Uniform price vs. pay-as-bid auctions in energy markets
I believe &quot;Auctions of Homogeneous Goods: A Case for Pay-as-Bid&quot; by Pycia and Woodward answers your questions theoretically. This is quite recent and their results are striking. They also briefly discuss some empirical insights.

The  pay-as-bid  (or  discriminatory)  auction  is  a  prominent
format  for  selling  homogenous goods such as treasury securities
and commodities.  We prove the uniqueness of its pure-strategy Bayesian
Nash equilibrium and establish a tractable representation of
equilibrium bids.  Building on these results we analyze the optimal
design of pay-as-bid auctions, as well as uniform-price auctions (the
main alternative auction format) allowing for asymmetric information.
We show that supply transparency and full disclosure are optimal in
pay-as-bid, though not necessarily in uniform-price; pay-as-bid
is revenue dominant and might be welfare dominant; and, under
assumptions commonly imposed in empirical work, the two formats are
revenue and welfare equivalent.