An Improved Time-Delay Grey Verhulst Model Optimized by Multi-Agent Reinforcement Learning for Electricity Market Forecasting

Document Type : Research Article

Authors

1 Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran

2 Alma Mater Studiorum – Università di Bologna, Bologna, Italy

Abstract

Accurate forecasting of electricity prices is still a difficult task because of the volatile and nonlinear nature of energy markets, as well as the limited availability of reliable data. Grey forecasting models are often used for this purpose, but they usually lack enough flexibility to capture delayed effects and complex interactions among several variables. To address these issues, this study aims to present an Improved Time-Delay Grey Multivariable Verhulst Model (ITGMVM), a new grey model developed for short-term electricity price prediction. The model introduces time-delay parameters that represent lagged relationships between variables, helping it respond better to dynamic market behavior. Two optimization methods are designed for parameter calibration: Partial Parameter Estimation (PPE) and Full Parameter Estimation (FPE), where the latter adjusts all parameters at the same time. These methods are supported by a new hybrid optimization framework called MARL-WOA, which combines Multi-Agent Reinforcement Learning (MARL) with Whale Optimization Algorithm (WOA). This combination improves the search process, leading to faster convergence and higher accuracy.  The model is evaluated using real-world data from Australia's National Electricity Market (NEM), specifically focusing on Sundays and Wednesdays between December 2023 and March 2024. Results show that ITGMVM, when optimized with FPE-MARL-WOA, outperforms six existing grey and hybrid models across multiple statistical metrics, achieving exceptional forecasting accuracy and robustness. The obtained results demonstrate the strength of integrating adaptive AI techniques with grey modeling to support decision-making in data-constrained energy environments.

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 10, Issue 2
December 2025
Pages 162-190

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History

  • Receive Date: 08 October 2025
  • Revise Date: 29 November 2025
  • Accept Date: 11 December 2025
  • Publish Date: 30 December 2025

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