# Efficiency Comparison of AC and DC Distribution Networks for Modern Residential Localities
## 4 Mathematical Model  
The entries of Table 2 form a matrix with 24 columns representing hours of the day and 12 rows representing loads in a typical modern home. The entries of the matrix are the power consumption per hour of each load. \[X_{i,j}=\left[ \begin{array}{cccc}x_{1,1}&x_{1,2}&\ldots&x_{1,24}\\ x_{2,1}&x_{2,2}&\cdots&x_{2,24}\\ x_{n1,1}&x_{n1,2}&\ldots&x_{n1,24}\\ \vdots&\cdot&\ldots&\vdots\\ x_{n3,1}&x_{n3,2}&\cdots&x_{n3,24}\end{array} \right]\in\mathbb{R}^{i\
eq j} \tag{1}\]  
where the power consumption of 'A' category loads is taken from \(x_{1}\) to \(x_{n1}\). Similarly, the power consumption of 'D' category loads is taken from \(x_{n1+1}\) to \(x_{n2}\), and 'I' category from \(x_{n2+1}\) to \(x_{n3}\). Here, '\(j\) defines hours of the day, from 1 to 24. In order to find the rating of the appropriate converter for the power/voltage level conversion, we need to find the maximum load that needs to be supplied by the converter. The maximum value of power consumption of load can be found from  
\[Y(i,1)=\max_{of}\bigl{(}x_{i,j}\bigr{)} \tag{2}\]  
where '\(Y\)' takes the maximum values of all the entries of \(X_{i,j}\) representing power consumptions of loads for each hour. A margin of 20% is further added to the maximum value of load consumption for practical reasons as  
\[R(i)=1.2\;\times Y^{T} \tag{3}\]  
The matrix is transposed to convert the column matrix formed in (2) to the row matrix for ease of calculations. The next step is to find the percentage loading. This value is important because it is utilized to find the operating efficiency of the converter. A matrix with the similar dimensions as \(X_{i,j}\) can be created  
\[P_{i,j}=\frac{100\times x_{i,j}}{R(i)} \tag{4}\]  
_For 'A' category loads:_  
The percentage loading is placed in a curve equation of the DC-AC converter that gives the efficiency of converter for each hour. A general equation may be  
\[\eta_{i,j}=f(l)\,;\,i=1\to n1\;and\;j=1\to 24 \tag{5}\]  
where '\(\eta\)' is the efficiency of the DC-AC converter as a function of percentage loading '\(l\)'. In (5), the efficiency of the converter is the function of percentage loading. The curves and their corresponding equations are presented in the subsequent section. _For 'D' category loads:_  
Similarly, the percentage loading is placed in the curve equation of the DC-AC converter, which gives the efficiency of converter for each hour. The same equation as (5), with required modifications, can be derived for converters driving 'D' category loads. \[\eta_{i,j}=f(l)\,;\,i=n1+1\to n2\;and\;j=1\to 24 \tag{6}\]  
_For 'D' Category loads:_  
There is no need for a conversion stage. Hence, the conversion efficiency is given as\[\eta_{i,j}=100\%;\;i=n2+1\to n3\;and\;j=1\to 24 \tag{7}\]  
To acquire the input matrix, we made use of the general formula of efficiency that equals 100% of output/input, which is depicted as  
\[Xin_{i,j}=\;\frac{X_{i,j}}{\eta_{i,j}}\times 100\% \tag{8}\]  
which contains the input power to a residential building for 24 h. The efficiency matrix is formed by combining (5)-(7), and the output matrix is the one formed in (1). All the entries of the input power matrix are added to give the total input to the building per hour as  
\[Xin_{j}=\sum_{i=1}^{n1}Xin_{i,j}+\sum_{i=n1+1}^{n2}Xin_{i,j}+\sum_{i=n2+1}^{n3 }Xin_{i,j} \tag{9}\]  
Similarly, the loadings of each home in each hour can be formed as  
\[X_{j}=\sum_{i=1}^{n1}X_{i,j}+\sum_{i=n1+1}^{n2}X_{i,j}+\sum_{i=n2+1}^{n3}X_{i,j} \tag{10}\]  
The overall efficiency of a building per hour can be computed by  
\[\eta_{BB}=\frac{X_{j}}{Xin_{j}}\times 100\% \tag{11}\]  
Suppose that we have a system with'm' buildings per DC-DC converter transformer (DC, XFMR) in a residential community. We can adopt the same procedure for the overall efficiency of the system as we did for one building.