# Optimal Subinterval Selection Approach for Power System Transient Stability Simulation

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## Abstract

**:**

## 1. Introduction

## 2. Preliminary

#### 2.1. Explicit Numerical Integration Method

Time Step (Cycle) | Range of Eigenvalue |
---|---|

1 | −120 < λ < 0 |

0.25 | −480 < λ < 0 |

0.1 | −1200 < λ < 0 |

0.05 | −2400 < λ < 0 |

#### 2.2. Multi-Rate Method

#### 2.3. SMIB System and Eigenvalue Analysis

**x**variables show the dynamic state variables such as generator rotor angle and speed. The

**y**variables show the algebraic variables such as the network bus voltage and angle:

- f: a vector of dynamic equations;
- g: a vector of algebraic equations;
- x
_{SMB}: a vector of dynamic state variables of SMIB system; - y
_{SMIB}: a vector of algebraic state variables of SMIB system.

- $A={\frac{\partial f}{\partial {x}_{SMIB}}|}_{({x}_{SMIB}^{o},{y}_{SMIB}^{o})};B={\frac{\partial f}{\partial {y}_{SMIB}}|}_{({x}_{SMIB}^{o},{y}_{SMIB}^{o})};C={\frac{\partial g}{\partial {x}_{SMIB}}|}_{({x}_{SMIB}^{o},{y}_{SMIB}^{o})};D={\frac{\partial g}{\partial {y}_{SMIB}}|}_{({x}_{SMIB}^{o},{y}_{SMIB}^{o})};{x}_{SMIB}^{o},\text{}{y}_{SMIB}^{o}$: SMIB system operating points.

_{sys}), eigenvalues corresponding to fast local modes can be identified. Real components of eigenvalues provide information about how fast the corresponding modes are varying. The required time step to prevent the numerical instability can then be determined.

## 3. Problem Definition and Proposed Approach

#### 3.1. Problem Definition

#### 3.2. Proposed Approach

## 4. Case Study

T_{r} = 0 | V_{i}_{max} = 10 | V_{i}_{min} = −10 | T_{c} = 1 |

T_{b} = 1 | K_{a} = 150 | T_{a} = 0.01 | V_{r}_{max} = 3.6 |

V_{r}_{min} = 0 | K_{c} = 0 | K_{f} = 0.04 | T_{f} = 0.4 |

T_{c}_{1} = 1 | T_{b}_{1} = 1 | V_{a}_{max} = 99 | V_{a}_{min} = −99 |

X_{e} = 0 | I_{lr} = 0 | K_{lr} = 0 |

#### 4.1. SMIB Eigenvalue Analysis

_{A}. Therefore, the required time step for the state V

_{A}to avoid numerical instability can be determined with the real part of eigenvalue information, which represents how much the dynamic state varies.

Bus Number | Generator ID | Max Eigenvalues | Bus Number | Generator ID | Max Eigenvalues |
---|---|---|---|---|---|

28 | 1 | −1602 | 54 | 1 | −44 |

28 | 2 | −1602 | 53 | 1 | −42 |

31 | 1 | −49 | 44 | 1 | −42 |

14 | 1 | −45 | 50 | 1 | −38 |

48 | 1 | −44 | – | – | – |

Real Part of Eigenvalues | Machine Angle | Machine Speed | Machine Eqp | Machine PsiDp | Machine PsiQpp | Exciter V_{A} | Exciter V_{F} |
---|---|---|---|---|---|---|---|

−1602 | 0 | 0 | 0.0001 | 0 | 0 | 1 | 0.0015 |

−45 | 0.0178 | 0.0183 | 0 | 0 | 0.9997 | 0 | 0 |

−32 | 0.0008 | 0.0007 | 0.0511 | 0.9987 | 0.0001 | 0.0001 | 0.0002 |

−22 | 0 | 0.0017 | 0 | 0 | 0.0002 | 0 | 0 |

−0.6 | 0.709 | 0.6983 | 0.0329 | 0.0056 | 0.0915 | 0 | 0.0007 |

#### 4.2. Subinterval Step Size

#### 4.3. Simulation Comparisons

**Figure 7.**Simulation comparison between four and eight subintervals: (

**a**) Voltage magnitude; (

**b**) Angle.

**Figure 8.**Simulation comparison with single-rate and multi-rate methods: (

**a**) Voltage magnitude; (

**b**) Angle.

Method Used | Time Step (Cycle) | Subinterval for Fast States | Computation Time (s) | Ratio of the Computation Time |
---|---|---|---|---|

Singlerate | 0.05 | – | 51.3 | 1 |

Multirate | 0.25 | 8 | 10.6 | 0.21 |

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Kim, S.; Overbye, T.J. Optimal Subinterval Selection Approach for Power System Transient Stability Simulation. *Energies* **2015**, *8*, 11871-11882.
https://doi.org/10.3390/en81011871

**AMA Style**

Kim S, Overbye TJ. Optimal Subinterval Selection Approach for Power System Transient Stability Simulation. *Energies*. 2015; 8(10):11871-11882.
https://doi.org/10.3390/en81011871

**Chicago/Turabian Style**

Kim, Soobae, and Thomas J. Overbye. 2015. "Optimal Subinterval Selection Approach for Power System Transient Stability Simulation" *Energies* 8, no. 10: 11871-11882.
https://doi.org/10.3390/en81011871