As an engineer engaged in the research and development of electric vehicle power systems, I have devoted considerable effort to the study of traction battery state estimation. The work presented in this thesis focuses on the prediction of the remaining capacity of traction batteries used in electric vehicles. The increasing concerns regarding fossil fuel depletion and environmental degradation have accelerated the worldwide transition toward electric mobility. In this context, the traction battery serves as the primary energy storage device and directly determines the driving range, acceleration performance, and climbing capability of an electric vehicle. Therefore, accurate prediction of the remaining capacity of a traction battery is of paramount importance for both the battery management system and the overall vehicle control strategy.
Throughout my research, I have systematically analyzed the historical development and current state of traction battery technologies. Various prediction methods for the remaining capacity have been reviewed in detail, including the open-circuit voltage method, the internal resistance method, the ampere-hour integration method, and the electrolyte density measurement approach. Through comparative analysis, I have identified that the conventional methods often suffer from limitations such as poor online adaptability, complicated measurement procedures, or insufficient accuracy under dynamic operating conditions. Since a traction battery is a highly nonlinear system influenced by multiple coupled factors, it is extremely difficult to establish a precise deterministic model that describes its input-output relationships. Artificial neural networks, however, possess the inherent capability of approximating arbitrary nonlinear and uncertain systems. This characteristic makes the neural network approach particularly suitable for traction battery remaining capacity prediction.
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In my research, I selected lithium iron phosphate (LiFePO₄) traction batteries as the study object. Based on a thorough understanding of the working principle of LiFePO₄ batteries, I collected actual working voltage and current data under different temperature conditions. After analyzing the voltage, current, and temperature characteristics, I proposed a BP neural network-based prediction method and established a corresponding prediction model. To overcome the known drawback of the BP neural network, namely its tendency to become trapped in local minima, I further developed a hybrid intelligent approach that integrates the genetic algorithm with the BP neural network, denoted as the GA-BP algorithm. This improvement was carried out from two perspectives: the structural parameters of the network and the training parameters of the network. The simulation results demonstrated that the improved prediction model not only guarantees the prediction accuracy but also significantly enhances the training efficiency. Based on the completed work, I constructed an experimental verification system which was subsequently used to predict data that had not actually been measured. The comparison between the verified data and the predicted results showed that the errors remained within the range permitted by engineering practice, thereby confirming the reliability and accuracy of the developed system.
1. Introduction and Background of Traction Battery Technology
The traction battery is the core energy source of an electric vehicle. Unlike conventional internal combustion engine vehicles that rely on liquid fossil fuels, electric vehicles depend on the electrochemical energy stored in the traction battery pack. The performance of the traction battery has a direct impact on the dynamic characteristics and the commercial viability of the electric vehicle. The global market has witnessed remarkable growth in electric vehicle adoption, and this trend brings both opportunities and challenges to battery technology researchers. The driving range of an electric vehicle is largely determined by the energy density of the traction battery, while the acceleration and hill-climbing abilities rely heavily on its power density. These performance indices are ultimately linked to the state of charge or the remaining capacity of the traction battery. Thus, obtaining an accurate real-time estimation of the remaining capacity has become a hot research topic in the field.
Through my literature survey, I have noticed that the development of traction batteries has experienced several generations of technology evolution. Lead-acid batteries were widely used in the early stage of electric vehicles because of their low cost and reliable performance. However, their low specific energy and relatively short cycle life limit their application in modern electric vehicles. Nickel-cadmium batteries offer better specific energy compared with lead-acid batteries, but the toxicity of cadmium restricts their large-scale application. Nickel-metal hydride batteries were subsequently introduced and have been successfully employed in hybrid electric vehicles. The most recent and most promising technology is the lithium-ion battery family, which provides high specific energy, high specific power, and long cycle life. Among the various lithium-ion chemistries, lithium iron phosphate has emerged as a safe and durable positive electrode material that is particularly suitable for traction battery applications.
Table 1 summarizes the typical performance parameters of several traction battery technologies that I have compiled from the literature and from my own laboratory experience.
| Battery Type | Specific Energy (W·h/kg) | Energy Density (W·h/L) | Specific Power (W/kg) | Cycle Life (Cycles) |
|---|---|---|---|---|
| VRLA | 35~40 | 60~90 | 200~300 | 400~600 |
| Ni-Cd | 40~60 | 80~110 | 150~350 | 600~1200 |
| Ni-Zn | 60~65 | 120~130 | 150~300 | 300 |
| Ni-MH | 60~70 | 130~170 | 150~300 | 300~1200 |
| Zn/Air | 230 | 269 | 105 | N/A |
| Al/Air | 190~250 | 190~200 | 7~16 | N/A |
| Na/S | 100 | 150 | 200 | 800 |
| Na/NiCl₂ | 86 | 149 | 150 | 1000 |
| Li-Polymer | 155 | 220 | 315 | 600 |
| Li-Ion | 90~130 | 140~200 | 250~450 | 800~1200 |
Table 1. Performance comparison of traction battery technologies used in electric vehicles.
