Quantitative comparison of power decompositions

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Electric Power Systems Research 78 (2008) 318–329

Quantitative comparison of power decompositions M. Erhan Balci, M. Hakan Hocaoglu ∗ Gebze Institute of Technology, Department of Electronics Engineering, Kocaeli 41400, Turkey Received 14 July 2006; received in revised form 12 January 2007 Available online 29 March 2007

Abstract This paper presents a comparison on power decompositions in a simple single phase circuit with nonsinusoidal waveforms of voltage and/or currents by giving particular emphasis to Reactive Power compensation. The experimental circuit is analysed by using exact analytical expressions for current and voltages determined via considering source impedance and nonlinearity, which is introduced due to supply side harmonics. Results demonstrate that; power decompositions proposed by Kusters and Moore, Fryze, Shepherd and Zakikhani, Sharon, and Czarnecki provide correct information regarding Power Factor improvement with passive compensation in nonsinusoidal voltage source–linear load, nonsinusoidal voltage source–nonlinear load and sinusoidal voltage source–nonlinear load cases. In these cases, Reactive component of Kusters and Moore’s power decomposition can completely be compensated when Power Factor is maximum if there is no resonance or significant changes on load voltages in the case of compensation capacitance is inserted. The Reactive components of Fryze, Shepherd and Zakikhani, Sharon, Czarnecki’s power decompositions attain minimum value when power factor is maximum. Furthermore, Kusters and Moore’s Reactive Power could directly be related to the power of optimum compensation capacitance. On the other hand, power decompositions proposed by Budeanu, Kimbark and Depenbrock do not provide any useful information about optimum Reactive Power compensation with a basic capacitance in the cases nonsinusoidal voltage source–linear load and nonsinusoidal voltage source–nonlinear load although they can completely be compensated in these cases. An important observation is that; Distortion Powers of Budeanu and Kimbark, and Depenbrock’s Residual Power have compensable parts; on the other hand, Kusters and Moore’s Residual Reactive Power, Sharon’s Complementary Power, Depenbrock’s In Phase Power and Czarnecki’s Scattered Power are almost constant when the power of compensation capacitor is varied. © 2007 Elsevier B.V. All rights reserved. Keywords: Power decompositions; Nonsinusoidal conditions; Power factor; VAr compensation; Harmonic distortion

1. Introduction In parallel with exploitation of alternating voltage for power delivery, the importance of Reactive Power and Power Factor for the system efficiency has become the subject of a number of studies [1]. As a result Classical Apparent Power decomposition, namely S2 = P2 + Q2 , has been largely employed to facilitate a tool for system efficiency improvement by means of Reactive Power compensation; nevertheless, the classical decomposition does not fulfil the conditions caused by the presence of nonsinusoidal voltage and/or current. Although, in classical manner, generated and distributed electric power must be sinusoidal with predetermined frequency and magnitude, voltages and currents

∗

Corresponding author. Tel.: +90 262 605 1706; fax: +90 262 653 84 90. E-mail addresses: [email protected] (M. Erhan Balci), [email protected] (M. Hakan Hocaoglu). 0378-7796/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2007.02.010

have nonsinusoidal characteristics, due to the nonlinear power system elements, such as generators, motors, transformers and various power electronic devices, etc. Consequently, a number of power definitions have been proposed for nonsinusoidal conditions to provide a tool for improvement in the system efficiency [2–12]. These definitions, which are based on various decompositions of Apparent Power, generally, bare the aims of accurate and easy determination of maximum Power Factor realizable with basic capacitive compensation and a criteria to measure the degree of loads’ nonlinearity. In this study, widely recognized power decompositions proposed by Budeanu, Fryze, Kimbark, Shepherd and Zakikhani, Sharon, Depenbrock, Kusters and Moore, and Czarnecki are analysed quantitatively. As a consequence of various propositions on power decompositions, comparison of different power components has been undertaken by number of researchers. In these studies, simple examples or load models like a constant current source

