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Power Notes 9

Power Notes 9 - 828 IEEE'I'RANSACI'lONS ON POWER...

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Unformatted text preview: 828 IEEE 'I'RANSACI'lONS ON POWER ELECTRONICS, VOL 19, NO. 3. MAY 2004 On Some Misinterpretations of the Instantaneous Reactive Power p—q Theory Leszek S. Czarnecki, Fellow, IEEE Abstract-—The main features of the instantaneous reactive power (IRP) p-q Theory, considered as a power theory of three-phase systems, are analyzed in this paper using the theory of the currents’ physical components (CPC).'This analysis shows that the p and q powers are not associated with separate power phenomena, but with multiple phenomena. Moreover, the results of the IRP p-q Theory contradict some common interpretations of power phenomena in three-phase circuits. Namely, according to the IRP p-q Theory the instantaneous reactive current can occur even if a load has zero reactive power, Q. Similarly, the instantaneous active current can occur even if a load has zero active power, P. Moreover, these two currents in circuits with a sinusoidal supply voltage can be nonsinusoidal even if there is no source of current distortion in the load. The analysis shows that a pair of values of instantaneous active and reactive p and q powers does not enable us to draw any conclusion with respect to the power properfies of three-phase unbalanced loads even in a sinusoidal situation. Thus, the Instantaneous Reactive Power p-q Theory does not identify power properties of such loads instantaneously. This conclusion may have an importance for control algorithms of active power filters. The paper reveals the relationship between the p and q powers and the active, reactive and unbalanced powers, P, Q, and D and specifies the required energy storage capability of activerpower filters operated under sinusoidal unbalaneed conditions. Index Terms—Active current, active power filters, apparent power, currents’ physical components (CPC) power theory, in- stantaneous active power, instantaneous reactive power, reactive current, switching compensators, unbalanced power, unbalanced systems. I. INTRODUCTION HE INSTANTANEOUS reactive power (IRP) p—q Theory, developed by Akagi, Kanazawa and Nabae, [l], [2], pro- vides mathematical fundamentals for the control of switching compensators, known commonly as “active power filters.” Although there is still substantial confusion [3] with respect to power phenomena in electrical systems and there are reports [4], [5] on some shortcomings of the IRP p—q Theory, it seems to be well established [6]—[9] in the electrical engineering commu- nity involved in switching compensator design. Moreover, there are attempts [10]—[l2] to provide deeper fundamentals for this Theory and it is becoming a theoretical tool for power proper- ties of three-phase system analysis [13]—[15] and instrumenta- tion [16]. Manuscript received March 14, 2003; revised October 15, 2003. Recom- mended by Associate Editor H. du T. Mouton. The author is with the Electrical and Computer Engineering Depart- ment, Louisiana State University, Baton Rouge, LA 70802 USA (e-mail: [email protected]; czameck®ee.lsu.edu). Digital Object Identifier 10.1109/I'PEL.2004.826500 When the IRP p—q Theory is considered as a theoretical fun- damental for a control algorithm design, it is irrelevant whether it interprets power properties of electrical circuits correctly or not. It is enough that it enables the achievement of the control objectives. However, when it is considered as a power theory one could expect that it does provide a credible interpretation of power phenomena in electrical systems. Having this expectation in mind, the following dilemma oc- curs. Power properties of three-phase, three-wire systems with only sinusoidal voltages and currents, i.e., even without any har- monic distortion, are determined by three independent features of the system. 1) permanent energy transmission and associated active power, P; 2) presence of reactive elements in the load and associmed reactive power, Q; 3) load imbalance that causes supply current asymmetry and associated unbalanced power, D. Thus, how can the IRP p—q Theory, based on only two power quantities, p and q, identify and describe'three independent power properties? Moreover, according to Akagi and Nabae, [2] who developed the Instantaneous Reactive Power p—q Theory, its development was a response to “...the demand to instantaneously compensate the reactive power. . ..” The adverb instantaneous in the name of this Theory and definitions of p and q powers in terms of instantaneous value of voltages and currents, suggest the possibility of instantaneous identification and compensation of the reactive power of a three-phase load. This is one of the main reasons for this theory’s attractiveness, both as a theoretical fundamental of control algorithms and as a power theory. Thus, the question occurs, is such an instanta- neous identification of power properties of three—phase systems, possible? The