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After giving it some thought, this issues deserves a thread for itself, even though it is tied to BEV competition, which has a thread.

If you want to know which OEM's will have a meaningful (say, several 100k EV sales) in the next 5 years you need to talk about battery capacity.
So, who is currently building capacity?

Lets look at the situation at the moment (2014):

(link: Top EV Battery Manufacturers -- 2014 Top 10 In Sales)

We know about the Panasonic/Tesla 35GWh by 2020 gigafactory,
There is also a 34GWh production capacity in several sites by BYD, again by 2020.
(Source: http://www.reuters.com/article/2015/03/13/byd-battery-idUSL1N0WE37P20150313)

One source claims Nissan might be planning to look outside AESC for batteries, and might turn to LG Chem:
Nissan's Electric-Car Battery Future: Will It Be With LG Chem?

Which brings up the question, what is the planned expansion for LG Chem?
Here is one plant:
Koreas LG Chem Constructing Chinese EV Battery Plant
"Once completed, the Nanjing battery plant will possess an annual production capacity of over 100,000 EV battery units a year."
This is clearly intended for the Chinese EV market.
From other sources, LG Chem currently has two existing battery plants. One in US, another in Korea.
So beyond the Chinese plant, will they also upgrade the existing plants?

This article claims they were at least at one point thinking about it:
"Car battery maker LG Chem to decide on capacity expansion in 3 months"

I couldn't find the follow up though.

As for what kind of capacity would they want, here is an interview Prabhakar Patil, CEO of LG Chem:
Battery maker shrugs off Tesla competition

Relevant takeaways:
1. They expect to have a 25% - 30% market share of capacity going forward (of what expected capacity??).
2. Mr. Patil has some doubts for the 500k per year capacity Tesla expects.
3. LG Chem's Holland, Mich., plant is working at 25 to 30 %of capacity, it needs to be at 80% to be profitable.
OK, so they probably won't be expanding it any time soon...

Altogether I don't see a plan for significant capacity expansion, excluding the new Chinese plant.

If there will be new suppliers, say, in China, that might be expanding capacity besides BYD, it would be an interesting story, but they would be starting from a low base point as the above graph shows.

In conclusion:
1. The big stories seem to be Panasonic/Tesla, BYD, and any future plans for expansion by LG Chem.
2. Currently there is one visible expansion plan for LG Chem, that is a new battery plant in China, for China.
3. There could be new other battery suppliers but they will be starting from a relatively low base point.

In any case, this is worth following since the future of EVs is tied completely into battery capacity. In other words, no batteries - no EVs.
2015 Q1 update on the numbers:
Taken from Cleantechnica ( 10 Biggest Electric Car Battery Manufacturers Are... | CleanTechnica )

Panasonic is actually gaining market share, faster than any other competitor:

Here you can see a more clear compare to the 2014 numbers:

This is even before taking into account the gigafactory, which will supply Tesla exclusively.
If you think Panasonic is all about Tesla though, think again, from this article (in French)
Les batteries dElon Musk arrivent en Europe sous la marque Panasonic
Panasonic is entering the residential storage business with its own product, the PowerStation. It is already offered in Japan for a few years now, since the Fukushima disaster.
With the massive advantage in economy of scale that Panasonic has, it will likely be a clear leader in its own right.
It will be interesting though to know what is are the prices that Panasonic will offer, and how they compare to the competition.
I've been trying to find some data on LG Chem's total battery build capacity. They've been chronically underutilized at their primary factories in Ochang, Korea. And yet they built a 1 GWh battery factory in Holland, MI for just over $300 million (half DoE loan, half LG money). According to this article:

LG Chem to Crank up Its Idle Battery Productions Lines | Korea IT Times

the Ochang factories have 3.2 GWh of battery production and the utilization rate at some point in 2013 was less the 20%. They had announced that they would invest roughly $1.8 billion through 2013, but unclear how much of that was actually spent. In any case, they are hurting with overcapacity.

The Holland, MI plant has been running with only 1 line out of 3 in operation. They are now expecting to build out the rest of the lines and hire the staff to run the plant at capacity, probably for the upcoming Volt 2, the Bolt, and the Ford EV products. See this article:

Global Expansion: LG Chem to Bring Holland Battery Plant Online This Year | BusinessKorea

I can't quite make sense of the last sentence though, as I am not sure what they mean by units. Even so, 400,000 units out of the Michigan plant seems unlikely even counting small battery packs for non-plug-in hybrids.

