesting performance is obtained when the control parameters have a bang-bang-type time evolution or are constant; (2) sizing the load is strongly constrained by the physiological characteristics of the backpack user. Specific result for resistive loads: (3) the amount of dissipated energy increases by increasing the walking frequency, for both controlled and un-controlled operation. Specific results for capacitive loads: (4) the optimal capacitance is constant in time and equals the smallest allowed value; (5) the amount of energy stored in a capacitor when fully charged depends on the average weight carried by the backpack user; (6) the stored energy increases by increasing the walking frequency.

Appendices

Appendix A

The solution of the following first order linear equation:

$$ \frac{dy}{{dx}} + P\left( x \right)y = Q\left( x \right) $$

(A1)

is given in [28, p.487]:

$$ y = u\left( x \right)\left[ {\int {\frac{Q\left( x \right)}{{u\left( x \right)}}dx} + C} \right] $$

(A2)

where $C$ is an integration constant and:

$$ u\left( x \right) = \exp \left( { - \int {P\left( x \right)dx} } \right) $$

(A3)

Equation (11) may be put under the form:

$$ \frac{{dv_{p} }}{dt} + \frac{a}{R}v_{p} = b\sin \left( {\omega t} \right) $$

(A4)

where

$$ a \equiv \frac{1}{{C_{p}^{eq} }} $$

(A5)

$$ b \equiv - \frac{{d_{33}^{eff} \omega F_{m} }}{{C_{p}^{eq} }} $$

(A6)

Comparison of Eqs. (A4) and (A1) allows an easy identification of the functions $y \to v_{p}$, $P \equiv a/R$ and $Q \equiv b\sin \left( {\omega t} \right)$. The function $u$ in Eq. (A3) is obtained by integration:

$$ u = \exp \left( { - \frac{a}{R}dt} \right) = \exp \left( { - \frac{at}{R}} \right) $$

(A7)

Replacing $y$, $P$, $Q$ and Eqs. (A7) in (A2) one finds:

$$ v = \exp \left( { - \frac{at}{R}} \right)\left[ {\int {\frac{{b\sin \left( {\omega t} \right)}}{{\exp \left( { - \frac{at}{R}} \right)}}dt + C} } \right] $$

(A8)

The integral in Eq. (A8) gives [28, p.357]:

$$ \int {\exp \left( \frac{at}{R} \right)\sin \left( {\omega t} \right)dt} = \frac{{\exp \left( \frac{at}{R} \right)\left[ {\frac{a}{R}\sin \left( {\omega t} \right) - \omega \cos \left( {\omega t} \right)} \right]}}{{\left( \frac{a}{R} \right)^{2} + \omega^{2} }} $$

(A9)

Replacing Eqs. (A9) in (A8) and using the initial condition:

$$ v_{p} \left( {t = 0} \right) = 0 $$

(A10)

one finds the value of the integration constant:

$$ C = \frac{\omega b}{{\left( \frac{a}{R} \right)^{2} + \omega^{2} }} $$

(A11)

Replacing Eqs. (A5),(A6), (A9), (A11) in Eq. (A8) gives the solution Eq. (13).

Appendix B

In many previous studies dealing with energy harvesters the systems of ODEs have constant coefficients. This allows using classical solution procedures such as that based on Laplacetransform, which has been widely used [23, 29,30,31]. However, optimal control approaches yield highly stiff ODEs with time-dependent coefficients and this requires usage of numerical solution procedures. The high drive frequency yields highly stiff ODEs and, as a consequence, suitable numerical procedures with small enough time step size should be used. For instance, the dimensionless time step in [15] is lower than $10^{ - 5}$ to avoid stability problems of the numerical solution and the discretization of the controls is made with time step of $2 \cdot 10^{ - 5}$.

When a constant value is adopted for the control $R$, the optimization problem reduces to an usual initial value problem which consists of solving the ordinary differential Eq. (11) with initial condition Eq. (12). Solving this equation has been performed by using an accurate solver (DDRIV3) [32]. The following values have been used for the parameters of this solver: EPS = 10^–6, HMAX = 10^–6, EWT = 10^–6. The results obtained by using DDRIV3 solver are used as a reference. A similar procedure has been use during the optimal control approach adopted in [15]. Simulations were performed in that paper with fixed control values by using LT-SPICE software and the results are used as reference solutions.

