An enhanced particle filter technology for battery system state estimation and RUL prediction

doi:10.1016/j.measurement.2022.110817

Measurement 测量

Volume 191, 15 March 2022, 110817
2022年3月15日，第191卷，110817

https://doi.org/10.1016/j.measurement.2022.110817 Get rights and content 获取权利和内容

Highlights 亮点

•
The particle filter can model the nonlinear degradation features of battery’s system.
粒子滤波器可以对电池系统的非线性退化特征进行建模。
•
The proposed enhanced particle technique can recognize and mitigate sample degeneracy.
提出的增强粒子技术可以识别和减轻样本退化。
•
The evolving fuzzy predictor can tackle the problem of lacking new battery measurements during the prognostic period.
进化模糊预测器可以解决在预测期间缺乏新电池测量的问题。
•
The developed evolving fuzzy predictor can characterize the input–output mapping for multiple-steps-ahead prediction.
发展中的演化模糊预测器可以描述多步预测的输入-输出映射。

Abstract 摘要

The particle filter (PF) technique can model nonlinear degradation features of battery’s system, and conduct battery state estimation based on noisy measurements. However, PF has some limitations in system state estimation related to sample degeneracy and impoverishment. In addition, its posterior probability density function cannot be updated during the prognostic period due to the absence of new battery measurements. In this work, an enhanced PF technology is proposed to deal with these problems so as to improve PF modeling accuracy for battery state-of-health monitoring and remaining useful life (RUL) prediction. Specifically, an enhanced particles method is proposed to reduce the impact of sample degeneracy and impoverishment in state estimation. An evolving fuzzy predictor is adopted and fused into the enhanced PF structure to deal with the lack of new battery measurements during the prognostic period. The effectiveness of the proposed enhanced PF technology is validated through simulation tests.
粒子滤波（PF）技术可以对电池系统的非线性退化特征进行建模，并基于噪声测量进行电池状态估计。然而，PF在系统状态估计方面存在一些限制，与样本退化和贫化有关。此外，由于缺乏新的电池测量数据，其后验概率密度函数无法在预测期间更新。在本研究中，提出了一种增强的PF技术来解决这些问题，以提高电池健康状态监测和剩余寿命（RUL）预测的PF建模精度。具体而言，提出了一种增强的粒子方法来减少状态估计中样本退化和贫化的影响。采用进化模糊预测器，并将其融入增强的PF结构中，以解决预测期间缺乏新的电池测量数据的问题。通过模拟测试验证了所提出的增强PF技术的有效性。

Keywords 关键词

Lithium-ion battery

Particle filter

Health monitoring

Remaining useful life prediction

Evolving fuzzy predictor

锂离子电池颗粒过滤器健康监测剩余寿命预测进化模糊预测器

1. Introduction 介绍

Lithium-ion (Li-ion) batteries are commonly used in various industrial and domestic applications, such as electric vehicles, industrial facilities, and portable communication devices [1], [2]. However, the Li-ion battery performance degrades over time due to problems such as capacity degradation and impedance growth over time, which will not only affect the battery’s performance but also result in possible system breakdowns or even safety issues, especially in vehicle and industrial applications [3], [4]. Therefore, reliable battery health monitoring and prognostics systems are very beneficial in industrial applications to diagnose the battery’s state-of-health (SOH) and predict the remaining-useful-life (RUL), in order to improve battery performance [5].
锂离子（Li-ion）电池广泛应用于各种工业和家用设备，如电动车辆、工业设施和便携通信设备[1]，[2]。然而，随着时间的推移，Li-ion电池的性能会下降，出现容量退化和阻抗增长等问题，这不仅会影响电池的性能，还可能导致系统故障甚至安全问题，尤其是在车辆和工业应用中[3]，[4]。因此，在工业应用中，可靠的电池健康监测和预测系统对于诊断电池的健康状态（SOH）和预测剩余可用寿命（RUL）非常有益，以提高电池性能[5]。

In general, RUL prediction can be performed by applying model-based methods and data-driven techniques [2], [3]. Data-driven techniques include neural networks (NN) [6] and fuzzy logic [7], along with their combination schemes such as neural fuzzy (NF) [8]. In those techniques, the datasets from battery testing are used for system training to identify the battery characteristics, monitor the system degradation behavior, and predict its RUL. The effectiveness of those data-driven techniques relies on the quality of training datasets. However, it is usually difficult to acquire accurate and representative battery datasets under variable battery conditions in industrial applications [1], [9]. In addition, the classical data-driven techniques also have limitations in parameters setting and adaptive capability to accommodate time-varying operating conditions such as in electric vehicles [3], [10].
通常情况下，可以通过应用基于模型的方法和数据驱动技术来进行剩余寿命（RUL）预测[2]，[3]。数据驱动技术包括神经网络（NN）[6]和模糊逻辑[7]，以及它们的组合方案，如神经模糊（NF）[8]。在这些技术中，从电池测试中获取的数据集用于系统训练，以识别电池特性，监测系统退化行为并预测其剩余寿命。这些数据驱动技术的有效性取决于训练数据集的质量。然而，在工业应用中，往往很难获得准确和代表性的电池数据集，因为电池条件会发生变化[1]，[9]。此外，经典的数据驱动技术在参数设置和适应时间变化的操作条件（如电动车）方面也存在局限性[3]，[10]。

In contrast, the model-based filtering methods could provide a unique insight in system characteristics; they can: (1) model the underlying physics of battery SOH degradation processes; (2) perform inferences for the hidden states in a dynamic system; and (3) represent the uncertainty in the estimated results [1], [3], [5], [11]. These properties make the model-based filtering methods more attractive in modeling Li-ion batteries that have electro-chemical properties that vary according to changes in operational conditions (e.g., humidity, temperature, speed and load).
相比之下，基于模型的滤波方法可以提供对系统特性的独特洞察力；它们可以：（1）对电池SOH退化过程的潜在物理进行建模；（2）对动态系统中的隐藏状态进行推断；以及（3）表示估计结果的不确定性[1]，[3]，[5]，[11]。这些特性使得基于模型的滤波方法在建模锂离子电池方面更具吸引力，因为锂离子电池的电化学特性会随着操作条件的变化而变化（例如湿度、温度、速度和负载）。

In general, a model-based filtering method uses some mathematical models to characterize the battery’s degradation and performance evolution during its lifetime [2], [12]. Model parameters can be determined by utilizing proper estimation/filtering methods [5], [13]. The Kalman filter (KF) and PF are the most common estimation techniques in this field, as they can recognize the model parameters in the monitoring process and perform inferences for the hidden states of the dynamic system [14], [15]. On the other hand, since the KF and its related variations (e.g., extended KF and unscented KF) cannot properly model systems with nonlinear/non-Gaussian features [3], [4], PF is more commonly used for battery health monitoring and RUL prediction [3], [12], [13]. For instance, PF is utilized in [16] to model the battery electrolyte and charge transfer resistances for state estimation and RUL prediction. An empirical capacity model is proposed in [17] using the coulombic efficiency factor and relaxation effect whereby PF is applied to identify the model parameters and the predicted future capacity values are extrapolated to estimate the RUL. The performance of PF for Li-ion battery prognosis is investigated in [18], and the research results show that PF outperforms the non-linear least squares estimator and unscented KF methods in battery RUL prediction. However, PF has some limitations in modelling application such as sample degeneracy and impoverishment, which will degrade its modeling and estimation accuracy [3], [13], [19]. In addition, PF posterior probability density function (PDF) cannot be updated during the prognostic process in which there are no new measurements. As a result, the PF prognostic performance could degrade significantly, especially for longer prediction horizons [13], [20].
一般情况下，基于模型的滤波方法使用一些数学模型来描述电池在寿命期间的退化和性能演变[2]，[12]。模型参数可以通过利用适当的估计/滤波方法来确定[5]，[13]。卡尔曼滤波器（KF）和粒子滤波器（PF）是该领域最常见的估计技术，因为它们可以在监测过程中识别模型参数并对动态系统的隐藏状态进行推理[14]，[15]。另一方面，由于KF及其相关的变种（如扩展KF和无损伤KF）不能正确地对具有非线性/非高斯特征的系统进行建模[3]，[4]，PF更常用于电池健康监测和剩余寿命预测[3]，[12]，[13]。例如，[16]中利用PF来模拟电池电解液和电荷传输电阻以进行状态估计和剩余寿命预测。[17]提出了一个基于经验的容量模型，利用库仑效率因子和松弛效应来估计模型参数，并推断出预测的未来容量值用于估计剩余寿命。在[18]中研究了PF在锂离子电池预测中的性能，并研究结果表明，PF在电池剩余寿命预测中优于非线性最小二乘估计器和无香KF方法。然而，PF在建模应用中存在一些限制，如样本退化和贫化，这将降低其建模和估计精度[3]，[13]，[19]。此外，在预测过程中，如果没有新的测量数据，PF后验概率密度函数（PDF）无法更新。因此，PF的预测性能可能会显著降低，特别是对于较长的预测时间[13]，[20]。

Several approaches have been suggested in the literature to reduce the impact of sample degeneracy and impoverishment in PF-based system state estimation. For example, the regularized PF [21], regularized auxiliary PF [22], and adaptive regularized PF [23] apply continuous distribution-based resampling to improve particle diversity and reduce sample degeneracy. However, if the high-likelihood region of the posterior PDF is not realized accurately, the resulting continuous distribution may not be accurate [3], [24]. In addition, some PFs apply artificial intelligence tools such as artificial fish swarm algorithm [25], mutation operation [3], and quantum particle swarm optimization [26] to locate the high-likelihood region in posterior space and optimize the distribution of the particles. However, these techniques usually have limitations related to over-fitting in parameter optimization; in addition, these methods are complex in implementation and computation, which make them difficult to use in real-time monitoring applications [19], [24].
文献中提出了几种方法来减少基于粒子滤波的系统状态估计中样本退化和贫化的影响。例如，正则化粒子滤波[21]、正则化辅助粒子滤波[22]和自适应正则化粒子滤波[23]应用连续分布的重采样来改善粒子多样性和减少样本退化。然而，如果后验概率密度函数的高似然区域没有准确地实现，得到的连续分布可能不准确[3]，[24]。此外，一些粒子滤波方法应用人工智能工具，如人工鱼群算法[25]、变异操作[3]和量子粒子群优化[26]来定位后验空间中的高似然区域并优化粒子的分布。然而，这些技术通常存在参数优化过拟合的限制；此外，这些方法在实施和计算上较为复杂，使其难以在实时监测应用中使用[19]，[24]。

On the other hand, another challenge facing battery RUL prediction is the lack of new measurements during the prediction period. To solve this problem, some researchers have combined the PF with some data-driven techniques in order to update the posterior PDF. For example, the regularized PF is combined with the nonlinear degradation autoregressive model in [13] to carry out battery RUL forecasting. A hybrid prognostic method using the unscented PF and an optimized multiple kernel relevance vector machine is proposed in [27] for battery RUL prediction. A hybrid method is suggested in [28] to investigate the combination of PF with different types of data-driven tools for system state forecasting. However, the reliability of these hybrid methods depends on the processing accuracy of each unit. For example, (1) PF may not constantly generate accurate state estimation when the posterior PDF is not properly characterized due to sample degeneracy. (2) The data-driven predictor could have limited capability in accommodating battery's electro-chemical characteristics, which vary with the system’s operating conditions.
另一方面，电池RUL预测面临的另一个挑战是在预测期间缺乏新的测量数据。为了解决这个问题，一些研究人员将PF与一些数据驱动技术结合起来，以更新后验概率密度函数。例如，在[13]中将正则化PF与非线性退化自回归模型相结合，进行电池RUL预测。在[27]中提出了一种混合预测方法，使用无损PF和优化的多核相关矢量机进行电池RUL预测。在[28]中建议使用混合方法，研究将PF与不同类型的数据驱动工具结合起来进行系统状态预测。然而，这些混合方法的可靠性取决于每个单元的处理精度。例如，(1) 当后验概率密度函数由于样本退化而得不到恰当的表征时，PF可能无法持续生成准确的状态估计。 (2) 数据驱动预测器可能在适应电池的电化学特性方面能力有限，这些特性随系统运行条件的变化而变化。

