Last time we talked about dangling participles and mentioned that they were different from absolute phrases or constructions. The latter contain a participle, but in this case the participle modifies a noun or pronoun instead of referring to the subject of the sentence. A frequently used absolute phrase is:
"All things considered, I would recommend that you buy the used one."
In this case, the participle ("considered") modifies a noun present in the clause ("things"). We are hence dealing with an absolute construction.
"The power being out, they decided to spend the night at a hotel."
Yet here the use of an absolute phrase makes for a slightly awkward sentence. It would be better to say:
"Since the power was out, they decided to spend the night at a hotel."
You may find a definition of absolute construction and some examples in The American Heritage Book of English Usage.