CN104267354B - A kind of peak power Forecasting Methodology of electrokinetic cell - Google Patents

Abstract

A kind of peak power Forecasting Methodology of electrokinetic cell, is related to the peak power Predicting Technique of electrokinetic cell.The present invention is the accuracy in order to improve the prediction of the peak power of electrokinetic cell.The method of the present invention includes two parts：The battery peak power Forecasting Methodology that A is decomposed based on zero state response and zero input response based on the electrochemical impedance spectroscopy equivalent-circuit model simplifying and the parameter On-line Estimation of fractional order federated Kalman filtering, B.The present invention has selected the reference model that the simplification impedance spectrum model comprising fractional order element is predicted as battery peak power, therefore proposed by the present invention not only can accurately predict the maximum output in short-term of battery it is also possible to peak power fan-out capability in longer period for the Accurate Prediction battery based on the electrokinetic cell peak power Forecasting Methodology of this model.The present invention is applied to the prediction of electric automobile online peak power.

Description

A kind of peak power Forecasting Methodology of electrokinetic cell

Technical field

The present invention relates to the peak power Predicting Technique of electrokinetic cell.

Background technology

The content of the relevant battery peak power Forecasting Methodology disclosed in domestic patent is less, in paper periodical both domestic and external The method having the prediction of some electrokinetic cell peak powers.

These methods mostly adopt the single order RC equivalent-circuit model of battery, using SOC, terminal voltage, electric current as its peak work The restrictive condition of rate prediction, the power forecasting method of early stage adopts the complete battery pair of offline battery model supplemental characteristic to export/return Feedback peak power prediction, but battery with the change of the environment such as aging or temperature, operating mode, model parameter also can change therewith Become, therefore, the power estimation method reliability based on off-line model supplemental characteristic is poor；

In the up-to-date paper delivered, battery model supplemental characteristic is obtained online by method for parameter estimation, can effectively carry The accuracy of high estimation, but because the error of single order RC model itself is larger, power prediction can only be confined to the shorter time It is inside effective.

In order to reduce discretization error when for power prediction for the state equation, when adopting less discretization in paper Between be spaced (such as taking time interval to be 1s or 0.1s), by the method for recurrence calculation, obtain battery 10s after current time Predictive value at the peak power predictive value at place or 60s, the shortcoming of this method is that there is substantial amounts of recurrence calculation, real-time Difference.In addition this method also cannot improve the power inaccurate problem of estimation causing due to model oneself error.

Content of the invention

The present invention is the peak power forecasting reliability in order to improve electrokinetic cell, reduces amount of calculation, thus providing one Plant the peak power Forecasting Methodology of electrokinetic cell.

A kind of peak power Forecasting Methodology of electrokinetic cell, it is realized by following steps：

Step A, the step of battery model parameter On-line Estimation, specially：

Step A1, when modeling to secondary cell, due to electric automobile operating condition frequency characteristic, therefore battery electrochemical The impedance operator of the medium frequency in impedance spectrum model can be by conventional purely resistive element R and the letter of normal phase element Q parallel circuit Turn to purely resistive element R to describe, the battery electrochemical impedance spectrum equivalent-circuit model after being simplified；

Electrochemical impedance spectroscopy equivalent-circuit model after this simplification includes open-circuit voltage OCV_e, ohmic internal resistance R_oWith weber resistance Anti- Z_W；

Step A2, according to step A1 obtain simplification after electrochemical impedance spectroscopy equivalent-circuit model set up fractional order karr State equation needed for graceful wave filter and observational equation, specially：

Take total current I flowing through secondary cell_LIt is on the occasion of data sampling period is 1s in electric discharge；

Wherein △^rFor differential operator, r is differential order, when r is for decimal, △^rRepresent fractional order differential operator, when r is During integer, △^rFor integer differential operator；

Take fractional order element Z_WBe both end voltage be U_WQuantity of state, have：

For battery model parameter, diffusion parameter X_W, open-circuit voltage OCV_eWith ohmic internal resistance R_oWith battery charge state (SOC) change is slow, therefore：

Aforementioned four equation is rewritten as matrix form, obtains the state equation of fractional order Kalman integrated filter：

Take U_LFor the observed quantity of system, then have：

U_L=OCV_e-I_LR_o-U_W

I_LThe total current representing and flowing through battery；

Take：

Y=U_L

Obtain the observational equation of fractional order Kalman integrated filter：

After equation discretization, have：

Wherein, w, v represent state-noise and the observation noise of system respectively；

Series definition (being also called the definition of Gr ü nwald-Letnikov fractional order differential) according to fractional order differential：

Wherein,

Separately take：Obtain the discretization recursion expression-form of Fractional Differential Equation by above formula：

Definition：

According to the series definition of fractional order differential, wherein：Amount of calculation will increase over time And constantly increase, this situation is not suitable for engineer applied, for this reason, above formula is rewritten as following form：

Step A3, the state equation needed for fractional order Kalman filter using step A2 structure and observational equation, right State, parameter and covariance matrix carry out time renewal and measurement updaue according to fractional order federated Kalman filtering algorithm：

