Download 1 Exploring methods for fast harmonic magnitude and phase

Тема Задатак
1
1. Истраживање метода брзог
израчунавања амплитуда и фаза
хармоника на потрошачу
2. Истраживање метода и
алгоритама за дефинисање
активне, реактивне и снаге
изобличења у условима
нелинеарних изобличења и
њиховог брзог израчунавања
3. Истраживање метода
дефиниције и израчунавања
фактора снаге изобличења
4. Дефиниције хардверских
решења имплементације
алгоритама израчунавања
хармоника, активне снаге,
реактивне снаге, снаге
изобличења и фактора снаге
изобличења.
Извршиоци
Носилац
х
М. Димитријевић,
Д. Стевановић
В. Литовски,
П. Петковић
х
Д. Стевановић,
М. Димитријевић
В. Литовски,
П. Петковић
х
Д. Стевановић
М. Димитријевић
В. Литовски,
П. Петковић
Д. Стевановић
М. Димитријевић
В. Литовски,
П. Петковић
2011
х
Тип
публ.
М61,
М51
Број
публ.
2
1 Exploring methods for fast harmonic magnitude and
phase calculation
1.1.1 Sinusoidal Voltage and Current in Single Phase Circuits
A sinusoidal voltage source
v(t )  2VRMSsin  ω0t 
supplying a linear load, will produce a sinusoidal current
(1.1)
i(t )  2 I RMSsin  ω0t  φ 
(1.2)
where VRMS is the RMS value of the voltage, IRMS is the RMS value of the current, ω is the angular
frequency, φ is the phase angle and t is the time. The instantaneous power is
p (t )  v(t )  i (t )
(1.3)
and can be represented as
p(t )  2V
I
RMS RMS
sin ω0t  sin  ω0t  φ   p p  pq .
(1.4)
Using appropriate transformations we can write:
pp  VRMS I RMS cos φ  1  cos  2ω0t   
 P  1  cos  2ω0t  
(1.5)
and
pq  VRMS I RMS sin φ  sin  2ω0t   Q sin  2ω0t 
(1.6)
where
P  VRMS I RMS cos φ, Q  VRMS I RMS sin φ
(1.7)
represent real (P) and reactive (Q) power.
It can be easily shown that the real power presents the average of the instantaneous power over
a cycle:
t +T
1 0 .0
(1.8)
P
v(t )  i (t )  dt
T t0
where t0 is arbitrary time (constant) after equilibrium, and T0 is the period (20 ms in European and
1/60 s in American system, respectively).
The reactive power Q is the amplitude of the oscillating instantaneous power pq. The apparent
power is the product of the root mean square value of current times the root mean square value of
voltage:
S  VRMS I RMS
(1.9)
or:
S  P2  Q2 .
Power factor is simply defined as the ratio of real power to apparent power:
PF  P .
S
For pure sinusoidal case, using equations (7), (10) and (11) we can calculate:
PF  cos φ.
(1.10)
(1.11)
(1.12)
1.1.2 Sinusoidal Voltage and Current in Polyphase Circuits
The power concepts developed for single-phase circuits with sinusoidal voltages and currents
can be extended to polyphase circuits. Equations (1.7) can be rewritten to define power terms
equivalent to the single-phase terms. In these equations, n represents a phase number, N is the
total number of phases, and φn phase difference:
N
P   I n ,RMSVn ,RMS cos φ n ,
n 1
N
Q   I n,RMSVn,RMS sin φ n
(1.13)
n 1
The apparent power is defined by equation (1.10).
Power factor is defined by equation (1.11). Note that the power factor is not always equal to the
cosine of the phase angle. In many three-phase balanced systems, the phase angles of all three
phases are equal and the cosine relationship holds. In unbalanced systems, each phase has a
different phase angle, the phase voltages and currents are not equal, and the cosine relationship is
no longer valid.
1.1.3 Nonlinear Loads in Single Phase Circuits
In the presence of nonlinear loads the system no longer operates in sinusoidal condition and use
of fundamental frequency analysis does not apply any more. Traditional power system quantities
such as effective value, power (active, reactive, apparent), and power factor need to be
numerically calculated from sampled voltage and current sequences by performing the FFT
algorithm.
The RMS value of some periodic physical entity X (voltage or current) is calculated according
to the well-known formula:
X RMS
1

