Lithium ion battery state-of-charge estimation method based on nonlinear FOM

Abstract

The invention discloses a lithium ion battery state-of-charge estimation method based on nonlinear FOM, and the method comprises the steps: building a nonlinear fractional order model of a lithium ion battery, and theoretically guaranteeing the stability of a fractional order system through employing a Mittag-Leffler method; aiming at parameter errors, interference and sensor noise, adopting a combination of a Luenberger observer and a sliding mode observer SMO to observe, and feeding back to a controller of the estimator; and finally, estimating the state of charge of the lithium ion battery by adopting a novel estimation algorithm based on a nonlinear fractional order model. By using the Lyapunov direct method, the stability of the estimator and the robustness of the estimator on the premise of some assumptions are ensured.

Description

Technical Field

[0001] The present invention relates to a technique for estimating the state of charge of a lithium-ion battery, specifically a method for estimating the state of charge of a lithium-ion battery based on a non-linear FOM, ensuring the stability and robustness of the state of charge estimation of the lithium-ion battery. Background Art

[0002] New energy electric vehicles are developing rapidly. However, compared with the number of fuel vehicles currently on the road, the number of electric vehicles is still very small. From the perspective of consumers, the main problems of electric vehicles include long charging time, high upfront cost, and range anxiety, etc. Currently, in order to maintain the battery health state and life, the charging rate needs to be conservatively limited. At the same time, the battery cost accounts for a large part of the total vehicle cost, and the driving range and battery life it can support still need to be improved. How to both increase the charging rate and ensure the battery life is a problem to be solved by the battery management system. Therefore, the battery management system is a key technology for the industrialization of new energy vehicles, and there is an urgent need to develop an advanced lithium-ion battery management system to solve these problems.

[0003] For battery management, one of the most important states is the state of charge (SoC) of the battery, which includes battery state monitoring and feedback control. Among them, the state of charge of the battery cannot be directly monitored, so real-time estimation is required. In the past 20 years, research on SoC estimation has been ongoing. The current main method, called Coulomb counting, can be easily implemented, but it is essentially an open-loop control method and has no ability to handle uncertainties.

[0004] To improve the estimation performance, closed-loop control should be adopted. Currently, model-based closed-loop control algorithms are usually used. Most of the established battery state estimators, such as the adaptive EKF, unscented Kalman filter (UKF), particle filter (PF), and Luenberger-type observer, rely on the equivalent circuit model. The equivalent circuit model based on the resistor-capacitor battery model is simple in calculation and parameter setting, so it is widely used for real-time monitoring of the available capacity. However, the ideal resistor and capacitor are completely linearized models, which may not accurately capture the non-linear characteristics of the battery dynamics, thus causing estimation errors and convergence problems; at the same time, the uncertainties caused by noise in the modeling process are ignored, and the battery stability and robustness cannot be guaranteed.

[0005] After a search of the prior art, it is found that Chinese Patent Document No. CN106405434A, with a publication date of February 15, 2017, discloses a method for estimating the state of charge of a battery. The method first establishes at least two estimation models under different battery SOHs based on the extended Kalman filter; then, for the estimation models under different battery SOHs, the extended Kalman filter algorithm is calculated to obtain the state-of-charge estimation values corresponding to the estimation models under different battery SOHs; finally, based on the state-of-charge estimation values, the weight values of the respective estimation models at a moment under different battery SOHs are obtained to obtain the optimal estimation of the state of charge of the battery at this moment. The method for estimating the state of charge provided by the present invention can overcome the problem of accuracy drift caused by estimating the state of charge of batteries in a battery pack under different battery SOHs using the same algorithm, so as to increase the stability and reliability of the state-of-charge algorithm. However, compared with the present invention, this prior art fails to consider modeling and parameter errors, disturbances, and sensor noise, and cannot theoretically prove the stability and robustness of the estimator. Summary of the Invention

