# A Multi-Slope Sliding-Mode Control Approach for Single-Phase Inverters under Different Loads

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Description

_{1}/dt denotes the time derivative of x

_{1}, ω

_{f}= 1/(LC)

^{0.5}, the time-varying term D(t) is considered as a disturbance, and ${v}_{oR}={V}_{Rm}\mathrm{sin}\text{\hspace{0.17em}}(\mathsf{\omega}\text{\hspace{0.17em}}t)$ is the reference for output voltage. The disturbance term is defined as:

## 3. Suggested Control Structure

#### 3.1. Conventional SMC Drawbacks

#### 3.2. SMC Using Non-Linear Function

#### 3.3. Multi-Slope (MS) Function

_{1}, α

_{2}and δ are adjustable coefficients of the function, which define the general form of the function and it is clear that MS function is a continuous and differentiable function. The variation of the slopes in Parts A and B and extent of these parts for different values of the multi-slope function coefficients (α

_{1}, α

_{2}, δ) are shown in Figure 3b. This clearly shows that the MS function includes two slopes and for small x values, the slope of Part A and for large value of x, the slope of Part B is more effective in this function.

#### 3.4. MS Function Coefficients Setting

_{1}, α

_{2}and δ and to achieve a desired and fast dynamic response with almost negligible overshoot in the output voltage, these coefficients must be adjusted wisely. The tasks of these coefficients in the MS function structure are as follows:

- (1)
- δ coefficient: This coefficient adjusts the slope of Part A and should be positive (δ > 0). On the other hand, by changing this coefficient, the gain of the function will be changed for small error values.
- (2)
- α
_{1}coefficient: This coefficient is utilized in MS function to adjust the slope of Part B and its effect on slope of Part A is negligible. In principle, this coefficient is utilized to adjust the gain of the MS function for large values of the error. - (3)
- α
_{2}coefficient: This coefficient adjusts the height of Part A. Therefore, by increasing this coefficient, the height of Part A will increase. It is worth mentioning that by increasing the value of this coefficient, low values of error in the input of the function creates more effect in the output of the function.

#### 3.5. Sliding Surface and Stability Considerations

_{m1,2}must be positive where m can be p, i, and d. The existence condition must be satisfied for stability of reaching mode as:

_{1}(x

_{1}) and ε

_{2}(x

_{2}) are defined as:

_{1}= 0 and A

_{2}= 0 are two lines in the x

_{1}-x

_{2}plane for the practical values of the inverter parameters. Note that the slopes of these lines are equal which means that these lines are parallel with each other. This slope can be obtained as:

_{1}and x

_{2}axes can be obtained from Equations (26)–(29).

_{1}(t) and k

_{2}(t) are defined as:

_{1}(0) increases to achieve a faster response, this large value of ε

_{1}(0) causes a reduction in sliding-mode existing region which can lead to an overshoot in output voltage. It is clear that for very large value of ε

_{1}(0), instability can be occurred in the system because of the excessive reduction of the existence regions. In order to increase the sliding-mode existing regions and to reduce the overshoots in output voltage for large value of ε

_{1}(0), it is clear that ε

_{2}(0) can be adjusted wisely so that the high speed response is yielded and the overshoot is eliminated from the output voltage. Therefore, the distance between the intersection points of the x

_{2}-axis and origin will be increased. In most cases, the following condition holds:

_{1}(x

_{1})/ε

_{2}(x

_{2}) value. In most cases, α

_{i}

_{1}and α

_{i}

_{2}are chosen so that the following condition holds:

_{1}(x

_{1})/ε

_{2}(x

_{2}) can be determined by:

## 4. Simulation Results

_{p}

_{1}= 3 × 10

^{6}, α

_{p}

_{2}= 6 × 10

^{5}, α

_{d}

_{1}= 30, α

_{d}

_{2}= 7.2 × 10

^{6}, α

_{i}

_{1}= 2 × 10

^{5}, α

_{i}

_{2}= 6 × 10

^{5}, δ

_{p}= 4000, δ

_{d}= 101,000, and δ

_{i}= 6 × 10

^{3}.

#### 4.1. Resistive Load (R = 3 Ω)

#### 4.2. Non-Linear TRIAC-Controlled Resistive Load (R = 3 Ω)

#### 4.3. Non-Linear TRIAC-Controlled Resistive Load (R = 4 Ω)

#### 4.4. Dynamic Change in the Load Resistance

#### 4.5. Load Type Change

#### 4.6. Rectifier Load

#### 4.7. State Trajectories

_{1}value) can be reduced using the proposed MSSMC method. When the current disturbance occurs and the trajectory moves toward the surface, the reaching mode can be reached faster than the conventional SMC because of the improved surface structure. Figure 11b shows a comparison between the proposed and conventional SMC with the same reaching time. As can be obtained, in both cases, the trajectories reach each other at Point A, but, as can be seen, the drop of the output voltage for the inverter, which is controlled by proposed method, is lower than the conventional SMC.

