28 d jovcic - analytical modeling of tcsc dynamics

7/29/2019 28 D Jovcic - Analytical Modeling of TCSC Dynamics

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AbstractThis paper presents an analytical, linear, state-

space model of a Thyristor Controlled Series Capacitor (TCSC).

Firstly, a simplified fundamental frequency model of TCSC is

proposed and the model results are verified. Using frequency

response of the non-linear TCSC segment, a simplified non-linear

state space model is derived, where the frequency of the

dominant TCSC complex poles shows linear dependence on the

firing angle. The non-linear element is linearised and linked with

the AC network model and the TCSC controller model that also

includes a Phase Locked Loop model. The model is implementedin MATLAB and verified against PSCAD/EMTDC in the time

and frequency domains for a range of operating conditions. The

model is sufficiently accurate for most control design applications

and practical stability issues in the subsynchronous range.

Index TermsModeling, Power system dynamic stability,

State space methods, Thyristor converters.

I. INTRODUCTION

Thyristor Controlled Series Capacitor (TCSC) is a series

FACTS device which allows rapid and continuous changes of

the transmission line impedance. It has great application

potential in accurately regulating the power flow on a

transmission line, damping inter-area power oscillations,

mitigating subsynchronous resonance (SSR) and improving

transient stability.

The characteristics of a TCSC at steady-state and very low

frequencies can be studied using fundamental frequency

analytical models [1],[2]. These particular models recognize

the importance of having different approach from SVC

modeling (assuming only line current as a constant) which,

although more demanding on the derivation, gives the most

accurate TCSC model.

The fundamental frequency models cannot be used in wider

frequency range since they only give the relationship among

fundamental components of variables when at steady-state.

Conventionally, the electromechanical transient programs likeEMTP or PSCAD/EMTDC are used for TCSC transient

stability analysis. These simulation tools are accurate but they

employ trial and error type studies only, implying use of a

large number of repetitive simulation runs for varying

parameters in the case of complex analysis or design tasks. On

the other hand, the application of dynamic systems analysis

This work is supported by The Engineering and Physical SciencesResearch Council (EPSRC) UK, grant no GR/R11377/01.

D. Jovcic and G.N. Pillai are with Electrical and Mechanical Engineering,University of Ulster, Newtownabbey, BT370QB, United Kingdom, (e-mail:[email protected], [email protected])

techniques or modern control design theories would bring

benefits like shorter design time, optimization of resources and

development of new improved designs. In particular, the

eigenvalue and frequency domain analysis are widely

recognized tools and they would prove invaluable for system

designers and operators. These techniques however always

necessitate a suitable and accurate dynamic system model.

There have been a number of attempts to derive an accurate

analytical model of a TCSC that can be employed in system

stability studies and controller design [3-6]. The model

presented in [3] uses a special form of discretisation, applying

Poincare mapping, for the particular Kayenta TCSC

installation. The model derivation for a different system will

be similarly tedious and the final model form is not convenientfor the application of standard stability studies and controller

design theories especially not for larger systems. A similar

final model form is derived in [4], and the model derivation is

improved since direct discretisation of the linear system model

is used, however it suffers other shortcomings as the model in

[3]. The modeling principle reported in [5] avoids

discretisation and stresses the need for assuming only line

current as an ideal sine, however it employs rotating vectors

that might be difficult to use with stability studies, and only

considers the open loop configuration. The model in [5] is alsooversimplified because of the use of equivalent reactance and

equivalent capacitance that might be deficient when used in

wider frequency range. Most of these reported models are

therefore concerned with a particular system or particular type

of study, use overly simplified approach and do not include

control elements or Phase Locked Loops (PLL).

In this study we attempt to derive a suitable linear

continuous TCSC model in state-space form. The model

should have reasonable accuracy for the dynamic studies in

the sub-synchronous range and it should incorporate common

control elements including PLL. To enable flexibility of the

model use with different AC systems, the model structure

adopts subsystem units interlinked in similar manner as withSVC modeling [6].

We also seek to offer complete closed loop model

verification in the time and frequency domains.

