Online estimation of electric bus energy consumption based on vehicle dynamics and improved FFRLS algorithm

doi:10.3969/j.issn.1674-8484.2025.05.009

Abstract

Abstract:

To improve the performance of electric bus energy consumption prediction models in terms of real-time capability, accuracy, and interpretability, this paper proposed a hybrid energy consumption prediction model that combined vehicle dynamics modeling and data-driven parameter identification for different operating conditions. The model established instantaneous power equations for acceleration, constant speed, and deceleration conditions, and calculated cumulative energy consumption through driving segment partitioning. The forgetting factors recursive least squares (FFRLS) method was introduced for online parameter identification, and the particle swarm optimization algorithm (PSO) was used to optimize the initial parameters and forgetting factors, resulting in the development of the real-time online predictive energy consumption model IFFRLS. The results show that the proposed FFRLS model performs excellently, achieving a maximum R-squared (R²) of 0.977 and a mean absolute percentage error (MAPE) of 11.16%, significantly outperforming the unmodified model.

Key words: electric bus, energy consumption, parameter identification, vehicle dynamics, forgetting factor recursive least squares (FFRLS)

CLC Number:

U121

ZHANG Xinfang, YAN Yiping, ZHANG Zhe, XU Zhigang, ZHANG Licheng. Online estimation of electric bus energy consumption based on vehicle dynamics and improved FFRLS algorithm[J]. Journal of Automotive Safety and Energy, 2025, 16(5): 747-756.

Figures/Tables 12

References 20

[1]	Flaris K, Gkritza K, Singleton P A, et al. Riders’perceptions towards transit bus electrification: Evidence from Salt Lake City, Utah[J]. Transport Res D, 2023, 117: No 103642.
[2]	Maybury L, Corcoran P, Cipcigan L. Mathematical modelling of electric vehicle adoption: A systematic literature review[J]. Transport Res D, 2022, 107: No 103278.
[3]	Weiss M, Dekker P, Moro A, et al. On the electrification of road transportation-a review of the environmental, economic, and social performance of electric two-wheelers[J]. Transport Res D, 2015, 41: 348-366. doi: 10.1016/j.trd.2015.09.007 URL
[4]	Torkey A, Zaki M H, El Damatty A A. Transportation electrification: A critical review of evs mobility during disruptive events[J]. Transport Res D, 2024, 128: No 104103.
[5]	Boeing G, Pilgram C, LU Yougeng. Urban street network design and transport-related greenhouse gas emissions around the world[J]. Transport Res D, 2024, 127: No 103961.
[6]	WANG Shiqi, LI Yuze, Chen A, ZHUGE Chengxiang. Electrification of a citywide bus network: A data-driven micro-simulation approach[J]. Transport Res D, 2023, 116: No 103644.
[7]	LI Wei, DING Haitao, XU Nan, et al. Toward carbon-neutral transportation electrification: a comprehensive and systematic review of eco-driving for electric vehicles[J]. IEEE Trans Transport Electrificat, 2023, 10(3): 6340-6360. doi: 10.1109/TTE.2023.3331727 URL
[8]	ZHANG Jin, WANG Zhenpo, LIU Peng, et al. Energy consumption analysis and prediction of electric vehicles based on real-world driving data[J]. Appl Energ, 2020, 275: No 115408.
[9]	YAO Enjian, YANG Zhiqiang, SONG Yuanyuan, et al. Comparison of electric vehicle’s energy consumption factors for different road types[J]. Discret Dyni Nat Soc, 2013, 2013(1): 328757.
[10]	ZHANG Rui, YAO Enjian. Electric vehicles’energy consumption estimation with real driving condition data[J]. Transport Res D, 2015, 41: 177-187. doi: 10.1016/j.trd.2015.10.010 URL
[11]	Fiori C, Ahn K, Rakha H A. Power-based electric vehicle energy consumption model: Model development and validation[J]. Appl Energ, 2016, 168: 257-268. doi: 10.1016/j.apenergy.2016.01.097 URL
[12]	Gallet M, Massier T, Hamacher T. Estimation of the energy demand of electric buses based on real-world data for large-scale public transport networks[J]. Appl Enery, 2018, 230: 344-56.
[13]	赵敬玉, 徐铖, 李晓宇. 基于实际行驶数据的电动汽车能耗分析与预测[J]. 机械工程学报, 2023(59): 263-274.
	ZHAO Jingyu, XU Cheng, LI Xiaoyu. Electric vehicle energy consumption analysis and prediction based on real-world driving data[J]. J Mech Engi, 2023(59): 263-274. (in Chinese)
[14]	CHUNG Yuwei, Khaki B, LI Tianyi, et al. Ensemble machine learning-based algorithm for electric vehicle user behavior prediction[J]. Appl Energ, 2019, 254: No 113732.
[15]	YAO Junlin, Moawad A. Vehicle energy consumption estimation using large scale simulations and machine learning methods[J]. Transport Res Part C Emerg Tech, 2019, 101: 276-296. doi: 10.1016/j.trc.2019.02.012 URL
[16]	JIANG Junyu, YU Yuanbin, MIN Haitao, et al. Trip-level energy consumption prediction model for electric bus combining Markov-based speed profile generation and Gaussian processing regression[J]. Energy, 2023, 263: No 125866.
[17]	Achariyaviriya W, Wongsapai W, Janpoom K, et al. Estimating energy consumption of battery electric vehicles using vehicle sensor data and machine learning approaches[J]. Energ, 2023, 16: No 6351.
[18]	HAO Xueyi, WANG Shunli, FAN Yongcun, et al. An improved forgetting factor recursive least square and unscented particle filtering algorithm for accurate lithium-ion battery state of charge estimation[J]. J Energ Stor, 2023, 59: No 106478.
[19]	ZHAO Bi, LIU Ruijun, SHI Dapai. Optimal control strategy of path tracking and braking energy recovery for new energy vehicles[J]. Processes, 2022, 10: No 1292.
[20]	LI Li, SONG Dongran, YANG Jian et al. Distributed and real-time economic dispatch strategy for an islanded microgrid with fair participation of thermostatically controlled loads[J]. Energy, 2022, 261: No 125294.

