CN104392080A - Lithium-battery variable fractional order and equivalent circuit model and identification method thereof - Google Patents

Abstract

The invention discloses a lithium-battery variable fractional order and equivalent circuit model and an identification method thereof. The lithium-battery variable fractional order and equivalent circuit comprises a run time circuit and a battery I-V characteristic circuit, wherein a capacitor in the battery I-V characteristic circuit is a variable fractional order capacitor. A second order RC circuit model is generalized to a non-integer order, and the model parameters and the number of fractional order of different SOC are identified based on a least square method, so that the fractional order and equivalent circuit varying order according to the SOC is obtained. The instruction of fractional order realizes the continuous change of the order number of the model, so that the model is relatively stable, good in dynamic property and high in precision. The variation of fractional order realizes more freedom and more flexibility and innovation of the model. As the number of RC networks is not increased, the fractional order model effectively solves the contradiction between the accuracy and practicality of the model, is suitable for various working conditions of batteries, and has high practical value. The invention provides a precise battery model easy to realize for precise estimation of SOC.

Description

A kind of lithium battery fractional order becomes rank equivalent-circuit model and discrimination method thereof

Technical field

The present invention relates to a kind of lithium battery fractional order and become rank equivalent-circuit model and discrimination method thereof.

Background technology

In order to tackle energy crisis and environmental pollution, electric automobile arises at the historic moment and has become the focus paid close attention in the whole world.Vehicle mounted dynamic battery is as the critical component of electric automobile, and its performance is most important to the dynamic property of car load, economy and security, is the key factor of restriction electric automobile scale development.Lithium battery prodigiosin metric density is high, long service life, cost performance are good and monomer voltage advantages of higher, progressively become the dynamic origin of hybrid vehicle or pure electric automobile.Accurate battery model to the appropriate design of onboard power lithium battery and safe operation significant, be the basis of battery SOC (state-of-charge), SOH (health status) evaluation method.

But, set up one accurately and the simple battery model of structure is by no means easy, this is because the chemical reaction of lithium battery interior relates to the complexity conversion of electric energy, chemical energy, heat energy, there is the non-linear and uncertain of height.At present, conventional battery model can be following five classes by the difference of modeling mechanism: 1. electrochemical model, 2. analytical model, 3. probabilistic model, 4. neural network model and 5. equivalent-circuit model.Wherein, equivalent-circuit model because of its simple, intuitive form and be suitable for the advantage such as electrical design and emulation and become a kind of new model be widely used.In equivalent-circuit model, Order RC model compares that other equivalent-circuit model physical significances are clear, identification of Model Parameters test easily execution, parameter identification method system, model accuracy are higher, can more accurately, the dynamic perfromance of simulated battery intuitively.But in battery charging and discharging initial stage and latter stage, because model order is lower, there is larger error of fitting in Order RC model, can not the static and dynamic performance of accurately simulated battery.Although the series connection exponent number increasing RC can improve the accuracy of battery model, the charge-discharge characteristic of electrokinetic cell can better be simulated, if but the exponent number of electrokinetic cell model is too high, the parameter in model is obtained by being unfavorable for, and also greatly can increase the calculated amount of model, system even can be caused to shake, so the exponent number of RC also should be limited on the other hand.Therefore, fixed structure equivalent-circuit model is difficult to describe the steep middle flat non-linear voltage characteristic in lithium battery two ends, can not solve the contradiction between the accuracy of model and practicality.

For this reason, Chinese invention patent application (application number 201410185885.7) and utility model (patent No. ZL201420226360.9) propose a kind of change rank RC equivalent-circuit model based on AIC criterion, by slightly increasing the complexity of model, the steep middle flat non-linear voltage characteristic in lithium battery two ends can be described more exactly, error is within 0.04V, efficiently solve the contradiction between model complexity and practicality, there is higher practical value.But this model is the battery model on integer rank, the switching of model can only be the change on integer rank, and therefore model order fluctuation is large, and do not meet the rule of development of occurring in nature gradual change, therefore model accuracy is very restricted.In fact, inside battery electrochemical reaction process is extremely complicated, comprises conductive ion transfer, internal electrical chemical reaction, discharge and recharge hesitation and concentration difference diffusional effect etc., shows stronger nonlinear characteristic, be more suitable for simulating with fractional model.Contrast integer model, fractional order battery model has more degree of freedom, larger flexibility and new meaning in design.Meanwhile, their introducing too increases many new phenomenons and rule, and what have that conventional integer rank battery model cannot realize is superior.

Summary of the invention

For solving the deficiency that prior art exists, the invention discloses a kind of lithium battery fractional order and become rank equivalent-circuit model and discrimination method thereof, according to the principle of electrochemical reaction of lithium ion battery, tradition Order RC equivalent-circuit model uses the RC network on two integer rank to describe polarization effect and the concentration difference effect of battery, two of model integer rank RC network are generalized to non-integral order (fractional order) by the present invention, and based on the model parameter at the different SOC place of least squares identification and order, thus obtain a fractional order equivalent-circuit model according to SOC change rank.The introducing of fractional order achieves the consecutive variations of model order, such that model is more stable, dynamic property is more excellent, precision is higher.Become rank parameter owing to adding fractional order, model obtains more degree of freedom, larger flexibility and new meaning.This model realizes on the basis of traditional Order RC model, do not increase the number of model RC network, efficiently solve the contradiction between model accuracy and simplicity, have higher practical value.

For achieving the above object, concrete scheme of the present invention is as follows:

A kind of lithium battery fractional order becomes rank equivalent-circuit model, comprises circuit and cell I-V characteristic circuit working time, described working time circuit and cell I-V characteristic circuit carry out Signal transmissions by CCCS and Voltage-controlled Current Source; Described working time, circuit comprised the self discharge resistance R of battery _dand electric capacity C _q, resistance R _dwith electric capacity C _qbe connected in parallel on the two ends of CCCS, one end ground connection of CCCS; The positive terminal of the Voltage-controlled Current Source of described cell I-V characteristic circuit is connected with one end of two branch roads be in parallel, negative pole end is connected with the negative pole end of battery model, and each branch road of described two RC network branch roads be in parallel includes two fractional order RC loops be in series and an internal resistance R _o, the described other end of two branch roads be in parallel is connected with the positive terminal of battery model.

