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The Geometric Algebra as a Power TheoryAnalysis Tool

1 Electrical Engineering Dept., 2Applied Mathematics Dept.,4 Electronic Technology Dept., 5 Applied Physic Dept.(University of Sevilla),3 Spanish Research Council (CSIC).

e-mail: castilla@us.es, Web site: http://11/wlv.irnase.csic.es/users/invespot/index.htm

Abstract-In this paper, a multivectorial decomposition ofpower equation in single-phase circuits for periodic n­sinusoidal !linear and nonlinear conditions is presented. Itis based on a frequency-domain Clifford vector spaceapproach. By using a new generalized complex geometricalgebra (GCGA), we define the voltage and currentcomplex-vector and apparent power multivector concepts.First, the apparent power multivector is defined as geometricproduct of vector-phasors (complex-vectors). This newexpression result in a novel representation andgeneralization of the apparent power similar to complex­power in single-frequency sinusoidal conditions. Second, inorder to obtain a multivectorial representation of anyproposed power equation, the current vector-phasor isdecomposed into orthogonal components. The powermultivector concept, consisting of complex-scalar andcomplex-bivector parts with magnitude, direction and sense,obeys the apparent power conservation law and it handlesdifferent practical electric problems where direction andsense are necessary. The results of numerical examples arepresented to illustrate the proposed approach to powertheory analysis.

I. INTRODUCTION

The analysis of power theory has been discussedextensively. The large number of papers publishedmotivated by the contributions of Budeanu [1] infrequency domain and Fryze [2] in time domain, suggestthat the work has not been finished. In n-sinusoidalconditions, research on power definitions [3]-[12] hasbeen carried out with very different objectives asmathematical meaning, physical meaning, power factorimprovement, distortionless conditions, etc. Moreover,some noteworthy progress has been made by thecontributions [5], [6], [11-12]. In particular [12] isconcerned with a representation of power equation in themathematical framework of Geometric Algebra.Therefore, it has been concluded that the typical nonlinearbehaviour of the distribution systems require, for itscomplete analysis, a new mathematical structure that canguarantee the multivectorial character of differentcomponents. In this sense, our work considers a newrepresentation of power theory deduced from generalizedGeometric Algebra [13-14]. It is based on adecomposition of apparent power into multivectorialcomponents in the frequency domain. The apparent powermultivector is derived in terms of the voltage and currentvector-phasors, and contains all power information(magnitude, direction, and sense).

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II. CLIFFORD SPACE-VECTOR THEORY: GENERALIZEDCOMPLEX GEOMETRIC ALGEBRA ( CCI" )

B. Generalized Complex Geometric Algebra:

New Geometric Product (g).Let us introduce vector-phasors (complex-vectors) inorder to analyze circuit power theory in nonsinusoidalconditions. To define these new phasors, we start from an

n-dimensional linear space V n, of elements that are

termed vectors. If {(J"J ,(J"2' (J"3' ...(J"n} is an orthonormal

basis of V n, (n is equal to the number of harmoniccomponents in periodic sinusoidal signals), the unitelement of this algebra is denoted by (j0 • The vector basis

for the Clifford algebra {C In} is generated by

{I, (J"k ' (J'k /\O"h , ... , O"J /\ 0"2 /\(J'3 ... /\O"n} (1)

s';;;;lar v~s ~ pselld~scalar I

(k:J, ... ,n) (k.h:l .....n;k:t:-h)

where " /\ " denotes the outer product and(J"k /\ (J'h = (J"k(J'h = (J'kh [10]. Each coefficient of a basic

vector (J"j replaces one of the orthonormal functions in the

Fourier decomposition. The elements in this geometricalgebra are termed multivectors [13]. But the electricalquantities voltage and current have no easy interpretationin classic Clifford Algebra. For this reason we will definea new geometric algebra - a generalization of the classicClifford Algebra, which we have termed "GeneralizedComplex Geometric Algebra" (GCGA) -. A morecomplete information can be seen in [13-14] andAppendix.

