ABSTRACT. Since the beginning of the power systems, harmonic distortion have been existed due to nonlinear equipments such as; generators, transformers ...
2004 11th International Conference on Harmonics and Quality of Power
COMPARISON OF POWER DEFINITIONS FOR REACTIVE POWER COMPENSATION IN NONSINUSOIDAL CONDITIONS M.Erhan BALCI
M.Hakan HOCAOGLU Electronics Engineering Department Gebze Institute of Technology, 41400. Gebze, TURKEY
ABSTRACT Since the beginning of the power systems, harmonic distortion have been existed due to nonlinear equipments such as; generators, transformers motors etc. Moreover, harmonic pollution on power systems has dramatically increased in parallel with large proliferation of power electronic devices. Consequently, the reactive power compensation in nonsinusoidal conditions became one of the most important problems in power systems. This paper presents a comparison on widely accepted power defmitions in a simple single phase circuits with nonsinusoidal waveforms of voltage and/or currents by~. giving.. particular emphasis to the reactive power compensation.
general, these comparisons neglect the line impedance and non linearity introduced due to distorted source voltage. In this work, widely accepted power d e f ~ t i o n shave been compared for a simple single phase circuit by giving particular emphasis to the reactive power compensations. Accordingly a closed form expression has been derived for nonsinusoidal situations taking into account source impedance and nonlinearity. Although comparisons have been undertaken for reactive power compensation, the results also give useful information on line loading. 2. SINUSOIDAL CONDITIONS
Instantaneous power for sinusoidal situations is given as
Keywords: Power definitions, noitsiiiusoidal conditions, VAr p ( t ) = " ( I ) . ;(I) = 2 . V .I . Sir~wr.Sin(ot - (D) (1) compensation, harmonic, power qua&. with instantaneous voltage; V(I) = JI.Y .sirrwc THE LIST OF PRINCIPLE SYMBOLS (2) and instantaneous current; V,: RMS value of nthharmonic voltage ;(l)=JI.l-S;n(Of-c) (3) Vm,: Maximum value ofn" harmonic voltage Instantaneous power can be divided into two components as; I.: R M S value of ufhharmonic current t : Time P(4 =PAt) +P N (4) (D" : Phase angle ofn" harmonic with instantaneous active power w. : Angular frequency of n" hqnonic p~(t)=v~I.cosy?-(l-coszwt) (5) It RMS value of voltage and instantaneous reactive power I: R M S value of current p , ( l ) = v . I . Siilp,-Sin2ot (6) p : Phase angle Instantaneous active, reactive and total powers for sinusoidal o : Angular frequency situations are shown in Figure 1. Tf:Period of hndamental waveform 1. INTRODUCTION
In classical manner, generated and distributed electric energy must be sinusoidal with predetermined frequency and magnitude. However, nonsinusoidal currents and voltages always exist in electric power systems due to nonlinear elements such as; transformers, motors, generators etc. Since the reference was made to the fact that oscillations of power between an alternating voltage source and load in 1888, a number of power defmitions have been discussed and compared for nonsinusoidal conditions in the literature [l]. It was repeatedly shown that classical definitions of electric Figure 1: Instantaneous active, reactive and total powers in power, namely; active, reactive and apparent power, does not sinusoidal situations. fulfill the conditions caused by harmonics. Consequently, various power definitions and calculation methods have been The average value ofthe instantaneous active power; proposed [ 1, 2, 31. Some of those definitions widely discussed = v.I,cos(D (7) and evaluated using simple analytical techniques [l, 2, 31. In defined as active Dower. The maximum value of the instantaneous reactive power;
0-7803-8746-5/04/$20.00 02004 IEEE.