It is specially：

Initialization：

Wherein, E [x] represents the mathematic expectaion of x, is experience preset value when method calculates,Represent x in initial time (k =0) estimated value,Represent the estimated value of the noise covariance in initial time (k=0) for the x；

The time of state, parameter and covariance matrix updates：

Wherein, Q_kIt is noise w_kCovariance,For k moment state and model parameter x_kPredictive value,During for k-1 Quarter state and model parameter x_k-1Correction value,Noise covariance matrix P for k moment x_kPredictive value,During for k-1 Carve the noise covariance matrix P of x_k-1Correction value；

The measurement updaue of state, parameter and covariance matrix：

Wherein, R_kIt is noise v_kCovariance, L_kIt is k moment Kalman filter gain size；

Step A4, terminal voltage U of collection battery_LWith total current I flowing through secondary cell_L, using the simplification of step A1 acquisition Rear electrochemical impedance spectroscopy equivalent-circuit model and step A3 update after the state equation needed for fractional order Kalman filter With observational equation, obtain open-circuit voltage OCV_e, ohmic internal resistance R_o, diffusion parameter X_WEstimated value, will obtain open-circuit voltage OCV_e, ohmic internal resistance R_o, diffusion parameter X_WEstimated value as battery estimated result, complete based on fractional order joint karr The secondary cell of graceful filtering simplifies impedance spectrum model parameter On-line Estimation；

Step B, the battery model parameter On-line Estimation result being obtained according to step A, the step carrying out power prediction：

Step B1, the weber impedance Z of derivation fractional order element_WWhether it is linear time invariant element：

If both end voltage U of fractional order element_WInitial value be 0, be I when applying amplitude_LStep current excitation When, both end voltage U of fractional order element_WVoltage response situation after Ns is calculated as follows, and N is positive number：

If the open-circuit voltage OCV of battery model_e, ohmic internal resistance R_o, diffusion parameter X_WNumerical values recited during power prediction It is constant；

Then U_WVoltage response in k=1s is：

Wherein, symbol ^ represents predictive value；

U_WVoltage response when k >=2 is：

Calculate：

After calculating：

By the voltage response that this recursion obtains 10s and 60s it is：

Thus, when battery model parameter X_WWhen constant or slowly varying, output valve U_WWith I_LFor linear relationship, infer and divide Number rank element Z_WFor linear time invariant element；

Then, the power forecasting method of fractional order element is：

In order to predict voltage response at k+ Δ Ts for the fractional order elementThis voltage response is divided into zero state ResponseAnd zero input response

Wherein zero state response is：

For k+10s moment, a=3.524；For k+60s moment, a=8.722；

Zero input responseDetermined by the data before the k moment, take time memory length L=of fractional order differential 60；

The zero input response in k+1 moment is：

The response of k+ Δ T moment fractional order element zero input voltage be can get by this recursion：

Wherein,It is battery model fractional order element Z_WEstimation between (k+1-L)～k moment for the two ends terminal voltage Value, when predicting k+10 moment, b=α, α are one group of constant coefficient matrix；

When predicting k+60 moment, b=β, β are another set constant coefficient matrix；

Thus, can get battery constant-current discharge electric current is I_maxWhen battery the k+ Δ T moment terminal voltage predictive value：

Electric discharge peak value power prediction method be：

If battery charge state (SoC or SOC) is the restrictive condition of battery limit working condition, SoC_minIt is battery discharge Terminate the minima of state-of-charge, then obtain maximum discharge current value now：

Capacity is battery capacity value, and unit is ampere * hour (Ah), SoC_kFor the battery charge state in k moment, SoC_minThe minimum state-of-charge limiting for battery discharge, the minimum state-of-charge that such as certain power battery for hybrid electric vehicle adopts For SoC_min=30%；

If battery terminal voltage U_LFor the restrictive condition of battery limit working condition, if now with maximum discharge current I_maxRight Battery discharge, battery model parameter OCV_ePredictive value in the k+ △ T moment is：

In above formula,It is by calculated battery model parameter OCV of fractional order federated Kalman filtering algorithm_e In the estimated value in k moment,For OCV_ePredictive value in the k+ △ T moment；I_maxIt is as battery limit work in terminal voltage Make maximum discharge current value during state limit condition, be value to be solved,It is the maximum discharge current of battery；OCV_kIt is in k The battery open circuit voltage values in moment, can be calculated by OCV (SoC) pre-defined function and obtain it is generally recognized that OCV and SoC has clearly Corresponding relation；The functional relationship of the pre-defined function of OCV and SoC can be obtained by handbook of batteries or test；For electricity Pond in the k+ △ T moment it is assumed that withCorresponding battery open circuit voltage values during constant-current discharge, equally can be predefined by OCV (SoC) Function calculates and obtains；Due in most cases, the change of OCV is all less and slow, it can thus be assumed that opening within the △ T period Road voltage variety is linear with discharge current；

And then calculate it is assumed that battery is I in discharge current_maxWhen, estimated value U of battery terminal voltage_L,k+△T：

