The cause of voltage phase shift in a power system

Let's call the 3 phases A, B and C and let's say we notionally have a neutral wire. Neutral is basically 0V in the system.

The "A" phase voltage (to neutral) is my chosen reference that all other voltage phase angles are measured from hence, V\$_B\$ is 120 lagging V\$_A\$ and V\$_C\$ is 120 degrees leading V\$_A\$.

OK so far?

What about the voltage between line A and line B (aka V\$_{AB}\$) - this is called line voltage (not to be mistaken with voltages between phase and neutral). Line voltages are \$\sqrt3\$ times bigger than phase voltages.

OK so far?

If you are not just examine what happens here: -

If you use trigonometry and resolve all the triangles you can find the length of V\$_{AB}\$ - it is \$\sqrt3\$ times bigger than either A or B to neutral.

It's also 30 degrees leading A and this is where the 30 degrees comes from.

So, a delta primary will receive primary line voltages of V\$_{AB}\$. V\$_{BC}\$ and V\$_{CA}\$.

Given that a transformer doesn't inherently phase shift anything (other than the trivial cases of 0 degrees and 180 degrees), any secondary winding voltage must be in phase with their respective primary voltage no-matter whether the secondary is connected delta or wye.

OK so far?

Then you have it because, a delta primary works with line voltages and these are 30 degrees shifted to their nearest phase voltage. The secondary outputs are also 30 degrees shifted and hence a wye secondary will produce a phase voltage that is 30 degrees shifted from the equivalent (but not directly connected to) phase voltage on the primary.

It's trivial to do the wye-delta transformer so I'll leave it to someone else.

Your question misses the point.

As there is a neutral conductor between the generator and the load, any imbalance between currents in the three phases is taken up by current in the neutral conductor.

Balance, when feeding multiple single phase loads, is purely statistical. If you power enough houses, then you will find that the current flow in neutral is noisy, generally small with the occasional peak.

Balance, when driving 3 phase loads, is part of the specification of the load. 3 phase motors and heaters will usually have a spec for what balance they are intended to achieve.

The connection of neutral to local ground is a different issue. If the whole system was connected to ground at a single point, say at the source, then of course there would be no current flow through that tie conductor, and no voltage difference between those points. However, the neutral conductor arriving at a distant point would have a voltage drop across it due to the current flowing, and a potential maybe a few volts from ground.

If there is a further local tie point, then that potential of a few volts will drive a current through that tie point. What is happening now is that the neutral current is being shared between the neutral conductor, and the ground, between source and load.

Generally, injecting significant currents into ground is deemed to be a BAD THING, and the many regulations around how to load 3 phase systems, how to size neutral conductors, when and where you can and can't tie neutral to local ground, are in pursuit of minimising ground currents.