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Power system transient stability analysis requires an appropriate integration time step to avoid numerical instability as well as to reduce computational demands. For fast system dynamics, which vary more rapidly than what the time step covers, a fraction of the time step, called a subinterval, is used. However, the optimal value of this subinterval is not easily determined because the analysis of the system dynamics might be required. This selection is usually made from engineering experiences, and perhaps trial and error. This paper proposes an optimal subinterval selection approach for power system transient stability analysis, which is based on modal analysis using a single machine infinite bus (SMIB) system. Fast system dynamics are identified with the modal analysis and the SMIB system is used focusing on fast local modes. An appropriate subinterval time step from the proposed approach can reduce computational burden and achieve accurate simulation responses as well. The performance of the proposed method is demonstrated with the GSO 37-bus system.

Modern power systems operate closer to their operation and stability limits. This is because of load demand growth, the open access of transmission network, and economic operation [

A transient stability simulator solves a set of nonlinear differential and algebraic equations (DAEs) representing machines and controllers placed in the power systems, and the power system network, respectively. It utilizes numerical integration methods to estimate dynamic states at the next time step and uses iterative techniques to solve the nonlinear algebraic equations. Numerical integration methods are divided into two main categories: implicit and explicit methods [

The multi-rate method provides an efficient integration technique for systems exhibiting a wide variety of time responses. The multi-rate method integrates different variables with different time steps [

As modern systems have become increasingly complex such as with the introduction of load dynamics, other power electronic models, and renewable energy sources, the system has become potentially more susceptible to numerical instability issues. Appropriate time steps should be carefully determined to prevent the numerical instability. However, commercial transient stability packages that utilize the multi-rate method often do not provide guidelines or a built-in function to select an appropriate subinterval time step. Instead, they use either a fixed value for all models of a type, or in the case of PowerWorld Simulator, heuristics based on model parameters. The tradeoff is between excessive computation if the interval is too short, and potential numeric instability if it is too long.

In this paper, a new subinterval selection approach is presented to determine the optimal numerical integration time step for the use with the multi-rate method. An appropriate subinterval can be determined by identifying how fast dynamic states are varying in the considered power system. The proposed approach uses modal analysis for single machine infinite bus (SMIB) models [

This paper is organized as follows:

Many commercial power system transient stability packages use explicit numerical integration methods, which estimate values for the next time step explicitly using present values. The second-order Runge-Kutta (RK2) method is one example of the explicit methods [

When the simplest test equation

As shown in

Based on Equation (3), minimum eigenvalues that the RK2 method can cover without numerical instability can be determined depending on the time step as shown in

Region of stability of the second-order Runge-Kutta (RK2) method.

Range of eigenvalues depending on the time step using the RK2 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 |

In practical power systems, only a small portion of the dynamic states are associated with fast dynamics and thus the use of very small time step for the entire simulation is not efficient in terms of computational time and storage. To avoid this computational burden, the multi-rate method that uses different time steps in a numerical integration scheme, has been commonly used [

Multi-rate method.

The SMIB system models the machine of interest in detail, while the rest of the system is represented with a Thevenin equivalent circuit of a voltage behind an impedance. The Thevenin equivalent impedance at bus

Single machine infinite bus (SMIB) system.

For small perturbations, it is sufficient to analyze the linearized power system model. Linear models are simpler to understand and have many useful tools for analysis. One such tool is eigenvalue analysis. Eigenvalues indicate the system stability and how close the system is to becoming unstable. It also shows what frequencies and modes exist in the system, as well as how the system states interact with these modes.