In described cell I-V characteristic circuit in two RC network branch roads be in parallel, discharge paths comprises the diode D connected successively _d, fractional order electric capacity FOE _1dwith resistance R _1dthe fractional order RC loop of composition, fractional order electric capacity FOE _2dwith resistance R _2dthe fractional order RC loop of composition and resistance R _od;

Charging paths comprises the reversal connection diode D connected successively _d, fractional order electric capacity FOE _1cwith resistance R _1cthe fractional order RC loop of composition, fractional order electric capacity FOE _2cwith resistance R _2cthe fractional order RC loop of composition and resistance R _oc.

Described working time circuit and I-V characteristic circuit set up contact by a CCCS and Voltage-controlled Current Source, when carrying out discharge and recharge to battery, load current i _batby CCCS to electric capacity C _qcarry out discharge and recharge, change C _qthe electricity stored, the change of characterizing battery SOC, C _qboth end voltage OCV also changes thereupon, and the controlled voltage source OCV of I-V characteristic circuit changes with the change of SOC.

Described electric capacity C _qrepresent the active volume of battery, C _q=3600C _ahf ₁f ₂, wherein, C _ahfor being the battery capacity of unit by ampere-hour, f ₁and f ₂the modifying factor of battery cycle life and temperature respectively.

The electric current of described CCCS is the end current i of battery _bat, the load current i when battery carries out discharge and recharge _batby CCCS to electric capacity C _qcarry out discharge and recharge, change electric capacity C _qthe electricity of middle storage, thus the change of characterizing battery SOC.

The voltage at the two ends of described CCCS is battery open circuit voltage OCV.

Two RC network branch roads be in parallel are RC network discharge paths and RC network charging paths respectively, and the electric capacity in two RC network branch roads is fractional order electric capacity.

Described RC network discharge paths fractional order element FOE _1dand FOE _2dexponent number α, β is different and change with battery SOC state, and meets 0≤α _d, β _d≤ 1.Work as α _d, β _dwhen=0, fractional order element FOE is equivalent to a resistance, works as α _d, β _dwhen=1, fractional order element FOE is equivalent to an integer rank electric capacity; As 0< α _d, β _dduring <1, fractional order element FOE is a fractional order electric capacity;

Described RC network charging paths fractional order element FOE _1cand FOE _2cexponent number α, β is different and change with battery SOC state, and meets 0≤α _c, β _c≤ 1.Work as α _c, β _cwhen=0, fractional order element FOE is equivalent to a resistance, works as α _c, β _cwhen=1, fractional order element FOE is equivalent to an integer rank electric capacity; As 0< α _c, β _cduring <1, fractional order element FOE is a fractional order electric capacity.

Lithium battery fractional order becomes a discrimination method for rank equivalent-circuit model, comprises the following steps:

Step one: write out the discharge process of lithium battery and the fractional order mathematics model expression of static condition;

Step 2: carry out constant current charge-discharge to lithium battery, obtains the active volume C of battery model _qwith self discharge resistance R _d;

Step 3: pulsed discharge test is carried out to lithium battery, obtaining the moment that different SOC places battery starts the moment drop-out value of battery terminal voltage when discharging, electric discharge terminates rear battery terminal voltage rises to the data such as the zero input response of value, discharge current and battery terminal voltage;

Step 4: the data obtained according to step 3, based on parameter and the exponent number of least squares identification model;

Step 5: the battery model parameter calculated according to step 4 calculates open-circuit voltage OCV, the ohmic internal resistance R at different SOC place _0d, activation polarization internal resistance R _1d, activation polarization fractional order electric capacity FOE _1d, concentration polarization internal resistance R _2dwith concentration polarization fractional order electric capacity FOE _2d;

Step 6: the model parameter obtained according to step 5, based on least squares identification open-circuit voltage OCV, ohmic internal resistance R _0d, activation polarization internal resistance R _1d, activation polarization fractional order electric capacity FOE _1d, concentration polarization internal resistance R _2dwith concentration polarization fractional order electric capacity FOE _2dand the relation between SOC;

Step 7: the parameter obtained according to step 2 to six, builds lithium battery fractional order and becomes rank equivalent-circuit model in Matlab.

In discharge process, the terminal voltage of lithium battery can be expressed as:

$U_{\mathrm{bat}}={\mathrm{OCV}}_d-i_{\mathrm{dis}}\&CenterDot;R_{0d}-U_{1d}(0+)\&CenterDot;(1-e^{-t^{\mathrm{\&alpha;}}/{\mathrm{\&tau;}}_{1d}})-U_{2d}(0+)\&CenterDot;(1-e^{-t^{\mathrm{\&beta;}}/{\mathrm{\&tau;}}_{2d}})---\left(1\right)$

In formula, U _batfor battery terminal voltage; R _0dfor ohmic internal resistance; OCV _dfor electric discharge open-circuit voltage; α, β are fractional order element FOE _1dand FOE _2dexponent number, meet 0< α, β≤1; i _disfor discharge current; τ _1d, τ _2dbe respectively the time constant of two RC network.

Work as α, during β=0, fractional order element FOE is equivalent to a resistance, works as α, and during β=1, fractional order element FOE is equivalent to an electric capacity;

U _1d(0+) and U _2d(0+) for battery discharge terminates the terminal voltage initial value of moment two fractional order RC branch roads, its value can be expressed as:

U _1d(0+)＝i _dis·R _1d(2)

U _2d(0+)＝i _dis·R _2d(3)

After battery discharge terminates, the terminal voltage of battery can be expressed as:

$U_{\mathrm{bat}}={\mathrm{OCV}}_d-U_{1d}(0+)\&CenterDot;(1-e^{-t^a/{\mathrm{\&tau;}}_{1d}})-U_{2d}(0+)\&CenterDot;(1-e^{-t^b/{\mathrm{\&tau;}}_{2d}})---\left(4\right)$

In formula, the polarizing voltage of battery with reduce gradually along with the growth of time, as t → ∞, with be tending towards 0, now battery terminal voltage U _batequal the open-circuit voltage OCV of battery.

