What are the advantages of using DC electric power over AC, as e.g. done in the EMB 135/145?
Now, DC is easier to produce than AC and easily to inverted to AC
To produce AC of a specific frequency, you will need one of below

An engine that rotates at a fixed rate.
A variable transmission box that normalizes the rotation rate.
Produce DC somehow then invert it to AC.

Method #1 is used by most American GE diesel locomotives. As a result, when the locomotive is providing AC power to the train cars, the engine needs to run at a fixed rpm, hence can not run at full rpm and full output. For this reason, a lot of other countries have switched to DC bus for train cars, essentially switched to method #3 (e.g. China uses DC600V for the train bus and it is inverted to 220V/380V 3-phase within each car).
Method #2 I believe is used by most airplanes, because fixing the engine at a certain RPM is simply not possible. The variable transmission is certainly not zero weight and is pretty complex and is certainly a point of failure. It seems like some airplane designs are switching to method #3 as well. They didn't switch to #3 earlier simply because aviation industry relatively is slow on new technology (so is the space industry).
Method #3 is basically what everyone else chooses since the beginning of this century, when semiconductor finally makes it cheap and simple and efficient to invert DC to AC. From small gasoline generators and home air conditioning, to electric cars and trains and ships, whenever the frequency of the power generator doesn't match the consumer, using DC and semiconductor power electronics to bridge the gap is the simple choice.
Most avionics are electronics, and most electronics runs on low voltage DC
Except the gyro, the CRT display, and environment control, everything else runs on DC internally. If you use an AC bus, then every device needs its own converter for it's internal working DC requirement. If you need a converter for each device anyway, then why not a DC-DC one and it's lighter and smaller and possibly cheaper?
Changing voltage used to be easier for AC than DC, now it's sort of the other way.
Before semiconductor, the only efficient way of converting voltage is via a transformer, hence AC. But transformers are heavy, which is why aviation uses 400Hz instead of 50/60Hz: transformers for higher frequencies are lighter and smaller. With modern semiconductor power electronics, you can basically convert DC voltages with near-zero weight, which is certainly desirable for airplanes. Also, semiconductor are made from sand, a lot cheaper than copper that makes a transformer.
DC-DC voltage conversion is more robust than AC-AC
An AC power transformer converts the voltage at a fixed ratio, so when the input side's voltage or frequency fluctuates, it propagates to the output, potentially affecting the downstream devices. DC-DC converters and DC-AC converters outputs at a fixed spec, so the input fluctuations are isolated from the output. This additional protection is certainly desirable for the aviation industry where safety, reliability and fault tolerance are the top of concerns.