2023 年 91 巻 3 号 p. 037001
The accuracy and connection mode of the equivalent model of retired power battery cells will affect the size of the combined capacity. To this end, this article analyzes series-parallel combination capacity of retired power batteries. First, because an accurate single equivalent circuit model of retired power batteries is necessary for the capacity analysis of the combination, the forgetting factor recursive least squares (FFRLS) algorithm is used to identify the parameters of the model, and the second-order RC equivalent circuit is selected as the single equivalent circuit model of retired power battery. Then the equivalent circuit model is used as the basis, respectively constructing first-parallel-before-series and first-series-before-parallel power battery combination, and fully and unequally arranged the retired power batteries under the two assemblies. Finally, the capacity and distribution characteristics of the two combinations under different arrangements are verified by simulation. The results show that no matter how the arrangement is changed, the maximum capacity of the first-parallel-before-series retired power battery combination is always better than the first-series-before-parallel combination, and the distribution position of the single retired battery in both combinations under the maximum combination capacity is obtained.
After a long-term use on the electric vehicle, the performance of the power battery will gradually decrease. When the power battery cannot meet the use requirements of the electric vehicle, it will be retired from the electric vehicle.1,2 If these retired power batteries are not properly disposed, they will not only waste a lot of valuable resources, but also cause serious pollution to the environment.3,4 In this context, the clean and safe disposal of retired power batteries is of great practical significance.
Among the retired power batteries, the vast majority of them also have a high residual capacity (70 %–80 % of the rated capacity). After detection, screening classification, and grouping, they can be used in low-speed electric vehicles, grid energy storage, and communication backup base stations with milder operating conditions to realize the echelon utilization of retired power batteries.5–7 Echelon utilization can give full play to its performance, improve resource utilization, and reduce environmental pollution.8,9
After the screening and classification of retired power batteries, their consistency has been greatly improved.10,11 However, due to the difference in characteristic parameters and arrangement among the cells, the specific energy of the battery system after grouping is much smaller than that of the battery cells, which will reduce the utilization rate of retired power batteries.12 Therefore, under the premise of ensuring safety, it is necessary to analyze the influence of the arrangement of retired power batteries on the combined capacity.
Song Ci and coworkers summarized the series-parallel topology design of battery systems, studied the design scheme of the balance structure of battery systems, and overviewed the current technical status and existing problems of battery system topology designs.13 Wang Shuai and coworkers synthesized six single cell battery packs into two topologies, series module and parallel module, through charge-discharge orthogonal experiments, and obtained the conclusion that the capacity of parallel module is more influenced by the internal resistance of single cell and the capacity of series module is more influenced by the capacity of single cell.14 Baronti F and coworkers used Gaussian distribution to generate cell capacities with different standard deviations, then grouped them, and selected some battery cell arrangements for comparison, and obtained the first-parallel-before-series power battery pack could obtain a larger average grouping capacity.15 Jiang Jun has realized the modeling simulation of series-parallel battery pack for typical battery pack series-parallel connection by equivalently transforming the complex connection and combining the series battery pack simulation calculation based on mathematical statistics and parallel branch unbalance current calculation.16 The above literatures improve the capacity and voltage of the battery system through series-parallel connection, but most of the series-parallel structure topologies are fixed topologies, and the influence of factors such as the difference in characteristic parameters between battery cells and the arrangement method on the group capacity of the battery system is not comprehensively considered.
Therefore, this article develops a study on the effect of the difference between retired power battery singles and the arrangement on the capacity of battery combinations. In order to improve the accuracy of the equivalent model of the retired power battery cell and reduce its complexity, this article uses the FFRLS algorithm to identify the parameters of the equivalent circuit model of the retired power battery, and selects the second-order RC equivalent circuit as the cell equivalent model. Based on the equivalent circuit model of the single unit, the power battery assemblies of first-parallel-before-series and first-series-before-parallel retired power battery assemblies are built through simulation, and the capacity and distribution characteristics of the two assemblies under different cell arrangements are analyzed, the arrangement and distribution positions of the retired power battery cells corresponding to the two combinations at the maximum capacity are obtained.
