Renewable and Sustainable Energy Reviews 57 (2016) 1428–1439

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Renewable and Sustainable Energy Reviews

http://d1364-03

n CorrE-m

journal homepage: www.elsevier.com/locate/rser

A review study of instantaneous electric energy transport theoriesand their novel implementations

Y. Beck a,n, N. Calamaro b, D. Shmilovitz c

a Faculty of Engineering, Holon Institute of Technology, Holon, Israelb Faculty of Engineering, University of Tel-Aviv, Energy Conversion Lab, Tel-Aviv, Israelc Faculty of Engineering, University of Tel-Aviv, Tel-Aviv, Israel

a r t i c l e i n f o

Article history:Received 10 May 2015Received in revised form19 September 2015Accepted 17 December 2015Available online 8 January 2016

Keywords:Active powerEnergy meteringHarmonic distortionPower transport theoryReactive power

x.doi.org/10.1016/j.rser.2015.12.08621/& 2015 Elsevier Ltd. All rights reserved.

espondence to: Faculty of Engineering, Holonail addresses: beck@hit.ac.il (Y. Beck), netzah@

a b s t r a c t

Electric energy transport theories are still a debatable subject regarding their validity and applicability invarious aspects. This paper includes a comparative study of the group of theories which are categorizedas instantaneous theories as opposed to periodic-averaged ones. As a preliminary review this paperpresents first the three apparent power definitions in order to emphasize the difference between them. Adiscussion regarding their accuracy for energy metering of various conditions such as multiphase andnon-sinusoidal environments is presented. Instantaneous theories are then reviewed and compared withperiodic averaged theories. The review focuses mainly on p-q theory (PQT) and its derivative, instanta-neous multi-phase theory. The legacy application of this theory for active filtering is briefly presentedand the new idea of using the theory for multi-load and multi-source theory is presented as well. Thecomparison identifies the similarities and differences between the instantaneous and periodic averagingtheories, and their inter-relations.

& 2015 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14292. Definitions of apparent power S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1430

2.1. General. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14302.2. The Buchholz definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14302.3. Arithmetic apparent power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14312.4. Geometric and vector definition of apparent power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14312.5. Comparison of the apparent power definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14322.6. Concluding discussion on apparent power definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1432

3. Modern instantaneous power theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14323.1. The p-q theory or instantaneous reactive power (IRP) p-q theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14323.2. PQT: formal theory foundation for three phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1432

3.2.1. The Clarke transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14323.2.2. justification of usage of Clarke transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14333.2.3. Definition of powers in PQT and the geometrical interpretation of the electric parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433

3.3. Generalization of PQT to include zero-phase sequence components for load imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14333.4. Generalization of PQT to include multi-phase systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434

3.4.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14343.4.2. Outline of multi-phase generalization for PQT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1434

3.5. Difference between PQT and periodic averaged time-dependent theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14364. PQT and multi-phase theory: areas of strength compared with other theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14365. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437

Institute of Technology, 51 Golumb st. 5810201, Holon, Israel.iec.ac.il (N. Calamaro), smilo@eng.tau.ac.il (D. Shmilovitz).

Y. Beck et al. / Renewable and Sustainable Energy Reviews 57 (2016) 1428–1439 1429

Appendix.Relations between Buchholz and geometric power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438

1. Introduction

Electric energy transport theories are one of the most impor-tant theories in grid analysis as well as in the understanding of thetransfer of energy in complex grids. The introduction of renewableenergy and the development of smart grids have increased theinterest in and the necessity of matching the appropriate theorywith the appropriate application.

These theories are interesting by virtue of their suitability for:1) grid monitoring and control, 2) grid energy management, 3)grid energy storage, 4) grid reactive power management, 5) gridmulti-source management. The issue of suitability of each theoryin terms of smart grid applications is an open debate. Recentpublications show renewed interest in the subject as well as var-ious applications of these theories [1–4].

Apparent power serves as a tool for grid generation andtransmission design. Therefore, prior to dealing with the variouspower theories the understanding of the various apparent powerdefinitions and their validity to multiphase and non-sinusoidalenvironments are essential. The paper starts with a short pre-sentation of the three apparent power definitions which reflectrecent changes in energy metering [5] and a discussion on theresemblance (not identity) of these various definitions is per-formed. These definitions yield different results, except for thecase when they are applied on a purely sinusoidal balanced sys-tem. The known definitions are therefore initially comprehendedthrough a survey of numerous sources of information and thefollowing questions are answered: what is apparent power? Whatis it used for? What is the initial definition? and what “physical”explanation was later added to justify the previous definitions?. Inthis paper we try to evaluate to what extent a unified approach ispossible for the various definitions of apparent power in non-sinusoidal environments. The understanding of the differences inapparent power definitions, especially in non-sinusoidal environ-ments, facilitates a suitable choice of definition. This paperattempts to show that similarity relation exists at least betweenthe Buchholz, geometric and vector apparent power definitions inthe single phase case and that in the multi-phase case there is abounded region of the difference.

There are two major groups of electric energy transport the-ories: 1) periodic averaged theories. These theories handle electricpower parameters as averaged over an a single network cycle andinclude the conservative frequency domain theory and the currentphysical components (CPC) theory, which are both discussed inlength at [2, 6–8]; 2) instantaneous theories. These theories handlethe electric power parameters in their instantaneous form in thetime domain. This group includes the p-q theory (PQT), the con-ventional Poynting vector theory (PVT) and the conservative timedomain theory (CPT time-domain). Instantaneous theories handleelectric power parameters in their instantaneous form in time.

In a previous paper we reviewed and dealt with the variousaspects of periodic averaged theories [8]. This paper is com-plementary to [5,8] and concludes the study of the power theoriesrange by covering instantaneous power theories as well as com-parison of the various theories with periodic averaged ones.

In Table 1, all the known active electric energy transport sub-theories are presented. It should be noted that there is a vast lit-erature on each theory and therefore only a few major

contributors are mentioned here. The highlighted rows in Table 1are the instantaneous theories which are the scope and main focusof this paper. After clarification of the various definitions of eachtheory, a theoretical comparative study approach is taken. Thecurrent state is that many researchers today regard these theoriesas essentially different and present examples where differenttheories and different apparent power definitions yield differentresults.

Power theories are ongoing specialization fields [1–3,9]. How-ever, if the theories are regarded as different perspectives of thesame phenomena, then acceptance of a future unifying theory isan option [10]. There has been some debate regarding theseinstantaneous theories' effectiveness compared with periodicaveraged theories and some works claim that the theories are lessuseful for energy measurement and active/reactive load char-acterization [7–14]. There are also other comparative works suchas [15,16] which deal with specific aspects of comparing betweenthe theory groups. The work presented in this paper demonstratesthat in-depth study of instantaneous theories' apparent powerdefinitions enables the development of new electric grid mon-itoring and energy management tools. Active filtering is a classicalapplication in which instantaneous theory is applied [11,17–19].Instantaneous theory is not limited to response time of more thanhalf a cycle and does not necessarily require storage components[17], but rather only switches. Grid monitoring and control is anapplication which is addressed in the paper as well. The paperinvestigates the extent of applicability of instantaneous theories topower conditioning [3,20,21].

