EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 14, No. 1, 2021, 234-247
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Markov Modeling of Battery Cell Behavior Taking into
account Pulsed Discharge Recovery

Victorien Konané1

1 Département de Mathématique, UFR-SEA, Université Joseph KI-ZERBO, Ouagadougou,
Burkina Faso

Abstract. In this work, we modeled the behavior of a battery. After having formulated a Marko-
vian model, we evaluated the delivered capacity as well as the gained capacity. We, likewise,
evaluated the mean number of pulses and studied the asymptotic behavior and the variance of this
mean number. As a last resort, we introduced an extension of the Markov model.

2020 Mathematics Subject Classifications: 60J20,41A60,37A50

Key Words and Phrases: Battery, Markov model, Gained capacity, Recovery mechanism

1. Introduction

In this work we are concerned with modeling the capacity level dynamics in battery cells
under discharge. Depending on usage patterns, batteries are commonly subject to varia-
tions in life-length and other measures of performance due to variable disload mechanisms
of the electrochemically stored charges. The nominal capacity of a battery is the amount
of charge which is delivered if the cell is put under constant use and drained of its load
under maximal discharge conditions. It is often the case, however, that the discharge pro-
cess occurs on slower time scales, which allows for recovery mechanisms to take place in
which bound charge becomes available and hence adds to the nominal charge level [2]. The
theoretical capacity of the battery is a measure of the maximal charge which in principle
could be obtained where the battery discharged arbitrarily slow, restricted only by the
total amount of electrochemical material contained in the cell.
Over the years, many battery models have been developed for different application areas
(see for instance [1, 3, 5–12] and references therein).
Chiasserini and Rao in [3] have introduced a battery model with recovery effect. Kaj
and Konané, in [8] have studied the so-called kinetic battery model, KIBAM with some
extensions like the spatial diffusion KiBAM and diffusive model KiDiBAM. Chen and al,
in [1] have studied an electric model of the battery capable of capturing the runtime of
the battery.

DOI: https://doi.org/10.29020/nybg.ejpam.v14i1.3881

Email address: kfourtoua@gmail.com (V.F. Konané)

http://www.ejpam.com 234 c© 2021 EJPAM All rights reserved.



V. F. Konané / Eur. J. Pure Appl. Math, 14 (1) (2021), 234-247 235

Our main approach to the modeling of simple batteries will be to focus on the interplay
between the remaining nominal capacity and the remaining theoretical capacity during
discharge evolution. In particular, we will study the discharge process by performing a
phase-plane analysis of nominal and remaining capacities. This will allow us to consider
performance measures such as gained capacity and delivered capacity as functions of basic
model parameters.
To introduce the main ideas of our approach, we consider a battery which is initially fully
loaded with nominal capacity N and which has the theoretical capacity T , at time t = 0.
Realistically, N ≤ T . For t ≥ 0, let x(t) denote the level of available charge and v(t) the
level of remaining theoretical capacity of the battery at time t. Then (v(0), x(0)) = (T,N).
Charge is drawn from the battery either continuously, or, such that the charge level drops
instantly from one discrete level to a lower level. The discharge process acts randomly
or in a deterministic fashion and it acts continuously in time or at discrete time epochs.
In either case, we may let λ denote an average discharge rate per time unit, and write
(vλ(t), xλ(t)) to emphasize the dependence on the discharge rate λ. The battery is empty
and stops functioning at the first instance of time t when either xλ(t) = 0 or vλ(t) = 0.
It is natural to consider the trajectory of the system (vλ(t), xλ(t)), t ≥ 0, as a path in

Figure 1: Phase-plane trace of nominal and theoretical capacities

the (v, x) phase-plane, which starts in (T,N) at time t = 0. Initially, the path moves
downwards and to the left in the (v, x) plane as the nominal, and hence the remaining
theoretical, capacity decreases. While the remaining theoretical capacity continues its
descent with the same average rate as the discharge process, it is reasonable to expect
that the battery may recover some nominal charge capacity because of chemical transport
in the electrolytes enabling previously stored material to become available, at least if λ
is not too large. This effect is likely to be less effective at lower levels of nominal charge.
The battery goes empty whenever x(v) hits one of the axis x = 0 or v = 0. If the path
exits the positive phase-plane through the axis v = 0 for a large x, this indicates that the
initial theoretical capacity is insufficient. If the path exits through x = 0 for large v, then