The detailed process of described step 5 is: due to the existence of battery ohmic internal resistance, when the cell is discharged, and battery terminal voltage can fall instantaneously, and its value is designated as Δ U ₁; When battery stops electric discharge, battery terminal voltage can rise to instantaneously, and its value is designated as Δ U ₂, therefore, the ohmic internal resistance R of battery ₀can be obtained by following formula:

$R_0=\frac{\mathrm{\&Delta;}{U}_1+\mathrm{\&Delta;}{U}_2}{2i_{\mathrm{bat}}}---\left(5\right)$

Activation polarization internal resistance R _1dcan be obtained by following formula:

$R_{1d}=\frac{U_{1d}(0+)}{i_{\mathrm{dis}}}---\left(6\right)$

Concentration polarization internal resistance R _2dcan be obtained by following formula:

$R_{2d}=\frac{U_{2d}(0+)}{i_{\mathrm{dis}}}---\left(7\right)$

Activation polarization fractional order electric capacity FOE _1dcan be obtained by following formula:

${\mathrm{FOE}}_{1d}=\frac{{\mathrm{\&tau;}}_{1d}}{R_{1d}}---\left(8\right)$

Concentration polarization fractional order electric capacity FOE _2dcan be obtained by following formula:

${\mathrm{FOE}}_{2d}=\frac{{\mathrm{\&tau;}}_{2d}}{R_{1d}}---\left(9\right)$

In described step 6: open-circuit voltage OCV and SOC exists nonlinear relationship, physical relationship formula is:

$\mathrm{OCV}=a_0+a_1\&CenterDot;\ln \mathrm{SOC}+a_2\&CenterDot;\ln (1-\mathrm{SOC})+\frac{a_3}{\mathrm{SOC}}+a_4\&CenterDot;\mathrm{SOC}---\left(10\right)$

In formula, a ₀-a ₄for constant, obtained based on least squares identification by experimental data.

Battery ohmic internal resistance R _odwith the relational expression of SOC be:

R _o(SOC)＝b ₀·e ^-SOC+b ₁+b ₂·SOC-b ₃·SOC ²+b ₄·SOC ³(11)

In formula, b ₀-b ₄for constant, obtained based on least squares identification by experimental data.

Activation polarization internal resistance R _1dwith the relational expression of SOC be:

R _1d(SOC)＝c ₀·e ^-SOC+c ₁+c ₂·SOC-c ₃·SOC ²+c ₄·SOC ³(12)

In formula, c ₀-c ₄for constant, obtained based on least squares identification by experimental data.

Activation polarization fractional order electric capacity FOE _1dwith the relational expression of SOC be:

FOE _1d(SOC)＝d ₀·SOC ⁵+d ₁·SOC ⁴+d ₂·SOC ³+d ₃·SOC ²+d ₄·SOC+d ₅(13)

In formula, d ₀-d ₅for constant, obtained based on least squares identification by experimental data.

Concentration polarization internal resistance R _2dwith the relational expression of SOC be:

R _2d(SOC)＝e ₀·e ^-SOC+e ₁+e ₂·SOC-e ₃·SOC ²+e ₄·SOC ³(14)

In formula, e ₀-e ₄for constant, obtained based on least squares identification by experimental data.

Concentration polarization fractional order electric capacity FOE _2dwith the relational expression of SOC be:

FOE _2d(SOC)＝f ₀·SOC ⁵+f ₁·SOC ⁴+f ₂·SOC ³+f ₃·SOC ²+f ₄·SOC+f ₅(15)

In formula, f ₀-f ₅for constant, obtained based on least squares identification by experimental data.

Activation polarization fractional order electric capacity FOE _1dthe relational expression of exponent number and SOC is:

α(SOC)＝g ₀·SOC ⁴+g ₁·SOC ³+g ₂·SOC ²+g ₃·SOC+g ₄(16)

In formula, g ₀-g ₄for constant, obtained based on least squares identification by experimental data.

Concentration polarization fractional order electric capacity FOE _2dthe relational expression of exponent number and SOC is:

β(SOC)＝h ₀·SOC ⁴+h ₁·SOC ³+h ₂·SOC ²+h ₃·SOC+h ₄(17)

In formula, h ₀-h ₄for constant, obtained based on least squares identification by experimental data.

Beneficial effect of the present invention:

1. traditional Order RC equivalent-circuit model is generalized to fractional order, and based on the model parameter at the different SOC place of least squares identification and order, obtains a fractional order equivalent-circuit model according to SOC change rank;

2. lithium battery is because of its special material and chemical characteristic, show fractional order dynamic behavior, describe its precision of battery behavior with integer rank to be very restricted, and Bian fractional calculus is when describing those object with fractional order characteristic itself, can the intrinsic propesties of description object and behavior thereof better;

3., owing to adding this unknown parameter of fractional order exponent number, model obtains more degree of freedom, larger flexibility and new meaning;

4. because fractional calculus has certain memory function, and more meet the simple philosophical viewpoint of the general continuous print of nature, fractional order becomes rank equivalent-circuit model thus obtains higher precision, better dynamic property and stability;

5. contrast and traditional Order RC model, owing to not increasing the number of RC network, the present invention efficiently solves the contradiction between model accuracy and practicality, there is higher practical value, and being applicable to the constant current charge-discharge of battery, pulse discharge and recharge and UDDS state of cyclic operation, the accurate estimation for SOC provides one accurately and the battery model easily realized.

Accompanying drawing explanation

Fig. 1 is that lithium battery fractional order of the present invention becomes rank equivalent-circuit model structural representation, and wherein c mark represents charging, and d mark represents electric discharge;

Fig. 2 is the response process figure of battery cell voltage under pulse charge of the present invention;

Fig. 3 is the response process figure of battery cell voltage under pulsed discharge of the present invention;

Fig. 4 is the recovery response comparison diagram of battery terminal voltage after change rank of the present invention fractional order, integer rank and the modeling pulsed discharge of fixed fraction rank, and wherein (a) is overall diagram, and (b) is partial enlarged drawing;

Fig. 5 is the graph of a relation of open-circuit voltage OCV and SOC under pulse charge of the present invention;

Fig. 6 is ohmic internal resistance R under pulse charge of the present invention ₀with the graph of a relation of SOC;

Fig. 7 is activation polarization internal resistance R under pulse charge of the present invention _1cwith the graph of a relation of SOC;

Fig. 8 is activation polarization fractional order electric capacity FOE under pulse charge of the present invention _1cwith the graph of a relation of SOC;

Fig. 9 is concentration polarization internal resistance R under pulse charge of the present invention _2dwith the graph of a relation of SOC;

