dc time constant estimation

8/12/2019 DC Time Constant Estimation

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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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dc time constant estimation

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