answer to this question is of a fundamental value for the power theory. It could be important also for control algorithms of active power filters. To answer such a question, this paper investigates how the IRP p—q Theory interprets and describes power phenomena in three-phase, three-wire systems. The Theory of the Currents’ Physical Components (CPC) developed in [17] by the author of this paper is used as a tool for the study. The IRP p—q Theory was deve10ped for three-phase systems with nonsinusoidal voltages and currents. To provide credible results for such systems, it should provide them for any sub-set of such systems. Three-phase, three-wire systems with sinu- soidal voltages and currents form just such a sub-set with rela- tively simple and easy to comprehend power phenomena. There- fore, properties of the IRP p-q Theory are verified using such simple circuits. 0885-8993/04S20.00 © 2004 IEEE CZARNECKI: MlSlNI‘ERPRETATlONS OF THE INSTANTANFDUS REACHVE POWER p-q THEORY 829 II. INSTANTANEOUS ACTIVE AND REACHVE CURRENTS The notions of the active and reactive currents have meanings that were established in electrical engineering long ago. The active current, defined by Fryze [18] in 1932, is the smallest load current that is necessary [10] if the load at the supply voltage u(t) has the active power, P, and has the same waveform as the supply voltage. This current was defined as . P za(t) = 8711.0) = G¢u(t) (l) where U is the supply voltage RMS value. For three-phase, three-wire systems, shown in Fig. l, Fryze’s definition of the active current is generalized [17] to the form ‘Ra P “R in“) = 1:5“ '- W Us = G¢u(t) (2) ’Ta “T where ||u|| denotes the three-phase RMS value of the supply voltage, namely ||u|| = ‘lUfi + U; + U72“ (3) The reactive current is the component of the supply current delayed by 1r / 2 with respect to the supply voltage and is defined insingle-phase systems with sinusoidal voltage and current as —Qd d W) = —U—2d(wt)u(t) = B‘aot) W)- (4) For three—phase, three-wire systems this definition is general- ized [17], [21] to . -Q d d 1,-(t) — wmua) — Be d(wt)u(t). Botluhe active and reactive currents have an explicit physical meaning. They are associated with the presence of the active and reactive powers, P and Q, and are related to the load equivalent conductance, Ga, and susceptance, Be. The concept of the ac- live current is also important for the design of compensators. Because it is the smallest supply current of the load that has the active power, P, this is the only current that should remain in the supply lines of the load after compensation of all useless current components. Now, let us compare the features of the active and reactive currents, as defined by (2) and (5), with the instantaneous active and reactive currents in the IRP p-q Theory. Observe however, that the adjective instantaneous in the name of these currents does not distinguish them from the common active and reac- tive currents, since definitions (2) and (5) specify instantaneous values of these currents. The IRP p-q Theory has evolved from the Fortescue, Park and Clarke Transforms of voltages and currents specified in natural, phase B, S and T coordinates, as shown in Fig. 1, into a pair of voltages and currents in orthogonal a and ,3 coordinates. The Clarke Transform of three-phase voltages has the form (5) UR ua_21_%—% 6 [“fi]— 50735—’§é :2: U Fig. l. Three-phase, three-wire system. For three-phase, three-wire systems as shown in Fig. l, with line voltages referenced to an artificial zero, so that 11.3 + as + W E 0, the Clarke Transform of the line voltages can be simplified to theform [:2] = [Q 35] [11:] =Cl:::l and similarly for the line currents, since in + is + i7» E 0 Jx/i: 0 l [“‘l - C [“‘l 1 . — . . l 7; V5 25 15 The line currents can be calculated from the currents in the a and [i coordinates with the inverse Clarke Transform [in] = fi’ 0 in = 0-1 i0 (9) ZS _ 713. 715 ig ig . With the voltages and currents transformed to the a and [3 co- ordinates, the instantaneous power of the load can be expressed as (8) p = uaia + 11,3753 (10) referred to as the instantaneous real or instantaneous active power in the IRP p—q Theory. The instantaneous imaginary power, q, is defined in [2] as (11) and is usually referred to [6]——[l6] as the instantaneous reactive power. With these two instantaneous powers, the instantaneous ac- tive and reactive currents are defined in [2]. The instantaneous active current, ip, is defined in the a and [i coordinates as q = uaip —— rigid, ua _ 11.3 i = — = -— up 11% + “33177 1/31; “(2' + “g? (12) while the instantaneous reactive current, iq, is defined as —ufi . “a z = — ’l, = — . all 11% + ugqi flq 11% + ugq (13) The instantaneous active and reactive currents in the supply lines can be calculated from these currents in the a and [i coordinates with the inverse Clarke Transform, (9), namely erode] Worm . zSp ‘fip , iSq ifiq Unfortunately, the reactive current has little in common with the reactive power, Q, of the load. This is shown in Illustration (14) \\ 830 82 Fig. 