[Paper Review] Mathematical representation of the WECC composite load model

Zixiao Ma, Zhaoyu Wang+3 more|arXiv (Cornell University)|Feb 23, 2019

Power System Optimization and Stability25 references4 citations

TL;DR

This paper presents the first complete mathematical representation of the WECC composite load model, including the novel DER_A model, derived from block diagrams using rigorous system dynamics and control theory. Verified via MATLAB and PSS/E simulations, the model achieves high accuracy with mean squared errors below 1.1×10⁻⁴ for real power and 7.3×10⁻⁵ for reactive power, enabling advanced dynamic analysis and parameter identification in power systems.

ABSTRACT

The Western Electricity Coordinating Council (WECC) composite load model is a newly developed load model that has drawn great interest from the industry. To analyze its dynamic characteristics with both mathematical and engineering rigor, a detailed mathematical model is needed. Although WECC composite load model is available in commercial software as a module and its detailed block diagrams can be found in several public reports, there is no complete mathematical representation of the full model in literature. This paper addresses a challenging problem of deriving detailed mathematical representation of WECC composite load model from its block diagrams. In particular, for the first time, we have derived the mathematical representation of the new DER_A model. The developed mathematical model is verified using both Matlab and PSS/E to show its effectiveness in representing WECC composite load model. The derived mathematical representation serves as an important foundation for parameter identification, order reduction and other dynamic analysis.

Motivation & Objective

• To address the lack of a comprehensive mathematical representation of the WECC composite load model in academic literature.
• To derive a detailed mathematical formulation for the newly introduced DER_A model, which replaces the complex PVD1 model in CMPLDWG.
• To enable accurate dynamic simulation and analysis of the WECC model using open-source and standard simulation tools.
• To support critical power system applications such as parameter identification, stability assessment, and model order reduction.
• To bridge the gap between commercial software implementations and academic research by providing a transparent, verifiable mathematical model.

Proposed method

• Reverse-engineered the WECC CMPLDWG composite load model from publicly available block diagrams and technical reports.
• Developed state-space and nonlinear differential equation representations for three-phase induction motors and the DER_A model.
• Formulated the DER_A model as a multi-stage dynamic system with voltage and frequency-dependent active/reactive power responses.
• Incorporated time constants, saturation functions, and protection logic (e.g., voltage ride-through, overcurrent blocking) into the mathematical structure.
• Validated the model using identical input signals (voltage, frequency) in both MATLAB and PSS/E, comparing output power trajectories.
• Quantified accuracy using mean squared error (MSE) between the derived model and PSS/E’s CMLDBLU2 and DERAU1 implementations.

Experimental results

Research questions

• RQ1 What is the complete mathematical formulation of the WECC composite load model, including the DER_A model, derived from its block diagram?
• RQ2 How accurately can the derived mathematical model replicate the dynamic power responses of the WECC model in standard simulation environments?
• RQ3 Can the proposed model be effectively validated using both MATLAB and PSS/E with consistent results?
• RQ4 What is the quantitative error margin between the derived model and commercial software implementations for real and reactive power outputs?
• RQ5 How does the model support downstream applications such as parameter identification and dynamic order reduction?

Key findings

• The paper successfully derives the first complete mathematical representation of the WECC composite load model, including the complex DER_A model, which was previously undocumented in academic literature.
• The derived model of the three-phase induction motor achieves a mean squared error (MSE) of 2.12×10⁻⁷ for real power and 2.13×10⁻⁵ for reactive power when compared to PSS/E’s CMLDBLU2.
• The DER_A model shows an MSE of 1.1053×10⁻⁴ for real power and 7.3079×10⁻⁵ for reactive power, confirming high fidelity to the PSS/E implementation.
• The model is validated across both MATLAB and PSS/E platforms, demonstrating consistent dynamic behavior under identical input conditions.
• The mathematical representation enables accurate simulation of voltage-dependent and frequency-dependent load dynamics, including delayed recovery and protection functions.
• The derived model provides a solid foundation for future research in parameter identification, stability analysis, and model order reduction in power system dynamics.