V. F. Konané / Eur. J. Pure Appl. Math, 14 (1) (2021), 234-247 236

a fraction of the initially stored capacity remains in the battery at the end of its life and
is hence wasted. In both cases the battery could be considered not properly designed. If
there exists a solution v0 > 0 of x(v) = 0, then v0 is the remaining capacity at the battery
charge expiration time. For this case we note that D = T − v0 is the delivered capacity of
the battery. We expect, based on the brief discussion above, that the delivered capacity
tends to N if λ → ∞ and to T if λ → 0, c.f. [13], Figure 6. Moreover, we define the
gain G of the battery to be the quantity G = T − N − v0. This is the capacity which is
gained during the life of the battery and measures the amount of bound charge that the
battery was able to convert into available charge and deplete during its time of operation.
Figure 1 indicates a typical trace in the phase-plane starting from (T,N) and ending in
(v0, 0). Clearly, the path can not visit the shadowed area in the lower right corner. In
some natural models, like the kinetic battery model (see [8]) the path x(v), 0 ≤ v ≤ T ,
can not cross the line x = cv, where c = N/T is the initial ratio of nominal to theoretical
capacity.

2. Markov chain model of pulsed discharge recovery

In this section, we formulate our Markov model and analyze the delivered and gained
capacities.

2.1. Model formulation

Consider a discrete time Markov process (Xn) with finite state space E = {0, . . . , N} and
jump transition probabilities

p0,0 = 1
pi,i−1 = q, 1 ≤ i ≤ N
pi,i = p(1− e−α(N−i)), 1 ≤ i ≤ N − 1

pi,i+1 = pe−α(N−i), 1 ≤ i ≤ N − 1
pN,N = p

where 0 ≤ p < 1, q = 1 − p. For example, the transition probability matrix for N = 4 is
given by

P =


1 0 0 0 0
q p(1− e−3α) pe−3α 0 0
0 q p(1− e−2α) pe−2α 0
0 0 q p(1− e−α) pe−α

0 0 0 q p


For this model, introduced in [3, 4], the state of Xn signifies the current nominal charge
level of a battery. α is a parameter that depends on the battery technology characteristics:
smaller α is, greater is the recovery capability of the battery. We assume that the level
may only change in steps of size one. Initially the battery is fully loaded in state X0 = N .
Each discrete time point a request is received with probability q to discharge one unit load.
With probability p the battery either remains in the same state or is able to recharge one



V. F. Konané / Eur. J. Pure Appl. Math, 14 (1) (2021), 234-247 237

load unit, as long it has not reached the minimum level 0, which is an absorbing state. Fix
an integer T , which is assumed to be the maximal theoretical capacity of the battery in the
sense that there is enough electrochemical material available in the battery to discharge
one unit of load a total of T times. The battery stops functioning either at the time of
absorption in state 0, or at the first time the battery has delivered T discharge events,
whichever occur first.

2.2. Delivered and gained capacities

Let Qn denote the number of discharge events after n steps of the Markov chain. Then
Qn =

∑n
k=1 Zi, n ≥ 1, is a random walk associated with the i.i.d. sequence (Zk) such

that P (Zk = 1) = 1 − P (Zk = 0) = q. Hence for each n, Qn has a binomial distribution
Bin(n, q) with parameters n and q. Also, Vn = T − Qn counts the remaining number
of units of charge in the battery at time n. In analogy with our previous analysis, we
consider now the path (Vn, Xn)n≥0, (V0, X0) = (T,N), evolving on the lattice points
{1 ≤ v ≤ T, 1 ≤ x ≤ N} until its exit time at the first passage of v = 0 or x = 0. To begin
we notice that EVn = T − qn and

E(Xn+1 −Xn|Xn = x) = −q + (1− q)e−α(N−x).