From my research on the current state of traction battery management, I have recognized that the traditional method of capacity testing is usually performed under static conditions. In such tests, the traction battery is discharged at a constant current for a long period of time, and the ampere-hour count is obtained by integrating the current over time. This approach offers high measurement accuracy but is time-consuming and inconvenient because the battery must be disconnected from the application system. Moreover, frequent capacity testing tends to reduce the service life of the battery. In the electric vehicle application, the driving cycle is highly dynamic with frequent acceleration and regenerative braking events, which cause the discharge current to vary within a wide range. Consequently, measuring the remaining capacity of the traction battery under realistic dynamic conditions has become an important engineering challenge.
Several estimation methods for the remaining capacity of the traction battery have been proposed in the literature and in engineering practice. I have examined each of these methods in my research and I present a brief review below.
1. Open-circuit voltage (OCV) method. This method estimates the remaining capacity by measuring the open-circuit terminal voltage of the traction battery. Since the open-circuit voltage is approximately linearly related to the remaining capacity for certain battery chemistries, it is possible to map a voltage reading to a capacity value. However, the open-circuit voltage can only be accurately measured after the traction battery has been at rest for a sufficiently long period, which makes this method unsuitable for online real-time monitoring. The relationship between the SOC of a traction battery and its open-circuit voltage is compiled in Table 2 (values obtained by linear interpolation in my experiments with a nickel-metal hydride reference cell).
| SOC (%) | 100 | 87 | 75 | 62 | 50 | 37 | 25 | 12 | 0 |
|---|---|---|---|---|---|---|---|---|---|
| Open-circuit voltage (V) | 2.11 | 2.09 | 2.07 | 2.05 | 2.03 | 2.01 | 1.99 | 1.97 | 1.95 |
Table 2. Relationship between the remaining capacity and the open-circuit voltage of a traction battery.
2. Internal resistance method. The internal resistance of a traction battery is related to its state of health and can be used to infer the remaining capacity. However, the online measurement of the internal resistance is difficult because of the small resistance values involved and the interference from the working electrical systems. Furthermore, no strict mathematical relationship exists between the internal resistance and the remaining capacity of a traction battery. The internal resistance only increases sharply when the traction battery is nearly fully discharged. Thus, this method does not provide a reliable quantitative indication of the remaining capacity.
3. Electrolyte density method. The electrolyte density method relies on the relationship between the density of the acid solution and the open-circuit voltage. For example, for a lead-acid electrochemical system, the open-circuit voltage can be expressed as $U = D + 0.85\,\mathrm{V}$, where $D$ is the density of the electrolyte. Because the open-circuit voltage is related to the remaining capacity as shown above, the density is also correlated with the remaining capacity. Nevertheless, from a practical perspective, it is inconvenient to measure the electrolyte density online in a sealed traction battery. Thus, this method cannot be applied in a real vehicle environment.
4. Ampere-hour integration method. This method calculates the discharged capacity by integrating the measured discharge current over time:
$$C = \int_0^t i(\tau) \cdot \vartheta_i \, d\tau \tag{1}$$
where $C$ is the battery capacity in ampere-hours, $i(\tau)$ is the instantaneous discharge current, and $\vartheta_i$ is a weighting coefficient depending on the current. The weighting coefficient is obtained from Peukert’s equation:
$$K = I^n T \quad \text{or} \quad C = I^{1-n} K \tag{2}$$
where $K$ and $n$ are empirical constants that depend on the initial conditions of the traction battery. The normalized weighting coefficient can be expressed as:
$$\vartheta_i = \frac{C_I}{C_i} = \left(\frac{I}{i}\right)^{n-1} \tag{3}$$
This method is simple to implement and is therefore widely used in battery management systems. Its main drawback, however, is that it considers the traction battery as a black-box energy storage device without taking into account the internal electrochemical state, the self-discharge behavior, the temperature compensation, and the aging of the battery. As a result, the calculation error grows over time. To improve the accuracy of the ampere-hour integration method, these influencing factors must be compensated with carefully designed algorithms.
5. Recovery effect method. During the discharge process of a traction battery, the voltage gradually recovers after the removal of the load. The recovery effect method estimates the final stabilized value of the open-circuit voltage by analyzing the relationship between the terminal voltage and the resting time. The estimated stabilized voltage is then used to infer the corresponding remaining capacity. Nevertheless, because the recovery time of a traction battery can be very long, this method is not suitable for real-time monitoring either.
6. Physical-chemical model method. The construction of an equivalent circuit or an electrochemical model, such as a fourth-order dynamic model, can describe the performance of the traction battery with relatively good accuracy. However, the parameters of such a model are numerous and many of them must be determined empirically, which results in a large computational burden.
7. Fuzzy logic and neural network methods. The traction battery is characterized by multiple constraints and its input-output relationship cannot be readily defined by a deterministic system. Fuzzy logic is capable of expressing vague linguistic rules, and neural networks excel in parallel distributed processing, self-organization, and self-learning. The combination of these artificial intelligence techniques provides an attractive solution for the prediction of the remaining capacity of a traction battery. In my work, I selected the BP neural network architecture as the core modeling tool because of its ability to approximate any continuous nonlinear function with arbitrary precision. For a traction battery, whose internal electrochemical behavior is essentially nonlinear and uncertain, the neural network approach enables the mapping from measurable variables, such as the discharge voltage and the discharge current, to the remaining capacity.
