CN115480166A - Lithium battery state-of-charge estimation method based on adaptive unscented Kalman filtering - Google Patents

Lithium battery state-of-charge estimation method based on adaptive unscented Kalman filtering Download PDF

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CN115480166A
CN115480166A CN202211174411.3A CN202211174411A CN115480166A CN 115480166 A CN115480166 A CN 115480166A CN 202211174411 A CN202211174411 A CN 202211174411A CN 115480166 A CN115480166 A CN 115480166A
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lithium battery
state
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王小利
吕杰超
蒋保臣
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Shandong University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/36—Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
    • G01R31/367—Software therefor, e.g. for battery testing using modelling or look-up tables
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/36—Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
    • G01R31/378—Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC] specially adapted for the type of battery or accumulator
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01R—MEASURING ELECTRIC VARIABLES; MEASURING MAGNETIC VARIABLES
    • G01R31/00—Arrangements for testing electric properties; Arrangements for locating electric faults; Arrangements for electrical testing characterised by what is being tested not provided for elsewhere
    • G01R31/36—Arrangements for testing, measuring or monitoring the electrical condition of accumulators or electric batteries, e.g. capacity or state of charge [SoC]
    • G01R31/382—Arrangements for monitoring battery or accumulator variables, e.g. SoC

Abstract

The invention relates to a lithium battery state of charge estimation method based on self-adaptive unscented Kalman filtering, which comprises the following steps: s1, acquiring a state equation and an output equation of a model based on a second-order equivalent circuit model of a lithium battery, and discretizing to obtain a state space equation of a discretization model; s2, identifying model parameters through a voltage response curve of battery discharge, a state equation of the model and an output equation; s3, substituting the identified parameters into a state equation and an output equation of the model, taking pulse discharge as model input, comparing the voltage of the output end of the model with the voltage of the actual end of the model, and verifying the precision of the model; s4, establishing a self-adaptive unscented Kalman filtering algorithm, and self-adaptively updating and calculating the system noise covariance and the observation noise covariance of the state space equation; and S5, estimating the terminal voltage value and the SOC value of the lithium battery by using a Kalman filter based on a self-adaptive unscented Kalman filtering algorithm. The method can accurately estimate the state of charge of the lithium battery, and has small error and high estimation precision.

Description

Lithium battery state-of-charge estimation method based on adaptive unscented Kalman filtering
Technical Field
The invention belongs to the technical field of battery management, relates to a lithium battery management technology, and particularly relates to a lithium battery state of charge estimation method based on adaptive unscented Kalman filtering.
Background
In recent years, with the increasing energy crisis and environmental problems, low-pollution and energy-efficient electric vehicles have become a focus of research in the automotive industry. The lithium ion battery has the characteristics of small size, light weight, high energy density, large output power and high safety new energy, and becomes the first choice of the energy storage device of the electric automobile. The State Of Charge (SOC) Of the battery directly reflects the remaining capacity Of the battery, and is an important basis for a vehicle control system to formulate an optimal energy management strategy. The SOC is an important battery performance parameter, and accurate estimation of the SOC has important significance for improving the safety performance of the battery, prolonging the service life of the battery and ensuring the reliable operation of a battery system.
At present, the common lithium battery SOC estimation methods include an ampere-hour integration method, an open-circuit voltage method, a neural network method, a Kalman filter algorithm (KF for short), and the like. The ampere-hour integration method estimates the SOC of the battery by accumulating the charged and discharged electric quantity, and meanwhile, certain compensation is carried out on the estimated SOC according to the discharge rate. The ampere-hour integration method is relatively simple and can dynamically estimate the SOC of the battery, but current integration needs to obtain an initial SOC value and accurately acquire the battery current, which causes the SOC estimation error to be continuously accumulated over time, and the estimation accuracy is low, so in practical application, the ampere-hour integration method is usually used in combination with other methods to improve the estimation accuracy. The open-circuit voltage method is to indirectly fit the corresponding relation between the open-circuit voltage of the battery and the SOC of the battery according to the change relation between the open-circuit voltage and the lithium ion concentration in the battery, then to measure the open-circuit voltage by standing the battery for a long time, and to obtain the current SOC of the battery according to the corresponding relation between the fitted open-circuit voltage and the SOC. The open-circuit voltage method requires the battery to be left standing for a long time to obtain the terminal voltage of a stable point, and cannot estimate the SOC of the battery on line in real time. The neural network method is a novel algorithm for simulating the human brain and the neurons thereof to process a nonlinear system, does not need to deeply research the internal structure of the battery, and can obtain the SOC of the battery only by extracting a large number of input and output samples which accord with the working characteristics of the target battery in advance and inputting the input and output samples into a system established by the method. The neural network method has a large workload, a large amount of comprehensive target sample data needs to be extracted to train the system, and the input training data and the training method influence the accuracy of SOC estimation to a great extent. The Kalman filtering algorithm is a novel optimized autoregressive data filtering algorithm, and the essence of the algorithm is to make optimal estimation on a complex dynamic system according to the principle of least mean square error. The Kalman filtering algorithm overcomes the serious defect that current integration depends on an initial value, a large amount of sample data is not needed, the SOC of the battery can be estimated on line, and the Kalman filtering algorithm has a remarkable application value in the SOC estimation of the power battery of the electric automobile with complex operation conditions and becomes a hotspot of battery SOC estimation algorithm research in recent years.
