NATIONAL POWER SYSTEMS CONFERENCE, NPSC 2002 368

T

Design of Dual Input Power System Stabilizer for Detailed Dynamic Model of Power System

Avdhesh Sharma and M.L.Kothari

Abstract-- The paper deals with design of dual input power system stabilizer for single machine infinite bus (SMIB) system considering detailed dynamic model of the system including turbine-governor dynamics. IEEE type PSS2B model of the dual input power system stabilizer, is considered for present study. It comprises of two blocks, e.g., signal processing block and lead-lag network block with crucial design parameters as stabilizing gain constant, Ks1, lead time constant, T1, and lag time constant, T2. These parameters are optimized using a new hybrid optimization technique. Studies were carried out to understand the effect of optimum dual input power system stabilizer on simplified as well as detailed dynamic models of SMIB system. Investigations reveal that the dual input power system stabilizer with optimum parameters obtained considering simplified dynamic model of SMIB system works well on the actual system i.e., detailed dynamic model of system. Index Terms-- Dual input Power System Stabilizer, Detailed Dynamic Model, SMIB System, Turbine-Generator Shaft Dynamics.

I. INTRODUCTION

he phenomenon of power system stability has received a great deal of attention in the past and is expected to receive increasing attention in the future also. As economies in the system design are achieved with larger unit sizes and higher per unit reactance generating and transmission equipment designs, more emphasis and reliance is being placed on controls to provide the required compensating effects with which to offset the reductions in stability margins inherent from these trends in equipment design. High initial response, highgain excitation systems equipped with power system stabilizers have been extensively used as an effective means of enhancing overall system stability. The power system stabilizer extends the system stability limits by modulating the excitation of the synchronous generator to provide damping to the oscillations of synchronous machine rotors relative to one another.

Delta Omega (input signal obtained from shaft speed deviation ��) and DeltaPOmega or dual input (input signals obtained from shaft speed deviation, ��and deviation of electrical power output, �Pe) stabilizers are most commonly used. The power system stabilizer models PSS1A and PSS2A have been described in IEEE standard 421.5“IEEE Recommended Practice for Excitation System Models for

Avdhesh Sharma is with Department of Electrical Engineering, M.B.M. Engineering College, J.N.V.University, JODHPUR-342011, INDIA (e-mail: avdhesh_2000@yahoo.com).

M.L.Kothari is with Department of Electrical Engineering, Indian Institute of Technology, Delhi; Hauz Khas, New Delhi-110 016, INDIA (e-mail: mohankothari@hotmail.com).

Power System Stability Studies” [1]. The modelPSS1A is of a single input PSS while the modelPSS2A is of a dual input PSS. Recently IEEE Digital Excitation Task Force has presented PSS2B, PSS3B and PSS4B models of the dual input PSS in the paper entitled, “Computer Models for Representation of Digital Based Excitation Systems” [2]. The IEEE type PSS2B model is a modified version of the type PSS2A model. Although sample data sets [1,2] for these models have been specified, the Task Force has clearly stated that the data sets are samples, not necessarily typical or representative. To the best of the authors’ knowledge, no attempt seems to have been made to optimize and analyze the performance of the dual input power system stabilizers. The phase compensation technique is generally used for optimizing the Delta-omega PSS. This is the most straightforward approach, easily understood and implemented in the field. However, the gain setting is obtained by trial and error approach. A hybrid optimization technique is proposed for optimizing the parameters of the PSS in the present work. Further it is seen that the studies have not been carried out to answer the pertinent question; whether a simplified dynamic model of the system is acceptable for optimizing the parameters of the conventional PSS? An attempt has been made to answer the above pertinent question in this paper. The main objectives of the work presented in this paper are:

1. To develop a comprehensive dynamic model of the SMIB system including governor, turbine, turbine-generator shaft dynamics, excitation system, and power system stabilizers.

2. To present a systematic approach for optimizing the parameters of the conventional as well as dual input PSS using a hybrid optimization technique.

3. To investigate whether the parameters of the conventional PSS obtained using simplified dynamic model are acceptable for the actual system, i.e., considering a detailed dynamic model of the system.

4. To study the dynamic response of the system with dual input power system stabilizer considering the detailed dynamic model of the system.

