# Lyapunov Stability and Performance Analysis of the Fractional Order Sliding Mode Control for a Parallel Connected UPS System under Unbalanced and Nonlinear Load Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Description

#### Proposed Control Scheme Based on Frictional-Order SMC

_{RL}${D}^{\alpha}$ is Riemann-Liouville (RL) fractional derivative of $\alpha \mathrm{th}$ order. If ${v}_{\mathit{odref}}{\text{}\mathrm{and}\text{}v}_{\mathit{oqref}}$ are assumed to be constant, hence ${\ddot{v}}_{\mathit{odref}}\text{}\mathrm{and}{\ddot{v}}_{\mathit{oqref}}$ are small values and can be neglected; also substituting values of ${\ddot{v}}_{\mathit{od}}\text{}\mathrm{and}\text{}{\ddot{v}}_{\mathit{oq}}$ from Equations (8) and (9) we get:

## 3. Stability Analysis

## 4. Parallel UPS System

#### 4.1. Droop Control

#### 4.1.1. Inductive Output Impedance Case: $Z=jX$

#### 4.1.2. Resistive Output Impedance Case: $Z=R$

## 5. Virtual Impedance Loop

## 6. Simulation Results

#### 6.1. Balanced Linear Load Condition

#### 6.2. Unbalanced Linear Load Condition

#### 6.3. Nonlinear Load Condition

#### 6.4. Step Load Change Condition

#### 6.5. Parallel Connected UPS Systems Operation

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${C}_{f}$ | Filter capacitor |

L | Filter inductor |

R | Filter resistor |

$\overrightarrow{{v}_{o}}$ | Three-phase inverter output voltage |

$\overrightarrow{{v}_{i}}$ | AC bus or capacitor voltage |

$\overrightarrow{i}$ | Inverter terminal current |

$\overrightarrow{{i}_{o}}$ | AC bus current |

$\overrightarrow{m}$ | Modulation index |

${V}_{\mathit{dc}}$ | Dc-link voltage |

$\theta $ | Transformation angle |

$\omega $ | Angular frequency |

${\beta}_{d}{\text{}\mathrm{and}\text{}\beta}_{q}$ | Model uncertainties in dq axis |

${e}_{d}{\text{}\mathrm{and}\text{}e}_{q}$ | Voltage tracking error in dq axis |

$\alpha ,\text{}\gamma \text{}\mathrm{and}\text{}\lambda $ | Design choice parameters |

