DYNAMIC STABILITY ANALYSIS (Small Signal Stability)

ByProf. C. Radhakrishna

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CONTENTSDYNAMIC STABILITY ANALYSIS (Small Signal Stability) – 1Small-Signal Stability of Multi-machine SystemsSpecial techniques for analysis of very large systemsCharacteristics of Small-Signal Stability ProblemsLocal problemsGlobal problemsDYNAMIC STABILITY ANALYSIS – 2IntroductionOverview of the Proposed MethodGenerating UnitSynchronous MachineCalculation of Equilibrium State ConditionsExcitation and Governor Control SystemsExcitation SystemTurbine-Governor SystemCombined Model of Generating Unit

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CONTENTS cont………Load RepresentationMulti-Component ModelsNetwork RepresentationState Space Model of the Overall SystemConclusions

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DYNAMIC STABILITY ANALYSIS (Small Signal Stability)-1

• Small-signal stability, is the ability of the power system to maintain synchronism when subjected to small disturbances.

• A disturbance is considered to be small if the equations that describe the resulting response of the system may be linearized for the purpose of analysis.

• The small-signal stability problem is usually one of insufficient damping of system oscillations.

Small-Signal Stability of Multi-machine Systems

Analysis of practical power systems involves the simultaneoussolution of equations representing the following:• Synchronous machines, and the associated excitation systems and prime movers.• Interconnecting transmission network.• Static and dynamic (motor) loads• Other devices such as HVDC converters, static var compensators

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• For system stability studies it is appropriate to neglectthe transmission network and machine stator transients.• The dynamics of machine rotor circuits, excitationsystems, prime mover and other devices are represented bydifferential equations.• The result is that the complete system model consists of alarge number of ordinary differential and algebraic equations.

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* Algebraic equations** Differential equations

Figure.1: Structure of the complete power system model

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• Analysis of inter-area oscillations in a large interconnectedpower system requires a detailed modelling of the entire system.• Special techniques have been developed that focus onevaluating a selected subset of eigenvalues associated with thecomplete system response.• AESOPS algorithm. It uses a novel frequency responseapproach to calculate the eigenvalues associated with the rotorangle modes.• The selective modal analysis (SMA) approach computeseigenvalues associated with selected modes of interest by usingspecial techniques to identify variables that are relevant to theselected modes, and then constructing a reduced-order model thatinvolves only the relevant variables.• The PEALS (Program for Eigenvalue Analysis of Large Systems)uses two of these techniques.• The AESOPS algorithm and the modified Arnoldi method.• These two methods have been found to be efficient and reliable,and they complement each other in meeting the requirements ofsmall-signal stability analysis of large complex power systems.

Special techniques for analysis of very large systems

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Characteristics of Small-Signal Stability ProblemsLocal problems

• Associated with rotor angle oscillations of a single generator or a single plant against the rest of the power system. Such oscillations are called local plant mode oscillations. • Most commonly encountered small-signal stability problems are of this category.• Local problems may also be associated with oscillations between the rotors of a few generators close to each other. • Such oscillations are called inter-machine or inter-plant mode oscillations. • The local plant mode and interplant mode oscillations have frequencies in the range of 0.7 to 2.0 Hz. • Analysis of local small-signal stability problems requires a detailed representation of a small portion of the complete interconnected power system. • The rest of the system representation may be appropriately simplified by use of simple models and system equivalents.

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Global problems

• Global small-signal stability problems are caused by interactions among large groups of generators and have widespread effects.

• They involve oscillations of a group of generators in one area swinging against a group of generators in another area. Such oscillations are called inter-area mode oscillations.

Large interconnected systems usually have two distinct forms of interarea oscillations:

(a)A very low frequency mode involving all the generators in the system. The frequency of this mode of oscillation is on the order of 0.1 to 0.3 Hz.

(b)Higher frequency modes involving subgroups of generators swinging against each other. The frequency of these oscillations is typically in the range of 0.4 to 0.7 Hz.

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DYNAMIC STABILITY ANALYSIS The analysis of dynamic stability can be performed by deriving a

linearized state space model of the system in the followingformp X = A X + B u

Where the matrices A and B depend on the system parametersand the operating conditions.

• The Eigen values of the system matrix A determine thestability of the operating point.

• The Eigen value analysis can be used not only for thedetermination of the stability regions, but also for the designof the controllers in the system.

The novel features of the proposed method :• It is not necessary to reduce the power system network to

eliminate non-generator buses. The same network used for load flowstudies can also be used for the dynamic stability calculations.

• The development of system model proceeds systematically by thedevelopment of the individual models of various components andsubsystems and their interconnection through the network model.

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Overview of the Proposed Method

At any bus k of an N-bus network the following equations apply

kIj

jj

kj

j

kk V

VPPP

kIj

jj

kj

j

kk V

VQQQ

where Ik is the set of buses that are connected to bus k. Also itwould be shown that for each bus, (P, Q) or (, V)

Fig. 1 Block diagram for power system network

can be eliminated depending on the type of bus. The A matrixformulation is based on identifying the interconnections among thevarious subsystems of the power system as shown in Figure 2.

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• Development of the system model is based on the formulation ofthe individual component models and identifying the variousinterconnections between the subsystems.

• The linearized network algebraic equations are solved in terms ofthe system state variables resulting in the final system model.

