Engineering, Technology & Applied Science Research Vol. 12, No.

6, 2022, 9737-9741 9737

Modeling and Analysis of Time Response Parameters

of a PMSM-Based Electric Vehicle with PI and PID
Controllers

M. Yerri Veeresh V. Naga Bhaskar Reddy

Department of EEE, Jawaharlal Nehru Technological Department of EEE
University Anantapur, Anantapuramu, India and Santhiram Rajeev Gandhi Memorial College of Engineering and
Engineering College, Nandyal, India Technology, Nandyal, India
saiveerushamu8@gmail.com chaitu.bhaskar@gmail.com

R. Kiranmayi
Department of EE
Jawaharlal Nehru Technological University Anantapur
Anantapuramu, India
kiranmayi0109@gmail.com

Received: 9 September 2022 | Revised: 26 September 2022 and 4 October 2022 | Accepted: 9 October 2022

Abstract-This paper presents the mathematical modeling of a tuning of the PI controller. The Ziegler-Nichols method is the
vector-controlled Permanent Magnet Synchronous Motor most popular, which dispenses with the need for a system
(PMSM) drive with either a Proportional Integral (PI) controller design and control parameters [5]. The PI and PID controllers
or a Proportional Integral Derivative (PID) controller as a are simple and effective, consequently, they are often used for
propulsion system for an Electric Vehicle (EV). Most commercial the control of the PMSM systems [6]. Controlling of a Battery
drives use a standard PI controller as a speed regulator. The EV (BEV) is not a simple task as the EV is essentially time-
vector control system model consists of the PMSM, a PWM variant. The EV’s primary limiting factor is the short running
inverter, the speed controller, and vehicle dynamics for speed distance (range) per battery charge. Another limiting factor is
control. The performance analysis of the drive is evaluated under
its acceleration time to reach the maximum speed limit [7]. The
transient conditions for settling time, rise time, steady state error
EV dynamics must be suitably selected to achieve better
of speed, and the vehicle’s acceleration at the wheel axle for
specifically designated values validated by MATLAB/Simulink. performance.
II. VEHICLE DYNAMICS
Keywords-Permanent Magnet Synchronous Motor (PMSM);
electric vehicle dynamics; Proportional Integral (PI) controller; The performance of vehicle modeling is initially obtained
Proportional Integral Derivative (PID) controller by the tractive effort equation. The force that drives the vehicle
forward and is transmitted to the ground through the drive
I. INTRODUCTION wheels is known as the tractive effort. This force has to
In recent years, several situations related to the environment overcome the rolling resistance force, aerodynamic drag force,
have led to reduced carbon emissions from vehicles. Electric and the elements of the vehicle’s weight (including payload)
Vehicles (EVs) can be alternatives to traditional SI or CI acting down to the slope. It also accomplishes the force
engine automotives [1-2]. PMSMs have been used in various required to accelerate the vehicle for the linear and angular
industrial applications like CNC machines, industrial robotics, motion of drive wheels [7-8]. The tractive force and the
air-conditioners, washing machines, wind power generation corresponding force equations depend upon the vehicle
systems, EVs, etc. The PMSMs are more suitable in EVs due to dynamics, shape, size, and structure parameters, such as rolling
their high efficiency, high power/torque density, smaller size, resistance coefficient, air density, drag coefficient, gear ratio
high torque/weight ratio, and maintenance-free operation [3]. and gear efficiency, wheel radius, etc., are illustrated in Figure
Better dynamic responsiveness and fewer torque ripples are 1 of [9]. The respective equations for Rolling Resistance Force

Acceleration Force F , and Angular Acceleration Force

provided by a vector-controlled PMSM drive, which needs a , Aerodynamic Drag Force , Hill Climbing Force ,
constant switching frequency. The outer loop speed control
significantly impacts system performance [4]. are respectively given by:
The PI controller performs well under steady-state = µ ∗ ∗ (1)
conditions. A lot of approaches have been proposed for the
Corresponding author: M. Yerri Veeresh
www.etasr.com Yerri Veeresh et al.: Modeling and Analysis of Time Response Parameters of a PMSM-Based Electric …
 Engineering, Technology & Applied Science Research Vol. 12, No. 6, 2022, 9737-9741 9738

= ∗ ∗ ∗ ∗ GA= > @ + − ( F?A

CHE
(2) A "
(12)
= ∗ ∗ sin (3) The stator flux linkages are given by:
= ∗ (4) F?A = I? @?A (13)

= (5) F A = I @ A + I0 @J (14)
!

