See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/224559053 The integral method to calculate the power states in electrical circuits Conference Paper in PRZEGLĄD ELEKTROTECHNICZNY · June 2009 DOI: 10.1109/CPE.2009.5156032 · Source: IEEE Xplore CITATIONS READS 5 2,259 1 author: Marek Hartman Gdynia Maritime University 43 PUBLICATIONS 150 CITATIONS SEE PROFILE All content following this page was uploaded by Marek Hartman on 14 April 2014. The user has requested enhancement of the downloaded file. Marek T. HARTMAN Gdynia Maritime University/ Department of of Marine Electrical Power Engineering, Gdynia, Poland The integral method to calculate the power states in electrical circuits Streszczenie. W artykule przedstawiono uwagi i propozycje dotyczące opisów stanów energetycznych w obwodach elektrycznych. Na podstawie obserwacji Emanuela, Erlickiego i Czarneckiego opisanych we wcześniejszych publikacjach, autor przeprowadza analizę matematyczną otrzymując zmodyfikowaną formułę mocy biernej QI zaproponowaną przez Iliovici. Formuła ta ma związek z energią pola elektrycznego i magnetycznego zgromadzonego lub występującego w otoczeniu elementów reaktacyjnych. Autor proponuje rozróżnienie przyczyn powstawania mocy biernej QI oraz mocy nierównokształtności K. Obie te moce sa składnikami mocy nieaktywnej N. Sugeruje się przeprowadzenie procesów redukcji lub eliminacji składników mocy nieaktywnej w dwóch etapach. (Metoda całkowa obliczenia stanów energetycznych w obwodach elektrycznych) Abstract. Some remarks on the power states calculation in the electrical circuits have been described. The new equation for Iliovici’s reactive power has been proposed. Based on Iliovici’s concept of the reactive power, Emanuel’s, Erlicki’s and Czarnecki’s observation concerning the reactive power properties, the new term of power based on equiformity of voltage and current waveforms has been introduced. The two steps of non-active power reduction or elimination has been also proposed. Słowa kluczowe: moc bierna, moc nieaktywna, analiza obwodów, teoria mocy Keywords: reactive power, non-active power, circuit analyze, power theory Introduction The problem of determining or defining the reactive power and the non-active power has existed for many years. An attempt to define these powers can be found in the American standard IEEE Std 1459-2000 [1], which gives the following: the definition of the reactive power Q according to Iliovici’ conception [2] for single-phase sinusoidal voltage u (t ) and current i (t ) waveforms (1) Q QI 1 2 udi 1 2 idu kT kT i[ udt ]dt the definition of the reactive power QB according to Budeanu’s conception for single-phase non-sinusoidal voltage and current waveforms (2) QB U h I h sin h h where: h is a shift-phase angle between the voltage and the current of h -th harmonic, h is the the definition of the non-active power N (3) harmonic order N S 2 P2 The standard [1] contains also the information that Czarnecki [3] and Lyon [4] questioned the usefulness of the power QB . In the author’s opinion there are still ambiguities concerning the notion of “reactive power” and “non-active power”, their mutual relations or interpretations. The lack of clear, unambiguous physical interpretation of both the powers has resulted in scientific polemics [5]. Reactive power and nonactive power Fryze [6][7] introduced the notion of ”multiplicity of the function” – he wrote that: “the power factor is equal to the unity, which means it reaches the maximum, when at every single moment the instantaneous current in the load is proportional to the instantaneous voltage of the load. That 194 is: when the function i (t ) is the multiple of the function u (t ) so that u (t ) Ri (t ) . Czarnecki [8] introduced the notion of “mutually proportional” characteristics: which means such y (t ) cx(t ) characteristics, for which we have and y c x . On the basis of Fryze’s and Czarnecki’s proposals it is possible to come up with the following definition of the “proportional characteristics of the voltage u (t ) and the current i (t ) ”: Def: The necessary and sufficient condition for the periodical current and voltage waveforms to be proportional is that the linear equality u (t ) R i (t ) occurs, where proportionality factor R is constant in the whole time interval [0 t T] and R is the natural number. Proportional waveforms are characterised by two properties: - there is no time-shift between the voltage characteristics u (t ) and the current characteristics i (t ) , therefore; u (t ) R i (t ) or u (t ) Ri (t ) - the voltage u (t ) and the current i (t ) waveforms have the same shape so they have the equiform waveforms. On the basis of the above, it is possible to consider a few special cases: a. Only the constancy of the factor R in the whole time interval [0 t T] causes the equality S P U RMS I RMS occurs and the power factor P / S 1 . b. If there appears a time-shift between the voltage and the current satisfying the following equality u (t ) A i (t ) or u (t ) A i (t ) , then the reactive power QI appears in the circuit. For the sinusoidal voltage and current characteristics it is QI U RMS I RMS sin and S 2 P 2 QI2 . c. If the load is resistive but non-linear or non-stationary and if its resistance is the time function Rt f (t ) the situation is more complicated. Author take into account Steinmetz’s remark included in [9]. For such a PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010 load characteristic the voltage and current are in phase but they have not the equiform waveforms. This means, in the author’s opinion that, the reactive power QI (1) is equal to zero, but S P . d. For any single-phase circuit with the load of the nonresistive character and for the nonsinusoidal voltage and current characteristics, except the reactive power, the nonactive power N characterizing also the shape of the voltage and the current waveforms. The conception of reactive power by Iliovici According to Ohm’s law, on the terminals of any twoterminal network representing any kind of load (linear, nonlinear, time - variant and so on), the fraction of voltage u (t ) and current i (t ) describes the load character. Let mark this fraction as z (t ) so that (4) (u 0 i 0) z (t ) If functions u (t ) and i (t ) are of the class C1 (have derivatives and are continuous in the whole range of the function domain) functions u (t ) and i (t ) are not of the class C1, but have finite derivatives in the points, where the function is not smooth (e.g. the characteristics of voltages and currents describing the circuits with converters, in which fast changes of the function of switches states occur), then it is possible to apply Green’s theorem to (10) as follows (11) Y 1 1 [ (udi idu )] { [ (i ) (u )]didu} u 2 M 2 F i Transforming the equality (11), we obtain : Y u (t ) i (t ) (12) during the time interval (0 < t < T). Derivative the both side of (4) as Majewskij’s proposed [10] we obtain du di u dz dt 2 dt dt i 1 1 du (t )di (t ) F where: F is the area of the surface inside the contour M . The equation (12) can be written in the following form : The equation (5) can be rewritten as (6) 1 { [(1) (1)]di (t )du (t )} 2 F F i (5) 1 [ (udi idu )] 2 M Y (13) dz 2 du di di du i i u (u i ) dt dt dt dt dt 1 F 1 [ (udi idu )] 2 M T Y (14) T dz 2 di du 0 dt i dt 0 (u dt i dt )dt (7) Divide the both side of (7) by 2 we get (8) 1 2 T dz 1 T di 0 du T T (0 < t < T) marks with its end a line l . If the line l is continues and constitutes a closed curve, i.e. a closed contour M , then it is possible to substitute the integration in time [in the equation (9)] with the integration over the curve l , that is: over the contour M . Treating i (t ) and u (t ) as the parametric equations of the curve l (the contour M ), the equation (9) can be written down as follow: T 1 di du 1 { [u (t ) i (t ) ]dt} dt dt 2 0 2 [udi idu ] M 1 udi idu 2Q I M M T 1 di du 1 di du (u i )dt [(u (t ) i (t ) ]dt dt dt dt 2 0 dt 2 0 Y 1 QI we can ask: what the Y stand for ? It can be concluded from the equality (9) that the vector Y on the plane [ u (t ), i (t ) ] while moving in the time interval (10) F 1 di du 1 { [u (t ) i (t ) ]dt} 2 0 dt dt 2 0 Marking the right hand side of (8) by letter "Y " (9) Y M In that way we obtained the answer on given question: the formula (9) marked by Y is equal to half of the reactive power QI dt i dt 2 (u dt i dt )dt 2 M Based on (1)(9) and (13) we can write Integrating (6) during the period of time [0, T], we obtain : T 1 udi idu [udi idu] M (15) Y 1 di du { [u (t ) i (t ) ]dt} 2 4 0 dt dt 1 4 [udi idu ] M F 1 2 2 1 udi 2 idu M M The equation (15), similar to proposed by Majewskij [10], includes the new formula of Illovici’s reactive power (16) QI 1 4 T T dz 2 1 di du 0 dt i dt 4 0 (u dt i dt )dt In the author’s opinion, limiting the formula (1) only to the sinusoidal voltage and current waveforms is unjustified. The equality (15) has the universal character and does not depend on the shape of the characteristics. So, the value of the reactive power Q QI can be calculated on the basis of (15) or (16). Emanuel and Erlicki [11][12] pointed out a possibility of practical use of (15) forty years ago. In their opinion PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010 195 “nonlinear resistance behaves like a reactive-power generator while having no energy-storing elements”. Emanuel and Erlicki’s observations were confirmed among others by Czarnecki, who wrote [8] that: ”the reactive