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Electric Power Systems Research 95 (2013) 140– 147

Contents lists available at SciVerse ScienceDirect

Electric Power Systems Research

jou rn al h om epa ge: www.elsev ier .com/ locate /epsr

rue power factor metering for m-wire systems with distortion, unbalance andirect current components

.T. Gaunt ∗, M. Malengretepartment of Electrical Engineering, University of Cape Town, Rondebosch 7701, South Africa

r t i c l e i n f o

rticle history:eceived 16 June 2011eceived in revised form 28 June 2012ccepted 31 July 2012

eywords:ower theory

a b s t r a c t

Two companion papers describe an internally consistent general power theory valid for instantaneousand average power for systems with any number of wires, under conditions of unbalance, distortionor dc components. Measuring apparent power and power factor under non-ideal conditions, even intypical three-phase four wire systems, needs the resistances of the wires, particularly the neutral wire,to be defined, and without this detail all conventional power theory approaches are inadequate andgive misleading measurement results. This paper illustrates the practicality of measurement of distorted

rue power factoreter

ompensation

power supplies, and describes laboratory tests showing the differences between measurements using thenew approach and a conventional instrument. Applications of the new approach in real power systems aredescribed, including hybrid ac/dc systems and smart grids, and in measuring quality of supply affectedby disturbing loads, and the non-active power in transformers subjected to geomagnetically inducedcurrents. The nature of changes required in international standards and the implications for furtherpractical research and development are identified.

. Introduction

Building on two companion papers [1,2], an approach to multi-hase power system measurement based on the general powerheory has been implemented in practice and the results illustratehe practical feasibility and significance of the approach. The keyoncepts and some implications are presented here in five sections.

First, in Section 2, a specification for power meters is defined inhe context of the need for accurate measurement and the prac-ical problems encountered by others. The specification is not anxploration of conventional metrology, which is covered in otherapers and standard specifications, but identifies the conditions forhich the meters must be appropriate and the parameters needed

s inputs to the meter for apparent and non-active power to beeasured under non-ideal conditions. In Section 3, a prototype

ardware and digital signal processor (DSP)-based true power fac-or meter that uses the general power theory is described. Theoncept of the prototype was extended to a computer based mea-urement system based on the transducers and processes of a

onventional power meter, so Section 4 describes laboratory testshat demonstrate practically the differences between measure-

ents with a conventional standard-compliant instrument and

∗ Corresponding author. Tel.: +27 21 6502810; fax: +27 21 6503465.E-mail addresses: ct.gaunt@uct.ac.za (C.T. Gaunt), Michel.Malengret@uct.ac.za

M. Malengret).

378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved.ttp://dx.doi.org/10.1016/j.epsr.2012.07.019

© 2012 Elsevier B.V. All rights reserved.

using the new approach. In Section 5 the benefits of the new meth-ods of measurement are discussed in four practical applicationareas. Finally, in Section 6 a possible approach to future standardsfor measurement of apparent power and power factor is proposed.

2. Specification of meters for accurate apparent powermeasurement

The IEEE standard 1459–2010 [3] clearly identifies various rea-sons why accurate measurement of distorted and unbalancedpower is needed. Briefly, loads on modern power systems havethe potential to disturb the delivery networks and the equipmentbeing used by customers, and there is a need to maintain suppliesof acceptable quality and apportion the costs of doing so to thosecausing the disturbances. In addition, meters and instruments areneeded for energy billing, energy quality evaluation, detection ofthe sources of distortion, and design of filters and compensators.The purpose of the standard is to provide “criteria for designingand using metering instrumentation”. Similarly, many authors haveaddressed the need for compensation of voltage unbalance anddistortion, although, in practice, all proposed solutions have beeninadequate in coping with combined distortion, unbalance and zerosequence or dc components.

Jeon [4] identified applications that require a theory suitable form-wire systems with wires of any resistance, including a multi-path transmission system and a three phase four-wire system inwhich the neutral conductor is of different resistance to the phase

C.T. Gaunt, M. Malengret / Electric Power Systems Research 95 (2013) 140– 147 141

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ig. 1. The TPF meter with input and output signals, display and selection buttons.

onductors. However, his solution for distortion due to harmon-cs was only valid for a balanced load. Similarly, Mishra et al.5] described the short-comings of load compensation approachesn power electronics proposed by other authors and formulatednother compensation technique, and they too found their schemeas inadequate, in this case working for unbalanced but not dis-

orted voltages.Montero et al. [6] compared four widely used strategies for con-

rolling shunt active power filters in 3-phase 4-wire systems andeported that only what they called the ‘perfect harmonic cancella-ion’ strategy “was capable of correct action under any conditionsf use” under the power definitions of IEEE Standard 1459, but ittill did not achieve complete compensation to unity PF.

