I have spent rather too long on this very question... on and off for two or three years.. or was it more? can't remember exactly. I came to the conclusion that, for the 12th dynasty (about 2000 - 1800 bc), Ulrich Luft's dating was the best offer of the expert Egyptologists. The other Egyptologists are Ludwig Borchardt, Richard Anthony Parker and Rolf Krauss all with "slightly" differing chronologies considered within the standard chronology which has scope for variation. Arriving at the real chronology for Egypt is like a huge jig-saw puzzle in which new pieces are being discovered by archaeology all the time. Some important pieces of the jig-saw have been found since the days of Borchardt relating to co-regencies and reign lengths, and astronomical accuracy has improved since Parker, which leaves just the views of Ulrich and Krauss.
The standard chronology of Egypt puts Userkaf at the beginning of the 5th dynasty about 2500 bc. This is too early for the life of Joseph. The only way he might have been Pharaoh is if the Egyptian standard chronology is wrong, and one of the unconventional chronologies, such as that of David Rohl, is correct.
The problem with finding who was Pharaoh is not with the Biblical chronology but with the Egyptian chronology.
However, by taking what we know of the Egyptian chronology and seeing synchronies with the Bible account in the life of Joseph we can have some confidence that Jacob first appeared before either Senusert II (Greek name - Sesostris II) or Senusert III or the Pharaoh immediately before Senusert II, Amenemhat II, in the last years of his reign. These pharaohs ruled during the 12th dynasty of the Middle Kingdom era. The latter half of the 12th dynasty is said to have been a golden age for Egypt. It is clear to me that Joseph played the central role in bringing about that golden age.
Astronomical data to help discover the reign years of the Pharaohs of the 12th dynasty include 1. the lunar observations from Il Lahun; 2. 40 other lunar observations; 3. a heliacal rising of the star Sirius on 4 Peret 16 in the 7th year of an unnamed Pharaoh (but who was either Senusert III or Amenemhat III).
The problem with dating based on lunar observational dates is that the moon follows a 19 year cycle so that dates tend to be close to repeating themselves every 19 years.
The problem with dating based on a heliacal rising of Sirius/Sothis is that though the most likely candidate was the city of Memphis, we cannot be sure from what location the observation was made, and if the observation had been made from Elephantine in southern Egypt then the year of observation is several decades later. Rolf Krauss favours observations from Elephantine, Ulrich Luft believes the observations were made from Memphis. The ancient Egyptians said that Sirius disappeared for 70 days in spring/early summer each year - exactly 70 days only happens at Memphis. The history of Moses only fits Egyptian history if Memphis was the city of observation in the 18th dynasty; in later history, in the 6th century AD we are told that the observations were done at Memphis ("Ancient Egyptian Chronlogy" edited by Hornung, Krauss & Warburton, 2006, p440); and Memphis best fits for a synchrony of Egyptian history with the life of Joseph as well.
Whatever the reasons why Krauss would want to shift the city of location (ibid, p441), if we are to believe the scriptural account of the life of Moses then we must hold to Memphis as the city of observation.
[In a nutshell: Moses returned to Egypt after 40 years living with his father-in-law Jethro, after the death of the pharaoh who sought to kill him (Exodus 2:15, 2:23, 4:19; Acts 7:23, 7:30). If the heliacal rising of Sirius was made from Memphis then this plainly fits with the long reign of Thutmoses III ending in 1450 bc. The other proposed city of observation puts the death of Thutmoses III at 1425 bc, which does not fit an Exodus date of 1446 bc: no other pharaoh of the 18th dynasty before Thutmoses III had a reign longer than 25 years.]
The Biblical Chronology for the Exodus
Jacob entered Egypt with the children of Israel in 1876 bc at the time that would later be called Passover. This can be calculated from Exodus 12:40,41. At least two independent methods establish the date of the Exodus as 1446 bc.
The first method was calculated first by Valerius Coucke and then independently by Edwin Thiele: they both calculated the beginning of the Temple construction by Solomon in 967 bc. Then 967 + 479 gives 1446 bc for the Exodus (1 Kings 6:1).
In the second method the Seder Olam, a Jewish chronology produced in the 1st century, tells us that Ezekiel is writing about a Jubilee Year in Ezekiel 40:1 and that this was the 17th Jubilee Year since entering the Promised Land. Compare this verse with Leviticus 25:9 - Rosh Hashanah, New Year's Day, fell on the 10th of the month only in a Jubilee Year. The author of the Seder Olam had access to other writings, probably written by the Temple priesthood. They were the ones recording the Sabbath Years and the Jubilee Years, together with recording which Jubilee Year it was since the entering of the Promised Land.
