Prior, numerous traditional (deterministic) optimization techniques have been successfully utilized, the most popular were: gradient based methods, Newton based method, simplex method, sequential linear programming, sequential quadratic programming, and interior point methods. Although, some of these deterministic techniques have excellent convergence characteristics and many of them are widely used in the industry however, they experience the ill effects of few shortcomings. Some of their disadvantages are: they cannot promise global optimality i.e. they may converge to local optima, they cannot readily handle binary or integer variables and finally they are developed with some theoretical assumptions, such as convexity, differentiability, and continuity, among other things, which may not be appropriate for the real power system conditions.
Moreover, the fast development of recent computational intelligence tools have motivated significant research in the area of non-deterministic that is, heuristic optimization methods. Some of these techniques are: Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Grenade Explosion Method (GEM), Artificial Bee Colony (ACO), Harmony Search (HS), Bacterial Foraging Algorithms (BFA), Differential Evolution (DE), Evolutionary Algorithms (EAs), Gravitational Search Algorithm (GSA). These methods are known for: their capabilities of finding global solutions and avoid to be trapped with local ones, their ability of fast search of large solution spaces and their ability to account for uncertainty in some parts of the power system.
But these evolutionary optimization techniques performance is dependent on algorithm specific control parameters (such as: PSO uses inertia weight, social and cognitive
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parameters; GA uses crossover rate and mutation rate etc.; GEM uses length of explosion, number of grenades thrown, pieces of shrapnel per grenade etc., ABC requires optimum controlling parameters of number of bees (employed, scout, and onlookers), limits etc., HS requires harmony memory consider rate, pitch adjusting rate etc.). Because of the difficulties in selecting the optimum algorithm specific control parameters these heuristic algorithms give near optimal solution for complex systems. An adjustment in algorithm parameter changes the viability of the algorithm.
Teaching-Learning-Based Optimization is a rising star among metaheuristic techniques with highly competitive performances [87] [88] [89]. This method is based on the impact of an educator on learners. Like other nature-inspired algorithms, TLBO is also a population-based method and uses a population of solutions to proceed to the global solution. The population is considered as a group of learners or a class of learners. The process of TLBO is divided into two sections: the initial segment consists of the ‘Teacher Phase’ and the second part consists of the ‘Learner Phase’. ‘Teacher Phase’ means learning from the teacher and ‘Learner Phase’ means learning by the interaction between learners. The basic philosophy of the TLBO method is explained in detail in next section.
4.3.1 Basic working procedure of TLBO method
Assume two different teachers, 𝑇1 and 𝑇2, teaching a subject with the same content to the same merit level learners in two distinct classes. Fig. 4.3 demonstrates the distribution of marks got by the learners of two different classes evaluated by the teachers. Curves 1 and 2 represent the marks acquired by the learners taught by teacher 𝑇1 and 𝑇2 respectively. It is seen from Fig. 4.3 that curve-2 represents better results than curve-1 and so it can be said that teacher 𝑇2 is superior to teacher 𝑇1 in terms of teaching. The main difference between both the results is their mean (𝑀2 for Curve-2 and 𝑀1 for Curve-1), i.e. a decent teacher delivers a better mean for the results of the learners. Learners additionally learn from interaction between themselves, which also helps in their results.
Consider Fig. 4.4, which demonstrates a model for the marks obtained for learners in a class with curve-A having mean 𝑀𝐴. The teacher is considered as the most proficient individual in the society, so the best learner is mimicked as a teacher, which is shown by 𝑇𝐴 in Fig. 4.4. The teacher tries to disseminate knowledge among learners, which will thus expand the knowledge level of the entire class and help learners to get good marks or grades. So a teacher increases the mean of the class according to his or her capability. In Fig. 4.4, teacher 𝑇𝐴 will try to move mean 𝑀𝐴 towards their own level according to his or her capability, thereby increasing the learners’ level to a new mean 𝑀𝐵. Teacher 𝑇𝐴 will put maximum effort into teaching his or her students, but students will acquire knowledge
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according to the quality of teaching delivered by a teacher and the quality of students present in the class. The quality of the students is judged from the mean value of the population. Teacher 𝑇𝐴 puts effort in so as to increase the quality of the students from 𝑀𝐴 to 𝑀𝐵, at which stage the students require a new teacher, of superior quality than themselves, and i.e. in this case the new teacher is 𝑇𝐵. Hence, there will be a new curve-B with new teacher 𝑇𝐵.
1
M
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2 Obtained Marks P ro b a b il it y D e n si ty Curve-1 Curve-2Fig. 4.3 Distribution of marks obtained by learners taught by two different teachers
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M
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B Obtained Marks P ro ba b il it y D en si ty Curve-A Curve-B BT
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0 10 20 30 40 50 60 70 80 90 100Fig. 4.4 Model for the distribution of marks obtained for a group of learners
In TLBO, different design variables will be analogous to different subjects offered to learners and the learners’ result is analogous to the ‘fitness’, as in other population-based optimization techniques. The teacher is considered as the best solution obtained so far.. The flowchart of TLBO method is shown in Fig. 4.5.
