# 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
components are arithmetically added and identify each power phenomenon present in any part of the system. In contrast, the squares of the apparent power components are added, as they are orthogonal quantities that quantify the effects of each phenomenon identified by the instantaneous power components. Proposed Methodology to Obtain Formulations of [PERSON]'s Apparent Power and a Possible Set of its Components  
Based on the similarities between the instantaneous and [PERSON]'s apparent power expressions, we developed a method to express [PERSON]'s apparent power and one possible set of components derived from the instantaneous power in three steps. This method was applied to a three-phase system as follows. 1. The instantaneous power, \(p(t)\), was expressed as a function of either of the following elements: 1. Its possible components (fundamental active (+\(a\)) and reactive (+\(r\)) of the positive-sequence, unbalanced (\(U\)), non-fundamental (\(H\)) system), i.e., \[p(t)=p_{+a}+p_{+r}+p_{U}+p_{H}\] (29) 2. The instantaneous voltages and currents (\(v_{z},\ i_{z}\)) in each phase (\(z=A,B,C\)), i.e., \[p(t)=\overrightarrow{v}\cdot\overrightarrow{i}=\left( \begin{array}{ccc}v_{A }&v_{B}&v_{C}\end{array} \right)\left( \begin{array}{c}i_{A}\\ i_{B}\\ i_{C}\end{array} \right)=\sum_{z=A,B,C}v_{z}\cdot i_{z}\] (30) 3. The instantaneous voltage and current components (\(\sum\limits_{n}v_{zn},\ \sum\limits_{m}i_{zm}\)) in each phase (\(z=A,B,C\)), resulting in the application of [PERSON]'s theorem and/or Fourier series to the part of the power system where the apparent power must be obtained, i.e., \[p(t)=\overrightarrow{v}\cdot\overrightarrow{i}=\left( \begin{array}{ccc}\sum \limits_{n}v_{An}&\sum\limits_{n}v_{Bn}&\sum\limits_{n}v_{Cn}\end{array}  \right)\left( \begin{array}{c}\sum\limits_{m}i_{Am}\\ \sum\limits_{m}i_{Bm}\\ \sum\limits_{m}i_{Cm}\end{array} \right)=\sum_{z=A,B,C}\left(\sum\limits_{n}v_ {zn}\cdot\sum\limits_{m}i_{zm}\right)\] (31)
2. The expressions were set for the square of the Buchholz's apparent power using the following method: 1. If the instantaneous power was expressed according to (29), the square of the apparent power (\(S^{2}\)) and the square of its possible components (\(P^{2}_{+},\ Q^{2}_{+},\ S^{2}_{U},\ S^{2}_{H}\)) were expressed by applying the changes \(p(t)\to S^{2}\), \(p_{+a}\to P^{2}_{+}\), \(p_{+r}\to Q^{2}_{+}\), \(p_  
\[S^{2}=\left|\overrightarrow{v}\right|^{2}\cdot\left|\overrightarrow{i}\right|^{2}= \left(V_{A}^{2}+V_{B}^{2}+V_{C}^{2}\right)\left(I_{A}^{2}+I_{B}^{2}+I_{C}^{2} \right)=\left(\sum_{z}V_{z}^{2}\right)\left(\sum_{z}I_{z}^{2}\right)\] (33) where the vertical bars (\(\parallel\)) represent the norm of a voltage or current vector;
2. If the instantaneous power was expressed according to (31) as a function of the components \((n,m)\) of the voltage and current vectors in each phase (\(z=A,B,C\)), the square of the apparent power was expressed as the products of the squares of the RMS voltages and currents, which were obtained as functions of the RMS voltage components (\(V_{2n}\)) and RMS current components (\(I_{zm}\)) in each phase (\(z=A,B,C\)), as follows: \[S^{2}=\left(\sum_{n}\left(V_{An}^{2}+V_{Bn}^{2}+V_{Cn}^{2}\right)\right)\left( \sum_{m}\left(I_{Am}^{2}+I_{Bm}^{2}+V_{Cn}^{2}\right)\right)=\left(\sum_{z} \sum_{n}V_{2n}^{2}\right)\left(\sum_{z}\sum_{m}I_{2m}^{2}\right)\] (34) A square root was applied to Expressions (32), (33), and (34) to formulate the apparent power or its components. The apparent power expressions derived from this methodology always agreed with [PERSON]'s apparent power [4].