Common battery cell equivalent circuit models include internal resistance model, Thevenin model, partnership for a new generation of vehicles (PNGV) equivalent circuit model, and general nonlinear (GNL) equivalent circuit model.17
Among them, the Thevenin second-order RC equivalent circuit model can well reflect the static and dynamic characteristics of the power battery, and the estimation error is not large. In addition, the model has a few parameters to be identified, and the complexity is moderate, which can meet the needs of practical applications. Therefore, considering the complexity and accuracy of the model, the second-order RC equivalent circuit is selected as the equivalent circuit model of the retired power battery. The second-order RC equivalent circuit model is shown in Fig. 1.
Second-order RC equivalent circuit model.
In Fig. 1, Uoc is the open circuit voltage. R0 is the internal resistance. R1 and C1 are used to simulate the electrochemical polarization characteristics of the battery, respectively. R2 and C2 are used to simulate the concentration polarization characteristics of the battery, respectively. The expression of terminal voltage U can be obtained from Kirchhoff’s Voltage Law as follows:
| \begin{equation} U = U_{\text{oc}}(t) - I(t)R_{0} - U_{1} - U_{2} \end{equation} | (1) |
| \begin{equation} \left\{ \begin{array}{l} \dfrac{U_{1}}{R_{1}} = I(t) - C_{1}\dfrac{\text{d}U_{1}}{\text{d}t}\\ \dfrac{U_{2}}{R_{2}} = I(t) - C_{2}\dfrac{\text{d}U_{2}}{\text{d}t} \end{array} \right. \end{equation} | (2) |
Common battery parameter identification methods include Least Squares method, genetic algorithm and maximum likelihood method and so on. Among them, the Least Squares method is simple and practical, but in practical applications, when the amount of data becomes larger, the amount of calculation will become particularly complex. Furthermore, the increase in observation data will also lead to recalculation of the data vector, which will increase algorithm complexity.18 Therefore, this article adopts the Forgetting Factor Recursive Least Square algorithm to identify the parameters of the second-order RC equivalent circuit model. By introducing the forgetting factor, the FFRLS algorithm changes the weight of the old and new data to prevent the old data from occupying a large proportion in the recursive process, which can alleviate the data saturation problem of the Recursive Least Squares algorithm, and can also improve the response speed and parameter identification accuracy of the algorithm.19,20
The zero-state response circuit of the second-order RC equivalent circuit model in the complex frequency domain is shown in Fig. 2.
Zero-state response circuit in complex frequency domain.
The state equation of the second-order RC equivalent circuit model is:
| \begin{equation} E(s) = -I(s)\left(R_{0} + \frac{R_{1}}{1 + R_{1}C_{1}s} + \frac{R_{2}}{1 + R_{2}C_{2}s}\right) \end{equation} | (3) |
Where, E(s) is the difference between the open circuit voltage Uoc(s) and the terminal voltage U(s), the transfer function of the model is:
| \begin{align} G(s) & = E(s)/I(s)\\ & = -\left(R_{0} + \frac{R_{1}}{1 + R_{1}C_{1}s} + \frac{R_{2}}{1 + R_{2}C_{2}s}\right) \end{align} | (4) |
Perform bilinear transformation on the Eq. 4, such that T is the sampling period yields, you can get:
| \begin{equation} s = 2(1 - z^{-1})/T(1 + z^{-1}) \end{equation} | (5) |
| \begin{equation} G(z^{-1}) = \frac{k_{3} + k_{4}z^{-1} + k_{5}z^{-2}}{1 - k_{1}z^{-1} - k_{2}z^{-2}} \end{equation} | (6) |