They have also built a new plant in Nanjing, China to supply 100,000 packs, but everytime they use the "packs" volume measurement, it is unclear how many kWh goes into a pack for that statement. Given the $500 million investment, I'm thinking it's around 1.5 GWh.

So I'm guessing LG has about 6 GWh of global lithium polymer battery production available. They are actually shipping less than 1 GWh. That means they can supply 120,000 x 50 kWh packs at full tilt. Not sure about the cost structure though, as presumably the Holland, MI plant output should end up being the cheapest for U.S. bound vehicles. The Nanjing plant is a joint venture to product packs for Chinese EVs, which usually have small packs. Also, as the energy density increases, the number of watts per cell increases which means the nameplate on these plants go up with battery chemistry changes.

Last year, LG Chem said they would invest $197.2 billion in battery plants for 2014 and they recently stated that they expect to invest $266 billion for 2015.
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I came across this article about NEC (which is part of the Nissan NEC JV, AESC) :
NEC to Deploy 60MW of Energy Storage in PJM : Greentech Media
"NEC to Deploy 60MW of Energy Storage in PJM"
Does capacity in grid/home battery storage transfer to capacity for EV batteries? My first intuition is partially yes. Tesla for example uses a NMC cathode for the 7 kwh battery, and NCA for all car batteries, yet they say it would be simple to keep both running in parallel at the gigafactory, or even switch to new chemistries in the future.
So even if NEC uses different cathode material for the grid storage packs there could be advantages for cars.
From a recent study both Nissan/NEC and Tesla/Panasonic enjoy a market lead with packs costing ~$300/kwh (probably less).

(link: http://www.nature.com/nclimate/journal/v5/n4/full/nclimate2564.html )
This might change the outlook for battery capacity for EVs worldwide: I'd like to have a list of all Li-Ion battery capacity being built or in existence, then look at the chemistry to see what would be most relevant to cars in the future.
Ok, I don't see much relevant news for EVs in the grid storage space.

There is a partnership between BYD and ABB, I couldn't find much information about the amount of storage they are planning but typical numbers for storage for BYD seem to be in the single digit MWh. Meh.
(Link: ABB partners with BYD to develop storage technologies: pv-magazine )

The leader seems to be AES (no connection to AESC), which has deployed 100 MWh so far and has near term plans for another 260 MWh. Interestingly their cost is at $250/kWh, same as Tesla's. (Link: AES Energy Storage Targets $30B Peak Power Substitution Market : Greentech Media)
They are definitely using Li-Ion (they say so), but I couldn't find what kind of chemistry they are using. Or if they would like to get into EVs in the future.
Either way all existing and planned capacity would put them with capacity less than Mitsubishi in 2014, this puts things in perspective.
I've been playing around in excel with the numbers, to try to get some estimate on how many electric cars should we expect in 2020.
First here is the graph, you can guess quite a bit of what I was trying to do from just seeing it, but I'll explain further:

I took the first quarter of 2015 and duplicated by 4, and extrapolated the 2014 to '2015' growth for another 4 years. Then I ignored the data for BYD, LG Chem and Panasonic, and inserted for each a different estimate: for BYD, they said they will do 34 GWh by 2020 so that is what I put. For LG, with the help of techmaven (thanks techmaven) I used an assumption that by 2020 they will use all available existing capacity plus the 1.5 GWh Chinese plant (a generous assumption). Finally for Tesla, they said they will do 500k a year. That works out to be all 2015 Panasonic capacity, plus 2/3 of 35 GWh Gigafactory. So that is what I used.

The figure I came up with is around 1.5M EVs per year, the 'unit' for EV being 50 KWh. Thats basically a model 3. There could be more hybrids of course, but I needed an apples to apples compare, so I picked one.
Last number is for perspective, what percent of the total car fleet would 1.5 M be. Not much...

EDIT: I see I messed up the AESC numbers. But the estimate for 2020 seems reasonable.
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Ok... I wanted to correct the data anyway.

The AESC data was way off. And I gave the small cell manufacturers a linear growth last time, which is an unfair assumption. For Tesla the growth rate is 50% a year. That is probably on the outer limit, so I just gave them a 25% growth rate. I also wanted to take into account an average improvement of up to 8% in cell density. It was published that Carlos Ghosn actually wants to cut production for AESC. So I assumed only 2015 production plus the 8% improvement per year.
(Link: http://www.reuters.com/article/2014/09/15/us-renault-sa-nissan-batteries-exclusive-idUSKBN0HA0CA20140915 - they also cite a research that claims there will be a little less than 1M EVs in 2020, which is definitely on the conservative side).