To tackle these aforementioned limitations, an enhanced PF (ePF) technology will be proposed in this paper for Li-ion battery health monitoring and RUL prediction. It aims to alleviate the PF limitations and improve the modeling and RUL prediction of battery’s degradation process. It has the following novel aspects: (1) the ePF will be proposed to reduce the impact of sample degeneracy and impoverishment in system state estimation. Different from the PF techniques proposed in [3], [22], [24], [29], the ePF technique can detect sample degeneracy in the posterior PDF, and process low-weight particles to better characterize the posterior PDF. (2) The ePF technique uses a novel implementation strategy to improve computational efficiency. (3) An evolving fuzzy predictor will be adopted to form a prognostic framework to tackle the lack of new battery measurements during the prognostic period. The evolving fuzzy predictor can improve adaptive capability of the ePF to deal with time-varying characteristics in battery systems.
为了解决上述限制，本文提出了一种增强型PF（ePF）技术，用于锂离子电池的健康监测和剩余寿命预测。它旨在减轻PF的限制，并改进电池退化过程的建模和剩余寿命预测。它具有以下新颖之处：（1）提出了ePF来减少系统状态估计中样本退化和贫化的影响。与[3]、[22]、[24]、[29]中提出的PF技术不同，ePF技术可以检测后验概率密度函数中的样本退化，并处理低权重粒子以更好地描述后验概率密度函数。（2）ePF技术使用一种新颖的实现策略来提高计算效率。（3）采用演化模糊预测器构建预测框架，以解决预测期间缺乏新电池测量数据的问题。演化模糊预测器可以提高ePF对电池系统中时变特性的自适应能力。

The remaining sections of this paper are structured as follows. The proposed ePF is introduced in Section 2. The developed prognostic framework is described in Section 3. The efficiency of the proposed ePF and prognostic framework is examined in Section 4 by simulation tests and battery RUL forecasting.
本文的其余部分结构如下。第二部分介绍了拟议的ePF。第三部分描述了开发的预测框架。通过模拟测试和电池寿命余量预测，第四部分检验了所提出的ePF和预测框架的效率。

2. Proposed ePF technique
2. 提议的ePF技术

The proposed ePF technique aims to reduce the effect of sample degeneracy and sample degeneracy so as to improve modeling accuracy. To facilitate discussion, a brief review of the classical PF will be given first.
拟议的ePF技术旨在减少样本退化和样本退化的影响，以提高建模准确性。为了便于讨论，首先将简要回顾经典PF技术。

2.1. PF overview 2.1. PF概述

PF employs the recursive Bayesian method by Monte Carlo simulation to perform inference in the state space [24], [29]. The state model in Eq. (1) describes the progress of the system state over time. The measurement or observation model in Eq. (2) correlates the noisy observations to the hidden state [30],(1) $x_{k} = f (x_{k - 1}, u_{k})$ (2) $y_{k} = h_{k} (x_{k}, v_{k})$ where $x_{k}$ denotes the hidden state to be estimated; $y_{k}$ denotes the observation (measurement) at the kth time instant; $u_{k}$ and $v_{k}$ are the respective process and measurement noise.
PF使用蒙特卡洛模拟的递归贝叶斯方法在状态空间中进行推断[24]，[29]。方程（1）中的状态模型描述了系统状态随时间的变化。方程（2）中的测量或观测模型将噪声观测与隐藏状态相关联[30]， (1) (2) 其中表示待估计的隐藏状态；表示第k个时间点的观测（测量）；和分别是过程和测量噪声。

The posterior PDF of the hidden state $x_{k}$ is characterized by N random samples (i.e., particles) ${x_{k}^{1}, x_{k}^{2}, . . ., x_{k}^{N}}$ and their related weights ${w_{k}^{1}, w_{k}^{2}, . . ., w_{k}^{N}}$ determined by the conditional likelihood of each particle [30], [31]. In general, PF algorithm undertakes state estimation in two steps:
隐藏状态的后验概率密度由N个随机样本（即粒子）及其相关权重所决定，这些权重是由每个粒子的条件似然确定的[30]，[31]。一般而言，粒子滤波算法包括两个步骤进行状态估计：

(1)
Prediction: the set of particles are randomly generated and propagated based on the information acquired from the previous iteration (i.e., prior PDF).
预测：粒子集合根据从上一次迭代（即先验概率密度函数）获取的信息进行随机生成和传播。
(2)
Correction: when a new observation is available, each estimated particle from the prediction phase is matched with the observation (measurement) utilizing the system state and measurement models [30], [31]. The weight is updated in accordance with the importance of the associated measurement. Specifically, the weight becomes larger if the mismatch between the prediction value and the measured value is reduced. However, after some iterations in particle propagation, the weight could concentrate only on a few particles (i.e., sample degeneracy) [19], [24], whereas the rest will have very small weights as illustrated in Fig. 1.
更正：当有新的观测可用时，预测阶段的每个估计粒子都会利用系统状态和测量模型[30]，[31]与观测（测量）进行匹配。权重将根据相关观测的重要性进行更新。具体来说，如果预测值和测量值之间的不匹配减少，权重将变大。然而，在粒子传播的一些迭代之后，权重可能只集中在少数几个粒子上（即样本退化）[19]，[24]，而其余粒子的权重将非常小，如图1所示。
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Fig. 1. Sample degeneracy and impoverishment problem representation [19].
图1. 样本退化和贫困问题表征[19]。

To reduce sample degeneracy, the general approach is to replicate these heavy-weight particles and replace them with light-weight particles at each iteration (i.e., resampling) [31]. However, it could generate another problem in resampling, or particles with significant weights would be selected repeatedly (i.e., sample impoverishment) [19], [31], as illustrated in Fig. 1.
为了减少样本退化，一般的方法是在每次迭代中复制这些重量级粒子，并用轻量级粒子替换它们（即重采样）[31]。但是，这可能会在重采样中产生另一个问题，即重量显著的粒子会被重复选中（即样本贫乏） [19]，[31]，如图1所示。

In this work, the ePF technique is proposed to deal with sample degeneracy and impoverishment, which will be discussed in the next section.
在这项工作中，将提出ePF技术来处理样本退化和贫乏问题，这将在下一节中讨论。

2.2. Proposed ePF technique
2.2. 提议的ePF技术

The proposed ePF has a new enhanced particles mechanism to reduce sample degeneracy and accelerate computation. Firstly, it will detect sample degeneracy in the posterior PDF space. When sample degeneracy is recognized, the ePF algorithm will generate new high-weigh particles to replace those existing light-weight particles, so as to characterize the high-likelihood region and improve particle diversity in the posterior PDF.
提议的ePF具有新的增强粒子机制，以减少样本退化并加速计算。首先，它将在后验概率密度函数空间中检测样本退化。当样本退化被识别出来时，ePF算法将生成新的高权重粒子来替换那些现有的轻量级粒子，以描述高可能性区域并改善后验概率密度函数中的粒子多样性。

In general, the number of efficiency particles (i.e., those with heavy weights) on the posterior PDF can be estimated using some measure [24], [32], such as the effective sample size $E_{S}$ , which is calculated by(3) $E_{S} (k) = \frac{{(\sum_{i = 1}^{N} w_{k}^{i})}^{2}}{\sum_{i = 1}^{N} {(w_{k}^{i})}^{2}} \times 100 %$ where $w_{k}^{i}$ are the importance particle weights at kth time instant. $E_{S}$ reaches its highest rate (i.e., 100%) when most particles located in the high-probability area given the observation (i.e., when $w_{k}^{i} ⩾ 1 / N$ ). If $E_{S}$ becomes smaller, it is more prone to sample degeneracy.
通常情况下，可以使用某些度量方法[24]，[32]来估计后验概率密度函数上的高效粒子（即具有较大权重的粒子）的数量，例如有效样本大小，其计算公式为 (3) 其中是第k个时间点的重要性粒子权重。当大多数粒子位于给定观测的高概率区域时（即），达到最高比例（即100%）。如果变小，样本退化的可能性更大。

The proposed enhanced particles mechanism will monitor the particle weights so as to detect sample degeneracy on the posterior PDF using $E_{S}$ from Eq. (3). The percentage threshold depends on specific applications; in this work, it is selected to be 60% to ensure most particles have significant weights. If $E_{S}$ < 60%, the enhanced particles method will be executed to improve the posterior distribution and reduce sample degeneracy. In processing, these recognized low-weight particles on the posterior PDF will be relocated to areas with high probability values so as to characterize these high probability areas more sufficiently. The following summarizes the related processing steps of the enhanced particles mechanism:
提议的增强粒子机制将监测粒子权重，以便使用方程（3）中的检测后验概率密度函数上的样本退化。百分比阈值取决于具体应用；在本研究中，选择为60%，以确保大多数粒子具有显著的权重。如果 < 60%，则执行增强粒子方法以改善后验分布并减少样本退化。在处理过程中，将这些在后验概率密度函数上被识别为低权重的粒子重新定位到具有高概率值的区域，以更充分地描述这些高概率区域。以下总结了增强粒子机制的相关处理步骤：

(1)
Specify the respective upper boundary ( $U_{b}$ ) and lower boundary ( $L_{b}$ ) of the posterior PDF for searching:
指定搜索的后验概率密度函数的相应上界（）和下界（）

(4)

U_{b} = \{\begin{matrix} x_{k}^{i} + σ_{k}, & if x_{k}^{i} ⩾ x_{B} \\ x_{B} + σ_{k}, & otherwise \end{matrix}

(5)

L_{b} = \{\begin{matrix} x_{k}^{i} - σ_{k}, & if x_{k}^{i} < x_{B} \\ x_{B} - σ_{k}, & otherwise \end{matrix}

where

σ_{k}

denotes the particles standard deviation; the particle weight

w_{k}^{i}

< 1/N;

x_{B}

is the particle value that has the greatest weight up to that point, and its weight is

w_{B} = max (w_{k}^{N})

.
(4)

(5)

其中

表示粒子的标准差；粒子权重

< 1/N；

是到目前为止具有最大权重的粒子值，其权重为

。

(2)
Generate a new particle within the high-probability area on the posterior PDF space:
在后验概率密度函数空间的高概率区域内生成一个新的粒子

(6)

x_{k}^{i} = L_{b} + (U_{b} - L_{b}) β

where

β \in [0, 1]

is a random number that will be used to facilitate exploring these boundaries with an almost uniform distribution, so as to recognize the high-likelihood region to generate a new particle with higher weight.
(6)

其中

是一个随机数，将用于在接近均匀分布的情况下探索这些边界，以识别生成具有更高权重的新粒子的高可能性区域。

(3)
Calculate the weight $w_{k}^{i}$ of the generated particle $x_{k}^{i}$ . If $w_{k}^{i} ⩽$ 1/N, a new particle $x_{k}^{i}$ will be generated utilizing Steps (2) and (3) until $w_{k}^{i}$ ≥ 1/N. If $w_{k}^{i} > w_{B}$ , update $x_{B}$ and $w_{B}$ by setting $x_{B} : = x_{k}^{i}$ , $w_{B} : = w_{k}^{i}$ .
计算生成的粒子的权重。如果 1/N，则使用步骤（2）和（3）生成新的粒子，直到 ≥ 1/N。如果，通过设置，来更新和。

