5.1.1. Transformer Parallelism Index

If transformers are under load and analysis, the Transformer Parallelism Index is determined and shows when it pairs with another transformer, and which part of the grid demand is suitable for this parallelism. This value can change from 0, where parallel transformers cannot work in parallel, or when they are not able to take charge of the connected equipment under the load, to 1, where transformers can respond to the demand. The interval 0 to 1 has illustrated a percentage value in the transformer under the load that can response the maximum load power. This index is accepted from 120% to 140% in Brazil in accordance with the ONS NT 038/2014 standard [29] and is illustrated in Figure 7.

**Figure 7.** Acceptable Over Load by ONS NT 038/2014.

The Transformer Parallelism Index value is illustrated in Equations (10)–(14);

$$\begin{cases} 1 & \frac{TR\_{Ny}}{TRP\_{Nx}} > \frac{f\_L}{L\_{max} - f\_L} \\ \frac{TRP\_{Ny}(L\_{max} - f\_L)}{TRP\_{Ny}f\_L}; & \frac{TRP\_{Ny}}{TRP\_{Nx}} \le \frac{f\_L}{L\_{max} - f\_L} \land 0 < f\_L \le L\_{max} \\ 0 & f\_L > L\_{max} \land f\_L = 0 \end{cases} \tag{10}$$

$$TRP\_{Ny} = \sum TRP - TRP\_{Nx} \tag{11}$$

$$f\_L \ge 0\tag{12}$$

$$TRP\_{Ny} \ge 0 \; , \; \forall \; y \in N \tag{13}$$

$$TRP\_{\text{Nx}} \ge 0 \,, \forall \, x \in N \tag{14}$$