Fig. 2 Block diagram showing the interconnections among the various subsystems of the power system

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The rotor circuit differential equations, including its motion, are given by p Xm = [Am] Xm + bme vfd + b mg Pm + [Bp] SgYm = [Cm] Xm

where, Xm = [Id Iq f k ]t

Ym = [Id Iq ]t

Sg = [Pg Qg]t

Also, the generator terminal bus voltage magnitude and phase angle are expressed in the formZg = [Dm] Ym + [Dp] Sg

where Zg = [g Vg]t

Id and Iq are state variables derived from the rotor flux linkages.

GENERATING UNITSynchronous Machine

Calculation of Equilibrium State Conditions

The values at the operating point (equilibrium state) of the power system are calculated from the load flow results of the system.

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• The excitation and governor control systems used in moderngenerators fall into standard categories compiled in IEEECommittee reports.

Excitation and Governor Control Systems

While it was initially thought that high gain voltage regulator loop witha fast acting static exciter would improve transient stability, thepractical experience was that it led to dynamic instability.

Power system stabilizer (PSS) which introduces supplementarystabilizing signal to suppress rotor oscillations has become adesirable part of any excitation system.

The change in the magnitude of the terminal voltage, Vg, is one of theinputs for the excitation system and this has to be expressed in termsof the state variables and is given in the equation (5).

The state space model of excitation system is represented in the formp Xe = [Ae] Xe + [Bem] Ym + [Bep] Sg + be ueye = [Ce] Xeye = Vfd; ue = Vref

where Xe, ue and ye are respectively the state, input and outputquantities; and the structures of the associated matrices areobtained for the IEEE Type 1 excitation system.

Excitation System

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The state space model of governor control system can berepresented in the formp Xg = [Ag] Xg + [Bgm] Ym + bg ugyg = [Cg] Xgyg = Pm; ug = Pmo

where Xg, ug and yg are respectively the state, input and outputquantities; and the structures of the associated matrices areobtained for an IEEE system model.

Turbine-Governor System

The following state space model is obtained, where all thecomponent elements are matrices.

Combined Model of Generating Unit

m mm me mg pe

e eem e epg

g ggm g

p X A B B X Bu

p X B A X Bu

p X B O A X Og e

g

O OO S b O

O b

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[Bme] = bme Ttel ; [Bmg] = bmg Tt

gl ;[B’em] = [Bem] [Cm] ; [B’gm] = [Bgm] [Cm] ;and Tel and Tgl are vectors containing only one non zero element

each equal to one and defined by the following equationsVfd = Tt

el Xe; Pm = Ttgl Xg

g

e

m

g

e

m

g

e

m

X X X

CC

C

yyY

The usual constant power, constant current and constantimpedance type loads and any other voltage dependentnonlinear loads can be represented in the general form

Load Representation

where consent coefficients kp, kq and the exponents np and nqdepend upon the type of load under consideration. Linearizing,we get

nqLqL

npLpL VkQVkP ;

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The nonlinear loads dependent on the bus frequency, if presentin the system, can also be handled without any difficulty, ifdesired.

where,

= [Al] Zl

= [PL QL ]t

= [L VL ]t

S

S

Z

1

1

][ nqLqq

npLpp

l VknOVknO

A

Multi-Component Models

The various subsystems described earlier can be assembledtogether for the analysis of large-scale power systemsincluding large number of machines and loads.

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[AM] = Block diag. [[Aml], [Am2],…………… [Amn]]and XM = [Xt

ml Xtm2…………… Xt

mn]t

g

eEP

P

G

E

M

GGM

EEM

MGMEM

G

E

M

uu

O B B

X X X

AOBAB

BBA

X pX pX p

G

EG

BOO

OBO

SO

G

E

M

G

E

M

G

E

M

X X X

CC

C

YY

Y

where all the components are matrices.Also,

ZG = [DM] YM + [Dp] SGSL = [AL] ZL

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The network equations are [SG

t SLt]t = [J] [ZG

t ZLt]t

where [J] is the Jacobian matrix of the network and is given by

where all the components are matrices. Substituting the equations (17) in (18) and simplifying, we get,

where [J’LL] = [JLL] + [AL]

Network Representation

• The network is represented by its Jacobian matrix in thepolar form.

• For a N-bus power system network the Jacobian is of (2N x2N) dimension and the identity of all the buses is preserved.

LLLG

GLGG

JJJJ

J ][

L

G

LLLG

GLGGG

ZZ

JJJJ

OS

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ZG = [DM] [CM] XM + [Dp] SG

MM

M

MLG

MGGM

L

G XCC

DJDJ

JZS

1][

LLPLG

GLPGGM JDJ

JDJUJ

')(

][where

State Space Model of the Overall System

State space model of the overall system p X = [A] X + [B] U

Y = [C] Xwhere X = [XM

t XEt XG

t] tU = [ue

t ugt ] t

Y = [YMt YE

t YGt] t

GGM

EEM

MGMEM

AOBAB'

BBA' ][ OA

G

E

BOO

OBO

B][

G

E

M

CC

C ][C

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• This state space model is amenable to the application of linearcontrol theory and eigenvalue analysis.

• This allows one to study the overall dynamic performance ofpower systems, including the interaction between machinecontrols.

CONCLUSIONS

REFERENCES :[1] Prabha Kundur: “Power System Stability and control”, The EPRI PowerSystem Engineering Series, McGraw-Hill, Inc., 1994.[2] C. Radhakrishna : “Stability Studies of AC/DC Power Systems” , Ph. D.

Thesis , submitted to Indian Institute of Technology Kanpur, India, 1980.

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THANK YOU