2. Example of a circuit with resistive unbalanced load. 1, where the IRP p-q Theory is applied to a circuit with zero reactive power, Q. Illustration 1: Let us assume that a resistive load, connected as shown in Fig. 2, is supplied from a symmetrical source of a sinusoidal, positive sequence voltage, with it}; = x/2U cos cult, U = 120 V. Given such assumptions, the supply voltage in the a and fl coordinates has the value an _ C fiUcoswlt ug _ «EU cos(w1t — 120°) Since the line currents are equal to fiU cos wlt =[\/§Usinw1t]' (15) 1,, = \/21cos(w1t + 30°) = 45, I = 103.9 A, iT o. The supply current in the a and ,8 coordinates has the value [in] :C[ in ] _ [filcos(w1t+30°)] i5 —’iR — -ICOS(w1t + 30°) (16) and consequently, the instantaneous active power of such a load is equal to p = uaia + «1,511; = \/§UI [1 + cos2(w1t + 30°)] (17) and the instantaneous reactive power is equal to q = uaig — 1152",, = —\/§UIsin 2(w1t + 30°). (18) Thus, the instantaneous active current of the load in the a and [3 coordinates, can be calculated. It is equal to . in, = fl}; = I [1 + c052(w1t + 30°)] coswlt, (19) u o . 1-6,, = mp = [[1 + c052(w1t + 30 )]smw1t (20) which in the phase-coordinates is equal to [21m] =I[1+cos2(w1t+3°°)]c_l[ coswlt] tsp sin out = £1 [1 + c052(w1t + 30°)] coswlt ] . (21) 0X [cos(w1t — 120°) The instantaneous reactive power, q, of the load considered in this Illustration is not equal to zero, hence, according to the IRP p-q Theory, a reactive current has to occur in the supply lines. Its value in the a and ,8 coordinates is equal to -1“, 2 q = Isin 2(w1t + 30°) sin cult, (22) i =— aq u§+ufi IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 19. NO. 3. MAY 2004 u ° 2 q = —Isin 2(w1t + 30°)cosw1t m <23) ifiq = and consequently, the inverse Clarke Transform results in the instantaneous reactive current in the supply lines [i39]=1siu2(w1t+ 30°)0-1[ ’qu 2 . = §Ism2(w1t+30°) X sin wlt sin(w1t — 120°) sin wlt — COS (4)1t (24) Equations (21) and (24) show that the names instantaneous ac- tive current and instantaneous reactive current were given in the IRP p-q Theory to currents that have nothing in common with the notion of the active and reactive currents as used in elec- trical engineering. Also the reactive current 2}, occurs in supply lines of the load in spite of its zero reactive power, Q. Moreover, both the active and reactive currents in systems with sinusoidal voltage and linear loads that do not generate harmonics, as con- sidered in the Illustration, are nonsinusoidal. For example, the active current in the line R is equal to i3? = % [2 coswlt + cos (wlt + 60°) cos(3w1t + 60°)} (25) thus, its contains the third order harmonic. This conclusion ob- tained from the IRP p-q Theory is in sharp contradiction to the notion of the active current that was introduced to electrical en- gineering by Fryze. The’suggestion of the IRP p-q Theory that line currents of a linear load, that does not generate harmonics in a system with a sinusoidal voltage, contain a nonsinusoidal com- ponent should be considered as a major misconception of elec- trical phenomena in three-phase circuits. Moreover, unlike the active current, in, defined by Fryze, the active current, ip, that results from the IRP p—q Theory is not the current that should remain in the supply lines after the load is compensated to unity power factor. It cannot be considered the compensation goal. Illustration 1 also shows that the Instantaneous Reactive Power p-q Theory does not identify power properties of the load instantaneously. Both powers are time-varying quantities, so that, a pair of their values at any single instant of time does not identify power properties of the load. The possibility of instantaneous identification of the active and reactive powers, 1) and q, does not mean that power properties of the load are identified instantaneously. For example, for t = 'r, such that (our + 30°) = 0, the instantaneous reactive power, q = 0, as it is in a circuit with purely resistive balanced load, while for t = 1', such that (wrr + 30°) = 90°, both the instantaneous active and reactive powers are equal to zero. Thus, power properties of the load cannot be identified without observation of the p and q powers over the entire cycle of their variability. However, even such an observation of these powers does not explain power properties of the load without additional analysis. For example, the instantaneous reactive power, q, has occurred in the circuit considered in Illustration 1 not because of the load reactive power, Q, but because of the load imbalance. How could this imbalance be identified having the values of the instantaneous reactive power q? Therefore, the IRP p-q Theory has no advantages with respect to the time interval needed for CZARNECKI: MISINTERPRETATIONS OF THE INSTANTANEOUS REACTIVE POWER p-q THEORY 831 Fig. 3. Example of three-phase circuit with purely reactive load. the identification of load properties over power theories based on the frequency-domain approach that require the system to be observed over one