Fuzzy logic is an extrapolation of classical set theory which provides mathematical framework for processing uncertainty in the information available by assigning membership values from zero to one [24]. In this section the incommensurable conventional objectives are fuzzified to form a single objective function.

### Fuzzification of S/S real power supply index

The S/S real power supply index is defined as the ratio of real power supply by the substation considering with DGs and shunt capacitors and without considering DGs and shunt capacitors placement.

Let us define,

$${\text{SPSI = }}\frac{{{\text{SPS}}^{\text{DGSC}} }}{{{\text{SPS}}^{\text{Base}} }}.$$

(3)

The fuzzification of S/S real power supply index (*SPSI*) is carried out considering the trapezoidal fuzzy set (\(\mu_{\text{SPSI}}\)) shown in Fig. 2a. The membership value of unity is given by the fuzzy set if the substation real power supply is limited to 40% of base case substation active power value and below by the DGs and shunt capacitors otherwise the value will be below unity. The mathematical expression for the fuzzy set can be derived from the fuzzy set shown in Fig. 2a. From Fig. 2a, \(\mu_{\text{SPSI}}\) can be described mathematically using Eq. (4):

$$\mu_{\text{SPSI}} = \left\{ \begin{array}{ll} 1 &\quad {\text{for SPSI}} \le {\text{SPSI}}_{ \text{min} } \\ \frac{{ ( {\text{SPSI}}_{ \text{max} } - {\text{SPSI)}}}}{{ ( {\text{SPSI}}_{ \text{max} } - {\text{SPSI}}_{ \text{min} } )}}& \quad {\text{ for SPSI}}_{ \text{min} } <{\text{ SPSI}} \le {\text{SPSI}}_{ \text{max} } \\ 0 &\quad{\text{ for SPSI}} > {\text {SPSI}}_{ \text{max} } \hfill \\ \end{array} \right.$$

(4)

In this work \({\text{SPSI}}_{ \text{min} }\) and \({\text{SPSI}}_{\text{max}}\) are taken as 0.4 and 1.0, respectively.

### Fuzzification of S/S reactive power supply index

The S/S reactive power supply index is defined as the ratio of real power supply by the substation considering with DGs and shunt capacitors and without considering DGs and shunt capacitors placement.

Let us define,

$${\text{SQSI = }}\frac{{{\text{SQS}}^{\text{DGSC}} }}{{{\text{SQS}}^{\text{Base}} }}.$$

(5)

The fuzzification of S/S reactive power supply index (*SQSI*) is carried out considering the trapezoidal fuzzy set (\(\mu_{\text{SQSI}}\)) shown in Fig. 2b. The membership value of unity is given by the fuzzy set if the substation reactive power supply is limited to 40% of base case substation reactive power value and below by the DGs and shunt capacitors otherwise the value will be below unity. The mathematical expression for the fuzzy set can be derived from the fuzzy set shown in Fig. 2b. From Fig. 2b, \(\mu_{\text{SQSI}}\) can be described mathematically using Eq. (6):

$$\mu _{{{\text{SQSI}}}} = \left\{ {\begin{array}{*{20}l} 1 \hfill & {\quad \quad \quad \quad {\text{for SQSI}} \le {\text{SQSI}}_{{{\text{min}}}} } \hfill \\ {\frac{{\left( {{\text{SQSI}}_{{{\text{max}}}} - {\text{SQSI}}} \right)}}{{\left( {{\text{SQSI}}_{{{\text{max}}}} - {\text{SQSI}}_{{\min }} } \right)}}} \hfill & {\quad \quad \quad \quad {\text{for SQSI}}_{{{\text{min}}}} < {\text{SQSI}} \le {\text{SQSI}}_{{{\text{max}}}} {\text{ }}} \hfill \\ 0 \hfill & {\quad \quad \quad \quad {\text{for}}\;{\text{SQSI}} > {\text{SQSI}}_{{{\text{max}}}} } \hfill \\ \end{array} } \right.$$

(6)

In this work \({\text{SPSI}}_{\begin{subarray}{l} { \text{min} } \\ \, \end{subarray} }\) and \({\text{SPSI}}_{\begin{subarray}{l} { \text{max} } \\ \, \end{subarray} }\) are taken as 0.4 and 1.0, respectively.

### Fuzzified active power loss index

The active power loss index is determined as the ratio of power loss with DG and shunt capacitors placement and without DG and shunt capacitors placement.