The SMIB system can be modeled with Equations (4) and (5) which represent the power system dynamics and the stator and network algebraic equations, respectively [

_{SMB}

_{SMIB}

The power system equations shown in Equations (4) and (5) are nonlinear. Hence, they need to be linearized around the operating point in order to find out the eigenvalues. The equations of the linearized power system are given by Equations (6) and (7):

Then, the system modal matrix is obtained by incorporating Equation (7) into (6) as following:

With the system modal matrix (_{sys}

As described in

In the approach presented here, the optimal subinterval time step is determined by analyzing the fast dynamics in each SMIB system. This works because the fast varying states are related to the local oscillations rather than inter-area oscillations. In the proposed method, the modal matrix is constructed by linearizing the SMIB system equations around an operating point. The SMIB eigenvalues can then be found, of which their magnitudes describe how rapidly the dynamics vary. By the use of the participation factor in linear system theory [

Finally, the subinterval step size for the fast dynamic states is determined by considering the region of stability equation of numerical integration method in use. For the RK2 method, explained in

Flowchart of the proposed approach.

The proposed approach was tested with the PowerWorld Simulator, which provides SMIB eigenvalue analysis and uses a multi-rate numerical integration method [

GSO 37-bus system.

Block diagram of EXST1 exciter.

EXST1 Exciter model parameters.

_{r} |
_{i}_{max} = 10 |
_{i}_{min} = −10 |
_{c} |

_{b} |
_{a} |
_{a} |
_{r}_{max} = 3.6 |

_{r}_{min} = 0 |
_{c} |
_{f} |
_{f} |

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

_{e} |
_{lr} |
_{lr} |

With the test case considered, SMIB systems for all generators were created and modal analysis of each SMIB system was then performed. The results in _{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.

Single machine infinite bus (SMIB) eigenvalue analysis with the GSO 37-bus case.

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 | – | – | – |

Participation factor of a generator at bus 28.

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 |

Due to the extremely fast modes (−1602), if a single rate approach was used, the time step for the entire system would need to be just 0.05 cycles to avoid numerical instability; this can be determined with Equation (3) and

PowerWorld Simulator has options that allow the user to override the built-in heuristics and directly specify the desired number of subintervals (in powers of two from two up to 128) for specific models. Hence, eight subintervals were chosen here to meet the minimum. In order to validate the performance of the proposed method, simulation comparisons were made by changing the subinterval step ratio. A three-phase bus to ground fault was applied at bus 28, which has the two EXST1 exciter generators. The fault was applied at 1.0 s and cleared at 1.1 s.

Simulation comparison between four and eight subintervals: (

Next, a comparison was made between single rate and multi-rate methods. With the single rate method, 0.05 cycles was used for the time step, which is determined based on

Simulation comparison with single-rate and multi-rate methods: (

Computational time comparison.

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 |

When extremely large negative eigenvalues exist with power system dynamics being integrated using explicit methods, special considerations should be required to avoid numerical instability during power system transient stability analysis. The multi-rate numerical integration method is a common approach to reduce this computational issue. For the use of the multi-rate approach, the ratio between the user-specified time step and the subinterval for fast variables should be determined. This task is not easily identified in commercial transient stability packages. In this paper, an optimal subinterval selection method has been proposed. The method utilizes SMIB eigenvalue analysis, which identifies fast modes and dynamic states related to those modes. Based on the region of stability of numerical integration method in use, the appropriate step size can then be determined by comparing a user-defined time step for slow variables with very small time steps for fast variables. Optimum step size can reduce computational demands as well as avoid numerical instability. Test simulations with the GSO 37-bus case confirm that the proposed method provides quite a good performance in terms of accuracy and computational speed.

The authors gratefully acknowledge the support for this work provided by Korea Electric Power Corporation (KEPCO), Seoul, Korea and the U.S. Department of Energy through award number DE-OE0000097.

Soobae Kim proposed the approach and performed simulation and wrote the vast majority of paper. Thomas J. Overbye overviewed the approach, made some additions, and proofread the manuscript.

The first author declares no conflict of interest. The second author is the original developer of PowerWorld Simulator, helped to develop the PowerWorld transient stability code, and is a co-owner of PowerWorld Corporation.