III. POWER MULTIVECTOR

A. Multivectorial Representation ofPeriodic Signals

Suppose that a nonsinusoidal voltage

u(t)=/2 L Upsin(pOJt+ap) (2)pELvN

is applied to a nonlinear load, where p is the harmonicorder of u(l). The resulting current has an instantaneousvalue given by

i(t) =/2 L Iq sin (qOJt + pq ) (3)qENvM

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where q is the harmonic order of i(t). It is assumed that agroup of voltage harmonics N exist that havecorresponding current harmonics of the same frequencies,that components L of the supply voltage exist withoutcorresponding current, and that components M of currentexist without corresponding voltages. In linearconditions, fJq == aq - qJq' qJq is the impedance phase angle

and L == {¢} ,M == {¢} . The capital Up and 1q represent

rms values of up (t) andiq (t).

In the {CCln , g} structure spanned by orthonormal

basis multivectors{lT j ,lT2 ,lT3 , ...lTn }, the associated p-th

harmonic voltage and q-th harmonic current can berepresented by the vector-phasors:

Note that, the squared value ISf in eqn.(5) may be

represented as

and is associated to linear (Dun) and/or nonlinear

(DNonlin) distortion power. It is seen from (6) that ILiI == \15\.

If a p == aq in linear operation, and a p == aq and/or

a p == 0 in non linear operation, eqn. (5) is now given by

(7)

(8)+ L Uplqe-jf3q a

pqpELuN,qEMpEL,qEN

S== LUp gI; == P+ jQ+DpENqEN

and eqn.(6) is given by

(4)U- -10 I .lap - U-p - p e lTp - pap

I == 11 Ie.ifiqa == 7 ITq q q q q

where lOp I == up' IIq 1== lq .Then,°== L Up,pELuN

1 == L 1 q , 1°12 == L 10pI2

and 1112 == L 11ql2qENuM pELuN qENuM 1-12 1- - 1

21-1

2\-1

2S == U g]* == U ] == p2 + Q2 + D2 (9)

B. Power Multivector

According to (Bl-B5), the apparent power at the

nonlinear load, can be obtained as a multivector S inCCln , generated by the geometric product "g" of the

voltage and conjugate current vector-phasors

,~= P'~L up gi; =(~u/ pCOS9'p +j~U/p sin9'p ) <To +qeNuM

The suggested apparent power multivector is veryimportant, and represents a new concept of apparent

power. The eqn. (9) is the squared value ofS, for linearand nonlinear networks under nonsinusoidal conditions.

This value lSi, is a consequence only of the multivector

Sand one of this paper's main contribution. In particular,

ISl2 is the sum of the squared values of the components of

S. It should be noted that whereas lSi is a simple value,

the multivector Shas magnitude, direction, and sense.

(5)

which consist of a complex-scalar and a complex­bivector. In eqn. (5), "g" is the new "generalized

complex geometric product" (B5), and (*) is the standard

"complex conjugate" operation (C2).

Clearly, Ipi =L UpIpcosrpp is the active power or averagepEN

value of the instantaneous power in the time domain.

101 =L UpIpsin rpp is the called reactive power and is not apEN

real physical quantity. It is merely the geometriccomplement of active component. Note from eqn. (5) that

(p + jQ) (To is the complex-scalar. The complex-bivector,

~ , named rotated distortion power, is given by

IV. BUDEANU'S, SHEPHERD'S, AND CZARNECKI'S

MULTIVECTOR POWER EQUATION

In this Section, widely accepted power equations havebeen analyzed in multivectorial form. Particular emphasisis given to the nonlinearity introduced by distorted sourcevoltage. In this sense, the resulting current vector- phasor

I eqn. (4), may be subdivided into three components:

~I denoted by "in phase", Il- "in quadrature" and lq "non

linear" currents

1== ~I + jll- + 1Nonlin. == L 1q" + jL 1 ql- + L 1 q,Nonlin (10)qEN qEN qEM

Then, in linear operation according to (EI-E2)

