Invited paper

Paolo TENTI1, Paolo MATTAVELLI1, Helmo K. MORALES PAREDES2 University of Padova, Italy (1), University of Campinas, Brazil (2)

Conservative Power Theory, Sequence Components and Accountability in Smart Grids Abstract. Smart grids offer a new challenging domain for power theories and compensation techniques, because they include a variety of intermittent power sources which can have dynamic impact on power flow, voltage regulation, and distribution losses. When operating in the islanded mode, smart micro-grids can also exhibit considerable variation of amplitude and frequency of the voltage supplied to the loads, thus affecting power quality and network stability. Due to the limited power capability of smart micro-grids, voltage distortion can also get worse, affecting measurement accuracy and possibly causing tripping of protections. In such a context, a reconsideration of power theories is required, since they form the basis for supply and load characterization and accountability. A revision of control techniques for harmonic and reactive compensators is also required, because they operate in a strongly interconnected environment and must perform cooperatively to face system dynamics, ensure power quality and limit distribution losses. This paper shows that the Conservative Power Theory (CPT) provides a suitable background to cope with smart grids characterization needs, and a platform for the development of cooperative control techniques for distributed switching power processors and static reactive compensators. Streszczenie. Sieci inteligentne (Smart Grids) są nowym wyzwaniem dla teorii mocy i kompensacji, gdyż sieci takie mają różnorodne źródła energii, mogące mieć dynamiczny wpływ na przepływ energii i jej straty oraz na zmienność napięcia. W sytuacji pracy izolowanej ineligentne mikro-sieci mogą zasilać odbiorniki napięciem o znaczącej zmienności amplitudy i częstotliwości, odziaływując na jakość energii oraz stabilność. Ograniczona moc mikro-sieci może pogłębiać odkształcenie napięcia, odziaływująć na dokładność pomiarów i wadliwe działanie zabezpieczeń. W takiej sytuacji niezbędna jest rewizja teorii mocy, gdyż teoria ta tworzy podstawy opisu odbiorników i rozliczeń energetycznych. Niezbędna jest także rewizja metod sterowania kompensatorów harmonicznych i mocy biernej, gdyż kompensatory takie działając w silnie odziaływującym na siebie środowisku, muszą współpracować zgodnie z dynamiką systemu tak, aby zapewnić jakość energii oraz ograniczać jej straty. Niniejszy artykuł pokazuje, że Konserwatywna Teoria Mocy (CPT) tworzy podstawy teoretyczne dla rozwiązywania różnych zagadnień w sieciach inteligentnych oraz platformę dla rozwoju metod sterowania rozłożonych przekształtników i kompensatorów mocy biernej. (Zachowawcza teoria mocy, składowe symetryczne i rozliczenia energetyczne w sieciach inteligentnych)

Keywords: Smart grids; Power Theories; Non-sinusoidal operation; Sequence components; Electrical accountability. Słowa kluczowe: Sieci inteligentne, teoria mocy, systemy niesinusoidalne, składowe symetryczne, rozliczenia energetyczne.

Introduction Smart grids represent one of the grand challenges at planetary level. The infusion of information technology throughout the electric grid creates new capabilities, with impact on environment, science and technology, economics and lifestyle. The term “smart grid” outlines the evolution of electrical grids and a change of paradigm in the electric market organization and management. In a global perspective, implementation of smart grids on a large scale will result in dramatic improvement of electrical services and considerable market increase. Technically speaking, smart grids include a number of distributed energy resources and electronic power processors, which must be fully exploited to reduce power consumption from the utility, improve power quality and increase distribution efficiency. The smart-grid paradigm is therefore different from the traditional one, based on the assumption of few power sources of large capacity and sinusoidal supply. In smart grids, energy sources can be small, distributed and interacting, and supply voltages can be asymmetrical and distorted. From the above considerations it follows that facing the problems of smart grids requires a revision of traditional power theories and a comprehensive approach to cooperative operation of distributed electronic power processors. This paper discusses a revision of the Conservative Power Theory (CPT) oriented to smart grids. In particular, the influence of frequency variation and voltage distortion are analyzed, and an accountability approach is presented, which allows separation of load and supply responsibility for reactive power, asymmetry and distortion and can set a basis for proper accounting of electrical supply. Conservative power terms under periodic nonsinusoidal operation The problem of defining power and current terms under non-sinusoidal conditions dates back some eighty years [1]. While definition of active power and current terms was

