An Approach to the Multivectorial Apparent Power in Terms of a Generalized Poynting Multivector

Progress In Electromagnetics Research B, Vol. 15, 401–422, 2009 AN APPROACH TO THE MULTIVECTORIAL APPARENT POWER IN TERMS OF A GENERALIZED POYNTING MULTIVECTOR M. Castilla and J. C. Bravo Electrical Engineering Department University of Sevilla Escuela Universitaria Polit´ecnica, Virgen de Africa 7 Sevilla 41011, Spain M. Ord´o˜nez Applied Mathematics Department University of Sevilla Escuela Universitaria Polit´ecnica, Virgen de Africa 7 Sevilla 41011, Spain J. C. Monta˜no Spanish Research Council (CSIC) Reina Mercedes 10, Sevilla 41012, Spain Abstract—The purpose of this paper is to explain an exact derivation of apparent power in n-sinusoidal operation founded on electromagnetic theory, until now unexplained by simple mathematical models. The aim is to explore a new tool for a rigorous mathematical and physical analysis of the power equation from the Poynting Vector (PV) concept. A powerful mathematical structure is necessary and Geometric Algebra offers such a characteristic. In this sense, PV has been reformulated from a new Multivectorial Euclidean Vector Space structure (CGn-R3) to obtain a Generalized Poynting Multivector (˜ S). Consequently, from ˜ S, a suitable multivectorial form ( ˜ Pand ˜ D) of the Poynting Vector corresponds to each component of apparent power. In particular, this framework is essential for the clarification of the connection between a Complementary Poynting Multivector (˜ D) and the power contribution due to cross-frequency products. A simple application example is presented as an illustration of the proposed power multivector analysis. Corresponding author: M. Castilla ([email protected]). 402 Castilla et al. 1. LIST OF SYMBOLS (NOMENCLATURE) n-sinusoidal = non-sinusoidal or multi-sinusoidal. R= real numbers E3= Euclidean vector space C= complex vector space Vn= linear space over real numbers Gn= Clifford algebra in n-dimensional real space CGn= complex Clifford Algebra Φ = operator Γ = time-domain frequency-domain transform CGt n-R3= time generalized Euclidean space CGn-R3= frequency generalized Euclidean space ~ 1X,~ 1Y,~ 1Z= Euclidean canonical basis ~ 1X,Y,Z = generic unitary vector of E3 σ1,...,k = Clifford algebra canonical basis IdC= identity operation ˜ z(t) = instantaneous geometric vector (˜ z∈ CGt n-R3) ˜ e(t) = instantaneous electric field geometric vector ˜ h(t) = instantaneous magnetic field geometric vector ˜ d(t) = instantaneous displacement field geometric vector ˜ b(t) = instantaneous magnetic induction field geometric vector ˜zX,Y,Z = components of ˜ z(t) ˜ zp=p-th harmonic component of ˜ z(t) ˜ ZX,Y,Z= spatial components of ˜ Z ˜ Z= spatial geometric phasor (˜ Z∈ CGn-R3) ˜ Zp= spatial p-th harmonic component of ˜ Z ˜ Z= geometric phasor ( ˜ Z∈ CGn) ˜ Zp=p-th harmonic component of ˜ Z ˜ Zpq = bivector component of ˜ Z ˜ E= electric field geometric phasor ˜ H= magnetic field geometric phasor ˜ D= displacement field geometric phasor ˜ B= magnetic induction field geometric phasor ˜ S= generalized Poynting multivector (GPM) ˜ P= Poynting multivector (PM) ˜ D= complementary Poynting multivector (CPM) Up=p-th harmonic voltage rms value Ip=p-th harmonic current rms value ⊗= classic geometric product ¯= generalized geometric product in CGn Progress In Electromagnetics Research B, Vol. 15, 2009 403 ◦= generalized geometric product in CGt n-R3 ·= inner product ∧= outer product ⊕= direct sum + = classic sum for scalars and also direct sum for multivectors j = imaginary unit ∗= conjugated operation †= reverse operation hi0= scalar part hi2= bivector part ˜ S= apparent power multivector k˜ Sk= norm, value or magnitude of multivector ˜ S ˜ Ω·= complex scalar ˜ Ω∧= complex bivector ωp, ωq= harmonic frequencies αp= phase angle of p-th voltage geometric phasor αq= phase angle of q-th current geometric phasor ϕq= phase angle between q-th voltage and q-th current geometric phasors ˜ δ= relative quality index multivector (RQI ) PF = power factor 2. INTRODUCTION 2.1. Motivation One of the fundamental issues in power system analysis is related with the electromagnetic theory in order to explain the energy transfer in an electric circuit. Hence, this paper establishes an electromagnetic foundation to the power equation representation. For this goal, a new Generalized Poynting Multivector is proposed. 2.2. Literature Review The electrical circuits in n-sinusoidal operation can be analyzed by means of mathematical tools that are much simpler to handle than electromagnetic theory based on Maxwell equations [1]. However, it is also true that these equations fully explain interactions between electric and magnetic fields and therefore explain every electromagnetic phenomenon, including the energy transfer in an electric system. In n-sinusoidal operation, the distorted electromagnetic fields can be represented as sums or series of harmonics. Each harmonic component 404 Castilla et al. of the field is governed by Maxwell equations and satisfies the Poynting Theorem. It is relevant to classify the contribution of these equations to the electric power theory into following lines of thought: a) Circuit theory analysis: First, is the most commonly used approach. It analyzes currents, voltages and circuit element properties. In this sense, circuit theory, ruled by simple equations based on Ohm’s law, can be regarded as a very particular case of electromagnetic theory, and power theory was developed mainly from circuit analysis. Electrical components of power systems are considered as elements of circuits and their electromagnetic behaviour is described by means of voltages and currents of element terminals. Circuit theory can explain only the power flows between components, and it is unable to reveal their spatial distribution. Nevertheless, no phenomena such as hysteresis losses or skin effects can be explained by circuit theory. These are phenomena of the electromagnetic field characteristics. For this first approach, Complex Algebra [2] provides an initial procedure to solve the problem, despite its limitation to the purely sinusoidal case. The n-sinusoidal operation imposes the substitution of the Complex Algebra approach with a new representation model and the reformulation of the energy balance. Considerable research efforts have been directed towards the representation of apparent power in various ways [3–9]. Specifically, in [9], the authors use Geometric Algebra to define a multivector power based on the decomposition of the instantaneous current into the active and reactive components. It should be noted that their approach does not distinguish between reactive and distortion power from a mathematical viewpoint. Furthermore, none of the aforementioned papers leads to a representation that could be considered universally satisfactory. b) Electromagnetic theory analysis: This second valid method analyzes the energy flow using the Poynting Theorem (PT), and therefore the Poynting Vector (PV) should be considered, since it represents the bridge between electromagnetic theory and circuit theory [10]. These tools are fundamental concepts of electromagnetic theory with respect to energy flow. The goal is to investigate the nature of the non-active power and some progress has undeniably been made. Numerous valuable contributions have appeared in the literature [11– 17], each shedding more light on some aspects of the problem. From among them, [12, 13] masterfully explain the physical mechanism of energy propagation in