The SoH characterizes the ability of the battery to store electrical energy relative to the ability of a new battery, which is an indicator that quantitatively describes the state of performance of tested battery. With increase in the number of charging/discharging operations, the SoH shows a downward trend. SoH estimation is not only important as a basis for calculating key parameters such as the SoC, but also a criterion for replacement of the lithium battery. The SoH may provide information on battery capacity, health, performance status, and longevity, which may be defined as follows:

S
⁢
o
⁢
H

=

C

max
.
p

C
N

(
3
)

Dynamic behavior of the lithium-ion battery depends on factors such as its current, SoC, capacity and voltage. In embodiments of the present disclosure, a knowledge-reinforced machine learning model integrated with a DEKF is provided to learn battery dynamics and predict capacity transitions in the battery. The RUL may be further predicted from estimated capacity transitions.

In various embodiments of the present disclosure, battery system modeling is described herein. The system of battery dynamics may be described as a discrete state-time and state-space model, which may be expressed as follows:

Transition
⁢

x
k

=

F
⁡
(

x

k
-
1

,

u

k
-
1

,

θ

k
-
1

)

+

w
k

,

θ
k

=

θ

k
-
1

+

r
k

⁢

Measurement
⁢

y
k

=

G
⁡
(

x

k
,

u
k

,

θ
k

)

+

v
k

(
4
)

- where x_k-1denotes a vector of system states, θ_k-1denotes a vector of system model parameters, u_k-1denotes a vector of observed inputs, and y_k-1denotes a vector of system observations or system measurements. The parameters w_kand r_kdenote vectors of process noise for the state and model parameters, respectively while v_kdenotes a vector of measurement noise. The functions F(·) and G(·) denote state transition and measurement functions, respectively. With above-mentioned system defined, estimating the system states x and model parameters θ from noisy observations may be proceeded. Mapping above defined model with the dynamic parameters of the battery system, x is referred as the SoC of the battery. The SoC of the battery, which is a volatile parameter, may vary between 100 and 0 percent within minutes. The system model parameter θ represents the cell capacity. The variation in the SoC over time may be modeled using the state transition function F(·), while the function G(·) may be configured to relate the voltage y with the SoC, capacity and measured input u.

In various embodiments of the present disclosure, prognostics of battery capacity and its KRML framework are described hereinafter. The integration of the DEKF with the ANN is graphically shown in a schematic diagram in FIG. 4. It may be observed that the system may include DEKFs running in parallel and an ANN model, where the top and bottom EKFs may adapt the system states x and system parameters θ while the ANN model may learn the measurement function GNN(·) of the dynamic system. It should be noted that the model parameters of ANN may be adaptively updated based on the DEKFs by continuously collecting data in system operation. By integrating the artificial neural network with the DEKF algorithm for lithium-ion battery online SoC estimation, entire KRML framework may involve training of machine learning models in the offline phase and synchronizing these models with the DEKF techniques in the online phase, which may contain two modules including (1) a diagnosis module (i.e., phase) and (2) a prognosis module (i.e., phase), as illustrated in FIG. 4. The diagnosis phase may provide estimation of the SoC by employing the ANN and DEKFs. The prognosis phase may learn the stochastic degradation model of battery capacity by employing the knowledge-reinforced Gaussian process regression.

In various embodiments of the present disclosure, Kalman filtering-based modeling of battery capacity is described herein. According to embodiments of the present disclosure, the ampere-hour counting approach may be employed to estimate SoC by dynamic integration of the current of the lithium-ion battery. The changes in internal parameters of the battery may not be considered, and SoC may be directly expressed as the integral of the current with respect to time. That is, SoC may be expressed as follows:

S
⁢
o
⁢

C
⁡
(
t
)

=

S
⁢
o
⁢

C
⁡
(

t
0

)

-

∫
0
t

η
C

⁢

I
⁡
(
t
)

⁢
dt

C

max
.
p

(
5
)

- where t denotes time, C_max·pdenotes maximum practical capacity, η_Cdenotes Coulombic efficiency (for discharge η_C=1) and I(t) denotes discharge current. The initial SoC(t₀) may be configured to 1 for the discharge cycle in the battery. In discretized form, equation (5) may be rewritten as follows:

S
⁢
o
⁢

C
k

=

S
⁢
o
⁢

C

k
-
1

-

η
k

*
Δ
⁢
t

C

k
-
1

⁢

i

k
-
1

(
6
)

Equation (6) represents iteration function of SoC, which may form the basis for describing battery dynamics via integration with the state-space model defined by the DEKF. Kalman filters may be sets of equations that provide efficient solutions for estimation of states using the least-squares method. The estimation process may be feasible even when the explicit nature of associated model is unknown. To estimate both SoC and SoH, the dual extended Kalman filter may be employed. In various embodiments of the present disclosure, the state variables and model parameters may be set as x=SoC and θ=C, respectively. The pseudo algorithm of the DEKF is briefly stated herein. In one embodiment, the pseudo algorithm of the DEKF may include following exemplary equations A1-A8.

