*2.4. Constraints*

The OF minimization is a primary task to obtain the optimal results. OF minimization is subjected to design the constraints so that the requirements of the EDS must be satisfied with DG operation. Thus, the constraints are discussed in the succeeding subsection [53,54].

#### 2.4.1. Equality Constraints

These constraints follow the Kirchhoff's current rule as the algebraic sum of powers in and powers out should be equal in an EDS [54,55]. Two of these constraints are described as follows.

$$\sum\_{i=1}^{N\_{\text{bus}}} AP\_{\mathcal{S}^{\text{r}n\_i}} = \sum\_{i=1}^{N\_{\text{bus}}} AP\_{dem\_i} + AP\_{loss} \tag{25}$$

where *APgeni* = AP generated by the generation units at *ith* bus, *APdemi* = AP demand at *ith* bus.

$$\sum\_{i=1}^{N\_{bus}} RP\_{\mathcal{S}^{\text{err}\_i}} = \sum\_{i=1}^{N\_{bus}} RP\_{dem\_i} + RP\_{loss} \tag{26}$$

where *RPgeni* = RP generated by the generation units at *ith* bus, *RPdemi* = RP demand at *ith* bus.

#### 2.4.2. Inequality Constraints

These constraints are associated with the limits applied to the system parameters for the operation of EDS. Some of these constraints are described as follows.

#### **A. Power flow**

To maintain the line capacity within limits, these constraints ensure the apparent power to be within limits at the ends of a line [53,54].

$$AP\_{a\_{ij}} \le AP\_{a\_{ij}}^{\max} \tag{27}$$

where *APmax aij* = the highest permissible apparent powers (*APa*) for lines *i* to *j*, *APaij* = the actual *APa* transmitted from *i* to *j*.

#### **B. DG capacity**

*Energies* **2020**, *13*, 5631

These limits ensure the non-reversal of power flow. The power from the substation is provided to the EDS must be greater than the DG power. Also, the DG has the minimum and maximum power generation boundaries [56].

$$\sum\_{i=1}^{n\_{D\Box}} AP\_{DG\_i} \le 0.85 \times \sum\_{i=1}^{n\_{bus}} AP\_{dem\_i} + AP\_{loss} \tag{28}$$

$$\sum\_{i=1}^{n\_{DG}} RP\_{DG\_i} \le 0.85 \times \sum\_{i=1}^{n\_{bus}} RP\_{dem\_i} + RP\_{loss} \tag{29}$$

$$AP\_{\mathrm{DG}\_p}^{\min} \le AP\_{\mathrm{DG}\_p} \le AP\_{\mathrm{DG}\_p}^{\max} \tag{30}$$

$$RP\_{DG\_p}^{min} \le RP\_{DG\_p} \le RP\_{DG\_p}^{max} \tag{31}$$

where *p* = 1, 2,. . . . . . , *nDG*, *APmin DGp* (set to zero) and *APmax DGp* (from Equation (28)) are the lower and upper *AP* outputs of *DG* unit *p*, respectively. *RPmin DGp* (set to zero) and *RPmax DGp* (from Equation (29)) are the lower and upper *RP* outputs of *DG* unit *p*, respectively. *nDG* = number of DGs present in the distribution network.

#### **C. Bus voltage**

The voltages at buses present in the EDS must be limited within minimum and maximum limits [57,58].

$$\mid V\_{i\_{minimum}} \mid \stackrel{\sim}{\sim} \mid V\_{\text{i}} \mid \stackrel{\sim}{\sim} \mid V\_{i\_{maximum}} \mid \tag{32}$$

where |*Viminimum* | and |*Vimaximum* | = the lower and upper boundaries of the bus voltage |*Vi*| which are set to 95% and 105%, respectively.