The cuneiform inscription VAT 4956 is an astronomical record of Nebuchadnezzar's 37th year and contains enough astron. information to confidently date it to the year starting 1st Nisanu 568 bc. The complete configuration of all the planets described on VAT 4956 only happens every few thousand years - eg Saturn's described position only happens every 29.5 years (so the Watchtower proposed year of planetary observations on VAT 4956 of 588 bc is not possible). VAT 4956 effectively puts the destruction of the city of Jerusalem by Nebuchadnezzar at 587 bc (Jeremiah 52:29) and the Year of Jubilee in Ezekiel 40:1 as 573 bc.... 573 bc + (49 * 17) + 40 = 573 + 833 + 40 = 1406 + 40 years in the wilderness = 1446 bc.
Also, in my opinion, the date of 1406 bc for the beginning of the conquest of the Promised Land is convincingly supported by the evidence found in the archaeological site at Jericho. (For more on this see: How do Christians reconcile archeology with the Bible in the account of the Battle of Jericho?)
The above pieces of data all synchronize beautifully, and 430 years before 1446 is 1876 bc (Exodus 12:40,41).
(What is most interesting is that the author of the Seder Olam struggled to make his data fit, because of the lengths of the reigns of the kings recorded in Kings and Chronicles are so complex: the fact then that he tells us Ezekiel 40:1 is talking about the 17th Jubilee is virtual proof he had access to other writings, because it fits so perfectly with the other data we now understand. For what it is worth, this synchrony also sinks the Wellhausen Documentary Hypothesis and all its derivatives since the Jubilee Cycle was instituted in the book of Leviticus.
[See http://rcyoung.org/articles/unexpected2.html at www.rcyoung.org for more on this.])
David Rohl's chronology - not Scriptural
Even though his date for the Exodus is about 1446 BC, David Rohl puts Joseph as ruling Egypt about 1667 bc and Jacob's descent into Egypt at about 1662 bc ("A Test of Time", p331). He does not take account of Exodus 12:40,41 in his chronology, so his chronology does not agree with Scripture. And his 200 years is hugely insufficient for the huge growth in population of the Israelites from about 70 people, when you consider it became over 2 million at the time of the Exodus.
Archaeological hints showing the era of Joseph
In the early part of the Twelth Dynasty the local "nomarchs" (barons) had enormous power undermining the power of the pharaoh. During the reign of Senusert III this power was broken and a huge centralised royal administration sprang up with great speed. Egyptologists recognise this transfer of power did not happen because of any civil war.
_The domestic policy of Senusert III "centred on the re-organization of the administrative system. Since the Old Kingdom (2686-2181 BC), the major threat to royal power had probably come from the nomarchs, the provincial governors; a shift in the funerary patterns of the elite (a decline in [the grandeur of] provincial tombs) may indicate that Senusert III reduced their authority drastically by removing many of their established privileges. The means by which this was achieved is unclear, but henceforth it was the king's 'viziers' who oversaw all branches of administration." ("British Museum Dictionary of Ancient Egypt", Shaw & Nicholson).
This mystery of the Middle Kingdom era fits neatly with the seven years of famine when Joseph bought Egypt for the Pharaoh.
For more on the amazing synchronies between the Bible and Egyptian history using the Egyptian standard chronology of Ulrich Luft see my answer here:- How we do we know that Joseph wasn't Hyksos?.
To those interested in a general overview of Egyptian history I recommend "British Museum Dictionary of Ancient Egypt" by Ian Shaw and Paul Nicholson (2nd edition).
Those who want to try and get to grips with the chronology of the 12th dynasty (2000 - 1800 bc) will sooner or later need to get to grips with "Die chronologische Fixierung des agyptischen Mittleren Reiches nach dem Tempelarchiv von Illahun" by Ulrich Luft, 1992. Though I cannot read German I found this indispensible for info on the "40 other lunar observations". If you can read German then please translate the work into English!
Also "Lunar and Sothic data from the archive of El-Lahun revisited"; A paper by Rita Gautschy in "Current Research in Egyptology 2010 - Proceedings of the Eleventh Annual Symposium - Leiden University 2010".
"Ancient Egyptian Chronology", edited by E. Hornung, Rolf Krauss, and D. Warburton, 2006.
Free Online Information
The Astronomical Evidence for Dating the End of the Middle Kingdom of Ancient Egypt to the Early Second Millennium: A Reassessment
Website: The Star Sirius in Ancient Egypt and Babylonia, by Rita Gautschy
Website: Last and First sightings of the lunar crescent, by Rita Gautschy (Egypt used the last sighting of old crescent, everyone else (Babylon, etc) used first sighting of new crescent to determine first day of each month)