# Microgrid Stability Controller Based on Adaptive Robust Total SMC

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## Abstract

**:**

## 1. Introduction

## 2. Proposed MSC-Based Microgrid

## 3. MSC Modeling

_{0a}, v

_{0b}, v

_{0c}are the AC bus voltages (per phase) and i

_{0a}, i

_{0b}, i

_{0c}are the AC currents (per phase) of the MSC; L

_{a}, L

_{b}, L

_{c}and C

_{fa}, C

_{fb}, C

_{fc}are the filter inductor and capacitor values, respectively; r

_{a}, r

_{b}, r

_{c}represent the equivalent series resistor (ESR) of the converter, inductor, and power line; r

_{fa}, r

_{fb}, r

_{fc}represent the ESR of the filter capacitor; and i

_{ma}, i

_{mb}, i

_{mc}represent the aforementioned disturbances in the microgrid.

_{n}, defined as:

_{PWM}= V

_{d}/v

_{tri}, and v

_{tri}is the amplitude of a triangular carrier signal.

#### 3.1. System Modeling in Islanding Mode

_{control1k}is the control signal, k = a, b, c.

**∆A**

_{k},

**∆B**

_{k},

**∆D**

_{k}represent the system parameter variations and they satisfy the matching condition which is

**∆A**

_{k}=

**B**

_{k}

**G**

_{k},

**∆B**

_{k}=

**B**

_{k}

**H**

_{k},

**∆D**

_{k}=

**B**

_{k}

**J**

_{k},

**F**

_{k}

**I**

_{mk}=

**B**

_{k}

**K**

_{k}. In order to analyze conveniently, define:

_{1k}and q

_{2k}are unknown positive constants and

**e**

_{k}is the voltage control error, which is defined as:

_{refk}is the output voltage reference, which is set to the nominal value.

#### 3.2. System Modeling in Grid-Connected Mode

**L**

_{g}= diag(L

_{ga}, L

_{gb}, L

_{gc}),

**i**

_{0}= diag(i

_{0a}, i

_{0b}, i

_{0c}),

**R**

_{g}= diag(R

_{ga}, R

_{gb}, R

_{gc}), ${\stackrel{\xb7}{i}}_{0}=diag({\stackrel{\xb7}{i}}_{0a}{\stackrel{\xb7}{i}}_{0b}{\stackrel{\xb7}{i}}_{0c})$,

**v**

_{0}= diag(v

_{0a}, v

_{0b}, v

_{0c}),

**v**

_{m}= diag(v

_{ma}, v

_{mb}, v

_{mc}), being

**u**

_{control2}= diag(u

_{control2a}, u

_{control2b}, u

_{control2c}) the control signal; L

_{ga}, L

_{gb}, L

_{gc}and R

_{ga}, R

_{gb}, R

_{gc}represent inductors and their respective ESRs; v

_{0a}, v

_{0b}, v

_{0c}and i

_{0a}, i

_{0b}, i

_{0c}are the AC voltage and current (per phase) respectively; and v

_{ma}, v

_{mb}, v

_{mc}represent disturbances or uncertainties in the microgrid. Equation (9) can be rearranged as:

**a**= diag(−R

_{ga}/L

_{ga}, −R

_{gb}/L

_{gb}, −R

_{gc}/L

_{gc}),

**b**= diag(k

_{PWM}/L

_{ga}, k

_{PWM}/L

_{gb}, k

_{PWM}/L

_{gc}),

**c**= diag(−1/L

_{ga}, −1/L

_{gb}, −1/L

_{gc}). Due to the parametric variations or external disturbances, Equation (10) should be modified as:

**∆a**,

**∆b**,

**∆c**represent the system parameter variations and they satisfy the matching condition which is

**∆a**=

**bf**,

**∆b**=

**bm**,

**∆c**=

**bn**,

**v**

_{m}

**/L**

_{g}=

**bg**. In order to analyze conveniently, define:

**w**is the uncertainty. Therefore Equation (11) is modified as:

**q**= diag(q

_{a}, q

_{b},q

_{c}) an unknown positive constant.