Li == ~ { ej(ap-aq ) (V J ejrpq

- VJ ejrpp)a } +~ pq qp pq

p<q ''-------v-----~

p,qEN DUn

1== L lq == ~I + jll- (11)qEN

+ ~ jap U J -jf3q == A + A ==~ e p qe a pq ilLin ilNonlin

pELuN,qEM ~pEL,qEN DNonlm

(6) and

- N,N - L,N - L,M - N,M== ~ Lin +~Nonlin +~Nonlin + ~Nonlin

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and Shepherd's power equation multivector is given by

[ ] [ ]

(19)- _ -" - _ - L,N - L,M - NMSD - I Up g I Iq - ~Nonlin + ~Nonlin + ~Nonlin

pEL qENpELuN qEM

In these conditions, we define the apparent powermultivector as

S= ug1* = (I Up +I Up Jg[I (1;11 +j1;l-) +I I;] =pEL pEN qEN qEM

= I Up g(I 1;11 +I 1;l-J+ Iup gI I; +pEN qEN qEN pEL qEM

+IUp gI1;+ IUp g(I1;,,+ Il;l-JpEN qEM pEL qEN qEN

(12)

- N,N - L,N - L,M - NMThe terms ~Act.,Lin' ~Nonlin' IJ.Nonlin '~Nonlin

in (EI0), (EI4)-(EI6).

can be seen

where the products

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C. Czarnecki's Multivector Power Equation

Czarnecki [4], distinguishes current components due tolinear load and nonlinearities. In this sense, the goal oftheir power equation is the identification of the physicalphenomena responsible of each current component,active, reactive, scattering and harmonic currents. Heproposed that the power equation related to this currentdecomposition as

are defined in Appendix.S~za = p2 +D; +Q~ +S~ (21)

A. Budeanu's Multivector Power Equation

Budeanu suggested that the apparent power consist ofthree components as in eqn.(5). The multivectorial form ofthis equation is given by

B. Shepherd and Zand's Multivector Power Equation

According to the current decomposition (11), theseauthors have proposed in [3] the power equation

(16)

where SR is the "apparent active power ", Sx is the

"apparent energy-storage reactive power" and SD the

"apparent distortion power". Applying (BI-B5) (EI0),the different power multivector components are found asfollows

In GCGA framework, Ds component coincides, for linear

operation, with "scattering" power defined in [5]

+ ~ j(ap-aq ) R (U I jCPq - U I jCPP)rr = AL,N + AN,NL.J e e p qe q pe v pq LJAct.Nonlin LJAct.Lin

p<q I

p,qEN Dpq .Lm

Reactive power is given by

(22)

+ L ej(ap-aq ) Im(UpIqejcpq )0"pq =pEL,qEN '-----v--"

Dpq,.Vonlin

and Sit apparent harmonic component, may be writtenSR =( I OpJg( I I';IIJ =

pEN qEN

=?+ I ej(ap-aq\UplqcosCfJq -UqlpcosCfJp)O"pq =p<q

- -N N= P + ~A~t,Lin

c~x =C~Up Jg(q~i;L J== jQ+ I ej(ap-aq)(Uplq sinq?q -UqIpsinCfJp)O"pq =

p<q

=jQ + :i~~~ct.,Lin

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(17)

(18)

3

- (~ -J"(~*J -LN -V,MSit = L.J Up g L.J Iq = AVonlin + L1Nonlin

pELu:V qEAI

Then,

where SCza is the Czarnecki's power multivector.

Thus, eqn.(9) can be written

(23)

(24)

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(25) an instantaneous value

Consequently, today's power equation approaches can beanalyzed from the proposed algebraic structure, but thosecannot explain the results obtained by this one.

v. NUMERICAL EXAMPLES

In this section, a numerical example is developed. Units ofphysical quantities are the standard ones of the MKSAsystem, and are thus omitted everywhere.

A. Example 1

Let a periodic n-sinusoidal voltage with instantaneous

value given by u1(t)=J2"[200sin(wt)+100sin(2wt)] be

applied to a non linear load. Note that a 1 =a 2 =0 phase

angles. The resulting current has an instantaneous value

. ~[40sin(mt - 53.1°) + 11. 7 sin(2mt - 69.4°) +], 1(1) =-v2

+10 sin(3mt + 30°)

The corresponding vector-phasors are respectively

V1= 200eJoa1+100eJOa2,IVI = 223.60

11

= 40e-J53.1a1 + 11.71e-J69.4a2 + 10eJ30a3,III = 42.86

From (5), and according to (8), the apparent powermultivector may be written

81 = vgI* = (5215.4 + j7494)ao+ (-1578 - jl006)a/2+

+(1732 - jl000)a13 + (866 - j500)a23

where 1J11 =5215.4 ,IQ11 =7494, ILl11=2916, IS/\ = 9584.15

and power factor P~=0.54 .Table I, illustrates the

simulation results.

TABLE I.

p=l, q=2 p=l, qc=3 p=2, qc=3

Li i1L~ = -1578 - jl006 L1L~ = 1732 - j j000 L1i;~ = 866 - j500

fJ Di2 = -1578 - jlO06 Di3 = 1732 - jlOOO D2j = 866 - j500

ILiI IL1L~I= 1871 I]L:, I = 2000 1]i;~1 = 1000

1

151

115121 = 1871 115131 = 2000 115231 = 1000

B. Example 2

Let a periodic n-sinusoidal voltage with instantaneousvalue given by

be applied to a non linear load. The resulting current has

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. ~[40Sin(mt-13.10)+ 11.7sin(2mt-39.40)+], 2 (1) = -v 2

+10sin(3mt + 30°)

In this case, 0.1= 40°,0.2= 30° , and the corresponding

vector-phasors are respectively

U2 = 200eJ4

0al + 100eJ3

0a2 , lUi = 223.60

i2 = 40e-J13

.1(Jj + 11.7e-

j39.4(J2 + 10e

J3°(J3' Iii = 42.86

From (5), and according to (6), the apparent powermultivector can be given by

32 = ugl* = (5215.4+ j7494)(Jo +(-1379- j1265)(Jj2 +

+(1970+ j347.3)(J13 +(1000+ jO)(J23

wherelP21 =5215.4 , IQ21=7494, ILl21=2916 , 1821=9584.15

and power factor PF;=0.54

Table II, illustrates the simulation results.

TABLE II.

p=l, q=2 p=l, q=3 p=2, q=3

Li L1L~ =-1379- j1265 L1~~ = 1970 + j3.J7.3 L1i;~ = 1000 + jO

15 Di2 = -1578 - j1006 Di3 = 1732 - jl000 D23

=c 866 - j500

ILiI I]L~I = 1871 lL1i~1 = 2000 lL1i~1 = 1000

115

1ID12 1 = 1871 ID13 1 = 2000 ID23 I = 1000

For possible compensation purposes, it is interesting tonote that on Budeanu's approach in Example 1 and 2, the

reactive power, Q/=Q2=j7494rro vector-scalars are the

same. From Shepherd's, and Czarnecki's multivectors,applying (18), (22), (EI0), (EI4)-(EI6), reactive powermultivectors are given by

Su:=j7494ao-j1006a12 ,

Q/r =j7494ao-j1006a/2

and consistent with (6), (19) and (23), distortion powerbivectors on the first example are found to be

