# Direct Power Compensation in AC Distribution Networks with SCES Systems via PI-PBC Approach

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## Abstract

**:**

## 1. Introduction

## 2. Dynamical Modeling

**Remark**

**1.**

**Lemma**

**1.**

**Proof.**

**Remark**

**2.**

## 3. Passivity-Based Control Design

#### 3.1. Bilinear Representation

**Definition**

**1.**

**Definition**

**2.**

#### 3.2. Lyapunov’s Requirements for Stability Analysis

#### 3.3. PI-PBC Design

## 4. Control Structure and Physical Constraint

#### 4.1. Control Law

**Remark**

**5.**

#### 4.2. Physical Operative Constraint

## 5. Test System and Simulation Scenarios

#### 5.1. The System Under Study

#### 5.2. Simulation Scenarios

**Scenario 1**(**S1**): Check the proposed controller to manage the active and reactive power independently in the SCES system.**Scenario 2**(**S2**): Evaluate the performance of the proposed controller applied to the SUCCESS system using the DPC model to relieve the oscillations of active and reactive power in the microgrid. This scenario employs a wind power generator located at bus 2, which provides active power and absorbs the reactive power shown in Figure 3. Additionally, the test system has two demands (see DL1 and DL2 loads in Figure 2), which draw active and reactive power, as depicted in Figure 4.

## 6. Results

#### 6.1. Scenario 1

#### 6.2. Scenario 2

#### Complementary Analysis

**Remark**

**6.**

## 7. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Wind generator: (

**a**) wind profile and (

**b**) active power provided and reactive power absorbed.

**Figure 5.**Dynamic behavior of the SCES system for S1: (

**a**) supercapacitor voltage; (

**b**) active power provided; and (

**c**) reactive power absorbed.

**Figure 6.**Dynamic behavior of the first SCES system for S2: (

**a**) supercapacitor voltage; (

**b**) active power provided; and (

**c**) reactive power absorbed.

**Figure 7.**Dynamic behavior of the second SCES system for S2: (

**a**) supercapacitor voltage; (

**b**) active power provided; and (

**c**) reactive power absorbed.

Scenario 1 | |||||
---|---|---|---|---|---|

${\mathrm{MAE}}_{p}$ [W] | ${\mathrm{MAE}}_{q}$ [var] | ${\mathrm{ITAE}}_{p}$ | ${\mathrm{ITAE}}_{q}$ | $\mathrm{THD}$ [%] | |

IDA-PBC | 43.39 | 30.93 | 10.69 | 7.72 | 1.57 |

PI-PBC | 23.71 | 20.65 | 5.94 | 5.35 | 1.55 |

Scenario 2 with${\mathrm{SCES}}_{\mathbf{1}}$ | |||||

${\mathrm{MAE}}_{p}$ [W] | ${\mathrm{MAE}}_{q}$ [var] | ${\mathrm{ITAE}}_{p}$ | ${\mathrm{ITAE}}_{q}$ | $\mathrm{THD}$ [%] | |

IDA-PBC | 28.26 | 23.79 | 7.01 | 5.98 | 0.99 |

PI-PBC | 24.68 | 18.59 | 5.82 | 4.64 | 0.98 |

Scenario 2 with${\mathrm{SCES}}_{\mathbf{2}}$ | |||||

${\mathrm{MAE}}_{p}$ [W] | ${\mathrm{MAE}}_{q}$ [var] | ${\mathrm{ITAE}}_{p}$ | ${\mathrm{ITAE}}_{q}$ | $\mathrm{THD}$ [%] | |

IDA-PBC | 21.39 | 18.76 | 5.28 | 4.66 | 1.82 |

PI-PBC | 18.38 | 14.60 | 4.36 | 3.63 | 1.80 |

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**MDPI and ACS Style**

Gil-González, W.; Martin Serra, F.; Montoya, O.D.; Ramírez, C.A.; Orozco-Henao, C.
Direct Power Compensation in AC Distribution Networks with SCES Systems via PI-PBC Approach. *Symmetry* **2020**, *12*, 666.
https://doi.org/10.3390/sym12040666

**AMA Style**

Gil-González W, Martin Serra F, Montoya OD, Ramírez CA, Orozco-Henao C.
Direct Power Compensation in AC Distribution Networks with SCES Systems via PI-PBC Approach. *Symmetry*. 2020; 12(4):666.
https://doi.org/10.3390/sym12040666

**Chicago/Turabian Style**

Gil-González, Walter, Federico Martin Serra, Oscar Danilo Montoya, Carlos Alberto Ramírez, and César Orozco-Henao.
2020. "Direct Power Compensation in AC Distribution Networks with SCES Systems via PI-PBC Approach" *Symmetry* 12, no. 4: 666.
https://doi.org/10.3390/sym12040666