Let a1 = R0, a2 = τ1τ2, a3 = τ1 + τ2, a4 = R0 + R1 + R2, a5 = R0(τ1 + τ2) + R1τ2 + R2τ1, where τ1, τ2 are time constants. The undetermined coefficients k1, k2, k3, k4, k5 of the difference equation can be obtained as:
| \begin{equation} \left\{ \begin{array}{l} k_{1} = \dfrac{-2T^{2} + 8a_{2}}{T^{2} + 2a_{3}T + 4a_{2}}\\ k_{2} = \dfrac{4a_{3}T}{T^{2} + 2a_{3}T + 4a_{2}} - 1\\ k_{3} = -\dfrac{a_{4}T^{2} + 2a_{5}T + 4a_{1}a_{2}}{T^{2} + 2a_{3}T + 4a_{2}}\\ k_{4} = \dfrac{-2a_{4}T^{2} + 8a_{1}a_{2}}{T^{2} + 2a_{3}T + 4a_{2}}\\ k_{5} = -\dfrac{a_{4}T^{2} - 2a_{5}T + 4a_{1}a_{2}}{T^{2} + 2a_{3}T + 4a_{2}} \end{array} \right. \end{equation} | (7) |
The discretized recurrence equation of the model obtained by Eq. 6:
| \begin{align} E(k) &= k_{1}E(k - 1) + k_{2}E(k - 2) + k_{3}I(k) \notag\\ &\quad+ k_{4}I(k - 1) + k_{5}I(k - 2) \end{align} | (8) |
In the Eq. 8, k is a discrete random variable, k = (0, 1, 2, …, n). Let the parameter vector θ(k) = [k1, k2, k3, k4, k5]T, φ(k) = [E(k − 1), E(k − 2), I(k), I(k − 1), I(k − 2)]T, y(k) = E(k). Then the least squares matrix form of the equation can be obtained as follows:
| \begin{equation} y(k) = \theta^{\mathbf{T}}(k)\varphi (k) \end{equation} | (9) |
To ensure τ1 and τ2 are unique, let $\tau_{1} = a_{3} + \sqrt{a_{3}{}^{2} - 4a_{2}} /2$, $\tau_{2} = a_{3} - \sqrt{a_{3}{}^{2} - 4a_{2}} /2$. The parameters to be identified, R0, R1, R2, C1, C2 can be obtained as:
| \begin{equation} \left\{ \begin{array}{l} R_{0} = a_{1}\\ R_{1} = [\tau_{1}(a_{4} - a_{1}) + a_{1}a_{3} - a_{5}]/(\tau_{1} - \tau_{2})\\ R_{2} = a_{4} - a_{1} - R_{1}\\ C_{1} = \tau_{1}/R_{1}\\ C_{2} = \tau_{2}/R_{2} \end{array} \right. \end{equation} | (10) |
The above is the implementation process of the FFRLS algorithm. The parameter identification process of the second-order RC equivalent circuit model based on the FFRLS algorithm is as follows:
Step 1: Convert the electrical expression of the equivalent circuit model to the least squares form shown in the equation.
Step 2: Calculate the gain matrix K(k), and the equation for its calculation is shown as:
| \begin{equation} K(k) = \frac{P(k - 1)\varphi (k)}{\lambda + \varphi^{\mathbf{T}}(k)P(k - 1)\varphi (k)} \end{equation} | (11) |
Where, P(k) is the covariance matrix. λ is the forgetting factor, which can assign weights to the parameter identification data and reduce the proportion of old data in the recursion process. Compared with the traditional Recursive Least Squares algorithm, the FFRLS algorithm requires less computation and has a faster response speed. The larger the λ, the slower the forgetting speed. If the value of λ is 1, FFRLS algorithm will degenerate into the Least Squares algorithm. The smaller the λ, the stronger the tracking ability of the FFRLS algorithm. The value of λ is usually taken between 0.95 and 1. Taking into account the forgetting speed and tracking ability of the algorithm, here we take λ as 0.99.
Step 3: Generally, through take a smaller real number matrix to initialize θ(0), and then update the parameter to be estimated θ(k).
| \begin{equation} \left\{ \begin{array}{l} e(k) = U(k) - U_{oc}(k) - \varphi^{\mathbf{T}}(k)\theta (k - 1)\\ \theta (k) = \theta(k - 1)K(k)e(k) \end{array} \right. \end{equation} | (12) |
Step 4: By recurring the parameters at moment k + 1 from step 2, and cycle until the FFRLS algorithm is recursively completed, so as to realize the parameter identification of the equivalent circuit model.