power in single-phase circuits with sinusoidal current and voltage characteristics can be associated with the energy accumulation in the reactance elements but it is not the general property of the reactive power Q and we that can interpret the reactive power Q as the measure of the influence of the phase-shift of the current relative to the voltage on the apparent power of the supplying source. This shift can be caused by the presence of reactance elements or simply by the periodical switch …….” In the author’s opinion, the quoted Authors are right with regard to the “reactive power”. It seems, however, that two causes of presence of the “non-active power” in the circuits should be clearly emphasized and interpreted. Nonequiformity power K In Budeanu’s conception it is possible to notice an attempt to identify the causes of non-active power generation. Not getting involved in the controversy over the rightfulness of Budeanu’s conception, it is necessary to notice that Budeanu suggested distinguishing ”the reactive power QB ” as the measure of the influence of the phaseshift of the current with regard to the voltage (2) and “the distortion power D ” as the measure of the mutual distortion of the nonsinusoidal voltage and current waveforms. Making use of the reactive power QI (15), it is possible to ask the question: in what conditions is the reactive power QI equal to zero, which means: when does the equality T T di (t ) 0 u (t )di(t ) 0 [u (t ) dt ]dt [U m sin t U m2 R sin t cos td t T i(t )du (t ) [i(t ) 0 0 R Um sin t ) R ]d t (18) dt T U m2 d( U d (U m sin t ) du (t ) ]dt [ m sin t ]d t dt R dt (19) sin t cos tdt This results from the comparison of the results of calculations of the integrals from (18) and (19) that the equality (17) is satisfied and the reactive power QI in the circuit is equal to zero ( QI 0 ). The circuit from figure 1 has only a resistive character. The resistive character of the circuit was reported earlier in the paper [5]. Based on (16) one can draw some specific conclusion: if the load is a linear, time-invariant resistor so z (t ) R , hence dz (t ) dR 0 and QI 0 , dt dt QI 0 appear ? On the basis of (15) and (9), it was found out that the reactive power QI 0 , only when the equality: (17) T T 0 0 u (t )di(t ) i(t )du (t ) 0 (a) is satisfied. Yet the condition of zeroing the reactive power QI (QI 0) is not the sufficient condition for the voltage and current characteristics to be the proportional characteristics, for which the equality S P occurs. The condition of the equiformity of the voltage and current characteristics is explained by the following example. Let the load in the circuit consist of a series connection of a triac and a resistor as in fig. 1 (b) Fig. 2 Electrical circuit with the nonlinear resistance load: a) characteristics of the voltage u (t ) and the current i (t ) , b) characteristics 196 if the load is a linear but time-variant resistor so z (t ) Rk in k – time interval, hence dz (t ) dRk 0 and QI 0 dt dt Fig. 1. Circuit with resistive load and triac The voltage u (t ) has the sinusoidal waveforms and the switching angle is (deg). The time characteristics of the voltage and current in addition to the characteristics on the plane [ u (t ), i (t ) ] are presented in figure 2b. For the given voltage and current waveforms, the values of the integrals from the equality (16) are : u (i ) if the load is a nonlinear resistor and voltage has antysimmetric waveform there is T 2 T dz 2 dz 2 0 dt i dt T dt i dt and QI 0 . 2 PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010 As the current i (t ) waveform from figure 2 does not have the same shape as the voltage u (t ) waveform, the nonequiformity power K will occur in the circuit. Therefore the question arises how to calculate the nonequiformity power K . The author advocates the use of the conception of current decomposition proposed by Fryze [6][7] and Czarnecki [8][14]. The current active component iaF (t ) according to the conception by Fryze, is the characteristics proportional to the voltage characteristics u (t ) . It determines the possible minimal effective value of the current, at which for a given voltage u (t ) the same active power P will dissipate in the circuit : iaF (t ) (20) P u 2 u (t ) T 1 where: P u (t )i (t ) dt is the active power, u T 0 ( K 0 ) to S 2 P 2 QI2 the phase-shift does not occur between the voltage and the current ( QI 0 ), but the waveforms are not equiform, then K 0 and S 2 P 2 K 2 the phase-shift does not occur between the voltage and the current ( QI 0 ) and moreover, the waveforms are equiform, then ( K 0 ) to S P . The author proposes to use : the word “compensation” to make reduction of reactive current component ir (t ) in