II. SYSTEM INVESTIGATED

A single machine infinite bus (SMIB) system with synchronous generator provided with IEEE Type-ST1 static excitation system is considered (Fig.1). Nominal parameters of the system are given in Appendix-1.

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369

A tandem compound single-reheat type steam turbine is assumed for present investigations. It consists of four cylinders e.g., high-pressure, intermediate pressure, and two low-pressure cylinders (Appendix-2). The steam turbine system utilizes governor-controlled valves at the inlet to the high pressure turbine to control steam flow.

Fig.1 Small perturbation transfer function block diagram of a single machine-infinite bus system and dual input power system stabilizer.

III. DUAL INPUT POWER SYSTEM STABILIZER

The transfer function model of the IEEE type PSS2B-dual input PSS is given in Ref. 3. The input signals to this PSS are the speed deviation ∆ω and terminal power deviation ∆Pe . The limits to the input signals, which represent the allowable ranges of the sensed values, depend on specific design parameters. For each input two washout blocks can be represented (Tw1 – Tw4 ) along with transducer time constant (T6 and T7). A torsional filter (Time constant T8 and T9) with indices m=5 and n=1 is provided. The VSMAX and VSMIN are the maximum and minimum limits of the stabilizer output respectively. The type PSS2Bdual input power system stabilizer may be considered as comprising two cascade connected blocks, i.e. (i) the processing block with ∆Pe and ∆ω as input signals, while ∆ωeq as the output signal and (ii) phase compensator or conventional PSS block, with ∆ωeq as the input signal and stabilizing signal as output.

IV. SMALL PERTURBATION DYNAMIC MODEL OF SMIB SYSTEM WITH DUAL INPUT POWER SYSTEM

STABILIZER:

A small perturbation transfer function block diagram of a single machine infinite bus system (neglecting governor, turbine-shaft dynamics) with Delta-Omega PSS is obtained. The small perturbation dynamic model of the system in state space form is obtained from the transfer function block diagram Fig.1 as:

where, x and p are the state and perturbation vectors respectively and are defined as:

x = [�� ����fd v1 v2 �Efdv3 v4 vs ] T

p = [∆Vref ∆Tm] T

A and p are the compatible matrices [4]. A detailed dynamic model of the system may be obtained considering the dynamics of governor and shaft system. The linearized equations of the complete system considering IP, LPA, LPB and HP turbines are given in Appendix-2.

V. OPTIMIZATION OF PSS PARAMETER USING HYBRID TECHNIQUE

The proposed hybrid technique for the optimization of PSS parameters comprises of the following steps:

1. Computation of the time constants of the lead-lag networks using phase compensation technique.

2. Computation of optimum stabilizer gain using ISE technique.

A step by step procedure for optimizing the time constants

of the lead-lag networks, is as follows: 1. For any given operating condition, compute K1 – K6 and

T3 constants of the Heffron-Phillips model. 2. Compute natural frequency of oscillation (ωn) of the

electro-mechanical mode neglecting damping

i.e., M

K o1n

ω=ω (2)

where, M=2H is the inertia constant and ωo is system frequency in rad./sec.

3. Compute ∠ GEP (at ωn) The transfer function relating the stabilizing signal vS

and ∆Ψfd (Fig.1) is: KA K3

GEP(s)= (3) ( 1 + s TA ) ( 1 + s T3 ) + KA K3 K6

where, KA = ( KAVR • TB / TC ). The time constant TR is very small and hence neglected. Let γ be the phase angle of GEP(s). i.e., ∠GEP(s)

=γ - tan-1

ω−+ω+

2n3A63A

n3A

TTKKK1)TT(

(4)

4. The phase lead compensator, Gc is designed to provide required degree of phase compensation. For 100% phase compensation:

∠ GC(jωn) + ∠ GEP(jωn) = 0 (5)

Assume two identical cascade connected leadlag networks, i.e. T1 = T3, T2 = T4. The transfer function of the phase compensator Gc of the PSS becomes:

2

2

1c sT1

sT1)s(G

++

= (6)

∆δ+ -

+

- +

-∆Vref

+

+∆Ψfd

+

∆Tm

∆ω

-

∆Te

5K

6K

1K

2K

C

BsT1sT1

++

A

AsT1

K+

3

3sT1

K+

dKMs1+

Σ

4K

Dual Input PSS ∆Pe

Σ

Σ Σ

ω0 s

. x = A x + Γ p (1)

