Geometric Algebra Power Theory (GAPoT): Revisiting Apparent Power under Non-Sinusoidal Conditions
Francisco Gil Montoya, Alfredo Alcayde, Francisco Arrabal-Campos, Raul Baños, Javier Roldán-Pérez
11 Geometric Algebra Power Theory (GAPoT):Revisiting Apparent Power under Non-SinusoidalConditions
Francisco Gil Montoya, Alfredo Alcayde, Francisco Arrabal-Campos, Raul Ba˜nos,and Javier Rold´an-P´erez,
Member, IEEE
Abstract —Traditional power theories and one of their mostimportant concepts –apparent power– are still a source of debateand, as shown in the literature, they present several flaws thatmisinterpret the power-transfer phenomena under distorted gridconditions. In recent years, advanced mathematical tools such asgeometric algebra (GA) have been applied to address these issues.However, the application of GA to electrical circuits requiresmore consensus, improvements and refinement. In this paper,power theories based on GA are revisited. Several drawbacks andinconsistencies of previous works are identified and modificationsto the so-called geometric algebra power theory (GAPoT) arepresented. This theory takes into account power componentsgenerated by cross-products between current and voltage har-monics in the frequency domain. Compared to other theoriesbased on GA, it is compatible with the traditional definitionof apparent power calculated as the product of RMS voltageand current. Also, mathematical developments are done in amulti-dimensional Euclidean space where the energy conservationprinciple is satisfied. The paper includes a basic example andexperimental results in which measurements from a utility supplyare analysed. Finally, suggestions for the extension to three-phasesystems are drawn.
Index Terms —Geometric algebra, non-sinusoidal power, Clif-ford algebra, power theory.
I. I
NTRODUCTION
Full understanding of power flows in electrical systems hasbeen a topic of interest during the last century. The mostrelevant efforts have been done in the frequency domain forsystems operating in steady state [1], and in the time domainby using both instantaneous and averaged approaches [2, 3].Outcomes from these studies are sometimes inconsistent andeven contradictory. For example, the well-known instanta-neous power theory can yield to incoherent results under spe-cific conditions [4]. Similar conflicting results have been foundfor well-established regulations such as the IEEE standard1459 [5]. Traditional techniques that are commonly appliedfor analysing power flows are based on linear-algebra toolssuch as complex numbers, matrices, tensors, etc, and theyare proven to be useful from the application point of view.However, none of them provide a clear overview of powerflows under disported and unbalanced grid conditions and thispoint is still an open discussion [6].Geometric algebra (GA) is a mathematical tool developedby Clifford and Grassmann by the end of the XIX centurythat has been refined by Hestenes in the last decade [7]. Thistool has brought new possibilities to the physics field suchas producing compact and generalised formulations [8]. Also,it can be easily used to manipulate integral and differential equations in multi-component systems [9, 10]. Even thoughGA is not widely known by the scientific community, it hasa great potential and has attracted interest in recent publi-cations [6]. GA has already been introduced to redefine theapparent power as the geometric product between voltage andcurrent, what is commonly written as M [11–14]. Comparedto the traditional definition of apparent power ( S = V I ), theuse of M has several advantages. A relevant one is that M is conservative in spite of S and this is of interest for itsapplication in distorted environments [15].In this paper, GA power theories proposed by differentauthors are reviewed in order to analyse some of the incon-sistencies raised so far, while additional ones not yet foundin the literature are also discussed [5, 11, 16]. Then, the GApower theory is redefined in order to solve these issues. Theproposed theory can be used to resolve electrical circuits, iscompatible with the traditional definition of apparent powerand provides a definition for its components that fulfils theprinciple of energy conservation. Numerical and experimentalresults are included in order to validate the main contributionsof this work. A brief introduction to GA and its terminologyis included in order to make the paper self-contained.II. G EOMETRIC A LGEBRA FOR P OWER F LOW A NALYSIS