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Fig. 4.5 Flowchart for Teaching Learning Based Optimization
4.3.1.1 Teacher Phase
As shown in Fig. 4.4, the mean of a class increases from 𝑀𝐴 to 𝑀𝐵 depending upon a good teacher. A good teacher is one who brings his or her learners up to his or her level in terms of knowledge. But in practice this is not possible and a teacher can only move the
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mean of a class up to some extent depending on the capability of the class. This follows a random process depending on many factors.
Let 𝑀𝑖 be the mean and 𝑇𝑖 be the teacher at any iteration ‘i’. 𝑇𝑖 will try to move mean 𝑀𝑖 towards its own level, so now the new mean will be 𝑇𝑖 designated as 𝑀𝑛𝑒𝑤. The solution is updated according to the difference between the existing and the new mean given by
𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 − 𝑚𝑒𝑎𝑛𝑖 = 𝑟𝑖(𝑀𝑛𝑒𝑤− 𝑇𝐹𝑀𝑖) (4.7) where 𝑇𝐹 is a teaching factor that decides the value of mean to be changed, and 𝑟𝑖 is a random number in the range [0, 1]. The value of 𝑇𝐹 can be either 1 or 2, which is again a heuristic step and decided randomly with equal probability as 𝑇𝐹 = round [1 + rand (0, 1) {2-1}].
This difference modifies the existing solution according to the following expression 𝑋𝑛𝑒𝑤,𝑖 = 𝑋𝑜𝑙𝑑,𝑖+ 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 − 𝑚𝑒𝑎𝑛𝑖 (4.8)
4.3.1.2 Learner Phase
Learners increase their knowledge by two different means: one through input from the teacher and the other through interaction between themselves. A learner interacts randomly with other learners with the help of group discussions, presentations, formal communications, etc. A learner learns something new if the other learner has more knowledge than him or her. Learner modification is expressed as
1 for 𝑖 = 1: 𝑃𝑛
2 Randomly select two learners 𝑋𝑖 and 𝑋𝑗, where 𝑖 ≠ 𝑗 3 if 𝑓(𝑋𝑖) < 𝑓(𝑋𝑗) 4 𝑋𝑛𝑒𝑤,𝑖 = 𝑋𝑜𝑙𝑑,𝑖+ 𝑟𝑖(𝑋𝑖 − 𝑋𝑗) 5 else 6 𝑋𝑛𝑒𝑤,𝑖 = 𝑋𝑜𝑙𝑑,𝑖+ 𝑟𝑖(𝑋𝑗− 𝑋𝑖) 7 End if 8 End for
9 Accept 𝑋𝑛𝑒𝑤 if it gives better function value 4.3.1.3 Pseudocode for TLBO
99 m: population size
MAXITER: maximum number of iterations Initialization ( )
while 𝐼𝑇𝐸𝑅 < 𝑀𝐴𝑋𝐼𝑇𝐸𝑅 Elite ← Select Best (P, Elite) for 𝑖 = 1: 𝑚 𝑇𝐹 = round (1 + rand) 𝑋𝑚𝑒𝑎𝑛 ← mean (𝑋𝑖) 𝑋𝑡𝑒𝑎𝑐ℎ𝑒𝑟 ← best (𝑋𝑖) 𝑋𝑛𝑒𝑤,𝑖 = 𝑋𝑖 + rand (𝑋𝑡𝑒𝑎𝑐ℎ𝑒𝑟 - (𝑇𝐹∙ 𝑋𝑚𝑒𝑎𝑛)) if 𝑓(𝑋𝑛𝑒𝑤,𝑖) < 𝑓(𝑋𝑖) 𝑋𝑖←𝑋𝑛𝑒𝑤,𝑖 end if j ← randi (m) if j ≠ i if 𝑓(𝑋𝑖) < 𝑓(𝑋𝑗) 𝑋𝑛𝑒𝑤,𝑖 = 𝑋𝑖 + rand (𝑋𝑖− 𝑋𝑗) else 𝑋𝑛𝑒𝑤,𝑖 = 𝑋𝑖 + rand (𝑋𝑗− 𝑋𝑖) end if end if if 𝑓(𝑋𝑛𝑒𝑤,𝑖) < 𝑓(𝑋𝑖) 𝑋𝑖←𝑋𝑛𝑒𝑤,𝑖 end if end for
P ← Replace worst with Elite (P, Elite) P ← Remove duplicate individuals (P) ITER = ITER + 1
end while