基于RLS的锂电池全工况自适应等效电路模型

郭向伟，王 晨，陈 岗，许孝卓

（河南理工大学电气工程与自动化学院，河南 焦作 454003）

Lithium batteries have gradually become the first choice of power batteries for new energy vehicles.The characteristics of lithium batteries are complex and variable, and are also coupled with one another, which brings many challenges to battery management.Battery modeling is a mathematical expression of battery characteristics.An accurate battery model can not only reflect the relationship between the battery characteristics and many other influencing factors but can also provide an important basis for accurate state estimation.The study of battery modeling is very important for improving the battery management level of new energy vehicles[1].

Common power battery models include the electrochemical model, the equivalent circuit model, the black box model, etc[2-3].ECMs have the characteristics of a simple model equation,convenient parameter identification and good realtime performance.They are widely used in various power battery state estimation methods.In recent years, a variety of ECMs have been proposed[4-7],such as the Rint model, the Thevenin model, the Partnership for a new generation of vehicles(PNGV) model, the DP model, the multi-orderRCmodel, etc.Theoretically, a multi-orderRCloop model has a higher accuracy, but in the application,the multi-orderRCloop needs to identify more parameters, the error of each parameter is greater,and its accuracy is even less than that of the DP model.Among the above models, the DP model can achieve a good balance between accuracy and computation[8], which is widely used.This paper mainly studies the DP model.

The parameter identification of the DP model in many literatures involves determining its ohmic resistance and the twoRCloops at the same time[9-11].In fact, the time-varying characteristics of each parameter of the model are different.Among them, the ohmic resistance is almost unchanged in a certain charging or discharging cycle under the same temperature and the same state of health (SOH)[12].To simulate the response characteristics of the power battery at different magnifications, the parameters of itsRCloop remain time-varying within a certain cycle.In the process of parameter identification, it is not advisable to identify each parameter with the same time-varying characteristics, which may easily lead to drastic changes in the ohmic resistance and adversely affect the accurate identification of theRCloop.

Currently, the most common method for power battery model parameter identification is the recursive least square method with a forgetting factor, which has the characteristics of being easy to understand, easy to apply in engineering, etc.In view of the situation where the system model's parameters are easily affected by the uncertainty in the application environment and cause major changes, the RLS can periodically optimize and update the parameters, which can overcome the uncertainty of the model parameters, and can accurately capture the real-time characteristics of the system[13].However, there are also several problems: according to its equation characteristics,it has a better identification effect for time-varying operating conditions, but a poor identification effect for time-invariant operating conditions, and may even diverge.In fact, the change in working conditions is very random.In the driving process of new energy vehicles, there are not only fast changing working conditions but also relatively stable working conditions at a constant speed,which leads to certain limitations in the application of the RLS method.Compared with the online model, the offline model identified by different current magnifications has a higher accuracy under constant current conditions.

Based on the time-varying characteristics of the DP equivalent circuit model parameters,this paper separates the ohmic resistance from the other two groups ofRCparameters in order to reduce the mutual influence of the model parameter identification process, and uses different identification methods for ohmic resistance andRCloops.An online R-DP model with a known ohmic resistance at a certain charging or discharging cycle is proposed, which can not only improve the accuracy of the model but also reduce its computation.On this basis, in view of the adaptability of RLS to different working conditions and the higher accuracy of offline modeling under constant current working conditions, a full working conditions ECM based on the adaptive output of the R-DP online model and the DP offline model is proposed to further improve the accuracy of the model.Finally, a model evaluation method is established based on accuracy and the operating speed to verify the superiority of the adaptive output model.

The remainder of this paper is organized as follows: In Section 2, the R-DP equivalent circuit model is established and the parameters are identified.In Section 3, the DP offline model parameters are identified.In Section 4, a full working conditions ECM based on the adaptive output of the R-DP online model and the DP offline model is established.The superiority of the adaptive output ECM in the full working conditions is verified in Section 4.Finally, conclusions are given in Section 5.