The geometric product was introduced by Clifford by theend of the XIX century, and it includes the external andinternal products of two vectors, namely a = α σ + α σ and b = β σ + β σ ∈ R [7]. The internal product can becalculated as follows: a · b = (cid:107) a (cid:107)(cid:107) b (cid:107) cos ϕ = (cid:88) α i β i (1)while the external product is: a ∧ b = (cid:107) a (cid:107)(cid:107) b (cid:107) sin ϕ σ σ (2)This operation does not exist in traditional linear algebra andits result is not a scalar nor a vector, but a new entity that iscommonly known as bivector [7]. Bivectors play a key role incalculations related to non-active power, as will be shown later.The external product is anticommutative, i.e., a ∧ b = − b ∧ a .The fundamental operation in GA is the geometric product: M = ab = ( α σ + α σ )( β σ + β σ )= ( α β + α β ) + ( α β − α β ) σ σ (3) a r X i v : . [ ee ss . S Y ] M a y where M consists of two elements. As these elements are ofa different nature, M is commonly referred to as multivector .The operator (cid:104)·(cid:105) k refers to k -grade component of a multivector.In (3), the term (cid:104) M (cid:105) is is a scalar, while the term (cid:104) M (cid:105) is abivector. Multivectors are classified according to their degree:scalars have degree zero, vectors one, bivectors two, etc. Thenorm of a multivector is: (cid:107) M (cid:107) = (cid:113) (cid:104) M † M (cid:105) (4)where M † is the reverse of M (see [7] for details).Considering a single-phase system operating under perfectsinusoidal conditions, it is possible to select an orthonormalbasis such as σ = { σ → √ ωt, σ → √ ωt } .Therefore, voltages and currents are transformed as follows: u ( t ) −→ u = α σ + α σ i ( t ) −→ i = β σ + β σ (5)The geometric product defined in (3) can be used to calculatethe geometric apparent power: M = ui = ( α β + α β ) (cid:124) (cid:123)(cid:122) (cid:125) P + ( α β − α β ) (cid:124) (cid:123)(cid:122) (cid:125) Q σ σ (6)This expression consists of two terms that can be clearlyidentified: P is a scalar and Q is a bivector. This result willbe extended to non-sinusoidal conditions in later sections.The geometric apparent power fulfils: (cid:107) M (cid:107) = (cid:104) M (cid:105) + (cid:104) M (cid:105) = P + Q = (cid:107) u (cid:107) (cid:107) i (cid:107) (7)III. GA-B ASED P OWER T HEORIES : O
VERVIEW
In this section, the main power theories based on GA arecritically discussed so that the main contributions of this papercan be better understood. • Menti . This theory was developed by Anthoula Menti etal. in 2007 [11]. This was the first application of GA toelectrical circuits. The apparent power multivector wasdefined by multiplying the voltage and current in thegeometric domain: S = ui = u · i + u ∧ i = (cid:104) S (cid:105) + (cid:104) S (cid:105) The scalar part matches the active power P , while thebivector part represents power components with a meanvalue equal to zero. It was demonstrated that the lat-ter holds for both sinusoidal and non-sinusoidal condi-tions, in steady-state. It was also demonstrated throughexamples that this theory can be applied to electricalcircuits already studied in the literature, in which thecomponents of the traditional apparent power were notdistinguishable. The reason is that bivector terms providesense and direction, while the traditional definition ofapparent power based on complex numbers does not.Unfortunately, the theory did not establish a generalframework for the resolution of electrical circuits underdistorted conditions. Also, the proposal was not appliedto decompose currents (for non-linear load compensation,for example), and it was not extended to multi-phasesystems. • Castilla-Bravo . This theory was developed by Castillaand Bravo in 2008 [12]. Authors introduced the con-cept of generalised complex geometric algebra . Vector-phasors were defined for both voltage and current: (cid:101) U p = U p e jα p σ p = ¯ U p σ p (cid:101) I q = I q e jβ q σ q = ¯ I q σ q Geometric power results from multiplying