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Designing of Power System Stabilizer and It Effects on Power System


Call for Papers Engineering Journal, May 2019

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Designing of Power System Stabilizer and It Effects on Power System

Vivek Rethinakaran. B,

Department of EEE, Parisutham Institute of Technology and Science, Thanjavur, India.

Varadarajan. M,

Department of EEE, Parisutham Institute of Technology and Science, Thanjavur, India.

Abstract The small signal instability is one of the major problems in power system operation caused by insufficient natural damping in the system. High values of external system reactance and high generator output with high response exciter can introduce negative damping even though synchronizing torque is increased and leads to system instability. The most cost-effective way of countering this instability is to use auxiliary controllers called power system stabilizers. It acts through the excitation system and enhances the dynamic stability by providing additional damping electrical torque. In the present work, to study local mode of power oscillation, Eigen values analysis and time domain analysis were carried out on a test system. To suppress the power oscillations the parameters of power system stabilizer were determined using a conventional design and simulation results shows its effectiveness.

Keyterms-Single Machine Infinite Bus, Power System Stabilizer, Small Signal Stability, Damping oscillations, Eigen Values.

I INTRODUCTION

Power system stability is the ability of an electrical power system, for given operating conditions, to regain its state of operating equilibrium after being subjected to a physical disturbance, with the system variables bounded, so that the entire system remains intact and the service remains uninterrupted. The rotor angle stability is the ability of the synchronous generator in an interconnected power system, to remain in synchronism after being subjected to disturbances. It depends on the ability of the machine to maintain equilibrium between electromagnetic torque and mechanical torque of each synchronous machine in the system. Instability

of this kind occurs in the form of swings of the generator rotor which leads to loss of synchronism.

Equations (1)-(3) along with other system states like exciter can be solved to determine the stability of the system..

III SMALL SIGNAL ANALYSIS OF SMIB

Fig.1 shows a single line diagram of a single machine system. For simplicity it is assumed that the synchronous machine is represented by a classical model. Further it is assumed that damper windings both in the d and q axes and armature resistance are neglected. The linearized model of the single machine infinite bus system is given in Fig 2

Fig.1. Single line diagram of SMIB

Fig.2. SMIB with classical model of synchronous generator

The state space model can be developed from equations (1),

(2) and (3) by taking Laplace transforms,

= B S

(4)

II PROBLEM FORMULATION

The objective of the present work is to design a power system stabilizer to increase system stability by damping the local oscillations. For a single machine infinite bus power system, the linearized equations are

d

dt = B Sm (1)

m

m m

m m

S = 1 ( S ) (5) 2

= 1 + 2 E (6)

Equation (4) and (5) can be written in matrix form as

= +

2Hd(Sm ) = S

+

(2)

m

D K1

Sm

dt m

= 2H 2H (7)

where,

Te = E iq xq (3)

B 0

It may be observed here that the elements of state matrix A

q

depend on D, H and the initial operating conditions.

The characteristic equation is given by,

0.04

KD=0

KD=1

Hs2 + Ds + 1 B

(8)

0.03 KD=-5

KD=10

Rotor Speed Deviation

Rotor Speed Deviation

0.02

For stability, both damping co-efficient (D) and K1

should be

0.01

positive. If D is negligible, the roots of the characteristic equations are

0

-0.01

,

= ± (1 B ) = ±j

(9)

-0.02

-0.03

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

1 2 2H n

The stability of the system for different cases has been analyzed by Eigen value analysis and time domain analysis. The Eigen values have conjugate pairs. If they have zero real parts the system here is marginally stable with sustained oscillations. If contain negative real parts the oscillations will be damped and system become stable and if contain positive real parts so the oscillation grows. The eigen value analysis are carried out for the above state matrix and the results are tabulated in the Table I and Table I. The results shows the characteristic of the system based on the damping co-efficient D for the different loading conditions. The eigen value analysis is validated by using the time domain analysis.

Time

Fig.3. Rotor Angle Deviation classical model high loading (Pg=1pu)

From Fig.3 it is clear that the system becomes unstable for the negative damping co-efficient. The damping of power oscillation increases with damping co-efficient D.

  1. SMALL SIGNAL ANALYSIS OF SMIB WITH FIELD CIRCUIT

    To study the effect of field winding flux linkage on system stability, the flux decay model of generator is considered. Fig 4 shows the linearized model of the SMIB system with the field flux variation.