electric power systems, [15] reconsiders the bases of electromagnetism in order to find a physical interpretation for the power equation, and [16] uses the PV to illustrate the nature of power flow in electric circuits using electromagnetic fields. However, Progress In Electromagnetics Research B, Vol. 15, 2009 405 critics of PV calculations [17] argue that electromagnetic theory is useless for practical applications of electric power theory. Against this reference, it is our view that the power equation can be based and interpreted through a new formulation of the Poynting Vector and that other aspects concerning the electromagnetic field in n-sinusoidal operation and their direct relation with power theory have yet to be thoroughly investigated. Thus, the purpose of this paper is to advance energy flow analysis by using a new mathematical structure for the representation of the power equation in single-phase circuits under nsinusoidal operation. In this way, a complete solution to the power equation analysis problem for linear/non-linear circuits based on a Generalized Poynting Multivector (˜ S), is presented. To this end, primarily our work introduces a CGt n-R3mathematical structure based on Clifford Algebra for the definition of the distorted electric and magnetic field intensities (˜ e,˜ h), which are time geometric fields associated to an Euclidean direction. From these definitions it is possible to obtain the quantities called spatial geometric phasors (˜ E,˜ H) in the CGn-R3structure in frequency domain. Consequently, our work is aimed at showing how an electric and magnetic field can be associated with the elements of Clifford Algebras [21, 22] for a new formulation and interpretation of power theory in this framework. This second approach is more general and fundamental that the first approach based on circuit theory, and it has the additional advantage of providing a physical insight into the spatial distribution of the power flow. Finally, this paper addresses the need to understand the multidimensional character of electric power theory and its relation to the electromagnetic theory. 2.3. Contributions The paper is concerned with a representation of the power equation under non-sinusoidal conditions from electromagnetic theory. The apparent power concept is better understood if a Clifford vector space is used for the representation of the distorted electric and magnetic field intensities. This generates a larger linear space called Generalized Euclidean Space CGt n-R3, which will be utilized in this paper for a new representation of the power equation. This objective cannot be reached on the Complex Algebra framework. 406 Castilla et al. 3. MATHEMATICAL FOUNDATIONS: GEOMETRIC EUCLIDEAN SPACES 3.1. Time Domain: Generalized Euclidean Space CGt n-R3 In order to introduce the instantaneous quantities of electric and magnetic fields, in this section we define a new structure for the time domain that we have named Generalized Euclidean Space,CGt nR3, whose coefficients belong to the Complex Geometric Algebra CGn constructed in [18]. Let n~ 1X,~ 1Y,~ 1Zobe the “canonic” basis of the Euclidean space E3. A generic element of CGt n-R3is given by ˜ z(t) = ˜zX~ 1X+ ˜zY~ 1Y+ ˜zZ~ 1Z(1) where each component in (1) is in the form ˜z(t) = kej[α(t)+θ]σa∈ CGt n, k≥0 and σais a basis element of CGnstructure [18]. Thus, the CGt n-R3structure is a CGt nvector space whose inner product is defined by ˜ z(t)·˜ w(t) = h˜zX,˜w∗ Xi0+h˜zY,˜w∗ Yi0+h˜zZ,˜w∗ Zi0(2) where, ˜ z(t) = ˜zX~ 1X+ ˜zY~ 1Y+ ˜zZ~ 1Z,˜ w(t) = ˜wX~ 1X+ ˜wY~ 1Y+ ˜wZ~ 1Z. Moreover, from (D1) the norm of ˜ z(t) is given by k˜ z(t)k=X i=X,Y,Z h˜zi,˜z∗ ii0(3) Now we define the outer product in this structure as ˜ z(t)∧(−˜ w(t)) =   ~ 1X~ 1Y~ 1Z ˜zX˜zY˜zZ −˜wX−˜wY−˜wZ  = (h−˜zY,˜wZi2+h˜zZ,˜wYi2)~ 1X + (h˜zX,˜wZi2+h−˜zZ,˜wXi2)~ 1Y + (h−˜zX,˜wYi2+h˜zY,˜wXi2)~ 1Z(4) Based on (3) and (4), the Geometric Algebra CGt n-R3is defined by the following geometric product ˜ z(t)◦˜ w(t) = ˜ z(t)·˜ w(t) + ˜ z(t)∧˜ w(t) (5) 3.2. Frequency Domain: Generalized Euclidean Space CGn-R3 Let Φ : CGt n→ CGn, Φ ¡k ej[α(t)+θ]σa¢=k ejθσa, be the operator that enables the transformation between time-domain and frequencydomain. We define CGn-R3as Φ(CGt n)~ 1X+ Φ(CGt n)~ 1Y+ Φ(CGt n)~ 1Z(6) Progress In Electromagnetics Research B, Vol. 15, 2009 407 where a generic element of this space is Φ(˜za) = ˜ Za. Note that CGnR3can also be seen as CGn-R3=n˜ ZX~ 1X+˜ ZY~ 1Y+˜ ZZ~ 1Z:˜ Z∈ CGno where ˜ Z=X p ¯ Zpσp,¯ Zp∈ C and σp∈ Gn Obviously, CGn-R3is a CGn(complex-geometric) vector space and the multiplication rule for two vectors ˜ Z,˜ W∈ CGn-R3is given by ˜ Z◦˜ W=˜ Z·˜ W+˜ Z∧³−˜ W´(7) where (7) is the restriction from CGt n-R3→ CGn-R3. The nesting of the geometric Euclidean spaces denoted by CGt n-R3, CGn-R3, and CGnare graphically illustrated in Fig. 1. Figure 1. Nested geometric Euclidean vector spaces. The fundamental concepts of Generalized Complex Geometric Algebra CGnare given in [18] and further research about Geometric Algebra can be found in [21, 22]. 4. DISTORTED PERIODIC ELECTRIC AND MAGNETIC FIELDS: BASIC CONCEPTS A periodic electromagnetic field is distorted if, simultaneously with the fundamental harmonic of the field, the highest harmonic components are present. In this way, if distorted vector field functions satisfy Dirichlet’s conditions, then they can be developed into Fourier series, namely: e(t)=X p ep(t),d(t)=X p dp(t),h(t)=X q hq(t),b(t)=X q bq(t) (8) 408 Castilla et al. where ep,dp,hq,bq, are harmonics of the field vectors. Each harmonic component in the frequency domain of a periodic electromagnetic field satisfies Maxwell’s equations ∇×Hp=Jp+jωpDp ∇×Ep=−jωpBp ∇·Dp=ρp ∇·Bp= 0 (9) where Ep,Hp,Dp,Bpare complex phasors of the p-th harmonic of the fields. One of the most important consequences of the first two Maxwell equations is Poynting’s theorem, which describes the flow of electromagnetic energy in space and for a volume venclosed by a surface s. This can be stated as ZZ −(e×h)nds =ZZZ e·jdv +ZZZ µh·∂b ∂t +e·∂d ∂t ¶dv (10) where nis the unit vector orthogonal to the infinitesimal surface ds,e and hare the instantaneous intensity of the electric and magnetic fields, dand bare the instantaneous flux densities of these fields respectively, and jis the instantaneous current density. The theorem simply means that the increase in stored energy in the fields plus the ohmic losses within a volume, equal the inflow of a vector e×hacross the surface bounding that volume. The vector e×his known as the Poynting Vector (PV), and gives the power density at a point on the surface in terms of the electric and magnetic fields at that point. Its physical meaning is also known [23]. The equivalent complex Poynting theorem for a system in linear media is given by −ZZ(Ep×Hp)nds =ZZZ EpJ∗ pdv+jωpZZZ[BpH∗ p−EpD∗ p]dv (11) The energetic interpretation of (11) is as follows ¯ Sp=Pp+jQp(12) where •¯ Spis a complex apparent power of the p-th harmonic received by the system enclosed in the surface “s”. •Ppis the active power of the p-th harmonic received by the system. •Qpis the reactive power of the p-th harmonic received by the system. Progress In Electromagnetics Research B, Vol. 15, 2009 409 One can readily observe that Pp=ZZZ EpJ∗ pdv (13) Qp= 2ωpZZZ ·BpH∗ p 2−EpD∗ p 2¸dv (14) where Pprepresents harmonic losses in Joules and Qpis associated to the average values of the p-th harmonic magnetic and electric energies accumulated in the volume v[15]. Another representation of (13) and (14) is given Pp= Re ·−I(Ep×H∗ p)¸nds (15) Qp= Im ·−I(Ep×H∗ p)¸nds (16) 5. POWER FLOWS IN DISTORTED ELECTROMAGNETIC FIELDS: GENERALIZED POYNTING MULTIVECTOR ( ˜ S) The following notation is adopted to define the electric and magnetic fields in the CGt n-R3framework: ˜ ep=|˜ ep|ej(ωpt+θp)σp~ 1X,Y,Z,˜ hq=|˜ hq|ej(ωqt+γq)σq~ 1X,Y,Z (17) where ˜ ep(t) and ˜ hq(t) are called instantaneous electric and magnetic complex-geometric fields respectively. Observe that the classic instantaneous fields can be derived from the real (or imaginary) part of the projections given by the scalar product (2) as follows: ep(t) = Im(˜ ep·σp) = Im{|˜ ep|ej(ωpt+θp)~ 1X,Y,Z}(18) hq(t) = Im(˜ hq·σq) = Im{|˜ hq|ej(ωqt+γq)~ 1X,Y,Z}(19) In order to obtain the geometric phasors, it is necessary to apply the Φ operator on these quantities (see Section 3.2). Φ(˜ ep) = |˜ ep|ejθpσp~ 1X,Y,Z =˜ Ep(20) Φ(˜ hq) = |˜ hq|ejγqσq~ 1X,Y,Z =˜ Hq(21) where ˜ Epand ˜ Hqare called harmonic “spatial geometric phasors” of the electric and magnetic harmonic fields respectively and verify 416 Castilla et al. By ignoring eddy currents, line impedance, fringing effects, then (32) and (33) can be expressed as ˜ E=1 lE (200ej0σ1+ 100ej0σ2)~ 1X(43) ˜ H∗=1 lH (10ej30σ1+ 5e−j45σ2+ 10e−j60σ3)~ 1Y(44) However, from (35) and (36), it follows that ZZ sX p ~ 1Z·˜ Pds = [(1732 + 353.5) + j(1000 −353.5)]σ0~ 1Z = (2085.5 + j646.5)σ0~ 1Z(45) Therefore, from (38), ZZ SX p6=q ˜ 1Z·˜ Dds=(−158.9−j1207.1)σ12 +(1000−j1732)σ13 +(500−j866)σ23 (46) This example states that Re{˜ Ω· 1}= 1732σ0, Re{˜ Ω· 2}= 353.5σ0, Im{˜ Ω· 1}=j1000σ0, Im{˜ Ω· 2}=−j353.5σ0and that the linear complex bivector component becomes ˜ Ω∧ 12 = (−158.9−j1207.1)σ12, as well as the nonlinear complex bivector components ˜ Ω∧ 13 = (1000 −j1732)σ13, ˜ Ω∧ 23 = (500 −j866)σ23 with their corresponding directions and senses. On the other hand, the rms values of voltage and current are given by k˜ Uk2= 2002+ 1002= 5 ·104and k˜ Ik2= 102+ 52+ 102= 225 respectively. The values of P=kRe{˜ Ω·}k2,kIm{˜ Ω·}k2,k˜ Ω∧k2are found to add up to k˜ Sk2=P2+kIm{˜ Ω·}k2+k˜ Ω∧k2= 11.25 ·106(47) Therefore, apparent volt-amperes k˜ Skat the terminals are found from the relation k˜ Sk2=k˜ Uk2k˜ Ik2= 11.25·106. Finally, from (41) and (42) we obtain the relative quality index and power factor respectively ˜ δ= 1 + j646.5σ0 2085.5σ0 +(−158.9−j1207.1) σ12+(1000−j1732) σ13+(500−j866) σ23 2085.5σ0 ° ° °˜ δ° ° °= 1.608 PF =1 ° ° °˜ δ° ° ° = 0.62 (48) Progress In Electromagnetics Research B, Vol. 15, 2009 417 The methodology in the above example differs greatly to that of circuit theory. Furthermore, unlike the circuit theory approach, it can be applied to solve and understand the operation of electric systems designed to work in the frequency domain. 8. CONCLUSION The suggestion that the power equation should be founded on electromagnetic theory is analyzed in this paper. This goal remains unexplained by simple mathematical models used in classical theory. To this end, we propose a Generalized Poynting Multivector (˜ S) based on Clifford Algebras, which is decomposed into a Poynting Multivector (˜ P) and a Complementary Poynting Multivector (˜ D). From Equations (34)–(40), both quantities are considered as the keystone of the bridge between electromagnetic theory and circuit theory. Thus, the real part of the flow of Poynting Multivector (˜ P) coincide with active power, and imaginary part of the complex scalar coincides with the power contribution due to like-frequency products. The Complementary Poynting Multivector (˜ D) is associated to the complex bivector or to the power contribution due to cross-frequency products. This analysis demonstrates that the power equation can be founded on the multivectorial concept of the Generalized Poynting Multivector (˜ S). Consequently, the apparent, active, and