the harmonicvoltage and conjugated harmonic current vector-phasors: (cid:101) S = (cid:88) p ∈ N ∪ Lq ∈ N ∪ M (cid:101) U p (cid:101) I ∗ q = (cid:101) P + j (cid:101) Q + (cid:101) D This proposal is able to capture the multicomponentnature of apparent power through the so-called complex-scalar (cid:101) P + j (cid:101) Q and the complex bivector (cid:101) D . However,this formulation requires the use of complex numbers,which could have been avoided by using appropriatebivectors [8]. Also, only definitions of powers werepresented and it was not extended to multi-phase systems. • Lev-Ari . This theory was developed by Lev Ari [13, 17],and it was the first application of GA to multi-phasesystems in the time domain. However, this work doesnot contain examples nor fundamentals for load compen-sation. Also, practical aspects required to solve electricalcircuits were not explained. • Castro-N ´u ˜nez . This theory was developed by CastroN´u˜nez in the year 2010 [18], and then extended andrefined in several later works [5, 15, 19, 20]. A relevantcontribution of this work consists on the resolution ofelectrical circuits by using GA (without requiring com-plex numbers). Also, a multivector called geometric ap-parent power that is conservative and fulfils the Tellegentheorem is defined [20]. As in Menti and Castilla-Bravoproposals, the results are presented only for single-phasesystems. Another contribution is the definition of a trans-formation based on k vectors that form an orthonormalbase: ϕ c ( t ) = √ ωt ←→ σ ϕ s ( t ) = √ ωt ←→ − σ ... ϕ cn ( t ) = √ nωt ←→ n +1 (cid:86)(cid:86)(cid:86) i =2 σ i ϕ sn ( t ) = √ nωt ←→ n +1 (cid:86)(cid:86)(cid:86) i =1 i (cid:54) =2 σ i (8)However, this basis presents some drawbacks. The mainone is the definition of the geometric power [21]. Inparticular, active power calculations do not match withthose obtained by using classical theories. Therefore,authors needed to include an ad-hoc corrective coeffi-cient [5]. Also, in the current version of the theory, itis not possible to establish optimal current compensationsince current decomposition did not consider the minimalactive current proposed by Fryze and supported by otherauthors [1, 3, 22]. Finally, the definition of the geometric Author ContributionMenti
Pioneer definition of geometric power in electrical circuits.
Castilla-Bravo
Generalized complex geometric algebra. Vector basis and complex-bivector power.
Lev-Ari
Time domain and multiphase power introduction.
Castro-N ´u ˜nez
Circuit analysis and definition of geometric impedance. Conservative geometric power demonstration.
Montoya
Corrections on definition of power components. Optimal current decomposition. Interharmonics.
Table IM
AIN CONTRIBUTORS TO
GA-
BASED POWER THEORIES . power does not follow the traditional expression (cid:107) S (cid:107) = (cid:107) U (cid:107)(cid:107) I (cid:107) , due to the transformation presented in (8). • Montoya . This framework was proposed by Montoya etal. [21], and it is an extension of Menti and Castro-N´u˜nez theories [5, 11]. It establishes a general frameworkfor power calculations in the frequency domain [23].Since it is the most recent work, it provides solutionsto some problems detected so far in other proposalsand the formulation is more compact and efficient. Theoptimal current for load compensation is presented, whatis helpful for power quality applications. Inter- andsub-harmonics can be easily modelled [24]. Also, thedefinition of apparent power is valid for distorted andnon-distorted voltages and currents [14]. However, thisframework is based on the use of k -vectors. Therefore,drawbacks related to the non-standardised definition ofapparent power and the fulfilment of the energy con-servation principle are inherit from previous theories.Also, harmonic power components cannot be easily de-composed since this would require inverting geometricvectors.The most relevant contributions to power theories based onGA are summarised in Table I.IV. GAP O T F
RAMEWORK AND M ETHODOLOGY
A. Circuit Analysis with GA