This enhanced mechanism will ensure that the set of N random samples (i.e., particles) ${x_{k}^{i}}_{i = 1}^{N}$ will be located in the high-probability region for better state estimation. Fig. 2(a) illustrates an example of the existence of sample degeneracy on the posterior distribution, whereby nearly all particles have almost zero weights. Fig. 2(b) shows how the proposed ePF technique can process those low-weight particles and reposition them into the high-probability region; it can generate better posterior PDF representation so as to improve particles diversity and reduce sample degeneracy.
这种增强机制将确保N个随机样本（即粒子）位于高概率区域，以实现更好的状态估计。图2(a)展示了后验分布中样本退化的一个例子，几乎所有粒子的权重几乎为零。图2(b)展示了所提出的ePF技术如何处理这些低权重粒子并将它们重新定位到高概率区域；它可以生成更好的后验概率密度函数表示，以改善粒子的多样性并减少样本退化。

Fig. 3 outlines the $E_{S}$ comparison of the posterior PDF over 50-time steps both with and without the use of the proposed ePF. It can be seen that the enhanced particles mechanism can detect the occurrence of the sample degeneracy on the posterior PDF. It can also process those low-weight particles and improve system state estimation.
图3概述了在使用和不使用提出的ePF的情况下，50个时间步骤上后验概率密度函数的比较。可以看到，增强粒子机制可以检测到后验概率密度函数中样本退化的发生。它还可以处理那些低权重的粒子并改善系统状态估计。

3. The proposed prognostic framework
提出的预测框架

In this section, a prognostic framework will be developed to integrate a data-driven evolving fuzzy (EF) predictor into the ePF structure. Its purpose is to tackle the problem of no new battery measurements during the RUL prediction period. As discussed in the Introduction, most data-driven techniques use a fixed reasoning architecture, which may not have sufficient adaptive capability in modeling significant variations in battery electro-chemical properties especially in the operation of electric vehicles. In this work, the EF predictor proposed in [33] will be adopted to predict the battery degradation trend during the prognostic process.
在本节中，将开发一个预测框架，将数据驱动的演化模糊（EF）预测器集成到ePF结构中。其目的是解决在剩余寿命（RUL）预测期间没有新的电池测量数据的问题。如介绍中所讨论的，大多数数据驱动技术使用固定的推理架构，可能在建模电池电化学特性的显著变化，特别是在电动车的运行中，缺乏足够的自适应能力。在本研究中，将采用[33]中提出的EF预测器来预测预测过程中的电池退化趋势。

The developed prognostics framework undergoes two phases for battery state estimation and RUL prediction, as discussed in the following subsections.
发展的预测框架经历了两个阶段，用于电池状态估计和剩余寿命预测，如下小节所讨论。

3.1. Phase 1: Degradation modeling
3.1. 阶段1：退化建模

The aim of Phase 1 is to capture and track the battery degradation characteristics based on the available battery datasets (e.g., capacitance and impedance) so far. The block diagram of degradation modeling processes is illustrated in Fig. 4, the ePF will use the degradation-prediction model (i.e., diagnosis model) to characterize the progress of the system state and battery degradation trend. It represents the battery’s health (i.e., SOH) as a function of battery conditions, time duration (i.e., elapsed cycles), and model parameters related to damage/aging behavior. The ePF conducts state estimation using the state transition model in Eq. (1) and observation (measurement) model in Eq. (2). The purpose is to model the relationship between the new measured degradation indicator values and the degradation model parameters. The ePF can estimate the posterior PDF of the hidden state (i.e., model parameters) through some random particles, where particle weights are adjusted according to the likelihood of each particle from the new observation. The updated/estimated posterior PDF can characterize the high probability area in the system space, which can be used to track the battery degradation and to model the fault propagation trend.
第一阶段的目标是基于目前可用的电池数据集（例如电容和阻抗）捕捉和跟踪电池的衰减特性。衰减建模过程的框图如图4所示，ePF将使用衰减预测模型（即诊断模型）来描述系统状态和电池衰减趋势的进展。它将电池的健康状况（即SOH）表示为与电池条件、时间持续（即经过的循环次数）和与损伤/老化行为相关的模型参数的函数。ePF使用方程（1）中的状态转移模型和方程（2）中的观测（测量）模型进行状态估计。目的是建立新测量的衰减指标值与衰减模型参数之间的关系。ePF可以通过一些随机粒子估计隐藏状态（即模型参数）的后验概率密度，其中粒子权重根据每个粒子从新观测中的似然度进行调整。更新/估计的后验概率密度函数可以描述系统空间中的高概率区域，可用于跟踪电池退化并建模故障传播趋势。

On the other hand, the EF predictor will be formulated in Phase 1 using the available battery data sets (e.g., capacity and impedance). In the adopted EF predictor, the fuzzy clusters/rules are in the form of Takagi-Sugeno-Kang (TSK) type-1 to characterize the input–output mapping for r-steps-ahead prediction, which can provide high dimensional modeling [34]. The fuzzy clusters will be formulated using the potential measure [33], and a new fuzzy cluster is generated if the new data point has a greater potential than the potentials of all current clusters. Based on the available battery degradation indicator values in Phase 1 ${y_{k}, y_{k - r}, y_{k - 2 r}, . . . . .}$ , the r-steps-ahead prediction can be described as:(7) ${\hat{y}}_{k + r} = g (y_{k}, y_{k - r}, y_{k - 2 r}, . . . .)$ where ${\hat{y}}_{k + r}$ is the forecasted indicator value at the (k + r)th time instant; and $g (∙)$ denotes the EF predictor.
另一方面，EF预测器将在第一阶段使用可用的电池数据集（例如容量和阻抗）进行制定。在采用的EF预测器中，模糊聚类/规则采用Takagi-Sugeno-Kang（TSK）类型-1的形式，用于描述输入-输出映射以进行r步预测，可以提供高维建模[34]。模糊聚类将使用潜力度量[33]进行制定，如果新数据点的潜力大于所有当前聚类的潜力，则生成新的模糊聚类。基于第一阶段可用的电池退化指标值，r步预测可以描述为：，其中 (7) 是（k + r）时刻的预测指标值；表示EF预测器。

3.2. Phase 2: RUL prediction
3.2. 第二阶段：寿命剩余预测

Fig. 5 shows the processing procedures in Phase 2. The moment to begin the battery RUL prediction is referred to as the starting point. The evolved EF predictor in Phase 1 will be applied to forecast the future measurement indicator values $({\hat{y}}_{k + 1} + {\hat{y}}_{k + 2}, {\hat{y}}_{k + 3}, . . . . . . {\hat{y}}_{k + r})$ , which will be used as predicted battery measurements during the prognostic process. The recognized ePF prediction model from Phase 1will then use these predicted values to update its posterior PDF and predict the battery degradation for SOH estimation. This is to forecast the battery RUL, or the time duration for the battery state to reach its threshold (i.e., 70% of the original battery SOH state).
图5显示了第2阶段的处理过程。开始进行电池剩余寿命（RUL）预测的时刻被称为起始点。第1阶段中演化的EF预测器将被应用于预测未来的测量指标值，这些值将在预测过程中用作预测的电池测量值。第1阶段中识别的ePF预测模型将使用这些预测值来更新其后验概率密度函数（PDF），并预测电池的退化情况以进行SOH估计。这是为了预测电池的剩余寿命，或者说电池状态达到其阈值（即原始电池SOH状态的70%）所需的时间。

The RUL prediction is usually conducted by modeling the fault propagation trend to estimate the time before failure. In Phase 2, the ePF model will firstly perform a one-step-ahead prediction to forecast the degradation state distribution (i.e., the predicted PDF) of the next time step (i.e., the ( $k + 1$ )th step) using the posterior PDF of the current state. The particles that shape the current posterior PDF will be propagated using the state transition model in Eq. (1) to form the predicted PDF. Then, the ePF will recursively update the predicted posterior PDF based on the likelihood of each particle, as illustrated in Fig. 6.
RUL预测通常通过建模故障传播趋势来估计故障发生前的时间。在第二阶段，ePF模型将首先进行一步预测，以预测下一个时间步骤（即第步）的退化状态分布（即预测的概率密度函数）。使用当前状态的后验概率密度函数来形成预测的概率密度函数的粒子将使用等式（1）中的状态转换模型进行传播。然后，ePF将根据每个粒子的似然性递归更新预测的后验概率密度函数，如图6所示。

The propagation operation in Fig. 6 will be repeated a number of times to forecast the degradation trend and to identify the moment when the battery SOH reaches its end-of-life level for RUL prediction. In general, the posterior PDF of the present state is considered to be the core element in Phase 2. The predicted indicator values by the EF predictor are used to update the posterior PDF so as to move particles to high-likelihood regions in order to improve modeling accuracy. The updated/estimated posterior PDF can not only characterize the high probability area of the system state, but it can also estimate the uncertainty in processing. In general, a PDF with a narrower-taller distribution would have a more accurate prediction than those with a wider PDF [1], [3], [32]. Fig. 7 illustrates a comparison of the PDF properties using a single model-based PF processing and using the developed prognostics framework.
图6中的传播操作将重复多次，以预测退化趋势并确定电池SOH达到寿命终点的时刻，用于RUL预测。一般来说，当前状态的后验概率密度函数被认为是第2阶段的核心要素。EF预测器预测的指标值用于更新后验概率密度函数，以将粒子移动到高可能性区域，以提高建模精度。更新/估计的后验概率密度函数不仅可以表征系统状态的高概率区域，还可以估计处理中的不确定性。一般来说，具有较窄-较高分布的概率密度函数比具有较宽分布的概率密度函数具有更准确的预测能力[1]，[3]，[32]。图7比较了使用单一基于模型的PF处理和使用开发的预测框架的概率密度函数属性。

4. Performance evaluation
性能评估

The performance of the proposed ePF technique and the prognostic framework is examined by simulation tests.
通过模拟测试来检验所提出的ePF技术和预测框架的性能。

4.1. Performance evaluation of the proposed ePF
4.1. 提出的ePF的性能评估

This testing will investigate the effectiveness of the proposed ePF in dealing with sample degeneracy and impoverishment in state estimation. For comparison, some well-accepted PF techniques will be utilized, including sampling importance resampling PF (i.e., SIR-PF) [31], regularized PF (i.e., RPF) [21], and mutated PF (i.e., MPF) [3]. The test will be conducted by applying the common benchmark model in this field [3], [21], [22], [24], [29], [30], [31], which has the following state and measurement equations:
此测试将研究所提出的ePF在状态估计中处理样本退化和贫化的有效性。为了比较，将使用一些被广泛接受的PF技术，包括采样重要性重采样PF（即SIR-PF）[31]，正则化PF（即RPF）[21]和变异PF（即MPF）[3]。测试将通过应用该领域中的常见基准模型[3]，[21]，[22]，[24]，[29]，[30]，[31]进行。该模型具有以下状态和测量方程：

(8) $X_{k} = \frac{1}{2} X_{k - 1} + \frac{25 X_{k - 1}}{1 + X_{k - 1}^{2}} + 8 cos [1.2 (k - 1)] + u_{k}$ (9) $Y_{k} = \frac{1}{20} X_{k}^{2} + v_{k}$

