A New Algorithm for Energy Measurement at Positive

Paper ID. 579

1

A New Algorithm for Energy Measurement at Positive Sequence of Fundamental Power Frequency, Under Unbalanced Non-Sinusoidal Conditions Roberto Langella, Member, IEEE, Alfredo Testa, Senior Member, IEEE Abstract - A new algorithm for energy measurement at positive sequence of fundamental power frequency under unbalanced non-sinusoidal conditions is presented. Extraction of a generalized positive sequence from the three-phase signals in time domain is performed first. Then, an accurate extraction of fundamental power frequency tones is obtained taking advantage of Hanning windowing together with interpolation of the FFT results. Active and reactive power for energy measurements are finally calculated via traditional formulas. Errors due to fundamental power frequency desynchronization are analyzed for the proposed algorithm and for two alternative traditional algorithms. Finally, numerical applications to simple case studies demonstrate the usefulness of the proposed approach and the importance of considering the positive sequence of the fundamental power frequency for a correct billing of the energy supplied to the costumers. Index Terms — Harmonic Pollution, Power Quality, Revenue Meters. I. INTRODUCTION

T

he analysis of the exchanges of energy among the distribution system, distorting and linear loads, is obviously the starting point for the choice of the metrics to adopt for a correct billing of the energy supplied to the costumers. Naturally, the problem is wide and has been object of remarkable interest in the international literature [1-9]. The recent work of the IEEE Working Group on Non-sinusoidal Situations, suggests a set of practical definitions for defining distortion power in terms of total fundamental and harmonic constituents, without additional cumbersome theory [9]. In [8], it was shown that by pricing of electric energy, presently based on the value of the integral of the load active power P measured by energy meters, the electric power utilities waste some revenues for the energy delivered to current harmonic generating customers and/or customers causing current asymmetry and that the customers that do not generate harmonics but are supplied with distorted and/or asymmetrical voltages are billed not only for the useful energy but also for the energy which may cause only harmful effects on their equipment. These two disadvantages of the present tariff could be eliminated if i) the energy account based on the value of the integral of the active power of only power system frequency component is used [10] and ii) in three-phase systems, if the energy account is based on the integral of the active power of the positive sequence components.

Moreover, the revenue meters of active and reactive energy, generally, use algorithms and techniques developed for the sinusoidal regimen and that, extended to the non-sinusoidal periodic regimen, introduce metrics “de facto” that sometimes correspond to the definitions of the powers introduced in the relevant literature and sometimes not [11-14]. In this paper, a new algorithm for energy measurement at positive sequence of fundamental power frequency under unbalanced non-sinusoidal conditions is presented. Extraction of the positive sequence from the three-phase signals in time domain is performed first, basing on the concept of the generalization of the symmetrical components technique to periodic non-sinusoidal voltages and currents [15]. Then, an accurate extraction of fundamental power frequency tones from the generalized positive sequence of voltage and current is obtained taking advantage of Hanning windowing together with interpolation of the FFT results. Active and reactive power for energy measurements calculations are finally calculated via traditional formulas, for sinusoidal quantities, in frequency domain. Errors due to fundamental power frequency desynchronization are analyzed for the proposed algorithm and for two traditional alternative algorithms based on the concepts of numerical integration of filtered signals and on the FFT of the phase signals, respectively. Finally, numerical applications to simple case studies demonstrate the usefulness of the proposed approach and the importance of considering the positive sequence of the fundamental power frequency for a correct billing of the energy supplied to the costumers. The paper is organized as it follows: basic definitions and metrics “de facto” adopted by commercial digital revenue meters are presented in Sections II and III, respectively; then, the proposed approach is presented; finally, the error sensitivity analysis and the numerical applications are reported. II. SOME BASIC DEFINITIONS For steady-state conditions, under the hypothesis that DC and interharmonics are not present, a non-sinusoidal instantaneous voltage or current has two distinctive components: the power system frequency components v1(t) and i1(t), and the remaining terms vH(t) and iH(t) that contain all the other harmonics: v(t ) = v1 (t ) + v H (t ) = 2V1 sin (ωt − α1 ) +

∞

∑

2Vh sin (hωt − α h ) , (1)

h ≠1

R. Langella and A. Testa are with Dipartimento di Ingegneria dell’Informazione, Seconda Università di Napoli, Via Roma, 29 - 81031 Aversa (CE) Italy, Ph. +39 081 5010239, Fax +39 081 5037042. (e-mail: [email protected]; [email protected]).

