I. Problem Formulation and Solution
.Optimal capacitor allocation problem in radial distribution system involves the maximization of savings from reduced energy losses and peak power losses by placing capacitor compensation in an optimal manner. The objective function is therefore given as,

                    (2-1)

Where Ke, Kp, and Kc are the constants for energy cost, peak power cost and capacitor cost, respectively. and are the power loss at any load level i with time duration T before and after compensation, respectively, and corresponds to the peak power loss before and after compensation, respectively, and Cj corresponds to the highest capacitor compensation to be placed at any location j. 

While optimizing capacitor compensation, the voltage magnitude at any location must not exceed the allowable limit.

GA is generally formulated for non-constrained optimization problem. However, constrained can be implemented as a penalty for the fitness function. Moreover, since genetic algorithm works on the principle of biological evolution and adaptation, and since over voltages was caused by over compensation which in effect, introduces additional losses on the system, the algorithm is expected to adapt and reject the solutions that can cause such over voltages. Therefore, such additional losses can act as a penalty for the fitness function.

We are extending the problem formulation of previous researches on capacitor optimization with their associated assumptions. In this research project, the simplifying assumptions employed by previous researchers was minimized to approach a more practical and real time solutions. In this project, we consider;
1. unbalanced three-phase network with unbalanced loads,
2. non-uniformly distributed loads,
3. capacitor placement on three-phase, two-phase and single-phase laterals,
4. varying load level.,
5. discrete fixed and switched capacitor sizes.,
6. non-uniform cross-section area of the feeder/wire sizes,
7. mutual coupling effect between conductors,
8. no pre-specified number and location of capacitors , and
9. non-constant voltage profile along the feeder.

Sensitivity Analysis
Sensitivity analysis was used to select the candidate locations for placing the capacitor in the distribution feeders. The estimation of these candidate locations basically helps reduce the search space for the optimization procedure. The sensitivity analysis is a systematic procedure to select those locations which has a maximum impact on the system real power loses, with respect to the nodal reactive power [3,4].

In order to determine the sensitivity of the bus, solve  in the following equation,
                               (2-2)  

Since the matrix above is just the system Jacobian relating nodal changes in voltage to nodal changes in power, an equivalent sensitivity for buses must be determined from these values

The maximum node sensitivity over each phase of a bus is taken to represent the entire bus sensitivity

Problem Solution
The problem solution for the optimal capacitor allocation is summarized as follows;
Part I. Determination of candidate locations for capacitor allocation.
1. Input the distribution system branch impedance values and the bus real and reactive power data.
2. Order the buses and select the buses with higher absolute value of sensitivity factors as a candidate locations
Part 2. Determination of optimal size, location, number and type of capacitors
1. discretize the load level into l different switching time.
2. Form an initial population of k strings each representing an encoded possible solution of nl variables (n candidate locations for l load levels). Representation is illustrated in table 3-1.

Table 2-1. Representation for any possible solution or individual.

For x bits of representation for capacitor size, the string length L will be xln.
3. Decode the genetic representation of each individual, then evaluate its fitness value which is the objective function value. 
4. In each generation, individuals are ordered according to their fitness values. A selection technique was employed to select individuals for reproduction and crossover.
As an elitist strategy, the highest individual will undergo crossover and will be passed on the next generation.
5. Repeat step 3 until maximum number of generation is reached.

Figure 2-1. The optimization process using GA

6. At any location, the minimum capacitor size that is required for any load level can be considered as the size of the fixed capacitors that could be placed on that location.

Introduction
Problem Formulation and Solution
Implementation, Testing and Results
Testing and Results (continued1)
Testing and Results (continued2)
Conclusion and References