Method to determinate over transient longitudinal reactance in a synchronous generator Spunei Elisabeta, Piroi Ion, Nedelea Valentin, Liuba Gheorghe
Piroi Florina Institute of Software Technology and Interactive Systems Vienna University of Technology Vienna, Austria
[email protected]
Departamentul Inginerie Electrică, FIEI Eftimie Murgu University of Reşiţa Reşiţa, Romania
[email protected],
[email protected]
Abstract—The paper presents a original method to determinate the over transient reactance after longitudinal axis of a synchronous generator, starting from its symmetric threephased sudden short-circuit. This method relies on the phenomena that appear in the statoric windings of the synchronous generator. It presents the general calculation equation and the experimental results of a sudden short-circuit for an existing generator. To know the phenomena that appear in the statoric winding, help engineers/designers a quick check of the dimension calculus. The suggested method is not found in the electric machines testing regulations. Keywords—synchronous generator, longitudinal supratransitorial reactance
I.
sudden
short-circuit,
INTRODUCTION
When a sudden short-circuit appears, high level currents appear in the statoric windings, in the excitation winding and damping winding, if there is one. The statoric currents that appear immediately after the short-circuit are due to the polar electromotive voltage that appears, UeE, caused by the resultant flux existing in the machine before the short-circuit appeared. Several parameters of the synchronous generator can be determined from the registration of the statoric currents variation. The most important are: the transient x'd and over transient reactance x''d, after longitudinal axis, the transient x'q and over transient reactance x''q, after transversal axis, the time constants of the damping winding of the two axis T''d and T''q, the time constant of the excitation winding T'd and the time constant of the statoric winding Ta [1], [2]. To determine the value of the over transient longitudinal reactance, from the sudden short-circuit test , using classic method, is toilsome and need to decompose the currents in periodic and non-periodic components [3]. The method suggested in this paper allows a more rapid determination of the reactance using the results at no-load running, in permanent symmetric three-phased short-circuit and the sudden threephased short-circuit test. When testing the sudden three-phased short-circuit, there are no restrictions imposed regarding the phase voltage value in which the three phases are short-circuited, the over transient longitudinal reactance not having saturated values. This thing is
benefic because the short-circuit can be produced in a way that when the highest values of the currents that appear do not jeopardize the synchronous generator. The only restriction imposed, requires that the three mentioned testing’s to be made at the same excitation current. II.
THEORY ELEMENTS OF THE SUDDEN SHORTCIRCUIT SYNCHRONOUS GENERATOR
In the field literature, the phenomena analyzed in the statoric phases having in view the transient currents variation that appear in them, for different values of the angle β0. This is the angle between the axis of the statoric phase, taken as reference, and axis d, according to which the statoric winding is arranged [4], [5],[6]. So, relations are established which show the transient currents variation in the statoric phases and in the excitation winding. With these equation we obtain the time constant that damp the respective currents, the transient and over transient reactance’s values, as well as the dispersion coefficient values in different windings. When the sudden three-phased short-circuit appears, because the statoric windings phases are in fact inductivities, the short-circuit currents determine a reaction flux with a longitudinal demagnetizing character. This flux tends to close through the rotor on the same route the resultant flux had closed, considered constant, in the moment previous to the short-circuit. This tendency could determine an abrupt lowering of the machine resultant flux, previous to the shortcircuit. To balance this tendency, in the rotor windings appear big transitory currents that oppose to the flux lowering, causing a conservation of the flux, previous to the short-circuit. Analysis of dynamic processes in symmetrical three-phase short circuit regime is based on synchronous machine equations in size reported in the stator:
d d q dt
ud Rs id
uq Rs iq d
d q dt
uE RE iE
d E dt
0 RD iD
d D dt
0 RQ iQ
d Q dt
Explaining the fluxes in accordance with the inductivities and the currents that produce these fluxes, it results:
d Ld id Ldh iE Ldh iD
(2)
Ld Ldh Ls ; LE Ldh LE ; LD Ldh LD (3)
In equation (1) and (2) the size ud, uq, uE represent voltage components of the statoric winding by axis d, q, respectively the excitation winding, id, iq represent the longitudinal and transversal components of the statoric current I, and iE, iD, iQ represent the currents in the excitation winding respectively the damping windings after axis d and q reported to the stator. The instantaneous electric angle speed of the rotor, which is considered constant, is identified as ω, Ψd, Ψq are the fluxes determined by the statoric windings corresponding to axis d and q, ΨE, ΨD, ΨQ are the fluxes determined by the excitation winding, respectively determined by the damping windings after axis d and q. In these equations are noted Ldh, Lqh the mutual principal inductivities after axis d, respectively q. LE is the inductivity of the excitation winding calculated as a sum of mutual principal inductivity after axis d, Ldh and the dispersion inductivities of the statoric winding Lsσ, respectively that of excitation LEσ, according to equation (3). The afferent inductivities to the damping’s windings LD, LQ are similarly calculated. In the case of sudden three-phased short-circuit, in the first two equations of (1), voltages ud and uq are null. Processing the first, third and fourth equation of (2) under these conditions, it follows:
Ldh
U eE X d I d
LQ Lqh LQ
did di di di Ls d Ldh E Ldh D dt dt dt dt
Because the tension U is zero in an sudden short-circuit, and the last element in equation (5) can be neglected due to the inductive character of the short-circuit current, it results:
where:
Ldh
U U eE j X d I d j X q I q
Q LQ iQ Lqh iq
Under the conditions neglect dispersion flux of the damping winding after axis d, that would be explainable for the machines with no damping winding, from equation (4) results that all the other dispersion fluxes are null, what is not in accordance with reality. This finding can be expressed as follows: if dispersion winding is neglected, all dispersions should be neglected. The conclusion is that in the calculus of the synchronous machines neglecting of any dispersion is not admitted, even when a damping cage is missing.
