Page 199 - 2024-Vol20-Issue2

195 |
TABLE I.
SOME BENCHMARK FUNCTIONS
#FUN Name Range Function
FUN1 De Jong’s sphere function [–5.12, 5.12] Unimodal
FUN2 Hyper-ellipsoid function [–5.12, 5.12] Unimodal
FUN3 Unimodal
FUN4 Sum of different power [–1, 1] Multimodal
FUN5 Ackley’s function [–32.768, 32.768] Multimodal
FUN6 Multimodal
Griewangk’s function [–600, 600]
Rastrigin’s function [–5.12, 5.12]
between CDOA and the above algorithms, the acceptable error
percentage should be determined. Equation (11) shows the
error percentage adopted in this work. The best solution is
accepted if its difference with the optimal solution is less than
or equal one percent.
|Bestcost - optimalcost|= 0.01 (11)
Due to the randomness of the initial values and the selection
of conditions, single run for the program is not a wise solution
to assess the performance of each algorithm. For this reason,
each algorithm is executed thirty times in this work. The
success rate of each algorithm is given in (12), where the
success rate (SR) is the percentage of the number of successful
runs to the total number of runs.
SR = number o f success f ul run × 100% (12)
number o f run
In Tables (II, III, IV, and V), the comparisons between the
algorithms that have already been mentioned above are rep-
resented for dimensions (5, 10, 15, and 20) respectively.
Through these tables, one can notice that the comparison
was made on the basis of the statistical results represented by
mean, standard deviation, best cost median, and success rate.
These statistical calculations were done through the program
(IBM SPSS Statistics 26) program version 20. The tests were
carried out by using a laptop with Core i7, 2.4 GHz processor,
Windows 10 Pro, a 512 Gb SSD and 16 Gb RAM.
After analyzing the statistical results, it is noted that the pro- where e(t) is the error signal that is corresponding to the
posed CDOA provides an excellent performance compared
to the other algorithms for the unimodal functions in all di-
mensions. For the multimodal function, the results were also
acceptable and can be relied on for the dimensions less than or
equal to 20. For dimensions larger than 20, the success rate is
low in some functions. This is a real interpretation of the No
Free Lunch theorem (NFL) [26] which stipulates that there
is no algorithm that can solve all equations and engineering
applications. Moreover, the algorithm can find the solution
for dimensions larger than 20 to a certain type of the equation
and fail in other types. It is worth mentioning here that the
proposed algorithm does not match the dimensions of more
than twenty variables of the multimodal functions, otherwise
Al-mtory, Alnahwi & Ali
CDOA is very efficient and reliable. Among the properties
of the algorithms that should be mentioned are the speed of
response, speed of implementation, and speed of convergence.
For convergence assessment purposes, Fig. 3. demonstrates
the convergence of the proposed algorithm toward the opti-
mum solution for some of the benchmark functions given
previously. A few iterations are required for CDOA to con-
verge to the optimal solution, and this is considered as one
of the main advantages of this algorithm. V lists the time
required to execute each algorithm. It is clear that CDOA
speed of execution is comparable with the fast modified CA
algorithm. Another important feature for the proposed algo-
rithm can be noticed in this work. When looking at CDOA
and comparing it with its counterparts, it can be seen that
there is no parameter in this algorithm that need to be set to
increase its convergence speed. In contrast, PSO requires four
setting parameters namely individual learning factor, maxi-
mum velocity, inertia weight, and social learning factor. CSA
requires setting adjustable parameters called flight, length,
and awareness probability. Modified CA parameters that need
to be set are the camel endurance and the camel visibility.
V. ENGINEERING APPLICATIONS
A. PID Controller
The term PID is an abbreviation of the words proportional,
integral, and derivative. Consequently, the controller has three
coefficients to be optimized one for the proportional gain, the
other for the integration gain, and the third is for the derivative
gain. Each coefficient has a certain effect on the system;
for example, it is possible to improve the steady state error
by controlling the proportional coefficient, or to eliminate
the steady state error through the integration coefficient. In
addition, the overshoot of the system is controlled through
the derivative coefficient [40]. The structure of PID controller
system is shown in, Fig. 4. The three coefficients of the PID
controller (KP, Ki, andKd) given in (13) can be optimized to
improve the performance of the entire system.
td
control signal = KPe(t) + Ki 0 e(t)dt + Kd dt e(t) (13)
difference between the input r(t) and the output of the system
y(t). The main function of PID controller is to keep the value
of the error signal as low as possible. As mentioned earlier,
the error signal can be reduce by optimizing the controller
coefficients KP, Ki, andKd . The transfer function L(s) of the
controller is given by [41]:
L(s) = KdS2 + KpS + Ki (14)
S