# Relationship between [PERSON]'s Apparent Power and Instantaneous Power in Three-Phase Systems
## 2 Formulations of Apparent and Instantaneous Power in Three-Phase Wye-Configured Systems
### [PERSON]'s Apparent Power
 \tag{20}\]  
Therefore, the source fundamental-frequency apparent power (16) can be decomposed into two components, similarly as expressed in reference [21], i.e.,  
\[S_{S1}=\sqrt{\sum\limits_{z}\left[\left(V_{zN1}^{+}\right)^{2}+\left(V_{zN1}^{ U}\right)^{2}\right]}\cdot\sqrt{\sum\limits_{z}\left[\left(I_{z1}^{+}\right)^{2}+ \left(I_{z1}^{U}\right)^{2}\right]}=\sqrt{S_{S+}^{2}+S_{SU}^{2}} \tag{21}\]  
where \(S_{S+}\) is the source fundamental-frequency positive-sequence apparent power,  
\[S_{S+}=\sqrt{\left[\left(V_{A\text{N1}}^{+}\right)^{2}+\left(V_{B\text{N1}}^ {+}\right)^{2}+\left(V_{C\text{N1}}^{+}\right)^{2}\right]\left[\left(I_{A1}^ {+}\right)^{2}+\left(I_{B1}^{+}\right)^{2}+\left(I_{C1}^{+}\right)^{2}\right] }=3\;V_{A\text{N1}}^{+}I_{A1}^{+} \tag{22}\]  
and \(S_{SU}\) is the source unbalanced power,  
\[S_{SU}=\sqrt{\sum\limits_{z}\left(V_{zN1}^{+}\right)^{2}\cdot\sum\limits_{z} \left(I_{z1}^{U}\right)^{2}+\sum\limits_{z}\left(V_{zN1}^{U}\right)^{2}\cdot \sum\limits_{z}I_{z1}^{2}}=\sqrt{\left(S_{SU}^{t}\right)^{2}+\left(S_{SU}^{t} \right)^{2}} \tag{23}\]  
where the source unbalanced power due to the current imbalances is represented by  
\[ \begin{array}{c}S_{SU}^{I}=\sqrt{\sum\limits_{z}\left(V_{zN1}^{+}\right)^{2 }\cdot\sum\limits_{z}\left(I_{z1}^{U}\right)^{2}}=V_{A\text{N1}}^{+}\sqrt{3 \cdot\left[\left(I_{A1}^{U}\right)^{2}+\left(I_{B1}^{U}\right)^{2}+\left(I_{ C1}^{U}\right)^{2}\right]}=\\ =3\;V_{A\text{N1}}^{+}\sqrt{\left(I_{A1}^{-}\right)^{2}+\left(I_{A1}^{0} \right)^{2}}\end{array}  \tag{24}\]  
where the superscripts and 0 denote the negative- and zero-sequence components of the fundamental-frequency line currents. The source unbalanced power determined by the voltage imbalance is represented by  
\[ \begin{array}{c}S_{SU}^{V}=\sqrt{\sum\limits_{z}\left(V_{zN1}^{U}\right)^{2 }\cdot\sum\limits_{z}I_{z1}^{2}}=\sqrt{\left[\left(V_{A\text{N1}}^{U}\right)^ {2}+\left(V_{B\text{N1}}^{U}\right)^{2}+\left(V_{C\text{N1}}^{U}\right)^{2} \right]\left(I_{A1}^{2}+I_{B1}^{2}+I_{C1}^{2}\right)}\\ =\sqrt{3\cdot\left[\left(V_{A\text{N1}}^{-}\right)^{2}+\left(V_{A\text{N1}}^{0 }\right)^{2}\right]\left(I_{A1}^{2}+I_{B1}^{2}+I_{C1}^{2}\right)}\end{array}  \tag{25}\]  
The source fundamental-frequency positive-sequence apparent power (22) can be also expressed as  
\[S_{S+}=3\;V_{A\text{N1}}^{+}\sqrt{\left(I_{A\text{1}a}^{+}\right)^{2}+\left(I _{A1\text{7}}^{+}\right)^{2}}=\sqrt{P_{S+}^{2}+Q_{S+}^{2}} \tag{26}\]  
where  
\[P_{S+}=3\;V_{A\text{N1}}^{+}I_{A\text{1}a}^{+}\text{ and }Q_{S+}=3\;V_{A\text{N1}}^{+}I_{A \text{1}r}^{+} \tag{27}\]  
represent the source fundamental-frequency positive-sequence active and reactive types of power, respectively. Finally, the source apparent power can be formulated as a function of its components, as follows:  
\[S_{S}=\sqrt{P_{S+}^{2}+Q_{S+}^{2}+S_{SU}^{2}+S_{SH}^{2}} \tag{28}\]The load apparent power and its components of the system in Figure 1 can be expressed similarly by using the values of the line currents and line-to-neutral voltages obtained by using the load neutral point (\(n\)) as a voltage reference.