II. TESTSYSTEM

The test system for the study is a long transmission system

compensated by a TCSC and connected to a firm voltage

source on each side. Figure 1 shows a single line diagram of

the test system where the transmission line is represented by a

lumped resistance and inductance in accordance with the

approach for sub-synchronous resonance studies [7]. Each

phase of the TCSC is composed of a fixed capacitor in parallel

with a Thyristor Controlled Reactor (TCR). The TCSC is

Analytical Modeling of TCSC Dynamics

D.Jovcic,Member, IEEEandG.N. Pillai


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controlled by varying the phase delay of the thyristor firing

pulses synchronized through a PLL to the line current

waveform. The controller is of a PI type with a feedback filter

and a series compensator.

Three different test systems are used in order to verify the

model accuracy with different system parameters and differentoperating points:

system 1 75% compensation, capacitive mode, system 2 40% compensation, capacitive mode, system 3 75% compensation, inductive mode,

The test system parameters are practically selected

according to the recommendations in [8] (regarding Xl/Xc and

natural resonance) and the test system data are given in the

Appendix.

- Calculated firing angle- PLL reference angle- Actual firing angle- Phase angle of- Mag. of voltage

PLL - Phase Locked loopPI - Voltage PI controllerTCR - Thyristor controlledreactor

TCR

TCSC

l

l

c

c

l

c

Figure 1. Test system configuration.

III. FUNDAMENTALFREQUENCYMODEL

The fundamental frequency model of a TCSC is derived

firstly to enable initialization of the steady-state parameters.

The voltage across the TCSC capacitor (vc) comprises an

uncontrolled and a controlled component [8], and it ispresumed that the line current is constant over one

fundamental cycle in accordance with [1-2],[5]. The

uncontrolled component v1, is a sine wave (unaffected by

thyristor switchings) and it is also constant over a fundamentalcycle since it is directly related to the amplitude of the

prevailing line current. The controlled component v2 is a non-

linear variable that depends on circuit variables and on the

TCR firing angle.

In this study, the controlled component is represented as a

non-linear function of the uncontrolled component and firing

angle, v2=N1(v1,,s), as shown in Figure 2. With this

approach, N1(v1,,s) captures the non-linear phenomena

caused by thyristor switching influence and all internal

interactions with capacitor voltage assuming only that the line

current and v1 are linear. We seek in our work to study

dynamics ofN1(v1,,s) in a wider frequency range and also to

offer a simplified representation for fundamental frequencystudies.

cs

1

-

+1v

1v

2v

2v

cv

cv

li

li - line current- linear component of TCSC voltage- non-linear component of TCSC voltage- TCSC voltage- firing angle

- Non-linear part dynamics

),,( 112 svNv =

),,( 11 svN

Figure 2. TCSC model structure.

The fundamental components of reactor current itcr and the

voltages v1 and v2 are selected as state variables and the non-

linear state-space model is presented as:

li

c

sv1

1 = (1)

tcric

gsv1

2 = (2)

21

11v

lgv

lgsi

tcrtcr

tcr = (3)

21 vvvc = (4)wheregrepresents the switching function:g=1 for thyristor in

conduction, andg=0 for thyristor in blocking state.

To derive the corresponding linear model at the nominal

frequency, it is postulated that the model has the following

structure:

lic

sv 11 = (5)

tcric

sv1

2 = (6)

2211 vkvksitcr = (7)

21 vvvc = (8)

where k1=k1() and k2=k2() are the unknown model

parameters dependent on the firing angle . The above model

structure is correct for zero firing angle (g=1,=0, i.e. full

conduction with measured from the voltage crest) and we

presume the same model structure but different parameter

values for(0,90).

Considering the two components of the thyristor current on

the right side of (7) we note that11vk produces the current

that is driven by a constant voltage v1 (over one fundamental

cycle) in accordance to the earlier assumption. As a result, the

configuration is similar to the shunt connection of TRC in a

SVC, and the constant1k can be calculated using the approach

of equivalent reactancealpl for the TCR current in a SVC

[6,8]:

)2sin(2/1 1

==

tcr

alp

l

lk(9)


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In order to determine the constant k2, we represent (6)-(7) as

a transfer function in the following form:

1/)(

)(

)(22

3

1

2

+==

d

fos

ksT

sv

sv

(10)

withc

kd

2= ,2

13

k

kk = (11)

