# A Global Tracking Sensorless Adaptive PI-PBC Design for Output Voltage Regulation in a Boost Converter Feeding a DC Microgrid

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## Abstract

**:**

## 1. Introduction

- The application of the PI-PBC design to regulate the output voltage in a boost converter feeding an unknown DC load that represents the possible consumption in a DC microgrid with constant current, resistance, and power loads, which is modeled as a DC load current.
- The integration of two external input estimators to the PI-PBC design, both with exponential convergence, allows one to find the expected values for the DC load current (immersion & invariance (I&I) method) and the voltage input (disturbance–observer (DO) approach), which makes our proposed control approach an adaptive- and sensorless-based design.

## 2. Mathematical Modeling and Problem Formulation

#### 2.1. Dynamical Modeling and Equilibrium Point

**Remark**

**1.**

#### 2.2. General Control Problem Definition

- To obtain a general feedback control law that allows for stabilizing of the output voltage ${x}_{2}$ to its desired reference ${x}_{2}$, thereby ensuring closed-loop stability and fast convergence.
- To determine the expected value of the load current ${i}_{DC}$ using an estimator with exponential convergence. The values of the constant resistance, power, and current that compose the loads are unknown, and these are impossible to calculate or estimate since they depend on the DC microgrid connected to the boost converter. The estimator of the load current will make the proposed controller work under a sensorless concept.
- To apply a disturbance–observer estimator to determine the expected value of the voltage source E with exponential convergence that permits one to obtain a sensorless-based controller approach.

## 3. Passivity-Based Control Theory

#### 3.1. PI-PBC Design

**Definition**

**1.**

**Lemma**

**1.**

**Proof.**

- Due to the skew-symmetric properties of the interconnection matrix, i.e., $\mathcal{J}\left(\mu \right)=-{\mathcal{J}}^{\top}\left(\mu \right)$, then, the component ${z}^{\top}\mathcal{J}\left(\mu \right)z$ is zero;
- Taking into account that $\mathcal{J}\left({x}^{\star}\right)=\mathcal{J}{x}^{\star}$ for bilinear systems, and defining the passive output as ${y}^{\top}={z}^{\top}\mathcal{J}{x}^{\star}$, then, (12) can be simplified as follows:

#### 3.2. Application to the Boost Converter

## 4. Sensorless Adaptive Design

#### 4.1. Estimation of the DC Load Current: Adaptive Control Design

**Lemma**

**2.**

**Proof.**

#### 4.2. Voltage Input Estimation

**Lemma**

**3.**

**Proof.**

## 5. Analysis and Experimental Results

#### 5.1. Simulation Results

#### 5.2. Effect of the Estimate $\widehat{E}$

#### 5.3. Performance of the Adaptive Proposed Controller

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Some classical converters employed to interface distributed energy resources and loads in DC networks.

**Figure 4.**Experimental setup: (

**a**) oscilloscope, (

**b**) DC electronic device in CPL mode, (

**c**) RT-Box with analog and digital breakout boards, (

**d**) current probe power supply, (

**e**) MOSFET driver power supply, (

**f**) DC-DC boost converter, (

**g**) current probes, (

**h**) input voltage power supply, and (

**i**) differential voltage probe.

**Figure 5.**Simulated comparison between the PI-PBC and SMC approaches: (

**a**) simulation results for the boost converter when the DC load is a 100 Hz square waveform that varies between 2 and 4 A with a 100 Hz square waveform and a duty cycle of $0.5$; (

**b**) input voltage changes from 10 to 12 V; (

**c**) Input voltage changes from 10 to 8 V.

**Figure 6.**Dynamical response of the proposed controller with and without the estimate $\widehat{E}$ when the ${i}_{DC}$ varies in a 100 Hz square waveform between 1 A and 2 A: (

**a**) the proposed controller measures the input voltage, and (

**b**) the proposed controller uses the estimate $\widehat{E}$. CH1: (3 V/div), CH2: (2 A/div), CH3: u (1 /div), and CH4: ${\widehat{i}}_{\mathrm{DC}}$ (1 A/div).

**Figure 7.**Dynamical response of the estimate $\widehat{E}$: (

**a**) input voltage changes between 10 and 12 V; (

**b**) input voltage changes between 10 and 8 V. CH1: (1 V/div); CH3: (1 V/div).

**Figure 8.**Dynamical response of the proposed controller when the input voltage varies from 10 to 12 V. (

**a**) Experimental results for the boost converter when the DC load is a 100 Hz square waveform that varies between 1 and 2 A, with a duty cycle of $0.5$. CH1: ${x}_{2}$ (3 V/div), CH2: ${x}_{1}$ (2 A/div), CH3: u (1/div), CH4: ${\widehat{i}}_{DC}$ (1 A/div), and time base of $5\phantom{\rule{0.166667em}{0ex}}$ ms. (

**b**) Input voltage changes from 10 V to 12 V. CH1: (1 V/div), CH3: (1 V/div), and time base of $10\phantom{\rule{0.166667em}{0ex}}$ ms.

**Figure 9.**Dynamical response of the proposed controller when the input voltage varies from 10 to 8 V. (

**a**) Experimental results for the boost converter when the DC load is a 100 Hz square waveform that varies between 1 and 2 A, with a duty cycle of $0.5$. CH1: ${x}_{2}$ (3 V/div), CH2: ${x}_{1}$ (2 A/div), CH3: u (1/div), CH4: ${\widehat{i}}_{DC}$ (1 A/div), and time base of $5\phantom{\rule{0.166667em}{0ex}}$ ms. (

**b**) Input voltage changes from 10 V to 8 V. CH1: (1 V/div), CH3: (1 V/div), and time base of $10\phantom{\rule{0.166667em}{0ex}}$ ms.

Component | Description | Type/Value |
---|---|---|

${Q}_{1}$ | Power MOSFET | IRFB4110 |

${D}_{1}$ | Schottky Power Diode | RURG8060 |

L | Inductor | Wurth Elektronik 74435584700, 47 $\mathsf{\mu}$H |

C | Multilayer Ceramic Capacitor | TDK C5750X7S2A106M230KB, $10\times 10\phantom{\rule{4pt}{0ex}}\mathsf{\mu}$F |

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## Share and Cite

**MDPI and ACS Style**

Gil-González, W.; Montoya, O.D.; Riffo, S.; Restrepo, C.; Muñoz, J.
A Global Tracking Sensorless Adaptive PI-PBC Design for Output Voltage Regulation in a Boost Converter Feeding a DC Microgrid. *Energies* **2023**, *16*, 1106.
https://doi.org/10.3390/en16031106

**AMA Style**

Gil-González W, Montoya OD, Riffo S, Restrepo C, Muñoz J.
A Global Tracking Sensorless Adaptive PI-PBC Design for Output Voltage Regulation in a Boost Converter Feeding a DC Microgrid. *Energies*. 2023; 16(3):1106.
https://doi.org/10.3390/en16031106

**Chicago/Turabian Style**

Gil-González, Walter, Oscar Danilo Montoya, Sebastián Riffo, Carlos Restrepo, and Javier Muñoz.
2023. "A Global Tracking Sensorless Adaptive PI-PBC Design for Output Voltage Regulation in a Boost Converter Feeding a DC Microgrid" *Energies* 16, no. 3: 1106.
https://doi.org/10.3390/en16031106