# Interval Type-II Fuzzy Rule-Based STATCOM for Voltage Regulation in the Power System

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## Abstract

**:**

## 1. Introduction

## 2. Background of IT2 Fuzzy Logic Systems

#### 2.1. IT2 Fuzzy Sets

_{Ã}(x) is the membership function (MF) of x in Ã. Notably, the MF represents the degree to which x belongs to Ã and the value of MF is normally between zero and unity. A high value of the MF indicates that x is very likely to be in Ã. Figure 1a shows an example of a T1 fuzzy set, Ã. Any number in the x domain corresponds to a membership value. The MF μ

_{Ã}(x) of a T1 fuzzy set can be chosen either based on the user’s experience or using algorithms.

**Figure 1.**Examples of (

**a**) a T1 membership function; (

**b**) IT2 membership functions (UMF Ā and LMF $\underset{\_}{\mathrm{{\rm A}}}$).

#### 2.2. IT2 Fuzzy Rules

_{1}is Ã

_{1n}and … and x

_{M}is Ã

_{Mn}then y is Y

_{n}. n = 1, 2, …, N.

_{in}(i =1, 2, …, M) are IT2 fuzzy sets and Y

_{n}=[y

_{n}, ${\overline{\text{y}}}_{n}$] is an interval. Assume that the input vector x* = (${x}_{1}^{*}$, ${x}_{2}^{*}$, …, ${x}_{\text{M}}^{*}$). The IT2 fuzzy reasoning can be implemented using the following steps [22].

_{in}, [${\text{\mu}}_{{\underset{\_}{\text{A}}}_{in}}$(${x}_{i}^{*}$), μ

_{Āin}(${x}_{i}^{*}$)], i = 1, 2, …, M, n = 1, 2, …, N.

_{n}(x*) ≡ [f

_{n}, ${\overline{\text{f}}}_{n}$], n = 1, 2, …, N.

_{n}(x*) and [y

_{n}, ${\overline{\text{y}}}_{n}$].

_{n}} and {${\overline{\text{y}}}_{n}$} are sorted in ascending order. The values of ${\text{y}}_{\ell}$ and y

_{r}in Equations (3) and (4) can be evaluated using Karnik-Mendel algorithms [23].

## 3. Proposed Method

_{bus}and ${\text{V}}_{\text{dc}}^{\text{st}}$ denote the phase angle of the controlled AC bus and the DC voltage across the DC capacitance C

_{m}, respectively The variables km and α are the modulation index and the phase angle of the STATCOM, respectively. From Figure 3, one can obtain Equations (8) and (9) as follows [1,2].

_{m}is the equivalent resistance of the STATCOM. ${\text{i}}_{\text{d}}^{\text{st}}$ and ${\text{i}}_{\text{q}}^{\text{st}}$ are the d- and q-axis currents that flow into the terminals of the STATCOM, respectively.

#### 3.1. Dynamic Equations of STATCOM

^{st}be the voltage at the STATCOM and Vd

_{bus}and Vq

_{bus}be the voltages of the d- and q-axes at the controlled AC bus, respectively. From the model of the STATCOM, the following dynamic equations are attained. Let the controlled bus voltage magnitude be Vt = $\sqrt{{\text{Vd}}_{\text{bus}}^{\text{2}}+{\text{Vq}}_{\text{bus}}^{\text{2}}}$. The following dynamic equations can be obtained:

_{ref}is a given voltage reference.

#### 3.2. IT2 Fuzzy Rules for Controllers

_{ref}− Vt in Equation (11) will be replaced by the output (modified ∆Vt) of the IT2 fuzzy rule base. The proposed IT2 fuzzy rule is expressed as

_{1}is Ã

_{1n}and x

_{2}is Ã

_{2n}then y is Y

_{n}. n = 1, 2, …, 25.

_{1}is the output of the PID controller (denoted as ΔV

_{t}’) and x

_{2}is the change rate of ΔV

_{t}’. The variable y is the modified ∆Vt, which replaces the term V

_{ref}− Vt in Equation (11).

_{1n}and Ã

_{2n}are expressed as trapezoid functions. Possible conditions of x

_{1}and x

_{2}could be NB, NS, ZR, PS or PB and the consequent actions y could be IB, IS, KV, DS or DB, as shown in Table 1.

Linguistic Variables of x_{1} and x_{2} | x_{1} = NB | x_{1} = NS | x_{1} = ZR | x_{1} = PS | x_{1} = PB |
---|---|---|---|---|---|

x_{2} = PB | KV | IS | IB | IB | IB |

x_{2} = PS | DS | KV | IS | IB | IB |

x_{2} = ZR | DB | DS | KV | IS | IB |

x_{2} = NS | DB | DB | DS | KV | IS |

x_{2} = NB | DB | DB | DB | DS | KV |

#### 3.3. Co-Simulation between NEPLAN and SIMULINK

_{in}and Q

_{in}). A constant time step for running the power flow program is specified in the NEPLAN package. V

^{st}, Vd

_{bus}, and Vq

_{bus}, computed by the NEPLAN package, are fed to Simulink.

#### 3.4. Discussions of the Proposed Method

_{p}over K

_{i}for both current and voltage control loops, where K

_{p}and K

_{i}denote the parameters of proportional and integral gains, respectively. Actually, the “ideal ratio” of K

_{p}over K

_{i}is not well-defined and may depend on the operating conditions of different power systems.