Let us define,

$${\text{PLI = }}\frac{{{\text{Ploss}}^{\text{DG}} }}{{{\text{Ploss}}^{\text{Base}} }}.$$

(7)

The conventional loss index (*PLI*) is fuzzified and the trapezoidal fuzzy set considered for active power loss (\(\mu_{\text{PLI}}\)) is shown in Fig. 3a. In the fuzzy set a membership value of unity is assigned if the loss is reduced to below 40% by the installation of DGs and shunt capacitors and if the loss is greater than 40% then the membership value between zero to one is given. From Fig. 3a, \(\mu_{\text{PLI}}\) can be described mathematically using Eq. (8):

$$\mu _{{{\text{PLI}}}} = \left\{ {\begin{array}{*{20}l} {1{\text{ }}} \hfill & {\quad \quad \quad \quad {\text{for PLI}} \le {\text{PLI}}_{{{\text{min}}}} } \hfill \\ {\frac{{\left( {{\text{PLI}}_{{{\text{max}}}} - {\text{PLI}}} \right)}}{{\left( {{\text{PLI}}_{{{\text{max}}}} - {\text{PLI}}_{{{\text{min}}}} } \right)}}} \hfill & {\quad \quad \quad \quad {\text{for PLI}}_{{{\text{min}}}} < {\text{PLI}} \le {\text{PLI}}_{{{\text{max}}{\kern 1pt} }} } \hfill \\ 0 \hfill & {\quad \quad \quad \quad {\text{for PLI}} > {\text{PLI}}_{{{\text{max}}}} } \hfill \\ \end{array} } \right.$$

(8)

In this work \({\text{PLI}}_{ \text{min} }\) and \({\text{PLI}}_{ \text{max} }\) are taken as 0.4 and 1.0, respectively.

### Fuzzified branch conductor current carrying ability limit index

The branch current carrying capacity limit is defined as the ratio of branch current to its current carrying capacity.

Let us define,

Now we define branch current capacity ratio as:

$${\text{BCI}}_{i} = \frac{{I_{i} }}{{{\text{IC}}_{i} }},\quad {\text{ for }}i = { 1},{ 2},{ 3}, \ldots ,{\text{ NB}} - 1.$$

(9)

The fuzzification of branch current carrying ability index (*BCI*_{i}) of individual branches of the distribution system can be determined using trapezoidal shape membership function considered (\(\mu_{{{\text{BCI}}_{i} }}\)) is shown in Fig. 3b. From Fig. 3b, \(\mu_{{{\text{BCI}}_{i} }}\) can be written as:

$$\mu _{{{\text{BCI}}_{i} }} = \left\{ {\begin{array}{*{20}l} {1{\text{ }}} \hfill & {\quad \quad \quad \quad {\text{for BCI}}_{i} \le {\text{BCI}}_{{{\text{min}}}} } \hfill \\ {\frac{{\left( {{\text{BCI}}_{{{\text{max}}}} - {\text{BCI}}} \right)}}{{\left( {{\text{BCI}}_{{{\text{max}}}} - {\text{BCI}}_{{{\text{min}}}} } \right)}}} \hfill & {\quad \quad \quad \quad {\text{for BCI}}_{{{\text{min}}}} < {\text{BCI}}_{i} \le {\text{BCI}}_{{{\text{max}}}} } \hfill \\ 0 \hfill & {\quad \quad \quad \quad {\text{for BCI}}_{i} > {\text{BCI}}_{{{\text{max}}}} } \hfill \\ \end{array} } \right.$$

(10)

The unity membership value is assigned if *BCI*_{i} has a value less than 0.4 and the membership value will be within zero to one if the *BCI*_{i} value is greater than 0.4 and hence in this work \({\text{BCI}}_{\begin{subarray}{l} { \text{min} } \\ \, \end{subarray} }\) and \({\text{BCI}}_{\begin{subarray}{l} { \text{max} } \\ \, \end{subarray} }\) are taken as 0.4 and 1.0, respectively.

In the present work, the average fuzzy membership functions of all the individual branch current capacity indices are considered as the fuzzified branch current capacity index of the distribution system.

Now we define fuzzy branch current capacity index of the distribution system as:

$$\mu_{\text{BCIT}} = \frac{1}{{{\text{NB}} - 1}}\sum\limits_{i = 1}^{{{\text{NB}} - 1}} {\mu_{BCI_i} } .$$

(11)

### Fuzzified min and max voltage limits at distribution system nodes

It is considered in this work that due to placement of DGs and shunt capacitors the node voltages of the distribution system must lie in within the specified voltage limits.

The following trapezoidal membership function is considered for fuzzified voltage limits.

### Fuzzification of voltage limits on node voltages

The fuzzy membership functions of all the individual node voltages are determined using the fuzzy set shown in Fig. 4 (\(\mu_{{V_{i} }}\)).

If the distribution system voltage is less than a specified voltage \(V_{\text{min} }\) or greater than \(V_{\text{max} }\), then membership value of less than unity is assigned. If it lies in between \(V_{\text{min} }\) and \(V_{\text{max} }\), unity membership value is assigned.

