# Energy-Based Model Representation for Design and Optimization of Piezoelectric Energy Harvesting Systems

Diana Angelica Torres Guzman, Betty Lemaire-Semail

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Diana Angelica Torres Guzman, Betty Lemaire-Semail. Energy-Based Model Representation for Design and Optimization of Piezoelectric Energy Harvesting Systems. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0291 . hal-05082136

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# Energy-Based Model Representation for Design and Optimization of Piezoelectric Energy Harvesting Systems

Diana Angelica Torres Guzman, Betty Lemaire-Semail

Univ. Lille, Arts et Métiers Institute of Technology, Centrale Lille, Junia, ULR 2697 - L2EP, F-59000 Lille, France

## Acknowledgements

This work has been carried out within the framework of the I-SITE project COMASYS (energy Continuum: from Materials to SYStems) of the University of Lille. The authors acknowledge IRCICA (Research Institute on software and hardware devices for information and Advanced communication, USR CNRS 3380) for providing equipment and facilities that were essential in advancing the project. The authors also wish to acknowledge the valuable contributions of Frederic Giraud, Anis Kaci, Christophe Giraud-Audine, Clément Maillet, Cédric Lapeyronie and Sophie Barrau.

## Keywords

« Energy harvesting», «Modelling», «Energy management», «Energetic Macroscopic Representation (EMR)».

## Abstract

This paper presents a modular and adaptive modelling and simulation tool for piezoelectric energy harvesting (PEH) systems, applicable to portable micro-systems, such as wearable devices, sensors, or bio-medical devices. The main distinguishing feature of this tool is that it is designed to optimize the design phase and the energy management strategies, thanks to a systemic approach. By employing Energetic Macroscopic Representation (EMR) and inversion-based control, the tool facilitates flexible analysis across varying levels of detail. Using this tool, a validated model is constructed, enabling the testing of power converter strategies that connect piezoelectric elements to loads, as well as the evaluation of two energy harvesting strategies for imposed displacement systems. This work demonstrates the potential of advanced modelling techniques to enhance the design and implementation of piezoelectric energy recovery systems.

## Introduction

Energy recovery systems using piezoelectric transducers offer a promising method for converting mechanical energy into electrical power. These devices generate electricity through the piezoelectric effect when subjected to mechanical stress, such as vibrations or impacts, producing electrical power that ranges from micro-watts to tens of milli-watts [2], [3], [4]. Their applications span urban, industrial, micro-systems, and remote environments [5]. Notably, some of these applications include wearable devices [6], [7], micro-power sensors and biomedical applications [8].

A significant area of research focuses on resonant piezoelectric energy harvesting (PEH) systems [9], [10], [11], designed to operate at specific frequencies for optimal energy recovery. Imposed displacement systems, which harvest energy from variable and non-resonant movements, encounter challenges related to efficiency and design complexity, as they must accommodate a broader range of frequencies and amplitudes, often resulting in a reduced quality factor and decreased energy harvesting capabilities [10], [12], [13]. One key point regarding micro energy harvesting is the optimization of the amount of energy which can be harvested, which requires control implementation, while minimizing the size and above all the energy consumption of the control system. Many control strategies are focused on this issue, which limits the degrees of freedom of the control.

While current modelling and optimisation approaches can focus independently in component analysis [14] or converter design [11], [15], this work aims to develop a modular and adaptable modelling and simulation tool that encompasses component-level details and system-level analysis.

The tool can evaluate different design architectures for PEH systems and help propose optimal energy management strategies, enabling a flexible, stepped approach for incremental refinements. The usual main concern about the
energy consumption of the optimization algorithms is not considered at the first step, in order to allow a systematic analysis of all the system parts, and to highlight their influence. This systematic analysis is grounded in *Energetic Macroscopic Representation* (EMR) and *inversion-based control* (see Section 1). EMR analysis and model inversion control, which has been used for PEH analysis [9], is formalized in this work.

Finally, we apply this tool to evaluate two energy harvesting strategies for imposed displacement systems.

