# Behavioral Modeling of DC/DC Converters in Self-Powered Sensor Systems with Modelica

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Simulation: Level of Detail vs. Time Scale

#### 1.2. Cycle-to-Cycle vs. Behavioral Modeling

#### 1.3. Power Class of Self-Powered Sensor Systems

## 2. Fundamentals of Self-Powered Systems and Power Converters

#### 2.1. Structure and Functions a Micro Power Management

**Energy extraction:**The output impedance of the harvester varies with the input power due to the environmental fluctuations. To extract as much energy from a harvester as possible, a continuous matching called maximum power point tracking (MPPT) is needed. The first power converter in Figure 1 is typically a step-up converter. For instance, a single solar cell only offers a voltage of about $0.5$ V at the MPP, but a Li-Ion battery has an open-circuit voltage of $3.7$ V.**Storage interaction (charge and protect):**Most secondary batteries (e.g., Li-ion) need a constant-current, constant-voltage charging scheme (CCCV). Due to power fluctuations, this scheme cannot be followed completely, but at least a voltage limiting to prevent under- and over-charge must be implemented in both converters.**Voltage supply:**A regulated constant voltage is required to supply the consumers. The second power converter is typically a step-down converter, as e.g., a low-power microcontroller needs a stable voltage of $1.8$ V, which is below the battery voltage.

#### 2.2. Power Management Integrated Circuits (PMICs)

#### 2.3. Structure of PMIC vs. the Presented Model

## 3. Related Work

**ModelicaStandardLibrary 4.0**[11] only contains switch-mode models of a DC/DC converters (e.g.,

`Modelica.Electrical.PowerConverters.DCDC.ChopperStepDown`), which is only mentioned here for the sake of completeness. Only the power transformation is modeled by ideal transistors and the PWM control is outsourced as another component (

`signalPWM`).

**Torrey et al.**suggest in [12] a behavioral model of a DC/DC converter using Modelica. By means of a proportional–integral (PI) controller and a commanded output voltage the output current is controlled. The corresponding load is then reflected to the input as a current sink by dividing ${P}_{\mathrm{out}}$ by ${V}_{\mathrm{in}}$ and a constant efficiency value. Another feature is a minimum operating input voltage ${V}_{\mathrm{min}}$, where the input current sink is set to zero for too low input voltages. The converter efficiency is modeled as a constant value and an additional input resistor representing quiescent currents. The model is intended to be used for high power applications (approx. 5000 W).

**PhotoVoltaics Modelica library**[13] includes a DC/DC converter to extract energy from solar cells (

`PhotoVoltaics.Components.Converters.DCConverter`). The input is modeled as voltage source, which is commanded externally by a MPP tracker, following the P&O concept. The input power is calculated and compared to the output power by means of an integral (I) controller. The controller controls the variable current source, which results in an output voltage that cannot be limited in any way. The model does not consider any losses. Furthermore, the converter is continuously running, and start-up or shutdown behavior is modeled.

**EDrives library**[14] includes three inverter models (DC/AC converters) to provide power to electric drives. The

`EDrives.PowerConverters.Averaging`model neglects switching effects. The output is represented by a voltage source, which is commanded externally. The output power is calculated by multiplying the commanded ${V}_{\mathrm{out}}$ and the resulting ${I}_{\mathrm{out}}$, which is then reflected to the input. An I controller compares ${P}_{\mathrm{in}}$ and ${P}_{\mathrm{out}}$ and controls the input current sink. The model assumes a constant efficiency of 100% and runs continuously.

**Oliver et al.**propose in [15] a behavioral model of a multi-output DC/DC converter. The input is represented by current sink and the outputs are represented by voltage sources with internal resistances. The focus is on modeling transient behaviors (start-up, load steps and output cross-over effects) with RCL networks without simulating the switching of the converter. A state-machine handles remote on/off an protections.