In this theory, different approaches already available inthe literature are unified and enhanced in order to analyseelectrical circuits in the geometric domain. The proposedmodifications give full physical meaning to basic principlesin electrical circuits. An orthonormal basis is used in orderto represent the multi-component nature of periodic signalswith finite energy: σ = { σ , σ , . . . , σ n } . For example, for avoltage signal u ( t ) : u ( t ) = U + √ (cid:80) nk =1 U k sin( kωt + ϕ k )+ √ (cid:80) l ∈ L U l sin( lωt + ϕ l ) (9)where U is the DC component, while U k and ϕ k are theRMS and phase of the k th harmonic, respectively. The set L represents possible sub- and inter-harmonics present in thesignal [24]. As in traditional circuit analysis based on complexvariables, a rotating vector (similar to e jωt ) can be defined.This would facilitate later analyses in the geometric domain.In addition, thanks to the linear properties of GA, it is possibleto define a single multivector that includes all the harmonicfrequencies present in the signal (this is not possible by usingthe traditional complex variable). A rotating vector n ( t ) in Figure 1. A vector multiplied by a rotor ( e ϕ σ ) rotates in clock- or counter-clock-wise direction depending on the type of multiplication. a two-dimensional geometric space G can be obtained asfollows [25]: n ( t ) = e ωt σ N e − ωt σ = RN R † = e ωt σ N = R N = N R † (10)where R = e ωt σ is a rotor [26] and N is a vectoror geometric phasor . In (10), left-multiplying produces op-posite effects compared to right-multiplying. Fig. 1 shows agraphical representation of a vector left-multiplied by a rotor(in green). This operation produces a rotation in clock wisedirection. Similarly, a vector right-multiplied by a rotor (inred), produces a rotation in counter-clock wise direction. Inorder to maintain the commonly accepted convention on signsin electrical engineering, vectors are left-multiplied by e ωt σ .Therefore, a positive sign in an angle refers to the clock-wisedirection. This implies that inductors impedance will havepositive angles while capacitors will have negative angles.However, the phase lead and lag changes its role: lag impliesrotation in counter-clock wise direction and lead in clock-wisedirection (see Fig. 1). It can be readily demonstrated that theprojection of a voltage u ( t ) over the basis σ yields the origi-nal voltage waveform, i.e., u ( t ) = √ α cos ωt + α sin ωt ) .This is the same as extracting the real part of the complexrotating vector, i.e. R e {√ (cid:126) V e jωt } . In fact, by using theorthonormal basis σ = { σ → √ ωt ; σ → √ ωt } , u ( t ) gets transformed into u = α σ + α σ . Therefore: u ( t ) = e ωt σ u = (cos ωt + sin ωt σ )( α σ + α σ )= ( α cos ωt + α sin ωt ) σ + ( α cos ωt − α sin ωt ) σ = 1 √ u ( t ) σ − H [ u ( t )] σ ) (11) u ( t ) i R L C + Figure 2.
RLC circuit used in Example 1. where H refers to the Hilbert transform of a signal [27].Therefore, u ( t ) = proj σ [ √ u ( t )] = √ u ( t ) · σ is theprojection of a rotating vector u ( t ) into σ . It is worthpointing out that the rotating vector u ( t ) is not the voltageitself, u ( t ) . This is a different interpretation compared to thatof other authors [5, 15]. This discrepancy will be analysed byusing the circuit depicted in Fig. 2.The time-domain equation that governs the circuit dynamicsis: u ( t ) = Ri ( t ) + L di ( t ) dt + 1 C (cid:90) i ( t ) dt (12)Time derivatives in GA should be calculated as follows [25]: d u ( t ) dt = ω σ u ( t ) (cid:90) u ( t ) dt = − σ ω u ( t ) If the source is sinusoidal and the circuit is operating in steady-state, (11) can be substituted in (12), yielding: proj σ (cid:104) √ e ωt σ u (cid:105) = √ proj σ (cid:2) Re ωt σ i + Lω σ e ωt σ i − σ Cω e ωt σ i (cid:105) (13)Equation (13) can be simplified, yielding u = R i + Lω σ i − σ Cω i (14)Rotors e ωt σ are cancelled out because they commute with σ . Therefore, it is not necessary to set any specific timeinstant, t , after performing the derivative, as suggested by CN.The result is an algebraic equation where only geometric pha-sors such as u and i are present. The geometric impedance can be obtained right-multiplying (14) by the inverse of thecurrent: Z = u i − = R + (cid:18) Lω − Cω (cid:19) σ = R + X σ (15)The geometric admittance can be defined as the inverse of thegeometric impedance: Y = Z − = Z † Z † Z = Z † (cid:107) Z (cid:107) = G + B σ (16)Both elements have