Initialization may be expressed as follows:

θ
0

^

=

E
[

θ
0

]

,

p
=

E
[

(

θ
0

-

θ
0

)

⁢

(

θ
0

-

θ
0

)

T

]

⁢

x
0

^

=

E
[

x
0

]

,

p
=

E
[

(

x
0

-

x
0

)

⁢

(

x
0

-

x
0

)

T

]

(

A
.1

)

For k∈(1, . . . , ∞), the state update for the weight filter may be expressed as follows:

θ
k
-

^

=

θ
k

⁢

p

θ
k

-

=

p

θ

k
-
1

+

p

r

k
-
1

(

A
.2

)

The state update for the state filter may be expressed as follows:

x
k
-

^

=

F
⁡
(

x

k
-
1

,

u

k
-
1

,

θ
k
-

^

)

⁢

p
=

A

k
-
1

⁢

p

x

k
-
1

⁢

A

k
-
1

T

+

p

w

k
-
1

(

A
.3

)

The measurement update for the state filter may be expressed as follows:

K
k
x

=

p

x
k

-

(

C
k
x

)

T

[

c
k
x

⁢

p

x
k

-

(

C
k
x

)

T

+

p

v
k

]

-
1

⁢

x
k

^

=

x
k
-

+

k
k
x

[

y
k

-

G
⁡
(

x
k
-

,

u
k

,

θ
k
-

)

]

⁢

p

x
k

=

(

I
-

K
k
x

⁢

C
k
x

)

⁢

p

x
k

-

(

A
.4

)

The measurement update for the weight filter may be expressed as follows:

K
k
θ

=

p

θ
k

-

(

C
k
θ

)

T

[

C
k
θ

⁢

p

θ
k

-

(

C
k
θ

)

T

+

p

v
k

]

-
1

⁢

θ
k

^

=

θ
k
-

+

K
k
θ

[

y
k

-

G
⁡
(

x
ˆ

k
-

,

u
k

,

θ
k
-

)

]

⁢

p
=

(

I
-

K
k
θ

⁢

C
k
θ

)

⁢

p

θ
k

-

(

A
.5

)

- where:

A

k
-
1

=

∂

F
⁡
(

x
,

u

k
-
1

,

θ
ˆ

k
-

)

∂
x

|
x

=

x
ˆ

k
-
1

(

A
.6

)

C
k
x

=

∂

G
⁡
(

x
,

u

k
-
1

,

θ
ˆ

k
-

)

∂
x

|
x

=

x
ˆ

k

(

A
.7

)

C
k
θ

=

∂

G
⁡
(

x
,

u

k
-
1

,

θ
ˆ

k
-

)

∂
x

|
θ

=

θ
ˆ

k
-
1

(

A
.8

)

By integrating the state-space model in equation (4) with equation (6), the stochastic capacity degradation model may be defined as follows:

Transition
⁢

{

x
k

=

F
⁢

(

x

k
-
1

,

u

k
-
1

,

θ

k
-
1

)

+

w

k
-
1

=

x

k
-
1

-

η
C

*
Δ
⁢
t

C

k
-
1

⁢

i

k
-
1

C
k

=

C

k
-
1

+

r

k
-
1

.

(
7
)

Measurement
⁢

y
k

=

G
⁡
(

x
k

,

u
k

,

θ
k

)

+

v
k

≈

G

N
⁢
N

(

x
k

,

i
k

,

C
k

)

+

v
k

- where G_NN(·) denotes measurement function derived from the ANN model. By integrating the ANN-based DEKF with knowledge-reinforced Gaussian process modeling described above, the stochastic degradation behavior of lithium-ion batteries may be learned effectively and configured to predict the remaining useful life of battery capacity.