Figure 11a shows the dependence of the average power, $P_{p,ave}$ (Eq. Table 1–6), on the load resistance $R$. $P_{p,ave}$ decreases by increasing the operation time $t_{f}$. This is explained as follows. $P_{p,ave}$ is the ratio between the energy $E$ dissipated on the load and $t_{f}$. The energy $E$, in turn, is the product between a constant factor, $P_{ss}$, and the factor $\psi \left( {t_{f} } \right)$ (see Eq. Table 1- 4c). $\psi \left( {t_{f} } \right)$ is obtained by integration of $\phi \left( t \right)$, whose time variation is rather periodical, except the term decreasing in time (see Eq. (15)). Therefore $\psi \left( {t_{f} } \right)$ does not increase linearly when $t_{f}$ increases and the consequence is that $P_{p,ave}$ decreases by increasing $t_{f}$. Notice that there is an optimum load resistance which makes the power a maximum and that optimum load resistance slightly depends on $t_{f}$.

Fig. 11

Dependence of several quantities on the load resistance $R$. a average power $P_{p,ave}$ for different values of the operation time $t_{f}$; b three solutions for the average power $P_{p,ave}$; c steady-state approximation for voltage $v_{ss}$ and power $P_{ss}$; d Amplitude $\Phi$ and phase difference $\lambda$ between the approximation voltage solution $v_{p,app}$ and input force $F$. Walking frequency $f = 5\;Hz$ and average input force $F_{m} = 83\;N$

Figure 11b shows the dependence of the average power $P_{p,ave}$ on the load resistance for $t_{f} = 1\;s$. Results are obtained by using the two analytical solutions (Eqs. Table 1–6 and Table 1–7) and the numerical solution obtained by using the DDRIV3 programming package. The numerical solution is in very good agreement with the “exact” analytical solution while the approximation analytical solution provides significantly different results. This shows the limitation of the results reported in literature based on the approximation solution.

Figure 11c shows the steady-state approximation for the voltage,$v_{ss}$(Eq. (14)), and the power dissipation, $P_{ss}$(Eq. Table 1–2). These quantities are often used in literature. They depend on the load resistance $R$ and each quantity is a maximum for a different value of $R$. The maximum of $P_{ss}$ is not shown in Fig. 11c.

The approximation value of the voltage, $v_{p,app}$ (Eq. (16) depends on the time dependent factor $\phi_{app} \left( t \right)$ which has the periodical variation shown in Eq. (17b). Figure 11d shows that the amplitude $\Phi$ and the phase difference $\lambda$ between the input force $F$ and the voltage $v_{p,app}$ are significantly large. They depend on the load resistance but do not depend on the operation time $t_{f}$.

Appendix C

Optimal Control Procedure

Several studies on vibration energy harvesting used indirect optimal control techniques which have the advantage that may be based on a powerful, rigorous, theoretical tool (the Pontryagin maximum mrinciple). Despite being usually less precise as the indirect optimal control methods, the direct methods have, however, some advantages. For instance, they do not need defining the Hamiltonian and to derive the adjoint equations, which are necessary steps when indirect methods are used. Also, they are easy to use in case of complicated switching structures coming from constraints on controls and state variables and they are more robust during the initialization phase. This explains the wide use of direct optimal control methods in industrial applications. For instance, the optimal control of the energy harvesting systems has been treated in [15] by using the open-source software tool CasADi with Python interface. The ODE system has been solved with CVODES from the SUNDIALS integrator suite while the obtained optimization problem has been solved with IPOPT.

The direct optimal control methods are based on ordinary differential equations (ODE) for the state variables. The ODE systems are solved by using appropriate boundary conditions while the state variables and controls are subjected to several constraints coming from the nature of the physical problem or from several space and time restrictions [33, chap. 4].