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A low value of T2 =0.05 sec. is chosen from the consideration of physical realization. Since the phase angle compensated by both the leadlag networks is equal i.e.γ/2. The parameters T1 is computed using following relation [4] :

T1 = ( ) .secTtan2

tan12n

1

n

ω+

γω

−

(7)

Here, it is desirable to reiterate that the PSS is initially designed excluding the dynamics of governor, turbine and the turbine-generator shaft systems. For the nominal operating condition and system parameters ∠ GEP = 86.6o and hence for 100% compensation T1 = 0.2825 seconds.

The optimum value of gain setting (KSTAB) is obtained using integral of squared errors (ISE) technique. The choice of a suitable performance index is extremely important for applying ISE technique for the optimization of KSTAB of PSS. In order to obtain optimum KSTAB, a quadratic performance index J as given below is evaluated for several values of KSTAB considering a step perturbation in mechanical torque. The optimum gain setting is the one that results in minimum value of J.

{ }∫∞

ω∆+δ∆−δ∆=0

22ss dt)t(.h))t((J (8)

where, h is the relative weighting factor. For the present studies, the value of h is obtained by a trial and error approach. The relative weighting factor h is set at a value for which the two error terms contribute equally to the value of J (h=400). ∆δss is the steady state value of ∆δ. ∆δss is computed by setting X& =0 and X = Xss . i.e., 0 = A Xss + Γp and hence Xss = - A-1Γ p (9)

where, Xss is the steady state value of X for constant p. For a step increase in Tm (i.e., ∆Tm=0.05 p.u.) assumed in the present studies, ∆δss is equal to 0.0471 rad.

Fig. 2 shows the plot of J versus KSTAB for T1* =

0.2825 sec., J first decreases with increase in KSTAB, attains a minimum value (Jmin) for KSTAB=27.48, and then increases with further increase in KSTAB. Hence, KSTAB= 27.48 becomes optimum gain setting of the PSS. Thus the optimum parameters of the Delta-omega PSS obtained using hybrid optimization technique are as follows:

KSTAB*=27.48, T1* =0.2825 sec., T2 = 0.05 sec.

The above PSS parameters are obtained for KAVR=400, TB=1.0 sec. and TC = 8.0 sec.

VI. ANALYSIS:

In order to investigate the performance of system with optimum Delta-Omega PSS, the dynamic responses considering 5% increase in reference voltage, were obtained and found satisfactory. When the same optimum Delta-Omega PSS applied to a detailed dynamic model (actual

system), one torsional mode of 16 Hz. was excited and response was highly oscillatory with increasing amplitudes. To mitigate this

Fig.2: Plot J vs KSTAB for T1 = 0.2825 sec. and T2 = 0.05 sec.

problem, the torsional filter were used. Figure3 shows the dynamic responses of the system for ∆ω considering 5 % step increase in reference voltage. When the optimum Delta-Omega PSS is applied to detailed dynamic model with torsional filter, the system become unstable. Further analysis of this responses of the system, shows that toirsional mode was stabilized however another (control) mode get excited. If stabilizer gain is reduced to 20.00, the system become stable. Hence it is clear that there is limit to increase the stabilizer gain when Delta-Omega is applied to detail dynamic model. Further it is explored whether optimum dual input power system stabilizer obtained considering simplified dynamic model is applicable for detailed dynamic model of power system.

Fig.3: Dynamic response of the system (using detailed dynamic model) for ∆ω with Delta-Omega PSS and torsional filter and detailed dynamic model considering ∆Vref = 0.05 p.u.

A. Optimization of the Parameters of the IEEE type PSS2B Model of Dual Input Power System Stabilizer

KSTAB →

J →

Pe = 0.9 p.u., Qe=0.2907 p.u., Vt = 1.0 p.u., and Xe = 0.65 p.u. KAVR=400, TB=1.0 sec., and TC=8.0 sec.

Time (seconds) →

∆ω (

p.u.

) →

Pe = 0.90 p.u., Qe=0.2907 p.u., Vt = 1.0 p.u. , and Xe = 0.65 p.u. KSTAB

* = 27.4794, T1=0.2825 sec., T2= 0.05 sec. KSTAB = 20.0, T1=0.2825 sec., T2 = 0.05 sec.

