# Of GTIs, battery sizing, and capacitor banks

__Introduction__I work for a fuel-cell R & D. We're building a 5kW (low-voltage) fuel-cell system that I wanted to tie into a grid using an XW6048, a model which requires a 48V-nominal battery bank to run properly.

Looking around on forums, I found that people sized their battery banks depending on a lot of factors. The "100Ah per 1kW" rule was thrown around a lot. I was not interested in running our 5kW GTI for long periods of time on batteries alone (full-load for only 10's of minutes at most) so I thought I might be able to get away with sizing a much smaller bank. But it was mentioned that the AC currents being pulled by the GTI are a concern, and the battery bank needed to be sized to accommodate these large AC currents. There's no rule-of-thumb for this as far as I can tell.

Reflecting on the AC currents further, I realized that it would be better not to place a significant portion of these currents onto our fuel-cells. Fuel-cells prefer a steady power draw.

Therefore, there were two technical challenges to solve:

(1) Save money --> a large, multi-$1k battery bank was not an option

(2) Minimize the AC currents drawn from the fuel-cell stacks

I wondered if placing a large capacitor bank in parallel with the battery bank might help. If a capacitor bank having significantly lower impedance were attached in parallel with a battery bank, AC currents more likely be pulled from the capacitors than the batteries. And if the batteries and capacitor bank were handling the majority of the AC currents, then very little would need to be pulled from the system.

I looked around on forums and asked some questions regarding input capacitors, and I never got any good answers. I posted on Solar-Guppy's forum, but he's shut it down now. I'm following up here instead so that people can learn from what I did.

__RMS current / charge analysis__I started with an ideal analysis in order to see what ballpark the input AC currents would be in. From previous experience, I knew that true-sine wave inverters pull in a sine-squared input current having a 120Hz frequency. The derivation is fun: just assume a constant input voltage and a 120V

_{RMS}/ 60Hz output waveform. Here's the set-up:

P

P

P

_{in}(t) = V_{IN}* I_{in}(t)P

_{out}(t) = V_{out}(t)^{2}/ R_{load}= V_{out}(t) * I_{out}(t)P

_{out}(t) = P_{in}(t)I'll let the reader take it from here. You're trying to find I

_{in}(t). Assuming 5kW in and 5kW out (100% efficiency) one can generate data for a single period of the input current waveform in Excel and carry out numerical analyses to get all sorts of relevant data. The datum I was most interested in was the RMS value of the current. I looked around on the web, and the conclusion I came to is that there is no analytical solution for integrating a sine-squared waveform. However, you can use Excel to compute the RMS numerically.

From my analysis, I found that for 5kW at 46V input (the minimum necessary for operation) ~77A

_{RMS}was demanded. 77A

_{RMS}is nothing to scoff at obviously: that's all heat-loss type current. One can see why people take the AC currents into account when sizing their battery bank.

The next thing I wondered was how much charge would be removed during a half-cycle of the 120Hz sine-squared current. Integration of the input current over one half-cycle gives this figure: Q = ~0.29C. Next, I calculated how much capacitance is required to hold 0.29C at 46V using the capacitance equation: C = Q / V = ~6.3mF. Finally, on a whim I decided that I didn't want this discharge / charge cycle to use up any more than 1% of the capacity of my capacitor bank. This would keep the ripple voltage to some minimum. Thus, I would need a 6.3mF / 1% = 0.63F bank. Rough math, I know, but it's probably close enough.

__Bank sizing / pricing__So, at a minimum I needed a capacitor bank of at least 0.63F that could handle at least 77A

_{RMS}.

Searching around on Digi-Key, I decided to make these my unit capacitor of choice: http://search.digikey.com/us/en/products/E36D800HPN104MEE3M/565-3332-ND/2095946. These 0.1F capacitors are only $36.88 apiece and can handle voltages up to 80V, nearly twice my required voltage. These capacitors also have a 21.42A

_{RMS}rating. This RMS capability adds in parallel, so a 0.7F bank would have a total RMS capability of ~150A

_{RMS}, double the computed necessary capability.