    Fig.4. Block diagram of SMIB with effect of field flux

    TABLE I

    EFFECT OF DAMPING CO-EFFICIENT FOR CLASSICAL MODEL (LOW LOADING)

    D

    0

    1

    10

    -5

    Eigen Values

    ±9.2128i

    -.05±9.2181i

    -.5 ±9.2047i

    .25 ±9.2148i

    Damping ratio

    0

    0.2698

    0.7335

    -0.3668

    Frequency(rad/sec)

    9.22

    9.22

    9.22

    9.22

    TABLE II

    EFFECT OF DAMPING CO-EFFICIENT FOR CLASSICAL MODEL (HIGH LOADING)

    D

    0

    1

    10

    -5

    Eigen Values

    ± 7.4772i

    -.05±7.4771i

    -.5 ± 7.4605i

    . 25 ± 7.4731i

    Damping ratio

    0

    .3331

    .9053

    -.4520

    Frequency(rad/sec)

    7.48

    7.48

    7.48

    7.48

    The field winding dynamics can be expressed as

    0

    E

    E

    = E + xq (10)

    0.025

    0.02

    Classical Mode

    Flux Decay

    Rotor Angle Deviation[p.u]

    Rotor Angle Deviation[p.u]

    By taking Laplace transform of equation (10)

    1 + 0 3 E = 3 3 4 (11)

    0.015

    0.01

    E

    0.005

    =

    4

    0

    3

    0

    (12)

    0

    0

    0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

    Time[sec]

    The dynamical behavior of the SMIB system in a compact form can be given as,

    Fig.5. Comparison of rotor deviation between classical model and flux decay model high loading (Pg=1pu)

  2. SMALL SIGNAL ANALYSIS OF SMIB WITH EXCITATION SYSTEM

    21 2 2 2

    = 0 0

    0 4 1

    (13)

    The main objective of the excitation system is to control the field current of the synchronous machine. The SMIB with excitation control is shown in Fig. 6. The field current is controlled so as to regulate the terminal voltage.

    0

    0

    0

    3

    Commonly used functions are the field-current limiter, maximum excitation limiter, terminal voltage limiter, volts-

    The eigen value analysis and time domain analysis is carried

    using expression (13) for different operating conditions.

    The resulting eigen values are tabulated in the Table III and Table IV. The results obtained for the various damping coefficients illustrates that negative real part of eigen values indicates the stable condition of the system. The system is partially damped due to the effect of field flux variation in both lo loading and high loading conditions. Fig.

    per-Hertz regulator and protection, and under excitation limiter. These are normally distinct circuits and their output signals may be applied to the excitation system at various locations as a summing input or a gated input. The static exciter IEEE TYPE ST1 is considered. The perturbation in the terminal voltage Vt can be expressed as

    5 shows the local oscillations are damped by flux decay model. The effect of field flux variation reduces the synchronizing torque and increase the damping torque component at the rotor oscillation frequency.

    V =

    +

    (14)

    TABLE III

    EFFECT OF DAMPING CO-EFFICIENT FOR FLUX DECAY MODEL (LOW LOADING)

    D

    0

    1

    10

    Eigen Values

    -.226

    -0.1185 ± 5.9314i

    -.2258

    -.1686 ±5.9322i

    -.2244

    -.6193 ± 5.9204

    Damping ratio

    1

    .2707

    1

    .3844

    1

    .3826

    Frequency(rad/sec)

    .8314

    5.93

    .8314

    5.93

    .824

    5.95

    TABLE IV

    EFFECT OF DAMPING CO-EFFICIENT FOR FLUX DECAY MODEL (HIGH LOADING)

    D

    0

    1

    10

    Eigen Values

    -.1225

    -.1703±6.45i

    -.1224

    -.2203±6.4529i

    -.1215

    -.6708±6.4456i

    Damping ratio

    1

    .3573

    1

    .4615

    1

    .3826

    Frequency (rad/sec)

    .4488

    6.45

    .4488

    6.46

    .4451

    6.48

    condition with exciter control the system is stable and the system becomes unstable hen the system is highly loaded. The positive real values of eigen values represent the unstable condition of the system due to heavy loading.