[0006] In view of the obvious deficiencies in the current battery parameter and state estimation technology, which cannot accurately capture the non-linear characteristics of the battery dynamics, thus causing estimation errors and convergence problems, cannot eliminate the uncertainty caused by noise in the modeling process, and cannot ensure the stability and robustness of the battery, etc., the present invention proposes a state-of-charge (SoC) estimation algorithm for lithium-ion batteries to ensure the stability and robustness of lithium-ion batteries. By using a combination of the Luenberger observer and the sliding mode observer SMO, an estimation scheme based on a non-linear fractional-order model (FOM) is proposed. Modeling and parameter errors, disturbances, and sensor noise are explicitly considered. Using the Lyapunov direct method, under certain assumptions, theoretical guarantees for the stability and robustness of the estimator are obtained. Specifically, the Lyapunov function theoretically guarantees stability, the Luenberger term guarantees the convergence of the nominal error, and the sliding mode term increases the robustness of the system under the condition of error uncertainty.

[0007] The present invention is realized through the following technical solutions:

[0008] The present invention relates to a method for estimating the state of charge of a lithium-ion battery based on a non-linear FOM, including: an estimation algorithm developed from a general non-linear fractional-order model and an estimator designed to ensure Mittag–Leffler stability. First, a non-linear fractional-order model of the lithium-ion battery is established, and the stability of the fractional-order system is theoretically guaranteed by using the Mittag–Leffler method; then, for parameter errors, disturbances, and sensor noise, a combination of a Luenberger observer and a sliding-mode observer SMO is used for observation and fed back to the controller of the estimator; finally, a new estimation algorithm based on the non-linear fractional-order model is used to estimate the state of charge of the lithium-ion battery. In addition, by using the Lyapunov direct method, the stability of the estimator and the robustness of the estimator under certain assumptions are theoretically guaranteed.

[0009] The fractional order is defined as the Grünwald–Letnikov (GL) definition:

[0010]

[0011] The non-linear fractional-order model can be represented by

[0012]

[0013] where i is the imaginary number and ω is the radian frequency. α = 0 indicates that the CPE is a resistor, and α = 1 and α = -1 indicate that the CPE is a pure capacitor and an inductor, respectively.

[0014] The non-linear fractional-order model of the lithium-ion battery is:

[0015]

[0016]

[0017]

[0018] y(t) = V ocp (x 1 (t)) + R 0 u(t) + x 2 (t) + x 3 (t)

[0019] In the dynamic system representation, u(t) is the input current, i.e., negative for discharge and positive for charge; the state vector is defined as: x = [SoC(t), V 1 (t), V 2 (t)] T ; y(t) represents the output voltage; C n is the nominal capacity of the battery, in ampere-hours (Ah); R0 , R 1 is a resistor; C i , is the CPE coefficient; the open-circuit voltage V ocp is a strictly monotonically increasing function of x i ; V 1 (t), V 2 (t) are the network voltage and the voltage of the Warburg-like element respectively; η is the coulombic efficiency.

[0020] For the estimator described above, its non-linear fractional-order system equation is as follows:

[0021] D α x(t) = Ax + Bu + H(x, u) + Δ(t, x, u)

[0022] y = h(x, u) + v(t, x, u)

[0023] where x is the unmeasurable state vector, and u and y are the measurable system input and output vectors respectively. Δ and v represent the uncertainties in the dynamic equation and the output equation respectively.

[0024] The parameter optimization constraint conditions in the estimation algorithm described above are as follows:

[0025]

[0026]

[0027] where L, E are positive definite, λ min and λ max are the minimum and maximum eigenvalues of L, E, M is a positive definite matrix, indicating that the Lyapunov stability described above is satisfied as follows:

[0028]

[0029]

[0030] where the state vectors z and Υ are defined as:

[0031]

[0032] ||Υ|| ≤ r||x||

[0033] where r is a positive constant.

[0034] The Mittag–Leffler stability described above is as follows:

[0035]

[0036]

[0037] where \(V(t, x)\) is the Lyapunov function; \(a\) 1 、\(a\) 2 、\(a\) 3 、\(b\) 1 、\(b\) 2 、\(b\) 3 are positive constants.