period, T, of its variability. Illustration 1 has demonstrated that the instantaneous reactive current has nothing in common with the load reactive power, Q. It also occurs that the instantaneous active current in the IRP p-q Theory has nothing in common with the load active power, P. This is shown in Illustration 2. Illustration 2: Let us consider a circuit with a purely reactive load as shown in Fig. 3, supplied like in Illustration 1, thus, with the voltage in the a and fl coordinates specified by (15). The line currents in such a circuit are equal to in = fll cos(w1t — 60°), I = 103.92 A is = '43, ir = 0. (26) thus, the line currents in the a and fl coordinates are equal to z“, _ 0 fit cos(w1t — 60°) i5 — -\/§Icos(w1t— 60°) _ fil cos(w1t — 60°) _ [ —Isin(w1t— 60°) ] ' (27) Consequently, the instantaneous real and_imaginary powers have the values p = uaia + um = JEUI cos(2w1t — 30°), (28) q = uaip — um}, = t/fiUI [1 + sin (2w1t — 30°)]. (29) In spite of the zero active power of the load, there is a nonzero active current in the circuit. Its value in the a and fl coordinates is equal to "a = u2+ufep = g [cos(w1t — 30°) + cos (3w1t — 30°)], (30) "fl i =——-— 5? ufi+u§p = —- g[sin(w1t — 30°) - sin (3w1t — 30°)] (31) and in the phase coordinates is equal to in m 7:5? ifip — 1 f 0 l " I 1 '75’ 75 % [cos (wlt — 30°) + cos (3w1t — 30°)] X ~12:[sin(w1t—30°)—sin(3w1t—30°)] ' (32) In particular, the instantaneous active current in the line B is equal to tap: %[cos(w1t + 30°) + cos(3w1t — 30°)]. (33) This means that according to the IRP p-q Theory an active cur— rent occurs even in purely reactive circuits. This current is non- sinusoidal even if there is no source of harmonics in the supply source and the load. Illustrations 1 and 2 show some features of the IRP p-q Theory when it is applied to systems with sinusoidal voltages and currents without any explanation of these features. They can be explained using the Theory of the Currents’ Physical Components. HI. APPARENT POWER RELATED AMBIGUI’I‘Y Power theory provides definitions of various powers in elec- trical circuits along with relations between them. Unfortunately, there is a major ambiguity with respect to one of the most com- monly used powers namely the apparent power, S, in three- phase systems. This ambiguity exists even at sinusoidal voltages and currents [3]. Namely, according to the IEEE Standard Dic- tionary of Electrical and Electronics Terms [19], the apparent power is defined as SA = URIR + UsIs + UTIT (34) and this power is referred to as arithmetical apparent power or as SG=W referred to as geometrical apparent power. There exists a third definition of the apparent power for three-phase three-wire sys- tems introduced for such systems under nonsinusoidal condi- tions in [17], but not referenced by the IEEE Standard Dictio- nary, namely (35) S = llull ' Hill- (36) For systems with sinusoidal voltages and currents this apparent power is equal to 5:53=‘,/U,2,+U§+U%-,/I},+U§+I; and was suggested [20] by Buchholz in 1922. In balanced three- phase systems with sinusoidal voltages and currents these three apparent powers have the same numerical value. The difference becomes visible when the load is unbalanced or waveforms are nonsinusoidal. This difference in a circuit with an unbalanced load is shown in Illustration 3. Illustration 3: For the system shown in Fig. 2, definitions (34), (35), and (37) result in (37) SA = 24.9 kVA, $0 = 21.6 kVA, $3 = 30.5 kVA and consequently, the power factor, /\ = P/ S, that means the rafio of the active and apparent power depends on the selection of the apparent power definition. Since the active power of the load in the Illustration considered is equal to P = 21.6 kW, 832 Fig. 4. Circuit with balanced resistive load and load active power, P = 100 kW. then, depending on the selection of the apparent power defini- tion, different values of the power factor are obtained, namely AA = 0.87, /\G = 1, AB = 0.71. Thus, it seems to be unclear what is the true value of the power factor. It is unclear as well, what is the power rating of a com— pensator, Sc, needed for the power factor improvement to unity value, that means Sc = \/ 52 — P2. The result depends on the selection of the apparent power defini- tion. Unfortunately, any relation between powers in three-phase circuits cannot be established without due clarification of this ambiguity related to the apparent power definition. (3 8) IV. SELECTION OF THE APPARENT POWER DEFINITION The apparent power is notassociated with a particular power phenomenon in electrical circuits. It is a conventional quantity, a figure of merit that describes the supply—equipment with respect to the voltage and current RMS values the supply is capable to provide customers. The same is with the power factor, A. It is the ratio of the active and apparent powers, P/ S. Its decline at a specified load active power, P, is associated with an increase in the RMS value of the supply current and consequently, with an increase...
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