To obtain a continuous approximation (vt, xt) of the average behavior of the path we apply
vt = T − qt and solve the equation

d

dt
xt = −q + (1− q)e−α(N−xt), x0 = N,

which yields

xt = N − 1

α
ln
(
eαqt − 1− q

q
(eαqt − 1)

)
.

Hence, the average nominal charge x as a function of remaining theoretical charge v obeys
the approximation

x(v) = N − 1

α
ln
(
eα(T−v) − 1− q

q
(eα(T−v) − 1)

)
. (1)



V. F. Konané / Eur. J. Pure Appl. Math, 14 (1) (2021), 234-247 238

Figure 2: Markovchain with T = 1000, N = 400, α = 0.005, q = 0.52; 10 simulated paths and associated
average discharge profile (1)

Figure 2 reveals that the discharge profile of the pulsed Markov model is subject to sub-
stantial fluctuations. Moreover, equation x(v) = 0 has a unique solution v0 > 0, if q > q0,
where

q0 = (1 +
eαT − eαN

eαT − 1
)−1 > 1/2.

In this case,

v0 = T − 1

α
ln
(qeαN − 1 + q

2q − 1

)
Thus, we have the following result:

Proposition 1. The delivered capacity and the gain capacity are given by

D(q, T, c) =
1

α
ln
(qeαcT + q − 1

2q − 1

)
and

G(q, T, c) =
1

α
ln
(qeαcT + q − 1

2q − 1

)
− cT. (2)

Proof. The proof of this proposition is based on the definitions of both Delivered and
gained capacity given in the introduction. We recall it: D = T − v0 and G = T − v0 −N
where v0 is solution of equation x(v) = 0.



V. F. Konané / Eur. J. Pure Appl. Math, 14 (1) (2021), 234-247 239

Figure 3: Gainedcapacity G(0.52, 1000, c), 0 ≤ c ≤ 1,and delivered capacity D(q, 1000, 0.4), q0 ≤ q ≤ 1,

In figure 3 we plot the gain and delivered capacity functions for different parameters value.

3. Analysis of the Mean number of discharge pulses

In this present section we provide some additional analysis of the original Markov chain
model, which we have not found in the literature. To this aim, we introduce

Di = number of downward jumps before absorption, if X(0) = i

Ai = number of steps before absorption, if X(0) = i

and put

di = E(Di) = E(number of downward jumps until absorption|X(0) = i)

and
ai = E(Ai) = E(number of steps until absorption|X(0) = i).

We have d0 = 0 and for 1 ≤ i ≤ N − 1, by conditioning on the first jump,

di = q(1 + di−1) + p (1− e−α(N−i))di + p e−α(N−i)d1+1.

Hence
q(di − di−1) = q + pe−α(N−i)(di+1 − di), 0 ≤ i ≤ N − 1.



V. F. Konané / Eur. J. Pure Appl. Math, 14 (1) (2021), 234-247 240

Clearly, dN−dN−1 = 1. Introducing the parameter κ = p/q > 0 and putting ui = di−di−1,
we obtain {

ui = 1 + κ e−α(N−i)ui+1, 1 ≤ i ≤ N − 1
uN = 1

The recursive system is readily solved to give

ui =

N−i∑
j=0

κje−α(2(N−i)−j+1)j/2, i = 1, . . . N.

Therefore

dk =
k∑
i=1

ui = k +
k∑
i=1

N−i∑
j=1

κje−α(2(N−i)−j+1)j/2 (3)

so

dk = k +

N−k∑
j=1

k∑
i=1

κje−α(2(N−i)−j+1)j/2 +

N−1∑
j=N−k+1

N−j∑
i=1

κje−α(2(N−i)−j+1)j/2.

Thus, dk − k equals

N−k∑
j=1

κjeα(j−1)j/2
e−αj(N−k) − e−αjN

1− e−αj
+

N−1∑
j=N−k+1

κjeα(j−1)j/2
e−αj

2 − e−αjN

1− e−αj
,

which yields

dk = k +
N−1∑
j=1

κje−α(j+1)j/2 e
−αj(max(N−k−j,0) − e−αj(N−j)

1− e−αj
.