More recently, Ustariz et al. [7] identified a significantly dis-orted 3-phase supply to a 6-phase, 12-pulse rectifier as a practicalnstallation requiring accurately calculated compensation, but theirolution did not allow for the supply wires to have different resis-ances. By contrast, Atefi and Sanaye-Pasand [8] studied the poweractor of a supply to an arc furnace with a neutral wire of negligibleesistance, which they found “has significant effect in the physi-al meaning of the apparent power”. While their solution providesigorous analysis it does not propose how compensation will beontrolled.

Clearly, practical combinations lead to measurement problemsnderlying any compensation control, apportionment of costs ornderstanding of “undesired losses, voltage drop and EMC prob-

ems” [9]. The generalised theory contributes to an improvednderstanding of all these practical problems and offers completeompensation, providing practical measurement of apparent andon-active power and power factor that is strictly accurate taking

nto account the wire resistances, is able to display the measureduantities, and generates the inputs needed by compensators.

. Initial true power factor meter for three-phase systems

The first meter conceived was for three equal resistance phaseires and a neutral wire with an assumed resistance of value

ero, infinite or equal to the phase conductors. This was termed true power factor (TPF) meter and a provisional patent was reg-stered. A prototype TPF meter was assembled around the Texasnstrument DSP TMS320F240 and used to measure unbalanced andistorted 3-phase supplies, comparing results with conventionaleters, without relying only on simulations. The meter displayed

ower, apparent power and power factor according to the selected

esistance of the neutral wire, and provided two signals suitable fornput into two compensators without and with energy storage.

Fig. 1 illustrates the TPF meter in a block diagram showinghe sequence of sensing the input parameters, processing the data

Fig. 2. Processor block diagram for the TPF meter.

according to the relative wire resistance values, and outputtinga result. The meter inputs consist of four currents and voltagesderived from transducers on the lines supplying the load. (Sub-sequently only m-1 currents were sensed as the m-th current isdefined by the others by Kirchhoff’s law.) The labels of the inputscorrespond to the load and compensators in the circuit connectiondiagram in the companion paper [2], without the apostrophe rep-resenting resistance weight or the (t) representing that they are afunction of time.

Fig. 2 illustrates how the instantaneous values of line currentsand voltages are processed by the DSP. The user of the instrumentcan select three conditions depending on the nature of the neutralwire. The instrument output is calculated according to the valueattributed to the neutral wire, assuming the resistances of all thephase wires are equal.

Clearly, this prototype instrument had the shortcomings ofassuming all phase wires had equal resistance and the limitationof the neutral resistance to only three values, but it demonstratedthe practical possibilities of the approach to metering and compen-sation.

The exercise also clarified an important issue distinguishinginstrument design and construction from the broader topic ofunderstanding the performance of power systems. It was still nec-essary, however, to assess the effects of the general power theoryapproach on practical power measurements and compare themwith the measurements made by conventional power instruments.

4. Experimental laboratory measurements

Instrument design and manufacture is a specialised area andthe risk of error in construction and calibration of a physical metermake it difficult to obtain strictly comparable results using a labo-ratory prototype TPF meter. Instead, a laboratory proof-of-conceptdemonstration was implemented by taking voltage and currentsamples with a good quality conventional (commercial) 3-phasepower meter. By adopting the sensing systems of a conventionalpower meter compliant with industry standards, all the conven-tional issues of metrology are removed from comparison by makingthem common to the conventional and novel approaches, and theeffect and practicality of the proposed new approach to measur-ing apparent power are demonstrated. The calculations can bemade by computer using the sampled signal data, and enable adirect comparison of the algorithms using a common base of phys-ical measurements for the conventional and new methods. Thiscomputer-calculation approach has an advantage of flexibility in

demonstrating the practicality of the new method for different con-ditions because alternative calculations can be carried out moreeasily by computer than by programming a physical DSP, but in apractical instrument the calculations would be carried out in a DSP.

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42 C.T. Gaunt, M. Malengret / Electric Po

Measurements were taken with a Yokogawa WT1600 Digitalower Meter connected as three-phase, four-wire system (3-P-W) with equal wire resistances. According to the Yokogawa man-al, power measurement complies with standard IEC76-1(1976).ithin the meter, the apparent power and non-active power are

erived from the sum of the three separately measured products ofMS voltages and currents.