519
Q = V.I.Siqo (8) defmed as reactive power. The vector sum of active and reactive powers; S' = P i+ Q' (9) defmed as apparent power for sinusoidal situations. The power factor @a which determines the line efficiency can he calculated as; P Pf (10)
3.2. FRYZE'S POWER EQUATION
Despite the fact that Budenau's power decomposition provides enough information on compensable reactive power, the calculated active and reactive powers merely give any data ahout the source efficiency. In addition, Budeanu's equation require harmonic domain calculations and sophisticated measurement devices. Accordingly, Fryze proposed a current based decomposition [4] in which current is divided into two orthogonal components namely; active and reactive currents. However in nonsinusoidal situation, the process is not The fust is calculated using load active power straightfonvard as depicted above. P iA(f)E-v(f) (17) 3. NONSINUSOIDAL, CONDITIONS V'
=s
The nonsinusoidal voltage and current generated by nonlinear and the second, loads could he analyzed using Fourier series. The i R ( t ) = i ( t ) - i A ( f ) The power equation as suggested by Fryze is nonsinusoidal voltage and current are expressed;
(18)
S2=P2+Q: (19) with Fryze reactive power Q, = V.I, (20) The main advantages of Fryze's decomposition are; to provide accurate information on source efficiency and to he As in sinusoidal condition the instantaneous power could determined using ordinary phasor measurement devices. simply be calculated with product of instantaneous voltages However calculated values are not suitable for reactive power compensator design, and currents 3.3. SHEPHERD AND ZAND'S POWER EQUATION It is clear that "power" in above d e f ~ t i o n sis actually in phase power which is converted to the work. However there is also harmonic in phase power in reverse direction i.e. from load to source. Despite the fact that the reverse power in phase, therefore, active in nature, it is treated as distortion power in Budeanu's decomposition and reactive power in Fryze's terms. The same observation can he made for Budeanu's reactive power in which reverse reactive power is considered as distortion power. This phenomenon was discussed by Shepherd and Zand in reference [ 5 ] . 3.1. BUDEANU'S POWER EQUATION Accordingly, the current is divided into three components Budeanu postulated that total power consists of two which are: active current; orthogonal components namely active and deactive powers (21) [3]. The former is simply the average power which could easily be calculated by averaging the instantaneous power in reactive current; time domain or by convolution in phasor domain:
Nevertheless, decomposition of instantaneous power in nonsinusoidal conditions is not a clear-cut process, as done for sinusoidal condition. 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 case). Consequently, a number of new defmitions have been proposed in the literature. They are briefly summarized here for convenience:
(14)
P = -p".I".cosq" I
Ix
=@Z
This component was chosen by Budeanu due to the fact that it and distortion current;
is actual power converted to the work. The later one was I D = JlqT (23) divided into two components Budeanu reactive and distortion Authors proposed a power equation related to this current powers. Budeanu reactive power is calculated by summing of decomposition: the individual harmonic reactive powers: sz = s; + s i +si (24) with active apparent power, Q, = cV,.I,.Sinpn (15) Distortion power is calculated by cross product of different harmonics voltages and currents; "'I
D'B -- S' - P' - Qj = cVn*:'Ji+ Vj:'J:- 2.VJa.In.Va.Cos(qn-qe)(16)
reactive apparent power, S,
=@*
Budeanu reactive power can be completely compensated with a simple capacitor. However this is not a case for the and distortion apparent power, distortion power, given in Equation 16. 520
and scattered current where z: the harmonic number of current in which there is no where k represents current harmonic numbers do not present voltage harmonics. in the set of voltage harmonic numbers N. The equivalent the harmonic number of voltage in which there is conductance of load defmed as; current harmonics. Therefore, active and reactive apparent powers provide =_ P (37) meaningful data about line loading, therefore, efficiency. * VI where n* harmonic admittance of load 3.4 SHARON'S POWER EQUATION = C,+ jB" (38) Although Shepherd and Zand's power equations give enough He proposed that power equation related to this current information on line loading, nevertheless can not be used to calculate power factor. Consequently, Sharon . proposed decomposition as: reactive component of power equation which is based on s2= p i + D; + g + D; (39) Shepherd and Zand's reactive apparent power. However he with reactive power chose the active component of his power equation as average Q, = V - 1 , (40) power i.e. P, thus; Sharon's power equation is defined [61 as; scattered power Ds = V .Is s2= P2+s; +si (41) (28) and generated harmonic power with reactive apparent power, I. Dx=V.I,, (42) The main property of Czamecki's power equation is (29) identification of the physical phenomena responsible for the source current increase.
(30) 4. ANALYSIS OF POWER DEFINITION FOR Sharon's power equation give data about line loading REACTIVE POWER COMPENSATION conditions which is made by reactive load current and power ne behavior of the reactive power compensation in factor easily be calculated. nonsinusoidal situations is analyzed for three different cases. These cases are nonsinusoidal voltage source-linear load, 3.5. KTMBARK'S POWER EQUATION nonsinusoidal voltage source-nonlinear load and sinusoidal Kimbark postulated that total power is consist of two voltage source-non~mearload, orthogonal components namely; active and deactive powers as done by Budeanu. Kimbark proposed active component as 4.1. NO"NtJSOIDAL VOLTAGE SOURCE AND averaxe power. The deactive Dower was divided into hvo LINEARLOAD components i.e. Kimbark reactive power and distortion powers The circuit used in the caSe of linear load with nonsinusoidal [7]. The first is calculated by harmonic phasors. voltage source is given in Figure 2. Q, = V,1,.Sky?, (31) 2 Line Thus, reactive power can easily be related to load parameters. On the other hand distortion power consists of individual harmonic reactive powers and cross products of different harmonics currents and voltages:
_ _
DE
=Jm
-
(32)
-
Figure 2: The circuit used for the analysis.