This paper shows in detail that instantaneous theory is differ-ent from periodic averaged theory with regard to physical active,reactive and distortion current/power physical components.Moreover, the paper studies whether multi-phase instantaneoustheory is suitable for analysis of multi-load/multi-source compu-tation for composition and decomposition of current/power inhierarchical grid transitions.

Because of the excessive size of a study covering all theories,this paper contains a comprehensive study of only instantaneoustheories: p-q theory (PQT) and multi-phase instantaneous theoryas a necessary addition to a previous study of periodic averagedtheories and multi-vector Poynting vector theory (PVT) [7]. Multi-phase formalism brings in several merits which are discussed inthis paper: (1) Physical comprehension. With it comes theunderstanding that the mathematical formalism of Clarke trans-form for PQT is not mandatory for instantaneous theories devel-opment [10,21]. (2) The generalized multi-phase theory is applic-able to all the specific cases: single-phase, double-phase, three-phase balanced and unbalanced cases. (3) Multi-phase formalismis shown to be suitable for describing the single phase and two-phase networks as well as three phase systems. (4) Multi-phasevector formalism enables comparison with other multi- vectortheories such as multi-harmonic, periodic averaging theories andinstantaneous PVT. Differences and similarities are highlighted.(5) It is shown that instantaneous theory is not equal to periodicaveraging in terms of optimization criteria for delivering activepower through minimal losses to load. Explicit multi-phaseformalism is investigated in this paper, because multi-phaseformalism is potentially suitable for describing multi-loads andmulti-sources, even if the network is a three-phase system.

Table 1A cluster of used theories.

# Theory Major contributors Years of publication

1 Conservative theory (CPT) Buchholz 1922– þTime domain Budeanu 1927– þFrequency domain Fryze 1932Fryze-Buchholz-Depenbrock (FBD) method (Quade, Rosenzweig) 1934, 1939

Depenbrock 1962Standards (knowledge solidification): IEEE 1459-2010, DIN 40110–{basic,1,2} A. E. Emanuel 2000, 2010

Tenti method (CPT) 2003Tenti method (CPT)2 Conservative-frequency domain M. D. Ili´c, J. L. Wyatt, N. LaWhite 1997

Reactive power vector decomposition V. Tipsuwanporn, F. Cheevawit, W. Piyarat, P. Thepsatorn,Y, Paraken

1999

3 PVT Poynting vector theory (PVT) A.E. Emanuel 1993-2010Multi-vector PVT single phase (periodic averaged theory) M. Castilla, J.C. Bravo 2009PVT for extraction of non-linear circuits, hidden information F. de León, J. Cohen, 2010

4 Instantaneous p-q theory (PQT) Akagi, Nabae, Kanazawa 1983PQT enhancement to unbalanced, and three-phase four wire Aeredes,þthe original authors 1996Generalized multi-phase instantaneous theory J. Willems 1994–2008

Peng, Lai et. all 1996X. Dai, G.Liu, R. Gretsch 2004

5 Current physical components (CPCT)-frequency domain L. S. Czarnecki 1988 effective: 19956 Critical studies that are not new theory Significant critical studies on PQT, PVT

counter-responsesL. S. Czarnecki, A.E. Emanuel 2006–2007

Instantaneous vs. periodic averaging J. Willems 2008Comparative studies H. K. M. Paredes, F. P. Marafao and L.C. P. da Silva 2009

Y. Beck et al. / Renewable and Sustainable Energy Reviews 57 (2016) 1428–14391430

2. Definitions of apparent power S

2.1. General

Apparent power is a figure intended to quantify the actualpower required from the source in order to provide a specificactive energy to the load [22]. This quantity is essential to trans-mission line design. An alternative definition is: the power mea-sured at the terminals of a load under test. Apparent power isusually not considered as an actual physical power [2,9]. In theliterature, three definitions of apparent power can be found [2,23].Each definition yields a different apparent power value. In thefollowing sections the various definitions are presented as well asa comparison between them. This, subject is still being debated bymany researchers and the following sections are intended to lay-out the definitions and shed some light at some of theseinequalities between the definitions.

2.2. The Buchholz definition

The Buchholz definition is based on the following equation [4,2]:

S¼ ‖V‖U‖I‖¼ VRMS U IRMS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiXLi

V2Li

sU

ffiffiffiffiffiffiffiffiffiffiffiffiXLi

I2Li

sð1Þ

where: i¼1,2,3,…,M is an integer and Li is the correspondingphase. The RMS value of the current and voltage of all phases iscalculated in accordance with the chosen method. A single phasetime-domain definition for the squared L2 norm of the voltage is:

‖V‖2 ¼ V2RMS ¼ vðtÞ; vðtÞ� �¼ Z T

0v2ðtÞdt ð2Þ

where v(t) is the instantaneous voltage. A frequency domainsingle-phase definition for the same voltage is:

‖V‖2 ¼ V2RMS ¼

XNn ¼ 0

V2n ð3Þ

where Vn is the harmonic voltage of order n. One possible meaningof the Buchholz definition formula for the single-phase case is thatthe load can be considered as a black-box with one port. At its

terminals the instantaneous voltages and currents are measured.These parameters are the only observable and measurable para-meters considered. In the three-phase case the RMS voltages andcurrents are defined as stated in [2,4]. The definition in (1) can beexplained by applying the ‘Buchholz–Goodhue apparent powerdefinition’. Namely, the RMS voltages and currents of each phaseare the cause of the actual three-phase system line active powerloss. This can be shown to be [2]:

ΔPV ffiXi ¼ Li

V2iN

Ri¼ 1Ref f

Xi ¼ Li

V2iN ;ΔPffi

Xi ¼ Li

RiI2iN ¼ Ref f

Xi ¼ Li

I2iN ð4Þ

There is no physical meaning for the Buchholz apparent poweras thought by major researchers [2,9]. There are however physicaljustifications such as (4). In [26] the Buchholz definition is derivedby maximizing the active power P according to Lagrange's multi-plier method in the case where: (1) phase voltages are given,(2) the phase currents are constant, (3) the net sum of the phasesand neutral currents is zero. The Budeanu three-power typerelation, active, reactive and distortion, requires the Buchholzdefinition. An alternative interpretation of the advantage ofBuchholz power to reflect the system’s power is suggested next. In[2,20] the multi-phase apparent power is derived from phasevoltages and currents (originally the operator in [20] was used tocriticize PVT for including useless non-active cross-phase productcomponents). Next we show how it leads to preservation of powerlaws in a multi-phase system using the Buchholz definition:

SStarðtÞ ¼∑Li

vLi ðtÞiLi ðtÞ; or SΔ ¼∑Li

vLi ;Lj ðtÞiLi ;Lj ðtÞ

‖S‖2 ¼ ∫T

0SðtÞSnðtÞdt≈∑

Li

V2Li∑Li

I2Li ¼ V2RMSI

2RMS ¼ S2Buchholz

‖ S!

geometric‖≈SBuchholz

9>>>>>>>=>>>>>>>;

ð5Þ

The result in (5) is not trivial and is detailed in the Appendix. Itis now shown that a simple product of v(t) and i(t) where v(t) and i(t) are the instantaneous voltage and current accordingly, repre-sented as complex phasors, can be expanded as Fourier series and

Y. Beck et al. / Renewable and Sustainable Energy Reviews 57 (2016) 1428–1439 1431

then all known power types can be extracted:

p¼ Ref v!ðtÞU i!ðtÞg ¼P

Li

vLi ðtÞiLi ðtÞ ¼ v!ph U i!

ph ¼ v!phTi!

ph

qBudeanu ¼ Imf v!ph U i!

phg;QD ¼ Ref v!ph � i

!phg ¼

Ppaq

Vp � Iq�Vq � Ip� �

9>>>>>>>=>>>>>>>;

ð6Þ

The results in (6) are obtained in [10] for instantaneous multi-phase theory, and in [9,27] for single-phase multi-harmonic(which equals to the periodic averaged) PVT. In [2,25] the apparentpower definitions are discussed in relation to the main interna-tional standards. The relevant standards on electric power andenergy formulations are DIN 40110 (2002) [30] and IEEE1459(2010) [31]. These standards adopt the FBD theory (Fryze-Buch-holz-Depenbrock) as shown in [5].