Several aspects of extracting the voltage and current samplesetermine their adequacy for the calculation of the derived values.ll readings must be simultaneous; in this instance the simulta-eous sampling is managed by the instrument at a sampling rate ofpproximately 12.5 kHz. (The required rate depends on the orderf harmonics that are significant in the distorted waveforms, andhis sampling rate in 50 Hz systems is close to the ‘9-2LE’ imple-

entation guideline of 256 samples/cycle used for power qualitypplications). Then, the accuracy and resolution of the sampled sig-als must comply with standard specifications for meters – thisequirement is exactly the same in the Yokogawa meter, whichenerates a digital value of the voltages and currents as the basis ofhe calculated outputs. The voltages are measured from the neutralire, and the neutral current can be calculated directly from the

hree line currents. The accuracy of the calculated values of powernd apparent power requires a sufficiently large number of sam-les to be added during a period T of a periodic voltage source (not

imited to one wavelength) so that an error of less than one samples not significant, while the summated truncation error is limited topproximately half the largest truncation error (per unit) of a sam-le measurement. The observation interval (the number of cyclesver which the measurement is made) can be defined in terms of

number of cycles or a time at a selected system frequency, andould be chosen differently – anything from one to many cycles –or special applications.

Four experiments were undertaken.

1) Using a nominally balanced supply from a three-phase variacconnected to the University’s laboratory supply and with bal-anced star-connected resistive loads each comprising three1 kW resistance heating elements with the star point connectedto the neutral wire.

2) Balanced supply and balanced star-connected reactive loadscomprising in each phase a 56 ohm resistance in series witha reactance of 16.80 ohms at 50 Hz.

3) Balanced supply and unbalanced star-connected resistive loads.4) Positive sequence supply and balanced star-connected resis-

tive load, with zero sequence voltage introduced by connectinga single phase voltage between the star point of the load andthe neutral wire using a step-down single phase isolation trans-former.

For each of the experiments a set of 1002 instantaneous values ofoltages and 1002 instantaneous values of current recorded by theokogawa power meter over four cycles were transferred to a com-uter. (To identify four cycles is easy in post-event processing but

n a practical meter the fundamental wavelength would be identi-ed by a Fourier transform). The instantaneous values of voltagesnd currents were used to calculate the apparent power and poweractor using three different methods: the conventional methodequivalent to that used by the meter itself) and two of the methodshat allow for a neutral wire of resistance equal to the phase wires,r zero resistance. The calculations with zero wire neutral resis-ances were included to illustrate the change in results if the voltageull reference is changed accordingly, given the same physical mea-

urements of current and voltage. (An equivalent calculation for aystem in which the neutral wire has infinite resistance, but usinghe same set of measurements would not be valid, because the cur-ent flowing in the neutral cannot be ignored). The results of the

ystems Research 95 (2013) 140– 147

three calculation methods were compared with the actual readingscollected from the power meter, resulting in four sets of data foreach experiment.

4.1. Calculations

The calculations in accordance with the conventional specifica-tion for the measurement of power are defined as follows. First,the RMS values of each phase current and voltage measured fromthe neutral are calculated from the 1002 sampled values. Then thepower associated with each phase (P1, P2 and P3) is calculated fromthe sum of the products of the sampled pairs of voltage and current,and the total power P is the sum of the three phase components.The apparent power of each phase is the product of the RMS currentand RMS voltage and the total apparent power S is the sum of thethree phase components. The power factor is the ratio P/S.

For the other calculations, the methods applied for each case aresummarised here from the companion paper [2], using the symbolsillustrated in Fig. 1 and with � to represent summation of the 1002sampled values. Note that in these three special cases (with rn = 0,r1 or ∞) S = ||V2

′|| ||I’|| = ||V2|| ||I||, but this is not generally true. Ineach case, S = ||V2

′|| ||I’|| and power factor � = P/S, but the derivationof the value ||V2

′|| and ||I’|| is different for each case as defined inthe companion paper on average power [2]. Clearly, the arithmeticprocesses are relatively simple for any DSP.

When rn = r1 = r2 = r3 then in = −(i1 + i2 + i3) and∥∥I ′∥∥ =

r1/2[(˙i21 + ˙i22 + ˙i23 + ˙i2n)/1002

]1/2.

Since v2 = (e − eref) = {e1 − eref, e2 − eref, e3 − eref, en − eref} whereeref = (e1 + e2 + e3)/4

Note that eref is a function of time and must be calculated atevery sample reading. Calculating the RMS vales and deducting eref(the null point RMS offset value) will lead to erroneous results.