From Fig. 4, we can write,

$$\mu _{{V_{i} }} = \left\{ {\begin{array}{ll} 0 &\quad {{\text{for }}V_{i} \le V_{{p_{1} }} {\mkern 1mu} } \\ {\frac{{(V_{i} - V_{{p_{1} }} )}}{{(V_{{{\text{min}}}} - V_{{p_{1} }} )}}} &\quad {{\text{for }}V_{{p_{1} }} < V_{i} < V_{{{\text{min}}}} } \\ {1.0} & {{\text{for }}V_{{{\text{min}}}} \le V_{i} \le V_{{{\text{max}}}} } \\ {\frac{{(V_{{{\text{max}}}} - V_{i} )}}{{(V_{{{\text{max}}}} - V_{{p_{2} }} )}}{\text{ }}} & {{\text{for }}V_{{{\text{max}}}} < V_{i} < V_{{p_{2} }} } \\ 0 &\quad {{\text{for }}V_{i} > V_{{p_{2} }} } \\ \end{array} } \right.$$

(12)

In this work, \(\, V_{{p_{1} }}\) = 0.93, \(V_{\text{min} }\) = 0.95, \(V_{\text{max} }\) = 1.05 and \(V_{{p_{2} }}\) = 1.07 are considered.

In the present work the average fuzzy membership functions of all the individual node voltages is considered as the fuzzy performance index of the distribution system.

Now we define fuzzy voltage limit of the distribution system as:

$$\mu {}_{VT} \, = \, \frac{1}{{{\text{NB}} - 1}} \, \sum\limits_{i = 2}^{NB} { \, \mu_{{V_{I} }} } .$$

(13)

### Fuzzified voltage stability index of distribution system nodes

A voltage stability index [23] is used and is described by Eq. (14). The Voltage stability index of node ‘*n*’ of distribution system is given by

$${\text{SI}}_{n} = \frac{{4\left\{ { (P_{n} \, x_{\text{mn}} - Q_{n \, } r_{\text{mn}} )^{2} + (P_{n} \, r_{\text{mn}} + Q_{n} \, x_{\text{mn}} ) { }V_{m}^{2} } \right\}}}{{V_{m}^{4} }},$$

(14)

where ‘*m*’ is the sending end node and ‘*n*’ is the receiving end node, and,

Node at which \({\text{SI}}_{n}\) (*n* = 2,3,…….,NB) is maximum, that node is most sensitive to voltage collapse. Therefore, voltage stability index of distribution network is given as:

$$u = { \text{max} }({\text{SI}}_{n} ),\quad {\text{ for }}n \, = { 2}, 3, \ldots ,{\text{NB}} .$$

(16)

Figure 5 shows the fuzzy membership function for maximum *n* voltage stability index (\(\mu_{u}\)). If the value of ‘*u*’ is less than a specified value \(u_{\text{min} }\), unity membership value is assigned and if ‘*u*’ is greater than or equal to \(u_{\text{max} }\), membership value of zero is assigned. If ‘*u*’ is lying in between \(u_{\text{min} }\) and \(u_{\text{max} }\) membership value less than one is assigned.

From Fig. 5, we can write,

$$\mu _{u} = \left\{ {\begin{array}{*{20}l} {{\text{ 1}}} \hfill & {\qquad {\text{ for }}u \le u_{{{\text{min}}{\kern 1pt} }} {\text{ }}} \hfill \\ {\frac{{\left( {u_{{{\text{max}}}} - u} \right)}}{{\left( {u_{{{\text{max}}}} - u_{{{\text{min}}}} } \right)}}} \hfill & {\qquad {\text{ for }}{\mkern 1mu} u_{{{\text{min}}}} < u \le u_{{{\text{max}}{\kern 1pt} }} } \hfill \\ 0 \hfill & {\qquad {\text{ for }}u > u_{{{\text{max}}}} } \hfill \\ \end{array} } \right.$$

(17)

In this work \(u_{\text{min} }\) = 0.04, and \(u_{\text{max} }\) = 1.0 are considered.

Since the fuzzified objectives are developed from normalized conventional objectives all the fuzzy objectives can be added to unique objective function through weighting factors.

$$F = w_{1} \mu_{\text{SPSI}} + w_{2} \mu_{\text{SQSI}} + w_{ 3} \mu_{\text{PLI}} + w_{ 4} \mu_{\text{BCIT}} + w_{ 5} \, \mu_{\text{VT}} + w_{ 6} \, \mu_{u}$$

(18)

Equal importance is considered for all the fuzzy objectives in Eq. (18) and the unity magnitude is taken for the weighting factors *W*_{1}, *W*_{2}, *W*_{3}, *W*_{4}, *W*_{5}, and *W*_{6}. However weighting factors can be varied according to the preferences of different operations. The fuzzy multiobjective function Eq. (18) is maximized subject to various operational constraints to satisfy the technical requirements of the distribution system for obtaining optimum DG capacity and shunt capacitor units. The authors have considered equal importance for all the fuzzy objectives and since the scaling of the fuzzy multiobjective function described by Eq. 18 does not effect the final optimization results, all the weighting factors of the objective function are considered unity magnitude value.