Observing (10) it is seen that d is the resonant frequencyof an un-damped second order system. The proposed

expression for this frequency is given below in (12) and it is

determined using the experimental frequency response of the

model for the non-linear TCSC segment as presented later in

section IV.

cltcr

d

2= (12)

Substituting thats= jo= j2fo (fo fundamental frequency)in (10), and using (11) in terms of input current and the output

voltage as given in (5)-(8) the TCSC gain at fundamental

frequency is derived:

( )( )[ ]1/

1/22

3

22

+

+=

doo

do

l

c

jcj

kj

i

v

(13)

and the TCSC fundamental frequency impedance is:

( )22

3

22

csc

/1

1/

doo

dot

c

kX

+= (14)

where

[ ]23 )2(

)2sin(2

=k (15)

and d is defined in (12).

Compared with the fundamental frequency TCSC model in

[1-2], the derived model (13) is considerably simpler, yet it

will be shown that the accuracy is very similar.

The above model is tested against PSCAD in the following

way. The system is operated in open loop, i.e. with a fixed

reference thyristor firing angle, in the configuration of Figure

1. The value of the reference thyristor angle is changed in

interval (0,90o) and for each angle the value of the TCSCvoltage is observed. Since all other parameters are constant,

the TCSC voltage is directly proportional to the TCSCimpedance and this is an effective way to obtain accurate

information on the fundamental TCSC impedance.

Figure 3 shows the steady-state TCSC voltage against the

range of firing angle values, where the above linear model

results are shown as model 2 and also the results fromresearchers [1,2] as model 1. It is seen that the proposed model

shows good accuracy across the entire firing angle range and

especially the results are very close to those from [1,2]. In fact

the above model differs negligibly from the model [1,2],

except in the less used low firing angles in range (20,30o)

where the difference is still below 5%. It is also evident thatthe two analytical models show small but consistent

discrepancy against PSCAD/EMTDC, and despite all

simplifications in PSCAD it was not possible to obtain better

matching.

Figure 3. Testing the fundamental frequency model. Steady-state TCSCvoltage for different values of firing angles.

IV. SMALL-SIGNALDYNAMICANALYTICALMODELA. TCSC model

The transfer function in (10) is accurate only at fundamental

frequency and it cannot be used for wider frequency studies.

The goal of the dynamic modeling is to derive a dynamic

expression forN1(v1,,s) of satisfactory accuracy in the sub-

synchronous frequency range and for small signals around the

steady-state operating point. The steady-state for an AC

system variable is a periodic waveform at fundamental

frequency with constant magnitude and phase.

The frequency response is of TCSC is studied assuming that

the input voltage is:

)2sin()2sin(01 tfAtfAv injinjo += (16)where the first sine signal denotes the steady-state operation

with all parameters constant. The second term is the input in

the experimental frequency response and it has small

magnitude Ainj


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Gain2

v1

v2

b1

a1

Switch2

Switch1

sine

cos

period

Source

PulseGenerator

1s

Int2

1s

Int1

1/(ct*ltcr)

Gain1

v2

sine

cos

period

a1

b1

Fourrieranalysis

0

Const.

1/(ct*ltcr)

Figure 4. Experimental frequency response set-up with the SIMULINK modelfor the non-linear TCSC segment.

magnitude and phase of the first harmonic of the output

voltage v2 is observed.

The pulse generator produces equally spaced conduction

intervals that are based on the fundamental frequency period

and which are dependent on the firing angle. Based on the

above assumption these firing instants are unaffected by theinjection signal.

Figure 5 shows the obtained time-domain responses for the

particular firing angle =72o and for two different frequencies

of the input signal. It is seen in Figure 5 b) that the output

trace for the 60Hz input frequency gives the familiar square

shape which is consistent with the results for the fundamental

frequency studies in [1,2,8]. The experimental frequency

response over the range of frequencies is repeated for all

values of the firing angle (0,90o) with a one-degree

increment.

