**3. Flower Pollination Technique**

Flower pollination is a process adopted by flowers and plants to reproduce. It is classified into biotic and abiotic. Biotic pollination is done by living organisms whereas nonliving accounts for abiotic pollination. The other way of classifying pollination is self and cross. Self-pollination means fertilization of the same plant and cross-fertilization happens for different plants. The long distance process is called global pollination, and implantation, happening over short distance, is called local pollination. Flower constancy also provides an assurance of nectar for the pollinators with minimum effort of learning, exploration and exploitation. The global pollination is carried out by the equation

$$a\_i^{m+1} = a\_i^m + L(a\_i^m - c\_\*) \tag{24}$$

where,

$$L \simeq \lambda \Gamma(\lambda) \frac{\sin(\frac{\pi \lambda}{2})}{\pi} \ast \frac{1}{s^{1+\lambda}} \tag{25}$$

*amii*th pollen at *m*th iteration

*L*—Levy weight-based size of each step (*s*; *s* > 0). *<sup>c</sup>*∗—Best current solution at current iteration.

Γ(λ)Gamma distribution function

> The pollination occurring locally is carried out by the following equation

$$a\_i^{m+1} = a\_i^m + \ \in (a\_o^m - a\_q^m) \tag{26}$$

*a<sup>m</sup> ii*th pollen at *m*th iterations

∈—takes a value of [0,1]

*a<sup>m</sup> o*and *a<sup>m</sup> q*—pollen from di fferent flowers from same plant.

Biotic, also called cross-pollinators, follows movement of step flight, which aids in attaining. A set of N flower population is generated with random solutions. Global pollination follows the rule of biotic and cross-pollination; the reproduction probability depends on flower constancy.

The two indispensable concepts of FPA are local and global pollination steps. Pollinators carry the pollens of the flower to far reaching places due to its custom manner. This helps in the exploration of the larger search space. Here the general tuning parameter of levy flight mechanism, which essentially incorporates the various distant step sizes carried out by the pollinator. Usually the nearby flower is pollinated by the pollens of the local adjacent flowers rather than the far-o ff flowers. Thus, the general probability tuning parameter using levy fight mechanism switches e ffectively between global pollination and local pollination ensures the e ffective exploration and exploitation of the learning with minimum learning e ffect.

A simple numerical example is illustrated here for the implementation of FPA, as given below.

Consider a simple objective function *f*(*z*) = *z*2 1 + *z*2 2 subject to *zg*,*<sup>i</sup>* = (0.3,0.3). The fitness value obtained is *f zg*,*<sup>i</sup>* = 0.18. Equation (26) is then applied and *zg*+1,*<sup>i</sup>* = (0.3, 0.3) and then updated to (0.1,1) for illustration. Then the newly updated. *zg*+1,*<sup>i</sup>* = (0.1, 0.3). As a result, the new fitness value solution *f zg*+1,*<sup>i</sup>* = 0.04. Here *f zg*+1,*<sup>i</sup>* < *f zg*,*<sup>i</sup>* . This infers that the old fitness value solution can be replaced by the currently obtained fitness value. For example, if the newly updated. *zg*+1,*<sup>i</sup>* = (0.9, 0.3), this results in the new fitness solution *f zg*+1,*<sup>i</sup>* = 0.9. Here *f zg*+1,*<sup>i</sup>* > *f zg*,*<sup>i</sup>* . This clearly indicates there is no progress to advance *zg*,*<sup>i</sup>* . Thus, this value should be discarded and proceeded for the updating the next fitness value as indicated in pseudocode.

### **4. Algorithmic Steps in FPA**

The sequential steps carried out in pseudocode of the flower pollination algorithm are presented as follows. The minimization objective function:

$$\min f(\mathbf{x}), \mathbf{x} = (\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_l) \tag{27}$$


*for i* = 1:*t if r and* < *p*,


•

	- end if

• end while

Managing the congestion is coded using the Flower Pollination Algorithm. The process of carrying the pollens of the flower to far reaching places assures the fittest population for survival in the search space. The e fficacy of FPA is implemented in terms of congestion cost minimization as shown in Figure 3. The parameters of FPA are λ, *s*, and size of population and iteration number. The criteria for optimal tuning obtained using FPA are λ = 1.6, *s* = 1, and size of population is 6 has been carried out for 25 iterations. Here the expedition between the global and local search using levy flight mechanism ensures the optimal output. Further FPA relieves congestion by suitable rescheduling of the real power of the generators. To validate its e ffectiveness, the obtained results are compared with other optimization algorithms already reported in literature.

**Figure 3.** Implementation of FPA for congestion management.