## 4. ARTSMC System

_{control1a}, u

_{control1b}and u

_{control1c}. The second part is the ARTSMC based current control loop, which produces the control signals u

_{control2a}, u

_{control2b}, and u

_{control2c}. When the microgrid operates in grid-connected mode, the current control loop determines the output power of the MSC by controlling its output current. In this scenario, its output voltage v

_{0a}, v

_{0b}, and v

_{0c}is equal to the grid voltage (v

_{ga}, v

_{gb}, v

_{gc}), which is assumed to be the nominal value, meaning that v

_{0a}, v

_{0b}, v

_{0c}equal to v

_{refa}, v

_{refb}, v

_{refc}, thus the voltage control loop is automatically disabled. When the microgrid operates in islanded mode, the voltage control loop enforces the output voltage of the MSC to track the reference value v

_{refa}, v

_{refb}, v

_{refc}and to keep the microgrid voltage constant. The output power of the MSC is determined by the power balance among DGs, loads, and losses in the microgrid. Under this circumstance, the current control loop is not necessary and therefore the switch S is open.

#### 4.1. ARTSMC Based Current Control Loop

_{0a}, i

_{0b}, i

_{0c}to track the output current reference i

_{refa}, i

_{refb}, i

_{refc}. Thus, a PLL structure is applied in order to generate the output current reference i

_{refa}, i

_{refb}, i

_{refc}, to achieve unity power factor. It is important to notice that the power factor can be adjusted by modifying θ

_{PLL}. The current control loop is synthesized as follows. The first step is to define a sliding surface [35,36]:

**e**

_{i}= diag(i

_{0a}−i

_{refa}, i

_{0b}−i

_{refb}, i

_{0c}−i

_{refc}),

**s**= diag(s

_{a}, s

_{b}, s

_{c}) and β is the state feedback coefficient. e

_{io}is the initial state of e

_{i}. The second step is to design the control to enforce the system state trajectories to go toward the sliding surface (15) and to stay on it:

**u**

_{1}the state feedback term,

**u**

_{2}the robust control term, and

**u**

_{3}the adaptive compensation term; ε is a small positive constant number;

**sign(s)**= diag(sign(s

_{a}), sign(s

_{b}), sign(s

_{b})) and sign(·) is the sign function; abs(·) is the absolute value function; $\widehat{q}$ is the estimated value of

**q**; the parameter deviation is defined as $\tilde{q}=\widehat{q}-q$; and the adaptive law is:

#### 4.2. ARTSMC Based for Voltage Control Loop

_{refa}, v

_{refb}, v

_{refc}, thus keeping the microgrid voltage constant when the microgrid operates in islanded mode. In order to eliminate the control error and to get a sliding motion through the entire state trajectory, let us define a sliding surface as:

**C**is a full rank constant matrix,

**CB**is nonsingular, and

**β**is state feedback control coefficient matrix, e

_{ko}is the initial value of e

_{k}. The system state trajectories are forced toward the sliding surface (26) and stay on it, by designing the control scheme as follows such that:

_{1k}is the state feedback term, u

_{2k}is the robust control term, u

_{3k}is the adaptive compensation term; ${X}_{rk}={\left[\begin{array}{cc}{v}_{refk}& {\dot{v}}_{refk}\end{array}\right]}^{\mathrm{T}}$; ε is a small positive constant and sign(·) is the sign function; ${\widehat{q}}_{1k}$, ${\widehat{q}}_{2k}$ are estimated values of ${q}_{1k}$, ${q}_{2k}$. The parameter deviations are ${\tilde{q}}_{1k}={\widehat{q}}_{1k}-{q}_{1k}$, ${\tilde{q}}_{2k}={\widehat{q}}_{2k}-{q}_{2k}$. Choose adaptive law as:

**β**, the robustness of sliding mode (35) can be determined. The strictly logical and rigorous proof illustrates the ARTSMC system is insensitive to parametric uncertainties and external disturbances.

## 5. Simulation Results

_{ref}is set to 220 V at 50 Hz, and the AC output current reference i

_{ref}is set according to the desired power exchange value at the PCC. Based on the sliding surfaces (15) and (26), the state feedback coefficients are designed to guarantee the robustness of the sliding modes (25) and (35), while determining the control performance and system stability, the parameters are given as: β = 30, β = [0.00270]. The DG

_{PV}system in Figure 1 is connected to the same AC bus with the MSC through a DC-to-AC inverter, which is in phase with the grid voltage, and injecting power to the microgrid.