For each firing angle, the gain and phase values across thefrequency range are plotted and Figure 6 shows an example

for the particular firing angle =76o

. Observing the frequencyresponse curves in Figure 6 and others for other firing angles,

it is concluded that the system shows higher order properties

and it becomes clear that it has dominant second order

dynamics. The following transfer function is proposed:

input v1

input v1

output v2

output v2

first harmonic v2

first harmonic v2

time [s]

a)

b)

time [s]

magnitude[kV]

m

g

itd

[

]

Figure 5. Experimental frequency response ofN1(v1,,s) using SIMULINK

model. Firing angle =72o, input frequency a)finj=12Hz. b)finj=60Hz.

gain

[dB]

phase[

deg]

frequency [rad/s]

fo

linear model

non-linear model

fund. frequency model

fo

Figure 6. Frequency response for the non-linear TCSC segment. Test system 1

with the firing angle =76o.

system 1 experimental

system 2 experimental

system 1 model

system 2 model

frqu

n

y[rd

/s]

firing angle [deg]

Figure 7. Linear relationship between the denominator characteristic

frequency (d) and the firing angle .

12

12

)(

)(

2

2

2

2

1

2

++

++=

d

d

d

n

n

n

w

s

w

s

w

s

w

s

sv

sv

(17)

The characteristic peak in the gain frequency response is

caused by the dominant complex poles and each of the troughs

by a pair of complex zeros. Observing the frequency response

across the range of firing angles, the location of the dominant

peak is recorded against the firing angle values and the circuit

parameters, and the final plot is shown in Figure 7. It soon

emerges that the denominator characteristic frequency d is a

linear function of the firing angle where the actualmathematical formula can be readily derived as given in (12).


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The linear formula shows very good accuracy and it reflects

well the variation in circuit parameters as seen in Figure 7.

Analysing (12) it is seen that the resonant frequency of a

TCSC starts at the natural resonance of L-C circuit (for=0)

and it linearly reduces with the increase in the firing angle.

The formula has importance for the practical design since itpoints to the TCSC firing angles that should be avoided if the

connected system is sensitive to resonance in a particular

frequency range (SSR problems for example).

There are four unknowns in (17) and they are determined in

a similar manner, by matching the frequency response and the

final step responses, and the proposed final formulae are given

below.

clw

tcr

d

2= (18)

)cos(38.0 =d (19)

+=

)(tan107.110

4

3

cl

fw

tcr

on (20)

)cos(2.0 =n (21)

The simulated frequency response with the above transfer

function is shown in Figure 6 where satisfactory matching is

observed, except at higher frequencies. It is emphasized here

that the model zeros fall in higher frequency region, implying

influence of sampling system phenomena, and the numerator

parameters in (17) are open for further research and

improvements.

It would be beneficial to use the simpler model as given in

(13) for the dynamic study in a wider frequency range. If wereplace the fundamental frequency in (13) with an independent

variable for frequency domain studies, jo=s, we can test themodel assuming the frequency variable represents deviations

around the nominal frequency. The resultant frequency

response curves are shown in Figure 6 labeled as the

fundamental frequency model. It is evident that such model

matches the frequency response only atfo; it has incorrect gain

for low frequencies, incorrect damping of the dominant mode

and it can not be used for wider frequency studies.

We point further that the complex poles of the transfer

function (17) have finite damping which is for capacitive

mode typically 0.07


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The filter time constant is of the order of2ms, which is in line

with studies in [6],[8] and [12]. It should be noted that the

voltage phase angle also affects the firing angle because the

actual firing angle is measured with respect to the voltage

curve. Here, we have direct correlation across each phase, and

the sampling frequency is 1/3 offs and therefore an additionallag is introduced represented by the delay filter with Td2.

Simulation results demonstrate improvement in the response

with the introduction of this delay element. The second order

filer with vc reduces the harmonics on the feedback signal.

PLL

Vref

Vref

cV

cV

+

+

+

+

+

PI controller Lead-lagFeedback filter Delay filter

Delay filter

/2

s

kk ip +22

2

2 fff

f

wsws

w

++

1

1

1 +sTd

1

1

2 +sTd

d

d

- TCSC voltage magnitude- reference voltage- voltage angle- delayed voltage angle

- line current angle- PLL reference angle- controller reference angle- actual firing angle

1

1

2

1

+

+

sT

sT

Figure 8. The controller model and firing angle calculation.