similar definitions to those ofimpedance/admittance in the complex domain. However,here both are multivectors because they consist of a scalarpart plus a bivector. The use of this criterion allows toovercome the drawbacks of other theories in which inductivereactance was negative while capacitive reactance waspositive [18].In order to transform the signal (9) from the time to thegeometric domain, the following basis can be defined: ϕ dc = 1 ←→ σ ϕ c ( t ) = √ ωt ←→ σ ϕ s ( t ) = √ ωt ←→ σ ... ϕ cn ( t ) = √ nωt ←→ σ n − ϕ sn ( t ) = √ nωt ←→ σ n (17)In addition, l sub- and inter-harmonics can be added byincreasing the number of elements in the basis by l afterthe highest-order harmonic ( n ) [24]. The voltage u ( t ) in (9)can be transformed to the geometric domain: u = U σ + (cid:80) nk =1 U k e − ϕ k σ (2 k − k ) σ (2 k − + (cid:80) lm =1 U m e − ϕ m σ (2 n +2 m − n +2 m ) σ (2 n +2 m − = U + (cid:80) nk =1 U k σ (2 k − + U k σ (2 k ) + (cid:80) lm =1 U m σ (2 m − + U m σ (2 m ) (18)where U k = U k cos ϕ k and U k = U k sin ϕ k . The sametransformation can be applied to i ( t ) in order to calculate i . Itis worth noting that i may include harmonics not present in thevoltage. By using the same rationale presented in (12)-(14),the geometric impedance can be defined for each harmonic as: Z k = u k i − k = R + (cid:18) kLω − kCω (cid:19) σ (2 k − k ) (19)where u k and i k are geometric phasors for the harmonic k .This proposal overcomes some drawbacks of previousGA power theories. First, it can readily accommodate DCcomponents in voltages and currents. Second, the traditionaldefinition of apparent power based on the product of the RMSvoltage and current is preserved, and this does not happens inother proposals [5]. These are contributions of this work. B. Power Flow in GA
There exist different definitions for apparent power inpower theories based on GA. Menti and Castro-N´u˜nez chose S = U I , while Castilla-Bravo chose S = U I ∗ . All of themare compatible with the energy conservation principle dueto the multi-component nature of GA [19]. However, resultsmight be inconsistent if the orthonormal basis that expandsthe geometric space is not carefully chosen. For example, inthe proposal of CN, k -vectors are used for the basis [18].Therefore, the geometric power calculation should be adaptedso that power components are correctly computed, as alreadymentioned in Section III. Also, non-active power calculationscan lead to erroneous results since the geometric power is notcalculated as M = U I † [21]. In order to prove it, the apparentgeometric power can be defined as: M = ui = u · i + u ∧ i (20)The value of (cid:107) M (cid:107) is the product of the voltage and currentmodules, provided that u and i are vectors: (cid:107) M (cid:107) = (cid:113) (cid:104) M † M (cid:105) = (cid:113) (cid:104) ( ui ) † ( ui ) (cid:105) = (cid:113) (cid:104) ( i † u † ) ( ui ) (cid:105) = (cid:112) (cid:107) u (cid:107) (cid:107) i (cid:107) = (cid:107) u (cid:107)(cid:107) i (cid:107) (21) where the property a † = a has been applied for vectors. Theapplication of this property is the key to overcome a definitionbased on the complex conjugate current. This feature cannot beapplied in other power theories based on GA since, in general, A † (cid:54) = A for any k -vector A with k > [18].In (20), several terms of engineering interest can be identi-fied. On the one hand, the scalar term (cid:104) M (cid:105) = u · i matches theactive power P , and it will be called active geometric power,or M a . On the other hand, the bivector term (cid:104) M (cid:105) = u ∧ i will be called non-active geometric power, or M N . C. Current Decomposition in GA
