CN114487879A - SOC estimation method based on adaptive square root center difference Kalman filtering - Google Patents

SOC estimation method based on adaptive square root center difference Kalman filtering Download PDF

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CN114487879A
CN114487879A CN202111611150.2A CN202111611150A CN114487879A CN 114487879 A CN114487879 A CN 114487879A CN 202111611150 A CN202111611150 A CN 202111611150A CN 114487879 A CN114487879 A CN 114487879A
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lithium battery
square root
time
covariance matrix
soc
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万佑红
达杨阳
谷康伟
何韦唯
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Nanjing University of Posts and Telecommunications
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Abstract

The invention discloses an SOC estimation method based on self-adaptive square root center difference Kalman filtering, which is characterized in that a second-order RC equivalent circuit model is established according to the internal mechanism of a lithium battery, and a recursive least square method (FFRLS) with forgetting factors is used for parameter identification; the method is improved on the basis of Central Difference Kalman (CDKF) filtering, self-adaptation and square root factors are introduced, the problem of noise uncertainty can be effectively solved by self-adaptation, the algorithm covariance matrix normality can be kept by square root, and the SOC estimation precision and the noise robustness can be effectively improved by combining the two factors. The invention provides a certain reference thought for estimating the SOC state of the lithium battery.

Description

SOC estimation method based on adaptive square root center difference Kalman filtering
Technical Field
The invention belongs to the technical field of state control and control optimization of complex systems, and particularly relates to an SOC estimation method based on adaptive square root center difference Kalman filtering.
Background
The lithium battery is a common power battery in new energy vehicles due to the characteristics of high single working voltage, long service life, large specific energy, long cycle life and wide working temperature range. The SOC of the lithium battery is used as an important parameter index in the battery management system, and has important significance in monitoring the safety state of the battery, estimating the service life, improving the working efficiency and the like. Therefore, accurate estimation of the SOC of the lithium battery is a very important part of the battery management system.
Since the SOC is an internal state quantity of the battery, it cannot be directly measured, and can only be estimated by measuring the output voltage and current of the battery. Two currently most commonly used non-model SOC estimation methods are ampere-hour integration and open-circuit voltage. The ampere-hour integration method estimates the SOC value by time integration of current, and although this method is easy to implement, it is easy to accumulate errors based on open loop, and the longer the time, the larger the error, the more often it needs to be corrected to eliminate the error. The open-circuit voltage method can obtain the SOC of the battery by determining the relationship between the OCV and the SOC of the battery and measuring the OCV of the battery, and has high precision. However, obtaining OCV requires separate measurements to isolate the battery from external circuitry, which is clearly impractical in a practical operating environment.
Therefore, in order to obtain an accurate SOC value, a model-based SOC estimation strategy is receiving attention, which may be classified into an electrochemical impedance model, an equivalent circuit model, and a neural network model. The electrochemical impedance model has higher precision, but the research on the internal chemical reaction mechanism of the battery leads the model to have extremely high complexity and difficult application. The neural network model needs a large amount of data sample training, and the data type, precision and training method selection have certain influence on the estimation result. The equivalent circuit model compromises the precision and the complexity of the model, the battery is equivalent to a circuit, and the SOC value is estimated by a circuit architecture state space equation.
The most commonly used SOC estimation algorithms based on the equivalent circuit model include Kalman filtering, H ∞ filtering, Particle Filtering (PF), and the like. The typical algorithm of Kalman filtering comprises Extended Kalman Filtering (EKF) and Unscented Kalman Filtering (UKF), and the like, wherein the extended Kalman filtering carries out first-order Taylor expansion linearization processing on a nonlinear function, the error is obvious when the nonlinear intensity of the system is large, the unscented Kalman filtering adopts sigma point approximation to replace first-order Taylor expansion, sigma point weighting and combination of approximate state quantities are adopted to improve the precision to second-order approximation, but the UKF parameter setting has large influence on the estimation result, and the setting parameter depends on experience. H infinity filtering reduces the negative influence of noise on state estimation by designing a proper cost function J, and the estimation result is conservative, because the noise is always assumed to be in an extreme condition, so that the estimation is a non-optimal estimation. The particle filter is based on a Monte Carlo method, a probability density function of particle distribution is approximated by searching a group of random particles which are propagated in a state space, and when the number of examples is enough, the approximation precision of a nonlinear and non-Gaussian system is high, but the practical application of the particle filter is limited due to the excessive calculation amount.
Disclosure of Invention
The purpose of the invention is as follows: aiming at the problems, the invention provides an SOC estimation method based on self-adaptive square root center difference Kalman filtering, which solves the problems of noise interference condition and calculation precision in an actual system.