∞

i(t ) = i1 (t ) + iH (t ) = 2 I1 sin (ωt − β1 ) + ∑ 2 I h sin (hωt − β h ) . h ≠1

(2)

Paper ID. 579 In three-phase, three-wire systems, denoting the phases by the subscripts a, b, and c, the fundamental power period by T1, the fundamental power frequency by f1 = 1 / T1 , the total active power can be defined as:

1 P= T1 =

∫ [v (t )i (t ) + v (t )i (t ) + v (t )i (t )]dt = a

a

b

b

c

c

τ

⎛ ⎜⎜V j1I j1 cosθ j1 + j = a ,b, c ⎝

∑

⎞ V jh I jh cosθ jh ⎟⎟ = P1 + PH h ≠1 ⎠ H

∑

[

(3)

]

⎛ ω1 τ +T ⎞ ⎜ ∫ i j1 (t ) ⋅ ∫ v j1 (t )dt dt + QBjH ⎟ = ⎜T τ ⎟ j = a ,b, c ⎝ 1 ⎠

∑

1

H ⎞ ⎛ = ∑ ∑ ⎜⎜V j1I j1 sin θ j1 + ∑V jh I jh sin θ jh ⎟⎟ = Q1 + QBH j = a ,b, c j = a ,b, c ⎝ h ≠1 ⎠

1 Pˆd = N

N

∑ ∑ v (k ) ⋅ i (k ) , j

(4)

P1+ = 3V1+ I1+ cos θ1+ ,

(5)

Q1+ = 3V1+ I1+ sin θ1+

(6)

S1+ = ( P1+ ) 2 + (Q1+ ) 2

(7)

The IEEE Standard 1459 states that the power factor definition: P+ PF+1 = 1+ (8) S1 plays the same significant role that the fundamental power factor has in non-sinusoidal single-phase systems giving reliable information about the lines utilization. Expressions similar to (5-8) apply to define and calculate negative and zero sequence powers, P1-, Q1-, P1o, Q1o. METRICS “DE FACTO” ADOPTED BY COMMERCIAL DIGITAL METERS

For the digital meters fs is the sampling frequency of the instrument and N=T1/fs the number of points in a period of the fundamental frequency f1=1/T1. Here, it is assumed, moreover, the absence of causes of uncertainty, that is to say ideal transducers, perfect synchronization between fs, N and T. What remains is the uncertainty related only to metric adopted.

j

(9)

j = a , b , c k =1

It is evident that the results of (9) coincide with those of (3), so digital meters are not able to separate the system power frequency contribute from harmonic contributes. As for the reactive power measurement, let us consider as Qˆ d the power reading of the digital meter: N 1 Qˆ d = (10) ∑ ∑ v j (k + Δ) ⋅ i j (k ) , N j = a ,b , c k =1 with Δ=N/4 which corresponds the numerical implementation of: τ +T 1 Q= v j ⎛⎜ t + T1 ⎞⎟ ⋅ i j (t )dt , (11) ∑ ∫ 4⎠ τ T1 j= a, b,c ⎝ 1

which is a definition valid only in sinusoidal regimen. It is possible to demonstrate by means of simple mathematical manipulations [14] that (11) can be seen as: Qˆ = Q + ε (12) d

The symmetrical voltage components V +, V -, V o and current components I +, I -, I o with the respective phase angles θ +, θ -, θ o yield the following definitions for the fundamental positive sequence active power (W), reactive power (VAr) and apparent power (VA):

III.

of the digital meter:

τ + T1

where τ is the instant in which the calculation of P starts and being H=N/2 the maximum harmonic order according to the sampling theorem and θh=βh-αh the phase angle difference between the phasors Vh e Ih. On the other hand, the Budeanu’s definition of reactive power is widely adopted, QB, even if it was demonstrated [6] that the Budeanu’s Resolution “cannot be used to design or to specify ratings of equipment in any situation and its use in industry must be abandoned in total”:

QB =

2 ˆ For the measure of the active power, let us call Pd the reading

1

d

that is to say the summation of Q1 with an error term, εd, which is function of the odd harmonic active powers and of the even harmonic reactive powers. As for the power factor, it is: Pˆd P PˆFd = = . (13) 2 2 P + (Q1 + ε d ) Pˆd2 + Qˆ d2 Finally, it is important to observe that, with the instrumentation considered, only the active power measurement corresponds with the definitions introduced in literature while, the reactive power measurement and, as a consequence, the power factor introduce incorrect metrics “de facto” at least meaningless. IV.