M. Erhan Balci, M. Hakan Hocaoglu / Electric Power Systems Research 78 (2008) 318–329

are generally used for analysis [1,13–16]. Furthermore, some cases mentioned in these studies, neglect the line impedance, which is an important issue to show the problem of Reactive Power compensation in nonsinusoidal conditions. In one of the most important analysis, Emanuel reviewed the power definitions and their physical meanings [1] by a conclusion he points out that Budeanu’s Reactive Power provides useless information for Power Factor improvement. On the other hand, the meters, which measure Fryze’s Reactive Power, is recommended in the case of the impact of source side harmonics on the load performance is non-consequential. In another study, Czarnecki shows how controversial the problem of the choice of power quantities in nonsinusoidal conditions and the choice of a power quantity may depend on a specific application [13]. He later compares two famous power theory developed by Fryze and Budeanu [14] and points out that Budeanu’s power decomposition misinterprets the phenomena by giving a detailed analysis on why it does not provide a tool for the Power Factor improvement in nonsinusoidal conditions. On the other hand, according to the author, Fryze’s decomposition are very important for developing power theory due to the fact that a number of concepts introduced in his decomposition, such as Instantaneous Imaginary Power are used for the controlling of active harmonic filters. He also mentions that Fryze’s power theory has a number of serious limitations on failing to define the power components in frequency domain and erroneously labelling Active Power as useful power. In his later study, he concluded that the power properties of circuits with nonsinusoidal conditions should not be characterised in terms of Distortion Power [15]. Sun and Xiang in [16] compare Distortion Power definitions proposed by Czarnecki and Slonim and Van Wyk [10], which is treated as Budeanu’s Distortion Power by using a novel concept, which is based on distortionless conditions of the system. They found that Czarnecki’s Scattered Power is more reasonable for defining the waveform distortion of current with respect to voltage. In all studies summarised above demonstrates that a comprehensive and detailed analysis should be undertaken to recap the power theory. Due to the fact that there is no consensus on the power decompositions and particularly failure of the frequency domain decomposition which properly explains the various phenomenon, such as resistive paradox, Classical Apparent Power concept is revisited and redescribed by some researchers. Firstly, Philipski discusses the problem in nonsinsoidal condition and addresses the theoretical weaknesses of Classical Apparent Power, S = V·I* [17]. Secondly, Ghassemi questioned the philosophy of electric power theory [18]. According to the idea presented in these studies, some papers focused on the modification of Classical Apparent Power concept. Wilczynski in [19] defines Apparent Power as the RMS value of Instantaneous Power so that Distortion and Scattered Powers are eliminated. Due to the fact that the definition of Reactive and Classical Complex Powers show the discrepancy in time and frequency domains, Ghassemi present a New Apparent Power by using RMS value of the Complex Instantaneous Power [20]. However, Classical Apparent Power is still enforced by IEEE Std

319

1459-2000 [21] and no other alternatives are accepted as more useful. Thus, in this study, Classical Apparent Power is used for the comparison. In this work, power decompositions have been compared for a simple single phase circuit by giving particular emphasis to Reactive Power compensation. Accordingly, closed form expressions for an ac chopper circuit current and terminal voltage, feeding a constant impedance load, have been derived for nonsinusoidal situations taking into account source impedance and nonlinearity introduced due to source side harmonics. Results show that although Fryze, Shepherd and Zakikhani, Sharon, Kusters and Moore, and Czarnecki give accurate information about passive Reactive Power compensation, only Kusters and Moore power decomposition is useful for obtaining the power of compensation capacitance. Moreover, it is shown from the analyses that Distortion Powers of Budeanu and Kimbark and Residual Power of Depenbrock’s power decompositions have compensable parts. 2. Power decompositions under nonsinusoidal conditions The nonsinusoidal voltage and current generated by nonlinear loads could be analyzed using Fourier series. The nonsinusoidal voltage and current are expressed;  √ 2 · Vn · cos(ωn t − αn ) (1) v(t) = V0 + n ∈ N+

i(t) = I0 +

 √

2 · In · cos(ωn t − δn )

(2)

n ∈ N+

where N+ is the positive integer numbers, V0 and I0 are the mean values of voltage and current. As in sinusoidal condition the Instantaneous Power could simply be calculated with the product of instantaneous voltages and currents:  √ 2 · In · cos(ωn t − δn ) p(t) = V0 · I0 + V0 · n ∈ N+

+ I0 ·

 √

2 · Vn · cos(ωn t − αn )

n ∈ N+

+

 √

2 · Vn · cos(ωn t − αn ) ·

n ∈ N+

×

 √ 2 · In · cos(ωn t − δn )

(3)

n ∈ N+

Nevertheless, decomposition of Instantaneous Power in nonsinusoidal conditions is not a clear-cut process, as done for sinusoidal conditions. In order to obtain information regarding phase angle, source efficiency and line voltage drop, it is necessary to decompose Instantaneous Power, preferably in phasor domain, as done for sinusoidal cases. Consequently, a number of new decompositions have been proposed in the literature. They are summarized here for convenience.

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2.1. Budeanu’s power decomposition

and Fryze’s Reactive Power

Budeanu postulated that Apparent Power is consist of two orthogonal components, namely Active and Deactive Powers [2]. The former is simply Average Power which could easily be calculated by averaging the Instantaneous Power in time domain or by convolution in phasor domain:  T  1 P= · v(t) · i(t) dt=V0 · I0 + Vn · In · cosϕn (4) T 0 +

Qf = V · Ir

n∈N

This component was chosen by Budeanu due to the fact that it is actual power converted to the physical work. The later one, Deactive Power, is divided into two components; Budeanu’s Reactive and Distortion Powers. Budeanu’s Reactive Power is calculated by summing of the individual harmonic Reactive Powers:  Qb = Vn · In · sinϕn (5) n ∈ N+

The main advantages of Fryze’s decomposition are; to provide accurate information on source efficiency and to be determined by using ordinary phasor measurement devices. However, calculated values are not suitable for Reactive Power Compensator design by simple capacitor. 2.3. Kimbark’s power decomposition Kimbark postulated that Apparent Power is consist of two orthogonal components, namely; Active and Deactive Powers as done by Budeanu. Kimbark proposed that active component as Average Power. The Deactive Power is divided into two components, i.e. Kimbark Reactive Power and Distortion Power [4]. The first is calculated by harmonic phasors. Qk = V1 · I1 · sinϕ1

Distortion Power is calculated by cross product of different harmonics voltages and currents;  Db = S 2 − P 2 − Q2b  = Vn2 · Ik2 + Vk2 · In2 − 2Vn · Vk · In · Ik cos(ϕn − ϕk ) n=k

(11)

(12)