Here the optimal control problem (OCP) is solved in several steps, which are shortly explained in the following. First, the dynamics of the energy harvester is described in terms of ODEs for the state variables and controls. Next, the objective function is defined. The objective function has to be extremized and the ODEs constitute constraints during the extremization procedure. At this stage the OCP is infinite dimensional since it involves functionals. Next, the state and control variables, as well as the dynamics equations, are discretized in the space of the independent variable. This way, the infinite dimensional OCP is transformed into a finite dimensional non-linear problem (NLP). This process is performed here by using the BOCOP package [34]. Further details on direct transcription methods and NLP optimization algorithms are given in [35, 36].The IPOPT package performs the optimization outside BOCOP, which constitutes the interface for other packages written in different programming languages (MUMPS for linear algebra procedures, ADOL-C for automatic differentiation and COLPACK for Graph Coloring Algorithm Package).

A few practical aspects follow. Several optimization methods are available in BOCOP and the Midpoint method is recommended and used here. Constraints have been considered here for the controlled resistance and capacitance of the load. A tolerance value $tol = 10^{ - 14}$ is adopted during numerical resolution. By default, the objective function is minimized by BOCOP. When maximization of a specific objective function is needed, a new objective function is defined, which is the negative of the default objective function. The convergence of the optimization algorithm is slower or faster, depending on the initial guess distributions of state variables and control. These distributions are usually found by trial procedures. The accuracy of the solution depends on the finesse of the discretization, which is described by the number $n_{steps}$ of steps to divide the interval $\left[ {0,t_{f} } \right]$. Several tests have been performed in order to find the most suitable number of steps. In most cases $n_{steps} = 500$.

Optimal control model for resistive load

The objective #1 is to maximize the harvested energy given by Eq. (20) under the constraints of the first order differential Eq. (11). This constitutes a Bolza problem, which may be transformed into a Mayer problem in two steps [33, chap. 4]. First, a new dependent variable $E$ is defined by using the following equation:

$$ \frac{dE}{{dt}} = \frac{{v_{p}^{2} }}{R} $$

(C1)

which comes from Eq. (20). The initial condition is:

$$ E\left( {t = 0} \right) = 0 $$

(C2)

The following dimensionless notation is used:

$$ \hat{t} \equiv \frac{t}{{t_{ref} }} $$

(C3)

$$ \hat{v}_{p} \equiv \frac{{v_{p} }}{{v_{ref} }} $$

(C4)

$$ \hat{R} \equiv \frac{R}{{R_{ref} }} $$

(C5)

$$ \hat{E} \equiv \frac{E}{{E_{ref} }} $$

(C6)

By using notations Eqs. (C3)-(C6) from Eqs. (1) and (20) one finds:

$$ \frac{{d\hat{v}_{p} }}{{d\hat{t}}} = C_{1} \frac{{\hat{v}_{p} }}{{\hat{R}}} + C_{2} \sin \left( {\omega t_{ref} \hat{t}} \right) $$

(C7)

$$ \frac{{d\hat{E}}}{{d\hat{t}}} = C_{3} \frac{{\hat{v}_{p}^{2} }}{{\hat{R}}} $$

(C8)

where:

$$ C_{1} \equiv - \frac{{t_{ref} }}{{C_{p}^{eq} R_{ref} }} $$

(C9)

$$ C_{2} \equiv - \omega F_{m} \frac{{t_{ref} }}{{v_{ref} }}\frac{{d_{33}^{eff} }}{{C_{p}^{eq} }} $$

(C10)

$$ C_{3} \equiv \frac{{t_{ref} v_{ref}^{2} }}{{E_{ref} R_{ref} }} $$

(C11)

The initial conditions used to solve Eqs. (C7) and (C8) are:

$$ \hat{v}_{p} \left( {\hat{t} = 0} \right) = 0 $$

(C12)

$$ \hat{E}\left( {\hat{t} = 0} \right) = 0 $$

(C13)

Equation (C12) comes from Eqs. (12) and (C3) while Eq. (C13) comes from Eqs. (20) and (C6).

The reference values used to obtain the dimensionless formulation of the optimal control problem are shown in Table

Table 4 Reference values

4. They are not of interest by themselves but proper reference values are needed to ensure the convergence of the numerical procedures. These reference values were obtained by a trial procedure.