THE PROPOSED ALGORITHM

In general, the evaluation of positive sequence powers can be obtained basically by means of two techniques. The former (called SUM in the following sections) is based on: - a proper filtering action of the 6 original phase signals, va, vb, vc, ia, ib, ic, to eliminate harmonic components obtaining v1a, v1b, v1c, i1a, i1b, i1c; - extraction of positive sequence components in time domain, v1+(t) and i1+(t), averaging the 3 phase voltages and 3 phase currents, v1a, v1b, v1c, i1a, i1b, i1c, lagging over T/3 the current (or voltage) in phase b and over 2T/3 the current (or voltage) in phase c; - processing of the extracted positive sequence components to implement formulas similar to 9) and 10). The latter (called FFT in the following sections) is based on: - FFT of the 6 signals, va, vb, vc, ia, ib, ic, to obtain the fundamental power phase components in frequency domain, V1a, V1b, V1c, I1a, I1b, I1c; - extraction of positive sequence components V1+ and I1+ by means of traditional frequency domain formulas; - processing of the extracted components to implement formulas 5) and 6).

Paper ID. 579 Inconvenient characterize both techniques: - the first suffers for difficulties typical of the filters; - the second requires a remarkable computational burden; - both present sensitivity to power frequency time variations which can be only in part compensated by the use of PLL. A contribution to solve these problems was given in [14]. In what follows, a simple algorithm (called INT in the following sections) which seems to solve the drawbacks of the aforementioned techniques is presented. It is based on three fundamental steps: 1. generalized positive sequence, v+(t) and i+(t), extraction in time domain as described in sub-section II.A; 2. positive sequence fundamental power frequency components, Vˆ1+ and Iˆ1+ , accurate estimations in frequency domain as described in sub-section II.B; 3. processing of the extracted components to implement formulas 5) and 6) using Vˆ1+ and Iˆ1+ . So doing, no filtering actions is needed, the number of needed FFT is reduced from 6 to 2 and the sensitivity to power frequency variations is compensated. A. Generalized Positive Sequence Extraction In [15] the generalization of the symmetrical components technique to periodic non-sinusoidal three-phase currents and voltages is derived both in time and frequency domain. The main conclusion of the paper is that an orthogonal decomposition of periodic non-sinusoidal three-phase signals into positive sequence, negative sequence and zero-sequence components is not possible, but that an additional current and voltage component should be introduced which is called the residual component. Considering a periodic three-phase signal, fa, fb and fc (a threephase current or a three-phase voltage, with or without neutral conductor), this signal can be decomposed in the following way: + − o ra ⎤ ⎡ f a (t )⎤ ⎡ f (t ) + f (t ) + f (t ) + f (t ) ⎢ ⎥ ⎢ ⎥ + − o rb ⎢ f b (t ) ⎥ = ⎢ f (t − T 3) + f (t + T 3) + f (t ) + f (t ) ⎥ (14) ⎢⎣ f c (t ) ⎥⎦ ⎢ f + (t − 2T 3) + f − (t + 2T 3) + f o (t ) + f rc (t )⎥ ⎢⎣ ⎥⎦

being f o, the generalized zero-sequence component: f o (t ) = 1 3 ( f a (t ) + fb (t ) + f c (t ) )

(15)

f +, the generalized positive-sequence component: ⎛ f a (t ) − f o (t ) + f b (t + T 3) − f o (t + T 3) ⎞ ⎟= f + (t ) = 1 3 ⎜ ⎜ + f (t + 2T 3) − f o (t + 2T 3) ⎟ (16) ⎝ c ⎠ ~ ~ ~ = 1 3 f a (t ) + f b (t + T 3) + f c (t + 2T 3)