The technical scheme is as follows: in order to realize the purpose of the invention, the technical scheme adopted by the invention is as follows: a SOC estimation method based on adaptive square root center difference Kalman filtering specifically comprises the following steps:
step 1, establishing a second-order RC equivalent circuit model based on the internal mechanism of the lithium battery, and deducing a state equation and an output equation of the lithium battery system according to the second-order RC equivalent circuit model;
step 2, based on the actually measured voltage and current data, performing parameter identification on the second-order RC equivalent circuit model by a recursive least square method with forgetting factors;
step 3, introducing a self-adaptive mechanism and a square root method to a central difference Kalman filtering algorithm based on a state equation and an output equation of the lithium battery system, and establishing a self-adaptive square root central difference Kalman filter;
and 4, taking the SOC as an estimator, and performing iterative estimation by using a self-adaptive square root central difference Kalman filter.
Further, the SOC estimation method further includes:
and adding noise in the SOC estimation process, and testing the noise immunity performance of the self-adaptive square root central difference Kalman filter.
Further, step 1 establishes a second-order RC equivalent circuit model based on the internal mechanism of the lithium battery, specifically as follows:
two RC circuits are utilized to simulate the electrochemical polarization reaction and concentration polarization reaction in the lithium battery, and the electrochemical polarization reaction and the concentration polarization reaction are obtained by kirchhoff's law:
Ut＝Uoc+U1+U2+U0 (1)
Figure BDA0003434936850000021
Figure BDA0003434936850000022
U0＝iR0 (4)
in the formula of UtTerminal voltage of lithium battery, UocFor the open circuit voltage, R, of a lithium battery1And R2Respectively electrochemical polarization resistance and concentration polarization resistance, C1And C2Respectively electrochemical polarization capacitance and concentration polarization capacitance, R0Is internal resistance, i is battery current, U1Is a resistance R1Terminal voltage, U2Is a resistance R2The terminal voltage of the terminal is detected,
Figure BDA0003434936850000034
is a voltage U1The instantaneous derivative of the signal is determined,
Figure BDA0003434936850000035
is a voltage U2Instantaneous derivative, U0Is an internal resistance R0A terminal voltage.
Further, in step 1, a state equation and an output equation of the lithium battery system are derived according to the second-order RC equivalent circuit model, and the method specifically includes the following steps:
calculating the SOC of the lithium battery by using an ampere-hour integration method, wherein the ampere-hour integration method is represented as follows:
Figure BDA0003434936850000031
in the formula, StRepresents SOC, S of the lithium battery at the time tt0Represents t0SOC, C of lithium battery at any momentnRepresenting the capacity of the battery, wherein eta is a coulombic efficiency coefficient;
discretizing the formulas (1) to (5) based on a nonlinear discrete system to obtain a state equation and an output equation of the lithium battery equivalent circuit model; the nonlinear discrete system model is represented as follows:
Figure BDA0003434936850000032
where k denotes the corresponding discrete time, xk+1Representing the state vector at time k +1, xkRepresents the state vector at the k-th time, ukIndicates the input amount at the k-th time, ykRepresents the output quantity at the k-th time, vkAnd nkRespectively representing process noise and measurement noise at the kth moment, wherein f () and h () are nonlinear functions corresponding to a state equation and an output equation respectively;
the state equation and the output equation of the lithium battery equivalent circuit model are expressed as follows:
Figure BDA0003434936850000033
in the formula of U1(k) Resistance R at time k1Terminal voltage, U2(k) Resistance R at time k2Terminal voltage, SOC (k), is an estimated value of SOC at the k-th time, U1(k-1) is the resistance R at the time of k-11Terminal voltage, U2(k-1) is the resistance R at the time of k-12Terminal voltage, SOC (k-1) is an SOC estimation value at the k-1 time, i (k-1) is a battery current at the k-1 time, v (k-1) and n (k-1) are process noise and measurement noise at the k-1 time respectively, and Ut(k) The terminal voltage of the lithium battery at the moment k, delta t is a sampling period, and tau1And τ2Is the time constant of the RC network, tau1＝R1*C1，τ2＝R2*C2，Uoc() Is UocFunction on SOC.
Further, the method of step 2 specifically includes:
based on a recursive least square method with forgetting factors, a pulse transfer function is obtained through a state equation and an output equation of the lithium battery system and is expressed as follows:
Figure BDA0003434936850000041
make U equal to Ut-UocThe pulse transfer function is converted into a difference equation by an impulse response invariant method:
U(k)＝a1U(k-1)+a2U(k-2)+b1I(k)+b2I(k-1)+b3I(k-2) (9)
in the formula of UtTerminal voltage of lithium battery, UocThe lithium battery open-circuit voltage is provided, U is the voltage difference between the lithium battery terminal voltage and the lithium battery open-circuit voltage, and G (z) is a transfer function of a state space equation corresponding to a z domain; u (z) is the z-domain transform of the voltage difference U; i (z) is a z-domain transform of current i; z is a complex variable; a is1、a2、b1、b2、b3The parameters to be identified are respectively external autoregressive models, specifically R in a state equation and an output equation1、C1、R2、C2And R0(ii) a U (k) is the voltage difference between the terminal voltage of the lithium battery and the open-circuit voltage of the lithium battery at the kth moment, U (k-1) is the voltage difference between the terminal voltage of the lithium battery and the open-circuit voltage of the lithium battery at the kth moment, U (k-2) is the voltage difference between the terminal voltage of the lithium battery and the open-circuit voltage of the lithium battery at the kth moment, I (k) is the current I at the kth moment, I (k-1) is the current I at the kth moment, and I (k-2) is the current I at the kth moment.
Further, the recursive least square method with forgetting factor in step 2 is expressed as follows:
Figure BDA0003434936850000042
in the formula, θ (k) is the parameter vector to be identified,
Figure BDA0003434936850000043
for the estimated parameter vector at the time instant k,
Figure BDA0003434936850000044
the estimation parameter vector at the k-1 moment is shown, T represents vector transposition, lambda is forgetting factor, K (k) is gain vector, y (k) is system actual output value, h (k) is data vector, P (k) is estimation error covariance matrix at the k-1 moment, P (k-1) is estimation error covariance matrix at the k-1 moment, and I is unit matrix.
Further, in step 3, a square root method is introduced into the central difference Kalman filter algorithm, and then the covariance matrix is transferred by a square root factor, and is iteratively updated through matrix qr decomposition and a choleupdate function, which are specifically as follows:
Figure BDA0003434936850000051
Figure BDA0003434936850000052
Figure BDA0003434936850000053
in the formula (I), the compound is shown in the specification,
Figure BDA0003434936850000054
represents the square root of the prior estimate error covariance matrix,
Figure BDA0003434936850000055
representing the output prediction auto-covariance matrix, Sv,kRepresenting the square root of the covariance matrix of the noise during time k, Sn,kRepresenting the square root of the measured noise covariance matrix at time k,
Figure BDA0003434936850000056
for the a posteriori estimation of the error covariance matrix square root,
Figure BDA0003434936850000057
and
Figure BDA0003434936850000058
computing weights, χ, for the center difference Kalman filtering covariancexThe method comprises the steps of representing a state vector related part in a predicted updating state x obtained by time updating of a sigma point through a state equation of a lithium battery system, wherein gamma represents an output value obtained by measuring and updating the sigma point obtained by prior state estimation and a prior covariance matrix through an output equation of the lithium battery system, and subscript 1: l, L +1:2L and 0 respectively represent the 1 st to Lth, L +1 st to 2 Lth and start columns of the matrix, K | K-1 represents the prior estimate, KkRepresenting a gain vector.
Further, in step 3, after the central difference Kalman filter algorithm introduces the square root method, an adaptive mechanism is added, and then the process noise covariance matrix and the measurement noise covariance matrix are updated in a square root form after the current iteration is completed, so as to be used for the next iteration of the current iteration, which is specifically as follows:
Figure BDA0003434936850000059
Figure BDA00034349368500000510
Sn,k+1＝(1-ζ)Sn,k+ζchol(Pr) (16)
in the formula resiRepresenting the residual covariance matrix, res, at time ikRepresenting the residual covariance matrix at time k, ykRepresenting the actual measurement value at the k moment;
Figure BDA00034349368500000511
representing the output value S obtained by measuring and updating the sigma point obtained by the posterior state estimation and the posterior covariance matrix through the state equation of the lithium battery systemv,k+1Representing the square root of the covariance matrix of the process noise at time k +1, T representing the vector transposition, Sn,k+1Represents the measurement noise at the k +1 timeThe square root of the acoustic covariance matrix, ζ is the scaling factor, ζ ∈ [0,1]]When ζ is set to 0, it means no adaptation is performed, ζ is set to 1, it means full adaptation, Len denotes a fixed window size, PrRepresenting a residual covariance matrix;
and when the noise statistical characteristic is uncertain, observing the square root of the noise covariance matrix and the measured noise covariance matrix in the process of adaptively updating the information through the fixed window length.