The Kalman filtering algorithm is an algorithm which utilizes a linear system state equation, outputs observation data through system input and outputs and performs optimal estimation on the system state. For the problem that kalman filtering cannot solve a nonlinear system, the document P é rez, g.; garmendia, m.; reynaud, j.f.; crego, J.; viscarat, U.S. enhanced closed loop State of Charge estimator for less-ion bases based on Extended Kalman Filter. Appl. Energy 2015,155,834-845, doi. Document He, h; xiong, r.; peng, J.real-time estimation of base state-of-charge with Unscented Kalman Filter and RTOS mu COS-II platform.appl.energy 2016,162,1410-1418, doi. But does not take into account uncertainty in the battery model and system noise. Uncertainties in model noise and system noise can lead to increased errors, slow convergence speed, and filter divergence. Literature "Sun, f.; hu, x.; zou, y.; li, S.Adaptation unscented Kalman filtering for state of charge estimation of a little-ion basis for electric fields. Energy 2011,36,3531-3540, doi.
In addition, in the conventional unscented kalman filter algorithm, the system noise covariance and the observation noise covariance are usually set as constants, which cannot truly reflect the dynamic characteristics of noise, and have a certain influence on the SOC estimation accuracy, thereby reducing the estimation accuracy.
Disclosure of Invention
Aiming at the problems of low estimation precision and the like in the prior art, the invention provides a lithium battery state of charge estimation method based on adaptive unscented Kalman filtering.
In order to achieve the purpose, the invention provides a lithium battery state of charge estimation method based on self-adaptive unscented Kalman filtering, which comprises the following steps:
s1, acquiring a state equation and an output equation of the model based on a second-order equivalent circuit model of the lithium battery, and discretizing to obtain a state space equation of the discretized second-order equivalent circuit model;
s2, identifying model parameters through a voltage response curve of battery discharge, a state equation of a second-order equivalent circuit model and an output equation;
s3, substituting the identified model parameters into a state equation and an output equation of a second-order equivalent circuit model, using pulse discharge as input of the second-order equivalent circuit model, comparing the output end voltage and the actual end voltage of the second-order equivalent circuit model, and verifying the precision of the second-order equivalent circuit model;
s4, establishing a self-adaptive unscented Kalman filtering algorithm, calculating variance of innovation and residual errors through a moving window method, and self-adaptively updating and calculating system noise covariance and observation noise covariance of a state space equation;
and S5, estimating the terminal voltage value of the lithium battery and the SOC value of the lithium battery by using a Kalman filter based on the established adaptive unscented Kalman filtering algorithm.
The finite, second-order equivalent circuit model has the equation of state:
Figure BDA0003864646080000041
in the formula I bat For the open circuit current of lithium battery, the discharge is positive; q bat The rated capacity of the lithium battery; r is si Ohmic internal resistance of the lithium battery; r is tf The polarization resistance of the lithium battery; c tf A polarization capacitor of the lithium battery; r ts Is a concentration polarization resistance; c ts Is a concentration polarization capacitor; u shape tf Is a polarization capacitor C tf The voltage across; u shape ts Polarizing the capacitance C for concentration ts The voltage across;
the output equation of the second-order equivalent circuit model is as follows:
U bat ＝U OC (SOC)-R si I bat -U tf -U ts (2)
in the formula of U bat Is the battery terminal voltage; u shape OC (SOC) is the open-circuit voltage of the battery associated with SOC.
Preferably, according to the second-order equivalent circuit model of the lithium battery, the state space equation of the discretized second-order equivalent circuit model obtained by combining the formula (1) and the formula (2) is as follows:
Figure BDA0003864646080000042
in the formula, τ tf ＝R tf C tf For a fast time constant, τ ts ＝R ts C ts Is a slow time constant;
order to
Figure BDA0003864646080000051
I bat (k)＝u k ，U bat (k)＝y k ，C＝[-1 -1 0]The state space equation of the discretized second-order equivalent circuit model is simplified as follows:
Figure BDA0003864646080000052
preferably, in step S2, the identified model parameter includes ohmic internal resistance R of the lithium battery si Lithium battery polarization resistance R tf Polarization capacitor C of lithium battery tf Concentration polarization resistance R ts Concentration polarization capacitor C ts And lithium battery open-circuit voltage U with functional relation oc (SOC) expression.
Preferably, in step S2, 10 lithium batteries are connected in parallel, 1C pulse discharge is adopted, each discharge time is 3min, then the lithium batteries are left standing for 2h, and the lithium batteries are discharged to cut-off voltage in a circulating manner to obtain a pulse discharge voltage curve and a pulse discharge current curve, so as to obtain the SOC and the open-circuit voltage U of the lithium batteries oc Performing least square fitting by using MATLAB to obtain open-circuit voltage U oc And a function relation curve of the lithium battery SOC.
Preferably, in step S2, the resistance-capacitance is identified by analyzing a voltage response curve of the battery pulse discharge and combining the resistance-capacitance characteristics; the specific process of identifying the resistor and the capacitor comprises the following steps:
the cell voltage response curve is divided into four phases:
A-B section: the battery begins to discharge from standing, the terminal voltage drops suddenly, and the polarization capacitor C tf Voltage U across tf Sum concentration polarization capacitance C ts Voltage U across ts Sudden change can not occur, and sudden drop of the voltage of the A-B section is caused by ohmic internal resistance R si And (4) causing.
Sections B to C: during the continuous discharge, the electrochemical polarization and the concentration polarization act together to reduce the voltage in an exponential change mode; before the B-C section, the voltage U tf Sum voltage U ts Zero, then the B-C section is zeroAnd (6) responding to the state.
C-D stage: the discharge current disappears, the voltage of the battery rebounds, the same as the A-B section, the C-D section voltage rebounds due to the ohmic internal resistance R si And (3) causing.
D-E section: the battery is kept stand, the voltage is slowly increased due to the action of electrochemical polarization and concentration difference, no current is discharged at the time, and the D-E section is zero input response.
According to the A-B section and the C-D section, the ohmic internal resistance R is obtained by the following formula si ：
Figure BDA0003864646080000061
In the formula of U A Is the voltage of the lithium battery terminal corresponding to the point A in the voltage response curve, U B Is the voltage of the lithium battery terminal corresponding to the point B in the voltage response curve, U C Is the voltage of the lithium battery terminal corresponding to the C point in the voltage response curve, U D And the voltage is the voltage of the lithium battery corresponding to the D point in the voltage response curve.