(

)

f -, the generalized negative-sequence component: ~ ~ ~ f − (t ) = 1 3 f a (t ) + fb (t − T 3) + f c (t − 2T 3)

(

and f rj, with j=a,b,c, the residual components: ~ ~ ⎧ f ra (t ) = 1 3 ~ f a (t ) + f a (t + T 3) + f a (t + 2T 3) ⎪⎪ ~ ~ ~ rb ⎨ f (t ) = 1 3 f b (t ) + f b (t + T 3) + fb (t + 2T 3) ~ ~ ~ ⎪ rc f (t ) = 1 3 f c (t ) + f c (t + T 3) + f c (t + 2T 3) ⎩⎪

( ( (

)

) ) )

(17)

(18)

3 The orthogonality of the components enables one to calculate powers, rms values, etc, for the components separately and adding them afterwards in the appropriate way. In [15] it is also demonstrated that, in presence of distorted threephase voltages and currents, the harmonic content of (16) is: f + (t ) = f1+ (t ) + f 2− (t ) + f 4+ (t ) + f5− (t ) + f 7+ (t ) + f8− (t ) + ... (19) that is to say a combination of positive sequence components (h = 1, 4, 7,…) and negative sequence components (h = 2, 5, 8,…), which justify the use of the adjective “generalized” for the positive sequence f +(t). Anyway, the fundamental positive power frequency components remain captured in f +(t). The application of generalized sequence concept in our case gives: v + (t ) = v1+ (t ) + v2− (t ) + v4+ (t ) + v5− (t ) + v7+ (t ) + v8− (t ) + ... (20)

i + (t ) = i1+ (t ) + i2− (t ) + i4+ (t ) + i5− (t ) + i7+ (t ) + i8− (t ) + ... (21) B. Accurate Fundamental Tones Extraction The extraction of the fundamental component in frequency domain from the positive sequence time series obtained by (20) and (21) can be conducted as follows. The sampled and windowed time domain generalised positive sequence signal is considered:

f w+(k) = f +(k) ⋅ w(k) with k = 0,1,….,N-1 (22) being f + the sampled positive sequence voltage or current signal, v+(k) or i+(k), and w the adopted window and N=T1/fs. The signal can be represented by the sum of two contributes, one at fundamental power frequency and the other at harmonic frequencies:

[

]

f w+(k) = f1+(k) + f H+(k) ⋅ w(k) with k = 0,1,….,N-1.(23)

In presence of desynchronization between the actual signal period T1a and fs, the estimation of the fundamental tone amplitude and phase angle based on the number of samples N can lead to inaccurate results. + ˆ+ An high accuracy estimation of frequency, νˆ1 , amplitude, F1 , + and phase angle, ψˆ1 , of the fundamental component can be accomplished by means of a proper interpolation of the spectrum samples calculated by FFT [16] giving:

(

fˆ1+ = Fˆ1+ sin 2πνˆ1+k +ψˆ1+

)

with k = 0,1,.....,N -1. (24)

The interpolation formulas are reported in Appendix with reference to the Hanning [17]. The application of the accurate fundamental tones extraction in our case gives: ˆ+ ˆ + Vˆ + = Vˆ +e− j(2πν1 k +α1 ) 1

1

ˆ+ ˆ + Iˆ1+ = Iˆ1+e− j (2πν1 k +β1 ) θˆ+ = βˆ + −αˆ + 1

1

(25)

1

C. Active and Reactive Powers Calculation Once extracted positive sequence fundamental power frequency voltages and currents, the application of formulas (5-8) using parameters values obtained starting from (25) is trivial. Moreover, if the quantities P and QB are known, an indication about the global distorting and unbalancing contributes can be easily calculated subtracting (5) and (6) form (3) and (4).