**Behrmann et al.**present in [16] a toolbox for energy analysis of self-powered sensors written in MATLAB Simulink. The toolbox features several blocks, which communicate via power ports. Thus, voltage and current levels are not considered and start-up and MPPT cannot be analyzed. However, the converter efficiency is modeled as a look-up table.

## 4. Model Scope and Discussion Based upon Modeling Aspects

#### 4.1. Electrical Representation

#### 4.2. Causal Connection of Input and Output

**forward definition**(approach of energy availability), where the power into the converter is controlled which then results in a certain output [13] or, (2) a

**backward definition**(approach of energy demand), where the output is controlled, which results in a certain power draw at the input of the converter [12,14,15].

#### 4.3. Efficiency Function and Power Losses

#### 4.4. Feedback Control

**energy extraction**, the input voltage ${V}_{\mathrm{in}}$ is controlled by the MPP tracking algorithm (FOCV or P&O). For

**storage interaction**with CCCV charging, a combination of ${I}_{\mathrm{out}}$ and ${V}_{\mathrm{out}}$ is used. And for

**voltage supply**, the output voltage ${V}_{\mathrm{out}}$ is controlled. (2) The converter model is embedded into a complete system of energy sources and sinks. The above voltages and currents are setpoints that can deviate from the actual value. They are no fixed values, since this would lead to contradictions within the system model.

#### 4.5. Converter Start-Up and Shutdown

## 5. The Proposed Behavioral Model

#### 5.1. Overview of the Power Path

#### 5.2. Electrical Representation and Working Principle

#### 5.3. Determination of the Output Current

#### 5.4. Efficiency Calculation Based on Power Losses

- The input current sweep shows a plateau with a flat maximum (i.e., ≈ 1 mA for the ADP5090), whereas the input voltage sweep is monotonically increasing without a clear maximum.
- The curves in the current sweep show a similar shape, but the higher Vin, the higher the efficiency. This can be explained by the fact that the ${V}_{\mathrm{out}}$-to-${V}_{\mathrm{in}}$ ratio moves closer to 1, which causes fewer losses.
- In both graphs, the efficiency drops rapidly for very small input powers. The behavior is similar even if ${V}_{\mathrm{in}}$ and ${I}_{\mathrm{in}}$ are swept independently of each other.
- Only in the ${I}_{\mathrm{in}}$ sweep, the efficiency decreases for high input currents. It will be especially dominant if the input voltage is small. There is no comparable slope at the ”right side” in the ${V}_{\mathrm{in}}$ graph.

#### 5.4.1. Power Loss Proportional to ${I}_{\mathrm{in}}$

#### 5.4.2. Power Loss Proportional to ${I}_{\mathrm{in}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\sqrt{{V}_{\mathrm{in}}}$

#### 5.4.3. Constant Power Loss

#### 5.4.4. Power Loss Proportional to ${I}_{\mathrm{in}}^{2}$

#### 5.5. Overview of the Control Path

#### 5.6. Minimum Start-Up and Working Voltage

`isOn`is introduced (see Figure 7). It indicates whether the converter is running or not.

#### 5.7. Feedback Error Determination and Setpoints

#### 5.8. Closed-Loop Feedback by a PI Controller

#### 5.9. Operation at the Maximum Power Point

`extMPP`or by the parameter

`fixedMPP`and with

`useExternalMPP`being disabled. To disable the MPP feature completely, the voltage ${V}_{\mathrm{mpp}}$ can be set to 0 V. To determine ${V}_{\mathrm{mpp}}$, several approaches are used in reality (FOCV and P&O) [7,8]. The different techniques are not part of this DC/DC converter model. However, the MPP tracking is implemented by means of an external tracker, which follows the FOCV method and a pilot cell. The setpoint ${V}_{\mathrm{mpp}}$ is calculated as the fraction of the open-circuit voltage ${V}_{\mathrm{oc}}$ according to

## 6. Simulation Setup

`EnergyHarvestingWSN.PowerConverter.UnitTests.SolarADP5090`.

