balancing.txt

Balancing algorithm




Preface

The purpose of (top) balancing is to make so that all cells hit 100% SoC at exactly the
same time when charging, i.e., they are full together. When capacity differences occur
as in any real system, the SoC levels do not match at bottom, i.e., when the pack is empty.

The primary purpose of top balancing is to maximize pack energy storage capability, since the 
cells provide their charge at maximum possible voltage. (Balancing at any other point at the 
SoC line provides a little bit less energy stored in the pack, although the same amount of
charge (coulombs).)

In a balanced pack, the smallest cell capacity defines the pack capacity. When discharging a
top balanced pack, the smallest cell reaches the low-voltage cutoff level first.

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TOP-BALANCED PACK

      Full                           Empty

   SoC          Charge Voltage        SoC          Charge Voltage     
|||||||||| 100%  10Ah    3.45V     ||--------  20%  2Ah    3.10V
|||||||||| 100%   9Ah    3.45V     |---------  11%  1Ah    3.00V
|||||||||| 100%  11Ah    3.45V     |||-------  27%  3Ah    3.12V
|||||||||| 100%  10Ah    3.45V     ||--------  20%  2Ah    3.10V
|||||||||| 100%   8Ah    3.45V     ----------   0%  0Ah    2.80V

                         ---> delta Charge = 8Ah (the capacity of the smallest cell)

UNBALANCED PACK

      Full                           Empty

   SoC          Charge Voltage        SoC          Charge Voltage     
||||||---- 60%    6Ah    3.28V     |||-------  30%  3Ah    3.14V
|||||||||- 89%    8Ah    3.35V     ||||||----  55%  5Ah    3.27V
|||------- 27%    3Ah    3.12V     ----------   0%  0Ah    2.80V
|||||||||| 100%  10Ah    3.45V     |||||||---  70%  7Ah    3.30V
|||||----- 50%    4Ah    3.25V     |---------  13%  1Ah    3.02V

                         ---> delta Charge = 3Ah.

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An unbalanced pack does not have a point where all cells share a common SoC%. Such a pack may have
severely reduced capacity, because the highest cell (cell with most charge) defines when to stop
charging and the lowest cell (cell with least charge) defines when to stop discharging, and
this difference may be smaller than even the smallest cell in the system.

Pack can be (top) balanced by charging every cell separately so that they are at 100%, before connecting
in series.

Assuming ideal battery technology, charging and discharging series-connected cells itself does not 
change the balance, because the series connection simply forces exactly the same current through every 
cell and hence, exactly the same amount of charge (coulombs (As), or Ah) is consumed from every cell.

So the only way the pack can drift away from a balanced state is an excess external connection
over some of the cells; or it can be an internal mechanism in the cell.

An external way to cause unbalance is to design a BMS that uses too much current so that it's significant.
(Actually, only the _difference_ between the current consumed between the cell modules is relevant, but 
one typically leads to other.) 

This BMS is designed to have mimimal current consumption which is well below the internal cell self discharge.

Assuming no unbalanced external connections, the battery technology itself has two internal mechanisms
causing uneven charge drain: differences in Coulombic Efficiency, and differences in Self Discharge (i.e.,
Leakage Current).

Because the cell consists of a very large surface area of the electrodes with very low distance
from each other, only separated by a thin, porous plastic sheet, impregnated with electrolyte solution,
this construction will inevitably conduct some electricity, hence causing self discharge. Because of the
very large variations measured between self discharge effect, manufacturer tend to specify a lot worse
than average conditions. Hence, for some Chinese prismatic LiFePO4 cells, 3% of capacity per month
is often specified but rarely measured in real life. Most manufacturers do not specify self-discharge at
all. 

It is indeed typical that the self discharge is so low that it is difficult to measure at all and causes
only minimal drift, hence advocating the use of non-balancing (only monitoring) BMS or no BMS at all.
A small portion of cells however can exhibit considerable self discharge, especially after aging, and
therefore some balancing mechanism is a good-to-have in the BMS designer's toolbox. 
In addition, it isn't guaranteed that all future products or different chemistries would have as good 
self discharge features as li-ion cells currently available.





Balancing

A theoretically optimum way to balance a pack is to provide charge to those cells that lack behind at the end of
charging. However, this requires expensive isolated converters that can supply power to the cells
independently even when they are series-connected by fixed connections. This is the preferred way with 
lead-acid batteries as they have quite a bit of balance drift.

However, as lithium ion needs only very little balancing action, an easier way of dissipating "excess" charge
from full cells as heat (called "shunting") is typically used. If the shunting current equals to 
charging current, the net effect at the cell is zero, and the charging can be continued as long as it takes
to get the lowest cell full, without danger of overcharging the highest cells - after all, all the 
charging current is turned into heat; you could say the current "skips" the cells through a resistor.