This paper is organized as follows: Section 1 introduces EMR and inversion-based control. Section 2 outlines the key components of a PEH system and presents a modular system representation along with a validation experiment performed in the resonant case. In Section 3, we propose and assess two strategies for imposed displacement PEH systems. Conclusions and future perspectives are discussed at the end.

# 1. Energetic Macroscopic Representation (EMR)

Energetic Macroscopic Representation (EMR) is a graphical formalism for modeling and control of energetic systems initiated in 2000s [16], [17], [18]. EMR is a tool built to organize the system model and to develop the control scheme based on three principles. The system model is represented, i.e., organized, based on the two principles:

* *Principle of causality:* the inputs and the outputs of an energetic system must follow the integral causality; in which the outputs are functional integrals of the inputs. The outputs are delayed with respect to the inputs [18].

* *Principle of interaction:* the subsystems of an energetic system interact with each other via pairs of action and reaction variables; in which the product of these variables is the instantaneous power exchanged between these subsystems [17].

One of the advantages of using EMR is to respect the physical causality and to be able to connect the different elements of the system, regarding their physical inputs and outputs. As a consequence, even if the modelling of one element has to be improved to take into consideration specific characteristics, its role in the whole system will be represented in the same manner, allowing the granularity choice for the analysis. Moreover, the control scheme is deduced from the model

representation based on the principle of inversion, in which, the control is considered as a functional inversion of the model (see annex 1 for detail).

# 2. Model of PEH Systems

<figure><img src="image_2.png" /><figcaption>Figure 1. General architecture of PEH</figcaption></figure>

PEH systems can take on various structures, but they typically consist of four common modules, each defined by one or several sub-components. These modules are the *source*, the *piezoelectric module*, the *converter* and the *load*. From an energy flow perspective, we consider that each module "imposes" a physical variable on the system and receives a corresponding reaction. This general architecture is illustrated in Figure 1. The action and reaction variables will be explained throughout the paper.

## A. Source module

The mechanical source imposes a velocity $u$ (i.e. a deformation) on the piezoelectric material and experiences the piezoelectric force $f_p$ (see: figure 1 numeral a). We distinguish two types of sources, depending on the type of the structure. It can either be from a resonant or a non-resonant (imposed displacement) structure.

Non-resonant (imposed displacement) structure model

Directly bent by a source of motion (typically elastic). Velocity $u$ is *directly imposed on the material*, regardless of its stiffness. Typical examples include wearables equipped with piezoelectric textiles [19]. Non-resonant sources can be modelled as a source of motion. The EMR, consequently, is a velocity source $u$, which receives a reaction force $f_p$.

<figure><img src="image_3.png" /><figcaption>Figure 2. Model and EMR of the imposed displacement source.</figcaption></figure>

## Resonant systems

These are the most common type of PEH. Typical examples include beams and other structures
submitted to periodic or repeated action forces, such as sinusoids and pulses [15].

<figure><img src="image_4.png" /><figcaption>Figure 3. Mechanical model of the resonant source: second order spring damper</figcaption></figure>

These systems use a resonator to amplify the motion produced by an action force $f_{acc}$. This amplified motion is described by a shape and frequency, defined as a ‘mode’. It is the reaction displacement of the resonator that imposes velocity $u$. The resonator is then attached to the piezoelectric transducer. The dynamic model for a single mode on the structure can be expressed using the second order ‘Mass-spring-damper’ model (1) illustrated in Figure 3. M, K and D are the modal mass, elasticity and dampening, respectively. $f_{acc}$ and $f_p$ represent the action and piezoelectric forces.

$$f_{acc} - f_p = M\dot{u}(t) + Du(t) + K \int u(t) \quad (1)$$

The EMR and related equations for each element can be found in figure 4. To create the EMR, the dynamic equation is separated into two coupled impedances. The inertial impedance accounts for the energy stored in the modal mass. The damped elastic impedance accounts for the energy stored by the elasticity of the resonator, which is damped by the modal dampening component. It is the inertial element that imposes the displacement $u$.