^{®}Core

^{™}i7 running at $2.5$ GHz. As numerical solver, the program Open-Modelica 1.18.0-dev was used and an interpolation interval of 100 s was chosen.

## 7. Results

#### 7.1. Simulation Performance

#### 7.2. Behavior Discussion

## 8. Discussion

## 9. Conclusions

- The model implements a complete start-stop behavior with minimum working voltage ${V}_{\mathrm{min}}$ and the cold-start voltage ${V}_{\mathrm{start}}$.
- The converter efficiency is modeled as a function, which is based on losses and depends on ${V}_{\mathrm{in}}$ and ${I}_{\mathrm{in}}$. The loss terms are carefully selected to easily extract the parameters from the manufacturer’s datasheet.
- The closed-loop controller allows three modes of operation: CV (constant-voltage output), CC (constant-current output) and MPP (following the maximum power point by regulating Vin).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

CCCV | constant-current, constant-voltage (charging scheme) |

DAE | differential algebraic equation |

DC | direct current (opposite of AC = alternating current) |

EH | energy harvesting |

FOCV | fractional open-circuit voltage (method) |

MPP | maximum power point |

PMIC | power management integrated circuit |

PI | proportional–integral (controller) |

P&O | perturb and observe (method) |

WSN | wireless sensor nodes |

## References

- Tang, X.; Wang, X.; Cattley, R.; Gu, F.; Ball, A.D. Energy harvesting technologies for achieving self-powered wireless sensor networks in machine condition monitoring: A review. Sensors
**2018**, 18, 4113. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Magno, M.; Brunelli, D.; Sigrist, L.; Andri, R.; Cavigelli, L.; Gomez, A.; Benini, L. InfiniTime: Multi-sensor wearable bracelet with human body harvesting. Sustain. Comput. Inform. Syst.
**2016**, 11, 38–49. [Google Scholar] [CrossRef] - Kim, J.; Lynch, J.P. Experimental analysis of vehiclebridge interaction using a wireless monitoring system and a two-stage system identification technique. Mech. Syst. Signal Process.
**2012**, 28, 3–19. [Google Scholar] [CrossRef] - Raghunathan, V.; Kansal, A.; Hsu, J.; Friedman, J.; Srivastava, M. Design considerations for solar energy harvesting wireless embedded systems. In Proceedings of the 4th International Symposium on Information Processing in Sensor Networks, IPSN, Boise, ID, USA, 15 April 2005; Volume 2005, pp. 457–462. [Google Scholar]
- Kokert, J.; Beckedahl, T.; Reindl, L.M. Evaluating Micro-Power Management of Solar Energy Harvesting using a Novel Modular Platform. J. Phys. Conf. Ser.
**2016**, 773, 12042. [Google Scholar] [CrossRef] - Kokert, J.; Beckedahl, T.; Reindl, L.M. Medlay: A reconfigurable micro-power management to investigate self-powered systems. Sensors
**2018**, 18, 259. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dondi, D.; Bertacchini, A.; Brunelli, D.; Larcher, L.; Benini, L. Modeling and optimization of a solar energy harvester system for self-powered wireless sensor networks. IEEE Trans. Ind. Electron.
**2008**, 55, 2759–2766. [Google Scholar] [CrossRef] - Mellit, A.; Rezzouk, H.; Messai, A.; Medjahed, B. FPGA-based real time implementation of MPPT-controller for photovoltaic systems. Renew. Energy
**2011**, 36, 1652–1661. [Google Scholar] [CrossRef] - Stoecklin, S.; Yousaf, A.; Gidion, G.; Reindl, L.; Rupitsch, S.J. Simultaneous power feedback and maximum efficiency point tracking for miniaturized rf wireless power transfer systems. Sensors
**2021**, 21, 2023. [Google Scholar] [CrossRef] [PubMed] - Jirgl, M.; Bradac, Z.; Fiedler, P. Testing the E-PEAS Energy Management circuit for Embedded Systems. IFAC-PapersOnLine
**2018**, 51, 432–437. [Google Scholar] [CrossRef] - Modelica Association. The Modelica Standard Library v4.0. 2020. Available online: https://modelica.org/ (accessed on 4 July 2021).
- Torrey, D.; Selamogullari, U. A Behavioral Model for DC-DC Converter using Modelica. In Proceedings of the 2nd International Modelica Conference, Oberpfaffenhofen, Germany, 18–19 March 2002; pp. 167–172. [Google Scholar]
