A 50 bus power system Ybus has 80% sparsity. The total number of transmission lines will be ___________

The sparsity of a bus admittance matrix (Ybus) refers to the proportion of zero elements in the matrix, which in a power system context usually relates to the absence of direct electrical connections (transmission lines or transformers) between buses. A system with N buses would have a Ybus matrix of size N×N. The total possible number of elements in the matrix is N^2, representing all possible connections including self-connections (line impedance to the ground at each bus).

For a 50 bus power system, the total possible number of elements in the matrix is 50^2 = 2500. Since the matrix has 80% sparsity, it means 80% of these 2500 elements are zero, implying no direct electrical connection. It also means that 20% of these elements are non-zero, representing existing electrical connections or lines.

Total number of non-zero elements (which corresponds to the number of existing connections including self-connections) = 20% of 2500 = 0.20 × 2500 = 500.

However, this count of 500 includes the self-connections at each bus, which are the diagonal elements of the matrix. Since there are 50 buses, there are 50 such self-connections (one per bus).

Therefore, the total number of transmission lines will be:

Number of non-zero off-diagonal elements (representing transmission lines) = 500 – 50 = 450.

So, the