I've done some calculations on a 5 battery parallel string and here are my results.
I'm using a value of 1 milliohm for the battery IR, and .1 milliohm for the link resistances which I think is more representative of what one gets with busbars rather than cable links. The load current is 100 amps. I'll reference this image:
The red and black load connections are the standard diagonal connection, and that connection gives me the following theoretical battery currents in amps:
Example 26
23.5
18.2
16.6
18.2
23.5
Some have suggested moving the load connection toward the middle of the string. If the black load connection is moved to the point on the busbar adjacent to the negative terminal of battery 2, and the red load connection is moved to the point on the busbar adjacent to positive terminal of battery 4, we get these theoretical currents in amps:
Example 27
18.3
21.8
19.8
21.8
18.3
This is a substantial improvement. In some of the earlier hookups it was possible to get a perfect balance by connecting the load to the right place on the busbars between the last and next to last batteries in the string. Of course, I have tried to do that for this 5 battery string by varying the connection point. It isn't possible to get perfect balance, but it is possible to get improvement. I had to consider how to measure the improvement, and what seems reasonable is to get the least variation in the battery currents. I had my calculations include the standard deviation of the battery currents, and tried all the connections between the end batteries and the next-to-the-end batteries in 5% increments. For this 5 battery string, the connection which is 66% of the way from the last battery to the next battery inward gives the minimum variance in the battery currents. This would be a connection like the yellow/blue connection shown in the image above, at the 66% point. Here is the result of the 66% connection theoretical battery currents in amps:
Example 28
20.0
20.6
18.7
20.6
20.0
I haven't given up on the possibility of a perfect balance for this string, but no joy so far.