If you suffer from a blackout on Shabbath, but the power returns on Shabbath, and you have certain food elements cooking, is there a problem of either Nolad or some other issue relating to food that was entirely or mostly cooked on Shabbath after the power was restored? What about water in an urn - would that be considered different from, say, Cholent, because we hold Yesh Bishul Ahar Bishul for water?
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Reb Moshe Feinstein (O.C. 4 Siman 74 Dinei Bishul 38) and Reb Shlomo Zalman Auerbach (Shmiras Shabbos Kehilchasa 1:69) say that returning to a different blech is Muttar (not in a case of powercut). The Orchos Shabbos 2:50 says that the same Din applies to a powercut. His proof is from the Chazon Ish 38:2 that says a person is allowed to place a fully cooked pot on a blech that will switch on on Shabbos. He brings from there that there is no Issur when a powercut occurs!
The reasoning they say is that one is allowed to cook foods on Shabbos provided that he do it in a way that he started the cooking on Erev Shabbos and is Garuf VeKatum. I assume your case is indeed the latter because if not it is Ossur even without a powercut. The issue that would therefore apply is has you previous cooking disappeared and is it considered a new cooking on Shabbos? To this they explain seeing as if you would have another hot plate to move it on to you do not want the Hefsek and it is considered a continuation and consequently Muttar. (They bring the similar to the case of the Mishnah Brurah 25:14 by Tefilling that slipped is not a Hefsek)
EDIT: In response to DoubleAA about Nolad (although the Orchos Shabbos still is matir regardless):
In my opinion the reason why nolad does not apply even if there is Nolad on a dovor Sheino Be'ein is because you did nothing wrong. Even if there is an issur of Nolad that is only to go and use the Nolad but you are being Shev VeAl Taaseh. The only Issurim on Shabbos that I can think of off-hand is Shehiyah and perhaps Chazarah and Hatmanah where the Chazon Ish holds even if a Monkey put it on you gotta take it of. (Note there may be a problem of Muktze!) But by Nolad Chazal never made this specific type of Gezera that even when you dont do an action you must still stop the action from happening. So there is no Issur in leaving it on the hob, the food definitely is not Nolad, it was always there so what can possibly be Nolad.
Being that no action is done, the question here is not a question of whether one would be transgressing a prohibition, but rather whether one would be permitted to derive benefit from the items which were affected by an apparent violation (הנאה).
In a broad sense, the halachot of deriving benefit from a transgression are fairly clear (see Shulchan Aruch 276 and Mishneh Torah Shabbos 6:2), if a gentile performs an action which violates Shabbos solely for the benefit of a Jew it is forbidden to derive benefit from the the results of his action until after Shabbos.
If, however, the gentile performed an action for himself or another gentile which violates Shabbos, the results of his action are permitted.
Therefore, in our case (in which the power goes out on or before Shabbos and is subsequently restored during the course of Shabbos), provided that the majority of people for whom the power company is restoring power are gentiles and the majority of people who work for the power company on Shabbos are gentiles (like in most parts of America), there should be no issue with deriving benefit from food which was heated or cooked on Shabbos even if it was completely raw or cold prior to the return of power (provided that the heating element is properly covered so that a Jew would not be tempted to adjust it - pursuant to the Rabbinic restriction of grufa ve'katuma for shehiya and chazara). In a situation where the majority of people involved are Jewish (like in Israel) there may issues.
The crucial point to note here is that there is no difference between benefiting from a light bulb which was turned on with the return of power and benefiting from food which was heated or cooked - if one would not derive benefit from the food, one should also leaves one's home so as not to benefit from the air conditioning, light, fans, etc. In this case, no gentile was asked to perform an action which would be a Shabbos violation and the action was not done specifically for a Jew, therefore there is no restriction on benefit.
I hope this answer helped clarify some of the complex issues involved with the situations you raised. It is not meant as a practical halachic answer, you should seek out a local competent Halachic authority for this purpose.