In particular,

dN = N +
N−1∑
j=1

κje−α(j+1)j/2 1− e−αj(N−j)

1− e−αj
. (4)

Similarly,
q(ai − ai−1) = 1 + pe−α(N−i)(ai+1 − ai), 0 ≤ i ≤ N − 1,

and aN − aN−1 = 1. Hence ũi = ai − ai−1 satisfies{
ũi = 1/q + κ e−α(N−i)ũi+1, 1 ≤ i ≤ N − 1
ũN = 1

It follows that ũk = uk/q hence ak = dk/q for 1 ≤ k ≤ N − 1. Thus,

dN = 1 + dN−1 = 1 + q aN−1 = q aN + p.



V. F. Konané / Eur. J. Pure Appl. Math, 14 (1) (2021), 234-247 241

3.1. Asymptotic behavior of the mean number of pulses

We study, in this section, the asymptotic behavior of dN .

Proposition 2. For fixed N and α→ 0,

dN → N +

N−1∑
j=1

κj(N − j) =


N 1

1−κ −
κ

(1−κ)2 (1− κN ), κ < 1

N(N + 1)/2, κ = 1
κ

(κ−1)2 (κN − 1)−N 1
κ−1 , κ > 1,

with equality for the case α = 0.

Proof. From relation (4), we have

dN = N +

N−1∑
j=1

κj(1− αj(j + 1)

2
)
αj(N − j)

αj

→ N +
∞∑
j=1

κj(N − j).

From this relation we conclude that κ = 1, that is p = q = 1/2, represents a transitional
regime. If the arrival probability is high in the sense q > 1/2, and so κ < 1, then the num-
ber of recharge events behaves on average linearly with N . Under low utilization, meaning
q < 1/2 and κ > 1, then in contrast dN has exponential increase as a function of N . It
appears unrealistic however to expect the pulsed discharge mechanism to run effectively
on batteries with such a large theoretical capacity T that lnT ∼ N represents normal
operation. At the critical parameter value reaches κ = 1 from below we see that the typi-
cal duration of battery operation before absorption given by dN , and disregarding for the
moment the role of T , changes from linear in N to quadratic in N . This suggests that in
order to achieve recharge events and keep the battery running until all T unit loads have
been discharged, one should try to operate the battery with an arrival probability q ≤ 1/2.

We also have the following result.

Proposition 3. For fixed α > 0 and N →∞,

dN −N →
∞∑
j=1

κje−α(j+1)j/2

1− e−αj
,

which is finite for any κ, α > 0.

Proof. To establish the result, we need to apply the dominated convergence in relation
(4).

Now we consider the simultaneous convergence of α→ 0 and N →∞, which allows us to
analyze the onset regime of non-linear increase of the expected absorbtion time.



V. F. Konané / Eur. J. Pure Appl. Math, 14 (1) (2021), 234-247 242

Proposition 4. (i) Fix β > 0 and take α = β/N → 0 as N →∞. Then, if κ < 1,

dN −N
N

→ 1

β

∞∑
j=1

κj

j
(1− e−βj) =

1

β
ln
(1− κe−β

1− κ

)
.

(ii) For γ > 0 and δ a real number, consider α = γ/N2 → 0 and

κ = 1 +
δ

N
, q =

1

2 + δ/N

as N →∞. For small α and large N ,

dN = N +
N−1∑
j=1

(1 +
δ

N
)je−α(j+1)j/2 1− e−αj(N−j)

1− e−αj

∼ N2 e
γ − 1− γ
γ2

− αN4
((eγ − 1− γ − γ2

2 )(γ − 2)

2γ3
+

1

4

)
+ o(N6α2).