[0038] Compared with the prior art, the beneficial effects of the present invention are as follows: 1) Perform equally accurate estimation of the state of charge (SoC) of a lithium-ion battery on the premise of ensuring the stability and robustness of the lithium-ion battery. 2) The estimator is framed by a two-term output error injection structure, including a Luenberger part and a sliding mode part. Under certain assumptions, it has been proven using Lyapunov's direct method that error convergence can be achieved in the presence of model uncertainties and noise. This general algorithm is used to estimate the state of a lithium-ion battery, and the basic assumptions of the estimator are reasonable for the fractional-order battery model. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] Figure 1 is a schematic flow chart of a method for estimating the state of charge of a lithium-ion battery based on a nonlinear FOM. DETAILED DESCRIPTION OF THE INVENTION

[0040] The present invention will be further described below in conjunction with the drawings and embodiments, but the protection scope of the present invention should not be limited thereby.

[0041] The present invention relates to a method for estimating the state of charge of a lithium-ion battery based on a nonlinear FOM, including: an estimation method developed from a general nonlinear fractional-order model as shown in Figure 1 , and an estimator designed to ensure Mittag–Leffler stability. First, a nonlinear fractional-order model of a lithium-ion battery is established, and the stability of the fractional-order system is theoretically guaranteed by using the Mittag–Leffler method; then, for parameter errors, disturbances, and sensor noise, a combination of a Luenberger observer and a sliding mode observer SMO is used for observation and fed back to the controller of the estimator; finally, a new estimation algorithm based on the nonlinear fractional-order model is used to estimate the state of charge of the lithium-ion battery. In addition, the stability of the estimator and the robustness of the estimator under certain assumptions are theoretically guaranteed by using Lyapunov's direct method.

[0042] The fractional order is defined by the Grünwald–Letnikov (GL) definition as:

[0043]

[0044] The described non - linear fractional - order model can be represented by

[0045]

[0046] where \(i\) is the imaginary number and \(\omega\) is the radian frequency. \(\alpha = 0\) indicates that the CPE is a resistor, \(\alpha = 1\) and \(\alpha=-1\) indicate that the CPE is a pure capacitor and an inductor respectively.

[0047] The non - linear fractional - order model of the lithium - ion battery is as follows:

[0048]

[0049]

[0050]

[0051] y(t)=V ocp (x 1 (t)) + R 0 u(t)+x 2 (t)+x 3 (t)

[0052] In the dynamic - system representation, \(u(t)\) is the input current, i.e., negative for discharging and positive for charging; the state vector is defined as: \(x = [SoC(t),V 1 (t),V 2 (t)] T ; \(y(t)\) represents the output voltage; \(C n \) is the nominal capacity of the battery, with the unit of ampere - hour (Ah); \(R 0 \), \(R 1 \) are resistors; \(C i \), \(\) are the CPE coefficients; the open - circuit voltage \(V ocp \) is a strictly monotonically increasing function of \(x i \); \(V 1 (t)\), \(V 2 (t)\) are the network voltage and the Warburg - like element voltage respectively; \(\eta\) is the coulombic efficiency.

[0053] For the described estimator, its non - linear fractional - order system equations are as follows:

[0054] D α x(t)=Ax + Bu+H(x,u)+\Delta(t,x,u)

[0055] y = h(x,u)+v(t,x,u)

[0056] where x is an unmeasurable state vector, and u and y are measurable system input and output vectors respectively. Δ and v represent the uncertainties in the dynamic equation and output equation respectively.

[0057] The parameter optimization constraints in the described estimation algorithm are as follows:

[0058]

[0059]

[0060] where L and E are positive definite, λ min and λ max are the minimum and maximum eigenvalues of L and E respectively, and M is a positive definite matrix. It is shown that the Lyapunov stability is satisfied as follows:

[0061]

[0062]

[0063] where the state vectors z and Υ are defined as:

[0064]

[0065] ||Υ|| ≤ r||x||

[0066] where r is a positive constant.

[0067] The Mittag–Leffler stability is as follows:

[0068]

[0069]

[0070] where V(t, x) is the Lyapunov function; a 1 、a 2 、a 3 、b 1 、b 2 、b 3 are positive constants.

