1072 PIERS Proceedings, Prague, Czech Republic, July 6–9, 2015

Estimation of Equivalent Model Parameters for LiFeO4 Batteries

Based on Particle Swarm Optimization
L. Chen1 , T. Geng1 , Q. Zhang1 , G. C. Wan2 , C. H. Jiang2 , and M. S. Tong2
1
School of Electrical and Electronic Engingeering, Shanghai Institute of Technology, Shanghai, China
2
Department of Electronic Science and Technology, Tongji University, Shanghai, China

Abstract— State of charge (SOC) is an important parameter of battery management system

(BMS) which reflects the reliability, safety, and lifetime of batteries. However, SOC cannot
be measured directly and we have to estimate it via analyzing some other parameters such as
voltage, current, and temperature. An accurate estimation strategy of SOC is necessary for the
BMS and we propose a novel model based on particle swarm optimization (PSO) in this paper.
The PSO is an optimization method which originated from artificial intelligence and evolutionary
computation. It is a simple, effective, and universal theory which solves problems by seeking their
individually best and globally best solutions. We apply this theory to optimally estimate the SOC
parameters for LiFeO4 Batteries and the simulated and experimental results have demonstrated
its effectiveness.
1. INTRODUCTION
Power batteries are very important components in electric vehicles and they have a significant
impact on the performance and security of a whole vehicle. To ensure the safe and reliable operation
of batteries, designing an optimal battery management system (BMS) is very essential. The BMS
manages the batteries based on their states and instructions from a vehicle management system
(VMS). The voltage, current, and temperature of batteries can be measured directly by sensors
while other parameters such as the state of charge (SOC) has to be estimated indirectly by certain
algorithms which usually rely on the battery model and its parameters. The battery model and
parameters are essential because they form the basis of loop simulations for hardware and SOC
estimation, and also provide a reference for the control of BMS.
In this paper, we propose a unified form of equivalent circuit model for LiFeO4 batteries based on
the analysis of similarity between the internal mechanism and external characteristic of a chemical
power source. A two-order equivalent circuit [1] of LiFeO4 batteries is built and their resistance
and capacity are estimated with a least square method (LSM) [2] and particle swarm optimization
(PSO) [3]. By comparing the results from different estimation methods, we find that the relationship
between the SOC and the electric motive force (EMF) or EMF-SOC can be significantly affected by
the temperature. We also build a backward-propagation (BP) neural network [4] in which the EMF
and temperature are taken as inputs while the SOC works as an output and the results are used
to estimate the SOC. The simulated and experimental results show that the proposed estimation
method can accurately estimate the SOC for LiFeO4 batteries.
2. PARTICLE SWARM OPTIMIZATION (PSO)
The PSO was first proposed by Kennedy and Eberhart in 1995 [5]. It is an optimized computa-
tional method based on a swarm intelligence and simulates the internal action between individuals,
group, and environment. Nowadays, the PSO is widely used in process optimization, parame-
ter estimation, and system control. In the PSO theory, the position of a particle represents a
solution of optimization problem. Each particle has a position and velocity that decide its di-
rection and speed. The state of particles depends on adaptive values that are determined by
objective function. Suppose in a D-dimension objective search space, a group is formed by m
particles which are randomly initialized. Then the position and flying velocity of the ith particle
can be represented by Xi = (xi1 , xi2 , . . . , xiD )T and Vi = (vi1 , vi2 , . . . , viD )T , respectively. Particles
will track two extremum values in a solution space. On each iteration, the individual extremum
P bi = (pbi1 , pbi2 , . . . , pbiD )T represents the best position of the particle while the global extremum
Gb = (gb1 , gb2 , . . . , gbD )T represents the best position of the group. The velocity and position are
updated each time according to the following formulas:
³ ´ ³ ´
k+1 k
vid = vid + c1 r1k pkid − xkid + c2 r2k gbkd − xkid (1)
k+1 k+1
vid = xkid + vid (2)
Progress In Electromagnetics Research Symposium Proceedings 1073

where vid k represents the velocity of the ith particle in the Dth dimension in the kth iteration and
k
xid represents its current position. Also, c1 and c2 are the acceleration factors or learning factors
which adjust the maximum step size of the fight to the globally best particle and individually best
particle. The adaptive values of c1 and c2 can accelerate the convergence and help the particles to
avoid falling into a local optimum. In addition, r1 and r2 are the random numbers in the range
[0, 1], pid represents the position or coordinate of the individual extremum point of the ith particle
in the Dth dimension, and gd represents the position of the global extremum point in the Dth
dimension [6].
3. PSO PARAMETER IDENTIFICATION BASED ON SECOND-ORDER RC
EQUIVALENT
3.1. LSM
Mathematical model of a battery should reflect and take into account its self-discharge phenomenon,
capacity, and resistance over potential and environmental temperature. In this paper, a second-
order RC equivalent circuit model is used to simulate the battery. In this model, E represents the
battery’s voltage which is affected by the SOC and R0 refers to its ohmic internal resistance. Also,
R1 and C1 denote its electrochemical polarization resistance and capacitance, respectively, and they
constitute a RC parallel circuit, while R2 and C2 indicate its consistent polarization resistance and
capacitance, respectively, and they constitute another RC parallel circuit. The test data of a study
on a 60AH-LiFePO4 battery by means of a constant discharge is shown in Figure 2. The battery
starts a discharge after 100 s, and stops the discharge when reaching the point B. The RC network
is in a zero input response from the point A to point B and the voltage rebounds rapidly as soon
as the circuit is broken. In the circuit, the loop current is zero and the ohmic internal resistance
drop is also zero.
UD − UB
R0 = (3)
I
At the same time, the RC network is in a zero input response. The voltage from the point B to
point C is
− t − t
UB (t) = EC (t) − I × R1 × e R1 ×C1 − I × R2 × e R2 ×C2 (4)
where EC(t) refers to the voltage at the point C. We utilize the test data and LSM [7] to figure out
the parameters including R1 , R2 , C1 , and C2 . The result is shown in Figure 3.
3.2. Battery’s Parameter Identification Based on the PSO
We initialize 10 particles and each particle’s position is Xi = (R1i , R2i , C11i , C12i )T while its velocity
is Vi = (vi1 , vi2 , vi3 , vi4 )T . All initial values are randomly chosen in the range of [0, 1]. On each
iteration, the parameter
t t
−R −R
Y1 (t) = EC (t) − I × R1 × e 1 ×C1 − I × R2 × e 2 ×C2 (5)
is calculated while Y2 (t) is its counterpart obtained by an experiment. The fitness function is
X |Y2 (t) − Y1 (t)|
f= (6)
t
Y2 (t)