Then∥∥V2

∥∥ = [(˙(e1 − eref)2 + ˙(e2 − eref)

2 + ˙(e3 − eref)2

+˙(0 − eref)2)/1002]

1/2and

∥∥V2

∥∥=[(˙(e1−eref)2+˙(e2 − eref)

2

+˙(e3 − eref)2 + ˙(0 − eref)

2)/1002]1/2

/r1/2, P = ˙(e1 − e ref)i1 +˙(e2 − eref)i2 + ˙(e3 − eref)i3 + ˙(0 − eref)in

Similarly, when rn = 0 then∥∥I ′∥∥ = [(˙i21 + ˙i22 + ˙i23)/1002]

1/2

and the neutral wire is the null point so v2 = (e − eref) whereeref = 0

so∥∥V2

′∥∥ = [(˙i21 + ˙i22 + ˙i23)/1002]1/2

/r1/2

and P = ˙e1i1 + ˙e2i2 + Se3i3For completeness, a set of measurements on a system without

a neutral, with rn = ∞, would be calculated as:

∥∥I ′∥∥ =[

(˙i21 + ˙i22 + ˙i23)1002

]1/2

Since v2 = (e − eref) = {e1 − eref, e2 − eref, e3 − eref, en − eref}whereeref = (e1 + e2 + e3)/3

then∥∥V2

∥∥ = [(˙(e1 − eref)2 + ˙(e2 − eref)

2 + ˙(e3 − eref)2

+˙(0 − eref)2)/1002]

1/2

P = ˙(e1 − eref)i1 + ˙(e2 − eref)i2 + ˙(e3 − eref)i3 + ˙(0 − eref)in

A spreadsheet of the arithmetic calculations is available in theonline version of the paper. It is limited to a small number of sam-ples to illustrate the processes, but can be extended to any numberof samples and cycles.

4.2. Results of laboratory measurements

The results of the laboratory measurements and ensuing cal-culations produce four sets of data for each the four experiments,presented in Tables 1–4.

C.T. Gaunt, M. Malengret / Electric Power Systems Research 95 (2013) 140– 147 143

Table 1Balanced resistive load.

Conventional theory Urms Irms S P �

Yokogawa WT 1600power meter readings

Phase 1 230.12 4.1549 955.73 955.63 0.99989Phase 2 232.2 4.22867 995.96 995.90 0.99994Phase 3 230.99 4.1047 948.02 947.98 0.999953-phase 231.1 4.1755 2899.7 2899.5 0.99993

Calculated values:conventional definition

Phase 1 230.25 4.1600 957.77 957.45 0.99967Phase 2 232.36 4.2885 996.47 994.92 0.99844Phase 3 230.90 4.1060 948.07 947.90 0.999823-phase 2902.3 2900.3 0.99930

General theory: calculated values ||V|| || I || S P �

Actual circuit: (Group 1 Rn = R1 = R2 = R3 = R3) 3-phase 400.02 7.2829 2913.3 2900.3 0.99552If Rn neglected: (Group 2 Rn = 0) 3-phase 400.40 7.2829 2916.1 2900.3 0.99457

Table 2Balanced reactive load.

Conventional theory Urms Irms S P �

Yokogawa WT 1600power meter readings

Phase 1 230.08 2.8125 647.11 451.35 0.69749Phase 2 232.05 2.9624 687.43 479.58 0.69764Phase 3 230.93 2.7546 636.11 436.02 0.685453-phase 231.02 2.8432 1970.7 1367.00 0.69365

Calculated values:conventional definition

Phase 1 230.13 2.8168 648.23 451.98 0.69726Phase 2 232.21 2.9663 688.80 480.43 0.69748Phase 3 230.98 2.7588 637.22 437.05 0.685873-phase 1974.3 1369.5 0.69366

General theory: calculated values ||V|| || I || S P �

Actual circuit: (Group 1 Rn = R1 = R2 = R3 = R3) 3-phase 400.02 4.9905 1996.3 1369.5 0.68600If Rn neglected: (Group 2 Rn = 0) 3-phase 400.30 4.9905 1997.7 1369.5 0.68554

Table 3Balanced supply and unbalanced resistive load.

Conventional theory Urms Irms S P �

Yokogawa WT 1600power meter readings

Phase 1 227.26 2.2336 507.62 507.13 0.99904Phase 2 228.19 4.2214 963.29 963.06 0.99976Phase 3 228.12 4.0536 924.71 924.60 0.999883-phase 227.86 3.5029 2395.6 2394.8 0.99966

Calculated values:conventional definition

Phase 1 227.30 2.2378 508.63 507.68 0.99814Phase 2 228.39 4.2249 964.91 963.79 0.99884Phase 3 228.15 4.0565 925.49 925.24 0.999733-phase 2399.0 2396.7 0.99903

General theory: calculated values ||V|| || I || S P �

Actual circuit: (Group 1 Rn = R1 = R2 = R3 = R3) 3-phase 394.52 6.5818 2596. 7 2396.7 0.92300If Rn neglected: (Group 2 Rn = 0) 3-phase 394.81 6.5818 2598. 6 2396.7 0.92231

Table 4Balanced resistive load with zero sequence voltage.

