Calculation of parallel string battery currents

[Solarod]

Hi, everybody! I'm a retired EE. I didn't want my EE skills to fade away after retirment, so I exercise them when I can. Whenever I see a non-trivial circuit analysis problem on one of various forums I frequent, I challenge myself to solve it. Recently I watched Will's video about current sharing in a parallel battery string, and it occurred to me that the string is just a circuit. It could be solved using standard circuit analysis techniques giving theoretical values for the battery currents. I wondered if this has been done before and posted on the web. I found a link on this forum to an article that gives numerical values: http://www.smartgauge.co.uk/batt_con.html This article solves a parallel string of 4 batteries with load connected in two ways--one case with the load connected at one end of the string and the other case using a diagonal connected load.

I decided to solve those two cases myself and compare my results with the results in the article. I drew up a model for the string and it turns out that
to completely solve the circuit takes 11 simultaneous equations, The article says its solution was done in about 1980, and the author says "the
calculation is incredibly difficult". Nowadays we have some very powerful mathematical software, so I knew that setting up the equations would be
the hard part; the actual number crunching would be easy since a computer would do it.

I set up the 11 equations and got a solution which I compared to the results in the article. My result was not the same at first, but having the
article's result to compare helped me track down my errors. I have got the equations right now. I don't want this thread to be a full on propellor head
thread, so I'm not going to post anything about the math; I'm only to give numerical results for a number of hookups and initial conditions. If anyone
wants to know about the math, ask and I'll do another thread.

The values for the initial parameters for the 4 battery string used by the article are 20 milliohms for the battery internal resistance, 1.5 milliohms for
the link cable resistance, and a 100 amp load. For the first case with the load connected to one end of the string I got the following values for the
battery currents in amps:

35.8
26.1
20.4
17.7

These are the very same values given in the article except for 3 single digit differences; these tiny discrepancies are no doubt due to rounding errors
in the mathematical calculations. I changed my setup to solve for the battery currents for the second case--the load connected diagonally. Using the same initial parameters I got the following values for the battery currents in amps:

26.7
23.3
23.3
26.7

Again, I got the same values as the article with a couple of small rounding errors. Getting this good agreement with the values in the article gives me confidence that I have got my equations right!

 

[Solarod]

Readers should understand that the results I'm going to post are theoretical values obtained with mathematical circuit analysis. In the real world
things will be different since battery IR is quite variable due to temperature, age, SOC, etc., whereas in the mathematical analysis the parameters are exact and don't vary. But the mathematical results are a useful starting point.

Now I want to calculate results for a number of variations of the 4 battery parallel string. All the calculations assume the 4 batteries are identical
with identical IR. This is an important point.

Edit: I want to make sure it's understood that the battery IR I'm referring to in all these calculations is the under load DC IR, not the IR measured with a 1 kHz stimulus like the low cost meters use.

The calculations in the article were done 32 years ago, and maybe back then a battery IR of 20 milliohms was typical, but modern LFP batteries of the size used in solar power installations are more nearly in the neighborhood of 1 milliohm or less. I'm going to use a battery IR of 5 miliohms as the high end of values found today in smaller batteries. I will use a value of 1 millohm for the link cable resistance since nowadays people are using larger link cables and probably better quality lugs.

For the first case I'm going to calculate the battery currents with the load connected to the end of the string, with my new initial parameters. Here's an image of the connection (image borrowed from Victron):

I've numbered the batteries 1 through 4 (top to bottom). I'll show my results for the battery currents in the order of the battery number, top to
bottom. With battery IR of 5 milliohms, link resistance of 1 milliohm, and load current of 100 amps, here are the theoretical battery currents in amps:

Example 1
11.1
15.5
26.2
47.3

The connection gives a large imbalance, as is well known for this connection. It's amazing to me that this connection is recommended at sites all over the Web.

Let's see what happens when the battery IR is reduced to 2 milliohms; maybe we got some better batteries. Here's what the theoretical currents in amps would be for this case:

Example 2
4.76
9.52
23.8
61.9

The imbalance has gotten worse. Continuing, I'm reducing the battery IR to 1/2 milliohm, and here are the theoretical currents in amps with that change:

Example3
0.49
2.45
14.2
82.8

The imbalance here is extreme. Clearly, for this hookup, the imbalance becomes worse as the battery IR becomes smaller. This example is typical of what modern LFP batteries would give us

 

[Solarod]

Now I'll obtain some results for the same 4 battery string, but with the load connected in the well known diagonal manner.

The battery IR is reset to 5 milliohms, with link resistance still at 1 milliohm, and 100 amp load current. The calculations show (I'm showing 6 digits for a reason needed later) that in this case the theoretical battery currents in amps are:

Example 4
29.1667
20.8333
20.8333
29.1667

The balance is much improved due to the diagonal connection, but this is with a relatively high battery IR of 5 milliohms. Let's see what happens if the IR is reduced to 2 milliohms. Here are the theoretical battery currents in amps for this case:

Example 5
33.3
16.7
16.7
33.3

OK, continuing with further reduced IR of 1/2 milliohm. This example is typical of what modern LFP batteries would give us The theoretical currents in this case are:

Example 6
41.7
8.33
8.33
41.7

We see the same effect we saw with the load connected to the end of the string. As the battery IR decreases, the imbalance gets worse. The imbalance in Example 6 is very bad; what can we do to improve it? Let's reduce the link resistance from 1 milliohm to 1/10 milliohm, with battery IR remaining at 1/2 milliohm. This 10 times reduction of the link resistance could be done with busbars. It probably wouldn't be possible to get such a low resistance link made of cable to fit on the battery terminals. Here is the result of the calculation; the battery currents in amps are:

Example 7
29.2
20.8
20.8
29.2

These are the same currents we got in Example 4. It's not great, but it's a vast improvement over Example 6.

 

[Solarod]

I've seen comments on the forum suggesting that, given a string with load connected in the diagonal manner, there might be a benefit to making the load connections more toward the middle of the string rather than at the very corners. For the 4 battery parallel string that would be done like this:

Setting the initial parameters back to: battery IR of 5 milliohms, link resistance of 1 milliohm, load current of 100 amps, the calculation gives this result for the theoretical battery currents in amps:

Example 8
20.8333
29.1667
29.1667
20.8333

Compare this to Example 4. The values of the currents are exactly the same, but distributed among the batteries in kind of an anti-symmetric way. When I saw this, I thought "I wonder if there is a way to make a connection halfway between the two methods?". Consider the link cable between the negative terminals of battery 1 and battery 2. At a point halfway between the ends of the cable strip away an inch of insulation and expose the copper conductor. Connect the negative load cable at that point. I know this is impractical but bear with me. Now in a similar manner expose the conductor at a point halfway between the ends of the link cable connecting the positive terminals of battery 3 and battery 4; connect the positive load cable at that point. It should be as illustrated in this image:

Now with 4 identical batteries having indentical IR of 5 milliohms, link cable resistance of 1 milliohm and load current of 100 amps the calculation shows that the battery currents in amps are:

Example 9
25.0
25.0
25.0
25.0

We have (theoretically) perfect balance! Is this a previously unknown connection giving perfect balance, like the "Halfway" connection shown in the Victron document? Does anyone know if this been published anywhere?

A very practical way of making this connection is possible when busbars are being used. Just drill a couple of holes at points on the busbars at the pertinent points halfway between the appropriate battery terminals and make the load connection there. This image shows what I mean:

The usual diagonal connection would be as shown with the red and black wires. The new connection would be as shown with the yellow and blue wires.

Even more useful are the following properties. For this new connection, it doesn't matter what the battery IR is!! The calculation of the battery currents remains (theoretically) perfectly balanced for any value of IR as long as all 4 batteries have the same IR. This is very handy because as the batteries are discharged, their IR will change but this change will not upset the (theoretical) perfect balance. Also, the resistance of the links has no effect on the (theoretically) perfect balance! This means that the busbar need not be very thick copper; it can be thinner and higher resistance than would normally be needed for good balance. The only restriction is that its resistance shouldn't be so high that it gets too hot.