Solving a differential equation according to a state equation of the model shown in formula (1) yields:
Figure BDA0003864646080000062
in the formula of U tf (0) Is a polarization capacitor C tf Initial voltage at both ends, U ts (0) Polarising the capacitance C for concentration ts Initial voltage across.
According to the condition that the discharge current of the D-E section is zero and zero input response is achieved, the point D is used as the time t =0, and the zero input response expression of the RC loop is obtained as follows:
Figure BDA0003864646080000063
and (3) combining the output equation of the model shown in the formula (2) to obtain the output equation of the lithium battery under the zero input response, wherein the output equation of the lithium battery under the zero input response is as follows:
Figure BDA0003864646080000064
order to
Figure BDA0003864646080000065
U tf (0)＝b 1 ，U ts (0)＝b 2 Equation (8) reduces to:
Figure BDA0003864646080000066
using MATLAB to take the formula (9) as a fitting function, performing least square fitting on the DE section to obtain b 1 ,b 2 ,λ 1 ,λ 2 To obtain τ tf ,τ ts The value of (c).
And obtaining a zero state response expression of the RC loop by taking the point B as the time of t =0 according to the condition that the section B-C is a zero state response:
Figure BDA0003864646080000071
and (3) combining the output equation of the model shown in the formula (2) to obtain the output equation of the lithium battery under the zero state response, wherein the output equation is as follows:
Figure BDA0003864646080000072
τ obtained by fitting equation (9) tf ,τ ts Substituting into formula (11), using formula (11) as fitting function, performing least square fitting on B-C section to obtain a 1 ,a 2 Is obtained by the equation (12) to obtain the resistance R tf ,R ts Is expressed as:
Figure BDA0003864646080000073
then according to tau tf ＝R tf C tf ,τ ts ＝R ts C ts Calculating the capacitance C tf ,C ts The value of (c).
Preferably, in step S4, an adaptive unscented kalman filter algorithm is established, and the process of adaptively updating the noise covariance and the observation noise covariance of the computing system is as follows:
for a nonlinear system, the state equation and observation equation for adding system noise and observation noise is expressed as:
Figure BDA0003864646080000074
wherein k is the current time, F (x) k-1 ,u k ) For the nonlinear system state transition equation, G (x) k-1 ,u k ) For non-linear system observation equations, x k Is a state variable, u k For known input, y k Is the observation signal, w is the system noise, v is the observation noise;
according to the KF principle, combining a state space equation shown in a formula (4) and a state equation and an observation equation shown in a formula (13), solving a first derivative of the nonlinear observation equation at the current state value to obtain an observation matrix H k Comprises the following steps:
Figure BDA0003864646080000081
determining initial values of state values
Figure BDA0003864646080000082
Sum state error covariance initial value P 0 Comprises the following steps:
Figure BDA0003864646080000083
Figure BDA0003864646080000084
calculate Sigma point:
Figure BDA0003864646080000085
wherein L is the length of the state vector;
calculating the weight:
Figure BDA0003864646080000086
wherein α =0,k i ＝0，β＝2；
Updating a predicted state value
Figure BDA0003864646080000087
Figure BDA0003864646080000088
Figure BDA0003864646080000089
Updating the predicted observations
Figure BDA00038646460800000810
Figure BDA00038646460800000811
Figure BDA00038646460800000812
Updating system covariance prediction value P xx|k ：
Figure BDA00038646460800000813
Calculating innovation d k And innovation d k Variance of (2)
Figure BDA0003864646080000091
Figure BDA0003864646080000092
Figure BDA0003864646080000093
Wherein, W is the length of the movable window;
updating system noise covariance Q k ：
Q k ＝K k-1 C dk K k-1 T (26)
Updating the observed covariance prediction value P yy|k ：
Figure BDA0003864646080000094
Updating the covariance P xy|k ：
Figure BDA0003864646080000095
Calculating the Kalman gain K k ：
Figure BDA0003864646080000096
Updating an estimated state value
Figure BDA0003864646080000097
Figure BDA0003864646080000098
Updating an estimated observation
Figure BDA0003864646080000099
Figure BDA00038646460800000910
Updating error covariance value P k ：
P k ＝P xx|k -K k P yy|k K T (32)
Calculating residual value r k Variance of sum residual
Figure BDA00038646460800000911
Figure BDA00038646460800000912
Figure BDA0003864646080000101
Updating the observation noise covariance R k ：
Figure BDA0003864646080000102
Compared with the prior art, the invention has the advantages and positive effects that:
according to the method, a second-order equivalent circuit model is established for the battery, parameter identification is carried out through a voltage response curve of battery discharge by using a least square method, the identified parameters are substituted into the model and verified by using a discharge pulse experiment, the result shows that the error of the model is not more than 0.8%, and an accurate battery model is provided for an SOC estimation experiment. Secondly, the SOC of the battery is estimated by adopting the adaptive unscented Kalman filtering algorithm, when model system noise and observation noise are uncertain, the adaptive unscented Kalman filtering algorithm updates the system noise covariance and the observation noise covariance in real time by monitoring the change of innovation and residual error in a filter, adjusts the filtering gain, corrects the estimated value in time, and has more accurate estimation result, error not more than 0.7%, high estimation precision and good robustness.