Has the advantages that: compared with the prior art, the technical scheme of the invention has the following beneficial technical effects:
the invention relates to an SOC estimation method based on adaptive square root central difference Kalman filtering, which introduces an adaptive and square root mechanism in a traditional central difference Kalman filtering algorithm and applies the adaptive and square root mechanism to an equivalent circuit model, aims at the problem of noise interference, inhibits the influence of noise fluctuation on SOC estimation precision through the adaptive mechanism, ensures that a covariance matrix keeps positive during each step of iterative updating through square root, avoids the problem of filtering divergence, does not need excessive tests for setting a step length h, and can adjust the convergence speed and robustness of a filter by taking a reasonable value near an optimal value under the condition of Gaussian distribution.
Drawings
FIG. 1 is a schematic diagram of an SOC estimation method based on adaptive square root center difference Kalman filtering according to an embodiment;
FIG. 2 is a second-order RC equivalent circuit model of a lithium battery according to an embodiment;
FIG. 3 is a graph of SOC versus OCV charge and discharge for an exemplary embodiment;
FIG. 4 is a graph comparing the state estimation curves of the algorithm of the present invention and the prior art algorithm under one embodiment;
FIG. 5 is a graph comparing the absolute value of the state estimation error of the algorithm of the present invention with that of the prior art in one embodiment;
FIG. 6 is an enlarged comparison of the state estimation local interval of the algorithm of the present invention with the prior art algorithm in one embodiment;
FIG. 7 is a graph comparing the convergence rate of state estimation for the algorithm of the present invention and a prior art algorithm under one embodiment.
Detailed Description
The technical solution of the present invention is further described below with reference to the accompanying drawings and examples.
Referring to fig. 1, the SOC estimation method based on adaptive square root center difference Kalman filtering specifically includes the following steps:
step 1: establishing a second-order RC equivalent circuit model of the lithium battery, and deducing a state space equation according to the model:
the second-order RC equivalent circuit simulates the electrochemical polarization reaction and concentration polarization reaction inside the battery by using two RC circuits, as shown in the attached figure 2, and can be obtained by kirchhoff's law:
Ut＝Uoc+U1+U2+U0 (1)
Figure BDA0003434936850000071
Figure BDA0003434936850000072
U0＝iR0 (4)
in formulae (1) to (4), UtTerminal voltage of lithium battery, UocFor the open circuit voltage, R, of a lithium battery1And R2Respectively electrochemical polarization resistance and concentration polarization resistance, C1And C2Respectively electrochemical polarization capacitance and concentration polarization capacitance, R0Is internal resistance, i is battery current, charging is positive, U1Is a resistance R1Terminal voltage, U2Is a resistance R2The terminal voltage of the terminal is detected,
Figure BDA0003434936850000073
is a voltage U1The instantaneous derivative of the signal is determined,
Figure BDA0003434936850000074
is a voltage U2Instantaneous derivative, U0Is an internal resistance R0A terminal voltage.
Under the condition of no disturbance of current and voltage, the SOC of the lithium battery can be obtained according to an ampere-hour integration method:
Figure BDA0003434936850000075
wherein S istRepresents SOC, C of the lithium battery at the time tnAnd eta represents the capacity of the battery, and is a coulombic efficiency coefficient.
Secondly, discretizing the formulas (1) - (5) to obtain a state space equation of the lithium ion battery as follows:
Figure BDA0003434936850000076
Figure BDA0003434936850000077
wherein, CnRepresenting the capacity of the battery, eta is the coulombic efficiency coefficient, delta t is the sampling period, v and n respectively represent the process noise and the measurement noise, and tau1And τ2Is the time constant of the RC network, tau1＝R1*C1，τ2＝R2*C2，Uoc() Is UocFunction on SOC.
The system matrix, the input matrix, the output matrix and the direct transfer matrix in the state space equation of the lithium battery are respectively as follows:
Figure BDA0003434936850000081
step 2: fitting OCV-SOC curves
In the experiment of this embodiment, a lithium ion battery with model number LG 18650HG2 is adopted, the rated voltage is 3.6V, the rated capacity is 3Ah, and the battery is charged and discharged at 0.5C magnification in an experimental environment at 25 ℃, so as to obtain corresponding curves of open-circuit voltage OCV and SOC in the charging and discharging states respectively. In order to reduce experimental errors, high-order fitting is carried out on the mean value of the SOC-OCV curves of charging and discharging.
The OCV-SOC curve is obtained by a battery static test, which is a method of obtaining the OCV by standing for a certain time after charging or discharging. The principle is as follows:
since SOC is not a direct observed quantity and the accuracy of experimental data is quite high, SOC is calculated by the most common ampere-hour integration method.
