TUNING OF PID CONTROLLER USING ZEIGLER NICHOLS
AND PARTICLE SWARM OPTIMIZATION IN AVR SYSTEM
Smriti Yadav
Vijay Bhuria
Electrical Engineering Department
Madhav Institute of Technology & Science
Gwalior, India
nandniy23@gmail.com
Electrical Engineering Department
Madhav Institute of Technology & Science
Gwalior, India
Vijay.bhuria@rediffmail.com
Abstract-A proportional–integral–derivative (PID) controller is a
parameters which is first found by Ziegler Nichols tuning. In
general, it is often hard to determine optimal PID parameters
with the Ziegler-Nichols formula in many industrial plants.
Several new intelligent optimization techniques have been arisen
in the past two decades: Genetic Algorithm (GA), Particle
Swarm Optimization (PSO), Ant Colony Optimization (ACO),
Simulated Annealing (SA) and bacterial Foraging (BF). PSO
[16-17] is a population based stochastic optimization algorithm.
Swarm intelligence based optimization [18] is also applied to
find optimized values. There are many literatures to find the
optimum tuning of PID controllers [19-20]. PSO is
computationally efficient compared to other optimization
techniques. In the PID controller design, the PSO algorithm is
applied to search a best PID control parameters. In this paper,
PSO based approach to optimal designing of PID controller to
AVR is presented. Particle Swarm Optimization (PSO) was
inspired from the social behavior of bird flocking. Particle
Swarm Optimization (PSO) Algorithms are effective and
intelligent choice at finding the best solution among the space of
all feasible solutions. PSO algorithms were used to evaluate the
optimum PID controller gain values where performance index
Integral square error (ISE) was used as the objective functions.
generic feedback controller widely used in industrial control
systems, process control, motor drive, and instrumentation. Despite
the popularity, the tuning aspect of PID coefficients is a challenge for
researchers and plant operators. In this paper, a Ziegler Nichols
tuning Proportional-Integral-Derivative (PID) controller is designed
for an Automatic Voltage Regulator (AVR) system, so that faster
settling to rated voltage is ensured and the instability is avoided.
AVR is a closed loop control system compensated with a PID
controller. The PID control method is most flexible and simple
method. This Paper presents a tuning method based on evolutionary
computing approach to determine the Proportional Integral
Derivative (PID) controller parameters in Automatic Voltage
Regulator (AVR) system. The main objective is to increase the step
response characteristics and reduce the transient response of AVR
systems. This paper described in details how to employ Particle
Swarm Optimization Technique (PSO) method to determine the
optimal PID controller parameters of an AVR system. The proposed
Algorithm can improve the dynamic performance of AVR system.
The proposed algorithm can improve the dynamic performance of
AVR system. Compared with Ziegler Nichols (Z-N) tuning method,
the proposed PSO method has better control system performance.
Keywords- AVR system, Proportional-Integral Derivative Controller,
Ziegler Nichols method Particle swarm optimization(PSO)
I.
INTRODUCTION
The main function of an AVR system is to hold the magnitude of
terminal voltage of a synchronous generator at a specified level.
Thus, the stability of the AVR system would seriously affected
the security of the power system. The Proportional integral
Derivative (PID) [1] controller is chosen compared to other
controllers because of its uncomplicated and robust behavior. A
simple AVR consist of amplifier, exciter, generator and sensor.
The block diagram of AVR with PID controller is shown in
Figure (2) [5].The step response of this system without control
has oscillation which will reduce the performance of the
regulation. Thus, a control technique must be applied to the
AVR system. For this reason, the PID block is connected in
series with amplifier. Several tuning methods have been
proposed for the tuning of control loop performance of the
system tuning of PID controllers [4] is imperative. There are
many literatures to find the optimum tuning of PID controllers
[12].The PID controller is a feedback mechanism widely used in
a variety of applications. Industrial plants [13]. To maintain the
stability and performance of the system tuning of PID controllers
[4] is Imperative. Due to, large range of tuning techniques, the
optimum performance cannot be achieved. Ziegler Nichols
Method (ZN) is one of the best conventional methods of tuning
available now .Several methods for determining the PID
II.