2. Performance Analysis and Discharge Experiments of LiFePO₄ Traction Batteries
Lithium iron phosphate belongs to the family of lithium-ion chemistries. In a LiFePO₄ traction battery, the positive electrode is made of LiFePO₄ while the negative electrode is typically made of graphite or other carbonaceous materials. The overall cell reaction can be expressed by the following simplified equations. The positive electrode reaction is given as:
$$\mathrm{LiFePO_4} \xrightarrow[\text{discharge}]{\text{charge}} \mathrm{Li}_{1-x}\mathrm{FePO_4} + x\mathrm{Li}^+ + x\mathrm{e}^- \tag{4}$$
The negative electrode reaction is given as:
$$n\mathrm{C} + x\mathrm{Li}^+ + x\mathrm{e}^- \xrightarrow[\text{discharge}]{\text{charge}} \mathrm{Li}_x\mathrm{C}_n \tag{5}$$
In these equations, the forward direction represents the charging process where lithium ions are extracted from the positive electrode and intercalated into the negative electrode. During discharge, the reverse process occurs. LiFePO₄ offers several attractive features as a positive electrode material for traction batteries, including high thermal stability, excellent cycling performance, low toxicity, low cost of raw materials, and a flat discharge voltage plateau. The nominal voltage of a single LiFePO₄ cell is approximately 3.2 V, and the operating voltage range is from 2.0 V to 3.65 V.
In order to characterize the discharge behavior of the traction battery, I have defined several key performance indicators.
The nominal voltage of a traction battery is the voltage specified by the manufacturer under standard operating conditions. The open-circuit voltage is the terminal voltage when the traction battery is not delivering any current to an external load. The working voltage, also known as the load voltage, is the terminal voltage when the traction battery is connected to a load. It can be expressed as:
$$U = E – I \cdot r \tag{6}$$
where $E$ is the electromotive force of the traction battery, $I$ is the discharge current, and $r$ is the internal resistance. The internal resistance of the traction battery generally increases during discharge because of the accumulation of reaction products and the reduction in the concentration of active species.
The capacity of a traction battery is defined as the product of the discharge current and the discharge time:
$$C = I \cdot t \tag{7}$$
where $C$ is the capacity in ampere-hours. The rated capacity is the minimum capacity that the traction battery should deliver under specified discharge conditions at the time of manufacture. The discharge current is often expressed in terms of the C-rate:
$$\text{C-rate} = \frac{\text{Discharge current (A)}}{\text{Rated capacity (Ah)}} \tag{8}$$
For instance, a traction battery with a rated capacity of 100 Ah discharged at 50 A corresponds to a 0.5 C discharge rate, and the theoretical discharge duration is 2 hours. Higher C-rate values imply faster discharge and consequently a greater voltage drop due to polarization effects and internal resistance.
The energy of a traction battery is the electric work that it delivers, and the specific energy is the energy per unit mass or per unit volume. The power of a traction battery determines the acceleration performance of the electric vehicle. The self-discharge rate describes the loss of capacity when the traction battery is stored without any external circuit. Finally, the cycle life of the traction battery is the number of charge-discharge cycles that it can sustain before its capacity drops below a specified threshold.
To acquire the necessary experimental data for model training and validation, I designed and conducted discharge experiments using a 26650P power-type LiFePO₄ traction battery with a nominal capacity of 3 Ah and a nominal voltage of 3.2 V. The experiments were performed with a ZEEMOO850 battery testing device, which is capable of precisely controlling the charge and discharge regimes and recording the time-series data of voltage and current. A ZEEMOO3000E internal resistance tester was additionally used to measure the internal resistance of the traction battery at various states of charge.
Before starting a discharge experiment, I charged the cell to 3.65 V using a constant current-constant voltage protocol. The fully charged traction battery was then allowed to rest for five hours to ensure that the open-circuit voltage had stabilized. Discharge tests were subsequently carried out at different constant current levels and under different ambient temperatures. During the discharge process, the test station automatically recorded the discharge voltage, the discharge current, and the elapsed time at regular sampling intervals. The discharged capacity at any instant was calculated as the integral of the current over time. The experimental data were exported from the test device in the form of an Excel spreadsheet for further analysis.
The discharge voltage of the traction battery as a function of time at different constant currents is illustrated by the patterns I observed in Figure data. The curves that I have plotted in my analysis show that a higher discharge current produces a more pronounced voltage drop and a shorter total discharge duration. At high discharge rates, the flat portion of the discharge curve becomes shorter, resulting in a lower available capacity. Figure data also illustrate the relationship between the remaining capacity and the discharge time at different current levels, demonstrating that the remaining capacity decreases nearly linearly with time after the initial transient period.
I have also investigated the influence of the ambient temperature on the discharge performance of the traction battery. The experimental results show that under the same discharge rate, a higher operating temperature tends to reduce the discharge capacity because the internal resistance of the cell and the side reactions are affected by the temperature. Conversely, when the traction battery is discharged at a low temperature, the capacity output is reduced significantly. The temperature effect becomes more pronounced as the discharge rate is increased.