Because li-ion balance drifts only very little, this kind of "shunting" fixes the unbalance
in just seconds, without wasting much energy as heat at all.

In practice, having a balance current that matches the charging current would not only be horribly expensive
but it would also pose a serious problem of heat removal in case the battery unbalance is not minor after all,
or there is any design error that leaves the resistor on, so balancing with a current more than, say, 
500 mA is definitely out of question.

So in practice, BMS's need to request the charger to lower their charge current for the balancing stage, or utilize
the end of the CV charging stage where the current is low. But even C/20 stopping current is a lot more than
practical dissipative shunting can support. Therefore, practically any BMS can not fully rebalance an unbalanced pack
"in real time" during the charging.

Therefore, the integrated balancing function in most if not all li-ion BMS's is only there to compensate
for very low-level drift caused by self-discharge and BMS's own consumption and cannot provide gross balancing.
In any case, the "shunting" current is lower than charging current at the same point so the "shunting" becomes,
strictly speaking, a misleading term. The purpose is to remove a small bit of energy from some of the cells so
that the SoC%s match more closely at 100%. A gross unbalance is only fixed a little bit at a time, and there
is always a negligible unbalance left.

So, given the low self-discharge of the cells, and the very low discharging action of our BMS design,
while agreeing that some balancing is good to have, we questioned the need for even the typical 200-500 mA range
of balancing currents. Even 500 mA is NOT enough to provide gross balancing anyway, and it cannot fully balance
in real time. Additionally, it can cause thermal design problems and at least increases the BMS cost. The only 
benefit of such a large balancing current is the ability to use higher-discharge ("almost faulty") cells, 
those that have self-discharge in the range of 1-3% per month.

However, typical shunting BMS's still don't implement much intelligence in how they remove energy from the cells, so they
do shunting only during the end of the charging stage, which can be very short in time. We felt that we could
remove the same amount of energy from the cells by using 1/10th of the current for 10 times longer. Note that
it is irrelevant when this is done; it doesn't need to be at the end of the charging stage. Li-ion doesn't need
to end up at 100%; it can end up at 99.9% with no problem. This means that due to unbalance, charging stops when
first cell hits 100%, while the lowest cell is at 99.9%; then, energy is removed from all other cells to match 99.9%.

This actually makes more sense than carefully charging at very low currents for a long time to exactly match 100%,
and saves energy because a high-power charger is inevitably very inefficient at nanoscopic currents. Having the cells
lack the amount of charge equaling the differences in smallest and largest self discharge * time interval between
two charges, will be minimal and will correspond to maybe a few inches of actual drive range in an EV. Lengthening
the charge process to carefully top off at 100% every time is not worth it.

An additional benefit in our approach is that no system error can accidentally cause large amount of shunting, causing
heat problems or quick discharging of the cell (which would destroy the cell).

But while other BMS's have just thrown in the dissipative balancing that happens at a certain voltage during the top-off
period, our approach needs a little bit more intelligent algorithm consideration to be able to handle the same amount
of dissipation of charge (or even more!) with high self-discharge cells, if those exist in the system.

Note that these algorithms apply to almost all BMS's, excluding those with powerful redistributive (isolated charging)
features or very strong (separately cooled) dissipative shunting. Most still do not implement these. We just have 
an stronger reason to think about this because we really lack the "raw power" in shunting.




The Balancing Algorithm

The first revision still uses only the time between the finishing of the charge and the startup of the car for
balancing, like most BMS's; the difference is that the charger doesn't need to stay on at low power 
(or pulsed at low duty cycle) during the balancing, which, in practice, increases the amount of time available
for balancing in most cases (e.g., chargers use timers to terminate the CV phase eventually, etc.)

What we actually need to do is to know the self-discharge of all cells Ileak_0...Ileak_n, and time spent after
last balancing t, and then remove charge C_i = t*(min(Ileak) - Ileak_i) from all cells i=0...n.

In practice however, we do not know the self-discharges, they are very difficult to measure, and even if we did,
they could change as the cells age. The process would be error prone. That's why no BMS uses this kind of approach;
they just approximate the SoC from the voltage and remove "some" amount of charge from the higher voltage cells.
As long as the charge removed is more than C_i defined above and at sane levels, the cell unbalance is not increased
over a certain level. (Let's remember that the balance is only there to maximize energy; a small unbalance is not
worth fixing if that costs too much or increases complexity, as the actual gain would be a few more inches of range).