30

developed in early 30’s [2], definition of power and current terms related to “reactive” and “harmonic” phenomena [210] is still under discussion and different solutions are proposed depending on application field [11-39]. The Conservative Power Theory (CPT), presented in [9,33,38], provides a possible background to approach the problem. However, it was developed under the assumption of constant operation frequency, which is not generally true in smart grids and, in particular, in smart micro-grids. Moreover, the effects of load unbalance and voltage asymmetry were not explicitly represented and quantified in the first stage [38]. A revised version of the CPT is presented here, which holds for whatever periodic operating condition. A. Unbiased integral and derivative of periodic variables and their properties For periodic quantities (period T, frequency f = 1/T, angular frequency ω=2πf) we define the operators: Average value (1.a)

x= x =

1 T

∫

T

0

x dt

Time derivative (1.b)

( dx x= dt

Time integral (1.c)

x ∫ = ∫ x(τ ) dτ t

0

Unbiased time integral (1.d)

) x = x∫ − x∫

Internal product (1.e)

x, y =

1 T

∫

T

0

x ⋅ y dt

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Norm

X = x =

(1.f)

To characterize the network utilization at the given port we define:

x, x

Power factor

Orthogonality (1.g)

(6.b)

x, y = 0

For vector quantities

x and y of size N we define:

Scalar product

Resistor R

n =1

x = xo x =

(2.b)

N

∑x n =1

Inductor L

2 n

Capacitor C

N

x, y = x o y = ∑ x n , y n

(7.c)

n =1

Norm

X= x =

N

∑ n =1

xn , x n =

N

∑X n =1

2 n

The vector norm is also called collective rms value. The above quantities have the following properties: ( ) (3.a) x, x = 0 x, x = 0

) ) x, y = − x , y ( ) ) ( x, y = − x , y = − x , y

( ( x, y = − x , y

(3.c)

B. Conservative power and energy terms Considering network Π with L branches, a set of voltages {ul }lL=1 and currents {il }lL=1 is said to be consistent

with the network if they satisfy the Kirchoff’s Law for Voltages (KLV) and Currents (KLC), respectively. It is easy to show that if branch voltages ul are consistent with the network, the same happens for quantities u(l and u)l .

(

Similarly for branch currents il and related quantities il and ) il . According to the Tellegen’s Theorem we can therefore affirm that every scalar product of KLV-consistent terms ( ) ( ) ul , ul , ul and KLC-consistent terms il , il , il is a conservative quantity. In the following, we will make reference to some conservative quantities which play a prime role in the Conservative Power Theory. In the general case of a polyphase network, let u and i be the voltage and current vectors at a generic port, we define: Instantaneous power (4.a)

p = uoi

Instantaneous reactive energy

)

(4.b) w = uoi We also refer to the corresponding average values: Active power (5.a)

P = p = u, i

Apparent power (6.a)

C. Selection of voltage reference In poly-phase networks, power and energy terms defined by (4) and (5) do not depend on the voltage reference. On the contrary, apparent power (6.a) is affected by this choice. In our approach, irrespective of neutral wire, we select a voltage reference which provides unity power factor for balanced resistive load. This gives a physical meaning to the apparent power and allows load characterization by means of the power factor as a global quality index. •

In absence of neutral wire, we assume the virtual center of the voltages as the voltage reference. Thus:

A=U I

N

∑u

(8.a)

n =1

n

=0

It is easy to show that this solution minimizes the voltage norm. Moreover, in case of resistive balanced load we get unity power factor, which is consistent with the physical meaning of the apparent power. The phase voltages can easily be computed from lineto-line voltage measurement, according to the relation: (8.b)

un =

1 N

N

∑ j =1

j≠n

un j

Also the amplitude (and consequently the norm) of the voltage vector can easily be evaluated, considering that: (8.c)

2

u =

1 N N ∑∑ 2 N n=1 j =1

j≠n

u n2 j

As concerns the phase voltage amplitude, we have:

) W = w = u, i

Another relevant, not conservative, power term is:

( iC = C uC ⇒ PC = 0, WC = −C U C2

For inductors and capacitors, irrespective of voltage and current waveforms, reactive energy W is therefore proportional to the average value of stored energy. In a passive network, due to conservation property, the total active [reactive] power absorption is simply obtained by adding the power [energy] consumption of each component.

