I am trying to find the most optimal configuration of cells/connections in a battery pack of a certain size/shape in order to minimize the voltage drop for each cell in the pack.
I have a grid (triangle shaped) with n batteries (aka cells)(represented by circles) in it. The exact dimensions of the triangle are yet to be determined (the amount of batteries that fit into the triangle and how they are arranged depends on the size of the triangle... ).
I need to form x groups of y batteries each. These groups (1 to x) are numbered (group 1,2,3,4,5,6,7...x)
The groups need to be connected in sequence (group n to group n+1) ( group 1 to group 2, group 2 to group 3 .... group (x-1) to group x )
Each battery within a group is connected to all it's neighboring batteries within the same group (4 connections/branches per cell in the example configuration below, 1 going to each neighboring cell).
The "shortest path" from each cell in the battery to the Battery- (Point B) should be minimized for all cells. So the path (total path length) from each cell within the pack to the final group (group x) in the pack should be minimized. [This way the minimum amount of wire/connection length can be used which will result in less voltage drop in the battery pack (the longer the wires/connections the higher the voltage drop) ] (an example of the "shortest path" is drawn in one of the images below, if the definition of "shortest path" is unclear from my explanation above please take a look at the image below).
The "shortest path" from each cell in the battery to the Battery- (Point B) should be minimized for all cells.
x = number of groups
y = number of cells per group
Path_Length = no_cells_in_path * connection_length_per_cell
Connection_length_per_cell = 18
No_cells_per_group = x/n
AVG_Path_Length_group = SUM(∀ cells in group)( Path_Length) /y
Group_Length_Factor = SUM(∀ groups) ( AVG_Path_Length_group ) / x
Number of connections between groups needs to be maximized.
Length of each connection between groups needs to be minimized.
- Length_conn_betw_groups = Sum of all connections' length between group n and group (n+1)
- AVG_length_conn_betw_groups = Length_conn_betw_groups / no_conn_betw_groups
- Parallel_loss_factor = SUM(∀ groups) (AVG_length_conn_betw_groups /no_conn_betw_groups )
Final goal:
minimize (Group_Length_Factor + Parallel_loss_factor )
I have made an image of a possible configuration. In this image I have indicated each group with uneven index (1,3,5...) with red and each group with even index (2,4,6...) with blue. In this example x=20 and y=13. (total cells = xy = 260) This is definitely not the best configuration, just an example to show what it (a configuration) might look like.
NOTE: The black lines drawn in between some of the groups is there so that the cells in for example group 4 won't get confused with the cells in group 6 (since they are both colored in blue but are different groups), the black lines are there to try to make the image more clear.
Here is an image of the grid/triangle (that the batteries should fit into)(for this example):
In the image below the example configuration can be seen:
In the image above both all connections have been drawn (connections within a group drawn in yellow/orange lines and connections between individual groups drawn in green lines).
The image below shows only the parallel connections (for the same example configuration):
For this example configuration I have (for 2 batteries/cells) in the grid determined the "shortest path" :
Shortest path 1 (group 1, cell 1 --> group 20 (battery minus ):
Shortest path 2 (group 9, cell 11 --> group 20 (battery minus ):
(shortest path in images above = grey arrow with black outline)
( For this example I've drawn the "shortest path" for 2 cells... however the "shortest path" should be determined for all cells and the sum of the shortest paths for each cell in the grid should be minimized )
Questions:
- Is there a way to solve such a problem (find configuration of groups/connections where (Group_Length_Factor + Parallel_loss_factor) is minimized (lowest possible value)?
- If there is an analytical way to solve it, could you briefly explain it/where to possibly find more info on this method?
- Or should this be solved by a computer trying all possible configurations, and calculating (Group_Length_Factor + Parallel_loss_factor) and choosing the configuration where it's the lowest value?
- If I should indeed do it with the computer: how can/should I express the "shortest path length" for each cell in the pack in a formula (for a certain configuration of groups/connections)?
If anything in my explanation is unclear/incomplete please let me know and I'll do my best to try and clarify it.
Thank you very much for any input!
