Lithium battery math, better than you may think
There’s a lot of talk about installing lithium batteries in boats, but the enthusiasm often comes to a crashing halt when boat owner hears the cost. I’ve just installed a Mastervolt lithium system in Have Another Day and I’m over the price shock. But as I work to understand the results of my upgrade I’m amazed by the benefits I hadn’t anticipated. Follow along with my math and see if you reach the same conclusions I do about the benefits of upgrading to a lithium house bank.
Deep cycle batteries are typically sold based on their voltage (v) and capacity measured in amp hours (Ah). For the sake of this article I’m going to compare an inexpensive flooded lead acid (FLA) battery and one of the most popular drop-in lithium iron phosphate (LiFePO4) batteries. I’ve selected a 100 Ah Duracell Group 31 12-volt deep cycle battery as the FLA exemplar. There are better batteries out there, but at $100 for a 100 Ah battery this is a value-leader. For comparison I’m going to use a BattleBorn 100 Ah 12-volt LiFePO4 battery that currently sells for $900.
I’m going to focus on cost not because it’s the only way to view the benefits, but because it’s probably the easiest way to understand what you’re getting. However, there are lots of other important benefits I touched on in part one of my Mastervolt install article, like faster charging allowing reduced generator runtime. Plus LiFePO4 batteries are lighter, smaller, sealed, better monitored, and have more safety systems than comparable FLA batteries.
To start, we’re going to look at the cost per amp hour of each battery. This math is pretty simple, the $100, 100 Ah Duracell is $1/amp hour. The $900, 100 Ah BattleBorn is $9/Ah. Right now the LiFePO4 battery is looking pretty darn expensive. However, I’ve been advised that it’s way more accurate to look at the cost per watt-hour (Wh) instead of the cost per amp hour, so let’s look at that instead.
In direct current (DC) calculating watts is very easy, it’s just volts times amps. So, a 12-volt 100-amp hour battery is a 1200 watt-hour battery. For the FLA battery the math will be $100 ÷ 1,200 = $0.083 / Wh and for the LiFePO4 it’s $900 ÷ 1200 = $0.75 / Wh. But, wait, that’s not a true comparison due to usable energy and nominal voltage factors, plus Peukert’s Law.”
The life of a battery is affected by how much it is discharged. The less you discharge the battery, the more discharge-charge cycles it will survive. The general wisdom seems to be that it’s best not to discharge an FLA battery past 50 % depth-of-discharge (DOD) and a LiFePO4 battery past 80%.
So, now, let’s take our watt-hour numbers and compare again, but this time based on useable energy. For FLA we have 1,200 watt-hours but only half are usable, so it’s 1,200 Wh * 0.5 = 600 usable watt-hours. At a cost of $100 for 600 usable Wh, we are at 0.17 per Wh. For LiFePO4 we also have 1,200 watt-hours, but 80% of those are usable, so 1,200 Wh * 0.8 = 960 usable Wh. At $900 for 960 usable Wh each watt-hour costs $0.94. We’re down to only a little over five times the cost for LiFePo4. That’s still a huge difference, but stick with me, there’s more to consider.
A 12-volt lead-acid battery is typically composed of six, 2-nominal-volt cells. Nominal voltage measures the typical voltage of a fully charged cell while being discharged with a load. So, the six cells combine to create a 12-nominal-volt battery. A 12-volt LiFePO4 battery is composed of four 3.3-nominal-volt cells meaning the battery actually has a 13.2 nominal voltage.
The difference in nominal voltage means that a 12-volt, 100-amp hour, LiFePO4 battery (13.2-nominal-volts) has 10-percent more watt-hours available than a 12-volt, 100-amp hour, flooded-lead-acid battery. Let’s take a look at the math behind this. For LiFePO4 we have a 13.2-volt battery with 100 amp hours of capacity. This yields: 13.2 v x 100 Ah= 1,320 watt-hours. For FLA we have a 12-volt battery with 100-amp hours of capacity. This gives: 12 v x 100 Ah = 1,200 watt-hours.
We can redo our math for the cost of usable watt-hours based on the new capacities. For FLA, nothing has changed so we remain at $0.17. For lithium the math is now 1,360 Wh * 0.8 = 1,088 usable watt-hours and $0.83 per usable watt-hour. We’re down below five times the cost, but we’re still not done.