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

DC voltage (Energy storage equipment) | 800 V |

filter capacitance | 3 μF |

filter inductance | 1.5 mH |

output voltage (RMS) (phase) | 220 V |

output frequency | 50 Hz |

_{PV}is set to 40 kW and the critical load inside the microgrid is 30 kW. At 0.5 s, the output power of DG

_{PV}increases to 50 kW and at 1 s the microgrid disconnects from the utility, thus operating in islanded mode. At 1.5 s the output power of DG

_{PV}decreases to 20 kW, and at 1.75 s the 30 kW critical load is connected to the microgrid.

_{PV}decreases to 20 kW at 1.5 s, the MSC starts injecting 10 kW of active power to the microgrid, and its output power raise to 40 kW as the critical load in the microgrid increases to 60 kW at 1.75 s, while the output power of the DG

_{PV}remains constant.

_{PV}, the microgrid voltage remains stable in all situations.

**Figure 7.**(

**a**) Voltage waveform of the MSC between 0.35 s and 0.65 s; (

**b**) Voltage waveforms of the MSC between 0.85 s and 1.15 s; (

**c**) Voltage waveforms of the MSC between 1.35 s and 1.65 s; (

**d**) Voltage waveforms of the MSC between 1.6 s and 1.9 s.

**Figure 8.**(

**a**) Current waveforms of the MSC between 0.35 s and 0.65 s; (

**b**) Current waveforms of the MSC between 0.85 s and 1.15 s; (

**c**) Current waveforms of the MSC between 1.35 s and 1.65 s; (

**d**) Current waveforms of the MSC between 1.6 s and 1.9 s.

_{PV}. Figure 10 and Figure 11 present the output voltage and current waveforms of the DG

_{PV}. Their output voltage keep steady, while the output current increases at 0.5 s and reduces at 1.5 s with no inrush current. The simulation results indicate smooth transaction and stable operation of the DG system. It shows that MSC provides DG

_{PV}additional islanded operation functionality without changing their inner control strategies conceived for grid-connected mode.

**Figure 10.**(

**a**) Output voltage waveforms of DG

_{PV}between 0.35 s and 0.65 s; (

**b**) Output voltage waveforms of DG

_{PV}between 0.85 s and 1.15 s; (

**c**) Output voltage waveforms of DG

_{PV}between 1.35 s and 1.65 s; (

**d**) Output voltage waveforms of DG

_{PV}between 1.6 s and 1.9 s.

**Figure 11.**(

**a**) Output current waveforms of DG

_{PV}between 0.35 s and 0.65 s; (

**b**) Output current waveforms of DG

_{PV}between 0.85 s and 1.15 s; (

**c**) Output current waveforms of DG

_{PV}between 1.35 s and 1.65 s; (

**d**) Output current waveforms of DG

_{PV}between 1.6 s and 1.9 s.

## 6. Experimental Results

**Figure 13.**(

**a**) Experimental results from grid-connected to islanding modes; (

**b**) Experimental results from islanded to grid-connected modes.

**Figure 14.**(

**a**) Experimental results in grid-connected mode; (

**b**) Experimental results in islanded mode.

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Su, X.; Han, M.; Guerrero, J.M.; Sun, H.
Microgrid Stability Controller Based on Adaptive Robust Total SMC. *Energies* **2015**, *8*, 1784-1801.
https://doi.org/10.3390/en8031784

**AMA Style**

Su X, Han M, Guerrero JM, Sun H.
Microgrid Stability Controller Based on Adaptive Robust Total SMC. *Energies*. 2015; 8(3):1784-1801.
https://doi.org/10.3390/en8031784

**Chicago/Turabian Style**

Su, Xiaoling, Minxiao Han, Josep M. Guerrero, and Hai Sun.
2015. "Microgrid Stability Controller Based on Adaptive Robust Total SMC" *Energies* 8, no. 3: 1784-1801.
https://doi.org/10.3390/en8031784