# Energy Conservation Law in Industrial Architecture: An Approach through Geometric Algebra

^{*}

## Abstract

**:**

**CGn**), where

**Gn**is the Clifford algebra in n-dimensional real space and

**C**is the complex vector space. To conclude, a numerical example illustrates the clear advantages of the approach suggested in this paper.

## 1. Introduction

_{rms}I

_{rms}, the principle of conservation of energy does not apply to this quantity [12]. Then, the question that arises immediately is: why?. The correct answer is based on the fact that, in sinusoidal conditions and with linear and/or nonlinear loads, the traditional apparent power definition is erratic, except for resistive loads. This is a direct consequence of having only magnitudes for currents and voltages in a circuit branch instead of an expression composed by signed quantities; with this limitation, network analysis involving all the harmonics simultaneously cannot even be performed. Moreover, Kirchhoff’s circuit laws become simply inapplicable because the addition of quantities that represent the time signal of different frequencies has been not defined until now. Thus, the principle of conservation of energy cannot be corroborated for unified systems where sources and loads work simultaneously. This principle states: The instantaneous rate of instantaneous volt-amperes at the input terminal is equal to the sum of the instantaneous volt-amperes at each load component.

## 2. Geometric Algebra Foundations

**G**algebra is the product of all three vectors:

_{3}**G**algebra the trivector corresponds to the highest grade element, usually named pseudoscalar (

_{3}**J**), and this top grade coincides with the dimension of the underlying vector space.

**G**is spanned by the basis set:

_{3}_{3}system, $a={\mathsf{\lambda}}_{1}{\mathsf{\sigma}}_{1}+{\mathsf{\lambda}}_{2}{\mathsf{\sigma}}_{2}+{\mathsf{\lambda}}_{3}{\mathsf{\sigma}}_{3}$ and $b={\mathsf{\mu}}_{1}{\mathsf{\sigma}}_{1}+{\mathsf{\mu}}_{2}{\mathsf{\sigma}}_{2}+{\mathsf{\mu}}_{3}{\mathsf{\sigma}}_{3}$, is given by

**G**), higher dimensional oriented subspaces are also called blades. Therefore, the k-blade term is used for a k-dimensional homogeneous subspace. Thus, a vector is a 1-blade, a bivector is a 2-blade, and so on, up to the pseudoscalar n-blade. The combination of any of these objects configures a multivector. The k-grade part of a multivector is obtained from the grade operatorc, [18]. Scalar, vectors, bivectors, trivectors, and, in general, multivectors are called simply “geometric objects”.

_{n}**M**in

**G**can be written in the expanded form [18],

_{3}**a**is a vector,

**B**is a bivector, and

**J**is the pseudoscalar.

**M**is defined by the obviously positive definite expression,

## 3. Industrial Building Loads: Two Types of Harmonic Sources

#### 3.1. Current-Source Load Type (Harmonic Current Source)

#### 3.2. Voltage-Source Load Type (Harmonic Current Source)

## 4. Multivector Energy Conservation Law in Industrial Architecture

## 5. Numerical Example

## 6. Conclusions

- The traditional apparent power concept for industrial loads is unreliable: only in certain situations does it agree with the principle of conservation of energy, and as a result, it can lead to erroneous conclusion. This power concept must be revisited. The classical tools based simply on complex numbers are not enough to achieve this goal.
- In contrast to the classical concept of the apparent power, the proposed definition as a power multivector considers the net flow of all the power components present in industrial electric loads.
- The distinct nature of power components is easily differentiated in this algebra. This permits obtainment of a general proof of the principle of conservation energy for linear and nonlinear loads in non-sinusoidal conditions: the volt-amperes at the input terminal equals the sum of the volt-amperes at each load component.
- The proposed GCGA framework permits an easy geometric interpretation of the law of conservation of energy based on complex geometric objects.
- The distortion power achieves the additivity property.
- The multivector power theory generalizes the Tellegen´s theorem for LTI power systems operating in the AC non-sinusoidal steady state.