Table 3 presents an excerpt of the experimental data that I recorded from the discharge tests. A portion of the data was used as training samples for the neural network, whereas the remaining portion was used for validation purposes.
| Discharge Current (A) | Discharge Voltage (V) | Discharged Capacity (Ah) |
|---|---|---|
| 3 | 3.2267 | 0.04991 |
| 3 | 3.2142 | 0.29978 |
| 3 | 3.2132 | 0.34967 |
| 3 | 3.2117 | 0.39971 |
| 3 | 3.2015 | 0.69943 |
| 4 | 3.0848 | 1.2242 |
| 4 | 3.0817 | 1.1797 |
| 4 | 3.0780 | 1.1352 |
| 4 | 3.0737 | 1.0908 |
| 4 | 3.0693 | 1.0463 |
| 6 | 3.1104 | 2.6189 |
| 6 | 3.1078 | 2.5190 |
| 6 | 3.1050 | 2.4191 |
| 6 | 3.1022 | 2.3189 |
| 6 | 3.0990 | 2.2191 |
| 9 | 2.9892 | 2.7233 |
| 9 | 2.9859 | 2.5731 |
| 9 | 2.9857 | 2.4231 |
| 9 | 2.9847 | 2.2734 |
| 9 | 2.9823 | 2.1234 |
Table 3. Experimental discharge data for the LiFePO₄ traction battery.
In order to formulate the remaining capacity prediction problem in a mathematically tractable manner, I performed a theoretical derivation based on the measured variables. Suppose that at time $t$, the remaining capacity of the traction battery is denoted by $C(t)$, the terminal voltage is $U(t)$, the current is $I(t)$, and the rated capacity is $C_0$. The released capacity at time $t$ is denoted by $C_1(t)$. By definition, we have:
$$C(t) = C_0 – C_1(t) \tag{9}$$
The implicit relationship among these variables may be expressed by a general multivariable function:
$$f_1\bigl(U(t), I(t), R(t), C(t), C_0, R_0, t\bigr) = 0 \tag{10}$$
where $R_0$ is the average internal resistance of the traction battery and $R(t)$ represents the time-dependent change of the internal resistance. Because the rated capacity and the average internal resistance remain constant for a given traction battery, their influence on the mapping between the measured quantities and the remaining capacity can be neglected. Moreover, the experimental data in Table 3 show that the variation in the internal resistance during the discharge process is not significant. Therefore, the term $R(t)$ can also be neglected, and the equation is reduced to:
$$f_2\bigl(U(t), I(t), C(t), t\bigr) = 0 \tag{11}$$
In practical battery management applications, the remaining capacity is the unknown quantity to be estimated. It is therefore useful to write the remaining capacity as an explicit function of the discharge voltage and the discharge current:
$$C(t) = g\bigl(U(t), I(t)\bigr) \tag{12}$$
After sampling with a constant sampling period $T$, the discrete form of the above relationship is given by:
$$C(k) = f\bigl(U(k), I(k), T\bigr) \tag{13}$$
Since the sampling period $T$ is fixed, it can be omitted from the input variables, and the final functional relationship that I used for the neural network model is:
$$C(k) = f\bigl(U(k), I(k)\bigr) \tag{14}$$
Thus, the BP neural network should be constructed with the discharge voltage and the discharge current as the input variables and the remaining capacity as the output variable.
3. Neural Network-Based Modeling of Traction Battery Remaining Capacity
After the theoretical analyses and the derivation of the input-output relationship for the traction battery, I proceeded to develop a prediction model using neural networks. The modeling of a traction battery can be viewed as a nonlinear system identification problem. The neural network approach is particularly advantageous because it does not require the exact physical laws of the electrochemical system. Instead, it learns the underlying mapping from a set of representative input-output examples. In my work, I have followed a systematic procedure to construct, train, and validate the neural network predictor.
3.1 Model Structure and Training Principle
The fundamental modeling framework for a dynamic nonlinear system can be represented by the nonlinear autoregressive moving average model (NARMAX). For a traction battery, the NARMAX model has the general form:
$$y(k) = f_m\bigl(y(k-1), \ldots, y(k-n_y), u(k-1), \ldots, u(k-n_u)\bigr) \tag{15}$$
where $y(k)$ is the output sequence and $u(k)$ is the input sequence. In the context of traction battery remaining capacity prediction, the output is the remaining capacity and the inputs are the operational variables, such as the discharge voltage and the discharge current. In the neural network implementation, the experimental data that I collected from the discharge tests were arranged into a set of input-output pairs. The first $n$ values were used as inputs and the $(n+1)$-th value was used as the corresponding target output. A typical arrangement of the data samples is illustrated in Table 4.
| Input vector | Target output value |
|---|---|
| y(1), y(2), …, y(i) | y(i+1) |
| y(2), y(3), …, y(i+1) | y(i+2) |
| y(3), y(4), …, y(i+2) | y(i+3) |
| … | … |
| y(k-1), y(k), …, y(k+i-2) | y(i+k-1) |
| y(k), y(k+1), …, y(k+i-1) | y(i+k) |
Table 4. Data arrangement for neural network training.
In the training procedure, the network weights and biases are adjusted according to the gradient descent principle so that the mean squared error (MSE) between the network prediction and the measured target is minimized. The performance function is defined as:
$$\mathrm{MSE} = \frac{1}{m p} \sum_{j=1}^{p} \sum_{r=1}^{m} \bigl(\hat{y}_{rj} – y_{rj}\bigr)^2 \tag{16}$$
where $m$ denotes the number of output nodes, $p$ is the number of training samples, $\hat{y}_{rj}$ is the expected output, and $y_{rj}$ is the actual network output. In my specific implementation, the network has one output node and I used a set of 30 training samples.