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The dual input or Delta-P-Omega PSS has been developed [5] for overcoming the limitations of the Delta-Omega PSS. The PSS2B model of the dual input power system stabilizer [3] has been considered in the present studies. It comprises of two cascade connected blocks, i.e., (i) the processing block with ∆Pe and ∆ω as input signals, and ∆ωeq as the output signal and (ii) lead-lag network block, with ∆ωeq as the input signal and stabilizing signal as output. The sample parameters of the processing block as given by the IEEE task force [3] are assumed (Appendix-3). The parameters of the second block (i.e., lead-lag networks) are optimized using hybrid optimization technique. The optimum time constants T1 = T3 = 0.2825 sec. and T2 = T4 = 0.05 sec. as obtained for Delta-Omega PSS are acceptable for dual input PSS also. The gain KS1 is optimized using ISE technique considering performance index, J (Equation-8). The performance index J is evaluated for 5% step increase in mechanical torque i.e., ∆Tm = 0.05 p.u. in the simplified dynamic model of the system (Fig.1).

Fig.4 shows the plot of J versus Ks1. It is seen that J first decreases with increase in Ks1, attains a minimum value (Jmin) for a gain setting, Ks1=19.48, hence optimum gain setting Ks1

*=19.48.

Studies are now carried out in order to understand whether the optimum dual input PSS obtained considering a simplified dynamic model of the system is acceptable for the actual system, i.e., with detailed dynamic model of the system.

Table-1 shows the eigenvalues of the detailed dynamic model of the system with optimum dual input PSS. It is clearly seen that all the modes of the system are stable. This fact is further corroborated by plotting the dynamic response of the system with optimum dual input PSS considering detailed dynamic model of the system (Fig.5). From the studies presented above it may be inferred that the optimum gain setting of the dual input PSS based on simplified model, obtained using ISE technique is acceptable with the actual detailed dynamic model of the system.

Fig.4: Plot J vs KS1 for ∆Tm=0.05 p.u. (T1 = 0.2825 sec. and T2 = 0.05 sec.)

TABLE-1 EIGENVALUES OF THE DETAILED SYSTEM MODEL

WITH OPTIMUM DUAL INPUT PSS (KS1

*= 19.4794, T1* = 0.2825 sec., T2 = 0.05 sec.)

Eigenvalues Frequency of the

oscillatory mode (Hz)

ξ of the oscillatory modes

−0.09 −0.1 −0.1 −0.11 −0.11 −0.15

−0.20 ± j 102.32 −0.21 ± j 190.45 −0.30 ± j 151.40 −0.33 ± j 276.44

1.18 1.47

1.93± j 3.58

2.33± j 15.54 3.22

4.02± j 0.78 4.75

7.21± j 5.78

16.14± j 6.53 37.85 ± j 15.28

51.26 53.14

16.3740 24.1085 30.3157 43.9966

0.6465

2.5012

0.6517 (Electromechanic

al mode) 1.4706

(Control mode) 2.77 6.5

1.95 x10-3 1.10 x10-3 1.98 x10-3 1.20 x10-3

0.47

0.15

0.98

0.78

0.93 0.93

Fig.5: Dynamic response of the system for ∆ω with dual input PSS and detailed dynamic model considering ∆Vref = 0.05 p.u.

VII. CONCLUSION

The significant contributions of the work presented in this paper are:

1. A detailed dynamic model of the system in state-space form is obtained considering governor, turbine, turbine-generator shaft dynamics.

Pe = 0.9 p.u., Qe=0.2907 p.u., Vt = 1.0 p.u., and Xe = 0.65

KS1*=19.4794

KS1 →

J →

Time (seconds) →

∆ω (

p.u.

) →

Pe = 0.90 p.u., Qe=0.2907 p.u., Vt = 1.0 , p.u. and Xe = 0.65 p.u.

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2. A new approach for optimizing the parameters of a PSS using hybrid optimization technique based on phase compensation and ISE techniques, has been presented.