- Brkic, J.; Ceran, M.; Elmoghazy, M.; Kavlak, R.; Haumer, A.; Kral, C. Open Source PhotoVoltaics Library for Systemic Investigations. In Proceedings of the 13th International Modelica Conference, Regensburg, Germany, 4–6 March 2019; Volume 157, pp. 41–50. [Google Scholar]
- Haumer, A.; Kral, C. The New EDrives Library: A Modular Tool for Engineering of Electric Drives. In Proceedings of the 10th International Modelica Conference, Lund, Sweden, 10–12 March 2014; Volume 96, pp. 155–163. [Google Scholar]
- Oliver, J.A.; Prieto, R.; Romero, V.; Cobos, J.A. Behavioral modeling of multi-output DC-DC converters for large-signal simulation of distributed power systems. In Proceeding of the 37th IEEE Power Electronics Specialists Conference, Jeju, Korea, 18–22 June 2006. [Google Scholar]
- Behrmann, T.; Budelmann, C.; Bosse, S.; Lehmhus, D.; Lemmel, M.C. Tool chain for harvesting, simulation and management of energy in Sensorial Materials. J. Intell. Mater. Syst. Struct.
**2013**, 24, 2245–2254. [Google Scholar] [CrossRef] - Gragger, J.V.; Giuliani, H.; Kral, C.; Bauml, T.; Kapeller, H.; Pirker, F. The SmartElectricDrives Library—Powerful Models for Fast Simulations of Electric Drives. In Proceedings of the 5th International Modelica Conference, Vienna, Austria, 4–5 September 2006. [Google Scholar]
- Shrivastava, A.; Calhoun, B.H. A DC-DC Converter Efficiency Model for System Level Analysis in Ultra Low Power Applications. J. Low Power Electron. Appl.
**2013**, 3, 215–232. [Google Scholar] [CrossRef] [Green Version] - Aloisi, W.; Palumbo, G. Efficiency model of boost dc-dc PWM converters. Int. J. Circuit Theory Appl.
**2005**, 33, 419–432. [Google Scholar] [CrossRef] - Texas Instruments. BQ25570 Datasheet—Nano Powerboost Charger and Buck Converter for Energy Harvester Powered Applications; Texas Instruments: Dallas, TX, USA, 2013. [Google Scholar]
- Kokert, J. A Modelica Library to Model and Simulate Energy Harvesting Powered Wireless Sensor Nodes (EH-WSN). 2021. Available online: https://github.com/jankokert/EnergyHarvestingWSN (accessed on 4 July 2021).
- Analog Devices. ADP5090 Datasheet—Ultralow Power Boost Regulator with MPPT and Charge Management; Analog Devices: Norwood, MA, USA, 2015. [Google Scholar]
- Tremblay, O.; Dessaint, L.A.; Dekkiche, A.I. A generic battery model for the dynamic simulation of hybrid electric vehicles. In Proceedings of the IEEE Vehicle Power and Propulsion Conference, Arlington, TX, USA, 9–12 September 2007; pp. 284–289. [Google Scholar]
- Tiller, M.; Winkler, D. A Searchable Index of All Known Freely Available Modelica Libraries. 2021. Available online: https://modelica.org/ (accessed on 4 July 2021).
- Felgner, F.; Exel, L.; Nesarajah, M.; Frey, G. Component-oriented modeling of thermoelectric devices for energy system design. IEEE Trans. Ind. Electron.
**2014**, 61, 1301–1310. [Google Scholar] [CrossRef]

**Figure 2.**Internal structure of the proposed power converter model implemented in Modelica. The model uses components such as a conductor and a current source and new components depicted as gray blocks. The green labels are global variables that are available in all components. The blocks itself contain only equations and no further Modelica components.