The capacity rating of a battery is typically given assuming a 20-hour discharge cycle; meaning, that a 100-amp hour battery will be discharged at 5 amps per hour for 20 hours. But, for lead-acid batteries, the faster you discharge the battery, the less capacity is available. Peukert’s component is a measure of the reduction in capacity resulting from larger loads. Mastervolt says their LiFePO4 batteries aren’t impacted by faster discharging and will maintain their rated capacity (scroll to the section on Peukert’s Law in the linked article).
As you can see in the chart above, discharging FLA batteries above their 20-hour rate can significantly decrease the amount of actual power they can deliver. For example, a 100 Ah FLA battery discharged at 25 amps (five times its 20-hour rate) will only deliver 71.3 amp hours.
Up to now, all of our cost comparisons have been based on the 20-hour rate of 5 amps. But, instead, let’s compare at a discharge rate of 20 amps. At 20 amps of discharge, the previously 100 amp hour battery actually provides 74.75 amp hours. At 12-volts, that’s 897 Wh, but we only want to discharge the battery to 50-percent, so we have 449 useable Wh. $100 ÷ 449 Wh = $0.22 per usable watt-hour for FLA. LiFePO4’s cost doesn’t change since they’re not impacted by the rate of discharge.
Now we’re at $0.22 per usable Wh for FLA and $0.83 for LiFePO4. These numbers can vary a lot. If you look at the comparison at 10 amps per hour the cost for lead-acid declines to $0.19, on the other hand, if consumption increases to 40 amps the cost increases to $0.26.
Another big difference between lead-acid and lithium batteries is the behavior of the batteries under larger loads. Lead-acid batteries’ voltage drops significantly under heavy loads. On the other hand, LiFePO4 voltage barely moves. The chart above shows a period of huge draws from the oven onboard Have Another Day and the nearly perfect stability of the voltage under the load. This was an unintentional test that I’ll detail in the second installment about my refit, but it also provided a great test of how the batteries perform under big loads. If I put this same load on my prior FLA battery bank the inverter would have shut down due to low voltage moments after the load started.
Even under moderate loads, voltage stability can affect capacity. That’s because an electrical load that’s being serviced by lower voltage will draw additional amperage. If you have a 100-watt load, at 12 volts it will require 8.3 amps of current, but if the voltage should sag half a volt, you will now need 8.7 amps for the same load. This additional current means the battery will be depleted sooner.
I don’t have good measurements of voltage sag and I don’t suspect it will change our cost per watt-hour significantly. But, I can tell you that not having the voltage sag under big loads sure feels better. It doesn’t seem like the whole DC power system is struggling under the load.
Battery life versus cost
The last way to compare the two battery technologies is by comparing how much energy they can deliver over their life. Battle Born says their batteries have a life of 3,000 – 5,000 cycles. The chart above suggests around 400 cycles for the flooded lead acid. Duracell offers a one-year warranty on their batteries but doesn’t publish a rated spec.
If we assume the Battle Born battery will last 3,000 cycles and deliver 1,088 watt-hours per cycle that gives us a total of 3,264,000 watt-hours or 3,264 kilowatt-hours (kWh). At 400 cycles for the Duracell and 449 watt-hours per cycle, we have 179,600 watt-hours or 179 kWh. This is when we see a big change in the cost comparison. These numbers mean that over their life the FLA battery costs $0.55 per kWh and the LiFePO4 costs $0.27 per kWh.
The cost per kilowatt-hour is a lot lower for LiFePO4, but I suspect that many lithium battery owners will never cycle them that often and are also unlikely them down to 80-percent DOD each time. With that in mind, the cost per kWh of LiFePO4 batteries is more likely to be comparable with FLA.
But let me be clear, if the costs of the two technologies are even close I think that LiFePO4 is a huge upgrade well worth your investment. It may take years for that investment to pay off, but during those years you will enjoy the many lithium benefits like faster charging, better voltage stability, more advanced safety systems, and reduced weight and size. For me, all these benefits and similar total costs make this upgrade a no-brainer.
Lastly, please let me know if you see any logical or mathematic errors in what I’ve outlined. I believe I’ve correctly applied all the considerations but I certainly may have made a wrong turn somewhere.