T0
t 0 +T0
  x(t ) 
2
dt
(1.14)
t0
where x(t) represents time evolution, T is the period and t0 is arbitrary time. For any periodic
physical quantity x(t), we can produce a Fourier representation:

x(t )  a0    ak  cos  kω0t   bk  sin  kω0t  
(1.15)
k 1
or

x(t )  c0  ck  cos  kω0t  ψ k 
(1.16)
k 1
ck  ak2  bk2
where c0  a0 represents DC component,
the magnitude of kth harmonic,
bk
2π
the phase of the kth harmonic, and ω0 
the angular frequency.
T0
ak
The Fourier coefficients ak, bk are calculated as:
ψ k  arctan

1
a0 
T0

T0

2

T
2
ak 
T0
x(t )dt ,
0
2

T0

T
0
2
 2kπt 
x(t )  cos 
 dt
 T0 
(1.17)
2
and

2
bk 
T

T0

T
0
2
 2kπt 
x(t )  sin 
 dt.
 T0 
(1.18)
2
th
The RMS value of the k harmonic is
X k, RMS 
ck
2
.
(1.19)
We can calculate total RMS value as
M
X
X RMS 
2
k , RMS
2
 X1,2 RMS  X H,
RMS
(1.20)
k 1
where M is highest order harmonic taken into calculation. Index “1” denotes first or fundamental
harmonic, and index “H” denotes contributions of higher harmonics.
Equations (1.14) – (1.20) need to be rewritten for voltage and current. Practically, we operate
with sampled values and integrals (1.17) and (1.18) are transformed into truncated series.
For a single-phase system where k is the harmonic number, φk phase difference between voltage
and current of kth harmonic, and M is the highest harmonic, the total active power is given by:
M
P   I k ,RMS  Vk ,RMS  cos φ k  P1  PH .
(1.21)
k 1
The first addend in the sum (1.21), denoted with P1, is the fundamental active power. The rest
of the sum, denoted with PH, is harmonic active power.
The total reactive power is given by:
M
Q   I k ,RMS  Vk ,RMS  sin φ k  Q1  QH .
(1.22)
k 1
It should be noted that the actual contribution of harmonic frequencies to active and reactive
power is small (usually less than 3% of the total active or reactive power). The major contribution
of higher harmonic to the power comes as distortion power D:
D2 
M

j k
j 1, k 1
I 2 j ,RMS V 2 k ,RMS
The apparent power is given by formula (1.9) and it can be written:
(1.23)
S 2  I 21,RMS V 21,RMS  I 21,RMS V 2 H,RMS 
S12
DV2
(1.24)
 V 21,RMS  I 2 H,RMS  V 2 H,RMS  I 2 H,RMS
DI2
SH2
where S1 represents the fundamental apparent power, DV the voltage distortion power, DI the
current distortion power and SH the harmonic apparent power. S1 and SH are
S1  P12  Q12 , SH  PH2  QH2  DH2
(1.25)
where DH represents harmonic the distortion power. The total apparent power is
S  P2  Q2  D2 .
It is obvious that for sinusoidal voltages and currents, (1.26) reduces to (1.10).
The total harmonic distortions, THD, are calculated from the following formulas:
I H, RMS
THDI 

I 1, RMS

1
I 1, RMS
M
I
2
j , RMS
j 2

2
I RMS
 I 2 1, RMS
(1.26)
(1.27)
I 2 1, RMS
and
THDV 

1
V1, RMS
VH,RMS
V1,RMS

M
V
k 2
2
k , RMS

(1.28)
2
VRMS
 V 21, RMS
V 21, RMS
where Ij,Vk j,k=1, 2, …, M stand for the harmonic of the current or voltage. It can be shown that:
DI  V1, RMS  I H, RMS  S1  THDI
DV  VH, RMS  I1, RMS  S1  THDV
(1.29)
SH  S1  THDI  THDV .
The fundamental power factor or displacement power factor is given by the following formula:
P
PF1  1  cos φ1.
(1.30)
S1
The total power factor DPF, defined by equation (1.11), taking into account (1.21) and (1.24),
becomes
P1  PH
P
TPF  
(1.31)
S
S12  DI2  DV2  SH2
and substituting (1.29) and (1.30):
TPF 
 PH 
1   cos φ1
P1 

1  THDI2  THDV2  THDI  THDV 
2
(1.32)
In real circuits, PH<<P1 and voltage is almost sinusoidal (THDV<5%), leading to a simpler
expression for TPF:
cos φ1
TPF 
1  THDI2
.
(1.33)
1.1.4 Nonlinear Loads in Polyphase Circuits
For polyphase systems, active, reactive and distortion powers are defined for each phase and
total powers are sum of all phase powers:
N
M
P   I n ,k ,RMS Vn ,k ,RMS  cos φ k
n 1 k 1
N
M
Q   I n ,k ,RMS Vn,k ,RMS  sin φ k
(1.34)
n 1 k 1
N
D2  
n 1
M

j k
j 1, k 1
I 2 n , j ,RMS  V 2 n ,k ,RMS .
Voltages Vn,k,RMS are phase-to-neutral voltages. Indexes n and k denotes phase number and
harmonic order, respectively. N is number of phases and M is number of harmonics taken into
calculation.
The apparent power can be defined as sum of apparent powers per phase – arithmetic apparent
power (SA) or as vector apparent power (S):
N
SA  Vn,RMS  I n,RMS
(1.35)
n 1
S  P Q  D
Note that Vn,RMS and In,RMS are total RMS values per phase, and n is a phase index.
SA uses arithmetic addition of vector magnitudes and is equal to apparent power S only if the
polyphase voltages and currents have equal magnitudes and equal phase difference; a situation
that often exists in balanced three-phase systems. The two alternative definitions of apparent
power, S and SA, give rise to two possible values for power factor, calculated using equation
(1.11).
2
2
2
1.1.5 Acquisition and discrete Fourier transform
Signal acquisition can be mathematically represented by sum of convolution of continual function
x  t  in time domain and Dirac delta function:
xS  t  
 
  x t    t  nT  dt
S
n  
(1.36)
where TS represents sampling period,  (t ) Dirac delta function defined as of the sequence of
Gaussians
0, t  0
,  t   

(1.37)
a 
, t  0
is continual function with non-zero values in discrete time points nTS. The
  t   lim
a 0
xS  t 
1
t 2
e
a2
frequency of sampling fS  1 TS must fulfill Nyquist–Shannon criterion, fS  B where B
represents signal bandwidth. If x  t  is periodical signal, with period T0 – which is also period of
elementary harmonic – bandwidth is B  f max  M  f 0 , where M represents order of the highest
harmonic taken into account.
Discrete Fourier transform maps time domain function into frequency domain, can be
derived from continual transform equation:
X  f   F  x t  

 x t  e
2 jft
dt
(1.38)

where f is frequency,   2 f , x  t  is time domain function and X  f  
frequency domain
representation. Discrete Fourier transform (DFT) maps discrete values of xs  nTS  into discrete
spectrum Xk:
N 1
X k   xs  nTS   e 2 jkn N
(1.39)
n 0
where k=0,1,…,N-1 is index of discretized spectrum, n sample index and N total number of
samples calculated into DFT. The calculated spectrum is made of individual components at the
frequency bins which are expressed as:
1
(1.40)
N  Ts
where Δf denotes frequency bin quantum. Exact DFT is possible if two conditions are fulfilled:
f k  k  f  k 