Drawings
Fig. 1 is a flowchart of a lithium battery state of charge estimation method based on adaptive unscented kalman filtering according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of a second order equivalent circuit model according to an embodiment of the present invention;
FIG. 3a is a schematic diagram of a pulse discharge voltage according to an embodiment of the present invention;
FIG. 3b is a schematic diagram of a pulse discharge current according to an embodiment of the present invention;
FIG. 4 shows U according to an embodiment of the present invention oc -SOC function diagram;
FIG. 5 is a partial discharge voltage plot of a battery pulse discharge in accordance with an embodiment of the present invention;
FIG. 6 is a schematic diagram showing comparison between the real and model values of the terminal voltage of the battery according to the embodiment of the present invention;
FIG. 7 is a graph illustrating model error values for battery terminal voltages in accordance with an embodiment of the present invention;
FIG. 8 is a flowchart of the AUKF algorithm adaptively updating the computational system noise covariance and the observed noise covariance, in accordance with an embodiment of the present invention;
FIG. 9 is a diagram illustrating Q values in the AUKF algorithm according to the embodiment of the invention;
FIG. 10 is a diagram illustrating R values in the AUKF algorithm according to an embodiment of the present invention;
FIG. 11a is a schematic diagram illustrating comparison between the UKF algorithm and the estimated terminal voltage of the AUKF algorithm according to the embodiment of the present invention;
FIG. 11b is a schematic diagram illustrating comparison of the UKF algorithm and the AUKF algorithm estimated terminal voltage error in the embodiment of the present invention
FIG. 12 is a schematic diagram of UKF algorithm and AUKF algorithm Kalman gain comparison in accordance with an embodiment of the present invention;
FIG. 13a is a schematic diagram of the UKF algorithm and the AUKF algorithm estimation SOC comparison according to the embodiment of the present invention;
FIG. 13b is a schematic diagram illustrating the comparison between the UKF algorithm and the AUKF algorithm for estimating SOC errors according to the embodiment of the present invention.
Detailed Description
The invention is described in detail below by way of exemplary embodiments. It should be understood, however, that elements, structures and features of one embodiment may be beneficially incorporated in other embodiments without further recitation.
Referring to fig. 1, an embodiment of the present invention provides a lithium battery state of charge estimation method based on adaptive unscented kalman filtering, including the steps of:
s1, acquiring a state equation and an output equation of the model based on a second-order equivalent circuit model of the lithium battery, and discretizing to obtain a state space equation of the discretized second-order equivalent circuit model.
A schematic diagram of a second-order equivalent circuit model of a lithium battery is shown in fig. 2. According to kirchhoff's law, there are:
the state equation of the second-order equivalent circuit model is as follows:
Figure BDA0003864646080000121
in the formula I bat For the open circuit current of lithium battery, the discharge is positive; q bat The rated capacity of the lithium battery; r si Ohmic internal resistance of the lithium battery; r is tf The polarization resistance of the lithium battery; c tf The polarization capacitor of the lithium battery; r is ts Is concentration polarization resistance; c ts Is a concentration polarization capacitor; u shape tf Is a polarization capacitor C tf The voltage across; u shape ts Polarizing the capacitance C for concentration ts The voltage across it.
The output equation of the second-order equivalent circuit model is as follows:
U bat ＝U OC (SOC)-R si I bat -U tf -U ts (2)
in the formula of U bat Is the battery terminal voltage; u shape OC (SOC) is the open-circuit voltage of the battery associated with SOC.
According to a second-order equivalent circuit model of the lithium battery, combining a formula (1) and a formula (2) to obtain a state space equation of a discretized second-order equivalent circuit model, the state space equation is as follows:
Figure BDA0003864646080000122
in the formula, τ tf ＝R tf C tf Is a fast time constant, τ ts ＝R ts C ts Is a slow time constant;
order to
Figure BDA0003864646080000123
I bat (k)＝u k ，U bat (k)＝y k ，C＝[-1 -1 0]The state space equation of the discretized second-order equivalent circuit model is simplified as follows:
Figure BDA0003864646080000131
and S2, identifying model parameters through a voltage response curve of battery discharge, a state equation of a second-order equivalent circuit model and an output equation.
The parameter identification technique is a technique for combining a theoretical model with experimental data for prediction. Parameter identification determines the parameter values of a set of models from a model built from experimental data, so that the data results calculated by the models better simulate the test data, and thus the position process can be predicted.
In this embodiment, the model parameters are identified through the battery discharge response curve, the state equation of the second-order equivalent circuit model and the output equation, and the model parameters to be identified include the ohmic internal resistance R of the lithium battery si Polarization resistance R of lithium battery tf Polarization capacitor C of lithium battery tf Concentration polarization resistance R ts Concentration polarization capacitor C ts And lithium battery open-circuit voltage U with functional relation oc (SOC) expression.
Specifically, 10 lithium batteries (the lithium battery adopts the SAMS type as the battery model number)UNG 30Q INR18650 power lithium battery, the specific parameters of which are shown in Table 1. ) Parallel connection, adopting 1C pulse discharge, discharging for 3min, standing for 2h, circularly discharging to cut-off voltage to obtain pulse discharge voltage curve (see figure 3 a) and pulse discharge current curve (see figure 3 b), and obtaining lithium battery SOC and open-circuit voltage U oc Lithium battery SOC and open circuit voltage U oc The correspondence is seen in table 2.
TABLE 1
Parameter(s) Numerical value
Cell type SAMSUNG 30Q INR18650
Rated capacity 3000mAh
Rated voltage 3.6V
Discharge cut-off voltage 2.5V
Weight (D) 48.1±1.5g
Size of 18.2mm(D)×65.0mm(H)
TABLE 2
U oc (v) SOC U oc (v) SOC
4.1617 1 3.7317 0.5034
4.0913 0.9503 3.6892 0.4537
4.0749 0.9007 3.6396 0.4040
4.0606 0.8510 3.5677 0.3543
4.0153 0.8013 3.5208 0.3046
3.9592 0.7517 3.4712 0.2550
3.9164 0.7020 3.3860 0.2053
3.8687 0.6524 3.2880 0.1556
3.8163 0.6027 3.2037 0.1059
3.7735 0.5530 3.0747 0.0563
Performing least square fitting by using MATLAB to obtain open-circuit voltage U oc The function relation with the SOC of the lithium battery is as follows:
U OC (SOC)＝122.4786*SOC 8 -401.4734*SOC 7 +485.6818*SOC 6 -239.2806*SOC 5 +3.7304*SOC 4 +44.9020*SOC 3 -19.8057*SOC 2 +5.0932*SOC+2.8341
obtaining the open-circuit voltage U by fitting oc See fig. 4 for a plot of the function versus the SOC of the lithium battery.