1 R-DP online model establishment and parameter identification

The DP equivalent circuit model is shown in Fig.1, whereiis the current (The sign is positive when discharging),Uocis the open circuit voltage(OCV) of the power battery,Ris the ohmic resistance,RpandRsare the polarization internal resistances,CpandCsare the polarization capacitances.The polarization effect is simulated by theRpCploop and theRsCsloop[14].

Fig.1 DP equivalent circuit model

The R-DP online model is a DP online model with a known ohmic resistance during charging or discharging process.Compared with the entire life cycle, under certain temperature conditions,the ohmic resistance of a lithium battery is almost unchanged during a certain charging or discharging[15].Based on this characteristic, the ohmic resistanceRis first identified, and then the remaining four parameters are identified online using the RLS method with a forgetting factor.To identify theRCloop parameters using RLS with a forgetting factor, a reliable OCV-SOC curve is required.In this section, the off-line identification of the ohmic resistanceRis first performed, and then the OCV-SOC curve and the parameters identification of theRCloops is derived in detail.For a certain charge and discharge process, the factors affecting the ohmic resistance include battery SOH and temperature.In this paper, SOH is taken as 1 and the ambient temperature is taken as 25 ℃ as an example to explain the offline identification of ohmic resistanceR, and calibrate the OCV-SOC curve.

1.1 Offline identification of ohmic resistance R

The measured terminal voltage response curve of the Sanyo lithium battery with a rated capacity of 3.2 Ah at a certain state (SOC = 0.9 with a constant current discharge of 320 mAh at 0.5 C) at the end of discharge is shown in Fig.2.Fig.3 is a schematic diagram of the terminal voltage response curve at the end of lithium battery discharge.

Fig.2 Terminal voltage response at the end of discharge

Fig.3 Terminal voltage response cruve

In Fig.3,V0represents the voltage at the last moment of discharge, andV1represents the jump voltage of the battery at the end of discharge.Erepresents the stable voltage after the battery polarization effect disappears, which is approximately the open circuit voltage.The jump of the voltageV0toV1reflects the static characteristics of the battery, indicating the process of losing the voltage drop on the ohmic resistanceR, corresponding to the process i in Fig.2.Therefore, according to Ohm's law, the formula for calculating the ohmic resistanceRcan be obtained,R=(V1-V0)/I.

1.2 OCV-SOC calibration experiment

The test system consists of a host computer,an EBC-X eight-channel battery divider cabinet(discharging current is 0.1—10.00 A, charging current is 0.1—5.00 A), and a Tektronix TDS2024C oscilloscope (200 MHz bandwidth, 2 Gs/s sampling rate, 4 channels) along with other components,as shown in Fig.4.Calibrate OCV-SOC curves under constant current and constant capacity intermittent discharge conditions of 0.2 C, 0.3 C,0.4 C, 0.5 C, 0.6 C, 0.75 C and 1 C[8].The specific calibration process of each group is as follows, Fig.5 shows the sixth-order polynomial fitting curve.

Fig.4 Bettery test system

Fig.5 OCV-SOC curve at different rates

(1) Firstly, the battery is charged with constant current (0.2 C), and then the battery is charged with constant voltage (cut-off voltage 4.25 V).After charging is completed, leave the battery for one hour to eliminate polarization effect;

(2) The battery is discharged with constant current and constant capacity (1/10 of the total capacity, 260 mAh);

(3) After the discharge is completed, leave the battery for one hour, and then measure the OCV;

(4) Repeat step (2) and (3) until the battery is completely discharged.

It can be seen from the Fig.5 that when the SOC is greater than 0.1, the OCV-SOC curves of different discharge rates are very close and almost the same, indicating that different discharge rates have little effect on the OCV-SOC curve under the same conditions of SOH and temperature[16].The smaller current, the smaller the influence on the polarization of the battery.In this paper, the OCV-SOC curve corresponding to 0.2 C is selected as the reference curve, and the fitting equation is shown in equation (1).