Wherein, the estimated value of terminal voltage be k moment open-circuit voltage estimated value, open circuit voltage variations value, ohm in the △ T period Internal resistance voltage difference, fractional order element Z_WZero state response magnitude of voltage, zero input response magnitude of voltage sum；

Thus release, when with terminal voltage U_LAs the extreme working position of battery constraints when, the maximum work of battery As electric current it is：

When the k+ Δ T moment reaches crest discharge power, consider above-mentioned restrictive condition, maximum discharge current value is：

The crest discharge power in k+ Δ T moment is：

The Forecasting Methodology of feedback current peak power of battery is with the Forecasting Methodology of above-mentioned discharge power in the same manner：

If I_LBe on the occasion of；

If SoC is the restrictive condition of battery limit working condition, SoC_maxIt is the maximum SOC of battery, then obtain Minimum feedback current value now：

If U_LFor the restrictive condition of battery limit working condition, then obtain minimum feedback current value now：

When the k+ Δ T moment reaches peak value feedback power, consider above-mentioned restrictive condition, minimum feedback current value is：

Thus, obtain the peak power of battery current feedback：

Complete the peak power prediction of electrokinetic cell.

In step A, due to the electrochemical impedance spectroscopy test data based on battery for the battery model, battery mould used in method Shape parameter has clear and definite physical significance, ohmic internal resistance R_oPhysical significance be：

R_o≈R_Ω+R_SEI+R_ct

Wherein, R_ΩFor high frequency ohmage, R_SEIFor SEI membrane impedance, R_ctFor Charge-transfer resistance；

In addition, weber impedance to be defined by below equation：

Wherein, W is ionic diffusion coefficient, for the ease of impedance parameter On-line Estimation, takes：

Obtain：

Beneficial effects of the present invention：

1st, the present invention has selected the ginseng that the simplification impedance spectrum model comprising fractional order element is predicted as battery peak power Examine model, because fractional order element has the memory characteristic of long period, the power based on this model therefore proposed by the present invention Battery peak power Forecasting Methodology not only can accurately predict the maximum output in short-term of battery it is also possible to Accurate Prediction is electric Peak power fan-out capability in longer period for the pond.

2nd, the method for the present invention, under multiple electric automobile operating conditions, can keep preferable peak power prediction energy Power, the adaptability for working condition of the inventive method is better than the online power prediction algorithm based on single order RC model for the tradition.

3rd, the method adopts zero state response and the method for zero input response decomposition to predict in k+10 moment and k+60 moment Voltage response situation, eliminate k～k+10, the state recursion link between k～k+60, greatly reduce peak power prediction Amount of calculation it is adaptable to electric automobile online peak power prediction.

Brief description

Fig. 1 is the battery impedance spectroscopy equivalent-circuit model simplifying；

Fig. 2 is that the current value emulation schematic diagram collecting when work operating mode is run simulated by electrokinetic cell battery；

This simulated condition is made up of the United States Federal's city operating mode (FUDS operating mode) and discharge and recharge pulse operation, for authentication The reliability of method.

Fig. 3 is the comparison diagram that battery terminal voltage predicted the outcome and surveyed terminal voltage；

Wherein UL surveys terminal voltage for battery, and ULd10 assumes that battery reaches during electric discharge peak value working condition in 10s Terminal voltage predictive value, ULd60 assumes that battery reaches terminal voltage predictive value during electric discharge peak value working condition, ULr60 in 60s Assume that battery reaches terminal voltage predictive value during energy feedback peak value working condition in 10s.Contrast is it is found that ULd10 With ULd60 respectively at 10s discharge pulse operating mode and 60s discharge pulse operating mode with UL closely, thus it can be extrapolated that the present invention Method has preferable battery discharge characteristic predictive ability.

Fig. 4 is 10s electric discharge peak power and the 60s electric discharge peak value power prediction value of battery.

As shown in Figure 4, the inventive method has preferable electric discharge peak value power prediction ability.

Specific embodiment

Specific embodiment one, a kind of peak power Forecasting Methodology of electrokinetic cell,

The invention discloses a kind of peak estimation method of electrokinetic cell, the method includes two parts：A is based on the electricity simplifying Chemical impedance composes the parameter On-line Estimation of equivalent-circuit model and fractional order federated Kalman filtering, B is based on zero state response and The battery peak power Forecasting Methodology that zero input response is decomposed.

The method be specially：Based on the electrochemical impedance spectroscopy equivalent-circuit model simplifying, the filter of fractional order federated Kalman The power of battery Forecasting Methodology that ripple, zero state response and zero input response are decomposed.It is realized by following steps：

Step one, the electrochemical impedance spectroscopy test result according to battery, due in impedance spectrum, by electrochemical impedance spectroscopy etc. Effect circuit model has done and has simplified further, the electrochemical impedance spectroscopy equivalent-circuit model after being simplified, as shown in figure 1, U_tAnd I_L Represent the terminal voltage of battery and the total current flowing through battery respectively.

The impedance spectrum equivalent-circuit model of this simplification includes OCV_e、R_oAnd Z_WThree elements.