Figure 10 is concentration polarization fractional order electric capacity FOE under pulse charge of the present invention _2cwith the graph of a relation of SOC;

Figure 11 is activation polarization fractional order electric capacity FOE under pulse charge of the present invention _1cthe graph of a relation of exponent number and SOC;

Figure 12 is concentration polarization fractional order electric capacity FOE under pulse charge of the present invention _2cthe graph of a relation of exponent number and SOC;

Figure 13 is the graph of a relation of open-circuit voltage OCV and SOC under pulsed discharge of the present invention;

Figure 14 is ohmic internal resistance R under pulsed discharge of the present invention ₀with the graph of a relation of SOC;

Figure 15 is activation polarization internal resistance R under pulsed discharge of the present invention _1dwith the graph of a relation of SOC;

Figure 16 is activation polarization fractional order electric capacity FOE under pulsed discharge of the present invention _1dwith the graph of a relation of SOC;

Figure 17 is concentration polarization internal resistance R under pulsed discharge of the present invention _2dwith the graph of a relation of SOC;

Figure 18 is concentration polarization fractional order electric capacity FOE under pulsed discharge of the present invention _2dwith the graph of a relation of SOC;

Figure 19 is activation polarization fractional order electric capacity FOE under pulsed discharge of the present invention _1dthe graph of a relation of exponent number and SOC;

Figure 20 is concentration polarization fractional order electric capacity FOE under pulsed discharge of the present invention _2dthe graph of a relation of exponent number and SOC;

Figure 21 becomes rank fractional order cell equivalent-circuit model voltage output map under pulsed discharge of the present invention;

Figure 22 becomes rank fractional order cell equivalent-circuit model voltage output map under pulse charge of the present invention;

Figure 23 becomes rank fractional order cell equivalent-circuit model voltage output map under constant-current discharge of the present invention;

Figure 24 becomes rank fractional order cell equivalent-circuit model voltage output map under constant-current charge of the present invention.

Embodiment:

Below in conjunction with accompanying drawing, the present invention is described in detail:

Build battery model and refer to that applied mathematics theory goes to describe response characteristic and the bulk properties of actual battery all sidedly as far as possible.So-called response characteristic refers to the terminal voltage of battery and the corresponding relation of load current; Bulk properties refer to the relation between the built-in variable ohmic internal resistance of battery, polarization resistance and polarizing voltage and SOC, temperature.

Be illustrated in figure 1 lithium battery fractional order disclosed by the invention and become rank equivalent-circuit model, comprise circuit and I-V characteristic circuit working time, wherein, I-V characteristic circuit comprises two-way branch road, and each branch road comprises the fractional order RC loop that a two groups one fractional order electric capacity FOE and resistor coupled in parallel forms.Described working time, circuit comprised the self discharge resistance R of battery _d, electric capacity C _qwith CCCS circuit, resistance R _dwith electric capacity C _qbe connected in parallel on the controlled source two ends of CCCS, one end ground connection of independent current source.

I-V characteristic circuit comprises ohmic internal resistance R ₀, activation polarization internal resistance R ₁, activation polarization fractional order electric capacity FOE ₁, concentration polarization internal resistance R ₂, concentration polarization fractional order electric capacity FOE ₂with CCCS, Voltage-controlled Current Source circuit, wherein:

The positive pole of the controlled source of Voltage-controlled Current Source circuit connects two-way, and a road connects diode D _drear contact resistance R _1d, resistance R _2d, resistance R _odthe positive pole of rear connection battery model, a road reversal connection diode D _crear contact resistance R _1c, resistance R _2c, resistance R _octhe positive pole of rear connection battery model.Fractional order electric capacity FOE _1dbe connected in parallel on resistance R _1dtwo ends; Fractional order electric capacity FOE _1cbe connected in parallel on resistance R _1ctwo ends; Fractional order electric capacity FOE _2dbe connected in parallel on resistance R _2dtwo ends; Fractional order electric capacity FOE _2cbe connected in parallel on resistance R _2ctwo ends; Voltage between the controlled source positive and negative electrode of Voltage-controlled Current Source circuit is battery open circuit voltage OCV.

Working time circuit and I-V characteristic circuit set up contact by a Flow Control current source and voltage controlled voltage source, when carrying out discharge and recharge to battery, load current i _batby Flow Control current source to electric capacity C _qcarry out discharge and recharge, change C _qthe electricity stored, the change of characterizing battery SOC, C _qboth end voltage OCV also changes thereupon, and the controlled voltage source OCV of I-V characteristic circuit changes with the change of SOC.

Electric capacity C _qrepresent the active volume of battery, C _q=3600C _ahf ₁f ₂, wherein, C _ahfor being the battery capacity of unit by ampere-hour, f ₁and f ₂the modifying factor of battery cycle life and temperature respectively.

The electric current of the controlled source of CCCS is the end current i of battery _bat, the load current i when battery carries out discharge and recharge _batby CCCS to electric capacity C _qcarry out discharge and recharge, change electric capacity C _qthe electricity of middle storage, thus the change of characterizing battery SOC.

The voltage at the controlled source two ends of described CCCS is battery open circuit voltage OCV.

Apply the discrimination method that above-mentioned lithium battery fractional order becomes rank equivalent-circuit model, for battery discharge, charging discrimination method is identical with electric discharge, does not repeat them here.Comprise the following steps:

Step one: write out the discharge process of lithium battery and the fractional order mathematics model expression of static condition;

Step 2: carry out constant current charge-discharge to lithium battery, obtains the active volume C of battery model _qwith self discharge resistance R _d;

Step 3: pulsed discharge test is carried out to lithium battery, obtaining the moment that different SOC places battery starts the moment drop-out value of battery terminal voltage when discharging, electric discharge terminates rear battery terminal voltage rises to the data such as the zero input response of value, discharge current and battery terminal voltage;

Step 4: the data obtained according to step 3, based on parameter and the exponent number of least squares identification model;

Step 5: the battery model parameter calculated according to step 4 calculates open-circuit voltage OCV, the ohmic internal resistance R at different SOC place _0d, activation polarization internal resistance R _1d, activation polarization fractional order electric capacity FOE _1d, concentration polarization internal resistance R _2dwith concentration polarization fractional order electric capacity FOE _2d;

Step 6: the model parameter obtained according to step 5, based on least squares identification open-circuit voltage OCV, ohmic internal resistance R _0d, activation polarization internal resistance R _1d, activation polarization fractional order electric capacity FOE _1d, concentration polarization internal resistance R _2dwith concentration polarization fractional order electric capacity FOE _2dand the relation between SOC;

Step 7: the parameter obtained according to step 2 to six, builds lithium battery fractional order and becomes rank equivalent-circuit model in Matlab.