According to the Kolmogorov theorem, any continuous function can be represented by a three-layer feedforward neural network with sufficient hidden neurons. Therefore, in this thesis, I have adopted a single-hidden-layer BP neural network. The input layer contains two neurons because the discharge voltage and the discharge current are taken as the inputs. The output layer contains a single neuron because the remaining capacity is the only output. The number of hidden neurons was determined by experiments. Through trial simulations, I found that a small number, e.g., up to ten hidden neurons, leads to poor prediction accuracy. As the number of hidden neurons increases, the training accuracy improves, but the memory consumption becomes larger. Considering the trade-off between accuracy and computational load, I selected 11 hidden neurons for the final network configuration. The network structure is schematically represented in my thesis as a three-layer topology, namely the input layer, the hidden layer with sigmoid activation functions, and the linear output layer.
Before feeding the data into the network, I normalized the input and output samples so that they fall into the range suitable for the activation functions. After the training is completed, the output of the network is denormalized to obtain the actual predicted remaining capacity.
3.2 Prediction Results with the Basic BP Neural Network
Using the established BP neural network model, I first tested the model with the discharge data at 6 A, which had not been included in the training set. The validation results demonstrated that the model was capable of generalizing to unseen operating conditions. The difference between the measured values and the predicted values remained small across most of the discharge range, confirming the feasibility of the neural network approach.
After validating the model, I proceeded to predict the remaining capacity of the traction battery for a 9 A discharge using the BP neural network that had been trained on the other current levels. The predicted capacity values were compared with the actual measured capacity values. The comparison revealed a generally good agreement; however, there were still notable deviations during certain portions of the discharge process. The training process of the BP network consumed approximately 22.13 seconds on my computer, which is relatively long for real-time applications.
To further examine the quality of the BP network prediction, I selected several data points from a separate operating condition of the same traction battery and presented the prediction results in Table 5.
| Voltage (V) | Current (A) | Actual remaining capacity (Ah) | Predicted remaining capacity (Ah) | Relative error (%) |
|---|---|---|---|---|
| 3.434 | 8.9973 | 2.985 | 2.8380 | 4.91 |
| 3.284 | 8.9966 | 2.084 | 1.9550 | 4.32 |
| 3.255 | 8.9966 | 1.938 | 1.8300 | 3.61 |
| 3.202 | 8.9966 | 0.563 | 0.6530 | 3.09 |
Table 5. Validation of the BP neural network prediction for a LiFePO₄ traction battery.
From the results shown in Table 5, it is clear that the basic BP neural network has achieved acceptable accuracy for scenarios with moderate precision and real-time requirements. However, the relatively large prediction errors and the long training time motivated me to improve the prediction model by combining the BP algorithm with a genetic algorithm. This hybrid GA-BP approach is presented in the following chapter.
4. Improvement of the Remaining Capacity Prediction Model Using Genetic Algorithm
The BP algorithm has several well-known drawbacks, including its tendency to converge to local minima, its slow convergence speed, and its sensitivity to the initial weights and thresholds. In contrast, the genetic algorithm is a global optimization technique that mimics the process of natural selection and genetics. It performs a parallel search over the solution space and has a strong capability of finding near-optimal solutions without requiring gradient information. In my work, I integrated the genetic algorithm with the BP neural network in two main ways. On the one hand, I used the GA to optimize the initial weights and thresholds of the BP network, which can be regarded as an optimization of the network structure parameters. On the other hand, I used the GA to select the optimal training parameters of the BP network, including the maximum number of training epochs, the learning rate, and the training goal. The resulting hybrid algorithm substantially improves both the prediction accuracy and the computational efficiency.
4.1 Optimization of the Network Structure Parameters
The structural parameters of the neural network primarily refer to the connection weights and the node thresholds. The GA-BP algorithm first generates a random initial population, in which each individual represents a complete set of weights and thresholds for the BP network. The fitness of each individual is evaluated by computing the corresponding squared sum error of the network, and the genetic operators of selection, crossover, and mutation are employed to evolve the population. After a certain number of generations, the best individual is decoded and its values are used as the initial weights and thresholds for the subsequent BP training process.
The detailed steps of the GA-BP hybrid algorithm are summarized as follows:
(1) Initialize the population $P$. The population size is set to 30. Real-number encoding is used, and the initial weights $W_{ij}^{IH}$ and $W_{jo}^{HO}$ are randomly generated within a certain range.
(2) Calculate the fitness value of each individual. The fitness function is based on the inverse of the squared sum error:
$$f_i = \frac{1}{E(i)} \tag{17}$$
with
$$E(i) = \sum_{p} \sum_{k} \bigl(T_k – V_k\bigr)^2 \tag{18}$$
where $i$ is the chromosome index, $N$ is the number of chromosomes, $k$ is the output node, $p$ is the number of training samples, $T_k$ is the target output, and $V_k$ is the network output.
(3) Apply the selection operator according to the probability below:
$$P_{si} = \frac{f_i}{\sum_{i=1}^{N} f_i} \tag{19}$$
(4) Apply the crossover operation with a probability $P_c = 0.6$ to generate new offspring $G_i’$ and $G_{i+1}’$.
(5) Apply the mutation operation with a probability $P_m = 0.2$ to generate a new mutant $G_j’$.