Based on the second-order RC equivalent circuit model, two retired power battery combinations, namely, first-series-before-parallel and first-parallel-before-series combinations, are built respectively. The schematic diagrams of the two combinations are shown in Fig. 3.
(a) First-series-before-parallel retired power batteries combination. (b) First-parallel-before-series retired power batteries combination.
In Fig. 3a, m single retired power cells are connected in series to form a series cell unit (SCU), and then n SCUs are connected in parallel to form a first-series-before-parallel retired power batteries combination. In Fig. 3b, n single retired power cells are connected in parallel to form a parallel cell unit (PCU), and then m PCUs are connected in series to form a first-parallel-before-series retired power batteries combination.
The initial capacity of the retired power battery cell is set as Cij. Under the actual application conditions, the actual amount of electricity released by the battery, that is, the integral of the discharge current I(t) and the discharge time t, is calculated as follows.21
| \begin{equation} C_{ij} = \int_{0}^{T}I(t)dt \end{equation} | (13) |
The main factors affecting the capacity of retired power battery combinations are initial capacity Cij, initial state of charge (SOC), internal resistance R, open circuit voltage Uoc, coulomb efficiency, internal resistance, change rate, and capacity decay rate and so on. Among them, coulomb efficiency, capacity decay rate and internal resistance change rate mainly affect the capacity decay characteristics of the battery after a long period of use.22 This article focuses on the analysis of the capacity of the retired power battery series-parallel combination at the beginning, so the influence of coulomb efficiency, internal resistance change rate, and capacity decay rate is not considered for the time being. In addition, there is a certain functional relationship between the open circuit voltage Uoc and SOC. Therefore, the main parameters selected to affect the capacity of the combination are the initial capacity, the initial SOC and the internal resistance R.
For the above two retired power battery combinations, let the internal resistance be Rij, and the initial state of charge be SOCij, where i is the serial number of the single battery, i $ \in $ [1, m]. j is the parallel number of the single battery, j $ \in $ [1, n]. From this, the calculation equation for the capacity of the two combinations can be obtained as:
| \begin{equation} C_{1} = \sum_{j = 1}^{n}\{\min (SOC_{ij}C_{ij}) + \min((1 - SOC_{kj})C_{kj})\} \end{equation} | (14) |
| \begin{equation} C_{2} = \min \left\{\sum_{j = 1}^{n}(SOC_{ij}C_{ij})\right\} + \min \left\{\sum_{k = 1}^{n}(1 - SOC_{ik})C_{ik}\right\} \end{equation} | (15) |
In the above equation, C1 is the capacity of the first-series-before-parallel retired power battery combination, C2 is the capacity of the first-parallel-before-series retired power battery combination.
3.2 Influence of monomer arrangement on the capacity of combinationAfter the screening and classification of retired power batteries, the consistency between cells has been greatly improved, but there are still differences. These differences and the arrangement of cells have an impact on the capacity of the series-parallel combination of retired power batteries. In general, there are two ways to arrange single retired power batteries, which are full arrangement and unequal arrangement.
(1) Full arrangement, the retired power batteries are arranged according to the arrangement and combination. For n battery cells, under the combination of first-parallel-before-series or first-series-before-parallel combination, the full arrangement is n!.
(2) Unequal arrangement, according to the full arrangement, theoretically there are n! full arrangements, but some arrangements are essentially equal, and it is only necessary to analyze the arrangements that are unequal. This arrangement is called an unequal arrangement. From the perspective of permutation and combination, it can be concluded that the unequal permutations of the first-series-before-parallel and first-parallel-before-series retired power battery combinations are as follows:
| \begin{equation} N_{1} = [C_{m \times n}^{n} \times C_{m \times n - 1}^{n} \times \cdots \times C_{n}^{n}]/m! \end{equation} | (16) |
| \begin{equation} N_{2} = [C_{\text{m} \times n}^{m} \times C_{m \times n - m}^{m} \times \cdots \times C_{m}^{m}]/n! \end{equation} | (17) |
Where, N1 and N2 are respectively the number of unequal permutations of first-series-before-parallel and first-parallel-before-series retired power battery combinations.