#### 4.8. Comparative THD

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**Suggested sliding-mode control structure: (

**a**) multi-slope and linear sliding surfaces and trajectory of states during the reaching mode; (

**b**) multi-slope function for different slopes of Part A and Part B; and (

**c**) existence regions of sliding-mode in the x

_{1}-x

_{2}plane.

**Figure 4.**Output voltage of the inverter for MSSMC and proposed controller in [42]: (

**a**) output voltages of the inverter controlled by fast controller and proposed MSSMC; and (

**b**) output voltages of the inverter controlled by proposed MSSMC and a fast controller when loading is performed at the peak of the output voltage.

**Figure 5.**Simulation results for a resistive load: (

**a**) output voltage and load current waveforms for a resistive load; (

**b**) the spectrum of the output voltage for a resistive load; and (

**c**) the spectrum of the output voltage for a resistive load.

**Figure 6.**Simulation results for a TRIAC-controlled resistive (R = 3 Ω) load: (

**a**) simulated waveforms of v

_{o}and i

_{Load}for a TRIAC-controlled resistive load; (

**b**) the spectrum of the load current for a TRIAC-controlled resistive load; (

**c**) the spectrum of the output voltage for a TRIAC-controlled resistive load; and (

**d**) output voltage and sliding function for a TRIAC-controlled-resistive load.

**Figure 7.**Simulation results for a TRIAC-controlled resistive (R = 4 Ω) load: (

**a**) simulated waveforms of v

_{o}and i

_{Load}for a TRIAC-controlled resistive load; (

**b**) the spectrum of the load current for a TRIAC-controlled resistive load; and (

**c**) the spectrum of the output voltage for a TRIAC-controlled resistive load.

**Figure 8.**Simulation results for a dynamic change in the load resistance: (

**a**) output voltage and load current for a dynamic change in the load resistance; and (

**b**) the spectrum of the output for a dynamic change in the load resistance.

**Figure 9.**Simulation results for a sudden change of load: (

**a**) output voltage and load current when the load suddenly changes from a resistive type to a capacitive-resistive type; and (

**b**) the spectrum of the output voltage when the load suddenly changes from a resistive type to a capacitive-resistive type.

**Figure 10.**Simulation results for a bridge-rectifier load: (

**a**) output voltage and load current for a bridge-rectifier load for the inverter controlled by MSSMC; (

**b**) the spectrum of the load current for a bridge-rectifier load for the inverter controlled by MSSMC; (

**c**) the spectrum of the output voltage for a bridge-rectifier load for the inverter controlled by MSSMC; and (

**d**) simulated results of output voltage and sliding function for a diode rectifier load.

**Figure 11.**State trajectories in the phase plane: (

**a**) state trajectories in the phase plane obtained by conventional and proposed controller; and (

**b**) state trajectories in the phase plane obtained by conventional and proposed SMC controller.

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

Fundamental Frequency, ω | 2π50 ${\scriptscriptstyle \raisebox{1ex}{$rad$}\!\left/ \!\raisebox{-1ex}{$s$}\right.}$ |

Output Filter Inductance, L | 250 µH |

Output Filter Capacitance, C | 100 µF |

ESR of the filter Inductance and Capacitor, r_{L}, r_{C} | ≈0 |

Dc-link voltage | 460 V |

**Table 2.**THD (total harmonic distortion) and harmonics of the output voltage for three control methods.

Comparison Category | Proposed Method in [42] k _{p} = 25, k_{i} = 60, k = 30 | Proposed MSSMC | Single Slope SMC (s = λx _{1 +} x_{2}), λ = 75,000 |
---|---|---|---|

THD (%) for non-linear load case | 3 | 2.85 | 3.22 |

Output Voltage Fundamental (v) | 308.4 | 306.2 | 308.6 |

2nd harmonic (% of fundamental) | 0.036 | 0.08 | 0.005 |

3rd harmonic (% of fundamental) | 0.63 | 0.3 | 0.69 |

4th harmonic (% of fundamental) | 0.042 | 0.078 | 0.001 |

5th harmonic (% of fundamental) | 0.75 | 0.74 | 0.71 |

Robustness | Good | Very good | Very good |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Khajeh-Shalaly, B.; Shahgholian, G. A Multi-Slope Sliding-Mode Control Approach for Single-Phase Inverters under Different Loads. *Electronics* **2016**, *5*, 68.
https://doi.org/10.3390/electronics5040068

**AMA Style**

Khajeh-Shalaly B, Shahgholian G. A Multi-Slope Sliding-Mode Control Approach for Single-Phase Inverters under Different Loads. *Electronics*. 2016; 5(4):68.
https://doi.org/10.3390/electronics5040068

**Chicago/Turabian Style**

Khajeh-Shalaly, Babak, and Ghazanfar Shahgholian. 2016. "A Multi-Slope Sliding-Mode Control Approach for Single-Phase Inverters under Different Loads" *Electronics* 5, no. 4: 68.
https://doi.org/10.3390/electronics5040068