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DOI: http://dx.doi.org/10.20534/ESR-16-9.10-224-226
Kholiddinov Ilkhombek Khosilzhonovich Senior research worker, applicant for a degree, Department of «Electrical stations, networks and systems», Tashkent State Technical University, E-mail: holiddinov_ilhombek@mail.ru Shaismatov Sayfulla Ergashevich, Director of JSC Teploelectroproekt
About the method of determining the symmetrical components of voltages and currents
Abstract: The article presents the methods of determination of the symmetrical components of voltages and currents. The existing methods of calculation of the symmetrical components of voltages and currents are analyzed. The biggest attention is paid to the analytical method of calculation. Accurate and universal equations are presented. Keywords: method of symmetrical components, asymmetry of current and voltage.
With the development and wide spread of three-phase systems of alternating current, the need in accurate description of their asymmetrical regimes, quite diverse in their nature, has come up. Currently, asymmetrical regimes are becoming not only emergency, but also operating modes. Asymmetrical regime can be comparatively easily calculated with the use of method of symmetrical components. Moreover, only the use of the method of symmetrical components provides an opportunity to determine currents and voltages in the rotating machines of alternating current, i. e. in such installations, the input resistances of which depend on the order of alternation of phases of supply line.
According to the method of symmetrical components, the linear symmetrical components of voltages and currents are defined as follows:
U =-
U =-
(uab + a ■ubc + a2 ■uca)>
(uAB+a 2-UBC + a-UCA),
s, =-
U
U
„ifl
=3+«
■= 3 ■(IA + a
In
Ib + «2 ■ Ic),
■ h + a ■ [c)>
+ IB + >
(1) (2)
(3)
(4)
(5)
(6)
where U1, I1 — complex voltage and current in the phase A of liner symmetrical components of direct sequence; U., I2 — complex voltage and current in the phase A of linear symmetrical components of inverted sequence; I0 — current of symmetrical components of zero sequence; UAB, UBC, UCA, IA, IB, IC — complex values oflinear voltages and currents; i2 — complex coefficient of asymmetry ofvoltage on the inverted sequence; q>2 — argument of the coefficient of asymmetry on the inverted sequence; a — phase operator ( a = e,uo ).
Phase symmetrical components of voltages and currents are defined according to the equations:
¡v = - .(( + a-U„ + a
U.
2$
■ic
U
=
uc),
Uc),
+UB+UC),
+ a -U„ + a ■
.U± u l
(7)
(8) (9)
(10)
■3 ■(
+ a ■ I„n + a ■ I,
0 '
(11)
where U, , I
1 9
I« = 1 ■
3
+ a ■ hr + a■ Ir
- 3 + a ' IBC + a ' 1CA) , (12)
— complex voltage and current (in the phase A) of symmetrical components of direct sequence; U2 ^ I2 ^ — complex voltage and current (in the phase A) of symmetrical components of inverted sequence; U0 — voltage of symmetrical components of zero sequence; UA, UB, UC, IAB, IBC, I — phase voltages and currents; i0 — complex coefficient of asymmetry ofvoltage on the zero sequence; y — argument of the coefficient of asymmetry on the zero sequence.
The relation of the modules of linear symmetrical components to the phase ones is
The calculation of each of symmetrical components ofvoltages and currents of three-phase network in the equations (1)—(12) presupposes preliminary measurement of three vectors (three modules and three phases), i. e. six values, which presents significant difficulties in their measurements and leads to bigger errors of measurement and determination of symmetrical components.
In the practice of exploitation of electrical networks, linear and phase voltages and currents are measured with the help of voltmeters and ampere-meters. Herewith, they take the form of real numbers, and not complex ones, as it is specified in (1)-(12), thus, it is reasonable to consider the ways of determination of symmetrical components according to the results in the real electrical networks.
The existing ways of determination of symmetrical components can be divided into the analytical methods of calculation, definition with the help of graphical constructions, nomograms and tables. Analytical methods of calculation are divided into accurate, requiring the use of the means of computing technology in most cases, and approximate.
The accurate analytical ways of calculations include formulas recommended in [1]. Thus, the existing values of voltages of the direct U and inverted U2 sequences, according to [1], are recommended to be calculated according to the equations (1) (2), which can be unified:
(13)
About the method of determining the symmetrical components of voltages and currents
where UAB, UBC, UCA — applicable values of interphase voltages.
The presented equation provides accurate decision, but the sphere of its application is limited: in the case of extreme asymmetry at UAB=0 and UBC=UCA the fractions in the radical expression give an uncertainty in the form of
U2 -U2 0
UAB 0 ( )
In addition, the equation (13) refers to the private case of the method of symmetrical components, when the initial vectors oflin-ear voltages form a closed triangle, i. e. the conditions are performed:
uca +ubc * uab :
Uab +Ubc * Uca ,
uca +uab * ubc >
which should be taken into account, because, in practice, inaccuracies in the metering of measurement devices are possible.