**Figure 3.**Icons of the components in the power path: input and output representation (

**left**), determination of output current (

**middle**) and efficiency calculation (

**right**).

**Figure 4.**Substitution of ${V}_{\mathrm{out}}$ with ${V}_{\mathrm{out}}^{\prime}$ by adding an exponential decaying term.

**Figure 5.**Efficiency w.r.t ${I}_{\mathrm{in}}$ and ${V}_{\mathrm{in}}$ of the ADP5090 boost converter. The output voltage ${V}_{\mathrm{out}}$ is $3.0$ V.

**Figure 6.**Icons of the components in the control path: start-stop mechanism(

**left**), determination of feedback error (

**middle**) and the PI controller (

**right**).

**Figure 9.**Simulation results showing ${I}_{\mathrm{in}}$, ${V}_{\mathrm{in}}$ and ${V}_{\mathrm{mpp}}$ in the top graph, where ${V}_{\mathrm{in}}$ follows most of the time ${V}_{\mathrm{mpp}}$ (covered by blue line). In the bottom graph, the efficiency of the DC/DC converter ${\eta}_{\mathrm{DCDC}}$ and of the solar cell ${\eta}_{\mathrm{sol}}$ is shown besides the battery voltage ${V}_{\mathrm{out}}$, which is increasing monotonously.

Authors | Electrical Representation | Input Behavior and Start-Up | Feedback Control | Efficiency | Operation at MPP |
---|---|---|---|---|---|

Torrey et al. [12] | In: current sink Out: current source | ${V}_{\mathrm{min}}$; but no start-up | commanded ${V}_{\mathrm{out}}$; limited ${I}_{\mathrm{out}}$ | const. value + input resistor | n/a |

Brkic et al. [13] | In: voltage source Out: current source | not modeled | commanded ${V}_{\mathrm{in}}$ | const., 100% | by external vDCRef |

Haumer et al. [14] | In: current sink Out: voltage source | not modeled | commanded ${V}_{\mathrm{out}}$ | const., 100% | n/a |

Oliver et al. [15] | In: current sink Out: voltage source | remote on/off by state diagram | n/a | look-up table | n/a |

Behrmann et al. [16] | In/Out: generic power ports | not modeled | n/a | look-up table | n/a |

this work | In: conductance Out: current source | ${V}_{\mathrm{min}}$ and ${V}_{\mathrm{start}}$ | commanded ${V}_{\mathrm{mpp}}$, ${V}_{\mathrm{out}}$, ${I}_{\mathrm{out}}$ | function based on power losses | by external ${V}_{\mathrm{mpp}}$ |

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**MDPI and ACS Style**

Kokert, J.; Reindl, L.M.; Rupitsch, S.J.
Behavioral Modeling of DC/DC Converters in Self-Powered Sensor Systems with Modelica. *Sensors* **2021**, *21*, 4599.
https://doi.org/10.3390/s21134599

**AMA Style**

Kokert J, Reindl LM, Rupitsch SJ.
Behavioral Modeling of DC/DC Converters in Self-Powered Sensor Systems with Modelica. *Sensors*. 2021; 21(13):4599.
https://doi.org/10.3390/s21134599

**Chicago/Turabian Style**

Kokert, Jan, Leonhard M. Reindl, and Stefan J. Rupitsch.
2021. "Behavioral Modeling of DC/DC Converters in Self-Powered Sensor Systems with Modelica" *Sensors* 21, no. 13: 4599.
https://doi.org/10.3390/s21134599