As can be seen from the formula (6), when the battery is left standing for a certain time after the completion of charging or discharging, R0，R1，R2Gradually decreases to 0, and the measured voltage is the open-circuit voltage OCV.
The measured open-circuit voltage OCV and the SOC calculated by the ampere-hour integration method are plotted in the same graph under the charge-discharge rate of 0.5C, as shown in fig. 3, the upper curve and the lower curve are OCV-SOC curves in the charge state and the discharge state, respectively, and the mean value of the charge-discharge OCV-SOC curves is fitted, wherein the fitting formula is as follows:
Figure BDA0003434936850000082
the fitted curve is shown in fig. 3.
And step 3: parameter identification by adopting recursive least square method with forgetting factor
Because parameters in the lithium ion battery model can be influenced by the ambient temperature and the aging degree of the battery, and the parameters can change along with the change of the SOC, the least square method is adopted for parameter identification. The least square method is widely applied to parameter identification due to the advantages of simple principle, high convergence rate and the like. The conventional least square method has a data saturation phenomenon, so that a forgetting factor is introduced to solve the problem. The recursive least square method (FFRLS) with the forgetting factor introduces the forgetting factor lambda on the basis of the recursive least square method (RLS), thereby effectively reducing the influence of old data on parameter estimation and aggravating the influence of data at the current moment on parameter estimation. The smaller the value of lambda is, the stronger the algorithm tracking time-varying parameter capability is, but the noise interference influence becomes larger, and the estimation error variance becomes larger, otherwise, the tracking time-varying parameter capability is reduced, and the estimation error variance is reduced. In order to balance between the tracking capability and the estimation error variance, the value of λ is generally between 0.9 and 1, and the value is 0.96 here. The expression of the recursive least square method with forgetting factors is as follows:
Figure BDA0003434936850000091
in the formula, θ (k) is the parameter vector to be identified,
Figure BDA0003434936850000092
for estimating the parameter vector, T represents the vector transposition, K (k) is the gain vector, y (k) is the actual output value of the system, h (k) is the data vector, P (k) is the estimation error covariance matrix, and I is the identity matrix.
The pulse transfer function is obtained by equation (6):
Figure BDA0003434936850000093
make U equal to Ut-UocEquation (10) is converted to a difference equation by the impulse response invariant method:
U(k)＝a1U(k-1)+a2U(k-2)+b1I(k)+b2I(k-1)+b3I(k-2) (11)
in the formula of UtTerminal voltage of lithium battery, UocG (z) is a state space equation corresponding to a z-domain transfer function; definition of U ═ Ut-UocThen U (z) is the z-domain transform of the voltage difference U; i (z) is a z-domain transform of current i; z is a complex variable; a is1、a2、b1、b2、b3The parameters to be identified are respectively external autoregressive models, specifically R in a state equation and an output equation1、C1、R2、C2And R0The relational expression (c) of (c).
And 4, step 4: a state estimation filter is established based on an adaptive square root central difference Kalman filtering algorithm, on the basis of the Central Difference Kalman (CDKF) filtering theory, an adaptive and square root theory is introduced to form an Adaptive Square Root Central Difference Kalman (ASRCDKF) filtering algorithm, and the problems of noise interference and calculation accuracy in an actual system are solved. The Square Root Center Difference Kalman (SRCDKF) filter construction process is as follows:
1) calculating central difference weight
Figure BDA0003434936850000101
In the formula, h is the central differential step length, and for Gaussian distribution, the optimal value of h is
Figure BDA0003434936850000102
And L is a state quantity dimension, and the weight is applied when the weight of the estimated value and the covariance matrix is calculated.
2) Initializing a filter
Figure BDA0003434936850000103
3)
Figure BDA0003434936850000104
In order to set the initial value of the state,
Figure BDA0003434936850000105
to set the estimation error covariance matrix square root, Sv,0And Sn,0The initial process noise and the metrology noise covariance matrix square root. Iteration starts to end of discharge (k ═ 1, …, end)
1. Calculate sigma point for time update:
Figure BDA0003434936850000106
in the formula (I), the compound is shown in the specification,
Figure BDA0003434936850000107
estimating the error covariance matrix square root for the time k-1;
2. the time update equation:
χk|k-1＝f(χk-1,uk-1) (15)
Figure BDA0003434936850000108
Figure BDA0003434936850000109
wherein f () is a time-updated nonlinear equation corresponding to the state equation, χ, of the state space model of equation (6)k|k-1Deriving a prior state set u for sigma point by time updating equationkIs the input quantity at the time point k,
Figure BDA00034349368500001010
weighting the prior state set to obtain a prior state estimate, χxRepresenting the relevant part of the prediction update state x obtained by the sigma point through the time update of the state equation of the formula (15),
Figure BDA00034349368500001011
square root, S, of a covariance matrix representing a priori estimation errorv,kRepresenting the square root of the process noise covariance matrix at time k.
