# State-Feedback Control of Interleaved Buck–Boost DC–DC Power Converter with Continuous Input Current for Fuel Cell Energy Sources: Theoretical Design and Experimental Validation

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## Abstract

**:**

^{®}/Simulink

^{®}, and by experimental tests using DS 1202 MicroLabBox.

## 1. Introduction

_{2}emissions and reducing global warming effects [4]. In this study, fuel cell energy sources are focused on. A fuel cell is an electrochemical generator, whose electrodes are continuously supplied with fuel and oxidant. In electric vehicles, proton exchange membrane fuel cell (PEMFC) technology is used [5]. Accordingly, proton exchange membranes supply hydrogen, while oxygen is obtained from the air. In most applications, the electrical energy produced by fuel cells is not well shaped for immediate use (e.g., the provided voltage is not constant). To be usable with several loads of different nature, a fuel cell needs to be associated with one or more power converters with appropriate topologies. The converters are required to shape the provided electric energy (see Figure 1).

_{fc_min}(minimum fuel cell voltage). However, they can be used in other applications.

## 2. Modelling and Analysis of Fuel-Cell in Association with Buck–Boost Converter

#### 2.1. System Presentation

_{k}with its equivalent series resistance ${r}_{k}$, a filtering capacitor C in parallel with the switches and diodes, a static switch S

_{k}controlled by the binary input signal u

_{k}, and a diode D

_{k}(k = 1, …, N). Each diode anode is connected to the same point with the load represented by a pure resistance R, according to the input impedance of the DC-bus. This impedance is actually unknown because it depends on the power demand. This uncertainty, together with other parameter uncertainties, will be investigated in the next section.

#### 2.2. System Modelling

_{k}denotes the binary control signal, taking values 1 or 0, N is the number of the parallelly connected Buck–Boost modules composing the IBBC; ${\tau}_{fc}={C}_{fc}{R}_{ac}$ is the fuel cell electrical time constant. This model is useful for circuit simulation purposes but is not suitable for the controller design because it involves binary control inputs u

_{k}. For the control design purpose, the following averaged model is obtained using the averaging technique [38], which will prove to be useful:

_{k}), x

_{2}and x

_{3}designate the average values of the capacitor voltage (v

_{c}) and the FC internal voltage (v

_{i}), respectively. The quantity μ

_{k}∈ [0, 1], which denotes the duty ratio function of the PWM control signal ${u}_{k}$, acts as the control input for each IBBC module. The quantities ${\overline{v}}_{fc}$, ${\overline{i}}_{fc}$ and ${\overline{v}}_{dc}$ respectively denote the average values of the fuel cell voltage ${v}_{fc}$, the fuel cell current ${i}_{fc}$, the DC-bus voltage ${v}_{dc}$. For simplicity, we assumed the IBBC modules to be identical, leading to equal inductances and their ESR in (7), i.e., ${L}_{k}=L$ and ${r}_{k}=r$, $k=1,\dots ,N$.

#### 2.3. System Steady State Analysis

**Remark**

**1.**

**Remark**

**2.**

## 3. Nonlinear State Feedback Controller

- (i)
- Tight regulation of output DC-bus voltage despite load uncertainty.
- (ii)
- Equal current sharing between IBBC branches, i.e., the inductor currents should be equal to each other in order to avoid overloading one of the modules, especially when supplying heavy loads. This property entails the reduction in the current ripple, which is beneficial for fuel cells.
- (iii)
- Asymptotic stability of the closed-loop system.

_{dc}to track its desired value V

_{d}. The point is that the Buck–Boost converter is of a non-minimum phase nature [39] and so perfect tracking of the arbitrary reference signals is not achievable. To avoid this issue, we seek the achievement of the above objective indirectly. Specifically, we consider the inductor currents ${x}_{1k}$, in IBBC modules, as output signals and aim at enforcing them to track reference signals, denoted I

_{d}. The latter is chosen so that if ${x}_{1k}={I}_{d}$ then ${v}_{dc}={V}_{d}$. From (13), (14), we obtain the following relationship between the desired current value ${I}_{d}$ and the desired voltage ${V}_{d}$:

**Theorem**

**1.**

- (1)
- The closed-loop system with state variables$({e}_{1k},{e}_{2})$is globally asymptotically stable around the origin;
- (2)
- The tracking errors${e}_{1k}$converge asymptotically to zero, implying proper current sharing between the modules;
- (3)
- The estimation error$\tilde{\theta}=\theta -\widehat{\theta}$converges to zero and, consequently, the estimated reference current${\widehat{I}}_{d}$converges to its real value,${I}_{d}$. It turns out that the tracking error$\epsilon ={v}_{dc}-{V}_{d}$converges to zero, ensuring tight regulation of the DC-bus voltage.