errP = −

sin(2πf1a / f1e ) cos(2πf1a / f1e + α + β ) 100 , f1a / f1e

(26)

where f1a is the actual frequency value, f1e is the estimated frequency value, α is the voltage phase-angle and β the current phase-angle. The corresponding error for the reactive power is: errQ = −

sin( 2πf1a / f1e ) cos(2πf1a / f1e + π / 2 + α + β ) 100 . (27) f1a / f1e

For the cases of FFT and INT algorithms, the error values have been evaluated by means of numerical simulations. Moreover, the effects on the errors of the starting instant of the measurement, which determines the value of α, and of the phase angle shift between voltages and currents, θ = β-α are considered. The results are reported in Figs. 1 and 2. It is easy to observe how it is not possible to distinguish from zero the errors of the proposed approach in the whole range considered, in a scale in which the maximum absolute values are the errors of FFT and SUM that reach amplitudes of about 0.03 % in presence of an uncompensated frequency error of 0.03 %. alfa 0 deg beta 0 deg alfa 90 deg beta 90 deg 0.04 0.02 0 -0.02

α=0°, β=0°

b) Active Power Error [%]

Active Power Error [%]

a)

0.04

α=90°, β=-90°

0.02 0

0 -0.02

Active Power Error [%]

Active Power Error [%]

0.02

-0.04 49.98 49.99 50 50.01 50.02 Fundamental power frequency [Hz]

x 10

alfa= 0 α=0°, deg beta= β=0° 0 deg

0 -1 -2 -3 49.98 49.99 50 50.01 50.02 Fundamental power frequency [Hz]

0.04

alfa= 0 α=0°, deg beta= -45 deg β=-45°

0.02 0 -0.02

Reactive Power Error [%]

c)

-0.04 49.98 49.99 50 50.01 50.02 Fundamental power frequency [Hz]

Reactive Power Error [%]

1

-3

1

x 10

alfa= 90 deg beta= α=90°, β=-90°-90 deg

0 -1 -2 -3 49.98 49.99 50 50.01 50.02 Fundamental power frequency [Hz]

d) 0.04

alfa= 90α=90°, deg beta= -135 deg β=-135°

0.02 0 -0.02 -0.04 49.98 49.99 50 50.01 50.02 Fundamental power frequency [Hz]

Fig. 2. Reactive power percentage error for three different algorithms (FFT: dash-dotted line; SUM: dashed line and INT: solid line) for different starting instants of the measurement and of the phase angle shift between voltages and currents: a) α=0, β=0; b) α= π/2 , β= π/2; c) α= 0 , β= −π/4; d) α= π/2 , β= −3π/4.

VI.

NUMERICAL APPLICATIONS

In this section, the results of two applications are reported. The former refers to a single-phase phase system with the presence of a non linear load, the latter to the case of a threephase system in presence of two linear loads, one balanced and the other unbalanced. The aim of these two applications is to show what is fair to bill in non ideal conditions and how the proposed algorithm can contribute to this. A. Case-Study 1 The numerical case-study developed refers to the simplified scheme reported in Fig.3. The non-linear load is an “Integral Cycle Controlled”, that is to say a circuit operated by a system that is based on alternations of intervals of logon and intervals of logoff, whose duration is an entire multiple of the fundamental period. The values of the simulation parameters are reported in Table 1, while the simulation results are summarised in Table 2 with reference to the main electrical quantities.

-0.02

-0.04 -0.04 49.98 49.99 50 50.01 50.02 49.98 49.99 50 50.01 50.02 Fundamental power frequency [Hz] Fundamental power frequency [Hz] alfa= 0 deg beta= -45 deg alfa= 90 deg beta= -135 deg 0.04 c) 0.04 d) α=0°, β=-45° α=90°, β=-135°

b)

-3

Reactive Power Error [%]

A complete analysis of the new algorithm performances in terms of accuracy and required computational burden is not developed for the sake of brevity. Attention is devoted only to the robustness of the proposed algorithm versus the fundamental power frequency desynchronization. The power fundamental frequency has been linearly varied from 49.985 to 50.015 Hz, with steps of 0.0075 Hz, which corresponds to the interval of maximum permissible synchronization error (PLL) for IEC Class A instruments [4]. The percentage errors on active and reactive power evaluation have been compared with reference to three methods, SUM, FFT and INT described in Sect. IV, properly implemented. The percentage error of the digital revenue meter SUM have been evaluated by means of analytical expressions derived in [18] with reference to the active power:

4 a)

Reactive Power Error [%]