All in all I made all the assumptions to maximize capacity ("best case"). Here is the data:


Here is the distribution by market share based on that capacity:

A few minor corrections
1. As suggested, I took the average of 350 60 kWh (model 3's) and 150 85 kWh (S and X) for Tesla.
2. I googled all of the relatively small manufacturers. GS Yuasa seems to be incurring debt from the batteries, so I don't see them expanding. I extrapolated only 8% improvement from battery energy density improvement.
3. Samsung is actually expanding, they are building a new $600M battery plant in China. So I gave them another 1.5 GWh over 2015 capacity.
(Link: Samsung SDI is Going to Build Chinas Largest Lithium-ion Battery Plant | Lux Populi)
4. For Wangziang, BPP, Tianneng and SB LiMotive I gave as before an average 25% growth rate. I didn't find anything on google about expansion but can't rule it out.
5. As before, for LG Chem I used all capacity today and capacity being build, and added 8% improvement per year (given the current 30% utilization rate that is generous).
6. As before, for AESC I only assume 2015 capacity plus 8% improvement per year (given plans to cut production that is also generous).

Here's the corrected data:


Following a conversation with techmaven (and giving the issue some thought), it makes much more sense to predict 2017 capacity than 2020, since there is a lot going on in new battery technology and if it takes 2 years to build new capacity, that fits quite well into the 2020 timeframe. I still want to do 2020, as a thought experiment, but it should be taken with a grain of salt.

The gigafactory will not be in full production by 2017, but the Panasonic Osaka plant will, and Panasonic will have ~7GWh capacity instead of the current ~3GWh, excluding the gigafactory.
That works out to be only ~87k model S and X worth until somewhere mid 2016 when the gigafactory starts producing, and there will need to be a ramp up.
So call it 120k model S and X? Or 9.6 GWh overall. This is guesswork. Anyway it seems likely to me that they will still be production constrained by end of 2016.
Link: **Still looking for public sources to back this up.** And @techmaven - is this a good understanding of what you said?

BYD plans to add 6GWh each year from 2015 until it reaches 34GWh. So from 10GWh by the end of 2014 (of capacity, not sales), they will have 28GWh by the end of 2017.
Link: http://www.reuters.com/article/2015...MZ20150313?feedType=RSS&feedName=businessNews
When talking about BYD it is important to keep in mind that they use a chemistry called LFP or Lithium-Iron-Phosphate. It has one of lowest energy density cells at up to 120 Wh/kg.
When they plan to sell hundreds of thousands of EVs, these are not pure EVs but plug-in hybrids!
Links: Types of Lithium-ion Batteries – Battery University
Technical Service | Energy Storage | BYD
The Qin (their best selling car) has a battery pack of only 13 kWh... Lets say the average BYD car sells 20 kWh, that would be 1.4M cars! Impressive if they come close. But somewhat far from the "Green Car" image.
BYD Qin | ChinaAutoWeb

LG Chem:
The big news within the next two years is an advancement from 200 Wh/kg to 220 Wh/kg in LG Chem's NMC batteries. **
Link: Types of Lithium-ion Batteries – Battery University
(The website says the Volt uses 30% LMO with 150 Wh/kg and 70% NMC NMC with 220 Wh/kg, which gives an average of 200 Wh/kg??**)
**Please take with a grain of salt. @techmaven / @anyone else - is this correct?**
LG has about 6 GWh full capacity (excluding China), I'll keep that as a round number to avoid false accuracy. What will they need for 200 mi range?
Link: Real-World Nissan LEAF Fleet Data Reveals...
From Nissan data we get around 300 Wh/mi average? (most of population is in better or equal to 300 Wh/mi from map), so 200 mi means 60 kWh?
For some perspective that means 100k 200 mi cars! Not so much given LG Chem has that for not just GM's bolt but all those other OEMs as well.
Chinese 1.5 GWh plant will do Chinese cars. And might very well be doing low density LFP batteries, not NMC. So we are talking about hybrids. Lets say 30 kWh per car, that works out to be 50k hybrid cars.

It now makes more sense why Nissan would want to work with LG Chem instead of AESC -
The Nissan leaf Manganese spinel battery gets only 140 Wh/kg! Compare that to the 220 Wh/kg of new NMC cells from LG Chem...
Link: http://www.electricvehiclewiki.com/?title=Battery_specs
It seems likely to me that Nissan would want to phase out the AESC batteries altogether. But will that happen by 2017? Not sure.
Lets say with high confidence that AESC will not produce more than the current ~1.4GWh 2015 (to be clear, the ~1.4 GWh is cars sold, not overall capacity).