**Proof.**

**Part 1**: Now, substituting the right side of (34) in (33), one obtains the following derivative of the Lyapunov function:

**Part 2**: After applying LaSalle’s invariance theorem [41], it further follows that the state vector $({e}_{1k},{e}_{2},\tilde{\theta})$ converges to the largest invariant set of (40–42) contained in the set $\left\{({e}_{1k},{e}_{2},\tilde{\theta})\in I{R}^{N+2}/{\dot{V}}_{1}=0\right\}$. Given (38), the invariant set denoted M is contained in ${M}_{0}\stackrel{\scriptscriptstyle\mathrm{def}}{=}\left\{({e}_{1k},{e}_{2},\tilde{\theta})\in I{R}^{N+2}/({e}_{1k},{e}_{2})=0\right\}$, which shows that

**Part 3**: We have just shown that ${\widehat{I}}_{d}$ converges towards its true value ${I}_{d}$. We will now demonstrate that the DC-bus voltage ${v}_{dc}$ converges to its desired value ${V}_{d}$.

**Remark**

**3.**

## 4. Simulation and Experimental Results

^{®}/Simulink

^{®}SimPower toolbox, and the experiments are performed using a laboratory prototype based on the Dspace DS1202 card.

#### 4.1. Simulation Results

_{dc}and its reference signal V

_{d}. In this figure, one can observe that the controller behavior is satisfactory. Indeed, the DC-bus v

_{dc}perfectly tracks its reference V

_{d}. The overshoot is 0 at t

_{0}and 5% of V

_{d}at the instant of change in the load, the system response time is less than 5 ms.

_{1k}between the inductance current in each IBBC branch and its reference. The figure clearly shows that the error e

_{1k}converges to zero, despite load variations. The signal ripple is tolerable, as it is less than 0.12A.

_{fc}and current i

_{fc}of the fuel cell in the presence of load variations. We can observe that the current of the fuel cell is continuous, which is beneficial for the fuel cell.

_{c}.

_{2}between the capacitor voltage x

_{2}and its desired value x

_{2d}. Clearly, e

_{2}is well regulated to zero, despite the variation in the load.

_{i}; one should note that the value of v

_{i}is low because its charge rate is very high. The value of v

_{i}also represents the discharge of hydrogen H

_{2}in the fuel cell.

#### 4.2. Experimental Results

^{®}/software

^{®}. The testbed consists essentially of the following elements:

- -
- Power supply from BK Precision.
- -
- Power resistors.
- -
- Programmable DC electronic load from BK Precision.
- -
- MicroLabBox-dSPACE DS1202 with Control Desk
^{®}/software^{®}plugged in a Pentium 4 personal computer. - -
- Semikron IGBT module (SEMITEACH).
- -
- Power card together with measurement card.
- -
- Two ferrite inductance.
- -
- Two hall effect current sensors.
- -
- Two voltage sensors.
- -
- A digital scope.

_{dc}and its reference signal V

_{d}. In this figure, one can observe that the controller behavior is satisfactory. Indeed, the DC-bus v

_{dc}perfectly tracks its reference V

_{d}.

_{d}was well estimated.

_{fc}of the fuel cell in the presence of load variations.

_{c}.

_{1k}and the PWM signals, with a switching frequency of 20 kHz.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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FC-IBBC system | $\frac{d{x}_{1k}}{dt}=-\frac{r}{L}{x}_{1k}-\frac{1}{L}\left(1-{\mu}_{k}\right){x}_{2}+\frac{1}{L}{\overline{v}}_{fc}$ | (7) |

$\frac{d{x}_{2}}{dt}=\frac{1}{C}{\displaystyle {\displaystyle \sum}_{k=1}^{N}}\left(1-{\mu}_{k}\right){x}_{1k}+\frac{1}{RC}\left({\overline{v}}_{fc}-{x}_{2}\right)$ | (8) | |

$\frac{d{x}_{3}}{dt}=-\frac{1}{{\tau}_{fc}}{x}_{3}+\frac{1}{{C}_{fc}}{\overline{i}}_{fc}$ | (9) | |

where $k=1,\dots ,N$; | ||

Adaptive control laws | $K=\frac{{V}_{d}}{N}\left({\eta}_{0}\frac{{V}_{d}}{{E}_{0}}+1\right)$ | (23) |