Paper ID. 579 V. ERRORS DUE TO FUNDAMENTAL POWER FREQUENCY DESYNCHRONIZATION

I1

E1

LR RR

Pulse

LL

RL

LD

0.02 0

Fig. 3. Case-Study 1: simplified electrical scheme. -0.02 -0.04 49.98 49.99 50 50.01 50.02 Fundamental power frequency [Hz]

Fig. 1. Active power percentage error for three different algorithms (FFT: dash-dotted line; SUM: dashed line and INT: solid line) for different starting instants of the measurement and of the phase angle shift between voltages and currents: a) α=0, β=0; b) α= π/2 , β= π/2; c) α= 0 , β= −π/4; d) α= π/2 , β= −3π/4.

TABLE 1 CASE-STUDY 1: SIMULATION PARAMETERS Network Linear Distorting Load Load 0.066 5.29 2.64 R [Ω] L [mH] 0.080 16.83 10.26 230 E1 [V] Duty Cycle Ton=400 ms e Toff=400 ms

RD

Paper ID. 579

5

TABLE 2 CASE-STUDY 1: SIMULATION RESULTS Linear Distorting Supply Generator Load Load Network P [W] 9385 9082 1146 19613 P1 [W] 9372 9404 836 19613 13 -323 310 0 Ph [W] 9369 7414 409 17192 QB [VAr] 9360 7494 339 17192 Q1 [VAr] 9 -80 71 0 Qh [VAr] S1 [VA] 13245 12024 26081 0.708 0.782 PF1 9385 9082 Pˆd [W] ) Qd [VAr]

9356

7581

PˆFd

0.708

0.768

R [Ω] L [mH] E1 [V]

The analysis of the results evidences what follows: - a 0.7% more of active power, P, is attributed to the linear load instead of the demanded power P1 without taking into account further detrimental effects due to extra losses at harmonic frequencies; of course, this power increment will be integrated in the time by the energy revenue meter; the measured PˆFd coincides with PF1; - a 3.6% less of active power, P, is attributed to the distorting load instead of the demanded power P1; - the generator obviously must supply a PG=PG1; - the indications of the traditional instruments are Pˆd =P for both loads, Qˆ < QL1 < QLB and Qˆ > QD1 > QDB, Dd

Ld

respectively for the linear load and for the distorting load. B. Case-Study 2 The numerical case-study developed refers to the simplified scheme reported in Fig.4, in which a three-phase balanced linear load and a single-phase unbalanced linear load are present. The values of the simulation parameters are reported in Table 3, while the simulation results are summarised in Table 4 with reference to the same electrical quantities evaluated for the first case-study. Ea Eb Ec

Ia LR RR Ib LR RR

Ic LR RR LB

RB

LU

RU

LN RN

Fig. 4. Case-Study 2: simplified electrical scheme.

Result analysis evidences that: - the balanced load is charged by a small amount of extra power (1 %), in comparison with the required power P+ and this without taking into account further negative effects related to the presence of negative sequence voltages; - unbalanced load is charged with a discount of about 20% on the basis of power P, in comparison with the required power P +; - the traditional instrument indications are Pˆd =P and Qˆ = Q for both loads. d

P [W] P + [W] P - [W] P o [W] Q [VAr] Q + [VAr] Q - [VAr] Q o [VAr] S + [VA] PF P+F Pˆd [W] Qˆ [VAr]

TABLE 3 CASE-STUDY 2: SIMULATION PARAMETERS Network Balanced Unbalanced Load Load 0.066 1.76 0.132 0.080 5.61 0.16 230 TABLE 4 CASE-STUDY 2: SIMULATION RESULTS Balanced Unbalanced Network Load Load 28489 22146 6119 28172 26075 3385 7.65 -543 308 -3389 -28489 -22146 -2318 -28172 -23830 -7.65 210.30 -308 1474 39841 31319 0.7071 0.7071 0.7071 0.7071 28489

22146

d

-28489

-22146

PˆFd

0.7071

0.7071

Neutral conductor 0.132 0.16

Generator 56754

-52953

Finally, it is worthwhile to observe that the use of some traditional instruments, that is to say the metrics de facto reported in Section III, for the measurement of the energy consumed by the customers can have economic consequences that do penalize the electrocommercial utilities and the non disturbing customers and do reward the customers that pollute, that, instead of being penalized, receive a prize in terms of reduction of the costs of the energy. VII.