<figure><img src="image_5.png" /><figcaption>Figure 4. EMR of the resonant source</figcaption></figure>

## B. Piezoelectric module

This module represents the electromechanical transformation and associated effects. It receives the deformation $u$, and imposes a piezoelectric

voltage $v_p$ (see: figure 1. Numeral b). The ‘Mason’ model [20] is used as a first approach due to its ability to effectively describe the electromechanical transformation under the assumptions of linearity, homogeneity, small deformations, and defined boundary conditions. This allows for a simplified representation of the system (however, the proposed tool can accommodate a greater granularity, enabling the inclusion of hysteresis and non-linearities if required by the study). Consequently, the model consists of three key components: a modal electro-mechanical transformation factor $N$, a ‘blocked capacitance’ $C_0$, and a resistor $R_0$ to account for dielectric losses. These components are represented in figure 5, with their corresponding model equations. Currents $i_m$ and $i_{conv}$ represent the motional current and the converter current, respectively.

<figure><img src="image_6.png" /><figcaption>Figure 5. ‘Mason’ model of the piezoelectric module</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6. EMR of the piezoelectric module</figcaption></figure>

The EMR is illustrated in figure 6, with the model equations. The electromechanical transformation is represented by a multi-physical transformation element, and the RC circuit is represented by an accumulation element.

## C. Validation of Source and Piezoelectric Modules

To ensure comprehensive validation of the constructed modules, we conducted an experiment to assess their behaviour under a variable load profile. The system is tested for the resonant case. By analysing various load properties, we aimed to confirm that the model
accurately reflects real system performance
across a wide range of loading conditions.

## Experimental Setup:

<figure><img src="image_8.png" /><figcaption>Figure 7. PEH with Variable load experimental validation setup</figcaption></figure>

As depicted in Figure 7, the experimental setup comprises a mass, a resonator, and an array of two piezoelectric ceramics connected to an activation voltage $V_{act}$. These ceramics serve as the force source $f_{acc}$. Another array of two identical recovery ceramics is attached to the resonator, functioning as the piezoelectric module. The recovery array is connected to a controlled AC voltage source, $V_{LOAD}$, that serves as a programmable resistive/reactive load emulator. The frequency of $V_{LOAD}$ is intentionally set to be slightly different from $V_{acc}$, thus creating a sliding phase effect that enables testing of a range of phase angles. Active power $P_{act}$ is measured for each phase angle and $V_{LOAD}$ amplitude.

## Simulation of the experimental setup:

The EMR of the setup is illustrated in Figure 8. In this configuration, the load module is directly attached to the piezoelectric. The parameters of the model are identified, and the model is implemented in MATLAB® Simulink for simulation.

To replicate the experimental conditions in the simulation model, the 'load' module is designed as a sinusoidal AC voltage connected to an RL input circuit. The results from the simulation and the experimental measurements are presented together in figure 9 for comparison.

We observe that, with correctly fine-tuned parameters, the curves not only display a similar

overall shape, but their orders of magnitude also align closely.

<figure><img src="image_9.png" /><figcaption>Figure 8. EMR of the setups used for validation: the source and piezoelectric are connected directly to a programmable load</figcaption></figure>

A region around a phase angle of 200° and $V_{LOAD} = V_{acc} = 15V$, provides the maximum power recovery of the measured setup in both cases. Slight differences between the simulated behaviour and the behaviour in reality could be explained by non-linearities observed in these types of systems [10].

<figure><img src="image_10.png" /><figcaption>Figure 9. ER with variable load experimental validation results. Model simulation results are plotted as a solid surface. Measurements are plotted over as a transparent mesh for comparison</figcaption></figure>

Finally, we note that the model results are highly sensitive to parameter variations, with significant differences observed even for changes as small as 1%, so parameter finetuning is important

## D. Converter Module

The converter is the element in the system which has a tuning input, and as a consequence, which allows the control of the system. The primary objectives of the converter are two: maximizing energy recovery from the electromechanical system and formatting the output signal for
optimal compatibility with the battery and load (see Figure 1c). To achieve the first objective, the literature recommends semi-passive solutions such as Synchronized Switch Harvesting on Inductor (SSHI) and Synchronized Energy Transfer (SECE) [12], which are particularly suited for resonator structures.