In this section, the current consumed by a load is decom-posed by using the proposed power theory. Simplifying (20)and taking into account that for any vector a − = a / (cid:107) a (cid:107) : M = ui −→ u − M = u − u (cid:124) (cid:123)(cid:122) (cid:125) i = ii = u − M = u (cid:107) u (cid:107) ( M a + M N ) = i a + i N (22)where i a is the active or Fryze current, while i N is the non-active current. This decomposition procedure has not beenused before for GA power theories in the frequency domain,and it is a novel contribution of this work. Also, in previouspower theories based on GA current decomposition was notguaranteed since multivectors might not have inverse and, inany case, its calculation is not straightforward [28].Each of the currents presented above has a well-establishedengineering meaning. The current i a is the minimum currentrequired to produce the same active power to that consumedby the load, while the non-active current i N is the currentthat does not affect the net active power. Therefore, the lattercan be compensated by using either passive or active filters.The current i N can be decomposed in two terms for practicalengineering purposes. The first one is related to transientenergy storage and leads to the reactive current. The secondone does not include storage and leads to the scattered currentintroduced by Czarnecki [29]. In addition, by using (16), (18)and Ohm’s law, the current i demanded by a linear load canbe calculated as: i = n (cid:88) k =1 Y k u k = n (cid:88) k =1 (cid:0) G k + B k σ (2 k − k ) (cid:1) u k = i p + i q where i p is commonly known as parallel current while i q as quadrature current : i p = n (cid:88) k =1 G k u k , i q = n (cid:88) k =1 B k σ (2 k − k ) u k (23)It can be readily demonstrated that they are orthogonal.Therefore, by comparing (22) and (23): i = i a + i N = i p + i q = i a + i s + i q (24)where i s = i p − i a is the scattered current, which can onlybe compensated by using active elements, while i q can becompensated by using passive elements [29]. There have beendifferent attempts to give physical meaning to these currentcomponents. For that purpose, the scattered power was defined as M s = ui s , while reactive power as M q = ui q . However,it has already been demonstrated that this decomposition hasno physical meaning, even though is useful for the engineeringpractice [30, 31]. In addition, the component i G is included tomodel current components whose frequencies are not presentin the voltage: i = i a + i s + i q + i G (cid:124) (cid:123)(cid:122) (cid:125) i N (25)The power factor can be defined in the geometric domain as: pf = (cid:104) M (cid:105) (cid:107) M (cid:107) = P (cid:107) M (cid:107) (26)V. E XAMPLES AND D ISCUSSION
Two examples will be given in order to validate the theoret-ical developments. The first one is the resolution of an
RLC circuit under distorted conditions, while the other one consiston the analysis of experimental data. The results obtained withthe proposed theory will be compared to those obtained byother theories. All the results have been obtained by usingMatlab and the Clifford Algebra toolbox [32].
A. Example 1. Non-Sinusoidal Source
The
RLC circuit presented in Fig. 2 has been used as anexample and benchmark for the different theories based onGA. First, Menti, Castilla-Bravo and Lev-Ari theories cannotaddress it since they do not offer the tools for analysing circuitsin the geometric domain. For these cases, it would be requiredto solve the circuit by using other techniques (such as complexalgebra), and then transform the results to the geometricdomain in order to analyse the power flow. Therefore, thecircuit will only be solved by using the theory proposed inthis paper (GAPoT), CN [18] and CPC (Czarnecki) [1]. Allof them allow to decompose the current.In the circuit, R = 1 Ω , L = 1 / H and C = 2 / F. Thesource voltage is u ( t ) = 100 √ ωt + sin 3 ωt ) . Kirchhofflaws can be applied in the time domain in order to obtainEquation (12). Then, GAPoT theory is used to transform it tothe geometric domain: u + u = R ( i + i ) + L ( ω σ i + 3 ω σ i ) − C (cid:18) σ i ω + σ i ω (cid:19) It can be seen that the superposition theorem is embedded inthe proposed formulation since all components are operated atthe same time. This is a clear difference compared to theoriesbased on complex numbers.By using (17), the geometric voltage turns into: u = u + u = 100 ( σ + σ ) (27)while impedances and admittances are calculated with (19): Z = 1 − σ −→ Y = 0 . . σ Z = 1 + σ −→ Y = 0 . − . σ Therefore, the current becomes: i = i + i = Y u + Y u = 50 σ + 50 σ − σ + 50 σ The geometric apparent power is calculated by using (20): M = ui = 10 (cid:124)(cid:123)(cid:122)(cid:125) M a = P − σ + 5 σ − σ − σ (cid:124) (cid:123)(cid:122) (cid:125) M N kVAThe active power consumption is 10 