In summary, the optimal control problem for the resistive load is defined as follows:

• independent variable: time $t$;

• control: the load resistance $R$

• state variables: voltage $v_{p}$, harvested energy $E$;

• objective function: one of objective #1 Eq. (21), objective #2 (Eq. (22)) or objective #3 Eq. (23).

The objective function (either Eq. (21), Eq. (22) or Eq. (23)) is maximized/minimized under the constraints of the ordinary differential eqs. (C7),(C8), which are solved by using the initial conditions eqs. (C12),(C13).

The optimal distributions of the state variables obtained by using the three objectives are generally different.

Verification of the optimal control procedure

Figure 12 shows the dependence of the energy dissipated $E$ on the load resistance $R$. Results obtained by using the analytical solution (Eq. Table 1−4a) and the numerical solution obtained by using the BOCOP package are shown. For shorter operation times $t_{f}$ (e.g. 1 s and 2 s) there is a very good agreement between the two solutions. The BOCOP solution provides slightly larger values at longer operation time (10 s).

Fig. 12

Dependence of the energy dissipated $E$ on the load resistance $R$. Analytical solution and solution obtained by using BOCOP package. Walking frequency $f = 5\;Hz$ and average input force $F_{m} = 83\;N$

Calibration of the optimal control model

The model proposed in [3] is based on second-order differential equations since it takes into account both the mechanical and the electrical responses of the piezoelectric stack. Here, due to the low-frequency of the source of energy, only the electrical response is considered, and, therefore, the stack model consists of a first-order differential equation. The model should be calibrated against the experimental results reported in [3].

Several tests have been performed in [3], as follows. Five tests were used to verify that the model predicts with reasonable accuracy the power generation for a system with a load resistance that was not tuned to the impedance of the stack. Two tests were used to analyze the effect of the frequency of the energy source on the performance of the harvester.

The present first-order model is calibrated here by using the following procedure. One assumes an oscillatory input force of frequency $f = 5\;Hz$ and a load resistance $R = 9.72\;k\Omega$ (test 2 of Table 2 of [3]). One gradually changes the average value of the input force with the aim to obtain a time distribution of the generated voltage similar with that obtained in test 2 of [3]. After calibration, the test 4 of [3] has been simulated and compared the voltage variation with that experimentally measured.

Figure 13 shows the time variation of the voltage $v_{p}$ for several input forces. The best agreement with the experimental results is obtained for $F_{m} = 180\;N$ and 200 N. These values are close to the values 176 N and 220 N used in [3]. When $F_{m} = 180\;N$, the voltage ranges between -4 V and 4 V. This is in good agreement with the results presented on Fig. 9 (bottom-left) of [3]. The average generated power obtained by using the present model is about 0.88 mW, which is close to the experimental value 0.85 mW shown in Fig. 11a of [3].

Fig. 13

Calibration of the optimal control model. Voltage $v_{p}$ as a function of the dimensionless time $\hat{t}$ for several values of the average input force $F_{m}$

Next, the calibrated model with $F_{m} = 180\;N$ was applied for an oscillatory input force of frequency $f = 5\;Hz$ and a load resistance $R = 28.3\;k\Omega$ (test 4 of Table 2 of [3]). The voltage ranges between -7 V and 7 V, in good agreement with the results presented on Fig. 10 (bottom-left) of [3]. The average generated power obtained by using the present model is about 1.05 mW while the experimental value is 0.88 mW as shown in Fig. 11a of [3]. The deviation between the simulated and experimental values is about 16% which is close to the accuracy of the second-order model developed in [3], which predicted the power output within 12% of the actual value.

Optimal control model for capacitive load

To prepare the Mayer problem, one defines the new dependent variable $E$ by using Eq. (C2):

$$ \frac{dE}{{dt}} = \frac{{q_{c} }}{C}\frac{{dq_{c} }}{dt} $$

(C14)

Equation (C14) is to be solved by using the initial condition Eq. (C13):

$$ E\left( {t = 0} \right) = 0 $$

(C15)

The following dimensionless notation is used:

$$ \hat{q}_{c} \equiv \frac{{q_{c} }}{{q_{ref} }} $$

(C16)

$$ \hat{C} \equiv \frac{C}{{C_{ref} }} $$

(C17)