Among them,b1,b2…b7are the coefficients of the sixth-order polynomial fitting,b1=-5.6944,b2=23.7660,b3=-39.4557,b4=32.9612,b5=-14.0483,b6=3.5610, andb7=3.1117.

1.3 RC parameter identification

Consider the following nonlinear dynamic system:

wheree(k) is the stationary zero-mean white noise,y(k) is the output variable of the system.For the DP model,φ(k)=[y(k-1) …y(k-n)I(k) …I(kn)]Tis the data matrix of the system,y(k) denotes the voltage at momentk,I(k) denotes the current at momentk.andθ(k)=[a1,a2,a3…a2n] is the parameter matrix to be identified for the system.

φ(k)=[y(k- 1)y(k- 2)I(k)I(k- 1)I(k- 2)]T(3)

This is the data matrix of the system.

This is the system's parameter matrix to be identified.For the battery system, the least squares method will cause data saturation in the process of parameters identification, which will not accurately reflect the characteristics of the new data, resulting in inaccurate identification results[17].To avoid the above situation, the forgetting factorλis introduced to form the RLS method with forgetting factor, the smallerλis, the stronger the tracking ability of identification but also the greater the fluctuation of parameter estimation.Generally, 0.95＜λ＜1, and in this paperλis 0.98[18].

To separate the identification of the ohmic resistance from the parameter identification of the entire model, for the DP model shown in Fig.1,the functional relationship can be expressed as:

Then, the transfer function is expressed as:

The bilinear transformation is used to map the system from the s-plane to thez-planein equation (7).

whereTis the system sampling interval time.The equation based on thez-plane is expressed as:

a1,a2,a3,a4anda5are the coefficients related to the model parameters.

Converting (8) into a difference equation:

whereI(k) is the system input, andy(k) is the system output.The expression is:

According to (9),a3=a4-a5, and the parameter matrix to be identified can be expressed as:

According to equations (3) and (12), equation(10) can be abbreviated as:

Equation (13) can use the RLS method with the forgetting factor to identify the parameter matrix.Substituting the bilinear inverse transformation factor shown in equation (14) into equation (10),equation (15) can be obtained.

From equations (6) and (15), the coefficients are correspondingly equal, so it can be obtained that:

So far, based on the RLS with the forgetting factor and equation (16), the four parameters of theRCloops can be solved by the four equations.Compared with the ordinary RLS-based identification process, the identification object is changed from five unknowns to four unknowns, which not only improves the identification accuracy but also reduces the amount of calculations.

2 DP offline model parameter identification

The ohmic resistanceRcan be identified offline from section 2.1.The DP model simulates the polarization process of the battery by superimposing the twoRCloops.In combination with Fig.1, assuming that the time constant of theRCparallel circuit composed ofRpandCpis small, it can be used to simulate the process of rapid voltage changes (V2-V1) when the current changes suddenly.Assuming that the time constant of the parallel circuit ofRsandCsis relatively large, it can be used to simulate the process of slowly changing the voltage (E-V2).

Suppose the battery is discharged for a period of time during (t0-tr), and then the remaining time is in a static state, wheret0,tdandtrare the discharge start time, discharge stop time, and static stop time, respectively.During this process,theRCnetwork voltage is expressed as:

Among them,τp=RpCpandτs=RsCsare the time constants of the twoRCparallel circuits.During the discharging of the battery, the polarizing capacitorsCsandCpare in the charging state.The voltages of theRCparallel circuits rise exponentially.After the battery enters the resting state from the discharged state, the capacitorsCsandCpare discharged to their respective parallel resistors, and the voltage decreases exponentially.The size of the resistors and capacitors in the model are related to the SOC of the battery and the charging or discharging current.The (E-V1) phase voltage change is caused by the disappearance of the polarization effect of the battery.The voltage relationship of the battery during this process is expressed as:

Equation (19) can be simplified as:

Among them,Rs=a/I,Rp=b/I,Cs=1/RsC, andCp=1/Rpd.Based on this, the values ofR,Rs,Cs,RpandCpcan be identified.