LJ/=D1= (-1578-j1006)a/2+(1732-j1000)a/3+

+ (866-j500)a23

SiD =(1732-j1000)a13 + (866-j500)a23

S/Il =(1732-j1000)a/3+ (866-j500)a23

On the second example, these components are given by

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S2x=j7494(Jo-.j1265(J/2 ,

Q2r =j7494(Jo-.j1265(J/2

where the coefficients Zl...k E C and the basis (jl...k E Cln •

It is trivial that CCln is a vector space over R . According

to (A 1) definition, in complex-vectors case we obtain the

and distortion power bivectors are given by

il2 = (-1379-j1265)(JI2+(1970+j347.3)(J/3 +

+ (1000+jO)(J23

S2D =(1970-j347.3)(J/3 + (1000+jO)(J23

S2h =( 1970-j347.3)(J/3 + (1000+jO)(J23

n

vector subspace [CCln1=LZp(J'p , where =p E C andp=l

(jp E Cl". The generic element zp(jp, is a p-th complex-

vector,. In complex-bivectors case, we obtain the vector

subspace [CClnL= LZpq(J'pq. The generic elementp:t=q

Zpq(jpq , is a pq-th complex-bivector.

B. Generalized Complex Geometric Product: g

Let {(jt' ... , (jn} a vector basis of CCln • For two vectors

Zp = zp(jp (p E 0) and Z; = z~(jq (q E \{I) where

0, \{I ~ {I, 2, ... ,n}, with associated complex numbers

zp =1 zp leJap

and ~ = Iz~ le Jl1q = Iz~ leJ(aq- iPq), we define a

new geometric product termed "generalized complexgeometric product", g:

As can be seen, these two examples cannot bedistinguished in terms of classical power approaches.

Both, SI and S2 power multivector decompositions,

have the same active power Ipi =5215.4 , reactive power

IQI = 7494, rotated distortion and distortion power

ILiI = 1151 = 2916, apparent power lSi = 9584.15, thus they

have the same power factor PF=O.54. It is logical toassume, that any difference between them must be basedon power multivector concept, which takes into accountvoltage phase angles, and the attributes: magnitude,direction and sense.

g: (91 a a 0 g)p' q

(BI)

and the basis transposition holds that

Note that the transposition operation is involutive.In particular, for two complex-vectors

(B3)

(B2)

The letter "g" represents the usual geometric product

and map ,aq

is an application in the complex planes

associated to any multivector product when; it is given by

where N =On \{I.

This new product for vectors i p and i q is given by

VI. CONCLUSIONS

In this paper a generalized geometric complex algebra(GCGA) approach for power equation representationunder n-sinusoidal conditions has been proposed.Adopting this tool all the theoretical bases of circuittheory keep their validity. In the mathematical framework

of our theory, the multivectorial apparent power § , powermultivector (magnitude, direction and sense), isdetermined in a natural way, from a new generalizedgeometric product of voltage and conjugated currentvector-phasors. This is the main and original contributionof our formulation. Therefore, the classic apparent powerS, (9), is simply a consequence of multivectorial apparent

power, S. In addition, by means of the GCGA structure, ageometric representation of apparent power components isobtained with a complex-scalar and a complex-bivectorialparts, that can be applied to the any classic approach. Thesuggested representation can provide a new vision anddetailed information for power quality and power factorimprovement, by means of new possible devices,strategies and control algorithms. The extension of thismathematical framework to power theory in multi- phasenetworks is possible.

VII. ApPENDIX

A. Generalized Complex Clifford Algebra

We denote as C the complex-vector space and Cln the

Clifford algebra generated from the n-dimensional realvector space Vn

• We define the set

CCln = {,_L ZI...k(J'l...k} (AI)k-l,2 ...n

where p,q EN, the product (BI) can be written

ZiZ'= Llzpllz;,I/9'p +L/(ap-aq)lzpll~I/9'qapq+p p<q

+LeJ(,xq-ap)IZq liz;, IeJ9'p aqp = Llzpllz;,I/9'p +q<p P

+L {i(,xp-aq)Izpll~ leNq - ~ap'aq i(aq-ap)\Zq liz;, IeJ9'p }apqp<q

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(B5)where

fun = IOpgII; =IOpgICI;" + jI;1-) =pE,V qEl\; P

=IOpgI~~1 + IOpgIIq*1- (E3)p q p q'---v- '----v--"

rLm ,1I rUn,;

where

I U Ij(a -a ) mc j(a -a )= e p q cos m (5 - ~l a - a )U I e q p cos m (5 =p q 't'q pq P q q P 't' P pq

p<q

fLin,11 = IOpgII;" =IOp' I 1:11 + IOp A Ii:"

(E4)pES qEV p q P q

'---v- '--v----'inner product outer product

(E5)Inner product:P= IOp' I 1:11 = IUpI pcosrpp(5ppEN qEN p=q

Outer product: I 0p A I 1:11 =pES qEe\'

(CI)

where (t) is the "reverse" operation.

