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Design an Optimal Fractional Order PID Controller for Speed Control of Electric Vehicle
Nahida Naji Kadhim1, Layla H. Abood1, Yousra Abd Mohammed2*
1 Department of Control and System Engineering, University of Technology, Baghdad 10066, Iraq
2 Communication Engineering Department, University of Technology, Baghdad 10066, Iraq
Corresponding Author Email: yousra.a.mohammed@uotechnology.edu.iq
https://doi.org/10.18280/jesa.560503
ABSTRACT
Received: 25 July 2023
Revised: 13 October 2023
Accepted: 24 October 2023
Available online: 31 October 2023
In recent years, electric vehicles have garnered significant attention due to their
environmental and economic advantages compared to conventional vehicles, including
reduced emissions and lower fuel costs. This study proposes an optimal fractional-order
PID (FOPID) controller to regulate electric vehicle (EV) speed. The FOPID controller is
advantageous due to its ability for stabilizing the system, managing parameter variations,
and mitigating potential disturbances. The tuning of this controller's gains is achieved
through an intelligent Ant Colony Optimization (ACO) algorithm. The selection of the
gain values is strategically based on minimizing error, thereby ensuring a robust system
response without overshoot or undershoots. The performance of the proposed controller is
analyzed and compared to a classical PID controller for demonstrating its superior
performance. Simulation results illustrate the efficiency of the proposed controller, which
exhibits no fluctuation or oscillation in its response (zero overshoot) and fast settling and
rise times of 0.0476 and 0.0297, respectively. By using the optimal gains determined by
the smart ACO, the proposed controller achieves a satisfactory and robust system response
in controlling EV speed.
Keywords:
electric vehicle, fractional-order PID
(FOPID), speed control, sunflower
optimization
1. INTRODUCTION
The motor is regarded as the significant part of an EV
framework, beside to the regulator, drive train, charger and
power supply. Regulator is the core of an EV, and the key for
the acknowledgment of an elite presentation EV with an ideal
equilibrium of most extreme speed lately, in view of
worldwide fuel supply, contamination issues and an Earth-
wide temperature boost; zero-dirtying types of electric
vehicles are a quickly developing innovation for power the
board and ecological saving issue [1]. Electrical vehicles have
been presented in the industry marketing [2]. Moreover, the
DC motors can give best job and regarded as a braking tool
due to their facilities in its fast torque behavior [3].
Various studies are utilized in the survey for adjusting its
power, the price and raising driving levels in order to enhance
the power maintaining issue [4, 5]. A linear quadratic
controller that controls the position of the throttle was
presented in the study [6] for hybrid electric vehicles.
However, the closed-loop system’s performance reduces when
a linear type of controller is used with the nonlinear plants. A
fuzzy logic controller is adopted for controlling a permanent
magnet synchronous motor’s speed was developed and
experimentally tested in the study [7]. A robust cruise control
system for a DC motor in EV was proposed in the study [8].
To compensate for the effects of changes in vehicle weight and
the road grade, numerical optimization is used with the
suggested. In the study [9], the EV has two inputs. The linear
velocity and the steering angle. The linear velocity is enhanced
by a Permanent magnet motor type and the steering angle is
presented with a stepper motor type. In cruise control, a steady
speed was achieved. In the study [10], a neural network-based
PI is proposed for an EV driven by induction motor drives. In
fact, power electronics and motor drives are actively
researching neural network concepts, and in the study [11], an
adaptive neural network approach is used to control an EV
based on an interior permanent magnet synchronous machine
(IPMSM), the adaptive scheme was presented depending on
the Lyapunov relation for achieve stability to ensure
robustness, and best tracking. In the study [12] for optimizing
the control of an electric vehicle system, new technique was
created depending on stochastic drive cycles. A sliding mode
controller for EV was presented in the study [13]. In the study
[14], a feedback linearization method was used to the Light
Weight EV (LWEV) system. In this method, the linear system
was adjusted by using LQR controller.
