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Hadaeghi & Abdollahi | 14
following equations [15]: C89:,) = C<=>,) + D8C)092? - E@F): (6)
'45 = 6'!+"# × ?')*!+"#89) - 9,:% (2) E@ = DGHIJ[1 + D] (7)
9, = + × ?)-*!+"# 9) (3) Where D is a random number between [0, 1], E@ is teaching
'!"# factor and its value can be either 1 or 2 and it is obtained
randomly by (7), C)092? is the best member of the population at
Where VDI is the voltage deviation index, v. is the voltage of iteration > that it is considered as a teacher, F) is the class
ith bus, v/ is the average bus voltage and N012 is the bus number. average at iteration >, C<=>,) is a member that needs training and
C89:,) is a trained member.
C. Bus Voltage Constraint
2) Learner phase: In this phase, each learner randomly
The voltage fluctuations in distribution systems are very
limited and the standards usually allow only minor changes exchanges information with another learner and thus increases
around the nominal value. Therefore, the voltage of the buses
should always be within a permissible range, which is expressed their knowledge. For the ith member, a member is randomly
according to (4): selected from the population (jth member). Then, if K(C)) <
K8CA:, member > is trained according to (8), otherwise it is
9)$)- = 9) = 9)$34 > = 1, 2, … , 2(5# (4) trained according to (9). This step is done for all members of the
population.
Where v. is the voltage of ith bus, 9)$)- and 9)$34 are the X89:,B = X<=>,B + r8XB - XC: (8)
minimum and maximum permissible voltages of ith bus,
respectively and N012 is the number of buses. X89:,B = X<=>,B + r8XA - X): (9)
D. Line Current Constraint Where D is a random number between [0, 1], C<=>,) is a member
that needs training and C89:,) is a trained member. If C89:,) is
To prevent overload of the lines, the current of each branch better than C<=>,), it will replace C<=>,) .
must be kept below or equal to its maximum capacity. This is
expressed by the following relation: B. Black Hole Optimization Algorithm
|5)| = |5)$34| > = 1, 2, … , 26)-7 (5) The Black Hole (BH) Algorithm is a population-based
method that is based on concept of the mechanism of black
Where |5)| is the absolute value of current in ith line, |5)$34| hole phenomenon. The BH algorithm directs the generated
are the maximum permissible current of ith line and 26)-7 is population towards the optimal response [18]. The process of
the number of lines. the black hole algorithm in this paper is as follows:
E. Radial Configuration of the System and Isolation 1) A random initial population (stars) is generated.
Constraints 2) The fitness value of each star is evaluated and the best
candidate in the population, which has the best fitness value,
The most severe constraint in the problem of distribution is selected as the black hole (XBH).
network reconfiguration is that distribution system 3) The new position of each star is determined according to
configuration must be radial and all buses must be contained. In the previous star and the position of the black hole as follows.
this paper, the system configuration is verified using the method
proposed in [16].
III. BASIS OF THE METHODS C)(O + 1) = C)(O) + DPIJ × 8CDE - C)(O): > = 1,2, … , 2
(10)
A. Teaching-Learning-Based Optimization
Where C)(O) and C)(O + 1) are the locations of the >Oh star
The Teaching-Learning-Based Optimization (TLBO) is an at iterations O and O + 1, respectively. CDE is the location of
optimization algorithm that introduced in 2012 by Rao et al. the black hole in the search space. N is the number of stars
[17]. This algorithm is based on concept of teaching and learning (solutions).
process in a classroom. In this algorithm, the population is
considered as students of a class and the best member is selected 4) While moving towards the black hole, a star may reach
as a teacher. The teacher tries to increase their knowledge by a location with lower cost than the black hole. Therefore, their
educating the students. Students also learn through location is exchanged and that star is considered as a black hole
communication with each other and increase their level of in the next round and all the stars move towards it.
knowledge. This algorithm has two phases, which include the
teacher phase and the learner phase. 5) The distance of each star from the black hole is
calculated. If its distance is less than the radius of the event
1) Teacher phase: In this phase, the teacher increases horizon, that star is removed and a star is randomly placed in
learner’s knowledge by teaching them. The relationships of this the search space instead. The radius of the event horizon in the
step are as follows: black hole algorithm is calculated using the following
equation:
