# 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
### Similarities between Formulations of Apparent Power and Instantaneous Power and their Components  
Table 1 displays the expressions for instantaneous power and Buchholz's apparent power and their respective components in a three-phase, non-sinusoidal, unbalanced subsystem. A visual scan of these expressions reveals formal similarities between instantaneous and Buchholz's apparent power, which are specified in the following principles:  
1. The instantaneous power \(p(t)\) in each part or subsystem of a z-phase power system is defined by the inner products of the voltage (\(\overrightarrow{v}\)) and current (\(\overrightarrow{i}\)) vectors [22; 23; 24; 25; 26; 27; 28] formed by the instantaneous values of the voltages and currents in each phase of the subsystem (\(p(t)=\overrightarrow{v}\cdot\overrightarrow{i}\)). Similarly, the instantaneous power components (active, reactive, unbalanced, non-fundamental, or distorted) are formed by the inner products of these voltage and current vector components. 2. The apparent power \(S\) in each part or subsystem of a z-phase power system is determined by the products of the norms (RMS values) of these voltage (\(\overrightarrow{v}\)) and current (\(\overrightarrow{i}\)) vectors in the subsystem (\(S=\left|\overrightarrow{v}\right|\left|\overrightarrow{i}\right|\)). Moreover, the apparent power components represent products between the norms (RMS values) of the components (active, reactive, unbalanced, non-fundamental, or distorted) of the voltage and current vectors. 3. The instantaneous and apparent types of power have the same components, which contain terms formed by the same voltages and currents in these components. The instantaneous power \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Components**} & \multicolumn{2}{c}{**Instantaneous Power**} & \multicolumn{2}{c}{**Apparent Power**} \\ \hline \multirow{4}{*}{Total} & \(p(t)=p_{a}^{+}(t)+p_{r}^{+}(t)+p_{ll}(t)+p_{H}(t)\) & \(S=\sqrt{p_{+}^{2}+Q_{+}^{2}+S_{ll}^{2}+S_{H}^{2}}\) \\ \cline{2-4}  & \(p(t)=\sum\limits_{z=A,B,C}v_{z}\cdot i_{z}\) & \(S=\sqrt{\sum\limits_{z}V_{z}^{2}}\cdot\sqrt{\sum\limits_{z}I_{z}^{2}}\) \\ \hline \multirow{4}{*}{\(\overrightarrow{\text{\tiny{Total}}}\)} & \multirow{2}{*}{Total} & \(p_{1}(t)=\sum\limits_{z=A,B,C}v_{z1}\cdot i_{z1}\) & \(S=\sqrt{\sum\limits_{z}V_{z1}^{2}}\cdot\sqrt{\sum\limits_{z}I_{z1}^{2}}\) \\ \cline{2-4}  & & Active & \(p_{a}^{+}(t)=\sum\limits_{z=A,B,C}v_{z1}^{+}\cdot i_{z1a}^{+}\) & \(P_{+}=3\;V_{A1}^{+}\cdot I_{A1a}^{+}\) \\ \cline{2-4}  & & Reactive & \(p_{r}^{+}(t)=\sum\limits_{z=A,B,C}v_{z1}^{+}\cdot i_{z1r}^{+}\) & \(Q_{+}=3\;V_{A1}^{+}\cdot I_{A1r}^{+}\) \\ \cline{2-4}  & \multirow{2}{*}{Total} & \multirow{2}{*}{Total} & \(p_{ll}(t)=p_{ll}^{l}(t)+p_{ll}^{V}(t)\) & \(S_{sll}=\sqrt{\left(S_{ll}^{l}\right)^{2}+\left(S_{ll}^{V}\right)^{2}}\) \\ \cline{2-4}  & & Current imbalance & \(p_{ll}^{l}(t)=\sum\limits_{z=A,B,C}v_{z1}^{+}\cdot i_{z1}^{ll}\) & \(S_{ll}^{l}=\sqrt{\sum\limits_{z}V_{z1}^{2}}\cdot\sqrt{\sum\limits_{z}\left(l_{ z1}^{l}\right)^{2}}\) \\ \cline{2-4}  & & Voltage imbalance & \(p_{ll}^{V}(t)=\sum\limits_{z=A,B,C}v_{z1}^{ll}\cdot i_{z1}\) & \(S_{ll}^{V}=\sqrt{\sum\limits_{z}\left(V_{z1}^{l}\right)^{2}}\cdot\sqrt{\sum \limits_{z}I_{z1}^{2}}\) \\ \hline \multirow{4}{*}{\(\overrightarrow{\text{\tiny{Total}}}\)} & \multirow{2}{*}{Total} & \(p_{H}(t)=p_{H}^{l}(t)+p_{H}^{V}(t)\) & \(S_{H}=\sqrt{\left(S_{H}^{l}\right)^{2}+\left(S_{H}^{V}\right)^{2}}\) \\ \cline{2-4}  & & Current distortion & \(p_{H}^{l}(t)=\sum\limits_{z=A,B,C}v_{z1}\cdot i_{zH}\) & \(S_{H}^{l}=\sqrt{\sum\limits_{z}V_{z1}^{2}}\cdot\sqrt{\sum\limits_{z}I_{zH}^{2}}\) \\ \cline{2-4}  & & Voltage distortion & \(p_{H}^{V}(t)=\sum\limits_{z=A,B,C}v_{z1}\cdot i_{z}\) & \(S_{H}^{V}=\sqrt{\sum\limits_{z}V_{zH}^{2}}\cdot\sqrt{\sum\limits_{z}I_{z}^{2}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of the instantaneous and apparent power expressions and their components in three-phase sources and loads according to our proposed approach.