D. Model Connections

The final model consists of three independent state-space

models: AC system, TCSC model and the controller model,

which includes PLL. The linearised TCSC model is linked

with the firing angle which is the output of the controller

model, using the artificial rotating variables as presented in

[6].The three subsystem models are linked through the coupling

variables to create the final state space model in a similar

manner as with the SVC model in [6], and the final model has

the following structure:

outsssout

outssss

uDxCy

uBxAx

+=

+=o

(25)

where s labels the overall system, and the model matrices

are:

=

actcacactccoacacco

actctcactcctcco

accocoactccocco

s

ACBCB

CBACB

CBCBA

A

**

**

**

cot

cot

(26)

[ ] [ ]0,, ==

= sacouttcoutcoouts

acout

tcout

coout

s DCCCC

B

B

B

B

All the subsystems D matrices are assumed zero in (26)

since they are zero in the actual model and this noticeably

simplifies development. The Bs is the external input matrix,

i.e. it links with the reference step uout=Vref whereas Cs

specifies the monitored outputsyout

=vc.

The matrix As has the subsystem matrices on the main

diagonal, with the other sub-matrices representing interactions

between subsystems. The following indices are used ac AC

system, tc TCSC, co controller. As an example the

matrix Btcco is the TCSC model input matrix that takes input

signals from the controller. The model in this form hasadvantages in flexibility since, if the TCSC is connected to a

more complex AC system only the Aac matrix and the

corresponding input and output matrices need modifications.

Similarly, more advanced controllers can be developed using

modern control theory (H, MPC,..) and implemented directly

by replacing the Aco matrix. The above structure enables the

model to be readily interfaced with the MATLAB model of

other FACTS elements, HVDC or other power system units.

V. MODEL VERIFICATION

A. Time domain

The linear model is coded in MATLAB and it is verified in

time domain by comparing the reference step responses withPSCAD model for the three test systems. All the parameters in

the PSCAD model are identical to those in MATLAB model

(AC system, controller and PLL parameters) and model for

switches includes typical snubber circuits. Figures 9.10 and 11

show the three responses of systems 1,2 and 3 respectively.

We can conclude that the model accurately reflects the system

Figure 9. Test system 1 response following a 1kV voltage reference stepchange.

Figure 10. Test system 2 response following a 0.5kV voltage reference stepchange.


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Figure 11. Test system 3 response following a 0.05kVvoltage reference stepchange.

structure since with different circuit parameters and operating

points we consistently have good response matching. In

inductive mode, shown in Figure 11, it is most difficult toobtain high accuracy since in this mode v2 is larger than v1 and

the system is highly non-linear, but this mode is seldom used

in practice.

B. Frequency domain

The time domain testing clearly indicates the dominant

oscillatory mode and also it verifies model for different

operating parameters and inputs. Step input testing however

cannot indicate the frequency range for the model validity.The experimental frequency response is performed with the

PSCAD software in the frequency range 1Hz


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Using the root locus and frequency domain design

techniques with MATLAB, a lead-lag compensator is

designed with the parameters as shown in the Appendix,

system 1. The root locus for the compensated system is shown

in Figure 13 b). It is seen that the damping of the dominant

mode is much improved. Also, the eigenvalue location is notsignificantly changed with two times increased transmission

system impedance, as demonstrated by curves m=2, and m=3.

Our studies show that a fast capacitor voltage feedback control

with a suitable lead-lag compensator can give a very robust

stabilizing controller for a TCSC.

It is emphasized here that the above design is a

multidimensional problem involving simultaneous variation of

controller gains, system strength and the compensator

parameters. Such problem would require a very large number

of trial-and-error simulation runs with conventional time-

domain simulators. The design also involves higher frequency

dynamics requiring an analytical model with high accuracy.

VII. CONCLUSIONS

The fundamental frequency model of a TCSC is developed

in a fairly simple form and the model shows good accuracy.

The TCSC model is segmented into linear and non-linear part

and the non-linear model is analysed in frequency domain

using SIMULINK. A simplified linear-continuous model for

the non-linear dynamics is developed by matching the

frequency response. The analysis of the non-linear dynamics

concludes that the dominant oscillatory mode is a linear

function of the firing angle. The linear TCSC model is

implemented in MATLAB and tested against non-linear

digital simulation PSCAD/EMTDC and good accuracy is

observed across a range of system parameters and operating

conditions. The model is also tested in frequency domain and

it is concluded that accuracy is good at frequencies f

28 d jovcic - analytical modeling of tcsc dynamics

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