Tempered Fractional Deterministic Learning for Online Battery State-of-Health Estimation: A Cycle-Level Benchmark Study

Abstract

Accurate battery state-of-health (SOH) estimation is essential for safe and reliable electric-vehicle operation. This paper presents a Tempered Fractional Deterministic Learning (TF-DL) framework that combines deterministic learning with a tempered fractional adaptation law to introduce tunable memory and graceful forgetting into SOH estimation. Two realizations are considered: an exact truncated variant (TF-DL-T) and an embedded low-memory variant (TF-DL-E). The framework is evaluated on a NASA-derived cycle-level battery aging benchmark with capacity-based SOH labels and a battery-level train/test split. After filtering, 14 batteries were retained, of which 9 were used for training and 5 unseen batteries were used for testing. Random Forest achieved the best overall performance (MAE = 0.0436, RMSE = 0.0496), while LSTM was the strongest sequence baseline (MAE = 0.0757, RMSE = 0.0920). Among the online RBF-based methods, TF-DL-E achieved the best performance (MAE = 0.0966, RMSE = 0.1077), outperforming GD-DL and TF-DL-T. Unlike offline methods, TF-DL-E operates online with constant memory, which makes it suitable for embedded battery management systems. The results indicate that TF-DL-E is the more robust and practically relevant tempered variant, whereas TF-DL-T remains more fragile and parameter-sensitive.

Keywords: battery state of health; tempered fractional calculus; deterministic learning; electric vehicles; remaining useful life; battery management system