# 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

## References

- Fouad, A.A.; Aboytes, F.; Carvalho, V.F.; Corey, S.L.; Dhir, K.J.; Vierra, R. Dynamic security assessment practices in North America. IEEE Trans. Power Syst.
**1998**, 3, 1310–1321. [Google Scholar] [CrossRef] - Kundur, P. Power System Stability and Control; McGraw-Hill: New York, NY, USA, 1994. [Google Scholar]
- Sauer, P.W.; Pai, M.A. Power System Dynamics and Stability; Prentice Hall: Upper Saddle River, NJ, USA, 1998. [Google Scholar]
- Griffths, D.F.; Higham, D.J. Numerical Methods for Ordinary Differential Equations: Initial Value Problems; Springer: London, UK, 2010. [Google Scholar]
- Stott, B. Power system dynamic response calculation. Proc. IEEE
**1979**, 67, 219–241. [Google Scholar] [CrossRef] - PowerWorld Corporation. Available online: http://www.powerworld.com/ (accessed on 1 September 2015).
- Simens PSS/E. Available online: http://www.siemens.com/ (accessed on 1 September 2015).
- General Electric Company. PSLF Version 18.0 User Manual; General Electric Company: Schenectady, NY, USA, 2011. [Google Scholar]
- Gear, C. Multirate Methods for Ordinary Differential Equations; Technical Report; University of Illinois at Urbana-Champaign: Champaign, IL, USA, 1974. [Google Scholar]
- Crow, M.L.; Chen, J.G. The multirate method for simulation of power system dynamics. IEEE Trans. Power Syst.
**1994**, 9, 1684–1690. [Google Scholar] [CrossRef] - Crow, M.L.; Chen, J.G. The multirate simulation of FACTS devices in power system dynamics. IEEE Trans. Power Syst.
**1996**, 11, 376–382. [Google Scholar] [CrossRef] - Chen, J.G.; Crow, M.L. A Variable Partitioning Strategy for the Multirate Method in Power Systems. IEEE Trans. Power Syst.
**2008**, 23, 259–266. [Google Scholar] [CrossRef] - Haque, M.H. A fast method for determining the voltage stability limit of a power system. Electr. Power Syst. Res.
**1995**, 32, 35–43. [Google Scholar] [CrossRef] - Haque, M.H. Novel method of assessing voltage stability of a power system using stability boundary in P–Q plane. Electr. Power Syst. Res.
**2003**, 64, 35–40. [Google Scholar] [CrossRef] - Glover, J.D.; Sarma, M.S.; Overbye, T.J. Power System Analysis and Design; Cengage Learning: Stamford, CT, USA, 2012. [Google Scholar]

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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