Problem understandin per unit system

Question

• I having trouble understanding PU system especially "briding" the gap from PU system -> to Actual system
• To illustrate this here is an example where I am given the one-line diagram for the three phases system:

The first objective is to find the current going from 2 to 3 in pu. When I am given the line voltage at bus 3 and the complex power of the load: $$V_3 = 400\ kV\require{enclose}%\enclose{phasorangle}{a+b}$$ $$S_3 = (77+j14) MV\!A$$ in the following calculations I will use: $$S_{base}=100~~MV\!A$$ $$V_{base}=400~kV$$ using that the formula for pu power in a three-phase per unit system is: $$S_3=V_3\cdot I_{23}^*$$ $$I_{23}= (\frac{S_3}{V_3})^* $$

$$ I_{23} =(0.77-j0.14) pu = 0.78\enclose{phasorangle}{-10.3^\circ}~pu $$ In the above calculation I let \$V_3\$ be the reference angle. If I wanted to find the actual current I would need to time the per unit value by base value: $$ I_{base} =\frac{S_{base}}{\sqrt{3} \cdot V_{base}}$$ $$ I_{base} =144.3\ A$$ Hence: $$ \{I_{23}\}_A = \{I_{23}\}_{pu}* \{I_{base}\}_A = 112.9\enclose{phasorangle}{-10.3^\circ}~A$$ Question 1: Is it correct that \$\{I_{23}\}_A\$ is the current flowing form bus 2 to bus 3 with \$V_3\$ (Line-Line) as reference angle set to 0? (of course the angle could be +120 and -120 also)

Now let's say I would like to first find the actual current then convert to per unit value. Using the line voltage for the red phase as reference (same as before) angle, assume a star load:

Line current equals phase current in star connection, for the red phase: $$ \{I_{23}\}_A =I_{RN}= (\frac{S_{3\phi}}{3\cdot V_{RN}})^* $$ $$ V_3 = V_{RN} \cdot\sqrt{3} \enclose{phasorangle}{30^\circ}\ V $$ $$ \{I_{23}\}_A = (\frac{S_{3\phi}}{\sqrt{3}*V_{3}\enclose{phasorangle}{30^\circ}\ V})^* $$ $$ \{I_{23}\}_A = 112.96\enclose{phasorangle}{-40.3^\circ}\ A $$ Getting per unit value: $$ \{I_{23}\}_{pu} = \frac{\{I_{23}\}_A}{I_{base}}$$ $$ \{I_{23}\}_{pu} = 0.78 \enclose{phasorangle}{-40.3^\circ}\ pu$$ Question 2 Why the difference in result (angle) from method 1 and method 2. I get the same magnitude but I get a difference in 30 degrees in the angle, why do I get that difference? What have I done wrong? My thoughts on question 2: I can see that in the second case if I used \$V_{RN}\$ as reference angle I would get the same angle, but on the other side in both cases, I used the line voltage for the red phase as the reference angle.

Sorry for my poor English, but thanks for the help in advance:) Literature suggestions to get a good understanding of per unit systems would also be greatly appreciated:D

Answer 1

You must note that the per unit system is not consistent with phase shifts through transformers, or star or delta connected loads.

The actual phase shift is only calculated in the end if it matters.

Answer 2

Simple clear explanation on how to solve per unit problem in power system.

It is important to note that phase angles have no effect when calculating your per unit impedances, they are however important for illustration, as you could be dealing with inductive or capacitive impedances.

The image and link below are from my YouTube channel:

https://www.youtube.com/watch?v=ydtmzQhv--Q

Answer 3

First I would notice that you got difference in phase angles and this has no relation to the per unit system as angles are expressed in degrees/radians regardless of using per units or actual values.

Then, when you draw a one-phase diagram you draw it for one single phase. So voltages are phase-to-ground voltages; you just multiply them by \$\sqrt 3\$ for convenience. In your second example you put line-to-line voltage as reference and introduce 30 degree shift in phase-to-ground voltage. This phase shift just propagated to the current.

So, actually $$ V_3=\sqrt 3 V_{RN} $$ is correct transform from voltages in real 3-phase system to voltages in one-phase equivalent.