This is my solution. And for the heck of it, I just bought 10 of them to catch the price break and make a 1F capacitor bank. I wasn't sure how accurate my analysis was, so I played it safe. I used copper bar to connect them all in parallel, I connected them to the GTI through a separate circuit breaker, and I wired up a charge / discharge circuit to be able to charge them from the battery bank.

__Testing__This is what I found. At 5kW / 50V, a sine-squared waveform of ~195A

_{pp}/ ~90A

_{RMS}is demanded by the GTI. With the 5kW power system, the 48V battery bank, and the 1F capacitor bank all connected in parallel, the AC current waveform is divided between the three as follows: ~9% from the system, ~14% from the batteries, and a whopping 77% from the capacitors!

I consider this a success.

__Was it worth it?__In total, the capacitor bank probably cost me between $400 and $500, and the batteries I'm using are semi-truck batteries for about $125 apiece = $500. The batteries I'm using are this kind: http://www.interstatebatteries.com/cs_eStore/Products/RT/PID-31-MHD%28Commercial%29.aspx. I think they are used in semi-trucks and such. I had the fellow I purchased them from call the company to ask what the effective capacity was of one of these batteries, and it comes to around 80Ah @ 25A discharge. Not much.

So, here are my options.

(1) I could have purchased a top-of-the-line, 48V / 500Ah battery bank for around $3,750, assuming the pricing here: http://www.wholesalesolar.com/battery-banks.html. Maybe I could have gotten away with a 400Ah bank for $3000. Maybe I could have purchased batteries individually and shaved a little of that price. Whatever. It's a lot of money for something I don't need. It's also a lot of space, as these batteries are typically quite large and you typically need more of them because they are 6V batteries, not 12V.

OR

(2) I built a battery / capacitor bank for around $1000, and it takes up less space. The batteries are also more readily available, so if one happens to go

*kaput*, no problem.

Option #2 is still the best option for me.

__Design criteria revisited__Let's look at the design criteria again for the sake of academics.

(1) One should design based on necessary RMS current capability. At 5kW, 27% more RMS current was demanded by the GTI than calculated (because I didn't take into account losses, rippling voltages, etc.). I propose a 1/3 correction factor as a good rule-of-thumb. Redoing the calculations, at 6kW (full-load capability of the XW6048) and 46V, the supposed RMS amperage pulled by the GTI is ideally ~85A

_{RMS}. Using the 1/3 correction factor, this comes to ~113ARMS. I like round numbers, so let's round up to 120A

_{RMS}for 6kW. In fact, dare we say

**20A**? I think this is sound, because I took other data at loads from 500W to 4000W in 500W increments, and the RMS current demand scales linearly.

_{RMS}per 1kW(2) One could choose to scale back on RMS requirements based on the fact that the capacitor bank will see only about 3/4 of the predicted AC current. Yet, why not give yourself some room to breathe? Thus, the

**minimum**rule is 15ARMS per 1kW, but the safe rule is 20ARMS per 1kW.

(3) One should design based on necessary capacitance to achieve either or both of a few constraints: (a) maximum voltage ripple, and (b) maximum power loss. Remember how I chose an arbitrary 1% charge / discharge number as my indication of minimum capacitance? Well, one could get a lot more detailed. Capacitors have certain ESR specs at 120Hz. This ESR contributes to power loss in the capacitor, but it also contributes to voltage ripple. The voltage ripple is simply the ESR multiplied with the RMS current (V = I * R). The power loss is the voltage ripple multiplied with the RMS current (P = I^2 * R). Honestly, I think voltage ripple is more important, as it has more of an effect on the performance of the GTI. For a capacitor bank built from 6 of the capacitors I selected above, I calculate only 13W lost over the whole bank (< 2W per capacitor) for 120A

_{RMS}. I doubt the temperature rise in the capacitor would be detectable.