Specifically, the voltage response curve of battery pulse discharge is analyzed, and the resistance-capacitance characteristics are combined to identify the resistance-capacitance. The specific process of identifying the resistor and the capacitor comprises the following steps:
partial discharge voltage diagram for battery pulse discharge referring to fig. 5, the battery voltage response curve is divided into four phases:
A-B section: the battery starts to discharge from standing, the terminal voltage drops suddenly, and the polarization capacitor C tf Voltage U across tf Sum concentration polarization capacitance C ts Voltage U across ts The sudden change can not occur, and the sudden drop of the voltage of the A-B section is caused by the ohmic internal resistance R si And (3) causing.
B-C section: during the continuous discharge, the electrochemical polarization and the concentration polarization act together to reduce the voltage in an exponential change mode; before the B-C section, the voltage U tf Sum voltage U ts And if the response is zero, the B-C section is a zero state response.
C-D stage: the discharge current disappears, the voltage of the battery rebounds, the same as the A-B section, and the voltage rebounds in the C-D section is caused by the ohmic internal resistance R si And (4) causing.
D-E section: the battery is kept stand, the voltage is slowly increased due to the action of electrochemical polarization and concentration difference, no current is discharged at the time, and the D-E section is zero input response.
According to the A-B section and the C-D section, the ohmic internal resistance R is obtained by the following formula si ：
Figure BDA0003864646080000151
In the formula of U A Is the voltage of the lithium battery terminal corresponding to the A point in the voltage response curve, U B Is the voltage of the lithium battery terminal corresponding to the point B in the voltage response curve, U C Is the voltage of the lithium battery terminal corresponding to the C point in the voltage response curve, U D The voltage is the voltage of the lithium battery corresponding to the D point in the voltage response curve;
solving a differential equation according to a state equation of the model shown in formula (1) yields:
Figure BDA0003864646080000152
in the formula of U tf (0) Is a polarization capacitor C tf Initial voltage at both ends, U ts (0) Polarising the capacitance C for concentration ts An initial voltage across;
according to the condition that the discharge current of the D-E section is zero and zero input response is achieved, the D point is used as the time t =0, and the zero input response expression of the RC loop is obtained as follows:
Figure BDA0003864646080000153
and (3) combining the output equation of the model shown in the formula (2) to obtain the output equation of the lithium battery under the zero input response, wherein the output equation of the lithium battery under the zero input response is as follows:
Figure BDA0003864646080000154
order to
Figure BDA0003864646080000155
U tf (0)＝b 1 ，U ts (0)＝b 2 Equation (8) reduces to:
Figure BDA0003864646080000156
using MATLAB to take the formula (9) as a fitting function, and performing least square fitting on the DE section to obtain b 1 ,b 2 ,λ 1 ,λ 2 To obtain τ tf ,τ ts A value of (d);
and according to the condition that the section B-C is zero state response, and taking the point B as the moment of t =0, obtaining a zero state response expression of the RC loop as follows:
Figure BDA0003864646080000161
and (3) combining the output equation of the model shown in the formula (2) to obtain the output equation of the lithium battery under the zero state response, wherein the output equation of the lithium battery under the zero state response is as follows:
Figure BDA0003864646080000162
τ obtained by fitting of formula (9) tf ,τ ts Substituting into formula (11), using formula (11) as fitting function, performing least square fitting on B-C section to obtain a 1 ,a 2 Is obtained by the equation (12) to obtain the resistance R tf ,R ts Is expressed as:
Figure BDA0003864646080000163
then according to tau tf ＝R tf C tf ,τ ts ＝R ts C ts Calculating the capacitance C tf ,C ts The value of (c).
The parameter values for the model were identified by applying the least squares method in MATLAB see table 3.
TABLE 3
R si (Ω) R tf (Ω) R ts (Ω) C tf (F) C ts (F)
0.0037 0.0019 0.0035 23340 501270
And S3, substituting the identified model parameters into a state equation and an output equation of the second-order equivalent circuit model, using pulse discharge as input of the second-order equivalent circuit model, comparing the output end voltage and the actual end voltage of the second-order equivalent circuit model, and verifying the precision of the second-order equivalent circuit model. The real value of the battery terminal voltage is compared with the model value, see fig. 6, the model error value of the battery terminal voltage is seen in fig. 7, and the relevant parameters of the model error are seen in table 4.
TABLE 4
Type of error MAE RMSE
Error value 0.51％ 0.8％
It can be seen from fig. 6 that the terminal voltage outputted by the model is almost consistent with the actual terminal voltage, and the difference between the terminal voltage and the actual terminal voltage of the model is within 0.05V as shown in fig. 7. Table 4 shows the error calculation results of the terminal voltages of the battery models, and the Mean Absolute Error (MAE) of the terminal voltages of the battery models was 0.51%, and the Root Mean Square Error (RMSE) of the terminal voltages of the battery models was 0.8%. The data show that the established second-order equivalent circuit model and the identified parameters are very reasonable and can be used for subsequent SOC estimation.
And S4, establishing a self-adaptive unscented Kalman filtering algorithm, calculating the variance of innovation and residual errors by a moving window method, and self-adaptively updating and calculating the system noise covariance and the observation noise covariance of the state space equation.