kW, while the rest is non-active power. The reactive power consumed by each harmonicis included in σ (2 k − k ) . Therefore, the reactive power of thefirst harmonic is − σ , while that of the third one is σ .This result is in good agreement with traditional analysesin the frequency domain where the reactive power of eachharmonic is the same, but with opposite sign. However, theterm − σ − σ cannot be obtained by using complexalgebra since it involves the cross-product between voltagesand currents of different frequencies. This is one of theadvantages of GA.The module of the geometric power is: (cid:107) M (cid:107) = (cid:113) (cid:104) M † M (cid:105) = (cid:107) u (cid:107)(cid:107) i (cid:107) = 141 . ×
100 = 14 , VAIf the CN theory is applied, the apparent power becomes: M CN = 10 + 10 σ + 10 σ kVA (28)The value of active power is 10 kW. However, the factor f = ( − k ( k − / has been used for the calculations. Also,it can be seen that it is not possible to distinguish reactivepower components generated by each harmonic since all ofthem are grouped in the term σ . Finally, it can be observedthat (cid:107) M CN (cid:107) (cid:54) = (cid:107) u (cid:107)(cid:107) i (cid:107) .By using the CPC theory, it is not possible to generate acurrent vector in the frequency domain nor a power multivec-tor. Also, the instantaneous value of currents should be usedto describe independent terms of power. The results are: P = 10 . W Q r = 10 . VAr D s = 0 VA S = 14 . VAThe value of active power calculated by the CPC theoryis, of course, correct. However, this theory cannot fullydescribe harmonic interactions between voltage and currentcomponents. The module of the total reactive power yields10 kVAr. However, it is not possible to calculate the individualcontribution of each harmonic nor its sign (sense).Regarding current decomposition, by using (22), it follows: i = 50 σ + 50 σ (cid:124) (cid:123)(cid:122) (cid:125) i a + 50 σ − σ (cid:124) (cid:123)(cid:122) (cid:125) i N Also, if (23) is applied, an identical result is obtained: i = 50 σ + 50 σ (cid:124) (cid:123)(cid:122) (cid:125) i p + 50 σ − σ (cid:124) (cid:123)(cid:122) (cid:125) i q If a harmonic compensator is to be designed, its susceptanceat each harmonic would be the same as that of the load, butwith the opposite sign: B cp = − B B cp = − B All the current will be compensated by using passive elementssince no scattered current is present (see [33] for more details).This means that i a = i p . Therefore, i N would be zero. σ σ σ σ σ σ (cid:107) · (cid:107) i a i s i p i q i Table IIC
URRENT DECOMPOSITION FOR THE CIRCUIT IN F IG . 2 AND
C=2/7 F.
Consider now a value of C = 2 / F in Fig. 2. This set ofparameters has been used in other scientific works since powercomponents cannot be distinguished if the classical conceptof apparent power is applied [19, 34]. For the voltage valuepresented in (27), the current becomes: i = 30 σ + 10 σ − σ + 90 σ and the geometric power is: M = 10 − σ + 3 σ − σ − σ + 8 σ kVAActive power consumption is the same to that obtained withother theories (10 kW). However, the rest of terms are differ-ent. Reactive power consumption for each harmonic has beenreduced. The term σ has appeared due to the interactionbetween in-phase components in the first voltage harmonic andthe third current harmonic. This term highlights that the systemcannot be fully compensated by using only passive elements.Despite the changes in various terms in current and powers,the module of the geometric power remains unchanged: (cid:107) M (cid:107) = (cid:107) u (cid:107)(cid:107) i (cid:107) = 141 . ×
100 = 14 . kVAThe current decomposition for this case is given in Table II.If the CN theory is applied, the power becomes: M = 10 + 6 σ + 6 σ + 8 σ kVA (29)where (cid:107) M (cid:107) = 15 . kVA. This value differs from thatobtained in the previous case, even though voltages andcurrents have not changed. Therefore, the proposed theorycaptures effects that others cannot (e.g. CN theory). B. Example 2. Measurements Analysis