With α = γ/N2 as N →∞, Then

dN −N
N2

→
∫ 1

0
eδxe−γx

2 1− e−γx(1−x)

γx
dx

∼ eδ − 1− δ
δ2

− δ + 2 + eδδ − 2eδ

2δ3
γ +O(γ2)

Proof. The proof of Proposition 4 is done by using the Taylor series expansion of order
1 in the neighborhood of 0 and the approximations method of a sum by an integral. From
relation 4, we have

dN −N
N

=
1

N

N−1∑
j=1

κje−α(j+1)j/2 1− e−α(N−j)j

1− e−αj

=
1

N

N−1∑
j=1

κj(1− α(j + 1)j/2)
1− e−

β
N
(N−j)j

1− (1− β
N j)

=
N−1∑
j=1

κj(1− β

N
(j + 1)j/2)

1− e−β(1−
j
N
)j

βj

=
1

β

N−1∑
j=1

κj

j
(1− β

N
(j + 1)j/2)(1− e−β(1−

j
N
)j).

As N goes to infinity, we get

dN −N
N

→ 1

β

∞∑
j=1

κj

j
(1− e−βj)



V. F. Konané / Eur. J. Pure Appl. Math, 14 (1) (2021), 234-247 243

=
1

β

∞∑
j=1

κj

j
− 1

β

∞∑
j=1

(κe−β)j

j
.

The series expansion of x 7→ ln(1− x) allow us to get point (i) of proposition 4, which is

dN −N
N

→ 1

β
ln
(1− κe−β

1− κ

)
.

We have

dN −N
N2

=
1

N2

N−1∑
j=1

κje−α(j+1)j/2 1− e−α(N−j)j

1− e−αj

=
1

N

N−1∑
j=1

(1 +
δ

N

j

)e−γ
(j+1)
N

j
N
/2 1− e−γ(1−

j
N
) j
N

γ j
N

=
1

N

N−1∑
j=1

eδ
j
N e−γ

(j+1)
N

j
N
/2 1− e−γ(1−

j
N
) j
N

γ j
N

→
∫ 1

0
eδxe−

γ
2
x2 1− e−γ(1−x)x

γx
dx.

The Maclaurin expansion of order 2 of maps γ 7→ e−
γ
2
x2 and γ 7→ e−γx(1−x) leads to get

dN −N
N2

∼ eδ − δ − 1

δ2
− δ + 2 + eδδ − 2eδ

2δ3
γ +O(γ2).

3.2. Variance of the discharge number of pulses

In this section, we study the variance of the discharge number.
Let

σ2i = Var(Di), 1 ≤ i ≤ N.

Set t0 = 0 and we introduce

ti = E(D2
i ), yi = ti − ti−1, 1 ≤ i ≤ N.

By conditioning on the first jump,{
yi = 1 + 2di−1 + κ e−α(N−i)yi+1, 1 ≤ i ≤ N − 1
yN = 1 + 2dN−1

so

yi = ui + 2

N−i∑
j=0

di−1+jκ
je−α(2(N−i)−j+1)j/2, i = 1, . . . N.



V. F. Konané / Eur. J. Pure Appl. Math, 14 (1) (2021), 234-247 244

Thus, for 1 ≤ k ≤ N ,

vk =

k∑
i=1

yi = dk + 2

k−1∑
i=0

di + 2

k∑
i=1

N−i∑
j=1

di−1+j κ
je−α(2(N−i)−j+1)j/2.

We will focus on the case k = N , where

vN = dN + 2

N−1∑
i=0

di + 2

N∑
i=1

N−i∑
j=1

di−1+j κ
je−α(2(N−i)−j+1)j/2.