(8.d)

Reactive energy (5.b)

( u L = L iL ⇒ PL = 0, WL = L I L2

(7.b)

Internal product (2.c)

u R = R iR ⇒ PR = U R2 R , WR = 0

(7.a)

Vector magnitude

(3.b)

For basic passive components, the application of definitions (5.a) and (5.b), considering properties (3), gives:

N

x o y = ∑ xn y n

(2.a)

(2.d)

P A

λ=

•

u n2 =

1 N

⎛ N ⎜∑ ⎜ ⎝ j =1

j ≠n

2⎞ u n2 j − u ⎟⎟ , n = 1 ÷ N ⎠

In presence of neutral wire, the neutral voltage u0 is taken as reference. In this case the phase voltages and currents do not sum zero, thus we define the zerosequence terms (see sequence component section) as:

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31

(9.a)

uz =

1 N

N

∑u n =1

and i z =

n

1 N

N

∑i n =1

n

=−

io N

where i0 is the neutral current. Since u0=0, the neutral wire does not contribute to power and energy terms. On the other hand, the zero sequence components of voltages and currents are accounted in the phase quantities and increase the collective rms values, and thus affect the apparent power too. The collective rms values of the phase voltages and currents are defined by: (9.b)

N

∑U

U=

n =1

N

∑I

I=

2 n

n =1

2 n

According to these definitions, the power factor is unity in case of resistive balanced load. Note that definitions (9.b) are formally identical to those used for N-phase systems without neutral wire and keep the same properties. In the following discussion, unless differently specified, we will therefore refer to phase quantities only and the results will be applicable irrespective of the neutral wire. Current terms under periodic non-sinusoidal operation A. Basic current terms In the general case of an N-phase network, let u and i be the phase voltage and current vectors at a generic port, we define the following basic current terms. The neutral current is considered apart, and is not affecting power and energy terms. • Active currents are the minimum phase currents (i.e., with minimum rms value) needed to convey active power terms Pn absorbed at the network port. They are given by: (10.a)

i a = {ia n }n=1 , ia n = N

u n , in 2

un

un =

Pn u n = Gn u n U n2

Gn is the equivalent phase conductance. Note that: N

n =1

n =1

u , i a = ∑ u n , ia n = ∑ Pn = P

(10.b)

(10.c)

N

N

∑I

Ia = ia =

n =1

2 an

=

⎛ Pn ⎜⎜ ∑ n =1 ⎝ U n N

Void currents are the remaining current terms:

iv = i − ia − ir

(12.a)

The neutral void current, if any, is determined by: N

iv o = −∑ iv n

(12.b)

n =1

As shown in [33], void currents can be split into scattered currents (due to different values of the equivalent phase conductance and susceptance at different harmonics) and generated currents (harmonics that do not exist in the voltage spectrum). In fact, such splitting is merely theoretical and can be severely affected even by small errors in the computation of voltage harmonics. For this reason we will consider the void current as a whole. •

Orthogonality. All the above phase current terms are orthogonal, thus:

I 2 = I a2 + I r2 + I v2

(13)

In the following, we further decompose active and reactive currents into balanced and unbalanced components. B. Balanced current terms • Balanced active currents are the minimum port currents (i.e., with minimum collective rms value) needed to convey total active power P absorbed at the network port. They are given by: (14.a)

ia = b

N

ia o = −∑ ia n

(11.a)

) u n , in ) Wn ) ) i r = {ir n }n=1 , ir n = ) 2 u n = ) 2 u n = Bn u n Un un N

Bn is the equivalent phase susceptance. Note that: (11.b)

N N ) ) u , i r = ∑ u n , ir n = ∑ Wn = W n =1

(11.c)

Ir = ir =

N

∑I n =1

2 rn

=

(14.b)

u, i a = P b

P U

(14.c)

I ab =

The balanced determined by:

neutral

2

) ⎛ Wn ⎞ ⎟⎟ ⇒ U I r ≠ W n =1 ⎝ n ⎠ N

∑ ⎜⎜ U) N

ir o = − ∑ ir n

⇒ U I ab = P active

current,

if

any,

is

N

iabo = −∑ iabn

(14.d)

n =1

•

Balanced reactive currents are the minimum port currents needed to convey total reactive energy W absorbed at the network port. They are given by:

) u, i ) W ) ) b i r = ) 2 u = ) 2 u = Bb u U u

Term Bb is the equivalent balanced susceptance and holds for all phases. Note that: ) b (15.b) u, ir = W

W I rb = ) U

(15.c)