Wherein, OCV_eFor being combined open-circuit voltage, main reflection battery open circuit voltage characteristic, because equivalent-circuit model simplifies Many processes of cell dynamics processes, and have ignored each dynamic (dynamical) boundary condition of battery charge and discharge process, because This is due to this battery model error of itself, OCV_eIt is the approximation of OCV, numerically mainly contain OCV and least a portion of Other chemical reaction potential value such as ion diffusion polarization potential.

OCV_e≈OCV

R_oFor being combined ohmic internal resistance, this parameter mainly reflects the medium-high frequency ohm impedance characteristic of battery electrochemical impedance spectrum (frequency is more than 0.5Hz), this parameter numerically approximates high frequency ohmage (R_Ω), SEI membrane impedance (R_SEI), electric charge transfer Impedance (R_ct) impedance sum.

R_o≈R_Ω+R_SEI+R_ct

Z_WIt is used to describe the weber impedance (Warburg) of the ion diffusion polarization characteristic of battery, U_WFor weber impedance two ends Voltage.Many phenomenons of nature meet fractional order characteristic, and the ion diffusion property process of battery charge and discharge process is especially such as This.It is that knowable to Qwest's figure, ion diffusion process meets fractional order differential characteristic, and this characteristic is normal from the electrochemical impedance spectroscopy of battery Represented with fractional order physical component weber impedance.Weber impedance to be defined by below equation：

Wherein, W is ionic diffusion coefficient, for the ease of impedance parameter On-line Estimation, takes：Obtain：

The feature of this impedance spectrum equivalent-circuit model is a simplified Conventional impedance and composes the high frequency (frequency in equivalent-circuit model More than 1kHz) and intermediate frequency impedance (frequency is more than 0.5Hz, and is less than 1kHz), from the associated description of test data and other paper From the point of view of, above-mentioned simplification impedance spectrum model can effectively reduce model parameter quantity, is suitable for the On-line Estimation of model parameter.

Step 2：State equation according to needed for above-mentioned equivalent-circuit model sets up fractional order Kalman filter and observation Equation：

Quantity of state based on fractional order Kalman integrated filter estimating circuit and parameter value, specific method is as follows.

First, take I_LIt is on the occasion of data sampling period is 1s in electric discharge.

1st, row write state equation and the observational equation of fractional order Kalman integrated filter：

Wherein：△^rFor differential operator, r is differential order, when r is for decimal, △^rRepresent fractional order differential operator, when r is During integer, △^rFor integer differential operator.

Take fractional order element Z_WIt is both end voltage U_WFor quantity of state, have：

For parameter X_W, OCV_e, R_oChange with battery charge state (SoC) is slow, therefore：

Aforementioned four equation is rewritten as matrix form, has：

Take U_LFor the observed quantity of system, then have：

U_L=OCV_e-I_LR_o-U_W

Take：

Y=U_L

Then have：

After above-mentioned equation discretization, have：

Wherein, w, v represent state-noise and the observation noise of system respectively, generally it will be assumed that both are independent noise.

Defined according to Gr ü nwald-Letnikov fractional order differential：

Wherein：

Separately take：

Arrive the discretization recursion expression-form of Fractional Differential Equation as available from the above equation：

Definition：

According to Gr ü nwald-Letnikov fractional order differential definition, whereinAmount of calculation will be with The increase of time and constantly increase, this situation is not suitable for engineer applied, for this reason, above formula is rewritten as following form：

Step 2, utilize fractional order Kalman integrated filter estimated state and parameter value

Initialization：

Wherein, E [x] represents the mathematic expectaion of x, is experience preset value when method calculates,Represent x in initial time (k =0) estimated value,Represent the estimated value of the noise covariance in initial time (k=0) for the x.

The time of state, parameter and covariance matrix updates：

Wherein, Q_kIt is noise w_kCovariance,For k moment state and model parameter x_kPredictive value,For the k-1 moment State and model parameter x_k-1Correction value,Noise covariance matrix P for k moment x_kPredictive value,For k-1 moment x Noise covariance matrix P_k-1Correction value.

The measurement updaue of state, parameter and covariance matrix：

Wherein, R_kIt is noise v_kCovariance, L_kIt is k moment Kalman filter gain size.

Power prediction part：

The basic thought of power prediction：

In use, the size of output power value is determined electrokinetic cell by load characteristic.Different load classes Type, load operating mode all can lead to the power of battery to export the difference of situation.The purpose of maximum output prediction is in reliability While protection battery system, predict the maximum power output ability in certain moment for the battery, to have given play to the maximality of battery Energy.The premise of peak power prediction is reliably to protect battery system, prevents its super-charge super-discharge.

Key point has two：The reference operating mode of the constraints of cell operating status and power prediction；

The limiting constraint of cell operating status：The instruction manual of reference battery, the major constraints of cell operating status Condition has：Maximum charging current, maximum discharge current, charge cutoff voltage, discharge cut-off voltage, SoC operation interval etc..At this In invention, obtaining current, voltage, SoC are as constraints.