Thus, Reactive Power can easily be related to load parameters. On the other hand, Distortion Power is consist of individual harmonic Reactive Powers without fundamental harmonic and cross products of different harmonics currents and voltages:  (13) Dk = S 2 − P 2 − Q2k

(6) Budeanu’s Reactive Power can be completely compensated with a simple capacitor. However, this is not a case for Distortion Power, given in Eq. (6). 2.2. Fryze’s power decomposition Despite the fact that Budeanu’s power decomposition provides a completely compensable Reactive Power, the calculated Reactive Power gives no data about the source efficiency. In addition, Budeanu’s decomposition require harmonic domain calculations and sophisticated measurement devices. Accordingly, Fryze proposed a current based decomposition [3] in which current is divided into two orthogonal components, namely; Active and Reactive Currents. The first is calculated by using Active Power ia (t) =

P v(t) V2

(7)

and the second, ir (t) = i(t) − ia (t)

(8)

The power decomposition as suggested by Fryze is S =P 2

2

+ Q2f

(9)

2.4. Shepherd and Zakikhani’s power decomposition Although Budeanu’s Reactive Power can be completely compensated by a basic capacitor, optimum compensation capacitance could not be determined by using his decomposition. This is the main motivation behind the work presented in [5,6], which aims to develop a formulae to give the maximum Power Factor and the minimum Reactive Voltamperes achievable by passive compensation. Accordingly, Shepherd and Zakikhani divide the current into three orthogonal components, which are: Active Current;  Ir = In2 · cos2 ϕn

(14)

Reactive Current; and  Ix = In2 · sin2 ϕn

(15)

Distortion Current;  Id = I 2 − Ir2 − Ix2

(16)

n ∈ n1

n ∈ n1

with Active power P = V · Ia

(10)

where n1 is defined as the common harmonics of voltage and current.

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Authors proposed a power decomposition related to these current components: S 2 = Sr2 + Sx2 + Sd2

(17)

321

which is the lack of direct determination of optimum compensation capacitance’s power, from Shepherd and Zakikhani’s decomposition. 2.6. Depenbrock’s power decomposition

with Active Apparent Power,   Sr = Vn2 · Ir2

(18)

Reactive Apparent Power, and   Sx = Vn2 · Ix2

(19)

n ∈ n1 ∪n2

n ∈ n1 ∪n2

Distortion Apparent Power, 

      Sd =  Vn2 · In2 + Vn2 . In2 + In2 n ∈ n1

n ∈ n3

n ∈ n2

n ∈ n1

n ∈ n3

(20) where n2 is the harmonic numbers of voltage in which there are no current harmonics and n3 is the harmonic numbers of current in which there are no voltage harmonics. This power decomposition provides formulae to calculate Reactive Power, which attains the minimum value by simple capacitor. However, this decomposition does not provide any information leading to determination of Power Factor due to the fact that Active Apparent Power (Sr ) is different from Active Power (P). In addition, it could not be directly used to determine the power of optimum compensation capacitance. 2.5. Sharon’s power decomposition Due to disadvantages of Shepherd and Zakikhani’s power decomposition, Sharon proposed new decomposition in which Active Apparent Power (Sr ) is replaced by Average Power (P) thus; Sharon’s power decomposition is defined [7] as; S =P 2

2

+ Sq2

+ Sc2

(21)

with Reactive Apparent Power, and  In2 . sin2 ϕn Sq = V ·

(22)

Complementary Apparent Power,  Sc = S 2 − P 2 − Sq2

(23)

n ∈ n1

where n1 is defined as the common harmonics of voltage and current. Sharon’s power decomposition calculates Reactive Power, which can be minimum by capacitive compensation. Unfortunately, Sharon’s decomposition inherits the same problem,

The necessity of defining power components, which is caused by non-fundamental harmonic part of source voltage and load nonlinearity motivated Depenbrock’s work [8]. Accordingly, he firstly divided source voltage into two components as: Fundamental Voltage; and √ vg (t) = 2 · V1 · cos(ω1 t + α1 )

(24)

Residual Voltage vnf (t) = v(t) − vg (t)

(25)

secondly, current is divided into five components as: the Active Current component caused by fundamental harmonic of source voltage and equivalent conductance; ig (t) = Geq · vg (t)

(26)

the Active Current component caused by fundamental harmonic of source voltage and the difference between fundamental harmonic’s conductance and equivalent conductance; ige (t) = (Gg − Geq ) · vg (t)

(27)

the Active Current component caused by residual part of source voltage and equivalent conductance; inf (t) = Geq · vnf (t)

(28)

the Active Current component caused by residual part of source voltage and the difference between non-fundamental harmonic part of conductance and equivalent conductance; ine (t) = (Gnf − Geq ) · vnf (t)

(29)

the Reactive Current caused by fundamental harmonic of source voltage; T

(1/T ) 0 i(t) · vg (t−(T/4)) dt T t− iq (t)= (30) · v g 2 Vg 4 That simply can be identified as Reactive Current of fundamental harmonic. Lastly, Residual Current, which is Nonactive Current without fundamental harmonic’s Reactive Current; id (t) = i(t) − ig (t) − ige (t) − inf (t) − ine (t) − iq (t)

(31)

where Active Power is expressed as the arithmetic sum of fundamental and non-fundamental harmonic’s Active Powers (Pg and Pnf ). P = Pg + Pnf Geq =

P V2

(32) (33)

322

Gg = Gnf =

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Pg Vg2

(34)

Pnf Vnf2

(35)

o

S =P

2

+ Q2q

o

V denotes the RMS value of instantaneous voltage’s derivative.