Be illustrated in figure 2 the response process figure of battery cell voltage under pulse charge of the present invention; Be illustrated in figure 3 the response process figure of battery cell voltage under pulsed discharge of the present invention; In process of pulse discharge, the terminal voltage of battery can be expressed as:

$U_{\mathrm{bat}}={\mathrm{OCV}}_d-i_{\mathrm{dis}}\&CenterDot;R_{0d}-U_{1d}(0+)\&CenterDot;(1-e^{-t^{\mathrm{\&alpha;}}/{\mathrm{\&tau;}}_{1d}})-U_{2d}(0+)\&CenterDot;(1-e^{-t^{\mathrm{\&beta;}}/{\mathrm{\&tau;}}_{2d}})---\left(1\right)$

In formula, U _batfor battery terminal voltage; R _0dfor ohmic internal resistance; OCV _dfor electric discharge open-circuit voltage; α, β are fractional order element FOE _1dand FOE _2dexponent number, meet 0< α, β≤1.Work as α, during β=0, fractional order element FOE is equivalent to a resistance, works as α, and during β=1, fractional order element FOE is equivalent to an electric capacity.U _1d(0+) and U _2d(0+) for battery discharge terminates the terminal voltage initial value of moment two fractional order RC branch roads, its value can be expressed as:

U _1d(0+)＝i _dis·R _1d(2)

U _2d(0+)＝i _dis·R _2d(3)

After battery discharge terminates, the terminal voltage of battery can be expressed as:

$U_{\mathrm{bat}}={\mathrm{OCV}}_d-U_{1d}(0+)\&CenterDot;(1-e^{-t^a/{\mathrm{\&tau;}}_{1d}})-U_{2d}(0+)\&CenterDot;(1-e^{-t^b/{\mathrm{\&tau;}}_{2d}})---\left(4\right)$

In formula, the polarizing voltage of battery with reduce gradually along with the growth of time, as t → ∞, with be tending towards 0, now battery terminal voltage U _batequal the open-circuit voltage OCV of battery.

The concrete grammar of described step 5 is: due to the existence of battery ohmic internal resistance, when the cell is discharged, and battery terminal voltage can fall instantaneously, and its value is designated as Δ U ₁; When battery stops electric discharge, battery terminal voltage can rise to instantaneously, and its value is designated as Δ U ₂.Therefore, the ohmic internal resistance R of battery ₀can be obtained by following formula:

$R_0=\frac{\mathrm{\&Delta;}{U}_1+\mathrm{\&Delta;}{U}_2}{2i_{\mathrm{bat}}}---\left(5\right)$

Activation polarization internal resistance R _1dcan be obtained by following formula:

$R_{1d}=\frac{U_{1d}(0+)}{i_{\mathrm{dis}}}---\left(6\right)$

Concentration polarization internal resistance R _2dcan be obtained by following formula:

$R_{2d}=\frac{U_{2d}(0+)}{i_{\mathrm{dis}}}---\left(7\right)$

Activation polarization fractional order electric capacity FOE _1dcan be obtained by following formula:

${\mathrm{FOE}}_{1d}=\frac{{\mathrm{\&tau;}}_{1d}}{R_{1d}}---\left(8\right)$

Concentration polarization fractional order electric capacity FOE _2dcan be obtained by following formula:

${\mathrm{FOE}}_{2d}=\frac{{\mathrm{\&tau;}}_{2d}}{R_{1d}}---\left(9\right)$

The concrete grammar of described step 6 is: open-circuit voltage OCV and SOC exists nonlinear relationship, and physical relationship formula is:

$\mathrm{OCV}=a_0+a_1\&CenterDot;\ln \mathrm{SOC}+a_2\&CenterDot;\ln (1-\mathrm{SOC})+\frac{a_3}{\mathrm{SOC}}+a_4\&CenterDot;\mathrm{SOC}---\left(10\right)$

In formula, a ₀-a ₄for constant, obtained based on least squares identification by experimental data.

Battery ohmic internal resistance R _odwith the relational expression of SOC be:

R _o(SOC)＝b ₀·e ^-SOC+b ₁+b ₂·SOC-b ₃·SOC ²+b ₄·SOC ³(11)

In formula, b ₀-b ₄for constant, obtained based on least squares identification by experimental data.

Activation polarization internal resistance R _1dwith the relational expression of SOC be:

R _1d(SOC)＝c ₀·e ^-SOC+c ₁+c ₂·SOC-c ₃·SOC ²+c ₄·SOC ³(12)

In formula, c ₀-c ₄for constant, obtained based on least squares identification by experimental data.

Activation polarization fractional order electric capacity FOE _1dwith the relational expression of SOC be:

FOE _1d(SOC)＝d ₀·SOC ⁵+d ₁·SOC ⁴+d ₂·SOC ³+d ₃·SOC ²+d ₄·SOC+d ₅(13)

In formula, d ₀-d ₅for constant, obtained based on least squares identification by experimental data.

Concentration polarization internal resistance R _2dwith the relational expression of SOC be:

R _2d(SOC)＝e ₀·e ^-SOC+e ₁+e ₂·SOC-e ₃·SOC ²+e ₄·SOC ³(14)

In formula, e ₀-e ₄for constant, obtained based on least squares identification by experimental data.

Concentration polarization fractional order electric capacity FOE _2dwith the relational expression of SOC be:

FOE _2d(SOC)＝f ₀·SOC ⁵+f ₁·SOC ⁴+f ₂·SOC ³+f ₃·SOC ²+f ₄·SOC+f ₅(15)

In formula, f ₀-f ₅for constant, obtained based on least squares identification by experimental data.

Activation polarization fractional order electric capacity FOE _1dthe relational expression of exponent number and SOC is:

α(SOC)＝g ₀·SOC ⁴+g ₁·SOC ³+g ₂·SOC ²+g ₃·SOC+g ₄(16)

In formula, g ₀-g ₄for constant, obtained based on least squares identification by experimental data.