Specifically, referring to fig. 8, an adaptive unscented kalman filter algorithm (hereinafter, referred to as the "AUKF algorithm") is established, and the process of adaptively updating the noise covariance of the computing system and the observed noise covariance is as follows:
for a nonlinear system, the state equation and observation equation for adding system noise and observation noise are expressed as:
Figure BDA0003864646080000171
where k is the current time, F (x) k-1 ,u k ) For the nonlinear system state transition equation, G (x) k-1 ,u k ) For non-linear system observation equations, x k Is a state variable, u k For known input, y k Is the observation signal, w is the system noise, v is the observation noise;
according to the KF principle, combining the state space equation shown in the formula (4) and the state equation and the observation equation shown in the formula (13), solving a first derivative of the nonlinear observation equation at the current state value to obtain an observation matrix H k Comprises the following steps:
Figure BDA0003864646080000172
determining initial values of state values
Figure BDA0003864646080000181
Sum state error covariance initial value P 0 Comprises the following steps:
Figure BDA0003864646080000182
Figure BDA0003864646080000183
calculate Sigma point:
Figure BDA0003864646080000184
wherein L is the length of the state vector;
calculating the weight:
Figure BDA0003864646080000185
wherein α =0,k i ＝0，β＝2；
Updating predicted state values
Figure BDA0003864646080000186
Figure BDA0003864646080000187
Figure BDA0003864646080000188
Updating the predicted observations
Figure BDA0003864646080000189
Figure BDA00038646460800001810
Figure BDA00038646460800001811
Updating system covariance prediction value P xx|k ：
Figure BDA00038646460800001812
Innovation at time k k Defined as the actual observed value y k And the predicted observed value
Figure BDA00038646460800001813
The difference and innovation d k The expression of (c) is:
Figure BDA0003864646080000191
calculating innovation d according to a moving window method k Variance of (2)
Figure BDA0003864646080000192
Figure BDA0003864646080000193
Wherein W is the length of the moving window;
updating system noise covariance Q k ：
Figure BDA0003864646080000194
Updating an observed covariance predictor P yy|k ：
Figure BDA0003864646080000195
Updating the covariance P xy|k ：
Figure BDA0003864646080000196
Calculating the Kalman gain K k ：
Figure BDA0003864646080000197
Updating an estimated state value
Figure BDA0003864646080000198
Figure BDA0003864646080000199
Updating an estimated observation
Figure BDA00038646460800001910
Figure BDA00038646460800001911
Updating error covariance value P k ：
P k ＝P xx|k -K k P yy|k K T (32)
Residual r at time k k Defined as the actual observed value y k And estimating the observed value
Figure BDA00038646460800001912
Difference of, residual r k The expression of (c) is:
Figure BDA00038646460800001913
computing residual r according to a moving window method k Variance of (2)
Figure BDA0003864646080000201
Figure BDA0003864646080000202
Updating the observation noise covariance R k ：
R k ＝C rk +H k P k H k T (35)。
In this embodiment, the length of the state vector L =3, and since the length of the system state vector is 3,q k Is a 3 x 3 symmetric matrix. Then Q is k Is shown as
Figure BDA0003864646080000203
Wherein Q is 12 ＝Q 21 ,Q 13 ＝Q 31 ,Q 23 ＝Q 32 . Q in SOC estimation using AUKF algorithm in MATLAB k Values are shown in figure 9. R in SOC estimation using AUKF algorithm in MATLAB k Values are shown in figure 10.
It should be noted that when the system noise covariance Q is k When the estimated state value is too large, the system covariance predicted value is increased, the state predicted value is increased, the estimated state value is increased, and finally the SOC estimation error is increased. Therefore, the value of the system noise covariance has an important influence on the estimation result of the SOC. When the observation noise covariance increases, the filter gain decreases, so that the influence of the observation error on the state estimation value becomes small. Conversely, as the observed noise covariance decreases, the filter gain increases, which increases the proportion of the observed error in the estimated state values. Therefore, the observation noise covariance plays an important role in correcting the influence of the observation error on the estimation result. The adaptive unscented Kalman filtering algorithm established by the invention monitors the change of innovation and residual error in the filter in real time, calculates the variance of the innovation and the variance of the residual error by a moving window method, corrects the system noise covariance by the variance of the innovation in real time, corrects the observation noise covariance by the variance of the residual error in real time, and has higher convergence speed and more accurate estimation result.
And S5, estimating the terminal voltage value of the lithium battery and the SOC value of the lithium battery by using a Kalman filter based on the established adaptive unscented Kalman filtering algorithm.
Specifically, the initial value of the AUKF algorithm is set: p 0 ＝diag([10 -5 ,10 -5 ,10 -3 ])，R＝1，x 0 ＝[0 0 0.6] T ，Q＝10 -7 ×eye(3)，W =1180. Then, SOC estimation is carried out by an AUKF algorithm under the UDDS working condition. And finally, carrying out SOC estimation by adopting an UKF algorithm, and comparing and analyzing the result of the SOC estimation by adopting the UKF algorithm. The results of the battery terminal voltages estimated by the UKF algorithm and the AUKF algorithm are shown in fig. 11 (fig. 11a is a terminal voltage versus voltage error graph, fig. 11b is a terminal voltage error graph). The SOC gain ratio of the UKF algorithm and the AUKF algorithm is shown in fig. 12. The SOC results of the UKF algorithm and the AUKF algorithm are shown in fig. 13 (fig. 13a is an SOC comparison map, and fig. 13b is an SOC error comparison map).