The "conjugated" operation ( *) is given by

The structure {CCln,g} IS a complex geometric

algebra because the properties associative, distributivewith respect to the sum, and contraction, are fulfilled.

C. Reverse and Conjugated Operations

We define the bivector reverse element as

(C2)=~ ej(CXp-CXq)CU I cos m -U I cos m )(5 =~N,N ,~ P q '1'q q P '1'p pq Act,Lm,p<q

(E6)D. Norm Definition.

The norm, value or magnitude, of a multivector i is a

unique scalar Iltll calculated by

(DI)

and 9tCap - aq) = e2j(a

p-a

q) is a rotation operator.

The "in quadrature" product is given by

f Lin ,1- = IOpgII;1- = IOp' I 1;1- + IOp A I 1;1- (E7)pElV qEiV pEN qEN pEN qEN

'--v----' '--v----'inner product ollter product

where we apply (*) in C ,and (t) in Cln •Inner product:Q= IOp' I 1;1- = IUpl psinrpp(5p

pEN qEN p=q(E8)

E. Current Complex- Vector Decomposition

1) Linear Group ofPower ComponentsLet a j = '"' I eJ(aq-(('q)a + '"' I e.i(f/q}a a current

LJq q LJq qqEN qEM

Outer product: I 0p A I 1;1- =pE}..' qEN

I U Ij(a -a) . o""C ) j(a -a ) .= e p q SIn (Jl (5 - J\ a - a U I e q p SIn m (5 =p q i'q pq P q q P 'IF P pq

p<q

multivector, where lq harmonic current for q E N can be

decomposed as follows

I = I e Jaq e - jtPq(j =q q q

- ~ j(ap-aq)CU I' U I' ) - ~N,N- ~ e p qSIn rpq - q pSIn rpp (5pq - React"Lin.p<q

(E9)Combining outher product expressions, eqn. (E6) and

(E9), we obtain

= Iqe-J(('q(jq = Iq(cos((Jq - jsin((Jq)(jq = (E1)

=lq cos ((Jq(jq - jIqsin((Jq(jq = Iq,,(jq - jIqi-

The subscripts "II" and "1-" indicates "in phase" and

"in quadrature" respectively, being Iq'l

= Iq cos qJq and

Iqi- = I q sin qJq the q-th current harmonic vector-phasors

in phase and in quadrature correspondingly. On the other

hand, by introducing conjugate operation (*), the J;harmonic is written as

The total linear group of power components is found tobe

and the generalized complex geometric product [; g j* is

given by

I: =1:11+ j1:1- = ~~I(5q + jJq*1-(5q =

=Iq*COS ((Jq(5q + jlq*sen((Jq(5q(E2)

f Lin =[11 +[1- = IUplpllCYo + jIUpl p1-(5o +~~~~v=p p

=IUplpcosqJp(5o + iIUplpsinqJp(5o + l1~;INp p

(Ell)

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2) Nonlinear Group ofPower Components

Let I = L I qej(/l

q)O'q a nonlinear current multivector,

qEM

where 1q is the harmonic current for q EM, the total

nonlinear group of bivectorial components are given by

ACKNOWLEDGMENT

We would like to thank the Ministry of Education andScience for supporting this work as part of a researchthrough project DPI-2006-17467-C02-01.

(E12)

(E13)

REFERENCES

[1] C. I. Budeanu, "Puisances reactives et Fictives", Instytut Romainde I'Energie. Bucharest, Romania 1927.