Conventional theory Urms Irms S P �

Yokogawa WT 1600power meter readings

Phase 1 208 3.85 800 800 1Phase 2 209 3.97 830 830 1Phase 3 289 5.16 1490 1490 13-phase 235 4.33 3120 3120 1

Calculated values:conventional definition

Phase 1 207.58 3.8602 801.2944 801.04 0.99968Phase 2 209.09 3.9734 830.8086 831.47 1.0008Phase 3 288.74 5.1650 1491.327 1491.1 0.999873-phase 3123.43 3123.6 1.0001

General theory: calculated values ||V|| || I || S P �

Actual circuit: (Group 1 Rn = R1 = R2 = R3 = R3) 3-phase 403.17

If Rn neglected: (Group 2 Rn = 0) 3-phase 412.53

8.1763 3296.4 3123.6 0.947598.1763 3373.0 3123.6 0.92608

1 wer Systems Research 95 (2013) 140– 147

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44 C.T. Gaunt, M. Malengret / Electric Po

In each table, the power meter reading is the actual readingiven by the Yokogawa instrument and the calculated value usinghe conventional method is the result of the computer calculationsing the 1002 sets of voltage and current values derived fromhe Yokogawa instrument. The instrument’s transducer accuracyffects the power meter reading and the calculated values equally.he accuracy of the calculated values is further affected by theadence and resolution of the signal samples, the cumulative errorn the summation over a period and truncation.

The largest difference between the actual and calculated val-es of effectively 12 measurements of power, apparent power andower factor is 0.18%, including the effects of truncation and accu-ulated errors during the calculation.Correctly taking into account the relative wire resistances as

equired in the calculations based on the general theory, the appar-nt power differs from the conventional measurement by up to 8%n these experiments. Even greater differences were found with

ore distorting loads.

.3. Interpretation of the results

Although in these experiments all the phase wire resistancesere equal and, therefore, the conditions are special cases of the

eneral theory of average power, several characteristics can bebserved from the results.

1) The apparent power and power factor calculated using the con-ventional calculations correspond closely with those obtainedfrom the Yokogawa WT1600 power meter in all four experi-ments. This step confirms that the extraction of the primarymeasurements from the good quality commercial instrumentand calculation of the derived values in a separate computer isconsistent with the process incorporated in the meter, withinthe limits of the transferred values.

2) In this particular experiment all four supply wires had the sameresistances and therefore only the calculations appropriate forthe special case of Group 1 are valid and give the true apparentpower and power factor for the actual circuit.

3) If different null points are used, as if the neutral wire resistanceswere not the same as the phase wire resistances, different val-ues of apparent power and power factor are calculated. Withthe same currents and voltages for different neutral wire resis-tances, the calculated power factor decreases with the neutralwire resistance. This is consistent with an understanding thatthe maximum power that can be transmitted (i.e. the apparentpower) increases as the resistance of the neutral wire decreasesfrom ∞ to 0, and that the amount of power that can be trans-mitted for the same line losses increases as the neutral wireresistance decreases.

.4. Implementation in a practical meter for m-wire systems

The above description of metering refers to three-phase sys-ems, with or without a neutral wire. The extension to m-wireystems requires only the additional inputs to the meter; the prin-ipal algorithms remain the same. Of course, most power systemsre three-phase systems, but there are many examples of two- andix-phase systems. A completely versatile meter would provide forurrent and voltage inputs for multiple phases, as well as a processor entering the relative wire resistances. The identification by theser of the wire resistances should prompt the instrument to selecthe voltage reference and the appropriate calculation algorithm

s defined in Table 1 of the companion paper [2]. Additionally,he design and manufacture of such an instrument must meethe needs for simultaneous extraction of the voltage and currentnputs with adequate resolution and accuracy, the identification

Fig. 3. 3-wire system with DC voltages.

of the integrating period T, and the selection of a suitable datasampling rate. The output of the instrument will be readings for ameter or control signals for a compensation controller.

5. Applications of measurement based on the generaltheory of power

The development of a general theory has been driven byneeds for solutions to particular power systems problems. Most ofthe time, most power systems operate with sinusoidal, balancedsupplies, for which existing definitions of power are adequate.However, at other times distortion, unbalance and dc or zerosequence current components do upset systems and the conven-tional definitions would give misleading results. The followingexamples illustrate the benefits of the rigorous general theory infour novel practical applications beyond the scope of the powersupply compensation such as was discussed in Section 2.

5.1. Application to mixed ac and dc systems

One such application for m-wire systems might be the optimalcontrol of parallel ac and dc inputs to a system, for which m-wiremetering will give a complete measurement of the apparent powerand introduce alternatives ways of controlling it.