3 Establishment of ECM under full working conditions

According to the derivation process of the RLS method, the data matrix must have full rank before the inversion calculation can be performed.In other words, the values of each column must be unequal, which requires that in the time of 0—N, there is at least one moment to satisfy.Moreover,the convergence of the RLS method takes a certain amount of time.When the speed of the current change is slow or the current is constant for a long time, the convergence effect is often not achieved, or even divergence occurs[19].Therefore,the prerequisite for RLS to be applied to online parameter identification is that the current of the battery is time-varying, and the identification accuracy of the constant current conditions will be greatly affected.

The DP offline model is based on different constant current rates to identify the model parameters, and has a higher accuracy than the online model for constant current conditions.The offline model parameter application process is based on the look-up table method or the function fitting method.These two methods are only based on the independent change process of each parameter for table look-up or fitting,without considering the relationship between the parameters.The parameter identification based on RLS can consider the relationship between the parameters at all times.In theory, for variable current conditions, RLS online identification has a higher accuracy than offline identification[20-21].

It can be seen from the above analysis that different parameter identification methods have different accuracies for different working conditions.Based on this, this paper establishes an equivalent circuit model based on the online model and offline model adaptive output under full working conditions.When the working condition current is constant, the offline model identification result is output; when the working condition current changes, the online model identification result is output.The specific adaptive output process is shown in Fig.6.

Fig.6 Adaptive output for all working conditions

This section first analyzes the model accuracy and running speed, and then establishes a model evaluation method based on the accuracy and speed.

4 Model advancement verification

4.1 RC Model accuracy verification

For estimation of the battery state, the accuracy of the equivalent circuit model plays an important role.This section first verifies the accuracy of the equivalent circuit model.The model accuracy verification mainly includes two contents, namely,the accuracy comparison verification of the R-DP online model based on the RLS and the DP online model, the verification of the full-condition adaptive output model and the R-DP online model based on the RLS and DP offline model.To simulate the constant current and variable current charging and discharging process of the battery at the same time, the model verification adopts two working conditions, the first one is a custom working condition, as shown in Fig.7(a), the current greater than zero indicates that the battery is discharged, Less than zero means the battery is charging, and the total time is 4200 s.As shown in Fig.7(b), the working condition of Economic Commission for Europe (ECE) are scaled down to suit the experimental object in this paper, also referred to as ECE working condition in this paper,with a single cycle of 200 s, the simulation duration of the model verification process is 10 cycles.The working condition test experimental platform is shown in Fig.8, which is composed of a host computer with control software, a middle computer,Neware battery test equipment and a battery.The test accuracy of the battery test equipment can reach 0.05%, the maximum current rise time is 1 ms, and the maximum data recording frequency is 10 Hz.It should be noted that in the process of collecting data using the test platform shown in Fig.8, the temperature change of the battery is less than 1 ℃, so the internal resistance of the simulation process is set to a fixed value, which is obtained by offline identification.

Fig.7 Model input conditions

Fig.8 Working condition test platform

4.1.1 Comparsion of the R-DP online model and DP online model based on RLS

Figs.9 and 10 show the simulation results of the R-DP online model and the DP online model based on the RLS.It can be seen from Fig.9 that both models can better track the changes in the measured terminal voltage.It can be seen from Fig.10 and Table 1 that, overall, the output based on the R-DP online model is closer to the real measured value.The main reason for this result is that the R-DP online model is compared with the ordinary DP online model.On one hand, theRidentification result is closer to the actual situation, more reliable and more accurate; on the other hand, the R-DP model only needs to identify four parameters, and theoretically has a higher identification accuracy.

Table 1 Average absolute error of each model

Fig.9 Simulation results of R-DP online model and DP online model

Fig.10 Error comparison of R-DP and DP online model

4.1.2 Comparison of the adaptive output model,R-DP online model, and DP offline model

Fig.11 shows the simulation results of the RDP online model and the DP offline model.Fig.12 shows the simulation results of the adaptive output model of the full working conditions.Fig.13 shows the error curve of each model, and Table 2 shows the average absolute error value of each model.