At the end of the charge, we know which cells are higher than others, and can simply remove some charge from them. This
can be done according to the voltage differential, e.g., if a cell is 100 mV higher than the lowest cell, we remove
double the charge from that cell compared to another that is only 50 mV higher. Doing this after charging ends would
leave the pack a bit undercharged, as energy is removed. This is however the simplest algorithm that doesn't need a 
memory element. Let's call this Algorithm 1.

A better way would be storing the voltage readings at the end of the charge in (the master's) EEPROM. Then, we could
start the shunting at the beginning of the next charging, instead of doing it after the fact. So, at the end of the
charging, the unbalance would already be fixed and the pack could remain at balanced 100% SoC. Let's call this 
Algorithm 2. This is effectively the same as Algorithm 1 in how it balances the pack, but it leaves the pack at 
full state without the need of starting the charger up again (or running it at very low current).


Algorithm 1:

3.55V   -> +0.10  -> Shunt for 10 minutes
3.65V*  -> +0.20  -> Shunt for 20 minutes
3.60V   -> +0.15  -> Shunt for 15 minutes
3.45V** -> +0.00  -> Do not shunt
3.50V   -> +0.05  -> Shunt for 5 minutes

*) Hits HVC, save the voltages and then stop charging.
**) Lowest, use as the reference level (no shunting).

Algorithm 2:

Same, but save the values in non-volatile memory and start shunting the next time charger is run.


Algorithm 3:

But what if "20 minutes" is not enough to fix the voltage differential of 0.20V, or if it is too much?
A self-tuning multiplier is introduced that tunes the algorithm to the curve steepness of the actual battery
(which is the function of battery chemistry, cell capacity, internal resistance and charging current just before
stopping, which, hopefully, are all more or less constants but unknown).

For this purpose, we would store the voltage differentials of the two latest charging endpoints.

 First charging                Second charging

3.55V    +0.10  -> Shunt for 10 minutes  ->  3.58V   +0.07
3.65V*   +0.20  -> Shunt for 20 minutes  ->  3.65V*  +0.14
3.60V    +0.15  -> Shunt for 15 minutes  ->  3.62V   +0.11
3.45V**  +0.00  -> Do not shunt          ->  3.51V** +0.00
3.50V    +0.05  -> Shunt for 5 minutes   ->  3.55V   +0.04

Now we would see that, for every cell, the differential has reduced to about 70% of the original. If we
wanted to remove the differential (i.e., reduce it to 0%) for a fully balanced pack, we would increase
the multiplier to 3.33 times the current multiplier, i.e., 20-minute shunting would become 66.7-minute
shunting, hence:

 First charging                Second charging               Third charging:

3.55V    +0.10  -> Shunt for 10 minutes  ->  3.58V   +0.07  -> Shunt for 23 minutes
3.65V*   +0.20  -> Shunt for 20 minutes  ->  3.65V*  +0.14  -> Shunt for 47 minutes
3.60V    +0.15  -> Shunt for 15 minutes  ->  3.62V   +0.11  -> Shunt for 37 minutes
3.45V**  +0.00  -> Do not shunt          ->  3.51V** +0.00  -> Do not shunt
3.50V    +0.05  -> Shunt for 5 minutes   ->  3.55V   +0.04  -> Shunt for 13 minutes.


This basically means a P controller for balancing, and a PI controller to control the P term.

We can have separate multiplier (P term) for every cell, or share the term because the conditions
(cell chemistry, capacity etc.) should be same for the pack so this would help averaging possible errors
out.

Probably, to avoid oscillations, we wouldn't want to try to fully fix the unbalance for the next charge,
so if the differential was reduced to 70% of the original, we wouldn't increase the multiplier to 3.33
times the current multiplier, but maybe to 1.5 or 2 times the current multiplier. This way, the multiplier
would slowly reach the values really needed to balance the pack as quickly as possible.

Multiplier as well as shunting times naturally need limits.

Also, when the algorithm has balanced the pack, it's expected that the voltage differentials will be close
to zero (after all, that's the purpose of the algorithm). In this case, it's impossible to accurately
calculate a new multiplier. But this is easily addressed by using some kind of threshold in the multiplier
control loop. This may automatically result in from integer (fixed point decimal) calculation.

Note that time between the charges doesn't need to be taken into account, because we use voltage differentials.
If the cell has more time to self-discharge, it will inevitable develop even lower voltage. This is the beauty of
the simplicity of the P controller.

Now, if the car would be left unused for some time and some of the cells would therefore self discharge
more than others, the multiplier would be already tuned so that the realized voltage differential would
give proper shunting times.

The problem here is of course the strong non-linearity of cell voltage vs. charge. Therefore, the
algorithm only works on a small portion of the curve which is very close to the endpoint.