The parameter identification of single retired power battery requires dynamic acquisition of its voltage and current. The voltage and current data of the dynamic stress test (DST) experiment is used as the input data for parameter identification.23 Taking the collected current and voltage as the input data for parameter identification, the FFRLS algorithm is used to identify the parameters of the second-order RC equivalent circuit model of the retired power battery. Set the time of parameter identification as t, the process of changing the coefficients k of the difference equation is shown in Fig. 4.
Change process of coefficient k of difference equation.
It can be seen from Fig. 4 that k1 and k2 fluctuate greatly in the early stage, while k3, k4, and k5 do not fluctuate much. As time increases, they all tend to be stable in the end.
Entering the DST experimental data, the parameter identification results of the second-order RC equivalent circuit model of the retired power battery can be obtained, as shown in Fig. 5.
Parameter identification results. (a) Identification result of internal resistance R0, (b) Identification result of electrochemical polarization internal resistance R1, (c) Identification result of electrochemical polarized capacitor C1, (d) Identification result of dense polarization internal resistance R2, (e) Identification result of dense polarization capacitor C2.
After obtaining the parameter identification results, input the experimental current data to estimate the voltage, and compare it with the actual voltage for error analysis. The real value of the U is compared with the estimated value as shown in Fig. 6, and the error of U is shown in Fig. 7.
Comparison between the actual value and the estimated value of U.
The error of U.
It can be seen from the above Fig. 6 that the actual value of the terminal voltage is basically consistent with the estimated value, indicating that the identification effect of the second-order RC equivalent circuit model of the retired power battery is better. It can be seen from the above Fig. 7 that the terminal voltage error fluctuates around 10 mv, the error is very small, and the accuracy of parameter identification is higher. Therefore, the selected model can be further analyzed for combinatorial capacity.
In order to expand the consistency difference between retired power battery cells, based on the second-order RC equivalent circuit model, 8 retired power batteries with a stepped distribution of available capacity were taken as the research objects. Their available capacity distribution is shown in Fig. 8.
Distribution of available capacity of retired power batteries.
The above retired power battery cells with a stepped distribution of available capacity are respectively formed into first-parallel-before-series and first-series-before-parallel retired power battery combinations. After combining them in full arrangement, both combinations have 8! = 40320. The capacity and proportion of the two retired power battery combinations under the full arrangement and combination can be obtained as shown in Fig. 9.
The capacity and proportion of the two combinations.
Among them, the proportion refers to the ratio of the total number of arrangements under the capacity of generating the combination to the sum of all arrangements. As can be seen from Fig. 9, for the first-series-before-parallel power battery combination, there are 4 kinds of combination capacities arranged according to the full arrangement. For the first-series-before-parallel power battery combination, there are 13 combination capacities. The proportion of the capacity of the two combinations is shown in Table 1.
| Type | First-parallel before-series (%) |
First-series before- parallel (%) |
Type | First-parallel before-series (%) |
First-series before- parallel (%) |
|---|---|---|---|---|---|
| #1 | 13.3 | 56.1 | #8 | 4.7 | 0 |
| #2 | 13.3 | 27.6 | #9 | 2.8 | 0 |
| #3 | 13.3 | 0 | #10 | 2.8 | 0 |
| #4 | 10.4 | 10.4 | #11 | 1.9 | 0 |
| #5 | 10.4 | 0 | #12 | 0.5 | 0 |
| #6 | 10.4 | 0 | #13 | 0.5 | 0 |
| #7 | 4.7 | 1.9 |
From the analysis and calculation of the above Fig. 9, it can be seen that the capacity distribution of first-parallel-before-series retired power battery combination is more dispersed than that first-series-before-parallel retired power battery combination. The maximum capacity of first-series-before-parallel retired power battery combination is 3.8329 Ah, and the maximum capacity of first-parallel-before-series retired power battery combination is 3.9326 Ah, the ratio of the capacity of the first-parallel-before-series retired power battery assembly is greater than the maximum capacity of the first-series-before-parallel assembly is 13.2 %. At the same time, it can also be obtained that the average capacity of the first-parallel-before-series retired power battery assembly is 3.5282 Ah, and the average capacity of the first-series-before-parallel retired power battery assembly is 3.5021 Ah. Therefore, comprehensive analysis can be obtained, the capacity performance of the first-parallel-before-series retired battery combination is better in the full arrangement and combination mode.