${S}_{d}$ and ${S}_{q}$ | Sliding surface in dq axis |

${}_{RL}D^{\alpha}$ | Riemann-Liouville (RL) fractional derivative |

${m}_{d}$ and ${m}_{q}$ | Feedback control law in dq axis |

${K}_{d}{\text{}\mathrm{and}\text{}K}_{q}$ | Sliding gain in dq axis |

## Appendix A

#### Appendix A.1. Basic Fractional Calculus

**Definition**

**A1.**

**Definition**

**A2.**

**Definition**

**A3.**

#### Appendix A.2. Fractional Differentiator Approximation

## References

- Aamir, M.; Kalwar, K.A.; Mekhilef, S. Uninterruptible power supply (ups) system. Renew. Sustain. Energy Rev.
**2016**, 58, 1395–1410. [Google Scholar] [CrossRef] - Aamir, M.; Mekhilef, S. An online transformerless uninterruptible power supply (ups) system with a smaller battery bank for low-power applications. IEEE Trans. Power Electr.
**2017**, 32, 233–247. [Google Scholar] [CrossRef] - Loh, P.C.; Newman, M.J.; Zmood, D.N.; Holmes, D.G. A comparative analysis of multiloop voltage regulation strategies for single and three-phase ups systems. IEEE Trans. Power Electr.
**2003**, 18, 1176–1185. [Google Scholar] - Guerrero, J.M.; Loh, P.C.; Lee, T.-L.; Chandorkar, M. Advanced control architectures for intelligent microgrids—Part II: Power quality, energy storage, and ac/dc microgrids. IEEE Trans. Ind. Electr.
**2013**, 60, 1263–1270. [Google Scholar] [CrossRef] - Hasanzadeh, A.; Onar, O.C.; Mokhtari, H.; Khaligh, A. A proportional-resonant controller-based wireless control strategy with a reduced number of sensors for parallel-operated upss. IEEE Trans. Power Deliv.
**2010**, 25, 468–478. [Google Scholar] [CrossRef] - Kawabata, T.; Miyashita, T.; Yamamoto, Y. Dead beat control of three phase pwm inverter. IEEE Trans. Power Electr.
**1990**, 5, 21–28. [Google Scholar] [CrossRef] - Mattavelli, P. An improved deadbeat control for ups using disturbance observers. IEEE Trans. Ind. Electr.
**2005**, 52, 206–212. [Google Scholar] [CrossRef] - Wang, M.; Li, F.; Liu, Y.; Huang, L.; Sakane, M. Distributed parallel operation of modified deadbeat controlled ups inverters. In Proceedings of the Power Electronics Specialists Conference, Orlando, FL, USA, 17–21 June 2007; pp. 1727–1732. [Google Scholar]
- Zhang, K.; Kang, Y.; Xiong, J.; Chen, J. Direct repetitive control of spwm inverter for ups purpose. IEEE Trans. Power Electr.
**2003**, 18, 784–792. [Google Scholar] [CrossRef] - Escobar, G.; Valdez, A.A.; Leyva-Ramos, J.; Mattavelli, P. Repetitive-based controller for a ups inverter to compensate unbalance and harmonic distortion. IEEE Trans. Ind. Electr.
**2007**, 54, 504–510. [Google Scholar] [CrossRef] - Deng, H.; Oruganti, R.; Srinivasan, D. Analysis and design of iterative learning control strategies for ups inverters. IEEE Trans. Ind. Electr.
**2007**, 54, 1739–1751. [Google Scholar] [CrossRef] - Cortes, P.; Rodriguez, J.; Vazquez, S.; Franquelo, L.G. Predictive control of a three-phase ups inverter using two steps prediction horizon. In Proceedings of the 2010 IEEE International Conference on Industrial Technology (ICIT), Vina del Mar, Chile, 14–17 March 2010; pp. 1283–1288. [Google Scholar]
- Kim, D.-E.; Lee, D.-C. Feedback linearization control of three-phase ups inverter systems. IEEE Trans. Ind. Electr.
**2010**, 57, 963–968. [Google Scholar] - Cortés, P.; Ortiz, G.; Yuz, J.I.; Rodríguez, J.; Vazquez, S.; Franquelo, L.G. Model predictive control of an inverter with output LC filter for ups applications. IEEE Trans. Ind. Electr.
**2009**, 56, 1875–1883. [Google Scholar] [CrossRef] - Low, K.-S.; Cao, R. Model predictive control of parallel-connected inverters for uninterruptible power supplies. IEEE Trans. Ind. Electr.
**2008**, 55, 2884–2893. [Google Scholar] - Yaramasu, V.; Rivera, M.; Narimani, M.; Wu, B.; Rodriguez, J. Model predictive approach for a simple and effective load voltage control of four-leg inverter with an output LC filter. IEEE Trans. Ind. Electr.
**2014**, 61, 5259–5270. [Google Scholar] [CrossRef] - Hamzeh, M.; Emamian, S.; Karimi, H.; Mahseredjian, J. Robust control of an islanded microgrid under unbalanced and nonlinear load conditions. IEEE J. Emerging Sel. Top. Power Electr.
**2016**, 4, 512–520. [Google Scholar] [CrossRef] - Baghaee, H.R.; Mirsalim, M.; Gharehpetian, G.B.; Talebi, H.A. A decentralized robust mixed H
_{2}/H_{∞}voltage control scheme to improve small/large-signal stability and frt capability of islanded multi-der microgrid considering load disturbances. IEEE Syst. J.**2018**, 12, 2610–2621. [Google Scholar] [CrossRef] - Abrishamifar, A.; Ahmad, A.; Mohamadian, M. Fixed switching frequency sliding mode control for single-phase unipolar inverters. IEEE Trans. Power Electr.