CONCLUSIONS

A new algorithm for energy measurement at positive sequence of fundamental power frequency under unbalanced non-sinusoidal conditions has been presented. Extraction of the positive sequence from the three-phase signals in time domain is performed first, basing on the concept of the generalization of the symmetrical components technique to periodic nonsinusoidal voltages and currents. Then, an accurate extraction of fundamental power frequency tones from the generalized positive sequence of voltages and currents is obtained taking advantage of Hanning windowing together with interpolation of the FFT results. Active and reactive power for energy measurements calculations are finally calculated via traditional formulas in frequency domain. Errors due to fundamental power frequency desynchronization have been analyzed for the proposed algorithm and for two traditional alternative algorithms based on the concepts of numerical integration of filtered signals and on the FFT of the phase signals, respectively. Finally, numerical applications to simple case studies have demonstrated the usefulness of the proposed approach and the importance of considering the positive sequence of the fundamental power frequency for a correct billing of the energy supplied to the costumers. Further research work will consider the analysis of the computational burden, smoothing effects and techniques to take into account the presence of interharmonics.

Paper ID. 579

6 APPENDIX

REFERENCES

A sampled and windowed single tone signal is considered: ⎛ ⎞ k f (k ) = F sin⎜⎜ 2πf1 +ψ ⎟⎟ ⋅ w(k ) with k=0,1,…,N-1 (A1) fS ⎝ ⎠

being A the tone amplitude, f1 the frequency, ψ the phase angle, fS the sampling frequency and w a generic window of length TW=N/fS. Thus, the signal spectrum evaluated by means of the DFT on N points and neglecting the negative frequency replica is: F (k ) =

F ⋅ exp( jψ ) ⎛ k ⎞ ⋅ W ⎜ −ν ⎟ with k=0,1,…,N-1 (A2) 2j L ⎝ ⎠

where ν = f1/fS is the tone frequency normalized to the sampling frequency. In presence of a small desynchronization between the tone period and sampled time window, none of the DFT components matches the actual tone frequency as it is shown in Fig. A1, where M is the order of the M-th DFT component and δ its normalized frequency deviation from the actual normalized frequency. Spectral Amplitude

δ

[2]

[3]

[4]

[5] [6]

[7]

[8]

[9]

[10]

1 N

[11]

[12]

M −3 N

M −1 M ν M +1 N N N Normalized frequency

[13]

Fig. A1. Example of the spectrum (--) and of DFT components (•) of a signal.

The interpolation of a given tone is based on the assumption of negligibility of the spectral leakage effects caused by the negative frequency replica, the other harmonic and interharmonic tones. These three latter conditions occur with a good approximation with the use of a proper window. The authors have selected the Hanning window because of its good spectral characteristics and the easiness of the interpolation formulas. Approximated expressions for the interpolated tone amplitude, Fˆ , frequency, νˆ , and phase angle ψˆ , are: δˆ 1 − δˆ 2 Fˆ = π S (M ) , sin πδˆ

(

) ( )

M ˆ +δ, L

νˆ = ψˆ = being

δˆ = with

[1]

π

2

(A3)

+ ∠S (M ) − M ⋅ π ⋅ δˆ,

S (M ) 2 −α ,α = 1+α S M + sign δˆ

()

(

( ))

sign δˆ = sign( S (M + 1) − S (M − 1) ) .

(A4)

(A5)

[14]

[15]

[16]

[17]

[18]