Concentration polarization fractional order electric capacity FOE _2dthe relational expression of exponent number and SOC is:

β(SOC)＝h ₀·SOC ⁴+h ₁·SOC ³+h ₂·SOC ²+h ₃·SOC+h ₄(17)

In formula, h ₀-h ₄for constant, obtained based on least squares identification by experimental data.

1. experiment is set up

For sea special 10 and 16 string column type 26650 lithium iron phosphate dynamic batteries carry out testing and emulating, nominal capacity is 23Ah, and nominal voltage is 51.2V.Battery testing platform by the AVL battery of advanced person simulate/test cabinet, AVL InMotion hardware-in―the-loop test platform, AVL switch board, temperature control box and and AVL Lynx control software design form.The voltage of experimental record battery, the operating mode value such as electric current and SOC, sample frequency is set to 1Hz.

Consider the difference of discharge and recharge parameter, by HPPC mixed pulses test (Hybrid Pulse Power CharacterizationTest, HPPC) the mixed pulses test in makes unidirectional pulse test into, i.e. the test of electrokinetic cell pulse charge and pulsed discharge test.So-called pulsed discharge, namely under room temperature 25 degree, by be full of electricity battery with the current discharge of 0.2C to SOC for 95%, stop the standing 45min of electric discharge, then with same current discharge to SOC for 90%, by that analogy, until terminate when SOC is 0%.Pulse charge process and process of pulse discharge similar, do not repeat them here.

2. model order and parameter identification

As shown in Figure 4, be the recovery response comparison diagram of battery terminal voltage after change rank of the present invention fractional order, integer rank and the modeling pulsed discharge of fixed fraction rank.As can be seen from the figure, fractional order change rank equivalent-circuit model simulation precision is the highest.

(1) open-circuit voltage OCV model

According to different SOC (3.45%, 5%, 10% ... 90%, 95%, 98.89%) battery open circuit voltage that records of place, matching can obtain charging open-circuit voltage and the electric discharge open-circuit voltage OCV of battery, as shown in Fig. 5 and Figure 13 respectively.And according to formula (10), application Matlab cftool tool box can pick out parameter a ₀-a ₄, as shown in table 1.

The open-circuit voltage OCV parameter that Matlab cftool Fitting Toolbox obtains applied by table 1

Parameter	a 0	a 1	a 2	a 3	a 4
						Charging identifier	-1.015	1.915	-0.02047	-0.006009	0.03582
Electric discharge identifier	2.588	-1.499	-0.007343	-0.01424	0.04207

(2) ohmic internal resistance R ₀model

According to different SOC (3.45%, 5%, 10% ... 90%, 95%, 98.89%) the battery ohmic internal resistance that records of place, matching can obtain charging ohmic internal resistance and the electric discharge ohmic internal resistance R of battery respectively ₀, as shown in Fig. 6 and Figure 14.And according to formula (11), application Matlab cftool tool box can pick out parameter b ₀-b ₄, as shown in table 2.

The ohmic internal resistance R that Matlab cftool Fitting Toolbox obtains applied by table 2 ₀parameter

Parameter	b 0	b 1	b 2	b 3	b 4
						Charging identifier	0.5612	-0.5588	0.5546	-0.2568	0.05642
Electric discharge identifier	-0.638	0.6397	-0.634	0.2967	-0.0663

(3) activation polarization internal resistance R ₁model

According to different SOC (3.45%, 5%, 10%, 15% ..., 95%) the battery electrochemical polarization resistance that records of place, can the rechargeable electrochemical polarization resistance of battery that obtains of matching and electrochemical polarization resistance R respectively ₁, as shown in Fig. 7 and Figure 15.And according to formula (12), application Matlab cftool tool box can pick out parameter c ₀-c ₄, as shown in table 3.

The activation polarization internal resistance R that Matlab cftool Fitting Toolbox obtains applied by table 3 ₁parameter

Parameter	c 0	c 1	c 2	c 3	c 4
						Charging identifier	3.891	-3.887	3.87	-1.835	0.4279
Electric discharge identifier	10.26	-10.24	10.1	-4.607	0.9788

(4) activation polarization fractional order electric capacity FOE ₁model

According to different SOC (3.45%, 5%, 10% ... 90%, 95%, 98.89%) the activation polarization fractional order electric capacity that records of place, matching can obtain rechargeable electrochemical polarization fractional order electric capacity and the electrochemical polarization fractional order electric capacity FOE of battery respectively ₁, as shown in Fig. 8 and Figure 16.And according to formula (13), application Matlab cftool tool box can pick out parameter d ₀-d ₅, as shown in table 4.

The activation polarization fractional order electric capacity FOE that Matlab cftool Fitting Toolbox obtains applied by table 4 ₁parameter

Parameter

d 0

d 1

d 2

d 3

d 4

d 5

Charging identifier

-3.775e 6

8.913e 6

-7.623e 6

2.976e 6

-5.542e 5

9.351e 4

Electric discharge identifier

1.387e 8

-5.26e 8

7.008e 8

-4.023e 8

9.138e 7

-3.683e 6

(5) concentration polarization internal resistance R ₂model

According to different SOC (3.45%, 5%, 10%, 15% ..., 95%) the battery concentration polarization internal resistance that records of place, can the charging concentration polarization internal resistance of battery that obtains of matching and electric discharge concentration polarization internal resistance R respectively ₂, as shown in Fig. 9 and Figure 17.And according to formula (14), application Matlab cftool tool box can pick out parameter e ₀-e ₄, as shown in table 5.

The concentration polarization internal resistance R that Matlab cftool Fitting Toolbox obtains applied by table 5 ₂parameter

Parameter	e 0	e 1	e 2	e 3	e 4
						Charging identifier	3.267	-3.265	3.271	-1.594	0.3971
Electric discharge identifier	-4.582	4.584	-4.535	2.102	-0.4632

(6) concentration polarization fractional order electric capacity FOE ₂model

According to different SOC (3.45%, 5%, 10%, 15% ..., 95%) the battery concentration polarization fractional order electric capacity that records of place, can the charging concentration polarization fractional order electric capacity of battery that obtains of matching and electric discharge concentration polarization fractional order electric capacity FOE respectively ₂, as shown in Figure 10 and Figure 18.And according to formula (12), application Matlab cftool tool box can pick out parameter f ₀-f ₅, as shown in table 6.