In various embodiments of the present disclosure, artificial neural network for battery system modeling is described herein. ANNs have been widely employed in various fields for applications such as optimization, function approximation, image processing and the like. As shown in FIG. 5, the ANN model, including an input layer, three hidden layers, and an output layer, may be configured to simulate complex dynamics of lithium-ion batteries. The numbers of neurons in the input and output layers may be determined by the physical system, where the number of neurons in the hidden layer may be predefined to reduce computational complexity and enhance network accuracy. A logistic function may be employed as the activation function between the layers of the ANN, which may be expressed as follows:

φ
⁡
(
t
)

=

1

1
+

e

-
β

⁢
t

(
8
)

- where β denotes a slope parameter. The network output may include the battery voltage, and the network inputs may include SoC (k), current I(k) and capacity (k) (maximum practical capacity). Therefore, the input vector may be defined as q_k=[{circumflex over (x)}_k⁻, i_k, Ĉ_k⁻], where k denotes a time step. The values of the hidden layer may be calculated based on the activation function as follows:

h
k
j

=

φ
⁢

(

∑

i
=
1

l

⁢

q
k
i

⁢

a
k

i
⁢
j

+

a
k

0
,
j

)

(
9
)

- where i denotes an index of an i^thinput node, j denotes an index of a j^thhidden node, a_k^ijdenotes a weight connecting the i^thinput node and the j^thhidden node at time step k, and a_k^0,jdenotes a bias of the j^thhidden node at time step k. Similarly, the output of the ANN model may be calculated as follows:

z
k

=

φ
⁢

(

∑

j
=
1

J

⁢

h
k
j

⁢

b
k
j

+

b
k
0

)

(
10
)

- where b_k^jdenotes a weight connecting the j^thhidden node with the output node at time step k, and b_k⁰denotes a bias of the output node at time step k. According to equations (9) and (10), the voltage may be obtained as follows:

V
k

=

φ
⁢

(

∑

j
=
1

J

⁢

(

∑

i
=
1

l

⁢

q
k
i

⁢

a
k

i
⁢
j

+

a
k

0
,
j

)

⁢

b
k
j

+

b
k
0

)

(
11
)

The ANN may be further integrated with DEKF for enhanced SoC prediction. The ANN model is nonlinear, so that for iterative purposes, the linearization of the ANN model may be needed.

In various embodiments of the present disclosure, knowledge-reinforced machine learning for battery prognostics is described herein. With the battery capacities estimated by the DEKF in embodiments of the present disclosure, the knowledge-reinforced Gaussian process regression (GP) method may be configured to train the stochastic capacity degradation model. GP is a powerful tool for Bayesian nonparametric function estimation. One of main advantages of GP is that the prediction interpolates the observation, and the prediction is probabilistic, so that the predictions by GP method (i.e., GP model) may be computed in empirical confidence intervals. To enhance accuracy of RUL prediction, in embodiments of the present disclosure, prior knowledge in battery capacity degradation may be integrated with the GP method to predict the RUL while reducing prediction uncertainty. The prior knowledge may include: 1) the battery capacity should be a positive value and lower than the battery maximum capacity, and 2) the battery capacity should be a monotonic decreasing function over time. Leveraging the prior knowledges as constraints in machine learning processes may enhance accuracy of the capacity prediction and reduce uncertainty of the RUL prediction.

Various models have been developed for reinforced Gaussian process regression. The constraints may be linear operators, which may be written as single linear operators. Let f˜GP(m(x), K(x, x′)) is a GP with mean m(x) and covariance function K(x, x′), which may be expressed as follows:

m
⁡
(
x
)

=

E
[

f
⁡
(
x
)

]

(
12
)

k
⁡
(

x
,

x
′

)

=

E
[

(

f
⁡
(
x
)

-

m
⁡
(
x
)

)

⁢

(

f
⁡
(

x
′

)

-

m
⁡
(

x
′

)

)

]

(
13
)

With the mean and covariance functions, the GP may be denoted as follows:

f
⁡
(
x
)

∼

GP
⁡
(

m
⁡
(
x
)

,

k
⁡
(

x
,

x
′

)

)

(
14
)

With a set of training data, the degradation model using Gaussian process regression may be trained to predict response as a Gaussian distribution at any new sampling location x*, which is expressed as follows:

f
(
x

*)

=

N
(

μ
(
x

*)

,

σ
(
x

*)

)

(
15
)

With a set of observations y={y_i}_i=1^Nand the test locations y={y_i*}_i=1^M, the joint distribution may be expressed as follows:

[

y

y
*

]

∼

N
⁡
(

0
,

[

K
X

K
(

X
,
X

*)

K
⁡
(

X
*

,
X

)

K
(

X
*

,
X

*)

]

)

(
16
)

- where K (X′, X″) denotes a covariance matrix, X and X* denote the training and testing locations, respectively, and K_X=K(X, X)+σ_n²I. The predictive distribution over the test locations may be given by:

y
*

❘
X

,
y
,

X
*
∼

N
(

μ
*

,
∑

*)

(
17
)

μ
*=

K
⁡
(

X
*

,
X

)

⁢

K
X

-
1

⁢
y

(
18
)

∑

*=

K
(

X
*

,
X

*)

-

K
⁡
(

X
*

,
X

)

⁢

K
X

-
1

⁢

K
(

X
,
X

*)

(
19
)

Furthermore, for each test location x*∈X*, the response may be predicted by the GP as a normal distribution with mean and variance, which may be expressed as follows:

μ
(
x

*)

=

K
⁡
(

x
*

,
X

)

⁢

K
X

-
1

⁢
y

(
20
)

σ
(
x

*)

=

K
(

x
*

,
x

*)

-

K
⁡
(

x
*

,
X

)

⁢

K
X

-
1

⁢

K
(

X
,
x

*)

(
21
)

In various embodiments of the present disclosure, the battery working cycles and working condition (e.g., temperature) X and capacities Y represent N observations as training pairs. To make sure that the predictions of battery capacities by the GP are within a certain range, two functions may be formulated, such that a(x)≤Lf(x)≤b(x), where L denotes a linear operator on the implementation of Gaussian process. To enforce the upper and lower bounds for the GP modeling, it is assumed that the constraint may only hold at a finite set of virtual observation locations X^v. In addition, it is assumed that virtual observations of the transformed process, Lf(x), may contain additive white noise with variance σ_v². Let C*(X^v) represents the matrix with rows (C*(X^v))_i=Lf(x^v_i)+ε^v_ifor i.i.d. (i.e., independent identical distribution) where ε^v_i˜N (0, σ²_vI), and let C (X^v) denotes the event C(X^v): =∩^s_i=1{a(x^v_i)≤(C(X^v))_i≤b(x^v_i)}, where S denotes a number of virtual observation locations. C(X^v) represents the function where the constraint a(X^v)≤Lf(X^v)+∈^v≤b(X^v) is satisfied for the virtual locations X^v. With the upper and lower bounds as constraints, the GP may be reformulated as follows:

f
❘
X

,
Y
,

X
v

,

C
⁡
(

X
v

)

:=

f
❘

f
⁡
(
X
)

+
ε

,

a
⁡
(

X
v

)

≤

Lf
⁡
(

X
v

)

+

ε
v

≤

b
⁡
(

X
v

)

(
22
)

- where f denotes a Gaussian process, X and Y denote the training data including working cycles and battery capacities estimated by the DEKF, and X^vdenotes virtual locations where Lf(X^v)+ε^vis bounded. a(X) and b(X) may be determined by the lower and upper bounds of battery capacities. The additive noise ε and ε^vmay be multivariate Gaussian with diagonal covariance matrices of elements σ²and σv²respectively.

To predict the battery capacity for future working cycles, the posterior predictive distribution f*|X*,X, Y, X^v, C(X^v) may be obtained for new inputs X* conditioned on the observed data Y=f(X)+ε and the constraint a(X^v)≤Lf(X^v)+ε^v≤b(X^v). The posterior predictive distribution may be then expressed as follows:

{

p
⁡
(

f
*

,

C
❘
Y

)

=

p
⁡
(

f
*

❘
C

,
Y

)

⁢

p
⁡
(

C
❘
Y

)

p
⁡
(

f
*

❘
C

,
Y

)

=

∫

a
⁡
(

X
v

)

b
⁡
(

X
v

)

p
⁡
(

f
*

❘

C
*

,
Y

)

⁢

p
⁡
(

C
*

❘
Y

)

⁢
dC
*

(
23
)

- where C* denotes a latent variable, p(C|Y) denotes the probability that the constraints hold for the virtual locations X^v.

