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The performance behavior of the lithium-ion battery can be simulated by the battery model and thus applied to a variety of practical situations. Although the particle swarm optimization (PSO) algorithm has been used for the battery model development, it is usually unable to find an optimal solution during the iteration process. To resolve this problem, an adaptive random disturbance PSO algorithm is proposed. The optimal solution can be updated continuously by obtaining a new random location around the particle’s historical optimal location. There are two conditions considered to perform the model process. Initially, the test operating condition is used to validate the model effectiveness. Secondly, the verification operating condition is used to validate the model generality. The performance results show that the proposed model can achieve higher precision in the lithium-ion battery behavior, and it is feasible for wide applications in industry.

A battery management system (BMS) is an electronic system used to manage a rechargeable battery or battery pack, and it is widely applied to many applications that use a battery or batteries, such as portable electronic devices, electric vehicles, and power grids [

Currently, there are three general types of battery models available in the literature: the electrochemical battery model, artificial neural network model, and equivalent circuit model (ECM) [

Parameter identification is an essential step in battery modeling, and its results directly affect the accuracy and reliability of the model. Identification methods are usually divided into two types [

Online identification methods adjust the parameters of the model in real time based on the condition and current state of the battery. Furthermore, the BMS makes use of such parameters and other information such as current, voltage, and temperature to evaluate the state of charge (SOC), state of health (SOH), and so on [

At present, there are two major types of offline identifications. The first type of methods is generally called traditional identification methods, like fitting method based on Least Squares [

For the development of the battery model using PSO algorithm, the particles may hover around local optimal solution during the iteration process without reaching the real optimal location. For this reason, an adaptive random disturbance PSO (ARDPSO) is proposed and its performances are validated by using a classical single model and a multiplex model as target optimization functions. Also, this algorithm can be used for identifying the parameters of the battery model thus achieving higher calculation precision.

The remaining contents of this paper are arranged as follows: Section

Particle swarm optimization (PSO) is an evolutionary computation technique, especially searching for optimization for continuous nonlinear, constrained and unconstrained, and nondifferentiable multimodal functions [

In each iteration, every particle updates its position and velocity in search space according to its individual best solution

For a particle swarm optimization, a better global search is needed from a starting phase to help the algorithm converge to a target area quickly, and then a stronger local search is used to get a high precision value. Therefore, the modification of improved standard PSO introduces the inertia weight

For getting a high precision solution,

in which

Although some related PSO algorithms in convergence speed and inertia weight control have been reported in the literature, the particles may still encounter problems such as hovering around the real optimal location but being unable to locate it. It means that the local optimal solution of the particle is not updated, then resulting in the global optimal solution not to be updated. Consequently, the distance between the searched solution and the real optimal solution will not become closer. For illustrating such a problem, a typical experiment uses the standard PSO and LPSO algorithms to find a solution that minimizes the single mode target function

Figures

Optimization procedure: (a) the motion curve of the global optimal solution of the standard PSO; (b) the fitness curve of the global solution of the standard PSO; (c) the motion curve of the global solution of the LPSO; (d) the fitness curve of the standard LPSO.

From Figure

(1) The global optimal solution (^{th} to 15^{th} iteration, the global optimal solution stops updating, while the global optimal solution continues to update after the 16^{th} iteration.

(2) During the iteration process, the global optimal solution of the standard PSO and LPSO gradually tends to the real optimal position (0, 0). However, it is noted that the standard PSO algorithm stops updating the global optimal solution at the 24^{th} iteration, and its fitness is 1.702324344665708e-05. On the other hand, the LPSO algorithm stops updating the global optimal solution at the 78^{th} iteration, and its fitness is 1.6916e-12.

(3) The inertia weight

However, both standard PSO and LPSO would encounter such problems:

(1) During the iterations, the global optimal solution may not be updated for each iteration.

(2) The global optimal solution may stop updating before the maximum number of allowable iterations is achieved.

For this reason, the ARDPSO algorithm is proposed to give the particle more opportunities to continue to update its local optimal solution and thus to find the global optimal more accurately. The global optimal solution updating from the ARDPSO algorithm is shown in Figure

The principle of the ARDPSO: (a) the global optimal solution updating; (b) the convergent procedure of ARDPSO.