**2012**, 27, 2507–2514. [Google Scholar] [CrossRef] - Matas, J.; De Vicuna, L.G.; Miret, J.; Guerrero, J.M.; Castilla, M. Feedback linearization of a single-phase active power filter via sliding mode control. IEEE Trans. Power Electr.
**2008**, 23, 116–125. [Google Scholar] [CrossRef] - Komurcugil, H. Rotating-sliding-line-based sliding-mode control for single-phase ups inverters. IEEE Trans. Ind. Electr.
**2012**, 59, 3719–3726. [Google Scholar] [CrossRef] - Saetieo, S.; Devaraj, R.; Torrey, D.A. The design and implementation of a three-phase active power filter based on sliding mode control. IEEE Trans. Ind. Appl.
**1995**, 31, 993–1000. [Google Scholar] [CrossRef] - Pichan, M.; Rastegar, H. Sliding-mode control of four-leg inverter with fixed switching frequency for uninterruptible power supply applications. IEEE Trans. Ind. Electr.
**2017**, 64, 6805–6814. [Google Scholar] [CrossRef] - Chang, E.-C.; Su, Z.; Xu, Z.; Wu, R.-C. Particle swarm optimization tuned fuzzy terminal sliding mode control for ups inverters. J. Intell. Fuzzy Syst.
**2015**, 29, 2483–2488. [Google Scholar] [CrossRef] - Choi, Y.; Choi, H.; Jung, J. An adaptive sliding-mode control technique for three-phase ups system with auto-tuning of switching gain. Electr. Eng.
**2014**, 96, 373–383. [Google Scholar] [CrossRef] - Baghaee, H.R.; Mirsalim, M.; Gharehpetian, G.B.; Talebi, H.A. A decentralized power management and sliding mode control strategy for hybrid ac/dc microgrids including renewable energy resources. IEEE Trans. Ind. Inf.
**2017**. [Google Scholar] [CrossRef] - Baghaee, H.R.; Mirsalim, M.; Gharehpetian, G.B.; Talebi, H.A. Decentralized sliding mode control of wg/pv/fc microgrids under unbalanced and nonlinear load conditions for on-and off-grid modes. IEEE Syst. J.
**2017**. [Google Scholar] [CrossRef] - Pan, I.; Das, S. Chaotic multi-objective optimization based design of fractional order piλdμ controller in avr system. Int. J. Electr. Power Energy Syst.
**2012**, 43, 393–407. [Google Scholar] [CrossRef] - Orlowska-Kowalska, T.; Dybkowski, M.; Szabat, K. Adaptive sliding-mode neuro-fuzzy control of the two-mass induction motor drive without mechanical sensors. IEEE Trans. Ind. Electr.
**2010**, 57, 553–564. [Google Scholar] [CrossRef] - DeCarlo, R.A.; Zak, S.H.; Matthews, G.P. Variable structure control of nonlinear multivariable systems: A tutorial. Proc. IEEE
**1988**, 76, 212–232. [Google Scholar] [CrossRef] - Pei, Y.; Jiang, G.; Yang, X.; Wang, Z. In Auto-master-slave control technique of parallel inverters in distributed ac power systems and ups. In Proceedings of the Power Electronics Specialists Conference, Aachen, Germany, 20–25 June 2004; pp. 2050–2053. [Google Scholar]
- Mahmood, H.; Michaelson, D.; Jiang, J. Accurate reactive power sharing in an islanded microgrid using adaptive virtual impedances. IEEE Trans. Power Electr.
**2015**, 30, 1605–1617. [Google Scholar] [CrossRef] - Han, H.; Hou, X.; Yang, J.; Wu, J.; Su, M.; Guerrero, J.M. Review of power sharing control strategies for islanding operation of ac microgrids. IEEE Trans. Smart Grid
**2016**, 7, 200–215. [Google Scholar] [CrossRef] - Guerrero, J.M.; Matas, J.; de Vicuna, L.G.; Castilla, M.; Miret, J. Decentralized control for parallel operation of distributed generation inverters using resistive output impedance. IEEE Trans. Ind. Electr.
**2007**, 54, 994–1004. [Google Scholar] [CrossRef] - Guerrero, J.M.; Vasquez, J.C.; Matas, J.; De Vicuña, L.G.; Castilla, M. Hierarchical control of droop-controlled ac and dc microgrids—A general approach toward standardization. IEEE Trans. Ind. Electr.
**2011**, 58, 158–172. [Google Scholar] [CrossRef] - Delghavi, M.B.; Yazdani, A. Sliding-mode control of ac voltages and currents of dispatchable distributed energy resources in master-slave-organized inverter-based microgrids. IEEE Trans. Smart Grid
**2017**. [Google Scholar] [CrossRef] - Delghavi, M.B.; Shoja-Majidabad, S.; Yazdani, A. Fractional-order sliding-mode control of islanded distributed energy resource systems. IEEE Trans. Sustain. Energy
**2016**, 7, 1482–1491. [Google Scholar] [CrossRef] - Zheng, L.; Jiang, F.; Song, J.; Gao, Y.; Tian, M. A discrete-time repetitive sliding mode control for voltage source inverters. IEEE J. Emerg. Sel. Top. Power Electr.
**2018**, 6, 1553–1566. [Google Scholar] [CrossRef]

