## 3 Apparent Power Difference Due to the Selected Voltage Reference Point: Neutral-Displacement Power
### Neutral-Displacement Power Expression  
Buchholz's expressions of the apparent powers measured at the PCC of the load and source of the system in Figure 2, using the neutral points of the load (n) and source (N) as voltage reference points, respectively, are as follows [20]:  
\[ \begin{array}{l}S_{l}=\sqrt{\left(V_{An}^{2}+V_{Bt}^{2}+V_{Cnt}^{2}\right) \left(I_{A}^{2}+I_{B}^{2}+I_{C}^{2}\right)},\\ S_{s}=\sqrt{\left(V_{AN}^{2}+V_{BN}^{2}+V_{CN}^{2}\right)\left(I_{A}^{2}+I_{B}^ {2}+I_{C}^{2}\right)},\end{array}  \tag{19}\]  
where the sets (\(V_{An},~{}V_{Bn},~{}V_{Cn}\)) and (\(V_{AN},~{}V_{BN},~{}V_{CN}\)) represent the root-mean-square (RMS) values of the load and source line-to-neutral voltages, respectively, and the set (\(I_{A},I_{B},~{}I_{C}\)) represents the RMS values of the line currents. From Equations (18) and (19), we obtain  
\[S_{n}^{2}=\left[\left(V_{An}^{2}+V_{Bt}^{2}+V_{Cn}^{2}\right)-\left(V_{AN}^{2} +V_{BN}^{2}+V_{CN}^{2}\right)\right]\left\{\left(I_{A}^{2}+I_{B}^{2}+I_{C}^{2} \right). \tag{20}\]  
Equation (20) was applied to the following two well-known properties of [PERSON]'s theorem [31]:  
(i) For the fundamental-frequency (\(h=1\)) and each harmonic (\(h\
eq 1\)), the sum of the squares of the RMS line-to-neutral voltages satisfies  
\[V_{Ah}^{2}+V_{Bt}^{2}+V_{Ch}^{2}=3\cdot\left(V_{Ah+}^{2}+V_{Ah-}^{2}+V_{Ah0}^ {2}\right), \tag{21}\]  
where the subscripts \(+\), \(-\), _and_\(0\) respectively indicate the positive-, negative-, and zero-sequence components. (ii) The positive- and negative-sequence line-to-neutral voltages have the same values for directly connected loads and sources, i.e.,  
\[V_{An+}=V_{AN+}~{}~{}~{}~{}~{}V_{An-}=V_{AN-}. \tag{22}\]  
Based on the application, Equation (20) can be simplified as  
\[S_{n}^{2}=3\cdot\left(V_{An0}^{2}-V_{AN0}^{2}\right)\left(I_{A}^{2}+I_{B}^{2} +I_{C}^{2}\right), \tag{23}\]where \(V_{An0}\) and \(V_{AN0}\) are the RMS values of the load and source zero-sequence line-to-neutral voltages, respectively, determined by the application of a Fourier series. \[V_{An0}=\sqrt{\sum_{y=1}^{\infty}V_{An0y}^{2}},\quad\ V_{AN0}=\sqrt{\sum_{x=1}^{ \infty}V_{AN0x}^{2}}. \tag{24}\]  
It can be observed that Equations (18) and (23), which express the square of the neutral-displacement power, are formally similar to Equation (8). The similarities are as follows:  
(a) The square of the neutral-displacement power relates the same quantities using the same terms as the neutral-displacement instantaneous power. (b) Equation (8) involves the product of instantaneous voltages and currents, and Equation (23) involves the product of the squares of the same voltages and currents. In our opinion, these similarities indicate that the neutral-displacement power (\(S_{n}\)), determined by Equations (18) and (23), measures the effects of the neutral-point displacement phenomenon on the apparent powers, whereas Equation (8) identifies the phenomenon responsible for the effect. According to Equations (18) and (24), the neutral-displacement power derived from Buchholz's apparent power can be expressed as follows:  
\[S_{n}=\sqrt{S_{l}^{2}-S_{s}^{2}}=\sqrt{3\big{(}V_{An0}^{2}-V_{AN0}^{2}\big{)} \big{(}I_{A}^{2}+I_{B}^{2}+I_{C}^{2}\big{)}}. \tag{25}\]  
Equation (25) indicates that two components of the neutral-displacement power can be defined, namely, the load component (\(S_{nl}\)) and source component (\(S_{ns}\)). The neutral-displacement power components satisfy  
\[S_{n}=\sqrt{S_{nl}^{2}-S_{ns}^{2}}. \tag{26}\]  
The first neutral-displacement power component,  
\[S_{nl}=V_{An0}\cdot\sqrt{3\cdot\big{(}I_{A}^{2}+I_{B}^{2}+I_{C}^{2}\big{)}}, \tag{27}\]  
is a measure of the portion of the load apparent power that is directly caused by the load neutral-point displacement with respect to an ideal voltage reference point.