Figure 1: A model for an FeLiPO4 battery. Figure 2: Discharge curve of an FeLiPO4 battery.
1074 PIERS Proceedings, Prague, Czech Republic, July 6–9, 2015

Figure 3: Parameters figured out by a LSM. Figure 4: EMF-SOC relationship in different tem-
peratures.

and the iteration is performed by the following steps:

Step 1: When g = 1, the position Xi1 and velocity Vi1 of the searched points are initialized with
the random numbers in the range of [0, 1], and Pi1 = Xi1 . Then we calculate the individual ex-
tremum values and the global extremum value is the best one among the individual extremum
points. Record the index of the best particle and set G as the position of that particle.

Step 2: Update the new position Xi2 and velocity Vi2 for each particle by Equations (1) and (2)
and calculate the particle’s new fitness value. If the new value is better than the previous
value, then let Pi = Xi2 . If there is an individual value that is better than the current global
value, the position of that particle will be set as G and the index of that particle will be
recorded. The global extremum value is then updated correspondingly.

Step 3: If the number of iterations reaches the maximum value that is set in advance, then the
iteration will stop and the best solution will be output. Otherwise, go back to Step 2.
3.3. EMF-SOC Model of LiFePO4 Battery Based on BP Neural Network
The initial capacitance should have been known before an ampere-hour method is used to calculate
the residual capacitance of a battery. Currently, the relation of SOC = f (SOC) is mostly used to
get the initial SOC. If the battery has rested for a long time, its terminal voltage can be regarded
as the EMF. For different temperatures, the same EMF corresponds to different SOCs as shown
in Figure 4. The BP network is a one-way-communication and multilayer-feed-forward neural
network and the BP means the backward propagation of errors whose main idea is the gradient
descent. Using the gradient search techniques can make the actual output value of network have
an expected error of square minimum value. The initial capacitance is affected by terminal voltage
and temperature. The EMF-SOC BP neural network can be obtained by setting the EMF and
temperature of the battery as an input while setting the SOC as an output.
3.4. Construction of State Space for Battery
We can get the state space for a battery according to the following equations
 
      − η∆t
SOCk 1 SOCk−1 Q
 
 Uk0   0   Uk−10   ³ R0 ∆t ´
 U1  =   − ∆t
e R1 ×C1
×
  Uk−11
+ 
 R1 1 − e R1 ×C1  × ik−1 + wk−1 (7)
−
k  ³ ´
Uk2 2
∆t
− Uk−1 − ∆t
e R1 ×C1 R2 1 − e R2 ×C2
1 2
Uk−1 = OCV (SOCk−1 ) − ik−1 × R0 − Uk−1 − Uk−1 + Vk−1 . (8)
Progress In Electromagnetics Research Symposium Proceedings 1075

In the equations, SOCk is the value of the SOC at the kth moment, Uk0 is the terminal voltage
of ohmic internal resistance, Uk1 and Uk2 are the terminal voltages of polarization resistance, ∆t is
the sampling time, η is the coulomb coefficient, Q is the nominal capacitance of the battery, i is
the current, and Uk−1 is the voltage of the battery. The random signal wk−1 and vk−1) refer to an
incentive noise and observation noise, respectively. We assume that they are white noises which
are independent of each other and follow a normal distribution.

4. SIMULATION RESULTS
We present the simulation results for the battery parameter identification based on the PSO method.
Assuming that c1 = c2 = 2 and r1 and r2 are the random numbers in the range of [0, 1]. The
dimensional speed vd is clamped between [vd min , vd max ] to prevent particles from moving away
from the search space. The number of iterations is selected as 10 and the identification results with
corresponding errors are shown in Figure 5. On the other hand, if the number of hidden layers
of the neural is selected as 5, the output of hidden layers of the neuron is selected as a “tansig”
function, the transfer function of the output is a “purelin”function, and the training function is a
“trainlm” function, then the EMF-SOC BP neural network can be established by setting the EMF
and temperature of the battery as an input while setting the SOC as an output. The result is
shown in Figure 6.

Figure 5: Result of the PSO. Figure 6: Result of the EMF-SOC based on the BP
network.

5. CONCLUSION
We use the LSM to identify the parameters of battery model. We also establish a second-order
RC network model for the LiFePO4 battery and figure out the values of some parameters such as
capacitances and resistances by the LSM and the PSO. Comparing these two algorithms, we find
that the PSO can reach the same accuracy as the LSM. The structure of EMF-SOC neural network
based on the BP neural network can be obtained via setting the EMF and temperature as an input
and the SOC as an output. The accuracy of calculating the SOC can be enhanced by considering
the influence of temperature. We present some simulated and experimental data to demonstrate
the estimation method and good results have been observed.
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