    Fig 6 The Block diagram of SMIB with exciter control

    0.025

    0.02

    Rotor Angle Deviation

    Rotor Angle Deviation

    0.015

    0.01

    Classical Model

    Flux Decay Exciter

    By linearizing the expression(14),

    0.005

    0

    0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

    V = + (15)

    Time

    5

    5

    Fig.7. Comparison of Rotor angle deviation for Classical model vs Flux Decay model vs Exciter low loading

    The system equation can be derived as

    (Pg=.5pu)

    = + (V

    K

    (16)

    0.1

    5

    6 Classical Model

    =

    1

    1

    2 0

    0.08

    Rotor Angle Deviation

    Rotor Angle Deviation

    0.06

    0.04

    0.02

    0

    -0.02

    Flux Decay Exciter

    2

    2

    0

    2

    0 0

    -0.04

    0 4

    0

    1 3

    0

    (17)

    -0.06

    0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

    Time

    0

    0

    0 5

    6

    1

    1

    Fig.8. Comparison of Rotor angle deviation for Classical

    The r lt for the sys

    with ex

    is tabulated in

    model vs Flux Decay model vs Exciter high loading

    esu

    tem

    citer

    (Pg=1pu)

    Table V and Table V. They infers that under low loading

    TABLE V

    EFFECT OF DAMPING CO-EFFICIENT WITH EXCITATION CONTROL (LOW LOADING)

    D

    0

    10

    Eigen Values

    -2.41±3.39i

    -1.1919 ±5.98i

    -2.41 ± 3.39i

    -1.38±5.990i

    Damping ratio

    2.13

    0.7308

    2.13

    .8458

    Frequency(rad/sec)

    4.16

    5.99

    4.16

    6.0

    TABLE VI

    EFFECT OF DAMPING CO-EFFICIENT WITH EXCITATION CONTROL (HIGH LOADING)

    D

    0

    10

    Eigen Values

    -2.88 ±3.77i

    .5518 ±6.18i

    -2.88 ±3.78i

    1.3412± 6.18i

    Damping ratio

    2.2330

    -.3275

    2.2294

    -0.2165

    Frequency (rad/sec)

    4.75

    6.18

    4.75

    6.18

    Fig.7 depicts that the system is stable when the system is lightly loaded and Fig 8 depicts that system becomes unstable

    under the heavy loaded condition. The system becomes unstable due to negative damping produced by exciter at high loading condition.

  3. DESIGNING OF POWER SYSTEM STABILIZER

    A cost efficient and satisfactory solution to the problem of oscillatory instability is to provide damping for

    =

    (19)

    generator rotor oscillations. This is conveniently done by providing power system stabilizers (PSS) as shown in Fig 10 which are supplementary controllers in the excitation systems. The signal Vs in is the output from PSS which has input signal derived from rotor velocity, frequency, electrical power or a combination of these variables. The power system stabilizer provides stabilizing signals to the voltage regulator to damp out oscillations in the power system. It consists of a washout circuit, dynamic compensator, torsional filter and limiter. The major objective of providing PSS is to increase the power transfer in the network, which would otherwise be limited by oscillatory instability. The PSS must also function properly when the system is subjected to large disturbances. The washout circuit is provided to eliminate steady-state bias in the output of PSS which will modify the generator terminal voltage. The PSS is expected to respond only to transient variations in the input signal (say rotor speed) and not to the dc offsets in the signal. The washout circuit acts essentially as a high pass filter and it must pass all frequencies that are of interest.

    The zeros of D(s) should lie in the left half plane. They can be complex or real. Some of the zeros of N(s) can lie in the right half plane making it a non-minimum phase. For design purposes, the PSS transfer function is approximated to T(s), the transfer fuction of the dynamic compensator. The effect of the washout circuit and torsional filter may be neglected in the design but must be considered in evaluating performance of PSS under various operating conditions. There are two design criteria. The time constants, T1 to T4 in (18) are to be chosen from the requirements of the phase compensation to achieve damping torque. The gain of PSS is to be chosen to provide adequate damping of all critical modes under various operating conditions.

    where,

    = ( 1 + 1 1 + 3 )

    /( 1 + 2 1 + 4 (18)

    Ks is the gain of PSS and the time constants, T1 to T4

    Fig 8 Block diagram of SMIB with PSS

    Fig 8 Block diagram of SMIB with PSS

    Fig.10. Block diagram of system with PSS

    \

    \

    are chosen to provide a phase lead for the input signal in the range of frequencies that are of interest (0.1 to 3.0 Hz). With static exciters, only one lead-lag stage may be adequate.

