Mixed Discrete SPBO
In general, the optimization algorithm is used to optimize (minimize) the objective function by obtaining the optimum value of the variable vector X. To optimize the variable vector X, consist of some continuous and discrete variables, a mixed discrete version of SPBO may be used. For an n-dimensional problem that includes continuous and discrete variables, the variable vector may be represented as in (Equation 1.5).
where [Xcont] and [Xdisc] are the continuous and discrete variable vectors, respectively. For the scenario of m continuous variables and remaining (n-m) discrete variables, the [Xcont] and [Xdisc] maybe expressed as in (Equation 1.6) and (Equation 1.7), respectively.
As mentioned earlier the mixed discrete SPBO is capable to handle both the continuous and discrete variables. In the mixed discrete SPBO, the continuous variables are updated as the conventional SPBO. The updating process of the continuous variables is the same as discussed in the previous section using four categories of students (namely best student, good student, average student, and students who want to improve randomly) with the help of (Equations 1.1–1.4). For the discrete variables, the discretization may be done with the help of the nearest vertex approach (NVA). The NVA method is normally based on finding out the Euclidean norm in the design space. The discrete variables may be expressed in terms of a hypercube, which are represented by the sets of ordered pair and can be represented as in (Equation 1.8)
where,
where, ℤ+ is the set of integers. In the hypercube H, the closest vertex of the discrete variables can be determined using the NPV method as expressed in (Equation 1.11).
where,
1.3 Problem Formulation
1.3.1 Objective Functions
The selection of the locations and the proper sizes of the RDGs (biomass and solar PV) and shunt capacitors in the distribution networks depend on the selection of the proper objective functions. Improper placement of the RDGs and shunt capacitors leads to the maloperation of the distribution networks which includes the increment in active power loss, poor voltage profile, and huge installation cost. In this book chapter, the active power loss, voltage deviation, and the effective annual installation cost of RDGs and shunt capacitors are considered as the main objective functions. The multi-objective function is converted to the single objective function by using the weighted sum approach, where the weights for the objective functions are selected based on their preferences. The objective function with the weights is stated in (Equation 1.12).
(a) Power Loss Index (PLI)
PLI is the ratio of the active power losses after and before the placement of RDGs and the shunt capacitors. PLI is defined as in (Equation 1.13).
The
The active power loss at any time instant t can be defined as in (Equation 1.14)
where NBr is the total number of branches, Zi is the branch impedance and |Iij| is the magnitude of the branch current, connected between bus-i and bus-j, defined as in (Equation 1.15).
After the RDGs and shunt capacitors connected to the distribution networks, the branch current magnitude |Iij| is modified by (Equation 1.16).