3.6. CZARNECKI'S POWER EOUATION ~
The need of identification decreasing source efficiency due to nonsinusoidal waveforms was motivated Czamecki's work. In his studies author clearly distinrmishes current comuonents due IO load and source nonllncantlcs Accordlngly, he employed a current bawd decomposltlon In which current divided into four components i.e. active, reactive, scattered and harmonic currents 171. source current Can he exuressed as: I' = I; + rf12 + I; + I,' (33) These components are defined as reactive current
-
A passive impedance model is used in the circuit for presentation of load. The current and voltage are calculated by super position theorem in phasor domain.
NONSIKUSOIDAL \'OLTAGE YONI.INEAR I.OAD 4.2.
SOURCE AND
The circuit used in the case of nonlinear load with
(34) generated harmonic current 18
=p
(35)
Figure 3: The circuit used for the analysis
521
The circuits operate in two modes as triac conduction and triac by ATP version of EMTP [8] for 90' triac conduction angle, cut off. During triac cut off mode, equations of the current and as depicted in Figure 4. voltage in circuit are expressed using super position theorem ..EMTP in phasor domain. During triac conduction mode, equations of ANALYTICAL current and voltage in the circuit are found by solving the system ofequations given below
4,"*vm(t) RLw -i,Jt) + Lh .-+ dt
= v(r)
(43)
Equations (4346) are solved for the initial conditions line current, load voltage and load current. The solutions of the system for line current iLine(t),load voltage vhd(t) and load current i-(t) is ilinc(f)=
2
[ A . . S i n ( w , . t t a.)+ B , . c o s ( w . . t + $7.)1+
Figure 4 The time domain comparison of the results with EMTP.
The analytical computation and EMTP results are in very (47) close agreement.
5. RESULTS
To show the effect of passive compensation capacitance over the power definitions; power components for each definitions are calculated in harmonic domain via FFT of sampled currents and load voltages of analyzed circuits which has triac controlled load. Three different cases have been considered. These are nonsinusoidal voltage source-linear load, nonsinusoidal voltage source-nonlinear load, and sinusoidal voltage source-nonlinear load. The variations of Bndeanu, Kimbark, Fryze, Shepherd and Zand, Sharon, and Czarnecki reactive powers with power of compensation capacitance for three different cases are shown in Figure 5. + t [ ( B n-Vm.. C - 0 ,+R,
. C . q -L,
+ t [ ( I + R - - C - Q +L,.C.D:).eln
+ t [ ( i + R , .C.D, +L,, + t [ ( l + R , .C.D, +L,
.eq']+
C.D,') .eZ.
..Sinusoidalvoltage sourceNonllnearload -Nonsinusaidal voltage souree-Nonlinear load --Nonsinusoidal voltage source-Linear load ......................................... ...
.B. .C.,'). G s ( q .I +qJ]+
,.......
cli
O.*l
4
08
""1
'I+
0.84
0.0
.C.0,') .c3. .e"']
........................................ ........ ...... ..~ .... ...... .......
......... ......
0.z
(49)
where & and B, are the coefficients for n* harmonic of the voltage source and c l , c2., c3. are the coefficients for n* harmonic of the voltage source. These coefficients and initial values of line current, load voltage and load current are given in Appendix. Triac cut-off time &4a is solution of the following equation
Ld (tuf-d)= 0 for interval ~ . d , i . . < ~ ~ , ~ ( ~ " d ~ ~ "/.V +Tf
L
.m
1
0.75
$w m%0.25
.........................
(50) Figure 5: The variations of Budeanu, Kimbark Flyze, Shepherd and Zand, Sharon, and Czamecki reactive powers.
The sinusoidal voltage source and nonlinear load case could easily he obtained by substituting n=l in the equations from In nonlinear load cases triac is conducted during 90". SX,Se and Qn reactive powers attain the same numerical values for (43) to (50). all compensation capacitance in three different voltage source In order to demonstrate accuracy of the solutions the results and load can be Seen that QF, sx, se and Q n can not Obtained by this method 'Ompared with the results produced be completely compensated by means of simple capacitance. 522
Distortion power calculated according to Budeanu and Kimbark still he compensated hy means of simple capacitance. This shows that Bndeanu and Kimbark‘s distortion powers consist of not only distortion power but also reactive power which can be compensated. Shephard’s distortion power So and Czamecki’s harmonic generated power DE are equal to zero for all cases due to load voltage and line current do not The variations Shepherd and Zand’s active apparent power, include different harmonics. average power and fundamental harmonic active power for 6. CONCLUSION three different voltage source and load cases are shown in In this study, some of the well known power definitions have Figure 6. been analyzed in a simple generic circuit for three voltage ..Sinusoidalvoltage source-Nonlinearload source and load cases.