AUTOMATIC VOLTAGE REGULATOR
The role of an AVR is to hold the terminal voltage magnitude of
a synchronous generator at a specified level. A simple AVR
system comprises four main components, namely amplifier,
exciter, generator, and sensor. For mathematical modeling and
transfer function of the four components, these components must
be linear zed, which takes into account the major time constant
and ignores the saturation or other nonlinearities. The reasonable
transfer function of these components may be represented,
respectively. The generator excitation system maintains
generator voltage and controls the reactive power flow using an
automatic voltage regulator (AVR). The role of an AVR is to
hold the terminal voltage magnitude of a synchronous generator
at a specified level. Hence, the stability of the AVR system
would seriously affect the security of the power system. The
problem of dynamic stability of power system has challenged
power system engineers recently. In a synchronous generator,
the electromechanical coupling between the rotor and the rest of
the system causes it to behave in a manner similar to a spring
mass damper system, which exhibits an oscillatory behavior
around the equilibrium state, following any disturbance, such as
sudden change in loads, change in transmission line parameters,
fluctuations in the output of turbine and others. Synchronous
generator excitation control is one of the most important
measures to enhance power system stability and to guarantee the
quality of electrical power it pro vides. Essentially, an AVR is to
hold the terminal voltage magnitude, V t(s), of a synchronous
generator at a specified level.
In the linear zed model, the transfer function relating the
generator terminal voltage to its field voltage can be represented
by a gain Kg and a time constant Tg. The generator transfer
function as Kg / (Tg S +1), where Kg depends on load (0.7–1.0)
and 1.0 s ≤ Tg ≤ 2.0 s. The same model has been taken in this
work. A simplified AVR system comprises four main
components, namely amplifier, exciter, generator, and sensor.
Block diagram is shown in the figure (1) and the transfer
function and their of the component of the AVR is shown in the
table (1).
Table 1: Transfer Function and Parameter Limits of AVR System
Parameters
Transfer
parameter
Parameter
function
limits
Limits value
PID
controller
Amplifier
Exciter
kp 
0.2  k p ,
ki
 kd s k
d,
s
ki  2.0
ka
1  sTa
ke
1  sTe
10  k a  40
0.02  Ta  0.1
1  k e  10
0.4  Te  1.0
Optimum
value
k p , k d , ki
Generator
ka  10
Ta  0.1
ke  1
Te  0.5
Kg depends on
Generator
kg
1  sTg
load
1.0  Tg  2.0
kg  1
Tg  1
Figure (1): Block Diagram of AVR System
sensor
ks
1  sTs
ks  1
0.001  Ts  0.06
Ts  0.02
III. AVR WITH PID CONTROLLER
The PID controller is used to improve the dynamic response as
well as to reduce or eliminate the steady-state error. The
derivative controller adds a finite zero to the open-loop plant
transfer function and improves the transient response. The
integral controller adds a pole at the origin, thus increasing
system type by one and reducing the steady-state error due to a
step function to zero. The AVR system with PID controller is
shown in a figure (2).
Figure (2): A Block Diagram of AVR System using PID controller
Figure (3): Block Diagram of PID Controller
After mathematical modeling of AVR system they obtained
transfer function is given
0.2S + 10
0.001S4 + 0.063S3 + 0.682S2 + 1.62S + 11
(1)
The step response curve of the given transfer function without
controller is shown in figure 6.
IV.
PID CONTROLLER
The PID controller is used to improve the dynamic response as
well as to reduce or eliminate the steady-state error. The
derivative controller adds a finite zero to the open-loop plant
transfer function and improves the transient response. The
integral controller adds a pole at the origin, thus increasing
system type by one and reducing the steady-state error due to a
step function to zero.
The transfer function of PID controller is described by the
following equation in the continuous s-domain (Laplace
operator)
(s) =
U(s)
Ki
= Kp + + Kds
E(s)
s
(2)
If we now rearrange that a little we come up with a more
conventional transfer function form:
GC (s) =
V.
PARTICLE SWARM OPTIMIZATION PID
Particle swarm optimization (PSO) algorithm is a population
based evolutionary computation technique developed by the
inspiration of the social behaviour in bird flocking or fish
schooling.
K p (Ti Td s2 + Ti s + 1)
Ti s
(3)
Where Kp is the proportional gain, Ti is the integral time
constant and Td is the derivative time constant. Such a
controller has three different adjustments (Kp, Ti, Td) which
interact with each other. For this reason, it can be very difficult
and time consuming to tune these three values in order to get the
best performance according to the design specifications of the
system.
Figure (4): PSO based PID Controller
It attempts to mimic the natural process of group communication
of individual knowledge, to achieve some optimum property. In
this method, a population of swarm is initialized with random
positions Si and velocities Vi. At the beginning, each particle of
the population is scattered randomly throughout the entire search
space and with the guidance of the performance criterion, the
flying particles dynamically adjust their velocities according to
their own flying experience and their companions flying
experience. In PSO, each single solution is a “bird” in the search
space; this is referred to as a “particle”. The swarm is modelled
as particles in a multidimensional space, which have positions
and velocities. These particles have two essential capabilities:
their memory of their own best position and knowledge of the
global best [14]. Each particle remembers its best position
obtained so far, which is denoted as pbest (Pit ). It also receives
the globally best position achieved by any particle in the
population, which is denoted as gbest (Git ).