2.3. Arithmetic apparent power

The second definition is referred to as the arithmetic apparentpower and is written as follows [2,24]:

S¼XLi

VLi U ILi� � ð7Þ

This definition implies that the total three-phase apparentpower relates to the arithmetic (scalar) sum of the apparent powerof all three phases. Because of energy conservation, that physicalquantity is considered as the total load energy. This definition doesnot equivalent to the Buchholz definition. Next, a heuristicexplanation of why this arithmetic definition is not necessarilyphysically correct is presented. It relies on proven theorems in thePoynting vector power theory [9]. If S is a vector it cannot bescalar. Most researchers agree that reactive power is a vectorpropagating with direction in the harmonics space (periodicaveraged) or generator phase space (instantaneous theories) orboth [9,23]. For example:

1) Reactive vector decomposition theory, which is a con-servative power theory in which:

Q!¼ V

,� I! ð8Þ

where: V!¼ V1;V2; :::;Vn½ �; I

!¼ I1; I2; :::; In½ � and therefore Q!

isnaturally a vector.

Then the apparent power S is represented according to therelation:

S!¼ Pþ jQ

!þ k D!

3

S2 ¼ P2þQ2BudeanuþD2

Budeanu

9=; ð9Þ

Where j; k are orthogonal unit vectors. It implies from (8) and (9)

that S is also a vector. Regarding the distortion power D!

in theBuchholz definition, this power is zero in the pure sine wave caseand is nonzero in otherwise.

2) In Poynting vector power transport theory, as in [9]. Themodel there, multi-vector PVT, uses the geometric apparent powerdefinition. This also yields vector reactive power, as does (8).Extending the description to a multi-phase system means that thetotal apparent power should naturally be a vector:

S!

Total ¼ S!

Li

h i¼ ½ S!L1 ; S

!L2 ; :::; S

!LN � ð10Þ

the voltage and current vectors are temporally phase-angle shifted

from one load phase to another. In general:

DTotalðtÞ ¼PLi

vLi ðtÞiLi ðtÞþPLi

PLj ;Li a Lj

ðvLi ðtÞiLjðtÞþvLj ðtÞiLi ðtÞÞ

DTotal

�� ��2aPLi

vLi ðtÞU iLi ðtÞ

8>><>>: ð11Þ

as (5) indicates.It is evident from (11) that the instantaneous distortion com-

ponents are generated by cross-product phase voltages and cur-rents. These components contribute net zero power to the periodicaveraged distortion power. The arithmetic apparent power defi-nition does not obey the orthogonal behavior of the current phy-sical components as in (9). In [28,29], a definition that maintainsorthogonality in the apparent power current components isdemonstrated. An alternative argument that the arithmetic defi-nition is to be less recommended is presented in [2], because ofviolation of parameters of the electric system line losses in theunbalanced case.

2.4. Geometric and vector definition of apparent power

The last definition of apparent power is the one in [2,9]:

S!¼ Pþ jQ

!

while : jQ!¼ jQ

!Bþ kDB

9=; ð12Þ

From (8), and (9) the following known result can be derived:

S2 ¼ P2þQ2 ð13ÞThis definition is also known as the geometric definition [24]

because it obeys Pythagoras' theorem, and includes orthogonalpower vectors (P ? Q). Eq. (13) implies orthogonally physicalcomponents, namely:

P UQ ¼ 0 and P � Q ¼ kP Q n������ ð14Þ

Orthogonal is meant not in the spatial sense (x, y, z) but withregard to inner products over harmonic components [11,9,23], orinner products over voltage verses current phase vectors [15]. Itshould be mentioned that not all works use the above terminologyfor geometric apparent power beyond the single phase case. In [2],for example, the above definition is included as a fourth typenamed vector apparent power. In this case the geometrical

apparent power for the three-phase case is: S,

1þS,

2þS,

3 and the

vector notation is jS,

1jþjS,

2jþjS,

3j. The similarity of these mathe-matical extensions to the Buchholz, in the three phase case, isconfined to limits as shown in the Appendix.

The completeness in terms of energy conservation of the activeand reactive powers, as shown above, does not mean that in othertheories and sub theories reactive power cannot be furtherdecomposed into additional orthogonal physical components, aswill be demonstrated.

Fig. 1(a), as in [2], shows decomposition of non-active power Qinto Budeanu reactive QB and Budeanu distortion power DB. Sa, Sb,Sc represents the phase apparent powers. Svector/geometric representsthe total system load vector or geometric apparent power.

Fig. 1(b) demonstrates a comparison between the arithmeticapparent power and the vector apparent power. Each phase'sapparent power is redrawn and the SArithmetic apparent power isshown along with the Svector/geometric apparent power.

There is no explicit mention of apparent power at PQT. In [3]which is a book on PQT (2007), an equivalent variable is named asthe complex instantaneous power vector S. It is obtained by takingthe total scalar power p and the total scalar amplitude of thereactive power q, and summing them as pþ jq. The complex powervector is similar to the geometric apparent power definition.

Fig. 1. (a) Vector/geometric apparent power for three-phase system. (b) Arithmetic apparent power compared with geometric [2].

Y. Beck et al. / Renewable and Sustainable Energy Reviews 57 (2016) 1428–14391432

2.5. Comparison of the apparent power definitions

The three apparent power definitions provide identical resultsonly if the line currents are purely sinusoidal and symmetrical,namely they are a linear balanced system. Otherwise these defi-nitions yield different results [2,7]. As long as Budeanu distortionpower exists (Da0), there can be no equality between the Buch-holz definition as derives from (9) and the geometrical apparentpower definition of (13). It is shown in [23] that the distortionpower is composed of the sum of cross-product harmonics VnxIm(where nam). Therefore it is evident that D¼0 only for a puresine wave. When asymmetry is considered, the three-phaseapparent power definitions should be treated carefully. Compar-ing the arithmetic apparent power with the geometric apparentpower, we obtain the vector sum of the three phases:

Sarithmetic ¼X4Li ¼ 1

VLi ILi

Sgeometric ¼X4Li ¼ 1

V!

Li �X4Lj ¼ 1

I!