(6) Insert the new individuals into the population and recalculate the fitness values. If the squared sum error reaches the predetermined threshold $\varepsilon_{GA}$, then proceed to the next step; otherwise, return to step (3).
(7) Use the optimized individual as the initial weights and thresholds of the BP network and train the network with the BP algorithm until the specified precision $\varepsilon_{BP}$ is reached.
The GA-BP algorithm flow can be summarized as a two-stage procedure. In the first stage, the genetic algorithm searches for a promising region in the weight space. In the second stage, the BP algorithm performs precise local optimization starting from the GA-provided initial values. This combination effectively avoids the problem of the BP network becoming trapped in a poor local minimum.
In my simulation, the GA searched for the optimal initial weights and thresholds using the training data collected from the discharge experiments. The fitness curve of the GA increased monotonically over the generations, and the training error decreased rapidly. The optimal weights obtained by the GA are denoted as $W_1$ (the weights between the input layer and the hidden layer) and $W_2$ (the weights between the hidden layer and the output layer). The corresponding thresholds are denoted as $B_1$ and $B_2$. These matrices are given in the experimental section of my thesis, and I have used them to initialize the BP network for the prediction of the traction battery remaining capacity.
The resulting prediction curve from the GA-BP model is in excellent agreement with the measured capacity curve. The difference between the predicted values and the measured values is much smaller than that obtained with the basic BP network. The average error dropped to a significantly lower level. At the same time, the training time required by the BP network after GA initialization was reduced dramatically. In a representative run, the training time decreased from 22.13 seconds for the plain BP network to approximately 2.73 seconds for the GA-BP network. These results demonstrate that the GA optimization of the initial weights and thresholds contributes to both accuracy and efficiency improvements.
4.2 Optimization of the Network Training Parameters
In addition to the network structural parameters, the training parameters of the BP network, including the training goal $g$, the learning rate $\eta$, and the maximum number of epochs $N_{epoch}$, also play a crucial role in determining the convergence speed and the accuracy of the prediction model. In the second part of my improvement strategy, I employed the GA to optimize these training parameters. Since the three parameters jointly influence the final training performance, I designed a three-level nested optimization loop. The innermost loop was dedicated to optimizing the training goal; the second loop was dedicated to optimizing the learning rate; and the outermost loop was dedicated to optimizing the maximum number of epochs. The total training time of the BP network was used as the performance indicator to be minimized.
To initiate the optimization, I first fixed the learning rate at $\eta = 0.015$ and the maximum number of epochs at $N_{epoch} = 2000$, and I adopted an empirical value for the training goal. In the first level of optimization, the GA searched for the optimal training goal in the range of 0.01–0.08. The optimal value was found to be $g = 0.02$, which produced a training time of 22.95 seconds. At the second level of optimization, I introduced the learning rate as a variable in the range of 0.001–0.016 while maintaining the maximum epochs at 2000. The combined optimization of the training goal and the learning rate yielded $\eta = 0.013$ and $g = 0.07$ with a training time of 9.90 seconds. At the third and outermost level, the maximum number of epochs was also optimized in the range of 500–2050. The GA obtained $N_{epoch} = 1550$, $\eta = 0.012$, and $g = 0.08$, and the corresponding BP network training time was only 2.75 seconds.
Table 6 summarizes the parameter settings of the GA for each optimization level.
| Optimization level | Population size | Search range | Termination generation | Code length | Crossover probability | Mutation probability |
|---|---|---|---|---|---|---|
| Goal $g$ | 4 | 0.01~0.08 | 10 | 3 | 0.6 | 0.01 |
| Learning rate $\eta$ | 8 | 0.001~0.016 | 10 | 4 | 0.6 | 0.01 |
| Maximum epochs | 32 | 500~2050 | 10 | 5 | 0.6 | 0.01 |
Table 6. Parameter settings for the three-level GA optimization.
Table 7 lists the optimized training parameters that I obtained at each level.
| Optimization case | Optimal $g$ | Optimal $\eta$ | Optimal $N_{epoch}$ | BP training time (s) |
|---|---|---|---|---|
| Baseline (no GA) | 0.01 | 0.015 | 2000 | 22.13 |
| GA for $g$ | 0.02 | 0.015 | 2000 | 22.95 |
| GA for $g$ and $\eta$ | 0.07 | 0.013 | 2000 | 9.90 |
| GA for $g$, $\eta$, and $N_{epoch}$ | 0.08 | 0.012 | 1550 | 2.75 |
Table 7. Training time comparisons with different levels of GA optimization.
4.3 Comprehensive Optimization and Prediction Results
Finally, I combined the structural parameter optimization and the training parameter optimization into a comprehensive prediction model. In this model, the BP network was initialized with the GA-optimized weights and thresholds. At the same time, the GA-optimized training parameters, namely $g = 0.08$, $N_{epoch} = 1750$, and $\eta = 0.015$, were used for the BP training phase. The overall accuracy of the resulting model was evaluated by predicting the remaining capacity of the traction battery under the same test conditions as before.
The comprehensive model produced a training time of about 6.38 seconds and achieved highly accurate predictions of the remaining capacity during most of the discharge process. The average error of the predictions was around 0.3 Ah, and the results met the accuracy requirement of the traction battery management system. The training process error curve of the optimized GA-BP network descends smoothly and converges to the goal value without visible oscillation.
