Atmospheric dust by itself shouldn't have any major impact on generation or distribution of electricity. The two are unrelated.
However, a fraction of the Earth's electricity production is based on current solar input, either directly or indirectly. There is the obvious: photovoltaic electricity production, which globally amounted to a total of 41.4 TWh in 2012. While this is a tiny sliver compared to global electricity production (also here), the global photovoltaic electricity producting sitting at almost exactly 1% of the US electricity production of 4,310 TWh, some countries have significantly higher proportions photovoltaic electricity production than others.
You will also face indirect effects due to changing weather patterns. I would expect wind patterns to change, which may or may not have an impact on wind turbines, and you may very well see changes in other energy generation techniques as well. Blocking out such a large amount of the sunlight as you describe is going to cause global temperatures to plummet, which in human societies would lead to an increased need to provide heating in buildings (wild animals would also suffer, obviously).
In summary, electricity supply from photovoltaic generation will obviously be directly impacted by sunlight being blocked (at this scale probably to the point of providing only negligible amounts of power, if indeed any at all), and other power sources may be affected as well as exemplified above. There is of course also the risk that the power grid has been damaged, but I would expect such damage to be primarily local or possibly regional in nature following an asteroid strike; the power grid is normally designed to be able to deal with failures without triggering cascade failures, but whole areas may be cut off from the larger grid to protect the rest of the grid. You may also see secondary effects from the strike itself, but that won't be about electricity per se.
The combined effect of all the above may be to force power rationing techniques in areas where affected power sources make up a noticable fraction of the total power mix. Rolling blackouts is a power rationing technique that has been used in practice, and likely would be employed in such a scenario, assuming enough people surviving the immediate effects for society to continue to function at all. Rolling blackouts were done in Japan after the 2011 Fukushima Daiichi disaster and more-or-less immediate shutdown of almost a quarter of Japan's nuclear reactors which led to a severe short-term power shortage. Venezuela is doing something similar in 2016 in response to drought.
Generally speaking, if you can't service everybody all the time, then it might be better to simply tell people that until further notice and for defined periods of time during the days they will simply have to make do without electricity service; the alternative may very well be grid overload, which at best is going to lead to a total shutdown and automatic reset, and at worst may very well cause severe equipment damage. Backbone electricity grid transformers, isolators and so on is not something you can just walk into a store and buy; such parts often need to be manufactured to order, and delivery times of months is far from unheard of even at the best of times. Even if things are otherwise good, this is a scenario the grid operator is going to want to avoid at (almost) all costs. With modern, "smart" metering, it might even be possible to reprogram the meters such that power (watts, not watt-hours) draw above a certain threshold for a given period of time shuts down service for that user for a period of time, to reduce the peak load on the grid. Even if such provisions aren't normally allowed for, I'm sure in such a situation someone would very seriously consider the possibility.
It's also worth remembering that even though electricity can be transmitted relatively easily over short to medium distances, moving electricity over large distances is a non-trivial problem, and one mostly avoided by having reasonably local power production. So for example, having enough electricity production in South America doesn't do much to help Canada cope, and the same goes with transmission across Eurasia.