Lj

9>>>>>>=>>>>>>;

ð15Þ

The phase vector product is defined in (8). Only if the voltageand current are orthogonal in phase, with no high-order harmo-nics and the phase reactive powers aligned, then the definitionsyield equal results. That is only for pure sine and balanced load.However, the Buchholz apparent power and the geometricalapparent power are related by (9) and (13), and the distortionpower has an explicit formula as shown in [2,23]. The theoriesmentioned in Table 1 are mapped by their apparent vector defi-nitions. Conservative theories support the Buchholz definition. PVTas represented by the sample paper [9] uses the geometricaldefinition, but as explained it relates to the Buchholz definition.CPCT [7] is operable with all the apparent power definitions, butits author/developer prefers the Buchholz definition.

2.6. Concluding discussion on apparent power definitions

It is important to perform a shift from theoretical computationsto real practice. It should be noted that geometrical apparentpower (instantaneous) leads to values that may seem not to reflectactive/reactive load characterization correctly [5,33]. It is knownthat even when restive load is considered the imbalance betweenthe phases results with cross-phase voltages and currents. Ananalogous issue regarding the Poynting vector was discussed in[14,20] where the cross-voltage current product was found todisplay the same phenomena. Another similar issue concernsinstantaneous power definitions of the active and reactive powersis discussed in [33]. As a result of [33], some of the scientificcommunity do not use instantaneous theories [14] for identifica-tion of active/reactive load characteristics. Geometric multi-phaseapparent power is not suitable for active/reactive load character-ization and consequently not for energy measurement either. It issuitable for the single-phase case. One might therefore ask what itis good for. Apparent power is not a physical parameter, but rather

some kind of mathematical vector sum of active and reactivepowers, reflecting the power at load boundary.

Moreover, it can be shown using simple manipulation that inthe single phase case, the geometrical instantaneous definition isequivalent to the Buchholz definition (see Appendix); Buchholzserves as the RMS of the geometric apparent power definition forthe single phase but not in the multiphase case.

Researchers who wish to avoid factious active and reactiveexpressions can use Buchholz apparent power which is alwayspositive. The Buchholz apparent power yields the RMS value of thegeometric apparent power in the single-phase case but not in themultiphase case.

3. Modern instantaneous power theories

3.1. The p-q theory or instantaneous reactive power (IRP) p-q theory

The p-q theory (PQT) was published in 1984 and described in[17,15]. The theory has evolved over the last three decades. Initi-ally formal theory foundations were established for the three-phase case [4,19] followed by implementation for active filtercontrol. Enhancement to multi-phase theory and usage of mod-ernized terminology is covered in [10,21,22]. A modern source onPQT reflecting its state of the art status is [3]. There, numerousenhancements of PQT are discussed, although the discussion isconfined to three phases. The first in-depth theoretical explanationof the interpretation of reactive power during unbalanced load orstar systemwith a neutral is given in [26]. Because of criticism [36]regarding its difference from other accepted theories, an in-depthanalysis of PQT physics was offered by Willems [10]. There, heproposed a new interpretation of PQT enabling its generalizationto a multi-phase system, with the special single-phase case whichwas not included in the original theory.

3.2. PQT: formal theory foundation for three phases

3.2.1. The Clarke transformLet a, b, c be the three-phase interface between source and

load. The Clarke transform [34] of phase voltages transfers thethree-phase balanced vector [a, b, c] to an [α, β, 0] reference frame.The instantaneous variables i(t), v(t), p(t), q(t) are denoted as i,v, p,and q. Applying the Clarke transformation to the vector of phasevoltages yields:

v

vαvβ

264

375¼

ffiffiffi23

q 1ffiffi2

p 1ffiffi2

p 1ffiffi2

p

1 �12 �1

2

0ffiffi3

p2 �

ffiffi3

p2

2664

3775U

vavbvc

264

375 ð16Þ

The phase current vector transformation is identical:

i

iαiβ

264

375¼

ffiffiffi23

q 1ffiffi2

p 1ffiffi2

p 1ffiffi2

p

1 �12 �1

2

0ffiffi3

p2 �

ffiffi3

p2

2664

3775

iaibic

264

375 ð17Þ

1

2

Fig. 3. Positions of instantaneous active and reactive powers in [α, β, 0] space.

Y. Beck et al. / Renewable and Sustainable Energy Reviews 57 (2016) 1428–1439 1433

The inverse voltage and current transforms are provided in [3].

3.2.2. justification of usage of Clarke transformationIn two important cases the Clarke transformation degenerates

the three-phase system into a two-phase system [35]:

1. In the three-phase four-wire system

vaþvbþvc ¼ 0 ð18ÞThe α,β,0 transformation separates the three-phase voltages

into voltages lying on the zero sequence axes. Condition (18)occurs for balanced loads, or at Δ connection without leakages.

2. A three-wire star (Y) system without a neutral (three-phasethree-wire) yields

iaþ ibþ ic ¼ 0; or : ineutral ¼ 0 ð19Þ

This means that one current variable is eliminated. The initialtheory is developed in favor of these two instances.

3.2.3. Definition of powers in PQT and the geometrical interpretationof the electric parameters

The active power of the three-phase [α, β,0] system is definedas:

p¼ vU i¼ vαiαþvβiβ ¼ vaiaþvbibþvcic ð20ÞThe reactive power is defined as a sum of vector products of the

phase variables:

q¼ vα � iβþvβ � iα ð21ÞAnother result discussed in multi-phase instantaneous theory

is reduced for the three-phase case as follows [22]:

q¼ v!� i!

; v!¼ va vb vc

; i!¼ ia ib ic

ð22Þ

The resulting power is formulated in matrix form as follows:

pq

" #¼

vα vβ�vβ vα

" #iαiβ

" #ð23Þ

where p is a scalar and q is the amplitude of the reactive power. Itshould be noted that:

) Instantaneous active power is a scalar product of the phasevoltage and current vectors. At [α, β, 0] space, the voltages andthe currents are parallel to each other and to the α and β axes.

) The instantaneous reactive power is a vector product of thethree-phase voltage and current vectors at the [α, β, 0] space.

Fig. 2 and Fig. 3 are inspired by [3,17]. In Fig. 2, there is a 120°difference between successive phases, whereas in the [α, β, 0]system the phases α, β are orthogonal. In Fig. 3, the power diagramis presented in the [α, β, 0] system. The active power is on a real

Fig. 2. Positions of voltage and current in [a,

plane consisting of voltages and currents which are in thesame phase.

The illustration is of a balanced resistive load, i.e. no phase shiftbetween voltage and current.

The reactive power is on an orthogonal vector, consisting onlyof the cross-products of the voltages and currents.

3.3. Generalization of PQT to include zero-phase sequence compo-nents for load imbalance

In the theory presented above, the third-phase element waszero in the [α, β, 0] space, since the voltage was balanced and didnot include a zero phase sequence voltage [36]. The generalizationfor the inclusion of a zero phase sequence component is handledin [3,15,35,36]. The enhancement of an unbalanced load system ora star (Y) system with a neutral line is owed to Aeredes et.al. in[15]. Suitable infrastructure was already introduced by maintain-ing a 3�3 matrix formulation in (16), (17) and (20). An additionalinstantaneous power is defined on top of the scalar active power,p, and the vector reactive power q. this power is named instan-taneous zero-phase sequence power, p0. When the load isbalanced only a positive sequence phase vector exists, similarly toCPCT [7].

p0 ¼ vU i¼ v0 U i0 ð24ÞEq. (24) reminds us that every general three-phase vector of

voltage/current waveforms is describable by the sum of threesymmetrical balanced sequences the positive, negative, and zerosequences.