    Fig.9. Block diagram of PSS

    In general, the dynamic compensator can be chosen with the following transfer function

    The eigen value analysis and time domain analysis are performed for the test system with low loading and high loading conditions. The result for the eigen value analysis for the system with power system stabilizer for the high loading and low loading conditions are tabulated in Table VII and Table VIII. The Table VII infers that the stability of the system is improved with the power system stabilizer and in Table VIII the system was unstable without the PSS under high loading condition due to effectiveness of the PSS the system becomes stable represented by negative real part of eigen values.

    TABLE VII

    EIGEN VALUE ANALYSIS WITH PSS AND WITHOUT PSS ON LOW LOADING CONDITION

    With PSS

    Without PSS

    Eigen Values

    -7.00±15.66i

    -2.174±4.89i

    -.5326

    -39.13

    -2.41±3.39i

    -.1919±5.98i

    TABLE VIII

    EIGEN VALUE ANALYSIS WITH PSS AND WITHOUT PSS ON HIGH LOADING CONDITION

    With PSS

    Without PSS

    Eigen values

    -5.39±13.67i

    -3.44±6.56i

    -.5294

    -39.82

    -2.88±3.77i

    .5518 ±6.18i

    0.025

    Rotor Angle Deviation[pu]

    Rotor Angle Deviation[pu]

    0.02

    0.015

    0.01

    0.005

    0

    Without PSS With PSS

    0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

    Time[sec]

  4. CONCLUSION

    In this paper, a comprehensive study on low frequency oscillations in a single machine infinite system is presented. Small signal analysis is carried out on the simple system to find the mode of oscillations. The simulation results shows that an electromechanical mode with low damping ratio and the introduction of exciter push the critical mode towards the imaginary axis by adding negative damping on it. Adding PSS on the machine with exciter is pulling the critical Eigen value by adding more damping. The time domain analysis also conducted to validate the small signal analysis and it shows the effectiveness of PSS on system stability.

    Fig.11. Effect of PSS on low loading condition

    Without PSS with PSS

    Without PSS with PSS

    0.1

    0.08

    Rotor Angle Deviation[p.u]

    Rotor Angle Deviation[p.u]

    0.06

    0.04

    0.02

    0

    -0.02

    -0.04

    -0.06

    REFERENCE

    1. K. R. Padiyar Power System Dynamics Stability and Control, BS Publications, 2008.

    2. R.RamanujamPower System Dynamics Analysis and Simulation, PHI Publications, 2009.

    3. E.V. Larsen. D.A. Swann, Applying power system stabilizers, Part- I:General concepts. IEEE Trans. Power Appar. Sysy. PAS-100(1981) 3017 3024.

[4]F.P. Demello, C.Concordia, Concepts of synchronous machines stability as affected by excitation control, IEEE Trans. Power Appar. Syst. PAS- 88(1969) 316-329.

[5] K. Bhattacharya, J. Nanda, M.L. Kothari, Optimization and performance analysis of conventional power system stabilizers, Int. J. Electr. Power Energy Syst. 19(7)(1997) 449-458.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time[sec]

Fig.12. Effect of PSS on high loading condition

Generator:

APPENDIX

0.1

CLASSICAL MODEL

0.08 FLUX DECAY MODEL

WITH EXCITER

PSS

Rotor Angle Deviation

Rotor Angle Deviation

0.06

0.04

0.02

0

-0.02

-0.04

Td0 = 6.0s; Xd = 1.6 pu; Xq = 1.55 pu; Xd = 0.32 pu; H = 5; D = 0;

Transmission:

Xe = 0.4 pu Excitation system:

KE = 200; TE = 0.05s

Operating condition:

(a)Pg = .5pu; Vt = 1.0pu; Eb = 1.0pu; fB = 60Hz

(b) Pg = 1pu; Vt = 1.0pu; Eb = 1.0pu; fB = 60Hz Parameters of PSS:

Tw = 2s; T1 = 0.078s; T2 = 0.026s; KS = 10

-0.06

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time

Fig.13. Effect of PSS on stabilization

In Fig.11, the system remains in stable condition and due to the effect of PSS, the system comes to stable condition much earlier than the system with excitation control. Fig.12 depicts that the system is unstable under heavily loaded condition without PSS and the system becomes stable when the PSS is added to the system. It does that by providing supplementary perturbation signals in a feedback path to the alternator excitation system. Fig 13 shows the effectiveness of power system stabilizer on the system stability in damping the oscillations. Thus a power system stabilizer improves the small signal power system stability by damping out the low frequency oscillations in the power system.

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