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2kw is not enough to do much but run a few lights and keep the fridge going.
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It'll do a lot more than that, particularly if you do some load-juggling.
Many loads only need to be operated intermittently - which means that you don't need to power all of them simultaneously.
The problem with big generators is that they're fuel-inefficient when powering small loads. A 6KW generator will use a LOT more gas powering a 200 watt load than a 2 KW model will.
Then, there's the extra purchase cost, size, weight and noise of a big generator.
The 80/20 Rule: During an extended power outage, maintaining 80 percent of your normal standard of living doesn't require a big generator or a lot of fuel. However, if you insist on also having that last 20 percent,
it will cost you dearly.