He proposed the Apparent Power as the vector sum of four different parameters; 2

where v(t) denotes the derivative of instantaneous voltage and

+U +N 2

2

(36)

with Reactive Power Qq = V · Iq

If the load has capacitive part, Inductive Reactive Current and Residual Reactive Current are defined as; Inductive Reactive Current is the current component, which can be compensated by the inductance, and T (1/T ) · 0 v¯ (t) · i(t) dt iql (t) = · v¯ (t) (43) V¯ 2

(37)

Residual Reactive Current;

In Phase Power, which is produced by the variation of load conductance, is given as in Eq. (38)  2 + I2 U = V · Ige (38) ne

iqlr (t) = i(t) − ia (t) − iql (t)

(44)

where v¯ (t) denotes the integration of instantaneous voltage and V¯ denotes the RMS value of instantaneous voltage’s integration.

And Residual Power, N = V · Id

(39)

The main property of Depenbrock’s power decomposition is separation of power components, which are occurred by the load conductance variation and voltage source distortion. Reactive Power (Qq ) can only be related to fundamental harmonic of current and total RMS value of voltages due to averaging operation, defined in Eq. (30). However, Residual Power (N) given in Eq. (39) also contains non-fundamental Reactive Power and could further be compensated by a simple capacitance.

Kusters and Moore proposed a power decomposition that provides the power system operator to determine if the possibility of improving power factor by means of shunt capacitance or inductance exists and to identify easily the proper value required to realize the maximum benefit. It is defined that current is consist of three components, namely; Active, Capacitive Reactive or Inductive Reactive and Residual Reactive Currents due to the fact that the need of identification capacitive or inductive improvement of Power Factor [9]. These current components are: Active Current; P · v(t) V2

(40)

Capacitive Reactive Current is the current component, which can be compensated by the capacitance in the case of inductive load; T o (1/T ) · 0 v(t) · i(t) dt o · v(t) (41) iqc (t) = o V2 Residual Reactive Current; iqcr (t) = i(t) − ia (t) − iqc (t)

S 2 = P 2 + Q2kus + Q2kusr

(45)

with Reactive Power for inductive load Qkus = V · Iqc

(46)

Residual Reactive Power for inductive load Qkusr = V · Iqcr

(47)

If the load has capacitive part, Reactive components of the proposed decomposition are:

2.7. Kusters and Moore’s power decomposition

ia (t) =

They proposed that power decomposition related to this current decomposition as:

(42)

Reactive Power Qkus = V · Iql

(48)

Residual Reactive Power Qkusr = V · Iqlr

(49)

Capacitive Reactive Power, in Eq. (46), can be expressed as  ∈ N + n · Vn · In · sinϕn Qkus = V · n (50)  2 2 n ∈ N + n · Vn and Inductive Reactive Power, in Eq. (48), can be expressed as  ∈ N + (Vn · In · sinϕn )/n Qkus = V · n (51)  2 2 n ∈ N + (Vn /n ) The main property of Kusters and Moore definition is decomposition of Fryze’s Reactive Power into two orthogonal components. One of them is the product of RMS values of voltage and current, which has the same wave shape with the derivative or integral of the voltage, and the second is the Residual Reactive Power. On the other hand, in reference [22], Czarnecki questions the validity of the claims on some special network conditions. He points out that in some conditions Kusters and Moore’s Reactive Power fails to achieve maximum

M. Erhan Balci, M. Hakan Hocaoglu / Electric Power Systems Research 78 (2008) 318–329

Power Factor available with a basic capacitance. It is apparent from the results that in extreme network conditions, where the load voltage significantly changes due to resonance, Kusters and Moore’s Reactive Power does not match the power of maximum Power Factor compensation capacitance. However, in our studies in most practical networks the decomposition successfully provides maximum Power Factor available with passive compensation.

The need of identification of the reason on the decreasing source efficiency due to nonsinusoidal waveforms, which is caused by source and/or load distortion, is the main motivation of Czarnecki’s work. In his studies, author clearly distinguishes current components due to load and source nonlinearities. Accordingly, he employs a current based decomposition in which current divided into four components, i.e. Active, Reactive, Scattered and Generated Harmonic Currents [11,12]. Source current can be expressed as; I 2 = Ia2 + Ir2 + Is2 + Ih2

(52)

These components are defined as; Active Current; P V Reactive Current;  Bn2 · Vn2 Ir = Ia =

with Reactive Power; Qr = V · Ir

(53)

(54)

n∈m

Scattered Current;  Is = (Gn − Ge )2 · Vn2

(55)

Generated Harmonic Current;  Ih = In2

(56)

n∈m

(60)

Scattered Power; Ds = V · Is

(61)

Generated Harmonic Power; Dh = V · Ih

2.8. Czarnecki’s power decomposition

323

(62)

The main property of Czarnecki’s power decomposition is identification of the physical phenomena responsible for the source current increase. The power decompositions, summarized above, are given in Table 1 in respect of their distinguishing features. In addition to these, the conventional reactive power and energy measurement method uses the average value of the product of the current samples and shifted voltage samples:  1 T Q= i(t)v(t − τ) dt (63) T 0 where τ is taken as a quarter of the fundamental period. Eq. (63) can be expressed in harmonic domain as [23];   π Vn In cos ϕn − n (64) Q= 2 + n∈N

