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settings depend on the short-circuit power and the DC timeconstants. Therefore, the actual value of the system timeconstant at the relay measuring point is important for thedesign of current transformers for protection issues.
III. CALCULATION OFTHVENINS IMPEDANCE INFREQUENCY DOMAIN
A. Proposed method
The considered positive sequence of a symmetrical powersystem consisting of two sub-systems U11, Z11 andU21, Z21connected by a line ZL1 is shown in fig. 2. The synchronous
U11
Z11
IA1
ZL1
U21
Z21
UA1
Fig. 2. Positive sequence system of two symmetrical active sub-systemsconnected by a line
sampled voltages uA and currents iA are measured at therelay installation node A. The fundamental frequency positivesequence phasors (symmetrical components) UA1 and IA1(fig. 2) are calculated form the measurements (e. g. disturbancerecord).
Applying Kirchhoffs voltage law to fig. 2 leads to
U11 = Z11 IA1+ UA1 (2)
with the voltage source
U11 = U11ej11. (3)
Under the assumptions
= const 11 = const (4)
U11= const Z11 = const (5)
Thvenins impedance Z11 can be calculated from two dif-ferent (quasi-)stationary system states (pre-fault, fault and/orpost-fault state) with a lag t= t2 t1 as
U11(t1) =U11(t2) ejt (6)
Z11 =UA1(t2) e
jt UA1(t1)
IA1(t1) IA1(t2) ejt
. (7)
From the imaginary and the real part ofZ11, the system timeconstant can be calculated by
TN = {Z11}
{Z11}. (8)
Because differences in voltage and current are evaluated, wecall the proposed method Delta method [6]. We proposeto estimate the system frequency fN and the fundamentalfrequency phasors UAandIAwith an iterative Pronys method[7], [8] because its results are more precise than by using aFFT algorithm.
B. Limitations of the Delta method
Beside the fulfilled assumptions (4) and (5) the networkfrequency fN must be calculated very precisly. For this, theproposed Pronys method for estimating phasors is suitable.Another requirement is, that the triggering transient systemevent between the analysed stationary states must not effectthe system voltage U11. Here, disturbance records with high
load changes are more applicable than those with short-circuits[6]. However, load changes normally wont trigger protectionrelays and therefore arent feasible for analysing disturbancerecords. By analysing fault current records the (in generally)changed network topology by switching circuit breakers in thecase of clearing fault currents as well as the non-stationarycurrents are challenging.
Another limitation of the Delta method is the need formeasurements of the total current infeed from the sub-systemto be analysed. In the case of recorded signals of a doublecircuit line only parts of the (fault) current are measured bythe relay depending on the line impedances and fault location.
The user of this method has to make sure, that total currentsare applied otherwise the results will not be reliable.
C. Verification of the Delta method
Tests on simulated and real disturbance records with typicalsampling frequencies fs = 1 kHz were performed to evaluatethe accuracy and the applicability of the proposed method.
1) Test on simulated signals: The accuracy of the Deltamethod was tested on simulated voltage and current wave-forms. The used network model in DIGSILENTPowerFactoryis shown in fig. 3.
R
ZNUN FA FB
A B
load
line
Fig. 3. Simulated single feeded network model with measuring relay R andtwo fault locations FA and FB
Simulated network parameters:
nominal voltage of system UN = 220kV short circuit power SN = 1000 MVA
length of line l= 50km variation of system time constantTN {50, 100} ms simulated load changes (LC)
inductive load Sload= 250 MVA; PF = 0.95 load change Sload {+10%,+100%} Sload
simulated short-circuits (SC)
fault type: k1E, k2, k2E bzw. k3 fault location at node A (0.1 % l) resp. B (99.9 % l) fault impedanceRf {0, 100}
Results of calculated system time constants for different sys-tem events and references are presented in table II. The relative
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V. SEPARATION OF MORE THAN ONE DC TIME CONSTANT
The previous assumptions of undistorted measurements ofthe short-circuit current ik(t) arent fulfilled in a typicalcircuitry as shown in fig. 4.
The recording device (relay or disturbance recorder) isattached to the secondary winding of the main protectioncurrent transformer ct, which transduces the primary current
ik(t). The internal current transformer R inside the relay alsoinfluences the sampled current signal. R1 in fig. 4 includes
R2
network
R1
protection current
transformer ct
protection relay / disturbance recorder
A/D converter#
ik= ipri
F
internal current
transformer R
isec
imeas
Fig. 4. Typical series connection of several current transformers for digitalrelays or disturbance recorders
the resistances of the main current transformer ct as well asthe wiring to the relay. The internal resistances (burden andsecondary winding of current transformer) are modelled withR2. The proposed method should be able to estimate the DCtime constant TDC ofik from measurements imeas.
Depending on the type of current transformers (closed ironcore or linearised with air gaps in magnetic core) the DCcomponents are transduced differently. For best results of the
estimated time constant TDC the DC component should betransduced ideally. However, decaying DC components effectthe protection algorithms (e.g. based on FFT) applied andtherefore linearised current transformers are widely used inrelays but also for the main protection transformers (TPY andTPZ types) [11], [4].
A. Enhanced signal model
A simplified model of a current transformer is shown infig. 5. Neglecting3 the leakage inductanceL and considering
L Rct
Lh
ipri' isec
ihLB
RB
Fig. 5. Simplified model of current transformer
3In general fulfilled, because L < LB Lh.
only ohmic burden LB 0the following equations are valid.
ipri= ih+ isec (15)
0 =dihdtLh isec(Rct+ RB) (16)
Applying (15) to (16) gives
0 =
dipridt
disecdt isec
Rct+RB
Lh (17)The signal model for an exponential decaying DC componentin the primary current with a time constant TDC can beexpressed as
ipri(t) =C1 exp
t
TDC
. (18)
Applying (18) to (17) gives the inhomogeneous differentialequation
0 =C1
TDCexp
t
TDC
+
disecdt
+isec
Tct(19)
with the time constant of the secondary current loop4
Tct = Lh
Rct+ RB. (20)
Considering the initial conditionsih(t0) = 0 and isec(t0) =ipri(t0) =C1 the solution of (19) is
isec(t) =C1
Tct TDC
TDCe
t
Tct Tcte
t
TDC
. (21)
The secondary current of the main current transformer ctisec is transduced by the internal current transformer R inthe relay (fig. 4) and therefore the signal model needs furtherextensions5. In (17) ipri becomes isec from (21) giving
0 =disec
dt
dimeasdt
imeas
TR(22)
with the relay time constantTR.As the solution of (22) the measured current is obtained
which is finally sampled and recorded by the device
imeas(t) =C2
TR Tct
TRe
t
Tct Tcte
t
TR
+ TR Tct
e
t
TDC e
t
TR
TDC TR
e
t
TDC e t
Tct
TDC Tct
(23)
Depending on the time constants Tct and TR a distortion ofthe DC signal component takes place.The curve fitting signal model depends on the assumed
number of DC time constants in the signal. Equation (11)fits this problem best, ifTct and TR can be neglected (i. e.Tct, TR ). If only one time constant can be neglected,(21) is valid. In all other cases, (23) should be adapted for thefitting model.
4In this paper we call it current transformer time constant. IfTct ,Tct can be substituted by TR in the following equations.
5Only necessary if more than one secondary time constant is relevant (seefootnote 4).
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