The total tractive force necessary to drive the vehicle is the TABLE II. PMSM SPECIFICATIONS

sum of all forces:

= + + + +
Parameter Value

"#
Stator phase resistance, Rs (Rd = Rq = Rs) 0.05Ω
(6) d-axis inductance, Ld 0.0007552H

Flux linkage, F J
The tractive torque corresponding to the tractive force with q-axis inductance, Lq 0.0008348H
wheel radius r, is given by: 0.192 Wb-turn

%& = ∗'
Inertia, J 0.011Kgm2
"# (7) Viscous damping, B 0.001417Nms
Pole pairs, P 4
The speed and torque at the axle of the wheel are given by: Static friction, Tf 0Nm

( =
*+
)# (8) The stator voltage equations in terms of electrical
% )# = ,- %#
parameters are obtained by substituting the flux linkages in the
(9)
stator voltage equations as:
N
G?A >? + D ( I @?A ( I0 @J
The vehicle equations are used to understand the vehicle
K L= M "
OK L + K L (15)
dynamics for different force inputs [10]. The mathematical
model of the speed-toque equation of the vehicle at the axle of GA −( I? > +
NH @ A 0
the wheels is given by: "

( )# = . 2
6% )# − %& 8 d: The electromagnetic torque is given by:
/0 1 45
(10)
%# = SF A @?A − F?A @ A T (16)
32 QR

The specifications of the vehicle chassis chosen for the

simulation are given in Table I. The torque equation in terms of inductances and currents
upon the substitution of flux linkages is given by [14]:
%# = VF J @?A + SI − I? T@?A @ A W (17)
TABLE I. VEHICLE CHASSIS SPECIFICATIONS QR
U

where F J is the rotor flux linkage that links the stator and is
Parameter Quantity
Net mass, m 1200Kg
Area of frontal surface, A 2.2m2 given by:
Rolling coefficient, µ F = I0 @J
Gear ratio, G 2
J
Gear efficiency, -
0.01 (18)
The angular speed (# and the instantaneous angle of the
0.9

rotor X of the machine are given by:

Wheel radius, r 0.2m
Drag coefficient, Cd 0.26

(# = Y [ 6%# − %0 8 (19)
R
Slope angle, 0o

4A1Z

X = ._ (# \:] ^: + X \0] (20)

III. MATHEMATICAL MODEL OF THE PERMANENT MAGNET "
SYNCHRONOUS MOTOR
A three-phase salient-pole sinusoidal wave shape back The stator q and d axes currents for a balanced three-phase
EMF PMSM with 87.75Nm, 560Vdc, and 3000rpm preset operation are given by:

cos X cos bθd − f cos bX + f @

e e
model parameters was designed mathematically. The model
@?A
K L= M Q
O g@h i
Q
specifications are given in Table II. The mathematical model of
@ A Q
sin X sin bθd −
e
f sin bX + f @
e
the PMSM is given in (11)-(22). The rotor reference frame is (21)
chosen because the rotor magnet's position will be determined Q Q
independent of the stator phase voltages, instantaneous induced The abc-to-dq transformation equations for stator voltages
phase EMFs, stator phase currents, and torque of the machine are given by:
cos X cos bX − f cos bX + f v
[11]. For synchronous motors, the rotating speed of the rotor
e e

rotating rate of the reference frame ( = ( = ( [12-13]. Y [ = QM O gvk i (22)

?A Q Q
and the space vector of the rotor flux are both equal to the
sin Xd sin bX − f sin bX + f vl
A e e
Q Q
In the rotor reference frame, the stator q and d axes voltage
equations in terms of flux linkages are:

V<= = >? @?A + +( F

BCDE
B" A (11)

www.etasr.com Yerri Veeresh et al.: Modeling and Analysis of Time Response Parameters of a PMSM-Based Electric …
 Engineering, Technology & Applied Science Research Vol. 12, No. 6, 2022, 9737-9741 9739

IV. PROPOSED SYSTEM MODEL WITH PI AND PID

CONTROLLER
For applications in control theory, the Proportional (P), the
PI, and the PID controllers are the most commonly used
controllers. As illustrated in Figure 1, the PI controller will
affect the performance of the system by increasing the system's
order by one and by reducing steady state error, disturbance
signal rejection, and relative stability. The system's sensitivity
with respect to parameters also decreases [15]. The transfer Fig. 2. Model analysis diagram of the speed control of the EV with
function of the PI controller is given by: PMSM drive using a PI controller.