In this study, an augmented and enhanced controller is
adopted, its depend on fractional calculus by adding two
parameters in fraction form to the integral and differentiator
gains, its named as a FOPID controller which is adopted for
controlling speed of EV based on using a unique and
intelligent ACO algorithm for finding the gains controller to
improve the EV system dynamic behavior and obtain a stable
and robust behavior. This study is organized as: Section 2
explains the EV system modeling. Section 3 indicates the
adopted controller used. Section 4 indicated the ACO
algorithm, Section 5 demonstrates the simulation results then
and then in Section 6, a conclusion is presented.
2. MODEL DETAILS
This section of the paper is presented an electronic speed
control schematic graph which employing a DC servo motor
Journal Européen des Systèmes Automatisés
Vol. 56, No. 5, October, 2023, pp. 735-741
Journal homepage: http://iieta.org/journals/jesa
735
appears in Figure 1, the vehicle’s dynamics are modeled using
the leader-follower configurations [6]. The electric vehicle
(EV) depicted in Figure 1 that utilizes an electronic throttle
control method is also known as a single-mode power split,
either in series or parallel, or both. A planetary gear system
starts the power supply to the wheels in both series and parallel
configurations. While the series flow technique gets electricity
from the engine to the battery and then back from the electrical
system to the wheels, the parallel flow method uses two paths:
one from the engine to the wheels and another from the battery,
to the motors, and back to the wheels. This construction
improves overall performance, reduces pollution, and has high
speed levels, among other advantages. The throttle plate is
rotated by a DC servo motor in this electronic throttle control
system, and its rotation is managed by the voltage provided to
the motor [15].
Figure 1. The electronic throttle control diagram
Eqs. (1)-(3) can be used to calculate the relationship
between follower vehicle’s acceleration, propulsion force,
and drag forces: [6, 16] where Fe (Engine force, a throttle
position function), Fg (Gravitational Force, A road grade
function: is 30% of weight of vehicle), θ (throttle position),
v (EV speed), and τe (Engine time constant commonly lie
between 0.1 to 1sec., here is taken 0.2s). The variables
adopted for work is presented in Table 1.
(1)
(2)
(3)
Table 1. Numerical parameters values
Constant
Notation
Value (SI unit)
Vehicle mass
1000Kg
Aerodynamic drag coefficient
4N/(m/s)2
Engine force coefficient
12500N
Engine idle force
6400N
Eq. (1) through (3) are used to build the Simulink model of
the vehicle seen in Figure 2 [17]. Eq. (4) through (7) display
the state variable representation of the vehicle, and Eq. (8)
displays the transfer function.
(4)
(5)
(6)
(7)
(8)
The Eigenvalues of the open-loop system are λ1=0 and
λ2=−5, which are derived from the characteristic equations of
the system expressed in Eqs. (4)–(7).
Ẋ(t)=Ax(t)+Bu(t) is the equation for a linear time invariant
(LTI) system. The system matrix A is represented by n×n, the
control matrix B by n×r, and the input vector matrix u by r×1
dimensions. A controlled and observable system exists when
the rank of matrix M = [B AB A2B ... An−1B] is n. Given that
matrix M and N's order equals the matrix's rank of 1 [18];
M=829000
, N=
.
Figure 2. EV simulink model
3. FOPID CONTROLLER
736
PID controllers is regarded as a simple and classical
controllers that adopted for improving system behavior.
Nowadays, studies are made with a different changes for PID
controller like combining it with a neural network [19, 20] or
changing its structure to reach to a best stability and robustness
as in [21, 22] or changing it by add a fractional variables
(integral & derivative) to the classical PID to improve system
output [23, 24], this type of change is an augmented type PID
controller. These parameters (μ for derivative variable and λ
for the integral variable) make the gains of controller be five
variables. In the automation and control fields, the Fractional
Order Controllers (FOCS) achieve more accurate and stable
performances, it is classified into four types: CRONE
controller, Tilt and Integral (TID) controller type, FOPID
controller and fractional-order lead-lag compensator, it is
consists of five variables: three normal gains and named
proportional, integral, and derivative while there is a two
fractional variables for integral and derivative, representing
the FOPID can be expressed using a graph of PID controller
that explained by the plane of the μ and λ variables that
appeared in Figure 3 [25].