Samsung SDI:
Samsung actually made this very easy for everyone to follow, they wrote all their plans in their website:
Link: Samsung SDI: From 130 Wh/kg to 250 Wh/kg By 2019 and 300 Wh/kg in ≈2020
Basically, until 2019 - they will do low density NCM with 130 Wh/kg (NCM is similar but not same as NMC). 2019 - new chemistry will come with 250 Wh/kg at scale. This is close to what Tesla is getting,
and this is the kind of stuff that make 2020 predictions so hard. So no expectation of capacity change till 2019. And since BMW will not do a new car with this kind of timeline before 2019, current 25 kWh, they said I remember demand is 30k for the i3... so capacity in 2017 should be around 750MWh.

The others are too small to bother. I googled to see if there is news of any of them building some massive factory, and nothing came up.

So there you have it, 2017 market based on capacity, with somewhat better data to back it up:

This is still a simplification though...
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By when will Tesla have 100000 run rate? 2016? So that 87000 is probably false. And it is clear that not having any information about the production rate at the gigafactory in 2016 makes this estimate a hand-waving exercise. For BYD, AESC, Samsung in 2015 I took producted MWh as an approximation for capacity. This is also hand waving, but it should affect 2017 numbers.
I realise this is about EV battery and not PHEV battery, but Mitsubishi does sell a similar amount of plugin vehicle as Tesla.
I expect Mitsubishi next PHEV (ASX) to outsell its current PHEV (Outlander) by around a factor of 3, due to price point reduction.
I realise this is about EV battery and not PHEV battery, but Mitsubishi does sell a similar amount of plugin vehicle as Tesla.
I expect Mitsubishi next PHEV (ASX) to outsell its current PHEV (Outlander) by around a factor of 3, due to price point reduction.

Regarding GS Yuasa, that supplies Mistsubishi, I made a rather banal simplification in the calculations: I took the currently produced supply data and used that as an approximation for their entire capacity, which like in the case of LG Chem could be much much bigger (LG is at 1/6 of their capacity). The excuse is that I don't know what factories GS Yuasa has, and what is their capacity. It would be great to get that information.
The excuse is that I don't know what factories GS Yuasa has, and what is their capacity.

Mitsubishi has said more than once the reason they have not imported the Outlander PHEV to the US is because they do not have enough batteries. All they get from GS Yuasa is consumed by Japan and Europe.

An increase in supply plus an ending of generous incentives for PHEVs in The Netherlands has the American introduction of the Outlander PHEV coming "soon."
Mitsubishi/GS Yuasa has 4 factories, for vehicle li ion batteries.

Kusatsu June/2009 110MWh pa capacity
Kyoto Dec/2010 180MWh pa capacity
Ritto I April 2012 800MWh pa capacity
Ritto II Spring 2013 1,200 MWH pa capacity

my guess is that the Ritto plants are based on Nissan's NEC cathode strip, and the pre Ritto plants are based on GS Yuasa's in house coated cathode. Due to the 2013 battery issues (which in my opinion were merely due to a rushed commissioning), I suspect iMiev batteries were still made at Kyoto because iMiev's were not affected by the bad batteries. As part of the improvement in battery manufacture process, some steps were removed, and others lengthened, if the revised aging/curing process became the new constraint for production, then annual production capacity will be less than planned in 2011 (note Japanese land is expensive).

All in all, Mitsubishi now seems production constrained/balanced for Outlander PHEV but only just, I would guess the arrival of the Outlander PHEV in the USA coincides with the start of demand constraint.
Mitsubishi's ASX PHEV has the price potential to sell about 3 vehicles per every Outlander PHEV sold (ie a quadrupling of PHEV production) I would guess that Mitsubishi has a realistic capacity to ramp up with for ASX PHEV, to perhaps 2-3x current PHEV production. That is to say, for the next couple of years, Mitsubishi can sell a 1 PHEV/EV at around the same rate as Tesla can sell an EV.

Some more color on LG Chem's Nanjing facility:


Published at the end of October, 2015, it indicates that the plant "could theoretically produce 50,000 batteries for advanced pure EVs a year, meaning the plant could also be used to supply up to 180,000 battery packs for plug-in hybrid EVs (PHEVs)"

There is a problem that the business and automotive press often uses "battery packs" as a metric which can be anything from 0.7 kWh to 95 kWh. But this says something different. Given that most PHEVs have a battery pack around 6-10 kWh, and if we take 8.5 kWh as the average, then the 180,000 x 8.5 kWh = 1.53 GWh. Which means their advanced battery pack metric would mean 30.6 kWh, which is a good bet. This is also corroborated by the earlier Korea Times articles on Ochang. I don't see it being much bigger and it might be slightly smaller (1.2 GWh). It's also pretty clear that it isn't at nameplate capacity.