3. Calculate sigma point for metrology update:
Figure BDA0003434936850000111
4. measurement update equation:
Figure BDA0003434936850000112
Figure BDA0003434936850000113
Figure BDA0003434936850000114
Figure BDA0003434936850000115
Figure BDA0003434936850000116
Figure BDA0003434936850000117
Figure BDA0003434936850000118
where h () is the measurement update nonlinear equation, corresponding to the output equation of the state space model, γk|k-1The sigma point is derived by a measurement update equation to obtain a prior output set,
Figure BDA0003434936850000119
representing the output predictive auto-covariance matrix,
Figure BDA00034349368500001110
weighting the a priori output sets to obtain a priori output estimates,
Figure BDA00034349368500001111
as a cross-covariance matrix, KkIn order to be a matrix of gains, the gain matrix,
Figure BDA00034349368500001112
for a posteriori estimate, ykThe actual measured value at the time k is,
Figure BDA00034349368500001113
is new.
The adaptive estimation principle is introduced into the SRCDKF algorithm, and compared with the SRCDKF algorithm, the ASRCDKF algorithm has the square root S of the process noise covariance matrixvSum-measure noise covariance matrix square root SnChanging the set initial fixed value into a change value adaptive to the iteration process, and making the residual error as follows:
Figure BDA00034349368500001114
in the formula reskRepresenting the residual covariance matrix at time k, ykRepresenting the actual measurement value at the k moment;
Figure BDA00034349368500001115
represents the output value obtained by measuring and updating the sigma point obtained by the posterior state estimation and the posterior covariance matrix through the state equation of the lithium battery system, wherein,
Figure BDA00034349368500001116
and
Figure BDA00034349368500001117
the calculation mode is the same, except that the sigma point body is composed of
Figure BDA00034349368500001118
Change to
Figure BDA00034349368500001119
For the measurement noise SnUpdating and improving, calculating residual covariance matrix P by sliding windowr：
Figure BDA00034349368500001120
Setting the fixed window size as Len, windowWhen the size of the mouth is less than or equal to k, kwWhen the window size is larger than k, kw＝Len。
Thus to process noise SvAnd measure the noise SnComprises the following steps:
Figure BDA0003434936850000121
Sn,k+1＝(1-ζ)Sn,k+ζchol(Pr) (29)
where ζ is the scaling factor, ζ ∈ [0,1], and is set to 0 when the initial iteration does not reach the window size Len, meaning that adaptation is abandoned when ζ is set to 0, and to 1 means that adaptation is completely done, regardless of the smoothness. When the noise statistical characteristics are uncertain, the noise covariance matrix in the process of adaptively updating the observation information with the fixed window length and the square root of the measured noise covariance matrix are used for reducing the influence of the uncertain model and noise statistical characteristics on filtering.
The updated results of the formulas (28) and (29) directly influence the singular decomposition results of the formulas (17) and (21), and the robustness to uncertain noise is stronger. While equation (25) becomes:
Figure BDA0003434936850000122
and 5: noise was added to the measured data to test the performance of the ASRCDKF filter on SOC estimation
Gaussian white noise of X-N (0,0.01) is added into the measured voltage, the SOC estimation is carried out on the UDDS working condition data of the same lithium battery by using the traditional EKF, CDKF and the proposed ASRCDKF filter, and the same initial covariance matrix, process noise and measured noise covariance matrix are set in all the methods. In the formula (29), the value of the adjustment factor ζ affects the self-adaptation and the smoothness effect, the self-adaptation makes the SOC estimation result less interfered by noise, the robustness is enhanced, the smoothness reduces the fluctuation of the SOC estimation value, and for balancing the two, ζ is taken as 0.7 for experiment. The estimation result verifies that after a self-adaption mechanism and a square root mechanism are introduced, the anti-interference performance of the algorithm is enhanced, the SOC estimation deviation can be controlled in a smaller range, the accuracy is higher, as shown in figures 4, 5 and 6, and meanwhile, the fast convergence property is still provided, as shown in figure 7.

Claims (8)

1. An SOC estimation method based on adaptive square root center difference Kalman filtering is characterized by comprising the following steps:
step 1, establishing a second-order RC equivalent circuit model based on the internal mechanism of the lithium battery, and deducing a state equation and an output equation of the lithium battery system according to the second-order RC equivalent circuit model;
step 2, based on the actually measured voltage and current data, performing parameter identification on the second-order RC equivalent circuit model by a recursive least square method with forgetting factors;
step 3, introducing a self-adaptive mechanism and a square root method to a central difference Kalman filtering algorithm based on a state equation and an output equation of the lithium battery system, and establishing a self-adaptive square root central difference Kalman filter;
and 4, taking the SOC as an estimator, and performing iterative estimation by using a self-adaptive square root central difference Kalman filter.