The generalized matrix formulation relating the voltage andcurrent as inputs and the power as output [18,3,35] is:

p0p

q

264

375¼

v0 0 00 vα vβ0 �vβ vα

264

375

i0iαiβ

264

375 ð25Þ

In (25) the zero phase sequence voltage is an incremental factorof the previous system described by (23). It provides furtherinterpretation of Fig. 1, where the 2�2 sub-matrix in the

b, c] system and in the [α, β, 0] system.

Y. Beck et al. / Renewable and Sustainable Energy Reviews 57 (2016) 1428–14391434

right-hand corner represents both positive and negative compo-nents. Next the currents are calculated from the power and thevoltages as follows:

i0iαiβ

264

375¼

v0 0 00 vα vβ0 �vβ vα

264

375

�1p000

264

375þ

0p

0

264

375þ

00q

264

375

0B@

1CA ð26Þ

Therefore:

i0iαiβ

264

375¼

i000

264

375þ

0iαpiβp

264

375þ

0iαqiβq

264

375 ð27Þ

Eqs. (26) and (27) separate the zero-phase sequence compo-nent from the rest of the system (positive sequence phase vector).When transferring from the [α, β, 0] plane to the [a, b, c] plane, thephase currents are introduced as follows:

iaibic

264

375¼

ffiffiffi23

q 1ffiffi2

p 1 0

1ffiffi2

p �12

ffiffiffi32

q1ffiffi2

p �12

ffiffiffi32

q266664

377775

i000

264

375þ

0iαpiβp

264

375þ

0iαqiβq

264

375

0B@

1CA ð28Þ

Next, the current components where the zero-phase sequenceis separated can be presented as follows:

iaibic

264

375¼

ia0ib0ic0

264

375

|fflfflffl{zfflfflffl}zero sequence

current

þiapibpicp

264

375

|fflfflffl{zfflfflffl}active current

þiaqibqicq

264

375

|fflfflffl{zfflfflffl}reactive current

ð29Þ

The same formulation can be derived with the instantaneousphase power:

papbpc

264

375¼

va U ia0vb U ib0vc U ic0

264

375þ

va U iapvb U ibpvc U icp

264

375þ

va U iaqvb U ibqvc U icq

264

375 ð30Þ

where:

papbpc

264

375¼

pa0pb0pc0

264

375

|fflfflfflffl{zfflfflfflffl}zero sequence

power

þpappbppcp

264

375

|fflfflfflffl{zfflfflfflffl}active power

þpaqpbqpcq

264

375

|fflfflfflffl{zfflfflfflffl}reactive power

ð31Þ

3.4. Generalization of PQT to include multi-phase systems

3.4.1. GeneralA multi-phase generalization to PQT was initially presented

in [21] and [37]. Later on, in [10] a significant comparativework on the difference between instantaneous and periodicaveraged theories is performed. The state-of-the-art general-ization of multi-phase instantaneous power theory is reflectedin [35,38]. Willems in [21] was the only work of the timedealing with generalizing PQT and has become a foundation ofmodern multi-phase PQT. Akagi et al. [3] provide an extensivestudy of PQT. The development of multi-phase PQT was gra-dual: initially developed for multi-phase, but still not gen-eralized to cases such as unbalanced load and neutral line andvery brief formulations were developed [21], then a separatebranch of three-phase unbalanced load was developed [38],followed by three-phase four-wire explicit formulation [39].Finally, a detailed formulation of modern multi-phase PQT [22]appeared, and then a critical comparison of multi-phase

instantaneous with other theories [10]. The motivation formulti-phase theory development is that the single-phase casecannot be derived from the three-phase case, and also a singleformulation of all three-phase variants. The multi-phase(m¼1,2,3,…,M) system cannot be derived as well. This isbecause the generalization requires comprehension of theinstantaneous electrical parameters. The zero-phase sequencetreatment is different from the early work performed in [24,40]and will be described ahead. The comparative work of Willemsis described in [10,21,37,41].

3.4.2. Outline of multi-phase generalization for PQTThe generalization is summarized next, with emphasis on

more elaborated proofs which are only briefly mentioned in anoriginal paper [21]. In [21,22] it is shown that the Clarketransformation is not required, for manifesting PQT general-ization ideas. In [3] Akagi et al. accepts this idea. Willems alsorelates to the difference between instantaneous power theoriesand power averaged theories. The original PQT development[34,15,17] presents Clark's transformation as a key element ofPQT. The knowledge developed by Willems is a new inter-pretation of PQT. The extension performed in [21,22] is accep-table to the scientific community as shown in [3]. In [10,21] amodern terminology of current and power decomposition ofcomponents is used. This terminology resembles CPCT [7] andconservative reactive vector space decomposition theory [23].This terminology was used rarely in the eighties when PQT wasinitially developed from the FBD method [2,32,42]. The initialdefinitions of multi-phase voltage vector and phase currentsvector are [3]:

vðtÞ ¼ vL1 ðtÞ; vL2 ðtÞ; :::; vLM ðtÞ

and iðtÞ ¼ iL1 ðtÞ; iL2 ðtÞ; :::; iLM ðtÞ ð32Þ

where: Li¼1, 2,…, M are the phase indices. [43] defines activepower current:

pðtÞ ¼ v!ðtÞh i

U i!ðtÞh i

¼ vðtÞT iðtÞ ¼¼ vðtÞT ipðtÞ )) ipðtÞ ¼ vðtÞT iðtÞ

vðtÞj j2 vðtÞ; where : vðtÞ ¼ pTotalðtÞvðtÞj j2 vðtÞ

8>>>><>>>>:

ð33Þ

The active power function is defined as the sum of phaseactive powers. It is similar to periodic averaged definition inthe sense that it has the same arithmetical operation but it isnot equivalent. The instantaneous reactive current is definedas:

iqðtÞ ¼ iðtÞ� ipðtÞ ð34Þ

The significance of (34) [21], is repeated here. The reactivepower is defined as:

qðtÞ�� ��¼ vðtÞ

�� ��U iqðtÞ�� �� ð35Þ

In the three-phase case, the definition in [α,β,0] space is of avector product as in (21), but that is not iq(t) as in (35). Themajority of the scientific community regards reactive power as avector. There is no violation of that assumption here. The followingrelations hold:

iðtÞ ¼ ipðtÞþ iqðtÞ ð36Þ

Orthogonality is also obtained:

ipðtÞ�� ��2 ¼ p2ðtÞ

vðtÞj j2

iðtÞ�� ��2 ¼ ipðtÞ

�� ��2þ iqðtÞ�� ��;

iqðtÞ�� ��2 ¼ q2ðtÞ

vðtÞj j2

9>>>>>=>>>>>;

ð37Þ

Y. Beck et al. / Renewable and Sustainable Energy Reviews 57 (2016) 1428–1439 1435

The proof is given by analogy to a proof from conservativetheories [25] on the orthogonality of active and reactive powers. Inthe periodic averaging theory:

S2 ¼ VTVIT�

I

P2 ¼ VT IV�

I

Q2 ¼ S2�P2

9>>>>=>>>>;

ð38Þ

From (33) and (36), for the M-phase instantaneous theory wecan derive:

ip ¼vT ivj vj2; vj j2 ¼ vTv; analogic� to : P2 ¼ VTIV

� I ð39Þ

and:

iq ¼ i� ip ¼ i�vT ivvj j2 ¼ ¼ vTv

� �i� vT iv

� �vj j2 ; analogic� to

:S2�P2

� I�1

vj j2 ð40Þ

Therefore, by analogy with (38), where it implies that Q and Pare orthogonal (ip ? iq).