#### **D. Branch current**

It refers as thermal capacity of the EDS lines. The current in the distribution lines must be within limits and should exceed the maximum current as given in Equation (33) [20].

$$I\_i \le I\_i^{\max} \tag{33}$$

#### *2.5. Constriction Factor-Based PSO (CF-PSO) Technique*

PSO is a novel progression computational technique which is in the frame since 1995. The use of this method is seen in RP dispatch [59], generation scheduling [60], renewable source integrated power system [61], and cost analysis [62]. In basic PSO method, the candidate solution is improved iteratively under any given constraint. The PSO algorithm is shown in Figure 5. Due to the reduction in computational time and requirement of less memory, PSO has overtaken many algorithms including the Genetic algorithm (GA) as PSO is mutation free. It searches the optimized value globally with the help of several particles present in a swarm based on specific constraints. As all particles have its local and global best values because of its own and global positions. This method updates the particle position and velocity as described in Equations (34) and (35).

$$V\_n^{p+1} = \mathcal{W}' \times V\_n^p + \mathcal{C}\_1' \times \mathcal{R}\_1' \times \text{(Personal}\_{BEST\_i} - X\_n^p) + \mathcal{C}\_2' \times \mathcal{R}\_2' \times \text{(Global}\_{BEST} - X\_n^p) \tag{34}$$

$$X\_n^{p+1} = X\_n^p + \mathbb{C}\_f \times V\_n^{p+1} \tag{35}$$

where *Vp*+1 *<sup>n</sup>* <sup>=</sup> *<sup>n</sup>th* particle velocity at (*<sup>p</sup>* + 1)*th* iteration, W' = particle inertial weight, *<sup>V</sup><sup>p</sup> <sup>n</sup>* = *nth* particle velocity at *pth* iteration, *C* <sup>1</sup>, *C* <sup>2</sup> = constants (0, 2.5), *R* <sup>1</sup>, *R* <sup>2</sup> = numbers generated randomly (0, 1), *PersonalBESTi* = the *<sup>n</sup>th* particle's best position considering its own property, *GlobalBEST* = the *<sup>n</sup>th* particle's best position considering the whole population, *Xp*+1 *<sup>n</sup>* , *<sup>X</sup><sup>p</sup> <sup>n</sup>* = *nth* particle position at (*p* + 1)*th* and *pth* iterations, respectively. *Cf* = constriction factor (CF) assures efficient convergence [63,64].

*Energies* **2020**, *13*, 5631

Due to faster convergence to the global point, the basic PSO faces the difficulty of premature convergence. The particles have started oscillating around the optimal point without providing any type of restriction to the highest velocity of the particles available in swarm. Therefore, the optimal global solution is rare to obtain. The use of properly defined CF is briefly described for advance convergence of the PSO [65]. This can also be applied for the DG siting and DG size in the EDS. It reduces the computation time and requires little memory. Although this technique suffers from partial optimization, by altering its parameter during problem solving will produce an improved result [66–68]. To obtain the improved result, a constriction factor (CF) is used and thus, the method is known as CF-based PSO technique. The parameters set for the CF-PSO are as follows. The values of initial weight, final weight, *<sup>C</sup>*1, *<sup>C</sup>*2, *<sup>R</sup>*1, *<sup>R</sup>*2, and CF are considered to be 9 × <sup>10</sup><sup>−</sup>1, 4 × <sup>10</sup><sup>−</sup>1, <sup>201</sup> × <sup>10</sup><sup>−</sup>2, 201 × <sup>10</sup><sup>−</sup>2, 0 to 1, 0 to 1, and 729 × <sup>10</sup><sup>−</sup>3, respectively. A flowchart is provided in Figure <sup>5</sup> to obtain the DG location, DG sizing and system reliability of the DGs in 33 bus EDS.

**Figure 5.** Algorithm implemented for the research work.

### **3. Reliability Assessment of Distribution System**

The RA of an EDS is as important as contrasted to other components and parts of the EDS. The IEEE guide for EDS reliability is given by standard number 1366-2012 [69]. According to given standard, reliability of an EDS can be analyzed using some reliability indices. The reliability indices considered for EDS reliability assessment include EENS, AENS, SAIDI, SAIFI, and ASAI. These indices are mainly classified in two categories which are elaborated in Equations (39)–(46c).