This testing has used the following conditions: the particle number N = 50, initial state $X_{0}$ = 0.1, the time steps k = 100, the variance of the measurement noise $v_{k}$ = 1. Also, four variances of process noise are $u_{k}$ = 10, 20, 30 and 40, respectively, which are used in simulation tests to examine the modeling reliability under different levels of process noise.
这次测试采用了以下条件: 粒子数量 N = 50, 初始状态 = 0.1, 时间步长 k = 100, 测量噪声方差 = 1。另外，分别设置了四种过程噪声的方差值 = 10, 20, 30 和 40，在模拟测试中用于检验在不同过程噪声水平下的建模可靠性。

The average root-mean-squares error (RMSE) is computed over 20 runs, between the real states and the approximated states. Table 1, Table 2 summarize the related mean and standard deviations for each test scenario over 20 times. Fig. 8 outlines the test results over 20 random runs using the data generated from Eqs. (8), (9) with process noise $u_{k}$ = 10. Findings reveal that the average RMSE increases with the rise of the process noise in all four PFs. At the same time, the proposed ePF has the ability to adjust itself to these changes and perform better than other related PFs under all testing conditions (i.e., with the smallest RMSE in terms of average mean and standard deviation).
平均均方根误差（RMSE）是在20次运行中，真实状态和近似状态之间计算得出的。表1和表2总结了每个测试场景在20次测试中的相关均值和标准差。图8概述了使用从方程（8）、（9）生成的数据进行的20次随机运行的测试结果，其中过程噪声 = 10。研究结果表明，所有四种粒子滤波器中，随着过程噪声的增加，平均RMSE也增加。同时，所提出的ePF具有自适应能力，在所有测试条件下表现优于其他相关的粒子滤波器（即在平均均值和标准差方面具有最小的RMSE）。

Table 1. Averaged mean of rmse over 20 runs.
表1. 20次运行的均方根误差的平均值。

Noise value 噪声值	Averaged mean of RMSE RMSE的均值
Noise value 噪声值	SIR-PF	RPF	MPF	ePF
10	6.009	5.638	5.375	4.768
20	7.760	7.085	6.694	6.119
30	9.036	8.324	7.885	6.508
40	9.518	8.624	8.388	7.864

Table 2. Comparison of standard deviation of rmse over 20 runs.
表2. 在20次运行中，均方根误差的标准差比较。

Noise value 噪声值	Averaged standard deviation of RMSE 均方根误差的平均标准差
Noise value 噪声值	SIR-PF	RPF	MPF	ePF
10	1.311	0.834	0.825	0.357
20	1.106	1.123	0.911	0.523
30	1.572	1.157	1.377	0.579
40	1.844	1.342	1.478	0.762

As an example, when the process noise $u_{k}$ = 30, the proposed ePF has RMSE of approximately 30%, 22%, and 18% lower than SIR-PF, RPF, and MPF techniques, respectively, and has about 40% less standard deviation than other related techniques. This is because the proposed ePF can effectively detect sample degeneracy into the posterior PDF and process those low-weight particles to maintain a higher $E_{S}$ for better performance. Furthermore, the robustness of ePF can be demonstrated by its lower standard deviation under different simulation test conditions.
例如，当过程噪声 = 30时，所提出的ePF的RMSE约比SIR-PF、RPF和MPF技术分别低30%、22%和18%，标准差也比其他相关技术低约40%。这是因为所提出的ePF能够有效地检测样本退化到后验概率密度函数中，并处理那些低权重的粒子以保持更高的以获得更好的性能。此外，ePF的鲁棒性可以通过在不同的模拟测试条件下较低的标准差来证明。

4.2. Performance evaluation for battery RUL prediction
4.2. 电池剩余寿命预测的性能评估

This testing will be conducted applying the Li-ion battery experimental data from the National Aeronautics and Space Administration Ames Prognostic Center of Excellence [35]. Further information about the data acquisition and experimental setup can be obtained from [24], [36]. The effectiveness of the developed prognosis framework, denoted as ePF-EF, will be examined in this section for battery SOH monitoring and RUL forecasting. It will investigate the framework's capability to alleviate the impact of sample degeneracy in battery state estimation, and to deal with the problem of no measurements during the prognostic process in Phase 2. For comparison, the following related PF methods will be utilized:
此测试将应用来自美国国家航空航天局阿姆斯预测卓越中心的锂离子电池实验数据[35]。有关数据采集和实验设置的更多信息可从[24]，[36]获得。本节将检验开发的预测框架（称为ePF-EF）在电池SOH监测和RUL预测方面的有效性。它将研究该框架在电池状态估计中减轻样本退化影响的能力，并解决第二阶段预测过程中没有测量的问题。为了比较，将使用以下相关的PF方法：

(1)
the quantum particle swarm optimization-based PF, denoted as QPSO-PF, which is a population-based swarm intelligence algorithm [26];
基于量子粒子群优化的PF，简称QPSO-PF，是一种基于种群的群体智能算法[26]；
(2)
the hybrid method of QPSO is integrated with an adaptive neuro-fuzzy inference system (ANFIS) [8], denoted as QPSO-ANFIS in this test.
QPSO的混合方法与自适应神经模糊推理系统（ANFIS）[8]集成在一起，在本次测试中被称为QPSO-ANFIS。

In investigating battery SOH for RUL forecasting, the battery capacity is usually used as a degradation indicator [3], [13], [16], [17], [22], [24], [25], [26], [27], which can be estimated by integrating the battery current over time. In this work, the empirical degradation model in Eq. (10) will be utilized to model the Li-ion battery physics, which considers the reduction in battery capacity as well as the battery’s self-recharge behavior [2], [25], [26]. The battery capacity can be converted to the SOH in a unified form in Eq. (11):(10) $C_{k + 1} = η_{c} C_{k} + b_{1} exp (- \frac{b_{2}}{Δ t_{k}})$ (11) $S_{k + 1} = \frac{C_{k + 1}}{C_{0}} \times 100$ where $η_{C}$ is the Coulombic coefficient ( $η_{C}$ =0.997 in this work); $C_{k}$ is the charging capacity at the kth cycle; $C_{0}$ is the initial capacity at the time k = 1; $b_{1}$ and $b_{2}$ are the parameters to be estimated; $S_{k}$ is the battery SOH at the kth cycle; and $Δ t_{k} = t_{k + 1} - t_{k}$ is the rest time interval from the kth cycle to the ( $k + 1$ )th cycle ( $Δ t_{k}$ =1 in this case).
在研究电池剩余寿命（RUL）预测的电池状态健康（SOH）时，电池容量通常被用作衰减指标[3]，[13]，[16]，[17]，[22]，[24]，[25]，[26]，[27]，可以通过对电池电流随时间的积分来估计。在本研究中，将利用方程（10）中的经验性衰减模型来建模锂离子电池的物理特性，该模型考虑了电池容量的降低以及电池的自充电行为[2]，[25]，[26]。电池容量可以通过方程（11）以统一的形式转换为SOH： (10) (11) 其中是库仑系数（本研究中 =0.997）；是第k个循环的充电容量；是时间k = 1时的初始容量；和是待估计的参数；是第k个循环的电池SOH；是从第k个循环到第（）个循环的休息时间间隔（在本例中 =1）。

This test uses 200 particles, which is similar to that QPSO-PF used in [26] to ensure a reasonable comparison. To compare the performance of the related methods, testing is performed over 50 times, using data of battery #5, which reaches its failure threshold at cycle 162. The comparison will be in terms of the accuracy of degradation modeling and RUL forecasting. The time moments to start the forecasting are chosen at cycles 86, 106, 126, and 146, respectively, which could represent the long-term, medium-term, and short-term predictions. Table 3 summarizes the average mean and standard deviation values of RMSE over 50 test runs. Fig. 9 shows the comparison results, at different prediction starting points (i.e., 86, 106, 126, and 146) over 50 random runs for the related techniques.
该测试使用了200个粒子，与[26]中使用的QPSO-PF类似，以确保合理比较。为了比较相关方法的性能，进行了50次测试，使用电池#5的数据，该电池在第162个循环时达到故障阈值。比较将从降解建模和剩余寿命预测的准确性方面进行。选择的预测起始时间点分别为第86、106、126和146个循环，分别代表长期、中期和短期预测。表3总结了50次测试运行中RMSE的平均值和标准偏差。图9显示了相关技术在不同预测起始点（即86、106、126和146）上进行的50次随机运行的比较结果。

Table 3. Average mean and standard deviation of RMSE over 50 runs.
表3. 50次运行的均值和标准偏差的平均RMSE。

Prediction starting point 预测起点	Technique 技术	Averaged mean of RMSE RMSE的均值	Standard deviation of RMSE 均方根误差的标准差
86	QPSO-PF	0.035	0.003
	QPSO-ANFIS	0.017	4.827 × 10⁻⁵ 4.827 × 10的0次方
	ePF-EF	0.015	5.576 × 10⁻⁵ 5.576 × 10的0次方
106	QPSO-PF	0.019	0.002
	QPSO-ANFIS	0.014	5.117 × 10⁻⁵ 5.117 × 10的0次方
	ePF-EF	0.013	2.244 × 10⁻⁵ 2.244 × 10的零次方
126	QPSO-PF	0.016	0.001
	QPSO-ANFIS	0.013	3.688 × 10⁻⁵ 3.688 × 10的0次方
	ePF-EF	0.008	2.259 × 10⁻⁵ 2.259 × 10的0次方
146	QPSO-PF	0.012	4.584 × 10⁻⁴ 4.584 × 10的0次方
	QPSO-ANFIS	0.006	7.003 × 10⁻⁵ 7.003 × 10^0
	ePF-EF	0.004	2.479 × 10⁻⁵ 2.479 × 10的0次方

The RMSE can be reduced as the prediction period becomes shorter using all of the related techniques. The QPSO-ANFIS outperforms the QPSO-PF, because the ANFIS predictor keeps updating the model parameters over the prediction period. The developed ePF-EF framework performs the best under all testing conditions; for example, its RMSE is approximately 60%, 30%, 50%, and 65% lower than QPSO-PF, as well as 10%, 8%, 40%, and 30% lower than the QPSO-ANFIS, corresponding to prediction starting points at 86, 106, 126 and 146, respectively. This is because the proposed ePF-EF framework can effectively merge the strengths of both the model-based ePF and data-driven EF techniques in modeling the underlying physics of battery health degradation. In this case, the ePF can alleviate the impact of sample degeneracy and impoverishment and improve system state estimation accuracy. In addition, it can properly characterize the high-likelihood area of the posterior PDF to track battery dynamic behavior and forecast the degradation state distribution. The evolving mechanism of the EF predictor can effectively capture the battery dynamic characteristics, even when using the limited available battery data in Phase 1, due to its effective adaptive capability to adjust its reasoning structures and parameters to accommodate the impact of time-varying test conditions. The ePF in Phase 2 can properly update its posterior PDF using the indicator values predicted by the EF and improve RUL prediction performance. Furthermore, the low standard deviation of the RMSE using the proposed ePF-EF framework can demonstrate its robustness under different operating conditions.
随着预测期限缩短，使用所有相关技术可以减小RMSE。QPSO-ANFIS优于QPSO-PF，因为ANFIS预测器在预测期间不断更新模型参数。在所有测试条件下，开发的ePF-EF框架表现最佳；例如，其RMSE约比QPSO-PF低60％，30％，50％和65％，比QPSO-ANFIS低10％，8％，40％和30％，对应于预测起始点分别为86，106，126和146。这是因为所提出的ePF-EF框架能够有效地将基于模型的ePF和数据驱动的EF技术的优势合并起来，对电池健康退化的基本物理进行建模。在这种情况下，ePF可以减轻样本退化和贫化的影响，并提高系统状态估计的准确性。此外，它可以正确地描述后验概率密度函数的高可能性区域，以跟踪电池的动态行为并预测退化状态分布。 EF预测器的演化机制能够有效捕捉电池的动态特性，即使在第一阶段使用有限的可用电池数据时，也能通过其有效的自适应能力来调整其推理结构和参数，以适应时变的测试条件的影响。第二阶段的ePF可以使用EF预测的指标值适当更新其后验概率密度函数，并提高剩余寿命预测性能。此外，使用所提出的ePF-EF框架的RMSE的低标准差可以证明其在不同工作条件下的稳健性。