It should be appreciated that the conventional method is easy to implement for practical point of view. However, none of the reactive components of power decompositions, summarised in this section and given in Table 1, is compatible with the conventional reactive power measurement method. On the other hand, it could be seen from the expressions of the power decompositions, apart from the components of Fryze’s power decomposition, all need extra computational efforts and could not be directly measured by the classical phasor measurement devices. Moreover, they need costly computational techniques and devices based on digital signal processing methods [24]. The accuracy of this type measurement device is mainly related to the sampling rate and technique.

n∈k

where k represents current harmonic numbers do not present in the set of voltage harmonic numbers and m denotes that voltage harmonic numbers. If dc component exists in voltage, it is not considered in the calculation of Reactive Current. The equivalent conductance of load defined as; P V2 where nth harmonic admittance of load Ge =

(57)

Yn = Gn + jBn

(58)

He proposed that power decomposition related to this current decomposition as: S 2 = P 2 + Q2r + Ds2 + Dh2

(59)

3. Case studies The behaviour of power decompositions, on Reactive Power compensation process in nonsinusoidal conditions, is analyzed for three different cases. These cases are nonsinusoidal voltage source–linear load, nonsinusoidal voltage source–nonlinear load and sinusoidal voltage source–nonlinear load. In all cases, loads are modelled as constant impedance and nonlinear loads are simulated as ac chopper circuit, given in Fig. 1. For the case of nonsinusoidal voltage source–linear load, current and voltage are calculated by Super Position theorem in phasor domain. In all cases, Reactive Power is compensated by a simple capacitance. The system can be analyzed into two operating modes as triac conduction and triac cut-off. During triac cut-off mode, the equations of the current and voltage in the system are determined by Super position theorem.

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Table 1 The power definitions above summarized in respect of their properties Power decomposition

Active component

Reactive component

Distortion component

Budeanu

Average Power

Cross product with different harmonics

Fryze Kimbark

Average Power Average Power

Shepherd and Zakikhani

Sharon

Apparent Active Power (it is not Average Power) Average Power

Depenbrock

Average Power

Completely compensable with a passive capacitance Not related to stored energy Reactive Power of Fundamental Harmonic Takes into account Reactive Power, which can be minimum by capacitance compensation Takes into account Reactive Power, which can be minimum by capacitance compensation Product of Fundamental Harmonic Reactive Current and total RMS voltage

Kusters and Moore

Average Power

Czarnecki

Average Power

Inductive or Capacitive Reactive Power that is compensable by optimum capacitance or inductance Takes into account Reactive Power, which can be minimum by capacitance compensation

– Sum of Cross product with different harmonics and nonfundemantal Reactive Powers It is equal to zero in all cases

It is the in phase power produced by the variation of load conductance In Phase Power (U) Residual Power (N) contains produced by the Nonfundamental Reactive Power and variation of load the power produced by the variation conductance of load susceptance The vector differences between Apparent power and vector sum of Active and Reactive Powers

Scattered Power (Ds ) is the in Phase Power produced by the variation of load conductance

Harmonic Generated Power (Dh ) is equal to zero in all cases

During triac conduction mode, the equations of current and voltage in the circuit are found by solving the system of equations given below  vs (t) = Vmn · sin(ωn · t + ϕn ) (65) n ∈ N+

RLine · iLine (t) + LLine ·

diLine + vLoad (t) = vs (t) dt

(66)

diLoad − vLoad (t) = 0 dt

(67)

RLoad · iLoad (t) + LLoad · iLine (t) − iLoad (t) − C ·

dvLoad =0 dt

(68)

Eqs. (65)–(68) are solved for the initial conditions of line current, load voltage and load current. The solutions of the system for line current iLine (t), load voltage vLoad (t) and load current iLoad (t) is  iLine (t) = [An · sin(ωn · t + ϕn ) + Bn · cos(ωn · t + ϕn )] n ∈ N+ 3  

+

[ck,n · eDk ·t ]

(69)

n ∈ N + k=1

 vLoad (t) =



[(Vmn − RLine · An + LLine · Bn · ωn )·

n ∈ N+

× sin(ωn · t + ϕn )] + Fig. 1. The system used in power comparison analysis.

 n ∈ N+

[(−RLine · Bn

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− LLine · An · ωn ) · cos(ωn · t + ϕn )] 3  

−

iLoad (t) =





[(RLine + LLine · Dk ) · ck,n · eDk ·t ]

n ∈ N + k=1



325

(70)

[(An − C · RLine · Bn · ωn

n ∈ N+

− C · LLine · An · ωn2 ) · sin(ωn · t + ϕn )]  + [(Bn − Vmn · C · ωn + RLine · An · C · ωn n ∈ N+

− LLine · Bn · C · ωn2 ) · cos(ωn · t + ϕn )] +

3  

[(1 + RLine · C · Dk

n ∈ N + k=1

Fig. 2. Waveforms of load voltage, line current and load current when Power Factor is maximum.