Figure

(1) In each iteration process, for each particle, if the fitness of its current solution (that is,

If

During the evolution of the global optimal solution, the distance between the global optimal solution and the real optimal solution decreases with increasing iteration time, and the coverage of the random disturbance thus becomes less. In Figure

A new random location generation function can be defined as formula

If the fitness of its current solution (that is,

(2) By comparing the fitness of the randomly obtained new solution to that of the abovementioned current solution, the better solution is selected as the new current location of the particle.

(3) If the new current location of the particle is superior to its local optimal solution, thus the local optimal solution is updated.

As above, the process of the ARDPSO is concluded as follows.

At

Get the moving speed:

Get the new location:

If the fitness of the current location

Get

And then, for all of the particles, we have the following.

Get

In order to verify the availability and wide applicability of the proposed ARDPSO algorithm, nine benchmarks from the BBO repository are adopted to test its performance, shown in Table

Benchmarks.

No. | function | formula | minimum |
---|---|---|---|

| Sphere Model | | 0 |

| Schwefel’s Problem 2.22 | | 0 |

| Schwefel’s Problem 1.2 | | 0 |

| Schwefel’s Problem 2.21 | | 0 |

| Rosenbrock Function | | 0 |

| Step Function | | 0 |

| Rastrigin Function | | 0 |

| Ackley Function | | 0 |

| Griewank Function | | 0 |

Without loss of generality, the parameters setting for each benchmark in Table

The dimensions of these functions are all set as 2.

The solution range of the benchmarks is based on Table

Generate the initial position of the particles randomly within its solution range.

In addition, we have the following.

For LPSO and ARDPSO, the maximum inertia weight is set as 0.9, while the minimum one is 0.5.

For ARDPSO, the parameter

Following the above rules, in the MATLAB 2015 (b) environment, standard PSO, LPSO, and ARDPSO are used to test 9 benchmark functions in Table

Average updating times of the historical local optimal solution.

Algorithm | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|

Average times | PSO | 4.81 | 4.64 | 4.63 | 4.80 | 5.42 | 4.47 | 3.32 | 3.45 | 3.42 |

LPSO | 18.95 | 18.70 | 18.77 | 18.52 | 19.22 | 6.92 | 15.69 | 18.06 | 16.01 | |

ARDPSO | 65.74 | 63.08 | 65.47 | 63.69 | 65.52 | 5.06 | 38.49 | 64.20 | 35.28 |

Statistical results diagram of historical local optimal updating (in which the red point presents the standard PSO, the green point presents the LPSO, and the blue point presents ARDPSO; the abscissa values 1-9 correspond to benchmarks 1-9, and the ordinate value indicates the number of updating times).

Statistical results diagram of global optimal updating (the abscissa values 1-9 correspond to benchmarks 1-9, and the ordinate value indicates the number of updating times).

The performance results as above indicate the following:

(1) For local optimal solution (

(2) For global optimal solution (

Average updating times of the global optimal solution.

Algorithm | | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|

Average times | PSO | 7.6 | 8.23 | 6.5 | 8.03 | 11.07 | 4.83 | 5.7 | 6.07 | 6.5 |

LPSO | 31.73 | 31.07 | 32.3 | 28.53 | 33.87 | 5 | 29.73 | 30.5 | 30.53 | |

ARDPSO | 64.8 | 63.73 | 64.13 | 63.87 | 64.7 | 2.93 | 36.5 | 61.63 | 32.27 |

Comparison of performance results.

| | _{ 1 } | _{ 2 } | _{ 3 } | _{ 4 } | _{ 5 } | _{ 6 } | _{ 7 } | _{ 8 } | _{ 9 } |
---|---|---|---|---|---|---|---|---|---|---|

Standard | Minimum Error | 0.001238535 | 0.015541741 | 0.000359473 | 0.012069324 | 0.00312616 | 0 | 0.002253 | 0.015926 | 3.47E-08 |