Fig.11 Simulation results of R-DP online model and DP offline model

Fig.12 Simulation results of the adaptive output model under full working conditions

Fig.13 Error curves of each model

It can be seen from Table 1 and Table 2 that the offline model error is the largest overall,reaching about 50 mV.The main factor affecting the accuracy of the offline model is the accuracy and quantity of the actual sampled data during the offline identification process.Compared with the ordinary DP online model, the R-DP online model based on RLS has higher accuracy due to the more reliable identification of ohmic resistance and fewer online identification parameters.The full-condition adaptive output model combines the advantages and disadvantages of the R-DP online model and the DP offline model, and has smaller errors than the R-DP online model.Itshould be noted that the size of the model error is greatly affected by the changes in the working conditions.If the working conditions are changed,the proportion of model accuracy improvement may change.However, the effects of the two methods proposed in this paper to improve the model accuracy are definite.Based on the above analysis, the RLS based full-condition adaptive equivalent circuit model improves the accuracy of DP equivalent circuit by improving the on-line parameter identification process of DP model and combining the online model with the offline model, which is of great significance to improve the state estimation accuracy of new energy vehicle power battery.

4.1.3 Model speed verification

In order to verify the reliability of the running speed of the above models, five different computers were used to simulate the two working conditions of the above models.The average running time of each model is shown in Table 3.

Table 3 Simulation time of each model

It can be seen from Table 3 that the running time of the DP offline model is the shortest, which is consistent with its least calculation equation.The R-DP online model has a shorter running time than the DP online model, which is consistent with the theoretical analysis.The adaptive output model runs longer than the R-DP online model,but the overall difference is not much.

4.2 Model evaluation

According to the model verification results,the four models have their own advantages and disadvantages for the battery state estimation.In the identification process, the adaptive output model has a higher accuracy than the other models, and the DP offline model has a faster speed than other models.In practical applications,it is often necessary to consider both the accuracy and speed at the same time in order to obtain a good balance between them.In this paper, the model selection factor is defined based on the weight coefficients of accuracy and speed set in the model selection process, as shown in the following equation, whereSis the selection factor of each model.

whereAandBare the selection weight coefficients of the accuracy and speed subjectively set, and their sum is 1.xreflects the multiple of the average absolute error of each model relative to the adaptive output model, andyreflects the multiple of the average simulation time of each model relative to the DP offline model.The larger the selection factor of each model is, the better the balance between precision and speed that can be achieved.When the model accuracy and speed weight coefficients are both 0.5, the selection factors of each model are shown in Fig.14.

Fig.14 Selection factors of each model

Compared with other models, the adaptive output model can achieve a better balance between accuracy and speed.Based on the above analysis, the full condition adaptive output ECM based on RLS improves the DP model online parameter identification process and combines the on-line model and off-line model.Under the premise of little change in the operation time, the model accuracy is improved, and a better balance is achieved between accuracy and speed.

5 Conclusions

An accurate lithium-ion battery model is of great significance for accurate estimation of the battery state.Currently, the most common method for battery model parameter identification is the RLS method based on forgetting factors.This paper is based on this method.First, according to the time-varying characteristics of the model parameters, the ohmic resistance was separated from the online identification process, and the R-DP online ECM with a known ohmic resistance was proposed, which reduced the online identification object based on RLS from five parameters to four, thus improving the accuracy and reducing the computation.Second, it can be determined from the characteristics of the RLS equation that it is not suitable for constant working conditions.Based on this, an ECM with an adaptive output based on online and offline models of different working conditions was proposed to further improve the accuracy of the model.Finally, based on the model accuracy and operating speed, a reliable model evaluation method was established.When the model accuracy and speed weight coefficients are the same, the adaptive output model can achieve a better balance between the accuracy and speed than other models.