signal x  t  is periodic, period is T0, x  t   x  t  nT0  , n Z

the observation time NTs is integer multiple of period T0 corresponding to signal x  t  .
DFT implementation defined by equation (1.39) requires N2 complex number calculations.
Optimized FFT algorithm requires as small as Nlog2  N  operations.
Main disadvantage of DFT is failing to accomplish the second condition in practice: if
frequency of x  t  varies, observation time will not be integer multiple of the period. In this case,
the result of DFT differs from expected result. The difference is characterized by wider signal
spectrum and an amplitude error. In order to eliminate this disadvantage, one can implement other
algorithms, such as Goertzel algorithm or Adaptive Real Time Monitoring (ARTM) algorithm [1],
or utilize iterative methods for harmonic amplitude and phase calculations, like Newton method
[2,3] or STLS algorithm [4].
Fourier transform (1.39) maps real, or complex number sequence xs  nTS  into complex
sequence. Inherent property of transform is
X N k  X  k .
(1.41)
X Nk  X k .
(1.42)
If sequence xs  nTS  is real, values Xk i X-k are complex-conjugate:
RMS value and phase for kth harmonic are:
X k ,RMS 
Xk
2
(1.43)
k  arg  X k  .
Reference
[1] Petre Minciunesc, Gabriel Antonesei, “Novel harmonic analysis method for smart metering,”
EE Times Europe, 2011
[2] Vladimir V. Terzija, Vladimir Stanojević, Marjan Popov, Lou van der Sluis, “Digital Metering
of Power Components According to IEEE Standard 1459-2000 Using the Newton-Type
Algorithm,” IEEE Transactions on Instrumentation and Measurement, Vol. 56, No. 6,
December 2007, pp. 2717-2724
[3] Vladimir V. Terzija, Vladimir Stanojević, “Two-Stage Improved Recursive Newton-Type
Algorithm for Power-Quality Indices Estimation,” IEEE Transactions on Power Delivery, Vol.
22, No. 3, July 2007, pp. 1351-1359
[4] Vladimir V. Terzija, Vladimir Stanojević, “STLS Algorithm for Power-Quality Indices
Estimation,” IEEE Transactions on Power Delivery, Vol. 23, No. 2, April 2008, pp. 544-552
Published papers
1.1 (Task 1.1) (category M63) Dimitrijević, M., Jovanović, B.: „Određivanje THD faktora
bazirano na integrisanim kolima Teradian 71M6533 i IMPEG“, Zbornik LV konferencije
ETRAN, Banja Vrućica, Bosna i Hercegovina, 06.06.-09.06., 2011, EL 3.4, ISBN 978-8680509-66-2
1.2 (Task 1.1) (category M23) Dimitrijević, M., Litovski, V.: „Power Factor and Distortion
Measuring for Small Loads Using USB Acquisition Module“, Journal of Circuits Systems and
Computers, World Scientific Publishing Co. Pte. Ltd., Singapore, DOI
10.1142/S0218126611007657, August, 2011, Vol. 20, No. 5, pp. 867-880, ISSN 0218-1266
1.3 (Task 1.1) (category M63) Dimitrijević, M., Litovski, V.: „Virtual Instrument for ThreePhase Power Quality Analysis,“ VIII Simposium on Industrial Electronics INDEL 2010,
Banja Luka, Bosnia and Herzegovina, 04.11.-06.11., 2010, pp. 189-194, ISBN 978-99955-4603-8
1.4 (Task 1.1) (category M33) Dimitrijević, M., Litovski, V.: „Computer Based Power Factor
and Distortion Measuring for Small Loads,“ Proceedings of the Small Systems Simulation
Symposium 2010, Niš, 12.02.-14.02., 2010, pp. 81-85, ISBN 987-86-6125-006-4