, \m] = no +
pq
r
(23)
The PI controller reduces rise time and minimizes the
steady-state error [16]. Peak overshoot, settling time, order, and
type of the system will be increased [15]. It functions as a low
pass filter. The output equations of the PI and PID controllers
are given by:
s\:] = no t\:] + nu . t\:] ^: (24)

s\:] = no t\:] + nu . t\:] ^: + n t\:] (25)

"
*
A command speed ω is compared with the motor's speed
ωr and provides the change or error in speed Δω, which is
given to the PI or PID controller to obtain the command torque
component of stator current Iq*. A limiter is used ahead of the
controller to avoid decreased stability. Fig. 3. Simulink model of dq to abc reference signals.

Fig. 1. PI controller block diagram.

The complete model block diagram of the PI controller of

the PMSM-based BEV is illustrated in Figure 2. The command
speed has been compared with the actual speed of the PMSM.
An error speed signal is given to the PI controller to tune the
signal with Kp and Ki values and provide the torque-producing
current reference component Iq*, which is essential to control Fig. 4. Simulink model of the PWM inverter.
the error speed by calculating the three-phase reference
currents Ia*, Ib*, Ic*. A Li-Ion battery with 100C capacity was
used as the supply, which is converted into a three-phase
supply voltage using a PWM inverter. The Li-Ion battery is
operated with a current-controlled device with the feedback
current provided by the PMSM. A sinusoidal PWM approach
with the reference currents Ia*, Ib*, Ic* corresponding to
command speed and the actual currents Ia, Ib, Ic produced by the
PMSM are collectively compared with a triangular reference
signal to produce the gate pulses. Such desired gate pulses
drive the three phase two-level inverter and will produce the
three phase voltages for the PMSM. The characteristic
equations of the PMSM building blocks are operated with

position, speed ωw , and torque Tw of the motor. The PMSM

inverter voltages and produce the instantaneous current,

will drive the vehicle's wheels through the transmission and

differentials by considering the vehicle's dynamic losses. Fig. 5. Simulink model of PMSM.

www.etasr.com Yerri Veeresh et al.: Modeling and Analysis of Time Response Parameters of a PMSM-Based Electric …
 Engineering, Technology & Applied Science Research Vol. 12, No. 6, 2022, 9737-9741 9740

Finally, the vehicle is operated smoothly with the required TABLE III. PERFORMANCE RESULTS OF THE PROPOSED SYSTEM
MODEL
speed and torque. The performance results are discussed below.
Axle side -
In MATLAB Simulink version R2017a, the proposed work Motor side - speed
speed
was simulated with the ode 23tb solver in continuous time Rise time Settling time
Time to
mode. The design sub models like dq to abc reference, PWM Controller Applied required to needed to get
Steady reach
inverter, PMSM model, and vehicle dynamic models are shown type and its load reach 90% steady-state
state error 100Km/hr
in Figures 3-7. gain values (Nm) of final value with 2%
(ms)
value (ms) tolerance (ms)
PI 0.0075rad/s
(Kp=0.35 80 50.1 71.0 or 48.1
Ki=16) 0.0716rpm
PID
0.0075rad/s
(Kp=0.35
80 54.8 70.1 or 53.0
Ki=16
0.0716rpm
Kd=0.01)

Fig. 6. Simulink model of PMSM internal characteristics.

Fig. 8. Simulated speed responses with PI and PID controllers.

Fig. 9. Simulated torque responses with the PI controller.

Fig. 7. Simulink model of vehicle dynamics (motor to wheels).