The concentration polarization fractional order electric capacity FOE that Matlab cftool Fitting Toolbox obtains applied by table 6 ₂parameter

Parameter

f 0

f 1

f 2

f 3

f 4

f 5

Charging identifier

-1.842e 5

4.507e 5

-4.116e 5

1.704e 5

-2.857e 4

4.572e 4

Electric discharge identifier

-4.529e 5

1.188e 6

-1.153e 6

5.072e 5

-9.649e 4

8.68e 4

(7) activation polarization fractional order electric capacity FOE ₁exponent number model

According to different SOC (3.45%, 5%, 10%, 15% ..., 95%) the battery electrochemical polarization fractional order electric capacity exponent number that records of place, can the rechargeable electrochemical polarization fractional order electric capacity FOE of battery that obtains of matching respectively ₁exponent number and electrochemical polarization fractional order electric capacity FOE ₁exponent number, as shown in Figure 11 and Figure 19.And according to formula (16), application Matlab cftool tool box can pick out parameter g ₀-g ₄, as shown in table 7.

The activation polarization fractional order electric capacity FOE that Matlabcftool Fitting Toolbox obtains applied by table 7 ₁order parameter

Parameter	g 0	g 1	g 2	g 3	g 4
						Charging identifier	21.24	-46.26	33.74	-8.854	0.9861
Electric discharge identifier	-15.45	27.57	-14.44	1.862	0.6595

(8) concentration polarization fractional order electric capacity FOE ₂exponent number model

For according to different SOC (3.45%, 5%, 10%, 15% ..., 95%) the battery concentration polarization fractional order electric capacity exponent number that records of place, can the charging concentration polarization fractional order electric capacity FOE of battery that obtains of matching respectively ₂exponent number and electric discharge concentration polarization fractional order electric capacity FOE ₂exponent number, as shown in Figure 12 and Figure 20.And according to formula (17), application Matlabcftool tool box can pick out parameter h ₀-h ₄, as shown in table 8.

The concentration polarization fractional order electric capacity FOE that Matlab cftool Fitting Toolbox obtains applied by table 8 ₂order parameter

Parameter	h 0	h 1	h 2	h 3	h 4
						Charging identifier	2.752	-6.434	5.085	-1.319	0.4986
Electric discharge identifier	4.813	-10.91	8.4	-2.468	0.6779

3. emulation and experimental verification

In order to verify the accuracy of battery model, constant current charge-discharge and pulse charge-discharge test are carried out to battery.As shown in fig. 21-22, be the contrast of the battery terminal voltage experimental result that obtains under pulse charge and discharge and model emulation result.As can be seen from the figure, the fractional order that the present invention proposes becomes rank equivalent-circuit models can the pulse charge and discharge process of reaction cell preferably, illustrates that this model is accurately.Wherein, the error that the error ratio produced at quiescent phase produced in the constant current charge-discharge stage is larger.

As shown in figs. 23-24, be the contrast of the battery terminal voltage experimental result that obtains under constant current charge and discharge and model emulation result.As can be seen from the figure, in initial stage and the latter stage of charge and discharge, model error is larger; And in the mid-term of charge and discharge, model output valve almost overlaps with experiment value.In the middle of the two ends of this and lithium battery are steep, flat non-linear voltage characteristic fits like a glove.

In sum, the simulation result that change rank disclosed by the invention fractional order equivalent-circuit model obtains meets experimental data substantially, and model maximum error within 0.05V, and is applicable to the operating mode such as pulse discharge and recharge and constant current charge-discharge of electric automobile.Demonstrate the validity becoming rank fractional model.

By reference to the accompanying drawings the specific embodiment of the present invention is described although above-mentioned; but not limiting the scope of the invention; one of ordinary skill in the art should be understood that; on the basis of technical scheme of the present invention, those skilled in the art do not need to pay various amendment or distortion that creative work can make still within protection scope of the present invention.

Claims (10)

1. lithium battery fractional order becomes rank equivalent-circuit models, it is characterized in that, comprises circuit and cell I-V characteristic circuit working time, described working time circuit and cell I-V characteristic circuit carry out Signal transmissions by CCCS and Voltage-controlled Current Source; Described working time, circuit comprised the self discharge resistance R of battery _dand electric capacity C _q, resistance R _dwith electric capacity C _qbe connected in parallel on the two ends of CCCS, one end ground connection of CCCS; The positive terminal of the Voltage-controlled Current Source of described cell I-V characteristic circuit is connected with one end of two RC network branch roads be in parallel, negative pole end is connected with the negative pole end of battery model, and each branch road of described two RC network branch roads be in parallel includes two fractional order RC loops be in series and an internal resistance R _o, described two other ends of RC network branch road be in parallel are connected with the positive terminal of battery model.

2. a kind of lithium battery fractional order as claimed in claim 1 becomes rank equivalent-circuit model, and it is characterized in that, two RC network branch roads be in parallel are RC network discharge paths and RC network charging paths respectively, and the electric capacity in two RC network branch roads is fractional order electric capacity.

3. a kind of lithium battery fractional order as claimed in claim 2 becomes rank equivalent-circuit model, it is characterized in that, described RC network discharge paths fractional order element FOE _1dand FOE _2dexponent number α, β is different and change with battery SOC state, and meets 0≤α _d, β _d≤ 1, work as α _d, β _dwhen=0, fractional order element FOE is equivalent to a resistance, works as α _d, β _dwhen=1, fractional order element FOE is equivalent to an integer rank electric capacity; As 0< α _d, β _dduring <1, fractional order element FOE is a fractional order electric capacity;

Described RC network charging paths fractional order element FOE _1cand FOE _2cexponent number α, β is different and change with battery SOC state, and meets 0≤α _c, β _c≤ 1, work as α _c, β _cwhen=0, fractional order element FOE is equivalent to a resistance, works as α _c, β _cwhen=1, fractional order element FOE is equivalent to an integer rank electric capacity; As 0< α _c, β _cduring <1, fractional order element FOE is a fractional order electric capacity.

