SPBO is presented. The problem formulation is discussed in Section 1.3. Section 1.4 presents the optimum DG placement in the distribution networks using the proposed mixed-discrete SPBO algorithm. Finally, the conclusions are drawn in Section 1.5.
1.2 Mixed Discrete SPBO
1.2.1 SPBO Algorithm
SPBO algorithm has been proposed by Das et al. [66]. Similar to other meta-heuristic methodology, SPBO also uses a set of populations. The population considered in SPBO is analogous to a group of students present in the class. In general, the performance of a student is analyzed based on the marks/grade obtained by the student in the examination. The student with the maximum marks/grade is considered as the best one in the class and the student is awarded accordingly. It is also very clear that in the test the students will aim to achieve the highest marks/grade for which they need to provide more effort in their study. The overall grade is the final score of the students in the examination. The overall grade of a student depends on the cumulative effort given by the student in each subject offered to them. The students enhance their performance in the examination by paying more attention and effort in their studies. SPBO is based on the psychology of the students who are trying to improve their performance in the examination as well as trying to get the highest marks/grade in the examination.
For the improvement of the overall performance, the students need to enhance their performance in each subject which are offered to them. However, the improvement of the students’ performance is not the same for all the students. Improvement of the students’ performance depends upon a few factors like the capability and efficiency of the student as well as how the student gets interested in the subject. If a student is interested in a subject, he/she will be more involved in that subject. As a result, the performance of the student will be improved in that subject. As the overall grades/marks depend on each subject, improvement in any subject will improve the student’s overall performance. To improve performance, some students use to give similar or better kind of effort as given by the best student while some use to give more effort than that of the average student. If a student is less interested in a subject, then he/she will try to give the subject an average effort to improve his/her overall performance in the exam. The effort of a student can’t be measured directly. Marks obtained by the student in a subject are the outcome of her/his effort given to that subject. So, it may be considered, in general, that the effort given by the student is equivalent to the marks/grade obtained. Considering all these facts and the psychology of the students, the students can be divided into four different categories (such as the best student, good student, average student, and the students trying to improve randomly) [66].
(a) Best student: The best student is to be considered as the student who scored the overall highest marks/grade in the examination. It is an obvious fact that the best student will always try to secure his/her position in the class. For securing his/her position, the best student should have to provide more effort towards each subject than the same given by the rest of the students in the class. Therefore, it can be concluded that the best student will always provide more effort than the same given by other students in the class. Mathematically, the student of this category can be visualized with the help of (Equation 1.1) [66]
where, Xbest indicates the marks/grades obtained by the best student, Xj indicates the marks of the student, randomly selected, in the same subject, and rand generates a random number between 0 and 1. In (Equation 1.1), the D parameter randomly takes the value either 1 or 2. Marks obtained in the subjects by the best student may be increased or maybe decreased depending upon the student and the subjects. However, the primary objective of the best student in the class is to secure his/her position by continuously scoring the highest overall marks in the examination.
(b) Good student: If a student is interested in a subject, then he or she may strive to make more effort to enhance his or her performance in that subject. As a result, their overall performance in the examination will be improved. The students of this kind can be classified as subject wise good student. The effort given by all the students in this category may not be the same because the psychologies of the students are different. Some students try to give a similar or even better effort than the best student, where as; some students try to improve their performance by giving more effort after considering the effort given by the best student as well as the average effort given by the students of the class. This group of students may be further divided into two categories. The first category of this group of students is those who try to give similar or even better effort as given by the best student. This category of students gives effort to the subjects in which they are interested. So, their performance in that subject, as well as their overall performance, gets improved. Improvement of these kinds of students in a subject can be explained using (Equation 1. 2a) [66]. The other category of good students is those students who give effort considering the effort given by the best student as well as try to give more effort than the average effort of the class so that their performance in that subject, as well as overall performance, gets improved. Mathematically, the student of this category can be represented as (Equation 1. 2b) [66].
Here, Xnewi is the improved performance of the ith student in that subject; Xbest and Xi are the marks/grade obtained in that subject by the best student and the ith student, respectively; mean represents the average marks obtained in the class in that particular subject and rand variable produces a random number between 0 and 1.
(c) Average student: In order to increase their overall success in the test, students with less interest in a subject aim to offer an average effort. The students of this category can be named as subject-wise average student. While giving average effort in a particular subject, students will try to improve their overall performance by paying more effort to the other subjects which are offered to them. The performance of this category of students may be explained with the help of (Equation 1. 3) [66]
where, Xi indicates the marks/grade of the ith student of the class in a subject, mean indicates the average marks obtained in that particular subject, and rand generates a random number between 0 and 1.
(d) Students who try to improve randomly: Except the three categories of students discussed above, there are certain students who strive to enhance their results without recognising the effort offered by rest of the students in the class. The students of these kinds always try to improve their performance randomly to some extent up to their limitations, depending upon