Recall

dk = k +
k∑
i=1

N−i∑
j=1

κje−α(2(N−i)−j+1)j/2

Thus

2

N−1∑
k=1

dk = 2

N−1∑
k=1

k + 2

N−1∑
k=1

k∑
i=1

N−i∑
j=1

κje−α(2(N−i)−j+1)j/2

so

vN = N +
N∑
i=1

N−i∑
j=1

κje−α(2(N−i)−j+1)j/2

+N(N − 1) + 2
N−1∑
k=1

k∑
i=1

N−i∑
j=1

κje−α(2(N−i)−j+1)j/2

+2
N∑
i=1

N−i∑
j=1

(i− 1 + j)κje−α(2(N−i)−j+1)j/2

+2

N∑
i=1

N−i∑
j=1

i−1+j∑
r=1

N−r∑
s=1

κse−α(2(N−r)−s+1)s/2 κje−α(2(N−i)−j+1)j/2

= N2 +
N∑
i=1

N−i∑
j=1

κje−α(2(N−i)−j+1)j/2

+2
N∑
i=1

N−i∑
j=1

(N − 1 + j)κje−α(2(N−i)−j+1)j/2

+2

N∑
i=1

N−i∑
j=1

i−1+j∑
r=1

N−r∑
s=1

κse−α(2(N−r)−s+1)s/2 κje−α(2(N−i)−j+1)j/2

This yields for σ2i = ti − d2i ,

σ2N =
N∑
i=1

N−i∑
j=1

κje−α(2(N−i)−j+1)j/2



V. F. Konané / Eur. J. Pure Appl. Math, 14 (1) (2021), 234-247 245

+2
N∑
i=1

N−i∑
j=1

(j − 1)κje−α(2(N−i)−j+1)j/2

+2
N∑
i=1

N−i∑
j=1

i−1+j∑
r=1

N−r∑
s=1

κse−α(2(N−r)−s+1)s/2 κje−α(2(N−i)−j+1)j/2

−
( N∑
i=1

N−i∑
j=1

κje−α(2(N−i)−j+1)j/2
)2

=
N∑
i=1

N−i∑
j=1

(2j − 1)κje−α(2(N−i)−j+1)j/2

+
N∑
i=1

N−i∑
j=1

i−1+j∑
r=1

N−r∑
s=1

κse−α(2(N−r)−s+1)s/2 κje−α(2(N−i)−j+1)j/2

−
N∑
i=1

N−i∑
j=1

N∑
r=i+j

N−r∑
s=1

κse−α(2(N−r)−s+1)s/2 κje−α(2(N−i)−j+1)j/2.

A better writting is

σ2N =
N∑
i=1

N−i∑
j=1

(2j − 1)κje−α(2(N−i)−j+1)j/2

+
( N∑
i=1

N−i∑
j=1

κje−α(2(N−i)−j+1)j/2
)2

−2
N∑
i=1

N−i∑
j=1

N∑
r=i+j

N−r∑
s=1

κse−α(2(N−r)−s+1)s/2 κje−α(2(N−i)−j+1)j/2.

The cases studied above in proposition 2 for E(DN ) suggests that for case (i) σ2N ∼
constN2 and case (ii) σ2N ∼ constN4. It is better to approximate the above sums directly
with integrals. Checking cases such as α→ 0, κ = 1 might give some insights.

4. Variations of the Markov chain model

In this section we introduce variations of the Markov chain model.
Recall that the Markov chain model has the defining properties

(Xn+1|Xn = x) =


x− 1 with probability q

x+ 1 with probability (1− q)e−α(N−x)
x with probability (1− q)(1− e−α(N−x)))



REFERENCES 246

It is natural to consider the extended, bivariate model (Vn, Xn)n≥1,with dynamics specified
by

(v, x)→


(v − 1, x− 1) with probability q

(v, x+ 1) with probability (1− q)e−α(N−x)−β(T−v)
(v, x) with probability (1− q)(1− e−α(N−x)−β(T−v))

which yields

xt = N +
β(T − v)

α
− 1

α
ln
(
eαqt − 1− q

q

(
eαqt − 1

))
.

Hence,for q ≥ 1/2 the average nominal charge x as a function of remaining theoretical
charge v obeys the approximation

x(v) = N − 1

α
ln
(
e−α(T−v) − 1− q

q

(
e−α(T−v) − 1

))
(5)

Conclusion

In this paper, we presented a Markov modeling approach of the battery based on the
work in [3]. We derived results on the delivered and gained capacities. These quantities
where studied in [8] for the kinetic battery models. Moreover, we investigated the mean
and variance of the number of pulses. We also looked at the asymptotic behavior of the
mean number of pulses. We end up the paper by the extended version of the Markov
model.

Acknowledgements

The authors would like to thank International Science Program for their partial finan-
cial support, the editor for accepting that we submit our paper in this journal and the
reviewers for their helpful comments.

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