For battery degradation modeling, it is assumed that the battery capacity decreases monotonically over the working cycles. Similarly, a set of virtual points may be utilized to enforce the monotonicity constraint while developing data-driven battery degradation models. In one embodiment, by treating partial derivatives as GPs with covariance kernel functions, the monotonicity constraint may be realized by enforcing at a discrete set of S virtual points, which may be expressed as follows:

∂
f

∂
x

⁢

(

x
i

)

≤
0

,

i
=
1

,
…

,
S

(
24
)

Then, with estimated battery capacity data for training purpose, the predictor of the battery capacity for following working cycles may be obtained by maximizing marginal likelihood function, which may be expressed as follows:

LF
⁡
(
θ
)

=

p
⁡
(

Y
,

C
❘
θ

)

=

p
⁡
(

Y
❘
θ

)

⁢

p
⁡
(

C
❘
Y

,
θ

)

(
25
)

- where θ denotes a vector of trainable parameters of knowledge-reinforced GP. It should be noted that disclosed method may adaptively learn complex battery degradation by continuously observing new online monitoring data. In the online phase, the diagnostics may be employed to estimate the battery capacity from online operating signals, and estimated online capacity data may be added as additional training data to update the knowledge-reinforced GP. As more online diagnostic estimations are added to update the knowledge-reinforced GP, the knowledge-reinforced GP may map capacity transition more comprehensively, and prediction capability of the model may become increasingly accurate.

The linearization procedure of the ANN model is described herein. According to various embodiments of the present disclosure, recursive routine derivative computation may include following exemplary equations B.1-B.12.

From equation (18), the following equation may be obtained:

A
k

=

∂

F
⁡
(

x
,
u
,
θ

)

∂
x

❘

=
1

(

B
.1

)

By setting q¹=SoC, the following equations may be obtained:

C
k
x

=

∂

G
⁡
(

x
,

u
k

,

θ
^

k
-

)

∂

q
k
1

=

∂

V
k

∂

q
k
1

=

z
k

(

1
-

z
k

)

⁢

∑

j
=
1

J

⁢

b
k
j

⁢

h
k
j

(

1
-

h
k
j

)

⁢

a
k

1
⁢
j

(

B
.2

)

C
k
θ

=

dG
⁡
(

x
^

k
-

,

u
k

,
θ

)

d
⁢
θ

=

∂

G
⁡
(

x
k
-

,

u
k

,
θ

)

∂
θ

+

∂

G
⁡
(

x
k
-

,

u
k

,
θ

)

∂

x
^

k
-

⁢

d
⁢

x
^

k
-

d
⁢
θ

(

B
.3

)

d
⁢

x
^

k
-

d
⁢
θ

=

∂

F
⁡
(

x
^

k
-
1

,

u

k
-
1

,
θ

)

∂
θ

+

∂

F
⁡
(

x

k
-
1

,

u

k
-
1

,
θ

)

∂

x
^

k
-
1

⁢

dx

k
-
1

d
⁢
θ

(

B
.4

)

d
⁢

x
^

k
-
1

d
⁢
θ

=

d
⁢

x
^

k
-
1

-

d
⁢
θ

-

K

k
-
1

x

⁢

dG
⁡
(

x
^

k
-
1

-

,

u

k
-
1

,
θ

)

d
⁢
θ

+
 

∂

K

k
-
1

x

∂
θ

[

y
k

-

G
⁡
(

x
^

k
-
1

-

,

u

k
-
1

,
θ

)

]

(

B
.5

)

By setting q³=C, the following equations may be obtained:

∂

F
⁡
(

x
^

k
-
1

-

,

u

k
-
1

,
θ

)

∂
θ

=

[

-

ηΔ
⁢
t

C
2

⁢

i

k
-
1

,
0
,
...
,
0

]

T

(

B
.6

)

∂

G
⁡
(

x
^

k

,

u
k

,
θ

)

∂
θ

=

[

∂

V
k

∂

q
k
3

,

∂

V
k

∂

a
k
ij

,

∂

V
k

∂

a
k
oj

,

∂

V
k

∂

b
k
j

,

∂

V
k

∂

b
k
0

]

,

(

i
=
1

,
…

,

I
;

j
=
1

,
…

,
J

)

(

B
.7

)

∂

V
k

∂

q
k
3

=

z
k

(

1
-

z
k

)

⁢

∑

j
=
1

J

⁢

b
j

⁢

h
k
j

(

1
-

h
k
j

)

⁢

a
k

3
⁢
j

(

B
.8

)

