## 6 Conclusions  
In this paper, a new bivectorial concept has been presented to be applied in control energy systems for sustainable constructions to the distortion power under periodic \(n\)-sinusoidal linear/nonlinear operation. Clifford theory turned out to be a suitable mathematical framework to introduce this new representation of distortion power bivector. It is derived in terms of geometric phasors from a multivectorial analysis of the apparent power (18) and not only as formal cross product of the rms values of voltages and currents. Moreover, the new power components are geometrically interpreted to distinguish the _rotated distortion_ and _distortion power_ bivectors. This geometric interpretation permits a vectorial direct sum of all the components into complex planes, (\(\mathcal{C}\)), defined for each multivectorial element of Clifford basis (\(\mathcal{G}_{n}\)). In classical approaches this goal is not possible, and any definition of distortion power is only a secondary result of this work. Both bivectors, _rotated distortion_ and _distortion power_ concepts and their relation to the optimization of energy control in building heating and cooling loads are the main subject of this paper.  
The potentiality of the power distortion bivector concept compared to any distortion power definition is important. Thus, analysing the results in Tables 1 and 2, the main findings of this investigation may be summarized as follows:  
* _Rotated distortion_ power bivector \(\widetilde{\Delta}\) incorporates in a single expression, Equation (21), and three attributes, namely magnitude, direction, and sense.
* In linear and nonlinear operation, if the voltage phase angles are the same or zero, _rotated distortion_ power \(\widetilde{\Delta}\) coincides with _distortion power_ bivector \(\widetilde{D}\) (Figure 6).
* In a general case, _rotated distortion_ power multivector depends of the harmonic voltage phase angles (Figure 8).
* For all cases, it can be seen that \(\|\widetilde{\Delta}\|=\|\widetilde{D}\|\).
* _Rotated distortion_ power \(\widetilde{\Delta}\) is an entirely mathematical and fictitious component and nonphysical quantity.  
In general, the multivectorial apparent power concept and the bivector distortion power concept (magnitude, direction and sense) are fundamental for a correct identification of harmonic source locations, power factor improvement, power quality indexes, dependency of the voltage phase angles and their physical interpretation, etc.  
Their application to electronic loads in today's microgrid allows one to control the energy of the load in vector form, considering its real and imaginary components when the phase angles present in the voltage harmonics differ from zero. In our case, residential building heating and cooling loads can inject higher harmonic currents than those comparable in industrial loads. Consequently, the harmonic spectra of these two types of loads are quite different where the 3 rd harmonic is highly noticeable for residential feeders.  
The suggested representation can provide a new language for the design of compensator systems, optimization algorithms or power quality indexes, that will help the design of more efficient energy networks in buildings and sustainable architecture. In this sense, the study of new applications is open it could be a task that deserves further research.  
Both authors have contributed equally to this work. Both authors wrote, reviewed and commented on the manuscript. Both authors have read and approved the final manuscript.  
The authors received no specific funding for this work.  
Not applicable.  
The authors declare no conflict of interest.