${\widehat{I}}_{d}=K\widehat{\theta}$ | (24) | |

${e}_{1k}={x}_{1k}-{\widehat{I}}_{d}$ | (25) | |

${e}_{2}={x}_{2}-{x}_{2d}$ | (28) | |

${x}_{2d}=\frac{1}{s+{c}_{2}}\left[{\displaystyle {\displaystyle \sum}_{k=1}^{N}}{c}_{1k}{e}_{1k}+{c}_{2}{x}_{2}+\frac{1}{C}{\displaystyle {\displaystyle \sum}_{k=1}^{N}}\left(1-{\mu}_{k}\right){x}_{1k}+\frac{\widehat{\theta}}{C}\left({\overline{v}}_{fc}-{x}_{2}\right)\right]$ | (36) | |

${\mu}_{k}=1+\frac{L}{{x}_{2}}\left(-{c}_{1k}{e}_{1k}+{e}_{2}+\frac{r}{L}{x}_{1k}-\frac{1}{L}{\overline{v}}_{fc}+\frac{K\gamma}{C}\left({\overline{v}}_{fc}-{x}_{2}\right){e}_{2}\right)$ | (37) | |

Adaptive law | $\dot{\widehat{\theta}}=\frac{\gamma}{C}\left({\overline{v}}_{fc}-{x}_{2}\right){e}_{2}$ | (35) |

Design parameters | ${\eta}_{0}\ge 1$;${V}_{d}0$; $\gamma >0$; ${c}_{1k}>0$;${c}_{2}0$; |

Parameter Designation | Value | |
---|---|---|

Fuel Cell | FC open circuit voltage | ${E}_{0}=28.3\mathrm{V}$ |

FC internal capacitor | ${C}_{fc}=130\mathrm{F}$ | |

Association of the activation and concentration resistances | ${R}_{ac}=0.155\mathsf{\Omega}$ | |

Ohmic resistance | ${R}_{O}=2.89\mathrm{m}\mathsf{\Omega}$ | |

IBBC | Number of IBBC | $N=3$ |

Filtering inductance | ${L}_{1}={L}_{2}={L}_{3}=1\mathrm{mH}$ | |

Filtering capacitor | $C=68\mathsf{\mu}\mathrm{F}$ | |

ESR of the inductance | ${r}_{1}={r}_{2}={r}_{3}=0.2\mathsf{\Omega}$ | |

Switching frequency | ${f}_{s}=20\mathrm{kHz}$ |

Parameter | Value |
---|---|

C_{11}= C_{12}= C_{13} | 6000 |

C_{2} | 10,000 |

$\gamma $ | 0.0025 |

η_{0} | 1 |

Parameter Designation | Value | |
---|---|---|

Fuel Cell | Ballard Nexa 1200 fuel cell module the fuel cell has a rated power of 1.2 kW | |

IBBC | Number of IBBCs | $N=2$ |

Filtering inductance | ${L}_{1}={L}_{2}=4\mathrm{mH}$ | |

Filtering capacitor | $C=110\mathsf{\mu}\mathrm{F}$ | |

ESR of the inductance | ${r}_{1}={r}_{2}=0.3\mathsf{\Omega}$ | |

Switching frequency | ${f}_{s}=20\mathrm{kHz}$ |

Parameter | Value |
---|---|

C_{11} = C_{12} | 2000 |

C_{2} | 90,000 |

$\gamma $ | 0.002 |

η_{0} | 1.077 |

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## Share and Cite

**MDPI and ACS Style**

Koundi, M.; El Idrissi, Z.; El Fadil, H.; Belhaj, F.Z.; Lassioui, A.; Gaouzi, K.; Rachid, A.; Giri, F.
State-Feedback Control of Interleaved Buck–Boost DC–DC Power Converter with Continuous Input Current for Fuel Cell Energy Sources: Theoretical Design and Experimental Validation. *World Electr. Veh. J.* **2022**, *13*, 124.
https://doi.org/10.3390/wevj13070124

**AMA Style**

Koundi M, El Idrissi Z, El Fadil H, Belhaj FZ, Lassioui A, Gaouzi K, Rachid A, Giri F.
State-Feedback Control of Interleaved Buck–Boost DC–DC Power Converter with Continuous Input Current for Fuel Cell Energy Sources: Theoretical Design and Experimental Validation. *World Electric Vehicle Journal*. 2022; 13(7):124.
https://doi.org/10.3390/wevj13070124

**Chicago/Turabian Style**

Koundi, Mohamed, Zakariae El Idrissi, Hassan El Fadil, Fatima Zahra Belhaj, Abdellah Lassioui, Khawla Gaouzi, Aziz Rachid, and Fouad Giri.
2022. "State-Feedback Control of Interleaved Buck–Boost DC–DC Power Converter with Continuous Input Current for Fuel Cell Energy Sources: Theoretical Design and Experimental Validation" *World Electric Vehicle Journal* 13, no. 7: 124.
https://doi.org/10.3390/wevj13070124