Figure 3. Fractional PID controller plane
In 1695, L'Hopital used the phrase fractional order calculus
to illustrate how some systems may be properly described
using fractional order differential equations. On this, Lurel,
Riemann, Laplace, Able, and Euler act. Calculus in fractional
order is studied more quickly starting in 1884. Differential is
a key variable in fractional order calculus. Since the two
fractional order type’s derivative and integrator may be
expressed by a single operator, this name has become popular.
The following is an explanation of the differintegral [26, 27]:
(9)
The operator's boundaries in Eq. (9) are "a" and "t.", "α"
represents the operation's order and is linked to R, (any rational
number) it might even be a complex number. Specifically, the
Riemann-Louville (RL) expression and the Grunwald-
Letnikov (GL) expression are chosen to represent the basic
fractional differintegral. The definition of GL is
(10)
The fractional differintegral represented by RL is:
(11)
For (n-1<α<n) and Γ (.) is the Gamma function.
In this study, a FOPID controller is adopted [28, 29] to
controlling the speed in EV system its structure is indicated in
Figure 4 and its transfer function is indicated in Eq.(12) below:
+
(12)
Figure 4. FOPID structure
4. ANT COLONY OPTIMIZATION (ACO)
Figure 5. ACO flow chart
ACO is a population-based strategy that translates the
combinatorial method of optimization; it emulates the
mechanism by which actual ants locate the shortest path when
conducting searches inside their colony. Each ant will secrete
a chemical material called pheromone; it will dispose this
material behind it and each ant will follow other ants
depending on the concentration of pheromone which indicate
the short way among other far routes of their way. In ACO, a
finite number of artificial ants are initiated. Each one will take
a decision to solve the problem. During this, each ant gives its
decision based on the problem utilized and on its own behavior.
The best decision is adopted based on its fitness function then
their decisions are represented by the path selected. ACO must
find an optimal way either locally or globally. The path details
737
found during their trip is saved in the pheromone attempts
relayed to their different paths. Pheromone attempts will be the
memory for all ants’ trips. They are decided their final decision
and change their ways depending on the pheromone trials and
update it to the optimal path [30, 31], the flow chart of ant
colony is indicated in Figure 5 and the steps of ACO algorithm
are given as follows:
Step 1: Initialize the ACO parameters like dimension of the
problem (), population size (), maximum number of
iterations (), alpha (), beta (), Evaporation rate (),
pheromone Matrix (), and change of pheromone ().
Step 2: Find the probability for each ant in solving the way
according to Eq. (13).
(13)
Each ant constructs its own tour utilizing a transition
probability.
Step 3: Calculate the minimum Integral Time Absolute
Error (ITAE) cost function [32, 33] indicated in Eq. (14) below
at each tour depending on the best ode minimum value. At
each tour a test is done to find the best ant that find the optimal
decision.
n
(14)
Step 4: Each ant will have a pheromone hormone on her
way and it is calculated as indicated in Eq. (15) below:
(15)
where,
is the pheromone value in the way of the kth ant
in the iteration tour and the amount of pheromone value will
be calculated using Eq. (16).
(16)
where, L is the road length during tour iteration. Based on step
3 the updated process will do just to the ant that find the
optimal path in which it should be allowed to lead elite.
Step 5: Now when the evaporation is completed then the
pheromones updated according to Eq. (17) as shown:
(17)
where, 0<<1 is the ‘evaporation factor’.
All of the ants update their data and determine whether or
not the maximum iteration was achieved after calculating the
pheromone and finishing the evaporation.
Step 6: Check the maximum iterations () value, if it is
reached it will stop, otherwise, Step 2 to Step 5 is repeated.
The FOPID controller is adopted to enhance the dynamical
behavior by reduce error between desired and actual values
by continuously compute the (ITAE) fitness function
suggested, the issue is how to select the optimal values the
FOPID variables based on ACO by choosing these values in
each iteration then check the fitness function when it is
minimized to a suitable value the process will stop in the
suitable iteration to measure the tuned values of FOPID
controller gains.