It can be seen from fig. 11a that the value of the battery terminal voltage estimated using the AUKF algorithm is closer to the true value than the value of the terminal voltage estimated using the UKF algorithm. It can be seen from fig. 11b that the value of the terminal voltage error estimated using the AUKF algorithm is smaller than the value of the terminal voltage error estimated using the UKF algorithm, and the value of the terminal voltage error estimated using the AUKF algorithm is more stable. As can be seen from fig. 9, the value of Q varies greatly at the beginning due to a systematic error caused by the initial value of SOC. As can be seen from fig. 10, the value of R fluctuates within a certain range due to an observation error. It can be seen from fig. 12 that the kalman gain of the AUKF algorithm is smaller than that of the UKF algorithm, because the change of Q improves the influence of system noise on the estimation result, and the change of R improves the influence of observation noise on the estimation result. From FIG. 13a, it can be concluded that the SOC value estimated using the AUKF algorithm is closer to the true value. From 13b, it can be seen that the SOC estimation error of the AUKF is smaller than the SOC estimation result of the UKF, and the convergence rate of the AUKF algorithm is faster.
In order to further explain the accuracy of the AUKF algorithm, the error analysis is carried out on the terminal voltage value and the SOC estimated value estimated by the UKF algorithm and the AUKF algorithm, the error analysis result of the terminal voltage is shown in a table 5, and the error analysis result of the SOC is shown in a table 6.
TABLE 5
Figure BDA0003864646080000211
Figure BDA0003864646080000221
TABLE 6
Figure BDA0003864646080000222
According to the table 5 and the table 6, the voltage value estimated by the AUKF algorithm is improved by 1.05 percent than the voltage value estimated by the UKF algorithm; for SOC estimation, the AUKF algorithm is more accurate than the UKF algorithm, and the precision is improved by 2.6%. Therefore, the SOC estimation is carried out by using the AUKF algorithm, the problems of model errors and uncertain observation errors can be effectively solved, the convergence rate of the filter is greatly improved, the estimation result is more accurate and stable, and the robustness is good.
The above-described embodiments are intended to illustrate rather than to limit the invention, and any modifications and variations of the present invention are possible within the spirit and scope of the claims.

Claims (7)

1. A lithium battery state-of-charge estimation method based on adaptive unscented Kalman filtering is characterized by comprising the following steps:
s1, acquiring a state equation and an output equation of a model based on a second-order equivalent circuit model of a lithium battery, and discretizing to obtain a state space equation of a discretized second-order equivalent circuit model;
s2, identifying model parameters through a voltage response curve of battery discharge, a state equation of a second-order equivalent circuit model and an output equation;
s3, substituting the identified model parameters into a state equation and an output equation of a second-order equivalent circuit model, using pulse discharge as input of the second-order equivalent circuit model, comparing the voltage of the output end of the second-order equivalent circuit model with the voltage of the actual end, and verifying the precision of the second-order equivalent circuit model;
s4, establishing a self-adaptive unscented Kalman filtering algorithm, calculating the variance of innovation and residual errors by a moving window method, and self-adaptively updating and calculating the system noise covariance and the observation noise covariance of the state space equation;
and S5, estimating the terminal voltage value of the lithium battery and the SOC value of the lithium battery by using a Kalman filter based on the established adaptive unscented Kalman filtering algorithm.
2. The lithium battery state-of-charge estimation method based on adaptive unscented kalman filter according to claim 1, characterized in that the state equation of the second order equivalent circuit model is:
Figure FDA0003864646070000011
in the formula I bat The current is the open circuit current of the lithium battery, and the discharge is positive; q bat The rated capacity of the lithium battery; r is si Ohmic internal resistance of the lithium battery; r is tf The polarization resistance of the lithium battery; c tf The polarization capacitor of the lithium battery; r ts Is a concentration polarization resistance; c ts A concentration polarization capacitor; u shape tf Is a polarization capacitor C tf The voltage across; u shape ts Polarizing the capacitance C for concentration ts The voltage across;
the output equation of the second-order equivalent circuit model is as follows:
U bat ＝U OC (SOC)-R si I bat -U tf -U ts (2)
in the formula of U bat Is the battery terminal voltage; u shape OC (SOC) is the open-circuit voltage of the battery associated with SOC.
3. The lithium battery state-of-charge estimation method based on adaptive unscented kalman filter according to claim 2, characterized in that, according to the second order equivalent circuit model of the lithium battery, the state space equation of the discretized second order equivalent circuit model obtained by combining formula (1) and formula (2) is:
Figure FDA0003864646070000021
in the formula, τ tf ＝R tf C tf For a fast time constant, τ ts ＝R ts C ts Is a slow time constant;
order to
Figure FDA0003864646070000022
I bat (k)＝u k ，U bat (k)＝y k ，C＝[-1 -1 0]The state space equation of the discretized second-order equivalent circuit model is simplified as follows:
Figure FDA0003864646070000023
4. the adaptive unscented kalman filter-based lithium battery state of charge estimation method of claim 3, wherein in step S2, the identified model parameter comprises ohmic internal resistance R of the lithium battery si Lithium battery polarization resistance R tf Polarization capacitor C of lithium battery tf Concentration polarization resistance R ts Concentration polarization capacitor C ts And lithium battery open-circuit voltage U with functional relation oc (SOC) expression.
5. The lithium battery state-of-charge estimation method based on adaptive unscented kalman filter of claim 4, characterized in that, in step S2, 10 lithium batteries are connected in parallel, 1C pulse discharge is adopted, 3min of each discharge is performed, then the lithium batteries are left to stand for 2h, and the lithium batteries are circularly discharged to cut-off voltage to obtain a pulse discharge voltage curve and a pulse discharge current curve, thereby obtaining the SOC and the open-circuit voltage U of the lithium batteries oc Performing least square fitting by using MATLAB to obtain open-circuit voltage U oc And a function relation curve of the lithium battery SOC.