It is possible to rapid assessment of these reactance, if one starts from the equation:
D LD iD Ldh id Ldh iE
Lq Lqh Ls ;
di diD di di LD E Ldh d Ldh E dt dt dt dt
In such conditions the value of the over transient longitudinal reactance can be determined, which by its way of definition has the character of a dispersion inductivity and a close value to the mentioned dispersion reactance.
q Lq iq Lqh iQ
E LE iE Ldh id Ldh iD
Ldh
di diE di di LE E Ldh d Ldh D dt dt dt dt
The relation allows the calculus of the polar electromotive voltage UeE, determined by the resultant flux supposed to be constant, previous to the short-circuit, when the synchronous longitudinal reactance Xd is known, and Id is determined from the test of permanent short-circuit. At the sudden short-circuit, the reactance Xd becomes X''d and Id becomes I''d3. This current is determined from the shortcircuit current registrations, on the three phases, as being the average of the three peaks appeared immediately after the short-circuiting of the statoric phases [1]. Consequently, the value of the over transient longitudinal reactance X''d, is deducted from relation:
X d
U eE I d3
We noticed the suggested method to determine the over transient longitudinal reactance is quick and easy to apply. III.
PRACTICAL APPLICATION OF THE METHOD
The testing’s were made on a three-phased synchronous generator with the following data: Sn – 353 kVA, Un – 400 V, In – 509 A, n – 300 rot./min, cos φ – 0,85, f – 50 Hz, connection Y. For the application of the proposed method following stages are necessary: a. He got up idle characteristic, U=f(IE) by known methods;
b. Shall be registered variation permanent phase short circuit currents „Fig. 1” and calculate their mean value for each value of the excitation current;
From the same figure it was determined the value of the longitudinal synchronous reactance xd, in [u.r.] with the relation
xd
In I Esc 0.9 [u.r.] I scn I E 0
(8)
or Xd = 0.4086 Ω. Introducing these values in the relation (6) we obtain the polar electromotive voltage value:
U eE
X d I d 79.94 V
d. Shall be registered variation of the phase current in sudden short circuit for the same amount of excitation current IEp.
Fig. 1 The current variation permanent symmetric three-phased shortcircuit of the synchronous generator
From the registration of the three statoric currents at the sudden short-circuit “Fig.3”, made at the excitation current IEp = 28.6 A, these peak values are obtained IR = 1163.718 A, IS = 1350.275 A, IT = 1318.535 A. They leading to an average value of the sudden three-phased short-circuit current I''d3 = 1277.51 A.
c. He got up permanent short-circuit functioning characteristics I=f(IE) and it represents on the same graph with the characteristic of no load running “Fig. 2”; [u.r.] U, I
C 3
Iscn B
1
2
Fig. 3 Time variation of the sudden three-phased short-circuit currents
Apply the equation (7) the over transient longitudinal reactance value is obtained:
1
X d
Isc3p
79.94 0.0626 [Ω] 1277.51
(7)
what in relative units is x''d = 0.138 [u.r.]. 20 IEp
40
IEδ
A IEsc IE0
100
120
140
IE 160 [A]
Fig. 2 The characteristics of the no load and short-circuit synchronous generator
All the testing’s and calculi made in order to determine the over transient longitudinal reactance were made on the same excitation current IEp = 28.6 A. From the testing at no-load running, and permanent symmetric three-phased short-circuit and the interpretation of the registration resulted the no-load running, and short-circuit characteristics shown in “Fig 2”. In this figure, 1 is noted for right air gap, 2 is no-load functioning and 3 is permanent short-circuit functioning characteristics. From this figure results the permanent short-circuit current Isc3 = 195 A.
If we apply the method known to attenuate the continuous current in the stator windings, when the rotor's axis d is oriented after the axis of two short-circuited phases, and the excitation winding is short-circuited [7], [2] it resulted for the over transient longitudinal reactance the value of x''d = 0.160 [u.r.]. The method of attenuate a continuous current implies a lot of constructions and graph work, things that can induce errors. So, the difference of 15.94% can be a result of these errors. If we apply the experiment to the sudden three-phased short-circuit, using decomposition of currents in periodic components “Fig 4” and non-periodic components “Fig 5” and operate the results with exponentials method [7], was obtained
for the over transient longitudinal reactance with value x''d = 0.142 [u.r.].
as the value, to the more accurate result of the known by now methods, namely the testing to sudden three-phase short circuit currents in decomposition of periodic and non-periodic components. The method is useful because it allows rapid determination supratranzitorii reactance value from usual tests undergone by any synchronous generator before putting into service and during checks after various repairs.
Fig. 4 Variation of the periodic components of sudden three-phased shortcircuit currents
This observation reinforces the authors' confidence in the truthfulness of the proposed method. The simplicity of the proposed method can lead to other researchers and designers to synchronous generators, to apply. REFERENCES [1] [2]
Fig. 5 Variation of the non-periodic components of sudden three-phased short-circuit currents
This method also needs many constructions and graph work, but not as complicated as the previous method, which can induce errors, too. This time the difference between the value determined by the suggested method, taken as reference, and the actual value is much smaller, that is 2,9%. IV.
CONCLUSION
From chapter III results that over transient longitudinal reactance value, determined by proposed method, is very close,
[3] [4] [5] [6] [7]
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