What is a power double? Nominally, PDs show 15+ HCP and tend to be balanced. Since NTOs are not 100% forcing, a PD can also be made on very strong takeout doubles (15-16+ with a doubleton in their suit, 18+ with shortness), on strong jump overcalls, or extremely strong two-suiters inappropriate for Michaels. A PD neither confirms nor denies a stopper in their suit. We'll have plenty of time to find one below 3NT.
We can make a PD on hands that would not overcall 1NT, either because they are a little bit too weak, or because they don't have a stopper.
Minimum power doubles do not have singletons. Very strong hands can be most any shape, unless this is played in conjunction with Overcall Structure, in which case, they tend not to be one-suited or two-suited. If the hand has a little extra, a stiff honor on the side is OK. If the hand has a lot extra, a small singleton is OK. I've never seen a PD with a void.
Power doubles apply even if the 1
or 1
opening
is semi-artificial, as in a Precision 1 or a
1 = natural or 15-17 or a Polish club (usually
11-13 balanced). They do not apply to big clubs.
They do not apply above the one-level. They do
not apply by passed hands, although partner can
have passed. They are very effective against
light third-seat openings.
After (1X)-X-(pass)-?:
Axx 
Kxx xxx QJ9x doubled.
It doesn't
demand a trump lead, but does require an alert,
"doesn't promise as much in trumps as a pass of
a normal takeout double." Against five-card
major openings, passes usually look a lot like
normal passes of takeout doubles, but don't
need quite as good trumps. Assume partner has
Jx in their suit and go from there. Pass creates
a force throughout.
Simple suit responses: show about 0-7 points, four-
card suits at the one-level, usually 5-card
suits at the 2-level. Not forcing. Balanced
hands will have a choice between a suit and 1NT.
Use judgment: with
KQ10x xxx xxx xxx to a power double of
1. With
K10xx xxx xxx Jxx
1NT: doesn't show anything, not even a stop in
their suit. Denies as much as a decent 7-count.
One may often have a choice between one of a suit
and 1NT. Location of high cards and length in
their suit should influence that choice. With
truly atrocious hands, try to bid the cheapest
suit. With stuff and/or length in their suit,
tend to bid 1NT. (Remember, partner tends to
have length there, so adverse ruffs are threatened
in a suit contract.) I once held xxxx Jxxx K9x Jx. That
caused a swing; I was +150 in 1NT, and my opponent,
playing 1NT from the other side, went down.
2NT: 7+-9- HCP, balanced. Usually at least half a stop in their suit.
2X: (Cue) game forcing, about a good 7-count and up. This can be done agressively, since the cards rate to be well-placed. We can get out in four of a minor.
Jump suit responses: invitational, 7-9ish, five-card suits.
4-level responses are natural, except that a 4-level
double-jump cue is namyats-like, in either major.
Responses to that are: 4 = interest in slam,
4 = P/C, 4 = huge hand for hearts, not so huge
for spades. In mid-chart events, all 4-bids are
namyats.
3-level triple jumps are preemptive.
After (1X)-X-(xx)-?
,
J/M: throughout). Otherwise, it's "nothing to say."
Others: same. I've seen a few psyched redoubles here.
After (1X)-X-(1Y or 2X)-?
Dbl: takeout. Can be done on shortness in the bid suit. If they have bid only one major, the double shows exactly four cards in the other major and shows nothing more than tolerance for the unbid suits. Double does not promise much in the way of values. At the 3-level, it promises values. At the 2-level, forcing partner to bid vulnerable at the 3-level requires something, but less than you think---partner has a good hand, and a weak advancer has to help out. When advancer passes, minimum PDers usually pass, too, so most of the responsibility to compete is upon advancer. Repeat, at the one-level, this does not promise any values, just the unbid major.
New suits: natural, some values, competitive, not forcing, not invitational
Cue of X: game forcing
Cue of Y: natural, forcing one round. They can psych here, and it's effective without this treatment. Later "cues" of responder's "suit" are natural, also.
Jump cue of X: natural, forcing, good six-card suit.
Jumps in new suits: invitational
1NT: some values, some stoppers. Usually about 4-7- HCP (Rule: Free bid of 1NT shows values. Free bid of suit or shows shape.)
2NT: Natural, invitational
Rebids by the PDer:
2 after a new suit is artificial and shows 19-21
or so. (Good 18s shade up sometimes.) Responses
to 2 are natural and non-forcing if not a jump or cue.
2 is not forcing.
2 after a NT response is Stayman.
Response to a cue-bid: New suits by doubler tend to be five-card suits except for the cheapest suit. NT promises a stopper; return cues tend to deny one.
Jumps in new suits show good strong jump overcalls.
(In OS, this only happens in /, / or /.)
Jump cue shows a mammoth takeout double with a void.
Jump to 2NT is a huge hand, 22-23 or so.
Raises show extra values, either by way of fit, shape, or high card. They promise at least 3-card support, usually four.
New suits after a simple suit response are five-card suits and don't promise any extra values. They are not forcing. (Yes, sometimes you'll be stuck. Thus, the cheapest new suit can (rarely) be a four-bagger.)
Cues after simple suit or 1NT responses are very strong. Herbert Negatives apply.
,
1, or
1.
The relay shows roughly 0-4 HCP. That means all other
minimum bids, including 1NT, show about 5-7. With 8,
cue to force to game. Double jumps become natural and
game forcing with 7-card suits. For example,
(1)-Dbl-(Pass)-3 shows something like
KQJ10xxx Axx xx A convention showing 19-21
is no longer needed in most sequences; doubler's cue is a game force
after a semi-positive response.
If the opponents open one of a minor, we are now using a jump cue to show very big hands with shortness in their suit, so power doubles pretty much guarantee two or more trumps.