6. The lithium battery state-of-charge estimation method based on adaptive unscented kalman filter of claim 4, wherein in step S2, the resistance-capacitance identification is performed by analyzing a voltage response curve of battery pulse discharge and combining resistance-capacitance characteristics; the specific process of identifying the resistor and the capacitor comprises the following steps:
the cell voltage response curve is divided into four phases:
A-B section: the battery begins to discharge from standing, the terminal voltage drops suddenly, and the polarization capacitor C tf Voltage U across tf Sum concentration polarization capacitance C ts Voltage U across ts The occurrence of a mutation is not allowed,
the sudden drop of the voltage in the A-B section is caused by ohmic internal resistance R si Caused by (a);
sections B to C: during the continuous discharge, the electrochemical polarization and the concentration polarization act together to reduce the voltage in an exponential change mode; before the B-C section, the voltage U tf Sum voltage U ts If the response is zero, the B-C section is in zero state response;
C-D stage: the discharge current disappears, the voltage of the battery rebounds, the same as the A-B section, and the voltage rebounds in the C-D section is caused by the ohmic internal resistance R si Caused by (a);
D-E section: the battery is kept stand, the voltage is slowly increased due to the action of electrochemical polarization and concentration difference, no current is discharged at the moment, and the D-E section is zero input response;
according to the A-B section and the C-D section, the ohmic internal resistance R is obtained by the following formula si ：
Figure FDA0003864646070000041
In the formula of U A Is the voltage of the lithium battery terminal corresponding to the point A in the voltage response curve, U B Is the voltage of the lithium battery terminal corresponding to the point B in the voltage response curve, U C Is the voltage of the lithium battery terminal corresponding to the C point in the voltage response curve, U D The voltage is the voltage of the lithium battery corresponding to the D point in the voltage response curve;
solving a differential equation according to a state equation of the model shown in the formula (1) to obtain:
Figure FDA0003864646070000042
in the formula of U tf (0) Is a polarization capacitor C tf Initial voltage, U, across ts (0) Polarising the capacitance C for concentration ts An initial voltage across;
according to the condition that the discharge current of the D-E section is zero and zero input response is achieved, the D point is used as the time t =0, and the zero input response expression of the RC loop is obtained as follows:
Figure FDA0003864646070000043
and (3) combining the output equation of the model shown in the formula (2) to obtain the output equation of the lithium battery under the zero input response, wherein the output equation of the lithium battery under the zero input response is as follows:
Figure FDA0003864646070000044
order to
Figure FDA0003864646070000045
U tf (0)＝b 1 ，U ts (0)＝b 2 Equation (8) reduces to:
Figure FDA0003864646070000046
using MATLAB to take the formula (9) as a fitting function, performing least square fitting on the DE section to obtain b 1 ,b 2 ,λ 1 ,λ 2 To obtain τ tf ,τ ts A value of (d);
and according to the condition that the section B-C is zero state response, and taking the point B as the moment of t =0, obtaining a zero state response expression of the RC loop as follows:
Figure FDA0003864646070000047
and (3) combining the output equation of the model shown in the formula (2) to obtain the output equation of the lithium battery under the zero state response, wherein the output equation of the lithium battery under the zero state response is as follows:
Figure FDA0003864646070000051
τ obtained by fitting of formula (9) tf ,τ ts Substituting the formula (11) into the formula (11) as a fitting function, performing least square fitting on the B-C section to obtain a 1 ,a 2 Is obtained by the equation (12) to obtain the resistance R tf ,R ts Is expressed as:
Figure FDA0003864646070000052
then according to tau tf ＝R tf C tf ,τ ts ＝R ts C ts Calculating the capacitance C tf ,C ts The value of (c).
7. The lithium battery state-of-charge estimation method based on adaptive unscented kalman filtering of claim 4, wherein in step S4, an adaptive unscented kalman filtering algorithm is established, and the process of adaptively updating and calculating the system noise covariance and the observation noise covariance comprises:
for a nonlinear system, the state equation and observation equation for adding system noise and observation noise are expressed as:
Figure FDA0003864646070000053
where k is the current time, F (x) k-1 ,u k ) For the nonlinear system state transition equation, G (x) k-1 ,u k ) For non-linear system observation equations, x k Is a state variable, u k For known input, y k Is the observation signal, w is the system noise, v is the observation noise;
according to the KF principle, combining the state space equation shown in the formula (4) and the state equation and the observation equation shown in the formula (13), solving a first derivative of the nonlinear observation equation at the current state value to obtain an observation matrix H k Comprises the following steps:
Figure FDA0003864646070000054
determining initial values of state values
Figure FDA0003864646070000055
Sum state error covariance initial value P 0 Comprises the following steps:
Figure FDA0003864646070000061
Figure FDA0003864646070000062
calculate Sigma point:
Figure FDA0003864646070000063
wherein L is the length of the state vector;
calculating the weight:
Figure FDA0003864646070000064
wherein α =0,k i ＝0，β＝2；
Updating predicted state values
Figure FDA0003864646070000065
Figure FDA0003864646070000066
Figure FDA0003864646070000067
Updating a predictive observation
Figure FDA0003864646070000068
Figure FDA0003864646070000069
Figure FDA00038646460700000610
Updating system covariance prediction value P xx|k ：
Figure FDA00038646460700000611
Calculating innovation d k And innovation d k Variance of (2)
Figure FDA00038646460700000612
Figure FDA00038646460700000613
Figure FDA0003864646070000071
Wherein, W is the length of the movable window;
updating system noise covariance Q k ：
Figure FDA0003864646070000072
Updating an observed covariance predictor P yyk ：
Figure FDA0003864646070000073
Updating the covariance P xy|k ：
Figure FDA0003864646070000074
Calculating the Kalman gain K k ：
Figure FDA0003864646070000075
Updating an estimated state value
Figure FDA0003864646070000076
Figure FDA0003864646070000077
Updating an estimated observation
Figure FDA0003864646070000078
Figure FDA0003864646070000079
Updating error covariance value P k ：
P k ＝P xx|k -K k P yy|k K T (32)
Calculating residual value r k Variance of sum residual
Figure FDA00038646460700000710
Figure FDA00038646460700000711
Figure FDA00038646460700000712
Updating the observation noise covariance R k ：
Figure FDA0003864646070000081
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