Most of the framework of "powers" were established during the "reign" of Octavian/Augustus. After the Battle of Actium in 31 BCE, there was no one left among the Roman elite in a position to challenge the authority of Octavian in the capital. Unlike his predecessor Caesar, Octavian was much more careful to "ease into" totalitarian rule over a series of years.

In January of 27, the Roman Senate and Octavian held a meeting. In it, Octavian made the gesture of "giving back" the Empire which he now nominally controlled. I'm not sure exactly how many legions were under his direct control by 27, since he had settled thousands of veterans already, but the number isn't important: he controlled virtually all of the human beings under arms in the Mediterranean, save for a few rebellious pockets, and he would soon deal with those as well. In this meeting, there was an elaborate show of him "giving back" the provinces of the Empire. Then, in turn, the Senate begged him to take that authority back. In the end, he retained direct control of Spain, Gaul, Syria, Cyprus, and Egypt, for a period of five years plus five additional years (the same as Caesar's imperium in Gaul). The only way the Roman state could conceive of such unprecedented and broad authority was via the office of consul, and so he was pre-selected as consul for every year down to 23 BCE. This meeting, usually called the 'First Settlement,' solidified the early conception of the imperial power: control over the provinces which contained the bulk of the legions, west to east, and an uninterrupted imperium via contiguous consulships. At this meeting Octavian also received the honorary title Augustus, and a whole slew of gifts loaded with symbolic power in Roman society. Octavian preferred the term princeps ("First among equals"), but now also had Augustus and imperator.

The nuance of the First Settlement is that it allowed the Senate room for dignity, whereas the dictatorship of Caesar had not. The Republican machinery was still running, sort of, and the senate still had business with which to busy themselves--just not in Spain, Gaul, or Syria! There were tax contracts to be awarded, envoys to be interviewed, proconsular appointments to be distributed, work contracts, etc etc etc. Immediately after the meeting in January of 23, Augustus left the capital to wage war on uncooperative tribes in Gaul and Spain, and he did not come back until 24 BCE. This gave the senate, and the capital, and Italy, and the world time to get used to the arrangement.

Then in 23 comes the "Second Settlement." His pre-arranged consulships were to expire, and now Augustus decided that he did not need the imperium implicit in the consulship any longer, which was by its very nature renewable but temporary. Instead, he took tribunicia potestas (the power of a tribune of the plebs) for life, giving him implicit veto in the senate, the ability to call a meeting, etc. His commands in his provinces (Gaul, Spain, Syria etc) were then renewed, giving him imperium proconsulare in those places (and there only, technically). This meant that, technically, he could not tell armies what to do in Italy (or anywhere else outside of his provinces). By this point it did not matter, because any potential rival who could wrest away control of the legions was immediately and quietly dealt with. That he was able to leave off the consulship in 23 suggests how entrenched his authority had become. The Roman world was weary, worn out by a century of civil wars, and Augustus was the man left standing.

There were some further changes in 19. Most sources say he was awarded imperium proconsulare maius in 19, which would have given him the ultimate authority over any army anywhere in the world. Just an afterthought by then, but it completed the basics of the emperor's power: imperium maius over everything; limitless power of the tribune of the plebs; permanent control over the armed provinces; and some other random symbolic possessions, like the office of Pontifex Maximus.