The reference operating mode of power prediction：If prediction battery, in the power output capacity in k+ Δ T moment, needs to know k～k+ Battery applying working condition between Δ T, but this operating mode is unknown.Conventional standard charged/discharged operating mode has：Constant current, constant voltage, Constant power load operating mode.Wherein, the amount of calculation minimum of battery peak power is solved based on constant current operating mode and battery can be reflected Power output capacity, therefore takes constant current operating mode can effectively prevent over-charging of battery mistake as the work operating mode between k～k+ Δ T Put.

1st, the Z of derivation fractional order element_WWhether it is linear time invariant element：

Assume U_WInitial value be 0, be I when applying amplitude_LStep current excitation when, voltage after Ns for the Uw rings Situation is answered to be calculated as follows：

Assume battery model parameter OCV_e、R_o、X_WDuring power prediction, numerical values recited is constant.

U_WVoltage response in k=1s is：

U_WVoltage response when k >=2 is：

Calculate：

Can obtain after calculating：

By the voltage response that this recursion obtains 10s and 60s it is：

Thus it is easy to infer Z_WElement is linear time invariant element.

Power prediction：

Due to Z_WElement is linear time invariant element, and response under current excitation for the element can be according to the superposition theorem of circuit Decomposed.In order to predict Z_WVoltage response at k+ Δ Ts, is classified as zero state response and zero input response：

Wherein：

Wherein zero state response is：

For k+10s moment, a=3.524；For k+60s moment, a=8.722.

Zero input responseDetermined by the data before the k moment, take time memory length L=of fractional order differential 60.

The zero input response in k+1 moment is：

K+ Δ T moment battery model element Z be can get by this recursion_WLocate zero input voltage response：

Wherein, when the k+10 moment, b=α, α are constant coefficient matrix.

When the k+60 moment, b=β, β are constant coefficient matrix.

Thus, can get battery constant-current discharge electric current is I_maxWhen battery the k+ Δ T moment voltage prediction value.

For the peak power prediction of energy type electrokinetic cell, take SoC bound, bound blanking voltage, maximum charge and discharge Electric current is as the extreme working position of battery.

When wherein, any one parameter value preferentially reaches during its restrained boundary it is believed that battery is assigned for this numerical value in this parameter To extreme working position, the now peak power predictive value in calculated input/output performance number as k+ Δ T moment.

The step of peak power prediction is as follows：

Electric discharge peak value power prediction：

If SoC is the restrictive condition of battery limit working condition,：

If U_LFor the restrictive condition of battery limit working condition, then：

Thus release, when terminal voltage U_tAs the extreme working position of battery constraints when, operating current is：

When the k+ Δ T moment reaches crest discharge power, discharge current value is：

The crest discharge power in k+ Δ T moment is：

Similar with electric discharge peak value power method, the feedback current peak power of battery can be calculated：

Now I_LBe on the occasion of.

If SoC is the restrictive condition of battery limit working condition,：

If U_LFor the restrictive condition of battery limit working condition, then：

When the k+ Δ T moment reaches peak value feedback power, feedback current value is：

Thus, obtain the peak power of battery current feedback：

If it follows that battery is run with constant current mode in k to k+ Δ T time, the electric discharge its peak work in k+ Δ T moment Rate, current feedback peak power are obtained by above formula prediction respectively.

If battery operated under other mode of operations, such as constant voltage or constant power mode, made with above-mentioned peak power Extreme working position for battery can effectively prevent from over-charging of battery from crossing putting.

Claims (2)

1. a kind of peak power Forecasting Methodology of electrokinetic cell, is characterized in that：It is realized by following steps：

Step A, the step of battery model parameter On-line Estimation, specially：

Step A1, when modeling to secondary cell, due to electric automobile operating condition frequency characteristic, therefore battery electrochemical impedance The impedance operator of the medium frequency in spectrum equivalent-circuit model is by conventional purely resistive element R and the letter of normal phase element Q parallel circuit Turn to purely resistive element R to describe, the battery electrochemical impedance spectrum equivalent-circuit model after being simplified；

Battery electrochemical impedance spectrum equivalent-circuit model after this simplification includes open-circuit voltage OCV_e, ohmic internal resistance R_oAnd fractional order Element Z_W；

Open-circuit voltage OCV_ePositive pole and ohmic internal resistance R_oOne end connect；Described ohmic internal resistance R_oThe other end and fractional order unit Part Z_WOne end connect；Open-circuit voltage OCV_eNegative pole and fractional order element Z_WThe other end as cell voltage U_LTwo ends；

Step A2, according to step A1 obtain simplification after battery electrochemical impedance spectrum equivalent-circuit model set up fractional order joint State equation needed for Kalman filter and observational equation, specially：

Take total current I flowing through secondary cell_LIt is on the occasion of data sampling period is 1s in electric discharge；

${\mathrm{\Δ}}^r=\frac{d^r}{{\mathrm{dt}}^r},r>0$

Wherein Δ^rFor differential operator, r is differential order, when r is for decimal, Δ^rRepresent fractional order differential operator, when r is integer When, Δ^rFor integer differential operator；

