## 3 Apparent Power Difference Due to the Selected Voltage Reference Point: Neutral-Displacement Power
### Neutral-Displacement Power Expression
#### 3.1.2 Expressions of the Neutral-Displacement Powers Using a Real Voltage Reference Point  
A real voltage reference point (\(G^{\prime}\)) may be the neutral point of three star-connected quasi-balanced large but not infinite impedances, or a grounding point with a non-zero voltage. The grounding points in power systems are usually real.  
A zero-sequence line-to-real grounding point voltage (\(V_{AG^{\prime}0}\)) generally has a small but not zero value.  
If the voltage reference point (\(G^{\prime}\)) is a real grounding point, the apparent power measured by the power analyzer at the PCC of the system in Figure 3 can be expressed as follows, according to [PERSON] [20]:  
\[S_{pcc}^{r}=\sqrt{\left(V_{AG^{\prime}}^{2}+V_{BG^{\prime}}^{2}+V_{CG^{\prime }}^{2}\right)\left(I_{A}^{2}+I_{B}^{2}+I_{C}^{2}\right)}, \tag{33}\]  
where the superscript \(r\) indicates that the apparent power is determined using a real grounding point as the voltage reference point. If the voltage reference is either a real grounding point (\(G^{\prime}\)) or an ideal grounding point (\(G\)), the quadratic difference between the apparent powers measured by the power analyzer at the PCC of the system in Figure 3 would satisfy the following, based on Equations (30) and (34):  
\[\left(S_{pcc}^{r}\right)^{2}-\left(S_{pcc}^{i}\right)^{2}=3\cdot\left(V_{AG^{ \prime}0}^{2}-V_{AG0}^{2}\right)\left(I_{A}^{2}+I_{B}^{2}+I_{C}^{2}\right). \tag{34}\]Because the zero-sequence line-to-ideal grounding point voltage is zero (\(V_{AG0}=0\)), whereas the zero-sequence line-to-real grounding point is non-zero (\(V_{AG^{\prime}0}\
eq 0\)), we have  
\[\left(S_{pcc}^{r}\right)^{2}-\left(S_{pcc}^{i}\right)^{2}=3\cdot V_{AG^{\prime} 0}^{2}\left(I_{A}^{2}+I_{B}^{2}+I_{C}^{2}\right)=S_{n,pcc}^{2}. \tag{35}\]  
The parameter \(S_{n,pcc}\) is the neutral-displacement power measured at the PCC using a real grounding point (\(G^{\prime}\)).  
From Equations (19) and (33), the quadratic difference between the load apparent powers of the system in Figure 3 measured by the power analyzer using the load neutral point (\(n\)) and the real grounding point (\(G^{\prime}\)) as the voltage reference points, respectively, satisfies  
\[S_{l}^{2}-\left(S_{pcc}^{r}\right)^{2}=3\left(V_{Anl0}^{2}-V_{AG^{\prime}0}^{ 2}\right)\left(I_{A}^{2}+I_{B}^{2}+I_{C}^{2}\right)=S_{nl}^{2}-S_{n,pcc}^{2}. \tag{36}\]  
The load apparent power (\(S_{l}\)) can thus be obtained in terms of the PCC apparent power (\(S_{pcc}^{r}\)), the load neutral-displacement power (\(S_{nl}\)), and the PCC neutral-displacement power (\(S_{n,pcc}\)), as follows:  
\[S_{l}=\sqrt{\left(S_{pcc}^{r}\right)^{2}+S_{nl}^{2}-S_{n,pcc}^{2}}. \tag{37}\]  
In the same way, the quadratic difference between the apparent powers of the source of the system in Figure 3 measured by the power analyzer using the source neutral point (\(N\)) and a real grounding point (\(G^{\prime}\)), respectively, can be obtained from Equations (19) and (33) as follows:  
\[S_{s}^{2}-\left(S_{pcc}^{r}\right)^{2}=3\left(V_{AN0}^{2}-V_{AG^{\prime}0}^{2} \right)\left(I_{A}^{2}+I_{B}^{2}+I_{C}^{2}\right)=S_{ns}^{2}-S_{n,pcc}^{2}. \tag{38}\]  
Hence, the source apparent power (\(S_{s}\)) can be expressed as a function of the PCC apparent power (\(S_{pcc}^{r}\)) as follows:  
\[S_{s}=\sqrt{\left(S_{pcc}^{r}\right)^{2}+S_{ns}^{2}-S_{n,pcc}^{2}}, \tag{39}\]  
where \(S_{ns}\) is the source neutral-displacement power, and \(S_{n,pcc}\) is the neutral-displacement power measured at the PCC.