Problems | Number of Fuzzy Rules | Types of Fuzzy Rules | Hardware | References |
---|---|---|---|---|

Maximum power-point tracker for photovoltaic arrays | 16 | Mamdani | Infineon TriCore TC1796 | [25] |

Design of wide-area damping controller to damp the inter-area oscillations | 30 | Mamdani | - | [26] |

Improvement of transient stability using FACTS devices | 30 | Mamdani | - | [27] |

Improvement of transient stability using bang-bang controller | 49 | Mamdani | - | [28] |

Control of the inverter for utilization of the wind energy | 49 | Mamdani | PC with DT2821 Data Card | [29] |

Design of power system stabilizer | 49 | Mamdani | - | [30] |

Design of power system stabilizer | 49 | Mamdani | - | [31] |

Power management of energy of storage systems | 60 | Takagi-Sugeno | DSP TMS320F2812 | [32] |

## 4. Simulation Results

_{p}, K

_{i}and K

_{d}in Figure 4 are 10, 10 and 0, respectively. Suppose that the variation of irradiation is linear. The MVA base is 100. The Appendix provides the parameters of IT2 fuzzy rules as well as parameters of the STATCOM model.

#### 4.1. Scenario 1: Switching a Heavy Load

**Figure 7.**Voltages at bus 3 obtained using three methods: (

**a**) traditional PID; (

**b**) T1 fuzzy rules; (

**c**) IT2 fuzzy rules (Scenario 1).

**Figure 8.**MVARs from STATCOM obtained using three methods: (

**a**) traditional PID; (

**b**) T1 fuzzy rules; (

**c**) IT2 fuzzy rules (Scenario 1).

#### 4.2. Scenario 2: Increasing and then Decreasing Irradiations

**Figure 10.**Voltages at bus 3 obtained using three methods: (

**a**) traditional PID; (

**b**) T1 fuzzy rules; and (

**c**) IT2 fuzzy rules (Scenario 2).

**Figure 11.**MVARs from STATCOM obtained using three methods: (

**a**) traditional PID; (

**b**) T1 fuzzy rules; and (

**c**) IT2 fuzzy rules (Scenario 2).

#### 4.3. Scenario 3: Varying Irradiations

^{2}at t = 0 s. The irradiation increases to 1400 W/m

^{2}at t = 10, and thereafter linearly changes to be 1000, 1400, and 1000 W/m

^{2}at t = 12, 15 and 18 s, respectively, in the third scenario. The variations of MW generations from PV are shown in Figure 12.

**Figure 13.**Voltages at bus 3 using obtained three methods: (

**a**) traditional PID; (

**b**) T1 fuzzy rules; (

**c**) IT2 fuzzy rules (Scenario 3).

**Figure 14.**MVARs from STATCOM obtained using three methods: (

**a**) traditional PID; (

**b**) T1 fuzzy rules; (

**c**) IT2 fuzzy rules (Scenario 3).

#### 4.4. Unscheduled Photovoltaic Outage

**Figure 15.**Voltages at bus 3 obtained using three methods: (

**a**) traditional PID; (

**b**) T1 fuzzy rules; (

**c**) IT2 fuzzy rules (Scenario 4).

**Figure 16.**MVARs from STATCOM obtained using three methods: (

**a**) traditional PID; (

**b**) T1 fuzzy rules; and (

**c**) IT2 fuzzy rules (Scenario 4).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix

_{m}= 20.3 p.u., C

_{m}= 0.123 p.u., R

_{st}= 0.00813 p.u., X

_{st}= 0.0325 p.u., Kc = 30, Tc = 0.001 s, Ks = 2, Ts = 0.01 s. The bases of power and voltage are 100 MVA and 23 kV, respectively.

Linguistic Variables | Upper MF [a, b, c, d] | Lower MF [a’, b’, c’, d’] |
---|---|---|

NB | [−2, −2, −1.7, −1.5] | [−2, −2, −1.5, −1.2] |

NS | [−1.5, −1.2, −1, −0.5] | [−1.2, −1.1, −1.1, −1] |

ZR | [−1, −0.5, 0.5, 1] | [−0.5, 0, 0, 0.5] |

PS | [0.5, 1, 1.2, 1.5] | [1, 1.1, 1.1, 1.2] |

PB | [1.2, 1.5, 2, 2] | [1.5, 1.7, 2, 2] |

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**MDPI and ACS Style**

Hong, Y.-Y.; Hsieh, Y.-L.
Interval Type-II Fuzzy Rule-Based STATCOM for Voltage Regulation in the Power System. *Energies* **2015**, *8*, 8908-8923.
https://doi.org/10.3390/en8088908

**AMA Style**

Hong Y-Y, Hsieh Y-L.
Interval Type-II Fuzzy Rule-Based STATCOM for Voltage Regulation in the Power System. *Energies*. 2015; 8(8):8908-8923.
https://doi.org/10.3390/en8088908

**Chicago/Turabian Style**

Hong, Ying-Yi, and Yu-Lun Hsieh.
2015. "Interval Type-II Fuzzy Rule-Based STATCOM for Voltage Regulation in the Power System" *Energies* 8, no. 8: 8908-8923.
https://doi.org/10.3390/en8088908