Same applies to the calculation of applicable values of current of the direct and inverted sequence according to the equations identical to (13):
' 12
V3 ■ IA ±. 4 ■ I2B-
12 -12 1 b ± r
IA
-+ I.
12 - 12
±B_t£
Ia
(15)
where IA, I^ IC — applicable values of the phase currents.
To determine the applicable value of the voltage of zero sequence in [1], it is recommended to use the equation:
Ubc -Uca
Ub
-3 •-
U-u,
(16)
4-U2-I U
ujc -UC
Uab
3, 4 -Ui
U
Ui -Ui
where UA, UB, UC — applicable values of phase voltages.
During the determination of U0 in four-phase networks according to (16) at UAB=0 and UBC=UCA (two-phase short circuit), the equation gives uncertainty identical to (14) and, thus, it is not used in extreme cases of asymmetry. Moreover, the equation (16) is notable for crockhood, which makes it difficult to use it for practical purposes.
[1] presents the equations for tasks, where only the calculation of modules of symmetrical components and coefficients of asymmetry is required, but there are not expressions to determine phase angles f2 and q>0
[2] presents the equations to determine symmetrical compo-
nents presupposing the measurement of absolute values of three,
and to determine the component of zero sequence — six (three linear and three phase) voltages:
U = Uab
1 3
. 1 + x2 - y2 (n 1 + y2 - x2
1 + x■cos] arccos-----1+ y■cos]--arccos
2 ■ x 3 ) 1 3
2 ■ y
1 + x2 - y2 . (n 1 + y2 - x2
x ■ sin\ arccos-----1 + y■ sin\--arccos—--
2 ■ x 3 ) 1 3 2 ■ y
U2 = Uab 2 3
where
, 1 + X2 - y2 n (n 1 + y2 - x2
1 + x• cos\ arccos--—I— 1 + y•cos\ —+ arccos—--
2 • x 3 ) ^ 3 2 • y
1 + x2 - y2 . (n 1 + y2 - X2
x •stnl arccos--—I— I- y •sin\ —+ arccos—--
2 • x 3 ) ^ 3 2 • y
U0 = i/3 ■UA (l+x 2+y 2 )-U2AB (l+x2 + y2), (19)
Vab = arctg
Vab = arctg
y
S+V2-X
■ y + 2 ■ x + 2 ■ y
2 2 x - y
-, 4 4
-1 - x - y
-n-n,
S-yf2~>
= -n-n,
(21)
(22)
■ x" ■ y2 + 2 ■ x2 + 2 ■ y2 -1 - x4 - y4 where n is established according to the sign of numerator and denominator of equations (21)-(22).
As it is seen from (17)-(22), in the event of equality of any linear voltages to zero (for Ua f'AB> f'AB —UAB=0), the equations give the division by zero and, consequently, are limited in use. Moreover, (17)-(20), (21), (22) represent complex mathematical expressions and require significant calculations.
According to [3], the algorithm with the highest accuracy of calculation (with the procedure error Am=0) of symmetrical components Ut and U2 is based on the correlations of the triangle [4]. In the obtaining the expressions [4], vector UAB was accepted as real axis relative to which, phases ofvectors U and U2 were determined.
At the same time, the algorithm [4] is difficult to use for quick evaluation of the values of U and U2 (it contains 19 arithmetical operations and two operations of square rooting), hence, in [3], its analytical simplification at the expense of the fact that real axis is directed along the vector of direct sequence is done. In this case, the relation of linear expressions with voltages U and U2 is expressed with the system of equations
UAB = J(U1 + U2 ■ cosp2 )2 + U2 ■ sin2p2, UBC =yj[(U1 + U2 ■ cos((p2 +120°)]2 + U22 ■ sin2(p2 +120°), I (23)
UCA =yj[(U1 +U2 ■ cos(p2 -120°)]2 +U22 ■ sin2(p2 -120°),
the solution of which, after further simplifications, made in [6], leads to the result:
^ =J 6
(24)
Uab + Ubc +U2CA ±_
_±V3 V4 UAB Ubc - (Uab +Ubc + U2ca )2 As calculations showed, the equations (24) give accurate solutions in all cases, do not have limitations of the sphere of use like equations (13), (17), (18) and are simpler in calculations than the expressions from [1].
Considering a known correlation of modules oflinear and phase symmetrical components from (24), one can obtain equations for phase symmetrical components, which were also received in the work [5]:
UU* =J~
UAB + U2BC +U2CA ± ±S -V4 UAB UBC
-(Uab +Ubc +Uca )2
Such equations can also be used to determine linear and phase symmetrical components of currents identically with (15):
(17) 11,2 6 [i2a+IB+IC ^ 'V 4 UAU - (UA+U2B+U2C)2 ],
i-,, =J — h
1C$ '
18 L'A +Isc +IcA
±V3 V 4 ■ IA B ■ I2BC-
(12 +12 \1AB T 1BC
+iCA )2 ].