输入：来自第i个驾驶段的数据D(t)，其中v(t)和a(t)分别表示在时间步t时刻的速度和加速度；通过粒子群优化的初始化的参数： ω(0)、λ(0)、C(0) 输出：待估计的模型参数ω(t) 初始化： ω(0)：通过PSO初始化模型参数 λ(0)：通过PSO初始化遗忘因子 C(0)：通过PSO初始化协方差矩阵对于每个时间步t在D(t)中：
1.	n = len(D)
2.	X = input vector
3.	For i∈[1, n] do
4.	if a(i) > 0.005 then
5.	X = [v(i), v(i)³, v(i) · a(i), 1]
6.	Else if a(i) < -0.05 then
7.	X = [v(i), v(i)³, v(i) · a(i), a(i)², 1]
8.	Else
9.	X = [v(i), v(i)³, 1]
10.	End if
11.	$\hat{P}(i)=\boldsymbol{X}(i) \cdot \boldsymbol{\omega}^{\mathrm{T}}(i)$
12.	$e(t)=P(i)-\hat{P}(i)$
13.	K(i) = C(i - 1) · X(i) / λ(i) + X(i) · C(i - 1) · X^T(i)]
14.	ω(i) = ω(i - 1) + K(i) · e(i)
15.	C(i - 1) = [C(i - 1) - K(i) · X^T(i) · C(i - 1)] / λ(i)
16.	Objective Function: Minimize\|P(i) -$\hat{P}(i)$\|with respect to λ(i)
17.	Optimize λ(i) using PSO
18.	End for
19.	ω(t) = {ω(1), ω(2), ω(3), ……ω(n)}
20.	Return ω(t)

输入：来自第i个驾驶段的数据D(t)，其中v(t)和a(t)分别表示在时间步t时刻的速度和加速度；通过粒子群优化的初始化的参数： ω(0)、λ(0)、C(0) 输出：待估计的模型参数ω(t) 初始化： ω(0)：通过PSO初始化模型参数 λ(0)：通过PSO初始化遗忘因子 C(0)：通过PSO初始化协方差矩阵对于每个时间步t在D(t)中：
1.	n = len(D)
2.	X = input vector
3.	For i∈[1, n] do
4.	if a(i) > 0.005 then
5.	X = [v(i), v(i)³, v(i) · a(i), 1]
6.	Else if a(i) < -0.05 then
7.	X = [v(i), v(i)³, v(i) · a(i), a(i)², 1]
8.	Else
9.	X = [v(i), v(i)³, 1]
10.	End if
11.	$\hat{P}(i)=\boldsymbol{X}(i) \cdot \boldsymbol{\omega}^{\mathrm{T}}(i)$
12.	$e(t)=P(i)-\hat{P}(i)$
13.	K(i) = C(i - 1) · X(i) / λ(i) + X(i) · C(i - 1) · X^T(i)]
14.	ω(i) = ω(i - 1) + K(i) · e(i)
15.	C(i - 1) = [C(i - 1) - K(i) · X^T(i) · C(i - 1)] / λ(i)
16.	Objective Function: Minimize\|P(i) -$\hat{P}(i)$\|with respect to λ(i)
17.	Optimize λ(i) using PSO
18.	End for
19.	ω(t) = {ω(1), ω(2), ω(3), ……ω(n)}
20.	Return ω(t)

车号	MSE / 10^-3	MAE	RMSE	R²	MAPE / %
Bus1	3.74	0.035 6	0.061 1	0.979 0	13.2
Bus2	1.57	0.027 0	0.039 6	0.967 0	11.5
Bus3	0.64	0.017 8	0.025 4	0.972 0	11.3
Bus4	1.85	0.031 2	0.043 0	0.986 0	10.3
Bus5	1.64	0.029 0	0.040 4	0.977 0	10.9
Bus6	0.71	0.018 1	0.026 6	0.984 0	12.5
……	……	……	……	……	……
平均值	1.95	0.057 5	0.092 1	0.977 2	11.16

车号	MSE / 10^-3	MAE	RMSE	R²	MAPE / %
Bus1	3.74	0.035 6	0.061 1	0.979 0	13.2
Bus2	1.57	0.027 0	0.039 6	0.967 0	11.5
Bus3	0.64	0.017 8	0.025 4	0.972 0	11.3
Bus4	1.85	0.031 2	0.043 0	0.986 0	10.3
Bus5	1.64	0.029 0	0.040 4	0.977 0	10.9
Bus6	0.71	0.018 1	0.026 6	0.984 0	12.5
……	……	……	……	……	……
平均值	1.95	0.057 5	0.092 1	0.977 2	11.16

模型	MSE	MAE	RMSE	R²	MAPE / %
FFRLS	0.3976	0.2505	0.3895	0.8454	33.18
IFFRLS	0.0195	0.0575	0.0921	0.9772	11.16