From Eqs. 16 and 17, it can be obtained that there are 105 unequal arrangements of the first-parallel-before-series retired power battery combination, and there are 35 arrangements in the first-series-before-parallel retired power battery combination. Figure 10 is a schematic diagram of the capacity distribution characteristics of the retired power battery assembly under the unequal arrangement.
Combination capacity under unequal arrangement.
It is not difficult to find from Fig. 10 that no matter how the arrangement changes, the capacity of the first-parallel-before-series retired power battery combination is always greater than that of the first-series-before-parallel retired power battery combination. Both combinations are in the first arrangement, and the group capacity reaches the maximum value. Figure 11 is a schematic diagram of the position distribution of retired power battery cells corresponding to two combinations in the first arrangement.
(a) First-parallel-before-series retired power battery combination. (b) First-series-before-parallel retired power battery combination.
Analysis of Fig. 11 shows that no matter how the retired power battery cells are distributed, there is always an arrangement that maximizes the capacity of the retired power battery assembly. For the first-parallel-before-series retired power battery combination, when the capacity of each parallel unit is close to the average value of all parallel units, the maximum capacity of the assembly is reached.
For the first-series-before-parallel retired power battery combination, the single retired power battery shall be arranged in ascending order from small to large, and then divided into two, with each half as a series unit. Finally, the maximum capacity of the combination can be obtained by paralleling them. Among them, the battery cell with the smallest capacity in the second series unit is always larger than the battery cell with the largest capacity in the first module.
In this article, the Forgetting Factor Recursive Least Square (FFRLS) algorithm is used to identify the parameters of the second-order RC equivalent circuit model for retired power batteries, and an equivalent model with small parameter error and high accuracy is obtained. Based on this, the first-parallel-before-series and first-series-before-parallel retired power battery combination are constructed respectively. Then, the size and distribution characteristics of the capacity of the two combinations are analyzed under the full and unequal arrangements, and it is obtained that the capacity of the first-parallel-before-series retired power battery combination was better than that of the first-series-before-parallel retired power battery combination in terms of distribution law, maximum value and average value, and the optimal single arrangement of the first-parallel-before-series retired power battery combination was obtained. Through the analysis of the capacity of retired power battery assembly, compared with the traditional first-series-before-parallel power battery assembly, the capacity utilization rate of first-parallel-before-series retired power battery assembly increased by 2.6 %, which improved the single utilization rate of retired power battery, and laid a certain foundation for the step utilization of the retired power battery. The capacity performance of retired power battery assembly is the basic performance characteristic after its combination and is also an important reference basis affecting the selection of combination mode, but it is not the only basis. Therefore, it is necessary to study the related performance advantages and disadvantages of the two series-parallel combinations in the future from many aspects, such as the remaining life, failure rate and state estimation difficulty of series-parallel retired power battery combinations.
Xing Gui Wang: Conceptualization (Equal), Formal analysis (Lead), Supervision (Lead)
Jie Wen Liu: Data curation (Lead), Software (Lead), Writing – review & editing (Lead)
Hai Liang Wang: Formal analysis (Equal), Investigation (Lead), Project administration (Equal)
Ying Jie Ding: Resources (Equal), Visualization (Lead)
Yong Ji Guo: Validation (Equal)
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.
State Grid Gansu Province Electric Power Company Science and Technology Project Foundation of China: W21FZ2730244
X. G. Wang: Xing Gui WANG (1963–), male, from Tianshui, Gansu Province, Professor, his main research directions are renewable energy power generation system and control, power electronics and electric drive, etc.
J. W. Liu: Jie Wen LIU (1997–), male, from Wuwei, Gansu Province, Master’s Degree, his main research direction is distributed energy grid-connected and energy storage technology.