Take fractional order element Z_WBe both end voltage be U_WQuantity of state, have：

${\mathrm{\Δ}}^{0.5}{U}_W=\frac{1}{W}{I}_L=X_W{I}_L$

For battery model parameter, diffusion parameter X_W, open-circuit voltage OCV_eWith ohmic internal resistance R_oWith battery charge state SoC's Change is slow, therefore：

$\left\{\begin{array}{c}{\mathrm{\Δ}}^1{X}_W\≈0\\ {\mathrm{\Δ}}^{1}OC{V}_e\≈0\\ {\mathrm{\Δ}}^1{R}_o\≈0\end{array}\right.$

Will：

${\mathrm{\Δ}}^{0.5}{U}_W=\frac{1}{W}{I}_L=X_W{I}_L$

$\left\{\begin{array}{c}{\mathrm{\Δ}}^1{X}_W\≈0\\ {\mathrm{\Δ}}^{1}OC{V}_e\≈0\\ {\mathrm{\Δ}}^1{R}_o\≈0\end{array}\right.$

It is rewritten as matrix form, obtain the state equation of fractional order Kalman integrated filter：

${\mathrm{\Δ}}^{\left[\begin{array}{c}0.5\\ 1\\ 1\\ 1\end{array}\right]}\left[\begin{array}{c}{U}_W\\ X_W\\ {\mathrm{OCV}}_e\\ R_o\end{array}\right]=\left[\begin{array}{cccc}0& I_L& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{U}_W\\ X_W\\ {\mathrm{OCV}}_e\\ R_o\end{array}\right];$

Take U_LFor the observed quantity of system, then have：

U_L=OCV_e-I_LRo-U_W

I_LThe total current representing and flowing through battery；

Take：

Y=U_L

Obtain the observational equation of fractional order Kalman integrated filter：

$\left\{\begin{array}{c}{\mathrm{\Δ}}^{N}x=\left[\begin{array}{cccc}0& I_L& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]x\\ y=\[\begin{array}{cccc}-1& 0& 1& -I_L\end{array}\]x\end{array}\right.$

After the observational equation discretization of this fractional order Kalman integrated filter, have：

$\left\{\begin{array}{c}{\mathrm{\Δ}}^N{x}_k=\left[\begin{array}{cccc}0& I_{L,k-1}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]x_{k-1}+w\\ y_k=\[\begin{array}{cccc}-1& 0& 1& -I_{L,k}\end{array}\]x_k+v\end{array}\right.$

Wherein, w, v represent state-noise and the observation noise of system respectively；

Series definition according to fractional order differential：

${\mathrm{\Δ}}^N{x}_{k+1}=\underset{j=0}{\overset{k}{\Σ}}{(-1)}^j\left(\begin{array}{c}N\\ j\end{array}\right)x_{k-j}$

Wherein,

$\left(\begin{array}{c}N\\ j\end{array}\right)=diag\[\begin{array}{cccc}\left(\begin{array}{c}0.5\\ j\end{array}\right)& \left(\begin{array}{c}1\\ j\end{array}\right)& \left(\begin{array}{c}1\\ j\end{array}\right)& \left(\begin{array}{c}1\\ j\end{array}\right)\end{array}\],$

$\left(\begin{array}{c}r\\ j\end{array}\right)=\left\{\begin{array}{cc}1& forj=0\\ r(r-1)...(r-j+1)/j!& forj>0\end{array}\right.,$

Separately take：Obtain the discretization recursion expression-form of Fractional Differential Equation by above formula：

Definition：

$A_{k-1}=\frac{\∂f(x_{k-1},I_{L,k-1})}{\∂x_{k-1}}{|}_{x_{k-1}={\hat{x}}_{k-1}^{+}}=\left[\begin{array}{cccc}0& I_{L,k-1}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],$

$C_k=\frac{\∂g(x_k,I_{L,k})}{\∂x_k}{|}_{x_k={\hat{x}}_k^{-}}=\[\begin{array}{cccc}-1& 0& 1& -I_{L,k}\end{array}\]$

According to the series definition of fractional order differential, wherein：Amount of calculation will increase over time and not Disconnected increase, this situation is not suitable for engineer applied, for this reason, above formula is rewritten as following form：

$\underset{j=1}{\overset{k}{\Σ}}{(-1)}^j{\mathrm{\γ}}_j{x}_{k+1-j}=\underset{j=1}{\overset{L}{\Σ}}{(-1)}^j{\mathrm{\γ}}_j{x}_{k+1-j},\begin{array}{c}k\≤64,L=k\\ k>64,L=64\end{array}$

Step A3, the state equation needed for fractional order Kalman integrated filter using step A2 structure and observational equation, right State, parameter and covariance matrix carry out time renewal and measurement updaue according to fractional order federated Kalman filtering algorithm：

It is specially：

Initialization：

${\hat{x}}_0=E\[x\],P_0^{+}=E\[(x-{\hat{x}}_0){(x-{\hat{x}}_0)}^T\]$

Wherein, E [x] represents the mathematic expectaion of x, is experience preset value when method calculates,Represent the estimation in initial time for the x Value,Represent the estimated value of the noise covariance in initial time for the x；