Processing efficiency (i.e., execution time) has a significant role in system monitoring applications, which can be an indicator of the computing complexity of the related techniques. Table 4 summarizes the average execution time using the related techniques, which are measured under the same testing conditions over 50 random runs using the same observation datasets. Fig. 10 schematically compares the average execution time using the related methods, corresponding to different prediction starting points (i.e., 86, 106,126 and 146), over 50 random runs.
处理效率（即执行时间）在系统监控应用中起着重要作用，可以作为相关技术计算复杂性的指标。表4总结了使用相关技术的平均执行时间，这些时间是在相同的测试条件下，使用相同的观测数据集进行50次随机运行测量得出的。图10以示意方式比较了使用相关方法的平均执行时间，对应于不同的预测起始点（即86、106、126和146），在50次随机运行中。

Table 4. The average execution time of the related methods over 50 runs.
表4. 相关方法的平均执行时间（50次运行）。

Prediction starting point 预测起点	Technique 技术	Time (sec) 时间（秒）
86	QPSO-PF	5.794
	QPSO-ANFIS	10.579
	ePF-EF	0.726
106	QPSO-PF	7.469
	QPSO-ANFIS	11.196
	ePF-EF	0.735
126	QPSO-PF	8.600
	QPSO-ANFIS	11.720
	ePF-EF	0.673
146	QPSO-PF	9.813
	QPSO-ANFIS	13.404
	ePF-EF	0.671

It is clear that the execution time increases considerably for both the QPSO-PF and the QPSO-ANFIS as the prediction period becomes shorter. This is because the extra available data have to be processed for modeling and forecasting; the QPSO-PF duplicates its particle numbers using a wave function to reduce sample degeneracy and impoverishment, which takes longer time for processing. The QPSO-ANFIS framework has an even longer execution time than the QPSO-PF in PF modeling and ANFIS training. In contrast, the developed ePF-EF framework can provide the most efficient processing, which is about 10 times faster than the QPSO-PF (0.726 sec vs. 5.794 sec, 0.735 sec vs. 7.469 sec, 0.673 sec vs. 8.600 sec, 0.671 sec vs. 9.813 sec), and about 15 times faster than the QPSO-ANFIS (0.726 sec vs. 10.579 sec, 0.735 sec vs. 11.196 sec, 0.673 sec vs. 11.720 sec, 0.671 sec vs. 13.404 sec). This is because the ePF will engage to process small-weight particles, only when the sample degeneracy is detected on the posterior PDF as discussed in Section 2.2. Such a strategy can not only maintain a higher number of $E_{S}$ on posterior PDF (to reduce sample degeneracy), but also reduce computational costs (i.e., execution time). In addition, the EF predictor has the ability to progressively evolve its reasoning formation to map the input–output spaces, which in turn can further accelerate the training process.
当预测时间变短时，QPSO-PF和QPSO-ANFIS的执行时间明显增加。这是因为要处理额外的可用数据进行建模和预测；QPSO-PF使用波函数复制其粒子数，以减少样本退化和贫化，这需要更长的处理时间。相较之下，开发的ePF-EF框架可以提供最高效的处理，其速度约为QPSO-PF的10倍（0.726秒对5.794秒，0.735秒对7.469秒，0.673秒对8.600秒，0.671秒对9.813秒），比QPSO-ANFIS快约15倍（0.726秒对10.579秒，0.735秒对11.196秒，0.673秒对11.720秒，0.671秒对13.404秒）。这是因为当在后验概率分布函数上检测到样本退化时，ePF只会处理小权重粒子，这一点在第2.2节中讨论过。这样的策略不仅可以保持后验概率密度函数上更高的数量（以减少样本退化），还可以减少计算成本（即执行时间）。此外，EF预测器具有逐步演化其推理形成的能力，以映射输入-输出空间，从而进一步加速训练过程。

Table 5 summarizes the outcomes of the battery RUL prediction using the related techniques, which also includes the relative errors and prediction starting points. Fig. 11 outlines the performance of the related methods for SOH estimation and RUL prediction with prediction starting at cycle 86 (over 80 cycles). Fig. 12, Fig. 13 depict the zoomed results for the medium-term and short-term predictions, starting at cycles 106 and 126, respectively.
表5总结了使用相关技术进行电池剩余寿命预测的结果，还包括相对误差和预测起始点。图11概述了相关方法在循环86开始（超过80个循环）的SOH估计和剩余寿命预测的性能。图12和图13分别展示了中期和短期预测的放大结果，起始循环分别为106和126。

Table 5. The RUL prediction results of the used techniques.
表5：所用技术的RUL预测结果。

Prediction starting point 预测起点	Technique 技术	Prediction result (cycle) 预测结果（周期）	Absolute error (cycles) 绝对误差（周期）	Relative error 相对误差
86	QPSO-PF	140	22	13.58%
	QPSO-ANFIS	152	10	6.17%
	ePF-EF	158	4	2.469%
106	QPSO-PF	146	16	9.88%
	QPSO-ANFIS	155	7	4.32%
	ePF-EF	159	3	1.85%
126	QPSO-PF	148	14	8.64%
	QPSO-ANFIS	154	8	4.94%
	ePF-EF	155	7	4.32%
146	QPSO-PF	152	10	6.17%
	QPSO-ANFIS	164	2	1.24%
	ePF-EF	161	1	0.617%

Test results indicate that the QPSO-PF has the lowest prediction accuracy (with the largest errors) under all testing conditions because it cannot adaptively update its model parameters through the prediction process, even though its performance can be improved for short-term predictions. The QPSO-ANFIS performs better than the QPSO-PF because the ANFIS can adaptively predict the degradation indicator values during the prognostic process (Phase 2) to update the degradation model in RUL prediction. However, the QPSO-ANFIS could not generate clear improvement even the prediction horizon becomes shorter and more data are used in modeling in Phase 1; this is because the ANFIS predictor has limited adaptive ability due to its fixed reasoning structure, which could limit its ability to handle the time-varying electro-chemical battery system dynamics.
测试结果表明，在所有测试条件下，QPSO-PF具有最低的预测准确性（具有最大误差），因为它无法通过预测过程自适应地更新其模型参数，即使其在短期预测中的性能可能会得到改善。QPSO-ANFIS的表现优于QPSO-PF，因为ANFIS可以在预测过程中自适应地预测故障指示值（第2阶段）以更新RUL预测中的故障模型。然而，尽管QPSO-ANFIS在模型建立的第1阶段中对预测时间跨度缩短并使用更多数据，但无法明显改善；这是由于ANFIS预测器由于其固定的推理结构而具有有限的自适应能力，这可能会限制其处理时变的电化学电池系统动态性能。

In contrast, the developed ePF-EF framework outperforms other predictors in terms of RUL prediction under all testing conditions, which is approximately 55%, 50%, and 13% more accurate than QPSO-ANFIS with prediction starting points at 86, 106, and 126, respectively. The EF predictor in the proposed ePF-EF framework can accommodate dynamic battery conditions by adaptively updating not only its parameters like the ANFIS, but also its reasoning architecture.
相比之下，发展的ePF-EF框架在所有测试条件下的剩余寿命预测方面表现优于其他预测器，其准确度分别比以86、106和126为起始点的QPSO-ANFIS预测器高出约55%、50%和13%。在提出的ePF-EF框架中，EF预测器可以通过自适应地更新其参数和推理架构来适应动态电池条件。

In battery health management and prognostics, reliable RUL prediction information can be used to schedule the recharging or repair operations [32], [37]. In this paper, the confidence interval associated with the RUL estimation will be represented in the form of PDF distribution (i.e., PDF interval), whereby a lower interval indicates less uncertainty and more reliability [3], [26]. In other words, a more accurate RUL prediction (with less uncertainty) corresponds to a PDF with a narrower and taller distribution, as illustrated in Fig. 7. The processing uncertainty of each technique can be characterized by using the PDF in the RUL prediction as the state reaches the battery’s end-of-life threshold. Fig. 14, Fig. 15 illustrate the PDFs generated by the related techniques for predictions starting at cycles 106 and 146, respectively, using the kernel density method [26]. It is clear that the PDF of the proposed ePF-EF framework has narrower and taller distributions than other related techniques, which can attest to its better performance and modeling efficiency to characterize the high-likelihood area of the posterior PDF.
在电池健康管理和预测中，可靠的剩余寿命（RUL）预测信息可用于安排充电或维修操作[32]，[37]。本文中，与RUL估计相关的置信区间将以PDF分布的形式表示（即PDF区间），其中较低的区间表示较少的不确定性和更可靠性[3]，[26]。换句话说，更准确的RUL预测（较少的不确定性）对应于具有较窄且较高分布的PDF，如图7所示。每种技术的处理不确定性可以通过在电池达到寿命阈值时使用RUL预测中的PDF来表征。图14和图15分别使用核密度方法[26]生成了从循环106和146开始的预测所产生的PDF。显然，所提出的ePF-EF框架的PDF比其他相关技术具有更窄且更高的分布，这可以证明其更好的性能和建模效率，以表征后验PDF的高概率区域。

In comparison, the QPSO-PF method has a wide PDF distribution, which becomes even larger as the forecasting horizon increases. This is because it duplicates particles to represent a broader region of the posterior PDF, without a mechanism to guide its particles to the high-probability region of the posterior PDF. Although the QPSO-ANFIS framework has a narrower PDF distribution than that of QPSO-PF, its distribution is slightly skewed because it selects particles with high weights many times, resulting in more particles concentrated to some areas. This degrades the diversity of the PDF distribution and could affect the RUL prediction accuracy.
相比之下，QPSO-PF方法具有广泛的概率密度函数分布，随着预测时间的增加，其分布变得更大。这是因为它复制粒子以表示更广泛的后验概率密度函数区域，但没有机制引导粒子进入后验概率密度函数的高概率区域。虽然QPSO-ANFIS框架的概率密度函数分布比QPSO-PF的要窄，但它的分布略微倾斜，因为它多次选择具有高权重的粒子，导致更多的粒子集中在某些区域。这降低了概率密度函数分布的多样性，可能会影响剩余寿命预测的准确性。

5. Conclusions 5. 结论

In this paper, a new ePF technique has been proposed to reduce the impact of sample degeneracy in system state estimation by monitoring the particle weights on the posterior PDF in order to detect and process sample degeneracy. A prognostic framework has been developed to combine the merits of the model-based ePF and data-driven EF to reduce modeling uncertainty and enhance the accuracy of SOH and RUL prediction. It involves two phases of operation. In Phase 1, the battery health degradation is modeled by the ePF, and the EF predictor is formulated using the available battery measurement indicator values. The RUL prediction is undertaken in Phase 2, where the ePF will perform successive one-step-ahead prediction. The forecasted indicator values by the EF predictor will then be used by the ePF to recursively update its posterior PDF to reduce modeling uncertainty. This prognostic framework can describe the evolution of the battery degradation state to predict when the battery SOH will reach its end-of-life threshold. The effectiveness of the proposed ePF and the prognostic framework has been examined using simulation tests. Test results reveal that the proposed ePF technique can reduce sample degeneracy and impoverishment. The prognostic framework can enhance the SOH estimation and RUL prediction accuracy, and facilitate computation efficiency.
本文提出了一种新的ePF技术，通过监测后验概率密度函数中的粒子权重来减少系统状态估计中样本退化的影响，以便检测和处理样本退化。开发了一种预测框架，结合了基于模型的ePF和数据驱动的EF的优点，以减少建模不确定性并提高SOH和RUL预测的准确性。它包括两个操作阶段。在第一阶段，电池健康退化由ePF建模，使用可用的电池测量指标值制定EF预测器。在第二阶段进行RUL预测，其中ePF将进行连续的一步预测。EF预测器预测的指标值将由ePF用于递归更新其后验概率密度函数以减少建模不确定性。这种预测框架可以描述电池退化状态的演变，以预测电池SOH何时达到其寿命阈值。使用模拟测试验证了所提出的ePF和预测框架的有效性。测试结果显示，所提出的ePF技术可以减少样本的退化和贫化。预测框架可以提高SOH估计和RUL预测的准确性，并提高计算效率。

