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PI-Controlled Variable Time-Step Power System Simulation Using an Adaptive Order Differential Transformation Method

Kaiyang Huang, Yang Liu, Kai Sun, Feng Qiu

TL;DR

The paper tackles the computational burden of time-domain power-system transient stability analysis by developing two adaptive semi-analytical DT-based solvers: VS-DT with fixed order and VSOO-DT with adaptive optimal order. By embedding PI-controlled step sizing and a complexity-driven order selection, the methods achieve large time steps without sacrificing stability, even for large-scale systems. The authors provide stability analysis, complexity formulations, and a three-scenario adaptive-order algorithm, validating the approach on IEEE 9-bus, IEEE 39-bus, and Polish 2383-bus systems, including N-1 contingencies, with substantial speedups and high accuracy (mean-max errors near $10^{-7}$ or smaller). The results demonstrate robust, scalable performance and indicate broad applicability to other SAS or adjustable-order numerical schemes beyond DT, offering a practical pathway toward real-time dynamic security assessment of large power grids.

Abstract

Dynamic simulation plays a crucial role in power system transient stability analysis, but traditional numerical integration-based methods are time-consuming due to the small time step sizes. Other semi-analytical solution methods, such as the Differential Transformation method, often struggle to select proper orders and steps, leading to slow performance and numerical instability. To address these challenges, this paper proposes a novel adaptive dynamic simulation approach for power system transient stability analysis. The approach adds feedback control and optimization to selecting the step and order, utilizing the Differential Transformation method and a proportional-integral control strategy to control truncation errors. Order selection is formulated as an optimization problem resulting in a variable-step-optimal-order method that achieves significantly larger time step sizes without violating numerical stability. It is applied to three systems: the IEEE 9-bus, 3-generator system, IEEE 39-bus, 10-generator system, and a Polish 2383-bus, 327-generator system, promising computational efficiency and numerical robustness for large-scale power system is demonstrated in comprehensive case studies.

PI-Controlled Variable Time-Step Power System Simulation Using an Adaptive Order Differential Transformation Method

TL;DR

The paper tackles the computational burden of time-domain power-system transient stability analysis by developing two adaptive semi-analytical DT-based solvers: VS-DT with fixed order and VSOO-DT with adaptive optimal order. By embedding PI-controlled step sizing and a complexity-driven order selection, the methods achieve large time steps without sacrificing stability, even for large-scale systems. The authors provide stability analysis, complexity formulations, and a three-scenario adaptive-order algorithm, validating the approach on IEEE 9-bus, IEEE 39-bus, and Polish 2383-bus systems, including N-1 contingencies, with substantial speedups and high accuracy (mean-max errors near or smaller). The results demonstrate robust, scalable performance and indicate broad applicability to other SAS or adjustable-order numerical schemes beyond DT, offering a practical pathway toward real-time dynamic security assessment of large power grids.

Abstract

Dynamic simulation plays a crucial role in power system transient stability analysis, but traditional numerical integration-based methods are time-consuming due to the small time step sizes. Other semi-analytical solution methods, such as the Differential Transformation method, often struggle to select proper orders and steps, leading to slow performance and numerical instability. To address these challenges, this paper proposes a novel adaptive dynamic simulation approach for power system transient stability analysis. The approach adds feedback control and optimization to selecting the step and order, utilizing the Differential Transformation method and a proportional-integral control strategy to control truncation errors. Order selection is formulated as an optimization problem resulting in a variable-step-optimal-order method that achieves significantly larger time step sizes without violating numerical stability. It is applied to three systems: the IEEE 9-bus, 3-generator system, IEEE 39-bus, 10-generator system, and a Polish 2383-bus, 327-generator system, promising computational efficiency and numerical robustness for large-scale power system is demonstrated in comprehensive case studies.

Paper Structure

This paper contains 15 sections, 36 equations, 7 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Differential Transformation Method
  3. PI Control-Based Step Strategy
  4. A typical step change algorithm
  5. A PI Control Based Step Strategy
  6. Stability of the PI Controller and Selection of Parameters
  7. Optimal Order Selection
  8. The Complexity of Power System Simulation Using DT
  9. Algorithm of the Adaptive Order Strategy
  10. Selecting Parameters
  11. Numerical Results
  12. Screening of Comprehensive Scenarios
  13. Comparing with the Variable-Step DT Method with a Fixed Order
  14. Comparison with Classical Numerical Method
  15. Conclusion

Figures (7)

  • Figure 1: Simulation results for the IEEE 9-bus system. (a) Relative rotor angles in the stable case. (b) maximum errors in the stable case. (c) Step size during the simulation in the stable case. (d) Optimal order of DT during the simulation in the stable case. (e) Relative rotor angles in the unstable case. (f) maximum errors in the unstable case. (g) Step size during the simulation in the unstable case. (h) Optimal order of DT during the simulation in the unstable case.
  • Figure 2: Simulation results for the IEEE 39-bus system. (a) Relative rotor angles in the stable case. (b) maximum errors in the stable case. (c) Step size during the simulation in the stable case. (d) Optimal order of DT during the simulation in the stable case. (e) Relative rotor angles in the unstable case. (f) maximum errors in the unstable case. (g) Step size during the simulation in the unstable case. (h) Optimal order of DT during the simulation in the unstable case.
  • Figure 3: Simulation results for the Polish 2383-bus system. (a) Relative rotor angles in the stable case. (b) maximum errors in the stable case. (c) Step size during the simulation in the stable case. (d) Optimal order of DT during the simulation in the stable case. (e) Relative rotor angles in the unstable case. (f) maximum errors in the unstable case. (g) Step size during the simulation in the unstable case. (h) Optimal order of DT during the simulation in the unstable case.
  • Figure 4: Stability comparison for the DT, VS-DT, and VSOO-DT methods for the 2383-bus system. (a)Errors comparison. (b)Execution time comparison.
  • Figure 5: Simulation errors of the 2383-bus system using different methods
  • ...and 2 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2