+ LLine · C · Dk2 ) · ck,n · eDk ·t ]

(71)

where An , Bn and ck,n are the coefficients for nth harmonic of the source voltage (vs (t)). These coefficients and initial values of line current, load voltage and load current are given in Appendix A. Triac cut-off time (tcut-off ) can be determined by solving Eq. (72). iLoad (tcut-off ) = 0

(72)

for interval tconduction < tcut-off < (tconduction + Tf /2). The conduction angles of the triac are selected according to the zero crossing and period of source voltage’s fundamental harmonic (Tf ). Accuracy of the solution is compared with ATP-EMTP and the results are in very close agreement [25]. 4. Results Power components for each decompositions are calculated in harmonic domain via FFT of sampled currents and load voltages. In analysis, nonsinusoidal voltage source has 3rd, 5th and 7th harmonics and its THDV is 40%. In nonlinear load cases, triac conduction angle is chosen as 90◦ . In nonsinusoidal voltage source–nonlinear load case, load voltage, load current and line current waveforms are shown in Fig. 2. Base voltage and current values are assumed as 125 V and 8 A in calculations. The variations of Budeanu (Qb ), Fryze (Qf ), Kimbark (Qk ), Shepherd and Zakikhani (Sx ), Sharon (Sq ), Depenbrock (Qq ),

Kusters and Moore (Qkus ), and Czarnecki (Qr ) Reactive Powers with power of compensation capacitance for three different cases are shown in Fig. 3. Power of compensation capacitance in the case of Power Factor attains its maximum that is pointed by the vertical line. Base power value is assumed as 1000 VA in calculations. It is clear from Fig. 3 that Shepherd and Zakikhani, Sharon and Czarnecki’s Reactive Powers attain the same numerical values for all compensation capacitance in three different voltage source and load cases. It can be seen that they and Fryze’s Reactive Power can reach their minimum values by means of a simple capacitance when Power Factor is maximum. Kusters and Moore’s Reactive Power becomes zero when Power Factor is maximum in all cases. On the other hand, for sinusoidal voltage source–nonlinear load case, the Reactive Powers of Budeanu, Kimbark and Depenbrock’s power decompositions obtain zero values when Power Factor is maximum. However, this is not the case for nonsinusoidal voltage source–linear load and nonsinusoidal voltage source–nonlinear load cases. Furthermore, the value of Kusters and Moore’s Reactive Power, which is in the case that compensation capacitance does not exist, is equal to the power of optimum compensation capacitance. Reactive Components of Fryze, Shepherd and Zakikhani, Sharon and Czarnecki could not be directly used to determine the power of optimum compensation capacitance in all cases. For three cases, the value of Kusters and Moore’s Reactive Power (Qkus ), which is in the case that compensation capacitance

Table 2 Comparison of Kusters and Moore’s Reactive Power and optimum Power Factor, and the power of optimum compensation capacitance and realizable maximum Power Factor Cases

Qkus

Scap

Sinusoidal voltage source–nonlinear load (conduction angles 90–270) Nonsinusoidal voltage source–nonlinear load (THDV = 40%, conduction angles 90–270) Nonsinusoidal voltage source–linear load (THDV = 40%)

0.5339

0.5440

0.3398 0.4196

(Scap − Qkus )/Scap

(pfmax − pfopt )

pfopt

pfmax

0.0186

0.7967

0.8006

0.003

0.3584

0.0519

0.6456

0.6476

0.002

0.4133

−0.0152

0.9596

0.9594

−0.0002

326

M. Erhan Balci, M. Hakan Hocaoglu / Electric Power Systems Research 78 (2008) 318–329

Fig. 3. Variations of Reactive Powers with power of compensation capacitance for three different cases: (a) nonsinusoidal voltage source–linear load, (b) nonsinusoidal voltage source–nonlinear load and (c) sinusoidal voltage source–nonlinear load.

Fig. 4. Variations of Distortion Powers with power of compensation capacitance for three different cases: (a) nonsinusoidal voltage source–linear load, (b) nonsinusoidal voltage source–nonlinear load and (c) sinusoidal voltage source–nonlinear load.

M. Erhan Balci, M. Hakan Hocaoglu / Electric Power Systems Research 78 (2008) 318–329

does not exist, the value of optimum compensation capacitance’s power (Scap ), optimum Power Factor (pfkus ), calculated by Eq. (73), according to Kusters and Moore’s power decomposition pfkus = 

P P 2 + Q2kusr

(73)