) ⇒ U I rb = W

The balanced neutral reactive current, if any, is determined by:

n =1

The neutral reactive current, if any, is determined by: (11.d)

u

P u = Gb u U2

Term G is the equivalent balanced conductance and holds for all phases. Note that:

(15.a)

Reactive currents are the minimum phase currents needed to convey reactive energy terms Wn absorbed at the network port. They are given by:

u=

2

b

n =1

•

u, i

2

⎞ ⎟⎟ ⇒ U I a ≠ P ⎠

The neutral active current, if any, is determined by: (10.d)

•

N

irbo = −∑ irbn

(15.d)

n =1

C. Unbalanced current terms • Unbalanced active currents are given by: (16.a)

ia = ia − ia u

b

⇒

{i }

u N an n =1

{(

) }

= Gn − G b u n

N

n =1

n =1

32

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Clearly, these currents exist only if the equivalent phase conductances differ from each other. Note that:

U = U 2f + U h2 = U f 1 + (THD (u ))

(16.b)

) ) ) ) ) 2 U = U 2f + U h2 = U f 1 + (THD (u ))

u, i a = 0 ⇒

i a ,i a = 0

u

(16.c)

I = I −I

u

b

2

⎛ P ⎞ ⎛P⎞ = ∑ ⎜⎜ n ⎟⎟ − ⎜ ⎟ ⎝U ⎠ n =1 ⎝ U n ⎠ N

b2 a

2 a

u a

where

2

N

u iao = −∑ iaun

Unbalanced reactive currents are given by:

ir = ir − ir

(17.a)

u

{i }

u N r n n =1

⇒

b

=

{(B

) }

) − B b un

n

N

n =1

Clearly, these currents exist only if the equivalent phase susceptances differ from each other. Note that:

) u u, i r = 0 ⇒

(17.b)

b

2

⎛ Wn ⎞ ⎛ W ⎞ ⎜⎜ ) ⎟⎟ − ⎜ ) ⎟ ∑ ⎝U ⎠ n =1 ⎝ U n ⎠ N

2

(17.c)

ir ,ir = 0 u

I ru = I r2 − I rb =

2

The unbalanced neutral reactive current, if any, is determined by: N

irou = −∑ irun

(17.d)

n =1

D. Complete current decomposition In conclusion, the phase currents can be split as:

i = ia + ir + ia + ir + iv b

(18.a)

b

u

I=

2

2

2

2

I ab + I rb + I au + I ru + I v2

Power terms under periodic non-sinusoidal operation From (6.a) and (18.b) we decompose the apparent power as: (19)

b2 a

b2 r

u2 a

u2 r

A =U I +U I +U I +U I +U I = 123 123 123 123 123 2

2

2

Q2

P2

2

2

N r2

N a2

2 v

V2

= P + Q + N + N +V = P + Q + N 2 +V 2 1 424 3 2

2

2 a

2 r

N

2

2

total

harmonic

distortion.

Since

1 + (THD (u )) U Q = ) W = ωW ) 2 U 1 + (THD (u )) 2

Sequence Components The above definitions offer a clear description of the physical meaning of the various current and power terms. In practice, however, they do not allow an easy separation of supply and load responsibility on the generation of reactive power, unbalance and distortion. This is particularly true in smart micro-grids, where line frequency can vary considerably and voltage distortion can be non negligible. The problems related to load characterization and power metering must therefore be addressed carefully, to avoid improper penalization of customers. For this purpose, in case of 3-phase networks, the CPT can be extended to sequence components, which allow more precise analysis on supply and load accountability. According to [39], the sequence components of a generic triplet of periodic phase quantities x1, x2, x3 (voltages or currents) can be expressed as: • Positive sequence (21.a)

Each current component has a precise physical meaning also under non-sinusoidal operation. Moreover, active and reactive currents refer to power and energy terms which are conservative in every network and keep their meaning also in presence of distortion, voltage asymmetry, and load unbalance.