However, they all attain their minimum value when power factors reach their maximum values for all voltage source and load cases. On the other hand, for sinusoidal voltage sourcenonlinear load QB and QK obtain zero values when power factor maximum. However, this is not the case for nonsinusoidal voltage source-linear load and nonsinusoidal voltage source-nonlinear load cases.
-Nonrinusoidalvoltare source-Nonlinearload I
Power decomposition according to Fryze, Shepherd and Zand, Sharon, Czamecki could be used to accurately estimate source efficiency. Reactive power calculated accordmg to Fryze, Shepherd and Zand, Sharon, and Czamecki could not be completely compensated by a passive capacitance. Optimum compensation capacitance could not be determined by using any of power definitions directly. On the other hand optimum compensation capacitance could be calculated by taking derivative of to Fryze, Shepherd and Zand, Sharon, Czamecki reactive powers.
Figure 6: The variations Shepherd and Zand’s active apparent power, average power and fundamental harmonic active As a future work, studies will be extended to cover constant power. power and voltage sensitive harmonic load models. As depicted in Figure 6; average power may attain value which is smaller than fundamental harmonic active power. REFERENCES This phenomenon demonstrates that harmonic active power [l] Emanuel, A.E.; “Powers in Nonsinusoidal Situations a might be in distortive nature. Therefore, average power does Review of Definitions and Physical Meaning”, IEEE Trans. not provide any meaningful information on loading conditions on Power Delivery, Vo1.5, No: 3, July 1990, pp.1377-1389. of the line. [2] IEEE Working Group on Nonsinusoidal Situations: Effect The variations of Budeanu, Kimbark, Shepherd and Zand, of Meter Performance and D e f ~ t i o n sof Power; “Practical Sharon and Czamecki power equation’s distortion components Definitions for Powers in Systems with Nonsinusoidal for three different voltage source and load cases are shown in Waveforms and Unbalanced Loads: A Discussion”, IEEE Figure 7. Trans. on Power Delivery, Vol.11, No.1. January, 1996, ..Sinusoidalvoltage source-Nonlinearload pp.79-101. -Nonsinusoidal voltage sourceNonlinear load -Nonrinusoidal voltage source-Linearload
[3] IEEE Std 1459-2000 IEEE Trial-Use Standard Defmitions for The Measurement of Electric Power Quantities Under Sinusoidal, Nonsinusoidal, Balanced, or Unbalanced Conditions, 21 June 2000. [4] Fryze, S.; “Active-, Reactive-, and Apparent Pow’& in Networks with Nonsinusoidal Waveforms of Voltage and Current”, (in Polish) Przegl. Elektr., No.7,8,1931, (in German) ETZ, Bd.53,1932.
[5] Shepherd, W. and Zand, P.; Energy Flow and Power Factor in Nonsinusoidal Circuits, (Cambridge University Press, 1979).
[6]Sharon, D.; “Reactive Power Definition and Power Factor Improvement in Nonlinear Systems”, Proc. IEE, 120, 1973, pp. 704-706. Figure 7: The variations of Budeanu, Kimbark, Shepherd and Zand, Sharon and Czamecki power equation’s distortion components. 523
[7] Czamecki, L. S.; "Comparison of Power Defmitions for [SI CANiAM EMTP User Group "Alternative Transient Circuits with Nonsinusoidal Waveforms", IEEE Tutorial Program (ATP) Rule Book", (CanadidAmerican EMTP Course 90EHO327-7-Pm pp.43-50. User Group, 1992).
for fmt cycle and t, =tondustion+ Tf /2 for second cycle )
( t,
Initial values of load current are zero at t, for fmt and second cycle, The characteristic equation is
+ La D ' + La
D + 3 = 0 where a, = LmLL&C a, a, The roots of characteristic equation are D'
, a2= RwL,,C
+ R,,LmC
, a,= RmR,,C
a,
DI -
6a,
3a,p
3a,
where ~ = ~ 3 6 a , a , a , - 1 0 8 a ~ a , - 8 a ~ + 1 2 a , ~ -3a:a:-54a,a,a,a,+Sla:a: l2a,a~
524
+12aia,
t L,_
+ Lid
, a4= Rbc
+ R-