The updated velocity of each particle can be calculated using the
present velocity and the distances from pbest and gbest as given
by the following equations:
Vit+1 = W t ∙ Vit + C1 ∙ R1 ∙ (Pit − Sit ) + C2 ∙ R 2 ∙ (Git − Sit )
(4)
Sit+1 = Sit + Vit+1
(5)
W t = (Wmax − Iter) × [
(Wmax −Wmin )
Itermax
]
(6)
Figure.6 Shows the optimized values of Kp, Ki and Kd for
different iterations
The updated velocity and the position are given in (4) and (5),
respectively. Equation (6) shows the inertia weight.
B. PSO-Based PID Controller Optimization
1)
PSO Tuning Parameters
The values in the Table III describe the PSO settings used for
this work
TABLE 2: PSO Tuning Parameters
PARAMETERS
Lower bound [Kp Ki Kd]
Upper bound [Kp Ki Kd]
Max Iterations
Population Size
Inertial weight [Wmax, Wmin]
VALUES
[0 0 0]
[150 150 150]
10
90
[0.8,0.4]
Acceleration coefficients [c1, c2]
[2 2]
2)
Steps in PSO-Based PID Controller Optimization
Step 1 Assign values for the PSO parameters
Initialize: swarm (N) and step size; learning rate (C1, C2) ;
inertia Weight (W);
Figure (5): Convergence of Kp , Ki and Kd for PSO tuned PID using
ISE
Step 2
Initialize Swarm Velocities and Position
Step 3
Evaluate the objective function of every particle and
record each particle’s Pit andGit .Evaluate the desired
optimization fitness function in D dimension variables.
Pit
Step 4
Compare the fitness of particle with its
replace the local best value as given below.
for i=1: N
If current fitness (i) < local best fitness (i);
Then local best fitness = current fitness;
local best position = current position (i);
end
and
Step 5
Change the current velocity and position of the
particle group according to (4) and (5).
Step 6
Steps 2–5 are repeated until the predefined value of
the function or the number of iterations has been
reached. Record the optimized Kp, Ki and Kd
values
Step 7
Perform closed-loop test with the optimized
values of controller parameters and calculate
the time domain specification for the system.
If the values are within the allowable limit, consider the
current Kp, Ki and Kd values. Otherwise perform the
retuning operation for Ki, by replacing the optimized
numerical values for Kp and Kd.
VI.
SIMULATION WORK
To improve the performance of the AVR system under transient
and steady state condition, a PID controller is inserted into the
forward path as shown in Figure 2. The parameters of the PID
controller are now adjusted by using evolutionary technique i.e.
Particle Swarm Optimization method and by Ziegler Nichols
method and the response obtained for the AVR system is
evaluated. The controller gains obtained from the methods are
listed in Table 3.
Figure (6): Close loop Step Response of AVR System without
Controller
TABLE 3: Comparison of Steady State Responses of AVR
System
Step Response With PID Controller
1.6
1.4
Without PID
ZN-PID
PSO_PID
Kp
0
4.19
0.8379
Ki
0
7.07
123.7462
Kd
0
0.62
80.6982
Rise Time(sec)
0.2837
0.1080
0.1449
Settling
Tim(sec)
9.2395
2.7525
0.5226
% Overshoot
68.5448
53.9320
10.7519
Peak Time
(sec)
0.8233
0.4636
0.3262
Parameter
1.2
Amplitude
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time
Figure (7): Close loop Step Response of AVR System with ZN tuned
PID controller
VII.
Figure (8): Close loop Step Response of AVR System with PSO
tuned PID controller
Figure 8 shows the corresponding step responses of without PID
controller and PSO-based PID systems. It can be clearly seen
that PSO tuned system shows improved response with respect to
the overshoot as compared to that of ZN tuned PID controller
system. Rise time and setting time are obtained by PSO tuned
PID Controller are a bit on higher side but are in acceptable
limit. The comparative output responses of the system tune using
PSO-based PID controller and ZN tuned PID controller is
shown in Table (3). The PSO tuned system shows greatly
reduced overshoot.
CONCLUSION
There are several methods such as Z-N,and PSO algorithms for
designing the parameters of the PID controller. The aim of this
paper is to find the optimum parameters of PID controller using
the PSO algorithm. The PID parameters searched by this method
results in better optimal parameters and accuracy compared to
other methods. The proposed algorithm is compared with Z-N
tuning method, the simulation results of AVR system validates
that the PSO algorithm is more superior compared to other
optimization techniques and has better performance also requires
less time to be performed. It can be observed from the extracted
result that PSO tuned PID Controller gives more improved
response with respect to the rise time , settling time , overshoot
and peak time as compared to the ZN tuned PID controller
system.
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