V. SIMULATION RESULTS
Simulations have been carried out with the PI and PID
speed controllers for a 3-phase, 550Vdc, 1500rpm, 97.96Nm
PMSM at a load torque of 80Nm and operating speed of
1500RPM or 157.079rad/sec. It can be observed that the rise
time (90% of the command speed) is attained in 50.1ms with Fig. 10. Simulated torque responses with the PID controller.
the PI controller, which is a better response than that of the PID
controller. The motor speed stabilizes at a reference speed The torque available at the wheel axle is 72.5Nm, with net
(with 2% tolerance) of 157.079rad/s in 70.1ms with the PID tractive resistance loss of 7.5Nm due to rolling resistance,
controller, which is slightly better performance than the one of aerodynamic, and hill-climbing torque losses in both PI and
the PI controller and with the steady state error of 0.0075rad/s PID controllers. The initial transient is higher in the PID
or 0.0716rpm in both PI and PID controllers, as shown in controller than in the PI controller, as seen in Figures 9-10. The
Figure 8. The vehicle's acceleration (the speed at the wheels or pulsating ripple torques are noticed due to the residual flux
the axle) was measured, and the time taken to reach the speed variation. The data of the mathematical model of the
of 100Km/h is 48.1ms with the PI controller, which is superior performance of the PMSM-based electric vehicle with PI and
to the performance of the PID controller, as shown in Figures PID controller through MATLAB simulations are given in
11-12. In the steady state, for a command torque of 80Nm, the Table III.
torque developed by the PMSM is 80Nm.