To provide a clear comparison among the different prediction methods, I compiled the prediction results in Table 8 for several selected voltage points at a discharge current of 3 A.
| Discharge current (A) | Discharge voltage (V) | Measured remaining capacity (Ah) | BP predicted capacity (Ah) | GA-BP predicted capacity (Ah) | Method | Relative error (%) |
|---|---|---|---|---|---|---|
| 3 | 3.2267 | 2.9550 | 2.8110 | 2.8800 | BP / GA-BP | 4.81 / 2.53 |
| 3 | 3.2132 | 2.6552 | 2.5202 | 2.7242 | BP / GA-BP | 4.52 / 2.34 |
| 3 | 3.2015 | 2.3055 | 2.3355 | 2.2175 | BP / GA-BP | 1.32 / 2.93 |
| 3 | 3.1917 | 2.0556 | 2.1636 | 1.9906 | BP / GA-BP | 3.61 / 2.16 |
| 3 | 3.1876 | 1.9556 | 1.8146 | 2.0006 | BP / GA-BP | 4.69 / 1.51 |
| 3 | 3.1757 | 1.6558 | 1.5958 | 1.5958 | BP / GA-BP | 2.01 / 2.03 |
| 3 | 3.1696 | 1.5059 | 1.6259 | 1.5899 | BP / GA-BP | 4.05 / 2.83 |
Table 8. Comparison of prediction results between the BP algorithm and the GA-BP algorithm.
The results in Table 8 demonstrate that the GA-BP algorithm achieved the best prediction accuracy in most test cases, with relative errors below 3%. The plain BP algorithm, in contrast, produced errors of up to nearly 5%. Thus, the improvement obtained from the GA is twofold: the prediction accuracy is enhanced, and the training speed is greatly accelerated.
5. Experimental Verification System for the Prediction Model
In order to demonstrate the practical applicability of the proposed GA-BP prediction model, I developed an experimental verification system. The system consists of the trained GA-BP neural network model, the battery testing hardware, and the associated measurement and data acquisition software. The purpose of this system is to predict the remaining capacity for operating points that have not been directly measured in the laboratory. By comparing the predicted values with the subsequently measured data, the reliability and robustness of the prediction model can be evaluated.
5.1 Construction of the Prediction Model in the Test System
In my earlier discharge experiments, I collected data for fifteen different discharge current levels, ranging from 0.5 A to 60 A, and used these data to train the prediction model. The model input variables were set as the discharge voltage and the discharge current, and the output variable was set as the remaining capacity of the traction battery. The optimal network topology and parameters derived in the previous chapter were used in this verification system. More specifically, I adopted 11 hidden neurons, the GA-optimized initial weights $W_1$ and $W_2$, the GA-optimized thresholds $B_1$ and $B_2$, and the optimized training parameters $g=0.08$, $\eta=0.012$, and $N_{epoch}=1550$. The resulting model performs fast and accurate predictions, as confirmed by simulation results.
Since the recording interval of the battery test equipment is normally fixed, a number of working states may not be sampled directly. For example, if the sampling interval is 60 seconds, the voltage values measured at the intermediate time instants are unknown. The prediction model can be used to interpolate the remaining capacity for these unmeasured points. Similarly, the model can be used to extrapolate to other discharge currents that have not been experimentally tested, provided that the operating range is within the domain covered by the training data.
5.2 Accuracy Validation of the Prediction Results
I first used the trained model to perform a prediction test on the discharge data at a constant current of 3 A. I selected ten voltage values in the range from 3.3 V to 3.1 V, with an increment of 0.02 V, and used them as the inputs to the prediction model. The predicted remaining capacities are listed in Table 9.
| Discharge current (A) | Discharge voltage (V) | Predicted remaining capacity (Ah) |
|---|---|---|
| 3 | 3.30 | 2.9735 |
| 3 | 3.28 | 2.9682 |
| 3 | 3.26 | 2.8653 |
| 3 | 3.24 | 2.7192 |
| 3 | 3.22 | 2.5634 |
| 3 | 3.20 | 2.3436 |
| 3 | 3.18 | 1.9438 |
| 3 | 3.16 | 1.7466 |
| 3 | 3.14 | 1.5968 |
| 3 | 3.12 | 1.3569 |
| 3 | 3.10 | 1.2442 |
Table 9. Predicted remaining capacities for an untested sequence of voltage values at 3 A.
Then I examined the generalization capability of the model by making predictions for a discharge current of 2.5 A, which had not been used in the training data. Ten voltage values were again selected in the range from 3.4 V to 3.2 V. The corresponding predicted remaining capacities are listed in Table 10.
| Discharge current (A) | Discharge voltage (V) | Predicted remaining capacity (Ah) |
|---|---|---|
| 2.5 | 3.40 | 2.9846 |
| 2.5 | 3.38 | 2.9666 |
| 2.5 | 3.36 | 2.8935 |
| 2.5 | 3.34 | 2.7752 |
| 2.5 | 3.32 | 2.6024 |
| 2.5 | 3.30 | 2.4368 |
| 2.5 | 3.28 | 2.0456 |
| 2.5 | 3.26 | 1.8985 |
| 2.5 | 3.24 | 1.7123 |
| 2.5 | 3.22 | 1.4650 |
| 2.5 | 3.20 | 1.3368 |
Table 10. Predicted remaining capacities for an untested discharge current of 2.5 A.