[2] S. Fryze, "Wik-,Blind, un Scheinleitung in ElektrischenStromkreisen mit nichtsinusoidalem Verlauf von Strom undSpanung" Elekt. Z., voI.53,pp.596-599,625-627,700-702,1932.

[3] W Shepherd and P. Zhakikhani. " Suggested definition of reactivepower for nonsinusoidal systems". Proc. Inst. Elect. Eng.VoI.I19,pp.l361-1362, Sep.1972.

[4] L.S. Czarnecki "An Orthogonal Decomposition of the Current ofNon-Sinusoidal Voltage Sources Applied to Non-Linear Loads".Circuit Theory and Applications. Vol. 11,235-239. 1983

[5] L. S. Czarnecki, "Considerations on the reactive power in non­sinusoidal situations" lEE Trans.on Instr. and Meas. 1985, IM-34,pp.399-404.

[6] L. S. Czarnecki:," Distortion power in systems with nonsinusoidalvoltage". lEE Proceeding-B,VoI.139,No.3 May 1992.

[7] D. Sharon. "Reactive power definitions and power factorimprovement in nonlinear systems". Proc lEE, Vol. 120, N°6, June1973.

[8] M.A Slonim J.D. Van Wyk. " Power components in a system withsinusoidal and nonsinusoidal voltages and/or currents". lEEProceedings, Vol. 135, Pt.B,No.2, March 1988.

[9] M.A Slonim. "Distortion power in linear and non-linear systems"Int. J. Electron., vol. 68,no 5,pp.769-778,1990.

[10] F.Ghassemi," New Apparent Power and Power Factor with Non­Sinusoidal waveforms". Power Engineering Society WinterMeeting, 2000.IEEE.Volume 4, 23-27 Jan. 2000, pp.:2852 - 2857

[11] A.M. Sommariva, " Power Analysis of One-Ports Under PeriodicMulti-Sinusoidal Operation", IEEE Trans. On Circuits andSystems.-I: Regular Papers, VoI.53.N°9. September 2006.

[12] A. Menti, T. Zacharias, J. Milias-Argitis: "Geometric Algebra: APowerful tool for Representing Power Under NonsinusoidalConditions". IEEE Trans. on Circuits and Systems, Vo1.54, N°3,March 2007.

[13] D. Hestenes, G. Sobczyk: " Clifford Algebra to GeometricCalculus: a unified language for Mathematics and Physics".(Kluwer Academic, Dordrech/Boston 1986).

[14] Ch. Doran, A. Lasenby:" Geometric Algebra for Physicists".Cambridge University Press 2005.

(E14)• ;5.~~~NOlllill = L ej(ap-aq

) ReUplqejrpq(}pif

pEL,qE}'; ~Dpq

-L:I.f (~-J"(~-*J ~ja -jf'~l\~nlin = ~ Up g ~ I q =~e p Uplqe q(J"pq

pEL qEM p,q ~Dpq

t Nm,/i" = P'~NOpg~ 1, + ~Opg(~ 1;11 + ~.1;~) =

+P~NOp.~ 1; +P~NOp A ~ 1; +~Opg(~I;1I +~1;~)

+~ jap U I j{3q _ AL,N +AL,M +~N,A[~ e p qe (J"pq - D.Nonlin D.Nonlin lVonlin

pEN,qEM ~Dpq

= I ej(ap-aq )

pEL,qEN

(IO J"(I1*J (IO J"(I1*J+(IO J"(I1*J=pEL p g qEN q + pE LPg qE;\[ q pE N P g qEM q

Similarly decompositions to eqn. (E4), (E5) Y (E6), thedifferent terms of t Nonlin can be developed to obtain

Eqn. (E12) represents an identifiable and separate groupof non linear power components. Thus,

[Nonlin =Og1;onlin =

(E15)

• ~L,M =~ jap ImU 1 e-j{3q(J"

React.,Nonlin. ~e p q pq

p,q ~

• AN,A[. = ~ j a p R U I e- j jlqD.Act.,i\onlin. ~e e p q (J"pq

p,q ~Dpq

(E16)

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