The time of state, parameter and covariance matrix updates：

${\hat{x}}_k^{-}=f({\hat{x}}_{k-1}^{+},I_{L,k-1})$

$P_k^{-}=(A_{k-1}+{\mathrm{\γ}}_1)P_{k-1}^{+}{(A_{k-1}+{\mathrm{\γ}}_1)}^T+Q_k+\underset{j=2}{\overset{L}{\Σ}}{\mathrm{\γ}}_j{P}_{k-j}^{+}{\mathrm{\γ}}_j^T$

Wherein, Q_kIt is noise w_kCovariance,For k moment state and model parameter x_kPredictive value,For k-1 moment state With model parameter x_k-1Correction value,Noise covariance matrix P for k moment x_kPredictive value,For making an uproar of k-1 moment x Sound covariance matrix P_k-1Correction value；

The measurement updaue of state, parameter and covariance matrix：

$L_k=P_k^{-}{\left(C_k\right)}^T{\[C_k{P}_k^{-}{\left(C_k\right)}^T+R_k\]}^{-1}$

${\hat{x}}_k^{+}={\hat{x}}_k^{-}+L_k^x\[y_k-g({\hat{x}}_k^{-},I_{L,k})\]$

$P_k^{+}=(I_k-L_k{C}_k)P_k^{-}$

Wherein, R_kIt is noise v_kCovariance, L_kIt is k moment Kalman filter gain size；

Step A4, terminal voltage U of collection battery_LWith total current I flowing through secondary cell_L, using after the simplification that step A1 obtains The state needed for fractional order Kalman integrated filter after battery electrochemical impedance spectrum equivalent-circuit model and the renewal of step A3 Equation and observational equation, obtain open-circuit voltage OCV_e, ohmic internal resistance R_o, diffusion parameter X_WEstimated value, will obtain open-circuit voltage OCV_e, ohmic internal resistance R_o, diffusion parameter X_WEstimated value as battery estimated result, complete based on fractional order joint karr The secondary cell of graceful filtering simplifies impedance spectrum model parameter On-line Estimation；

Step B, the battery model parameter On-line Estimation result being obtained according to step A, the step carrying out power prediction：

Step B1, derivation fractional order element Z_WWhether it is linear time invariant element：

If both end voltage U of fractional order element_WInitial value be 0, be I when applying amplitude_LStep current excitation when, point Both end voltage U of number rank element_WVoltage response situation after N s is calculated as follows, and N is positive number：

If the open-circuit voltage OCV of battery model_e, ohmic internal resistance R_o, diffusion parameter X_WDuring power prediction, numerical values recited is not Become；

Then U_WVoltage response in k=1s is：

${\hat{U}}_{W,1}={\hat{X}}_{W,0}{I}_L-{(-1)}^1\left(\begin{array}{c}0.5\\ 1\end{array}\right)U_{W,0}$

Wherein, symbol ^ represents predictive value；

U_WVoltage response when k >=2 is：

${\hat{U}}_{W,k+1}={\hat{X}}_{W,0}{I}_L-\underset{j=1}{\overset{L}{\Σ}}{(-1)}^j\left(\begin{array}{c}0.5\\ j\end{array}\right){\hat{U}}_{W,k+1-j}+w$

Calculate：

${\hat{U}}_{W,1}={\hat{X}}_W{I}_L$

${\hat{U}}_{W,2}={\hat{X}}_{W,0}{I}_L-({\left(-1\right)}^1\left(\begin{array}{c}0.5\\ 1\end{array}\right){\hat{U}}_{W,1}+{\left(-1\right)}^2\left(\begin{array}{c}0.5\\ 2\end{array}\right)U_{W,0})$

After calculating：

${\hat{U}}_{W,2}=1.5{\hat{X}}_{W,0}{I}_L$

By the voltage response that this recursion obtains 10s and 60s it is：

${\hat{U}}_{W,10}=3.524{\hat{X}}_{W,0}{I}_L$

${\hat{U}}_{W,60}=8.722{\hat{X}}_{W,0}{I}_L$

Thus, when battery model parameter X_WWhen constant or slowly varying, output valve U_WWith I_LFor linear relationship, infer fractional order Element Z_WFor linear time invariant element；

Then, the power forecasting method of fractional order element is：

In order to predict voltage response at k+ Δ Ts for the fractional order elementThis voltage response is divided into zero state responseAnd zero input response

${\hat{U}}_{W,k+\mathrm{\Δ}T}={\hat{U}}_{W,k+\mathrm{\Δ}T}^{zs}+{\hat{U}}_{W,k+\mathrm{\Δ}T}^{zi}$

Wherein zero state response is：

${\hat{U}}_{W,k+\mathrm{\Δ}T}^{zs}=a{\hat{X}}_{W,k}{I}_{\max }$

For k+10s moment, a=3.524；For k+60s moment, a=8.722；

Zero input responseDetermined by the data before the k moment, take time memory length L=60 of fractional order differential；