CRediT authorship contribution statement
CRediT作者贡献声明

Mohamed Ahwiadi: Methodology, Software, Validation. Wilson Wang: Supervision, Validation.
Mohamed Ahwiadi: 方法论，软件，验证。Wilson Wang: 监督，验证。

Declaration of Competing Interest
竞争利益声明

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
作者声明他们没有已知的竞争性财务利益或个人关系，可能会影响本文所报道的工作。

Acknowledgement 确认

This work is supported in part by Natural Sciences and Engineering Research Council of Canada (NSERC), eMech Systems Inc., and Bare Point Water Treatment Plant in Thunder Bay, ON, Canada.
本项工作得到加拿大自然科学与工程研究理事会（NSERC）、eMech Systems公司和加拿大安大略省雷湾市Bare Point水处理厂部分支持。

References 参考资料

[1]
M.F. Ge, Y. Liu, X. Jiang, J. Liu
M.F.Ge，Y.Liu，X.Jiang，J.Liu
A review on state of health estimations and remaining useful life prognostics of lithium-ion batteries
锂离子电池健康状态估计和剩余寿命预测的综述
Measurement, 174 (2021), Article 109057
测量，174（2021年），文章109057
View PDF View article View in Scopus Google Scholar
查看PDF在Scopus中查看文章在谷歌学术中查看
[2]
D.A. Pola, H.F. Navarrete, M.E. Orchard, R.S. Rabie, M.A. Cerda, B.E. Olivares, J.F. Silva, P.A. Espinoza, A. Perez
D.A.Pola，H.F.Navarrete，M.E.Orchard，R.S.Rabie，M.A.Cerda，B.E.Olivares，J.F.Silva，P.A.Espinoza，A.Perez
Particle-filtering-based discharge time prognosis for lithium-ion batteries with a statistical characterization of use profiles
基于粒子滤波的锂离子电池放电时间预测，具有使用特征的统计描述
IEEE Trans. Rel., 64 (2) (2015), pp. 710-720
IEEE Trans. Rel., 64 (2) (2015), pp. 710-720 《IEEE可靠性期刊》，第64卷第2期，2015年，第710-720页
View in Scopus Google Scholar
在Scopus中查看Google学术
[3]
D. Li, W. Wang, F. Ismail D.李，W.王，F.伊斯梅尔
A mutated particle filter technique for system state estimation and battery life prediction
系统状态估计和电池寿命预测的突变粒子滤波技术
IEEE Trans. Instrum. Meas., 63 (8) (2014), pp. 2034-2043
IEEE Trans. Instrum. Meas.，63（8）（2014），pp. 2034-2043
View in Scopus Google Scholar
在Scopus中查看Google学术
[4]
Y. Liu, et al. 刘Y等人。
A nonlinear observer SOC estimation method based on electrochemical model for lithium-ion battery
基于电化学模型的锂离子电池非线性观测器SOC估计方法
IEEE Trans. Ind. Appl., 57 (1) (2021), pp. 1094-1104
IEEE交易对工业应用,57(1)(2021)，pp. 1094-1104。
CrossRef View in Scopus Google Scholar
在Scopus中查看CrossRef的内容
[5]
Y. Li, B. Xiong, D.M. Vilathgamuwa, Z. Wei, C. Xie, C. Zou
李阳，熊波，D.M.维拉特加姆瓦，魏志，谢超，邹超
Constrained ensemble kalman filter for distributed electrochemical state estimation of lithium-ion batteries
约束集合卡尔曼滤波器用于分布式锂离子电池电化学状态估计
IEEE Trans. Ind. Informat., 17 (1) (2021), pp. 240-250
IEEE工业信息学报，17卷1期（2021年），第240-250页。
CrossRef View in Scopus Google Scholar
[6]
D. Yang, Y. Wang, R. Pan, R. Chen, Z. Chen
A neural network-based state-of-health estimation of lithium-ion battery in electric vehicles
Energy, 105 (2017), pp. 2059-2064
View PDF View article View in Scopus Google Scholar
[7]
M. Landi, G. Gross
Measurement techniques for online battery state of health estimation in vehicle-to-grid applications
IEEE Trans. Instrum. Meas., 63 (5) (2014), pp. 1224-1234
View in Scopus Google Scholar
[8]
H. Dai, P. Guo, X. Wei, Z. Sun, J. Wang
ANFIS based online SOC (State of Charge) correction considering cell divergence for the EV (electric vehicle) traction batteries
Energy, 80 (2015), pp. 350-360
View PDF View article View in Scopus Google Scholar
[9]
T. Sutharssan, D. Montalvao, Y.K. Chen, W.-C. Wang, C. Pisac, H. Elemara
A review on prognostics and health monitoring of proton exchange membrane fuel cell
Renew. Sustain. Energy Rev., 75 (2017), pp. 440-450
View PDF View article View in Scopus Google Scholar
[10]
I. Škrjanc, J.A. Iglesias, A. Sanchis, D. Leite, E. Lughofer, F. Gomide
Evolving fuzzy and neuro-fuzzy approaches in clustering, regression, identification, and classification: a survey
Inf. Sci., 490 (2019), pp. 344-368
View PDF View article View in Scopus Google Scholar
[11]
X. Hu, L.e. Xu, X. Lin, M. Pecht
Battery lifetime prognostics
Joule., 4 (2) (2020), pp. 310-346
View PDF View article View in Scopus Google Scholar
[12]
M. Kordestani, M. Saif, M.E. Orchard, R. Razavi-Far, K. Khorasani
Failure prognosis and applications–a survey of recent literature
IEEE Trans. Reliab., 70 (2) (2021), pp. 728-748
CrossRef View in Scopus Google Scholar
[13]
D. Liu, Y. Luo, J. Liu, Y.u. Peng, L. Guo, M. Pecht
Lithium-ion battery remaining useful life estimation based on fusion nonlinear degradation AR model and RPF algorithm
Neural Comput. Appl., 25 (3-4) (2014), pp. 557-572
CrossRef View in Scopus Google Scholar
[14]
Y. Ma, Y. Chen, X. Zhou, H. Chen
‘Remaining useful life prediction of lithium-ion battery based on gauss-hermite particle filter’
IEEE Trans. Control Syst. Technol., 27 (4) (2019), pp. 1788-1795
CrossRef View in Scopus Google Scholar
[15]
Y. Wang, C. Zhang, Z. Chen
On-line battery state-of-charge estimation based on an integrated estimator
Appl. Energy, 185 (2017), pp. 2026-2032
View PDF View article Google Scholar
[16]
B. Saha, K. Goebel, S. Poll, J. Christophersen, An integrated approach to battery health monitoring using Bayesian regression and state estimation, in: Proc. IEEE Autotestcon, 2007, pp. 646–653.
Google Scholar
[17]
B. Saha, K. Goebel, Modeling Li-ion battery capacity depletion in a particle filtering framework, in: Proc. Annu. Conf. Prognost. Health Manage. Soc., San Diego, CA, USA, 2009, pp. 1–10.
Google Scholar
[18]
E. Walker, S. Rayman, R.E. White
Comparison of a particle filter and other state estimation methods for prognostics of lithium-ion batteries
J. Power Sour., 287 (2015), pp. 1-12
View PDF View article CrossRef View in Scopus Google Scholar
[19]
T. Li, S. Sun, T. Sattar, J. Corchado
Fight sample degeneracy and impoverishment in particle filters: a review of intelligent approaches
Expert Syst. Appl., 41 (8) (2014), pp. 3944-3954
View PDF View article View in Scopus Google Scholar
[20]
L. Liao, F. Köttig
A hybrid framework combining data-driven and model-based methods for system remaining useful life prediction
Appl. Soft Comput., 44 (2016), pp. 191-199
View PDF View article View in Scopus Google Scholar
[21]
C. Musso, N. Oudjane, F. LeGland
Improving Regularized Particle Filters
Sequential Monte Carlo Methods in Practice, Springer-Verlag, New York, NY, USA (2001)
Google Scholar
[22]
J. Liu, W. Wang, F. Ma
A regularized auxiliary particle filtering approach for system state estimation and battery life prediction
Smart Mater. Struct., 20 (7) (2011), pp. 1-9
Google Scholar
[23]
S. Hong, Z. Shi, L. Wang, Y. Gu, K. Chen
Adaptive regularized particle filter for synchronization of chaotic Colpitts circuits in an AWGN channel
Circ. Syst. Signal Process., 32 (2) (2013), pp. 825-841
CrossRef View in Scopus Google Scholar
[24]
M. Ahwiadi, W. Wang
An enhanced mutated particle filter technique for system state estimation and battery life prediction
IEEE Trans. Instru. Meas., 68 (3) (2019), pp. 923-935
CrossRef View in Scopus Google Scholar
[25]
Y. Tian, C. Lu, Z. Wang, L. Tao
Artificial fish swarm algorithm-based particle filter for Li-ion battery life prediction
Math. Probl. Eng., 2014 (2014), pp. 1-10
Google Scholar
[26]
J. Yu, B. Mo, D. Tang, H. Liu, J. Wan
Remaining useful life prediction for lithium-ion batteries using a quantum particle swarm optimization-based particle filter
Quality Engi., 29 (3) (2017), pp. 536-546
CrossRef View in Scopus Google Scholar
[27]
X. Sun, K. Zhong, M. Han
‘A hybrid prognostic strategy with unscented particle filter and optimized multiple kernel relevance vector machine for lithium-ion battery
Measurement, 170 (2021), Article 108679
View PDF View article View in Scopus Google Scholar
[28]
J. Liu, W. Wang, F. Ma, Y.B. Yang, C.S. Yang
A data-model-fusion prognostic framework for dynamic system state forecasting
Eng. Appl. Artif. Intell., 25 (4) (2012), pp. 814-823
View PDF View article View in Scopus Google Scholar
[29]
M. Ahwiadi, W. Wang
An adaptive particle filter technique for system state estimation and prognosis
IEEE Trans. Instru. Meas., 69 (9) (2020), pp. 6756-6765
CrossRef View in Scopus Google Scholar
[30]
H. Zhang, E. Liu, B. Zhang, Q. Miao
RUL prediction and uncertainty management for multi-sensor system using an integrated data level fusion and UPF approach
IEEE Trans. Ind. Informat., 17 (7) (2021), pp. 4692-4701
CrossRef View in Scopus Google Scholar
[31]
M. Arulampalam, S. Maskell, N. Gordon, T. Clapp
A tutorial on particle filters for online nonlinear /non-Gaussian Bayesian tracking
IEEE Trans. Signal Process., 50 (2) (2002), pp. 174-188
View in Scopus Google Scholar
[32]
D. An, J.H. Choi, N.H. Kim
Prognostics 101: a tutorial for particle filter-based prognostics algorithm using Matlab
Rel. Eng. Syst. Safety, 115 (1) (2013), pp. 161-169
View PDF View article View in Scopus Google Scholar
[33]
W. Wang, D.Z. Li, J. Vrbanek
An evolving neuro-fuzzy technique for system state forecasting
Neurocomputing, 87 (2012), pp. 111-119
View PDF View article Google Scholar
[34]
D. Ge, X.J. Zeng
A self-evolving fuzzy system which learns dynamic threshold parameter by itself
IEEE Trans. Fuzzy Syst., 27 (8) (2019), pp. 1625-1637
CrossRef View in Scopus Google Scholar
[35]
B. Saha, K. Goebel, Battery Data Set NASA Ames Prognostics Data Repository, NASA Ames, CA, USA, 2007 [Online]. https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository/.
Google Scholar
[36]
NASA Prognostics Center of Excellence, Battery Prognostics, Laboratory Setup [Online]. Available: https://ti.arc.nasa.gov/tech/dash/groups/pcoe/battery-prognostics/lab-setup/.
Google Scholar
[37]
S. Sankararaman
Significance, interpretation, and quantification of uncertainty in prognostics and remaining useful life prediction
Mech. Syst. Signal Process., 52-53 (2015), pp. 228-247
View PDF View article View in Scopus Google Scholar