order to decrease reactive 2 U 2 RMS where ir (t ) is the current reactive component and ik (t ) is the current nonequiformity component. When the reactive power in the circuit is equal to zero QI 0 so ir (t ) 0 , the current in the circuit i (t ) , apart from the active iaF (t ) , will have only the nonequiformity * k component i (t ) : (22) In special cases when : the phase-shift occurs between the voltage and the current ( QI 0 ), but the waveforms are equiform, then power QI , Based on Czarnecki’s CPC concept of current decomposition, the supply current in a circuit with any load (linear, nonlinear, time-variant etc.) according to the author’s different interpretation, can be expressed as: (21) i (t ) iaF (t ) ir (t ) ik (t ) component the word “equiformisation” to make reductions of nonequiform current component ik (t ) so that to decrease nonequiformity power K . Traditionally passive shunt compensator (e. g capacitors) can be used to carry out full compensation ( QI 0 ) but only to decrease equiformity power K . This can be the first step to achieve total non-active power N elimination. In the second step of this procedure (when QI 0 ), the nonequiformity power K should be reduced. This can be done by the active filter or in some cases by the special kind of passive filter. Proposed methodology can be explain by using the power factor PF definition [15] where: i (t ) iaF (t ) i (t ) * * k PF where * means under condition QI 0 . Non-active power N On the basis of the analyses presented in chapter Nonequiformity power K, it is possible to formulate unambiguously the components of the non-active power with regard to the causes of its occurrence. Hence : the reactive power QI is the measure of the energy accumulation in the reactance elements (the measure of the influence of the phase-shift of the current with regard to the voltage). the non-equiformity power K is the measure of nonlinearity of the load without the energy accumulation (the measure of the current non-equiform with regard to the voltage). Both the powers QI and K can be compensated and/or corrected in the same way. This phenomenon has been firstly reported by Czarnecki [13]. In a given circuit, situations occur during which the powers QI and K are not simultaneously equal to zero [ QI 0 , K 0 ]. On the basis of the IEEE Standard, which introduced the notion of the non-active power N (3), it is possible to write : (23) N f (QI , K ) and thus, we obtain the square equation of the power in the form : (24) PF DPF DF (25) P P is so called the true power factor, S U RMS I RMS DPF U1RMS I1RMS cos cos U1RMS I1RMS is commonly known as the displacement power factor (for the purely sinusoidal voltage/current waveforms), is the power factor angle, DF 1 1 THD 2 u 1 THDi2 is so called the distortion power factor. If the voltage has the sinusoidal waveform, the distortion power factor DF is simplified to the relation DF 1 1 THDi2 The displacement power factor DPF is related to the reactive power QI , the distortion power factor DF is related to the equiformity power K . The practical example of the specific passive filter is Lineator TM. “Lineator” is a wide spectrum filters, tri-limbed reactor fitted with a small capacitor bank as illustrated in Fig 3. [16]. The Lineator filter is connected to the load as per a standard AC line reactor [i.e. between the mains supply and the rectifier(s)]. This makes the current equiformisation to the voltage shape as is shown at Figures 4 and 5. It means that Lineator reduces mainly the the distortion power factor DF . S 2 P 2 N 2 P 2 f (QI , K ) 2 PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010 197 Conclusions The presented conception of the description of the power states in electrical circuits with any voltage and current waveforms does not contain any limitations connected with the analysis method (the convergence of a Fourier series, the orthogonalization of the function) or the kind of load (linear, nonlinear or time-dependent) and it can be used in any single-phase circuit. Particular power state values are calculated with the help of integrals, so the active power P T P Fig. 3 Lineator wide spectrum filter schematic [16] T 1 1 p(t )dt u (t )i (t )dt T 0 T 0 the reactive power QI dz 2 1 di du 0 dt i dt 4 0 (u dt i dt )dt the apparent power S T 1 4 S 1 2 1 2 u (t )dt i (t )dt T o T o T T QI T the non-active power N N S 2 P2 Fig. 4 The power supply voltage and current waveforms with 150kW AC PWM drive load (with 3% DC bus reactor) [16] The figure 5 illustrates that Lineator not reduce the reactive current component significantly. The phase-shift between power supply voltage and current still exists. Actually Lineator does increase the displacement power factor DPF slightly (nothing like active filter). Fig. 5 The power supply voltage and current waveforms after TM application and with 150kW AC PWM drive load (with Lineator 3% DC bus reactor )[16,17] Due to that fact, the reactive power compensation should be repeated and detuned. It is worth emphasizing that the displacement power factor DPF can be improvement only by e. g adding shunt capacitors or active reactive power compensator via active filter or not static VAR compensator. If the active filter is applied in the second step of the non-active power elimination, the reference current as ik (t ) can be calculate from equation (22). 