IEEE Working Group on Nonsinusoidal Situations, “Practical Definitions for Powers in Systems with Nonsinusoidal Waveforms and Unbalanced Loads,” IEEE Transactions on Power Delivery, Vol. 11, No. 1, Jan. 1996, pp.79–101. Emanuel, A. E., “Apparent Power Definitions for Three-Phase Systems,” IEEE Transactions on Power Delivery, Vol. 14, No. 3, July 1999, pp. 767–72. A. Emanuel, ”Some challenges for the IEEE standard 1459 Power Engineering Society General Meeting, 2005. IEEE 12-16 June 2005 Page(s):2319 - 2320 Vol. 3. A. Emanuel, J.L. Willems, J.A. Ghijselen, ”The apparent power concept and the IEEE standard 1459-2000”, Power Delivery, IEEE Transactions on Volume 20, Issue 2, Part 1, April 2005 Page(s): 876 – 884. A. Ferrero, “Definitions of electrical quantities commonly used in nonsinusoidal conditions”, ETEP, Vol. 8, 1998 pp.235-40. Czarnecki, L. S., “What is Wrong with Budeanu’s Concept of Reactive and Distortion Power and Why it Should be Abandoned,” IEEE Transactions on Instrumentation and Measurement, Vol. IM-36, No.3, Sept. 1987. D.Gallo, R.Langella, A.Testa, "On the Processing of Harmonics and Interharmonics: Using Hanning Window in Standard Framework”, IEEE Transaction on Power Delivery. Vol. 19 Issue: 1, Jan. 2004. L. S. Czarnecki, “Comments on Active Power Flow and Energy Accounts in Electrical Systems with nonsinusoidal Waveforms and Asymmetry”, IEEE Transaction on PD. Vol. 11 Issue: 3, Jul. 1996. IEEE Standard 1459-2000, “Definitions for the measurement of electric power quantities under sinusoidal, nonsinusoidal, balanced or unbalanced conditions,” IEEE PES, PSIM Committee, June 2000. L.S. Czarnecki, "Two algorithms of the fundamental harmonic complex RMS value calculation," Archiv fur Elektrotechnik75 (1992) pp. 163168. M.D. Cox, T. B. Williams, “Induction varhour and solid-state varhour meter performance on nonsinusoidal loads,” IEEE Trans. On Power Delivery, Vol.5, No.4,Nov.1990, pp1678-1686. P. S. Filipski, P. W. Labaj, “Evaluation of reactive power meters in the presence of high harmonic distortion,” IEEE Trans. on Power Delivery, Vol.7, No.4, Oct. 1992, pp. 1793-99. R. Arseneau , B. Hughes, “Selecting revenue meters for harmonic producing loads,” IEEE International Conference on Harmonics and Quality of Power, Paper hpq 054, Lake Placid NY, Sept. 12-15, 2004. D. Gallo, C. Landi, N. Pasquino, N. Polese, “A New Methodological Approach to Quality Assurance of Energy Meters Under Non-Sinusoidal Conditions”, IMTC 2006 Sorrento, Italy 24-27 April 2006. P. Tenti, J. L. Willems, P. Mattavelli, E. Tedeschi,:”Generalized Symmetrical Components for Periodic Non-Sinusoidal Three-Phase Signals”, 7th International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions, Cagliari, July 10-12, 2006. D.Gallo, R.Langella, A.Testa, "Desynchronized Processing Technique for Harmonic and Interharmonic Analysis”, IEEE Transaction on Power Delivery. Vol. 19 Issue: 3, Jul. 2004. P.Langlois & R.Bergeron, "Interharmonic analysis by a frequency interpolation method", Proceeding of the 2nd International Conference on Power Quality, Atlanta, USA, September 1992, pp. E-26:1-7. L. Peretto, J. L. Willems, A. E. Emanuel: “The effect of the integration interval on the measurement accuracy of rms values and powers in systems with nonsinusoidal waveforms”, 7th International Workshop on Power Definitions and Measurements under Non-Sinusoidal Conditions, Cagliari, July 10-12, 2006.

Roberto Langella (S’00–M’01) was born in Naples, Italy, on March 20, 1972. He received the degree in electrical engineering from the University of Naples, in 1996, and the Ph.D. degree in electrical energy conversion from the Second University of Naples, in 2000. He is currently an Assistant Professor in electrical power systems at the Second University of Naples. Dr. Langella is a member of IEEE Power Engineering Society. Alfredo Testa (M’83–SM’03) was born in Naples, Italy, on March 10, 1950. He received the degree in electrical engineering from the University of Naples, in 1975. He is a Professor in electrical power systems at the Second University of Naples, Aversa, Italy. He is engaged in research on electrical power systems reliability and harmonic analysis. Dr. Testa is a Senior Member of IEEE Power Engineering Society and of the Italian Institute of Electrical Engineers (AEI).