Usage of notations Eqs. (C3),(C4), (C6), (C16) and (C17) in Eqs. (31), (32) and (C14) yields, respectively:

$$ \frac{{d\hat{v}_{p} }}{{d\hat{t}}} = \left\{ \begin{gathered} sign\left( {\hat{v}_{p} } \right)D_{1} \left( {D_{0} \left| {\hat{v}_{p} - \frac{{\hat{q}_{c} }}{{\hat{C}}}} \right|} \right) + D_{F} \sin \left( {\Omega \hat{t}} \right)\quad if\quad D_{0} \left| {\hat{v}_{p} } \right| - \frac{{\hat{q}_{c} }}{{\hat{C}}} \ge 0 \hfill \\ 0\quad otherwise \hfill \\ \end{gathered} \right. $$

(C18)

$$ \frac{{d\hat{q}_{c} }}{{d\hat{t}}} = \left\{ \begin{gathered} D_{2} \left( {D_{0} \left| {\hat{v}_{p} - \frac{{\hat{q}_{c} }}{{\hat{C}}}} \right|} \right)\quad if\quad D_{0} \left| {\hat{v}_{p} } \right| - \frac{{\hat{q}_{c} }}{{\hat{C}}} \ge 0 \hfill \\ 0\quad otherwise \hfill \\ \end{gathered} \right. $$

(C19)

$$ \frac{{d\hat{E}}}{{d\hat{t}}} = \left\{ \begin{gathered} D_{3} \frac{{\hat{q}_{c} }}{{\hat{C}}}\left( {D_{0} \left| {\hat{v}_{p} - \frac{{\hat{q}_{c} }}{{\hat{C}}}} \right|} \right)\quad if\quad D_{0} \left| {\hat{v}_{p} } \right| - \frac{{\hat{q}_{c} }}{{\hat{C}}} \ge 0 \hfill \\ 0\quad otherwise \hfill \\ \end{gathered} \right. $$

(C20)

where:

$$ D_{0} \equiv \frac{{v_{ref} C_{ref} }}{{q_{ref} }} $$

(C21)

$$ D_{1} \equiv - \frac{1}{{C_{p}^{eq} }}\frac{{t_{ref} }}{{v_{ref} }}\frac{1}{{R_{c} }}\frac{{q_{ref} }}{{C_{ref} }} $$

(C22)

$$ D_{2} \equiv \frac{1}{{C_{ref} }}\frac{{t_{ref} }}{{R_{c} }} $$

(C23)

$$ D_{3} \equiv \frac{{t_{ref} }}{{E_{ref} }}\frac{1}{{R_{c} }}\frac{{q_{ref}^{2} }}{{C_{ref}^{2} }} $$

(C24)

$$ D_{F} \equiv - \frac{{t_{ref} }}{{v_{ref} }}\frac{{d_{33}^{eff} }}{{C_{p}^{eq} }}\omega F_{m} $$

(C25)

$$ \Omega \equiv \omega t_{ref} = 2\pi ft_{ref} $$

(C26)

Therefore, one obtains a system of just three differential equations, (C18), (C19) and (C20), with three unknown: $\hat{v}_{p}$, $\hat{q}_{c}$ and $\hat{E}$. These equations are solved by using the initial conditions:

$$ \begin{gathered} \hat{v}_{p} \left( {\hat{t} = 0} \right) = 0 \hfill \\ \hat{q}_{c} \left( {\hat{t} = 0} \right) = 0 \hfill \\ \hat{E}\left( {\hat{t} = 0} \right) = 0 \hfill \\ \end{gathered} $$

(C27 a,b,c)

Equations (C27a) and (C27b) come from Eqs. (33a) and (33c) while Eq. (C27c) comes from Eq. (C15).

In summary, the optimal control problem for the capacitive load is defined as follows:

• independent variable: time $t$;

• control: the load capacitance $C$

• state variables: voltage $v_{p}$, electric charge in the condenser, $q_{c}$, harvested energy $E$;

• objective function: objective #1 defined by Eq. (35).

The objective function Eq. (35) is maximized under the constraints of the ordinary differential eqs. (C18),(C19),(C20), which are solved by using the initial conditions eqs. (C27a,b,c).