∂

V
k

∂

a
k
ij

=

∂

z
k

∂

h
k
j

⁢

∂

h
k
j

∂

a
k
ij

=

z
k

(

1
-

z
k

)

⁢

b
k
j

⁢

∂

h
k
j

∂

a
k
ij

=

z
k

(

1
-

z
k

)

⁢

b
k
j

⁢

h
k
j

(

1
-

h
k
j

)

⁢

a
k
i

(

B
.9

)

∂

V
k

∂

a
k
oj

=

∂

z
k

∂

a
k

0
⁢
j

=

∂

z
k

∂

h
k
j

⁢

∂

h
k
j

a
k

0
⁢
j

=
 

z
k

(

1
-

z
k

)

⁢

b
k
j

⁢

∂

h
k
j

a
k

0
⁢
j

=

z
k

(

1
-

z
k

)

⁢

b
k
j

⁢

h
k
j

(

1
-

h
k
j

)

(

B
.10

)

∂

V
k

∂

b
k
j

=

∂

z
k

∂

b
k
j

=

z
k

(

1
-

z
k

)

⁢

h
k
j

(

B
.11

)

∂

V
k

∂

b
k
0

=

∂

z
k

∂

b
k
0

=

z
k

(

1
-

z
k

)

(

B
.12

)

Equations B.4-B.12 may be substituted into equation B.3 to compute C_k^θ.

In various embodiments of the present disclosure, case study results are used to demonstrate the method for diagnostics and prognostics of lithium-ion batteries. The battery experimental dataset is provided by the Prognostics Center of Excellence in NASA. A set of lithium-ion batteries (#5, 6 and 7) were run through 3 different operational profiles (charge, discharge and impedance) at room temperature. Charging was carried out in a constant current (CC) mode at 1.5 A until the battery voltage reached 4.2 V and then continued in a constant voltage (CV) mode until the charge current dropped to 20 mA. Discharge was carried out at a constant current (CC) level of 2 A until the battery voltage fell to 2.7 V, 2.5 V, and 2.2 V for batteries 5, 6 and 7, respectively. Impedance measurement was carried out through an electrochemical impedance spectroscopy (EIS) frequency sweep from 0.1 Hz to 5 k Hz. Repeated charge and discharge cycles result in accelerated aging of the batteries while impedance measurements provide insight into the internal battery parameters that change as aging progresses. The experiments were stopped when the batteries reached end-of-life (EOL) criteria, which was a 30% fade in rated capacity (from 2 Ahr to 1.4 Ahr). As capacity was measured only in discharge cycles, the discharge dataset may be utilized for prognostics of lithium-ion batteries capacities in various embodiments of the present disclosure. To validate above-mentioned method, the dataset of battery 05, including 138 discharge cycles and 44501 sample points, may be utilized for training, and datasets of batteries 06 and 07 may be used for validation.

According to various embodiments of the present disclosure, the inputs of the ANN may be battery SoC, capacity, and current; and the output may be the battery terminal voltage. With the dataset collected from the baseline lithium-ion battery, the ANN may be trained, and final weights matrix obtained from the ANN model may describe complex interactions within batteries. It should be noted that the weights and bias of the ANN model may be updated iteratively using the DEKF to handle batch to batch variations between lithium-ion batteries.

With the battery capacities estimated by the DEKF in real time, the knowledge-reinforced machine learning method may construct the GP method to predict the stochastic degradation of battery capacities under different working conditions such as charging/discharging rate and temperature. The connection of the ANN, DEKF and knowledge-reinforced GP method is illustrated in FIG. 6. The stochastic degradation model may be adaptively updated by adding more data to improve accuracy for the capacity prognostic of lithium-ion batteries and reduce prediction uncertainty. To make comprehensive comparison, existing GP and knowledge-reinforced GP methods may be implemented, based on same set of data, to predict the capacity of lithium-ion batteries. With initial ANN model trained based on the dataset extracted from the battery 05, the knowledge-reinforced machine learning framework may be utilized to perform online health monitoring and prognostics of battery 06. With the DEKF algorithm, the battery capacity and state of the charge may be captured in real time, which allows development of the stochastic degradation model for the projection of battery capacity over cycles. The existing GP regression may be first utilized to build the mapping relationship between the number of working cycles and the battery capacities. Then, the knowledge-reinforced machine learning method may be employed to enforce the prior knowledge in development of machine learning models which may reduce prediction uncertainty and increase accuracy for the prognosis of battery capacities.