and realizable maximum Power Factor (pfmax ) are given in Table 2. It is shown from Table 2 that Kusters and Moore’s Reactive Power gives accurate information for the power of optimum compensation capacitance and realizable maximum Power Factor. However, if the systems are in the resonance conditions where the terminal voltage significantly changes with the compensation, information regarding optimum compensation could not be determined directly as mentioned in reference [22]. The variations of Distortion Powers proposed by Budeanu (Db ) and Kimbark (Dk ), Kusters and Moore’s Residual Reactive Power (Qkusr ), Sharon’s Complementary Power (Sc ), Depenbrock In Phase and Residual Powers (U and N), and Czarnecki’s Scattered Power (Ds ) for three different voltage source and load cases are shown in Fig. 4. Due to the fact that Shepherd and Zakikhani’s Distortion (Sd ) and Czarnecki’s Harmonic Generated (Dh ) Powers are equal to zero in the systems as a result, they are not shown in Fig. 4. A close look to Fig. 4 reveals that for all three cases, Distortion Powers of Budeanu and Kimbark, and Depenbrock’s Residual Power have compensable parts. Sharon’s Complementary Power and Czarnecki’s Scattered Power attain the same numerical values for all compensation capacitance’s powers in three different voltage source and load cases. Moreover, for all cases, Kusters and Moore’s Residual Reactive Power, Sharon’s Complementary Power, Depenbrock’s In Phase Power and Czarnecki’s Scattered Power are almost constant during the variation of capacitance. 5. Conclusion In this study, widely recognised power decompositions have been analyzed in a simple generic circuit for three voltage source and load cases, using exact analytical solutions of line current and terminal voltage by accounting line impedance and source nonlinearity, that covers the most of the practical applications for the nonsinusoidal conditions. From the results it can be concluded that: The power decompositions, which are proposed by Budeanu, Kimbark and Depenbrock, could not be used for the maximisation of Power Factor. The power decompositions of Fryze, Shepherd and Zakikhani, Sharon, Kusters and Moore, and Czarnecki provide information on the system efficiency. Thus, they could be used for the maximisation of Power Factor. They could be used for the calculation of optimum compensation capacitance’s value with some mathematical efforts. Furthermore, only Kusters and Moore’s decomposition can be

327

implemented for direct provision of optimum compensation capacitance’s power when there is not any resonance and terminal voltage is almost constant during the compensation. Appendix A  Δ  1,n   Δ2,n An =   Δ3,n   Δ4,n

 Δ5,n   Δ6,n  , Δ5,n   Δ6,n 

 Δ  3,n   Δ4,n Bn =   Δ3,n   Δ4,n

 Δ1,n   Δ2,n   Δ5,n   Δ6,n 

where Δ1,n = LLoad · C · Vmn · ωn2 − Vmn Δ2,n = −RLoad · C · Vmn · ωn Δ3,n = (RLine · LLoad · C + RLoad · LLine · C) · ωn2 − RLoad − RLine Δ4,n = LLoad · LLine · C · ωn3 − (RLoad · RLine · C + LLine + LLoad ) · ωn Δ5,n = −LLoad · LLine · C · ωn3 + (RLoad · RLine · C + LLine + LLoad ) · ωn Δ6,n = (RLine · LLoad · C + RLoad · LLine · C) · ωn2 − RLoad − RLine    ε1,n δ2,n δ3,n       ε2,n λ2,n λ3,n     ε3,n γ2,n γ3,n   , c1n =    δ1,n δ2,n δ3,n     λ1,n λ2,n λ3,n     γ1,n γ2,n γ3,n     δ1,n δ2,n ε1,n       λ1,n λ2,n ε2,n     γ1,n γ2,n ε3,n   c3n =    δ1,n δ2,n δ3,n     λ1,n λ2,n λ3,n     γ1,n γ2,n γ3,n 

  δ1,n    λ1,n   γ1,n  c2n =   δ1,n   λ1,n   γ1,n

ε1,n ε2,n ε3,n δ2,n λ2,n γ2,n

 δ3,n   λ3,n   γ3,n  , δ3,n   λ3,n   γ3,n 

where ε1,n = iLine,n − An · sin(ωn · ti + ϕn ) − Bn · cos(ωn · ti + ϕn ) ε2,n = {vLoad,n − (Vmn − RLine · An + LLine · Bn · ωn ) · × sin(ωn · ti + ϕn ) + (RLine · Bn + LLine · An · ωn ) · × cos(ωn · ti + ϕn )

328

M. Erhan Balci, M. Hakan Hocaoglu / Electric Power Systems Research 78 (2008) 318–329

ε3,n = {−[An − ωn · C · (RLine · Bn + LLine · An · ωn )] · × sin(ωn · ti + ϕn ) +[−Bn +C · ωn · (Vmn −RLine · An +LLine · Bn · ωn )] · × cos(ωn · ti + ϕn )} δ1,n = eD1 ·ti ,

δ2,n = eD2 ·ti ,

δ3,n = eD3 ·ti

λ1,n = (−RLine − LLine · D1 ) · eD1 ·ti λ2,n = (−RLine − LLine · D2 ) · eD2 ·ti

β 3α1 α3 − α22 α2 + − 12α1 3α1 β 3α1

√ 6α1 α3 − 2α22 β 3 +j + 2 6α1 3α1 β

D2 = −

β 3α1 α3 − α22 + 12α1 3α1 β

√ α2 3 6α1 α3 − 2α22 β − −j + 3α1 2 6α1 3α1 β

D3 = −

where  36α1 α2 α3 − 108α21 α4 − 8α32 + 12α1 12α1 α33 − 3α22 α23 − 54α1 α2 α3 α4 + 81α21 α24 + 12α32 α4