www.etasr.com Yerri Veeresh et al.: Modeling and Analysis of Time Response Parameters of a PMSM-Based Electric …
 Engineering, Technology & Applied Science Research Vol. 12, No. 6, 2022, 9737-9741 9741
[8] J. Larminie and J. Lowry, Electric Vehicle Technology Explained, 2nd
ed. Wiley, 2012.
[9] M. Y. Veeresh, V. N. B. Reddy, and R. Kiranmayi, "Range Estimation
of Battery Electric Vehicle by Mathematical Modelling of Battery’s
Depth-of-Discharge," International Journal of Engineering and
Advanced Technology, vol. 8, no. 6, pp. 3987–3992, Aug. 2019,
https://www.doi.org/10.35940/ijeat.F8800.088619.
[10] M. Yildirim, M. C. Catalbas, A. Gulten, and H. Kurum, "Computation of
the Speed of Four In-Wheel Motors of an Electric Vehicle Using a
Radial Basis Neural Network," Engineering, Technology & Applied
Science Research, vol. 6, no. 6, pp. 1288–1293, Dec. 2016,
Fig. 11. Simulated vehicle acceleration pointing at the top speed of https://doi.org/10.48084/etasr.889.
100Km/hr with the PI controller. [11] P. T. Giang, V. T. Ha, and V. H. Phuong, "Drive Control of a Permanent
Magnet Synchronous Motor Fed by a Multi-level Inverter for Electric
Vehicle Application," Engineering, Technology & Applied Science
Research, vol. 12, no. 3, pp. 8658–8666, Jun. 2022, https://doi.org/
10.48084/etasr.4935.
[12] P. Pillay and R. Krishnan, "Modeling of permanent magnet motor
drives," IEEE Transactions on Industrial Electronics, vol. 35, no. 4, pp.
537–541, Aug. 1988, https://doi.org/10.1109/41.9176.
[13] C. Bowen, Z. Jihua, and R. Zhang, "Modeling and simulation of
permanent magnet synchronous motor drives," in ICEMS’2001.
Proceedings of the Fifth International Conference on Electrical
Machines and Systems, Shenyang, China, Dec. 2001, vol. 2, pp. 905–
908, https://doi.org/10.1109/ICEMS.2001.971825.
Fig. 12. Simulated vehicle acceleration pointing at top the speed of [14] A. Mansouri and T. Hafedh, "Torque Ripple Minimization and
100Km/hr with the PID controller. Performance Investigation of an In-Wheel Permanent Magnet Motor,"
Engineering, Technology & Applied Science Research, vol. 6, no. 3, pp.
987–992, Jun. 2016, https://doi.org/10.48084/etasr.644.
VI. CONCLUSION
[15] J. C. Basilio and S. R. Matos, "Design of PI and PID controllers with
In this paper, the mathematical model of a vector controlled transient performance specification," IEEE Transactions on Education,
PMSM drive with PI and PID controllers as a propulsion vol. 45, no. 4, pp. 364–370, Aug. 2002, https://doi.org/10.1109/
system for an electric vehicle is developed, and its simulation TE.2002.804399.
results are presented. The results demonstrate that the PI [16] S. Mikkili and A. K. Panda, "SHAF for mitigation of Current harmonics
with p-q and Id-Iq control strategies using both PI and Fuzzy
controller achieves more robust tracking response of the Controllers," in International Conference on Sustainable Energy and
command speed with less steady-state error than the PID Intelligent Systems (SEISCON 2011), Chennai, India, Jul. 2011, pp. 358–
controller. Good execution time and time are noted in the 362, https://doi.org/10.1049/cp.2011.0389.
performance analysis. The vehicle acceleration at the wheel
axle reached the desired value within a minimal time. The AUTHORS PROFILE
overall output responses of the model show that the vehicle can
M. Yerri Veeresh was born in Kurnool, India. He received the B.Tech
run smoothly with good static and dynamic performance (Electrical and Electronics Engineering) degree, M.Tech (Power Electronics
characteristics with the PI controller. and Electric Drives) from Jawaharlal Nehru Technological University
Anantapur, India in 2009 and 2013 respectively. Presently, he is pursing his
REFERENCES Ph.D. in the Department of Electrical Engineering, JNTU Anantapur and is
[1] C. C. Chan and K. T. Chau, "An overview of power electronics in also working as an Assistant Professor in the Santhiram Engineering College,
electric vehicles," IEEE Transactions on Industrial Electronics, vol. 44, Nandyal. His fields of interest include Electric Vehicles, Power Electronics
no. 1, pp. 3–13, Oct. 1997, https://doi.org/10.1109/41.557493. and Drives, and Energy Sources.
[2] Global Electric Vehicle Outlook 2022. France: International Energy
Agency, 2022. V. Naga Bhaskar Reddy was born in Kurnool, India. He received the B.Tech
(Electrical and Electronic Engineering) degree from the Bangalore University,
[3] S. Morimoto, "Trend of permanent magnet synchronous machines," Bangalore in 2000, M.Tech (Power Electronics and Drives) from the Bharath
IEEJ Transactions on Electrical and Electronic Engineering, vol. 2, no. Institute of Higher Education Research (BIHER), Chennai, India in 2005. He
2, pp. 101–108, 2007, https://doi.org/10.1002/tee.20116. attained his Doctoral degree from the Jawaharlal Nehru Technological
[4] R. N. Hajare and A. G. Thosar, "Modeling and Simulation of Permanent University, Kakinada in 2012. He is currently a professor and the Head of the
Magnet Synchronous Motor using MATLAB," vol. 7, no. 3, pp. 413– Department of Electrical and Electronic Engineering, R.G.M College of
423, 2014. Engineering and Technology, Nandyal. His areas of interest are Power
[5] V. V. Patel, "Ziegler-Nichols Tuning Method," Resonance, vol. 25, no. Electronics, Microcontrollers, and Power Electronic converters.
10, pp. 1385–1397, Oct. 2020, https://doi.org/10.1007/s12045-020-1058-
z. R. Kiranmayi received the B.Tech (Electrical and Electronic Engineering)
[6] E.-C. Shin, T.-S. Park, W.-H. Oh, and J.-Y. Yoo, "A design method of degree, M.Tech., and Doctoral degree from Jawaharlal Nehru Technological
PI controller for an induction motor with parameter variation," in University Anantapur, India in 1993, 1995, and 2013 respectively. She is
IECON’03. 29th Annual Conference of the IEEE Industrial Electronics presently working as a professor the Head of the Department of Electrical
Society, Roanoke, VA, USA, Aug. 2003, vol. 1, pp. 408–413, Engineering at JNTUA, Anantapuramu. She has published more than 40
https://doi.org/10.1109/IECON.2003.1280015. research papers in international and national conferences and journals. Her
[7] "Electric and Hybrid Vehicles, Design Fundamentals [Book Review]," areas of interest include Electrical Power Systems and Photo Voltaic Systems.
IEEE Circuits and Devices Magazine, vol. 21, no. 5, pp. 26–27, Sep.
2005, https://doi.org/10.1109/MCD.2005.1517392.

www.etasr.com Yerri Veeresh et al.: Modeling and Analysis of Time Response Parameters of a PMSM-Based Electric …