2

(20.b)

u

Since all terms are orthogonal, we have: (18.b)

means

Equation (20.b) shows that, unlike reactive energy W, reactive power Q is not conservative. In fact, it depends on line frequency and (local) voltage distortion.

n=1

•

THD

) U f U f = ω , we have:

The unbalanced neutral active current, if any, is determined by: (16.d)

2

•

Negative sequence

(21.b)

•

x1n (t ) =

x1 (t ) + x2 (t − T 3) + x3 (t − 2 T 3) 3

x2n (t ) = x1n (t + T 3), x3n (t ) = x1n (t + 2 T 3)

Zero sequence

(21.c)

x1z (t ) = x2z (t ) = x3z (t ) =

x1 (t ) + x2 (t ) + x3 (t ) 3

In presence of distortion, the above sequence components are not sufficient to recompose the original waveforms. Another term is needed, i.e.: • Residual components

2

2

x1 (t ) + x2 (t + T 3) + x3 (t + 2 T 3) 3 x2p (t ) = x1p (t − T 3), x3p (t ) = x1p (t − 2 T 3) x1p (t ) =

(21.d)

xnr (t ) =

xn (t ) + xn (t − T 3) + xn (t + T 3) , n = 1÷ 3 3

•

P is active power

•

Q is reactive power

•

Note that, unlike sequence components, residual terms are not symmetrical and must be computed separately for each phase. In conclusion, we can decompose any three-phase quantities as:

N is unbalance power (split into active and reactive components)

(21.e)

•

V is void power Looking in particular at the reactive power we have:

It is easy to demonstrate that terms of different sequence are orthogonal, thus:

Where:

(20.a)

U Q = U I rb = ) W U

(21.f)

)

If we decompose the collective rms values U and U into fundamental and harmonic components we get:

x = x p + xn + x z + xr

2

2

2

X = X p + Xn + Xz + Xr

2

If we separate the fundamental and harmonic components of voltages and currents we get:

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33

u = u f + uh

(22)

i = i f + ih

U n = U fp , n = 1 ÷ 3 ⇒ U = 3 U fp

⇒ U = U 2f + U h2 ⇒

(25)

I = I 2f + I h2

Let’s now split the fundamental quantities into sequence components (residual terms do not exist for sinusoidal variables), i.e.:

U fp 3 U fp ) ) , n =1÷ 3 ⇒ U = Un =

ω

u = u f + u f + u f + uh p

(23.a)

n

z

(26.a)

i = i f + i f + i f + ih p

n

z

2

(23.b)

p2

2

I = If +I

n2 f

2

+I

z2 f

+I

2 h

(26.b)

(24.a)

(24.b)

n

n

z

z

+ u h , i h = Pfp + Pfn + Pfz + Ph ) )p p )n n )z z W = u, i = u f , i f + u f , i f + u f , i f + ) + u h , i h = W fp + W fn + W fz + Wh

In general terms, it is very difficult develop a theory which is able to separate the contributions of source and load to distortion and unbalance terms, when only the measurements at PCC are available. There are some previous works for the separation of the load and source distortion [40-43], but even in these cases it was not possible to establish a general theoretical background which is valid in any operating conditions. The CPT and the related decompositions are a viable tool to establish an accountability approach which gives some meaningful information under practical operating conditions. If we consider a three-phase network and measure the power and energy absorption at the connection point between utility and load (Point of Common Coupling, PCC), only a portion of it is generally accountable to the load. In fact, the negative sequence, zero sequence, and harmonic contents of the supply voltage are usually under responsibility of the utility, especially if short-circuit power at PCC is much larger than the load power, and their effect should not be ascribed to the load. Note that the effects of a non-ideal voltage supply are twofold: first, the active and reactive current terms track the voltage waveform, according to (10.a) and (11.a), and are therefore affected by the same non-ideality. • second, the active and reactive power terms are augmented in presence of negative-sequence, zerosequence and harmonic voltages, according to (24); A proper accountability approach must depurate the power and current terms from the effects of voltage nonidealities, ensuring that the load is charged only for its reactive, unbalance and harmonic impact. Of course, we don’t know the currents that the load would absorb under sinusoidal and symmetrical supply. For our accountability approach we consider only the portion of the active and reactive phase currents which are related to the fundamental voltages of positive sequence. Obviously, if a model of the load is available it can be used for a more accurate evaluation. If the supply voltages were sinusoidal with positive sequence ( u pf ) we had:

34

U

⇒

p f

Pl n = u fp n , ia l n = Pn I al =

1 U fp

3

U fp

2

U n2

∑P

2 ln

n =1

ir l n

) 2 U fp )p ⇒ Wl n = u f n , ir l n = Wn ) 2 Un

) = Bn u fpn

⇒

I rl =

3

1 U fp

∑Q

2 ln

n =1

The total power terms accountable to the load and the corresponding equivalent balanced conductance and susceptance are: 3

Pl = ∑ Pl n

⇒ Glb =

n=1

(27.a)

