# Finding Nodal equation for a node having an index different from its line outflows

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Hello everyone,

Could someone please help me look at a way to find the nodal balance equation at a node for inputs having different indices?

More explanation: Refer to the diagram below. Assume nodes 1,2, and 3 are set of buses indexed b. Then connecting lines between the buses are Set with index domain (i,j). To find the nodal balance equation at node 2 such that incoming is made equal to outgoing is somehow difficult for me because of difference in indices. For example say node 2, the sinusoidal sine is generator Pg with index b, outgoing arrow is Demand with index b and line flow (LF) indexed (i,j). The index domain (i,j) only shows the sending end bus and the receiving end bus. Hence the nodal balance equation is:

Pg(b) = Demand(b) + LF(i,j). How can I find this sum over different indices knowing that number of buses with index b are not equal to the number of line flows?

Secondly, the line flow (LF) is made of two characteristics of different indices. the characteristics are Y indexed (i,j) and Angle indexed b. How can I multiply two characteristcs not having the same characteristcs.

I know my question is really long but I am sorry. Just trying to make it clear. Thanks

I await prompt responses.Thank you all

Userlevel 5
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Hello @seglove , instead of having different indices (i, j) for the line flows, you should use the same indices as your bus nodes.

For example, set_bus with index b is {1, 2, 3}. Declare multiple indices for this set (as described here: Multiple indices for a set — AIMMS How-To) - i, j can be indices in the same set_bus. Then LF(i, j) will represent the line flow from bus_i to bus_j. You balance equation can then be

`Pg(i) = Demand(i) + LF(i, j)`

or

`Pg(b) = Demand(b) + LF(b, j)`

Indices that belong to the same set can be used interchangeably, i.e., Pg(i) is equivalent to Pg(b) (if i and b are indices in the same set)).