Average Error | 0.07380834 | 0.331487833 | 0.061015048 | 0.139138918 | 0.362969219 | 1 | 0.846391 | 0.103188 | 0.004895 | |

Maximum Error | 0.681954989 | 0.808067334 | 0.24427567 | 0.446676649 | 2.771664211 | 0.133333 | 2.130416 | 0.41696 | 0.010275 | |

LPSO | Minimum Error | 3.93E-14 | 3.03E-07 | 2.63E-14 | 8.81E-08 | 7.55E-14 | 0 | 1.53E-13 | 2.42E-08 | 7.77E-16 |

Average Error | 7.59E-12 | 3.48E-06 | 1.15E-11 | 2.90E-06 | 6.72E-10 | 0 | 0.1990 | 3.71E-07 | 0.0043 | |

Maximum Error | 4.22E-11 | 1.01E-05 | 1.05E-10 | 9.85E-06 | 1.35E-08 | 0 | 0.994959 | 1.58E-06 | 0.0074 | |

ARDPSO | Minimum Error | 8.89E-33 | 1.26E-16 | 2.61E-32 | 3.08E-17 | 1.23E-32 | 0 | 0 | 8.88E-16 | 0 |

Average Error | 3.22E-29 | 1.98E-15 | 1.70E-29 | 2.05E-15 | 3.90E-27 | 0 | 0.0099 | 4.09E-15 | 0.0023 | |

Maximum Error | 5.11E-28 | 7.08E-15 | 1.18E-28 | 1.78E-14 | 7.31E-26 | 0 | 0.994959 | 1.15E-14 | 0.0074 |

(3) From the error statistics in Table

The following can be concluded:

(1) For step function

(2) From formula (

(3) When the particle could not update its

In this section, the ARDPSO algorithm is used to identify the parameters of the ECM.

The following experiments use NEWARE BTS-4008 as the power battery test system and 18650 NMC batteries (the positive material is LiCoxNiyMnzO_{2}) as the experiment object. The selected 18650 NMC battery was manufactured in Tianjin with a rated capacity of 2000 mAh and a rated voltage of 3.7 V. The rated charging and discharging cut-off voltages are 4.2 V and 2.5 V, respectively.

The open circuit voltage- (OCV-) state of charge (SOC) curve of the above tested battery is obtained according to the method described in [

The error criterion is a crucial factor to determine the precision of parameters identification so that the root square error is commonly used for battery models parameters identification [

At present, the simple model, the first-order RC model, and the second-order RC model are the most applied battery models [

For the three ECMs, the dynamic behavior of the battery cannot be effectively modeled by the simple model due to the lack of consideration in the battery polarization effect. For example, when the battery is at rest after it is charged/discharged for a certain time, its SOC value is a constant with 0 current. Then, the simple model will output a constant terminal voltage, but in fact its terminal voltage is variable. The first-order RC model is based on the connection with one RC network in series with the simple model, and the delay characteristic of the first-order RC network is used to simulate the polarization effect of the battery. Therefore, during the battery rest, the first-order RC model can output a variable. However, it is unsuitable for the voltage transient process of the battery. The second-order RC model that connects two RC networks in series with the simple model can simulate the electrochemical polarization and concentration polarization separately. Accordingly, it could better model the dynamic behavior of the battery. In this study, the influence from different parameter identification methods is mainly considered.

The RDPSO algorithm evaluated with the fitness function shown in (

Flow diagram of system identification based on ARDPSO.

Based on the known capacity and the OCV-SOC curve of the battery, the LPSO and ARDPSO are used to identify the parameters of the abovementioned three ECM models. In this section, two different operating conditions are selected for experiments. The first one is the test operating condition shown in Figures

Test operating condition: (a) battery current; (b) measurement voltage; (c) SOC; (d) OCV.

Verification operating condition: (a) battery current; (b) measurement voltage; (c) SOC; (d) OCV.

In these figures, we have the following.

Figure

Figure

Figure

Figure

The simple, first-order RC, and second-order RC models were used to test the least square method [

Error result of least square, LPSO, and ARDPSO in test operating conditions.