Since the general theory is applicable to any waveforms on sys-tems with any number of wires, it can also be applied to multi-wiredc systems. This is examined by application to a 3-wire dc systemsupplying two resistive loads and which, for simplicity, has all threewires with equal resistances, as illustrated in Fig. 3.

Since one wire can be identified as a neutral and all the wireresistances have equal resistance r, this circuit belongs to the spe-cial case Group 4 identified in [2], so measuring from the neutralwire (wire 3) is appropriate and the null point calculation does notneed resistance weighting but is shown to illustrate the general fivecalculation steps of Section 4.5 in [2] with e = {2, 1, 0}, i = {2, 1, -3},p = e • i = 5.

eref = (2 + 1 + 0)/3 = 1, giving v2 = e − eref = {1, 0, −1}, and v2′ =

1/√

r{

1, 0, −1}

and ||v2′||2 = 2/r,

g = p

||v2′||2 = 5

(2/r)= 2.5r

ia′ = gv2

′ = 2.5√

r{1, 0, −1} = √r{2.5, 0, −2.5} and ||ia

′||2 = r12.5

Since p =∥∥v2

′∥∥ ·∥∥ia

′∥∥ = ((1/√

r)√

2)(√

r√

12.5) = 5 and

s =∥∥v2

′∥∥ ·∥∥ia

′∥∥ =√

2/r ·√

r14 = 5.29 the power factor� = p/s = 0.945.

The dc example can be extended to the average domain byconsidering the current i1 in Fig. 3 to be a square wave of value±2 V, while the voltage in the second ‘phase’ is a constant dc volt-age of 1 V. It is obvious that the optimum reference will not be thesame throughout a period T.

For the first half of T: e(t ) = {2, 1, 0}, i(t ) = {2, 1, −3} and the

1 1calculations are the same as above.

For the second half of T: e(t2) = {−2, 1, 0}, i(t2) = {−2, 1, 1}∥∥i(t2)∥∥2 = 6,

∥∥i(t2)∥∥ =

√6

wer Systems Research 95 (2013) 140– 147 145

st

sr

i

1

k

a

1

P

√

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Apparent power after inter- wire corr ection, Sa

Apparent power, S Qa, corrected without storage

Q, tota l non -active power

QA, Co rre cted on ly with ener gy storage

Powe r, P

C.T. Gaunt, M. Malengret / Electric Po

The power during the second half p(t2) = e(t2)•i(t2) = 5 is theame as during the first half, but the reference point has changedo eref (t2) = (−2 + 1 + 0)/3 = −0.33. Following the same calculation

teps as above it is found that∥∥v2

′(t2)∥∥2 = 4.666/r and

∥∥ia′(t2)

∥∥2 =5.369 and again � = 0.945.

However, taking the average over the whole period, as described

n Sections 4 and 5 of [2],∥∥V2

′∥∥2 =∥∥v2

′(t1)∥∥2 +

∥∥v2′(t2)

∥∥2/2 =

/r (2 + 4.666) /2 = 3.333/r giving∥∥V2

′∥∥ = 1.825/√

r

A = P/∥∥V2

′(t2)∥∥2 = 5/3.33/r = r1.5

Ia′(t) = kAv2

′(t) where v2′(t) = √r{2.5, 0, −2.5} from t = 0 to T/2

nd√

r{−1.66, 1.33, 0.33} from t = T/2 to T,∥∥Ia

′∥∥ = kA ·∥∥V2

′∥∥ =.5

√3.333r = 2.7384

√r.

=∥∥V2

′∥∥ ·∥∥Ia

′∥∥ = 1.825 × 2.7384 = 5

S =∥∥V2

′∥∥ ·∥∥I ′∥∥ = 1.825 × 3.027 = 5.525 where

∥∥I ′∥∥ =r((

√14 +

√6)/2)

1/2 = 3.027 giving � = 5/5.525 = 0.905.It can be seen that the true power factor in the average domain

s less than the average of the power factors during the two halferiods and, therefore, that a compensator with energy storageould decrease the losses by approximately 10% and approx 5%sing compensation without energy storage.

In extending the concept of apparent power and power factor toc circuits it is clear these parameters can have nothing to do withhase angle between voltage waveforms and current waveformsut simply confirms that power factor is a figure of merit reflectinghe effective use of a particular supply, with the intention of min-mizing transmission losses. Therefore, instruments for measuringpparent power and power factor in ac systems should measurehe same parameters for dc or mixed systems, provided the inputsre sensed correctly. This concept opens up new approaches to theperation and control of mixed ac and dc power systems.