(3a) Therefore, design capacitance based on maximum voltage ripple. This is a more interesting constraint. One contributor is the aforementioned ESR, is calculated easily, but requires an actual ESR to calculate. The second contributor is much more important: the voltage ripple due to charge / discharge of the bank. Recall: dQ = C(V) * dV. Thus, choose a peak-to-peak ripple-voltage (dV), and calculate a minimum capacitance based on the dQ seen during a half-cycle of the current waveform. Once you have a chosen capacitor, take the ESR into account by adding the two contributors.

__Closing__It's late at night on a Friday evening, and I'm still at work writing this. I know, my friends, I know... I am a nerd.

0

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## Comments

26,204adminYou can try sending

Solar Guppya PM--He has not been here for a month and a half...Looks like you did some interesting homework there. Very cool.

One other issue that may or may not affect your work. With MPPT Solar Charge controllers, they seem to have a fairly long Time Constant (10's of seconds?) and some do not appear to respond very fast to quickly changing battery bank / voltage conditions.

From what I have read here--One problem I see with a "small" bank is if you have a steady state condition (5kW from the "energy source") and the load changes quickly (clouds cover array, GT Inverter shuts down due to line fault, if off grid, loads change dramatically)--At least with Solar MPPT controllers, there have been problems where they continue pumping current into the battery bank--And if the bank is too small, it can exceed 72 Volts and fault the XW inverter (hard reset or temp shutdown--depending on firmware revision?) before the MPPT controller shuts down the current producing the excess voltage...

A 100 AH @ 48 Volts per 1kW of energy source/inverter sink seems to be necessary to sink the excess current until the controller's TC allows it to ramp back on the current. Even AGM's which have a very low impedance when sourcing current and (some brands?) appear to go into a "chemical" high impedance state when fully charged when presented with "excessive" charging currents.

So--depending on how quickly your fuel cell will drop its output current (without an external/internal current shunt or dump) may limit your minimum AH rating for your battery bank.

My guess anyway.

-Bill

10,311✭✭✭✭very interesting for sure. i would like to hear the reaction to this from a few people.

396✭✭✭I would like to the the P-P reactive current flow waveform from the battery and fuel-cell to the cap during the off period of the demand waveform. This one of the papers I used for research on battery models. It has a very good description of hybrid battery-ultracap systems interactions with solar and wind. http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=10&ved=0CGgQFjAJ&url=http%3A%2F%2Fnanomech.ems.psu.edu%2FPDF%2FCharith_Tammineedi_Thesis.pdf&ei=2PpkT-GhJKe0iQfa28DpBQ&usg=AFQjCNH38j0Qrm6yImArqdOW0zg6OpPr2w&sig2=CkqV4Ol06VagpLpAVzj-dA

7,022✭✭✭✭What is the expected lifetime of the capacitor bank, @ 120 hz ? months, weeks, years? I ran these calcs some time ago, and came up with a capacitor lifetime of 2-3 years. Then their fuses blow. You put fuses (and their resistance) in the bank right ??

Xantrex has a fancy input capacitor (one of the fault codes is Input Cap Overheat) and I wonder what it is, and what if adding a couple more of them would do. [ note to self, next time I open up the case, see if it's visible, note mfg & model # ]

Also super caps may not be the right critters, regular low ESR caps may have even lower internal resistance.

|| Midnight Classic 200 | 10, Evergreen 200w in a 160VOC array ||

|| VEC1093 12V Charger | Maha C401 aa/aaa Charger | SureSine | Sunsaver MPPT 15A

solar: http://tinyurl.com/LMR-Solar

gen: http://tinyurl.com/LMR-Lister ,

1,959✭✭✭You have touched on most of the key points, ESR , whether its a battery or capacitor is your primary concern.