Algorithm | Error | Simple model | First-order RC | Second-order RC |
---|---|---|---|---|

Least Square | RMS Error | 0.0472 | 0.0451 | 0.0426 |

Max Error | 0.7836 | 0.7045 | 0.7264 | |

Acc Error | 578.7310 | 413.8880 | 383. 9528 | |

LPSO | RMS Error | 0.0472 | 0.0443 | 0.0434 |

Max Error | 0.7836 | 0.7196 | 0.7494 | |

Acc Error | 578.5120 | 386.2990 | 347.1315 | |

ARDPSO | RMS Error | 0.0461 | 0.0421 | 0.0397 |

Max Error | 0.7836 | 0.7139 | 0.7217 | |

Acc Error | 544.5102 | 323.9048 | 313.6077 |

Note that each variable in formula (

The above three models based on the same parameters have similar output voltages and error distributions, and only the results of the second-order RC model based on LPSO and ARDPSO are shown in Figure

Testing results of the second-order RC model: (a) the measured voltage of the battery and the simulation results of the battery model; (b) simulation errors of the battery model.

From the experiment results, the following can be concluded:

(1) When the three abovementioned models are identified under the same operating conditions, the global optimal location is further optimized by the ARDPSO with the added random disturbance function, as compared to the LPSO. As a result, the output voltage obtained from the ARDPSO is much closer to the true battery voltage than the LPSO.

(2) Under the same operating conditions, the same identification method is used to obtain the parameters of the three models. The second-order RC model that uses two RC networks to simulate the electrochemical polarization and concentration polarization has more precision than the first-order RC model with one RC network. However, the simple model does not consider the polarization characteristics; comparing with the simple model, the first-order RC model is more precise.

(3) From Figure

(4) Table

(5) From Table

To validate the generality of the ARDPSO for each of ECM models, the parameters identified under the test operating condition are used as the input to obtain the output voltage of ECM model. Figure

Error distribution of the battery models with the parameters identified by ARDPSO: (a) simple model; (b) first-order RC model; (c) second-order RC model.

Error results of the battery model parameters by ARDPSO.

The following can be seen from Figures

(1) For the three abovementioned models, the results of the verification operating condition (as shown in Figure

(2) Among the three models, the second-order model can better simulate the static and dynamic behavior of the battery. Thus the second-order RC model has smaller maximum error, RMS error, and average error than simple model and the first-order RC model.

(3) The parameters identified by the proposed ARDPSO under one operating condition are suitable for other operating conditions for ECM models, verifying the effectiveness and generality of the ARDPSO. That is, the battery model can be used by BMS to predict the states of the battery.

The PSO has been widely used in many applications like identification of ECM model. It and its extended algorithm such as LPSO could update both local and global optimal solutions by moving particles to achieve the target. However, it is found that the solution either local or global optimum may not keep updating for a period of time during the particles movement. The ARDPSO algorithm is proposed to continue to update the optimal solutions. Test results from multiple benchmark functions have verified that the ARDPSO can improve the updating process for both local and global optimal solutions. Accordingly, the ARDPSO can reach higher solution precision than the standard PSO and LPSO.

As ECM model parameters can affect the static and dynamic behaviors of the battery model, the ARDPSO therefore introduces a new weighted fitness function to identify ECM parameters. Based on the evaluation tool using the maximum error, RMS error, and the average error, it is obvious that the parameters of ECM model have been identified accurately under the test operating condition. Besides, it indicates the ARDPSO promises a better performance than the LPSO. For future work, the black box algorithms such as neural network and support vector machine will be used to model the battery and further compared with the ARDPSO algorithm in the state of charge (SOC) and state of health (SOH).

The authors declare that they have no conflicts of interest.

This research work was partly supported by the National Science and Technology Ministry [no. 2015BAA09B01], the National Natural Science Foundation of China [Project no. 51377044], the Natural Science Foundation of Hebei Province [Project no. E2017202284], and the Youth Foundation of Hebei Education Department [Project nos. QN2017314, QN2017316].

_{4}-based lithium ion secondary batteries