Accontability

•

Pl n

Wl n Ql n I rln = ) p = p Uf Uf

P = u, i = u f , i f + u f , i f + u f , i f + p

⇒

Similarly, for the reactive terms we have:

Moreover: p

ia l n = Gn u fp n I aln =

Owing to orthogonality we have:

U = U pf + U nf + U zf + U 2h

ω

The portions of active current and power accountable to the load in each phase are:

3

Wl = ∑Wl n

Pl 3U fp

⇒ Blb =

n=1

2

Wl Ql ) p2 = ω 2 3U f 3U fp

The balanced current terms accountable to the load are:

i a l = Glb u f

⇒ I abl =

1 Pl p 3Uf

)p b i r l = Blb u f

⇒ I rbl =

1 Wl 1 Ql )p = p 3Uf 3Uf

b

(27.b)

p

The unbalanced current terms accountable to the load are:

( (

) )

iaul n = ia l n − iab l n = G n − Glb u fp n ) irul n = ir l n − irbl n = Bn − Blb u fp n

(28.a) Thus: (28.b)

I aul =

∑ (G

I rul =

∑ (B

3

n =1 3

n =1

)

2

2

b U fp = n − Gl

n

)

1 U fp

2 ) 2 1 − Blb U fp = p Uf

Pl2 3

3

∑ Pl2n − n =1 3

∑Q n =1

2 ln

−

Ql2 3

The void current terms must also be revised. In fact, void currents i v satisfy the condition:

u, i v = 0 ⇒ u pf , i v + u nf + u zf + u h , i v = 0 ) ) ) ) ) u , i v = 0 ⇒ u pf , i v + u nf + u zf + u h , i v = 0 This shows that void currents are not orthogonal to

)p

p

voltages u f and their integrals u f owing to the presence of non-ideal voltage terms. We can however depurate the void currents from those components which are not p

)p

orthogonal to u f and u f , thus achieving the void current terms i v l which can be accounted to the load: (29.a)

ivl = iv −

u pf , i v

3U

u pf p2 f

) u pf , i v ) − ) 2 u pf 3U fp

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In this way, we have restored the orthogonality condition: (29.b)

u pf , i v l

=0

) u pf , i v l = 0

In conclusion the currents accountable to the load are: (30.a)

i l = i a l + i r l + i v l = i ba l + i br l + i ua l + i ur l + i v l

However, the load is balanced and is not responsible for such additional power consumption. In fact, the unbalance power accounted to the load is zero. Moreover, in absence of distortion, there is no void power. a) Balanced load.

All current terms are orthogonal, thus: (30.b)

I l = I abl2 + I rbl2 + I aul2 + I rul2 + I v2

We can finally define the apparent power accountable to the load as: (31) Al = U fp I l = Pl2 + Ql2 + N a2l + N r2l + Dl2

1424 3

b) Unbalanced load.

N l2

Application Examples In order to test the accountability approach presented in the previous section, the circuits of Fig.1 and Fig.2 (threewires and four-wires) were considered. Different cases of voltage supply were tested, namely: • Case I – Symmetrical sinusoidal voltages; • Case II – Asymmetrical sinusoidal voltages; • Case III – Symmetrical non-sinusoidal voltages; • Case IV – Asymmetrical non-sinusoidal voltages. The supply voltages applied in Case I and Case II are given in table 1.

c) Distorting load.

Table 1: Voltages for case I and II Case I U1 = 127∠0º Vrms U2 = 127∠-120º Vrms U3 = 127∠120º Vrms

Figure 1: Application example for accountability test in three-phase circuit without neutral wire

Case II U1 = 127∠0º Vrms U2 = 113∠-104,4º Vrms U3 = 147,49∠144º Vrms

The voltages for cases III and IV are the same of cases I and II with the addition of 10% of 5th and 7th harmonics for rd th th th the three-wires circuit and 10% of 3 , 5 , 7 and 9 harmonics for the four-wires circuit. The line parameters are RL0 = RL1= RL2= RL3= 1mΩ and LL0 = LL1= LL2= LL3= 10 μH. Three different loading conditions were considered: • • •

Balanced load (Fig. 1a and Fig. 2a): R1=R2=R3=1Ω, L1=L2=L3=2 mH; Unbalanced load (Fig. 1b and Fig. 2b): Same load of the previous example plus phase-to-phase resistor R13= 1Ω. Distorting load (Fig. 1c): L1=L2=L3=0,25mH, R=2Ω, C=250μF