A mathematician would say that the (theoretically) perfect balance is invariant with respect to battery IR and link resistance. These properties are crucially dependent on the batteries having identical IR.

I expect that some reading about this may be skeptical. I was skeptical myself at first, but I've checked it several times. I invite verification by members of the community. Perhaps someone will do a simulation with Spice. The ultimate proof would be a hardware proof. If someone already has a string of 4 identical batteries in parallel with busbars, they could drill the two new holes and connect the load there, subsequently checking the balance.

 

[WindWizard]

Very nice write up. Thanks for the information.

 

[timselectric]

Now with 4 identical batteries having indentical IR of 5 milliohms, link cable resistance of 1 milliohm and load current of 100 amps the calculation shows that the battery currents in amps are:

Example 9
25.0
25.0
25.0
25.0

We have (theoretically) perfect balance! Is this a previously unknown connection giving perfect balance, like the "Halfway" connection shown in the Victron document? Does anyone know if this been published anywhere?

I don't see how this is possible, since #1 and #4 have more wire resistance in their path. But, I don't have any way of real world testing. So, I can't debate the numbers you have come up with.
I will say that it was an interesting read. So, thanks for posting.

 

[Solarod]

I don't see how this is possible, since #1 and #4 have more wire resistance in their path. But, I don't have any way of real world testing. So, I can't debate the numbers you have come up with.
I will say that it was an interesting read. So, thanks for posting.

Remember that the full diagonal connection gives more current in 1 and 4, less in 2 and 3. This new connection adds more resistance for 1 and 4 which reduces their current, and it add less resistance for 2 and 3, which leads to more current in 2 and 3. Thus, the currents are balanced for the overall connection.

 

[timselectric]

Remember that the full diagonal connection gives more current in 1 and 4, less in 2 and 3. This new connection adds more resistance for 1 and 4 which reduces their current, and it add less resistance for 2 and 3, which leads to more current in 2 and 3. Thus, the currents are balanced for the overall connection.

Right, I get how you got there.
It just doesn't seem right. But, you learn something new every day.
Thanks

Edit: I guess that I was more surprised by the full diagonal results. Not your final results.

 

[HRTKD]

Solarod, we've definitely seen out-of-balance situations when people use the diagonal method with their LiFePO4 batteries. It's good to see that the math backs up what we're seeing in real life.

I chose to wire each battery directly to the common bus bars. My testing, as seen from the BMS app, shows that my batteries share the load pretty close to equal.

 

[WYtreasure]

Hi, everybody! I'm a retired EE. I didn't want my EE skills to fade away after retirment, so I exercise them when I can. Whenever I see a non-trivial circuit analysis problem on one of various forums I frequent, I challenge myself to solve it. Recently I watched Will's video about current sharing in a parallel battery string, and it occurred to me that the string is just a circuit. It could be solved using standard circuit analysis techniques giving theoretical values for the battery currents. I wondered if this has been done before and posted on the web. I found a link on this forum to an article that gives numerical values: http://www.smartgauge.co.uk/batt_con.html This article solves a parallel string of 4 batteries with load connected in two ways--one case with the load connected at one end of the string and the other case using a diagonal connected load.

I decided to solve those two cases myself and compare my results with the results in the article. I drew up a model for the string and it turns out that
to completely solve the circuit takes 11 simultaneous equations, The article says its solution was done in about 1980, and the author says "the
calculation is incredibly difficult". Nowadays we have some very powerful mathematical software, so I knew that setting up the equations would be
the hard part; the actual number crunching would be easy since a computer would do it.

I set up the 11 equations and got a solution which I compared to the results in the article. My result was not the same at first, but having the
article's result to compare helped me track down my errors. I have got the equations right now. I don't want this thread to be a full on propellor head
thread, so I'm not going to post anything about the math; I'm only to give numerical results for a number of hookups and initial conditions. If anyone
wants to know about the math, ask and I'll do another thread.

The values for the initial parameters for the 4 battery string used by the article are 20 milliohms for the battery internal resistance, 1.5 milliohms for
the link cable resistance, and a 100 amp load. For the first case with the load connected to one end of the string I got the following values for the
battery currents in amps:

35.8
26.1
20.4
17.7

These are the very same values given in the article except for 3 single digit differences; these tiny discrepancies are no doubt due to rounding errors
in the mathematical calculations. I changed my setup to solve for the battery currents for the second case--the load connected diagonally. Using the same initial parameters I got the following values for the battery currents in amps:

26.7
23.3
23.3
26.7

Again, I got the same values as the article with a couple of small rounding errors. Getting this good agreement with the values in the article gives me confidence that I have got my equations right!

Welcome to the party Solarod.

I look forward to being able to comprehend what you have written. Trust me, I will work at it and ask questions about what I do not understand.

 

[chrisski]

would you mind sharing the 11 simultaneous calculations you mentioned?

 

[Solarod]

would you mind sharing the 11 simultaneous calculations you mentioned?

I don't mind, but I wonder where to put such a thread; it will be full-on propellorhead.

Perhaps @Will Prowse or one of the moderators has a preference.

 

[MisterSandals]

I don't mind, but I wonder where to put such a thread; it will be full-on propellorhead.

Why not add it here? Its your thread to do as you please.

And thanks! I learned something today. Looking forward to more...

It's got good potential to become a resource in that area.

 

[Solarod]

Having seen that simply moving the load connection to the 4 battery parallel string slightly away from the far corners of the diagonal, more toward the center of the string, can give a theoretical perfect balance, I can't help but wonder if this would work with parallel strings with other numbers of batteries.

I'm going to try it with a string of 3 batteries simply because that's less work than a 5 battery string would be.

Starting with the standard diagonal connection to the string:

With the initial parameter settings of 5 milliohm IR for all 3 batteries, 1 milliohm link resistance and 100 amp load, here are the calculated battery currents in amps:

Example 10
35.3
29.4
35.3

Now let's see what happens as the battery IR decreases. Setting the IR to 2 milliohm the currents are then:

Example 11
37.5
25.0
37.5

And next set the battery IR's to 1/2 milliohm. The calculated theoretical currents are then:

Example 12
42.9
14.3
42.9

Now let's leave the battery IR at 1/2 milliohm, but reduce the link resistance from 1 milliohm to .1 milliohm. This gives us these theoretical battery currents in amps:

Example 13
35.3
29.4
35.3

It's an improvement over Example 12, but it's still not very good.

Now let's try the trick that worked so well with the 4 parallel battery string in Example 9. Moving the connection of the load cables to a point halfway down the link at the top, and halfway up the link at the bottom, and restoring the battery IR to 5 milliohms and the link resistance to 1 milliohm, we get these battery currents (hoping they will be perfectly balanced ):

Example 14
32.4
35.3
32.4

Darn! It's pretty good, but it's not perfect. But I'm not one to give up. I tried adjusting the amount by which the load connections are moved toward the middle of the string, and with a few tries I found the golden ticket. If the load connections are moved 1/3 (???) of the way toward the middle of the string in the top link and the bottom link, with the battery IR of 5 milliohms and link resistance of 1 milliohm, the battery currents in amps are:

Example 15
33.3
33.3
33.3

Perfect theoretical balance just like the 4 parallel battery case in Example 9!!

 

[Solarod]

Why not add it here? Its your thread to do as you please.

And thanks! I learned something today. Looking forward to more...

It's got good potential to become a resource in that area.

I said in the beginning I didn't want this thread to be a propellor head thread. The community is not all EE's and I don't to scare anyone away from reading this thread all the way through.

 

[Solarod]

I have checked the 3 parallel battery string with the 1/3 offset of the two links giving perfect theoretical balance. This hookup has the same very worthwhile property the 4 battery hookup had, that if the IR of the 3 batteries changes any amount (within reason, of course), the perfect theoretical balance is not disturbed; same with changes in the link resistance. These properties theoretically only exist if the IR of the 3 batteries remain identical however they change due to temperature or SOC. In the real world, of course, they won't ever be identical, but if they're within normal ranges it's good.