**Figure 4.**Hierarchical control block diagram of UPS system. SPWM: sinusoidal pulse width modulation.

**Figure 7.**Configuration of parallel-connected UPS system. SMC: Sliding mode control; EMI filter: electromagnetic interference filter; PQ: active and reactive power.

**Figure 8.**Droop control scheme for parallel-connected UPS system: (

**a**) inductive output impedance; (

**b**) resistive output impedance.

**Figure 9.**Equivalent model of two parallel-connected inverters to an ac common bus with: (

**a**) output inductive impedance; (

**b**) output resistive impedance.

**Figure 12.**Waveform of voltage and current of proposed FOSMC under balanced linear load current. (

**a**) phase-voltage ${v}_{oabc}$ (upper row) and current ${i}_{oabc}$ (lower row); (

**b**) voltage d-axis and q-axis.

**Figure 13.**Waveform of output voltage and current under unbalanced linear load current: (

**a**) phase-voltage ${v}_{oabc}$ (upper row) and current ${i}_{oabc}$ (lower row) when proposed FOSMC is employed; (

**b**) phase-voltage ${v}_{oabc}$ (upper row) and current ${i}_{oabc}$ (lower row) when conventional PI controller is employed.

**Figure 14.**Waveform of output voltage and current under nonlinear load current: (

**a**) phase-voltage ${v}_{oabc}$ (upper row) and current ${i}_{oabc}$ (lower row) when proposed FOSMC is employed; (

**b**) phase- voltage ${v}_{oabc}$ (upper row) and current ${i}_{oabc}$ (lower row) when conventional PI controller is employed.

**Figure 15.**Waveform of voltage ${v}_{oabc}$ (

**upper row**) and current ${i}_{oabc}$ (

**lower row**) under step load change current when proposed FOSMC is employed.

**Figure 16.**Waveform of voltage ${v}_{oabc}$ (

**upper row**) and current ${i}_{oabc}$ (

**lower row**) under linear load for parallel operation of UPS-1 and UPS-2.

**Figure 17.**Waveform of voltage ${v}_{\mathit{oabc}}$ (

**upper row**) and current ${i}_{\mathit{oabc}}$ (

**lower row**) under nonlinear load for parallel operation of UPS-1 and UPS-2.

**Figure 18.**Waveform of active power P (

**blue**) and reactive power Q (

**red**) under linear load for parallel operation of UPS-1 and UPS-2.

Output Impedance | $\mathit{Z}\text{}=\text{}\mathit{jX}\text{}(\mathrm{Inductive}\text{}\varnothing =90\xb0)$ | $\mathit{Z}\text{}=\text{}\mathit{R}\text{}(\mathrm{Resistive}\text{}\varnothing =90\xb0)$ |
---|---|---|

Active Power | $P\text{}=\text{}\frac{\mathit{EV}}{X}\mathit{sin}\varnothing \text{}\cong \text{}\frac{\mathit{EV}}{X}\varnothing $ | $P\text{}=\text{}\frac{\mathit{EVcos}\varnothing -{V}^{2}}{R}\text{}\cong \text{}\frac{V}{R}(E-V)$ |

Reactive Power | $Q\text{}=\text{}\frac{\mathit{EVcos}\varnothing -{V}^{2}}{X}\text{}\cong \text{}\frac{V}{X}(E-V)$ | $Q\text{}=\text{}-\frac{\mathit{EV}}{R}\mathit{sin}\varnothing \text{}\cong \text{}-\frac{\mathit{EV}}{R}\varnothing $ |