4. a kind of lithium battery fractional order as claimed in claim 1 becomes rank equivalent-circuit model, and it is characterized in that, in described cell I-V characteristic circuit in two RC network branch roads be in parallel, discharge paths comprises the diode D connected successively _d, fractional order electric capacity FOE _1dwith resistance R _1dthe fractional order RC loop of composition, fractional order electric capacity FOE _2dwith resistance R _2dthe fractional order RC loop of composition and resistance R _od;

Charging paths comprises the reversal connection diode D connected successively _c, fractional order electric capacity FOE _1cwith resistance R _1cthe fractional order RC loop of composition, fractional order electric capacity FOE _2cwith resistance R _2cthe fractional order RC loop of composition and resistance R _oc.

5. a kind of lithium battery fractional order as claimed in claim 1 becomes rank equivalent-circuit model, it is characterized in that, described working time circuit and cell I-V characteristic circuit set up contact by a CCCS and Voltage-controlled Current Source, when carrying out discharge and recharge to battery, load current i _batby CCCS to electric capacity C _qcarry out discharge and recharge, change C _qthe electricity stored, the change of characterizing battery SOC, C _qboth end voltage OCV also changes thereupon, and the controlled voltage source OCV of I-V characteristic circuit changes with the change of SOC.

6. a kind of lithium battery fractional order as claimed in claim 1 becomes rank equivalent-circuit model, and it is characterized in that, the voltage at the two ends of described CCCS is battery open circuit voltage OCV.

7. a kind of lithium battery fractional order as described in as arbitrary in claim 1 to 6 becomes the discrimination method of rank equivalent-circuit model, it is characterized in that, comprises the following steps:

Step one: write out the discharge process of lithium battery and the fractional order mathematics model expression of static condition;

Step 2: carry out constant current charge-discharge to lithium battery, obtains the active volume C of battery model _qwith self discharge resistance R _d;

Step 3: pulsed discharge test is carried out to lithium battery, obtaining the moment that different SOC places battery starts the moment drop-out value of battery terminal voltage when discharging, electric discharge terminates rear battery terminal voltage rises to the data such as the zero input response of value, discharge current and battery terminal voltage;

Step 4: the data obtained according to step 3, based on parameter and the exponent number of least squares identification model;

Step 5: the battery model parameter calculated according to step 4 calculates open-circuit voltage OCV, the ohmic internal resistance R at different SOC place _0d, activation polarization internal resistance R _1d, activation polarization fractional order electric capacity FOE _1d, concentration polarization internal resistance R _2dwith concentration polarization fractional order electric capacity FOE _2d;

Step 6: the model parameter obtained according to step 5, based on least squares identification open-circuit voltage OCV, ohmic internal resistance R _0d, activation polarization internal resistance R _1d, activation polarization fractional order electric capacity FOE _1d, concentration polarization internal resistance R _2dwith concentration polarization fractional order electric capacity FOE _2dand the relation between SOC;

Step 7: the parameter obtained according to step 2 to six, builds lithium battery fractional order and becomes rank equivalent-circuit model.

8. a kind of lithium battery fractional order as claimed in claim 7 becomes the discrimination method of rank equivalent-circuit model, and it is characterized in that, in discharge process, the terminal voltage of lithium battery can be expressed as:

$U_{\mathrm{bat}}={\mathrm{OCV}}_d-i_{\mathrm{dis}}\&CenterDot;R_{0d}-U_{1d}(0+)\&CenterDot;(1-e^{{-t}^a/{\mathrm{\&tau;}}_{1d}})-U_{2d}(0+)\&CenterDot;(1-e^{{-t}^{\mathrm{\&beta;}}/{\mathrm{\&tau;}}_{2d}})---\left(1\right)$

In formula, U _batfor battery terminal voltage; R _0dfor ohmic internal resistance; OCV _dfor electric discharge open-circuit voltage; α, β are fractional order element FOE _1dand FOE _2dexponent number, meet 0< α, β≤1; i _disfor discharge current; τ _1d, τ _2dbe respectively the time constant of two RC network; U _1d(0+) and U _2d(0+) for battery discharge terminates the terminal voltage initial value of moment two fractional order RC branch roads.

9. a kind of lithium battery fractional order as claimed in claim 8 becomes the discrimination method of rank equivalent-circuit model, it is characterized in that, U _1d(0+) and U _2d(0+) its value can be expressed as:

U _1d(0+)＝i _dis·R _1d(2)

U _2d(0+)＝i _dis·R _2d(3)

After battery discharge terminates, the terminal voltage of battery can be expressed as:

$U_{\mathrm{bat}}={\mathrm{OCV}}_d-U_{1d}(0+)\&CenterDot;e^{{-t}^a/{\mathrm{\&tau;}}_{1d}}-U_{2d}(0+)\&CenterDot;e^{{-t}^b/{\mathrm{\&tau;}}_{2d}}---\left(4\right)$

In formula, the polarizing voltage of battery with reduce gradually along with the growth of time, as t → ∞, with be tending towards 0, now battery terminal voltage U _batequal the open-circuit voltage OCV of battery.

10. a kind of lithium battery fractional order as claimed in claim 9 becomes the discrimination method of rank equivalent-circuit model, it is characterized in that, the detailed process of described step 5 is: due to the existence of battery ohmic internal resistance, when the cell is discharged, battery terminal voltage can fall instantaneously, and its value is designated as Δ U ₁; When battery stops electric discharge, battery terminal voltage can rise to instantaneously, and its value is designated as Δ U ₂, therefore, the ohmic internal resistance R of battery ₀can be obtained by following formula:

$R_0=\frac{{\mathrm{\&Delta;U}}_1+{\mathrm{\&Delta;U}}_2}{{2i}_{\mathrm{bat}}}---\left(5\right)$

Activation polarization internal resistance R _1dcan be obtained by following formula:

$R_{1d}=\frac{U_{1d}(0+)}{i_{\mathrm{dis}}}---\left(6\right)$

Concentration polarization internal resistance R _2dcan be obtained by following formula:

$R_{2d}=\frac{U_{2d}(0+)}{i_{\mathrm{dis}}}---\left(7\right)$

Activation polarization fractional order electric capacity FOE _1dcan be obtained by following formula:

${\mathrm{FOE}}_{1d}=\frac{{\mathrm{\&tau;}}_{1d}}{R_{1d}}---\left(8\right)$

Concentration polarization fractional order electric capacity FOE _2dcan be obtained by following formula:

${\mathrm{FOE}}_{2d}=\frac{{\mathrm{\&tau;}}_{2d}}{R_{2d}}---\left(9\right).$