With some of the other hookups, if the IR of the batteries changes but they remain identical (real world differences within normal ranges), the balance will be disturbed; for example the normal diagonal hookup to the extreme corners of the string. The balance of these standard hookups varies with changes in battery IR even if they track together, whereas the new hookups I've described in this thread are invariant with respect to tracking changes in IR. This is a new thing.

 

[chrisski]

Wherever this ends up I think this thread and the spin off that contains the math is ’pin worthy.’ I have had the smart gauge parallel battery in my signature block for a year and would love to see the math behind it.

IMO, the math portion could get posted in the resource section, perhaps in the planning and sizing subsection. https://diysolarforum.com/resources/categories/planning-and-sizing-tools.5/

the downloadable pdf file would not have the content lost in pages of back and forth as threads sometimes do.

 

[time2roll]

Thank you for this. The entire time I thought the lower battery resistance would help keep the battery load balanced. Of course that is flat wrong.

This actually says with LFP or similar low resistance batteries the balanced connection is more important than ever.

 

[chrisski]

Thank you for this. The entire time I thought the lower battery resistance would help keep the battery load balanced. Of course that is flat wrong.

This actually says with LFP or similar low resistance batteries the balanced connection is more important than ever.

I don’t have a real imeadance measrement device for my batteries like a yr1035+, but I do have a milliohm reading on my batteries as I do a capacity check. Ican’t speak to the accuracy, but the internal resistance is significantly more empty than full, and it builds resistance quickly. On my 25 ah cells, they started with 1 - 4 milli ohms, but went up to 25 to 45 milliohms. I would think if resistance goes up as batteries drain, it would make them a bit more balanced than the lower resistance math shows.

 

[Solarod]

Wherever this ends up I think this thread and the spin off that contains the math is ’pin worthy.’ I have had the smart gauge parallel battery in my signature block for a year and would love to see the math behind it.

IMO, the math portion could get posted in the resource section, perhaps in the planning and sizing subsection. https://diysolarforum.com/resources/categories/planning-and-sizing-tools.5/

the downloadable pdf file would not have the content lost in pages of back and forth as threads sometimes do.

I was thinking I could put it in the "Danger Zone" sub forum.

I haven't heard from any moderators or Will about any preference.

 

[Solarod]

Thank you for this. The entire time I thought the lower battery resistance would help keep the battery load balanced. Of course that is flat wrong.

This actually says with LFP or similar low resistance batteries the balanced connection is more important than ever.

That article I referenced with calculations made 32 years ago set the battery IR at 20 milliohms; really? The LFP batteries everyone on this forum are using have IR typically less than 1 milliohm, and those numbers I posted for the 1/2 milliohm IR in various hookups like Example 6 are terrible. You can't get cable and lugs big enough to balance that which wouldn't be too big to fit on a battery terminal; I think bus bars are a must especially since these modern setups are often pulling multiple 100's of amps, and may very well overheat a cable with lugs small enough to fit on a big LFP battery.

 

[SunbeltLocked]

Wherever this ends up I think this thread and the spin off that contains the math is ’pin worthy.’ I have had the smart gauge parallel battery in my signature block for a year and would love to see the math behind it.

IMO, the math portion could get posted in the resource section, perhaps in the planning and sizing subsection. https://diysolarforum.com/resources/categories/planning-and-sizing-tools.5/

the downloadable pdf file would not have the content lost in pages of back and forth as threads sometimes do.

I remember studying something like this in my EET courses at DeVry:
nodal analysis

 

[Solarod]

I don’t have a real imeadance measrement device for my batteries like a yr1035+, but I do have a milliohm reading on my batteries as I do a capacity check. Ican’t speak to the accuracy, but the internal resistance is significantly more empty than full, and it builds resistance quickly. On my 25 ah cells, they started with 1 - 4 milli ohms, but went up to 25 to 45 milliohms. I would think if resistance goes up as batteries drain, it would make them a bit more balanced than the lower resistance math shows.

Quite true. Conversely, as the batteries are being charged, the IR will go down leading to worse balance.

The remarkable property of these new connections I've discovered is that the balance doesn't change when the IR changes, as long as all the batteries' IR tracks under all conditions, temperature, SOC, etc.