.2. Smart grids under fault conditions

The development of power systems in the direction of smartrids with less predictable sources of energy and loads had led var-ous commentators to speculate that the key enabling technologiesor transforming power delivery may include wide area meter-ng, power electronics and demand management, all enabling fastesponse to system variations. For example, voltage control andault ride-through of small turbines can be assisted by reactiveower management, to use the older term. It is essential that thealculation of the non-active power and the compensating currentseeded to bring systems to optimum conditions should be accuratend fast, even, or especially, when normal conditions are disturbed.

Most high voltage power systems operate with neutrals effec-ively earthed (grounded) at all transformers, so the resistance ofhe neutral wire is usually a low, non-zero value determined byhe substation earthing (grounding) of the power transformers, oran be neglected completely in terms of the costs of losses. Onhe other hand, many low voltage power systems are built witheutral conductors that are not the same resistance as the phaseires, especially where French or Scandinavian bundle conductors

re installed. Although under perfect conditions there is no currentn the neutral of either group of power systems, in practical cases

ith harmonics, unbalance, zero sequence or dc components theurrent in the neutral cannot be ignored. Therefore the definition of

on-active power will be significantly different for large wind gen-rators connected to high voltage networks (rn low or zero) thanor small wind generators or inverter-connected PV panels on lowoltage systems (rn > r).

Fig. 4. Relationship between apparent power Sa with inter-wire correction andcorrection towards power P only possible with energy storage components.

The maximum apparent power that can be transmitted in thehigher voltage network exceeds the value indicated by conven-tional theory, but in the lower voltage networks is lower thanindicated by the conventional theory, and under non-ideal con-ditions the current in the neutral affects the fault response andstability of the power system.

5.3. Quality of supply affected by disturbing loads

The quality of power supplies is regulated in most countries bytariffs and quality of supply specifications referring separately toharmonics, unbalance and reactive (or non-active) power compo-nents. Many disturbing loads affect all three of these parameterssimultaneously and only a valid measure of the apparent powerand power factor, taking into account the unbalance and distor-tion, will give a correct indication of the efficiency of use of thetransmission network. Thus, correctly measured non-active powercould be an appropriate basis for tariffs designed to recover thecosts of disturbing loads and encourage the improvement of theirperformance.

It is not implied in this work that power systems or individ-ual transmission links should operate at unity power factor. Theydo not at present, and a more rigorous definition of apparentpower and power factor will not change the optimum economicoperation in future. However, the proposed metering is capableof identifying how much compensation requires storage capac-ity or can be corrected merely with energy switching betweenwires, as illustrated in Fig. 4, from which the relevant costs ofcorrection can be estimated even if complete correction is notimplemented.

5.4. Power networks in the presence of geomagnetically inducedcurrents

Another condition that spurred the research leading to thederivation of a general power theory valid for distorted voltageconditions was the need to understand through more accuratemodelling the response of power systems to geomagneticallyinduced currents (GICs). Quasi-dc GICs, with frequencies below1 Hz, are induced in transmission lines by geomagnetic dis-turbances initiated by solar activity. The GICs flow throughtransformers and, by biasing the magnetic circuit and causinghalf-wave saturation, generate harmonics and voltage collapse.Therefore, the sinusoid is distorted by harmonics, unbalancedbecause of the half-wave saturation and offset by the GIC flow-ing in the neutral and shared between all phases. It has beenreported that geomagnetic storms are associated with “massiveswings of reactive power” [10]. However, since that reactive powerwas measured by conventional methods that cannot take accountof the dc component, unbalance or the harmonics generated in

the transformers, the real behavior of the system is not describedaccurately. Work on laboratory models [11] shows that non-activepower measurement in a transformer with an unbalanced com-plex load under distorted conditions is significantly higher when

146 C.T. Gaunt, M. Malengret / Electric Power S

0

50

100

150

200

250

300

350

0 0.2 0.4 0.6 0.8 1

Injected DC [A]

AP

, Q

[V

A, V

Ar]

S (ge n t h)

Q (gen th)

S (c onv)

Q (conv)

Fig. 5. Higher apparent and non-active power (S and Q) associated with injectedDu

mtTcsr

6

pbdDmatafs

amtrd

(

(

7

ga

C in transformer, when calculated with general power theory (gen th) instead ofsing conventional definition (conv).

easured according to the general power theory than according tohe conventional method of measurement, as illustrated in Fig. 5.hese results indicate that the reported increase in reactive poweronsumption in transformers subjected to GICs [12,13] and theystem voltage drop need to be re-assessed and further work isequired.

. Introducing a new measurement specification

The experimental results of the prototype meter and theroof-of-concept demonstration illustrate significant differencesetween measurements of non-active power and power factor inisturbed or distorted conditions according to the approach used.ifferences of more than 8% have been shown between measure-ents using the conventional approach of existing specifications

nd the general theory that includes the resistances of the conduc-ors to accurately determine the apparent power, non-active powernd power factor. These differences have significant implicationsor those publishing, using or complying with standard meteringpecifications.