Three-phase circuits without neutral wire According to (8), only two line-to-line voltages and two phase currents need to be measured in three-wire circuits, as shown in Fig.1. Table 2, 3 and table 4 summarize, in per unit, the power terms computed according to the definitions given in (19) (power absorbed at the PCC), and the corresponding terms ascribed to the load according to the accountability approach given in (31). In all cases, the apparent power at PCC is set as the reference value (1 pu). Balanced load – Case I : All power terms coincide, for PCC and load. In fact the voltage is sinusoidal and symmetrical with positive sequence, thus the load is entirely responsible for the consumption of active and reactive power; there is no unbalance power nor void power because load is balanced and there is no distortion in voltages and currents. Balanced load – Case II : Although the power factors are identical, the power terms are not equal for PCC and load. In fact, the voltage is asymmetrical, and there are power terms associated to the negative sequence.

Table 2: Power terms (pu) for balanced load in three-wires circuit Case I Case II Case III Case IV PCC LOAD PCC LOAD PCC LOAD PCC LOAD A 1,0000 1,0000 1,0000 0,9634 1,0000 0,9840 1,0000 0,9538 P 0,7985 0,7985 0,7985 0,7693 0,7913 0,7758 0,7945 0,7556 Q 0,6020 0,6020 0,6020 0,5800 0,6023 0,5962 0,6022 0,5770 N 0,0000 0,0000 0,0000 0,0000 0,0002 0,0002 0,0056 0,0061 V 0,0000 0,0000 0,0000 0,0000 0,1054 0,1038 0,0782 0,0760 λ 0,7985 0,7985 0,7985 0,7985 0,7913 0,7885 0,7945 0,7922

Balanced load – Case III : The power factor is slightly lower than in previous cases, due to the presence of the void power, which appears due to voltage distortion and is partially accounted to the load since its impedances are frequency-dependent. The active and reactive power on the load side are lower than those at PCC because the harmonic power terms are not accounted to the load. The unbalance power is zero because the voltages are symmetrical and the load is balanced. Balanced load – Case IV : Similarly to cases II and III, the load is accounted for less active, reactive and void power than the PCC. The unbalance power is not zero due to an apparent unbalance generated by asymmetrical voltage harmonics (remember that phase conductance Gn and susceptance Bn are affected by distortion power). The data related to the case of unbalanced load are given in Table 3. Unbalanced load – Case I : In this case, we note that the apparent, active and reactive power accounted to the load are slightly higher than those measured at PCC. This is the effect of the line impedances that, due to the negative sequence component of the load currents, cause a negative sequence voltage to appear at the load terminals, thus affecting the computation of parameters Gn and Bn and the corresponding power terms accounted to the load. Unbalanced load – Case II : In this case, the power terms at PCC and load are different. Note in particular that the load, for the same reason mentioned in the previous case, is slightly penalized for the reactive and unbalance power. Instead, the active power accounted to the load is

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definitely lower than that at PCC, since the effects of supply asymmetry have been deducted.

a) Balanced load.

Table 3: Power terms (pu) for unbalanced load in three-wires circuit

A P Q N V

λ

Case I PCC LOAD 1,0000 1,0009 0,8275 0,8280 0,2432 0,2451 0,5060 0,5060 0,0000 0,0000 0,8275 0,8273

Case II PCC LOAD 1,0000 0,9050 0,8841 0,7668 0,2135 0,2245 0,4156 0,4250 0,0000 0,0000 0,8841 0,8473

Case III PCC LOAD 1,0000 0,9832 0,8254 0,8098 0,2422 0,2417 0,5055 0,4980 0,0674 0,0666 0,8254 0,8237

Case IV PCC LOAD 1,0000 0,8973 0,8808 0,7570 0,2135 0,2232 0,4185 0,4231 0,0581 0,0566 0,8808 0,8437

Unbalanced load – Cases III and IV : In these cases we recognize the different effects of voltage asymmetry and distortion. Note that the apparent and active power accounted to the load are always lower than those computed at PCC, due to the depuration of the effects of voltage asymmetry and distortion. The data related to the case of distorting load are given in Table 4. Table 4: Power terms (pu) for distorting load in three-wires circuit

A P Q N V

λ

Case I PCC LOAD 1,0000 1,0001 0,9328 0,9329 0,2399 0,2399 0,0004 0,0004 0,2690 0,2690 0,9328 0,9328

Case II PCC LOAD 1,0000 0,9608 0,9331 0,8896 0,2198 0,2298 0,0970 0,0992 0,2677 0,2628 0,9331 0,9259

