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(1.16)

      Here, are the voltage magnitude and the load angle after inserting the DGs and shunt capacitors.

      (b) Voltage Deviation Index (VDI)

      Improvement in the voltage profile in the distribution network is measured by voltage deviation from the nominal voltage of 1.0 pu. The voltage deviation index is measured as the ratio of the voltage deviations (VDs) after and before the placement of the RDGs and shunt capacitors which are defined for the time instant t as

      (1.17)

      where VDDG is defined as

      (1.18)

      (1.19)

      Here, are the voltage of ith bus after and before placement of DG to the distribution network. The value of voltage deviation near zero indicates an improved voltage profile.

      (c) Installation Cost Index (ICI)

      Total installation cost includes the cost of the RDGs and the shunt capacitors. Lifetime (LT) of the RDGs and the shunt capacitors are used to obtain the annual installation cost. The ICI is defined as

      (1.20)

      Here, CostDG(t) is the effective annual installation cost of the DGs and shunt capacitors at time instant t, and can be defined as

      (1.21)

      and the installation cost, considering the maximum sizes of DGs and shunt capacitors, is defined as

      (1.22)

      In this book chapter, equal preference is given to each objective function in the weighted sum approach. Therefore, the weights for the three objective functions are

      1.3.2 Technical Constraints Considered

      The minimization of the weighted average of the three objective functions is subjected to the fulfilment of certain equality and inequality constraints as stated below [68],

      (a) Equality Constraints

      At any time instant t, the active and the reactive power balance for the distribution network should be maintained by the equation as follows

      (1.23)

      This equality constraint is fulfilled by the load flow solution of the distribution networks. Putility, Qutility are the active and reactive power taken from the utility, respectively, PDG, QDG are the active and reactive power supplied by the RDGs, in order, and shunt capacitors, and is the percentage load level expressed in per unit.

      (b) Inequality Constraints

       (i) The bus voltages of the distribution networks at any time instant must be within the minimum (Vmin) and maximum (Vmax) voltage level as in(1.24)The placement of RDGs and the shunt capacitors may increase the bus voltage beyond the limit at low voltage level and decrease the bus voltage at high load level. Therefore, bus voltages are kept within the prescribed limit by introducing the penalty function defined as(1.25)

       (ii) The active and reactive power generation of the DGs should be within the specified limit given as(1.26)where are the minimum and the maximum limit of the active power injection of the DGs and are the minimum and the maximum limit of the reactive power injection of the DGs which are to be maintained at any time instant. These inequalities are maintained by introducing the penalty function as below(1.27)

       (iii) The total active and reactive power injections by the RDGs and shunt capacitors at any time instant should be less than the total active and reactive power load demand of the distribution network at the same instant. This inequality constraint is defined as(1.28)These inequality constraints are considered by introducing the two penalty functions as(1.29)

       (iv) Branch currents, after the placement of DGs and shunt capacitors, should be within the maximum allowable level to maintain the feeder thermal limit. The branch current limit should be(1.30)This limit is considered in the objective function with the help of a penalty function described as(1.31)After considering all the penalty functions, the objective function, to be minimized, is(1.32)The values of the penalty factors τfeed are to be selected as very high value for a minimization problem. In this book chapter, the penalty values are selected as 1016 [68]. If any of the inequality constraints violates then the corresponding penalty function adds a very high value to the objective function.

1.4 Comparison of the SPBO Algorithm in Terms of CEC-2005 Benchmark Functions

For the comparison purpose, different algorithms (viz. PSO [69], TLBO [70], CS, and SOS) are considered in this book chapter and the parameters for the different algorithms considered are tabulated in Table 1.2 as below. The study has been carried out to analyze the performance of SPBO to solve CEC-2005 benchmark functions [67, 71]. The analysis has been done considering 30-dimensional problems. The comparison performance study of SPBO with PSO, TLBO, CS, SOS has been shown in Table 1.3. It may be noticed from Table 1.3, that the SPBO’s performance is better compared to that given by the remaining optimization algorithms considered. SPBO stands 1^st rank in nine of benchmark functions of CEC-2005 according to the optimum solution achieved as well as based on converging mobility. The mean of the result obtained in nine benchmark functions is better as compared to the mean achieved by PSO, TLBO, CS and SOS. In only one function (i.e. F8) SPBO stands 3^rd. Compared to the other algorithms considered in this analysis, it can be found in the study that the performance of the proposed SPBO algorithm is better.

Table 1.2 Parameters of different algorithms for benchmark functions.

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