We propose a novel stepped approach to the converter design. In the initial stage, our focus will be on implementing and testing a specific energy management strategy using a closed-loop control system, assuming the converter components are ideal. Losses and energy consumption will be addressed in subsequent design phases. Central to this effort is the development of an optimal energy management strategy to be integrated into the converter.

The second objective will necessitate the implementation of a battery charging strategy following the completion of controlled conversion, which will be addressed in future studies.

<figure><img src="image_11.png" /><figcaption>Figure 10. MCS of the imposed displacement structure</figcaption></figure>

The converter module is represented in EMR as a controlled conversion block, with a modulation function, such that $-1<m<1$, allowing for control over the amplitude and phase of the selected mechanical variable.

The Maximal Control Structure of the whole system is obtained by inversing step by step the different elements. The structure and EMR of an imposed displacement system, including converter with the MCS, is illustrated as an example in figure 10. The voltage control in the piezoelectric RC circuit is achieved with a feedforward PI controller. The strategy will vary depending on the design requirements. A few examples are examined in the next section.

### 3. PEH Energy Management strategies for the converter

In light of the proposed stepped approach, strategies can be evaluated based on their effectiveness in recovering piezoelectric energy. Key considerations for developing these strategies include whether the circuit operates in resonant mode or under imposed displacement, the nature of the expected excitation (such as periodic, pulse, sinusoidal—monotonic or otherwise—or human motion like walking or limb flexing), and the characteristics of measurable variables. These factors are critical for optimizing energy recovery and ensuring efficient system performance.

To illustrate the process, two strategies are evaluated for imposed displacement under non-sinusoidal excitation, a scenario that still presents significant challenges.

#### A. Energy Management Strategies for Imposed Displacement PEH

In imposed displacement PEH, the energy provided by the mechanical source over a period of time $t1$ to $t2$ can be expressed in terms of force and velocity (2).

$$E = \int_{t1}^{t2} u. fp \, dt \qquad (2)$$

##### Strategy 1

With nearly periodic inputs, the bibliography [12] suggests that an alignment between $fp$ and $u$ at the piezoelectric level can help maximize energy recovery, while allowing the least energy requirements from the converter. This strategy, described by (3) is in line with existing techniques such as SSHI. $k$ is limited by the energy available and the limitations of the components.

$$fp_{ref} = ku \qquad (3)$$

##### Strategy 2

This second strategy (4) is based on similar principles to Strategy 1, i.e., aiming to minimize reactive power while maximizing the piezoelectric force $fp$ at all times. It operates within the constraints of energy availability and the physical limits of the system.

$$fp_{ref} = fp_{max} \cdot sign(u) \qquad (4)$$
While strategy 1 focuses on minimizing implementation complexity, strategy 2 aims to maximize energy recovery by operating the piezoelectric device at its maximum force during excitation. Assuming this is feasible, and setting aside critical factors such as losses and component energy consumption, strategy 2 is expected to deliver superior results. The extent of this improvement will inform decision-making in subsequent stages of the design process, when additional factors are taken into consideration.

## B. Simulation

### Source of motion:

The displacement $u$ can represent different types of excitations, depending on the desired simulation, e.g. pulses, a ramp, a slow (about 1 Hz) sinusoid or even recorded human motion. For this case we use accelerometer measurements from a light walk, converted to a velocity profile (figure 1).

<figure><img src="image_12.png" /><figcaption>Figure 11. Recording of the walking motion. For testing, imposed velocity of deformation is assumed to be proportional to the integral of the accelerometer measurements. (Up) over a period of 150 seconds, (down) detail over 5 seconds.</figcaption></figure>

## C. Tests with an ideal load

The energy recovery at the load, which can be defined as in (6), is tested in simulation for both strategies. For this purpose, an ideal load, represented as a DC current source of 6mA is assumed.