In various embodiments of the present disclosure, the following constraints: (1) the battery capacity should be within a boundary defined as [0, 2.5], and (2) the battery capacity should be decreasing over time, may be used as a piece of prior knowledge to facilitate the learning of capacity degradation behaviors. These constraints may be expressed as follows:

0
<

f
⁡
(
x
)

<
2.5

,

df
⁡
(
x
)

dx

<
0

(
26
)

- where f(x) denotes the capacity function over working cycles, x denotes the number of cycles. To illustrate above-mentioned method in a clear fashion, referring to FIG. 7, the prediction process may be decomposed into three steps, namely, data processing, model training, and capacity predicting.

A comparison between above-mentioned knowledge-reinforced GP method and existing machine learning methods is presented to evaluate performance of the knowledge-reinforced GP method for RUL prediction of lithium-ion batteries. An existing knowledge transfer based neural network (KTNN) for RUL prediction had been developed. FIGS. 8-12 illustrate the comparison results of the battery 06 RUL predictions at different working cycles between the knowledge-reinforced GP method and the KTNN method, respectively. For instance, in FIG. 8, the training data points of the GP method may include X=[x₁, . . . , x₅], and Y=[y₁, . . . , y₅]; and the testing data may contain the rest of the cycles and actual discharging capacities. It can be observed that accuracy of capacity prediction for future cycles based on the existing GP method, the knowledge-reinforced GP method and the KTNN method may be improved by adding more training data. Nevertheless, the knowledge-reinforced GP method may achieve a consistently more accurate projection of battery capacity with much less uncertainties, especially for the cases with less training data in the beginning of battery life cycle. For the existing GP method, it may be seen that prediction of battery capacity for future working cycles may have chances to be increasing over cycles as demonstrated by the upper bound of GP predictions. With the prior knowledge, the knowledge-reinforced GP method may be able to guarantee that the battery capacity function is a monotonic decreasing function over time. Furthermore, the existing GP method may predict the battery capacities as negative values while the knowledge-reinforced GP method may make sure that all predictions of battery capacities are within a particular range from 0 to 2.5. To evaluate the prediction accuracy of the model quantitatively, the root mean square error (RMSE) of the lithium-ion battery RUL prediction may be used for comparison purpose. With actual capacity f(i) at ith cycle, the RMSE may be defined as follows:

RMSE
=

(

1
n

⁢

∑

i
=
1

n

⁢

(

f
⁡
(
i
)

-

f
*

(
i
)

)

2

)

(
27
)

- where f*(i) denotes capacity prediction.

The RMSEs for different methods are provided in Table 1. As shown in Table 1, the knowledge-reinforced GP method may achieve a lower RMSE than the existing GP method under different training conditions. In the scenario with only 5 training data points (i.e., 5 cycles), the RMSE of the existing GP method may be about 0.4004, while the RMSE of the knowledge-reinforced GP method may be about 0.0666. The comparison results may demonstrate that, by incorporating knowledge-based constraints, the knowledge-reinforced GP method may outperform the existing GP method.

TABLE 1

Comparison between knowledge-reinforced

method and existing GP method

Training conditions

5
20
50
100
150

Prediction method
cycles
cycles
cycles
cycles
cycles

Existing GP method
0.4004
0.3823
0.3540
0.1199
0.0334

Knowledge-reinforced
0.0666
0.0627
0.0569
0.0552
0.0201

GP method

The knowledge-reinforced GP method had been also utilized to predict capacity fading for the battery 07. FIGS. 13-14 demonstrate the stochastic degradation models using the existing GP method, the knowledge-reinforced GP method and the KTNN method at working cycles 60 and 100 respectively. Similarly, the knowledge-reinforced machine learning method may provide more accurate prediction of battery capacities with less amount of uncertainties, especially for the cases when the battery capacities data is very limited at the beginning of battery life cycle. It is clearly demonstrated that adding knowledge-based constraints in the machine learning process may greatly reduce uncertainty for battery capacity prediction for future working cycles. By adaptively estimating the battery capacities using the DEKF algorithm, the stochastic degradation model of battery performance may be updated to reduce the uncertainty associated with capacity prognosis of lithium-ion batteries.

To further verify the effectiveness of the knowledge-reinforced machine learning method, the knowledge-reinforced GP method may be compared with existing battery RUL prognostics methods such as the support vector machine (SVM) method, the linear regression method, and the KTNN method. The battery