[0071] For the state of charge (SoC) estimation algorithm of the lithium-ion battery, the main task is to derive the estimation gain. In the linear matrix inequality M is related to the feasibility of the problem and also affects the convergence rate obtained from . A four-step procedure is used to derive the estimation gain:

[0072] S1: Initialize a positive value λ, M = λ / n + λ, and increase it until It is feasible. Then, calculate the value of the target parameter L of the lithium-ion state of charge.

[0073] S2: Further increase and repeat the first step to obtain a feasible L library.

[0074] S3: According to the obtained L and calculate the gain L s .

[0075] S4: Adjust the value of the gain L within the candidate set L until the desired estimation performance is achieved in terms of the convergence rate and steady-state error. s The values are adjusted until the desired estimation performance is achieved in terms of the convergence rate and steady-state error.

[0076] Compare different estimation algorithms, including the Luenberger estimator, SMO estimator, EKF estimator, and the estimator under the biased parameter FUDS test proposed by us. The designed estimator can capture the SoC information with an error bounded by 0.03. Although the initial error is large and the noise persists, under the conditions of parameter bias and aging, it is proven that the estimation algorithm is superior to other estimators in terms of error recovery ability. Specifically, the combination of the Luenberger observer and the SMO observer improves the maximum error recovery ability, which is better than EKF and SMO.

[0077] In summary, the present invention establishes a non-linear fractional-order model of a lithium-ion battery, theoretically guarantees the stability of the fractional-order system by using the Mittag – Leffler method, and adopts a new estimation algorithm based on the non-linear fractional-order model to estimate the state of charge of the lithium-ion battery. In addition, by using the Lyapunov direct method, the stability of the estimator and the robustness of the estimator under certain assumptions are theoretically guaranteed.

[0078] The above specific implementation can be locally adjusted by those skilled in the art in different ways without departing from the principles and purposes of the present invention. The protection scope of the present invention is subject to the claims and is not limited by the above specific implementation. All implementation solutions within its scope are subject to the present invention.

Claims

1. A method for estimating the state of charge of a lithium-ion battery based on a non-linear FOM, characterized in that, it includes the following steps: S1. Construct a non-linear fractional-order model of the lithium-ion battery, and use the Mittag–Leffler method to ensure the stability of the fractional-order system; S2. For parameter errors, disturbances, and sensor noise, a combination of a Luenberger observer and a sliding mode observer SMO is used for observation and fed back to the controller of the estimator; S3. Use the state of charge estimation algorithm of the lithium-ion to derive the estimation gain, and then calculate the state of charge parameter. Rapidly cycle this process until the state of charge parameter converges, that is, obtain the state of charge of the lithium-ion battery based on the non-linear fractional-order model.

2. The method for estimating the state of charge of a lithium-ion battery based on a non-linear FOM according to claim 1, characterized in that, in the step S1, the non-linear fractional-order model of the lithium-ion battery is based on the battery equivalent circuit model (ECM), introducing a fractional capacitor and introducing a constant phase element from the frequency domain perspective.

3. The method for estimating the state of charge of a lithium-ion battery based on a non-linear FOM according to claim 1, characterized in that, the step S3 specifically includes: S3.1 Derive the estimation gain: Use a four-step procedure to derive the estimation gain: S3.2 Initialize any positive value λ, and let M = λ / n + λ, which is a positive definite matrix, and increase λ until is feasible, where c 2 is the Lipschitz constant, determined by the fractional-order model; L l is the initial value of the lithium-ion state of charge parameter; is the difference between the observed current and the actual current; Then, calculate the value of the lithium-ion state of charge target parameter L obtained; Loop through this step to obtain the feasible library L; S3.3 Calculate the estimated gain L according to the feasible library L s , and the formula is as follows: where |Δ| is the absolute value of the observer estimation error; N is a positive constant to ensure the stability margin, determined by the model; S3.4 Adjust the estimated gain L within the feasible library L according to the inequality in 3.3 s until the two reach an ideal range in terms of the convergence rate and the steady-state error.

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