<figure><img src="image_13.png" /><figcaption>Figure 12. Illustration of energy recovery strategies and results. Up: Strategy 1, $fp$ and $u$ details over 10 seconds, and energy recovery over 150 seconds. Down: Strategy 2, $fp$ and $u$ details over 10 seconds, and energy recovery over 150 seconds.</figcaption></figure>

<caption>Figure 13. .</caption>

For both strategies, it is also assumed that the piezoelectric can deliver a maximum force of about 0.12N (thus, for strategy 1 $k = 10$).

$$E = \int v_p \cdot i_{conv} dt \quad (6)$$

The simulation results are displayed alongside the scaled $fp$ and $u$ measurements in Figure 12 for Strategies 1 and 2. Over a continuous excitation period of 150 seconds, Strategy 1 allows a recovery of about 0.06 Watts-second, while Strategy 2 allows about 0.09 Watts-second. As expected, this result suggests that Strategy 2 may be the more efficient solution, by a factor upto 50% higher.

## Conclusions and perspectives

In this work, we built and tested a tool for the development and testing of devices and various control strategies for piezoelectric energy harvesters (PEH), facilitating a modular and adaptable design process, thanks to a systemic approach. For that purpose, EMR formalism is used.

To demonstrate this approach, we propose a validated model used to evaluate two energy management strategies in a structure with imposed displacement. The results indicate that, under ideal conditions and ignoring component
energy consumption, a strategy that maximizes piezoelectric force in phase with deformation velocity can increase the energy recovery by upto 50%, compared to a typical strategy that applies piezoelectric force as a multiple of the deformation velocity. These findings may evolve in later design phases.

We recommend several improvements for subsequent phases. The model can be refined by enhancing granularity and specificity of component parameters with expert input.

Non-linear effects, such as hysteresis and amplitude-dependent frequency responses, can significantly impact real-world performance. Additionally, incorporating losses at both mechanical and electrical levels—such as resistor losses and efficiencies—is essential.

Future steps will involve exploring different converter architectures and developing detailed component models, for innovative applications such as the testing of new piezoelectric materials and the design of microgenerators for portable and medical devices.

## Annex 1.

<caption>Table 1. illustrates the EMR elements, their pictograms, and their descriptions.</caption>

<table><caption>Table A.1: EMR Elements</caption><thead><tr><th>Element</th><th>Pictogram</th><th>Description</th></tr></thead><tbody><tr><td>Power variable</td><td>Pair of action and reaction variables</td><td></td></tr><tr><td rowspan="2">Signal variable</td><td>Mandatory signal variable</td><td></td></tr><tr><td>Optional signal variable</td><td></td></tr><tr><td>Measurement</td><td>Measurement of variables</td><td></td></tr><tr><td>Source</td><td>Terminal of the system, which supplies or dissipates energy</td><td></td></tr><tr><td>Accumulation</td><td>Accumulation of energy; introduces a delay; represents system dynamics</td><td></td></tr></tbody></table>

<table><tr><td>Mono-physical conversion</td><td>Mono-domain conversion of energy; optional tuning input</td></tr><tr><td>Multi-physical conversion</td><td>Multi-domain conversion of energy; optional tuning input</td></tr><tr><td rowspan="2">Coupling</td><td>Mono-domain coupling or distribution of energy between more than two subsystems</td></tr><tr><td>Multi-domain coupling or distribution of energy between more than two subsystems</td></tr><tr><td>Accumulation inversion</td><td>Indirect inversion of the accumulation elements (controller)</td></tr><tr><td>Conversion inversion</td><td>Direct inversion of energy conversion element (gain)</td></tr><tr><td>Coupling inversion</td><td>Direct inversion of coupling element</td></tr><tr><td>Estimation or model</td><td>Model copy of the system implemented in the control;</td></tr><tr><td>Strategy</td><td>To impose reference, distribution, or weighting factors to the control</td></tr></table>
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