## Author Contributions

## Conflicts of Interest

## Abbreviations

AC | Alternating Current |

ASD | Adjustable Speed Drive |

${V}^{n}$ | n-dimensional real vector space |

$C$ | complex vector space |

GA, ${G}_{n}$ | Geometric Algebra |

GCGA, $C{G}_{n}$ | Generalized Complex Geometric Algebra |

LTI | Linear Time Invariant |

m-phasor | multivector phasor |

PCC | Point of Common Coupling |

RMS | Root Mean Square |

var | reactive volt-amperes |

vad | distortion volt-amperes |

## References

- Wagner, V.E.; Balda, J.C.; Barnes, T.M.; Emannuel, A.E.; Ferraro, R.J.; Griffith, D.C.; Hartmann, D.P.; Horton, W.F.; Jewell, W.T.; McEachern, A.; et al. Effects of harmonics on equipment. IEEE Trans. Power Delivery
**1993**, 8, 672–680. [Google Scholar] [CrossRef] - Henderson, R.D.; Rose, P.J. Harmonics: The effects on power quality and transformers. IEEE Trans. Ind. Applicat.
**1994**, 30, 528–532. [Google Scholar] [CrossRef] - Budeanu, C.I. Puisances Reactives et Fictives; Instytut Romain de l´Energie: Bucharest, Romania, 1927. [Google Scholar]
- Shepherd, W.; Zhakikhani, P. Suggested definition of reactive power for nonsinusoidal systems. Proc. Inst. Elect. Eng.
**1972**, 119, 1361–1362. [Google Scholar] [CrossRef] - Czarnecki, L.S. Considerations on the reactive power in non-sinusoidal situations. IEEE Trans. Instr. Meas.
**1985**, 34, 399–404. [Google Scholar] [CrossRef] - Sharon, D. Reactive power definitions and power factor improvement in nonlinear systems. Proc. IEE
**1973**, 120, 6. [Google Scholar] [CrossRef] - Ghassemi, F. New Apparent Power and Power Factor with Non-Sinusoidal waveforms. Power Eng. Soc. Winter Meet.
**2000**, 4, 2852–2857. [Google Scholar] - Slonim, M.A.; Van Wyk, J.D. Powers components in a system with sinusoidal and nonsinusoidal voltages and/or currents. Proc. Inst. Elect Eng.
**1988**, 135, 76–84. [Google Scholar] [CrossRef] - LaWhite, N.; Ilic, M.D. Vector Space Decomposition of Reactive Power for Periodic Nonsinusoidal Signals. IEEE Trans. Circuits Syst.
**1997**, 44, 4. [Google Scholar] [CrossRef] - Sommariva, A.M. Power Analysis of One-Ports Under Periodic Multi-Sinusoidal Operation. IEEE Trans. Circuits Syst.
**2006**, 53, 9. [Google Scholar] [CrossRef] - Castilla, M.; Bravo, J.C.; Ordoñez, M.; Montaño, J.C. Clifford Theory: A geometrical interpretation of multivectorial apparent power. IEEE Trans. Circuits Syst.
**2008**, 55, 10. [Google Scholar] [CrossRef] - Shepherd, W.; Zand, P. Energy Flow and Power Factor in Nonsinusoidal Circuits; Cambridge University Press: London, UK, 1979. [Google Scholar]
- Castilla, M.; Bravo, J.C.; Ordoñez, M.; Montaño, J.C. An Approach to the multivectorial apparent power in terms of a generalized poynting multivector. Prog. Electromagnet. Res.
**2009**, 15, 401–422. [Google Scholar] [CrossRef] - Castilla, M.; Bravo, J.C.; Ordoñez, M.; Montaño, J.C. The geometric algebra as a power theory analysis Tool. Przeglad Elektrochniczny
**2009**, 1, 202–207. [Google Scholar] - Menti, A.; Zacharias, T.; Milias-Argitis, J. Geometric Algebra: A powerful tool for representing power under nonsinusoidal conditions. IEEE Trans. Circuits Syst.
**2007**, 54, 601–609. [Google Scholar] [CrossRef] - Castro-Núñez, M.; Castro-Puche, R. Advantages of geometric algebra over complex numbers in the analysis of Networks with nonsinusoidal Sources and Linear Loads. IEEE Trans. Circuits Syst.
**2012**, 59, 2056–2064. [Google Scholar] [CrossRef] - Clifford, W.K. On the classification of Geometric Algebras. In Mathematical Papers; Tucker, R., Ed.; Macmilliam: London, UK, 1882; pp. 397–405. [Google Scholar]
- Doran, C.; Lasenby, A. Geometric Algebra for Physicists; Cambridge University Press: London, UK, 2003. [Google Scholar]
- Ventkastesh, C.; Srikanth, D.; Siva Sarma, D.V.S.S.; Sidulu, M. Modelling of nonlinear loads and estimation of harmonic in industrial distribution system. In Proceedings of the Fifteenth National Power Systems Conference, Bombay, India, 16–18 December 2008.
- Pomilio, J.A.; Deckmann, S.M. Characterization and compensation of harmonics and reactive Power of residential and commercial loads. IEEE Trans. Power Delivery
**2007**, 2, 2. [Google Scholar] [CrossRef] - Castilla, M.; Bravo, J.C.; Ordoñez, M. Geometric algebra: A multivectorial proof of Tellegen’s theorem in multiterminal networks. IET Circuits, Devices Syst.
**2008**, 2, 4. [Google Scholar] [CrossRef]

**Figure 1.**Basis sets of

**G**: scalar as point, vectors as directed line segments, bivectors as oriented planes, and trivector as oriented volume.

_{3}**Figure 6.**(

**a**,

**b**) Load fed by rectifier and harmonic voltage source equivalent, both with shunt filter.

**Figure 9.**(

**a**) ${\mathsf{\sigma}}_{0}$-plane: active and reactive power conservation; (

**b**) ${\mathsf{\sigma}}_{31}$-plane: distortion power conservation.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Bravo, J.C.; Castilla, M.V.
Energy Conservation Law in Industrial Architecture: An Approach through Geometric Algebra. *Symmetry* **2016**, *8*, 92.
https://doi.org/10.3390/sym8090092

**AMA Style**

Bravo JC, Castilla MV.
Energy Conservation Law in Industrial Architecture: An Approach through Geometric Algebra. *Symmetry*. 2016; 8(9):92.
https://doi.org/10.3390/sym8090092

**Chicago/Turabian Style**

Bravo, Juan C., and Manuel V. Castilla.
2016. "Energy Conservation Law in Industrial Architecture: An Approach through Geometric Algebra" *Symmetry* 8, no. 9: 92.
https://doi.org/10.3390/sym8090092