198 For these reasons the author suggests naming this proposal the integral method to calculate power states values in the single-phase circuit. The proposed decomposition of the instantaneous current is similar to the way Czarnecki described in His CPC power theory [8,14]. However, there are some differences between CPC and the author’s concept and the different interpretation are proposed: 1. It has been proposed to calculate the reactive power QI from (16). 2. The condition whether the reactive power QI is not equal to zero is based on (16) 3. The reactive power QI is the measure of the energy accumulation in the reactance elements (the measure of the influence of the time-shift of the current with regard to the voltage). 4. The non-active power N consists of two components: the reactive power QI and the nonequiformity power K . 5. The nonequiformity power K is the measure of nonlinearity of the load without the energy accumulation (the measure of the current non-equiform with regard to the voltage shape). 6. The compensation is the action required to reduce or eliminate the reactive power QI . 7. The equiformisation is the action required to reduce or to eliminate the nonequiformity power K . 8. It has been proposed to carry out the non-active power N reduction or elimination in two steps. In the first one, the reactive power QI can be compensated by the passive shunt filter e.g. capacitor. In the second step the nonequiformity power K can be eliminated by the active filter or in some cases by the special kind of passive filter. Acknowledgment The author thanks to Czesław Krawczyk, Ph. D, from Department of Mathematics, and to Tadeusz Piotrowski Ph. D. form Marine Power Engineering Department, both from Gdynia Maritime University, Gdynia, Poland for their assistance and fruitful discussion and as well as to Ian C. PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010 Evens from the Harmonic Solutions Co. UK for his assistance and information about the Lineator industrial application. REFERENCES [1] IEEE Std 1459-2000, “IEEE Trial-Use Standard Definitions for [2] [3] [4] [5] [6] [7] [8] the Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, Or Unbalanced Conditions” Illovici M., Definition of measure de la puissance de l’energie reactives, France Electr. 1925, pp.49-52, pp.931-954 (in French) 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 Lyon V., [Discussion to H.L.Curtis and F.B.Silsbee: Definitions of Power and Related Quantities], AIEE Transactions, Vol.54, No.4, April 1935, pp.394-404], Electrical Engineering, Oct. 1935, p.112 Hartman M.T., Why the new physical interpretations of the reactive power on terms of the CPC power theory is not true, 5th International Conference – Workshop, CPE 2007, Gdańsk, Poland, May 2007 Fryze S., Moc rzeczywista, urojona i pozorna w obwodach elektrycznych o przebiegach odkształconych prądu i napięcia. 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Part II – Practical Circuits, IEEE Transactions on Industry and General Applications, vol. IGA-4, no. 4, July-August 1968, pp. 447-455 [13] Czarnecki L.S., Distortion power in systems with nonsinusoidal voltage”, IEE proceedings-B, vol. 139, no. 3, May 1992 [14] Czarnecki L.S., Currents' Physical Components (CPC): Fundamental of Power Theory, Przeglad Elektrotechniczny (Proc. of Electrical Eng.) R84, No. 6/2008 (in Polish) [15] Grad W. M., Gilleskie R.J., Harmonics and how they relate to power factor, Proc. of the EPRI Power Quality Issues& Opportunities Conference (PQA’93), San Diego, CA, November 1993 [16] Evans I. C., Guidance Notes for the Control of Harmonics in Electrical Power Systems (ABS Pub 150) –, May 2007 [17] Evans I. C., Hoevenaars A.H., Meeting Harmonic Limits on Marine Vessels, IEEE Electric Ship Technologies Symposium (ESTS’07), Arlington, VA, USA, 21- 23 May 2007. Author: prof. dr hab. inż. Marek T. Hartman, Gdynia Maritime University, 81-87 Morska Str., 80-225 Gdynia, Poland, E-mail: mhartman@am.gdynia.pl PRZEGLĄD ELEKTROTECHNICZNY (Electrical Review), ISSN 0033-2097, R. 86 NR 3/2010 View publication stats 199