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Accurate capacity and remaining useful life prediction of lithium-ion batteries based on improved particle swarm optimization and particle filter
2024, Energy
Accurate prediction of capacity and remaining useful life (RUL) for lithium-ion batteries (LIBs) is crucial for ensuring safe and reliable operation of electric vehicles. However, the battery capacity degradation and external environmental disturbances make it still challenging to achieve this goal. In this article, an accurate capacity and RUL prediction method is proposed by combining improved particle swarm optimization (IPSO) with particle filter (PF) algorithms. First, the parameters of particle swarm optimization (PSO) algorithm are adjusted by adaptive weights to avoid the problem of local optimal solution. Subsequently, the optimal particle searched by IPSO is updated continuously by the PF algorithm to achieve a more accurate posterior estimation. Finally, the proposed IPSO-PF method is verified by two independent and public datasets of NASA and CALCE batteries. The results validate that the proposed method has high precision and generalizability in predicting the capacity and RUL of LIBs even at various charging rates and battery types.
A multi-stage time series processing framework based on attention mechanism for early life prediction of lithium-ion batteries
2024, Journal of Energy Storage
Lithium-ion batteries play a vital role in powering portable electronic devices and electric vehicles. In order to shorten the research time, it is significant to accurately predict the life of lithium-ion batteries in the early cycle. However, due to the nonlinear degradation pattern with inconspicuous capacity variation in early cycles, predicting the life of the battery remains challenging when only the limited early cycle samples are available. In this paper, a multi-stage time series processing framework based on attention mechanism is proposed. To extract divergent features, category based intra-cycle sequence feature extraction network (CICSN) is designed for battery voltage, current and temperature (V/I/T) in the cycle, and inter-cycle global feature extraction network (ICGN) is proposed for discharge capacity across cycles. The cross-attention mechanism is adopted to fuse features, further improving the performance of prediction. A comprehensive evaluation of the proposed method is conducted through a lithium-ion battery dataset collected by MIT. The results show that the values of MAPE, MAE and RMSE decrease to 6.4%, 45 cycles and 67 cycles, which are better than existing research methods and demonstrate the effectiveness and validity of the proposed model.
Particle filter-based fatigue damage prognosis using prognostic-aided model updating
2024, Mechanical Systems and Signal Processing
Fatigue damage prognosis studies typically involve updating a degradation model based on the collected measurements. The updated model can accurately depict the past degradation process, while it is always assumed to be capable of describing the future damage progression, and thus can be used for remaining useful life (RUL) prediction. However, this assumption cannot always hold true, due to the uncertainties arising from the degradation process. In this context, this paper proposes a scheme of prognostic-aided model updating in a particle filter (PF)-based prognostic framework, which empowers the updated model to capture both the historical and future degradation process, and consequently, enhances the prognostic performance. First, two degradation models are formulated using the physics-based damage state and the data-driven health indicator (HI), respectively. The former is then re-formulated in state space by incorporating a measurement model. Next, the state space model is updated by the latest measurement through PF, providing the RUL and lifetime percentage (LP) distributions. Finally, the particles of damage state and parameters are modified by comparing the LP distributions from the updated model and the latter degradation model. The feasibility and robustness of the proposed method are validated by an experimental study, where an aluminum lug joint subject to fatigue crack growth is monitored by a guided wave measurement system.
Improved planar near-field measurement based on data assimilation
2024, Measurement: Journal of the International Measurement Confederation
The planar near-field measurement is widely used for antenna radiation testing because of its compact test space. However, due to the inevitable truncation error in the practical test, the measured radiation pattern of the antenna under test (AUT) is accurate only within certain confidence (angular) interval. In order to improve the measurement accuracy outside the confidence interval, this paper proposes an improved method of planar near-field measurement based on data assimilation. A reasonable mathematical model for the AUT radiation pattern is established so that the simulation data and the practical test data are fused with improved accuracy. The effectiveness of the proposed method for parameter selection has been verified experimentally, compared with the conventional method. In addition, the proposed method indicates a universal approach of complementing the measurement data with simulations or other data sources for improved accuracy.
A weight-allocation-based ensemble remaining useful life prediction approach for a single device
2024, Measurement: Journal of the International Measurement Confederation
Accurate prediction of the remaining useful life (RUL) can improve device reliability and reduce maintenance costs. In real working conditions, the absence of historical data for devices of the same type and the unlabeled degradation data for a single device present significant challenges for predicting RUL. Furthermore, it is difficult to predict RUL accurately due to the randomness and unstable prediction accuracy of an individual model in different scenarios. To address these issues, this paper proposes an ensemble prediction method based on weight allocation. Consider the advantages and disadvantages of various prediction methods, Particle Filter (PF), Support Vector Regression (SVR), and Gaussian Process Regression (GPR) are selected as member models for the ensemble prediction method. The ensemble method divides observed unlabeled degradation data into a training dataset and a validation dataset, among which the training dataset trains the model, and the validation dataset is utilized to calculate the weights for each member model. Particularly, we conducted a data augmentation approach using uncertain processes to estimate the initial state parameters of the PF. A weight allocation method using distance measures is presented to assign weights to the prediction results of each member model and perform weighted prediction to obtain the final ensemble prediction results. This paper demonstrates the effectiveness of the proposed ensemble prediction method through two practical examples. Results show that the proposed prediction method has higher accuracy compared to an individual prediction model and mean weight prediction model.
A review of efficient applications of genetic algorithms to improve particle filtering optimization problems
2024, Measurement: Journal of the International Measurement Confederation
Particle filtering (PF) is a sequential Monte Carlo method that draws sample (particle) values of state variables of interest to approximate the posterior probability distribution function (PDF) of the state and to infer the true state from this PDF. The values of the state variables that belong to a particle can differ from those of the other particles and the evaluated importance weights of particles can be different as well. Resampling employs a selection process to eliminate low-weight particles and retain high-weight particles that are further used to construct the new prediction PDF (or prior PDF) for drawing particles at the next time step. Genetic algorithms (GAs) help in finding the optimal solution by stochastically identifying new solutions that have higher fitness values. Each solution is treated as an individual: the individuals that have high fitness survive and are allowed to create offspring. The individuals with low fitness are more likely to be eliminated from the population according to the principle of the selection process. Selection is employed in both PF and GAs. However, particles found after the selection process can be left trapped around local maxima of the objective function. We can integrate genetic operations (e.g., crossover and mutation) to find new offspring particles that are expected to have higher weights. We construct the new prediction PDF to draw new particles at the next time step after we obtain offspring particles. This paper reviews and discusses efficient techniques of integrating GAs to enhance the performance of PF. These techniques can open doors into research advances in state prediction and estimation, leading to new insights and innovations.

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[1] [1]
M.F. Ge, Y. Liu, X. Jiang, J. Liu
M.F.Ge，Y.Liu，X.Jiang，J.Liu
A review on state of health estimations and remaining useful life prognostics of lithium-ion batteries
锂离子电池健康状态估计和剩余寿命预测的综述
Measurement, 174 (2021), Article 109057
测量，174（2021年），文章109057
View PDF View article View in Scopus Google Scholar
查看PDF在Scopus中查看文章在谷歌学术中查看

[2] [2]
D.A. Pola, H.F. Navarrete, M.E. Orchard, R.S. Rabie, M.A. Cerda, B.E. Olivares, J.F. Silva, P.A. Espinoza, A. Perez
D.A.Pola，H.F.Navarrete，M.E.Orchard，R.S.Rabie，M.A.Cerda，B.E.Olivares，J.F.Silva，P.A.Espinoza，A.Perez
Particle-filtering-based discharge time prognosis for lithium-ion batteries with a statistical characterization of use profiles
基于粒子滤波的锂离子电池放电时间预测，具有使用特征的统计描述
IEEE Trans. Rel., 64 (2) (2015), pp. 710-720
IEEE Trans. Rel., 64 (2) (2015), pp. 710-720 《IEEE可靠性期刊》，第64卷第2期，2015年，第710-720页
View in Scopus Google Scholar
在Scopus中查看Google学术

[3] [3]
D. Li, W. Wang, F. Ismail D.李，W.王，F.伊斯梅尔
A mutated particle filter technique for system state estimation and battery life prediction
系统状态估计和电池寿命预测的突变粒子滤波技术
IEEE Trans. Instrum. Meas., 63 (8) (2014), pp. 2034-2043
IEEE Trans. Instrum. Meas.，63（8）（2014），pp. 2034-2043
View in Scopus Google Scholar
在Scopus中查看Google学术

[4] [4]
Y. Liu, et al. 刘Y等人。
A nonlinear observer SOC estimation method based on electrochemical model for lithium-ion battery
基于电化学模型的锂离子电池非线性观测器SOC估计方法
IEEE Trans. Ind. Appl., 57 (1) (2021), pp. 1094-1104
IEEE交易对工业应用,57(1)(2021)，pp. 1094-1104。
CrossRef View in Scopus Google Scholar
在Scopus中查看CrossRef的内容

[5] [5]
Y. Li, B. Xiong, D.M. Vilathgamuwa, Z. Wei, C. Xie, C. Zou
李阳，熊波，D.M.维拉特加姆瓦，魏志，谢超，邹超
Constrained ensemble kalman filter for distributed electrochemical state estimation of lithium-ion batteries
约束集合卡尔曼滤波器用于分布式锂离子电池电化学状态估计
IEEE Trans. Ind. Informat., 17 (1) (2021), pp. 240-250
IEEE工业信息学报，17卷1期（2021年），第240-250页。
CrossRef View in Scopus Google Scholar

[6] [6]
D. Yang, Y. Wang, R. Pan, R. Chen, Z. Chen
A neural network-based state-of-health estimation of lithium-ion battery in electric vehicles
Energy, 105 (2017), pp. 2059-2064
View PDF View article View in Scopus Google Scholar

[7] [7]
M. Landi, G. Gross
Measurement techniques for online battery state of health estimation in vehicle-to-grid applications
IEEE Trans. Instrum. Meas., 63 (5) (2014), pp. 1224-1234
View in Scopus Google Scholar

[8] [8]
H. Dai, P. Guo, X. Wei, Z. Sun, J. Wang
ANFIS based online SOC (State of Charge) correction considering cell divergence for the EV (electric vehicle) traction batteries
Energy, 80 (2015), pp. 350-360
View PDF View article View in Scopus Google Scholar

[9] [9]
T. Sutharssan, D. Montalvao, Y.K. Chen, W.-C. Wang, C. Pisac, H. Elemara
A review on prognostics and health monitoring of proton exchange membrane fuel cell
Renew. Sustain. Energy Rev., 75 (2017), pp. 440-450
View PDF View article View in Scopus Google Scholar