2. The SOC estimation method based on adaptive square root center differential Kalman filtering as claimed in claim 1, further comprising:
and adding noise in the SOC estimation process, and testing the noise immunity performance of the self-adaptive square root central difference Kalman filter.
3. The SOC estimation method based on adaptive square root center difference Kalman filtering according to claim 1, characterized in that, step 1, based on the internal mechanism of the lithium battery, a second-order RC equivalent circuit model is established, specifically as follows:
two RC circuits are utilized to simulate the electrochemical polarization reaction and concentration polarization reaction in the lithium battery, and the electrochemical polarization reaction and the concentration polarization reaction are obtained by kirchhoff's law:
Ut＝Uoc+U1+U2+U0 (1)
Figure FDA0003434936840000011
Figure FDA0003434936840000012
U0＝iR0 (4)
in the formula of UtTerminal voltage of lithium battery, UocFor the open circuit voltage, R, of a lithium battery1And R2Respectively electrochemical polarization resistance and concentration polarization resistance, C1And C2Respectively electrochemical polarization capacitance and concentration polarization capacitance, R0Is internal resistance, i is battery current, U1Is a resistance R1Terminal voltage, U2Is a resistance R2The terminal voltage of the terminal is detected,
Figure FDA0003434936840000013
is a voltage U1The instantaneous derivative of the signal is determined,
Figure FDA0003434936840000014
is a voltage U2Instantaneous derivative, U0Is an internal resistance R0A terminal voltage.
4. The SOC estimation method based on adaptive square root center difference Kalman filtering according to claim 3, characterized in that, in step 1, the state equation and the output equation of the lithium battery system are derived according to a second-order RC equivalent circuit model, and the method specifically comprises the following steps:
calculating the SOC of the lithium battery by using an ampere-hour integration method, wherein the ampere-hour integration method is represented as follows:
Figure FDA0003434936840000021
in the formula, StRepresents the SOC of the lithium battery at time t,
Figure FDA0003434936840000022
represents t0SOC, C of lithium battery at any momentnRepresenting the capacity of the battery, wherein eta is a coulombic efficiency coefficient;
discretizing the formulas (1) to (5) based on a nonlinear discrete system to obtain a state equation and an output equation of the lithium battery equivalent circuit model; the nonlinear discrete system model is represented as follows:
Figure FDA0003434936840000023
where k denotes the corresponding discrete time, xk+1Representing the state vector at time k +1, xkRepresents the state vector at the k-th time, ukIndicates the input amount at the k-th time, ykRepresents the output quantity at the k-th time, vkAnd nkRespectively representing process noise and measurement noise at the kth moment, wherein f () and h () are nonlinear functions corresponding to a state equation and an output equation respectively;
the state equation and the output equation of the lithium battery equivalent circuit model are expressed as follows:
Figure FDA0003434936840000024
in the formula of U1(k) Resistance R at time k1Terminal voltage, U2(k) Resistance R at time k2Terminal voltage, SOC (k), is an estimated value of SOC at the k-th time, U1(k-1) is the resistance R at the time of k-11Terminal voltage, U2(k-1) is the resistance R at the time of k-12Terminal voltage, SOC (k-1) is an SOC estimation value at the k-1 time, i (k-1) is a battery current at the k-1 time, v (k-1) and n (k-1) are process noise and measurement noise at the k-1 time respectively, and Ut(k) The terminal voltage of the lithium battery at the moment k, delta t is a sampling period tau1And τ2Is the time constant of the RC network, tau1＝R1*C1，τ2＝R2*C2，Uoc() Is UocFunction on SOC.