Watanabe et al. [43] demonstrate orthogonality with phasevectors and not with harmonic vector components. In the specificcase of a three-phase system the total current can be calculatedfrom the orthogonality of phase voltages:

iðtÞ�� ��2 ¼ iaðtÞ

�� ��2þ ibðtÞ�� ��2þ icðtÞ

�� ��2 ð41Þ

Eq. (41) is acceptable for special cases: balanced load, fromDe-Moivre's theorem, or star (Y) connection for multi-phasebalanced as well as unbalanced load from KCL. The definitionsdeveloped here [21] converge to the single-phase classicaldefinition. In the three-phase case, they converge to the ori-ginal PQT development [15]. They converge to the two-phasedefinition of phases α, β as defined by Clarke transformation. In[21] there is no explicit multi-phase mathematical formalismper phase M. it does provide explicit vector formalism and alsoreduced it to single phase, two-phase and three-phase the-ories. In [22] explicit variables appear, but for the three-phasecase only. Next a complementary development is presented.From (33):

ip;nðtÞ ¼vT ðtÞ i!ðtÞ

vðtÞ�� ��2 vnðtÞ ¼

vnðtÞPnv2nðtÞ

p!ðtÞ ¼ p!ðtÞPnv2nðtÞ

vnðtÞ ð42Þ

Since reactive power was defined in (35), it can be shown,based on analogy of (40), to (38) that distortion power is a sum ofcross-product phase voltage pairs. The harmonic vectors arereplaced with instantaneous phase vectors. According to (34), (36)and (40):

iqðtÞ ¼ iðtÞ�vT ivvj j2 ¼ vvT i�vT iv

vj j2 ð43Þ

Relating the reactive current to the reactive power:

s2=i¼ vvT i

p2=i¼ vT iv

iqðtÞiðtÞ ¼ s2ðtÞ�p2ðtÞvj j2 ¼ q2ðtÞ

vj j2

9>>>=>>>; ð44Þ

In order to comprehend the first equation of (44), it is neces-sary either to compare physical units or follow (38), originallydeveloped for periodic averaged theories. We have used aninstantaneous apparent power definition and equivalence to (38).Then, by using the identity to the last expression in (48) we

obtain:

iqðtÞiðtÞ ¼PN

j ¼ 1

PN

k ¼ jþ 1vnðtÞikðtÞ�vkðtÞinðtÞ½ �2PN

n ¼ 1v2nðtÞ

iq;ðn;kÞðtÞ ¼ 1iðtÞU

vnðtÞikðtÞ½ �2PN

n ¼ 1v2nðtÞ

¼ q2 ðn;kÞðtÞiðtÞ

PN

n ¼ 1v2nðtÞ

8>>><>>>: ð45Þ

The result of (45) is new here and show the relation betweenthe total system reactive power and the cross-phase decomposedpower elements:

iqðtÞ ¼PN

j ¼ 1PN

k ¼ jþ1 iq;ðk;nÞðtÞ�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiiq;ðn;nÞðtÞ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiiq;ðk;kÞðtÞ

qþ iq;ðk;nÞðtÞ

h iiqðtÞ ¼ 1

iðtÞPN

n ¼ 1v2nðtÞ

PNj ¼ 1

PNk ¼ jþ1 qðn;kÞðtÞ�qðk;nÞðtÞ

h i28>><>>:

ð46ÞResults (43) for the nth phase active current component and

results (45) for the nth, kth cross-phase reactive current compo-nent, and the nth phase reactive current component establishedequations similar to those of conservative theory, except that theywere instantaneous expressions. The analogy step that was per-formed is to the harmonic vector, while here it is a phase vector.This makes them energetically un-equivalent. The results are notsurprising, since it is known that the theory was developed on thebasic representation in (22), but in the time domain instead of thefrequency domain. Eqs. (42) and (45) are multi-phase explicitequations for the nth and kth cross-phase current component. Avery close result for multi-phase PQT is found in [22]. Moreinformation on three-phase three and four wires can be found in[38,39]. All these papers use the generalized result (43) to extractthree-phase, three-wire and four-wire special cases. The reactive(n, k) component represents a flow of energy between non-cross-product phases of voltage and current. Emanuel [44] shows, in apaper on PVT, that this power flow is owed to the interaction of theradiation of the electro-magnetic field from one phase with thecurrent and voltage of another phase. It’s net contribution overone period is zero. Eqs. (43) and (45) set PQT in a mathematicalframework which enables comparison with the periodic averagedtheories and with PVT. This result when reduced to the two-phasecase resembles (26):

ipðtÞ ¼ iαpðtÞþ iβpðtÞiqðtÞ ¼ iαqðtÞþ iβqðtÞ

9=; ð47Þ

The decomposition is identical to PQT active current (26):

iαpðtÞ ¼ vαðtÞv2α ðtÞþv2β ðtÞ

pðtÞ;

iβpðtÞ ¼ vβ ðtÞv2α ðtÞþv2β ðtÞ

pðtÞ

8><>: ð48Þ

and the original PQT reactive current and power from:

iq2ðtÞ ¼ vαðtÞiβ ðtÞ�vβðtÞiαðtÞð Þ2

v2α ðtÞþv2β ðtÞ¼

¼ q2ðtÞv2α ðtÞþv2β ðtÞ

¼ q2ðtÞ

v!��� ���2

qðtÞ ¼ vαðtÞiβðtÞ�vβðtÞiαðtÞ

8>>>>>><>>>>>>:

ð49Þ

The work presented in [22] is similar and is summarized here:

p¼ u!U i!¼ uT i; q¼ u!� i

!¼ u!� i!

pþ i!

q

� ; s¼ u!

��� ��� i!

ip ¼ p u!u!U u!

¼ p u!

u!��� ���2 ; iq ¼ q!U u!

u!U u!¼ q!U u!

u!��� ���2 ¼ qTu

uuT ¼u!� i

!� U u!

u!��� ���2

8>>>>><>>>>>:

ð50Þ

Y. Beck et al. / Renewable and Sustainable Energy Reviews 57 (2016) 1428–14391436

Applying (50) to multi-phase vectors, for any phase, M, yieldsidentical results to (42) and (46) which are driven here fromalgebraic manipulation of (32)–(37), and were developed in [3]. In[22] the explicit development is implemented for three phases. Daiet al. [22] suggest that its novelty compared with [21] is explicitreactive power definition, whereas Willems [10] provides anidentical reactive current definition. The two papers provideidentical results. The next discussion is whether the multi-phase

enhancement of instantaneous reactive power as a u!� i!

vectorproduct is physically meaningful. Initially, physical meaningfulnessis explained as an analogous harmonic vector cross-product in (50)which is beyond the requirement to design, a formalism whereactive and reactive power and current components are orthogonal.That requirement is the first justification for the usage of ortho-gonal power components: P and Q. The second reasoning is asfollows: if a simple product operator v(t) � i(t) is performed,expanding the variables to their Fourier series as in [23] indicatesfor (38), and then taking the real and imaginary parts of only the(n,n) product pairs, the active (P¼ RefPnVnIng) and reactive(Q ¼ ImfPnVnIng) power components arise. The reactive distor-tion component emerges as the harmonic vector of cross-harmonic product pairs (n,k) where nak (P¼ RefPn;mVnImg)which is a vector product Vn½ � � In½ � of the voltage and currentvectors of harmonics. A simple multiplication yields active, reac-tive and distortion power.