non-active powers can be expressed and differentiated in terms of ˜ S. The application of the proposed Generalized Poynting Multivector (˜ S) to power theory should indicate important advances for any real future research in this area. ACKNOWLEDGMENT We would like to thank the Ministry of Education and Science for supporting this work as part of a research through project DPI-200617467-CO2-01. APPENDIX A. GENERALIZED COMPLEX GEOMETRIC PRODUCT IN CGn We define as Cthe complex-vector space, and Gn, the Clifford algebra on n-dimensional real space Vn. We define the set CG= n  X k=1,2...n ¯ Z1...kσ1...k   (A1) 418 Castilla et al. where the coefficients ¯ Z1...k ∈ C and the basis σ1...k ∈ Gn. Obviously CGnis a vector space over R. According to (A1) definition, in the complex-vector case, we obtain the vector subspace [CGn]1= n P p=1 ¯ Zpσp, where ¯ Zp∈ C and σp∈ Gn. The generic element ¯ Zpσp, is a pth complex-vector, and can be represented by the geometric phasor ˜ Zp= (ap+jbp)σp. In the complex-bivector case, we obtain the vector subspace [CGn]2=P p6=q ¯ Zpqσpq. The generic element ¯ Zpqσpq, is a pq-th complex-bivector, and can be represented by ˜ Zpq = (apq +jbpq)σpq. In the most general form, complex-multivectors, we obtain the vector subspace [CGn]k=P¯ Z12...kσ12...k. The element ¯ Z12...kσ12...k, is the 12 . . . k-th complex-multivector, and may be represented by ˜ Z12...k = (a12...k +j b12...k)σ12...k. Therefore, CGn(A1), also can be represented as CGn=C |{z} complex scalar ⊕[CGn]1 |{z} complex vectors ⊕[CGn]2 |{z} complex bivectors ⊕···⊕ [CGn]n | {z } complex pseudoscalar The structure {CGn,¯} is a complex geometric algebra since the following properties are fulfilled: associative, distributive with respect to the sum and contraction. APPENDIX B. PARTICULAR CASE: GENERALIZED COMPLEX GEOMETRIC PRODUCT FOR COMPLEX VECTORS (GEOMETRIC PHASORS) Let {σ1, . . . , σn}be a vector basis of CGn. For two vectors ˜ Zp= ¯ Zpσp(p∈Ω) and ˜ Z0 q=¯ Z0 qσq(q∈Ψ) where Ω,Ψ⊆ {1,2, . . . , n}, and where complex numbers associated to each vector are ¯ Zp=Zpejαp ¯ Z0 q=Z0 qejβq=Z0 qej(αq−ϕq)(B1) we define a new geometric product termed “generalized complex geometric product”,¯: ¯:¡<αp,αq,⊗¢(B2) The symbol “⊗” represents the classic geometric product [21] and <αp,αqis an application in the complex planes associated to any multivector product when αp6=αq, and is given by <αp,αq¡¯ Z0 p,¯ Z0 q¢=½e−2j(αq−αp)if p > q, p, q ∈N 1 otherwise, p and/or q /∈N(B3) Progress In Electromagnetics Research B, Vol. 15, 2009 419 where N= Ω ∩Ψ. This new product for vectors ˜ Zpand ˜ Z0 qis given by ¯ Zpσp¯¯ Z0 qσq=¯ Zp¯ Z0 qσpq (B4) and the basis transposition states ¡¯ Z0 q¯ Zpσqp¢= (−1)<αp,αq¯ Zp¯ Z0 qσpq (B5) Note that the transposition operation is involutive. If αp=αq∀p,q∈N, then <αp,αp=IdC(B6) and “¯”, (B2), will then become the classic geometric product “⊗”. It should be noted that when Cis restricted to real numbers, the classic Clifford Algebra is obtained. In particular, for two complex vectors ˜ Z=X p Zpejαpσpand ˜ Z0=X q Z0 qej(−αq+ϕq)σq, where the angles αpand (−αq+ϕq) identify the phase of the p-th and q-th harmonics respectively, the generalized complex geometric product in linear operation (p, q ∈N), can be written ˜ Z¯˜ Z0=X p ZpZ0 pejϕp+X p<q ej(αp−αq)ZpZ0 qejϕqσpq +X q<p ej(αq−αp)ZqZ0 pejϕpσqp =X p ZpZ0 pejϕp +X p<qnej(αp−αq)ZpZ0 qejϕq−<αp,αqej(αq−αp)ZqZ0 pejϕpoσpq (B7) where <αp,αqej(αq−αp)ZqZ0 pejφpσq p =ej(αp−αq)ZqZ0 pejφpσqp APPENDIX C. REVERSE AND CONJUGATED OPERATIONS We define the bivector reverse element as ¡¯ Zq pσq p¢†= (−1) ¯ Zpqσpq (C1) where (†) is the “reverse” operation. The “conjugated” operation (∗) is given by ¡¯ Zpσp¢∗=¯ Z∗ pσp(C2) 420 Castilla et al. APPENDIX D. 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