5. The SOC estimation method based on adaptive square root center difference Kalman filtering as claimed in claim 1, wherein the method of step 2 specifically comprises the following steps:
based on a recursive least square method with forgetting factors, a pulse transfer function is obtained through a state equation and an output equation of the lithium battery system and is expressed as follows:
Figure FDA0003434936840000031
make U equal to Ut-UocThe pulse transfer function is converted into a difference equation by an impulse response invariant method:
U(k)＝a1U(k-1)+a2U(k-2)+b1I(k)+b2I(k-1)+b3I(k-2) (9)
in the formula of UtTerminal voltage of lithium battery, UocThe lithium battery open-circuit voltage is provided, U is the voltage difference between the lithium battery terminal voltage and the lithium battery open-circuit voltage, and G (z) is a transfer function of a state space equation corresponding to a z domain; u (z) is the z-domain transform of the voltage difference U; i (z) is a z-domain transform of current i; z is a complex variable; a is1、a2、b1、b2、b3The parameters to be identified are respectively external autoregressive models, specifically R in a state equation and an output equation1、C1、R2、C2And R0(ii) a U (k) is the voltage difference between the terminal voltage of the lithium battery and the open-circuit voltage of the lithium battery at the kth moment, U (k-1) is the voltage difference between the terminal voltage of the lithium battery and the open-circuit voltage of the lithium battery at the kth moment, U (k-2) is the voltage difference between the terminal voltage of the lithium battery and the open-circuit voltage of the lithium battery at the kth moment, I (k) is the current I at the kth moment, I (k-1) is the current I at the kth moment, and I (k-2) is the current I at the kth moment.
6. The SOC estimation method based on adaptive square root center difference Kalman filtering as claimed in claim 1, wherein the recursive least square method with forgetting factor in step 2 is expressed as follows:
Figure FDA0003434936840000032
in the formula, θ (k) is the parameter vector to be identified,
Figure FDA0003434936840000033
for the estimated parameter vector at the time instant k,
Figure FDA0003434936840000034
the estimation parameter vector at the k-1 moment is shown, T represents vector transposition, lambda is forgetting factor, K (k) is gain vector, y (k) is system actual output value, h (k) is data vector, P (k) is estimation error covariance matrix at the k-1 moment, P (k-1) is estimation error covariance matrix at the k-1 moment, and I is unit matrix.
7. The SOC estimation method based on adaptive square root center difference Kalman filtering as claimed in claim 1, wherein in step 3, a square root method is introduced to the center difference Kalman filtering algorithm, and then the covariance matrix is transferred by square root factor, and is iteratively updated by matrix qr decomposition and choleupdate function, specifically as follows:
Figure FDA0003434936840000041
Figure FDA0003434936840000042
Figure FDA0003434936840000043
in the formula (I), the compound is shown in the specification,
Figure FDA0003434936840000044
represents the square root of the prior estimate error covariance matrix,
Figure FDA0003434936840000045
representing the output prediction auto-covariance matrix, Sv,kRepresenting the square root of the covariance matrix of the noise during time k, Sn,kRepresenting the square root of the measured noise covariance matrix at time k,
Figure FDA0003434936840000046
for the a posteriori estimation of the error covariance matrix square root,
Figure FDA0003434936840000047
and
Figure FDA0003434936840000048
computing weights, χ, for the center difference Kalman filtering covariancexThe method comprises the steps of representing a state vector related part in a predicted updating state x obtained by time updating of a sigma point through a state equation of a lithium battery system, wherein gamma represents an output value obtained by measuring and updating the sigma point obtained by prior state estimation and a prior covariance matrix through an output equation of the lithium battery system, and subscript 1: l, L +1:2L and 0 respectively represent the 1 st to Lth, L +1 st to 2 Lth and start columns of the matrix, K | K-1 represents the prior estimate, KkRepresenting a gain vector.
8. The SOC estimation method based on adaptive square root center difference Kalman filtering as claimed in claim 7, wherein in step 3, after introducing the square root method, the center difference Kalman filtering algorithm adds an adaptive mechanism, and then the process noise covariance matrix and the measurement noise covariance matrix are updated in square root form after the current iteration is completed, so as to be used in the next iteration of the current iteration, which is as follows:
Figure FDA0003434936840000049
Figure FDA00034349368400000410
Sn,k+1＝(1-ζ)Sn,k+ζchol(Pr) (16)
in the formula resiRepresenting the residual covariance matrix, res, at time ikRepresenting the residual covariance matrix at time k, ykRepresenting the actual measurement value at the k moment;
Figure FDA00034349368400000411
representing the output value S obtained by measuring and updating the sigma point obtained by the posterior state estimation and the posterior covariance matrix through the state equation of the lithium battery systemv,k+1Representing the square root of the covariance matrix of the process noise at time k +1, T representing the vector transposition, Sn,k+1Represents the square root of the covariance matrix of the measured noise at the time k +1, zeta is an adjustment factor, and zeta is in the range of 0,1]When ζ is set to 0, it means no adaptation is performed, ζ is set to 1, it means full adaptation, Len denotes a fixed window size, PrRepresenting a residual covariance matrix;
and when the noise statistical characteristic is uncertain, observing the square root of the noise covariance matrix and the measured noise covariance matrix in the process of adaptively updating the information through the fixed window length.
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
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