In this example, the voltage and current waveforms of atypical house in Almer´ıa (Spain) are analysed. The open-platform openZmeter [35, 36] was used for analysing powerquality. Fig. 3 shows voltage and current measurements in atime window of 200 ms, taken with a sampling frequency of15.625 kHz (3125 samples). Several home appliances wereon, like a TV, LED lights and electronic appliances suchas a router, satellite receiver and other devices in stand bymode. The current waveform was highly distorted since thecurrent THD was 88.3%, while the voltage THD was 6.63%.Fig. 4 shows the voltage and current spectrum for the first fiftyharmonics (for the sake of clarity, the fundamental componentis not shown). Fifth and seventh harmonic voltage componentsare prominent while even harmonics are insignificant due tothe half-wave symmetry of the waveform. From Table III, itcan be concluded that most of the energy is concentrated in
Figure 3. Voltage and current measurements from a house in Almer´ıa, Spain.Figure 4. Voltage and current spectrum from a house in Almer´ıa, Spain.Figure 5. Instantaneous power waveform and active power P in Example 2. the first five odd harmonics. The RMS value of the voltage is234.00V while that of the current is 2.61A. Fig. 5 shows thepower waveform as well as the value of P that is 359.15 W.A geometric vector can be derived by using the data of Ta-ble III. A value of n = 5 has been considered (the fundamentalcomponent plus four harmonics). Therefore, the dimension ofthe geometric space has been set 10 ( n ). It is worth pointingout that in the proposed theory the dimension of the geometricspace can be chosen according to the requirements (numbers ofharmonics). This is an advantage compared to other theories.The voltage and current expressions in polar form are: u = 233 . e − . σ σ + 0 . e − . σ σ + 4 . e . σ σ + 4 . e − . σ σ + 0 . e − . σ (9)(10) σ i = 2 . e − . σ σ + 0 . e . σ σ + 0 . e − . σ σ + 0 . e . σ σ + 0 . e − . σ (9)(10) σ and the geometric power is: M = 359 .
21 + 408 . σ − . σ − . σ + 1 . σ + 0 . σ (9)(10) + O (30)where O includes the rest of bivectors that appears due tothe cross products. The value of M a is . W, whichis similar to that obtained by using the digital samples ofvoltages and currents. Results for the reactive power of each
Voltage CurrentOrder (cid:107) V (cid:107) (V) ϕ v (rad) (cid:107) I (cid:107) (A) ϕ i (rad)fund 233.92 -1.57 2.33 -0.723rd 0.46 -2.61 0.93 1.855th 4.74 1.28 0.45 -1.697th 4.02 -0.07 0.49 1.709th 0.42 -2.60 0.16 -1.44 Table IIIO
DD HARMONICS PRESENT IN THE WAVEFORMS OF E XAMPLE P i Q i Order ozm ozm GA fund 361.80 408.56 408.503rd -0.102 -0.426 -0.4255th -2.134 -0.346 -0.3467th -0.408 1.955 1.9559th 0.028 0.063 0.062
Total 359.15
Table IVH
ARMONIC ACTIVE (W)
AND REACTIVE (VA R ) POWER MEASUREMENTS . i p i a i s i q i N iσ -0.007 -0.007 0.000 1.746 1.746 σ σ -0.266 σ -0.108 0.001 -0.109 -0.789 -0.898 -0.897 σ -0.126 0.009 -0.135 0.070 -0.065 -0.056 σ σ -0.101 0.026 -0.127 0.036 -0.091 -0.065 σ -0.007 0.002 -0.010 -0.484 -0.494 -0.492 σ -0.057 -0.002 -0.055 0.077 0.022 σ (cid:107) · (cid:107) Table VC
URRENT COMPONENTS OBTAINED FROM CURRENT MEASUREMENTS .Figure 6. Total, active and non-active current for the measurements. harmonic are also similar. These values are shown in Table IV.Table V shows the current components presented in (22). Inorder to compute i p and i q according to (19), the geometricimpedances were calculated for each harmonic. The value ofthe total current was (cid:107) i (cid:107) = 2 . A, while i a = 1 . A. Thelatter is the minimum current that would produce the sameactive power. Fig. 6 shows the waveforms of i , i a and i N . VI. E
XTENSION TO M ULTI -P HASE S YSTEMS
It might be possible to extend the proposed theory to multi-phase systems thanks to the use of N -dimensional vectors. InGA, the different phases can be operated as vector arrays inthe geometric space. For example, for a three-phase system,arrays of dimension three can be used, while each element ofthe array would be a vector of dimension N , depending on thenumber of harmonics to be considered. In particular, voltagesand currents could be expressed as: [ u ] = (cid:2) u R u S u T (cid:3) , [ i ] = (cid:2) i R i S i T (cid:3) (31)so that the geometric power would be calculated as M = [ u ] [ i ] T (32)The development of this theory for three-phase and multi-phase systems is of interest for further research.VII. C ONCLUSION
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