[10] [10]
I. Škrjanc, J.A. Iglesias, A. Sanchis, D. Leite, E. Lughofer, F. Gomide
Evolving fuzzy and neuro-fuzzy approaches in clustering, regression, identification, and classification: a survey
Inf. Sci., 490 (2019), pp. 344-368
View PDF View article View in Scopus Google Scholar

[11] [11]
X. Hu, L.e. Xu, X. Lin, M. Pecht
Battery lifetime prognostics
Joule., 4 (2) (2020), pp. 310-346
View PDF View article View in Scopus Google Scholar

[12] [12]
M. Kordestani, M. Saif, M.E. Orchard, R. Razavi-Far, K. Khorasani
Failure prognosis and applications–a survey of recent literature
IEEE Trans. Reliab., 70 (2) (2021), pp. 728-748
CrossRef View in Scopus Google Scholar

[13] [13]
D. Liu, Y. Luo, J. Liu, Y.u. Peng, L. Guo, M. Pecht
Lithium-ion battery remaining useful life estimation based on fusion nonlinear degradation AR model and RPF algorithm
Neural Comput. Appl., 25 (3-4) (2014), pp. 557-572
CrossRef View in Scopus Google Scholar

[14] [14]
Y. Ma, Y. Chen, X. Zhou, H. Chen
‘Remaining useful life prediction of lithium-ion battery based on gauss-hermite particle filter’
IEEE Trans. Control Syst. Technol., 27 (4) (2019), pp. 1788-1795
CrossRef View in Scopus Google Scholar

[15] [15]
Y. Wang, C. Zhang, Z. Chen
On-line battery state-of-charge estimation based on an integrated estimator
Appl. Energy, 185 (2017), pp. 2026-2032
View PDF View article Google Scholar

[16] [16]
B. Saha, K. Goebel, S. Poll, J. Christophersen, An integrated approach to battery health monitoring using Bayesian regression and state estimation, in: Proc. IEEE Autotestcon, 2007, pp. 646–653.
Google Scholar

[17] [17]
B. Saha, K. Goebel, Modeling Li-ion battery capacity depletion in a particle filtering framework, in: Proc. Annu. Conf. Prognost. Health Manage. Soc., San Diego, CA, USA, 2009, pp. 1–10.
Google Scholar

[18] [18]
E. Walker, S. Rayman, R.E. White
Comparison of a particle filter and other state estimation methods for prognostics of lithium-ion batteries
J. Power Sour., 287 (2015), pp. 1-12
View PDF View article CrossRef View in Scopus Google Scholar

[19] [19]
T. Li, S. Sun, T. Sattar, J. Corchado
Fight sample degeneracy and impoverishment in particle filters: a review of intelligent approaches
Expert Syst. Appl., 41 (8) (2014), pp. 3944-3954
View PDF View article View in Scopus Google Scholar

[20] [20]
L. Liao, F. Köttig
A hybrid framework combining data-driven and model-based methods for system remaining useful life prediction
Appl. Soft Comput., 44 (2016), pp. 191-199
View PDF View article View in Scopus Google Scholar

[21] [21]
C. Musso, N. Oudjane, F. LeGland
Improving Regularized Particle Filters
Sequential Monte Carlo Methods in Practice, Springer-Verlag, New York, NY, USA (2001)
Google Scholar

[22] [22]
J. Liu, W. Wang, F. Ma
A regularized auxiliary particle filtering approach for system state estimation and battery life prediction
Smart Mater. Struct., 20 (7) (2011), pp. 1-9
Google Scholar

[23] [23]
S. Hong, Z. Shi, L. Wang, Y. Gu, K. Chen
Adaptive regularized particle filter for synchronization of chaotic Colpitts circuits in an AWGN channel
Circ. Syst. Signal Process., 32 (2) (2013), pp. 825-841
CrossRef View in Scopus Google Scholar

[24] [24]
M. Ahwiadi, W. Wang
An enhanced mutated particle filter technique for system state estimation and battery life prediction
IEEE Trans. Instru. Meas., 68 (3) (2019), pp. 923-935
CrossRef View in Scopus Google Scholar

[25] [25]
Y. Tian, C. Lu, Z. Wang, L. Tao
Artificial fish swarm algorithm-based particle filter for Li-ion battery life prediction
Math. Probl. Eng., 2014 (2014), pp. 1-10
Google Scholar

[26] [26]
J. Yu, B. Mo, D. Tang, H. Liu, J. Wan
Remaining useful life prediction for lithium-ion batteries using a quantum particle swarm optimization-based particle filter
Quality Engi., 29 (3) (2017), pp. 536-546
CrossRef View in Scopus Google Scholar

[27] [27]
X. Sun, K. Zhong, M. Han
‘A hybrid prognostic strategy with unscented particle filter and optimized multiple kernel relevance vector machine for lithium-ion battery
Measurement, 170 (2021), Article 108679
View PDF View article View in Scopus Google Scholar

[28] [28]
J. Liu, W. Wang, F. Ma, Y.B. Yang, C.S. Yang
A data-model-fusion prognostic framework for dynamic system state forecasting
Eng. Appl. Artif. Intell., 25 (4) (2012), pp. 814-823
View PDF View article View in Scopus Google Scholar

[29] [29]
M. Ahwiadi, W. Wang
An adaptive particle filter technique for system state estimation and prognosis
IEEE Trans. Instru. Meas., 69 (9) (2020), pp. 6756-6765
CrossRef View in Scopus Google Scholar

[30] [30]
H. Zhang, E. Liu, B. Zhang, Q. Miao
RUL prediction and uncertainty management for multi-sensor system using an integrated data level fusion and UPF approach
IEEE Trans. Ind. Informat., 17 (7) (2021), pp. 4692-4701
CrossRef View in Scopus Google Scholar

[31] [31]
M. Arulampalam, S. Maskell, N. Gordon, T. Clapp
A tutorial on particle filters for online nonlinear /non-Gaussian Bayesian tracking
IEEE Trans. Signal Process., 50 (2) (2002), pp. 174-188
View in Scopus Google Scholar

[32] [32]
D. An, J.H. Choi, N.H. Kim
Prognostics 101: a tutorial for particle filter-based prognostics algorithm using Matlab
Rel. Eng. Syst. Safety, 115 (1) (2013), pp. 161-169
View PDF View article View in Scopus Google Scholar

[33] [33]
W. Wang, D.Z. Li, J. Vrbanek
An evolving neuro-fuzzy technique for system state forecasting
Neurocomputing, 87 (2012), pp. 111-119
View PDF View article Google Scholar

[34] [34]
D. Ge, X.J. Zeng
A self-evolving fuzzy system which learns dynamic threshold parameter by itself
IEEE Trans. Fuzzy Syst., 27 (8) (2019), pp. 1625-1637
CrossRef View in Scopus Google Scholar

[35] [35]
B. Saha, K. Goebel, Battery Data Set NASA Ames Prognostics Data Repository, NASA Ames, CA, USA, 2007 [Online]. https://ti.arc.nasa.gov/tech/dash/groups/pcoe/prognostic-data-repository/.
Google Scholar

[36] [36]
NASA Prognostics Center of Excellence, Battery Prognostics, Laboratory Setup [Online]. Available: https://ti.arc.nasa.gov/tech/dash/groups/pcoe/battery-prognostics/lab-setup/.
Google Scholar

[37] [37]
S. Sankararaman
Significance, interpretation, and quantification of uncertainty in prognostics and remaining useful life prediction
Mech. Syst. Signal Process., 52-53 (2015), pp. 228-247
View PDF View article View in Scopus Google Scholar

Outline 大纲

Cited by (38) 引用（38）

Figures (15) 数字（15）

Tables (5) 表格（5）

Measurement 测量

Highlights 亮点

Abstract 摘要

Keywords 关键词

1. Introduction 介绍

2. Proposed ePF technique
2. 提议的ePF技术

2.1. PF overview 2.1. PF概述

2.2. Proposed ePF technique
2.2. 提议的ePF技术

3. The proposed prognostic framework
提出的预测框架

3.1. Phase 1: Degradation modeling
3.1. 阶段1：退化建模

3.2. Phase 2: RUL prediction
3.2. 第二阶段：寿命剩余预测

4. Performance evaluation
性能评估

4.1. Performance evaluation of the proposed ePF
4.1. 提出的ePF的性能评估

4.2. Performance evaluation for battery RUL prediction
4.2. 电池剩余寿命预测的性能评估

5. Conclusions 5. 结论

CRediT authorship contribution statement
CRediT作者贡献声明

Declaration of Competing Interest
竞争利益声明

Acknowledgement 确认

References 参考资料

Cited by (38)

Accurate capacity and remaining useful life prediction of lithium-ion batteries based on improved particle swarm optimization and particle filter

A multi-stage time series processing framework based on attention mechanism for early life prediction of lithium-ion batteries

Particle filter-based fatigue damage prognosis using prognostic-aided model updating

Improved planar near-field measurement based on data assimilation

A weight-allocation-based ensemble remaining useful life prediction approach for a single device

A review of efficient applications of genetic algorithms to improve particle filtering optimization problems

More documents by Mohamed Ahwiadi
莫哈迈德·阿赫瓦迪的更多文件

Measurement: Journal of the International Measurement Confederation, Volume 191, 15 March 2022
《测量：国际测量联合会杂志》，2022年3月15日第191卷

IEEE Transactions on Fuzzy Systems, Volume 30, Issue 3, 1 March 2022
2022年3月1日，IEEE模糊系统交易，第30卷，第3期

IEEE Transactions on Instrumentation and Measurement, Volume 69, Issue 9, September 2020
IEEE仪器与测量交易，第69卷，第9期，2020年9月

An enhanced particle filter technology for battery system state estimation and RUL prediction电池系统状态估计和剩余寿命预测的增强粒子滤波技术

Highlights 亮点

Abstract 摘要

Keywords 关键词

1. Introduction 介绍

2. Proposed ePF technique2. 提议的ePF技术

2.1. PF overview 2.1. PF概述

2.2. Proposed ePF technique2.2. 提议的ePF技术

3. The proposed prognostic framework提出的预测框架

3.1. Phase 1: Degradation modeling3.1. 阶段1：退化建模

3.2. Phase 2: RUL prediction3.2. 第二阶段：寿命剩余预测

4. Performance evaluation性能评估

4.1. Performance evaluation of the proposed ePF4.1. 提出的ePF的性能评估

4.2. Performance evaluation for battery RUL prediction4.2. 电池剩余寿命预测的性能评估

5. Conclusions 5. 结论

CRediT authorship contribution statementCRediT作者贡献声明

Declaration of Competing Interest竞争利益声明

Acknowledgement 确认

References 参考资料

An enhanced particle filter technology for battery system state estimation and RUL prediction
电池系统状态估计和剩余寿命预测的增强粒子滤波技术

2. Proposed ePF technique
2. 提议的ePF技术

2.2. Proposed ePF technique
2.2. 提议的ePF技术

3. The proposed prognostic framework
提出的预测框架

3.1. Phase 1: Degradation modeling
3.1. 阶段1：退化建模

3.2. Phase 2: RUL prediction
3.2. 第二阶段：寿命剩余预测

4. Performance evaluation
性能评估

4.1. Performance evaluation of the proposed ePF
4.1. 提出的ePF的性能评估

4.2. Performance evaluation for battery RUL prediction
4.2. 电池剩余寿命预测的性能评估

CRediT authorship contribution statement
CRediT作者贡献声明

Declaration of Competing Interest
竞争利益声明