According to [9] in the instantaneous reactive power, is thephase cross-product of non-active power fluctuations through theelectric and magnetic field radiation from one phase to theanother phase. Fig. 4 shows a vector scheme of a multi-phasesystem. These fluctuations contribute over a single period a zeronet power, and do not reflect oscillations between the source andthe load. Thus, usage of multi-phase instantaneous vector productformalism is correct for phases. It is not physically meaningful forload to source oscillations, but it is correct for phase to phaseradiation oscillations.

Furthermore (50) appears as a legitimate multi-phasedescription. This description is no worse than describing a multi-phase system only as a vector of independent phase behaviors

½Q!1;Q!

2;Q!

3�.

3.5. Difference between PQT and periodic averaged time-dependenttheories

In [23] it is demonstrated that there is a time-dependentreactive power operation equivalent to the periodic averagedreactive power in terms of energy balance. That operation isapplicable in multi-vector PVT, for example.

qðtÞ ¼ vðtÞdiðtÞdt

� iðtÞdvðtÞdt

ð51Þ

PVT by itself is an instantaneous theory. Multi-vector PVT isperiodic averaged since Fourier transform is applied. The

Fig. 4. multi-phase instantaneous system.

equivalent periodic averaged reactive power operation is, [25]:

Q!

Total ¼ Vn½ � � In½ � ð52Þ

Eq. (52) is derived from the Buchholz apparent power defini-tion in (8) as shown in [26]. The instantaneous reactive powerdefinition (21) is not equivalent to periodic averaged theories. PQTis thus not equivalent to the periodic averaged theories. This dif-ference does not rule it out and it is successful in active filtering.Buchholz apparent power definition is a formal definition withouta relation to a physical parameter. However, the definitions of (21)and (52) have the same formalism. The first in the time domain forthe multiphase case and the second relates to the frequencydomain with referring to each phase seperatly.

Opinions in the literature regarding the usage of instantaneousreactive power and instantaneous power theory are split. Somesay that power theory does not reflect well the active and reactiveload characteristics [21]. As a result, for applications requiring truereflection of load active and reactive characteristics, someresearchers do not use instantaneous theory [6]. Counter-opinionstates that the modern electronics with switching circuits imple-mented in converters require real-time active compensation,which instantaneous reactive power enables [3]. Overall, PQT doescompensate for non-linear waveforms successfully. It appears thatboth periodic and instantaneous theories groups are relevant.

4. PQT and multi-phase theory: areas of strength comparedwith other theories

There are several reasons why PQT is considered a dominantimplementation theory:

1. Cheaper and simpler active filter implementation: PQT providesthe option of implementing active filters, with energy storingdevices [3] such as capacitors, inductors, and current sources.PQT also enables implementation of active filters withoutenergy storage devices [15], using only switching circuits.Periodic averaged theories enable implementation only withenergy storage devices. Implementation without energy storageis cheaper and easier to maintain. That is a major factor in PQT'ssuccess.

2. Ability of instantaneous response to transient power conditioning:current switches in regulators and inverters insert transients tothe grid. It is hard to handle non-periodic waveforms withperiodic averaged theories for compensation with responsetime smaller than a single cycle. Periodic averaged theoriesare therefore referred to as quasi-periodic [41].

3. Black box load knowledge: there is no need to know loadstructure in order to break it down to the current/power phy-sical components. That property is shared by periodic averagedtheories such as CPCT, and multi-vector PVT [9].

4. Offers theoretical simplicity and sophistication: early PQT was asimple theory. Modern multi-phase PQT [3,22] and modernthree-phase unbalanced theory [3,39], offer a sophisticationcomparable to the rest of modern theories such as multi-vector PVT, CPCT, and reactive vector space decomposition.Simplicity is an advantage in terms of both comprehensionand implementation. Sophistication is important for modern-day applications.

5. Computation complexity: PQT is a time domain. It requires Ndigital samples per cycle per phase, and O(N) computationcomplexity. A periodic averaged theory operating at the fre-quency domain requires N∙logN operations to compute FFT,times four wire phases, times voltage and current separately.Czarnecki [7] suggests a look-up table to reduce computation

Table 2A functional comparison of instantaneous theories to periodic averaged ones.

Feature Instantaneous Periodic averaged

Active/reactive load characterization (according to pureresistive/reactive load test)

Less recommended by some scientists because of apparentmisinformation

Applicable

Energy measurement as supported by standards (IEEE-1459, DIN 40110)

Not recommended due to active/reactive load characterization Applicable

Equivalent active energy measurement Applicable and Equal to periodic averaged Applicable and Equal to instantaneousGrid monitoring and control, power conditioning Applicable ApplicableActive filtering applicable ApplicableResponse time As small as sample rate Slower than or equal to half a periodTransient analysis/recognition Applicable Applicable but less than InstantaneousRequires FFT not required RequiredActive filter required/not required, storage components Not necessarily require RequiresInitial formulation (basis for the entire theory) qðtÞ ¼ vðtÞ � iðtÞ ) s2ðtÞ ¼ p2ðtÞþq2ðtÞ Q ¼ ½Vn� � ½In� ) S2 ¼ P2þQ2

Y. Beck et al. / Renewable and Sustainable Energy Reviews 57 (2016) 1428–1439 1437

complexity to O(N). With current computation technology,performing FFT is not such an issue any more [45]. Thecomputational issue was a key factor historically in terms ofaccelerating PQT development as compared with CPCT [46].

For all these reasons PQT has been more dominant than othertheories over the past three decades. However, this review of pastpapers suggests that the last word has not yet been spoken.

5. Conclusion

In this paper a review of apparent power definitions andinstantaneous electric energy transport group of theories arepresented. The review is believed to cover most of today'sinstantaneous theories while other review papers on the subjectfocus on a partial group or different aspects of the theories. In theFirst part of the paper several different apparent power definitionswere presented: Buchholz, arithmetic, geometric and vectorpower (which is similar to geometric). The theories are fist shownto yield identical results in the linear balanced three phase system.It is shown that each definition results in different terms whennonlinear and unbalanced loads are concerned. In order to con-tribute to the knowledge in this field, this paper attempts to pointout similarities of terms as well as total inequalities between thevarious definitions. Some insights into a possible physical notationwere presented, although there is no proven physical meaning toany of the definitions. The Appendix shows expressions whichsupport that a single-phase geometric apparent power is equiva-lent to the Buchholz apparent power in the sense that the latter issimply the RMS value of the former. For a three-phase instanta-neous reactive power vector, it is shown that Buchholz apparentpower is bounded in a finite region of magnitude in relation to thegeometric apparent power and therefore for design purposes itcan be used though the terms are not equal. The problem of loadcharacteristic active/reactive identification was discussed in termsof vector power for three phases, and its physical source isexplained. The question of which definition to use was also han-dled by emphasizing the physical point of view of each definition,although apparent power is generally accepted as not being aphysical parameter.