 

[RCinFLA]

The 1 kHz impedance measurement only represents a smaller portion of cell terminal voltage slump. Overvoltage slump dominates at most used range of load current. 1 kHz impedance is mostly conductive resistance of cell metallic connections within cell, with a minor amount of ionic diffusion resistance.

Cell 1kHz impedance cell terminal voltage slump has an almost linear relation to cell current.

Overvoltage slump, also called polarization potential, is overhead energy used by cell to move lithium ions within cell to supply demanded current. Its terminal voltage variance value has a logarithmic relation to cell current. It is highly dependent on aging of cell and cell temperature.

As cell ages, overpotential slump will degrade 3x or more from its initial new cell overpotential value for moderate, 0.2-0.4 C(A) cell demand current. Overpotential cell terminal voltage slump at moderate cell current correlates well to cell matching and extractable capacity. It has an exponential time constant decay to equilibrium. In the 15% to 90% state of charge range for LFP, the time to equilibrium terminal voltage is in the 1 to 3 minute range. For given model cell, it is dominate effect for cell matching. 1 kHz impedance measurement is not a good indicator of cell matching.

Overvoltage slump is highly dependent on cell temperature. It rises at a greater rate below about +10 degrees C. This is why LFP performance drops off significantly at cold temps.

Many folks have trouble with bus bar connection resistance due to aluminum oxide build up on cell terminals. Clean terminals with solution of 50% - 70% water diluted white vinegar to remove oxide then clean then dry with 91-99% isopropyl alcohol. Aluminum oxide regrows quickly so terminal connections need to be clamped down soon after cleaned and dried. Keep salty fingers off of contact surfaces.

FYI, the slight bumps in SoC voltage curve is due to the way lithium ions are stored in the negative electrode graphite lattice layers. The lattice layers are like a multi-level parking garage that only opens particular levels when previous parking level is nearly full. The most prominent bumps are at about 55% and 10% state of charge points where a new parking level arrangement happens. There are four other minor bumps that are harder to detect by SoC open circuit voltage.

For LFP cells, the overall cell voltage is the positive LFP cathode electrode potential minus the negative graphite electrode potential. LFP cathode has a very flat voltage versus SoC until less than about 5% state of charge. Most of the SoC voltage profile is due to graphite negative electrode which ranges from about -0.25 vdc at totally discharged to about 0 volts at fully charged. LFP positive electrode potential is almost flat at 3.43 vdc from fully charge to about 5% state of charge where it starts to quickly collapse. This means fully charged cell is close to 3.43vdc rested open circuit voltage.

Cell open circuit voltage when absorb charged above 3.43v is due to surface charge, mostly in graphite. This has very little capacitance storage capacity, approximately 0.01% of cell AH capacity, and can be quickly dissipated with a short load. 0.01% of 280 AH cell is only 28 mAH's. This surface charge will bleed off on its own if cell is left open circuited in a few hours to a few days. Don't think because your top balanced cells have rested open circuit voltage between 3.45v and 3.65v they are not balanced. As long as their rested open circuit voltage stays above 3.45v they are full charged and balanced.

 

[Solarod]

Here is a comparison of two hookups of 4 batteries in parallel, with an image from Victron to illustrate:

The hookup on the left is the well-known "Halfway" connection which gives theoretical perfect balance. It depends on the IR of the 4 batteries remaining identical, and, of course, in the real world those parameters will not perfectly track. If the batteries are nominally identical, same manufacture date, etc., even if there is some lack of equality of battery IR the balance will be good.

The hookup on the right using busbars with the usual connection of the load (red and black cables) to the extreme corners can have good, but never theoretically perfect balance. If the connection is modified slightly as shown by the yellow and blue cables, the balance becomes theoretically perfect, with the requirement that the IR of the 4 batteries track. A particular advantage of the yellow/blue connection is that the balance does not require the busbar resistance to be very low (but it shouldn't be high enough to cause the busbar to overheat).

To change the usual connection as shown by the red and black to the yellow and blue connection is trivially easy. To implement the "Halfway" hookup with busbars will not be as easy. The choice seems slanted toward the busbar and yellow/blue hookup.