## 3 Apparent Power Difference Due to the Selected Voltage Reference Point: Neutral-Displacement Power
### Neutral-Displacement Power Expression
#### 3.1.1 Expressions of the Neutral-Displacement Powers Using Ideal Voltage Reference Points  
An example of an ideal voltage reference point is the neutral point of a balanced star-configured load with an infinite impedance in each phase, connected to an ideal ground with zero resistance (Figure 3). The main attribute of such an ideal grounding voltage reference point (G) is a zero-sequence line-to-ideal-reference point voltage of zero (\(V_{AG0}=0\)), according to [PERSON]'s theorem [31]. The apparent power measured by the power analyzer at the PCC of the system in Figure 3 when the voltage reference point is an ideal grounded point (G) can be expressed as follows, according to [PERSON] [20]:  
\[S_{pcc}^{i}=\sqrt{\big{(}V_{AG}^{2}+V_{BG}^{2}+V_{CC}^{2}\big{)}\big{(}I_{A}^ {2}+I_{B}^{2}+I_{C}^{2}\big{)}}, \tag{29}\]where the superscript \(i\) indicates that the apparent power is obtained using an ideal grounding point as the voltage reference point. The quadratic difference between the load apparent power (\(S_{l}\)) measured using the neutral point (\(n\)) as the voltage reference point and the apparent power measured at the PCC (\(S_{pcc}^{i}\)) can be determined as follows:  
\[\begin{split} S_{l}^{2}-\left(S_{pcc}^{i}\right)^{2}=\left[\left(V_ {An}^{2}+V_{Bn}^{2}+V_{Cn}^{2}\right)-\left(V_{AG}^{2}+V_{BG}^{2}+V_{CG}^{2} \right)\right]\left(I_{A}^{2}+I_{B}^{2}+I_{C}^{2}\right)=\\ =3\left(V_{An0}^{2}-V_{AG0}^{2}\right)\left(I_{A}^{2}+I_{B}^{2}+I_ {C}^{2}\right).\end{split} \tag{30}\]  
Because a line-to-ideal grounding point voltage does not have zero-sequence components (\(V_{AG0}=0\)), according to [PERSON]'s theorem, it is obtained using Equation (30).  
\[\sqrt{S_{l}^{2}-\left(S_{pcc}^{i}\right)^{2}}=\sqrt{3\cdot V_{An0}^{2}\left(I_ {A}^{2}+I_{B}^{2}+I_{C}^{2}\right)}=S_{nl}. \tag{31}\]  
Hence, the load apparent power difference in the system in Figure 3 measured by the power analyzer using either the load neutral point (\(n\)) or an ideal grounding point (\(G\)) as the voltage reference point can be determined from the load component of the neutral-displacement power (\(S_{nl}\)). Proceeding in the same way,  
\[\sqrt{S_{s}^{2}-\left(S_{pcc}^{i}\right)^{2}}=\sqrt{3\cdot V_{AN0}^{2}\left(I_ {A}^{2}+I_{B}^{2}+I_{C}^{2}\right)}=S_{ns}, \tag{32}\]  
the apparent power difference measured by the power analyzer at the source of the system in Figure 3 when the voltage reference point is either the source neutral point (\(N\)) or an ideal grounding point (\(G\)), and it is determined by the source component of the neutral-displacement power (\(S_{ns}\)).  
As it was observed in Section 3.1, a comparison of Equations (14) and (31) revealed some formal similarities.  
(1) Both equations contain the same quantities (voltages, currents, and powers).  
(2) The quantities are instantaneous (Equation (14)), and they are the square of the RMS voltages and currents or square of the powers (Equation (31)).  
These similarities also apply to Equations (16) and (32). In our opinion, they suggest that the neutral-displacement powers (\(S_{nl}\) and \(S_{ns}\)) given by Equations (31) and (32) quantify the effects of the neutral displacement phenomenon identified by the instantaneous powers in Equations (14) and (16).