3. Studies reveal that the optimum gain setting of the Delta-Omega PSS obtained using simplified dynamic model of the system is not acceptable for a realistic system, i.e., with detailed dynamic model including governor, turbine, turbine-generator shaft models even if torsional filter is incorporated. The gain setting of the Delta-Omega PSS needs to be restricted to a low value in order to ensure that none of the modes are adversely affected with the incorporation of the PSS.

4. Investigations clearly show that the gain setting and time constants of the dual input PSS obtained considering a simplified dynamic model are acceptable for the realistic system, i.e., including governor, turbine, turbine-shaft models.

VIII. ACKNOWLEDGMENT The authors gratefully acknowledge the financial support received from Department of Science and Technology, Power Grid Corporation of India Limited, ABB and AICTE (research project no. R&D/2001-02/8020-91)which made it possible to conduct this research.

IX. APPENDIX 1

The nominal system parameters and operating condition are:

H =3.5 sec, T′do=8 sec, Xd=1.81 p.u., X′d=0.3 p.u., Xq=1.76 p.u. KA=400, TR=0.02 sec., TA=0.05 sec., TB =1.0 sec., TC = 8.0 sec. Pe =0.9 p.u., Qe=0.3 p.u., Vt=1.0078 p.u., Xe = 0.65 p.u

XI. APPENDIX 2

The equations of the complete rotor system considering IP, LPA,LPB, HP turbines may readily be written as follows:

GEN: 2H1 dt

)(d 1ω∆ = K12 (∆δ2 - ∆δ1 ) - ∆Te - D1 (∆ω1)

(A2.1)

dt

)(d 1δ∆ = (∆ω1) ωo (A2.2)

LPA : 2H2 dt

)(d 2ω∆ =∆TLPA + K23 (∆δ3 - ∆δ2 ) –

K12 (∆δ2 - ∆δ1 ) - D2 (∆ω2) (A2.3)

dt

)(d 2δ∆ = (∆ω2) ωo (A2.4)

LPB : 2H3 dt

)(d 3ω∆ = ∆TLPB + K34 (∆δ4 - ∆δ3 ) –K23 (∆δ3 -

∆δ2 ) - D3 (∆ω3) (A2.5)

dt

)(d 3δ∆ = (∆ω3) ωo (A2.6)

IP: 2H4 dt

)(d 4ω∆ = ∆TIP + K45 (∆δ5 - ∆δ4 ) – K34 (∆δ4 - ∆δ3)

- D4 (∆ω4) (A2.7)

dt

)(d 4δ∆ = (∆ω4) ωo (A2.8)

HP: 2H5 dt

)(d 5ω∆ = ∆THP - K45 (∆δ5 - ∆δ4 ) – D4 (∆ω4)

(A2.9)

dt

)(d 5δ∆ = (∆ω5) ωo (A2.10)

Differential equations for a tandom compound single reheat type steam turbine:

gKUdt

)g(dT 1gEg ∆−ω∆−∆=∆ (A2.11)

HPHPHP

CH Tg.Fdt

)T(dT ∆−∆=∆

(A2.12)

IPHPHP

IPIPRH TT.

FF

dt)T(dT ∆−∆=

∆(A2.1) (A2.13)

LPAIPIP

LPALPACO TT.

FF

dt)T(dT ∆−∆=

∆ (A2.14)

where, LPALPA

LPBLPB T.

FFT ∆=∆

X. APPENDIX 3 The IEEE recommended settings of the processing block of the type PSS2B model of the dual input PSS are [3]:

Ks2=0.99, Ks3=1.0, T5=0.033 sec., T6=0.0 sec., T7=10 sec., T8= 0.5 sec., T9=0.1 sec., T10=0.0, n=1, m=5, Tw1= Tw2 = Tw3= Tw4= 10 sec.

XI. REFERENCES: [1] IEEE recommended practice for excitation system models for power

system stability studies, IEEE standard 421.5, 1992. [2] P. Kundur, “Power system stability and control”, McGraw Hill, Inc.,

1994. [3] IEEE Digital Excitaion System Sub-committee report, “Computer

models for representation of digitalbased excitation systems”, IEEE Transactions on Energy Conversion, Vol. 11, No. 3, September 1996, pp.607−615.

[4] Avdhesh Sharma, ”Artificial Neural Network and Fuzzy Logic System based Power System Stabilizers”, Ph.D. thesis, Indian Institute of Technology, Delhi, 2001.