# The Impact of Power Definitions on the Disaggregation of Home Loads for Smart Meter Measurements
## 2 Power Definitions
### [PERSON]'s Proposal  
Professor [PERSON] created new definitions of apparent power in non-sinusoidal systems and understood that this power has more than two components, which can be represented in a three-dimensional system. The squared RMS (Root Mean Square) harmonic currents can be divided into two orthogonal terms according to (1):  
\[I_{h}^{2}=\left(I_{h}\cos\theta_{h}\right)^{2}+\left(I_{h}\sin\theta_{h}\right) ^{2} \tag{1}\]  
where \(\theta_{h}\) is the phase angle between the harmonic voltage phasor \(V_{h}\) and the harmonic current phasor \(I_{h}\). From Equation (1), it is possible to obtain the expression of the apparent power squared (2):  
\[S^{2}=V^{2}I^{2}=\sum_{h=1}^{v}V_{h}^{2}\Bigg{[}\sum_{h=1}^{v}\left(I_{h}\cos \theta_{h}\right)^{2}\Bigg{]}+\sum_{h=1}^{v}V_{h}^{2}\Bigg{[}\sum_{h=1}^{v} \left(I_{h}\sin\theta_{h}\right)^{2}\Bigg{]} \tag{2}\]Applying [PERSON]'s identity, the apparent power squared can be simplified to Equation (3):  
\[ \begin{array}{c}S^{2}=V^{2}I^{2}=\left(\sum\limits_{h=1}^{v}{{V_{h}}{I_{h}}\cos {\theta_{h}}}\right)^{2}+\left(\sum\limits_{h=1}^{v}{{V_{h}}{I_{h}}\sin{\theta_ {h}}}\right)^{2}\\ +\sum\limits_{m=1}^{v-1}{\sum\limits_{n=m+1}^{v}}{\left[{{\left({{V_{m}}{I_{n}} }\right)}^{2}+{\left({{V_{n}}{I_{m}}}\right)}^{2}-2{V_{m}}{V_{n}}{I_{m}}{I_{n}} \cos({\theta_{m}}-{\theta_{n}})}\right]}\end{array}  \tag{3}\]  
Equation (3) shows that apparent power in non-sinusoidal systems can be decomposed into three orthogonal components, visualized as a right-angled parallelepiped in Figure 1. The diagonal of this parallelepiped, given by Equation (4), represents the apparent power. In the previous equation, \(\theta_{m}\) is the phase angle between the harmonic voltage phasor \(V_{m}\) and the harmonic current phasor \(I_{m}\) and \(\theta_{n}\) is the phase angle between the harmonic voltage phasor \(V_{n}\) and the harmonic current phasor \(I_{n}\). \[S^{2}=P^{2}+Q_{B}^{2}+D_{B}^{2} \tag{4}\]  
The first term is the total active power (5) and the second term is called reactive power (6), or Budeanu reactive power, \(Q_{B}\). In the previous figure, \(P_{1}\) is the active power at the fundamental frequency, \(P_{H}\) is the active power at harmonic frequencies, \(Q_{1}\) is the reactive power at the fundamental frequency and \(Q_{H}\) is the reactive power at harmonic frequencies. \[P=\sum\limits_{h=1}^{v}{{V_{h}}{I_{h}}\cos{\theta_{h}}} \tag{5}\]  
\[Q_{B}=\sum\limits_{h=1}^{v}{{V_{h}}{I_{h}}\sin{\theta_{h}}} \tag{6}\]  
The third term is the distortion power (7). Nevertheless, the usual computation of distortion power is through (8). \[D_{B}=\sqrt{\sum\limits_{m=1}^{v-1}{\sum\limits_{n=m+1}^{v}}{\left[{{\left({{ V_{m}}{I_{n}}}\right)}^{2}+{\left({{V_{n}}{I_{m}}}\right)}^{2}-2{V_{m}}{V_{n}}{I_{m}}{I_{n}} \cos({\theta_{m}}-{\theta_{n}})}\right]}} \tag{7}\]  
\[D_{B}=\sqrt{S^{2}-P^{2}-Q_{B}^{2}} \tag{8}\]  
Figure 1: [PERSON]’s proposal definition for apparent power: (**a**) three-dimensional representation and (**b**) power components (separated into fundamental and nonfundamental). ### [PERSON]'s Proposal  
[PERSON]'s power definition separates fundamental (50/60 Hz) active and reactive powers from non-fundamental components [45]. This definition can be derived from the RMS voltage and current decomposition shown in Equation (9). \[V^{2}=V_{1}^{2}+V_{H}^{2};\ V_{H}^{2}=\sum_{h\
eq 1}V_{h}^{2};\ I^{2}=I_{1}^{2}+I_{ H}^{2};\ I_{H}^{2}=\sum_{h\
eq 1}I_{h}^{2} \tag{9}\]  
From (9), the result is that the apparent power squared has four terms (10):  
\[ \begin{array}{l}S^{2}=V^{2}l^{2}=\left(V_{1}^{2}+V_{H}^{2}\right)\left(I_{1} ^{2}+I_{H}^{2}\right)=\left(V_{1}I_{1}\right)^{2}+\left(V_{1}I_{H}\right)^{2} +\left(V_{H}I_{1}\right)^{2}+\left(V_{H}I_{H}\right)^{2}=\\ =S_{1}^{2}+D_{1}^{2}+D_{V}^{2}+S_{H}^{2}\end{array}  \tag{10}\]  
The first term is the fundamental or 60/50 Hz apparent power (11) and the remaining three terms are the non-fundamental (non-60/50 Hz) apparent power (12).