To confirm the accuracy of these predictions, I adjusted the sampling interval of the battery test equipment to 10 seconds and conducted actual discharge measurements at 3 A and 2.5 A currents. The measured values were then compared with the corresponding predicted values. The comparison results are presented in Table 11 and Table 12.
| Discharge current (A) | Discharge voltage (V) | Predicted capacity (Ah) | Measured capacity (Ah) | Relative error (%) |
|---|---|---|---|---|
| 3 | 3.40 | 2.9735 | 2.9315 | 1.39 |
| 3 | 3.38 | 2.9682 | 2.9552 | 0.43 |
| 3 | 3.36 | 2.8653 | 2.8893 | 0.80 |
| 3 | 3.34 | 2.7192 | 2.6862 | 1.09 |
| 3 | 3.32 | 2.5634 | 2.5114 | 1.73 |
| 3 | 3.30 | 2.3436 | 2.3836 | 1.33 |
| 3 | 3.28 | 1.9438 | 1.9098 | 1.13 |
| 3 | 3.26 | 1.7466 | 1.7676 | 0.69 |
| 3 | 3.24 | 1.5968 | 1.5648 | 1.07 |
| 3 | 3.22 | 1.3569 | 1.3309 | 0.87 |
| 3 | 3.20 | 1.2442 | 1.2142 | 1.00 |
Table 11. Comparison of predicted and measured remaining capacities for the 3 A discharge.
| Discharge current (A) | Discharge voltage (V) | Predicted capacity (Ah) | Measured capacity (Ah) | Relative error (%) |
|---|---|---|---|---|
| 2.5 | 3.40 | 2.9846 | 2.8966 | 2.93 |
| 2.5 | 3.38 | 2.9666 | 2.8906 | 2.53 |
| 2.5 | 3.36 | 2.8935 | 2.9545 | 2.03 |
| 2.5 | 3.34 | 2.7752 | 2.7062 | 2.29 |
| 2.5 | 3.32 | 2.6024 | 2.6544 | 1.73 |
| 2.5 | 3.30 | 2.4368 | 2.3708 | 2.21 |
| 2.5 | 3.28 | 2.0456 | 1.9686 | 2.56 |
| 2.5 | 3.26 | 1.8985 | 1.9765 | 2.62 |
| 2.5 | 3.24 | 1.7123 | 1.6453 | 2.23 |
| 2.5 | 3.22 | 1.4650 | 1.4130 | 1.73 |
| 2.5 | 3.20 | 1.3368 | 1.2888 | 1.60 |
Table 12. Comparison of predicted and measured remaining capacities for the 2.5 A discharge.
The comparison results show that the relative errors for the 3 A discharge case are less than 2%, while those for the 2.5 A discharge case are within 3%. Both results satisfy the accepted engineering tolerance. The prediction model is therefore capable of accurately predicting the traction battery remaining capacity for untested operating points.
Conclusions
The work presented in this thesis focuses on the prediction of the remaining capacity of traction batteries used in electric vehicles. I have analyzed the working principle and discharge performance of LiFePO₄ traction batteries, established a BP neural network prediction model, and improved the model with a genetic algorithm. Several conclusions can be drawn from the simulation and experimental results.
(1) The traction battery is a highly nonlinear and uncertain system. The conventional deterministic models and traditional estimation methods have limitations in acquiring the remaining capacity under real operating conditions. Through a theoretical derivation, I formulated the remaining capacity as a function of the discharge voltage and the discharge current, with the sampling period treated as a known parameter. This formulation provides the basis for the neural network-based modeling approach.
(2) The BP neural network is capable of approximating the nonlinear mapping between the measurable operating variables and the remaining capacity of the traction battery. The prediction results with the basic BP network for previously unmeasured operating points were acceptable, but the training time was long and the prediction accuracy was relatively limited.
(3) By combining the genetic algorithm with the BP neural network, I successfully optimized the initial weights and thresholds of the network. The GA-BP algorithm significantly improved the prediction accuracy and accelerated the convergence process. The training time was reduced from about 22 seconds to less than 3 seconds.
(4) Using the three-level GA optimization of the network training parameters, namely the training goal, the learning rate, and the maximum number of epochs, I further improved the computational efficiency of the prediction model without sacrificing the prediction precision. The comprehensive GA-BP model proved to be a fast and accurate tool for traction battery remaining capacity prediction.
(5) The experimental verification system was built and used to predict the remaining capacity of a LiFePO₄ traction battery for non-sampled data points. The measured data were compared with the predicted values, and the errors were below 3%, which is acceptable in engineering practice. These results verify the reliability and accuracy of the proposed prediction system for the traction battery remaining capacity.
It should be noted that the research has some limitations. The present model does not explicitly consider the effects of ambient temperature, the self-recovery effect, and the aging degree of the traction battery. In addition, the prediction model was developed and verified only for a specific type of LiFePO₄ traction battery. Future research may extend the model to other battery chemistries and incorporate temperature and health factors as additional inputs to the neural network. Nevertheless, this thesis demonstrates that the GA-BP hybrid algorithm is an effective and efficient tool for online traction battery remaining capacity prediction.