Case III PCC LOAD 1,0000 0,9820 0,8955 0,8777 0,3915 0,3875 0,0000 0,0000 0,2116 0,2095 0,8955 0,8938

Case IV PCC LOAD 1,0000 0,9389 0,9259 0,8604 0,2563 0,2581 0,1217 0,1237 0,2496 0,2436 0,9259 0,9164

Distorting load – Case I: In this case, we note that all power terms accounted to load and PCC are nearly equal. This is because the effect of the distorting load at PCC voltages is very low. If the distorting load causes significant distortion of PCC voltages, its apparent, active and reactive power can be higher than those measured at PCC. In fact, parameters Gn and Bn are affected by the voltage distortion, and the corresponding increase of power terms is accounted to the load. Distorting load – Case II: An unbalance power appears due to the interaction between negative-sequence voltages and distorting load, which causes a negativesequence current absorption at PCC. For the same reasons mentioned before (balanced load – Case II and unbalanced load – case II), the reactive and unbalance power are not equal, but the active power accounted to the load is definitely lower than that at PCC. Distorting load – Case III : The power factor is slightly lower than in previous cases, due to voltage distortion. The active, reactive and void power on the load side are lower than those at PCC, because the distortion voltages are not accounted to the load. The unbalance power is zero because the load is balanced. Distorting load Case IV: Note again the effects of voltage asymmetry and distortion on the distorting load. The apparent and active power accounted to the load are always lower than those computed at PCC due to the depuration of the effects of voltage asymmetry and distortion. Three phase circuit with neutral wire In four-wire circuits, three voltage and current measurements are needed, as shown in Fig.2. The data interpretation is very similar to the previous cases (three-wire). Observe however that the balanced load is never penalized from voltage asymmetry (negative and zero sequence) in terms of unbalance power. Moreover, the active and reactive power accounted to the load are always lower than those measured at PCC, since the effects of voltage asymmetry and distortion are depurated.

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b) Unbalanced load.

Figure 2: Application example for accountability – Three-phase circuit with neutral wire

Tables 5 and 6 summarize the power terms (in pu) computed at PCC and accounted to the load. Table 5: Power terms (pu) for balanced load in four-wires circuit

A P Q N V

λ

Case I PCC LOAD 1,0000 1,0000 0,7985 0,7985 0,6020 0,6020 0,0000 0,0000 0,0000 0,0000 0,7985 0,7985

Case II PCC LOAD 1,0000 0,9576 0,7985 0,7646 0,6020 0,5765 0,0000 0,0000 0,0000 0,0000 0,7985 0,7985

Case III PCC LOAD 1,0000 0,9689 0,7849 0,7553 0,6028 0,5908 0,0000 0,0000 0,1431 0,1390 0,7849 0,7795

Case IV PCC LOAD 1,0000 0,9282 0,7849 0,7231 0,6028 0,5657 0,0000 0,0000 0,1433 0,1362 0,7849 0,7791

Table 6: Power terms (pu) for unbalanced load in four-wires circuit

A P Q N V

λ

Case I PCC LOAD 1,0000 1,0009 0,8275 0,8280 0,2432 0,2451 0,5060 0,5060 0,0000 0,0000 0,8275 0,8273

Case II PCC LOAD 1,0000 0,9242 0,8828 0,7886 0,2141 0,2310 0,4181 0,4229 0,0000 0,0000 0,8828 0,8533

Case III PCC LOAD 1,0000 0,9666 0,8183 0,7879 0,2427 0,2397 0,5029 0,4883 0,1363 0,1328 0,8183 0,8151

Case IV PCC LOAD 1,0000 0,8921 0,8729 0,7498 0,2137 0,2259 0,4153 0,4058 0,1407 0,1340 0,8729 0,8405

Conclusions A revision of the Conservative Power Theory (CPT) has been presented, which is particularly oriented to smart micro-grids, where supply voltage distortion and frequency variation can be non negligible. Current and power terms have been revised accordingly, and an extension to polyphase circuits has been discussed. Taking advantage of a definition of sequence components which holds also in case of distorted quantities, an accountability approach has also been introduced, which allows to separate the responsibility of load and supply in the generation of active, reactive, unbalance and void power. The approach has been tested by simulation in some cases of practical interest to show the properties of the proposed decomposition and the quantitative results in view of energy tariffation policies. REFERENCES [1] [2] [3] [4] [5]

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