Frequency Droop | ${\omega \text{}=\text{}\omega}^{*}-{k}_{P}P$ | ${\omega \text{}=\text{}\omega}^{*}{\text{}+\text{}k}_{P}Q$ |

Amplitude Droop | ${E\text{}=\text{}E}^{*}-{k}_{Q}Q$ | ${E\text{}=\text{}E}^{*}-{k}_{Q}P$ |

Droop coefficient | ${k}_{P}=\frac{\Delta \omega}{{P}_{N}}$ | ${k}_{P}=\frac{\Delta \omega}{{2Q}_{N}}$ |

Droop coefficient | ${k}_{Q}=\frac{\Delta E}{{2Q}_{N}}$ | ${k}_{Q}=\frac{\Delta E}{{P}_{N}}$ |

Parameters | Symbol | Values |
---|---|---|

Switching frequency | f | 10 kHz |

Filter inductor | L | 25 mH |

Filter capacitor | C_{f} | 600 µF |

Resistor | R | 4 mΩ |

Dc-link voltage | V_{dc} | 670 V |

Angular frequency | w_{o} | 314 rad/s |

Parameters | Symbol | Values |
---|---|---|

FOSMC parameter | $\gamma $ | 0.9 |

FOSMC parameter | $\lambda $ | 1500 |

Fractional-order Slide gains | ${K}_{d},{K}_{q}$ | 4.5 × 106 |

PI proportional gain | ${K}_{P}$ | 0.15 |

PI integral gain | ${K}_{I}$ | 42 |

Load Types | Proposed FOSMC Voltage THD (%) | Conventional PI Controller Voltage THD (%) |
---|---|---|

Balanced load | 0.44 | 1.83 |

Nonlinear load | 1.03 | 6.84 |

No Load | 0.37 | 1.53 |

Reference | Control Technique | %Voltage THD Linear Load | %Voltage THD Nonlinear Load | Controller Complexity |
---|---|---|---|---|

[3] | PI | 16 | 42 | Low |

[7] | Dead Beat | 2.1 | 4.8 | Medium |

[5] | PR | 1.4 | 4.6 | Low |

[14] | MPC (observer based) | 2.82 | 3.8 | Medium |

Proposed | Fractional-order SMC | 0.44 | 1.03 | Low |

Characteristics | Proposed FOSMC | [23] | [36] | [37] | [38] |
---|---|---|---|---|---|

Response time | Very High | High | High | High | High |

Tracking accuracy | Fast | Medium | Medium | Fast | Medium |

Robustness | Very High | Moderate | Moderate | High | High |

Chattering phenomena | No | No | Yes | No | No |

Harmonic compensation | Excellent | Good | Good | Excellent | Excellent |

Transient response | Very Fast | Fast | Fast | Very Fast | Fast |

THD% (Nonlinear load) | 1.03 | 1.7 | 1.9 | 1.3 | 1.0 |

THD% (Linear load) | 0.44 | 0.4 | 0.6 | 0.4 | 0.7 |

Reference tracking error (${e}_{v}\%)$ | 1.89 | 2.7 | - | - | 2 |

© 2018 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/).

## Share and Cite

**MDPI and ACS Style**

Ali, M.; Aamir, M.; Khan, H.S.; Waqar, A.; Haroon, F.; Jafri, A.R. Lyapunov Stability and Performance Analysis of the Fractional Order Sliding Mode Control for a Parallel Connected UPS System under Unbalanced and Nonlinear Load Conditions. *Energies* **2018**, *11*, 3475.
https://doi.org/10.3390/en11123475

**AMA Style**

Ali M, Aamir M, Khan HS, Waqar A, Haroon F, Jafri AR. Lyapunov Stability and Performance Analysis of the Fractional Order Sliding Mode Control for a Parallel Connected UPS System under Unbalanced and Nonlinear Load Conditions. *Energies*. 2018; 11(12):3475.
https://doi.org/10.3390/en11123475

**Chicago/Turabian Style**

Ali, Muhammad, Muhammad Aamir, Hussain Sarwar Khan, Asad Waqar, Faheem Haroon, and Atif Raza Jafri. 2018. "Lyapunov Stability and Performance Analysis of the Fractional Order Sliding Mode Control for a Parallel Connected UPS System under Unbalanced and Nonlinear Load Conditions" *Energies* 11, no. 12: 3475.
https://doi.org/10.3390/en11123475