Given the justifiable reluctance to adopt new definitions ofpparent power and non-active power in a mature industry with souch legacy instrumentation, control, metering and tariffs aligned

o what might become out-of-date standards [14], an appropriateesponse will be required especially from the international stan-ards organisations, including IEEE and IEC.

It is proposed that a two-step approach will be useful:

1) Identify and define when the loss of accuracy of simple, conven-tional instruments under disturbed conditions can be toleratedand, conversely, where the accuracies of measured parame-ters are sufficiently important to justify installing instrumentsthat are correct in the presence of distortion, unbalance andzero-sequence and direct current components.

2) Where accurate measurement under non-ideal conditions isrequired, such as for the applications described above and byseveral other authors, then a new specification for instrumentsfor metering and compensation control will be required, basedon suitable inputs (including relative wire resistances) andmeasurement algorithms.

. Conclusion

Linear algebra has provided an approach that integrates aeneral theory of instantaneous power with a general theory ofverage power. The theory provides easy ways of measurement

ystems Research 95 (2013) 140– 147

of power systems even in the presence of distortion, harmonics,zero sequence and dc components, and provides answers that areinternally consistent in all cases. Metering can distinguish betweenthe component of non-active power that can be corrected by inter-wire switching by power electronics and that which can only becorrected with energy storage components, giving practical inter-pretation to both components of non-active power.

The differences between various other approaches to powertheory and the reasons why they do not always give results thatare consistent with each other, or produce complete correction ofpower supplies to unity power factor, can be easily appreciated.While some of the objectives of compensation might be adequatelymet for some purposes by other approaches, the new theoreticallyrigorous and easy to apply practical measurement meets the need,identified by many authors including Emanuel and Czarnecki, foran unambiguous definition of apparent power and power factor.It is immediately evident that the new approach can provide abenchmark against which any other algorithms for power factorcorrection can be compared, recognising it may not necessary tocorrect to unity power factor in all cases and that some approachesto limited correction might be useful.

In addition to the issues raised in terms of practical power sys-tems measurement and compensation, the general power theoryhas wide implications for the teaching of apparent power, non-active power and power factor. Also, it provides a new tool formaking power calculations in a way that takes account of impor-tant conditions occurring in practice. However, it does not yet give“all the answers” and there is broad scope for further research.

From a practical perspective, power systems engineers will needways to determine the effective wire resistances of power networksto the point of metering; not necessarily the absolute values ofthe resistances but their relative values. The general power theoryopens up new ways of controlling the compensation of non-activepower, such as in grids with dispersed non-synchronous genera-tion, unbalanced loads or loads controlled by power electronics,and in communication-responsive networks, and these applica-tions will need to be developed. Whatever losses occur at unitypower factor, any lower power factor represents avoidable losses,requiring extra capacity and incurring costs. If tariffs are adopted todiscourage disturbing loads because they reduce the efficiency oftransmission, then accurate meters and tariffs for those loads mustbe developed.

From a theoretical perspective, Jeon [15] investigated systemswith frequency dependent resistances, such as might be affectedby harmonics. The present general approach to m-wire systemsonly considered the wire resistances to be constant, which is fullyvalid only after compensation. Further, it is recognised that the newapproach does not include the response of the load to differentvoltage conditions as would arise after correction. Therefore, theaction of voltage correction in response to a disturbance will needto be extended to the transient behavior of the load. The approachmight be extended to allow the effect of unsymmetrical voltagesto be distinguished from unsymmetrical loads. In a similar way,if voltages change because of the change in line losses achievedby compensation, iterative recalculation might be implemented,but this may change the basis of the definition of active current insystems supplying distorting loads.

Acknowledgement

The authors acknowledge the valuable contributions to all three

papers describing the development and application of the gen-eral theory of power that have been made by our colleagues andreviewers. The value of critical comment and review should neverbe under-estimated.

wer S

A

t

R

[

[

[

[

[power, (note), European Transactions on Electrical Power 3 (Jan/Feb) (1993)

C.T. Gaunt, M. Malengret / Electric Po

ppendix A. Supplementary data

Supplementary data associated with this article can be found, inhe online version, at http://dx.doi.org/10.1016/j.epsr.2012.07.019.

eferences

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[2] M. Malengret, C.T. Gaunt, General theory of average power for multi-phasesystems with distortion, unbalance and direct current components, ElectricPower Systems Research (2011), http://dx.doi.org/10.1016/j.epsr.2011.11.020.

[3] IEEE Power and Energy Society, Standard definitions for the measurement ofelectric power quantities under sinusoidal, nonsinusoidal, balanced or unbal-anced conditions, IEEE Standard, 1459–2010.

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