(18)
Based on the known correlations of the method of symmetrical components,
Ua = Ult + U2i +U0,
y0 = £ç V = UL, x = ^jk, y = ^a., (20)
U U U U
^ a ^ a ^ ab ab
There are also equations to determine phase angles of the vectors of voltages of the direct (f'AB) and inverted (f 'AB) sequences relative to the vector of voltages in the phase AB [2]:
UU
U
Ub = a2-Uit + a-U2^ +U0,\ (25)
C = a-U if + a2-U 2^ + U 0, and equations (7) — (9), the calculation of symmetrical component of voltage of zero sequence can be done according to the equation
U 0 3 (a +U2b +U2c-U 2-U 22), which, considering (24), takes form:
2
2
+
AB
1
6
AB
+
Uo = ( +UB + UC)-U2AB - U2BC - U2CA. (26)
The equation (26) gave accurate results in all cases during calculations, doesn't have limitations in the sphere of use like (16), (19) and has a simpler form than the equation from [1].
Based on (26), the equation to determine symmetrical component of current of zero sequence will be an equation:
I =1 h■if + f + f )-f - f - f
0 v AB BC ca) 1A 1B 1C-
For quick evaluation of U, U , U0 in [1], it is permissible to use approximate equations:
U = 3 ( +Ubc + Uca), (27)
U2 = 0,62-{UH6-U„), (28)
Uo = 0,62 -Uhm4) , (29)
where Uu6 (Utt6), Um (U— the most and least applicable values from three interphase (phase) voltages.
Unlike precise equations, the determination of symmetrical currents according to (28) and (29) is not permissible because I2 and I0 can reach significant values and these equations are based on the allowances: U2<<UI and U <<U, which are usually performed in the networks of electric supply.
The attempts to simplify the definition of symmetrical components with the help of graphic constructions [6], nomograms [2, 7], tables [2] have been made repeatedly.
Geometric constructions allow clearly determining the calculation equations but require significant time.
Nomograms, despite small accuracy defined by one-two signs, play quite important role in the analysis of asymmetrical processes, definition of maximal values of linear and phase voltages and currents, qualitative evaluation of non-balanced processes during the prediction of dynamics of multi-phase systems in the emergency regimes, calculations of short circuits.
Despite the simplicity and sufficient accuracy, the tables and nomograms are not convenient to use during the processing of a big amount of measurement data.
Due to the availability of means of computing technology during the processing of a big amount of information, which is related to the control of quality of electric energy, precise analytical methods of determination of symmetrical components are preferred. The calculation of symmetrical components of voltages should be done according to the equations (24), (26), which have advantages before the equations recommended by GOST 32144-2013: and are more universal; give an opportunity to determine complex values and are simpler in calculation. Proposed expressions can be used to calculate complex values of symmetrical components of currents.
References:
1. ГОСТ 32144-2013 Электрическая энергия. Совместимость технических средств электромагнитная. Нормы качества электрической энергии в системах электроснабжения общего назначения. 01.01.2014.
2. Шидловский А. К., Музыченко А. Д. Таблицы симметричных составляющих. - К.: Наукова думка, 1976. - 204 с.
3. Железко Ю. С., Артемьев А. В. Определение симметричных составляющих напряжений с помощью вольтметра. - Известия вузов. Энергетика, 1985, № 2. - С. 10-15.
4. РД.153-34.0-15.501-00//Методические указания по контролю и анализу качества электрической энергии в системах электроснабжения общего назначения. В 2-х ч. Ч. 1. -Ташкент: ГАК Узбекэнерго. - 2000.-59 с.
5. Цапенко Е. Ф., Юнис Камаль. К вопросу расчета симметричных составляющих фазных напряжений электрических сетей. -Известия вузов, 1992, № 2. - С. 31-33.
6. Рожавский С. М. Номограмма для расчета режима несимметрично нагруженной трехфазной линии с изолированной нейтралью. -Известия вузов. Энергетика, 1965, № 6. - С. 93-94.
7. Ковзан А. А. Оценка несимметрии в трехфазных системах с помощью номограмм. -Известия вузов. Энергетика, 1961, № 7. -С. 28-34.
DOI: http://dx.doi.org/10.20534/ESR-16-9.10-226-228
Xudoyberdiyev Tolibjon, doctor of technical sciences, professor of the Department.
Boltaboev Bohodir, Candidate of the technical sciences, assistant professor.
Muradov Rahimjon, senior scientific employee researcher, PhD in Technique.
Razzoqov Bohodir,
Researcher, assistant of faculty the Faculty of Agro-engineering, Agricultural Institute of Andijan, the Republic of Uzbekistan.
E-mail: mrahimjon@bk.ru
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