As for Holland MI:

Global Expansion: LG Chem to Bring Holland Battery Plant Online This Year | BusinessKorea
(note the typo in that article, it mentions 40,000 units as the capacity of the Holland plant, but then uses 400,000 at the bottom)
Which means if Ochang is 200,000 units @ 3.2 GWh (16 kWh @), then Holland is 0.64 GWh.

The article is definitive that LG has 3 plants, so we can figure out LG Chem's total nameplate capacity:

Ochang, SK: 3.2 GWh
Nanjing, CN: 1.6 GWh
Holland MI, US: 0.7 GWh

That's 5.5 GWh of total nameplace capacity. However, they are not yet at nameplate. Ochang, for instance, doesn't have all the lines built out. Holland, MI plant is getting a lot bigger soon.

The problem with GWh as a metric is that the batteries are getting denser. Which means each cell has more capacity, and the plant is actually limited by cells. So let's assume that the cells got 25% more dense so far (using the newer 2015/2016 chemistry).

So let's make the 25% assumption and the Holland MI plant expansion with a 4th line for a view in late 2016:

Ochang, SK: 4 GWh
Nanjing, CN: 2 GWh
Holland MI, US: 1.2 GWh

Total would then be 7.2 GWh, at full nameplate. Note that they haven't hit full nameplate ever, and Ochang has been severely underutilized.

In comparison, the Panasonic has been building out to handle Tesla's needs. Tesla will ship something like 3.5 GWh to 4 GWh of cells in 2015. Panasonic had to size their expansion to somewhere around 5.5 GWh in their Osaka plants to handle Tesla's original run rate expectation for the end of 2015, and somewhere around 7-8 GWh for 2016. Of course, when the first phase of the Gigafactory comes online, then we're talking another 4-6 GWh in 2016. So the end of 2016 looks like 11-14 GWh for Tesla + Panasonic (for Tesla).
LG Chem is also adding a battery factory in Europe soon, likely in Poland:

LG Chem to build a new battery factory for electric cars in Europe | Electrek

(Audi will be one of the new customers, dual sourcing with Samsung SDI: Audi developing EV batteries with LG Chem, Samsung SDI )

Samsung is talking about 35 GWh (!) until 2020 according to this report:

Samsung SDI has become an electric vehicle battery pure-play. It now owns 5 production lines in Korea and China with total capacity of 4.5GWh. But by 2020, it will bring its capacity to around 35GWh. Analysts are not missing a beat – this capacity is “interestingly the same ‘cell’ capacity” of Tesla‘s (TSLA) Gigafactory, noted Bernstein Research‘s Mark Newman.

Hello, Gigafactory! Samsung Moves To Compete With Tesla - Asia Stocks to Watch - Barrons.com

I haven't seen other sources / final confirmation for this huge capacity increase by Samsung - which would put it on par with Tesla's original cell outplan plan by the end of this decade.
LG Chem is also adding a battery factory in Europe soon, likely in Poland:

LG Chem to build a new battery factory for electric cars in Europe | Electrek

Right... so another ~1 GWh in Europe at some point, but I think it's immaterial for 2016 since start of construction hasn't happened yet. Basically, since the other plants aren't at nameplate, the new plant is small enough to be in the noise at that point.

Samsung is talking about 35 GWh until 2020 according to this report:

Hello, Gigafactory! Samsung Moves To Compete With Tesla - Asia Stocks to Watch - Barrons.com

I haven't seen other sources / final confirmation for this huge capacity increase by Samsung - which would put it on par with Tesla with Tesla's original cell outplan plan by the end of this decade.

Strange that they chose the same amount as the original Gigafactory amount. That's very, very curious. Makes it quite suspicious actually. In any case, that's good. Even if Samsung SDI hits 35 GWh as LG hits maybe 15-30 GWh in 2020 still means that Tesla has a huge portion of the EV market. The big key is that Samsung SDI, LG and others depend on the automakers to place the orders. Thus far, they have been too tepid and therefore their schedules are more likely to slip. With fast charging standard maybe in 2017, with the next gen of vehicles in limited production in 2017/2018/2019, the big ramp up for battery plants is probably starting in 2018 or 2019 as automotive executives assess the sales of this next generation of BEVs... which means actual capacity build is more like 2018/2019 through 2022. Again, Tesla will be more aggressive.