# Solar Photovoltaic Integration in Monopolar DC Networks via the GNDO Algorithm
## 3 Solution Methodology
### Generalized Normal Distribution Optimization Algorithm  
The GNDO approach is a mathematics-inspired algorithm based on the philosophy of Gaussian distribution, which allows describing multiple natural and physical phenomena with a high level of precision [27]. A Gaussian distribution function generally takes the form presented in Equation (13).  
\[f(x)=\frac{1}{\sqrt{2\pi\delta}}\text{exp}\left(\frac{(x-\mu)^{2}}{2\delta^{2}} \right), \tag{13}\]  
where \(x\) is a random variable that represents the probability of distribution, which is associated with the scaling (\(\delta\)) and the location (\(\mu\)) parameters [28].  
Note that parameters \(\mu\) and \(\delta\) determine the mean and standard deviation values of the random variables \(x\). To illustrate the effect of these parameters in the behavior of Equation (13), Figure 1 presents the Gaussian distribution for different values of \(\mu\) and \(\delta\)[27].  
In order to understand the behavior of the Gaussian distribution in Figure 1, note that (i) the movement of the Gaussian distribution is given by the direction of the increase in the \(\mu\) parameter. At the same time, its standard deviation and amplitude remain constant due to the fixed value of the \(\delta\) parameter (see Figure 1a); and (ii) the amplitude and standard deviation are modified when the \(\delta\) parameter is changed, whereas the location of the Gaussian distribution remains fixed since the \(\mu\) parameter is constant, as can be seen in Figure 1b).  
Considering the advantage of Gaussian distributions in describing multiple physical and natural phenomena with a high level of precision, the author of [27] proposed an efficient and reliable optimization approach that combines the advantages of distribution probabilities with combinatorial optimization methods (population-based optimization algorithms).  
The main features of the GNDO are listed below:  
* An initial population is generated with a normal distribution throughout the solution space. This initial population evolves through the solution space to explore and exploit its promissory sub-regions. During the first stages of the optimization process, the variances regarding the positions of the solution individuals show minimal variations,  
Figure 1: Variations in the Gaussian distribution as a function of the changes in the \(\mu\) and \(\delta\) parameters: (a) behavior of the Gaussian distribution when \(\delta\) is fixed, and \(\mu\) is varied; (b) behavior of the Gaussian distribution when \(\mu\) is fixed, and \(\delta\) is varied.  
and the location of the decision variables concerning the global optimal solution can be considered to be randomly distributed with a normal structure.
* As the evolution process through the solution space advances, the main position and the standard deviation are continuously decreased. This is done in order to pass from the exploration phase to the exploration of the solution space, i.e., refining the solution around the best solution reached.
* At the end of the optimization process, the variance of the positions between all the solution individuals and the distance between the mean position and the optimal solution reach minimum values.  
Considering the characteristics mentioned above of the GNDO approach, the local and global exploration features of this optimization approach are presented below [24].