As for whats better, its depends!. Having a wonker capacitor bank would only be useful for applications that have short peak demands, such as gridtie operation. For anything of significant load on your inverter, you will easily meet the 1kw/100 ah SG rule as for most off-grid systems that have 6kW inverters will be in need of battery banks in the 1000's of aHr range.

If you want just GT performance, run higher voltages you no longer need the external capacitor bank and that's what all the current crop of GT inverters are

Buffering a fuel cell is an interesting application and having capacitor bank is likely a better solution, unfortunately, I would guess 99.9999% of XW installs now and into the remainder of my life time will not be connected to fuel cells, for the rest of us, it will be some form of battery storage.

On a cost basis, its a wash, as your 1K capacitor bank would sell for ~3K retail, not to much different from an off the shelf battery bank

A little history, Trace sold a bunch of SW4048's for grid-tie operation only, and yes, they added a bank of capacitors, this was 1997-8 time-frame. In the end, low-frequency massive transformers used by these inverters just doesn't make sense economically compared to the GT high frequency inverters prolific in the market. even SMA dropped the big iron designs.

3To follow up...

First of all, I went back to the math and found that the RMS current into the GTI is actually a lot easier to calculate than I first thought. I worked out the math and validated it with the numerical analysis. Thus, my analysis still applies, but now is made easier for everyone because now it is more analytical than numerical. What I found was:

_{IN,RMS}= I_{IN,AVE}/ SQRT(2)where I

_{IN,AVE}= P_{IN}/ V_{IN}Taking into account the correction factor indicated by my measured data, I still stand by my 20A

_{RMS}per 1kW rule of thumb.Second, in some of the additional research I did (papers listed below), it was mentioned that the ESR is not meant as a model for voltage ripple. Thus, I was wrong to say what I did about ripple voltage determination and ESR as a factor. The capacitance equation is probably the best means for calculating a worst-case voltage ripple.

Third, to the comments.

(1) @ BB

The response time of our system depends entirely upon our system design. We use DC/DC converters to buck up an array of stacks to the same working voltage (~50V for our 48V-nominal bank). We are monitoring the battery current and the system current dynamically. Everything is controlled by a SCADA software which can achieve updates of our inputs / outputs at periods well below one second.

(2) @ nsaspook

I'm not sure what you are asking. However, since you bring up ultra-caps, I will say this: I looked into the possibility of using ultra-caps and concluded that the cost : benefit ratio was too high. A large bank of inverter-grade capacitors like those that I am using is probably sufficient.

(3) @ mike90045

Excellent question regarding lifetime. First of all, yes, the capacitors are mechanically fused.

I am not sure how to calculate theoretical lifetime without doing a lot more work. I have found these resources:

http://jianghai-europe.com/wp-content/uploads/JIANGHAI_Elcap_Lifetime_-_Estimation_AAL.pdf

http://www.edn.com/article/471688-Determining_end_of_life_ESR_and_lifetime_calculations_for_electrolytic_capacitors_at_higher_temperatures.php

http://www.chemi-con.com/index.php?option=com_content&view=article&id=12&Itemid=19

http://www.cde.com/tech/multipliers.pdf

http://www.cde.com/tech/thermalapplet.pdf

http://www.cde.com/tech/thermalmodel.pdf

http://www.cde.com/tech/selectinvcap.pdf

http://www.rubycon.co.jp/en/products/alumi/pdf/Life.pdf

What I get primarily from my research is this: keep your caps well below the rated temperature, avoid the rated voltage, and avoid the rated ripple. Now, if my calculations are correct, I am at least avoiding the rated voltage and rated ripple to a substantial degree. I have not yet been able to detect a temperature rise in the caps at full-load. The bank could, theoretically, last me for well over 2-3 years.

(4) @ solar guppy

We are looking at higher input voltages for our larger future systems. The XW6048 was our choice because we must currently work at low working voltages. And, yes, I do realize our system is unique. I only offer my work as an example for others who might be thinking about developing their own RE systems.

Best regards,

Tele