Figure 6. EV system with proposed controller based
on ACO algorithm
Figure 6 explain the proposed controller used for controlling
EV speed based on ACO algorithm.
5. SIMULATION RESULTS AND DISCUSSION
This section explains the simulation results for the
recommended optimal controller for EV speed control using
Matlab software version 2019. To improve system response,
the ACO algorithm is adopted; its initial parameters are chosen
based on the literature, and its iteration number and population
are chosen through trials to produce quick and accurate results.
The results are shown in Table 2 below.
Table 2. ACO parameters
Description
Value
ANT population
100
Number of iteration
30
Pheromone variable
1
Evaporation variable
0.05
Initial concentration
1.5
Heuristic variable
2
Pheromone
100
Initial uniform probability
0.5
Controller gains
5
In this study the, the ITAE fitness function equation is used
for monitoring the error during simulation and based on its
value the ACO algorithm will find a controller gains. system
response is shown in Figure 7.
FOPID's ability to provide the desired reaction quickly and
steadily is confirmed by a comparison with two traditional
controllers (optimal PID and classical PID). Figure 8 show its
superiority upon the two controllers used, Table 3 list the gains
of all controller tested and Table 4 explain the transit analysis
for each controller.
Table 4 makes it clear that the recommended FOPID
controller outperforms the two conventional controllers that
were employed (PID, Optimal PID) in its transient analysis
and try to reach its desired speed with best settling time (0.
0476) while in classical PID and optimal PID are slow and
equal to 0.235s and 0.093s respectively, also its faster in its
rise time with value equal to 0.0297s with respect to classical
PID and Optimal PID with a value equal to 0.163 and 0.062s
respectively. This deviation in classical PID is due to its simple
structure with a manually chosen gain values and its effect
reflected on system response while the optimal PID give a
response better than the classical PID but still not reach to the
FOPID controller stable response, then by analyzing system
738
response a stable behavior is shown clearly without any
overshoot and fluctuations in its response this is due to the
fractional parameters effect and the efficient tuning algorithm
used in selecting the more suitable and tunable controller gains
that achieve its best performance.
Figure 7. Response EV speed
Figure 8. System response for all controllers
Table 3. All controllers gains
Controller
Kp
KI
KD
λ
μ
Classical
PID
0.01
0.044
0.0016
-
-
Optimal
PID
0.008552
0.0633
0.0005
-
-
FOPID
0.00041
0.000067
0.00091
0.891
0.985
Table 4. Comparative results of response parameters
Controller
Max. Overshoot
(Mp%)
Peak
Time (tp)
Rise
Time (tr)
Settling
Time (ts)
Classical
PID
0.25
0.45
0.163
0.235
Optimal
PID
0.2
0.19
0.062
0.093
FOPID
0
0.08
0.0297
0.0476
6. CONCLUSION
In this study a FOPID controller is adopted for maintaining
speed in EV system, a smart optimization method is used for
finding the suitable values for controller gains and improves
system behavior by minimizing error level between desired
speed and actual speed also by tracking the suitable desired
speed with a stable output and without any fluctuating values.
A comparison analysis with the other controllers indicates that
the proposed controller is more efficient in achieving best
analysis values; it is faster than classical PID by 79.44% and
optimal PID by 48.81% in rise time and faster than classical
PID by 81.77% and optimal PID by 52.09% in settling time
with a stable response level, this controller can be
implemented in different embedded systems due to its efficient
structure and all gains is calculated offline which reduce any
delay may happen. The suggested controller with its fractional
variables and the best optimizing algorithm maintain a best
and efficient desired response with an optimal level values. In
future work, many ideas can be suggested like make a hybrid
system with intelligent methods like neural or fuzzy systems
to enhance system performance.
ACKNOWLEDGMENT
For help in finishing this study, the authors are grateful to
the University of Technology in Baghdad-Iraq.
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