The basic instantaneous theory PQT is presented. Then,enhancement to include zero phase sequence in order to gen-eralize the theory in the cases of three-phase four-wire, three-phase three-wire, is also described. Then, a multi-phase instanta-neous theory was studied. By using explicit formalism fromexisting papers, power and current physical components werewritten for the nth and kth cross-phase. Usually multi-phaseformalism is developed for three-phase systems, but here the nth

phase formulation was provided to enable cases such as multi-load

and multi-source usage. This presentation is not new, but theformalism in this paper offers a new perspective on multi-phaseusage, and also sheds some light on how orthogonality of currentcomponents and non-active power are obtained. A comparativestudy is presented, showing two distinct differences betweeninstantaneous theories and all other theories: (1) there is un-equivalence of calculating conduction losses of active power. (2)un-equivalence of energy formulas. It is shown that the study ofinstantaneous theory enables novel grid implementations such as:(a) Active grid filtering and power conditioning, (b) Grid powercomposition and decomposition of sub-branches into a mainbranch and vice versa (as shown in (46)). The use of instantaneousreactive current as grid indicator for inter-phase energy flow isdemonstrated as well. A comparative of instantaneous and peri-odic averaged theories as they were presented here is finallysummarized in Table 2.

Acknowledgment

This research was supported in part by the ISG (Israeli SmartGrid) Consortium, administrated by the Office of the Chief Scientistof the Israeli Ministry of Industry, Trade and Labor.

Appendix. Relations between Buchholz and geometric power

In the Single phase case the Buchholz apparent power defini-tion and the geometric apparent power definition or vector defi-nitions are equivalent. [9] discusses multi-vector PVT anddemonstrates periodic averaging instantaneous variables andexecutes a Fourier transform. The distortion power operatorobtained is:

DðtÞ ¼X

jak;linear

VjIkejϕk �VkIje

jϕj

� ej αj �αkð Þejðωj �ωkÞtσjk

þX

jak;non� linear

VjIkej αj �αkð Þejðωj �ωkÞtσjk ð53Þ

Taking the RMS value of (53) yields:

D2 ¼Z

DUD�dt ¼Xjak

VjIkejϕk �VkIje

jϕj

� 2

8<:

9=;

1=2

ð54Þ

Eq. (54) is the single-phase Buchholz apparent powerdefinition.

In the multi-phase case the suggested multi-phase apparentpower S based on geometric apparent power orientation yields a

Y. Beck et al. / Renewable and Sustainable Energy Reviews 57 (2016) 1428–14391438

Buchholz apparent power as order of magnitude:

SStarðtÞ ¼PLi

vLi ðtÞiLi ðtÞ ; or SΔ ¼XLi

vLi ;Lj ðtÞiLi ;Lj ðtÞ

‖S‖2 ¼Z T

0SðtÞS�ðtÞdt ð55Þ

Developing the apparent power operator further yields:

‖S‖2 ¼XLi

Z T

0vLi ðtÞiLi ðtÞ�� ��2dtþ X

Li ;Ljðia jÞ

Z T

ovLi ðtÞiLi ðtÞ� �

vLj ðtÞiLj ðtÞ� �

dt ¼XLi

Z T

0vLi ðtÞiLi ðtÞ�� ��2dtþ X

Li ;Ljðia jÞ

Z T

ovLi ðtÞv�Lj ðtÞ

� iLi ðtÞi�Lj ðtÞ

� dt

ð56ÞThere is no constraint regarding the magnitude relationships

between VL1

�� ��; VL2

�� �� IL1�� ��; IL1�� �� and therefore only order of mag-

nitude equivalence is obtained:

‖S‖2CPLi

VLi ILi� �2þ P

Li ;Ljðia jÞVLi ILj

� 2¼P

Li

V2Li

PLi

I2Li ¼ V2RMSI

2RMS ¼ S2Buchholz

‖S‖¼ ‖Sgeometric‖CSBuchholz

8><>:

ð57ÞThe upper and lower bounds of the difference between the

geometric and the Buchholz apparent powers are as follows: TheBuchholz power which is the upper bound, because it “assumes”all harmonics are simultaneously maximized, take values accord-ing to the following equation:

S¼ VRMSIRMS

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þy2þz2

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þu2þw2

p

where : f��y; z;u;wjr1g ð58Þ

where y, z are the normalized RMS S, T phase voltages as comparedwith the R phase with a value of 1 p.u. Also, u and w are thenormalized S, T phase currents. The region of existence is boundedin a product of two balls and is bounded by:

1rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þy2þz2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þu2þw2

pr3 ð59Þ

A ratio of 3 is not undermined but it is bounded.

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Yuval Beck was born in Tel Aviv, Israel, in 1969. He received a B.Sc degree inelectronics and electrical engineering in 1996, an M.Sc. degree in 2001, and a Ph.D.degree on the subject of ground currents caused by lightning strokes in 2007, allfrom Tel Aviv University. Since 1998, he has been with the InterdisciplinaryDepartment, the Faculty of Engineering, Tel Aviv University. In 2008 he joined HIT-Holon Institute of Technology, Holon, Israel, as a lecturer and since 2010 has actedas the head of the Energy and Power Systems Department in the Faculty of Engi-neering. His research interests include smart grid technologies, lightning dischargephenomena, lightning protection systems, power electronics, and photovoltaicsystems.

Netzah Calamaro was born in Tel Aviv, Israel in 1967. He received a B.Sc. in elec-trical and electronic engineering from the Technion Israel Institute of Technology in1990 and an M.Sc. in electrical engineering in 1998 from the Technion. From 1995to 2005 he worked at Intel RnD on chip design. From 2005 to 2008 he worked ondeveloping quality monitoring equipment, real-time power factor correction, and

SCADA systems. From 2005 until today, Netzah has worked at the MeteringDevelopment Laboratory, Israel Electric Company as a systems engineer and deputychief metrology officer. He is also conducting research at Tel Aviv University'sEnergy Conversion Lab. His research interests are control algorithms for activefilters, grid diagnostics, and load identification. More precisely, he works on electricpower transport theories' application to non-linear, time-variant loads, identifica-tion of multiple loads, and energy management. He is a member of the SmartMeters Standard Committee of the Israel Standard Institute and was involved inapproving Israel's energy metering standards.

Doron Shmilovitz was born in Romania in 1963. He received a B.Sc., M.Sc., and Ph.D. from Tel Aviv University in 1986, 1993, and 1997, respectively, all in electricalengineering. From 1986 to 1990, he worked in R&D for the IAF where he developedprogrammable electronic loads. From 1997 to 1999, he was a post-doctorate fellowat New York Polytechnic University, Brooklyn. Since 2000, he has been with theFaculty of Engineering, Tel Aviv University, where he has established a state-of-the-art power electronics and power quality research laboratory. His research interestsinclude switched-mode converters, topology, dynamics and control, power quality,and special applications of power electronics such as those for alternative energysources and powering sensor networks, implanted medical devices, and generalcircuit theory.