 β=

3

λ3,n = (−RLine − LLine · D3 ) · eD3 ·ti

References

γ1,n = (1 + RLine · C · D1 + LLine · C · D1 2 ) · eD1 ·ti

[1] A.E. Emanuel, Powers in nonsinusoidal situations a review of definitions and physical meaning, IEEE Trans. Power Deliv. 5 (3) (1990) 1377– 1389. [2] C.I. Budeanu, Reactive and Fictitious Powers, Publication No. 2 of the Rumanian National Inst. Bucuresti, 1927. [3] S. Fryze, Wirk-, Blind-, und Scheinleistung in Elektrischen Stromkreisen Mit Nichtsinusoidalformingem Verfauf von Strom und Spannung, Elektrotechnische Zeitschriji 53 (25) (1932) 596–599. [4] E.W. Kimbark, Direct Current Transmission, J. Wiley and Sons, 1971. [5] W. Shepherd, P. Zakikhani, Power factor correction in nonsinusoidal systems by the use of capacitance, J. Phys. D: Appl. Phys. 6 (1973) 1850– 1861. [6] W. Shepherd, P. Zand, Energy Flow and Power Factor in Nonsinusoidal Circuits, Cambridge University Press, 1979. [7] D. Sharon, Reactive power definition and power-factor improvement in nonlinear systems, Proc. Inst. Electr. Eng. 120 (1973) 704–706. [8] M. Depenbrock, Wirk- und Blindleistung, ETG-Fachtagung Blindleistung, Aachen, 1979. [9] N.L. Kusters, W.J.M. Moore, On the definition of reactive power under nonsinusoidal conditions, IEEE Trans. Power Apparatus Syst. 99 (5) (1980) 1845–1854. [10] M.A. Slonim, J.D. Van Wyk, Power components in a system with sinusoidal and nonsinusoidal voltages and/or currents, IEE Proc. Electr. Power Appl. 135 (2) (1988) 76–84. [11] L.S. Czarnecki, Powers in nonsinusoidal networks: their interpretation, analysis and measurement, IEEE Trans. Instrum. Meas. 39 (2) (1990) 340–345. [12] L.S. Czarnecki, Physical reasons of currents rms value increase in power systems with nonsinusoidal voltage, IEEE Trans. Power Deliv. 8 (1) (1993) 437–447. [13] L.S. Czarnecki, Comparison of power definitions for circuits with nonsinusoidal waveforms, IEEE Tutorial Course 90EH0327-7-PWR (1990) 43–50. [14] L.S. Czarnecki, C.I. Budeanu, S. Fryze, Two frameworks for interpreting power properties of circuits with nonsinusoidal voltages and currents, Electr. Eng. (Archiv fur Elektrotechnik) 80 (6) (1997) 359– 367. [15] L.S. Czarnecki, Distortion power in systems with nonsinusoidal voltage, IEE Proc. Electr. Power Appl. 139 (3) (1992) 276–280. [16] S.Q. Sun, Q.R. Xiang, Waveform distortion and distortion power, IEE Proc. Electr. Power Appl. 139 (4) (1992) 303–306. [17] P.S. Filipski, Apparent power—a misleading quantity in the non-sinusoidal power theory: are all non-sinusoidal power theories doomed to fail, Eur. Trans. Electr. Power 3 (1) (1993) 21–26.

γ2,n = (1 + RLine · C · D2 + LLine · C · D2 2 ) · eD2 ·ti γ3,n = (1 + RLine · C · D3 + LLine · C · D32 ) · eD3 ·ti and the initial values of line current and load voltage related to nth harmonic of voltage source iLine,n =

Vmn · sin(ωn · ti + ϕn − φn ) |RLine + j(ωn LLine − (1/ωn C))|

vLoad,n = −iLine,n · j φn = arctan

1 ωn C

(ωn LLine − (1/ωn C)) RLine

(ti = tconduction for first cycle and ti = tconduction + Tf /2 for second cycle). Initial values of load current are zero at ti for first and second cycle. The characteristic equation is α2 α3 α4 D3 + D2 + D + =0 α1 α1 α1 where α1 = LLoad LLine C α2 = RLoad LLine C + RLine LLoad C α3 = RLoad RLine C + LLine + LLoad α4 = RLine + RLoad The roots of characteristic equation are D1 =

α2 β 6α1 α3 − 2α22 − − 6α1 3α1 β 3α1

M. Erhan Balci, M. Hakan Hocaoglu / Electric Power Systems Research 78 (2008) 318–329 [18] F. Ghassemi, What is wrong with the electric power theory and how it should be modified, in: IEE 9th International Conference of Metering and Tariffs for Energy Supply-Mates 99, 1999, pp. 109–114. [19] E. Wilczynski, Total apparent power of the electrical system for periodic, deformed waveforms, IEE Proc. Electr. Power Appl. 147 (4) (2000) 281–285. [20] F. Ghassemi, New concept in ac power theory, IEE Proc. Gen. Transm. Distrib. 147 (6) (2000) 417–424. [21] IEEE Std 1459-2000, IEEE Standard Definitions for the Measurement of Electric Power Quantities under Sinusoidal Non-sinusoidal, Balanced or Unbalanced Conditions, IEEE Standard, September 2002.

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[22] L.S. Czarnecki, N.L. Kusters, W.J.M. Moore, Additional discussion to ‘reactive power under nonsinusoidal conditions’, IEEE Trans. Power Apparatus Syst. 102 (4) (1983) 1023–1024. [23] E.B. Makram, R.B. Haines, A.A. Girgis, Effect of harmonic distortion in reactive power measurement, IEEE Trans. Ind. Appl. 28 (4) (1992) 782–787. [24] J.C. Montano, et al., DSP-based algorithm for electric power measurement, IEE Proc.-A. 140 (6) (1993) 485–490. [25] M.E. Balci, M.H. Hocaoglu, Comparison of power definitions for reactive power compensation in nonsinusoidal conditions, in: Proceeding of 11th IEEE ICHQP, 2004, pp. 519–524.