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Evaluating the Impact of a Load Admittance Approximation in Transient Stability-Constrained Optimal Power Flow

Alex Junior da Cunha Coelho, Araceli Hernandez, Luis Badesa

TL;DR

This work tackles the computational burden of transient stability-constrained OPF by introducing a voltage-based load admittance approximation that allows precomputing Kron-reduced networks using a nominal bus voltage of $|V|=1.0$ p.u. for the pre-fault network. The TSC-OPF is formulated with an AC-based generator model and discretized swing dynamics, leveraging reduced admittance matrices for during- and post-fault periods. Case studies on the WECC 9-bus system show that rotor-angle trajectories during the first swings are well captured and that applying a voltage-corrected load admittance reduces mean absolute errors relative to time-domain benchmarks, even at coarser time steps. Overall, the proposed approach offers a scalable and robust path to implement TSC-OPF in larger networks with improved convergence properties.

Abstract

The Transient Stability-Constrained Optimal Power Flow (TSC-OPF) incorporates dynamic stability constraints into the OPF formulation to ensure secure and economical operation under disturbances. While discretizing system dynamics enables the use of nonlinear programming techniques, it significantly increases computational burden. To enhance scalability, many studies simplify the network by representing loads as constant admittances, allowing the use of Kron reduction. However, computing the Kron reduction outside the optimization requires a voltage-based assumption to convert loads from constant power to constant admittance. This paper proposes a practical voltage-based load admittance approximation and evaluates the errors it may introduce in rotor angle and speed deviation trajectories. Case studies on the WECC 9-bus system show that the proposed approach reproduces rotor dynamics consistent with time-domain simulations during the first few seconds while considerably reducing implementation effort and mitigating convergence issues. The proposed framework thus offers a simple and effective strategy for scalable TSC-OPF implementations.

Evaluating the Impact of a Load Admittance Approximation in Transient Stability-Constrained Optimal Power Flow

TL;DR

This work tackles the computational burden of transient stability-constrained OPF by introducing a voltage-based load admittance approximation that allows precomputing Kron-reduced networks using a nominal bus voltage of p.u. for the pre-fault network. The TSC-OPF is formulated with an AC-based generator model and discretized swing dynamics, leveraging reduced admittance matrices for during- and post-fault periods. Case studies on the WECC 9-bus system show that rotor-angle trajectories during the first swings are well captured and that applying a voltage-corrected load admittance reduces mean absolute errors relative to time-domain benchmarks, even at coarser time steps. Overall, the proposed approach offers a scalable and robust path to implement TSC-OPF in larger networks with improved convergence properties.

Abstract

The Transient Stability-Constrained Optimal Power Flow (TSC-OPF) incorporates dynamic stability constraints into the OPF formulation to ensure secure and economical operation under disturbances. While discretizing system dynamics enables the use of nonlinear programming techniques, it significantly increases computational burden. To enhance scalability, many studies simplify the network by representing loads as constant admittances, allowing the use of Kron reduction. However, computing the Kron reduction outside the optimization requires a voltage-based assumption to convert loads from constant power to constant admittance. This paper proposes a practical voltage-based load admittance approximation and evaluates the errors it may introduce in rotor angle and speed deviation trajectories. Case studies on the WECC 9-bus system show that the proposed approach reproduces rotor dynamics consistent with time-domain simulations during the first few seconds while considerably reducing implementation effort and mitigating convergence issues. The proposed framework thus offers a simple and effective strategy for scalable TSC-OPF implementations.

Paper Structure

This paper contains 10 sections, 5 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. TSC-OPF Formulation
  4. Load Modelling and Kron Reduction
  5. Comparison Procedure
  6. Case Studies
  7. Contingency 1
  8. Contingency 2
  9. Conclusion
  10. Acknowledgements

Figures (3)

  • Figure 1: Single-line diagram of the WECC 9-bus, 3-machine system.
  • Figure 2: Contingency 1 - Comparison of rotor angle and speed deviation trajectories for G3 obtained with TSC-OPF w/o and w correction using $\mathrm{\Delta t} = 10 \,\mathrm{ms}$ against the benchmark (i.e., ANATEM with $\mathrm{\Delta t} = 1 \,\mathrm{ms}$).
  • Figure 3: Contingency 2 - Comparison of rotor angle trajectories for G3 obtained with TSC-OPF w/o and w correction and ANATEM using $\mathrm{\Delta t} = 10 \,\mathrm{ms}$ against the benchmark (i.e., ANATEM with $\mathrm{\Delta t} = 1 \,\mathrm{ms}$).