The zero input response in k+1 moment is：

${\hat{U}}_{W,k+1}^{zi}=-\underset{j=1}{\overset{L}{\Σ}}{(-1)}^j\left(\begin{array}{c}0.5\\ j\end{array}\right){\hat{U}}_{W,k+1-j}^{zi}$

The response of k+ Δ T moment fractional order element zero input voltage be can get by this recursion：

${\hat{U}}_{W,k+\mathrm{\Δ}T}^{zi}=\underset{j=1}{\overset{L}{\Σ}}{b}_{k+1-j}{\hat{U}}_{W,k+1-j}$

Wherein,It is battery model fractional order element Z_WEstimated value between (k+1-L)～k moment for the two ends terminal voltage, When predicting k+10 moment, b=α, α are one group of constant coefficient matrix；

When predicting k+60 moment, b=β, β are another set constant coefficient matrix；

Thus, can get battery constant-current discharge electric current is I_maxWhen battery the k+ Δ T moment terminal voltage predictive value：

Electric discharge peak value power prediction method be：

If battery charge state is the restrictive condition of battery limit working condition, SoC_minIt is that battery discharge terminates state-of-charge Minima, then obtain maximum discharge current value now：

$I_{max,SoC}=\frac{({\mathrm{SoC}}_k-{\mathrm{SoC}}_{min})\×Capacity}{\mathrm{\Δ}T}$

Capacity is battery capacity value, and unit is ampere * hour (Ah), SoC_kFor the battery charge state in k moment, SoC_minFor The minimum state-of-charge that battery discharge limits；

If battery terminal voltage U_LFor the restrictive condition of battery limit working condition, if now with maximum discharge current I_maxBattery is put Electricity, battery model parameter OCV_ePredictive value in the k+ Δ T moment is：

In above formula,It is by calculated battery model parameter OCV of fractional order federated Kalman filtering algorithm_eIn k The estimated value carved,For OCV_ePredictive value in the k+ Δ T moment；I_maxIt is as battery limit working condition in terminal voltage Maximum discharge current value during restrictive condition, is value to be solved,It is the maximum discharge current of battery；OCV_kIt is in the k moment Battery open circuit voltage values；For battery in k+ Δ_TMoment it is assumed that withCorresponding battery open circuit electricity during constant-current discharge Pressure value；Due in most cases, the change of OCV is all less and slow, it can thus be assumed that the open-circuit voltage within the Δ T period becomes Change amount is linear with discharge current；

And then calculate it is assumed that battery is I in discharge current_maxWhen, estimated value U of battery terminal voltage_L,k+ΔT：

Wherein, the estimated value of terminal voltage be k moment open-circuit voltage estimated value, open circuit voltage variations value, ohmic internal resistance in the Δ T period Voltage difference, fractional order element Z_WZero state response magnitude of voltage, zero input response magnitude of voltage sum；

Thus release, when with terminal voltage U_LAs the extreme working position of battery restrictive condition when, the maximum operating currenbt of battery For：

When the k+ Δ T moment reaches electric discharge peak power, consider the restrictive condition of the extreme working position of above-mentioned battery, maximum Discharge current value is：

$I_{max}=min(I_{max,SoC},I_{max,U_L},I_{\max }^{\lim })$

The electric discharge peak power in k+ Δ T moment is：

The Forecasting Methodology of the Forecasting Methodology of current peak feedback power and above-mentioned electric discharge peak power is in the same manner：

If I_LBe on the occasion of；

If SoC is the restrictive condition of battery limit working condition, SoC_maxIt is the maximum SOC of battery, then obtain now Minimum feedback current value：

$I_{min,SoC}=\frac{({\mathrm{SoC}}_k-{\mathrm{SoC}}_{max})\×Capacity}{\mathrm{\Δ}T}$

If U_LFor the restrictive condition of battery limit working condition, then obtain minimum feedback current value now：

When the k+ Δ T moment reaches current peak feedback power, consider above-mentioned restrictive condition, minimum feedback current value is：

$I_{min}=max(I_{min,SoC},I_{min,U_L},I_{\min }^{\lim })$

Thus, obtain battery current peak value feedback power：

Complete the peak power prediction of electrokinetic cell.

2. a kind of peak power Forecasting Methodology of electrokinetic cell according to claim 1 is it is characterised in that in step A,

Due to the electrochemical impedance spectroscopy test data based on battery for the battery model used in method, battery model parameter has clearly Physical significance, ohmic internal resistance R_oPhysical significance be：

R_o≈R_Ω+R_SEI+R_ct

Wherein, R_ΩFor high frequency ohmage, R_SEIFor SEI membrane impedance, R_ctFor Charge-transfer resistance；

In addition, weber impedance to be defined by below equation：

$Z_W=\frac{1}{W{\left(jw\right)}^{0.5}}$

Wherein, W is ionic diffusion coefficient, for the ease of impedance parameter On-line Estimation, takes：

$X_W=\frac{1}{W}$

Obtain：

$Z_W=\frac{X_W}{{\left(jw\right)}^{0.5}}.$